Reinforcement Stability and Fracture of Cover Concrete

in Reinforced Concrete Members

Rajesh Prasad Dhakal1 and Koichi Maekawa2

Abstract: The main aim of this study is to propose a simple and reliable method to predict the buckling length of longitudinal
reinforcing bars and also to predict the spalling of cover concrete in reinforced concrete members. Stability analysis is conducted giving
due consideration to both geometrical and mechanical properties of the longitudinal reinforcing bars and lateral ties. The tie stiffness
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required to hold longitudinal reinforcing bars in different buckling modes is derived from energy principles, and it is compared with actual
tie stiffness to determine the stable buckling mode. The buckling length is computed as the product of the stable buckling mode and the
tie spacing. The proposed buckling length determination method is experimentally verified for various cases. A design method for lateral
ties to avoid buckling-induced strength degradation is also recommended. The effect of lateral deformation of longitudinal bars is
quantitatively evaluated and incorporated in the simulation of cover concrete spalling. Analytical prediction considering spalling and
buckling according to the proposed methods showed better agreement with the experimental result.
DOI: 10.1061/共ASCE兲0733-9445共2002兲128:10共1253兲
CE Database keywords: Buckling; Concrete, reinforced; Fracture; Stiffness; Deformation; Spalling; Reinforcement.

Introduction forced concrete members with a system of lateral ties, the bar
length used in such bare bar constitutive relations should be re-
During earthquakes, reinforced concrete members may experi- placed with the buckling length of the longitudinal reinforcing
ence significant lateral deformation of the longitudinal reinforcing bar. Hence, the potential buckling length should be predetermined
bars accompanied by the spalling of cover concrete due to large considering geometrical and mechanical properties of reinforcing
compressive strain. Analytical models neglecting these inelastic bars and cover concrete spalling that may also affect the rein-
material mechanisms cannot capture the post-peak softening be- forcement stability.
havior accurately, and will consequently overestimate the re- Longitudinal reinforcing bars inside reinforced concrete 共RC兲
sponse ductility 共Suda et al. 1996兲. Hence, average stress-strain members, when subjected to large compressive strain, undergo
relationships of concrete and reinforcing bar including spalling lateral deformation. This behavior is referred to as buckling, and
and buckling mechanisms are needed. is mainly associated with geometrical nonlinearity. Similarly, the
It is commonly assumed that the behaviors of the reinforcing writers have defined spalling as the detachment of a part of cover
bar in tension and in compression are similar. In reality, average concrete from the core concrete, finally losing its load-carrying
behavior in compression is different from that in tension. This capacity. Because of their interdependency, these two mecha-
difference is mainly attributed to the geometrical nonlinearity as- nisms should be considered simultaneously and separating them
sociated with large lateral deformation of buckled reinforcing may lead to an incorrect outcome. The formulation of an average
bars. Various average compressive stress-strain relationships in- compressive stress-strain relationship applicable to longitudinal
cluding buckling 共Monti and Nuti 1992; Gomes and Appleton reinforcing bars in RC members with lateral ties consists of three
1997; Rodriguez et al. 1999兲 have been proposed based on ex- parts: 共1兲 formulation of a bare-bar average model including
perimental and/or analytical studies of bare bar under axial com- buckling; 共2兲 incorporating the effect of lateral ties on the stability
pression. All of these relationships implicitly or explicitly suggest of the longitudinal reinforcing bar; and 共3兲 accounting for the
that the average compressive response of bare bar is a function of interaction between cover concrete spalling and reinforcement
the length-to-diameter ratio. For practical application in rein- buckling. An average stress-strain relationship for bare bar includ-
ing buckling is formulated and verified by the authors 共Dhakal
1
Research Fellow, School of Civil and Structural Engineering, Nan- 2000兲. Hence, this paper focuses mainly on the latter two parts.
yang Technological Univ., 50 Nanyang Ave., Singapore 639798; for-
merly, Graduate Student, Univ. of Tokyo. E-mail: cdhakal@ntu.edu.sg Determination of Buckling Length
2
Professor, Dept. of Civil Engineering, Univ. of Tokyo, Hongo 7-3-1,
Bunkyo-Ku, Tokyo 113, Japan. E-mail: maekawa@concrete.t.u-
tokyo.ac.jp Effect of Lateral Ties in Buckling Length
Note. Associate Editor: Julio A. Ramirez. Discussion open until
As mentioned earlier, one parameter that governs the average
March 1, 2003. Separate discussions must be submitted for individual
compressive behavior of the longitudinal reinforcing bar is the
papers. To extend the closing date by one month, a written request must
be filed with the ASCE Managing Editor. The manuscript for this paper buckling length. The assumption that the buckling length of lon-
was submitted for review and possible publication on April 28, 2000; gitudinal reinforcing bars inside an RC member is equal to the
approved on February 26, 2002. This paper is part of the Journal of spacing of lateral ties does not hold true except in some special
Structural Engineering, Vol. 128, No. 10, October 1, 2002. ©ASCE, cases, such as when: 共1兲 lateral ties are very stiff; 共2兲 longitudinal
ISSN 0733-9445/2002/10-1253–1262/$8.00⫹$.50 per page. reinforcing bars are very slender; and 共3兲 tie spacing is very large.

JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002 / 1253

J. Struct. Eng. 2002.128:1253-1262.

 ing bars inside an RC member, which is equal to the integral
multiple of tie spacing, is proposed based on stability analysis.

Assumptions and General Flow of Computation

The entire process of buckling length determination is illustrated
with a flow chart in Fig. 1. First, the actual stiffness of a lateral tie
effective to each longitudinal reinforcing bar connected to the tie
is computed. Next, the minimum transverse stiffness at the tie
locations required to hold a longitudinal reinforcing bar in differ-
ent buckling modes is determined using energy principles. Here,
Fig. 1. Flow-chart of buckling length determination buckling mode refers to the number of tie spacing covered by the
buckling length. If the effective tie stiffness is less than the re-
quired stiffness for mode n⫺1 but exceeds that for mode n, the
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provided system of lateral ties can hold the longitudinal reinforc-

Previous studies in this regard 共Bresler and Gilbert 1961; Scribner ing bars in the nth buckling mode. In other words, n is the stable
1986; Papia et al. 1988; Pantazopoulou 1998兲 were directed to- buckling mode and multiplying it by the tie spacing gives the
wards the stability-based design of transverse reinforcement in buckling length for the given combination of longitudinal and
flexural RC members based on some contradictory assumptions transverse reinforcement. In derivations based on force equilib-
regarding the relationship between tie spacing and the buckling rium, the complex kinematics of the system needs to be solved to
length. Bresler and Gilbert 共1961兲 assumed that the lateral ties are compute the forces on lateral ties. However, utilizing the
sufficiently rigid to prevent lateral displacement of the longitudi- stiffness-based approach described in this paper enables the com-
nal bar at the tie locations, and replaced the buckling length with plexities involved with the numerical implementation of the equi-
tie spacing in order to determine the size of the lateral ties. In librium equations to be removed, and does not require the simpli-
contrast, Scribner 共1986兲 reported that the plastic hinge in mem- fications commonly used in similar derivations.
bers subjected to repeated reverse inelastic flexure spans through The longitudinal reinforcing bar is simulated as a flexural
a length equal to overall beam depth. By assuming that the tie member fixed at both ends of the buckling length to represent the
spacing is equal to one-fourth of beam depth, it was concluded restraining mechanism of lateral ties at these locations. As shown
that the buckling length extends through three tie spacings. in Fig. 2, a cosine curve satisfying the fixed boundary condition is
However, axial compression tests of reinforced concrete adopted to define the deformational shape of the buckled bar. As
prisms 共Kato et al. 1995兲 showed that the buckling length varies the lateral deformation of the buckled bar increases, some sec-
from one to several times tie spacing depending on the geometri- tions within the buckling length undergo compression hardening
cal and mechanical properties of lateral ties and the longitudinal and some experience unloading 共Dhakal 2000兲. As the unloading
bar. If the size and spacing of lateral ties are properly designed to stiffness is equal to the elastic modulus E s , the flexural rigidity of
provide a rigid support to the longitudinal reinforcing bar at the the unloading sections is equal to the elastic rigidity E s I, where I
tie locations, buckling confines between two adjacent stirrups. It is the moment of inertia of the bar cross section. On the other
is to be noted that if the buckling length changes from one to two hand, flexural rigidity of hardening sections is significantly
times tie spacing, the length-to-diameter ratio will be doubled, smaller than the elastic rigidity due primarily to much reduced
and the average compressive stress-strain relationship of rein- stiffness in hardening. Hence, the average flexural rigidity within
forcement over the buckling length domain is significantly the buckling length is undoubtedly less than the elastic rigidity,
changed 共Dhakal 2000兲. As the average compressive response of but its accurate estimation is difficult due to complex mechanisms
the reinforcing bar is very sensitive to the buckling length, sim- involved. The average flexural rigidity of the longitudinal bar is
plified assumptions regarding buckling length might lead to sig- also influenced by its yield strength. As the associated plasticity is
nificantly inaccurate reinforcement behavior. Here, a theoretical less and the secant stiffness is higher in high-strength bars, the
method to determine the buckling length of longitudinal reinforc- average flexural rigidity increases with increase in yield strength

Fig. 2. Simulation and determination of required spring stiffness

1254 / JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002

J. Struct. Eng. 2002.128:1253-1262.

 and vice versa. Finally, considering the qualitative influence of Table 1. Required Spring Stiffness for Different Buckling Modes
nonlinear strain distribution and yield strength, the average Equivalent Required Stiffness k eq
flexural rigidity of the main bar is expressed as EI Stable Average
共Lateral Ties Eliminated in兲
⫽0.5E s I 冑 ( f y /400), where f y is the yield strength expressed in buckling of centrals L/2 and L/3,
mode, n Central L/2 Central L/3 Central L/4 k eq
megapascal. In fact, this approximation closely represents the ac-
tual behavior, as will be justified later through experimental veri- 1 0.7500 0.7500 0.7500 0.7500
fication. 2 0.1649 0.1649 0.1649 0.1649
The lateral ties are simulated by discrete elastic springs. In 3 0.0976 0.0976 0.0371 0.0976
reality, lateral ties show elastoplastic behavior and its tangential 4 0.0758 0.0137 0.0137 0.0448
stiffness reduces nearly to zero after yielding. Ties around the 5 0.0084 0.0084 0.0084 0.0084
middle of the buckling length are more likely to yield due to 6 0.0063 0.0063 0.0032 0.0063
larger lateral deformation of the longitudinal bars. Strictly speak- 7 0.0052 0.0022 0.0022 0.0037
ing, lateral deformation at each tie position should be exactly 8 0.0046 0.0016 0.0016 0.0031
computed and if it exceeds the yielding deformation, the corre- 9 0.0013 0.0013 0.0008 0.0013
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sponding spring should be eliminated from the system. However, 10 0.0011 0.0006 0.0006 0.0009
to obtain a reliable approximation, a parametric study is con-
ducted in this study with the springs in central L/2, L/3, and L/4

