THE BEHAVIOUR IN COMBINED BENDING; TORSION AND ...

THE BEHAVIOUR IN COMBINED BENDING;

TORSION AND SHEAR OF REINFORCED

CONCRETE BEAMS.

by

MICHAEL P. COLLINS.

A thesis submitted to the Professorial Board of the University of New South Wales as partial fulfilment of the requirements for the degree of Doctor of Philosophy.

1967.

I hereby declare that this thesis has not been previously presented by me, either in whole or in part, for the award of an academic degree .

. .

ACKNOWLEDGEMENTS

The results of this thesis were obtained in the course of

research on combined bending, torsion and shear in reinforced

concrete members, carried out under the sponsorship of the

Australian Road Research Board. The author wishes to express

his gratitude to the Board, both for their sponsorship and

encouragement.

The author also wishes to thank Associate Professors

F .E. Archer and A.S. Hall,. the supervisors of this thesis, for

their valuable guidance throughout this project,.

Thanks are also extended to Professor F .S. Shaw, Head

of the Department of Structural Engineeril.lg, without whose

co-operation this work would not have been possible.

The author is very grateful for the technical assistance

given to him by members of the staff of the Structures

Laboratory, in particular J . M .· Whitney and J. W. Taylor.

Finally the author would like to thank Mrs. T ~ Itsikson

for tracing the diagrams in this thesis and Miss J •. Howland for

typing the manuscript.

ABSTRACT

In this project the behaviour of reinforced concrete beams

loaded in combined torsion, bending and transverse shear has been

examined Over one hundred beams were tested under known

distribution of moment, shear and torsion to study their failure

characteristics with particular reference to possible failure

modes and the effect of reinforcement. It was found that for

beams containing only longitudinal reinforcement, the effects of

flexure tend to cancel each other and hence if a conservative value

of the pure torsional strength is chosen torsion-flexure interaction

can be ignored.

For beams containing both longitudinal and transverse

reinforcement, two different types of torsion-flexure interaction

were fo nd to be possible. For beams containing equal areas of

top and bottom longitudinal steel the presence of flexure reduced

the torsional capacity. On the other hand, for beams in which the

area of tension steel exceeds that of the compression steel, the

torsional capacity was considerably increased by the presence

of moderate bending moments.

For beams containing only longitudinal reinforcement it was

found to be satisfactory to assume linear interaction between the

ultimate strengths in transverse shear and torsion. For beams

containing both longitudinal and web steel the shear-torsion

interaction behaviour was found to depend upon the section

properties of the member.

Expressions for the failure loads of rectangular reinforced

concrete beams loaded in combined bending, torsion and shear

have been derived. These expressions were obtained in the main

from a study of the equilibrium situation of the observed modes of

failure. Good agreement between the predictions of these

expressions and the experimental results was found, not only for

beams of this investigation, but also for a large number of beams

reported in the literature.

Simple, and fairly direct, ultimate strength design

procedures were developed from the analysis equations. These

procedures are presented in this thesis.

TABLE OF CONTENTS

CHAPTER 1. - INTRODUCTION

l • l Background

Page No.

1. l

1. 2 Layout and Scope 1.3

CHAPTER 2. - HISTORICAL REVIEW

2.l Introduction 2. 1

2. 2 Beams Containing only

Longitudinal Reinforcement. 2. 1

(a) Pure Torsion 2. 1

(b) Bending and Torsion 2. 5

(c) Torsion, Bending and

Transverse Shear. 2. 8

2. 3 Beam Containing Both Longitudinal

and Transverse Reinforcement. 2. 13

(a) Pure Torsion. 2. 13

(b) Bending and Torsion. 2. 21

(c) Torsion, Bending and Transverse

Shear. 2.44

3.1

3.2

3.3

3.4

3.5

CHAPTER 3 -:. ~2-{PERIMENTAL WORK

Background

Description of Test Specimens

(a) Plain Concrete.

(b) Beams Containing Only

Longitudinal Steel.

(c) Beams Containing Both

Longitudinal and Transverse

Page No.

3.1

3.4

3.6

Steel. 3. 8

Materials. Fabrication and Curing

Method of Loading, and

Instrumentation.

General Behaviour of Test Beams

(a) Plain Concrete Beams.


Longitudinal Steel..

( c) Beams Containing Both

Longitudinal and Transverse

3.10

3. 15

3.21

3.21

3.26

Steel. 3. 36

4.1

4.2

4.3

4.4

4.5

5. 1

CHAPTER 4. - BEAMS WITHOUT WEB

REINFORCEMENT.

Page No.

Introduction. 4. 1

Beams Loaded in Pure Torsion. 4.2

Beams Loaded in Combined

Bending and Torsion. 4.12

(a) Plain Concrete Beams. 4.12


Longitudinal Reinforcement. 4.16

Beams Loaded in Combined

Transverse Shear and Torsion. 4.31

(a) Nominal Stresses at Failure 4.31

(b) Transverse Shear - Torsion

Interaction. 4.35.

_'":"f ective Shear Method. 4.40

CHAPTER 5. - WEB REINFORCED BEAMS

UNDER COMBINED TORSICN" AND FLEXURE.

Page No.

Introduction 5. 1

5.2

5.3

5.4

5.5

6.1

6.2

General Assumptions.

Analysis of Mode 1

( a) Equilibrium Equations.

(b)) Determination of Tan Q 1

(c) Determination of x1 .

(d} Simplification.

Analysis of Mode 3 •

(a) Equilibrium Equations.

(b) Determination of Tan Q3 .

Flexure - Torsion Interaction.

Page No.

5.2

5.4

5.4

5.8

5.9

5.10

5.12

5.·12

5.14

5.15

CHAPTER 6. - WEB REINFORCED BEAMS

UNDER COMBINED TRANSVERSE SHEAR

AND TORSION.

Background 6. 1

Predominantly Shear Failures. 6.2

(a) Geometry of the Failure Surface. 6.2

(b) Forces Acting at Failure. 6.8

(c) Equilibrium Equations. 6.9

(d) Simplification. 6.li

6.3

6.4

Mode 2 Failures.

(a) Equilibrium Equations

(b) Magnitude of .the Forces

(c) Determination of Tan 92.

Mode 3 Failures .

6. 14

6. 14

6.16

6.18

6. 19_

6 .5 Transverse Shear - Torsion

7.1

7.2

7 .. 3-

Interaction. 6. 20

CHAPTER 7. - EXPERIMENTAL VERIFICATION

OF THE THEORY FOR WEB REINFORCED BEAMS

Introduction.

Limitations on the Steel.

7.1

7.2

(a) Excessive Transverse Steel 7. 2

(b) Excessive Longitudinal Steel. 7. 4

(c} Longitudinal Steel out of

Proportion to the Transverse

Steel. 7._4

Interaction of Bending and Torsion. 7.7

7. 4 Ir. .·action of Transverse Shear

and Torsion-

7 .. 5 A.ccuraey of the Analysis Equations

7.9

7.14

Page No.

7.6 Simplified Analysis Procedure 7.22

CHAPTER 8. - DESIGN OF BEAMS SUBJECT

TO TORSION.

8.1 Introduction.

8. 2 Derivation of the Design Equations for

Web Reinforced Beams.

8.3

8.4

9. 1

9.2

Design Procedure.

Design Example.

CHAPTER 9. - CONCLUSIONS.

Conclusions.

Suggestions for Further Research.

Appendix A - REFERENCES.

Appendix B - EXPERIMENT AL DATA.

Appendix C - OPTIMUM VALUE OF r.

Appendix D - ANALYSIS OF TEST DATA.

8.1

8 .. 1

8.6

8.8

9. 1

9.7

LIST OF FIGURES

Figure Title Page Uo.

2. 1 A Comparison of Observed and Predicted Shear-Torsion Interaction for Beams Containing only Longitudinal Reinforcement. 2. 12

2.2 Ultimate Torque/Cracking Torque -Percentage of Reinforcement. 2.18

2.3 Torsional Strength - Amount of Transverse Reinforcement. 2.19

2.4 Interaction of Bending and Torsion as Suggested by Cowan. 2.23

2.5 Manner of Failure of Beam Loaded in Bending and Torsion. 2.24

2.6 Lessig's Mode 1 Failure Surface. 2.26 2,7 Bending-Torsion Interaction as

Predicted by the Theory of Lessig. 2.29 2.8 Accuracy of Lessig' s Theory

Versus K. 2.32 2.9 Accuracy of Lessig's Theory Versus

B!Jding-Torsion Interaction as 2.34.

2.10 Predicted by the Theory of Yu-'in. 2.38

2. 11 Bending-Torsion Failure Mode of Hollow Beams Proposed by Evans. 2.40

2. 12 Lessir's Mode 2 Failure Surface. 2.46 2. 13 Shear-Torsion Interaction for

Typical Beam as Predicted by Lessig' s Theory. 2.49

FiE 're Title Page No.

3.1 Details of Beams Containing · Longitudinal Steel. 3.7

3.2 Details of RE, RU and R Series. 3.9 3.3 Details of V. U and T Series. 3. 11 3.4 Loading Arrangement. 3.16 3.5 One point Loading Rig. 3.17 3.6 Bending Moments, Transverse Shear

Forces and Twisting Moments Produced in Test. 3.18

3.7 Rotation Gauge. 3.20 3.8 Failure View of Pure Torsional

Plain Concrete Specimen. 3.22 3·_ 9 Torque.;.Twist Curve for Plain

Concrete (Beam PI). 3.24 3.10 Torque-Twist Curve for Beam LI. 3.28 3. 11 Failure of Beam Containing Only

Longitudinal Steel Loaded Predominantly in Torsion. 3 .. 29

3.12 Failure of Beam Containing Only Longitudinal Steel Loaded Predominantly in Bending. 3,.30

3. 13 Torque-Twist Curve for Beam L2. 3.31 3. 14 The Effect of a/don Shear Failure. 3.34 3. 15 The: of a/ d on the Failure

Appearance. 3. 3 · 3. 16 ·-, the

~e at Failure. 3.3:

Figure Title Page Fo.

3.17 The Appearance of Beam S4 at Failure. 3.37

3.18 Developed Failure Surface Beam RUSA Mode 1 Failure. 3.43

3.19 Torque-Twist Curves for Beams Containing Web Reinforcement RE Series. 3.45

3.20 Developed Failure Surface Beam 36T4 Mode l Failure. 3.47

3.21 Developed Failure Surface Beam 36T4c Mode 3 Failure. 3.48

3.22 Developed Failure Surface of Beam 38TS Mode 2 Failure. 3.49

3. 23 Failure Surface of Beam 77 0 5 3.51 3.24 Torque-Twist Curves for Beams

of R Series. 3.52 3.25 The Effect of Torsion on the

Appearance of the West Sides of the Beams. 3.55

3.26 The Effect of Torsion on the Appearance of the East Sides of the Beams. 3.56

3.27 The Effect of Torsion on Appearance of the Tops of the Beams. 3.57

3.28 The Effect of Stirr:__ip Spacing on the Failure Appearance. 3.59

3.29 Strains in the West Legs of Ties. 3.60 3.30 Strain in Legs of Ties. 3.61 3.31 Strain in Top Longitudinal Bars

of Beam T5. 3.62

-:--··- ,·") Title 3=' a fie :~-r o . - --3.32 Beam T5 at Failure. 3.63 3.33 Beam T2 at Failure. 3.63 3.34 Strain in Top Longitudinal

Bars of Beam T2. 3.64

4, l Comparison of TL'-ory and Experiment for Plain Concrete Beams. 4. 11

4.2 Equilibrium Situation at a Cracked Section. 4. l 9

4.3 V,.riation of the Depth of Uncracked concrete, d •

with the Applied Moment~ 4.21 4.4 Bending-Torsion Interaction

Curves for Beams Containing only Longitudinal Steel. 4.26

4.5 Interaction of Torsion with Bending for Beams Containing only Longitudinal Steel. 4.28

416 The Average Shear Stresses 4. 34 4.7 Shear-Torsion Interaction for

:-- ,...., 't-..,_~ Car. 1g Only Longitudinal Steel. 4.37

5.1 General View of Mode 1 Failure Surface. 5.3

5.2 Forces Acting on the Mode J Failure Surface. 5.5

5.3 Forces Acting on the Mode 3 Failure surface. 5. 13

Figure Title Pa1:~_I.'Jo,

5.4 Interaction Diagram for Beams Failing in Mode 1. 5.17

5.5 Interaction Diagram for Beams Failing in Mode 3 . 5.19

• 6. 1 General View of Failure Surfaces. 6.3 6.2 Developed Failure Surfaces. 6 /.. 6.3 Values of u Versus T /Vb. 6.6 6.4 Forces Acting on Effective Shear

Failure Surface. 6.7 6.5 Accuracy of the Approximate

Equation for Predominantly Shear Failures. 6. 13

6.6 View of the Mode 2 Failure Surface. 6. 15

6.7 Shear-Torsion Interaction for Web Reinforced Beams. 6.21

7. 1 The Effect of the Nominal Shear Stress on the Accuracy of the Theory. 7.3

7.2 The Effect of the Parameter r/r 0

7.6 on the Accuracy of the Theory. 7.3 Bending-Torsion Interaction for

Web Reinforced Beams. 7.8 7.4 Bending-Torsion Interaction for

RU Series. 7.10 7.5 Shear-Torsion Interaction for V

Series. 7 .11

Figure Title Page No.

7.6 Shear-Torsion Interaction for T Series. 7.13

7.7 Frequency Histogram for the Accuracy of the Theory for Web Reinforced Beams Within Restrictions. 7.23

8.1 Eccentrically Loaded Spandrel Beam. 8.9

8.2 Design of Longitudinal Steel. 8.12

LIST OF TABLES

Table Title Page No.

z. 1 A Comparison of Experimental Results with the Theories of Nylander, Rr --nakrishnan and Vijayarangan, and Gesund and Boston. Z.9

z.z A Comparison of Experimental Results and Published Theories for Web Reinforced Beams Loaded in Bending and Torsion. 2.43

Z.3 A Comparison of Experimental Results and Published Theories for Web Reinforced Beams Loaded in Shear and Torsion. 2.51

3.1 Summary of Beams Tested. 3.3 l.Z Details of .Plain Concrete Beams. 3.5 3.3 Details of R Series. Beam

. Properties. 3.12 3_·4 Details of Steel Used. 3.14 3.5 Values of the Shear Modulus of

Elasticity for Plain Concrete Beams. 3.25

3.6 Failure Loads of Beams Containing Only Longitudinal Steel. 3.38

3.7 Experimental Results for Series RE, RU and R. 3.41

3.8 Experimental Results for Series V~ U and T. 3.53

Table Title Page No~ -4.-1 A Comparison of "Elastic"

and "Plastic" Failure Stresses. 4.4

4.2 Analysis of Pure Torsion Tests on Plain Concrete Specimens. 4.7

4.3 A Comparison of Theory and Experiment for lj3eams Containing only Longitudinal Reinforcement Loaded in Pure Torsion. 4.13

4.4 Computed Failure Stresses for Plain Concrete Rectangular Beams Subjected to Bending and Torsion. 4.17

4.5 A Comparison of the Theory with Experimental Results for Beams Containing only Longitudinal Steel Loaded in Bending and Torsion. 4.29

4.6 Nominal Shear Stress at Failure. 4.33 4.7 Percentage of Ultimate Load at

which Flexure Cracks were First Observed. 4.33

4.8 Average Shear Stress at Failure on Reduced Section. 4.36

4.9 A Comparison of the Theory with Experimental Results for Beams Containing Only Longitudinal Reinforcement Loaded in Shear and Torsion. 4. 39

Table Title Page No.

5. 1 Error Introduced By Ignoring the Last Term of Equation 5. 4 5. 11

6.1 Values of µ for Test Beams. 6.6

7. 1 Range of Parameters Covered by Available Test Data. 7.15

7.2 A Comparison of Experimental Results With the Proposed Theory for Web Reinforced Beams. 7.17

A w

al' a2

a3,a4

b

b'

=

=

=

=

=

=

=

=

NOTATION.

The cross-sectional area of longitudinal steel near the tension (bottom) face of the beam.

The cross-sectional area of longitudinal steel near the side face of the beam.

The cross-sectional area of longitudinal steel near the compression (top) face of the beam.

The cross-sectional area of one leg of hoop reinforcement.

The cover on the longitudinal steel (see Figure 3. 2).

The cover on the transverse steel ( see Figure 3. 2).

The width of the beam (minimum dimension).

The width of the hoops.

C , Cf C , = The couple, direct force and shear force c ' s acting on the compression zone.

D = The dowel force on the steel.

d

d'

d C

=

=

=

The effective height of the beam (h-a 1).

The !leight of the hoops.

The depth of the uncTf"'!ked concrete zone. ..

d 0

d3

F w

Fl

f' C

ft

f'' t G

h

jd

Kl

K2 kd

M

M u

=

=

=

=

=

= =

=

=

=

=

=

=

=

=

The distance from the shear centre of the uncracked concrete zone to the centre line of the beam.

The distance from the bottom of the beam to the centroid of the top .steel ..

A force exerted by the web steel.

A force exerted by the longitudinal steel,.

The cylinder compressive strength of the concrete.

= Tlie ·yield strength of steels AL 1, ALZ' AL3 and Aw.

The maximum principal tensile stress.

The tensile strength of the concrete.

The shear modulus of rigidity.

The height of the beam.

The level arm of the moment of resistance in flexure.

1 - kd/h.

l - b/3h.

The depth to the neutral axis.

The bending moment applied. to the section.

The computed ultimate flexural capacity of the section in the absence of torsion.

p = AL 1/bd

R = The ratio of AL3 f L 3 to AL 1 f L l .

r = A parameter relating transverse to longitudinal steel A f

b' w w s ALI fLl

r = The design value of r. 0

s = The tensile force on the steel.

s = The spacing of the hoops.

T 1: T 2, T 3 = The predicted torsional strength in modes 1, 2 and 3.

T

T'

T C

T 0

T s

V

V C

veff V

0

=

=

=

=

=

=

=

=

=

The twisting moment applied to the section.

T V 1 + 2 .:::_

The torque resisted by the concrete.

The pure torsional capacity of the section.

The torque resisted by the steel.

The transverse shear force applied to the section.

The shear capacity of tile concrete when torsion is absent.

An effective shear force.

The shear capacity of the section when torsion is absent.

=

=

T/½b2 (h-~).

V /bd-.

x 1, x2 , x3 = 'the depth to the centre of the compression zone in modes 1, 2 and 3 ..

=

=

=

6 =

4-' = O" =

't: =

µb =

The ratio of the height to the width of the section (h/b).

The ratio of the effective height to the effective ~idth of the section (h-a/b-a2).

J 1 + 2~,c

Vb/2T.

= The inclination of the compression zone to the cross-section of the beam in modes 1, 2 and 3.

The ratio of torque to moment (T/M).

The direct stress.

The shear stress.

The portion of the top face crossed by diagonal tension cracks.

1.1

CHAPTER 1

INTRODUCTION.

1. 1 Background.

The reinforced concrete frame as employed in multi- storey

building construction followed in form the earlier method of

construction incorporating load bearing brick walls and timber

floors supported on timber girders which in turn rested on timber

storey posts. In the early application of reinforced concrete to

such structures designers were slow to take full account of the

contimdty which was inherent in this form of construction, held back

no dovbt by the difficulties associated with the analysis of such

highly redundant structures.

Following the widespread dissemination of Professor

Hardy Cross's Moment Distribution method of analysis and more

recently the availability of computer methods of analyeing frames

it has become the generally accepted practice to treat the

1. 2

reinforced concrete frame as a series of continuous, alpiet plane,

frames.

Torsion, however, has generally been completely ignored

by designing engineers who have seen it as a secondary problem which

could be taken care of by the large factors of safety inherent in

flexural design methods. This attitude or a belief that insufficient

was known of the effects of torsion has been reflected in many codes

of practice which have made no reference to methods of design for

torsion. Those which have usually only proposed approximate

working stress rra thods.

In recent years interest has been reawakened in the subject

of ultimate strength methods of design and following considerable

theoretical and experimental studies methods of designing reinforced

concrete members for flexure are now fairly well established. With

the advent of these design methods which may be relied on to give

quite accurate results the tendency will be to reduce factors of safety.

Such a trend emphasises the need for a satisfactory solution to the

problem of torsion.

Reinforced concrete structures today frequently incorporate

members subject to significant torsional moments; freeways

and interchanges require members curved in plan; slabs and shells

have edge members which may be subjected to considerable torsion;

the spiral staircase is often used as a striking architectural feature.

In view of the above the necessity of understanding the behav:io -:-- '):;_

reinforced concrete members subjected to torsion is obvious.

1. 2 Aim and Scope.

An extensive investigation of the behaviour of reinforced

concrete elements subjected to torsion was iniated in t~e

Department of Str1.:ctltral Engineering, University of New So,;th

Wales at the start of 196 :, . As a first stage in this investigatio,1

isolated rectangular reinforced concrete beams were studied.

During the first two years an extensive literature survey was

conducted and the behaviour of beams subjected to combined flexvre

and torsion was studied. It is to be noted that this stage of the

investigation was conducted as a joint project with P .F. Walsh.

During the last eighteen months of the program the author devoted

his attention to beams loaded in combined bending, torsion and

transverse shear while Walsh (Ref. 1. 1) studied the deformation

properties of beams loaded in combined flexure and torsion.

The particular aim of this thes~s is to study the behaviOl,:r

of isolated rectangular beams loaded in combined torsion, bending

and transverse shear, and from this study to obtain expressions

for the ultimate load of such members. Although it is recognized

that bending mor.1.1.ent and shear force do not play independant roles

in relation to the strength of a reinforced concrete beam, the

generally accepted practice i-s, design of treating these actions

separate}_? has been followed in this work.

A subsidiary, but no less important aim of the thesis ::.s ,:o

prena:i.t relatively simple methods of design for combined acticns ir..

keeping with the results of the investigation.

CHAPTER 2

HISTORICAL REVIEW

2. 1. INTRODUCTION.

2. 1

It is the object of this chapter to summarise the results of reported

investigations into the behaviour of reinforced· concrete members subjected

to torsion, with and without other actions. In addition it is intended to

discuss the various strength theories which have been proposed and to

compare the published experimental results with predictions by these

theories.

Reinforced concrete beams may for convenience be divided into two

main groups: beams with longitudinal steel only and beams with both

longitudinal and transverse steel. This division has been maintained in

this chapter in discussing the effects of torsion.

2. 2. BEAMS CONTAINING ONLY LONGITUDINAL REINFORCEMENT.

a. Pure Torsion.

As early as 1911 it was recognized that beams reinforced with

longitudinal steel only and loaded in pure torsion fail immediately after

2.2

the appearance of the first diagonal tension crack. The twisting moment

to cause such cracking was found to be comparable with the maximum

twisting moment which can be resisted by a plain concrete section

similar to the reinforced member in all respects except for the

reinforcement. In some cases small increases in torsional strength,

with the addition of longitudinal steel, have been recorded (Ref. 2. 1).

Cowan (Ref. 2. 2) suggests that these increases can be explained if

consideration is given to the higher shear modulus of the steel which is

replacing the concrete. However tests by Young, Sagar and Hughes

(Ref. 2. 3) showed lower strengths for the reinforced sections than for

comparable plain concrete sections. This lowering of the torsional

capacity with the addition of longitudinal steel, was probably due to the

presence of stresses induced by restrained shrinkage.

While most investigators:·agree that the pure torsional strength

of a beam containing only longitudinal steel is reasonably close to

the strength of a plain concrete member, they do not agree on how the

strength of such a plain concrete beam may be calculated. It has been

generally agreed that a maximum principal stress criterion is the

appropriate failure criterion to employ. There has not, however, been

agreement as to the distribution of stress in such a member at the point

of failure.

Most of the early investigators advocated the use of the elastic

theory for computing the torsional stresses. In accordance with this

theory, the ultimate torsional moment of a rectangular concrete beam :i.s:

T = keb2 h f' , t ......... 2. 1.

where bis the width of the beam (the smaller dimension),

h is the depth

ft is the tensile strength of the concrete

and ke is a function of the ratio h/b.

2.3

Numerical values of this function have been calculated by St. Venant

(Ref. 2. 4) and published by Cowan (Ref. 2. 2) but a useful approximation

to these values is given by:

l ke = ----------2.6

3 + ......... 2. 2.

0. 45 + h/b

In 1934 Turner and Davies (Ref. 2. 5) drew attention to the fact

that the behaviour of concrete in torsion is to some degree "plastic".

This plasticity leads to a redistribution of stresses as the load

approaches its ultimate value. Turner and Davies suggested that this

effect might be allowed for, in calculating the ultimate torque, by

multiplying the elastic torque by a factor of l . 2.

Marshall (Ref. 2. 6), in 1944, and Nylander (Ref. 2. 7), in 1945.

both suggested that consistent results were best obtained if concrete wc.s

treated as an ideal plastic material. At failure the torsi.onal shear

stress would then be constant over the whole section and equal to the

ultimate tensile strength of the concrete. It followi:;, that for a

rectangular beam, the torsional strength would be given by:

T : .! b 2 (h - b) f I 2 3 2 3 t . . . . . . . . . . .

2.4

This equation can be r.ea~ranged to give:

. . . . . . . • . 2. 4.

where ~ = ½ (1 - 3h~b) . . . . . . . . . 2. 5.

Examination of equations 2. 2 and 2. 5 will show that for a wide range of

the ratio depth to breadth of a rectangular section, the ratio of kp to

ke will be 1. 66 + O. 06. This means that for all practical rectangular

beams, the ultimate torque as calculated from the plastic theory will be

1. 66 + 0. 06 times the ultimate torque as calculated from the elastic

theory.

A major problem in the use of either the elastic or the plastic

theory is the determination of the tensile strength of the concrete.

Ideally the tensile strength should be found by direct tensile test, but

the test is very troublesome and so it is rarely performed. Usually

the tensile strength is found by indirect test, either by a bending test

or a splitting test. The value of the tensile strength calculated from

such a bending or splitting test will depend upon the particular

assumption made regarding the behaviour of the concrete, e. g. linear

-elastic, non-linear elastic or plastic.

Kemp (Ref. 2. 8) in 1961, analysed the results reported in the

literature of plain concrete beams in pure torsion. He compared the

maximum tensile stress at failure, as calculated by the plastic theory,

with the compressive strength of the concrete. He concluded that a

conservative estimate of the maximum tensile stress at failure was

4~, where_f~ was the compressive strength of the concrete. In

accordance with this the torsional strength would be given by:

2 b ~ T = 2 b (h - -) f' . 3 C

. ........ 2. 6.

While it might be expected that equation 2. 6 would be less

accurate than equation 2,. 3 it has the advantage that the torsional

strength is expressed in terma of the most commonly specified

parameter of the concrete, its compressive strength.

2. 2. b. Bending and Torsion.

2.5

Except in very exceptional circumstances torsion will be

accompanied by bending in reinforced concrete construction. Very

frequently the bending moment will be much !anger than the twisting

moment and the designer may then elect to use a beam with longitudinal

steel only. It is therefore important that information regarding the

possible interaction of torsion and bending be available for such beams.

Unfortunately the problem of the behaviour and strength of beams with

longitudinal steel only subject to combined bending and torsion has

received very little attention.

Nylander (Ref. 2. 7) in 1945, conducted the first series of tests

on this type of beam. He considered that beams with an "ordinary

ratio" of reinforcement, failed by yielding of the longitudinal reinforcement.

By analysing a cracked section of a beam sustaining bending and torsion,

he deduced that the torsion was resisted partly in the uncracked concrete

zone and partly by a couple composed of a horizontal shear force in the

uncracked concrete zone and a dowel force associated with the steel.

The torsion resisted wholly by the concrete he expressed in the form J3 To,

where To is the pure torsional capacity of a plain section, and J3 is a

constant depending upon the shape of the beam and the percentage of

2.6

longitudinal reinforcement. Appropriate values of 13 for beams of

rectangular section, determined by means of test results, were given

by Nylander.

e. g. 13 = 0. 55 when d/b = 1. 5 and p = 0. 005.

He then computed the shearing stresses set up in the steel by the

remaining torque. Additionally he calculated the direct stresses in the

steel by the normal bending formula. Using the Huber-Beltramis

criterion of yield for steel given below:

f- = J f 2 + 3 l 2 , Ll X

where fx is the direct stress and 1:' the shearing stress, he obtained the

following formula for the amount of longitudinal steel required,

1 ALl =

fLl

2 + 3,T - 13To)

()..'5d .......... 2. 7.

The design equation given by Nylander rests on two main

assumptions; that a large portion (up to 85° / o) of the torque is resisted

wholly by the uncracked concrete zone and that the failure is governed by

yielding of the steel. These assumptions are not universally true.

Frequently the shear force which the steel can resist will be governed by

spalling of the concrete and not yield of the steel as has been assumed.

Furthermore, for high ratios of torsion to flexure, it is possible for a

tension crack to cross the top surface of the beam prior to failure, and

destroy the "uncracked" concrete zone.

Ramakrishnan and Vijayarangan (Ref· .. 2. 9) in 1963 published the

results of a series of tests on beams without web reinforcement. They

observed that the presence of a flexural moment did not noticably effect

2.7

the torsional strength of their beams. They concluded that the

torsional strength of such beams could be calculated by ignoring

both the longitudinal reinforcement and the applied bending moment, and by

using an elastic distribution of stress and a maximum tensile stress

criterion of failure. They proposed the following empirical

relationship for the limiting value of the tensile stress:

ft, = 2. 6 Cu 2 / 3 2.8

where Cu is the cube crushing strength of the concrete. As this

empirical relationship was based on only the few results from their

own test beams, all of which were made from the one concrete mix,

it is not surprising that good correlation between their experimental

results and their theoretical predictions '.was: obtained.

' .

Gesund and Bo~ton (Ref. 2. 10), in 1964, proposed a theory for

calculating the failure loads of the type of beams being considered here.

They analysed the case in which the torsional capacity of the beam is

governed by spalling of the concrete. They suggestoc'1 ·that.i:he torsion

failure will only be influenced by bending moment in as much as the

magnitude of the dowel force required to spall the concrete depends on

the spacing of the flexural cracks. By assuming that the dowel force on

any bar is proportional to its distance from the axis about which the

beam rotates at failure, they obtained the following formula for the

torsional capacity:

T = F (r + .! l::: r/) c c re

......... 2. 9

where F c is the dowel force on the critical bar, that is the bar at which

spalling of the concrete occurs, re is the distance of this bar from the

failure hinge and r. is the distance of the ith bar from the failure hinge. 1

2.8

To find the value of F c it is necessary to calculate the force

required to spall off a block of the concrete. The method proposed

by Gesund and Boston is a trial and error process and involves

making assumptions, not easily justified, regarding the shape of the

concrete spall, the spacing of the flexural cracks, the bond strength,

the average tensile stess in the concrete, the area of the compression

zone in the beam and the bearing-stress distribution along the bar.

In view of the number of assumptions oh which the method is

based it is not unexpected that agreement between the theory and

experimental results is only "fair". Furthermore, as the method

of calculating the failure torque is a cumbersome trial and error process

it does not commend itself as a design method.

To obtain more information about the three methods proposed

Table 2. 1 was prepared. This table contains the failure loads of all

beams, of the type under discussion, tested by previous investigators.

In this table the failure loads predicted for these beams, by the three

theories discussed above, are compared with the experimentally

determined failure loads. It can be seen that all investigators obtain

reasonable correlation only for their own test results. Both

Nylander' s and Ramakrishnan's theories give estimates of the ultimate

torque very much greater, in some cases, than the experimental values.

Gesund and Boston's theory, on the whole, seems the most satisfactory

of the three. It is considered that the standard of accuracy attained,

however, hardly iustifies the complex nature of their analysis.

2. 2. c. Torsion, Bending and Transverse Shear.

Despite the relative importance of the problem of shear force

combined with torsion and bending, very little experimental work has

TABLE 2.1

A COMPARISON OF EXPERIMENTAL RESULTS WITH THE THEORIES OF NYLANDER, RAMAKRISHNAN AND VIJARANGAN AND GESUND

AND BOSTON

Investigator Beam Failure Loads T exp/Tth

No. eor.

Torque Moment Nylander Rama- Gesund kip. in. kip. in. krishnan and

and Vija- Boston rangan

1 39.0 52.l 1. 12 0.72 1. 09 2 31. 2 52.l 0.97 0.72 0.88

Nylander 3 39.0 58.0 1. 15 0.79 1. 10 4 35. 1 58.0 1.07 0.79 0.99 5 54.6 75.7 0.85 0.86 1. 70 6 50.7 75.7 0.90 0.80 1. 61 7 50.7 110. 0 0.98 0.74 1. 51 8 54.6 110. O 1.03 0.80 1. 63 9 31. 2 58.0 1.07 0.89 0.89

10 19. 5 58.0 0.89 0.89 0.89

B4 17. 1 99.0 1. 36 1. 34 1. 34 BS 24.8 45.4 0.90 1.03 0.81

Ramakrishnan B6 10.7 108.0 1. 39 1. 39 1. 39 and C3 21. 7 111.0 1. 12 1. 06 1. 08

Vijarangan C4 20. 1 90.7 1.00 0.95 0.91 CS 23.2 105.0 1. 03 1. 01 0.99

3 58 58 1. 10 0.69 1. 09 4 64 64 0.82 0.77 1. 60 5 43 86 0.74 0.83 1. 48

Gesund 6 36 108 0.74 0.68 1. 24 and 7 59 177 1. 05 0.93 1. 31

Boston 8 49 195 1. 06 1. 02 1. 09 9 42 83 0.44 0.49 1. 50

10 39 156 0.63 0.63 1. 56

Summary Maximum 1. 39 1. 39 1. 70

Minimum 0.44 0.49 0.81

Mean 0.98 0.87 1. 23

21°/o 24°/o 0

Standard Deviation 24 /o

2 10

been carried out for beams reinforced with longitudinal

reinforcement alone. In fact Nylander (Ref. 2. 7). appears to

have been the only person who has experimentally investigated the

problem. As might be expected, he found that the effect of direct

shear force is to substantially reduce the torsional capacity of a beam.

Nylander tested a series of ten identical beams under an eccentric

concentrated load. The eccentricity of the load varied from zero, i. e.

no torsion, to infinity, i. e. pure torsion. He concluded that the load at

failure, in the case of combined shear and torsion, could be computed by

putting the combined shearing and torsional stress equal to the tensile

strength of the concrete. (In his investigation he conducted direct

tension tests to determine the tensile strength of the concrete). He

suggested that the stress due to transverse shear be computed from the

well known formula,

Vy = V

b"d' ] ......... 2. 10

and for this case he suggested that j be taken as equal to 0. 764. He

suggested further that the stress due to the torsion be calculated in

accordance with the plastic theory, i.e. :

= T ......... 2. 11 ½ b2 (h-b/3)

While Nylander's theory agrees fairly well with his test results

it ignors the complex nature of the shear failure. The theory implies

that the beam will fail in shear when the nominal shearing stress, vv,

equals the tensile strength of the c_oncrete. A. N. Talbot (Ref. 2. 11)

pointed out the fallacies of such a procedure as early as 1909. More

recently Keni (Ref. 2. 12) has shown that, for the same concrete, the

value of Vy may be of the order of 15 times greater for heavily

reinforced short beams than for long beams with a low percentage of

reinforcement.

2. 11

Kemp {Ref. 2. 8), after examining Ny lander's results, proposed

that a safe lower bound for the failure loads of the type of beams under

discussion is,

V Vo

+ = 1 ......... 2. 12.

where V is the ultimate shear for combined shear and torsion, T is the

ultimate torque for combined shear and torsion, VO is the ultimate shear

for zero torsion and T 0 is the ultimate torque for pure torsion. Of course

if VO and T O are both calculated on the basis that vv and vt equal the tensile

strength of the concrete, which is constant, then the above theory is

identical to Nylander's. However, Kemp recognized the "nominal"

nature of both these shearing stresses and so he suggested that V0 be

calculated on the basis of a nominal stress of 2/it while Tu be based C -

on 4.fo_. These values apparently were chosen to tie in with the value C

Kemp suggested for the case of pure torsion and the values currently

being used in shear design.

The available test results, i. e. Nylander's, for longitudinally

reinforced beams loaded in shear and torsion, have been plotted in

Figure 2. 1 in the form of a shear-torsion interaction diagram. It

appears that for the particular reinforcement ratio and shear span ratio

used by Nylander, the interaction between torsion and transverse

shear is approximately linear. Also plotted in Figure 2. I are the

theoretical interaction lines suggested by Nylander and Kemp. It can

be seen that for these beams both theories give safe results.

2.12

16,.--~--------~-------I I

tl\ NYLANDER S EXPERIMEN

' TAL RESULTS -• ' -.i ' . 12------t-' ----+-----+------t

~~i ~,~~,

,~ 1,-} C ,,. ·- 8 '9,. ci. ,i---..-----t-- <"'-\.~---+-------+-----~ ~~ ..

~ z 0 1/) a:: ~ ' ' 'I o-------------....--------2 4 e

TRANSVERSE SH r.·· ~R

FIG. 2.1 A COMPARISON OF OBSERVED AND PREDICTED SHEAR-TORSION INTERACT!ON FOR BEAMS CONTAINING O\ILY LONGITUDINAL STEEL.

2. 13

2. 3. BEAMS CONTAINING BOTH LONGITUDINAL AND TRANSVERSE

REINFORCEMENT.

a. Pure Torsion.

A great number of experimental investigations have been

undertaken, particularly in the past thirty years, to examine the effects

of combined longitudinal and transverse reinforcement on the torsional

capacity of concrete members.

For beams subjected to pure torsion there is agreement between

almost all investigators that, irrespective of the amount or disposition

of reinforcement, tensile cracks appear on the face of the specimens,

with an inclination to the longitudinal axis of the beam of approximately

45°, when the twisting moment reaches the value of the cracking moment

for a similar beam of plain concrete.

Once the member has cracked the torsional stiffness is reduced.

The behaviour beyond this stage and the value of the maximum twisting

moment which can be resisted then appears to depend upon the amount

and positioning of the reinforcement.

The early researchers, noting that reinforcement increased the

torsional capacity of a concrete member, proposed empirical formulae

for calculating the increased torsional resistance of reinforced members.

Turner and Davies (Ref. 2. 5) in 1934 proposed an empirical relation

which may be expressed as:

T = T (1 + 0.25p'), C

......... 2. 13.

where T is the maximum torsional capacity of the reinforced member,

Tc is the cracking torque or torsional capacity of a plain concrete

member, and p' is the percentage of reinforcement. The percentage

2. 14

of longitudinal steel is p' / 2 and the area and spacing of transverse

reinforcement is such that the volume of steel per unit length of the

beam is the same as for the longitudinal steel.

Turner and Davies recommended that p' should be not less than

unity for sections carrying considerable twisting moments.

Marshall and Tembe (Ref. 2. 13 )1 in 1941, agreed with Turner and

Davies' proposal for values of p' less than 1. 5, but recommended that

for higher values of p' the following expression should be used:

T = T (1. 33 + O. lp'). C

. ........ 2. 14.

Rausch (Ref. 2. 14) in 1929 attempted a rational solution to the

problem of estimating the torsional capacity of reinforced concrete beams.

Rausch considered the reinforced member as being analogous to a space

frame, in which tensile forces could be resisted by the reinforcement -

either spirals or combined hoops and longitudinal bars - and compressive

forces could be resisted by the concrete. In this way he developed the

following expressions for the required web and longitudinal steel.

s A = T,

w 2f b'd' ......... 2. 15.

w

and AL (b' + d') T, = f b'd' L

......... 2. 16.

where A is the cross-sectional area of one leg of the hoop reinforcement, w

AL if the area of longitudinal bars, b' d' is the area enclosed by the hoops,

and s is the spacing of the hoops.

2 11~

In Rausch's expressions fw and fL would have represented the

permissible stresses in the transverse and longitudinal steel

respectively. However, as his treatment of the problem ignored

completely any tensile {i. e. shear) strength of the concrete, his

expressions could equally well be regarded as relations between the

steel areas and the ultimate twisting moment, with fw and fL

representing the yield strengths of the steel. Indeed Rausch in discussing

certain published test results stated that the very high twisting of the

members led him to believe that the reinforcement yielded before failure

of the members.

Rausch' s expressions are still quoted in many modern codes for

reinforced concrete constructipn: Germany, Egypt, Hungary and Poland.

Andersen {Ref. 2. 15) carried out a series of tests on beams in

pure torsion in 19 35 and 19 3 7. He employed spiral reinforcement in the

majority of his test specimens as he considered such reinforcment to be

most effective in resisting torsion. His theory of ultimate strength is

based on the assumption that the concrete resists a portion of the

ultimate moment and the remainder is resisted by the spiral reinforcement.

His approach is to assume that the shear {tension) stresses in the concrete

vary linearly from zero at the centre to a maximum, equal to the concrete

tensile strength, at some distance from the centre and beyond this point

the stress in the concrete is constant at the maximum value. The

moment of resistance of the concrete can be calculated in terms of the

parameters, tensile strength, radius of the section and distance to the

point of maximum stress. The total moment of resistance in torsion is

calculated by assuming that the section is homogeneous and that the

extreme fibre stress is the value obtained by considering the same linear

2. 16

variation as described above, reaching a maximum (higher than the

concrete strength) at the edge of the section. The moment carried

by the reinforcement then is the difference between the total moment

and the moment carried by the concrete. The expression for the

moment carried by the reinforcement is exactly the same as Rausch' s

expression for a circular cross-section.

To deal with the case of square and rectangular cross- sections

Andersen relates such sections to equivalent circular sections. He

determines the radius of the equivalent circular section by equating

the expressions for maximum shear stress for the rectangle to that of

the circle for a given twisting moment.

Cowan (Ref. 2. 16), who has written extensively on the subject of

torsion in concrete, in 1950 proposed that the torsional resistance of a

concrete beam was provided partly by the concrete and partly by the

steel. He suggested a similar formula to Rausch for calculating the

contribution of the steel, but with the numerical value 2 replaced by a

parameter :l. . Making use of the St. Venant torsion solution for a

rectangular shaft, he deduced that ;l. would vary with the ratio of

depth to width of the cross-section, but that for wide variations of this

ratio it would differ only slightly from 1. 6.

Cowan's work forms the basis of the torsion provisions of the

present Australian Building Code (Ref. 2. 17).

All of the aforementioned proposals agree in one respect in

requiring equal volumes of steel per unit length of beam in both the

longitudinal and transverse directions. It has further been suggested

that where these volumes are not equal the smaller of the two will

govern the torsional capacity.

2 .17

To investigate the above theories, an analysis has been made of

those test beams in which there was sufficient or more than sufficient

longitudinal steel to satisfy the equal volume requirement. Where the

cracking torque, T , was required for the analysis it was obtained from C

the reported test results of companion beams without web reinforcement.

