AP04-Bittnar2.vp


1 Introduction
When a precast reinforced concrete (R/C) panel building is

demolished using controlled explosions, selection of appro-
priate sizes, placement, and timing of the charges is crucial in
order to ensure complete collapse of the structure while not
damaging surrounding objects. The design of safe and effi-
cient deconstruction procedures can be facilitated by means
of computer simulations. In contrast to the standard struc-
tural analysis, the main objective of such a simulation is to
predict the mechanical behavior of a structure during the
phase when it disintegrates and loses static stability. The me-
chanical phenomena to be dealt with include dynamic motion
(finite displacements and rotations) and interaction of debris
on the structural level (macroscale), and fracture and yielding
on the material level (mesoscale). Simultaneous treatment of
all these mechanisms would be computationally too costly;
thus, a computational methodology based on a sequential
multiscale approach has recently been proposed [1]. In order
to reduce the number of degrees of freedom involved in the
dynamic analysis at the macroscale, the entire structure is
modeled as an assembly of beam finite elements, which re-
present structural members (panels) and their joints. The
governing relations among bending moment, axial force, cur-
vature, and axial strain of the beam elements are formulated
by modeling the overall behavior of R/C panel sections or
joints that undergo local damage (mesoscale model).

During demolition, structural members and joints may be
exposed to loading states that are diametrically different from
those for which they were designed. Nevertheless, the load
bearing capacity of an R/C section exposed to an arbitrary
combination of axial force and bending is usually determined
by concrete crushing in compression and/or reinforcement
yielding in tension. Both of these phenomena can be well
modeled by plasticity. If a section is exposed to further defor-
mation upon reaching the load capacity, the load drops, but
usually not immediately to a zero value. The post-failure re-
sponse is often dominated by highly ductile behavior of the
reinforcing bars, which can typically sustain strain up to the
order of 10�1. The corresponding residual load carried by a

failed section or joint cannot be neglected when analyzing the
disintegration and dynamic motion of a structure during de-
molition. In tension, the behavior of reinforcing bars can still
be modeled by simple one-dimensional plasticity, but in com-
pression, plastic buckling has to be taken into account. The
present paper deals with modeling of the latter phenomena.

2 Physical phenomena
Let us consider an R/C member which is loaded by such a

combination of axial force and bending moment, that at least
on one side it is exposed to compression. Concrete crushing
starts when the maximum compressive strain attains the level
of roughly 0.002. As the crushed concrete spalls, it exposes
longitudinal reinforcing bars. Upon losing the support of
concrete cover, the reinforcing bars buckle outward Fig. 1.
The buckling length is limited either by transversal stirrups
or by the length of the spalled concrete zone. Eventually the
bars yield in plastic hinges, which form in the locations of ex-
tensive bending. Since the crushed and spalled concrete
ceases to contribute to the load bearing ability of the section,
its overall response is then dominated by the behavior of the
buckling bars.

3 Analytical model
3.1 Loading and boundary conditions

The behavior of a buckled bar is modeled by the relation-
ship between applied axial force P and the relative displace-
ment of its ends u, as shown in Fig. 2. Force P is positive when
compressive and similarly positive u means contraction. We

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Acta Polytechnica Vol. 44  No.5–6/2004

A Simplified Analysis of the
Post-buckling Behavior of a Compressed

Reinforcing Bar
P. Kabele

Recently, a computational methodology based on a sequential multiscale approach, which facilitates numerical simulation of an R/C
building demolition has been developed. In this type of analysis, it is necessary to capture the behavior of compressed reinforcement bars until
complete rupture, which occurs due to extensive bending in the post-buckling regime. To this end, a simplified analytical model of the
post-buckling behavior of a compressed bar is proposed. The simplification consists namely in considering rigid-plastic material behavior,
neglecting axial contraction of the central line, and approximating the shape of the deformed central line in the plastic hinges by a circular
arch. Consequently, the axial loading force, bar end displacement, and extreme strain can be expressed in relatively simple closed forms. The
results obtained with the proposed model show very close agreement with those obtained by a detailed and realistic finite element analysis,
which justifies the use of the simplifying assumptions.

Keywords: reinforcement bar, hardening plasticity, post-buckling behavior.

Fig. 1: Concrete cover spalling and reinforcement buckling in an
R/C member



consider that the bar is free to buckle in plane x-z within
length L. As the bar is in reality continuous, the in-plane rota-
tions and displacements in z-direction are fixed at the end
points of length L. Since the problem is symmetric about axis
z, we solve it only on one half of L, i.e. in the interval ��L/2, 0�.
Consequently, the displacement in x-direction and the rota-
tion are fixed on the symmetry axis.

3.2 Assumptions
We accept the following general assumptions:

(1) The bar is modeled as a Bernoulli-Euler beam, i.e. the pla-
nar cross-sections perpendicular to the central line prior
to deformation remain so also after deformation takes
place. This assumption is acceptable, since the length of
the bar, which is free to buckle, is usually much larger than
the cross-sectional size.