冕 冉 冊
n⫹1
regions eliminated from the system. The expansion of core con- 共 n⫹1 兲 s EI d 2 y n⫹1 2
k n is2
crete in compression, which induced tensile strain in lateral ties, U n⫹1 ⫽
0 2 dx 2
dx⫹c i 兺
i⫽1
y
2 n⫹1
is not explicitly considered in the formulation. The lateral force in
the tie, which counterbalances the outward thrust from the con-
fined core concrete, is accounted for by including the spring en-
ergy in the computation. Experimental and analytical investiga-
⫺ 冕 0
共 n⫹1 兲 s

2 冉 冊
P n dy n⫹1
dx
2
dx (3)

tions 共Irawan and Maekawa 1994兲 have proved that the tensile Here, c i ⫽coefficient to incorporate the plasticity of lateral ties
strain induced in the lateral ties due to core-concrete expansion is and its value is 0 for the eliminated springs and 1 for the rest.
less than yielding strain except for the axially compressed RC Similarly, k n and P n ⫽critical spring stiffness and the axial load
member with spiral hoops. Consequently, the stiffness of the lat- corresponding to the nth mode, respectively. Now, using the pre-
eral ties will remain elastic in spite of the core concrete expan- scribed deformational shapes and minimizing U with respect to
sion, thus justifying the simulation. each of the maximum amplitudes a n and a n⫹1 , Eqs. 共4兲 and 共5兲
can be obtained.

冉 冊
Formulation n
⳵U 2␲ 4 EI c i k n 2i␲ 2
P n␲ 2
As shown in Fig. 2, the derivation of the required spring stiffness ⳵a n
⫽0⇒ 3 3 ⫹
n s 4 i⫽1
1⫺cos
n 兺 ⫺
2ns
⫽0
k n corresponding to an arbitrary mode n should address two con- (4)
secutive deformational modes n and n⫹1, as it tries to avoid the

兺冉 冊
n⫹1
(n⫹1)th mode and to sustain the nth mode. The lower modes ⳵U 2␲ 4 EI c ik n 2i␲ 2
P n␲ 2
need not be considered because they are already checked in the ⫽0⇒ 3 3⫹ 1⫺cos ⫺
⳵a n⫹1 共 n⫹1 兲 s 4 i⫽1 n⫹1 2 共 n⫹1 兲 s
previous steps and proved not to exist. First, a term U, which is
expressed as the sum of the energies U n and U n⫹1 associated with ⫽0 (5)
the two buckling modes n and n⫹1, is introduced in Eq. 共1兲. As
shown in Eqs. 共2兲 and 共3兲, energy corresponding to each buckling These two simultaneous equations finally yield the required
mode includes the strain energy, energy stored in the springs, and spring stiffness k n and the corresponding load P n . The required
the energy due to shortening of reinforcing bar. spring stiffness, computed for different buckling modes corre-
sponding to the three different sets of eliminated springs, is
U⫽U n ⫹U n⫹1 (1) shown in Table 1. The equivalent stiffness k eq mentioned in the

冕 冉 冊 冕 冉 冊
n table is a dimensionless parameter and multiplying it by ␲ 4 EI/s 3
ns EI d2y n 2
k n is2 ns P n dy n 2
U n⫽
0 2 dx 2
dx⫹c i 兺
i⫽1 2 n
y ⫺
0 2 dx
dx gives the spring stiffness k n required to stabilize the longitudinal
bar in the corresponding buckling mode. As expected, the re-
(2) quired stiffness becomes smaller for higher-buckling modes. It