In Figure 2. 2 the ultimate torque of the test beams, expressed as

a ratio of their cracking torque, is plotted against the percentage of

reinforcement that they contained. Also plotted on this figure are the

equations suggested by Turner and Davies (Equation 2. 13) and Marshall

and Tembe (Equation 2. 14). As may be seen from Figure 2. 1, these

equations do not give estimates of the torsional capacity of reinforced

beams which agree well with the test results. Further in order ~o employ

these methods in practice, it would be necessary to estimate the

cracking torque of the beam and this can not be done with accuracy.

The theory proposed by Rausch (Equation 2. 15) is compared with

the experimental results in Figure 2. 3. In this figure the ultimate

torque of the test beams, expressed as a ratio of their cracking torque is A f

plotted against the dimensionless parameter b'd' w w which is a measure s Tc

of the amount of web steel in the section. Now Rausch's theory can be

expressed as

T T

C

= 2 b'd' A f

WW sT

C

......... 2. 16

and in Figure 2. 3 this equation is plotted. It would appear from this . A f

figure that for values of b'd' w w between 0. 5 and 1. 0, tolerable s Tc

agreement with experimental results is obtained. For higher values of

this ratio, however the theory may grossly overestimate the ultimate

T -Tc

3.0. I I I I I

Ernst • • 1 I Marsht 111 • Tembe 0

2.5f I I I ·1 I Anders,n X

Cowan * :1 I I I Turner ,. Davies 6 I I

2.0

I 1: I I 0

1.5 I ·- - I I LI ----- I I"

0

1. o...,_ - a

0.5-.-~, I I

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 p'

FIG. 2.2. ULTIMATE TORQUE/ CRACKING TORQUE - PERCENTAGE OF REINFORCEMENT

~ ~ CD

T Tc

3.0. . - - ' LEGEND:

• I Ernst • 2.51 I /l -~ I • I Marshall•Tembe o

2.01 Iv~ I-

1. 51 LJ: i I

0.50 0.5 1.0

I

I 0 0

1.5

b1d' fw Aw STt.

Andersen Cowan

I Turner a. Qavies

l 0 0

2.0 2.5

FIG. 2.3 TORSIONAL STRENGTH- AMOUNT OF

r RANS VERSE REINFORCEtv1ENT

X

• ~ -

20

t,.) ·.,

l.o

2.20

torque while for low values, that is for lightly reinforced sections,

Rausch's theory will underestimate the failure torque.

Cowan assumed that the torque on a reinforced section is

resisted partly by the concrete and partly by the steel. That is

T = T + T ......... 2. 17. C S

He obtained the following formula for the contribution of the

reinforcement to the torque:

f A T = 1. 6 b'd' ~ ......... 2. 18.

s s

where f s is the steel stress.

As Cowan was concerned with developing a working stress

design method the term f in his equation really refers to the maximum s

permissible steel stress. His method has however been used (Ref. 2. 18)

to compute the contribution of the web steel at ultimate load and for this

purpose fs would be replaced by fw, the yield strength of the steel.

Equation 2. 17 can be rearranged to give:

T T

C

T = 1 + s

T C

and substitution for T we have: s

T T

C

A f = 1 + 1. 6 b'd' w w

sT C

......... 2. 19.

2.21

This equation has been plotted on Figure 2. 3. Examination of this

figure reveals that equation 2. 19 overestimates the effect of the steel.

In the Australian Code the unconservative nature of Cowan's

formula for the "additional twisting moment due to the insertion of the

hoops" is offset by a very low estimate of the torque taken by the concrete

and by placing severe limitations on the amount of reinforcement permitted

in any section.

Andersen's method, although slightly more conservative than

Cowan' s is subiect to the same limitations. It has the additional

disadvantage of being extremely complex.

Ernst (Ref. 2. 19) in 195 7 conducted a series of tests to investigate

the effect on the torsional strength of variations in the ratio of longitudinal

to transverse reinforcement. He found that both the longitudinal and

transverse steel yielded at failure for wide variations in this ratio.

Further he found that increasing the percentage of longitudinal steel

increased the torsional capacity of the beams, even though the volume of

transverse steel per unit length was not increased. The theories outlined

above cannot account for this increase in strength.

2. 3. b. Bending and Torsion.

Beams subjected simultaneously to the combined actions of a

twisting and bending moment are of much more interest, from the

practical viewpoint, than members with torsion only. This subject has,

fortunately, received considerable attention of late years.

The first worker in this field was Cowan (Ref. 2. 20) who in

195 3 put forward a theory for reinforced concrete beams subjected to

the type of loading under discussion. He emphasized the fact that in

this theory he was concerned mainly with the behaviour up to the

"visco-elastic limit" (this point corresponds with a marked change in

the slopes of the load-deflection and torque-twist curves).

2.22

His theory therefore is not intended to refer, necessarily, to the

maximum load such members can sustain.

Cowan contends that it is necessary to distinguish between two

distinct types of failure. Under predominantly bending loads the concrete,

presumably on the top surface of the beams, crushes while under

predominantly torsional loading a cleavage failure, recognizable by the

clean appearance of the fracture, occurs. By making use of two possible

criteria of failure, a maximum principle stress criterion applying to

the cleavage failures and the internal friction theory for the crushing

failures, he concluded that "a moderate amount of bending does not

decrease the torsional strength, but on the contrary increases it"

(Ref. 2. 2). Cowan also concluded that the addition of a small amount of

torsion will reduce the bending strength only slightly. He therefore

decided that it was reasonable "to assume that no reduction in the

maximum permissible stresses is required when flexure and torsion

occur in combination". To illustrate the above conclusions Cowan drew

the diagram which is reproduced in Figure 2. 4.

It will be seen later that in the light of more recent investigations

the above statements by Cowan require modification.

In the last few years several investigators have proposed theories

to calculate the ultimate strength of beams containing web reinforcement

and loaded in combined bending and torsion. It is generally agreed that

failure of the beams takes place ,in the manner suggested in Figure 2. 5.

The beam fails when tension crrcks on three of the sides open allowing

the segments of the beam to rotate about an axis located near the

fourth side. At failure the concrete in this face crushes. In

analyzing this mechanism various investigators have made differing

--1

tz w 2'. 0 2 (!) z IV')

~ I-

BENDING MOMENT

FIG. 2. 4. COWAN1S SUGGESTED INTERACTION DIAGRAM FOR REINFORCED CONCRETE BEAMS.

I I I I

2.23

THEORETICAL INTERACTION-FU:....'.... LINE DESIGN SIMPLIFICATION - DOTTED LINE

/ ·--------------------·--·-- ,· ,,

·'

,.,

---·'

, , ..

FIG, 2. 5 MANNER OF FAILURE OF A BEAM LOADED IN BENDING AND TORSlON f\)

f\) ~

2.25

assumptions. For example the direction of the axis about which the

beam rotates has been taken as parallel to the longitudinal axis of the

beam, Gesund (Ref. 2. 21), joining the ends of the tension spiral,

Lessig (Ref. 2. 22) and Yudin (Ref. 2. 23), or at a certain angle to the

longitudinal axis of the beam, Evans (Ref. 2. 24). Attention will now be

given to the various theories produced by these, and other, differing

assumptions.

Lessig was the first worker to derive expressions for the ultimate

load of this type of beam. For the purpose of analysis she assumed that

the intersection of the failure surface with the beam faces are straight

lines and further, that the inclination of these lines on the three sides

corresponding to the tension cracks is constant. Her idealized failure

surface is shown in Figure 2. 6. This type of failure is called Mode 1 by

Lessig to distinguish it from the failure surface which she used for

combined shear and torsion and labelled Mode 2.

In her analysis, Lessig assumed that all steel traversed by the

failure crack yields. She recognized that this was not always the case

but she considered it the most common cause of failure. "Premature"

failures in which the steel does not yield are dealt with by empirical

formulae. These will be treated in detail later.

In the analysis of Mode 11 the forces in the vertical legs of the

hoops intersected by the failure crack, dowel forces and tensile stresses

in the concrete are ignored. As mentioned above, an inclined failure

hinge is assumed. This hinge has a projected length of q (see I

Figure 2. 6). Moments about this hinge are calculated for the tensile

forces in the longitudinal reinforcement and in the bottom horizontal

parts of the transverse steel. These moments are then equated to the

components of the external moments about this axis. The length of the

2.26

\ N-._.,_....-------+-------,.----\\ \

b

1 h

~ ~-------J FIG. 2.6 BENDING-TORSION FAILURE SURFACE PROPOSED

BY LESSIG. MODE 1

2.27

failure hinge which makes the external moment a minimum is then

determined. Finally the depth of the compress ion zone is found by

equating the compressive forces perpendicular to the failure surface

to the components of the tensile forces in the steel and by assuming that

stresses in the other directions will not effect the failure of the concrete.

The final expressions obtained by Lessig have been presented by

her in a number of slightly different forms. When incorporated into the

Russian Code (Ref. 2. 25) the expressions \\ere presented in the following

manner:

, ...... 2. 20.

b 2 AL l fL 1 s ( 2h + b) (-) i' + A f ...... 2. 21.

WW

but 2h + b

b

where T 1 is the failure torque in Mode 1 and a,p = T /M, the other

symbols being defined in Figure 2. 6. A full derivation of the above

equations is available in Ref. 2. 2.

Lessig does not explicitly discuss the interaction of bending and

torsion, and the form of her equations does not make it easy to discover

what she would predict. Upon examination it is found that her theory

suggests that the shape of the ''interaction curve" depends upon the

2 28

properties of the beam in question. However, for a large number of

beams, the shape predicted will be similar to that shown in Figure

2. 7. For all beams, Lessig's equations would predict that the

presence of bending decreases their torsional capacities, and the presence

of torsion their flexural capacities.

Tests by Chinenkov (Ref. 2. 26) verified that torsion does indeed

reduce a beam's flexural capacity. He reported that the following

reductions had been observed:

Ratio of loading.

T/M = 0. 1

T/M = 0.2

T/M = 0.3

T/M = 0.4

Reduction in Bending Moment at Failure.

0 - 1°/o.

7 - 19°/ 0.

23 - 2s0 /o. 29 - 33°/o.

Chinenkov concluded that Lessig's formulae predicted failure loads

agreeing well with the test results. Experimental values slightly higher

than the predicted values could, he thought, be accounted for by the tensile

stresses in the uncracked concrete which were ignored in the theory. In

connection with Lessig's assumption regarding the constant inclination of

the tension crack Chinenkov stated that this assumption did not agree with

observed facts.

It is to be noted that Lessig intended that the equations given above

should only be applied to those beams which fail by yield of both the 1

longitudinal and the transverse steel. If the theory was applied to beams

in which the reinforcement did not yield it would lead to unconservative

results. Limits must thus be placed on the amount and the distribution

of the reinforcement to exclude this case.

1-z w 2 0 2 (!) z 1-ll)

~ 1--

BENDING MOMENT

FIG. 2. 7 BENDING TORSION INTERACTION AS PREDICTED BY THE THEORY OF LESSIG.

2. 29

2.30

In the first instance the amount of reinforcement must be limited

so that the concrete near the top face of the beam will not crush before

the steel yields.

Lessig conducted a series of tests in 195 7-5 8 (Ref. 2. 2 7) in an

attempt to define this limit empirically.

She observed that even with high percentages of steel this mode of

failure rarely occurred for values of 4' in excess of 0. 2. From those

tests in which compression failure occurred she obtained the following

relation between the depth of the compression zone, xc, and the ratio

of loading,

h~: = 0.55 - 0.7fo (0~ 4'~ 0.2) ....... 2.23 2

For cases where x , from the above, was greater than or equal C

to twice the distance from the top face to the centre of the top steel, the

relation between steel areas and x would be: C

X C'

.... 2. 24.

whereas if the indicated value of x were less than twice this distance the C

steel area would then be given by:

0. 85 f I b ( 1 + 5 4' ) X . C C

. ........ 2. 25.

The number of tests on which the empirical equation 2. 23 was

based was rather limited and Lessig suggested there is a need for

further work on this aspect of the problem.

2.31

A much more common case of concrete compression failure

occurs when the ratio of twisting moment to bending moment is higher

than for the case mentioned above. In this case the concrete near the

side face crushes before the steel yields.

A number of beams tested by Lessig failed in this manner.

She assumed that the ultimate twisting moment of such a beam could be

expressed by a relation of the form:

T = K b 2 h f'. C

As K would appear dependent upon the relative values of twisting

moment and direct shear force, she examined the effect of a ratio involving

these two actions. For the type of failure being considered she concluded

that there was no correlation between this ratio and the value of K. As

K varied between 0. 07 and 0. 12 she suggested that such a compression

failure would not occur if:

T 0. 07. . ........ 2. 26.

To test the effectiveness of this limitation, i. e. its ability to

exclude unconservative results, a large number of test results ha.ve

been examined. For these results the failure torque predicted by the

theory, Tth , has been compared with the failure torque observed in eor

the tests, T , in the form of a ratio T /Tth exp. exp eor. If the value of

this ratio is less than one then the theory is unconservative. In Figure

2. 8 the ratio T /Tth has been plotted against the value of K, i. e. 2 exp eor

Tth /b h f' . The general trend of results shown in the figure suggests eor c

1.6--------,.----T"--~----T---~--'"'1"-------------, LEGEND This Invest • Chinenkov X

~

+ Ernst 0 Lessig +

1.2 .. t .... Lyalin • Yudin I -i-

- - - --+ -- --------- , CC1NOn 0 Evans e L: +I 1Gesund 0 0 Cl> .c ~

"-. 0.8 • • .. j+ I + +I + L-+- ~7- - r ·---- --- - - -+--- - - ·· -· -- · -ci. )(

I-Cl>

-0

Cl) Cl> :,

g

+ 1-------+------+-------1--- ----+------+--------1-------+---- --+--------I- -------+·--- - --- -

+

o.~: 1-- - - --- +----- - ----- +-- ·----- -------+---------- --- --1------ ·-------·-+--· -- -- -- -- -1----- - - - - -

-- +- - - ---+--- - - -+- - -- --~ - - ---+---- ---

0 0.02 0.04 0 .06 o.os 0.10 0 12 014 . 0.16 0.18 0.'20 0.22

T. Values of K = theor.

t' b2 h C

FIG 2.8 ACCURACY OF LESSIG1 S THEORY VERSUS K

I\)

w I\)

2.33

that the criterion is a satisfactory one. Thus if beams having values of K

higher than 0. 07 are ignored most of the unconservative results are

eliminated. It can also be seen from the figure that applied within the

limitation Lessig's theory gives reasonably accurate results.

Apart from excluding cases in which the total amount of

reinforcement was too great Lessig also felt that it was necessary to

exclude those cases in which the ratio of the transverse to the

longitudinal steel was inappropriate for the ratio of the loads applied to

the beam. This limitation was to ensure that both the longitudinal and

the transverse steel yield at failure.

Lessig and Lyalin (Ref. 2. 28) attempted to establish the limits of

the ratio of transverse to longitudinal steel for which yielding of both

steels could be guaranteed. They fixed limits as follows:

A f w w

o. 5 ~ A f Ll Ll

b s

( 1 + ! j 2hb + b) 6. 1 . 5. . . . . . . 2. 2 7.

Lessig further suggested that beams which do not satisfy this

restriction could be analysed by ignoring the excess guantity of

transverse or longitudinal steel.

The limits imposed by Equation 2. 27 are very severe and have the

additional disadvantage of being difficult to calculate. To investigate the

necessity for such a restriction Figure 2. 9 has been prepared. In this

figure the accuracy of the theory, i. e. the ratio T /Tth , is exp eor

compared with

a parameter R1 !o + ~ ~>-

L: 0

2.

1.

1.

1.

1.

l 1. +'

1-

0

8

6

A

2·

:)

b 0

0 A -

0 0(

u

"--o. d. 91',.J 0

X Cl)

1- 0.6 •u

0 VI Cl) :J

0. .

0 > 0. •

0 0

0

0.2

/\.

0

u~

")(.

X X

0

D

• I

- ------r-l-- -

..I.. I + I .. I • +

I I • H+ +a ., • ~ • .,. u ,.J • axo • 00 • I D I

I . I I I I

I 04 0.6 0.6 1.0

Aw fw b [ 2. Jb ] Values of R,= -------- • S 1+ 'fJ V ~b ALI f LI

FIG 2.9 ACCURACY OF LESSIG'S THEORY VERSUS R1

•o C

2.0 3.0

D

.... ---···-------

4.0 I\)

w ~

2 35

In plotting this figure those results which do not satisfy Equation 2. 26

have been eliminated.

In the preparation of Figure 2. 9 the theoretical torque has been

calculated without heed being paid to the limitations on the ratio of

transverse to longitudinal steel. As has been pointed out above, Lessig

considers that this approach is only valid when the value of R 1 lies

between 0. 5 and l. 5. Examination of Fig. 2. 9 shows that, while it is true

that the theory gives less accurate results when the value of this parameter

is less than 0. 5, the limitation imposed by Equation 2. 27 could safely be

liberalized.

Yudin (Ref. 2. 23) in 1962 criticised the method of analysis

proposed by Lessig. While he agreed in the main with the failure model

presented by Lessig (Figure 2. 6), he contended that it would be preferrable

to consider equilibrium about two axes, the longitudinal axis of the beam

and an axis perpendicular to it. This approach led him to seperate the

components of the load on the beam. The stirrups were to resist the

torsion while the longitudinal steel was primarily to resist the bending.

Further as he was mainly interested in producing design equations he

made two more simplifying assumptions. He assumed that the inclination

of the tension spiral may be taken as 45° irrespective of the ratio of the:

loads, and that the internal "lever arms" may be taken as the dimensions

of the "core'' of the cross section. Using these assumptions Yudin

obtained the following expressions for the ultimate loads: A f

T = 2 b'd' w w

2.28. s

(Note: This is the same as Rausch's formula) and

where

b' + d' c1 = 2b'

2. 36

......... 2. 29.

Gesund, Shuette, Buchanan and Gray (Ref. 2. 21) in 1964 proposed

a theory for reinforced beams subjected to combined bending and torsion.

Their approach is essentially the same as that proposed by Yudin in that

they consider moments about longitudinal and transverse axes. However,

whereas Yudin assumed that the failure crack crossed the bottom surface

of the beam at an angle of 45° Gesund et al assumed that it crossed at the -1

variable angle 9. They took the value of 9 as cot O. 5 when i.f.J ~ 0. 25 0

and 90 when 'f' <. 0. 25. Further, they assumed that the compression

"failure region" rather than being straight as assumed by Lessig and

Yudin was S- shaped and that at failure the beam pivots about the straight

central portion of the S.

The resulting expression for the maximum bending moment in the

presence of torsion is,

M= M

u ......... 2. 30.

where M is the calculated flexural capacity of the beam and: u

C = d' (h+b cot 9) 2 b'd' + (h-a )b'cot 9·

1

Gesund proposed that the torsional resistance could be taken as

the greater of two moments; one based on the consideration of dowel

forces and the other based on the premise that the hoop steel yields

before failure.

2.37

The general form of the equation which predicts the torsional

resistance by considering the effect of the dowel forces is:

T = Fe tc + r~ f Z(h~Za4) r/ -,.2:r/] J ......... 2.31

where rt is the average radius and the other terms have the meanings

given earlier in the discussion of beams without transverse steel (see

Equation 2. 9).

The general form of the other equation which predicts the

torsional resistance and which is obtained by taking moments about the

failure hinge of the forces in the vertical and bottom legs of the

transverse steel - assumed to be yielding - is:

T = A f

WW $

... 2. 32.

where 9 is the angle between the failure crack on the bottom of the beam

and the beam axis.

For the case of pure torsion it might be expected that cot 9

would be approximately unity. With cot 9 = 1 the above expression is

essentially the same as Rausch's formula.

The form of both Yudin's and Gesund's equations suggest that the

torsional strength of a beam is not effected by the addition of bending

moment, but that the flexural capacity is reduced by the presence of

twisting moment. The interaction curve predicted by these theories

would thus be similar to that shown in Figure 2. 10.

Both of these theories divide possible failures into torsional

failures and flexural failures. They conclude that the torsional failures

are not effected by bending moment nor by the amount of longitudinal

I I ' 1:- !

z w :I. 0 :I

2 38

TORSION FAILURE

BENDlNG MCf\.1ENT

FIG. 2. 10 BENDING TORSION 1NTERACTION AS PREDICTE:t) BY T~E THECPY C'F VUDIN.

2.39

steel in the beam. The first of these conclusions is contradicted by the

results of Cowan and the theory of Lessig while Ernst has shown that the

amount of longitudinal steel has a marked effect on the torsional strength.

Further they conclude that the amount of transverse steel in the bea~ does

not effect the "flexural" failures. Gesund's own results contradicted this

conclusion as when he decreased the spacing of the ties in his beams he

increased their strength in combined bending and torsion by up to 40° / o.

In 1965, Evans and Sarkar examined the case of rectangular beams

failing in combined bending and torsion. They were chiefly concerned

with hollow members but the theory proposed was applicable to solid beams.

The failure model proposed by Evans and Sarkar is shown in Figure 2. 11.

They assumed that the tension crack would form on the bottom surface at

' an angle cl:. , normal to the direction of the maximum tensile stress, and

that the crack would spread at this same angle till it was 6/ 10th of the

way up the sides, where it would change its direction to 45°. Further,

they assumed that when these cracks opened the beam would rotate about

a "compression hinge" which would form at 45° to the axis of the beam.

By equating the moments due to external forces and those due to

internal forces about this axis of rotation, and by assuming that the steel

crossed by the failure surface yields Evans set up the following equations

for the bending moment at failure:

1 [ 2 Awfw f , M = -- f' x b + A f (d-x) + -- (h-a -x)(b-2a )Cot oe I +'fl c LI LI s 3 4

- (h-Za3)(o. 6 Cot oc' + 0. 4)[ b cotoc' + h{0. 6 Cotat' + 0. 4)-b]}]

2. 33.

Where x, the depth of compression is to be found by equating the

FIG.2.11. BENDING-TORSION FAILURE MODE OF HOLLOW BEAMS PROPOSED BY EVA~S.

2.40

2.41

compressive and tensile forces along a plane normal to the axis of

rotation.

Thus

x= 2

A f \\' w

-$--

f' b. C

I (b - 2a 4 ) cotcC.

I

......... 2. 34

In finding the inclination of the crack, cl:. , Evans suggested that a fully

plastic distribution of torsional shear stress and a semi-plastic

distribution of normal flexural stess;. as used by Cowan (Ref. 2. 20),

be employed. For a solid rectangular beam this leads to the following:

where

' cot<,(=

f =

1

( f + F+1 ) ......... 2. 35.

1 1. 06 (1 - 3d) ......... 2. 36.

oC.4' Evan:'s equations, like Lessig's, suggest that the presence of

bending moment will decrease the torsional capacity of the beams and that

the flexural capacity will be reduced by the application of a twisting

moment. The shape of the interaction curve predicted by these equations

will thus be similar to that shown in Figure 2. 7.

A major drawback of the above theory is the complexity of the

equations produced. Evans and Sarkar suggested that a quick though

fairly inaccurate, assessment of the strength of a beam in combined

bending and torsion would be given by:

M = M

u 1 +'\'

......... 2. 37.

It will be noted that the equation is similar in form to the equations

proposed by Yudin and Gesund.

2 42

For a theory to be considered satisfactory it should not only

predict correctly the qualitative effect of various parameters on the

strength of the beams but should also lead to accurate quantitative

predictions. To test the various theories proposed for beams loaded in

bending and torsion by this criterion Table 2. 2 has been prepared. This

table contains the results of beams containing web steel, loaded in

bending and torsion. Full details of these beams may be found in

Appendix D. As well as recording the failure loads of the beams the

table lists the parameters T / Tth for the various theories. exp eor Although only Lessig gave equations to exclude over-reinforced

beams, all the ultimate strength theories have been based on the

assumption of the steel yielding at failure. For this reason Table 2. 2

does not contain the results of any over- reinforced beams.

Examination of the table reveals that the theory of Gesund et al

gives the most satisfactory results for the beams listed. Thus the

ratio of the observed failure load to the load predicted by Gesund' s

theory has an average value of 1. 29 with a coefficient of variation of

15°/o. Further, the least conservative result is only in error by 12°/o.

The theory therefore gives an accurate and conservative estimate of the

failure loads. It must be remembered, however, that the theory

involves a tedious trial and error approach and so is too cumbersome

for use in design.

The theory of Evans and Sarkar, while still being complex,

involves less calculations than the theory of Gesund et al. Unfortunately

it often seriously overestimates the strength of a beam. Examination

of Table 2. 2 shows that for this theory the ratio T /Th has an exp t eor

TABLE 2. 2

A Comparison of Experimental Results and Published Theories for Beams

Loaded in Bending and Torsion. ,.

T /T, e:xp t;ieor Investigator Beam Torque Moment Lessig Yudin Evans Gesund

kip.in. kip.in.

R5 75.4 75.4 3.86 3.47 0.67 0.88

Cowan and R2 79.0 158.0 3.84 3.64 1.04 1.27

Armstrong Rl 43.0 258.0 1.95 1.98 1.32 1.51

Sl 82.6 206.5 2.95 2.85 1.26 1.52

S4 64.6 258.4 2.23 2.23 1.41 1.63

1 79.0 79.0 0.90* 1.07 0.79 1.02

2 102.0 102.0 0.91 1.26 0.93 1.32

3 61.0 122.0 0.93* 1.09 0.90 1.13

4 67.0 134.0 0.95 1.20 0.89 1. 25

5 49.0 147.0 0.97* 1.16 0.97 1.19

Gesund and 6 56.0 168.0 1.10 1.33 1.00 1.37

Boston 7 43.0 173.0 1.07 1.25 1.07 1.29

8 44.0 176.0 1.08 1. 27 0.97 1.31

9 60.0 120.0 1.00 1.46 0.64 1.04

10 44.0 176.0 0.84* 1.07 0.77 1.21

11 68.0 138.0 0.78* 1.12 0.76 1.18

12 53.0 213.0 0.90* 1.26 0.94 1.46

B28 0.1 48.6 486.0 1.04* 1.14 1.08 1. 23

B28 O.la 46.9 469.0 1.00 1.11 1.05 1.20

B28 0.4 146.0 365.0 1.40 1.99 0.94 1.49

B28 0.4a 139.0 347.0 1.33 1.90 0.90 1.42

t:hinenkov B28 0.4b 146.0 365.0 1.40 1.99 0.93 1. 25

828 0.4c 153.0 382.0 1.44 2 .12 0.89 1.36

B28 0.4d 125.0 313.0 1.19 1.67 0.85 1.49

B28 0.4e 132.0 330.0 1.28 1.77 0.86 1.54

Mean 1.51 1. 70 0.95 1. 29

Standard Deviation 56% 40% 19% 15%

Number of Tests 25 25 25 25

2.44

average value of 0. 95 and a coefficient of variation of 190/o. For

some beams in this table, however, the theory overestimates the failure

load by about 50° /o.

Of the four ultimate strength theories discussed, only the theory

of Yudin is simple enough for use in design (see Equations 2. 28 and 2. 29).

Unfortunately the equations proposed by Yudin, while being simple and

conservative, are not very accurate, having an average value for

T /T h of l. 70 and a coefficient of variation of 40° / o. exp t eor

If the theory offered by Yudin is the most simple available for this

type of beam, then the theory offered by Lessig is the most comprehensive.

It is therefore rather surprising that Lessig's predictions are so

inaccurate. (Average value of T / Tth = 1. 51, coefficient of exp eor

variation= 56°/o.) One of the main reasons for these inaccuracies is the

unsatisfactory nature of the limits placed on the ratio of the longitudinal

to the transverse steel (Equation 2. 27), and the necessity of ignoring

"excess" steel to satisfy these limits. Tha. analysis of those beams for

which this procedure was not necessary (beams marked with an

asterisk in Table 2. 2), yielded much more accurate results.

2. 3. c. Torsion, Bending and Transverse Shear.

As in the case of beams not containing web steel, the important

problem of beams failing in combined torsion and transverse shear has

received less attention than the simpler cases of bending and torsion and

pure torsion.

Lessig (Ref. 2. 22) was the first to suggest a theory for

calculating the ultimate load of this type of beam. She observed that

when a beam is loaded in predominantly transverse shear and torsion,

cracks appear first on the side surface of the beam on which the

2 45

principal stresses due to torsion and shear are of the same sign.

These cracks then spread to the top and the bottom surface. On the side

where the torsion and transverse shear stresses oppose each other

cracks do not appear until just prior to failure. Based on these

observations Lessig proposed an idealized failure mode, which she

called Mode 2. This failure model is shown in Figure 2. 12.

By employing similar methods to those used in analysing the Mode

1 (bending and torsion) failure surface Lessig obtained the following

expressions for the failure torque in Mode 2:

where

and

but

and

(b-az-2- ) Aw f w q2 x2 [ j T 2 = q2(1+6) h AL2 fL2 + S(2b + h) ........ 2. 38.

6

q2

q2

Vb = 2T

= JALZ fLZ s(Zb + h)

A f ' w w

£ 2b + h,

h 2 2

0.85f~(q2 +h)

..........

A f q2 2 7 -s-7-2b_w_+_h_) _J

2. 39.

.. 2. 40.

For the given ratios of loading, ~ and 6, the predicted failure torques in

both Mode 1 and Mode 2 could be calculated and the lower of the two would

be the predicted failure torque.

As in the case of the Mode 1 equations, Lessig places limits on the

ratio of transverse to longitudinal steel outside which her theory does no-I:

2.46

-t Sl-1 I I ___ L.02,

~ ,...A_1/AL2

I

I h I I I _j_------+------~

FIG. 2.12 SHEAR- TORSION FAILURE SURFACE

PROPOSED BY LESSIG. MODE 2

apply. These are:

0.5 ~ h s

2.47

1. 5. . ........ 2. 41.

From her tests, Lessig found that for beams containing only

small amounts of transverse steel, and loaded with high transverse

shear forces in combination with torsion, a more usual shear failure

occurs. She empirically related the strength of such beams to their

shear capacity in the absence of torsion, given in the Russian Code by

V 0

0. 51 f' C

A f w w

s n

where n is the number of "legs" of the stirrups.

.... 2. 42.

She concluded that the magnitude of the transverse shear at

failure when torsion was also present could be found by the relationship:

V = V

0

1. 5 l + 0

......... 2. 43.

Lessig established this empirical relationship from only a very

small number of test results and so no great confidence can be placed

in it. Actually the form of Equation 2. 43 seems unsuitable. This

equation can be rearranged to give:

V = V 0

3T b

......... 2. 44.

2.48

While, equation 2. 38 for the Mode 2 failure, can be rearranged to give:

V = 2T

0

b 2T b

......... 2. 45.

where T is the pure torsional strength of the beam in Mode 2. Equations 0

2. 44 and 2. 45 have been plotted in Figure 2. l 3 for a hypothetical beam.

It can be seen that because of the slopes of the lines, Equation 2. 44, which

is supposed to be related to predominantly shear failures, tends to be

critical when the torsion is high. On the other hand, the Mode 2

equation, which is supposed to cover torsional failures, is critical when

the shear is very high. This situation is unsatisfactory.

Yudin (Ref. 2. 23) as in the case of bending and torsion, agreed

with the basic failure mechanism proposed by Lessig, but derived his

equations from different assumptions. He suggested that for beams

loaded in combined transverse shear and torsion the twisting moment

could be replaced by an effective twisting moment T', where

T' = Vb'

T + 2 . ......... 2. 46.

The area of hoop steel required to resist the combined action would then

be:

A = w

T's 2 f b'd''

w ......... 2. 47.

Further additional longitudinal steel is required in the side face of the

beam to balance the moment about a vertical axis of the forces in the

horizontal legs of the hoops. This quantity is:

To

..... z w ~ 0 ~

(!) z ..... V)

~ .....

2.49

TRANSVERSE SHEAR FORCE Vo

FIG. 2. 13. SHEAR - TORSION INTERACTION FOR TYPICAL

BEAM AS PREDICTED BY LESSIG' S THEORY.

2.50

= T' (b' + d')

......... 2. 48.

In 1964, Yudin (Ref. 2. 29) published the results of a series of

beams, designed by his method and tested in combined transverse shear

and tors"ion He concluded that the test results of the majority of the

beams were closer to the theoretical values computed using his formulae

than to those calculated on the basis of Lessig's. Unfortunately many

of his beams violate the limits placed on Lessig's theory.

The results of all under-reinforced beams loaded in combined

bending, torsion and shear have been collected in Table 2. 3. In the

table a comparison is made between the experimentally observed

failure loads and the failure loads predicted by the theories of Lessig

and Yudin. Study of this table shows that the simpler theory of Yudin

has an average value of T /T of 1. 57 with a coefficient of exp theor

variation of 34° / o and in thus more accurate than the theory of Lessig

{Mean of T /Tth = 1. 66, coefficient of variation= 44°/o). exp eor

Those beams for which Lessig's analysis did not involve ignoring

excess steel are marked with an asterisk. It can be seen that for these

beams the predictions of Lessig are fairly accurate.

TABLE 2. 3

Comparison of Experimental Results and Published Theories for Beams

Loaded in Shear and Torsion.

T /T exp theor

Investigator Beam Torque I Moment I Shear Lessig Yudin kip.in. I

kip.in,! kips

B1112 146.0 243.0 9.24 1.51 1.21

BIII2A 151.0 261.0 10.21 1.54 1.31

B1115 156.0 416.0 15.56 1.56 1.38

BIIISA 151.0 416.0 15.52 1.47 1.36

B1116 92.0 156.0 4.07 1.14* 1.62

BIII6A 83.4 156.0 4.16 1.11* 1.56

BIIl7A 90.4 313.0 8.06 1.20* 1.55

B1117 83.4 278.0 7 .16 1.09* 1.41

Lessig B1118 114.5 191.0 4.99 1.87 2.07

BIIl8A 111.0 191.0 4.99 1.80 1.85

B1119 78.0 313.0 4.63 1. 33 1.45

BIII9A 79.0 313.0 4.99 1.80 1.85

B1119 78.0 313.0 4.63 1.33 1.45

BIII9A 79.0 313.0 4. 71 1.39 1.55

WB 53.0 132.0 6.64 2.73 3.35

WBA 57.3 143.0 7.16 2.85 3.45

B8 0.1 52.0 520.0 12.52 1.04 1.18

B8 O. lA 55.5 555.0 13.36 1.04 1.17

B7 0.2 93.8 468.0 11.30 1.69 1.28

B2 139.0 694.0 16.65 1.14* 1.33

Lyalin B2A 139.0 694.0 16.65 1.06* 1.24

B2A 139.0 694.0 16.65 1.06* 1.24

B3 194.0 486.0 17.48 1.30 1.21

B3A 194.0 486.0 17.48 1.30 1. 21

B5 194.0 972.0 23.24 1.07* 1. 25

BSA 194.0 972.0 23.24 1.16* 1.36

B6 167.0 833.0 20.19 1.14 1.34

B6A 181.0 903.0 21.88 1.24 1.46

....

2. 3 (contd.) T /T . exp ti.eor

Investigator Beam Torque Moment Shear Lessig Yudin kip.in. kip.in. kips

7 9.9 49.3 2.51 3.95 1.49

18 7.9 39.4 2.00 2. 77 1.59

Yudin 19 7.9 39.4 2.00 2. 77 1.59

21 7.9 39.4 2.00 2. 77 1.59

Mean 1.66 1.57

Standard Deviation 44% 34%

Number of Tests 29

3. 1

CHAPTER 3

EXPERIMENTAL WORK

3 .1 BACKGROUND.

It is convenient to divide the discussion of the tests into three

sections; plain concrete beams, beams containing only longitudinal

steel and beams reinforced with both longitudinal and transverse steel.

It has been pointed out in the previous chapter that while most

investigators· assume that the ''maximum stress theory" is a suitable

failure criterion for plain concrete beams subjected to torsion,

disagreement exists as to how this stress should be calculated.

Further, it has been shown that for a wide range of ratios of depth

to breadth of a rectangular section the indicated tensile strength of

concrete, calculated on the basis of the failure torque with an "elastic"

stress distribution is 1. 66 times the tensile strength calculated on the

basis of a "plastic" stress distribution. Results of pure torsion tests

on beams of rectangular cross section _only, therefore, cannot provide

sufficient evidence to support a theory of "plastic" stress distribution

3.2

as opposed to an "elastic" stress distribution. To test the

consistancy of these two theories a series of beams having different

shaped cross sections was, therefore, cast.

Although plain concrete beams are not employed in practice, study

of their behaviour is still rewarding because the failure load of such

beams, usually corresponds to the cracking load of a fully reinforced

member. Thus, to determine whether the cracking load of a concrete

section in bending and torsion, could be satisfactorily predicted by

employing a maximum stress criterion for concrete strength, together

with a "plastic" stress distribution, a series of plain concrete beams

were cast and tested in combined bending and torsion.

To collect additional information on the relationship between

cylinder compressive strength and torsional "tensile" strength, a plain

concrete specimen was cast from each batch of concrete used during

this investigation. This specimen was then tested in pure torsion.

A study of the literature in the last chapter revealed that while

a large number of investigators have tested beams containing

longitudinal steel only, in pure torsion, very few have studied the more

practical case of combined bending and torsion, and only one has tested

beams in combined transverse shear and torsion. Attention was

therefore concentrated on these two problems. Two series of beams

containing only longitudinal steel, were tested in combined bending and

torsion, while four series of beams were tested in combined transverse

shear and torsion.

Previous investigators have disagreed about the effect that

flexure has on the torsional capacity of beams containing both longitudinal

and transverse steel. It was felt that this disagreement could be

explained by the fact that different investigators had tested different

Type

Plain

Longitudinally Reinforced

Fully Reinforced

TABLE 3.1 SUMMARY OF BEAMS TESTED

Series Number of Tests Purpose of Tests

REP, RUP, 10 To compare the Elastic and (HC, Rl, SC, CRPl) Plastic theories.

p 7 To predict failure loads in combined bending and torsion.

w 5 To find the "tensile" strength of concrete in torsion.

L, LB 9 To study the behaviour in combined bending and torsion.

S, LS, Q, 29 To study the behaviour in combined (LS, L6, SLl) transverse shear and torsion.

RE, RU 17 To study the effect of the ratio of top to bottom steel on the flexure torsion interaction.

R 11 To study the effect, on the behaviour, of various reinforce-ment patterns.

V, U, T. 29 To study transverse shear-torsion interaction behaviour.

vJ

vJ

3.4

types of beams. In particular, it was thought that the ratio of top

to bottom longitudinal steel might have a marked effect on the interaction

behaviour of the beams. Two series of beams were therefore

designed to examine the effect of this ratio.

A third series of beams, containing both longitudinal and transverse

steel, were also tested in combined flexure and torsion. This series was

primarily, designed to investigate the effect on the failure behaviour of

varying the ratio of transverse to longitudinal steel. Tests of this series

also compared tied and welded reinforcement, and open and closed stirrups.

Very few results are a_yailable in the literature, of fully

reinforced beams loaded in combined transverse shear and torsion. In

particular, no experimentally g.erived interaction curve between shear

and torsion has been published. Three series of beams were, therefore,

designed and tested to remedy this deficiency.

A list of the various series of beams tested and a brief summary of

the purpose of each series is given in Table 3 .1.

3. 2 DESCRIPTION OF TEST SPECIMENS.

a. Plain Concrete.

Details of all plain concrete specimens cast are given in Table 3. 2.

For completeness Table 3. 2 also contains the failure loads of the specimens

but these need not concern us at present.

The first four specimens HC, Rl, SC and CRPl were all cast

using a concrete mix containing 3 / 811 maximum size aggregate. This

fine aggregate was chosen to facilitate the casting of the hollow cylinder.

Specimens HC; Rl and SC had been designed to test the consistencies of

the "elastic" and the ''plastic" stress distributions. These three

specimens, which were three feet long, were tested in a conventional

Beam

HC

Rl

SC

CRPl

REP2

REP4

REPC

RUP2

RUP4

RUPL

Pl

P2

P3 ,.P4

PS P6 PC

Wl

W2

W3

W4 W5

TABLE 3. 2: DETAILS OF PLAIN CONCRETE BEAMS

Size and Shape Concrete Details Failure Moments (kip-ins)

f' (p. s. i.) ft (p. s. i.) Type Torque Moment C

(Brazil)

Hollow Cylinder O.D. 7,200 640 3 / 8" aggregate 13. 6 0.0 = 611 • I.D. = 4 11 • Pan Mix.

6" x 4" Rectangle 7,200 640 13. 4 0.0

611 Dia. Solid Cylinder 7,200 640 20.0 0.0

911 x 611 Rectangle 6,300 600 23.3 36,6

9 11 x 611 Rectangle 4,600 575 Transit Mix 29.6 20,3

1011 x 6½'' Rectangle 4,600 575 Pour 1 50.5 0,0

611 Dia. Solid Cylinder 4,600 575 18. 8 0.0

911 x 6" Rectangle 3,680 516 Transit Mix 19. 0 35.7

1011 x 6½'' Rectangle 3,680 516 Pour 2 66.3 0.0

9" x 6" L-shape 311 3,680 516 19. 8 0.0 thick

9" X 611 Rectangle 7,100 - 3 / 4" aggregate 37.4 24.7

9" x 6" Rectangle 6,350 Pan Mix 25.2 39.6 -9" x 6" Rectangle 6,550 970 47.6 5.5 9" x 6" Rectangle 6,363 945 43.0 5.5 9" x 6" Rectangle 7,450 880 0.0 51. 3 9" x 6" Rectangle 7,400 - 11. 1 51. 4 6" Dia. Solid Cylinder 6,660 - 20.8 0.0

10·. 4"x6. 8" Rectangle 4,630 - Transit Mix 73.2 0.0 10" x 6. 5 11 Rectangle 4,320 - Pour 3 74.5 0.0

10. 2" x6. 9" Rectangle 4,050 - Pour 4 77.4 0.0

10" x 6 .111 Rectangle Pour 5

5,030 - Pour 6 60.5 0.0 10. 211 x5 11 Rectangle 3,560 - Pour 7 31. 9 o.o

I.,

u

3.6

torsion machine (Tinius Olsen 300 kip-inch capacity). To hold the

test specimens in the grips of the machine, steel end plates were

attached to their ends with an epoxy glue.

To enable bending and torsion tests to be carried out,

rectangular beams ten feet long were cast. Th€se beams were

tested in one of the special rigs described later in this chapter. The

P series consisted of six rectangular beams to be tested in bending and

tors:ion and one cylindrical specimen to be tested in pure torsion. All

beams of the P series were cast from nominally the same concrete.

Both the REP and the RUP series comprised three specimens; two

of rectangular section, one to be tested in bending and torsion and

one in pure torsion. The third specimen was circular or L-shaped

in section and was to be tested in pure torsion. All three specimens

of each series were cast from the one pour of concrete.

As mentioned earlier at least one pla:in concrete specimen was

cast from each batch of concrete used during the investigation. These

rectangular beams, which were four feet long, comprise the W series.

