Experimental study on the fracture of lightly reinforced concrete elements subjected to eccentric...

$: Experimental study on the fracture of lightly reinforced concrete elements subjected to eccentric compression$
1 23

Materials and Structures ISSN 1359-5997 Mater StructDOI 10.1617/s11527-014-0476-3

Experimental study on the fracture oflightly reinforced concrete elementssubjected to eccentric compression

Rocío Porras, Jacinto R. Carmona, RenaC. Yu & Gonzalo Ruiz

1 23

Your article is protected by copyright and all

rights are held exclusively by RILEM. This e-

offprint is for personal use only and shall not

be self-archived in electronic repositories. If

you wish to self-archive your article, please

use the accepted manuscript version for

posting on your own website. You may

further deposit the accepted manuscript

version in any repository, provided it is only

made publicly available 12 months after

official publication or later and provided

acknowledgement is given to the original

source of publication and a link is inserted

to the published article on Springer's

website. The link must be accompanied by

the following text: "The final publication is

available at link.springer.com”.

ORIGINAL ARTICLE

Experimental study on the fracture of lightly reinforcedconcrete elements subjected to eccentric compression

Rocıo Porras • Jacinto R. Carmona •

Rena C. Yu • Gonzalo Ruiz

Received: 29 December 2013 / Accepted: 24 November 2014

� RILEM 2014

Abstract The objective of this work is to study the

structural response of lightly reinforced concrete

elements under eccentric compression from an exper-

imental perspective. Given the difficulty of testing

elements of a large slenderness ratio, an experimental

campaign was carried out on columns of reduced size

to facilitate material control and specimen handling, as

well as to minimize the data scattering. Fifty-four

micro-concrete specimens were tested to study the

influence of reinforcement amount, slenderness ratio

and load eccentricity. The applied load and the

horizontal displacement at mid-span of the specimen

(the additional eccentricity) were measured during the

entire loading process, including the post-peak stage.

At the same time, independent coupon tests were

conducted for material characterization. Based on this

study, a brittle–ductile classification according to the

slenderness ratio and the initial load eccentricity is

proposed. Meanwhile a methodology to evaluate the

minimum reinforcement amount for reinforced con-

crete elements subjected to eccentric compression is

formulated. In addition, the experimental results

provide a valid source for modeling the behavior of

lightly reinforced concrete elements subjected to

eccentric compression.

Keywords Lightly reinforced concrete �Slenderness ratio � Brittle–ductile transition �Additional eccentricity

List of symbols

A Cross-section area, A ¼ bh

As Cross-section area of the steel bars

b Cross-section width (dimension in z-axis)

d Loading point displacement

ea Additional eccentricity at the midspan of the

specimen

e0 Initial load eccentricity

eT Total load eccentricity, eT ¼ e0 þ ea

eal Eccentricity at the linearity limit

E Elastic modulus of concrete

Es Nominal elastic modulus of steel rebar

fc Concrete compressive strength

ft Tensile strength of concrete

fR Flexural strength of the element section

fy Steel yield strength

GF Specific fracture energy

h Specimen thickness (cross-section depth,

dimension in y-axis)

H Specimen height (dimension in the longitudinal

direction, x-axis)

Iy Moment of inertia of the section with respect to

the y-axis

‘ch Characteristic length ‘ch ¼ EGF

f 2t

Lp Effective length for calculating the slenderness

ratio, k

R. Porras � J. R. Carmona � R. C. Yu (&) � G. Ruiz

Universidad de Castilla-La Mancha (UCLM), Avda.

Camilo Jose Cela s/n, 13071 Ciudad Real, Spain

e-mail: [email protected]

Materials and Structures

DOI 10.1617/s11527-014-0476-3

Author's personal copy

m Slope of the straight line of ea=e0 as a function

of qp Perimeter of the steel bars

P Applied load

P� External load normalized by the section

capacity for compression PAfc

W Section elastic modulus Iy=ðh=2Þz Mechanical arm of the section

bH Hillerborg’s brittleness number, bH ¼ H‘ch

eu Steel ultimate strain

g Non-dimensional bond strength of the concrete-

steel interface

k Slenderness ratio

kinf Lower limit of the slenderness ratio, beyond

which secondary effects need to be taken into

account

q Reinforcement ratio q ¼ As

A

r0:2 0.2 % Offset yield strength

ru Steel ultimate strength

sc Bond strength of the concrete-steel interface

1 Introduction

The failure of reinforced concrete (RC) slender

elements, such as columns or panels, is a complex

phenomenon. This is due to the material and geomet-

rical nonlinearities involved [1], the bond deteriora-

tion between the concrete bulk and the steel rebar [2],

the effect of creep [3–5] as well as the cracking and

compressive damage in concrete [6]. All the men-

tioned factors can lead to stiffness reduction of the

column, which in turn exerts a great influence on its

buckling failure due to second-order effects [1, 7, 8].

Compressive failure in structural concrete elements

has been addressed experimentally in the literature by

several authors from different perspectives. In 1994,

Bazant and Kwon [9] studied the size effect in the

failure of columns. Shortly after that, in 1997, Kim and

Yang [10] tested 30 columns made of reinforced high

strength concrete (HSC). By changing the slenderness

ratio, concrete strength and reinforcement ratio, they

drew the main conclusion that the use of HSC improves

the capacity for more stocky columns but has almost no

influence on that of the slender ones. In the same year,

Foster and Attard [11] observed that the ductility of the

column is related to the transverse reinforcement, and

that the effectiveness of this reinforcement is strongly

influenced by the concrete strength.