Fig. 3. Values of n b and n l for common reinforcement arrangements

JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002 / 1255

J. Struct. Eng. 2002.128:1253-1262.

 Fig. 4. Buckling modes observed in experiment
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can be noticed that the range of eliminated springs does not in- beams and columns reinforced with normal strength and high-
fluence the result except for some special buckling modes, which strength steel bars. The experimental parameters and the compu-
is contributed by the ambiguity arisen due primarily to the exis- tation of the buckling mode are presented in Table 3. Comparative
tence of spring exactly at the boundary of the assumed yield zone. curves relating the equivalent stiffness and the predicted as well
Hence, an average of the equivalent stiffness corresponding to the as experimentally observed buckling modes are shown in Fig. 5.
former two cases 共springs within central L/2 and L/3 eliminated兲 Furthermore, the accuracy of the proposed method, in terms of
is recommended for further use 共Table 1兲. the difference between observed and predicted buckling modes, is
also depicted in Fig. 5. As can be seen in the illustrations, the
proposed method is in very good agreement with the experimental
Stiffness of Lateral Ties
observations, and even the higher buckling modes could be satis-
The buckling tendency of longitudinal bars induces axial tension factorily predicted. Hence, the approximation used in computing
in the tie legs along the buckling direction. Hence, the resistance average flexural rigidity of reinforcing bars is justified.
provided by the lateral ties against buckling of the longitudinal
bar is mainly contributed by the axial stiffness of these tie legs.
The axial stiffness of each tie leg is E t A t /l e , where E t , A t and l e Design Recommendation for Lateral Ties Against
are elastic modulus, cross-sectional area, and the leg-length, re- Buckling
spectively. Assuming that the total stiffness of n l tie legs along the
buckling direction contribute equally to n b longitudinal bars that Lateral ties are commonly designed to provide additional shear
are prone to simultaneous buckling, the restraining stiffness of the resistance and confinement to the core concrete. However, the
tie system effective against buckling of each longitudinal bar can contribution of lateral ties in resisting buckling of longitudinal
be calculated using Eq. 共6兲. bars is seldom considered. The authors believe that properly de-
signed lateral ties can successfully restrict buckling of longitudi-
E tA t n l
k t⫽ ⫻ (6) nal bars, and consequently improve the postpeak response and
le nb ductility. If the value of a parameter L/D 冑 ( f y /100) is less than or
The values of n l and n b for some common arrangements of equal to eight, premature buckling can be avoided and the average
longitudinal and lateral reinforcement are illustrated in Fig. 3. The compressive response of the longitudinal bar is stable 共Dhakal
values shown in Fig. 3 are for flexural loading, where only the 2000兲. Accordingly, lateral ties can be designed using the follow-
reinforcing bars in the compressive side are prone to simultaneous ing steps to avoid buckling-induced strength degradation in the
buckling. Nevertheless, in case of concentric axial compression, postpeak region.
all bars have equal strain and bars in both sides of the tie legs tend 1. Design lateral ties 共spacing s and the diameter D兲 according
to buckle at the same time. Hence, n b and n l along each axis to the existing shear strength criteria.
should be determined considering the tie legs parallel to the axis 2. Compute s/D 冑 ( f y /100). If it is more than eight, reduce the
and longitudinal bars in both sides of the tie legs. spacing so that s/D 冑 ( f y /100) is less than or equal to eight.

Comparison with Experimental Results

Table 2. Specimen Details
For experimental verification, an axially loaded prism and a lat-
erally loaded flexural column are considered. The specimen de- Properties Prism Column
tails are illustrated in Fig. 4, and the computations of stable buck- Cross section 20⫻20 cm 2.4⫻2.4 m
ling modes according to the proposed method are presented in Main bar 6-D13 72-D35
Table 2. Computations in Table 2 yield stable buckling modes of Stirrups D6@10 cm D19@30 cm
one for the prism and three for the flexural column, which re- Number of bars (n b ,n l ) 6, 2 19, 2
semble with the experimental observation shown in Fig. 4. Length of tie leg (l e ) 16 cm 220 cm
For further verification, the predictions according to the pro-
Young’s modulus (E t ) 200 GPa 200 GPa
posed method are compared with 45 experimental observations
Yield strength ( f y ) 355 MPa 424 MPa
共Bresler and Gilbert 1961; Scribner 1986; Kato et al. 1995; PWRI
Equivalent stiffness (k eq) 1.126 0.1015
1999兲 including some experiments by the writers. These cases
Mode, n 1 3
include compression tests of prisms as well as bending tests of

1256 / JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002

J. Struct. Eng. 2002.128:1253-1262.

 Table 3. Calculation of Buckling Mode and Comparison with Experimental Observation
Buckling Buckling
Flexural rigidity Normalizing Tie stiffness Equivalent mode mode
Bar diameter Young’s modulii Yield strength 共EI兲, GPa cm4 Tie spacing, stiffness, 共k兲 Tie area Tie length Number of (k t ), GPa cm stiffness computed observed
Test number 共D兲, cm (E s ,E t ), GPa ( f y ), MPa E s I/2冑 ( f y /400) 共s兲, cm ␲ 4 EI/s 3 , GPa cm (A t ), cm2 (l e ), cm bars (n l ,n b ) E t A t /l e n l /n b (k eq⫽k t /k) (n model) (n exp)
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1a 1.563 200 438 30.66 6.3 11.94 0.691 24.3 2,3 3.785 0.317 2 1
2a 2.188 200 478 122.98 7.6 27.29 0.691 27.98 2,3 3.293 0.121 3 3
3a 2.188 200 478 122.98 7.6 27.29 1.227 29.89 2,3 5.474 0.201 2 3
4b 1.563 200 445 30.86 20 0.38 0.307 16.25 2,4 1.888 5.024 1 1
5b 1.563 200 445 30.86 20 0.38 0.307 16.25 4,6 2.517 6.699 1 1
6c 1.59 200 342 29.01 9.3 3.51 0.713 13 2,4 5.487 1.562 1 1
7c 1.27 200 343 11.83 9.3 1.43 0.713 13 2,4 5.487 3.832 1 1
8c 0.953 200 379 3.94 9.3 0.48 0.713 13 2,4 5.487 11.49 1 1
9c 1.59 200 342 29.01 7 8.24 0.317 13 2,4 2.436 0.296 2 3
10c 1.27 200 343 11.83 7 3.36 0.317 13 2,4 2.436 0.725 2 2
11c 0.953 200 379 3.94 7 1.12 0.317 13 2,4 2.436 2.176 1 1
12c 1.59 200 342 29.01 4.7 27.22 0.317 13 2,4 2.436 0.089 4 4
13c 1.27 200 343 11.83 4.7 11.09 0.317 13 2,4 2.436 0.219 2 3
14c 0.953 200 379 3.94 4.7 3.70 0.317 13 2,4 2.436 0.659 2 3
15c 1.59 200 342 29.01 3.5 65.91 0.317 13 2,4 2.436 0.037 5 5
16c 1.27 200 343 11.83 3.5 26.87 0.317 13 2,4 2.436 0.091 4 4
17c 0.953 200 379 3.94 3.5 8.95 0.317 13 2,4 2.436 0.272 2 2
18c 1.59 200 342 29.01 4.7 27.22 0.126 13 2,4 0.967 0.035 5 5
JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002 / 1257