All five beams of this series were tested in pure torsion,

3. 2. b. Beams Containing Only Longitudinal Steel.

All test beams, in this group, were rectangular in section and

were ten feet long. Details of these beams are given in Figure 3 .1.

Where bending and torsion faiiures were required (bt:ams Ll, L2, L3,

L4, L7, L8, LBl and LB3) the shear spans were reinforced with 3/8"

diameter mild steel ties at 3" centre.

The L series, which was comprised of etght similar beams, was

primarily intended for the study of bending a!ld torsion failures though

two beams, LS and L6 were tested in combined shear and torsion.

, fr 6~i fT 1-r

! i II ii~ tl

2x ! Q> bars '

I I I ' I

C\I -·o 0) ..... 11= 5'' 3 x 8 <I> bars 7-:d==.-l .J. ,--!1

L, LS SERIES & SL 1

5" 1 I I L:1--:::::==- 2 >C 8 Q) bars

• I I I

S SERIES & LB3

tt 5~1 .11 , . IO

: 0 . ..... co

lb -~

Jx~<f> bars I [ , JJ; t2 2 3m bars~ 1 X4T -

LB1 &. LB2 Q SERIES

FIG- 3. 1. DETAILS OF BEAMS CONTAINING LO;,~(JITUDiNAL STEEL.

C-v

-.J

3.8

Beams of the LB series, LBl, LB2, and LB3, were also tested in

bending and torsion. This series was to study the effect on the

failure behaviour of adding top longitudinal steel.

Beams in the S series (5 beams), LS series (5 beams), and

Q series (12 beams) and beams LS, L6 and SLl were all tested in

combined transverse shear and torsion. To prevent possible anchorage

failures of beams of the Q series, 4.i x 511 x 3/4" steel plates were

welded to the ends of the longitudinal bars.

3. 2. c. Beams Containing Both Longitudinal and Transverse Steel.

All fully reinforced beams were 10" x 6½" in section and were

ten feet long.

Two series of beams, RE and RU, had been designed to study the

interaction between torsion and flexure. The first series, RE, comprised

five beams having equal top and bottom longitudinal steel. The specimens

were similar and details are given in Figure 3. 2. The second series RU,

comprised nine beams in which the area of top steel was only one quarter

of the area of the bottom longitudinal steel. Again these specimens were

all similar and details of these are shown in Figure 3. 2.

Beams of the third series, R, were designed to study the effect

on the failure behaviour of varying the ratio of transverse to

longitudinal steel, and of using open or closed stirrups. Thus the

eleven beams in this series were not all similar. Beams in this

series were given names such as 3.6T4 in which the first number

referred to the diameter of the top bars, the second the diameter of the

bottom bars and the last number the spacing of the stirrups. Further

if the third character was O rather than T it meant the specimen had

open stirrups. Thus beam 77T5 was a beam containing 7 /8" <p bars

3.9

FIG. 3.2. DETAILS OF RE, RU AND R SERIES.

---- ---·---6.5"---

10"

.t!

. -1 r-o.s~ r~-1 -_--1

jties I ---at 3•cc 10"

_l 1.6"

I 1------.

_J1.3"L i

RE SERlE?

j ~0.62" -,

RU SERIES

a~ . ! .

,-~-- -- -65"- § R SERIES

j ties

at s'CC

j ties at 4• cc

,¼•dia

1.8"

[;40'

3.10

top and bottom and closed hoops at 511 centres. Full details of beams of

the R series are given in Table 3. 3 and Figure 3. 2. c.

Three series of fully reinforced beams were tested in combined

transverse shear and torsion. Beams of the first series, V, were

designed so that in the absence of torsion they would fail in shear rather

than flexure. The seven specimens of this series were all similar and

details are given in Figure 3. 3. In the second series, U, the amount

of transverse steel was increased so that in the absence of torsion the

beams would fail in flexure. The three beams of this series were all

similar and details are given in Figure 3. 3. The six beams of the

third series, T, were primarily designed to fail in torsion. Once again

all specimens in the series were similar, details being given in Figure

3. 3.

3. 3. MATERIALS, FABRICATION AND CURING.

In the early stages of the investigation, concrete was mixed in a

six cubic ft. capacity, horizontal, non-tilting, pan-type mixer. Two

different mixes were used. For bE:ams HC, Rl, SC and CRPl a mix

containing 3 / 811 pea gravel and a fine beach sand was used. This

concrete mix had a cement factor of 8. 5 sacks per cu. yd. and a water

cement ratio of 0. 55. The second mix, which was used for beams of the

P and L series, contained 3/4" rounded river gravel, 3/8" crushed river

gravel and fine beach sand. This mix had a cement factor of 6 sacks per

cu.yd. and a water-cement ratio of 0.55. For both mixes a high early

strength cement was used.

Three 6 x 12 inch cylinders were cast along with each beam made

from the above concrete mixes. These control specimens were stored

under water and tested at the same time as their companion beams. In

some cases 4 x 8 inch cylinders were also cast. these being used for

I--/ Hoops butt weld d

r* 1081" "! e

ns~~-Ji 5• ~ - ;t,. =,::;.-----2-1:1 <I> cw5 bars , ,

• 6.5"7 LO N d .. 81 _l

~-1-----r £' 4> ties

--.1- at 4f cc

.. :..ri- .s-f 4> cw5 bars

'N a 3 cc b

~-3 {i:o~~- _u_ bars

Hoops butt welded

I t,.~ ~ -M L,-h

at 3i11 cc

bars

T

FIG. 3.3. DETAILS OF V, U AND T SERIES.

Beam al a2

36T4 1. 6 1. 3

36T4c 1. 6 1. 4

36T5. 5 1. 6 1. 4

77T5 1. 7 1. 4

7705 1.5 l. 3

77T4, 1. 7 l. 4

7704 l. 7 1. 4,

7703 1. 7 1. 6

24T3 1. 3 1. 4

38T5 1. 7 l. 6

I 3304 1.411.4 i

-:r.P....-=~::E; .3 ~ .3 L>::J:!:."7"..A..ZZ...S C>~ ~ S~:R.X~S

BEAM PROPERTIES

i Longitudinal Steel Transverse Steel I I I I

a3 a. IALl fLl AL3. fL3 s f ~- X sq.m. kips/sq. sq.m. kips/sq. ins. kips/sq.

in. in. in.

0.8 0.6 0.88 37.7 0.22 43.4 4 43.0

0.8 0 7 . ' 0.88 37.7 0.22 46.7 4 43.0

0.9 0.7 0.88 37.7 0.22 43.4 51.. 2 43.0

0.9 0.6 1. 20 38.8 1. zo 38.8 5 ~,3. 0

0. 7 0.5 1. 20 38.8 1. 20 38.8 5 (3.0

0.9 0.6 1. 20 33.8 1. 20 38.8 4 ~n. o O. 9 0.6 L 20 38.8 1. 20 38.8 4 1,3. 0

0.9 0.8 1. 20 38.8 1. 20 38.S 3 1,3. C

0.8 0.9 0.39 47.1 0.10 68.7 3 43.0

0. ~ 0.8 1. 57 38. 7. 0.22 45.2 5 -13.0

0. C 0.9 0.22 43.0 0.22 43.0 4 43.0

Type

Closed

Closed

Closed

Closed

Open

Closed

Open

Open

Closed

Closed

Open

w .N

3. 13

"Brazil" tests.

Later in the investigation it was found more convenient to

use commercial transit mixed concrete. This had the advantage that a

number of beams could be poured from the one batch of concrete and

thus the concrete strength could be held sensibly constant across any

series of beams. The aggregate used in this concrete consisted of

3/4" round river gravel, a coarse river sand and a fine beach sand.

High early strength cement was used and an air entraining agent was

added to give a slump of nominally 3 inches.

In all, seven pours of commercial concrete were used. Twelve

6" x 12" cylinders were cast from each pour, stored under water and

then tested at the same time as their companion beams. The cylinder

crushing strength of the concrete at the time of testing is recorded in

the tables along with the failure loads of the beams.

All reinforcing steel was thoroughly cleaned before assembly

into a reinforcing cage. In the case of the RE and RU series, the

transverse steel was spot welded to the longitudinal bars. All other

fully reinforced beams had their hoops tied to the longitudinal steel.

The transverse steel of the V, U and T series was formed into closed

hoops by flash-butt welding. The longitudinal steel used in series V

and U consisted of 5 / 8" diameter deformed, cold twisted bars. All

other steel used consisted of mild steel bars. Details of the steel

used in series R have already been given in Table 3. 3 and details of

the steel in the other series are given in Table 3. 4.

The beams were cast in steel forms. Two days after casting

most beams were removed from the forms and placed under water for

curing. In the case of beams of the RE, RU, LS, Sand LB series

curing took place under hessian, sand and polythene. The hessian

3. 14

TABLE 3.4. DETAILS OF STEEL USED

Beam Series Diameter of Type of Yield Strength Bar Bar of Steel

Kips/ sq. in.

LS, L, SLl 3/411 Plain 45.0

S, LB 5/8" Plain 41. 0

Q 3/4" Deformed 42.0

3/811 Plain 49.0

RE, RU 1/211 Plain 44.5

3/4" Plain 46.8

l / 411 Plain 58.6

3/8" Plain 50.0

V, U, T

5/8" Deformed 65.2 Twisted

l" Deformed 39.3

3. 15

and sand were kept continually damp and tests on companion cylinders

showed this method of curing to be just as efficient as ''under water".

All beams were cured for at least 28 days prior to testing,

most being removed from the water tanks just before testing. However,

beams of the V, U and T series were allowed to dry for at least six

hours prior to testing. These beams were then painted with a PVA

paint to facilitate the recording of cracks.

3. 4. METHOD OF LOADING AND INSTRUMENTATION.

The majority of beams were tested in a three-dimensional

reaction frame. Two loading schemes were employed. Figure 3. 4

indicates, diagrammatically, how the loads were applied in the first

scheme. Each beam was tested over an eight foot span, one end being

held against torsion and the other being free to twist. The bending load

was applied by means of a hydraulic jack and a spreader beam, while

the torsion was applied by another jack at the end of an outrigger arm.

The jacks were hydraulically interconnected so that during the test the

ratio of torsion to bending remained constant. This ratio could be varied

by changing either the jack sizes or the length of the outrigger. Special

roller bearings under the spreader beam and under the ends of the

specimen, ensured that the test beam was simply supported in bending

and was restrained against twisting only at the fixed end.

The second loading scheme is illustrated in Figure 3. 5. It

will be seen that in this scheme one jack is used to apply both the torsion

and the flexure. The eccentricity of this jack determines the ratio of

the twisting moment to the bending moment. As in the first system,

roller bearings under the ends of the specimen ensured that it was

simply supported in bending and restrained against twisting at only one

end.

·.,

TORSION / JACK

SUPPORT---

( 1>

- cc~ ROLLERr:=:=::0

ROLLERS

,L ~ BEARING A

BENDING JACK

PROVING

RING

~EARING D

'l>

BEAM

cc: J;' ~ ROLLERS

D

c( ,.j;:J ( D

( < f9:' BEARING C

§ =-:?'ROLLERS ~;:::

BEARING B

FIG 3.4 LOADING ARRANGEMENT --- ·---~" ..... _,. ---· .. --~- .. ---·---- ---

w ~

en

Bearing D

.,, r ~ --------Support

Torsion Bracket

FIG. 3.5 ONE POINT LOADING RIG. v-> ..... '-J

~11111111111111I~ BENDING MOMENTS

TRANSVERSE SHEAR FORCE

111111111111111 I I I I I I I I I I I I I I I I I I I I I I 11111 TWISTING MOMENTS

SCHEME I

3.18

BENDING MOMENT

TRANSVER!;,E SHEAR FORCE

1111111111111 I 1111111 TWISTING MOMENTS

SCHEME II

FIG. 3. 6. BENDING MOMENTS, TRANSVERSE SHEAR FORCES AND TWISTING MOMENTS PRODUCED IN THE TEST BEAMS.

3. 19

The bending moments, transverse shear forces and twisting

moments produced in the test beams by these two loading systems are

shown in Figure 3. 6. It will be noticed that in the first scheme the

centre portion of the test beams is subjected to only bending and torsion

while in the second scheme torsion is produced in only one half of the

beam.

In both schemes, load was applied to the beams in about ten

increments up to failure. Larger increments were applied in the

initial stages of a test anp smaller increments as the loads approached

the ultimate. The increments varied between about 15 ° / o and 5° / o

of the total load. After each increment, the load was held constant

while crack development, deflections and rotations, and for some beams

strains were recorded. The normal time to complete such a test was

about forty-five minutes.

For the earlier tests rotation measurements were obtained by

measuring the deflections, with the aid of dial gauges, at the ends of

rigid transverse arms. This method was not entirely satisfactory,

mainly because it was not possible to measure reliably the large

rotations which occur as the load approaches its ultimate value. To

overcome this difficulty a continuous recording rotation gauge was

designed. This gauge, which is similar in principle to that used by

Hsu (Ref. 3 .1), is illustrated in Figure 3. 7. The gauge made use of

the Mohr and Federhoff Precision Extensometer which is an

inductance transducer giving a multiplication factor of between 200

and 2000. This device drives the recording drum of the Mohr and

Federhoff loading equipment. The gauge consists, essentially, of a

shaft and an outrigger which was designed to convert a rotation into a

deflection.

F\G. 3 . 7

3.21

The gauge and its ancillary equipment were used as follows

to measure the angular rotation occuring in a beam. Two wooden

blocks were glued on the face of the beam a distance apart equal to

the length over which twist was to be measured. The gauge was then

screwed to one of these blocks and a steel angle to the other. The

gauge and the angle were then connected by a light brass tube, on each

end of which was a phosphor-bronze bellows. The bellows were very

stiff in torsion but extremely flexible in bending and longitudinal

extension, and acted as universal joints. The relative twist between

the two blocks was thus changed into a deflection by the gauge, this

deflection was amplified by the Precision Extensometer and recorded

on the drum of the machine. The sensitivity of the assembly was such

that a twist of 3.15 x 10-4 radians moved the pen on the recording drum

a distance of 1 cm.

During the tests of some beams, steel strains were recorded.

These were obtained by affixing SR-4 electric resistance strain gauges

to the steel, using a polyester based glue. The gauges were

waterproofed with an epo.xy resin prior to the casting of the beams.

3. 5. GENERAL BEHAVIOUR OF TEST BEAMS.

a. Plain Concrete Beams.

All plain concrete beams failed immediately on the formation

of a tensile crack. The failure crack usually had an inclination which

was normal to the direction of the principal tensile stress. Thus in

pure torsion a spiral crack at 45° to the longitudinal axis caused failure,

whilst under bending and torsion the tensile crack formed at a steeper

angle. In all cases the tensile, spiral crack had to "jump back" at

failure so that a failure surface could be formed. The appearance of

a typical failure surface is shown in Figure 3. 8. The loads at which

FAILlJR

PLAI J

VIE ./ OF PURE TORSIONAL

SP CIM _N

3.23

faHure occurred for the various beams are recorded in Table 3. 2.

For several of the plain concrete beams rotation measurements

were taken and torque-twist curves were drawn. It was found that the

curves are close to linear almost up to failure. However, a small

amount of inelastic behaviour seems evident before failure (see Figure

3. 9).

As the elastic deformations of a reinforced section are

approximately the same as those of a plain section, the stiffness of

plain. sections is of interest. In general the relationship between

the torsion applied to a section and the rotation produced in the

proportional range may be given by:-

TL = 3. 1

GJ

Where J = the elastic torsion constant of the section, i.e. 3 ¥ hb where

}/ is a function of h/b and can be found from the following table.

h/b 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 5.0

'If 0.141 0.166 0.187 0.204 0.217 0.229 0.249 0.264 0.291

L = the length of the beam,

and G = the shear modulus of elasticity of the concrete.

Thus one requires suitable values of G to predict the torsional

stiffness of the section. Table 3. 5 gives values of G derived from the

test results of those plain concrete beams for which rotation

measurements were available. This table indicates that a reasonable

approximation to the valu;:; of G is given by:

G = 250 (2000 + f' ) C

.......... 3. 2.

:io 11--r--r----r---.---,,. ----

30--------+-- ~ I ----+------------1

-. . & I

ci. 20 :JI: -., :::, 0-L,

10 0 J-

1 2 3 4 . -l Radians x 10

Twist over a' length

w FIG 3.9 TOROUE- TWIST CURVE FOR PLAIN CONCRETE I\)

~ ( BEAM P1 )

Beam

Pl

P2

P3

PS

REP4

RUP4

Wl

3.25

TABLE 3. 5. : VALUES OF THE SHEAR MODULUS OF ELASTICITY FOR PLAIN CONCRETE BEAMS

f' G G from C

Experimental Equation 3. 2 -p.s.i.

p.s.i. x106 p.s.i. x106

6,400 2.10 2.10

6,350 2.16 2.09

6,550 2.08 2.14

7,400 2.04 2.35

4,600 1. 71 1. 65

3,680 1. 50 1. 42

4,630 1. 86 1. 66

3.26

where both G and f' are expressed in terms of pounds per square C

inch.

3. 5. b. Beams Containing Only Longitudinal Steel.

During the earlier stages of loading, the behaviour of all beams

was essentially ''linear elastic". Consideration of the torsional

stiffness of the beams at this stage suggests that the stiffness is

unaffected by the reinforcement, and may be taken as that of a plain

concrete section. For instance the shear modulus of elasticity for

beams Ll and L2, if the reinforcement is ignored, is 1. 93 x 106 p. s. i.

and 2. 46 x 106 p. s. i. For these beams the formula given in the

previous section for plain beams, Equation 3. 2, leads to values

of 2.09 x 106 p.s.i. and 2.15 x 106 p.s.i. respectively.

For beams loaded in combined bending and torsion, two distinct

types of behaviour were observed. Under predominantly torsional

loading ( 'f' > 0. 5 approximately) a brittle failure occurred. This

type of failure took place in beams Ll, L3, L6, L 7, LBl, LB2 and

LB3. Cracks were visible only just before failure. The cracks were

inclined to the longitudinal axis of the beam, being approximately normal

to the direction of the principal tensile stress. Initially cracks

occurred in the lower portion of the cross section but were prevented from

opening by the bottom longitudinal steel. With increasing load the

torsional stresses probably increased more rapidly than the bending

compressive stresses following the reduction of the effective cross

section by cracking. A stage was eventually reached when an :inclined

tensile crack extended to the upper surface of the beam resulting in a

sudden drop in the torsional resistance. The torque on the section

was then resisted by dowel forces and further rotation took place at

greatly reduced loads. As the beam rotated a crushed zone became

3.27

visible on one side of the specimen. Figure 3 .10 shows the

torque-twist behaviour of a beam (Ll) failing in this manner, and

Figure 3 .11 the appearance of such a beam at failure.

If the test beam was loaded at lower ratios of torsion to

bending ( 4' < 0 . .3) a more gradual failure ensued which was similar

to a flexural failure. Beams L2 and LB failed in this manner. As

before, a cracked zone developed in the lower portion of the cross

section. In this case, however, the higher bending moment prevented

the cracks from spreading to the top surface of the beam. With increased

loading the tension cracks widened until the steel yielded, whereupon the

cracks opened further and extended upwards. Finally the concrete in

the compression zone crushed and precipitated failure. As there were

torsional stresses present, the compression zone was not normal to

the longitudinal axis of the beam. Rotation then took place about this

compression "hinge", with the two portions of the beam rotating

relative to each other. This rotation brought into play dowel forces

between the steel and the concrete and these caused segments of the

concrete to spall off. The appearance of a beam (L2) which failed in

this manner is shown in Figure 3.12. In Figure 3.13 the torque-twist

curve of this beam is given. Comparison with Figure 3 .10 shows that

beam L2 was considerably more ductile than beam Ll.

Of the four series of beams loaded in combined transverse shear

and torsion, series Q is the most instructive. Beams of this series,

unlike those of the three earlier and much smaller series, were

tested under the one point, eccentric load system of Scheme II. This

series was designed to investigate the effect on the failure behaviour

of varying both the shear span to depth ratio (a/ d) and the

eccentricity of the load. The eccentricity being expressed by its ratio

'"': C , ci .¥. -t,

so1 -,-----r---r---.---~---

501 I I :/1 II I I \ \ \ \

401 I iP I I ~:::.,, 'l.: I -----

301-1 ----;--:-,---t----~,-----+-----+-----1

5- 20 L.

~

't----1----t--------i--------4-------4-------~--- --·------··-

2 4 6 x 10·' Radians

Total Twist over 8 ft.

FIG. 3.10 TORQUE-TWIST CURVE FOR BEAM L 1

w I\) O>

.. A EA ,,A Al G O LY

TEE LOADED

0 S10

FIG 312 FAILURE OF BEAM CONTAINING ONLY LONGITUDINAL ST EEL LOADED IN PREDOMINANTLY BENDING.

330

.........

. £ d.

.::L. -() :J CT L 0 ~

70------------------------

'I ., I I soi I I / '<, ...... .._ ___ -

5or--1----Y~--+----t--==-=-_J 40

1---,.-------t----------+-----------+----------+------------·· --

5 15 x 10-=- Radians

Total Twist over 8 ft.

FIG. 3.13. TORQUE - TWIST CURVE FOR BEAM L 2.

t,J

t,J -

3.32

to the width of the beam (e/b).

Although the Q series consisted of only twelve beams,

eighteen separate tests were performed. This was possible because

for tests at low a/ d ratios only a small portion of the length of the

beam was loaded in any one test. Thus when the a/ d ratio was two,

which corresponds to a span of three feet, it was possible to perform up

to three separate tests on the ten foot length of the beam.

The effect on the failure behaviour of varying the a/ d ratio

when torsion is not present (e/b = 0. 0) is well known. At high values

of a/ d flexural cracks are prominent on the side surfaces of the beam

well before failure. These cracks, while being vertical near the bottom,

are approximately 45° to the vertical by the time they reach the middle

of the beam. With the addition of load these inclined flexural cracks

propagate, fairly rapidly, towards the load point. As failure approaches,

a horizontal crack, directed towards the adjacent support, forms at or

above the level of the longitudinal bars. The horizontal portion of the

crack commences near the point where the inclined crack intersects

the longitudinal bar. Failure is finally caused either by the compression

zone disintegrating or by a complete splitting of the concrete along the

longitudinal steel.

With low values of a/ d flexural cracks take no great part in the

failure. Some time before failure diagonal tension cracks appear In

the middle of the side surfaces of the beam. As the load :is increased

these cracks gradually spread towards both the top and the bottom of

the beam. A stage is reached when the beam resembles a tied arch,

with the longitudinal steel providing the tie. Failure finally occurs

when the diagonal crack reaches the load point and destroys the "arch".

3.33

The appearance of these two types of shear failures is

contrasted in Figure 3 .14. It is of interest that the arch action of beam

Qll (a/d = 2. 0) enabled it to sustain 12 / 3 times the shear force of beam

Ql 2 (a/ d = 4 . 1) .

The addition of torsion, as would be expected, modified the

failure behaviour. In general the failures were more sudden, with

fewer cracks forming prior to failure. Torsion tended to "open up"

the beam on the side where the torsion and shear stresses were additive

(that is the side under the eccentric load, the west side in the case of the

Q series). On the opposite side torsion held the concrete together.

The effect of small eccentricities of the load was more noticable for

small a/ d ratios. This point is demonstrated in Figure 3 .15 where the

east faces {torsion opposes the shear on these faces) of three beams

tested with the same eccentricity of load (e/b = 0. 3) but with different

a/ d ratios are shown. It will be seen that for beam Q9, tested with

a/ d equal to 5. 5, torsion has had very little effect on the appearance of

the failure (c. f. beam Ql2). On the other hand for beam Q6 (a/ d = 3. 0)

the presence of torsion has almost suppressed completely the

appearance of cracks on the east face, while for beam Qllb (a/d = 2. 0)

the torsion has actually caused crushing of the east face.

The increase in load which the beam could sustain after the

appearance of diagonal cracks was sharply reduced by the presence

of torsion. Thus with a high eccentricity of load the appearance of a

diagonal crack usually heralded failure. This point is demonstrated in

Figure 3 .16, which shows the west sides (combined torsion and shear)

of four beams loaded at the same a/ d ratio but different e/b ratios.

It will be seen that the number of cracks forming prior to failure

FIG. 3.14.

-~..;,-.. - -QI?,;,;·-- . ~

012

Q 11

THE EFFECT OF

09

Q6

•' °-, -~

e b= o.o a er= 4 .1

e a b =0.0 c1=2.0

~ Q\J SHEAR FAILURES.

e b =0.3

.ft =0.3 b

-,

a d = 5.5

---·~---

9-:3.0 d

Q 11 b ~ :0.3 , b .

.9.:2.0 d

FIG. 3 . 15. THE EFFECT OF THE RATIO

APPEARANCE.

g_ ON THE FAILURE d

l ,,

,1

012 ~=4.1 ~ =O.O b

a 5 a=4.1 ~=0.3 b

;,- ,1 Q7 o "" ' ' ,/ . ., , . " I\ . 41 / / i

cl - oi / i

.. ·. ··:,/<f ~~-LL 07a ~=4.1 ~=0.6

2p.a:::iq::ic:::=:::c;::::c:::::::::z.:::ai::::;::;;::::;==-:-=::::;::~~.,... ' '

. . ~.,,..,,-;., · 1/;· _: j ' C/7_-..J«T ' / . \ ~,, . . ' f

, ) . '

I ,,; I ':Id. =+·I . • , ' //, • I g'

- ·· I • I

~~.i......i:ii.....--:..-'7", . ...:...:....._..::..,~,.___..,.....,._,....JJ , - J

07 b=1.8

FIG. 3.16.. THE . EFFECT OF THE RATIO ~ ON THE

APPEARANCE AT FAILURE

,I

l . I

3 . 355

3.36

decreases as the value of e/b increases. Indeed, for the case of beam

Q7, (e/b = 1. 8) no cracks at all were seen before failure while for beam

Q7a (e /b = o:.6) diagonal cracks were observed only in the last increment

prior to failure.

The thirteen beams of the three earlier series (LS, S and beams

LS, L6 and LSl) were all tested under the two point load system of

Scheme I. All beams of this series were tested with a shear span to

depth ratio of about three. As was the case with beams of the Q series

tested at this value of a/ d, all specimens failed with the opening of a

tension spiral around three sides of the beam. On the fourth face, where

the torsional stresses opposed the transverse shear stresses, the ends of

the spiral were joined by an inclined crushed zone. Figure 3 .17 shows

the appearance of one of these beams at failure.

The results of all the test beams which contained only longitudinal

steel have been presented in Table 3. 6. The beams have been divided into

two groups, those tested under Scheme I (two point loading) and those

tested under Scheme II (one point loading).

3. 5. c. Beams Containing Both Longitudinal and Transverse Steel.

All beams of the RE and RU series were loaded under Scheme I.

The beams in these two series could be described as failing in one of

three modes. In each of the modes a cracked tensile zone intersected

three exterior faces of the beam in a helix while a compression zone near

the fourth face joined the two ends of this helix. The beam failed when

the steel intersected by the tensile crack yielded, allowing the beam to

rotate about an axis in the compression zone. Failure with this axis

near the top surf ace of the beams is referred to as a mode 1 failure,

near the side face as a mode 2 failure, whilst a mode 3 failure indicates

that the axis formed near the bottom surface.

-/

SIDE ON WHICH TORSION OPPOSES SHEAR

SIDE ON WHICH TORSION AND SHEAR ARE

ADDITIVE

FIG. 3.17 THE APPEARANCE OF BEAM S4 AT FAILURE

- 3,37

1 •I

TABLE 3. 6. : FAIL URE LOADS OF BEAMS CONTAINING ONLY LONGITUDINAL STEEL

3.38

PART 1. Beams subjected to two point loading.

Concrete Failure Loads

Beam fi C

Torque Moment Shear

p. s. i. kip. in. kip. in. kips.

Ll 6360 57.0 32.3 -L2 6580 60.1 205.7 -L3 7225 61.1 5.5 -L4 7260 0.0 276.0 -L7 6190 47.8 80.5 -L8 6470 49.6 263.5 -LBl 4050 54.1 141. 6 -LB2 4050 55.4 6.3 -LB3 4050 60.7 161. 5 ·-

L5 6400 43.4 144. 4 4.05

L6 6780 52.5 29.7 0.75

SLl 3300 46.2 77.3 2.20 LSl 4050 - 252.0 8.98

LS2 4050 52.5 5.2

LS3 4050 39.6 53.4 1.88

LS4 4050 25.7 117. 8 4.18

LS6 4050 37.6 101. 7 3.56

Sl 4050 41. 8 181.0 6.40

S2 4050 - 310. 0 11. 05

S3 4050 64.2 6.3 S4 4050 47.0 126.9 4.50 S5 4050 47.7 69.3 2.43

3. ; ;

TABLE 3. i

PART 2 BEAMS SUBJECTED TO ONE POINT LOADING

Beam f' Nominal Nominal Failure Loads C

a/d e/b T M V p.s.i.

kip. in. kip. in kips,

Ql 3,560 5.5 0.3 13.9 263 5.60

Q2 " 5.5 0.6 2<:1,. 1 193 4,20

Q3 " 5.5 o.o o.o 306 6.45

Q4 " 5.5 1.8 27.8 72 1. 58

Q4a " - <:>CJ 33,8 0 0,0

Q5 " 4. 1 0.3 16.5 205 5.75

Q6 II 3.0 0,3 19.4 162 6.05

Q6a " 3.0 0.6 24.4 114 4.27

Q7 " 4. 1 1.8 28.8 60 1. 71

Q7a " 4. 1 0.6 27.3 154 4.32

Q8 II 3.0 0.0 o.o 226 8.40

Q8a " 3.0 1.8 33.0 47 1. ') 0

Q9 " 5.5 0.3 18,3 237 5.

Ql0 " 2.0 1. 8 36.4 39 2.20

Qll II 2.0 0.0 o.o 225 12,55

Qlla " 2.0 0,3 26.5 72 4,05

Qllb II 2.0 0.6 27.2 180 10.05

Ql2 " 4. 1 0.0 0,0 271 7.55 ·.

3.40

In each of the test series RE and RU the first specimen was

tested with a high ratio of torsion to bending and this ratio was reduced

for each successive member of the series. The ratio (4') of torsion to

flexure is shown in Table 3. 7.

In all beams with high ratios of torsion to bending (i.e. specimens

REl, 2, 3 and RUl, 2, 3l3A), cracks were first visible on the side

surfaces. The cracks formed at about 45° to the axis and they gradually

extended, at almost constant inclination, to the top and bottom of the

member. Upon increasing the load the tensile cracks spread across the

bottom surface of the beam, and at still higher loads appeared on the top

surface. Failure of the beam ensued when the longitudinal steel yielded,

permitting opening of the tensile cracks on three sides of the beam and

rotation of the member about an axis near the fourth side.

Beams RE2 and RE3 (with equal top and bottom steel), failed by

yielding of the bottom longitudinal steel and opening of tensile cracks on

the sides and bottom of the beams. That is to say, these beams failed in

mode 1. The beams with more steel in the bottom (RUl, 2. 31 3A), on

the other hand, failed by yielding at the top steel and opening of tensile

cracks on the sides and top of the beams. This has been defined as a

mode 3 failure.

This behaviour would·. be anticipated for beams sustaining nearly

pure torsion. The maximum tensile stresses occur at the mid-he:ight

of the vertical faces (long sides of the rectangle) and initiate era.eking.

For a symmetrically reinforced beam (RE), even a small amount of

bending will cause the final failure to take the form of mode 1. If

the area of the top steel is much less than that of the bottom steel, the

top is more likely to yield and a mode 3 failure will result, with

compression at the bottom, unless counteracted by the presence of

Beam

REl

REZ RE3 RE4

RES RE4>:< REZ>:•

RUl RU3A,:< Ruz,:.,:.:

Ru3,:c,:.:

RU3A**' RU4 RU5 RU5A

RU6

RU7

36T4 36T4c,:•,:< 36T5. 5,:c,:.:

77T5

7705 77T4

7704>:C* 7703

24T3,:•*

38T5 3304

TABLE 3. 7. : EXPERIMENTAL RESULTS FOR SERIES RE, RU AND R

Experimental failure loads Observe

lp T M V Failure

(kip-ins.) (kip-ins.) (kips.) Mode

12. 90 Bl.4 -, 6.3 0.3 2 2. 61 83.4 32.0 0.9 1 1. 80 81. 5 45.3 1. 3 1 0.88 74.6 84.4 2.4 1

o. 61 66.0 108.2 3.1 1 0.28 38.0 134. 0 3.8 1 0.00 o.o 160. O 4.6 -11. 60 73.3 6.3 0.3 3

12.08 76.0 .6.3 0.3 3 1. 66 84.9 -51:1 - 3

1. 25 105.0 84.0 - 3

P.63 89.4 141. 3 - 1 0.59 85.5 145. 0 4.1 Indefinite 0.30 75.4 249.7 7.1 1 0.25 68.3 266.8 7.6 1

0. 21 59 .1 281. 2 8.0 1

0.00 0.0 304.0 8.7 -

0.26 62.6 240.4 7.5 1 1. 54 94.1 61.1 - 3 0.50 85.9 173.4 - 1

0.35 91. 6 262.4 8.2 Indefinite

0.35 96.4 278.4 8.7 Indefinite 0.48 107.6 223.4 7.0 Indefinite

0.49 99.2 201. 4 - Indefinite 0.34 74.6 218. 9 6.8 Indefinite

1. 52 70.8 46.6 - 3 0.37 89.1 216. 4 6.7 2 0.32 30.3 94.4 3.0 1

,:,: Retest. ,:,:* Shear Spans Clamped.

3.41

f' C

p.s.i.

4,600 4,600 4,600 4,600

4,600 4,600 4,600

3,680

4,630 3,680

3,610 4,630 3,680 3,680 4,400

3,610

3,680

4,400 4,340 4,630

4,630

4,630 4,630

4,630 3,830

4,340 3,830 3,830

3.42

quite high bending moments.

The specimen~ with lower ratios of torsion to bending (i. e. RE4,

5, and RU5, 6, 7) were clearly influenced primarily by flexure. Cracks

appeared first on the bottom faces of the beams. At higher loads these

cracks extended to the side surfaces where their angle changed from

almost vertical at the bottom to approximately 60° to the vertical near the

top. At still higher loads, one or more of the tension cracks began to

open. As the beam rotated, a shattered compression zone became

plainly visible on the top face. The load then began to drop of gradually,

showing that failure had occurred. With additional rotation, the main

cracks widened and horizontal 'dowel' cracks appeared about two inches

below the top face. The appearance of the failure surface at this stage

is shown in Figure 3.18. For beams of both series, the predominance

of bending forced the compression hinge to occur near the top face.

Beam REl, whose initial cracking was described previously,

failed with the compression hinge near a side face (mode 2). Usually

this type of failure is associated with the presence of shear force. In

this beam, the side cover to the main reinforcement was greater than

for the other beams of the series (1. 7 in. instead of 1. 3 in.), and the

specimen was subjected to almost pure torsion.

Beam RU4 failed in the shear span and its mode of failure was

not clearly defined. Subsequent calculation showed that the three failure

modes were nearly evenly balanced at this point. Because of this,

specimens RUZ, 3 and 3A (tested later than RU4) were provided with

additional external clamps in the shear spans to ensure their failure

in the central region.

Beams RE4, RE2 and RU3A were unloaded immediately upon their

reaching failure load. The ratio of torsion to flexure in each case was

~- . ~ - - \ ..1• • . .... -

------- - - ........-...... · _ .........

J

I" •

. "' · '

.... -~=··- -

IG. 3.18 DEVELOPED FAILURE SURFACE BEAM RU 5A

MODE 1 FAILURE.

3.44

then changed and, in the case of RU3A, the external clamps were shifted

from the shear spans to the central region. The specimens were then

reloaded. All three beams developed 'new' cracks during the reloading

and developed 'new' fracture surfaces at failure. The results of these

re-tests along with the earlier beams are given in Table 3 .. 7.

For some of the specimens of the RE series, twist measurements

were taken. Figure 3.19 shows torque-twist curves for three beams.

Each graph exhibits an initial straight portion, the slope of which

corresponds to the torsional stiffness of the uncracked section. It will

be seen that this stiffness is virtually independent of the. bending which

may be present and of the amount and distribution of the steel. If the

stiffness, GJ, is calculated on the basis that J is the elastic torsion

constant for the rectangular section (ignoring the effect of the steel), then

the slope of these curves corresponds to a value of 1. 76 x 106 p. s. i. for

G. This value agrees substantially with the value (1. 63 x 106 p. s. ::.. )

predicted by Equation 3. 2 which was obtained from tests on plain

concrete beams. It can, therefore, be concluded that the torsional

rigidity is provided almost entirely by the concrete before 'cracking'.

The load at which the beam 'cracks' (i.e. the load at which the

beam begins to lose its initial stiffness), and the behaviour of the bf':am

ai"ter this point, is seen to depend on the amount of bending present.

The three beams whose torque-twist curves are given in Figure

3 .19 all contained the same areas of steel. However, each was tE.sted

at a different ratio of torsion to bending. As would be expected, the onset

of cracking is hastened by the presence of bending. Thus, as the ratio

of torsion to bending decreases, the departure from 'uncracked'

behaviour takes place at progressively lower torques. Beam RE2, in

which the influence of bending was quite small (4' = 2. 60), exhibited

II) ., .c. u C

I

100

80

60

ci 40 .:.:.

RE 2 l '\l = 2.60) _x- -x-/

.,,,---,._- _'o'o,;--x--x-x -l~O;? I ~~x .. _<cf' __..,.x---4

¥ . \. "I _1/x ;,;.----

-------x--- -·

.,, :l u L

~

J ' /" ~~~ i ! I

20f-' I I l i

X

0 L 1 x 10-• J ~ Twist, Radians per inch

FIG. 3.19 TORQUE-TWIST CURVES FOR BEAMS CONTAINING WEB REINFORCEMENT RE SERIES

w ~ CJl

a second linear range up to failure. In other words, after cracking,

the beam settled down to a constant torsional stiffness, which was

considerably smaller than the initial stiffness. However, in beams

for which the ratio of torsion to bending was smaller {RE4 and RES),

cracking continued and the stiffness progressively decreased up to

failure. It is also of interest to note that as the ratio of torsion to

bending is decreased, the total twist of the beam before failure is

decreased.

Beams of the R series were also tested under the first loading

scheme and the results for this series are presented in Table 3. 7.

3.46

The beams in this series which contained relatively small

amounts of longitudinal steel failed in a similar manner to beams of

the RE and RU series. Thus beams 36T4, 36T5. 5 and 3304, which

were all tested predominantly in flexure failed in what has been called

mode 1 {see Figure 3. 20), while beams 36T4c and 24T3, tested under

predominantly torsion, failed in mode 3 {see Figure 3. 21). Beam ·

38T5 failed in the shear span in a typical mode 2 failure {see Figure

3. 22).

Those beams which contained large size top longitudinal bars

{77T5, 7705, 77T4, 7704 and 7703), on the other hand, failed in a

manner not previously described. All beams of this group were tested

with ~ at about 0.4. Three of these beams (7705, 7704 and 7703)

were reinforced with open stirrups, while the other two {77T5 and 77T4)

had closed ties. The two groups of beams behaved in a similar

fashion up to the point when a cleavage crack crossed the top surface

of a beam. At this point the beams with the open stirrups failed

suddenly with the opening of the top crack and the appearance of 'dowel'

cracks on the side surfaces. Beams with closed ties, on the ot'lier

,__ ____________ _

· .. . ~ / : . .

-----~---- __.___ - -·

=--=~' -~ ---~-=h..-=---- --

~--I ---~~ ... _ -~ -- ~ -~ --· - ---.-... ~

FIG. 3.2Ql DEVELOPED FAILURE SURFACE BEAM 36T 4

MODE FAILURE.

3.47

..I

\}J, s.

-r. s ,·

£"". s. .

FIG. 3.21· DEVELOPED FAILURE SURFACE OF BEAM

MODE 3 FAILURE

36T4c

t

'

3.48

3-4

9

.. ' . ~'

I ..

. .

!t .,.

,, l,

\

? y.

r ~ '\--l!)

0 f-

~

CX) ("')

• 2 <!

,I ' I

w

co I I.

1 .

l.L' L

y.

11 0

1! I'

w

I . } -

u

I ~

) {)

,

. I 0:: ::>

. f

V)

\ .I

w

w

.. ! et::

t :::,

u _

J <!

<! LL

l f L

LL 0::

.I :::,

I V

)

'I '1

0

I J

w

(\J f

0... '

·I ,·

I J

0 'I

_J

w

j "-../

w

0

.. 'lt. >

0

w

0 2

~

~ (\l

I "'

' :

M

I

.~

() •

--

I LL

·--

3.50

hand, could sustain further load after the appearance of the top crack.

The addition of this load caused the crack to gradually widen and large

'dowel' cracks would then appear on the upper parts of the side

surfaces. The appearance of such a beam at failure is shown in Figure

3.23.

In general, those beams which contained large areas of

longitudinal steel failed in a less ductile manner. Some indication of

this behaviour is given in Figure 3. 24 which shows the torque-twist

curves of two beams. Both beams were tested at similar ratios of

torsion to flexure. It will be noticed that the beams which had the

greater proportion of transverse steel and the smaller area of

longitudinal steel (beam 24T3) twisted through a much greater angle

before failure.

The fact that the transverse steel was tied to the longitudinal

steel in this series and not welded as in the earlier series, made no

appreciable difference to the behaviour of the beams. Thus beam

36T4 (tied stirrups), which was similar to Beam RU5A (welded

stirrups) and was loaded under similar conditions, exhibited a failure

crack pattern closely resembling that of beam RU5A (compare

Figures 3.18 and 3. 20).

Beams of the V, U and T series were all tested under the onf'!

point loading system (Scheme I). Various eccentricities of loads were

used, ranging 'from zero (simple shear) to infinity (pure torsion).

Values of the eccentricity used for each beam and the resulting failure

loads are recorded in Table 3. 9.

The V series was designed so that in the absence of torsion the

beams would fail in shear rather than flexure. The main purpose of

- ..... '\·- ,--.... : BOTTOM

" .' : . . .. . ,. ·. ' ' ~ ·, ·'·

. \' ::' / '_ •·. ~ ~~~,.- ---:~ --~ ¥""' -- - -~ - .-.._____,. . .. -

r. ,;: ·; :.. -~=- _;_:.,_.;: .:-:--~

·· '· · . . ... . ..... ,,.

:·;. ,,' ' ' . ·) ' ~- ' ..

• • .., , i • .i l ,- ....____,, ·. -~ . . .. l -/.. •. . ) ' . ;' . ·, ~

\ ' I - • .. - .v-~

~ - ~- f -~ .. ::::;==: ' J .

~; : • • I

I . ' ,

; :.·. .... . ·"' .