All the mentioned studies have served to understand

the failure in slender RC elements and variables

involved in it. However, given the difficulty of testing

elements with a large slenderness ratio, experimental

campaigns measuring the entire loading process,

including the post-peak stage as well as independent

material characterization for concrete and steel proper-

ties, at least to the authors’s knowledge, are not available

in the literature. Thus a need was felt for a complete

experimental program in which all material character-

istics, including those related to the steel-concrete

interaction, should be measured through direct testing.

In this work, we present the results on the tests

carried out on elements of reduced size to facilitate

material control and specimen handling, as well as to

minimize the data dispersion [12]. All the specimens

are weakly reinforced, this entitles us to employ the

framework based on fracture mechanics of concrete

[13–15] for results interpretation. This research is an

extension of the previous work on the study of lightly

reinforced concrete elements [16, 12] subjected to

bending forces. By focusing on the effect of an

eccentric compressive force, we aim at complement-

ing the study of the fracture behavior of lightly

reinforced concrete elements. Preliminary results of

the current work, presented in a Spanish confer-

ence [17] have been employed in the author’s recent

numerical model [14] for validation. In this paper, we

concentrate on the experimental aspects of the LRC

elements subjected to eccentric compression.

The rest of the paper is structured as follows: a brief

overview of the experimental program is given in

Sect. 2. The materials and specimens are described in

Sect. 3. Section 4 summarizes the experimental pro-

cedures. The experimental results are presented and

discussed in Sect. 5. Finally, in Sect. 6 some conclu-

sions are extracted.

2 Overview of the experimental program

The experimental program was designed to study the

fracture behavior of lightly RC slender elements. In

particular, we want to determine whether the structural

responses of these elements are dependent on the

specimen slenderness ratio, the initial load eccentricity

and/or the reinforcement amount. In addition, the



program had to provide an exhaustive material charac-

terization to allow a complete interpretation of the test

results.

As mentioned before, we designed tests on panels

of reduced size. This is based on the size effect law

described through the Hillerborg’s brittleness number

(see [18] for details)

bH ¼H

‘ch

; ‘ch ¼EGF

f 2t

ð1Þ

where H is the specimen height, ‘ch is the character-

istic length; E, GF and ft are respectively the elastic

modulus, the fracture energy and the tensile strength

of the concrete. According to the size effect law, two

geometrically similar structures will display a similar

fracture behavior if their brittleness numbers are equal

[19, 20]. For instance, the behavior of specimens of

300, 600 and 1200 mm in height, made from a micro-

concrete with a ‘ch of 90 mm will represent that of

specimens of 1, 2 and 4 m made from an ordinary

concrete with a ‘ch of 300 mm.

Therefore tests on fifty-four micro-concrete spec-

imens were carried out. All the specimens were

geometrically similar with the same rectangular cross-

section of 50 � 70 mm (h � b), see Fig. 1. Three

different heights of 30, 60 and 120 cm were chosen to

give slendernesses ratio k of 21, 42 and 83 respec-

tively, and k is calculated as follows

k ¼ Lpffiffiffiffiffiffiffiffiffi

Iy=Ap ; Iy ¼

1

12bh3; A ¼ bh ð2Þ

where Lp is the effective length, which is dependent on

boundary conditions, in this case, it is equal to the

height H.

The specimens were longitudinally reinforced with

1, 2 or 3 ribbed steel bars of 2.5 mm in diameter

(which resulted a reinforcement ratio of 0.14, 0.28 and

0.42 % respectively). The type of bars was also chosen

so that the non-dimensional steel-concrete bond

strength, defined by Ruiz, Elices and Planas [21] and

by Ruiz [22] as

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Es

E

sc

ft

p ‘ch

As

s

; ð3Þ

falls in the range of regular-size ribbed bars, which is

25� g� 50. Three settings of the initial load eccen-

tricity, 1, 25 and 50 mm (which we denominated as

setting type a, b and c), were applied, see Fig. 1.

Standard characterization and control tests were

performed to determine the compressive strength,

elastic modulus, tensile strength and fracture energy of

the concrete. Test samples were fabricated from the

same batch of concrete. Steel properties such as the

yield strength and the modulus of the steel were

obtained from tensile tests. The steel-to-concrete bond

characteristics were determined from pull-out tests in

which both the pulling force and the steel slip were

measured. The results of all these tests are shown in

the next section.

3 Material characterization

3.1 Micro-concrete

A single micro-concrete mix was employed throughout

the experiments. The maximum grain size of the

a=1mm b=25mm c=50mm

Reinforcement

Load eccentricity

Height andslenderness

e0initial eccentricity

eaadditionaleccentricity

L

M

S

120c

m

70mm

60c

m

30cm

50m

m

(3, 2 and 1 bars)

(a, b and c)

(S, M and L) λ λ λ

Cross-section

y

z b=70mm

h=50mm

ρ= 0 .42% ρ= 0 .28% ρ= 0 .14%

Py

z

x

SML

H =H =H =

30 cm 60 cm120 cm

SML

214283

Fig. 1 Geometry and load configuration of the tested specimens



siliceous sand was 4 mm and the portland cement was

of ASTM type II. All of the cement used was taken

from the same cement container and dry stored until

use. The mixing proportions by weight were 0.47:3.2:1

(water:aggregate:cement). The granulometric curve of

the aggregate follows the recommendations of ASTM

C33 and those of the Spanish Code EHE-08 [23].

Nine batches were necessary to cast all the panel

specimens plus the required characterization speci-

mens. The Abrams cone slump, measured immedi-

ately before casting, was of 1.8 cm in average. All the

specimens were cast in steel moulds, vibrated during

10 s while fastened to a vibrating table, wrap-cured for

24 h, de-moulded and stored in a moist chamber at

20� 2 �C until the test (about 1 month).