19c 1.27 200 343 11.83 4.7 11.09 0.126 13 2,4 0.967 0.087 4 4
20c 0.953 200 379 3.94 4.7 3.70 0.126 13 2,4 0.967 0.261 2 3
21c 1.59 200 342 29.01 3.5 65.91 0.126 13 2,4 0.967 0.014 5 6
22c 1.27 200 343 11.83 3.5 26.87 0.126 13 2,4 0.967 0.036 5 5
23c 0.953 200 379 3.94 3.5 8.95 0.126 13 2,4 0.967 0.108 3 3
24c 1.59 200 342 29.01 2.3 232.25 0.126 13 2,4 0.967 0.004 7 7
25c 1.27 200 343 11.83 2.3 94.67 0.126 13 2,4 0.967 0.010 5 5
26c 0.953 200 379 3.94 2.3 31.55 0.126 13 2,4 0.967 0.030 5 3
27c 1.59 200 739 42.64 14 1.51 0.713 13 2,4 5.487 3.624 1 1
28c 1.27 200 978 19.97 14 0.71 0.713 13 2,4 5.487 7.740 1 1
29c 1.27 200 978 19.97 7 5.67 0.317 13 2,4 2.436 0.429 2 2
30c 1.59 200 343 29.05 7 8.25 0.317 13 2,4 2.436 0.295 2 2
31c 1.27 200 356 12.05 7 3.42 0.317 13 2,4 2.436 0.712 2 2
32c 1.27 200 978 19.97 4.7 18.73 0.317 13 2,4 2.436 0.130 3 3
33c 1.27 200 356 12.05 4.7 11.30 0.317 13 2,4 2.436 0.215 2 3
34c 1.59 200 739 42.64 3.5 96.88 0.317 13 2,4 2.436 0.025 5 4
35c 1.27 200 978 19.97 3.5 45.37 0.317 13 2,4 2.436 0.054 4 3
36c 1.59 200 343 29.05 3.5 66.00 0.317 13 2,4 2.436 0.037 5 4
37c 1.27 200 356 12.05 3.5 27.37 0.317 13 2,4 2.436 0.089 4 3
38c 0.953 200 379 3.94 3.5 8.95 0.317 13 2,4 2.436 0.272 2 2
39c 1.27 200 978 19.97 4.7 18.73 0.126 13 2,4 0.967 0.051 4 6

J. Struct. Eng. 2002.128:1253-1262.

 Buckling Buckling

computed observed
3. If it is less than eight, find the largest possible buckling mode

(n exp)
mode

7
7
3
4
1
1
n so that the value of ns/D 冑 ( f y /100) is not more than eight.
4. Compute the axial tie stiffness k t effective to each longitu-
dinal bar according to Eq. 共6兲.
␲ 4 EI/s 3 , GPa cm (A t ), cm2 (l e ), cm bars (n l ,n b ) E t A t /l e n l /n b (k eq⫽k t /k) (n model)
mode
5. Compare the ratio k t /(␲ 4 EIs 3 ) with the equivalent required

5
7
3
5
1
1
stiffness k eq corresponding to mode n determined in step 3
from Table 1.
Tie stiffness Equivalent

6. If the ratio is greater than k eq corresponding to mode n, the

stiffness

0.021
0.006
0.101
0.025
4.013
1.126
current system of lateral ties is capable of avoiding prema-
ture buckling of longitudinal reinforcing bars.
7. If the ratio is less than k eq, either the diameter should be
increased or the spacing should be reduced. Step 2 to step 6
Tie area Tie length Number of (k t ), GPa cm

should be repeated until the ratio k t /(␲ 4 EIs 3 ) becomes

0.967
0.967
0.275
0.549
4.755
1.320

greater than k eq corresponding to mode n.

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Cover Concrete Spalling

2,19
4,19
2,4
2,4

2,3
2,6

Tension Softening due to Splitting Crack

When a flexural RC member is subjected to high moment, the
cover concrete in the compression side suddenly spalls off and
219.6
219.6

loses further load-carrying capacity. Hence, for analytical model-

13
13

20
16

ing, it is important to know when the cover concrete spalls so that

its contribution in the overall response is accounted for. In this
study, an analytical method to determine the compressive strain of
0.126
0.126
2.865
2.865
0.713
0.317

the nearby longitudinal reinforcing bar corresponding to cover

concrete spalling is proposed.
The tensile capacity of cover concrete in the transverse direc-
tion decreases with an increase in the splitting crack width, which
Normalizing
stiffness, 共k兲

can be described by the bilinear tension softening model 共Fig. 6兲.