(2) The bar undergoes finite displacements, therefore the
equilibrium equations are formulated on the deformed
configuration.

(3) In the post-buckling regime, the elastic deformations are
negligibly small compared to the plastic deformations.
Consequently material is modeled as rigid-plastic with lin-
ear hardening. Since we do not consider load reversals, we
will use the deformation theory of plasticity.

(4) When the bar buckles, the contribution of central-line ax-
ial contraction to the overall contraction is negligible in
comparison with the contribution due to bending (finite
deflection).

3.3 Moment-curvature relation
In the view of Assumption (1), the normal strain distribu-

tion is linear on the bar cross-section. Since axial straining of
the central line is neglected [Assumption (4)], the strain can
be expressed as:

� � �� � , (1)

where � denotes the local axis normal to and originating at
the central line, and � is the curvature of the central line.

Assumption (3) implies that the material constitutive law
can be written as:

� � �

� � � � � �

� �

� � �

0 ��
��

for
for

y

y h yEsgn( )
(2)

where � is the normal stress, �y is the yield strength and Eh is
the linear-hardening modulus.

By combining Eqs. (1) and (2) and considering the equiva-
lence of bending moment M and the moment due to normal
stress, the following relation is obtained:

M M K p� �sgn( )� �0 (3)
where

M Ay
A

0 � � � � d (4)

and

K E Ap h
A

� � �
2 d (5)

and A is the cross-sectional area of the bar. Note that Eq. (3)
holds only for those parts of the bar which are fully plastic
(plastic hinges), i.e. � �� y on the entire section, or, in terms
of the bending moment, M M� 0. Outside the plastic hinges,
the curvature is zero for any M.

3.4 Equilibrium on a deformed bar
Let us consider a symmetric half of a buckled bar, as shown

in Fig. 3. The maximum deflection is denoted as w. To satisfy

the equilibrium of forces in x and z directions, reaction R � P
and V � 0. The anti-symmetry of the deformed shape implies
that moment reactions �MA � MB. To maintain the equilib-
rium of moments, we require

2M P wB � . (6)
It is obvious from Fig. 3 and Fig. 4 that the bending mo-

ment induced by the load and the reactions is a linear func-
tion of coordinate z:

M M P zB� � � (7)
which in combination with Eq. (6) gives:

M P z
w

� �
	



�

�


�

2
. (8)

3.5 Geometry of a deformed bar
As discussed in section 3.3, the curvature of the deformed

bar outside the plastic hinges is zero, which means that the
corresponding part of the central line remains straight.

The shape of the central line within the plastic hinges can
in general be obtained by solving together Eqs. (3) and (8).

166 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No.5–6/2004

x

x

z

P P

PP

u/2 u/2

L/2 L/2a)

b)

Fig. 2: Configuration and loading of a reinforcing bar: a) unde-
formed, b) deformed

x

R

P

V

M
A

M
B

w

z

Fig. 3: Load and reactions on a deformed bar



However, solution of the differential equation is rather com-
plicated, since it involves elliptic integrals [3].

To simplify the problem, we approximate the shape of the
central line in plastic hinges by circular arches with constant
curvature �av (grey parts in Fig. 4). Due to the anti-symmetry,
the plastic hinges on both ends of the analyzed half of the bar
have the same length c. The arches are delimited by angle �.
Then

�
�

av
c

� . (9)

To ensure a smooth transition from the plastic hinges to
the straight rigid portion of the bar (black part in Fig. 4), the
latter has to be inclined at the same angle � from the horizon-
tal direction. While taking into account Assumption (4) (the
central line length remains constant and equal to L/2), the two
parameters c and � completely describe the shape of the de-
formed central line. Then it is obvious from Fig. 4 that maxi-
mum deflection w can be expressed as:

w
c L

c� � � �	


�

�


�2 1

2
2

�
� �( cos ) sin . (10)

Similarly, the relative displacement of the bar ends (Fig. 2)
is:

u
L c L

c� � � �	


�

�


�

�

�
�

�

�
�

�
�
�

�
�
�

2
2

2
2

2
�

� �sin cos . (11)

3.6 Derivation of P-u relationship
In the light of Eq. (3), the assumption of constant curva-

ture implies that the bending moment within a plastic hinge is
also constant. Thus, for the right-side hinge:

M M Kav p av� �0 � . (12)
We consider that this constant moment is also equal to the

average moment in the plastic hinge (Fig. 4), then

M M Mav B� �
1
2 0

( ) . (13)

Now we substitute equations in the following order:
(9) � (12) � (13), from which we express MB. This result is
substituted together with Eq. (10) into Eq. (6), which allows us
to express force P in terms of c and �:

P
M K

c
c L

c

p
�

�

� � �
	



�

�


�

2 4

2 1
2

2

0
�

�
� �( cos ) sin

. (14)

In order to eliminate the length of the plastic hinge c, we
consider that the bending moment from Eq. (8) must be equal
to –M0 at the right end of the left-side hinge, i.e. for

z
c

� �
�

�( cos )1 . (15)

Eq. (8) is then rewritten in the form:

� � � �
�

�
�

�

�
�M P

c w
0 1 2�

�( cos ) . (16)

After substituting Eqs. (10) and (14), the above equation is
solved for c. Of two roots, the following one is relevant:

c
K LM LM K

M

K

p p

p

�
� �

�

�

2 2

2 1

2

2
0 0

2

0

� � � � �

�

�

sin ( cos sin )

( cos )
2

02 1

sin

( cos )
.