. ".. . ·~ : ·.;~ .. :::• - .,, .. ' [, · .... , .,,

. ~·

. · •. -h ., .... ~

.. . .... '¥1 _ , _ _ . ..L. _ _ • • --- ·-

!="IG. 3.23 FAILURE SURFACE OF BEAM

..

TOP

I -

77 (15""

3.51

J

II)

~ u C

100

80 I

60 I

ci. 40 ~

~--C' L.

{:!. 20

01

~

x'

j

* I

j >.<

V

/ X I

I X I

' X I

,.._

l 1 I I I

/x-~---· ... _. 36 T 4C (r=0.1381 't' = 154)

/ 1

x/ I !

- !

' I I I I

I "' • 1. 521 _J .... --x-~

2.4,. 3 (r=0-444*_ .~ ~--- ~-----L-. ----- ------ ---

~-

! I I I

Twist, Radians per inch.

FIG. 3. 24 TORQUE-TWIST CURVES FOR BEAMS OF R SERIES w CJ? I\)

Beam

Vl v1,:c V2 v2-1., V3 y3,:,

V4 y4,:, V5 y5,:,

V6 V6':< V7 y7,:,

Ul u1,:, U2 u2,:c U3 u3,:,

Tl T2 T3 T4 T4* T5 T5,:, T6 T6,:•

TABLE 3. 8. : EXPERIMENTAL RESULTS OF SERIES V, U, AND T.

Eccentricity Concrete Failure Loads of

f' 1r;oadinr T(kip-ins.) M(kip-ins.) rn::hes· C

0.87 38.2 544 1. 80 60.6 421 1. 59 52.3 413 0.62 86.0 166 0.31 16.9 685 2.52 0 79.2 394 -4.70 0 91. 3 243 ...0

0.76 II 40. 4 664 0.00 s::: 0.0 723

0 00

•.-I 89.7 0 +-' Cll

0.46 •.-I 24.8 668 ~

2.98 Cll 82.9 348 :> 3.29 C+-1 83.9 298

0 5.75 +-' 103.4 210

s::: Q)

106.6 0 00 •.-I

CJ 6.30 •.-I 99.5 119 C+-1

C+-1

0.80 Q) 43.9 689 0

3.38 CJ 85.2 315 1. 94 82.7 533 1. 15

. 66.2 720 •.-I .

Ul 2.88 . 75.1 327

0. 11. 30 0 92.9 102 0.00 ~ 0.0 563 0

1. 26 - 53.0 523 U')

2.78 II 81. 3 366 1. 86 ~ 63.4 432

Cll oc, Q) 75.4 0

5.33 ~ 83.0 196 0.63 29.4 584

3.53

V(kips)

22.3 17.3 17.0

7. 1 27.8 16.2 10.2 27.0 29.4

0.0 27.2 14.4

9.3 12. 1

0.0 8.4

28,0 13. 1 21. 8 29.3

13.5 4. 1

23.0 21.4 15. 1 17. 7

0.0 .8. 3

23.9

3.54

this series was to study the effect of torsion on the shear failure.

The beam tested in simple shear (beam VS, ecc. = 0)

exhibited what is _usually called a "shear-compression" failure. That

is, after diagonal cracks had formed the web reinforcement and the

compression zon~ continued to carry load until the stress in the web

reinforcement reached the yield point. With further increases in load,

the additional shear force was resisted mainly by the concrete

compression zone. Failure finally occurred when the compression zone

was destroyed by the combined compressive and shear stresses.

The additior_ of torsion modified this behaviour. On the side of

the beam where the transverse shear stresses and the torsional stresses

were additive (west side) diagonal cracks appeared at lower loads and

were more prominent (see Figure 3. 25). On the east sides of the beams,

where the torsional stresses oppose the shear stresses, the angle at

which the "diagonal" cracks formed, depended on the eccentricity of the

load. Thus, if Figure 3. 26 is examined it will be seen that as the

eccentricity of _the load is increased the inclination of the cracks swings

through about 90°, from the direction of the "shear" to that of the

"torsion". Further it will be noticed, that on this side of the beams

as the eccentricity is increased the number of cracks forming prior to

failure decreases. It may also be noted that in the beams with higher

eccentricities a crack, apparently caused by "crushing", appeared at

failure on the east face.

The effect of torsion on the top faces of the beams was of some

interest.As can be seen in Figure 3. 27, the addition of torsion reduced

the amount of "crushing" on the top face. Indeed it seemed that as

the eccentricity was increased the area of crushing moved towards the

east face. In the limiting case of pure torsion a cleavage crack

1 .

·II.

J

I

' . . ,o :·· i\

,. -4..._

_ .. \· ... ·

i' .

,\, ,".~.--.r·\.;·,·--...

V5 ecc. = o.o"

'i-

V2 ecc. = 1. 5911

-~~,---·-~·,~=~~· ., .. '·-·· ... , , . l ,i V4.

V4 ecc. = 4. 10-

wnr .....

I '

.. - ··1

FIG. 3. 25 THE EFFECT OF TORSION ON THE APPEARANCE

·~ OF THE WEST SIDES OF THE BEAMS.

3.55

.)

..... ..

\ ...

., . , ... .....

. . ' . ~ .

V3 ecc.= O 31

Ve. E/QT

I-

~----- ··- ---;- ·. ·, . ·;--_ .. ..,.- .. _:---_ .,.. -12:·~, -T~": . ;. ·_. V4 . . . . • £:FIS. _. . . . .· : .,. . '- .-'·,--::--:,

. . . . T , I

' •i · I· . "·· '· -.-· I C

-~.) "' . . , . ii- . . "( .. . ,<" ~-~- .

' . t• '-... ~~-~

'.· ·~ ., . . . \ . ' . '··NI l . • •• <· ',,'0-k· .· ~,,:·· . .,~~- ./ ~

. . .. ("'""' . .. . '. \ . . ' • ' ,, I • • / ' ," "\ • .,. • ,. • ~ ,,,- :

. ~ ~ ·""' -. . ... , ,.... - .... ~ - .... .... - ;I;. •. r..- ~ • ~:. ), .. '. j ~ •

-· ~'f..·, .• ..,,.l .,:...-, .. .4·: .. ':to _ ••• -~~~--. --.- .... ,1.:-1.' - - •• ,. ·- --.i.-:.. .... ~.:'I ....

V4 ecc. =4. 70

FIG 3. 26 THE EFFECT OF TORSION ON THE APPEARANCE

OF THE EAST SIDES OF THE BEAMS.

• I

.;), 00

' ,,· ',

J

... ...

~~ >~:~kw .. l" '> . 'TOi"

• ~· .. l • . , .. , .... L. , .. "'.,:, • .:r. . ~··- __ ,:: __ _

! I I.

--I i •

V 5 ecc. = o.o•

' ... I '4()..l.. c:.,,

V2 ecc. = 1. 59"

... ----- ----·

i--

l . t

L.· 14t!tL .....

-~~;&· ' :.<'· -l ; . ,: :" .

'j .' ·"",. ; .

! ,t ·- .. \ ! ·,

'j,. -

i~ \ ;'-.,: .:. ~ '..,-:_. ....~ ........ ,.,.-.....,,_.""',.,_-__,, .. , .... s-.. ..... ,-::::...., ... ,_~

-.~x:,:: -. ; . ·~~~..::

I

i

---~

' i ···-' I

V 5:t- ~cc. = Gl'0

i oP

FIG. 3. 27 THE EFFECT OF TORSION ON APPEARANCE OF .THE TOPS OF THE BEAMS.

3.57

I ,,

3.58

extended completely across the top face. Beams of the U series were

similar to those of the V series excej:>t that the spacing of their hoops

was closer. As would be expected, they sustained higher loads than

beams of theV series, though their failure behaviour was very similar.

The higher failure loads of the U series were obtained at the expense of

extensive diagonal cracking. Thus beam U2, which was tested at a

similar eccentricity to beam Vl, carried about 25 ° / 0 more load than Vl

but was much more extensively cracked at failure (see Figure 3. 28).

For a number of beams of the V and U series, steel strains were

measured. Figure 3. 29 gives the measured strains in west legs of the

ties of three beams. (It will be recalled that on the west face torsion

and shear stresses are additive). In each case the strain in the tie was

only nominal until the cracking load was reached. As would be expected

increasing the eccentricity decreased this cracking load. What is of

interest is that at high eccentricity (beam V2) the strain increased much

more rapidly with the addition of load than at lower eccentricities.

On the east face (torsion opposes shear) of the beams the strain

in the ties remained at a lower value than on the west face (see

Figure 3. 30). However, as failure approached the strain in these ties

increased rapidly. In all cases the strain in the ties at failure exceeded

the yield strain.

Beams of the T series contained less longitudinal steel than

beams of the V and U series and in the absence of torsion they failed in

flexure. However, when loaded eccentrically beams of this seri&s

behaved in a similar manner to the earlier beams. Thus, beam T5

(eccentricity = 1;86") failed with opening of diagonal cracks on the west

face~ and with the appearance of crushing on the east face (see Figure

3. 31.). For this beam the strains in the top longitudinal bars wer1:;

• ,>', .-1: ~

-~

V 1 ( ecc. = Q. 87) 3" spacing = 4 4

'. ' ~ . - ·-.i.~.-

/~~\=i ---1----+---'"i''-"' - ~ ---- ..r,

U2 ( ecc.= Q. 81)

spacing = 3"

FIG. J. 28 · THE EFFECT OF STIRRUP SPACING ON THE . FAILURE APPEARANCE.

3.5~

J ,, \ '

j

3. 6

0

l )( -----------------

-----------

---

-----------~

------I-

X

l -------

---------------

I -en

I C

I

C

tn \

'-w

\

.fJ

I-en

\ 0

\ '-

l1. \

u 0

\ E

-

V)

z ffi ..J

<(

a:: I-

I-en V

)

w

' ~

z

~, ~

<(

a:: ~

V)

-0

) C

\I

8 M

IC

) (!)

_L

l1.

0 IC

) 0

M

C\I

C\I

.... ( s

d!)I)

~\f3HS

3S~

3AS

N\f~

l

3.6

1

I )(

-C

0 L ~

en 0 L

u tn

E

w

-~(J')

t-\0

~

lJ.. <

( 0

0:::: t-

V)

V)

~

l.!J ...J

~

~

<(

0:::: t-V

)

0 M

M

' 8

<.!) ii:

IO

J_

0

U')

0 IO

0

IO

0 I')

C\I

C\I

--

(Sd

!')f) ~

\i3HS

3

5~

3/\S

N\i~

l

2-----------------------.--,. Ill a. ~ - 15

w u a:: ft a:: 10 <( w I V)

w V) a:: w > V) z <( a:: 1--

-1500 -10CX)

BEAM T5 ecc.= 1.as·

X

-500 +500

STRAIN (microstrains)

FIG. 3. 31 STRAIN IN TOP LONGITUDINAL BARS OF BEAM T5

+1000

3.62

3.63

FIG. 3.32 BEAM T5 AT FAILURE

FIG 3.33 BEAM T2 AT FAILURE

5----------------.....-----

oEAST

-en .9-~ -

3 w u BEAM T2 0:: 0 lL. ecc. = 11. 3"

~ 2 w :I: U)

w U)

ffi 1 >-\/)

~ 0:: ~

-'too +500 +1000 +15CX) STRAIN Cmlcrostrain)

FIG. 3. 34 STRAIN IN TOP LONGITUDINAL BARS OF BEAM T2

+2000

3.64

3.65

recorded and these are presented in Figure 3. 31. It will be seen that

while the bar on the east face remains in compression that on the west

face goes into tension prior to failure.

When the eccentricity of the load was further increased the beam

failed with a cleavage crack crossing the top face and extending down both

the east and the west sides of the beam. The appearance of such a beam

at failure is shown in Figure 3. 33. For this case, beam T2, both the

east and the west top longitudinal bars were in tension prior to failure

(see Figure 3. 34). This type of failure where the steel which is

normally regarded as "compression" reinforcement is actually in

tension at failure, has been called a mode 3 failure.

CHAPTER 4.

BEAMS WITHOUT WEB REINFORCEMENT.

4. 1. INTRODUCTION.

4. 1

Wherever torsional loads are significant the prudent designer will

provide web reinforcement for his beams. It might therefore be asked

what justification there is for examining· .. beams without web reinforcement

loaded in torsion.

As has been pointed out in earlier chapters, the failure torque of

a beam not containing web reinforcement corresponds closely to the

diagonal cracking load of a web reinforced member and hence it provides

a good measure of the load that can be regarded as permissible if

diagonal cracking is to be avoided.

Further, the designer must have some estimate of the failure

torque of a beam not containing web reinforcement before he can decide

that the torque is significant and web reinforcement is needed. For this

purpose the designer requires simple but conservative equations to

4.2

predict the failure loads.

4. 2. BEAMS LOADED IN PURE TORSION.

As was pointed out in Chapter 2, the ultimate strength in pure

torsion of a beam containing only longitudinal steel is comparable with

the maximum twisting moment which can be resisted by a plain concrete

section. It is appropriate, therefore, to discuss the torsional strength

of plain concrete specimens.

In order to predict the failure of a plain concrete specimen it is

necessary (a) to know the criterion of failure of the concrete (b) to know

the distribution of stress or strain over the cross-section.

In regard to (a) a number of different failure theories have been

advanced for concrete (Ref. 4.1, 4. 2, 4. 3 and 4. 4), but the problem of

predicting failure for a combined stress state is far from being solved.

For the case of a plain concrete member subjected to pure torsion,

however, the stress situation at any point in the specimen is that of pure

shear. Thus all the stress parameters commonly used in concrete

failure theories bear a constant relationship to each other. For

convenience, the failure criterion may be designated as a maximum

principal tensile stress criterion, though it should be appreciated that

the value of this limiting tensile stress for the particular biaxial stress

situation of pure torsion, may not correspond to values obtained for

other stress situations. For a particular concrete, if scale effects can

be ignored, this limiting tensile stress should be constant for all torsion

specimens.

In regard to (b) it may be said that if the stress-strain relation

ship of the concrete either in tension or compressi.on becomes non-linear,

then the relationship in shear will likewise become non-linear.

4.3

Neve 'rthele ss, the supporters of the "elastic" theory assume that the

shear stress distribution on the cross section will be that given by the

usual elastic theory applied to a homogeneous material with a completely

linear stress-strain relationship. The supporters of the "plastic"

theory adopt the other extreme point of view and assume that, near

failure, the stress-strain curve of the concrete is horizontal; that is

to say, the stress is constant irrespective of the strain. This leads to

the well-known plastic distribution of stress, in which the shear stress

is constant over the whole cross-section. Having adopted either

hypothesis, and it is impracticable to consider distributions other than

those corresponding to either elastic or plastic behaviour, it is then

possible, for a given torque and given cross-section, to calculate the

stress at any point.

The fact that the maximum principal tensile stress at failure

should be a constant for torsion specimens made from the same concrete,

can be used to investigate the suitability of the elastic and the plastic

theories. For this purpose it is not sufficient that only rectangular

beams be tested because, as was shown in Chapter 2, for this type of

beam both theories will give equally consistent results. Thus for

each concrete mix at least two different shapes of beam were tested.

The results for the plain concrete specimens tested in pure

torsion have been analysed and presented in Table 4. 1. For each

specimen the maximum principal tensile stress at failure has been

calculated, firstly on the basis of elastic stress distribution and

secondly on the basis of plastic stress distribution. It may be seen

from Table 4.1 that the stress at failure calculated on the plastic basis

is approximately constant for any one concrete. On the other hand,

4.4

TABLE 4. 1. A COMPARISON OF "ELASTIC" AND "PLASTIC" FAILURE

STRESSES.

f' Failure Stresses (p. s. i.) Beam Shape C

(p. s. i. ) Elastic Plastic

HC Hollow Cylinder 7,200 389 329

R Rectangle 7,200 600 358

SC Solid Cylinder 7,200 473 352

REP4 Rectangle 4,600 513 304

REPC Solid Cylinder 4,600 443 332

P3 Rectangle 6,550 638 378

PC Solid Cylinder 6,660 492 367

4.5

the shape of the cross-section has a marked effect on the indicated

failure stress calculated on the elastic basis. The evidence of these

few tests therefore, clearly supports the plastic rather than the elastic

theory.

It may be concluded that a satisfactory approach for calculating

the torsional strength of plain concrete specimens is to assume that the

shear stress distribution is uniform and that the maximum

tensile stress has reached its limiting value. Thus for rectangular

sections,

...... 4. 1

where f is the limiting value of the maximum principal tensile stress. t The question of a suitable value for \ still remains. Ideally,

for each mix of concrete, f would be determined from a torsion test t

on a small specimen. This unfortunately, is not practical as it is

usual to specify the concrete by only one parameter, its compressive

strength. It is therefore more convenient to relate the value of f to t

the compressive strength e>f the concrete. It is of course recognized

that no exact or universal relationship exists between the tensile and

compressive strengths of concrete. A frequently employed approximation,

however, is that the tensile strength varies as the square root of the

compressive strength.

=

Thus an appropriate expression for f would be: t

....... 4. 2

The value of the parameter C, which defines the limiting tensile stress

for the particular biaxial stress situation of pure torsion, has been

evaluated from the results of torsi.on tests.

4.6

All available test results of plain cone rete beams tested in pure

torsion have been collected in Table 4. 2 For each specimen the

tensile stress at failure has been calculated on the basis of plastic

behaviour. Using this stress and the reported value of the cylinder

compressive strength (where cube strengths were given a conversion

factor of ·O. 8 was used) the value of C for each test result was determined.

These values are listed in Table 4. 2 and it will be seen that the average

value of this constant for the 97 test results is 5. 0 with a coefficient of

variation of 20° /o.

As the failure of beams not containing web reinforcement can take

place with little or no warning it would seem prudent to adopt a

conservative failure criterion. To this end a value of 3. 5 has been chosen

for C.

Thus,

4.3

The failure torque of a rectangular plain concrete beam may now

be obtained by substitution into Equation 4. 1. i. e.

T = 1. 75 b 2 (h - ~) K 4.4

The theoretical failure torque of each of the 80 plain rectangular

specimens has been calculated using Equation 4. 4. The theoretical

values, Tth , have been compared with the experimentally observed eor.

values, T , in both Figure 4. 1 and Table 4. 2. It will be seen from exp.

Figure 4. 1 that the correlation between experiment and theory is

satisfactory. In fact examination of Table 4. 2 will show that for these

4.7

TABLE 4.2

ANALYSIS OF PURE TORSION TESTS ON PLAIN CONCRETE SPECIMENS PART ;_ _ .. RECTANGULAR SPECIMENS

Investigator Beam Torque f' Plastic T kip. in,

C . Failure C

ex:e, p. S. L

T Stress theor.

Rl 13. 40 7200 358 4.23 1. 21 This REP4 50.50 4600 305 4.50 1. 29

Investigation RUP4 66.30 3680 400 6.60 1. 89 Wl 73.20 4630 389 5.72 1. 63 W2 74.50 4320 450 6.85 1. 96

Bach and Graf Ei0.50 2830 274 5. 15 1. 47 128.90 2830 274 5.16 1. 47

Young, Sagar and Al 13.90 1700 333 8.09 2.31 Hughes 2 20.20 1700 287 6,97 1. 99

3 37. 40 1700 448 10.88 3. 11

Turner and Davies S 1 11. 00 2400 263 5.39 1. 54 2 11.75 2400 281 5,76 1. 64

Rl 17,00 2400 318 6.51 1. 86 2 12.00 2400 224 4.59 1. 31

Andersen 3A 55,00 4100 322 5,03 L44 B 73,00 4100 311 4,86 L39

C 120,00 4100 401 6.27 L 79 4A 67,00 6900 392 4,73 L 35

B 88,00 6900 374 4.51 L 29 C 117, 00 6900 391 4.72 l, 35

Marshall and TembE 01 7.59 1430 182 4,82 L38 2 7.28 1430 174 4.62 1. 32 3 6,91 1430 165 4,39 l. 25 4 7.69 1430 184 4.88 1.39 5 8,70 2560 208 4, 13 1. 18 5 10.56 2560 253 5.01 1. 43 5 9. 24 3000 221 4,05 1. 16 8 8.70 3000 208 3.81 1. 09 8 9,22 3120 221 3.96 J. 13

10 9,22 3120 221 3.96 l. 13

4.8

Table 4.2 Contd.

Investigator Beam Torque f' Plastic T kip. in.

C . Failure C ex:e. p.s.1.

T Stress theor

Marshall and Tembe (contd.) Al 10.25 2560 274 5.43 1. 55

2 9.74 2560 260 5.16 1.47 3 10.25 2560 274 5.43 1. 55 4 10.30 2560 275 5.45 1.56 7 6.15 2560 288 5.70 1. 63 8 7.18 .2560 336 6.65 1. 90 9 9.52 2560 446 8,82 2.52

10 9.22 2560 432 8.54 2.44 11 6.86 2560 321 6.36 1. 82 12 7.06 2560 330 6.54 1. 87

Nylander I l 46.80 3580 288 4.81 1. 38 2 52.61 3580 323 5.41 1.55

II 5 41.60 3580 256 4.28 1. 22 6 36.40 3580 224 3.74 1.07

Cowan X 38.50 3390 267 4.59 1. 31 Tl 58.10 6200 461 5.86 1.67

Humphreys POA 20.20 7000 484 5.79 1. 66 B 19.40 7000 465 5.56 1. 59 C 20.10 7000 482 5.77 1. 65 D 19,70 7000 472 5.65 1. 61 E 1990 7000 477 5.71 1. 63

PROA 48.10 7000 461 5.52 1. 58 B 41 80 7000 401 4.80 1. 37 C 42,40 7000 407 4.87 1. 39

PRHA 4210 7000 404 4.83 1. 38 B 44,50 7000 427 5. 11 1.46 C 44.00 7000 422 5.05 1.44

PSOA 69.00 7000 413 4.95 1.41 B 68,40 7000 410 4.91 1.40 C 69,00 7000 413 4.95 1. 41

4.9

Table 4 .7. Contd.

Investigator Beam Torque f' Plastic T c. exE, kip. in. p.s.1. Failure C

Th Stress t eor

Humphreys PTOA 16.60 7000 461 5.51 1.57 B 15.90 7000 441 5.28 1.51 C 14.70 7000 408 4.88 1. 39

PUOA 22.20 7000 448 5.36 1. 53 B 21. 70 7000 438 5.24 L 50

C 22.60 7000 456 5.46 1. 56 RPl 28.55 6000 334 4.32 1. 23

2 26.36 6200 308 3.92 L 12 3 24.24 6350 284 3.56 L 02 4 27.44 6350 321 4.04 1. 15 5 28.04 6400 328 4. 11 1. 17 6 32,04 6400 375 4.69 1. 34 7 28.24 7180 330 3.91 1. 12 8 29.76 6500 348 4.33 1. 24 9 31. 24 6950 366 4.39 1. 25

Mean 5.20 1.48 Standard Deviation 20°/o 20°/o

No. of Tests 75 75

4. 10

PART 2. CIRCULAR SPECIMENS

Investigator Beam Torque f' Plastic T C ex:e kip. in. p.s.i. Failure C Comment

T Stress theor

This HC 13.6 7200 341 4.03 1. 15 Hollow Investiga- SC 20.0 7200 353 4.17 1. 18 Solid tion SCL 20.8 6660 367 4.51 1. 29 II

SCR 18.8 4600 332 4.90 1.40 II

lA 130.0 1780 167 3.97 1. 13 Hollow lB 140. 3 II 180 4.28 1.22 II

Graf and lC 108.2 II 139 3.30 0.95 II

Morsch 2A 216.5 II 211 5.02 1.43 Solid 2B 216.5 " 211 5.02 1.43 II

2C 173,2 II 169 4.01 1. 14 II

Miyamoto GRPl ll. 3 1821 277 6.49 1.85 Solid

Rl 25,2 2000 188 4.20 1.20 Solid R2 30.8 2100 229 5.01 1.43 II

R3 35.2 2980 262 4.81 1. 37 II

Andersen R4 29.8 3200 222 3.93 1. 12 II

R5 32.2 3590 240 4.01 1.14 II

R6 42.2 5200 314 4.37 1. 25 II

. Mean 4.52 l.59

Standard Deviation 15° /o 15 /o No. of Tests 17 17

Analysis of All Results T ex:e C

T theor

Mean l.44 5.0

Standard Deviation 20°/o 20°/o

No. of Tests 92 92

100----.-------,---------~----,---...,--~

X

0

40 • 0

• 30

-C

ci. ~ - 20

ci. ~ ,._: THIS INVESTIGATION • YOUNG, SAGAR• HUGHES O

TURNER a. DAVIES 0 10 ANDERSEN • MARSHALL It TEMBE +

NYLANDER 0 COWAN • HUMPHREYS X MIYAMOTO (!) ..

30 40 50 75

Ttheor ( kip. in.)

FIG. 4.1 COMPARISON OF THEORY AND EXPERIMENT FOR PLAIN CONCRETE BEAMS.

100

4. 12

specimens the average value of the ratio T /Th . exp t eor 1s 1. 48 with a

coefficient of variation of 20° /o.

All available test results of rectangular beams containing only

longitudinal steel have been collected in Table 4. 3. These 71 beams have

been analysed by means of Equation 4. 4, that is, they have been treated

as plain concrete beams. It will be seen from the table that in this case

the average value of T /Tth is l. 55 with a coefficient of variation of exp eor

18° / o. Comparing this value with the result obtained for plain concrete

specimens one will see that the assumption of ignoring the longitudinal

steel for members tested in pure torsion, is justified.

4. 3. BEAMS LOADED IN COMBINED BENDING AND TORSION.

a. Plain Concrete Beams.

To examine the use of the plastic theory and the maximum

principal stress criterion for predicting first cracking, several plain

concrete beams were tested in bending and torsion. The results of these

tests have been analysed in the following manner. It was assumed that at

failure the distribution of stress across the section is that corresponding

to plastic behaviour for both bending and torsion. In other words:

er = ....... 4. 5

and 2T

= b 2 (h-b/3).,

...... 4. 6

where ft is the normal stress and 'l: the shear stress. It is recognized

that a more accurate approach for calculating the bending stress would

have been to assume that the stress-strain relationship in the compression

zone is linear and in the tensile zone is parabolic. This approach was

4. 13

TABLE 43

A COMPARISON OF THEORY AND EXPERIMENT FOR BEAMS CONTAINING ONLY LONGITUDINAL REINFORCEMENT LOADED IN PURE TORSION

T /T exp theor

Investigator Beam Torque f' Plastic Eff. Shear C

kip. in. p.s.i. Eq. (4. 7 see Sect. 4. 4

156.0 2820 1. 52 1. 58 173.4 II 1. 70 1. 76 160.2 II 1. 58 1. 62 160.2 II 1. 58 1. 62 180.8 II 1. 76 1. 84

Bach and 173.4 II 1. 70 1. 76 Graf 130.0 II 1.49 1. 93

136.8 II 1. 57 2.02 136.8 II 1. 57 2.02 141.0 II 1. 61 2.09 130.0 II 1.49 1. 93 141.0 II 1. 61 2.09

Young Bl 14.0 1700 2.33 2.40 Sagar and B2 22.7 II 2.15 2.61 Hughes B3 36.7 II 2.40 3.14

Turner and S3 12.5 2400 l. 76 l. 81 Davies S7 12.0 II 1. 32 l. 70

B1 l 69.8 2100 l. 30 1. 36 2 77.l 2250 l. 39 1.45 3 80,l 2250 1.44 1.50 4 84.5 3600 l. 21 l. 25

Andersen 5 88.5 3600 1. 27 l. 31 6 97.6 3680 l. 37 1. 43 7 105.l 5000 1. 27 l. 33 8 109,2 5000 l. 32 l. 38 9 119. 9 5200 l.42 1.48

., .. ·---·

4. 14

Table 4. 3 Contd.

T /T exp theor

Investigator Beam Torque f' Plastic Eff. Shear C kip. in. p.s.i. Eq, (4, 7) see

Sect. 4. 4

lA 50.0 3900 1. 34 L 39 lB 73.0 3900 1. 43 1,62 lC 90.0 3900 1. 23 1.28

Andersen 2A 62.0 7000 1.24 L 29 2B 98.0 7000 l. 43 1. 63 2C 122.0 7000 L 25 1. 30

Bl 11. 8 2560 L 79 2.15 B2 11. 3 2560 l. 71 2.06

Marshall B3 11. 8 2560 L 79 2. 15 and Tembe Cl 11. 3 2560 1. 71 2,06

C2 11. 9 2560 1,80 2, 16 C3 10.8 2560 1. 65 1. 99

III lA 13,0 2859 1.50 L 99 III lB 13, O 2859 L 50 L 99

Nylander IV SA 15.6 3079 1. 73 2.30 IV SB 14.7 3079 1. 64 2.17

Cowan A 36.0 3380 l. 22 L52

3THU 37,6 3923 0.95 L 23 Ernst 4TRO 34,4 " 0,88 1,13

5TRO 33.8 " 0,86 1.11

4,15

Table 4.3 Contd.

T /T exp theor

Investigator Beam Torque f' Plastic Eff. Shear kip. in.

C Eq. (4. 7) p. s. L see SecL 4. 4

Pl A 19.7 7000 1. 62 1. 68 B 21. 5 II L 76 L82 C 22.2 II 1.82 L89 D 21. l II 1. 73 1.80 E 19,3 II 1. 57 1.65

Humphreys PRI A 44.8 II 1. 47 L90 B 46.l II 1. 52 1.96 C 45.8 II 1.52 L 95

PSI A 74.0 II 1. 52 2.10 B 73.5 II 1. 50 2.08 C 72.0 II 1. 4 7 2.04

PTI A 16.5 II 1. 56 2, 16 B 15.5 II 1. 4 7 2.03 C 14.9 II 1. 42 L 95

PUI A 24.7 II 1. 71 2.42 B 21, 2 II 1. 47 2.08 C 22.2 II L 53 2, 18 .

Gesund l 36.0 4379 0.91 0.95 and Boston 2 39.0 437 9 0.99 L 03

Ramakris- 28.8 3119 1.86 2,30

nan and 28.8 3099 1. 87 2,30 Vijarangan 26.l 2639 1. 83 2,28

23.2 2179 1. 79 2,23 20.l 1969 1. 63 2.03 21. 7 2000 1. 75 2,15

Mean l. 55 1.84

Standard Deviation 18°/o 22°/o

No. of Tests 71 71

4. 16

used by Cowan (Ref. 2. 20) and he obtained a factor of 4. 23 rather than

4 as used in Equation 4. 5. After both the normal and the shear stress at

failure had been calculated the maximum principal tensile stress was

found by using the well known formula,

f = t

er 2 , ....... 4. 7

The results of this analysis are presented in Table 4. 4. An

examination of the results shows that for any one concrete mix the

calculated principal tensile stress at failure is reasonably constant. It

may be concluded that the strength of a plain concrete beam loaded in

combined bending and torsion, can be calculated by employing a

maximum stress criterion for concrete strength together with a

plastic stress distribution.

b. Beams Containing Only Longitudinal Reinforcement.

Beams containing only longitudinal steel and loaded in combined

fleYUre and torsion have been observed to fail in several modes, the mode

depending in any particular case on the ratio of twisting moment to bending

moment and the section properties of the member ( Chapters 2 and 3).

In general. the sequence of events comprising failure may be

described as follows: Application of the bending moment cracks the

lower portion of the beam and reduces the effective cross-section;

further) it applies a compressive force to the uncracked concrete

?.one and a tensile force to the steel When the member is twisted

shearing stresses are induced in the uncracked concrete zone and dowel

forces on the steel If spalling of the concrete is ignored failure of th,:>

beam will be initiated either by the concrete failing under the combined

compressive and shearing stresses or by the steel yielding due to the

Beam

Pl P2 P3 P4 PS P6

TABLE 4 , 4 COMPUTED FAILURE STRESSES FOR PLAIN CONCRETE RECTANGULAR BEAMS SUBJECTED TO BENDJNG AND TCRSION

Failure Stress · Failure Stress Failure Stress

in p, s. i. in p , s .L in p . s . i.

(corrected to f' = 6 . 720 n , s ~i.)

Beam f' - 4 600 . ,.. - , p . S, l ,

Beam 1 ;, ::: 4 , f;j 0

l,;

416 423 412

REP2 340 RUP2 366

378 REP4 304 RUP4 398

403 420

mean = 409 p. s , i. mean = 322 p . s . L mean= 382 p. s. i.

mean devn . = 12p.s .i. mean devn . = 18p.s.i. mean devn . = 16p.s . i.

.i:,.

.... -J

4. 18

tensile and dowel forces.

In the following analysis we will examine the modes of failure

described above and thus attempt to determine the flexure-torsion

interaction behaviour of this type of beam.

b. (i) Equilibrium Equations.

The equilibrium situation at a cracked section of a member

subjected to a bending moment Mand a twisting moment T is

depicted in Figure 4. 2. Examination of this figure reveals that

the equilibrium equations for the section are as follows:-

M = C jd

T = T C

+ D jd

4.8

4. 9,

in which C is the compressive force on the uncracked concrete

zone, D the dowel force on the steel, T the twisting moment C

resisted in the uncracked concrete zone and jd the lever arm

for the force C about the tensile steel.

In considering failures of the concrete the conservative

assumption that D is equal to zero will be made.

Thus,

T = T . C

If the depth of the uncracked concrete zone is designated d then C

the average compressive and shear stresses in this region will

be: M

a- = "d bd J C

....... 4. 10

and

-r T if d b. = >

.!.b2 (d - £) C

2 C 3

. ...... 4. 11.

~ ~ ~r -jd

j__ ~ 0~1 D ~

FIGURE 4. 2. EQUILIBRIUM SITUATION AT A CRACKED SECTION.

---r j d

___ J_

~

......

...0

4.20

In considering failures of the steel the torsional

resistance of the uncracked concrete zone will be ignored.

Thus the dowel force on the steel is:

D = T

jd ......... 4. 12

The tensile force on the steel, S, will of course be:

M s = jd .

b. {ii) Reduction of Effective Section.

4. 13

The depth of the uncracked concrete zone will depend

upon the magnitude of the bending moment applied. It might

be expected that the depth of this zone, d , would vary with the C

applied moment, M, in a manner similar to that shown by the

full line in Figure 4. 3. Until the cracking moment is reachE-d

d is equal to the height of the beam, h. At the ultimate flexurd C

capacity, M , the depth of uncracked cone rete is very nearly u

equal to the depth to the neutral axis, kd. As a simplifying

approximation it will be assumed that d will vary in thE' mamLr· C

represented by the dotted straight line in Figure 4. 3. The

equation of this line is:

d C

M = h- -

M u

(h - kd).

b. (iii) Failure of the Concrete.

"". -.. "0 """ 4. 14.

When the ratio of torsion to flexure is high failure will be

iniated by cleavage of the concrete. Cowan (Ref. 4. l} h2.s

suggested that a suitable criterion for the cleavage fracture

4.21

"' h ' "--d '--C "-

'--. '--.

'--.

'--"-

7 !<d

_L_ 0 1.0

M -M

u

FIGURE 4. 3. VARIATION OF THE DEPTH OF UNCRACKED

CONCRETE, d , WITH THE APPLIED BENDING C

MOMENT.

4.22

of concrete is Rankine' s maximum stress theory. Due to its

simplicity this criterion will be adopted. That is, it will

be assumed that at failure the maximum principal tensile stress

will be a constant.

i. e.

- ½o- +

=

and so

= 1 +

Now from Equation 4. 1,

f t =

T 0

= f ' t

......... 4. 15

......... 4. 16.

where T is the pure torsional strength of the section. 0

Hence from Equations 4. 11 and 4. 16,

= T T

0

b h - 3

d - b C 3

and substituting for d from Equation 4. 14, C

= T

T 0

M 1 - -

M u

1 (h - kd) (h- b/3)

or

where

Kl =

K = 2

and ~ =

Similarly:

(T - = ft

=

=

1

T T

0

kd --h

1 1---3oC.

h/b.

M "d bd J C

M u

2T 0

,

b jd

1

1 M

M u

.!.b2 {h - b ) 2 3

T 0

K2

M u

Kl -M

"t and ft

When the expressions for

4.23

..... 4. 17

..... 4.18

..... 4. 19

..... 4. 20,

<r ft

are substituted

into Equation 4. 15 the following equation for torsion=flexure

interaction is obtained.

:u] j1+::: 1

~-~ M K2 ]

2 T b

= T K2 jd 0 u - K

M l

... , . 4, 21

4.24

b(iv) Failure of the Steel.

When the ratio of torsion to flexure is low failure will be

initiated by yielding of the steel. Adopting the huber-Beltramis

criterion for yield of the steel we have:

f =}cr2 + 3 ~ 2 Ll

where f L 1 is the yield strength of the steel.

Now Oi;. · the direct stress in the steel, will be:

<r = s

..... 4. 22

where S is the direct force in the steel and ALl is the area.

The shearing stress in the steel, 't' , will be

where D is the transverse force on the steel.

Thus from Equation 4. 22

= j sz + 3D2

Substituting from Equations 4. 12 and 4. 13 we have;

4.25

and so the interaction equation for this type of failure becomes;

2 ( M ) M

u +

T 2 2 3 (--2) (..l)

M T U 0

= 1

b. (v) Discussion of Torsion - Flexure Interaction.

..... 4. 23.

It is apparent from the equations derived above that the

shape of the torsion-flexure interaction diagram for a beam

containing only longitudinal steel will depend on the section

properties of the member. To illustrate this point interaction

diagrams for two different beams have been presented in

Figure 4. 4. From this figure it will be seen that for heavily

reinforced, slender beams of the type tested by Nylander, the

presence of a moderate bending moment will increase the

torsional capacity of the section. On the other hand, for lightly

reinforced, stout beams of the type tested in this investigation,

the presence of bending moment will decrease the torsional

capacity. It can be seen from Figure 4. 4 that these theoretical

predictions agree fairly well with the observed behaviour of the

test beams.

b. (vi) Simplification.

Although the above analysis was based on fairly arbitary

assumptions and ignored several possible modes of failure it

was seen to predict correctly the trend of observed experimental

results. Unfortunately the analysis equatio~s are fairly compltx

and furthermore they contain the term T (the pure torsional 0

strength of the section). If T has to be calculated large errors 0

will be introduced and so the resulting accuracy of the analys:~s

.. 0

1.2-----------

1.UI a I t::c I e I I

m 081 I I I I O • . 0.61 I I I I ~II

a!

' t-0.41 I I I I 11 .. 021 I I I I I 0.21 I I I I ~ II

O 02 04 06 0B 10 0 ....... ----...- ... --.... .- """ "".O

M/Mu M/Mu NYLANDER SERIES Ill THIS INVESTIGATION SERIES L

p = 2.69 °lo ,:1.. = 2.1 P= 1.860/o ~= 1.5

M~T o = 11. 6 M~To = 4.5

FIG. 4.4 BENDING - TORSION INTERACTION CURVES FOR BEAMS CONTAINING ONLY LONGITUDINAL STEEL.

4.27

will not justify the complexity of the calculations.

It would seem from the analysis equations that if a

conservative value for T were chosen torsion-flexure interaction 0

could be ignored. To investigate this possibility Figure 4. 5

was prepared. In this figure, the ratio of the observed failure

moment to the calculated flexural strength is plotted against the

ratio of the observed failure torque to the calculated pure

torsional strength for all available test results. Equation 4. 4

was used to calculate the pure torsional strengths (it will be

recalled that on average this equation is 35° / o conservative}.

A considerable range of section properties is encompassed in the.si::,

test results, the ratio of height to width varying from 1 to 2. l, th,::

percentage of steel from 0.4°/o to 2.4°/o, and the concrete

cylinder strengths from 2000 p. s. i. to 7, 200 p. s. i. It can be

seen from Figure 4. 5 that the assumption of no interaction is

satisfactory if Equation 4. 4 is employed as the criterion for the

torsional strength of the beam. That is the torsional strength

of a beam loaded in flexure and torsion may be found by

calculating the pure torsional strength:

The accuracy of this equation for the case of beams

loaded in combined flexure and torsion is demonstrated I:

Table 4. 5. It can be seen from this table that the mean v&,lue

of T /T h for the 43 available results if 1. 49 with a exp t eor

coefficient of variation of 18 ° / o.

2.0

1.8

1. 6

1. 4

1. 2

0 1. 0 I-

" I-0.8

.. 0

di 0.6 Cl J

~ 0.4

0.2

0

X i

~ X ~ ...

..&..

-~ • ·+ ,. 6 + 1

X ~ • ox • •6 ...

HX 6 i txx .oo 6

-• -0

0

0 6 h

0 + ~~

0 6 6 (

• 6

I

0 1.4 0.2 0.4 06 0.8 10 1.2

Values of M/Mu

LEGEND: This investigation • Gesund & Boston 0

Nylander Series m X

Series VIII 0

Ramakrishnan Vijayarangan +

FIG. 4.5 INTERACTION OF TORSION WITH BENDING FOR BEAMS CONTAINING ONLY LONGITUDINAL STEEL.

4.28

4.29

TABLE 4. 5

A COMPARISON OF THE THEORY WITH EXPERIMENTAL RESULTS FOR BEAMS CONTAINING ONLY LONGITUDINAL STEEL LOADED IN BENDING

AND TORSION

' 'f /T exp theor.

Investigator Beam Torque Moment Plastic Eff, Shear Eq. (4. 7) see

Sect. (4,4j

Ll 57.0 32.3 1. 62 2.12 L2 60. 1 205.7 1. 68 2. 19 L3 61. 1 5.5 1. 63 2, 13 L7 47.8 80.5 1. 38 1.80

This L8 49.6 263.5 1.40 1, 83 investiga- LS2 52.5 5.2 1. 87 2.54 tion S3 64.2 6.3 1. 74 2, 19

LBl 54.l 141. 6 1.47 1.90 LB2 55.4 6.3 1.50 1.91 LB3 60.7 161. 5 1. 65 2. 14

B4 17. 1 99.0 1. 36 L52 BS 24.8 45.4 1.85 2.56 B6 10.7 108.0 1. 39 1, 39 C3 21. 7 111. O 1. 69 2,34

Ramakrishnan C4 20.l 90.7 1. 63 2.26 and CS 23.2 105.0 1. 79 2.47 Vijarangan B6 * B

23.2 108.0 1. 70 2.35

Cl * B 23.2 108.0 1. 79 ·2,48

C6 * B 21. 7 90.7 1. 75 2.42

4, 30

Table 4. 5 Cont.

T /T exp theor.