159.86
45.37

21.64
2.70

1.18
1.17

The area enclosed by the curve gives fracture energy, which is

defined as the energy required to create a perfect crack of unit
area that cannot transfer the normal stress anymore. As the frac-
ture energy is considered to be a material constant depending on
Bar diameter Young’s modulii Yield strength 共EI兲, GPa cm4 Tie spacing,

many conditions, it should be determined by experiments. The

共s兲, cm
3.5
2.3
30
15
10
10

splitting crack width corresponding to zero tensile capacity can be

calculated using the fracture energy G F and the tensile strength f t
of cover concrete 共Fig. 6兲. A splitting crack of this width should
Flexural rigidity

be formed to cause complete spalling of the cover concrete.

E s I/2冑 ( f y /400)

749.77
749.77
19.97
19.97

12.16
12.03

Factors Influencing Cover Concrete Spalling

Two factors contribute to the formation of the splitting crack lead-
ing to cover concrete spalling. First, the compressive strain dete-
riorates the cover concrete creating some vertical cracks, due to
( f y ), MPa

which the tensile capacity in the transverse direction is reduced.

978
978
424
424
363
355

Next, the buckling tendency of the longitudinal bar widens these

cracks, finally reducing the tensile capacity to zero and separating
the cover concrete from the core concrete. Here, the equivalent
(E s ,E t ), GPa

damage due to axial compressive strain is represented by fracture

parameter K, which was originally proposed 共Maekawa and Oka-
200
200
200
200
200
200

mura 1983兲 to account for the stiffness degradation of concrete

due to fracture. For normal concrete, the fracture parameter is
calculated as in Eq. 共7兲, where E is the ratio of compressive strain
Bresler and Gilbert 共1961兲.

to the peak strain of concrete.

共D兲, cm

K⫽exp兵 ⫺0.73E 关 1⫺exp共 ⫺1.25E 兲兴 其 ,

1.27
1.27
3.49
3.49
1.27
1.27

1.0⭓K⭓0.25 (7)
Table 3. 共Continued兲

Kato et al. 共1995兲.

As shown in Fig. 6, the splitting crack width required to cause

Scribner 共1986兲.

cover concrete spalling consists of two parts: 共1兲 (1⫺K)G F / f t ,

PWRI 共1999兲.
Test number

representing the equivalent splitting crack width due to axial com-

Authors.

pressive strain and 共2兲 (4⫹K)G F / f t , representing the equivalent

crack widening due to reinforcement buckling. Assuming the
42d
43d
40c
41c

44e
45e

maximum lateral deformation of the longitudinal reinforcing bar

b

d
a

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Fig. 5. Comparison of proposed method with experimental observations

at the center of the buckling length a to be equal to the widening

of the splitting crack, the critical lateral deformation a cr required
to cause complete spalling of cover concrete can be expressed as
L⫽2 冕 冉
0
␭/2
1⫹
a 2 ␲ 2 2 2␲x
2␭ 2
sin
␭
dx⫽ 冊
4␭ 2 ⫹a 2 ␲ 2
4␭
(10)

in Eq. 共8兲. Replacing ␭ with L(1⫺␧ p ), and neglecting higher-order terms

of ␧ p gives Eq. 共11兲.
GF
a cr⫽ 共 4⫹K 兲 (8) a 2␲ 2
ft
␧ p⫽ (11)
4L 2
Determination of Spalling Strain When a in Eq. 共11兲 is replaced with its critical value a cr , we
get Eq. 共12兲.
Longitudinal reinforcing bars undergo axial shortening under
elastic compression, and the plastic compressive strain is accom- a 2cr␲ 2
panied by lateral deformation. Fig. 7 illustrates the relationship ␧ sp
p⫽ (12)
4L 2
between the plastic compressive strain ␧ p and the maximum lat-
eral deformation a. In fact, elastic shortening is excluded from the Here, ␧ sp
p is the plastic compressive strain of the longitudinal
buckling length L, and the plastic strain will be later added to the reinforcing bar required to cause cover spalling, and adding it
elastic part to obtain total strain. In the figure, ␭ is the projected with the yield strain gives the total compressive strain corre-
length of laterally deformed bar and can be computed as L(1 sponding to the complete spalling of cover concrete. Strictly
⫺␧ p ). Hence, the integration of the deformed length over the speaking, Eq. 共12兲 holds only for the monotonic case 共absolute
range ␭ gives the original length L, as in Eq. 共9兲. compressive strain兲, and the loading history must be considered

冕 冕冑 冉 冊
for cases in which tensile loading may have been applied.
␭ ␭ dy 2
L⫽ ds⫽ 1⫹ dx
0 0 dx
Application and Verification

冕 冑 冉⫽2
␭/2

0
1⫹
a␲
␭
sin
2␲x
␭ 冊 2
dx (9)
Experimental Setup and Specimen Details
Using Taylor’s expansion for the square-root term, and ne- A reinforced concrete cantilever column was tested to investigate
glecting higher-order terms of a 2 /␭ 2 , we obtain Eq. 共10兲. the reliability of the proposed models in predicting the postpeak

Fig. 6. Equivalent deterioration of cover concrete due to axial

compression Fig. 7. Effect of reinforcement lateral deformation in cover spalling

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J. Struct. Eng. 2002.128:1253-1262.