Material characterization was carried out on specimens

of the same age as the panel specimens. For each batch,

four compressive tests and four elastic modulus tests were

conducted according to ASTM C39 and ASTM C469

respectively, on cylinders of 75 mm in diameter and

150 mm in height to measure fc and Ec. Four Brazilian

tests were performed on cylinders of the same

dimension, following the procedures recommended

by ASTM C496 to characterize the tensile strength ft.

The specific fracture energy GF was measured

through three-point bending beams of 75 mm in depth,

50 mm in width and 300 mm in span. The testing

procedures followed were according to RILEM TC-50

[24] and with the improvements proposed by Elices,

Guinea and Planas [25–27]. Such tests were performed

in position control on notched beams resting on anti-

torsion supports. Three linear ramps at different

displacements rates were devised: 10 lm/min for the

first 25 min, 50 lm/min for the following 15 min and

250 lm/min until the end of the test.

The mechanical and fracture properties of the

micro-concrete measured are given in Table 1. The

characteristic length according to Eq. (1) is 93 mm. It

needs to be remarked that the mean values in Table 1

are the test results of 36 measurements (four from each

of the nine batches). The relative large scattering of the

compressive strength and elastic modulus is mainly

due to this fact.

3.2 Steel reinforcement

Since the tests were carried out in a reduced scale, in

order to achieve the desired reinforcement ratios, the

diameter of the steel bars had to be smaller than that of

standard ones. In this case, commercial ribbed wires

with a nominal diameter of 2.5 mm were used. The

measured characteristics of these wires, such as the

nominal elastic modulus, Es, the 0.2 % offset yield

strength, r0:2, the ultimate strength, ru, and the

ultimate strain, eu, are given in Table 2.

3.3 Steel-concrete interface

The stress transfer between steel and concrete was

measured through pull-out tests. Thirty-six pull-out

tests were performed, four for each batch. Pull-out

specimens consisted of 75 � 75 �150 mm prisms

with a wire embedded along their longitudinal axis.

The bonded length was 25 mm to allow a constant

shear stress at the interface [28, 29], see Fig. 2 for the

experimental setup. The concrete specimen was held

by the stiff frame fastened to the machine actuator

while the wire was pulled out through a hole in the

upper steel plate. The relative slip between the wire

and the concrete surface was measured at the bottom

end to eliminate the effect of the elastic deformation in

the wire. The tests were carried out at a constant

displacement of 2 lm/s. The bond strength, sc, is

calculated as the measured peak force divided by the

contact area. The non-dimensional bond-strength

according to Eq. (3) is 32.

4 Experimental procedure for eccentric

compression tests

As mentioned before, fifty-four micro concrete spec-

imens of three sizes (Small, Medium and Large), three

Table 1 Micro-concrete properties and the corresponding

characteristic length

fc Ec ft GF ‘ch

[MPa] [GPa] [MPa] [N/m] [mm]

36.5 � 6 26 � 5 3.9 � 0.5 51 � 5 93

Table 2 Reinforcement steel properties and the steel-concrete

bond strength

Es r0:2 ru eu sc g[GPa] [MPa] [MPa] [%] [MPa] [–]

188 � 5 458 � 25 603 � 12 4.3 � 0.8 3.8 � 1.1 32



reinforcement ratios (1, 2 and 3 rebars) and three

setting of initial eccentricities (a, b and c for 1, 25 and

50 mm respectively), two for each type, were tested.

We name each specimen according to this order, for

example, L2b-1 represents a large size specimen with

2 rebars, loaded with type b setting for the initial

eccentricity, and it is the first of the two columns

tested.

All specimens were tested in a vertical position, see

the experimental setup in Fig. 3, where the details of

the bottom support and the steel plate bolted to the

specimen for eccentric loading are also depicted. The

measured variables include the applied load, P, the

loading point displacement, d, and the additional

eccentricity at the midspan of the specimen, ea, see

Fig. 3. A servo-controlled testing machine, with a load

capacity of 1 MN, was employed for the load appli-

cation, although the load was measured with a 250 kN

load cell, with a precision of 0.5 % of the measured

value (minimum of 0.25 kN). An inductive trans-

ducer, with 20 mm of stroke and a precision of 0.5 %

of the measured value (minimum of 40 lm), was used

to quantify the additional eccentricity. The tests were

performed in position control. The experimental

results and discussion are presented next.

5 Experimental results and discussion

The experimental load versus additional eccentricity

curves, P–ea, for the RC slender elements are shown in

Fig. 4. These curves are gathered so that the load

eccentricity is kept constant for vertical columns and

the slenderness ratio fixed for horizontal rows. Each

subfigure depicts the curves for six specimens of the

same size, same load eccentricity and for the three

reinforcement ratios tested. Regrettably, two tests for

the smaller slenderness (S3b-2 and S3c-1) were not

recorded due to a failure of the bolts which joined the

steel plates, located at the ends of the specimen and the

concrete column. In addition, the exact value of the

initial load eccentricity was measured before the

initiation of each test, this measurement is also

included below each subfigure.

A typical P–ea curve in Fig. 4 starts with a linear

ramp-up, then a loss of linearity is observed before

reaching the peak load, which indicates the initiation

of fracture. Note that the peak load decreases with the

increase of either the slenderness ratio or the initial

load eccentricity. The influence of the slenderness,

initial eccentricity load and reinforcement ratio is

clearly noticeable.

