Moment-Rotation Relation of Top- and Seat- Angle Connections

Moment-Rotation Relation of Top- and Seat-Angle Connections

著者 KISHI Norimitsu, MATSUOKA Kenichi, CHEN Wai-Fah, NOMACHI Sumio

journal orpublication title

Memoirs of the Muroran Institute ofTechnology. Science and engineering

volume 37page range 163-172year 1987-11-10URL http://hdl.handle.net/10258/729

Moment-Rotation Relation of Top-and Seat-Angle Connections

Norimitsu KISHI， Ken-ichi G. MATSUOKA， Wai-Fah Chen* and Sumio G. NOMACHI**

Abstract

In this paper. the moment-rotation relationship of the top-and seat-angle steel heam-to-column building

connection is developed_ In this development. the initial elastic stiffness and ultimate moment capacity of the

connection are determined by a simple analytical procedure_ Using the initial stiffness and the ultimate mo-

ment capacity so obtained. a three-parameter power model similar to that of Richard and Abbott (1975). was

adopted here to represent the moment-rotation relationship of the connection. The analytical model is found

in a good agreement with the experimental results

* Professor and Head of Structure. School of Civil Engineering. Purdue University. West Lafayette. IN 47907 U.S.A

* * Professor and Associate Dean of Scientific Affair. Department of Civil Engineering. College of Industrial Technology， Nihon University. Narashino. Chiba

1. Introduction

In the analysis of steel frame structure it is customary to assume that the beam-to-column con-

nections are either perfectly pinned or perfectly rigid_ However， it is recognized that an actual

beam-to-column connection in a building frame always possesses some flexibility in its moment-

rotation behavior.

The newly published AISC/LRFD specification (1986) designates two types of construction in

its provision; Type FR (Fully Restrained) construction and Type PR (Partially Restrained) con

struction. If the type PR construction is used， the effects of the connection flexibi¥ity on the be-

havior and strength of these frame structures should be considered in the analysis and design pro

cedures_ The semi-rigid joints will have a destabilizing effect on the overall stability of frame

structures， since additional drift will occur in the joints as a result of the decrease in the etfective

stiffness of the members to which the connections are attached_ Such effect has been studied by

Lui and Chen (1986). and Goto and Chen (1987). among others.

The semi-rigid beam-to-column connections play a very important role in the LRFD procedure目

Though several researchers have published papers discussing the connection rigidity for all

163

Norimitsu Kishi， Ken-ichi G. Matsuoka， Wai-Fah Chen and Sumio G. Nomachi

connection types in steel frames， since C.R. Y oung performed experiments to estimate the rigidity

of steel frame connections in 1917; however， the connection behavior has not been standardized

yet At present， the significance of the data base， that is the collection of experiments for beam

to-column connections conducted worldwide， is much emphasized. Nethercot (1985) conducted a

literature survey for the period 1915-1985 and reviewed all steel beam-to-column connection test

data and their corresponding curve r:epresentations. Goverdhan (1983)， Kishi and Chen (1987)

collected extensively the available test 'data on moment-rotation characteristics and compared the

experimental results with various prediction equatioris

In this paper， an analytical procedure is developed to predict the moment-rotation characteris-

tics of the top-and seat-angle connections by determing first the initial stiffness and these ulti

-mate moment capacity of the connections_ The three-parameter elastic-plastic stress-strain model

proposed previously by Richard and Abbott (1975) is then used to represent the moment-rotation

behavior of the connection. The experimental results reported by Hechtman et al. (1947) are

used here to verify the procedure.

2. Formulation of the Prediction Equation

2.1 General

A typical top-and seat-angle steel connection is shown in Fig. 1. In the design of such con-

nections， the following assumptions are usually made: 1) the seat angle transfers only vertical

COLUMN

-回・

'

司E

a

COLUMN

「古「

叫ん，一寸

Fig. 1. Typical Top-and Seat-Angle Connection.

164

COLUMN

Fig. 2. Deflected Configuration of

Top-and Seat-Angles at the

Elastic _Condition.


2) the top angle is provided merely reaction and does not provide any restraining to the beam

However， for lateral stability of the beam and is not considered to carry any gravity loads

according to experimental results conducted by Hechtman et aL (1947)， Altman et aL (1982) and

Azizinamini et aL (1985)， it has been clearly shown that this connection rotates at the critical sec-

tion of the seat angle， and that the top angle provides resistance to the bending forces at the end of

Thus， the top-and seat-angle connection belongs to the Type PR the beam as shown in Fig. 2.

construction in the AISC/LRFD specification

Initial Stiffness 2.2

To determine the initial elastic stiffness Rki， we assume that the top-and seat-angle connec

tion behaves in the following manner:

Materials of the top and seat angles are linearly elastic and their displacements are smal!.