 Table 4. Experimental Parameters
Parameters Values
Cross section, mm 250⫻250
Main reinforcing bars 6-D13
Lateral ties, mm D10@100
Cover thickness, mm 30
Axial load, kN 250
Shear span, mm 1,200
Concrete strength f c⬘ , MPa 28.6
Fig. 8. Experimental setup and geometrical property of specimen Young’s modulus E s , GPa 202
共Unit: mm兲 Yield strength f y , MPa 360
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response. The experimental setup and the specimen layout are used for analyzing the tested RC column. The RC column is
shown in Fig. 8, and the geometrical and mechanical properties of represented by frame elements, and is analyzed by fiber technique
the specimen are listed in Table 4. The column is intentionally 共Menegotto and Pinto 1973兲. In fiber technique, the member cross
provided with significant axial compression 共14% of the axial section is divided into many cells, and each element consisting of
capacity兲 and thick cover concrete 共30 mm兲 to highlight the in- several longitudinal fibers is represented by its centerline. The
fluence of spalling and buckling. In order to avoid shear failure, strain of each fiber is calculated based on the Euler-Kirchoff’s
the column was designed so that the shear capacity is higher than hypothesis; i.e., plane section remains plane after bending. The
the bending capacity. The rigid footing, which was cast mono- response of each element is the integration of all fiber responses
lithically with the column, was tightly fixed to the base slab with that are computed based on the average constitutive models of the
prestressing tendons to ensure the cantilever mechanism. Two ac- materials in the corresponding fibers. Fiber technique and mate-
tuators of a triaxial loading machine were used to apply simulta- rial models used for concrete and reinforcing bar are schemati-
neously cyclic lateral displacement at 120 cm from the footing cally illustrated in Fig. 9. These material models are fully path
top and an axial compression at the top of the column. dependent and take into account the steel-concrete bond and load-
ing rate effect in the concrete response. The details of these mod-
els and their verification for RC members subjected to static and
Material Models and Analytical Simulation
dynamic loading are elaborated in a book by Okamura and
To verify the applicability of proposed methods in FEM analysis, Maekawa 共1991兲.
the computed buckling length is used in a bare-bar buckling The rigid footing is represented by a fixed support at the base
model 共Dhakal 2000兲, and the spalling criterion is supplemented of the column. The RC column is discretized into five elements
with the elastoplastic and fracture model 共Maekawa and Okamura and the cross section is divided into 221 cells. To simulate cover
1983兲. These enhanced models are installed in a finite-element spalling, the stress transferred by concrete fibers outside the lat-
analysis program COM3 共Hauke and Maekawa 1999兲, which is eral tie is reduced to zero once the spalling strain is reached in

Fig. 9. Fiber technique and material models used in FEM analysis

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Fig. 10. Experimental and analytical results 共load-displacement curve兲

nearby reinforcing bars. To incorporate the P-delta effect, geo- design, is also overestimated. This research is hence useful to
metrical nonlinearity is given due consideration in the analysis. practitioners engaged in seismic design, as it enlightens the
Due to substantial axial load, the pullout of reinforcing bars from mechanisms that may impair the seismic performance of RC
the column-footing joint was negligible in the experiment, and is structures.
hence neglected in the analysis.

Conclusions
Results and Discussions
As the column was designed to have comparatively higher-shear An analytical method to determine the buckling length of longi-
strength, no diagonal shear crack could be seen. In the experi- tudinal reinforcing bars inside reinforced concrete members was
ment, uniform flexural cracks appeared gradually, and cover spal- proposed. A comparison with several experimental observations
ling at the column base could be noticed after a few loading revealed ample evidence of the reliability of this method. More-
cycles when the applied displacement reached around 15 mm. over, an enhanced design method of lateral ties to avoid prema-
However, the buckling displacement could not be distinguished in ture buckling of the longitudinal bar was also proposed. The de-
the experiment although buckled bars were seen after scratching terioration of cover concrete due to axial compressive strain and
out the spalled cover concrete. The experimental and analytical widening of splitting cracks due to buckling tendency of the lon-
load-displacement curves with and without using buckling and gitudinal bar were separately considered in the derivation of spal-
spalling models are shown in Fig. 10. As indicated in the figure, ling criterion. The proposed buckling length computation and
the spalling displacement predicted in analysis is also close to 15 spalling criterion were used to carry out finite-element analysis of
mm. A gradual decrease of lateral load can be observed after a laterally loaded cantilever RC column under axial compression.
initiation of cover spalling in the experimental result. In contrast, The analytical results were in good agreement with the experi-
a sudden drop in the load is seen in the analytical result. This is mental results, which further verified the validity of the proposed
because the stress carried by cover concrete fibers is abruptly models.
neglected once the spalling criterion is fulfilled. Hence, this
abrupt reduction in the lateral load is the overall contribution of
spalling, and is carried over throughout the postspalling phase. In Acknowledgments
the analysis, buckling took place during the last loading cycle,
after which an additional reduction in the lateral load could be The writers gratefully acknowledge TEPCO Research Foundation
observed. The difference between the analytically predicted lat- and Grant-in-aid for scientific research No. 11355021 for provid-
eral loads minus the spalling-induced sudden drop is the contri- ing financial support to accomplish this research.
bution of buckling, which increases gradually with the applied
displacement.
Although the load-displacement curve predicted without con- Notation
sidering spalling and buckling exhibits mild postpeak softening
that is due primarily to the P-delta effect, the lateral load in the The followings symbols are used in this paper:
postpeak region is much higher than in the experimental result. A t ⫽ cross-sectional area of lateral tie;
Incorporating buckling and spalling models in the analysis sig- a ⫽ maximum amplitude at center of buckling
nificantly improved the agreement between the analytical and ex- length;
perimental results, and the enhanced FEM analysis could reliably a cr ⫽ critical value of a corresponding to cover
predict the postpeak softening behavior as well. The ultimate de- spalling;
formation, defined as the displacement in the postpeak region c i ⫽ coefficient to include plasticity of lateral tie;
corresponding to either yield load or 80% of the peak load de- D ⫽ diameter of longitudinal reinforcing bar;
pending on design codes followed, is significantly overestimated E ⫽ ratio of compressive strain to peak strain of
if these inelastic material mechanisms are overlooked. As yielding concrete;
displacement is not much influenced by spalling and buckling, the E s ⫽ Young’s modulus of longitudinal reinforcing
response ductility, which is an important parameter in seismic bar;