In order to complement the experimental data, we

also drew the crack patterns observed from each test

by copying it directly from both sides of the specimen.

In Fig. 5, the most characteristic ones are depicted for

Fig. 2 Experimental setup for pull-out tests

d

ea

P

P

(a) (b) (g)

(c)

(d)

(e)

(f)

Fig. 3 a Specimen during the test. b–c Top loading device.

d Steel plate bolted to the specimen for applying eccentric

loading. e–f Bottom support. g Schematic diagram of the

specimen and the measured parameters (applied load, P,

loading-point displacement, d, and the additional eccentricity,

ea)



each slenderness ratio and initial eccentricity tested.

Note that the crack patterns change from typical

compressive failure for S (small) elements to repre-

sentative bending fracture for L (large) ones.

As mentioned in the Introduction, these results were

used to validate a fracture mechanics model to

describe the buckling behavior of lightly reinforced

concrete columns [14].

S1a-1S1a-2S2a-1S2a-2S3a-1S3a-2

S

S1b-1S1b-2S2b-1S2b-2S3b-1

S1c-1S1c-2S2c-1S2c-2S3c-2

S S

0

20

40

60

80

100

0 2 4 6 8 10

P (k

N)

ea (mm)

0

5

10

15

20

25

0 2 4 6 8 10

P (k

N)

ea (mm)

0

2

4

6

8

10

12

0 2 4 6 8 10

P (k

N)

ea (mm)

e0

S1a-1

S1a-2

S2a-1

S2a-2

S3a-1

S3a-2

7.1 mm

6.7 mm

2.0 mm

2.1 mm

2.5 mm

3.2 mm

e0

S1b-1

S1b-2

S2b-1

S2b-2

S3b-1

S3b-2

30.3 mm

30.3 mm

25.5 mm

26.4 mm

26.1 mm

--- mm

e0

S1c-1

S1c-2

S2c-1

S2c-2

S3c-1

S3c-2

51.0 mm

50.4 mm

50.8 mm

49.8 mm

46.1 mm

47.4 mm

M1a-1M1a-2M2a-1M2a-2M3a-1M3a-2

M M M

P (k

N)

P (k

N)

0

20

40

60

80

100

0 5 10 15 20

P (k

N)

ea (mm)

M1b-1M1b-2M2b-1

M2b-2M3b-1M3b-2

0

2

4

6

8

10

12

0 5 10 15 20e

a (mm)

M1c-1M1c-2M2c-1

M2c-2M3c-1M3c-2

0

5

10

15

20

25

0 5 10 15 20e

a (mm)

e0

M1a-1

M1a-2

M2a-1

M2a-2

M3a-1

M3a-2

5.0 mm

6.9 mm

1.9 mm

2.0 mm

7.1 mm

6.5 mm

e0

M1b-1

M1b-2

M2b-1

M2b-2

M3b-1

M3b-2

29.7 mm

26.9 mm

28.0 mm

29.2 mm

26.0 mm

27.0 mm

e0

M1c-1

M1c-2

M2c-1

M2c-2

M3c-1

M3c-2

46.6 mm

47.0 mm

50.3 mm

51.4 mm

45.1 mm

45.0 mm

LL

P (k

N)

P (k

N)

P (k

N)

L1a-1L1a-2L2a-1L2a-2L3a-1L3a-2

L

0

20

40

60

80

100

0 5 10 15 20 25 30e

a (mm)

L1b-1L1b-2L2b-1

L2b-2L3b-1L3b-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30e

a (mm)

L1c-1L1c-2L2c-1

L2c-2L3c-1L3c-2

0

5

10

15

20

25

0 5 10 15 20 25 30e

a (mm)

e0

L1a-1

L1a-2

L2a-1

L2a-2

L3a-1

L3a-2

7.8 mm

2.9 mm

2.5 mm

2.9 mm

6.2 mm

8.0 mm

e0

L1b-1

L1b-2

L2b-1

L2b-2

L3b-1

L3b-2

25.0 mm

24.5 mm

29.6 mm

23.5 mm

29.5 mm

29.7 mm

e0

L1c-1

L1c-2

L2c-1

L2c-2

L3c-1

L3c-2

46.6 mm

51.2 mm

49.5 mm

47.2 mm

47.5 mm

47.7 mm

Fig. 4 Load versus additional eccentricity for small (first row),

middle (second row) and large (third row) size specimens loaded

with eccentricity of 1 mm (left column), 25 mm (central

column) and 50 mm (right column). Specific values of the

initial eccentricity for each specimen are given below each

subfigure



Next we focus our discussion on the shape of the P–

ea curves in Fig. 6. In addition, a method to determine

the mechanical reinforcement in lightly RC elements

subjected to eccentric compression based on tests

results is proposed.

5.1 Brittle, ductile or transitional behavior

The shape of a typical P–ea curve is dependent on

the slenderness ratio and the initial load eccentricity,

see Fig. 6. In this figure, we put together the P–ea

curves for small and large specimens, with different

settings of initial eccentricity e0, but all reinforced

with one single steel bar. It can be observed that, for

the same slenderness ratio k (or the initial eccentricity

e0), the increase of e0 (or k) leads to an increase of the

displacement and additional eccentricity ea at peak

load.