Investigator Beam Torque Moment Plastic Eff. Shear Eq. (4. 7) see

Sect. (4.4

III 2A 13.0 9.2 1.46 2.03 2B 13.0 9.2 1.46 2.03 3A 14.3 48.4 1. 65 2.28 3B 13.9 48.4 1. 60 2,22 4A 16.5 72.6 L94 2,69 4B 15.6 72,6 1.84 2.54

VII l 39.0 52.l 1. 19 L 30 2 3L 2 52.l 0.95 1.04

Nylander 3 39.l 58.0 1. 19 1. 30 4 35.2 58.0 L 07 1. 17 5 54.7 75.5 1. 70 1,86 6 50.7 75,5 L58 L 73 7 50.7 10.0 l. 51 L65 8 54.7 10.0 1. 63 L 78 9 31. 3 58.0 0.94 1. 03

10 19.5 58.0 0,96 0,96

3 58.0 58.0 l.47 L 70 4 64.0 64.0 1.62 1, 89

Gesund and 5 43.0 86,0 L49 L 74 Boston 6 36.0 108.0 1. 25 1.46

7 59.0 177.0 L 32 1.54 8 49.0 195.0 1. 10 1. 28 9 42.0 83.0 0.97 1.47

10 39.0 156,0 1. 17 L 77 --

Mean 1. 49 1.86 Standard Deviation 18°/o 0 24 /n No. of Tests 43 43

---

4.31

It may be recalled that the mean value of T / T for exp theor

rectangular beams containing only longitudinal steel and loaded

in pure torsion was 1. 55 + 18° / o. Comparison of these

values supports the assumption that on average the effect of

flexure on the torsional strength can be ignored.

4. 4. BEAMS LOADED IN COMBINED TRANSVERSE SHEAR AND

TORSION.

4. 4a. Nominal Stresses at Failure.

It has generally been assumed that a maximum principal stress

approach accounts satisfactorily for the strength of beams loaded in

combined transverse shear and torsion. Further, the sum of the

nominal transverse shear stress, V , and the torsional shear stress, V

Vt, has been taken as a measure of the diagonal tension and as such

has been related to the cylinder strength of the concrete, f 1 • The Q C

series of tests described in the previous chapter may be used to

investigate this approach.

For each beam of the Q series the nominal transverse shear

stress at failure was calculated from the usual formula: V

vv = bd'

while the torsional shear stress at failure was calculated from the

plastic theory, i. e.

T b

(h - -) 3

The value of V v + Vt for each beam was then tabulated against the nom:Ln.al

values of the shear span to depth ratio and the eccentricity to width rs:.t:.'.),

4.32

The results of this analysis are presented in Table 4. 6.

Examination of Table 4. 6 reveals that the value of the nominal

shear stress at failure (Vt + V v)' depends upon the value of the two

parameters a/ d and e/b. Thus although all beams of the Q series were

cast from the one batch of concrete (f' was sensibly constant), the value of C

V t + V v varied from 14 7 p. s. i. to 491 p. s. i. as the ratios of loading were

changed. Hence, to obtain consistent results from a "nominal shear

stress" design approach, the limiting value of the stress must be relatr.d

to the values of a/ d and e/b (or M/Vd and T /Vb) as well as to the value

off'. C

The increase in the value of the nominal shear stress at failure

with increasing e/b ratios and decreasing a/d ratios might be explained

by the fact that as e/b increases and a/ d decreases the moment on the

section at failure will be smaller and so flexural cracking will be reduced.

In support of this explanation Table 4. 7 lists the percentage of the

ultimate load at which flexural cracking was observed. It will be seen

that this percentage varies in the same manner as the nominal shear

stress.

Another explanation for the large variation in the nominal stress

at failure is that Vt + V vis the average combined shear stress over only

one portion of the cross-section. Thus if Figure 4. 6 ia examined it

will be seen that while V + V is the average intensity of shear stress t V

over one area of the section, other areas are at intensities Vt - V v

and/ Vt 2 + V v 2 . It is improbable that the maximum value V·t· applying

to only portion of the section would govern failure. On the other hand

it is possible that a more realistic approach would be to consider an

average intensity of shear stress. In view of the effects of flexural

A 2.0 3.0 4. 1 5.5 b

Qll Q8 Ql2 Q3

0.0 287 192 172 147

Qllb Q6 Q5 Ql Q9

0.3 491 324 289 275

Qlla Q6a Q7a Q2

0.6 346 331 361 327

Ql0 Q8a Q7 Q4

1. 8 399 358 315 303

oO Q4a 324

TABLE 4. 6. NOMINAL SHEAR STRESS

(V + V) AT FAILURE (p. s. i.) V t

~ 2.0 3.0 4.1 5.5 b

0.0 72°/c 77°/o 47°/o 40°/o

0.3 70°/o 75°/o 62°/o 61°/o

0.6 100°/o 95°/o 94°/o 75°/o

1. 8 1000/o 100°/o 100°/o 85°/o

oQ 100°/o

TABLE 4. 7. PERCENTAGE OF

ULTIMATE LOAD AT WHICH FLEXURE

CRACKS WERE FIRST OBSERVED.

.i:,.

v-l v-l

vt

vt

V t

TORSION

vt V V

V V

TRANSVERSE

SHEAR

FIGURE 4. 6. THE AVERAGE SHEAR STRESSES.

Vt +Vv

V 2 V

/v/+vv2

V - V t V

TORSION PLUS

TRANSVERSE SHEAR

~

\.,)

~

4.35

cracking it is reasonable to consider the average intensity of shear

stress in relation to the reduced section bd where d is given by C C

Equation 4. 14.

Provided that Vt is larger than V v' the overall average

intensity of shear stress is given by:

V av

=

!.bz/v 2 + V 2 a t V

bd C

where in this case,

and

V = V

V bd

C

T b - -) 3

if d C

..... 4. 24.

> b.

In Table 4. 8 the values of V at failure for the Q series of av

beams are listed. Comparison of these values with those given in

Table 4. 6 shows that for this series of beams the values of V at av

failure are much more consistent than the values of V v + Vt.

4. 4. b. Transverse Shear - Torsion Interaction.

While the above discussion suggests that a design procedure

might be formulated in which the value of the limiting shear stress would

be related to the values of a/ d and e/b, a simpler and more satisfactory

procedure can be developed if an empirical interaction between transverse

shear and torsion is employed.

The experimentally observed shear-torsion interaction behaviour

is shown in Figure 4. 7, where the ratio of failure torque to pure

TABLE 4. 8. AVERAGE SHEAR STRESS (V ) AT FAILURE av

ON REDUCED SECTION. (p .. s .. i..)

~ b 2.0 3.0 4. 1 5.5

0.0 457 312 336 358

0.3 505 327 377 435

0.6 315 303 417 443

1.8 385 ·355 325 320

oO 324

4.36

4 .. 37

1.2--------------~---..---------..... Nylander A This

Investigation I -·-c

0 I aa ~-- · -·-I

I !_ -i-I

I 7

06 _._ I _,_

I

~~ I -r

06

Q4 -I

02'1-----+-----h-----h------~----+--------~

QL----"'-----....L----~---~~--~~--~ 0.2 04 06 06 1.0 1.2 y_ v.

FIG. 4. 7. SHEAR TORSION INTERACTION FOR BEAMS CONTAINING ·ONLY LONGITUDINAL STEEL.

4. 38

torsional strength is plotted against the ratio of failure shear to the "pure"

shear strength. For the results plotted both the pure torsional and shear

strengths (T and V ) were determined from tests on companion 0 0

specimens. The range of parameters covered by the results plotted

in this figure is a/d varied from 2. 0 to 5. 4, h/b from l. 5 to 2. 1,

percentage of steel from l. 4° / o to 5. 1 ° / o, and concrete cylinder strengths

from 3, 000 p. s. i. to 6, 800 p. s. i.

It is evident from Figure 4. 7 that the assumption of linear

interaction between transverse shear and torsion will lead to satisfactory

results.

Hence,

T T

0

+ V V

0

= 1 ......... 4. 25

Before the strength of a beam loaded in combined transverse shear

and torsion can be calculated it is necessary to know the values of both

T and V . If T is calculated using Equation 4. 4, and V is obtained 0 0 0 0

from the A. C. I. equation (Reference 4. 6) that is:

V = (l.9r;;-o JI~ + 2,500 ~) bd M

........ 4. 26

then the results presented in Table 4. 9 are obtained. From this table

it can be seen that for the available results the parameter T /Tth exp eor

has an average value of l. 93 and a coefficient of variation of 15°/o.

This result reflects the conservative nature of both Equation 4. 4 and 4. 26.

4.39

TABLE 4.9

A COMPARISON OF THE THEORY WITH EXPERIMENTAL RESULTS FOR BEAMS CONTAINING ONLY LONGITUDINAL REINFORCEMENT LOADED

IN SHEAR AND TORSION

T /T exp theor

Investigator Beam Torque Shear M Plastic Eff. Shear -kip. in. kips. Vd eq. (4. 8) see

Sect. 4. 4

LS 43.4 4.0 4.52 1. 75 2.01 L6 52.5 0.8 5.01 1.54 1. 86 SLl 46.2 2.2 4.45 2.21 2.53 LS3 39.6 1. 9 3.74 1. 71 2.00 LS4 25.7 4.2 3.71 1. 59 1. 77

This LS6 37.6 3.6 3.76 1. 91 2. 19 investiga- Sl 41.8 6.4 3.07 1. 93 2.08 tion S4 47.0 4.5 2.97 1. 67 1.80

SS 47.7 2.4 3.10 1. 60 1. 77 Ql 13. 9 5.6 5.37 1. 68 1. 87 Q2 24. 1 4.2 5.25 1. 89 2.21 Q4 27.8 1. 6 5.21 1. 57 1. 94 Q4A 27.3 4.3 4.07 2.04 2.37 QS 16.5 5.7 4.07 1.80 2.00 Q6 19.4 6.0 3.06 1. 95 2. 15 Q6A 24.4 4.3 3.05 1. 87 2.12 Q7 28.8 1. 7 4.01 1. 63 1. 98 Q7A 27.3 4.3 4.07 2.04 2.37 Q8A 33.0 1.8 2.98 1. 83 2. 16 Q9 18.3 5.0 5.42 1. 77 2.02 Ql0 36.4 2.2 2.03 2.04 2.28 QllA 26.5 4.0 2.03 1.88 2.06 QllB 27.2 10.0 2.05 2.91 3.09 IV2A 6.9 4.3 4.4 2.34 2.66

2B 6.8 4.1 4.4 2.25 2.56 Nylander 3A 10.3 2.5 4.4 2.07 2.55

3B 12.7 2. 1 4.4 2.15 2.74 4A 13.4 1. 6 4.4 2.01 2.60 4B 13. 6 1. 6 4.4 2.00 2.61

Mean 1. 93 2.22 Standard Deviation 15°/o 16°/o No. of Tests 29 29

4.40

4. 5. EFFECTIVE SHEAR METHOD.

Although the equations derived above are relatively simple to use,

there are advantages to be gained by recasting them in a different form.

In particular it is convenient to relate the torsional strength, T , to the 0

shear strength, V . 0

Thus we may write,

or

T 0

C

=

=

V b 0

C

V b 0

T 0

........ 4. 27

........ 4. 28.

The shear-torsion interaction equation can now be written as:

V + CT b

= V . 0

....... 4. 29

This equation means that the torque on a member loaded in combined

bending, torsion and shear can be replaced by an equivalent shear

force of magnitude CT /b. Design is then carried out for the applied

bending moment and the effective shear force by the usual methods.

For the design of rectangular sections it is possible to derive

a simple approximation for C.

The A.C.I. Code (316-63) gives the following simplified

expression for V : 0

V = 2bdfi:_ 0 C

Hence,

Equation 4. 4 gives the expression for T as: 0

2 bA T = 1. 75 b (h - -) f' . 0 3 . C

C =

=

2bd~. b

1. 75 b 2 (h - ~) /-;;;

1. 14 d

(h - b) 3

4.41

If dis taken as 0. 9h and it is recalled that d:.. = h/b, the

above equation can be written as:

C = 1. 03

1 (1 - -)

30.:

......... 4. 30

For design purposes it would be more convenient if C were a

constant and hence as a conservative simplification we will take

C = 1. 6

which corresponds approximately with the case oC. = 1.

The expression for the pure torsional strength now becomes:

......... 4. 31.

When this equation is used to analyse beams tested in pure

torsion, the results presented in the last column of Table 4. 3 are

obtained. It will be seen that the average value of T /Tth for exp eor

4.42

the 75 tests listed is 1. 84 with a coefficient of variation of 22°/o. For

bending and torsion tests the mean value of T /Tth is 1. 86 with a exp eor coefficient of variation of 24°/o (see Table 4.4).

When C is taken as 1. 6, the equation for the shear-torsion

interaction becomes:

V + 1. 6T b

= V. 0

......... 4. 32.

In the last column of Table 4. 9 the values of T / Tth exp eor,

obtained by using the above equation are listed. It will be seen that

for these tests this parameter has an average value of 2. 22 and a

coefficient of variation of 16 ° / o.

As would be expected use of the constant value for C leads to

conservative results. However, it is felt that the resulting equations

are sufficiently accurate for design office use and they do have the

great advantage of simplicity.

CHAPTER 5

WEB REINFORCED BEAMS UNDER COMBINED

TORSION AND FLEXURE. THEORY.

5. 1. INTRODUCTION.

5.1

There is universal agreement among workers in the field, that

combinations of longitudinal and transverse steel increase the torsional

capacity of beams. Advantage may be taken of this fact to reduce the

overall dimensions of beams subjected to torsion in combination with

flexure.

In recent years several investigators have proposed theories to

calculate the ultimate strength of beams of this type. It is generally

agreed that failure of the beams takes place when tension cracks on

three sides open, allowing the segments of the beam to rotate about

a "hinge" located near the fourth side. The assumptions, concerning

the shape of the failure surface, made by the various investigators have

been discussed in Chapter 2.

5.2

5. 2. GENERAL ASSUMPTIONS.

Observation of the failure behaviour of web reinforced beams,

and study of the failure surfaces proposed by previous investigators

led to the adoption of the idealized failure mechanism shown in Figure

5. 1. The compression "hinge" of this failure mechanism has been taken

as a straight line joining the ends of the fracture surface and making an

angle of 9 with the normal cross section. Furthermore, it has been

assumed that the intersection of the fracture surface with the face

opposite the compression zone is a straight line whose inclination may

be defined by a spiral joining the ends of the compression zone.

Following Lessig, failure with the compression zone occurring

near the top face is referred to as a Mode l failure, whilst a Mode 2

failure indicates that this zone forms near a side face. This latter

type of failure normally occurs only in the presence of transverse shear

force and so it will not be dealt with in this chapter. In the present

investigation some specimens developed the compression hinge near the

bottom surface and failures of this type have been labelled Mode 3.

Expressions for the failure loads in both Mode 1 and Mode 3, will

now be derived. It should be noted, that in this analysis it will be assumed

that both the longitudinal, and the transverse steel yield at failure.

Under certain circumstances it is possible for the beam to fail without

yielding of the steel. It is therefore necessary to place limitations on

the theory to exclude this type of beam. These limitations will be

considered in Chapter 7.

Dowel forces are ignored in this analysis, and the contribution

of the tensile stresses in the concrete is also omitted. These

approximations lead to satisfactory results except for beams in which the

:---,,

:~. ---

-:., '

...,_. .

. -

, • <:;J

'. • •

• •

• J

• -

-~

--

• •1

._

• .,,.

"-A

,

: . -

\ : '

' .... •--':

. ••'4

) -." ..

-~--:.

"J. \

' •1

,)

--,I•;--

I·•

•

-< , •

• .-,

, ._p,..

' /

• I

I \

• .• 4

"/. ·,

,, ··.,.

''q

__ .

~

\ - ...:

' . -·

I '

..J..-'-.

-,

'\ •

• , I

/ •

, ~

: '-.

,I

I

5-3

w

u ~

a:: ::, V

)

z 0 V) ~

0 I-

(!) z 0 z w

CD

0 w

N

_J

<l

w

0 lO

(!)

lL

amount of transverse steel is very small. For such beams the theory

will frequently lead to a low estimate of the torsional capacity. This

deficiency in the theory will be referred to again in a later chapter.

5. 3. ANALYSIS OF MODE 1.

5.4

A general view of the first mode of failure is shown in Figure 5. 2.

In this figure an attempt has been made to represent the forces acting at

failure. Among the forces shown are, the force exerted by the web

steel on the sides, F , and the bottoms, F ·b, as well as the force ws w

exerted by the bottom longitudinal steel, F 1 b. The direct stresses

acting on the concrete "compression" zone have been represented by a

force, Cf, acting at a point labelled Owhich is a distance of x 1 from the

top surface. The shear stresses acting have been represented by a shear

force, C , and a couple, C , which acts in the plane of the compression S C

zone.

The length of beam which the failure surface occupies is equal to

b tan 91 (see Figure 5. 2). Some portion of this length will separate

the two forces exerted by the side branches of the web steel. This

portion will be called '?/. b tan 9 1.

For the simplicity, it is assumed that the bottom branches of the

stirrups are at the same level as the main longitudinal steel. Thus both

force F 1 b and F wb are a distance (d - x 1) from point O.

5. 3. a. Equilibrium Equations.

The sum of the moments of the external and the internal forces

about a transverse axis is equal to zero. For convenience the transverse

axis passing through point O will be chosen.

\b tan e,

\ \

b

I Fws 1 , •. ~ X

\ \

"C -l h

I Fib I

FIG. 5.2 FORCES ACTING ON THE MODE 1 FAILURE SURFACE.

M

-A,:-

OJ OJ

5.6

Thus,

M + Cc sin '\ - F lb ( d - x l) + F ws '/ . b. tan Q l = 0. . .. 5. 1.

The sum of the moments of the external and the internal forces

about a longitudinal axis is equal to zero. Once more the axis through

point O will be chosen.

Thus,

T - Cc cos e1 - Fwb (d-x1) - Fws b' = 0. . .. 5. 2.

where b' is the width of the hoops.

Eliminating Cc from Jlquations 5. l and 5. 2, and rearranging we

have:

+ F sin e1 (b' -Zf.L). ws

. .. 5. 3.

Now if both sides of equation 5. 3 are divided by cos e1 then the

expression becomes:

M + T.t = (F lb+ F wb· t) (d-x1) + F ws·t. (b' - 2{b) ... 5.4.

where, for convenience:

5.7

As the bottom longitudinal steel is assumed to be yielding,

where ALI is the area of the steel and fLl is its yield stress ..

If A is the cross-sectional area of one leg of the hoop w

reinforcement, f is its yield stress and s is the spacing of the hoops, w

then the force in the bottom branches of the transverse steel per unit

length of the beam would be A f / s. Now the length of the beam over WW

which these branches are intercepted by the failure surface may be taken

as:

Thus

b' 2h+ b

F = wb

b. t.

A f w w

s

b' bt 2h+ b"

If the relationship between transverse steel and longitudinal steel is

expressed by a parameter r where A f

w w r =

s

then Equation 5. 5 be comes

where

r F =

wb 1 + 2oC

oC = h/b.

b' ......... 5. 6.

Similarly the force in the side branches of the web steel is given by

F = ws

r 1 + ~

h b'

5.8

Now the length of the beam separating the two forces exerted by

the side branches of the web steel may be taken· as:

". b. t b+h

= b + 2h. b. t.

'2(. b = l + oC

l + 2oc. b.

From the above Equation 5. 4 can be rewritten as:

+ r

l+2oC.... h b'"

2 1+oc.. ALl fLl. t (b'- 1+2a:.· b).

If o/. = T /M .this equation can be rearranged to give:

T (1/"1 + t) AL/Ll (d-xl)

r = l + 1 + 2d:.

For convenience we may write:

2 T = A f (d - x) l + X. t

Ll Ll 1 + t 'I'

r where X =

l + ?<$.. h (1 -

d - x 1

5. 3. b. Determination of Tan 91.

~+

... 5. 7.

1 + (I;. 1 + 20C.

... 5. 8.

The inclination, Q 1, of the hinge will be such as to make the failure

load a minimum. This means that:

dT dt

= 0.

Therefore:

(..!.. + t) ""

For Equation 5. 9 to be satisfied:

( !. + t} 2

X.2t-(l+X.t) \fl

and so:

1 J (!.)2 + 1

t = - - + X "' "'

d dt

= o.

( 1 + xt2) - (1+xt2)

5.9

d 1 -(- + t) dt 'fl

. . . . . . . . . 5. 9.

......... 5. 10.

When this value is substituted into Equation 5. 7, the failure

torque for a Mode 1 failure is obtained as:

1 - "'). . .... 5.ll.

5. 3. c. Determination of x 1 .

= 0 .

The depth to the compression resultant, x 1 depends on the

distribution of the stress in the compression zone as well as the depth of

the zone. This depth, in turn depends on the strength of the concrete in

combined shear and compression. Because of the complexity and

indeterminacy of the factors involved, an accurate formulation for the

value of x 1 is not possible.

Fortunately it is not necessary that the value of x 1 should be

known exactly, as even considerable variation in this value will have

little effect upon the failure torque. Thus, in the interests of simplicity,

it has been arbitrarily assumed that the depth to the compression

5.10

resultant may be taken as the same as· it would be in pure flexure.

If M is the calculated ultimate capacity of the member in u

simple flexure, i. e.

Equation 5. 11 reduces to:

5. 4. d. Simplification.

1 + X

1 'f' ). .. ... 5. 12.

Of the three variables which determine the torsional strength of

a beam, Mu, X and t , X is the most arduous to calculate being given

by the expression:

(1 - 1 + (I:.. 1 + 2c:C

To simplify this expression we may make the conservative assumption that:

r X = 1 + 2oC. '

Equation 5. 12 now reduces to:

T 1 = 2r ( J (.~.) 2 + 1 : 20C:. M 1+2<C .,.

u

1 'f').

..... 5. 13.

. ... 5.14.

To demonstrate the order of the conservative error introduced by

Equation 5. 13, Table 5. 1 has been prepared. In this table the failure

torque calculated by using the simple expression for X, Equation 5. 13,

is compared with the failure torque calculated using the more complex

expression for X, Equation 5. 8,

5 (I

TABLE 5.1. Error Introduced by Ignoring the Last Term of

Equation 5. 4.

Beam Properties. b

1.25 11 = T(l 3)

<X r h T(8)

4' --d-x

1

1.2 0.97

0.25 0.075 1.4 0.97

1

1. 2 0.94

1.00 0.150

1.4 0.92

1. 2 0.93

0.25 0.100

1.4 0.92

2

1.2 0.90

1.00 0.175

1.4 0.88

5.12

by means of a ratio T{l 3) / T(S)' From Table 5. 1 it can be seen that

for a wide range of practical beams use of the more simple expression

will introduce a conservative error of about 10° /o. It may be noted

that the simplification introduced by Equation 5. 13 could have been

accomplished by ignoring the last term of the equilibrium expression

(Equation 5. 4).

5.4. ANALYSIS OF MODE 3.

A general view of the third mode of failure is shown in Figure

5. 3. Shown in this figure are the forces in the top longitudinal steel,

Flt , and the top branches of the web steel, F wt. Also shown are the

forces acting in the concrete compression zone and the side branches

of the stirrups.


The sum of the moments of the external and internal forces about

a transverse axis passing through point O is equal to zero. Thus

where d3 is the distance from the bottom of the beam to the centroid

of the top steel.

The sum of the moments of the forces about a longitudinal axis

through point O is equal to zero.

T - Cc cos 93 - Fwt (d3-x3 ) - Fws b' = 0. . .... 5, 16.

Eliminating C from Equations 5. 15 and 5. 16 we have:. C

T

-9-

b --~. . ' , y. ·. r- tt ' . - F \. . .

wt"" \.•. ,_......,,, r A' ' .. " D

' • ~ •\ ._ I • . ... \,, .. • :"c,_- ' ""-_ _,• • .._• I

• ,,,. " - "' ~ C .,. _,

h

FIG. 5.3 FORCES ACTING ON TrlE MODE 3 FAILURE SURFACE.

_..._

i< i

u ---

OJ -w

5. 14

+ F sin 8 3 (b' -1,b). . ..... 5. 17. ws

As in the case of Mode 1, the last term in this equation will be

ignored.

Now if Flt R ::

Fib ;:

then equation 5. 1 7 can be recast as:

r 2 R + 1 + 2ce tan 83

tan 8 3 - 1/'I' ...... 5. 18.

5.4. b. Determination of Tan 83.

The inclination, 8 3, of the hinge will be such as to make the

failure torque a minimum. If dT/dtan83 is equated to zero it is

found that T is a minimum when:

1 tan 8 :: -

3 o/ + J (~)2 + R(l+20C: )

r ....... 5. 19.

When this value is substituted into Equation 5. 18, the failure torque

for a Mode 3 failure is obtained as:

2r ( j (!)2 + (1 + 24C)R + .!.) T 3 :: ALl fLl (d3 - x3) 1+2~ f r 'f-'

...... 5. 20.

As a simplifyiµ.g assumption we will take:

Therefore Equation 5. 20 reduces to:

= 2r

1+2~

5. 15

... 5. 21.

For a given value of \.\J and known beam dimensions, the two

torques T 1 and T 3 can be computed from Equations 5. 14 and 5. 21. The

smaller of these two values will usually be the twisting moment at

failure if transverse shear force is absent. In general the strength

of a beam will be governed by the Mode 1 failure. Only in the case of

beams tested under high ratios of torsion to flexure and containing less

top than bottom longitudinal steel will the Mode 3 equation be critical.

5. 5. FLEXURE - TORSION INTERACTION.

For the purpose of studying interaction behaviour, Equation

5. 14 may be recast.

If it is recalled that 4' = T / M and, X = r / l + 2<C, Equation

5. 14 can be written as:

... 5. 22.

where M 1 is the failure moment associated with T 1

On rearranging and squaring to eliminate the radical sign the

5. 16

following relation is obtained:

M + 4x -1 ... x o Mu - "2: = . ... 5. 23.

Now if T is taken as the torsional capacity in Mode 1 when 0

M = 0, it can be seen from Equation 5. 21 that:

... 5. 24.

or M = T /2/x. U 0

Substituting for M in the first term of Equation 5. 23 and u

dividing both sides of the equation by 4X we have:

T ( __! )2 T

0

+ - 1 = 0. ... 5. 25.

In Figure 5. 4 the theoretical interaction behaviour of beams

for which Mode 1 is critical is shown. It can be seen that the theory

predicts substantial decreases in the flexural capacity with the application

of torsion and vice versa.

Contrary to the above, Cowan and Armstrong (Ref. 5. 1) found

that for their test beams the addition of bending moment increased the

torsional capacity. It is considered that this type of interaction is

peculiar to beams which have been designed primarily for flexure (that

is, containing more bottom than top longitudinal steel), but have been

0 1-

.............. I-

5.17

1. \------,------,-------:-----~----,

2 I ( I'( + M = 1

To/ Mu

0.

0.4L------+------+---~

Q2L-----1-----l-----+-----+------+-t

0.2 0.6 0.8 1.0

Fl3. 5.4 INTERACTION DIAGRAM FOR BEAMS FAILING tN MODE 1.

5.18

tested under predominantly torsional loads. For this type of beam a

Mode 3 failure will be critical. In this mode, failure is initiated by

yielding of the weaker top longitudinal steel. A moderate bending moment

opposes this action and thus increases the torsional capacity of the beam.

A larger bending moment will cause yielding of the bottom longitudinal

steel, that is, the beam will fail in Mode 1. It is to be noted that since

the Mode 1 failure mechanism for such a beam involves the yielding of

a larger area of longitudinal steel than does the Mode 3 failure, the

torsional capacity will still be greater than the pure torsional strength of

the beam, which is governed by the Mode 3 failure.

The Mode 3 equation can be written as:

= +

Again rearranging terms and squaring to eliminate the radical

sign the following relations is obtained:

M3 - 4 X M

u - 4 X R = 0.

Substituting for M and dividing by 4 X we have: u

T (-3/

M u

- R = 0.

... 5. 26.

. .. 5. 27.

It can be seen from this equation that the shape of the interaction

curve will depend on the value of R (the ratio of the top to the bottom

longitudinal steel). In Figure 5. 5 the interaction curves for R = I, R = ½ and R = ¾ have been drawn. Also plotted on this figure is the Mode I

5. 19

1.2---------------,------,---......,

02---------+---------+-----t--1

02 0.8

FIG. 5.5 INTERACTION DIAGRAM FOR BEAMS WHICH MAY FAIL IN MODE 3.

1.0

5.20

interaction curve. It is evident from Figure 5. 5 that if R(. I the beam is

governed by the Mode 3 equation for low values of M/M and so in this u

range bending increases the torsional capacity. Application of larger

bending moments will cause the Mode 1 equation to become critical.

Once this critical point is passed, corresponding to a change from a

Mode 3 to a Mode l failure, the torsional capacity decreases with

increasing bending moment.

Study of the two relevant equations (5. 25 and 5. 27) reveals that this

transition point will occur when:

l M M u

= ... 5. 21.

6. l

CHAPTER 6

WEB - REINFORCED BEAMS UNDER COMBINED

TRANSVERSE SHEAR AND TORSION. THEORY.

6. l BACKGROUND.

Torsion will almost invariably occur in association with

transverse shear. It is of some importance, therefore, to study the

failure behaviour of beams loaded in combined transverse shear and

torsion.

In the absence of torsion the mode of shear failure most likely

to occur in web-reinforced beams of normal design is the so - called

"shear-compression" failure.

The sequence of events occurring in this failure mechanism

has been described by ACI-ASCE Committee 326 as follows. (Ref.

4. 5 ):

"As the external load increases after diagonal cracking, the

web reinforcement and the compression zone continue to carry shear

until the stress in the web reinforcement has reached the yield point.

6.2

Further increase in external shear must then be resisted by the

compression zone alone. Failure occurs when the compression zone

is destroyed by the combined compression and shear stresses. "

Experiments conducted during this investigation (c. f. Chapter

3) have shown that the presence of torsion modifies this failure behaviour.

In the case of beams loaded in simple shear (T /Vb = 0), the

compression zone forms near to and parallel to the top face of the beam,

the diagonal tension cracks form only on the side faces of the beam,

while the tension crack on the bottom face is perpendicular to the

longitudinal axis of the beam (see Figure 6. 1. a.).

When the beam is loaded in combined transverse shear and

torsion at low eccentricities (T /Vb low), the compression zone moves

towards the side face of the beam on which the torsional and transverse

shear stresses oppose (see Figure 6. 1. b). The torsion also causes

diagonal cracking on both the top and the bottom faces of the beam.

When the eccentricity of the load is further increased (T /Vb

high) the compression zone covers all of the side face of the beam

while diagonal tension cracks cover the other three sides (see

Figure 6. 1. c.). This mode of failure has been described previously

( Chapter 2 and 3) and labelled Mode 2.

In the limiting case of pure torsion (T /Vb = oO ) it is possible

that the compression zone will form near the bottom surface of the

beam. This type of failure has been labelled Mode 3.

6. 2. PREDOMINANTLY SHEAR FAILURES.

a. Geometry of the Failure Surface.

The assumed shape of the failure surface for the case of

predominantly shear failures is shown in Figure 6. 2. Also shown on

(a) SIMPLE SHEAR

(b) HIGH SHEAR LOW TORSION

(c) LOW SHEAR HIGH TORSION

FIG. 6.1 VIEW OF FAILURE SURFACES. ·

6.3

BOTTOM

EAST (T - V)

TOP

WEST

{T + V)

SIMPLE SHEAR

(a)

HIGH SHEAR PLUS

TORSION

(b)

FIGURE 6. 2. DEVELOPED FAILURE SURFACES.

Ebr

HIGH TORSION PLUS

SHEAR

( C)

0'

,i:i,.

6.5

this figure are the developed failure surfaces for the case of "simple"

shear and for a Mode 2 failure.

In Figure 6. 2, #,lJ is the portion of the top face crossed by

diagonal tension cracks, ·72, d is the portion of the side face crossed by

the compression zone, while c b is the projection of the bottom crack

onto the longitudinal axis of the beam. All three parameters, µ, "'1,

and £,. , will vary from O to 1 as the ratio T /Vb increases.

In Table 6. 1 the values ofµ for the three series of web reinforced

beams tested in this investigation have been listed. In Figure 6. 3 these

values have been plotted against the values of the ratio T /Vb. From

this diagram it can be seen that a reasonable approximation toµ would

be:

µ = T/Vb

µ = l

(T/Vb .t::.. 1)

(T/Vb ~ 1)

. . . . . . . . . 6. I.

It will arbitarily be assumed that 1l. and c. will vary in the same manner

asµ.

Thus

'l, = t = µ = T /Vb (T /Vb < 1) ........ 6. 2

~ = t,, = µ = l (T/Vb~ 1).

An end view of the failure surface is shown in Figure 6. 4. In

this figure the "shear centre" of the uncracked concrete zone has been

labelled O, and the distance of this point from the centre line of the beam

has been called d . 0

For the case of simple shear this distance will he

equal to zero while for high values of T /Vb it will be some portion of b/ 2, 2 b

say 3 2· We will assume that d varies in the same manner as µb. 0

TABLE 6.1. VALUES OFµ FOR TEST BEAMS.

Beam

Vl

Vl* V2

V2* V3

V3*

V4 V4*

V5

V5*

µ o.5 ~--

0

0

FIGURE 6. 3.

T

vb µ

0.26 0.6

0.54 0.9 0.48 0.6

1. 86 1.0

0.09 0.2 0.75 0.9

1. 37 1.0 0.23 0.5

0 0

(P 1.0

0.25 0.50

Beam

V6

V6* V7

V7*

Ul

Ul*

U2

U2*

U3

U3*

T

vb

0.14

0.88 1.40

1. 31

00

1. 82

0.24 1.00

0.58

0.35

0.75

T Vb

T VALUES OFµ VERSUS Vb.

µ Beam

0.2 Tl

1.0 TZ 1.0 T3

1.0 T4

1.0 T4* 1.0 T5

0.8 T5* 1. 0 T6

0.8 T6* 0.8

1. 0

6.6.

T

vb µ

0.85 1.0

3.50 1.0 0 0.0

0.38 0.5

0.83 1.0 0.28 0.6 00 1.0 1. 54 1.0

0.19 0.1

1. 25 1.50

8 ~d J V I 07 I

F b < w

d'

---J

FIGURE 6.4. FORCES ACTING ON EFFECTIVE

SHEAR FAIL URE SURF ACE.

6.7

6.8

Thus,

d 2 b T b T (..'.I. < 1). = 2 . = 3 . 0 3 Vb Vb Vb

. ...... 6. 3.

d b (

T :;;?:, 1). =

0 3 Vb

6.2.b. Forces Acting at Failure.

In Figure 6. 4 the forces acting on the failure surface have been

represented. (For convenience, that side of the beam on which the

shear stresses due to torsion and those due to transverse shear are

additive , has been called the west side.) Amongthe forces shown in

Figure 6. 4 are, the force exerted by the web steel on the west side,

F , the east side, F , the top, F t' and the bottom, F b' as well ww we w w as the transverse shear V , and the torque T , taken by the uncracked

C C

concrete zone.

If it is assumed that at failure the web steel crossed by the

failure surface will be yielding and if it is recalled that A is the crossw

sectional area of one leg of the web steel, f is its yield stress and s is w

the spacing, then the force exerted by the web steel on any one side of

the beam per unit length would be:

A f /s. w w

Thus if the length of the failure surface on each side is obtained

from Figure 6. 2, it can be seen that:

F = WW

A f w w s

6.9

. d

F = we

A f w w . d (1 - µ) ........ 6.4 s

F = F = wt wb

A f w w s

µb.

In the limiting case of simple shear American and Australian

design procedures relate the shear carried by the concrete, V , to co

the shear capacity of a beam not containing web reinforcement. It

would seem reasonable, therefore, to relate V and T to the load carrying C C

capacity of such a beam. In Chapter 4 it was shown that a conservative

estimate of the strength of this type of beam was:

V + 1. 6T b

= V. 0

Hence for the case of a web reinforced beam we have:

V + C

1. 6T C

b = V co

6. 2. c. Equilibrium Equations.

........ 6. 5.

The sum of the moments of the external and internal forces

about a longitudinal axis of the beam is equal to zero. For conveniencP.

the axis passing through point O will be chosen (see Figure 6. 4).

Thus,

T + d V - T = 0. 0 C

Hence,

T =T+d V-e o

A f w w s

6.10

....... 6. 6.

Substituting forµ and d , for the case of T /Vb < 1, Equation 6. 6 becomes: 0

T T + b T V - A: f w . bd [ dd I + ~ + .! b I - .! C = 3 • Vb 3 2 b 3 .1:..J .1:..

Vb Vb

6. 7.

d' b' If d and b are both taken as equal to 0. 85 then Equation 6. 7 becomes:

T C

4 = -T -

~

A f w w s

bd G- 94 _ o. 3 3

T ..•. (Vb <.. 1)

T Vb.

........ 6. 8.

The sum of the internal and external forces in the vertical

direction is equ~l to zero.

Thus,

V - V - F - F = 0. C we WW

Substituting for F and F , we ww

V = V C

+ 2 A f

w w . d -s /J

But from Equation 6. 5,

1.6 T C

b

A f w w s

........ 6. 9.

d.

Hence,

V=V +2 co

A f w w d -s

1. 6 T C

b - µ

A f w w

s

6. 11

d. 6. 10.

The shear capacity of the section in the absence of torsion, V , is 0

given by the shear capacity of the concrete plus the shear capacity of the

web reinforcement as given by the truss-analogy.

That is,

V 0

= V + 2 co

A f w w

s

Equation 5. 10 thus becomes: I. 6 T

V = Vo--b~-c - µ.

d.

A f w w s

d. ........ 6. 11.

When the expression for T as given by Equation 6. 8 is substituted into C

the above equation it becomes:

+ I. 6 A f

w w s

d T

(1.32 - 0.33 Vb)

(.!. < 1) Vb .

T Vb

This Equation can be rearranged to give:

A f T T w w (1.31 -1 + 1. 6

s. V 0.33 Vb) Vb

V 0 =

V 1 + 2. 1

T 0

Vb

T (Vb < l). . ......... 6. 12

T By a similar process it is found that for Vb ~ 1,

V V

0

=

A f d 1 + 1. 58 w w

sV 0

T 1. 53 + 1. 6 Vb

6. 2. d. Simplification.

6. 12

........ 6. 13.

The use of the above expressions for the design of beams is A f

considerably complicated by the presence of the parameter w w d, s V

0

as this implies that the torsion- shear interaction behaviour depends on

the amount of web steel in the beam.

Equations 6. 12 and 6. 13 have been plotted in Figure 6. 5 for three Aw fwd

values of the parameter s V , namely O. 3 (curve a), O. 2 (curve b) 0

and 0. 1 (curve c). These values will encompass most web reinforced

beams covered by normal design practice. Also plotted in Figure 6. 5

is the equation:

V V

0

+ 1.6T bV

0

= 1. ........ 6. 14.

It can be seen from the figure, that for the regions where a predominantly

shear failure is likely (high values of V /V ) this equation is a reasonable 0

approximation to the more complex expressions derived above.

Use of Equation 6. 14 for the case of predominantly shear failures

has the additional advantage that then the expression for beams

containing web steel would be of the same form as that for beams with

only longitudinal steel (Equation 4. 32).

6.13

1.0 I

I ·I

j

0. 8.

V+ 1. 6T = I V bV

0 0

0.6

~I 0 I

~:> I ~ ,.a

I 0.4

.J

I T

I = I Vb

0.2

I I

/

0 0.2 0.4 0.6 0.8 I. 0

V V

0

FIGURE 6. 5. ACCURACY OF THE APPIDXIMATE EQUATION FOR

PREDOMINANTLY SHEAR FAILURES.

6. 14

That is,

V+ 1.6T =V b 0

........ 6. 15.

6. 3. MODE 2 FAILURES.

A general view of the second mode of failure is shown in Figure

6. 6. Shown in this figure are the forces in side longitudinal steel,

F 1 s' the top and bottom branches of the web steel, F wt' the side branches

of the web steel, F , and the forces in the concrete compression zone, ws

C , Cf and C . S C

The length of beam which the failure surface occupies is equal to

h tan 92 {see Figure 6. 6) and some portion of this length will separate

the two forces exerted by the top and bottom branches of the web steel.

This portion will be called 't h tan 9 2 .

In the following analysis the same assumptions as were used in

the analysis of Mode 1 and Mode 3 will be employed.


The sum of the moments about a longitudinal axis through point

O is equal to zero.

Thus:

......... 6. 16.

b

T Fis

-A-h

Fwt

I•_ h tan 91 ~

FIG. 6.6 VIEW OF THE MODE 2 FAILURE en _,a

UI

6. 16

The sum of the moments about a vertical axis through point O is

equal to zero.

Thus,

have:

........ 6. 17.

Eliminating C from Equations 6. 16 and 6. 17 and rearranging, we C

+ F wt sin 92 {d' - 'i h),

6. 18.

As in the case of Mode 1 and Mode 3, the last term in this

equation is negligible and will be ignored. Equation 6. 18 thus becomes:

x2 {l-2b) = {Flscot92+Fws){b-a2-x2) .

. . . . . . . . 6. 19.

As x2 will be much smaller than 2b it would be reasonable to make the

conservative assumption that the term xz is negligible. 2b

So Equation 6. 19 becomes:

........ 6. 20.

6. 3. b. Magnitude of the Forces.

As in the analysis of this mode it has been assumed that the

steel crossed by the failure surface yields the magnitude of F ls will

6.17

be:

........ 6. 21.

where AL2 is the area and fL 2 the yield stress of the longitudinal steel

near the side face.

For most beams we may take:

= ½ (1 + R) ALI fLl. . ....... 6. 22.

where R = AL3 fL3

ALI fLl

Now the length of the beam over which the side branches of the

transverse steel are intercepted by the failure surface may be taken as

d' 2 b+h. h. tan 92 .

Thus,

F ws

= A f

w w s

It is assumed that:

d' b' =

h b

= cJ:...

and if it is recalled that

r = A f

w w s

d' h 2 b+h

b'

........ 6. 23.

then Equation 6. 23 may be recast to give:

F = ws

oC. r tan Q 2

2 ALI fLl. 1 + oC

6. 18

. ...... 6. 24.

If the above expressions for F and F 1 are substituted into the ws s

equilibrium expression (Equation 6. 20) the following expression

is obtained:

or

T(l+o)=

where Vb

6 = 2T .

r.oC. 2

l+r:£

r .oe. Q 2 tan 2

1 +~

6.3.c. Determination of Tan 9 2

....... 6. 25.

The inclination, Q 2 , of the failure hinge will be such as to make

the failure loads a minimum.

This means that:

d + r.oC. t 9 ]

. 2 an 2_ 1 +-cC.

It is found that Equation 6. 26 is satisfied when:

j 2 l+-tan 9 = ___!!:_

2 r oC. 1 + R

2

= 0. .... 6. 26.

....... 6. 27.

6. 19

When this expression is substituted into Equation 6. 25, we obtain the

following expression for the Mode 2 failure torque~

r oC. .. . 6. 28. T 2 (1 + 6) = ALI fLI (b - a2 - xzj

It will be convenient to assume that:

2 . 2( 1 + R).

1 + cc.