The center of rotation for the connection is located at the leg adjacent to the compression 2.

beam flange at the end of the beam， (Point C in Fig. 2).

The top angle acts as a cantilever beam in which the fixed support is assumed to be at the 3.

fastener-hole edge near the beam flange in the leg adjacent to the column face as shown in Fig.

3.

The resisting momεnt at the center of rotation is so small that it can be neglected 4.

Based on these assumptions and considering the shear deformation in leg of the top angle， the hori

BOLT

NUT

FI XEO SUPPORT

TOP ANGLE

回目『ーー p

F罰--F一回

Cantilever Model of the Top Angle.

165

3 Fig

Norimitsu Kishi. Ken-ichi G. Matsuoka. Wai-Fah Chen and Sumio G. Nomachi

zontal displacementム ofthe heel of the top angle corresponding to the beam flange force P (Fig.

3) is

A P・(g，)3. 0.78・(t，)21=一一ーニー(1+一一一一τ-')

3・(EI)、. (gj)Z . (1)

in which

EI bending stiffness of the leg adjacent to the column face.

gj = gt3-D/2-t/2 (Fig. 3)

D db， the case using rivets as fasteners

W， the case using bolts as fasteners

db fastener's diameter

W nut's width across flats

tt thickness of the top angle (Fig. 2)

g/ gage distance from the top angle's heel to the center of fastener holes in the leg adja

cent to the column face (Fig. 1 or 3).

Here， the coefficient of shear deformaticm is taken as k=6/5 (Gere and Timoshenko， 1984).

The relationships between the horizontal displacement d. and the end rotation θr， and the

connection moment M and the beam force acting at the leg of the top angle P， are

.o =dj・()r

M =dj'p

in which

dl the distance between the centers of the top and bottom angles as shown in Fig. 2.

(d+t/2+t/2)

where

ts thickness of bottom angle， and

d the total depth of the beam section.

Substituting Eq. (1) into Eq. (3) and using Eq. (2)， the bending.moment M is given by:

M= _3・(E乃 (dj)2・一一一一一一一一一一~←台四一司叩f7

0.78・(t，>2. (g，)3 (1+一一ーで_"_) '~l

(gj )，Z

from which the initial connection stiffness Rki is determin巴das

166

(2)

(3)

(4)

Rki


3・(EI)

(078q)2) 1+一一一ーで←←)

(gY

(d 1)2 ・一一一←でァ σT

(gl )"

2.3 Ultimate Bending Capacity

(5)

Based on the experimental results by Altman et al. (1982)， we assume the collapse mechanism

for the top-and seat-angle connection as shown in Fig. 4. Since the distance between two plas

tic hinges is rather short compared with the top angle's thickness， we take into account the effect

of shear force on the yielding of the material

The work equation for the mechanism shown in Fig. 4 with the plastic moment Mp， and the

shear force in the top angle leg Vp， (force P in Fig. 3) is given by

2.Mp・() = Vp.g2・θ (6)

Using the Drucker's yield criteria (1956) for the combined bending moment Mp and shear

force Vp

MP. . . t争 1

(A瓦)+(j;否)'= 1 (7)

in which Mo and Vo are respectively the plastic bending moment capacity and the plastic shear

force capacity of the angle leg without coupling. Using the Tresca's yiεlding criterion， we have

TOP ANGlE

Fig. 4. Mechanism of the Top Angle at the

Ultimate Condition.

167


Mo = σy・lt・(tt)2/4 (8)

Vo = 6 Y・It.t/2 (9)

in which σy is the yield stress of the top angle. Substituting Eqs. (6)， (8) and (9) into Eq. (7) and

rearranging， (Vp/Vo) is obtained as

VP. gっ Vp(-::-)"+ーニ(一一)一1=0Vo - tt' Vo

(10)

The uitimate shear strength v争canbe determined by solving Eq. (10)

Taking the moment with respect to the center of rotation in the leg adjacent to the compress-

ion beam f1ange (point C in Fig. 2)， the ultimate moment capacity Mu is

Mu = Mos+Mp十 VP'd2 )

-l (

in which

Mos plastic moment capacity at point C of the seat angle in Fig. 2.