From Fig. 6, we can identify two types of structural

behavior in the specimens tested based on the global

shape of the P–ea curves. Two values of the additional

eccentricity are employed to characterize these curves,

the eccentricity at the linearity limit and that at the

peak load, which are respectively denominated as eal

and ea, see Fig. 7. The structural behavior is consid-

ered as brittle, if the ratio ea=eal is less than 5; it is

deemed as ductile if this ratio is more than 10; a

transitional region brittle–ductile is defined if ea=eal

falls between 5 and 10. The factor 10 is chosen by

referring to the ratio between the ultimate and yielding

strain in a metallic material. Brittle behavior is

characterized by an abrupt peak load whereas a ductile

element is portrayed by a smooth development of the

P–ea curve. It needs to be clarified that neither the

condition that the ultimate load is greater than the peak

load, nor the requirement for the ratio ea=eal more than

10 alone, is sufficient to define a ductile behavior. Both

conditions have to be simultaneously satisfied. This

explains that the scenario depicted in the first subfigure

in Fig. 7 will always be brittle, whereas the ones in the

second and third subfigures are subjected to the

amount of reinforcement in the structure element. If

there is sufficient reinforcement, the condition that

ea=eal falls between 5 and 10 will define a brittle–

ductile behavior, otherwise it will lead to a brittle one;

in the same way, the condition that ea=eal is greater

than 10 will define a ductile behavior only if there is

sufficient reinforcement, otherwise it will result in a

brittle failure.

We summarize the observed structural behavior in

Fig. 8a following the criterion just explained. The x-

axis represents the slenderness ratio and the y-axis

stands for the ratio between the initial eccentricity e0

and the column depth h. Each point represents a tested

specimen. Three regions are outlined according to the

specimen behavior.

It needs to be pointed out that, the reinforcement

ratio has a strong influence on lightly reinforced

concrete elements with ductile behavior. In these

elements we identify two types of failures, ductile or

brittle, depending on the relation between the cracking

load and the ultimate load [31], see Fig. 7. The

cracking load is defined as the load needed to reach

the maximum flexural stress in the un-cracked

concrete element and it is mainly dependent on

concrete and steel-concrete interface properties [22].

The ultimate load is reached when the reinforcement

fails and it is governed by reinforcement ratio and the

bond between concrete and steel. In elements present-

ing a brittle behavior, the influence of the reinforce-

ment is weak as it is shown in the first column of

Fig. 4.

a b cL

a b cM

a b c

S

Fig. 5 Typical cracking maps for small, medium and large size

(S, M and L), and for each three setting of initial load

eccentricities (a, b and c.)



P (k

N)

P (k

N)

Δλ Δλ

e0Δ

ac

e0Δ

c

a

P (k

N)

P (k

N)

0

1

2

3

4

5

0 5 10 15 20

S1-c

ea (mm)

01020304050607080

0 5 10 15

S1-a

ea (mm)

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

L1-a

ea (mm)

0

1

2

3

4

5

0 5 10 15 20 25e

a (mm)

L1-c

(a) (b)

(c) (d)

Fig. 6 Four typical behaviors at post peak, as a function of the initial load eccentricity and the slenderness ratio. The rows are of

different size, whereas the columns are of different settings of initial eccentricity

BRITTLE

P

< 5e

BEHAVIOR

S

M

P

5 -10

Frágil-Dúctil

ea

e =

P

>10

Dúctil

e

DUCTILE

DUCTILE-BRITTLE

ea l

ea l ea l

a

ea ea

a ea l

ea l ea l

a ea l

P

ρ= 0.14%

ρ= 0.42%

ea

Fig. 7 Classification criterion for brittle, ductile–brittle and ductile behavior according to the ratio between the additional eccentricity

at the linearity limit and that at the peak load, and the influence of the reinforcement ratio



Crack patterns also help to identify the element

behavior, see Fig. 5. For the classified brittle speci-

mens, cracks tend to be parallel or inclined to the

loading direction; whereas for the ductile specimens,

they are perpendicular to the longitudinal axis and

concentrated around the mid span. For both cases, the

reinforcement ratio is not seen to play a significant role

on the observed crack patterns.

Design of reinforced concrete elements subjected to

compression requires that the second order effects

should be taken into account. To avoid the second

order calculations, Marı and Hellesland [30] proposed

a limit for the slenderness ratio. The Spanish Code

EHE-08 [23] includes this lower limit for the slender-

ness ratio in columns. In the case of a constant initial

eccentricity along the compressed specimen, this

slenderness ratio limit can be written as:

kinf ¼ 35

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:24

P�1þ 0:24

e0=h

� �

s

; P� ¼ P

Afc

ð4Þ

where P� is the axial load normalized by the section

capacity under compression. In Fig. 8b we plot the

three areas defined in Fig. 8a and the results of kinf for

five different values of P�, from 0.1 to 1. It is observed

that for the lowest axial force P�, the structural

element response is more like a beam rather than a

column, kinf falls inside the region where a structural

element exhibits a ductile behavior. For this kind of

elements second order effects can hardly be avoided,

although ductility is always achievable by arranging

adequate reinforcement, see Fig. 7. As the axial force

increases, second order effects need to be covered in

the design process, whereas the reinforcement ratio

has a weaker influence on the element capacity. This

trend can be seen more clearly in Fig. 9, where we

have plotted the evolution of P� with respect to the

ratio between ea and e0.

These curves concur with the above experimental

observations: for higher values of slenderness ratio,

lower values of the peak load are obtained, and the

behavior can be classified as ductile. Furthermore, the

chart included in Fig. 8b can be employed by practic-

ing engineers to predict the behavior of lightly RC

elements, and to determine the necessity of including

secondary effects in their calculations.