JOURNAL OF STRUCTURAL ENGINEERING / OCTOBER 2002 / 1261

J. Struct. Eng. 2002.128:1253-1262.

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EI ⫽ average flexural rigidity of longitudinal rein- tionship of reinforcing bar including buckling.’’ Eng. Struct., 19共10兲,
forcing bar; 822– 826.
Hauke, B., and Maekawa, K. 共1999兲. ‘‘Three-dimensional modeling of
f t ⫽ tensile strength of concrete;
reinforced concrete with multi-directional cracking.’’ J. Mater. Concr.
f y ⫽ yield strength of longitudinal reinforcing Struct. Pavements, JSCE, 634共45兲, 349–368.
bar; Irawan, P., and Maekawa, K. 共1994兲. ‘‘Three-dimensional analysis of
G F ⫽ fracture energy of normal concrete; strength and deformation of confined concrete columns.’’ Concr. Li-
I ⫽ moment of inertia of longitudinal reinforc- brary Int., 24, 47–70.
ing bar; Kato, D., Kanaya, J., and Wakatsuki, K. 共1995兲. ‘‘Buckling strains of
K ⫽ fracture parameter in elastoplastic and frac- main bars in reinforced concrete members.’’ Proc., 5th East Asia and
ture model; Pacific Conf. in Structural Engineering and Construction EASEC-5,
k eq ⫽ equivalent required spring stiffness, k eq Gold Coast, Australia, 699–704.
⫽k n ⫻s 3 /␲ 4 EI; Maekawa, K., and Okamura, H. 共1983兲. ‘‘The deformational behavior and
k n ⫽ required spring stiffness for nth buckling constitutive equation of concrete using the elastoplastic and fracture
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mode; model.’’ J. Fac. Eng., Univ. Tokyo, 37共2兲, 253–328.

Menegotto, M., and Pinto, P. E. 共1973兲. ‘‘Method of analysis of cyclically
k t ⫽ axial stiffness of lateral tie;
loaded RC plane frames including changes in geometry and nonelastic
L ⫽ buckling length of longitudinal reinforcing
behavior of elements under normal force and bending.’’ Preliminary
bar; Rep. No. 13, Int. Association of Bridge and Structural Engineering
l e ⫽ effective length of tie leg; IABSE, Zurich, Switzerland, 15–22.
n b ⫽ number of main bars prone to simultaneous Monti, G., and Nuti, C. 共1992兲. ‘‘Nonlinear cyclic behavior of reinforcing
buckling; bars including buckling.’’ J. Struct. Eng., 118共12兲, 3268 –3284.
n l ⫽ number of tie legs along buckling direction; Okamura, H., and Maekawa, K. 共1991兲. ‘‘Nonlinear analysis and consti-
P n ⫽ axial compressive load corresponding to nth tutive models of reinforced concrete.’’ University of Tokyo, ISBN4-
mode; 7655-1506-0 C 3051, Gihodo Publication, Tokyo.
s ⫽ spacing of lateral ties; Pantazopoulou, S. J. 共1998兲. ‘‘Detailing for reinforcement stability in RC
U ⫽ total potential energy of system; members.’’ J. Struct. Eng., 124共6兲, 623– 632.
␧ p ⫽ plastic compressive strain of reinforcing bar; Papia, M., Russo, G., and Zingone, G. 共1988兲. ‘‘Instability of longitudinal
bars in RC columns.’’ J. Struct. Eng., 114共2兲, 445– 461.
␧ sp
p ⫽ compressive plastic strain required for cover PWRI 共1999兲. ‘‘Size effect in the seismic performance of reinforced con-
spalling; and crete piers.’’ Tech. Memorandum No. 234, Public Work Research In-
␭ ⫽ projected length of buckled reinforcing bar. stitute PWRI, Tsukuba, Japan 共in Japanese兲.
Rodriguez, M. E., Botero, J. C., and Villa, J. 共1999兲. ‘‘Cyclic stress-strain
behavior of reinforcing steel including effect of buckling.’’ J. Struct.
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