If it is recalled that:

h - a 1 = A. ,....

then Equation 6. 28 can be rearranged to give:

1 =

1 + 6

6. 4. MODE 3 FAILURES.

J 2(l+R)r 2 + oe..

......... 6. 29.

...... 6. 30.

With very high eccentricities of load it is possible for a Mode 3

failure to occur. In the previous chapter it was shown that the torsional

strength of a beam failing in this mode increases rapidly with the ratio

of moment to torque. When shear is present the moment varies along the

length of the beam and some difficulty may arise in estimating the flexural

moment acting on the failure mechanism. As all the specimens tested

in this investigation were simply supported in bending and subjected to

constant torque along the shear span an explicit formula for this

situation will be derived.

6.20

The length of beam which a Mode 3 failure surface occupies is

equal to b. tan 93 . As the strength of the beam in this mode is increased

with increasing moment the failure will occur as near to the support as

possible. It therefore follows that the distance of the centre of the

failure surface from the support will be½ b tan 93 . The moment at this

point will be of magnitude V. ½ b tan 0 3 and so the value of the ratio T / M

will be:

4' T = V ½ b tan 93

1 = 6. tan 93·

......... 6. 31.

This value for ~ can be substituted into the equilibrium

equation for the Mode 3 failure (Equation 5. 18). Following the same

steps as before the equation for the failure torque is obtained. This is:

= _2_ ~ 1 - 6 ./ l+20f ....... 6. 32.

6. 5. TRANSVERSE SHEAR - TORSION INTERACTION.

For the purpose of studying interaction behaviour, the Mode 2

equation may be recast. Thus Equation 6. 30 can be written as:

V 2T + -b

= 2M

u b ~) 2(1 + R) r

2 + ~ .

Likewise the Mode 3 equation can be written:

2T b

- V = 4M

u b

j r. R 1 + 20C..

..... 6. 33.

........ 6. 34.

6.21

EFFECTIVE

SHEAR MODE -~

TRANSVERSE SHEAR

FIGURE 6. 7. SHEAR-TORSION INTERACTION FOR WEB REINFORCED

BEAMS.

6.22

It will be recalled that the equation for effective shear failures was

V + 1.6 T b

= V. 0

....... 6. 35.

For any given beam the right hand sides of the above equations

will be constants, and so it can be seen that the shear-torsion interaction

behaviour will be made up of three linear parts. To demonstrate this

point the interaction diagram for a typical beam has been given in Figure

6. 7. It has been assumed that in the absence of torsion this beam will

fail in shear (Mode 1 not critical) and in pure torsion the beam will

fail in Mode 3 (R < 1). It will be seen that the shear-torsion interaction

diagram is composed of three straight lines. These correspond to

Equations 6. 33, 6. 34 and 6. 35.

CHAPTER 7

EXPERIMENTAL VERIFICATION OF THE THEORY

FOR WEB REINFORCED BEAMS

7. l. INTRODUCTION.

7.1

Reliance can be placed on any theory only if experimental

verification can be supplied. In this chapter a comparison will be made

between the behaviour predicted by the equations derived in the last two

chapters and the observed behaviour of test beams. An appropriate

method of making this comparison is to examine the ratio of maximum

load obtained in any test to the load predicted by the theory presented in

earlier chapters e. g. T /T h . This ratio should ideally be exp t eor

greater than or equal to unity. For this purpose the results obtained

in this investigation as well as those reported in the literature will be

employed. A brief summary of all experimental results is given in

Appendix B.

In the derivation of the analysis equations it was assumed that

web steel crossed by the failure surface yielded at failure. Further, in

7.2

the case of Mode 1, 2 or 3 it was assumed that the longitudinal steel

also yielded. If the beam contains excessive amounts of steel7 failure

may occur before yield of the steel and in this case the analysis equations

may lead to unconservative results. Limits must therefore be placed

on the range of validity of the theory to exclude these cases.

7. 2. LIMITATIONS ON THE STEEL.

a. Excessive Transverse Steel.

If the amount of web steel is increased a stage is eventually

reached where crushing of the concrete on the sides of the beam precedes

yielding of the reinforcement. In the case of transverse shear without

torsion this limit is often expressed in terms of the nominal shear stress.

e.g.

V bd

It is suggested that an appropriate limit for the case of combined

transverse shear and torsion would be:-

veff bd

= V + 1.6 T/b ~ bd -r 8~ ........ 7. l

In Figure 7. 1 the parameter T /Tth , which is a measure of exp eor V

the accuracy of the theory, is plotted against eff bd~

C

(The values of T /Tth for the various beams are listed in Appendix D). exp eor

It can be seen that for high values of the nominal "shear" stress, that is

for excessively reinforced beams, the theory predicts failure loads in

excess of the experimental values. Further it can be seen that if the

c.:

2.0

1. . I

1.

1.0

I

I

2 .c. a ._... B

" ci. X

I-CJ

• • + ' + . . _, -t-. _;~

• X I 11 ~ ex • • o. •

..... • 1·

• ,-

• I

'* -1- • + . •. J ~ .,,,._!_, .. _ ~-: ~ j

·~¼. I I •• a-;-:-a .... 0 ••••

0. _,, THIS INVESTIGATION • EVANS e

ERNST 0 YUDIN -L I

GESUND ET AL • LYALIN * CHINENKOV X COWAN 0 LESSIG +

t I -i- -i-

I I _i_i_

.

• . .,.

• +

• + + 1+

• +

00 2 4 6 8 10

VALUES· OF Vetf.

bd.J?;"

f

+ • -~ ~ .... • • ••• • I • • ..

• + •

+

12

FIG. 7, 1 · THE EFFECT OF THE NOMINAL SHEAR STRESS ON THE ACCURACY OF THE THEORY.

14

=" C,J

7.4

limitation implied by Equation 7. 1 is employed the majority of the

unconservative results are eliminated.

7. 2. b. Excessive Longitudinal Steel.

Lessig (Ref. 2. 27) reported that test specimens containing

excessive amounts of longitudinal reinforcement failed prior to yield of

the reinforcement, with crushing of the concrete near the top surface.

Consideration of the equilibrium of forces in the longitudinal direction

suggests that an appropriate criterion to eliminate this type of failure

would be the criterion often used in flexural design. That is:

ALI fLl - AL3 fL3

bd f' C

f' 0.4. . ....... 7. 2.

All beams tested during this investigation were designed to

satisfy the above criterion.

7. 2. c. Longitudinal Steel out of Proportion to the Transverse Steel.

In the development of the analysis equations for Modes 1, 2 and 3

it was assumed that both the longitudinal and the transverse steel yielded

at failure. However, tests in which the strains of the steel have been

measured (Ref. 2. 19 and 2. 28) show that for low values of the ratio

"amount of transverse steel" to "amount of longitudinal steel" (a

measure of this ratio is the parameter r) the longitudinal steel may not

yield. An investigation must thus be made of the range of r for which

the equations for Modes 1, 2 and 3 will be valid. As this range will

depend on the factors o:., and 'tJ a parameter r O incorporating these

factors is introduced. The value of the parameter r corresponds to 0

a desirable design value of the ratio r for given values of c;:. and 'Y . The value of r may be calculated from the equation below.

0

r = 0 4 +

1 4

4' Jl+2oc

The derivation of this equation is given in Appendix C.

7.5

....... 7. 3

In Figure 7. 2 the parameter T /Tth is plotted against r/r exp eor o

where r is the actual value of r for the test beam and r is the optimum 0

value of r as given by Equation 7. 3. Only those beams which were

governed by Modes l, 2 or 3 were used in the preparation of this figure

( see Appendix D). Further Figure 7. 2 does not include results of tests

where failure may have been initiated by crushing of the concrete (i. e.

those beams not satisfying Equations 7. 1 or 7. 2).

It will be noticed that for low values of r / r there is a wide 0

scatter of experimental points, but the theory is still generally conservative.

In this range, corresponding to beams with relatively small amounts of

transverse steel, the idealised modes of failure are no longer applicable.

Factors ignored in the development of the analysis equations, such as

tensile stresses in the concrete and dowel forces exerted by the steel,

are now of considerable importance. As the errors thus introduced

offset the usually smaller error involved in the assumption that the

longitudinal steel yields, the analysis equations still give usable, if not

completely reliable, results. For values of r/r greater than 0. 9, the 0

theory is both consistent and accurate. It is concluded that the

analysis equations should only be applied when:

r r

0

0.9. . ...... 7.4.

2.0 I ~ 1-

0

0 . I 0

~ 1-, • • 1.61 • l X0..!~- e • I X~ I .. ~ I L 0 4) s:;

,-!' 1.21 • o 010 o'it- x •• I ·><xr • I ' I -I •

" . . _,

<Z>~ §) I 0 ... _ 0 I 0 ~ Q ~ 1.0

~4' I I eol I I+ 1· + 08

~ V)

~ 0.41 j I ~, THIS INVESTIGATION • EVANS (:)

ERNSt· 0 YUDIN I -i-

,._jd GESUND ET AL • LYALIN • CHINENKOV X COWAN 0 LESSIG +

0 :--, 05 1.0 1.5 2.0 2.5 3.0 15 4.0 0)

VALUES OF ~/r0

FIG. 7., 2 THE EFFECT OF THE PARAMETER r/rtJ ON THE ACCURACY OF THE THEORY.

7.7

7. 3. INTERACTION OF BENDING AND TORSION.

For any theory of combined bending and torsion to be considered

satisfactory one should require, in addition to values of T / Tth ~ 1, exp eor ~

that the theory correctly predicts the effect of flexure on

torsional strength.

In Chapter 5 it was shown that for the most common mode of

failure (Mode 1) the flexure-torsion interaction equation predicted by

the proposed theory was:

M + M

u = 1. ....... 7. 5.

This form of interaction is illustrated in Figure 7. 3, where a

comparison has been made with the available experimental results for

which Mode 1 was critical (see Appendix D). The tests reported in

Figure 7. 3 are those which were subjected to uniform bending and

torsion along the failure length and which satisfy the restrictions

imposed on the theory (Equations 7. l, 7. 2 and 7. 4). It can be seen

from the figure that the theoretical interaction curve forms a

satisfactory lower bound to the experimental results. In particular

it can be observed that the theory correctly predicts the substantial

decrease in flexural capacity induced by the application of torsion and

vice versa,

Beams which contain more bottom than top longitudinal steel

(R ~ 1) and which are loaded predominantly in torsion, may fail in

the manner referred to as a Mode 3 failure. The interaction equation

for this type of failure was derived in Chapter 5 as :

7.8

1.2--------------i.-----------~--~

• • {J_ t + .M. • 1

• To M11 0.8

(:) x• • • X

06 • ~o

~ • • ~

0.4

V) w 3 ~ 0.2

~ 0 a 0.4 06 08 1.0 1.2

VALUES OF M/Mu LEGEND:

THIS INVESTIGATION • CHINENKOV X ERNST 0 LESSIG + GESUND • EVANS (!)

FIG.7. 3- BENDING TORSION INTERACTION FOR WEB REINFORCED BEAMS

q )2

0

M M

u

7.9

= R. ....... 7. 6

This equation predicts that for this type of failure the torsional

capacity will be increased by the presence of flexural moment.

To test the above equation a series of test beams which contained

more bottom than top longitudinal steel (RU series) were tested. The

results of these tests are compared with the theoretical expression in

Figure 7. 4. It is seen from the figure that for beams of this type the

ultimate twisting moment is increased by the addition of bending moment

up to some critical value as predicted by the Mode 3 equation. Further

that the portion of the torsion-bending interaction diagram corresponding

to Mode 3 behaviour forms a lower bound to the test results.

7. 4. INTERACTION OF TRANSVERSE SHEAR AND TORSION.

In the previous chapter it was shown that for a beam which in the

absence of torsion fails in transverse shear, and which has less top than

bottom longitudinal steel, the theoretical shear-torsion interaction

diagram is composed of three straight lines ( see Figure 6. 7).

To compare the theoretical predictions with an experimentally

obtained interaction diagram the V series of beams was tested. The

results for the series of tests have been plotted in Figure 7. 5, the

ordinates representing the twisting moment T and the abscinae the

shear force V. The three straight lines corresponding with the

theoretical torsion-shear interaction for these beams have also been

drawn on this graph. It will be seen from this figure that as these

beams contained less top than bottom longitudinal steel they failed

in Mode 3 (Equation 6. 34) when loaded predominantly in torsion. Further

1°20

100

80

i:: ..... . .e- 60

,!rid -E-4

40

20

G)

0 40 80 120 160 200 M (kip. in.)

FIGURE 7. 4. BENDING-TORSION INTERACTION FOR RU SERIES.

240 280 320 -...J -0

100

80 I / - 1V1Ul.JJ:!, > ~

I I 60

-i::

•r-1

p.. •r-1 ..!:tl 40 -E-t

20 I I

0 5

~ I

I

I

10

01 ,; ..,

MOOE 2

~i

I EFFECT!'

MODE

I

15

V (kips.)

-

.7'. 0

- - -

I

20

FIGURE 7.5. SHEAR-TORSION INTERACTION FOR V SERIES.

0

"l ®

25 30

-.J

......

......

7. 12

as they failed in transverse shear in the absence of torsion the effective

shear mode was critical for low values of torsion. (In plotting Equation

6. 35 the observed value of V was employed.) It can be seen from 0

Figure 7. 5 that the observed interaction behaviour agrees fairly well

with the theoretical predictions.

If a beam fails in flexure rather than transverse shear when

torsion is absent, then for low values of torsion the failures will be

flexure-torsion (Mode l) rather than shear-torsion failures. The

flexure-torsion interaction equation has been given in the previous

section as:

+ M M

u = 1,

where T is the pure torsional strength in Mode 1. 0

Because of the loading arrangement used, i. e. one and two

point loading1it is possible, for the beams tested in this investigation,

to relate the moment applied to the transverse shear force.

Thus,

M = V. a

where a is the shear span. The flexure-torsion interaction equation

can thus now be changed to a shear-torsion interaction equation.

+ V. a M

u = 1. . . . . . . . 7. 7.

Equation 7. 7 has been plotted in Figure 7. 6 for the T series of

beams (these beams failed in flexure in the absence of torsion). Also

plotted in this figure are the Mode 3 and Mode 2 equations for these

7. 13

100

0

0 80

60

s:: •.-t

. 0..

•.-t 40 .!:I.:

~

20

MODE l-4-

0 5 10 15 20 25

V (kips.)

FIGURE 7. 6. SHEAR-TORSION INTERACTION FORT SERIES.

7. 14

beams, as well as the observed test results. It will be seen that the

equations correctly predict the trend of the experimental results.

7. 5. ACCURACY OF THE ANALYSIS EQUATIONS.

In the preceding sections of this chapter it was shown that the

theory presented in Chapters 5 and 6 could accurately predict flexure

torsion and shear-torsion interaction behaviour. Further it could be

seen {c.f. Figure 7.1 and 7. 2) that within the restrictions imposed the

accuracy of the theory was not significantly effected by the amount of

web steel in the section or by the ratio of longitudinal to transverse

steel. The theory must therefore be regarded as satisfactory in

predicting the qualitative effect of various parameters on the behaviour

of the beams .

To test the "quantitative" accuracy of the theory, advantage has

been taken of the large number of test results reported in the literature.

In fact, in this analysis no distinction will be made between the results

of this investigation and the results reported in the literature.

A summary of the available test data is presented in Appendix

B. The range of parameters encompassed by the 249 test results

listed in this appendix is given in Table 7. 1. All of these test results

have been analysed by equations derived in Chapters 5 and 6 and a summary

of this analysis is given in Appendix D.

The equations employed in the analysis were:-

7. 15

TABLE 7.1. RANGE OF PARAMETERS COVERED BY AVAILABLE

TEST DATA.

Parameter Minimum Maximum

h/b 1.0 2.4

f' 680 p. s. i. 8,500 p. s. i. C

ALlfL1-AL3fL3 0.0 1.0

bd f' C

r 0.10 5.89

r 0

veff

bd~ 1.1 13.5

C

4-' 0.02 oO

6 0.0 4. 1

Mode l Eqµation (c. f. Equation 5. 14.)

2 r =

l + 20C. +

Mode 2 Equation (c.f. Equation 6.30)

1 + 2oC r

= 1

1 + 6 oC J 2(1 + R)r

(3 2+0C

Mode 3 Equation (c. f. Equation 5. 21)

2 r =

1 + 2oC.

or (c. f. Equation 6. 32)

= I: 6 / I:·: Effective Shear Equation (c. f. Equation 6.)

V + 1. 6 T

=V b 0

or bV

T 0

= 1. 6 + 26

1 - -) 'Y

7.16

...... 7. 8.

...... 7. 9.

...... 7. 10.

...... 7.11.

...... 7. 12.

Those test results which satisfied all of the restrictions imposed on

the theory (Equations 7. 1, 7. 2 and 7. 4) have been presented in Table 7. 2.

The table has been subdivided into three sections; pure torsion, bending

and torsion, and bending, torsion and shear. When the failure load was

TABLE 7. 2

A COMPARISON OF EXPERIMENTAL RESULTS WITH THE PROPOSED THEORY FOR WEB REINFORCED BEAMS

PART 1 - PURE TORSION

Investigator Beam Torque Moment Shear T

exp

7. 17

Mode kip. in. kip. in. kips T

theor

3TR15 61. 7 0.0 (). () 1. 12 l

Ernst 3TR30 76.0 0.0 0.0 0.97 1

4TR30 85.0 0.0 0.0 0.95 1

Evans HBl 44. l 0.0 0.0 1. 15 3

BKl 121. 0 0.0 0.0 0.98 2 Lessig BKlA 104.0 0.0 0.0 0.87 2

-Mean 1.01

Standard Deviation 10°/o

No. of Tests 6

7. 18

PART 2. BENDING AND TORSION


Mode ~ kip. in. kip. in. kips. T

theor

R4. 20 59.9 331. 0 0.0 1. 15 l

R4.24 56.5 264.0 0.0 0.96 l R3. 20 50.7 252.0 0.0 1.08 1

R3. 24 53.7 230.0 0.0 L 03 1

R3. 30 61. 6 207.0 0.0 1. 02 l

R2. 24 44.2 205.0 0.0 1. 24 1

Walsh R2.30 49.7 176.0 0.0 1. 18 1 R2. 38 53.4 138. O 0.0 1. 06 I

Rl. 30 41. 8 146.0 0.0 1. 07 1 REI 81. 4 6.3 0.0 0.88 2 RE2 83.5 32.0 0.0 1. 01 I

RE3 81. 5 45.0 0.0 1. 01 ]

RE4 74.6 84.4 0.0 1. 12 l

RE5 66.0 108.2 0.0 L 16 l

RE4* 38.0 134. 0 0.0 1. 10 1

RU3 105.0 84.0 0.0 1.26 3 This RU3A 89.4 149.3 0.0 1.00 1

Investigation RUZ 84.9 51. 1 0.0 1. 11 3 RU5 75.4 249.7 0.0 1. 17 1

RU5A 68.3 266.8 0 .. 0 1. 14 1

RU6 59.1 281.2 0.0 l. 15 l 36T4 62.6 240.4 0.0 1. 21 1 36T4C 94.1 61. 1 0.0 1. 23 3 36T5. 5 85.9 173. 4 0.0 1. 32 1

77T5 91. 6 262.4 0.0 1. 31 l

77T4 107.6 223.4 0.0 1. 26 l 24T3 70.8 46.6 0.0 0.92 2

1 79.0 79.0 o.o 1. 07 1

2 102.0 102.0 0.0 1. 01 1

3 61. O 122.0 0.0 1.06 1

Gesund 4 67.0 134.0 0.0 0.96 i d

Schuette 5 49.0 147.0 0.0 1.08 l

Buchanan 6 56.0 168.0 0.0 1. 08 l

and 7 43.0 173.0 0.0 1. 14 1

Gray 8 44.0 176. O 0.0 1. 05 1

11 68.0 138.0 0.0 0.98 J 12 53.0 213.0 0.0 L 06 I

7. 19

PART 2. BENDING AND TORSION (contd.)


Mode ~ kip. in. kip. in. kips. T

theor

HB2 33.9 66.8 0.0 1. 25 l HB3 20.4 75.3 0.0 1. 10 1 HB4 15.7 81. 6 0.0 1.05 l HB5 13.2 81. 5 0,0 l. 06 1

Evans HB8 21.4 79.6 0.0 1.05 1 and HB9 18.3 85.l 0.0 0.98 1

Sarkar HBl0 17.3 91. 3 o.o 0.99 ]

HBll 14. l 94.0 o.o 0.99 l HB14 41. 7 82.1 0.0 1. 03 1 HB15 29.9 111. 0 0.0 1.11 l HB16 23.5 129.0 0.0 1.05 1 HB17 19.4 137.0 o.o 1. 06 l

B28 0. l 48.6 486.0 0.0 1. 12 1 B28 0. lA 46.9 469.0 0.0 1.09 l

Chinenkov B28 0. 2 83.4 417.0 o.o 1. 14 I B28 0. 2A 83.4 417.0 o.o 1. 23 1 B28 0. 4F 139.0 347.0 o.o l. 15 I

Lessig BU6 69.5 173.6 0.0 0.87 1

Mean 1. 10

Standard Deviation 9°/o

No. of Tests 55

7.20

PART 3. SHEAR AND TORSION

Investigator Beam Torque Moment Shear T Mode ex:e

kip. in. kip. in. kips T theor

RU4 85.5 145.0 4. 13 1.21 3V

Walsh Rl. 30A 42.6 97.l 4. 13 0.89 VEF R3. 20B 59.0 78.9 3.41 1. 19 3V

V3 16.9 685.0 27.80 1. 65 VEF V6 24.8 668.0 27.20 1. 72 VEF

This U2 43.9 689.0 28.00 L53 VEF

Investigation U3* 66.2 720.0 29.30 1.80 VEF T4 53.0 523.0 21. 40 1.54 2 T5 63.4 432.0 17.70 1. 52 2 T6* 29.4 584.0 23.90 l. 36 VEF

6 4.6 45.7 2.32 l. 01 l 10 5.9 59.0 3.00 1. 31 l

Yudin 11 14.3 71. 7 3.65 l. 66 l 12 11. l 55.7 2.83 1.29 l 13 11. l 55.7 2.83 1. 29 1 22 6.5 32.6 l. 66 1. 28 1

BIII5 156.0 416.0 15.56 1. 40 2 BIII5A 151. 0 416.0 15.52 1. 33 1 BIII6 92.0 156.0 4.07 1. 31 l BIII6A 83.4 156.0 4.16 1. 27 1

Lessig BIII7A 90.4 313.0 8.06 1. 37 1 Blll7 83.4 278.0 7.16 1. 24 I BIII9 78.0 313.0 4.63 1. 50 1 BIT 19A 79.0 313.0 . 4.71 1. 59 1

PART 3. SHEAR AND TORSION (contd.)

Investigator Beam

B8 O. 1 B8 O. IA B8 O. 2A B8 O. 4A B7 O. 2 B7 O. 2A BlO 0. 2 BlO O. 2A Bl

Lyalin BlA B2 B2A B3 B3A BS BSA B6 B6A

Torque Moment Shear kip. in. kip. in. kips

52.0 520.0 12.52 55.5 555.0 13.36 97.0 486.0 11. 69

139. 0 347.0 8.38 93.8 468.0 11. 30 90.2 451.0 10.87

104.0 521.0 12.53 104.0 521. 0 12.53 90.3 452.0 10.82 90.3 452.0 12.75

139.0 694.0 16.65 139.0 6~4.0 16.65 194.0 486.0 17.48 194.0 486.0 17.48 194.0 972.0 23.24 194.0 972.0 23.24 167.0 833.0 20. 19 181. 0 903.0 21.88

Mean

Standard Deviation

No. of Tests

Summary of All Tests

Mean

Standard Deviation

No. of Tests

L 18 0

15 /o

103

7.21

T ModE ~

T theor

1.11 1 1.11 1 1. 20 1 1. 16 VEF 1. 31 VEF 1. 22 VEF 1. 26 l 1. 23 l 1. 33 l 1. 29 l 1. 29 l 1. 22 1 1. 25 VEF 1.24 VEF 1.22 l 1. 31 l 1. 31 1 1. 43 l

1. 33

13°/o

42

7.22

governed by Equation 7. 8 the mode has been designed as 1, similarly 2, 3,

3V and VEF refer to Equations 7. 9, 7. 10, 7. 11 and 7. 12 respectively.

It can be seen from the first part of Table 7. 2 that good agreement

between the experimental results and the predicted failure loads is

obtained for beams tested in pure torsion. The values of T /Tth exp eor

have a mean of 1. 01 and a coefficient of variation of 10°/o.

Examination of Table 7. 2 shows that the proposed theory accurately

predicts the ultimate strength of members loaded in combined bendbg and

torsion. For this case T / Tth has a mean of 1. 10 and a coefficient exp eor

of variation of 9°/o.

For the case of combined bending torsion and shear the mean value

of T /Tth is 1. 33 and the coefficient of variation 13° / o. To a large exp eor

extent this result reflects the conservative nature of the A. C. I. equations

which were used to predict the simple shear strength. When a more

accurate expression for the shear capacity of web reinforced beams is

derived and becomes generally accepted, the accuracy of the theory for

shear combined with torsion and flexure can be improved.

In Figure 7. 7 a frequency histogram of the accuracy of the theory,

for the 103 web reinforced beams which satisfy the restrictions, has been

drawn. It may be concluded from this figure that the proposed theory is

sufficiently accurate for safe and efficient design of structural members

to resist combined torsion, bending and shear loads.

7. 6. SIMPLIFIED ANALYSIS PROCEDURE.

In the following chapter the analysis equations will be rearranged

into a form suitable for rapid design. Considerable simplification of the

design procedure results if the equation relating to Mode 2 failures is

Q. ::,

15

e 10 (!)

C

II) ~ II)

t! 0

~ E ::, z

5

0

-- -

~

-

---I

0.8 1.0

Mean

Standard Devn.

- N8 of Tests

- ,-

-

1.2

Texp./ Ttheor.

-

n n 1.4 1.6

7.23

1.18 15 °/o 103

-n n

1.8

FIG. ,. 7 FREQUENCY HISTOGRAM FOR THE ACCURACY OF

THE THEORY FOR WEB REINFORCED BEAMS

WITHIN RESTRICTIONS

7.24

TABLE 7. 3. THE EFFECT OF IGNORING THE MODE 2 EQUATION.

T /T exp theor.

Investigator Beam l* 2*

This Investigation. REI 0.88 0.85

24T3 0.92 0.90

T4 1. 54 1. 50

T5 1. 52 1. 45

Lessig BII 15 1. 40 1. 35

l* Equations 7.8, 7.9, 7.10, 7.11 and 7.12 used

in evaluating T h t eor.

2* Equations 7. 8, 7. 10, 7. 11 and 7. 12 used.

7.25

ignored. If Table 7. 2 is examined it will be seen that the occurence of

Mode 2 failures is rare. Further if these beams are analysed by

ignoring the Mode 2 equation little loss of accuracy results. This

result can be seen in Table 7. 3, where the values of T IT h exp t eor,

for those beams failing in Mode 2, have been calculated in two ways;

by using all four equations and by using only the three equations pertaining

to Modes 1, 3 and Effective Shear.

In view of this result the Mode 2 equation will not be considereed

in the development of the design method.

CHAPTER 8

DESIGN OF BEAMS SUBJECT TO TORSION

8.1. INTRODUCTION.

8. 1

The theory developed in the foregoing chapters can be used to design

beams subjected to combined bending, torsion and shear. For this purpose

it is advantageous to rearrange some of the equations. In particular, the

form of the equations dealing with the analysis of beams containing web

steel and loaded in bending and torsion is not suitable for design. With

some modification and re-arrangement, however, straight forward design

procedures result.

8. 2. DERIVATION OF DESIGN EQUATIONS FOR WEB REINFORCED

BEAMS.

Provided that sufficient top longitudinal steel is present the failure

torque of a web reinforced beam loaded in bending and torsion is given by

8.2

Equation 5.14. That is

T Zr cJ <!>2 1 + 2d'. 1

= + - -) M 1 + 2~ r 'fl

. . . . . 8. 1 u

A f b'

where w w

r = s ALlfLl

..... 8. 2

The given bending moment and torque (i.e. T and 'V are known)

can be resisted by many different beams. As is usual in de sign a choice

is first made of some parameters, and the remaining dimensions are then

determined to satisfy the basic equations.

The parameter r may be chosen from within a wide range of values.

This means, physically, that there is a good deal of choice as to how the

load is shared between the longitudinal and the web steel. The design

can be considerably simplified, however, if r is chosen as an "optimum

value" r , w.hich gives something approaching a minimum volume of steel 0

(see Appendix C). This value is given by:

1 r =

0 4 + 4/(q, ../1 + 20: ) . ..... 8. 3

If this value is substituted for r in Equation 8. 1 and if for

convenience we write A 2 = 1 + 2oC. then we obtain:

~D -~ T 1 ( !. )2 4A

4 A 2 -- = + -- + M

2~...2 'I' ~ u + 4' -

1 =

:l ~ .!. ~

1 M =T(.i\.+-)

u 'f'

but T

= M

M = T:il. + M u

or M u

where T'

= M + T'

= T-;\. =

8. 3

...... 8. 4

. . . . . . 8. 5

Thus from the given values of M and T and the chosen value of OC.

the required flexural capacity of the section M can be found. The area u

of longitudinal steel, ALI, can now be determined from the usual flexure

equation,

M u

= . . . . . . 8. 6

where jd is the internal lever arm which may be found from the following

formula:

jd = d (1 - o. 59 p fLl ft ) . ...... 8. 7

C

The A.C.I. Building Code for Reinforced Concrete 318 - 63, and the

S.A.A. Code for Concrete CA2 - 1963 both limit the value of p to some

value less than the balanced reinforcement ratio so that compression

failures in flexure will not occur in the event of overload. These

restrictions should apply in the present situation, e. g. p shall not

exceed O. 4 f~/ fLl.

If the value of M as given by Equation 8. 4 is u

substituted into Equation 8. 6 we obtai.n:

M =

T' + fLl jd

8.4

8.8

A certain ratio of transverse to longi.tuclinal steel has already

been chosen (r = r ) and thi.s rati.o fixes the amount of transverse steel 0

required.

Wi.th r = r , r given by Equation 8. 2 and r by Equation 8. 3, the 0 0

following relation is obtained:

A f w w b'

= s

=

=

But from Equation 8. 8:

A f b' w w = s ALl fLl

and so:

A T's = w 4b' jd f

w

1

4 + _!_ If.I>.

T ;t 4(T + M)

T' 4(M + T').

T'

4 ALl fLl jd ,

....... 8. 9

Top longitudinal steel must be provided. It may be required to

prevent a Mode 3 failure, and : f not a nominal amount must in any case

be provided to support the hoops.

In Chapter 5 it was shown that the transi.tion from a Mode 1 to a

Mode 3 failure occurs when:

M M

u = ¼ (1 - R).

Thus a Mode 3 failure will not be critical if:

R;;, l _ 2M M

u

and M = M + T' u

= 1 2M

=

For design purposes:

hence

AL3 =

M + T'

T' - M

M+·T1

T' - M

jd fLl l

8.5

. • . . . . 8. 1 0

To prevent a shear and torsion failure it is necessary to design

the beam to resist an effective shear of:

V eff = V + 1.6T

b

If the A.C.I. code is employed this means:

V + 1. 6T b

= 2 bdµ_ + C

8 bd_r-;;: C

2A f w w

s d

where A is the area of one leg of the hoop reinforcement. w

8.6

...... 8. 11

Little information is available to assist the designer who wishes to

use bent-up bars to provide the shear strength. In fact, as far as the

writer is aware, no tests have ever been performed on beams loaded in

shear and torsion which contain bent;...up bars. However, a reasonable

approach would be to calculate the shear capacity of bent up bars which

are being employed and then to design stirrups for the remaining transvers~

shear together with the twisting moment expressed as an effective shear

force 1. 6 T /b.

8. 3. DESIGN PROCEDURE.

The design of a beam either with or without web reinforcement and

loaded in torsion combined with bending and shear may be carried out in

the following manner. T, M and V are the desired ultimate capacities in

torsion, flexure and shear.

Rule 1.

Rule 2.

Rule 3.

and

where

Rule 4.

8.7

The beam must be proportioned to resist an effective shear

force V eff where:

V eff 1. 6.T

= V + b .

If web reinforcement is not required to satisfy Rule 1, i.e.

if V eff ~ 2 bd f~, ~t is necessary to provide only

sufficient bottom longitudinal steel to resist M. That is:

M =

fLl jd.

If web reinforcement is required to satisfy Rule 1, then both

top and bottom longitudinal steel must be supplied:

= T' +M

fLl jd

T' - M

fLl jd

If web reinforcement is required to satisfy Rule l then a

minimum amount must be provided. This minimum is

or

given by the greater of:

A w

T 1·s =

A = w

4b' Jdf w

where A is the area of one leg of the hoop reinforcement. w

8. 4. DESIGN EXAMPLE.

8.8

The use of the above rules will be illustrated in a de sign example.

A spandrel beam, which is fully restrained at both ends, is

eccentrically loaded by a secondary beam. Figure 8.1 shows the

bending moments, twisting moments and transverse shear forces

present in the beam when the desired ultimate. load of 40 kips is acting

at an eccentricity of 5 inches.

It is desired to use a web reinforced beam and for the purposes

of this example the following choices have been made,

f' C

= 3, 0 00 p. s. i.

= f = 33,500p.s.i. w

It will be assumed that the width of the beam is 10 11 •

Rule 1.

Rule 2.

veff = V + 1. 6T

b

= 36 kips.

= 20+ 160 10

As web steel is being used Rule 2 does not apply.

~ <

1,440 kip. in.

40 kips.

l 24 1-011

1,440 kip.in.

BENDING MOMENTS

J 8.9

1,440 kip. in.

~::mJIIIJJIJJ[IIJlllll!lj. · , .

111111111111111 r 1111 ~~~. in.

TWISTING MOMENTS ..

~ps _ 11111111111111 1 11111 I _ . .. I I 111 I 1111111 I I l I I I [ 20 kips·

TRANSVERSE SHEAR

FIGURE 8.1. ECCENTRICALLY LOADED SPANDREL BEAM.

Rule 3.

Footnote:

Try h/b = d: = 2

= 224 kip. in.

Thus the greatest value of:

M + T' = 1440 + 224

= 1663 kip. in.

If p is chosen as 0. 015 (see footnote) then:

= (1-0.59.l(0.015JC33.5) = j 3

2 Now bd =

=

M + T'

p.j. fLl

1663 0.015x0.9x33.5

0.90.

= 3,680 in 3 .

Thus if b = 1011 , d = 19.5 11 and soh = 21.5 11 •

ThereforeoC.= h/b = 21.5/10 =2.15.

We now recalculate T' using the more exact value of d:..

T' = l 00 J 5. 30 = 230 kip. in.

Thus T' + M = 230 + 1440 = 1670 kip.in.

8.10

The value of p employed in the ultimate strength method of design for

flexure can vary within wide limits. In choosing any particular vah .. e-,

attention should be giv;en to the effect on deflections as well as economy.

In general it will be found economical to employ a value of p somewhat

greater than that indicated for balanced design by the permissible: stress

method (Ref. 8.1).

2 and so bd

Hence b = Now,

ALI

Similarly:

1670 =

1663 X

1011 and d =

=

=

T' +M

fLl jd

T' + M 588

T' - M 588

8.11

3680 = 3,700 in3

19.511 are satisfactory.

T' + M ....... 33.5JC 0. 9 J( 19.5

To proportion the areas of longitudinal steel we thus require the

values of T' + Mand T' - M. When consideration is given to

the change of sign of M at the point of contraflexure we obtain

the effective moment envelope shown in Figure 8. 2. This

diagram may be used in the calculation of the minimum

quantities of longitudinal steel {in both top and bottom faces)

necessary to resist the combined flexure and torsion.

At the mid-point of the beam:

T' + M = 1670 kip.in.

1670 hence ALl = = 2. 84 sq.in.

588

say 2 -1 11 4' + 2 - 7 /811 cj> bars in the lower face. Only

nominal areas of top steel are required at this position as

T' - Mis less than zero.

At the supports:

T ' - M = 1 6 7 0 kip. in.

f I

~ I

-G,-1

~, I I

N ~I

ro I

.0

. -e-1

~' I I

'u1

'dp

{ 0 L 9 I

ml

~I

.o I ~

~I

r-I

I

NI

N "-----'-

---_

__

.

fl ro .o I

:_9- I

-----

___ _

J_ .. -- -

··-·.

-I

I'll s., ~

.0

I

-9-I

:...1

8.

12

. ....l µ:l µ:l ~

en. ....l c:t: ~

0 ::> ~ -0 z 0 ....l ril ::r: ~ ~

0 z 0 - en. µ:l 0 . N

. 0

0

µ:l 0:: ::> 0 -~

Rule 4.

8.13

hence 2 - 1114> + 2 - 7 / 811 c:I> bars are required in the top

face with only nominal areas of steel in the lower face of the

beam. Bond and anchorage requirements will of course

influence the curtailment of longitudinal reinforcement.

A w T'

= s 4 b' jd f

w

taking b' = 8. 511 , we have

A w

s

or

A w s

= 230 4x 8.5x 0.9x 19.5 x 33.5

= 0 .. 0118 inl /in.

=

= p6 - 2Jtl0xl9.5>< 0.055] 2 l 19.5"33.5

0 0 3 . 2,. = .. 11 1n 1n.

i.e. not critical.

Hence

A w s

= 0. 0118 in2/in.

Say 3/ 8 11 ; ties at 9'' c. c.

8.14

Check V eff } 8 bd0c

36 l' 8d0~19.5,c0.055 = 86kips.

i.e. not critical.

This final check is made to ensure that the cross-section

of the beam is sufficiently large that a concrete compression

failure will not occur in the web under combined torsion and

shear.

9.1

CHAPTER 9

CONCLUSIONS.

9.1 CONCLUSIONS.

In this research project the behaviour of reinforced

concrete beams loaded in torsion, combined bending and torsion

and combined bending, torsion and transverse shear has been

examined. Analysis equations for the prediction of the ultimate

load of rectangular beams have been derived and a simple method

of design for these beams has been presented.

In particular the following conclusions were reached:

1. Beams reinforced with only longitudinal steel and loaded

in pure torsion fail immediately after the appearance of

the first diagonal crack at torques comparable with the

9.2

maximum twisting moment which can be resisted by a

plain concrete member.

2. A conservative estimate of the torsional capacity of a plain

concrete member may be obtained by assuming that the

shear stress is constant over the full section and that

failure occurs when this stress reaches a value of 3. sjr~. 3. Whether the presence of bending will increase or decrease

the torsional Cl\pacity of a beam containing only longitudinal

steel depends on the section properties of the member.

Flexural cracking will reduce the effective section re&is'fing

the torque but the flexural compressive stresses in the

uncracked concrete zone will increase the ability of this

zone to resist torsi~n. Heavily reinforced, slender beams

will have only a small portion of their cross_; section

destroyed by flexural cracks and hence flexure may increase

the torsional capacity of such beams. On the other hand.

lightly reinforced, squat beams will be extensively cracked

by flexure and hence their torsional capacity will be

reduced.

4. Because the effects of flexure tend to cancel each other,

if a conservative value for the pure torsional strength is

chosen, torsion-flexure interaction for beams containing

only longitudinal steel can be ignored.

9.3

5. The value of the nominal shear stress at failure for

beams containing only longitudinal reinforcement and loaded

in combined transverse shear and torsion, depends upon the

loading ratios. The value of the failure stress increases

with increasing values of the ratio T /Vb and decreasing

values of the ratio M/Vd.

6. While a design procedure might be formulated in which the

value of the limiting shear stress is related to the loading

ratios, it is found that a simpler and more satisfactory

procedure results if linear interaction between the strengths

in transverse shear and torsion is assumed.

7. Prior to the failure of any fully reinforced (i. e. containing

both longitudinal and transverse steel) member sustaining

torsion and bending,· a tensile crack developes forming a

warped surface which intersects three exterior faces of

the beam in a helix; the compression zone on the fourth

face joins the two ends of the helix and is consequently

inclined to the axis of the beam. At failure the steel

crossed by the tensile crack yields, permitting rotation

of the member about an axis in the compression zone.

Following Lessig (Ref. 2. 22), failure with the

compression zone occuring near the top face is ref erred

to as a Mode 1 failure, whilst a Mode 2 failure indicates

9.4

that this zone forms near a side face. In the present

investigation some specimens developed the compression

hinge near the bottom surface and failures of this type

have been labelled Mode 3.

8. Two different types of interaction between bending ~nd torsion

for beams containing both longitudinal and transverse steel

are possible. For beams in which the areas of top and

bottom steel are equal the presence of bending moment

reduces the torsional capacity. On the other hand for

beams in which the area of tension steel e:x;ceeds that of

the compression steel, the torsional capacity is

considerably increased by the presence of bending moment

'1p to a certain limiting value of the bending moment. For

bending moments beyond this limit the torsion capacity -6

decrease(\.

This interaction behaviour can be well explained by

considering the failure modes. For beams having

symmetrical reinforcement a Mode 1 failure will always

be :rr..Dre critical than a Mode 3 failure. For this reason

the beam will al ways fail in either Mode 1 or 2 and in

these modes the presence of flexure reduces the torsional

capacity.

On the other hand, when beams containing less top

9.5

than bottom longitudinal steel are loaded at high ratios of

torsion to bending, a Mode 3 failure will be critical. In

this mode failure is init.ted by the yielding of the top

longitudinal steel. A moderate bending moment tends to

oppose this action and thus increases the torsional capacity

of the beam. A larger bending moment will cause yielding

of the bottom longitudinal steel; that is, the beam will

fail in Mode 1 .

9. In the absence of torsion the mode of shear failure most

likely to occur in web-reinforced beams of normal design is t

the so-called "shear-compression" failure. In this mode

the compression zone forms near to and parallel to the

top face of the beam, the diagonal tension cracks form

only on the side faces of the beam (all stirrups crossed by

the diagonal cracks yield), while the tension crack on the

bottom face is perpendicular to the longitudinal axis of the

beam.

When the beam is loaded in combined transverse shear

and torsion at low eccentricities (T/Vb low), the

compression zone moves towards the side face of the beam

on which the torsional and transverse shear stresses

oppose. The torsion also causes diagonal cracking on

both the top and bottom faces of the beam. This type of

9.6

failure is referred to as an Effective [' 1-.f:'ar tlu:".'e ..

When the eccentricity of the load is further increased

the compression zone covers all of the side face of the

beam while diagonal tension cracks cover the other three

sides (Mode 2 failure).

At very high eccentricities it is possible that the

compression zone will form near the bottom surface of the

beam ( Mode 3 failure).

10. Expressions for the failure loads of fully reinforced beams

loaded in combined bending and torsion or combined

bending, torsion and transverse shear have been derived.