5.2 Minimum mechanical reinforcement in lightly

RC elements subjected to eccentric

compression

For elements with a ductile behavior, see Fig. 8, the

amount of reinforcement plays an important role [12,

21]. As mentioned before, when the ultimate load is

lower than the cracking load, the failure is abrupt, see

Fig. 7. Avoiding this sudden failure is of paramount

importance in structural design. Some studies on RC

elements under flexion have been carried out to define

a minimum reinforcement ratio to prevent such an

unwarned failure, see [31, 21]. However, there is no

such research available for slender RC elements under

eccentric compression. In this Section, we endeavor to

formulate a practical methodology to calculate the

minimum reinforcement for ductile elements under-

going accidental overloading perpendicular to the

element, including the effect of the load eccentricity,

the slenderness and the axial load.

λ

1.2

0

0.2

0.4

0.6

0.8

1

20 30 40 50 60 70 80 90 100

e /h 0

Brittle - Ductile

P =0.10P =0.25P =0.50P =0.75P =1.00

1.2

0

0.2

0.4

0.6

0.8

1

e /h 0

λ20 30 40 50 60 70 80 90 100

Brittle - Ductile

**

***

(a) (b)Fig. 8 a Brittle–ductile

transition with respect to the

slenderness ratio and the

normalized initial load

eccentricity, note that the

ductile region is dependent

on the reinforcement ratio.

b Lower limit of the

slenderness ratio according

to Marı and Hellesland [30]



The minimum amount of mechanical reinforcement

prescribed to avoid a sudden and brittle failure is

defined as that for which, the ultimate moment Mu and

cracking moment Mcr are equal [21]. This criterion,

including an eccentric compression load, takes the

form of the following equation:

Mu ¼ Mcr ! Asfyz ¼ WfR þ PW

A� eT

� �

ð5Þ

where the term on the left is the ultimate bending

capacity for a reinforced concrete section and the

second term represents the cracking moment, As is the

reinforcement amount, fy is the rebar’s yield strength

and z is the mechanical arm of the section, which in the

absence of more precise calculations, can take on the

value of 0:8 h [32]. In the term on the right, fR is the

concrete flexural strength, i.e. the strength provoking

rupture or cracking if attained in the outermost fiber,

W is the elastic section modulus which is defined as

W ¼ Iy=ðh=2Þ, where Iy is the moment of inertia and

h=2 is the distance from the neutral axis to the far most

fibre, P is the axial load, A is the cross section area and

eT is the total load eccentricity (the sum of the initial

eccentricity e0 and the additional eccentricity ea).

Equation 5 refers to both right or leftmost fibers of the

element and that is why eT can be added or detracted.

S1a-1S1a-2S2a-1S2a-2S3a-1S3a-2

S1b-1S1b-2S2b-1S2b-2S3b-1

S1c-1S1c-2S2c-1S2c-2S3c-2

S S S

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10e / e

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1e / e

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5e / e0 0 0a a a

P* P*

P*

M1a-1M1a-2M2a-1M2a-2M3a-1M3a-2

M M M

M1b-1M1b-2M2b-1M2b-2M3b-1M3b-2

M1c-1M1c-2M2c-1

M2c-2M3c-1M3c-2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10e / e

P*

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1e / e

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5e / ea 0 0 0a a

P*

P*

LL L

L1b-1L1b-2L2b-1

L2b-2L3b-1L3b-2

L1c-1L1c-2L2c-1

L2c-2L3c-1L3c-2

L1a-1L1a-2L2a-1

L2a-2L3a-1L3a-2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10e / e0

0

0.05

0.1

0.15

0.2

0.25

0 0,2 0,4 0,6 0,8 1e / e

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5e / e0 0a a a

P* P*

P*

Fig. 9 Normalised axial load versus additional eccentricity for small (first row), middle (second row) and large (third row) size

specimens loaded with eccentricity of 1 mm (left column), 25 mm (central column) and 50 mm (right column)



Assuming that the reinforcement has to be symmetric

leads to consider only the positive sign.

The flexural strength fR in reinforced concrete

elements is a function of the percentage of reinforce-

ment, the bond strength and the bar diameter as well as

the concrete, steel and steel-concrete interface

mechanical properties. This fact has been demon-

strated in several research on lightly reinforced

concrete beams [31, 33, 22, 21]. The flexural strength

is also affected by size effect [13, 31, 22, 21]. Some

expressions to its evaluation have been proposed in the

cited references. In the absence of available informa-

tion, it also can be evaluated through the expressions

included in the Model Code-2010 [34], or in the

Eurocode-2 [35].

For a rectangular section b�h, Eq. 5 can be

rewritten as:

Asfy ¼5

24AcfR þ P 1þ 6eT

h

� ��

ð6Þ

which can be used to determine the minimum amount

of reinforcement in slender RC elements previously

classified as ductile using the chart in Fig. 8. In this

equation only eT is unknown since the value of ea is

not a data of the problem. Next the calculation of ea is

explained.