6y・ls・(tl/4 (12)

Mp plastic moment capacity at point Hz of the top angle

dz d十t/2+k (13)

k distance from the top angle's heel to the toe of fillet as shown in Fig. 4

2.4 Modeling the M-() r Relationship

Using the initial connection-stiffness Rki and the ultimate moment capacity Mu of the connec-

tion， the moment rotation (M-θr) relationship can be represented adequately by the power model

M= R-θy

../fl .¥nl 1In +Rkp・θ711+ (θγ/()o)"f 11

(1唱

in which

Rkp plastic connection stiffness

Rt = Rki-Rkp

θ。 areference plastic rotation

n shape parameter

The connection stiffness Rk in Eq. (1品is

168


dM R1

Rk 'n' ，ム 1同 +Rkpdsr 11+(sr/θ0) n I I叶 1)/

(15)

For an elastic-perfectly plastic moment-rotation curve， Rkp=二0，Equations (14) and (15) reduce to

RM・θrM= 唱/

I1+UJγ/ e 0 )111 1In

dM Rki

11十 (e r/θ0)111 (叶1)/11Rk =

dsr

(16)

(1カ

in which e 0 = Mu/ Rki. Equations (16) and (1わ representthe M-e r relationship and the stiffness

of the top-and seat-angle connections， respectively. The power model was originaly proposed

by Richard (1961) and later applied by Goldberg and Richard (1963).

This power model is an effective tool for designers to execute the second-order nonlinear

structural analysis quickly and accurately. This is because the connection stiffness can be deter

mined directly from Eq_ (16) without iteration. For example， the equation for e r in Eq. (16) can be

represented as

M θγ

Rki・11-(M/Mu)" I 1In

(18)

3. Experimental Verifications

To verify the power model proposed here， for representing the M-θr curve of the top-and

seat-angle eonnections， the tests by Hechtman et al. (1947) are used. Rivets are used for fasten-

ers in these tests. The comparison on each level of ultimate moment capacity is done. The re

sults are shown in Fig. 5. to 8. In these figures， the experimental results are compared with the

analytical power model， the polynominal model proposed by Frye-Morris (1978) and the modified

exponential model as the curve-fitting method introduced by Kishi-Chen (1987). Selecting a suit-

able value for the shape parameter n， the results obtained by the power model agree rather well

with the experimental results similar to that of the polynominal and modified exponential models

It can therefore be concluded here that the proposed power model represents ad巴quatelythe mo

ment-rotation behavior of the top-and seat-angle connections

4. Conclusions

In this paper， the'moment-rotation relationships of the top-and seat-angle connections are de-

veloped. The initial stiffness and the ultimate moment capacity of the connections are determined

169


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analytically and used as two of the three parameters in the proposed power model. The prop-

osed power model is found in a good agreement with available results. The power model can be

easily implemented in a second-order nonlinear structural analysis_

References

AISC (1986)， Load and Resistance Factor Design Manual of Steel Construction， 1st. Ed.， AISC， Chicago.

Altman， W. G.， Jr.， Azizinamini， A.， Bradburn， J. H. and Radziminski， J. B. (1982)， Moment-Rotation Charac-

teristics of Semi-Rigid Steel Beam-Column connections， Structural Research Studies， Department of Civil

Engineering， University of South Carolina， Columbia， South Carolina

Azizinamini， A.， Bradburn， J. B.， and Radziminski， J. B. (1985)， Static and Cyclic Behavior of Semi-Rigid

Steel Beam-Column Connections， Dept. of C. E.， U. of South Carolina.

Drucker， D. C. (1956)， The Effect of Shear on the Plastic Bending of Beams， Journal of Applied Mechanics，

December， 509-514.

Gere， J. M. and Timoshenko， S. P. (1984)， Mechanics of Materials， 2nd Ed.， Wadsworth， Belmont， CA.， 660-

666

Frye， M. J. and Morris， G. A.， (1975)， Analysis of Flexibily Connected Steel Frames， Canadian Journal of

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Goldberg， J-E. and Richard， R. M. (1963)， Analysis of Nonlinear Structures， Journal of the Structural Divi

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Goverdhan， A. V. (1983)， A Collection of Experimental Moment-Rotation Curves and Evaluation of Predic

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Nethercot， D. A. (1985)， Steel Beam-to-Column Connections - a Review of Test Data， CIRIA， London

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to Purdue University， West Lafayette， IN.， in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

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Young， C. R. (1917)， Bulletin No. 104， Engineering Experiment Station， University of Illinois， Urbana， IL

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Moment-Rotation Relation of Top- and Seat- Angle Connections

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