These expressions were obtained in the main from a study

of the equilibrium sit;,ation of the modes of failure

described above. Good agreement between the predictions

of these expressions and the experimental resl'lts was found,

not only for the beams of this investigation, b: t also for a

large number of beams reported in the literature.

11. In the derivation of the analysis equations for web reinforced

beams it was assumed that web steel crossed by the failure

surface yielded at failure. Further in the case of Modes

1, 2 and 3 it was assumed thc,t the longitudinal steel also

yielded. If the beam contains excessive amounts of steel

failure may occur before the yield of the steel and in this

9.7

case the analysis equations may lead to unconservative

results. Empirical limitations must therefore be placed

on the range of validity of the theory to exclude these

cases.

I 2. It is possible to develop simple ultimate strength design

procedures from the analysis equations. Such a method

has been presented in this thesis.

9. 2 SUGGESTIONS FOR FURTHER RESEARCH.

As was stated in the introduction, the main aim of this

thesis was to study the behaviour of isolated rectangular beams

loaded in combined torsion, bending and transverse shear, and

from this study to obtain expressions for the ultimate strength of

such members. As a result of the completion of this work,

several topics now suggest themselves for further study. In

particular the ultimate strength of reinforced concrete grillages

and three dimensional frames in which torsion will almost always

occur in combination with flexure, might now be examined. Work

on this problem would derive some assistance from Walsh'n

study (Ref. I. I) of the stiffness of rectangular beams loaded in

bending and torsion. Another practical problem which could now

be studie J in some detail is ·. · behaviour of edge beams in

reinforced concrete slabs. A study of the failure characteristics

of these beams would reveal the effects of beam- slab interaction and

to what extent the edge beams may be treated as isolated rectangular

beams.

A.I.

APPENDIX A

REFERENCES.

1.1 Walsh, P.F.

"The Strength and Stiffness of Rectangular Reinforced Concrete

Beams in Combined Bending and Torsion".

Ph.D. Thesis, University of New South Wales, 1967.

2. 1 Bach, C. and Graf, 0.

"Tests on the Resistance of Plain and Reinforced Concrete

to Torsion".

Deutscher Ausschuss fur Eisenbeton, Berlin, Heft 16, 1912

(in German).

2. 2 Cowan, H. J.

"Reinforced and Prestressed Concrete in Torsion".

Edward Arnold, London, 1965.

2. 3 Young, C.R., Sagar, W. L., and Hughes, C. A.

"Torsional Strength of Rectangular Sections of Concrete,

Plain and Reinforced".

University of Toronto, School of Engineering,

Bulletin No. 3, 1922.

2. 4 Todhunter, I and Pearson, K.

"A History of the Theory of Elasticity and of the Strength

of Materials".

Dover Publications, New York, 1960.

A.2.

2. 5 Turner, L. and Davies, V. C.

"Plain and Reinforced Concrete in Torsion with Particular

Reference to Reinforced Concrete Beams".

The Institution of Civil Engineers, London,

Selected Engineering Papers. No. 165, 1934.

2. 6 Marshall, W. T.

"The Torsional Resistance of Plastic Materials with Special

Reference to Concrete".

Concrete and Constructional Engineering (London)

Vol. 39, No. 4, April 1944.

2. 7 Nylander, H.

"Torsion and Torsional Restraint by Concrete Structures".

Statens Kommittee for Byggnadsforskning, Stockholm.

Bulletin No. 3, 1945 (Swedish with English summary).

2. 8 Kemp, E. L., Sozen, M.A., and Siess, C .P.

"Torsion in Reinforced Concrete".

University of Illinois, Civil Engineering Studies,

Structural Research Series No. 226, Sept. 1961.

2. 9 Ramakrishnan, V. and Vijayarangan, B.

"The Influence of Combined Bending and Torsion on

Rectangular Beams without Web Reinforcement".

The Indian Concrete Journal, Vol. 37, No. 11, Nov. 1963.

A.3.

2. 10 Gesund, H. and Boston, L.A.

"Ultimate Strength in Combined Bending and Torsion of

Concrete Beams Containing only Longitudinal Reinforcement".

Journal of the American Concrete Institute,

Proceedings. Vol. 61, No. 11, Nov. 1964.

2. 11 Talbot, A.N.

"Reinforced Concrete Wall Footings and Column Footings".

University of Illinois Engineering Experiment Station,

Bulletin No. 67, March, 1913.

2. 12 Kani, G .N.

"The Riddle of Shear and its Solution".

Journal of the American Concrete Institute,

Proceedings, Vol. 61, No. 4, April, 1964.

2.13 Marshall, W.T. and Tembe, N.R.

"Experiments on Plain and Reinforced Concrete in Torsion".

Structural Engineer, London, Vol. 19, No. 11,

Nov. 1941,

2. 14 Rausch, E.

"Berechnung des Eisenbetons gegen Verdrehung".

Berlin, Julius Springer, 1929. (in German).

2. 15 Andersen, P.

"Rectangular Concrete Sections Under Torsion".

Journal of the A.C.I. Proceedings, Vol. 34, No. 1, 1937.

A.4.

2. 16 Cowan, H.J.

"Elastic Theory for the Torsional Strength of Rectangular

Reinforced Beams".

Magazine of Concrete Research, Vol. 2, No. 4, July, 1950.

2. 17 Standards Association of Australia.

"SAA Code for Concrete in Buildings".

A.S. CA2 - 1963.

2. 18 Zia, P.

"Torsional Strength of Prestressed Concrete Members".

Journal of the A.C.I. Proceedings, Vol. 58, April, 1961.

2. 19 Ernst, G. C .

"Ultimate Torsional Properties of Rectangular Reinforced

Concrete Beams".

Journal of the A.C.I., Proc. Vol. 54, No. 4, Oct. 1957.

2.20 Cowan, H.J.

"The Strength of Plain, Reinforced and Prestressed Concrete

Under the Action of Combined Stresses with Particular

Reference to the Combined Bending and Torsion of Rectangular

Sections" .

Magazine of Concrete Research Vol. 5, No. 14, Dec. 1953.

2.21 Gesund, H., Schuette, F.J., Buchanan, G.R. and

Gray, G.A.

"Ultimate Strength in Combined Bending and Torsion of

Concrete Beams Containing Both Longitudinal and Transverse

Reinforcement".

Journal of the A.C.I. Proceedings, Vol. 61, No. 12, Dec. 1964.

A.5.

2. 22 Lessig, N .N.

"Determination of the Load Bearing Capacity of Reinforced

Concrete Elements with Rectangular Cross-section Subjected

to Flexure and Torsion".

Concrete and Reinforced Concrete Institute, Moscow,

Work No. 5, 1959, pp 5-28.

Translated by Margaret Corbin as "Foreign Literature

Study No. 371 11 Portland Cement Association, Research

and Development Laboratories, Skokie. III.

2.23 Yudin, V.K.

"The Determination of the Load Carrying Capacity of

Rectangular Reinforced Concrete Sections Subject to

Combined Torsion and Bending".


Work No. 6, 1962, pp 265-8 (in Russian).

2. 24 Evans, R.H., and Sarkar, S.

"A Method of Ultimate Strength Design of Reinforced Concrete

Beams in Combined Bending and Torsion".

The Structural Engineer, Vol. 43, _No. 10, Oct. 1965.

2. 25 State Committee on Construction of the U. S.S. R. Council

of Ministers.

"Structural Standards and Regulations".

State Publishing Office for Literature on Structural Engineering,

Architecture and Structural Materials, Moscow, 1962,

(in Russian).

2.26

2.27

2.28

A.6.

Chinenkov, Y . V.

"Study of the Behaviour of Reinforced Concrete Elements

in Combined Flexure andTorsion".


Work No. 5, 1959, pp 29-53.


Study No. 370'.'. ·Portland Cement Association Research and

Development Laboratories, Skokie, III.

Lessig, N. N.

"Study of Cases of Failure of Concrete in Reinforced Concrete

Elements with Rectangular Cross-section Subjected to Combined

Flexure and Torsion".

"Design of Reinforced Concrete Structures,"

Edited by A.A. Gvozdev, Moscow, 1961, 322 pp.


Study No. 398". Portland Cement Association, Research

and Development Laboratories, Skokie, III.

Lyalin, I. M.

"Experimental Studies of the Behaviour of Reinforced Concrete

Beams with Rectangular Cross-section Subjected to the Comblned

Action of Transverse Force, Flexural and Torsional Moment".

Concrete and Reinforced Concrete Institute, Moscow, W c,rk

No. 5, 1959, pp 54-77.


Study No. 402". Portland Cement Association Research

and Development Laboratories, Skokie, III.

2.29

A. 7.

Yudin, V.K.

"Behaviour of Reinforced ConcreiE Beams with Rectangular

Cross-sections Subjected to Flexure and Torsion". Concrete

and Reinforced Concrete Institute, Moscow, Work No. 1, 1964,

pp 30 - 35. Translated by Margaret Corbin as "Foreign

Literature Study No. 402". Portland Cement Association,

Research and Development Laboratories.

3.1 Hsu, T.T.C., and Mattock, A.H.

"A Torsion Test Rig".

Journal of the P.C.A. Research and Development Laboratories.

Vol. 7, No. 1, Jan. 1965.

4. 1 Cowan, H. J.

"Strength of Reinforced Concrete Under the Action of Combined

Stresses and the Representation of the Criterion of F ailun-

by a Space Model".

Nature (London), Vol. 169, 1952, p. 663.

4. 2 Bresler, B., and Pister, K.

"Strength ofConcrete Under Combined Stresses".

Journal of the A.C.I. Proceedings, Vol. 55, Sept. 1958.

4. 3 Richart, F., Brandtzaeg, A., and Brown, R. L.

"A Study of the Failure of Concrete under Combined Compressiv2

Stresses".

University of Illinois Engineering Experiment Station,

Bulletin No. 195, Nov. 1928.

4. 4 Guralnick, S. A.

"Strength of Reinforced Concrete Beams".

American Society of Civil Engineers, Transactions,

Vol. 125, Part I, 1960, p. 603.

A.8.

4. 5 ACI - ASCE Committee 326, Shear and Diagonal Tension.

"Shear and Diagonal Tension - Pt. 2".

Journal of the A.C.I. Proceedings, Vol. 59, Feb. 1962.

5. l Cowan, H.J. and Armstrong, S.

"Experiments on the Strength of Reinforced and Prestressed

Concrete Beams and of Concrete Encased Steel Joists in

Combined Bending and Torsion".

Magazine of Concrete Research, Vol. 7, No. 19, March, 1955.

8. l Archer, F. E.

"Economics of Reinforced Concrete Designed by the Ultimate

Load Method" .

Constructional Review, Sydney, Nov. 1956.

APPENDIX B

This appendix contains a summary of all the experimental data

used in the comparisons of theory and experiment in Chapters 2 and 6

for beams with web reinforcement. Further details of these test results

can be found in the references cited in Appendix A. A discussion of the

test results of this investigation is, of course, given in Chapter 3.

In the table the concrete strengths have been expressed in terms of the

cylinder compressive strength. Where the investigator specified the

concrete used by its cube strength a conversion factor of O. 8 has been

employed.

APPENDIX B. EXPERIMENTAL DATA

Geometry (inches) Web Steel Longitudinal Steel Failure Loads

I Beam h b al a2 A f s ALl fLl R f' T M V +-' .... rn o w w C

kip. kip. kips (I) +-' No. sq. in. k.s.i . in. sq. in. k. s. i p. s. i. :> co i::::: tlD in. in.

~El 10.0 6.5 l .6 1.7 0.110 49.0 3.00 0.392 44.0, 1.000 4599 81.4 6.3 0.18 RE2 l'l.() 6.5 1.6 1.3 o.110 49.0. 3.00 0.392 44.0 1.000 4599 83.5 32.0 0.92 RE3 l'J.l"I 6.5 1.6 1.3 0.110 49.0 3.C'O 0.392 44.0, 1.000 4599 81.5 45.0 1.28 RE4 1 1 • n 6.5 1.6 1.3 n.110 49.(' 3.00 0.392 44.0 l .ooo 4599 74.6 84.4 3.26 RE5 10.() 6.5 1 .6 1.3 0.110 49.C 3.CO 0.392 44.0 1.000 4599 66.0 108.2 3.0 7 R E4>< 10.n 6.5 1.6 1.3 0.110 49.C" 3.00 0.392 44.0 1.000 4599 38.0 134.0 3.86 RUl 1n.o 6.5 1.0 1.4 0 .110 49.C 4.CO o. 880 46.8 0.239 3679 73.3 6.3 0. 18 RU3t* l ,,_ 0 6.5 1.0 1.4 0.110 49.r 4.00 0.880 46.8 0.239 4629 76.0 o.3 0. l CJ RUZ 10.0 6.5 1.8 l. 4 0.110 49.C" 4.00 C.880 46.8 0.239 3h19 84.9 51.l o.o RU3 1n.n 6.5 l .8 l .4 C'.110 49.0 4.CO o. 880 46. 8 0.239 3679 105.0 84.0 0 .0

i::::: RU3A 10.0 6.5 1.8 1.4 0.110 49.C 4.00 0.880 46.8 0.239 4629 89.4 149.3 o.o 0 ..... RU4 10. rt 6.5 1.8 1.4 0.110 49.0 4.00 0.880 46.8 0.239 3679 85.5 145.0 4.13 +-' co RU5 l "· r, 6.5 1.8 1.4 0.110 49.C 4.00 o. 880 46.8 0.239 3679 75.4 249.7 7.15 0.0 ..... RU5A 10.0 6.5 1.8 1.4 0.110 49.(" 4.CO 0.880 46.8 0 .239 4399 68.3 266.8 7.80 +-' rn RU6 1n.e 6.5 1.8 1.4 n.110 49.r 4.00 ('. 880 46.8 0.239 3679 59.l 281.2 7.95 (I)

:> 36T4 ll"\.O 6.5 1.6 1.3 0.110 43.0 4.0C c. 880 37.7 0.295 4399 62.6 240.4 7. 53 i::::: ..... 36T4C 10. 0 6. 5 1.6 1. 4 0.110 43.(' 4.00 o. 880 37.7 0.309 4339 94.l 6 l. l o.o rn 36T5.5 1n.n 6.5 1.6 1 .4 0.110 43.0 5. 50 0.880 37. 7 0.295 4629 85.9 173.4 o.o .....

..c: 77T5 10.0 6.5 1.7 1.4 f).110 43.n s.co 1.200 37.7 1.000 4629 91.6 262.4 o.o ~ 77T4 1'1.0 6.5 1.7 l. 4 0.110 43.0 4.00 1.200 37.7 1.000 4629 107 .6 223.4 o.o 24T3 11'.'.0 6.5 1.3 1.4 o. 110 43.r 3.00 0.392 4 7. l o. 372 4339 70.8 46.6 a.o 38T5 1r,.n 6.5 1.7 1.6 0.110 43.0 5.00 1. 5 70 38.7 o.156 3829 80.l 216.4 6. 73

tlJ N

Expected page number is not in the original print copy.


Geometry (inch,es) Web Steel Longitudinal Steel Failure Loads I

h b al a2 A f s ALl fLl R f' T M V +-' s... Cll 0 Beam w w C

kip. kip. kips Q) +-' sq. in. k. s. i. in. sq. in. k. s. i~ p. s. i. > (1j No. in . in. ..5 -~

Vl 1n.o 6.5 1.5 1.3 0.049 58.6 4.75 1.840 65.2 0.333 5029 38.2 544.Q 22.30 Vl* 10.0 6.5 1.5 1.3 0.049 58.6 4.75 1. 840 65.2 0.333 5029 60.6 421.0 17.30 V2 10.n 6.5 1.5 1.3 0.049 58.6 4. 75 1.840 65.2 0.333 5029 52 •. 3 413.0 11.00 V2* 10.0 6.5 1.5 1.3 0.049 58.6 4.75 1. 840 65.2 0.333 5029 86.0 166.0 7.11.) V3 10.0 6.5 1.5 1.3 0.049 58.6 4.75 1. 840 65.2 0.333 5029 16.9 685.0 27.80 VJ* l(l.'l 6.5 1.5 1.3 0.049 58.6 4.75 1.840 65.2 0.333 5029 79. 2 394.0 16.20 v,. 10.(' 6.5 1.5 1.3 0.049 58.6 4.75 1.840 65.2 0.333 5029 91.3 243.0 10.20

~ V4* 10.0 6.5 1.5 1.3 0.049 58.6 4.75 1. 840 65.2 0.333 5029 40.4 664.0 27.vO 0 V5* 10.'l 6.5 1.5 1.3 0.049 58.6 4. 75 1.840 65.2 0.333 5029 89. 7 o.o o.o •.-1 +-' V6 10.r, 6.5 1.5 1.3 0.049 58.6 4.75 1.840 65.2 0.333 5029 24.8 668.0 21.20 (1j bD V6=~ 10.f' 6.5 1.5 1.3 0.049 58.6 4. 75 1. 840 65. 2 0.333 5029 82.9 348.0 14.40 •.-1 +-'

V7 1 (). 0 6.5 1.5 1. 3 0.049 58.6 4.75 1.840 65.2 0.333 5029 83.9 298.D 9.30 Cll Q)

V7t: 10.n 6.5 1.5 1.3 0.049 58.6 ~.75 1.840 b5.2 C.333 5029 103.4 210.0 12.10 > ~ Ul 10.0 6.5 1.5 1.3 0.049 58.6 3.CO 1.840 65.2 o. 333 5029 106.6 o.o o.o H

Cll Ul* 10.0 6.5 1.5 1.3 'l.049 58.6 3.CO 1.840 65.2 o.333 5029 99. 5 199.0 8 .40 •.-1

..c:: U2 10.0 6.5 1.5 1.3 0.049 58.6 3.C'O 1.840 65.2 0.333 5029 43.9 689.0 28 .. 0Q ~ U2* 10.0 6.5 1.5 1.3 0.049 58.6 3.CO 1.840 65. 2 0.333 5029 85.2 315.0 13.10

U3 1 '). 0 6.5 1.5 1.3 0.049 58.6 3.00 1.840 65.2 0.333 5029 82. 7 533.0 21.80 U3* 10.0 6.5 1.5 1.3 0.049 58.6 3.CO 1.840 65.2 0.333 5029 66.2 120.0 29.3G Tl 1n.o 6.5 1.2 1.1 0.049 58.6 3.62 1.570 39.3 0.140 5029 75.l 327.0 13.50 T2 10.0 6.5 1.2 1.7 0.049 58.6 3.62 1.570 39.3 0.140 5029 92.9 102.0 4,.10 T4 10.0 6.5 1.2 1.7 ').049 58.6 3.62 1.570 39.3 o.; 140 · 5029 53.0 523.0 21.40 T4* H>.;O 6 .5 1.2 ·l. 7 0.049 58.6 3.62 1.570 39.3 0.140 5029 81.3 366.0 15.10 T5 10.0 6.5 1.2 1.7 0.049 58.6 3.62 l.570 39.3 0.140 5029 63.'t 432.0 17.70 T5* 10.0 6.5 1.2 1.7 0.049 58.6 3. 62 1.570 39.3 0.140 5029 75.4 o.o Q.O T6 l'l.'l 6.5 1.2 1.7 o.;049 58.6 3.62 1.570 39.3 0.140 5029 8l.O 196.0 8.30 T6* 10., 6.5 1.2 1.1 0.049 58.6 ~-62 1.570 39.3 0.140 5029 29.4 584.Q 23.90

t:d . ""


Geometry (inches) Web Steel

I +-' ~ h b al az A f s 00 0 Beam w w Q) +-' > ro No. sq. in. k. s. i . in. i::: b.O

>--< ......

R4.2(' l'l." 5. I) l.5 1.2 1).049 58.6 2. 59 R4.24 10.0 5.0 1.5 1.2 0.049 58.6 2.25 R3.2() 10.0 5. ') 1.5 1.1 0.049 58.6 3.12 R3.24 l'l.O 5. r) 1.5 1. 1 0.049 58.6 2.12 R3.31" 1n.r 5.,., 1 .5 l. 1 o.r49 58.6 2. 38 R2.24 1r.o 5. I') 1.7 1.2 0.049 58.6 3.80 RZ.30 10.') 5.0 1.7 1.2 0.049 58.6 3.32 RZ.38 l r,. n 5.1") 1. 7 1.2 0.049 58.6 2.84 R l. 30 11). 0 5. 'l 1.2 1.2 0.049 58.6 3.95 R4A l"l. ') 5.l'l 1.5 1.2 0.049 58.6 3. 22 R48 1,.,. "I 5."'l l .5 1.2 0.049 58.6 3.22 R4.2~A 11).n. s.o l .5 1.2 o.r,49 73.6 2.00 R4.2')8 10.r, 5.0 1.5 1.2 ().049 73.6 2.00 R4.24A 11).0 5.0 l .5 1.2 0.049 58.6 2.00 R3A l"l. 0 5.0 1.5 l. l "-049 73.6 3.00

.c:: R3B 1,. ('\ 5.0 1.5 l. l 0.049 73.6 3.00 00 R3.2"A l r').,., 5.0 1 .5 1.1 1).049 58.6 2. 50

r-1 R3.20B 11').n 5.(' 1.5 1. 1 0.049 58.6 2.50 ro

~ R3.24A 1 , • () 5.n 1.5 l. l O.t:'49 73.6 2. 50 R3.24B 1 "'· n 5.'l 1.5 1. 1 0.049 73.6 2. 50 R3.3CA 1". 0 5.0 1.5 1.1 0.049 73.6 2.00 R3.30B 10.0 5.0 1.5 1.1 0.049 73.6 2.00 R2A 1 I). 0 5.0 l. 7 1.2 l').049 58.6 3.00 R2B l'>.0 5.1') 1.7 1.2 0.049 58.6 3.CO R2.24A 10.0 5.1) 1.1 1.2 0.049 58.6 3.30 R2.30A 10.0 5.0 1.7 1.2 r).049 73.6 ~-00 R2.308 10.0 s.o 1.7 1.2 0.049 73.6 3.00 R2.38A 10.0 5.0 1.1 1.2 0.049 73.6 2. 50 R2.38B 10.0 5.0 1.7 1.2 0.049 73.6 ~.50 Rl.30A 10.1" s.o 1.2 1.2 0.049 58.6 3. 14 Rl.30B 10.n 5.0 1.2 1.2 0.049 58.6 r3. 14

Longitudinal Steel

ALI fLl f' R C

sq. in. k.s.i p.s.i.

1. 420 40.6 0.155 3474 1. 420 40.6 0.155 3034 1.220 40.6 o. 180 3444 1.220 40.6 0.180 3157 1.220 40.6 0.180 3414 0.830 42.4 0.265 3239 0.830 42.4 0.265 3314 0.830 42. 4 0.265 3474 0.710 41.3 o. 310 3279 1.420 40.6 0.155 2954 1.420 40.6 0.155 2954 1.420 40.6 0.155 3474 1.420 40.6 0.155 3474 1.420 40.6 0.155 3034 1.220 40.6 0.180 2919 1.220 40.6 0.180 2919 1.220 40.6 0.180 3444 1.220 40.6 0.180 3444 1.220 40.6 0.180 3157 1.220 40.6 0.180 3157 1.220 40.6 0.180 3414 1.220 40.6 0.180 3414 0.830 42.4 0.265 3199 0.830 42.4 0.265 3199 0.830 42.4 0.265 3239 0.830 42.4 0.265 3314 o. 830 42.4 0.265 3314 0.830 42.4 0.265 3474 0.830 42.4 0.265 3474 o. 710 41.3 0.310 3279 0.110 41.3 0.310 3279

Failure Loads

T M kip. kip. in. in.

59.9 331.0, 56.5 264.0 50.7 252.0 53. 7 230.0 61.6 201.0 44.2 205.0 49.7 176.0 53.4 138.0 41.8 146.0 50.5 92.l 51.6 166.0 72.0 220.0 70.8 151.0, 62.l 255.0 57.6 92.1 51.5 166.0 61.7 183.0 59.0 78.9 56. 5 92.9 54.9 158.0 62.2 203.0, 63.4 86.8. 55.7 111.0 50.q 78.7 62.3 140.0 50.3 104.Q 50.6 71.8 54.6 120.0 48.8 70.7 42.6 97.l 40.9 61.8

V kips

o.o o .• o o .• o o .• o o.o o.o o.o o.o o.o 3.94 6.92 9.08 6.30 8.21 3 .• 94 6.92 7.60 3.41 3.97 6.57 8.39 3.73 4.71 3.42 5.85 4.43 3.13 5.08 3.08 4.13 2.73

to

~


Geometry (inches) Web Steel Longitudinal Steel Failure Loads I

+-' ~ h b al a2 A . f s ALl fLl R fl T M V . 00 0 w w C Q) +-' Beam kip. kip. kips > Cll sq. in. k . .s·. j : . in. sq. in. k.s.i p.s.i. ~ Ol) No. in. in. - •ri

13 TR 3 12.0 6.0 1.3 1.0 0.049 55.5 l4.00 0.220 53.6 1.000 3922 34.3 o.o o .• o 13TR7 12.0 6.0 1.3 1.0 0.049 55.5 1.00 0.220 53,.6 1.000 3922 49. 7 o.o o .• o t3TR15 12.0 6.0 1.3 1.0 0.049 55.5 4.00 0.220 53.6 1.000 3922 61.7 o.o o.o BTR30 12.0 6.0 1.3 1 .o. 0.049 55.5 2.00 0.220 53.6 1.000 3922 76.0 o.o o.o ~TR3 12.0 6.0 1.4 1.1 0.049 55.5 14.00 0.390 41.0 1.000 3922 35.0 o.o o.o

+-' 4TR 7 12.0 6.() 1.4 1.1 O.C'49 55.5 1.00 0.390 41.0 1.000 3922 54.8 o.o o..o 00 ~ 4TR15 12.r 6.0 1.4 1.1 0~049 55.5 4.00 0.390 41.0 1.000 3922 74.0 o.o o.o i...

l4TR30 12.0 6.0 1.4 1.1 0.049 55.5 2.00 0.390 41.0 1.000 3922 85.0 o.o o.o ~ 5TR3 12.0 6.'l 1 .4 1.2 0.049 55.5 14.00 0.610 48.6 1.000 3922 43.0 o.o. o.o 15 TR7 12.0 6.() 1.4 1.2 0.049 55.5 1.co 0.610 48.6 1.000 3922 59.7 o.o o.o 5TR15 12.0 6.0 1 .4 1.2 0.049 55.5 4.co 0.610 48.6 1.000 3922 76.5 o.o o •. o 5TR3f'l 12.0 6.".) 1.4 1.2 0.049 55.5 2.00 0.610 48.6 1.000 3922 92.6 o.o o~o R3 9.0 6.0 o.a o.a 0.049 20.8 4.00 0.580 48.5 0.672 7699 71.8 o.o OAO

s:: R.5 9.0 6.0 o.a o.a o.;049 20.8 4.00 0.580 48.5 0.672 8500 75.4 75.lt o.o Cll R2 9.0 6.0 o.a o.;a 0.049 20.8 4.00 0.580 48.5 0.672 7299 79.0 158.0 o.o ~ Rl 9.0 6.0 o.a o.a 0.049 20.8 4.00 0.580 48.5 0.672 7299 43.0 258.0 o .• o 0 u Sl 9.0 6.0 o.a o.a 0.049 20.a 3.00 0.580 48.5 0.672 7199 82.6 206.5 o .• o

54 9.0 6.0 0.8 o.a 0.049 20.8 3.00 0.580 48.5 0.672 6969 64.6 258.4 Q.O 1 a.o a.o 1.5 1.5 0.110 50.0 5.00. o.5ao 51.0 0.672 5029 79.0 79.0 o.o 2 8. f) a.o 1.5 1.5 0.110 50.0 2.00 o.580 51.0 0.672 5299 102.0 102.0 0 .,o

,. 3 a.o 8.0 1.5 1.5 0.110 50.0 5.00 o. 580 51.0 0.672 5309 61.0 122.0 o.o .

rl 4 a.o 8.0 1.5 ·1 .5 0.110 50.0 2.00 0.580 51.Q 0.672 4679 67.0 134.0 0,.0

Cll 5 a.o 8.0 1.5 1.5 0.110 50.0 5.00 o.sso 51.0 0.672 4239 49.0 147.0 0..0 +-' 6 a.o a.o 1.5 1.5 0.110 50.;0 2.00 0.580 51.0. 0.672 405() 56.0 168.0 o..o Q)

'O 7 a. f\ 8.0 1.5 1.5 0.110 50~0 5.00 0.580 51.0 0.672 5279 43.0 173.0 0..0 ~ 8 0.0 8.o 1.5 1.5 0.110 50.0 2.00 o.sao 51.0 0.672 5739 44.0 176.0 o.o ;::l 00 9 12.0 6.0 2.0 1.8 0.110 50.0 a.co 0.580 51. 0. 0.672 4859 60.0 120.0 o.o Q)

c., Ito 12.0 6.0 2.0 1.a 0.110 50.0 a.co o. 580 51.0 0.672 3899 44.0 176.0 0..0 b. 1 12.0 6.0 2.0 , 1 .a 0.110 50.0 4.00 0.580 51.0 0.672 4859 68.0 138.0 o •. o

12 12.n 6.0 2.0 1.a 0.110 50.0 4.00 0.580 51.0 0.672 3899 53.0 213.0 o,.o

tj

APPENDIX B. EXPERIMENTAL DATA.


+-' ~ h b al a2 A f s ALl fLl R f' T M V ·~ 0 Beam w w C ' +-' kip. kip. kips > ro

No. sq. in. k. s. i in. sq. in. k.s.i p.s.i. i:: b.O in. in. i-j "M

BKl 11.8 7.9 1.0 1.2 0.122 49.l tt. 90 0.560 34.5 1.000 2009 121.0 o.o o.o BKlA 12.r, 8.1 1.0 1.4 0.122 49.l 4.90 0.560 34.5 1.000 1569 104.0 o.o o • .o BK2 12.0 7.9 1.4 1.2 0.122 49. l 3.90. 0.630 46.5 1.000 1709 147.8 o.o o.o BK2A 11.e 7.9 1.2 1.2 0.122 49.1 3.90. 0.630 46.5 1.000 1849 153.0 o.o o .• o BK3 15.8 6.7 1 .o 1.4 0.122 49. 1 4.90 0.630 46.5 1.000 190,9 175.5 o.o o.o BK3A 15.8 6.7 l.Cl 1.4 0.122 49. l 4.90 0.630 46.5 1.000 1569 149.3 o.o o..o BU4 12.0 8.1 1.4 1.6 0.122 49.1 14.90 l .940 54.0 0.273 909 111.0 556.0 a.o BU4A 11. 8 7.9 1.2 1.4 0.122 49.l ~-90 1.940 54.0 0.273 909 104.0 522.0 o.o BU5 11.8 7.9 1.2 l. 4 0.122 49. 1 14.90 0.990 49.7 1.000 679 69.5 86.8 o.o BUSA 11.8 7.9 1 .o 1.4 0.122 49.1 ~.90 0.990 49.7 1.000 714 67.6 174.0 o.o BU6 11.8 7.9 l .o l .4 0.122 49. 1 14.90 0.630 46.5 1.000 750 69.5 173.6 o.o BU6A 12.0 7.9 1.1 1.4 0.122 49.1 ~-90 0.630 46.5 1.000 789 73.0 104.2 o.o 8117 12.n 6. l 1.4 1.4 0.122 40.0 3.90 l.900 55.5 0 .. 163 1429 62.5 625.0 22.95 8117A 11.a 6.1 1.2 1. 4 0.122 40.0 3.90 1.900 55.5 0.163 1659 62.5 625.0 22.95 B118 12.2 5.9 1.6 1.2 0.122 40.0 3.90 0.980 50.0 0.347 1509 74.0 382.0 14.50 B1l8A 11.8 5.9 1.2 1.3 0.122 40.0 3.90 0.980 50.0 0.347 1779 74.0 382.0 14.35 B119 11.8 6.1 1.2 1.3 0.122 40.0 3.90 0.980 50.0 0.347 1559 88.6 157.0 6.04

b.O BI I 9A 11.8 6.1 1.2 1.4 0.122 40.0 3.90 0.980 50.0 0.347 1729 94.0 156.0 6.07 "M Ul BII 10 11.8 8.1 1 .o 1.3 0.122 40.0. 3.90 0.980 50.0 0.347 1639 137.0 3't8.0 12 .. 99 Ul Cl) B II ll')A 11.8 7.9 1.2 1.2 o. 122 40.0 3.90. 0.980 50.0 o.347 180.9 125.0 313.0 11.77 ~ B1111 12.0 7.9 1.2 1.1 0.122 40.0 f).90 0.980 50.0 0.347 1599 122.0 156.0 6.12

BllllA 11.8 7.9 1.4 1.1 0.122 40.0 3.90 0.980 50.0 0.347 1659 115.0 156.0 6.14 8[112 12.,:, 7.9 1.4 1.2 0.078 51.5 3.90 1.250 84.5 0.176 2779 146.0 243.0 9.24 B 1112A 12.0 7.9 1.2 1.2 0.078 51.5 3.90 1.250 84.5 0.176 2779 151.0 261.0 10.21 B1113 11.8 6.1 1.2 1.4 0.121 45.5 3.90. l.900 55.5 0.189 2129 93.5 520.0 19.16 81113A 11.6 6.1 1.2 1.4 0.121 45.5 3.90 1.900 55. 5 0.189 2579 125.0 625.0 23.16 81114 11.8 5.9 1.2 1.8 0 .121 45.5 3.90 1.900 55.5 o.1a9 2379 83.2 416.0 15.51 8 II 14A 12.n 5.9 1.2 2.0 0.121 45.5 3.90 1.900 55.5 o.1a9 2649 113.0 572.0 21.26 81115 12.2 7.9 1.2 1.6 0.121 45.5 3.90 o. 980 49.5 0.418 2539, 156.0 416.0 15.56 81115A 12.0 7.9 1.2 1.4 · 0.121 45.5 3.90 0.980 49.5 0.418 2759 151. 0 416.0 15.52 B 1116 12. 4 6.1 1.0 1.4 0.122 42.5 3.90 0.350 58.0 1.000 3259 92.0 156.0 4.07 81116A 12.0 6. l 1.0 1.4 0.122 42.5 3.90 0.350 58.0 1.000 3549 83.4 156.0 4.14

to



+-' S-1 h b al a2 A f s ALl fLl R fl T M V Ul 0 (IJ +-' Beam w w

in. C

kip. > ctl kip. kips i:: 00 No. sq. in. k. s. i sq. in. k. s. i p.s.i. - -~ in. in.

B1l17A 12.2 6.1 1.2 1.4 0.122 42.5 3.90 0.622 50.6 1.000 3969 90.4 31.3.0 8.06 B1117 12.2 6.1 1 .2 1.4 0.122 42.5 3.90 0.622 50.5 1.000 3799 83.4 278.0 7. 16 BI 118 12.2 6.1 1.~ 1. 4 o.rs9 41.5 B.90 0.933 50.5 0.667 3509 114.5 191.0 4.99 BIIlBA 12.2 6. 1 1 • ,, 1.2 0.('59 41.5 6.90 0.933 50.5 0.667 3699 111.0 191.0 4.99 B1119 12.0 6.1 o.a 1.4 0.059 41.5 6.90 0.622 50.5 1.000 3389 78.0 313.0 .4.63

00 BI 119A 12.ii 5.9 1.(1 1. 4 r.059 41. 5 3. 90 0.622 50.5 1.000 3589 79.0 313.0 4.71 -~ Ul Bil?(l 12.2 7.9 2.4 1 .4 n.122 42.5 3.90 1.592 49.0 0.240 1489 125.0 313.0 11.77 Ul (1) B ll 2"A 12.0 7.9

2 ·" 1.4 0.122 42.5 3.90 1.592 49.0 0.240 1609 130.2 339.0 12.69

~ 81121 12.0 7.9 2.('I 1.9 0.122 42.5 3.90 1. 592 49.0 0.240 1549 120.8 313.0 11.77 B1121A 12.2 7.9 2.0 1.6 0.122 42.5 3.90 l. 592 49.0 0.240 1629 99.0 313.0 11.78 11 B 11. 8 6.5 2.r, 1.2 0.('44 32.7 6.10 0.622 56.5 1.000 24Q9 41.7 206.5 10.20 l[BA 11.8 7. 1 2.3 1.2 0.044 32.7 6.10 0.622 56.5 1.000 28.89 57.3 2u6.5 10 .17 WB 11.8 6.1 2. ('. 1.2 0.044 32.7 6.10 0.622 56.5 1.000 2409 53.0 132.0 6.64 WBA 11.a 6.4 2.n 1.2 0.044 32.7 &.10 0.622 56.5 1.000 2889 57.3 143.0 7.16

B28 f'. l 11.e 7.9 1.4 1 .4 0.060 41.f' 3. 20 0.995 54.0 1.000 1223 48.6 486.0 o .• o 828 O.lA 12.0 7.9 1.4 l .4 0.060 41.C 3.20 0.995 52.5 1.000 1223 ·46.9 469.0 o .. o B28 n.2 12.2 7.9 1.4 1 .4 0.060 41.C 3.20 1.030 52.3 1.000 1023 83.4 417.0 o.o B28 n.2A 12.0 7.9 1.6 1.4 0.060 41.0 3.20 0.995 53.3 1.000 1223 83.4 417.0 o.o B2 8 r.4 12.('I 7.9 1.6 1.4 0.060 41.0 3.20 1.050 52.3 1.000 2783 146.0 365.0 o.o

> B28 r.4A 12.r 7.9 1.6 1 .4 O.C60 41.0 3.20 1.050 52.3 1.000 2783 13.9.O 347.0 o.o 0 ..!t: B28 r.4B 12. f' 7.9 1.6 1 .4 0.060 41.C 3.20 1.040 52.3 1.000 4319 · llt6.0 365.0 o.o i:: (1) B28 f'.4C 11.B 7.9 1.3 1.4 O.r"60 41.C 3.20 1.080 55.0 1.000 4319 153.0 .382.0 o.o i:: B28 0.4D 11.B 7.9 1 .4 l .4 0.060 41.(' 3.20 1.010 52.5 l .ooo. 2271 125.0 313.0 o •. o -~ ..s:: u B28 r.4E 12.0 7.9 1.4 1.4 0.060 40.C 3.20 l. 030 . 52.5 1.000 2271 132.0 330.0 o.o

828 C'.4F 11.8 7.9 1.4 1.4 0.121 40.C 3.20 1.050 53.0 1.000 2271 139.0 34 7 .0. o..o

to -.J


· Geometry (inc~_es) Web Steel Longitudinal St~el F allure Loads I

+> M h b al a2 K' f: s ALl fLl R f' T M V OJ 0 Beam ~1ti w w r• C kip. kip. kips

No. sq. in. k.s.i ~ 1n. sq. in. k. s. i p.s.i. .s.~ I in. in. 88 K 12.2 7.9 1.4 1.; 4 o. 07 9 44.5 13, 20 . 1.010 54.5 1.000 1359 125.0 o.n o..o 88 KA 12.2 7.9 1.4 1.4 0.019 44.5. 13.20: 1.010 54.5 1.000 2399 153.0 o.o Q.G 88 0.1 12.2 7.9 1.4 1.4 0.019 44.5 3.20: o.9ao 54.5 1.000 1759 52.0 520.0 12.52

· 88 O.lA 12.2 1.9 1.4 1.4 0.019 44.5 3.20. 1.030 55.8 1.000. 1759 ss.s 555.0 13.36 88 0.2 11.e 1.9 1.4 1.4 0.079 44.5 3.20 1.000 56.2 1.000. 1535 90.0 451.Q 10.85 88 0.2A 12.2 7.9 1.4 1.4 o.;019 44.5 3.20. 1.040 52.0 1.000 1759 97.0 486 .. 0 l·l.69

B8 0.4 12.2 7.9 1.4 1.4 0.079 44.5 3.20 1.040 56.4 1.000 1727 132.0 347.0 8.42 88 0.4A 12.0 7.9 1.4 1.4 0.079 44.5 3.20. 1.030 57.6 1.000 ·. 2223 139.0 347.0 8.38 87 0.2 12.0 1.q 1 •. 4 l .4 o.os9 4,l.6 3.20 1.040 55.1 1.000 153151 93.8 468.0 11.30 87 ·o .2A 12.0 7.9 1.4 1.4 0.059 41.6 3.20. 1.010 55 .• 5 · 1.000 1927 90.;2 451.0 1().,.87 810 0.2 12.0 7.9 1.4 1.4 0.111 40.8 3.20 1.020 5.2.5 1.000, 2215 104.0 521.0 l2.53 8tO·o.2A 12.2 7.9 1 .4 1.4 0.111 40.8 3.20. 1.020 53.0 1.000 - 2399 104-0 521.O 12..53 81 · 7.9 8.7 1.4 1 .4 0.074 63.5 3.20 1.440 53.3 1.000 4087 90.3 452..Q lG.82 BIA 7.9 8.1 "1-.4 1·;4 · o---tt-~ -6¼..-- 1 • .zo - 1.490 55.0 1_.000. 3935 90~3 452.Q 12 •. 75 82 11.a 8.7 1.4 1 .4 0.074 63.; 5 , 3. 20 l.490' 47.8 1.000 .· 4087 119.0 694.O 16-.65 BZA 11.e 8.1 1.4 1.4 0.074 63.5 3.20: 1.460 531110 l.;'QOO · 4095 · 139.0 694.0 16.65 83 11.a 8.7 1.4 l .·4 0.014 63.5 3.20 • 1.440 51.Q 1.000 3927 194.0 486.0 17.48 83A i.1.a 8.7 1 .4 1.4 0.074 63.5 1.20 1.490 51.0 1.000: 3921 194.0 486.0 11.48 85 15.7 8.7 1.4 1.4 0.074 63.5 3.20 · 1.460 54.6 1-.:000 370~ · 19!t.O 972.0 23.24 = 85A 15.7 8.1 1 .4 .l.4 0.074 63.5 3.21): 1.480 48.8 1.000 4135 194.0 972.0 23~_24 •.-1

ci! 86 15.7 6.7 1.4 1.4 0.074 63.5 3.20. 1.460 47.lt 1.:000 .· 4215 · 167.Q 833 •. 0 20..19 ~ 86A 15.7 6.7 1.4 1.4 0.014 63.5 3.20 · 1.460 47.0 1.000 .. 41)5 181.0 903.O. U.88 ...:i

87 11.8 7.9 1.4 1.4 0.270 37.0. 4 .. to : · 1.060 47.0 1.000 2655 221.0 o.o 4).;Q 87A 11.e 7.5 1.2 1.2 o..;21O 37.0. 4·.10: 1.060 47.0 1.-000 2655 20&.o o.o o.o' 88 11.8 1.5 1.2 1.2 0.010 68.l 4.10, 1.010 49.2 1.000. 2591 1S6.0 156.0 3.83 88A 11.e 1.9 1.4 1.4 · 0,010 68.7 4.to · 1.060 47.0 1.:000 - 2615 156.0 156.0 3.63 89 9.1 5.9