In Fig. 10, we present the observed dependence of

the ratio ea=e0 on the reinforcement ratio. These nine

graphs are ordered in the same manner as those in

Fig. 4. We approximate the straight lines which

followed by the experimental points as follows

ea

e0

¼ mqþ B ð7Þ

where m is the slope, B is the ratio ea=e0 corresponding

to that of a plain concrete element. When there is no

reinforcement, the structure fails at the point of

fracture initiation. Therefore the B coefficient in

Eq. 7 can be calculated as follows

B ¼ ea

e0

�

�

�

�

q¼0

¼ PH2

8EIy

¼ fc

8Ek2P� ð8Þ

where elastic beam theory has been employed to

obtain the maximum transversal displacement (in this

case ea) for a beam loaded with the bending moment

Pe0. Assuming the maximum ea for q ¼ 0 is the one

that the load P should reach the cracking moment Mcr,

we have

fR ¼ �P

Aþ Pe0

W) P ¼ fRAW

W � Ae0

�!A¼bh

P¼ bh2fR

6e0 � h

ð9Þ

Consequently the non-dimensional load for a rectan-

gular section b� h is

P� ¼ P

bhfc

¼ fR

½6e0=h� 1fc

ð10Þ

Insert the above equation to the expression for B in

Eq. 8,

B ¼ ea

e0

�

�

�

�

q¼0

¼ fR

8E

k2

½6e0=h� 1 ð11Þ

It needs to be pointed out that, in Eqs. 9–11, the

condition e0=h 1=6 is assumed a priori; otherwise,

the section is under compression, it will not make

sense to talk about minimum reinforcement. In

addition, it can easily be shown that if

e0

h\

1

6

fc þ fR

fc � fRð12Þ

the plain element reaches failure in compression

before failing in bending and, so, the condition in

Eq. 12 has to be fulfilled.

Moreover, with the experimental data obtained, we

carry out numerical approximation and non-linear

optimisation to obtain the slope as follows

m ¼ 4:3� 10�5 k2:4e0hð Þ

0:012

ð13Þ

Consequently, the additional eccentricity for lightly

reinforcement elements, normalized by the initial

eccentricity e0, can be written as:

ea

e0

¼ q 4:3� 10�5 k2:4e0hð Þ

0:012h i

þ fR

8E

k2

½6e0=h� 1ð14Þ

Therefore, our proposal for the minimum reinforce-

ment for concrete elements under eccentric compres-

sion is to employ Eqs. 6 and 14. The system of these

two equations could be simply solved in a spreadsheet.

In Fig. 11, we illustrate the influence of the

slenderness and the initial eccentricity on the mini-

mum reinforcement in a reinforced column of rectan-

gular section. The assumed material properties and

cross-section dimension are given in Table 3. In

Fig. 11a, the x-axis represents the slenderness, the



L

e a/e 0

(%)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5

e a/e

0

(%)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

S

0

(%)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

S

(%)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

M

e a/e

0

(%)

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

Ma/

e 0

(%)

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

M

e a/e

0

(%)

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

L

(%)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5

Le a

/e 0

(%)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5

S

Fig. 10 Ratio of additional and initial eccentricities versus reinforcement amount

P=100 kNP=500 kNP=1000 kNP=2000 kN

P=100 kNP=500 kNP=1000 kNP=2000 kN

0

0.5

1

1.5

2

75 100 125 150

(%)

0

0.5

1

1.5

2

0.5 1 1.5 2e

0/h

(%)

e0/h =0.4

= 75

(a) (b)

Fig. 11 The influence of a the slenderness, when e0=h is fixed at 0.4, and b the initial eccentricity, when k is kept at 75, on the minimum

reinforcement ratio



y-axis is the minimum reinforcement ratio, whereas

the initial ratio e0=h is kept constant at 0.4. For the

lowest axial force, the reinforcement ratio is around

0.2 % which is a normal value for beams. A larger

slenderness requires more minimum reinforcement as

load increases. In Fig. 11b the effect of the initial

eccentricity is demonstrated for several axial forces.

The x-axis stands for the initial eccentricity e0, again

normalized by the column depth h, the y-axis is same

as that of Fig. 11a, whereas the slenderness is kept

fixed as 75. As the initial eccentricity increases

minimum reinforced ratio also increases. The influ-

ence of axial force is represented clearly again, as

larger values of reinforcement are required for higher

values of the axial load.

This example illustrates the importance of defining

the minimum reinforcement percentage for slender

columns or panels with a high initial eccentricity.

6 Summary and conclusions

We have presented experimental results on lightly

reinforced concrete slender elements under eccentric

compressive load. Geometrically similar specimens

and characterizing coupons were cast from a single

micro-concrete. Tests were carried out for 54 speci-

mens of three heights, with three reinforcement ratios

and three positions of initial load eccentricity. Mate-

rial (concrete and steel rebars) and interface properties

were measured through independent tests. Concrete-

making and testing procedures were strictly controlled

to reduce experimental scatter. The following conclu-

sions can be drawn from the study:

– Through analyzing the load versus additional-

eccentricity curves, we have classified the struc-

tural behavior of the tested elements according to

their slenderness ratio and initial load eccentricity.

– The classification chart obtained provides a guide-

line to predict the structural behavior for lightly

reinforced slender elements.

– It has been observed that the reinforcement ratio

plays an important role on the failure, when the

behavior of the element is considered ductile, but it

barely influences the observed crack patterns.

– An expression to evaluate the additional eccen-

tricity and another to estimate the minimum

reinforcement amount for reinforced concrete

elements subjected to eccentric compression are

proposed.

– These experimental results can be used effectively for

modeling the behavior of lightly reinforced concrete

elements subjected to eccentric compression.

Acknowledgments The authors acknowledge the financial

support from the Ministerio de Economıa y Competitividad,

MAT2012-35416, and INDAGSA, Spain.