1 ·"' 1.4 0.015 50.7 4.10. 1.000 so.a 1.000: 1023- 39.0 '2G&.A 10.QS

89A 9.1 .5.9 1.4 l .4 0~015 50.7 4.10 - 1.010 52.6 1.000 1087 1t1.o 235.0.. 11:.39 B10. 9. 1 7.1 1.4 1.4 0.015 50.7 4.10; 1.060 47.G 1.000. 967 36.,4 182.0 8.82 BlOA 9.1 5.9 1 .4 :1.4 0.075 5C>.7 4.10. 1.060 47.0 1.000. lOil. ltl.8 209.0, m.13 811 9.1 5.9 1.4. 1.,.. 0.075 50.7 ,..10 0.101 57.3 1.000 1,055 ·36.6 209.0 10.10 811A 9.1 5.9 l.4 1 _,. 0.:015 so.1 1t.10 0.670 Slt.7 1.000. 1119 36.t6 209..0 16.lG 812 9. 1 5.9 1.4 1.4 O'J0l5 50.7 1t.1O 0.682 56.0. 1.000, 887 . 21t.2 1&.4 3.1a 812A 9.1 5.9 1.4 l _,. O.;O75 so.1 4.10 - ()~682 56.0. 1.000 921 Jl;.z, lOt-"4 s.01

' tt'. . ' 0


Geometry (inches} Web Steel Longitudinal Steel Failure Loads I t/l ~ h b al a2 A f s ALl fLl R f' T M V Q) .8 !Beam w w . in. C

~ ~ SQ. in. k.s.i sq. in. k. s. i p.s.i. kip. kip. kips

No . 1n. 1n. ...... ...... 1 6.3 3.5 1 .o 1.0 n.031 64.0 5.90 0.244 49. 8 0.422 1351 1.2 36.2 1.84 2 6.3 3.5 1.n 1.0 o.o 31 64.0 5.90 0.244 49.8 0.422 1351 1.2 36.2 1.84 6 6.3 3.5 1.r 1.0 0.031 64.r 5.90 0.244 49.8 0.422 1351 4.6 45.7 2.32 7 6.3 3.5 1.0 1.c 0.031 64.0 5.90 0.244 49.8 0.422 1351 9.9 49.3 2 .. 51

1 ri , 6.3 3.5 1.0 1.0 0.031 64.0 5.90 o.244 49.8 0.422 1351 5.9 59.0 3 .• oo 3 6.3 3.5 1.r 1.0 0.031 r64.C 2.95 0.244 49.8 0.422 1351 11. 9 59.4 3.02 4 6.3 3.5 l. C 1 .o O.f'31 i64.0 2.95 0.244 49.8 0.422 1351 11.9 59.4 3.02 A 6.3 3.5 1.0 1.0 O.f-31 64.0 2. 95 0.244 49.8 0.422 1351 13. l 65.3 3.32

::::: CJ I 6.3 . 3.5 1.n 1.n o .r 31 :64.o 2.95 0.244 49.8 0.422 1351 13. l 65.3 3.32 ...... "Cl 11 1.0 0.031 64.r 3.94 0.312 49.8 1.000 2319 14. 3 71.7 3.65 ;:I · 6.3 3.5 1.2 >i 121 6.3 3.5 1.2 l .o O.C'31 64.0 3.94 0.312 49.6 1.000 2319 11.1 55.7 2,.83

13 6.3 3.5 1.2 1.0 o.o 31 64.0 3.94 0.312 49.8 1.000 2319 11. 1 55.7 2.83 17 6.3 3.5 1.2 1.0 0.031 64.0 7.87 0.312 49.8 0 .500 _ 2175 7.2 36.0 1.83 18 6.3 3.5 1.2 1.0 0.031 64.0 1. 87 0.312 49.8 0.500 2175 7.9 39.4 2.00 1 g 6.3 3.5 1.2 1.0 0.031 64.0 7.87 0.312 49.a 0.500 2175 7. 9 39.lt 2.00 2,:, 6. 3 3.5 1.2 1.0 0.031 64.0 7.87 0.312 49.8 o.soo 2175 9.1 45.7 2.32 21 6.3 3.5 1.2 l .o 0.031 64.0 7.87 0.312 49.S: 0.500 2175 7.9 39.4 2.00 22 6.3 3.5 1.c 1.0 0.031 64.0 7.87 0.156 49.8 2.000 2175 6.5 32.6 l.b6 -HBl o.a

1o.a 7.6 6. 1 0.049 46.(' 4.00 0.220 59.0 0.759 6919 44.l o.o o.o

HB2 7.6 6. l 0.0 1 0.a 0.049 46.0 4.00 0.220 59.0 0.759 6463 33.9 66.8 o.o HB3 7.5 6.0 o.a-·0.0 0.('149 46.(l 4.CO 0.220 59.0 o. 759 6359 20.4 75.3 o.o

s... HB4 7.7 6.1 o.a o.a 0.(149 46.0 4.00 0.220 59.0, 0. 759 7000 15.7 81.6 o .• o Cll ~ HB5 7.5 6.0 o.a 0.8 0.049 46.0 4.00 0.220 59.0 0.759 6063 13.2 81.5 o..o s... Cll HB7 9.0 6.1) 0.8 0.0 0.049 40.8 4.00 0.220 54.5 0.895 5119 36.l o.o o.o

(/) HB8 9.0 6.0 o.a 0.8 0.049 40.8 4.00 o. 220 54.5 Q.895 5099 21.4 79.6 o.o 'O

::::: H89 9.0 6.0 o.a o.e 0.049 46.0 4.00 o. 22() 59.0 0~759 4031 18.3 85.1 o.o Cll HB 10 9.1 6.0 0.8 o.8 0.049 46.0 4.00 0.220 59.0 0.759 6439 17.3 , 91.3 o.o t/l

HB 11 9. ') 6.0 0.0 o.e 0~049 46.0 4.00 0.220 59.0 o.759 5199 14.1 94.0 o.o ::::: Cll HB 13 12.0 6.0 o.a o.a 0 .() 49 40.8 4.00 0.220 54.5 0.895 5159 51.3 o.o o.o ::> ~ HB 14 12.1 6.0 o.8 0.8 0.049 46.0 4.(0 0.220 59.0 0.759 5199 41.7 82.l o.o

HB 15 12.0 6.0 0.0 0 .8. 0.049 40.8 4.00 0.220 54.5 0.895 5159 29.9 111.0 o.o HB 16 12.1 6.0 o.a o.a 0.049 46.0 4.00 0.220 59.0 0.759 4031 23.5 129.0 o.o HB 17 12.0 6.0 o.8 0.8 I O.C,49 46.C, 4.00 0.220 59.0. 0.759 6439 19. 4 137.0 O.O'

b:: ,c

C.1.

APPENDIX C.

OPTIMUM VALUE OF r.

The total volume of reinforcement per unit length of beam is

given by:

= C l AL l ( 1 + kr) . . . . . . A. 1

where 2

( l+R)b ..... A.2

and

c 1 = l+R.

For a given size of beam both C 1 and k are constant if the ratio

of top to bottom steel remains constant. The area of steel ALI to

prevent the most common type of failure, mode l, is, from Equation

5. 14,

2r [/ 1 2 1 + 2 oc. ALI = T 1' (h-al - xl) 1+2oC. ('1_,) + r

When this is substituted for ALI' equation (Al) becomes:

W=C 2

where c 2 is a constant.

..... A . .3

C. 2.

This function is a minimum when:

1 r =

k+

The value of k ( see equation A2) is dependent upon <X... , R and the

ratio of cover on the steel to the width of the section. Hence, the

influence of these variables would need to be considered in any attempt

to find a minimum weight solution. An examination of these variables

for practical situations indicates that k might vary between 2 and 7.

Tests (Ref. 2. 19, 2. 20), show, however, that reinforced beams behave

in a relatively ductile manner when subjected to torsion and bending if

r is not unduly small. It is therefore advantageous to adopt a k value

somewhat less than the upper limit.

The design process is, of course, considerably simplified if a

constant value is adopted for k. In view of the above remarks the

value k = 4 has been adopted so that:

r = 0

1

4 4+

(J}fl + Zoe

D.l.

APPENDIX D.

The experimental data listed in Appendix B have been analysed,

and the results of the analysis are presented in this appendix. The

failure loads of each beam have been expressed in terms of the three

ratios T / T , M/ M and V /V . T is the pure torsional strength of the 0 U O 0

beam as calculated from the theory set out in Chapter 5 {see Equation

5. 24). M is the calculated flexural strength and V is the shear capacity U 0

of the beam as given by the A. C. I. code.

The parameters r/r , pf /f' and V ff/bd (7;., which are listed 0 y C e C

in the table, are related to the restrictions on the theory discussed in

Chapter 7.

i. e.

r/r t 0.9 {D. 1) 0

pf /f-' :t> 0.40 {D. 2) y C

V eff/bd /f~ 1>- 8. {D. 3).

Beams which do not satisfy the above requirements have been

included in the table, but they have been marked with the letters R, P

and V if they violate the limits set out in equations {D. 1), {D. 2) and

{D. 3) respectively.

For each beam the theoretical failure torque has been calculated

in four ways, from the Modes 1,2 and 3 equations and from the effective

shear formula. These values have been expressed in the table as the

ratios T /Tth . Also listed in the table is the critical value of exp eor

T /Tth and its associated mode. exp eor

APPENDIX D. ANALYSIS OF TEST DATA

T Critical ex:e

I T T +-' ~ pf V -· theor.

Ul 0 Beam T M V ~

Cl) +-' No.

r :J. eff Tth. Restrict-> ro - I- - -

~ 00 T M V r f' bdfI': 1 2 3 VEF Mode H •.-I 0 u 0 0 C C ions

-.;•-,;'~'••·I••

REl 0.83 0.05 0.01 2.12 o.oo 6.2 0~85 o.aa o. 82 0.68 0.88 2 RE2 0.89 0.23 0.03 2.23 o.oo 5.8 1.01 0.90 0.85 0.72 1.01 1 RE3 I) .83 0.32 0.04 2.60 o.oo 5.7 1.01 0.85 0.79 0.72 1.01 1 RE4 o.16 0.61 0.11 3.19 o.oo 5.2 1.12 o. 85 0.65 0.73 1.12 1 RE5 n.67 0.78 0.10 3.7'l O.f'O 4.5 1.16 o. 76 o.s1 0.65 1.16 1 RE4* 0.39 0.96 0.13 5.61 o.oo 3.2 1.10 0.50 0.26 0 • .45 1.10 1 RUl 0.60 0.02 0.01 0.12 0.16 4.6 0.61 0.15 1.22 o. 70 1.22 3V R R~3A* 0.60 0.02 0.01 0.72 0.13 4.2 0.62 0.75 1.23 0.65 1.23 3V R RU2 0.10 0.11 o.o 0.9'l 0.16 5.8 0.19 0.86 1.11 0.81 1.11 3

~ RU3 0.86 o.za o.o 0.97 0.16 6.3 1.01 1.06 1.26 1.00 1.26 3 0 ...... RU3A o. 71 0.49 o.o 1.21 0.13 6.1 1.00 o.ae 0.76 0.76 1.00 1 +-' ro RU4 0.70 0.49 0~16 1.28 0.16 6.4 0~99 1.00 1. 21 0.97 1.21 3V 00 ...... RU5 0.62 0.84 0.28 1.83 0.16 6.8 1.11 1.00 0.87 0~99 1.11 1 +-'

Ul (I.) RU5A 0.55 0.88 0.28 2.04 0.13 6.1 1.14 0.93 0.10 0~87 1.14 l ~ RU6 0.48 0.95 0.31 2.33 0.16 6.0 1.15 0.86 0.56 0.87 1.15 1

H 36T4 o.57 0.94 0.28 2.19 o. 10 5.3 1.21 0.96 0.64 0~84 1.21 1 36T4C 0.87 0.24 o.o 0.96 0.10 5.2 1.00 1.01 1.23 0.87 1.23 3

Ul 36T5.5 0.93 0.67 o.;o 1.06 0.09 4.3 1.32 1.15 0.;91 0.98 1.32 l ...... ..c: 77T5 0~83 o.;1e 0~0 1.01 o.oo 4. 7 1.31 o.;a2 0.52 o.;99 1.31 1 ~

7714 0.87 0.67 o.o 1.12 o.oo 5.7 1.26 0.86 0.60 1.00 1.26 1 24T3 0.73 o.;30 o.o 2.13 o.;os 5.1 0.90. 0.92 0.;86 0.58 0.92 2 38T5 0.70 0.52 0.29 o. 71 0.25 6.1 1.01 1.21 1.29 1.15 1.29 3V R

'

-t:,

N


I +' ~ pf v-00 0 Beam T M V r eff (l) +' - __J_ > c,j No. T M V r fl bd/F s:: 0.0

1--l •H 0 u 0 0 C C

Vl 0.30 0.65 1 .15 0.11 0.29 4.9 Vl* 0.47 0.50 0.89 0.43 0.29 4.9 V2 0.41 0.49 0.88 0.47 0.29 4.9 V2* 0.67 0.2n 0.37 0.19 0.29 5.0 V3 0.13 0.82 1. 44 2.02 0.29 4.9 V3* 0.61 0.47 0.84 0.33 0.29 4.9 V4 0.71 0.29 0.53 0.22 C.29 4.9 V4* 0.31 ').79 1.40 0.88 0.29 4.9 V5* r,. 1() o.o o.n 0.10 0.29 4.5 V6 () .19 o.ao 1.41 l .3 8 0.29 4.9

s:: V6* 0.64 () .41 o.74 0.30 0.29 4.9 0 V7 0.65 0.35 0.49 0.26 0.29 4.8 •H +' V7* o.ao 0.25 0.61 0.19 o. 29 5. 1 c,j bD Ul 0.66 o.o o.o 0.15 0.29 5.9 •H +'

0.33 0.3') 00 Ul* 0.61 0.24 n.29 6.5 (l)

> U2 0.21 0.82 1.11 1.33 0.29 6.5 s:: U2* 0.53 0.38 0.52 0.43 0.29 6.5 -00 U3 0.51 0.63 0.86 0.64 0.29 6.5 •H ..c:: U3* 0.41 0.86 1.16 0.97 0.29 6.5 E-t Tl 0.65 0.69 0.59 0.11 0.18 5.3

T2 0.80 0.22 0.18 0.38 0.1a 4.1 T4 0.46 1.11 0.93 1.44 0.18 5.5 T4• 0.70 0.78 0.66 0.79 0.18 5.3 T5 0.55 0.92. 0.11 1.01 0.18 5.4 TS* 0.65 o.o o.o. 0.24 0.18 3.0 T6 0.12 0.42 0.36 0.53 0.1a 5.1 T6* 0.25 1.24 1.04 2.65 0.18 5.7

T exp

T theor.

1 2 3

p. 76 1.04 0.13 p. 78 1.10 0.36 p~ 72 l .01 0.28 IQ.77 1.03 0.90 o. 84 1.01 0.02 K).89 1.24 0.57 ~.87 1.11 0.87 p.90 1.21 0.12 ~- 70 0.84 1. 21 o. 84 1.07 o.os ~.88 1.22 0.65 p.85 1.01 0.12 0.94 1.34 1.07 0.66 o.ao 1.14 ~- 74 0.95 0.11 p.90 1.01 0.09 p.15 0.96 0.51 p.92 1.15 o. 35 l .02 1.21 0.18 1.08 1.50 0.64 0.92 1.34 1.63 1.21 1.54 -o.18 1.19 1.64 0.66 1. 17 1.52 0.31 p.65 0.95 1.51 Q.96 1.38 1.15 1.29 1.35 o.os

VEF

1.64 1.67 1.54 1.46 l.65 l.85 1.69 1.91 1.24 1.12 1.80 1.58 1.88 1.11 1.30 1.53 · l .34, l.66 1.ao 1.39 1.1a. 1.~· 1.53 1.45 0.86 1.25 1.36

Critical - .

T ~

Tth. Mode

1.64 VEF 1.67 VEF 1.54 VEF 1.46 VEF 1.65 VEF 1.85 VEF 1.69 VEF l.91 VEF 1.24 VEF 1.12 VEF 1.80 VEF 1.sa VEF 1.88 VEF 1.14 3 1.30 VEF 1. 53 VEF 1.34 VEF 1.66 VEF 1.80 VEF 1.50 2 11-63 3V t."54 2 1.64 2 1.52 2 1.55 3 1.38 2 1.36 VEF

Restrict-ions

R R R R

R R R R

R R R R R

R R

R R

R

R R

t, ~

T Critical

I r.-+-' ~ pf V theor. T rJl 0 M V Q) +-' Beam T r LY.. eff , ex:e Restrict-> C1I No. T M V f' . bd/1' 1 2 3 VEF Tth. Mode i::: bJ) r 1-1 •r-4 0 u 0 0 C ., C ions

R4.20 0.61 0.82 o.;o 1.01 0.32 6.7 1.15 0.79 0.42 0.96 1.15 1 R4.24 o.;54 0.65 o.o 1.04 0.37 8.1 0.96 0.69 0.40 0.97 0.97 VEF V R3.20 0.62 o. 73 o.o 0.91 0.21 6.0 1.oe 0.78 0.46 0.81 1.08 l R3.24 n.61 0.66 o.o 0.94 0.29 7.0 1.02 0.11 0.48 0.90 1.02 l R3.30 0.65 0.60 o.o 0.93 0.21 7.8 1.02 0.82 0.59 0.99 1.02 l R2.24 0.71 0.84 o.o 1.1'0 0.18 4.9 1.24 o.;94 o.so o.;e4 1.24 1 R2.30 0.74 0.11 o.o 0.97 o. 18 5.7 1. 18 0.88 0.61 0.85 1.18 1 R2.38 ,. 73 0.56 (' .o 0.94 0.11 6.7 1.06 0.87 0.69 o.8a 1.06 1 t

Rl.30 0.67 0.65 o.o 0.97 0.13 5.0 1 .07 o.;a3 o.54 o.76 1.01 1 R4A 0.58 0.23 0.21 0.43 0.38 a.o o. 70 0.89 1.09 1.09 1.09 l3V R R4B 0.59 0.41 0.37 0.57 0.38 9.0 0.83 1.01 o.;92 1.21 1.21 \IEf V R °k4.20A 0.58 0.55 0.45 1.13 0.32 13.1 0.91 0.98 0.9.3 1.60 1.60 VEF V R4.208 0.57 0.37 0.31 0.93 0.32 11.2 o.79 o.;90 l.M 1.44 1.44 VEf V R4.24A 0.56 0.63 0.44 1.01 0.37 12.5 0.96 0.96 o. 88 1.50 1.50 VEF V R3A 0.61 n.21 0.21 0.63 C'l.31 8.9 0.76 0.90 1. 10 1.22 1.22 VEF V R R3B 0.55 0.48 0.38 0.90 0.31 ll.l 0.84 0.92 0.79 1.21 1.21 VEF V ; R3.20A 0.67 0.53 0.38 0.82 0.21 9.9 0.98 1.11 1.01 1.37 1.37 VEF V R

..c:: R3.21)8 0.64 0.23 0.11 0.56 0.21 7.5 0.76 0.93 1.19 1.12 1.19 3V R rJl R3.24A 0.55 0.21 0.21 0.11 0.29 9.4 0.10 0.81 0~98 1.15 1.·1s VEF V R ..... C1I R3.24B 0.53 0.4:6 (\.34 1.01 0.29 11.6 o. 81 0.87 0.81 1.26 1.26 VEF V ~ R3.30A 0.54 0.59 0.42 1.36 0.21 12.5 0.91 0.91 0.1s 1.42 1.42 VEF V

R3.30B 0.55 0.25 0.19 0.89 0.21 9.5 0.69 o.ao 1.02 1.21 1.21 VEF V R R2A 0.79 0.45 0~25 0.78 0.19 8.4 1.05 1.14 1.14 1.21 1.21 VEF V R R2B 0.12 0.32 0.18 0.10 0 .19 1.1 0.90 l .01 1. 10 1.06 1.-10 3\1 R R2.24A 0.93 o.s1 0.31 0.75 0.18 8.1 1.25 1.36 1.30 1.37 1~37 VEF V R R2.30A 0.63 0.42 0.23 1.00 0.18 9.4 0.88 0.92 0.90 1.oa 1.08 VEF V R2.30B 0.64 0.29 0.16 0.85 o. 18 8.3 o.so o.aa 0.99 1.02 1.02 VEF V R ' R2.38A 0.62 0.48 0.26 1.24 0.11 10. 1 0.91 0.92 0.87 1.16 1.16 VEf V RZ.388 0.56 0.28 0.16 1.03 o. 17 9.0 0.12 0.11 0.86 0.96 0.96 VEF V

. Rl.30A 0.61 0.43 0.21 0.96 0.13 7.5 0.86 0.94 o. 77 0.89 0.94 2 IU.308 0.59 0.21 o. 14 1).79 0.13 7.4 0. 74 o.as o. 82 0.19 o.as 2 R

t5 VJ

APPENDJX D. ANALYSIS OF TEST DATA

I I I I I T I Critical ~-I T +-' H pf veff theor.

T exp I I Restrict-C/l 0 M V a, +-' Beam T r :.J.... > ctl No. T M V f' bd/"f' 1 2 3 VEF Th Mode ions i::: bD r

1-t .,-1 0 u 0 0 C C t . 3TR3 1.16 ,, • 0 o.o 0.29 o.oo 2.0 1.16 1.11 1.16 0.77 1.16 1 1: 3T~7 1. 19 o.o o.o 0.58 o.oo. 2. 8 1.19 1.14 1.19 0.83 1.19 1 3TR15 1.12 n.o o.o l .o l o.oo 3.7 1.12 l .07 1.12 o,. 74 1.12 1 3TR3'J o. 97 o.o o.o 2.03 o.oo 5.2 0.97 0.93 0.97 0.63 0.97 1 4TR3 l .('4 n.o o.o 0.21 o.oo 2.3 1.04 l .oo 1.04 o.so 1.0,. l

+-' 4TR1 1.15 o.n o.o 0.43 o.oo 3.2 1.15 1.11 1.15 0.92 1.15 1 1: C/l i::: 4TR15 1. l 7 o.o o.o 0.1s 0.00 4.2 1.17 1.13 1.11 0.90 1.11 l H

µ:J 4 TR3 f' 0.95 o.c o.o 1.50 o.o, 6.0 (l.95 0.92 0.95 0.11 0.95 l 5TR3 C.97 o.c o.ii 0.12 0 .00 2.9 0.97 0.96 0.97 0.98 0.98 VEf R 5TR7 0.95 o.o o.o 0.23 o.oo 4.0 0.95 0.94 0.95 1.01 1.01 VEF R 5TR15 C.92 o.o o.o 0.40 o.oo 5.5 0.92 0.91 0.92 0.93 0.93 VEf R 5TR30 0.19 c.o c.o 0.81 o.oo 7.9 0.79 0.78 0.79 0.11 0.79 1 R

~·· R3 1.50 o.o o.o 0.19 0.02 2.4 1.50 1.61 1. 83 1.54 1.83 3 R R5 1.57 0.34 o.o 0.28 c.02 2.5 1. 75 1.69 1.68 l.57 1.75 1 R

@ I R2 1.65 0. 72 o.o 0.38 Q.03 2.4 2.05 1.78 1. 55 1.73 2.05 1 R

51 Rl 0.90 1.1 7 o.o 0.75 0.03 1.6 1.66 0.97 o. 53 0.94 1.66 1 .R Sl 1.50 0.94 o.o 0.57 0.03 2.6 2.04 1.61 1. 26 1.63 2.0,4 1 R S4 1.11 1.18 o.o 0.75 0.03 2.2 1.90 1.26 o. 80 1.29 1.90 1 R

1 0.82 0.44 o.o 1.36 0.04 4.0 1.07 0.90 o. 73 0.14 1.01 1 2 0.67 0.51 o.o 3.40 0.04 5.3 1.01 o. 73 0.50 0.67 1.01 1 3 0.63 0.68 o.o 1.86 0.04 3.1 1.06 0.69 0~42 0.57 1.06 1

. 4 0.44 o. 75. o.o 4.65 0.04 3.9 0.96 0.48 0.22 0.47 0.96 1 .---I 5 0.52 o. 83 o.o 2.36 0.04 2.7 1.08 o.56 0.26 0.47 1.08 1 ctl +-' 6 0.37 0.95 o.o 5.89 0.05 3. 1 1.08 0.41 0.,13 0.42 1.oa 1 a,

"Cl 7 0~45 0.96 o.o 2.87 0.04 2.0 1.14 0.49 0.18 0~40 1.14 1 i::: 8 0.29 0.97 o.o 7.14 0.03 2.1 1.05 0.31 o.oa 0.28 1.os 1 ;:::s

9 0.86 0.43 n.o 0.60 Q.03 3.5 1.10 1.00 0.11 0.74 1.10 1 1: C/l a, 1~ 0.64 0.64 r.o 0.88 0.04 3.C 1.04 o.74 0.44 0.56 1.04 1 0

11 0.69 0.5') o.o 1. 21 0.03 4.4 0.98 o.80 o.ss 0.54 0.98 l 12 0.54 0.78 o.o 1.11 0.04 3.6 l .06 0.63 0.30 0.47 1.06 1

t,

U1


T Critical ex:e.

I T +-' ~ pf T 00 0 Beam T M V veff theor.

Q,) +-' r :...J... ex:e :> co - -bd{fi

Restrict-i:: b.O No. T Ml\ V r f' 1 2 3 VEF Tth. Mode >-< •.-1

0 0 0 C C ions

BKl I). 97 o.c 0. I) l .65 o.oo 6.6 0.97 0.98 0.97 o.80 0.98 2 BKlA n.a4 o.o n.o 1.59 o.oo 6.7 0.84 0.87 0.84 0.73 0.87 2 BK2 r).93 o.o o.o l. 36 o.oo 9.3 0.93 0.91 0.93 1 .oa 1.08 VEF V BK2A r).95 C'.O o.o 1.36 o.oo 9.1 0.95 0.94 o. 95 1.08 1. 08 VEf V BK3 1.15 o.o r.o 0.82 o.oo 8.2 1.15 1.19 1.15 1.21 1.21 VEf V R BK3A r,.98 0.0 o.o 0.82 I).CO 9.0 0.98 1.01 0.98 1.13 1.13 VEF V R BU4 1).76 1.03 ". I) 1.03 0.97 5.9 1.43 0.98 0.49 1.06 1.43 l p

BU4A l"I. 70 0.97 c.o l.C7 1. 0:) 6.2 1.34 C.90 0.45 1.04 1.34 l p

BU5 0.38 ".:I .19 o.o 1.c2 o.oo 13.3 0.48 0.39 0.30 0.81 0.81 VEF V BUSA n.35 C-.36 ,, • 0 1.43 o.oo 10 .4 o.sa 0.37 0.22 0.75 0.1s VEF V BU6 0.48 0.60 n.o 2.29 o.oo 6.9 0.87 o.so 0.26 0.1s 0.87 1 BU6A n.51 0.36 o.o 1.74 o.oo 8.5 o. 72 C,.52 0.36 o.76 o. 76 VEF V B117 !).61 1. 2 9 1.11 1.23 o. 95 9.3 1.53 1.73 0.28 2.01 2.01 VEF V p

Bll7A 1.54 1.16 1.09 1.23 0.82 9.5 1.37 1.57 0.24 1.87 1.87 VEF V p b.O 8118 0.55 0.87 0.75 1.57 n.34 12 .6 1.13 1.02 0.39 1.78 1.78 VEF V •.-1 00 B118A n.s3 0.83 0.68 l. 5 2 G.29 12.0 1.09 1.05 0.38 1.63 1.63 VEF V 00 Q,) B 119 "-61 0.34 0.30 0.87 0.32 12. 4 0.81 0.93 o. 82 1.43 l .43 VEF V R ~ BII9A 0.66 0.34 r.2a 0.81 0.29 ll.3 0.85 1.02 0.91 1.43 l .43 VEF V R

BI 111 t'.68 0.12 C.46 1.56 c.22 9.6 1.13 1.18 0.71 1.41 1.41 VEF V 81 Il OA 0.66 0.68 0.41 1.54 0.21 9.6 1.oa 1 .09 0.10 1.30 1.30 VEF V 81111 0.62 0.33 0.22 1.15 0.23 10. l o. 81 0.88 0.85 1.13 1.13 VEF V 8111111 0.62 ('1.35 n.23 1.18 0.23 1c. o 0.82 o.aa 0.83 1.10 1.10 VEF V BI 112 0.60 C.25 ,,.29 0.46 0~37 7 .1 O.74 0.95 1.01 1.23 1.23 VEF R 81112A 0.59 C.26 0.32 0.47 O.37 7.1 0.74 0.97 1.04 1.21 1.21 VEF R 81113 0.62 ').11 o.ao 0.85 o. 62 10.6 1.12 1.38 0.54 1.83 l. 83 VEF V R p B II 13A 0.74 0.83 0.90 0.19 0. 52 11.0 1.21 1.58 0.74 2.17 2.11 VEF V R p 81114 0.6") '1.58 0.64 n.61 0.57 8.9 0.96 il.40 0.62 1.56 1.56 VEF V R p BI 114A 0.78 C.72 0.81 0.54 0.51 8.3 1.22 1 1.91 o.ao l.98 1.98 VEF V R p B1115 o.ao O.0a 0.44 1.59 0. 13 1.1 1.35 1.40 o. 75 1.35 1.40 2 81115A ,.,.._ 76 0.89 0.43 1.74 n.12 1.1 1.33 1.30 0.70 1.29 1.33 l

' 81116 0.89 0.11 n.13 l .98 o.oo 5.4 1.31 l .08 0.11 0.89 1.31 l j B1l16A n.a2 0.74 r.13 2.08 0.0() 5. l 1.21 1.01 0.70 0.81 1.21 l ,____ -

tl O'


T Critical eXQ,

I pf. veff T T

+> s.. Beam T M V r theor. eXQ Ul 0 :..J... Restrict-Q) +> No. T M V r f' bd/F .l 2 3 VEF T Mode > cd ions i::: bi> 0 u 0 0 C C th.

- •r-1

81117A 0.74 0.97 0.24 1.85 0~00 5.5 1.37 0.98 0.54 0.94 1.37 l BI 117 0.68 0.87 0.22 1. 81 o.oo 5.7 1.24 0.90 o.so .Q.88 1.24 1 81118 1.11 0.40 0.22 0.40 0.07 5.5 1.33 1.47 1.18 1.56 1. 56 VEF R 81118A 1.03 0.40 0.22 0.44 0.06 5.5 1.25 1.31 1.09 1.51 1. 51 VEF R

bi> B1119 0.90 0.96 0.21 0.96 o.oo 4.2 1. 50 . 1.16 o.74 1.15 1. 50 l •r-1 Ul 81119A 0.98 0.98 0.22 0.90 o.oo 4.2 1. 59 1.23 o. 80 1.22 1.59 1 R Ul Q) 81120 0.75 0.61 0.49 0.96 0.51 10.5 1.11 1.19 0.96 1.55 1.55 VEF \i p ~ 81120A 0.70 0.59 0.50 0.99 0.47 10.6 1.05 1.16 o. 81 1.54 1.54 VEF \I p

81121 o. 72 1).56 0.47 0~83 (1.48 8.9 1.06 1.30 0.91 1.46 1.46 \IEF \j R p 81121A 0.49 0.49 0.45 1.03 0.45 10.1 o.so 0.90 0.53 1.22 1.22 VEF \i p

118 0.81 0.69 0.90 0.45 o.oo 3.6 1.21 1.35 o.54 1.81 1.81 VEF R IIBA 1. r,2 0~69 0.84 0.42 o.oo 3.3 1. 42 1.48 0.13 1.91 1.91 VEF R WB 1.12 0.45 0.60 0.21 o.oo 3.7 1.36 1.43 0.91 1.87 1.87 VEF R WBA 1. 11 0.47 0.61 0.29 o.oo 3.5 1.37 1.44 0.90 1.e2 1.82 VEF R

828 0.1 0.36 1.01 o.o 1.82 o.no 3.0 1.12 0.37 0.12 Q.46 1.12 1 828 O.lA 0.35 0~98 o.o 1.86 o.oo 3.0 1.09 o .• 35 0.11 0.43 1.09 1 828 0.2 0.61 0.82 o.o 1.05 o.oo 5.4 1.14 0.61 0~32 0.11 1.14 1

> 828 0.2A 0.65 0.89 o.o 1.01 o.oo 4.8 1.21 0.64 0.34 .0.79 1.23 1 0 ~ 828 "~4 1.10 0.74 o.o 0.;67 o.oo 4.5 1.53., 1.08 0.19 1.22 1.53 1 R i::: 828 0.4A 1.os 0.11 o.o' 0.67 o.oo 4.5 1.46 1.03 0.1s 1.16 1.46 l R Q)

i::: 828 0.48 1.04 0.11 o.o 0.67 o.oo . 3. 8 1. 't5 1.02 0~ 75 1.13 1.45 l R •r-1 .Cl 828 0 .4C 1.06 0.68 o.o o.59 o.oo 3.9 1.45 1.01 0.11 1.11 1.45 1 R u

828 0.40 0.94 0.66 o.o 0.69 o.oo 4.9 1.11 0.95 0.67 1.oa 1.33 l R 828 0.4E 0.98 0.;66 o.o o.;66 o.oo 4.9 1.37 0.99 0.11 1.14 1.37 1 R B28 0.4F 0.73 0.:69 o.o 1.30 o.oo 6.2 1.15 0.13 0.46 0.90 1.15 1

a -.J


t ~ pf veff 00 0 Beam T M V r Cl) +" ~ > ~ No. T M V r f' bd/f' .E •.-I 0 u 0 0 C . C

B8 K 0.69 o.o o.o o.so o.oo 9.4 88 KA "l.84 o.o o.o 0.50 o.oo 7.6 B8 O.l 0.29 1.r,4 0.44 3.08 o.oo 5.8 88 O.lA 0.30 1.03 ("l.47 2.86 o.oo 6.2 88 ,.2 n. 51 0.89 0.42 1.73 o.o 8.1 88 O.ZA n.54 0.96 0.41 1.78 o.oo 7.3

88 D.4 '!.70 J.63 1.31) 1.09 o.oo a.a B8 0.4A ~.75 0.64 0.27 1 .05 o.oo 8.0 B 7 '.'l. 2 0.62 0.88 0.49 1.13 o.oo 7.1 87 0.2A 0.61 0.87 0.45 1.16 o.oo 6.5 810 n.2 0.51 1.06 0.40 2.45 o.oo 6.8 810 0.2A 0.50 1.03 o.37 2.42 o.oo 6.6 B1 0.53 l .13 0.41 2.04 o.oo 5.7

i::: BlA 0.52 1.08 0.48 1.91 o.oo 6.4 •.-I r-1

B2 o.ss 1 .06 0.39 1.98 o.oo 5.7 Cl! >,

B2A rt.53 0.98 0.39 1.83 o.oo 6.0 ~ 83 0.74 0.70 0.41 1.22 o.oo 7.5 83A o.74 0.10 0.41 1.23 o.oo 7.5 85 0.59 0.95 0.40· 1.60 o.oo 6.4 BSA 0.60 1.03 0.40 1.76 o.oo 5.6 B6 0.11 0.93 0.41 1.23 o.oo 7.4 86A o.;11 1.02 0.44 1.24 o.oo 7.4 87 n.as o.o o.o. 1.35 o.oo 12.4 87A o.ao o.o n.o · 1.21 o.oo 13.5 88 0.86 0.32 0.12 0.84 o.oo· a.2 BSA 0.90 0.35 0.11 0.;89 o.oo 1.1 89 0.44 0.65 o. 86 1.14 o.o 12.4 B9A o.;so 0.66 0.95 0.98 o.oo 12.1 B10, o.;34 0.58 0.65 1.49 o.oo 10.9 8HM 0.48 0.67 0.87 1.11 o.oo 12.4 811 0-41. 0.82 0.86 1.44 o.oo 12.0 BllA o.so. 0.91 0.83 1.59 o.oo 11.7 B12 0.32 o.;32 0.35 1.03 o.oo 12.9 812A 0.42 0.43 0.46 1.06 o.oo 12. 7

T exQ.

T . theor.

. l 2 3

0.69 0~&9 0.69 o. 84 0.84 0.84 1. 11 0.57 o.oa 1. 11 0.58 o.oa 1.12 0.75 0.21 1.20 o.eo 0.28 1.09 0.88 0.53 1.13 o.93 0.57 1.20 0.92 0.33 1.19 0.90 0.32 1.26 0.75 0.21 1.23 0.73 0.26 1.33 o.ao 0.25 1.29 0.84 0.21 1.29 0.83 0.26 1.22 o. 81 0.25 1.11 1.03 0.47 1.11 1.03 0;1t1 1.22 o.aa 0.28. 1.31 o.;91 0.29 1.31 1.00 0.42 1.43. 1.09 0.46 o.as o.as Q..;85 0.80 o.ao o.ao 1.03 0.93 0.1a 1.09 o.;-99 0.'82 0.87 o.eo 0.22 0.93 0.89 0.21 0.74 0.64 0.16 0.91 0.84 o.;25 1.04 0.89 o.; 22 1.13 0.93 0.22 0.52 0.49 0.20 0.68 0.63 0.25

VEF

1.01 0.98 0.81 0.86 1.13 1.09 1.24 1.16 1.31 1.22 1.01 1.00 1.03 1.11 0.99 0.99 1.25 1.24 1.02 1.01 1.21 1.32 1.32 1.35 1.15 1.01 1.11 2.01 1.25 1.8ft 1-.10, 1.65 0.95 1.22

Critical

T exo

T Mode th.

1.01 VEF V 0.98 VEF 1.11 1 1.11 l 1.13 VEF V 1.20 1 1.24 VEF V 1.16 VEF 1.31 VEF 1.22 VEF 1.26 l 1.23 1 1.33 l 1.29 l 1.29 1 1.22 1 1.2s VEF 1.24 VEF 1.22 l 1.31 1 1.31 l 1.43 l 1.32 VEF .v 1.35 VEF V 1.15 VEF V 1.09 l 1.11 VEF V 2.01 VEF V 1.25 VEF V 1.84 VEF V 1.10 VEF V 1.65 VEF V 0.95 VEF V 1.22 VEF V

Restrict-ions

R R

R R

t:r 00


T Critical eXQ

I T +-' ~ pf veff theor. T fl.l 0 Beam T M V r ":.J_ exp Cl) +-' - Restrict-::,. co No. T M V r f' bd/F 1 2 3 VEF T Mode s:::: bJ)

1-1 ..... 0 u 0 0 C C th. ions

l n.59 0.69 n.37 0.85 n.2a 1.0 1.03 1.oa o. 50 1.02 1.08 2 R 2 0.59 0.69 0.37 0.85 0.28 7.0 1.03 1 .08 o.so 1.02 1.08 2 R 6 0.37 t).87 C.46 1 .43 0.28 6.4 1.01 0.89 0.15 0.88 1.01 l 1 0.81 0.94 n.so 0.85 0.28 1.0 1.41 1.48 0.69 1 .40 1.48 2 R

10 0.48 1.13 0.60 1.44 0.28 6.4 1.31 1.15 0.19 1.13 1.31 1 3 0.68 1.14 0.55 1.69 0.20 8.5 1.46 1.25 0.59 1.55 1.55 VEF \I 4 n.68 1.14 0.55 1.69 0.28 8.5 1.46 1 .25 0.59 1.55 1.55 VEF \

s:::: 8 0.75 1.25 0.61 1.69 0.28 8.5 1.60 1.38 0.65 1.11 1.11 VEF \

..... 9 r-.75 1.25 ('.61 1.69 0.20 8.5 1.60 1.38 ~.65 1. 71 1.11 VEf \ "Cl ;:::s 11 0.91 1.16 C.53 1.co o.oo 7.1 1.66 1.36 ,.so 1.48 1.66 1 >i 12 0~71 0.90 r.41 1 .("(\ o.oo 7.1 1.29 1.05 0.39 1.15 1.29 l

13 0.11 0.90 0.41 1.00 o.oo 7.1 1.29 1.05 0.39 1.15 1.29 l 17 0.66 0.59 0.42 0.5".l 0.20 5.2 1.02 1.13 0.52 1.11 1.11 VEF R 18 0.12 ').65 0.46 o.so 0.20 5.2 1.12 1.24 o.. S.7 1.29 1.29 VEF R 19 0.72 0.65 0.46 0.50 o. 20 s.2 1.12 1.24 0.57 1.29 1.29 VEF R 2!" n.e3 0.75 0.53 0.50 o. 20 5.2 1.29 1.43 0.65 1.49 1.49 VEF R 21 0.72 0.65 0.46 0.50 o. 20 5.2 1.12 1.24 0.57 1.29 1.29 VEF R 22 0.10 o. 89 ().38 1.0, o. 19 4.2 1.20 0~88 0.32 1.05 1.28 l HBl 1.00 o.o c.o 0.92 0.01 2.9 1.00 1.06 1.15 0.81 1.15 3 HB2 0.11 0.78 o.o 1.89 0.01 2.1 1.25 0.82 0.51 0.63 1.25 1 HB3 0.48 ('~89 0 .r, 2.68 0.01 1.5 1.10 o.so 0.21 0.40 1.10 1

~ HB4 0.35 0.94 o.o 3.47 0.01 1.1 1.05 0.37 0.12 0.28 1.05 l

co HB5 0.31 0.97 o.o 3.88 0.01 1.1 1.06 0.33 0.09 0.26 1.06 l ~ HB7 0.81 o.o o.o 0.87 o.oo 3.2 0~ 81 0.82 o. 86 0.65 0.86 3 R ~ co HB8 0~48 0.83 o.o 2.48 o.oo 1.5 1.05 0.49 0.22 0.38 1.05 1 U)_

"Cl HB9 'l.38 o.83 o.o 3.00 0.02 1.6 0.98 0.39 0.15 0.32 0.98 1 s:::: HBlO 0~35 0.87 o.o 3.28 0.01 1.2 0.99 0.36 0.13 0.27 0.99 l co fl.l HBll 0.29 0.91 o.o 3.91 0.01 1.1 0.99 0.10 0~08 0.24 0.99 l s:::: HB13 0.94 o.o o.o 0.87 o.oo 2.9 0.94 0.93 0.99 0.67 0.99 3 R co ::,. H814 0.69 0.57 o.o 1.10 0.01 2.2 1.03 0.10 o.so 0.50 1.03 1

ri1 H615 0.55 0.84 o.o 2.31 o.oo 1.5 1.11 0.54 0.21 0.39 1.11 1 HB16 0.39 0.90 o.o 3.11 0.01 1. 4 1.05 0.40 0.15 0.30 1.05 1 HB17 0.32 0.96 o.o 3.75 0.01 0.9 1.06 0.13 o. 10 0.23 1.0-6 1 t,

"'

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