References

1. Bazant ZP, Cedolin L (2003) Stability of structures. Dover,

Mineola, New York

2. Yalcin C, Saatcioglu M (1995) Inelastic analysis of rein-

forced concrete columns. Computers and Structures

77(5):539–555

3. Kwak HG, Kim JK (2004) Ultimate resisting capacity of

slender RC column. Comput Struct 82:901–915

4. Kwak HG, Kim JK (2006) Nonlinear behaviour of slender

RC columns(1). Numerical formulation. Constr Build Mater

20:527–537

5. Kwak HG, Kim JK (2006) Nonlinear behaviour of slender

RC columns(2). Introduction of desing formula. Constr

Build Mater 20:538–553

6. Majewski T, Bobinski J, Tejchman J (2008) FE analysis of

failure behaviour of reinforced concrete columns under

eccentric compression. Eng Struct 30(2):300–317

7. Furlong RW (1993) Slenderness of columns in braced

frames. J Struct Div ASCE 119(11):3405–3415

8. McGregor JG, Breen JE, Pfrang EO (1970) Design of

slender concrete columns. ACI J 67(1):6–28

9. Bazant ZP, Kwon YW (1994) Failure of slender and stocky

reinforced-concrete columns, test of size effect. Mater

Struct 27(166):79–90

10. Kim KD, Yang JK (1997) Buckling behaviour of slender

high-strength concrete colums. Eng Struct 17:39–51

11. Foster SJ, Attard MM (1993) Experimental test on eccen-

trically loaded high-strength concrete colums. ACI Struct J

94:295–303

12. Ruiz G, Elices M, Planas J (1998) Experimental study of

fracture of lightly reinforced concrete beams. Mater Struct

31:683–691

13. Bazant ZP, Planas J (1998) Fracture size effect in concrete

and other quasibrittle materials. CRC Press, Boca Raton

14. Carmona JR, Porras R, Yu RC, Ruiz G (2013) A fracture

mechanics model to describe the buckling behavior of

lightly reinforced concrete columns. Eng Struct 49:588–599

Table 3 Material properties and cross-section geometry

assumed for the calculations shown in Fig. 11

fR Ec fy b h

[MPa] [GPa] [MPa] [mm] [mm]

3 28 500 500 500



15. Carmona JR, Ruiz G, del Viso JR (2007) Mixed-mode crack

propagation through reinforced concrete. Eng Fract Mech

74:2788–2809

16. Ruiz G, Carmona JR (2006) Experimental study on the

influence of the shape of the cross-section and of the rebar

arrangement on the fracture of lightly reinforced beams.

Mater Struct 39:343–352

17. Porras-Soriano R, Ruiz G, Carmona J, Yu R (2009) Estudio

experimental de paneles esbeltos de hormigon debilmente

armados. Anales de Mecanica de la Fractura 26(2):627–632

18. Hillerborg A, Modeer M, Petersson P (1976) Analysis of

crack formation and crack growth in concrete by means of

fracture mechanics and finite elements. Cem Concr Res

6(6):773–782

19. Elfgren L, Shah SP (eds) (1989): Analysis of concrete

structures by fracture mechanics. Proceedings of the inter-

national RILEM workshop dedicated to Professor Arne

Hillerborg. RILEM Committee on Fractures Mechanics of

Concrete, Abisko

20. Petersson PE (1981) Crack growth and development of

fracture zone in plain concrete and similar materials. Report

No. TVBM-1006, Division of Building Materials, Lund

Institute of Technology, Lund

21. Ruiz G, Elices M, Planas J (1999) Size effect and bond-slip

dependence of lightly reinforced concrete beams. In:

Carpinteri A (ed) Minimum reinforcement in concrete

members. Elsevier, London, pp 67–97

22. Ruiz G (2001) Propagation of a cohesive crack crossing a

reinforcement layer. Int J Fract 111:265–282

23. Ministerio de Fomento: EHE Instruccion del Hormigon

Estructural. Ministerio de Fomento (2008)

24. RILEM (1985) Determination of the fracture energy of

mortar and concrete by means of the three-point bend tests

on notched beams. Mater Struct 18:285–290

25. Elices M, Guinea GV, Planas J (1992) Measurement of the

fracture energy using three point bend test 3. Influence of

cutting the P-d tail. Mater Struct 25:327–334

26. Guinea GV, Planas J, Elices M (1992) Measurement of the


experimental procedures. Mater Struct 25:121–128

27. Planas J, Elices M, Guinea GV (1992) Measurement of the


bulk energy dissipation. Mater Struct 25:305–312

28. Losberg A, Olsson PA (1979) Bond failure of deformed

reinforcing bars based on the longitudinal splitting effect of

the bars. ACI J 76(1):5–17

29. RILEM/CEB/FIP (1970) Test and specifications of rein-

forcement for reinforced and prestressed concrete: four

recomendations of the RILEM/CEB/FIB,2: Pullout tes.

Mater Struct 3(15):175–178

30. Marı A, Hellesland J (2005) Lower slenderness limits for

rectangular reinforced concrete columns. J Struct Eng

131(1):85–95

31. Fantilli A, Ferretti D, Iori I, Vallini P (1999) Behavior of R/C

elements in bending and tension: the problem of minumum

reinforcement ratio. In: Carpinteri A (ed) Minimum rein-

forcement in concrete members. Elsevier, London, pp 99–125

32. Calavera J (2008) Proyecto y Calculo de Estructuras de

hormigon. Intemac, Madrid

33. Fantilli AP, Ferretti D, Rosati G (2005) Effect of bar

diameter on the behavior of lightly reinforced concrete

beams. J Mater Civil Eng 17(1):10–18

34. Federation Internationale du Beton (FIB) (2012) Model

Code 2010—Final draft, Vols 1,2. Bulletins 65 and 66,

Lausanne

35. Eurocode 2, design of concrete structures, Part 1–1: general

rules and rules for buildings (EN1992-1-1). European

Committee for Standardization, Brussels (2004)



Experimental study on the fracture of lightly reinforced concrete elements subjected to eccentric...

Documents

Transcript of Experimental study on the fracture of lightly reinforced concrete elements subjected to eccentric...