�

�M �

(17)

If we now substitute Eq. (17) into Eqs. (11) and (14), both P
and u depend only on parameter �. The desired P-u relation-
ship is thus obtained in a parametric form. Note that parame-
ter � has a clear physical meaning: it is the angle of inclination
of the straight portion of the buckled reinforcing bar. Note
also that the derived equations are valid only after the bar
cross sections at x L L� � 2 0 2, , have completely yielded due
to post-buckling bending.

3.7 Extreme strain
A complete rupture of the buckled bar is initiated when

the extreme strain reaches material strain capacity �u. This
first happens at the sections with extreme bending moments
MA or MB, that is, for x L L� � 2 0 2, , . On these sections, the
rupture criterion is first satisfied at the most distant points
with � �� ext . By using Eqs. (6), (14), (10), and (17) the ex-
treme moment is expressed in terms of parameter �. Then we
obtain the extreme strain from Eqs. (3) and (1) as:

� �ext ext
B

p

M M
K

�
� 0 . (18)

4 Validation
In order to validate the proposed analytical model, a typi-

cal problem of a buckling bar is solved and the results are
compared with those obtained with FEM.

The material and geometrical properties of the analyzed
reinforcement bar are listed in Table 1. They correspond to
standard steel no. 10 216, which was in the past often used in
precast R/C panels. For the sake of simplicity of the FE analy-
sis, the cross section has the shape of a square with side d.

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Acta Polytechnica Vol. 44  No.5–6/2004

x

w

z

r
av

r
av

c

c

j
j

j

L
/2

-2
c

1
av

av

c
r

k j
= =

M(z)

M
A
=-M

B

M
B

M
0

M
av

Fig. 4: Geometry and distribution of bending moment on a de-
formed bar

E*) � *) �y Eh �u L d

GPa – MPa MPa – m m

210 0.3 206 1387.5 0.24 0.2 0.01
*) Used only in FE analysis

Table 1: Material and geometrical properties of the analyzed re-
inforcement bar



The FE analysis was preformed assuming plane stress,
finite displacements and finite strains. The material was
modeled as elastic-plastic with linear hardening. The model
represented the entire length L. To facilitate buckling, the bar
was given a piecewise linear lateral imperfection with the
maximum value at the vertical symmetry line equal to 0.25 %
of L. The non-uniform FE mesh consisted of 100�20 9-noded
elements, and it was refined in the locations of the plastic
hinges.

According to Eq. (18), the bar rupture is initiated at
P � 4.34 kN and u � 0.0342 m, which correspond to
� � 0.804962 rad. Fig. 5 compares the post-buckling load-
-displacement curves obtained with the proposed model and
with FEM. A close agreement is evident for the entire relevant
range, i.e. up to u � 0.0342 m. Fig. 6 shows the values of
extreme strain calculated according to Eq. (18) and the corre-
sponding extreme values of logarithmic strains obtained with
FEM. It is obvious that the proposed model captures very well
the extreme compressive strain, while it overestimates the ten-
sile strain. Note that the difference between the strains on the
compressed and tensioned surfaces is a result of considering
the finite displacements and strains in the FE analysis.

5 Concluding remarks
A simplified approach to the analysis of the post-buckling

behavior of a compressed bar has been presented. The simpli-
fication consists namely in considering rigid-plastic material
behavior, neglecting axial contraction of the central line, and
approximating the shape of the deformed central line in plas-
tic hinges by a circular arch. Consequently, axial loading force
P, end displacement u, and extreme strain �u can be expressed
in relatively simple closed forms. This feature is particularly
desirable, since the formulation will be used in a multiscale
context to model the behavior of an R/C structure during de-
molition [2].

The results obtained with the proposed model show very
close agreement with those obtained by a realistic detailed fi-
nite element analysis, which justifies the use of the simplifying
assumptions.

6 Acknowledgment
The research presented in this paper was supported by

grant no. 103/02/0658 provided by the Science Foundation of
the Czech Republic and by Contract of the Ministry of Educa-
tion of the Czech Republic No. J04/98: 210000003.

References
[1] Kabele P., Pokorný T.: “Computational Analysis of Pre-

cast Concrete Building Deconstruction.” In “Proceed-
ings of the Fifth World Congress on Computational
Mechanics (WCCM V), July 7–12, 2002, Vienna, Austria”
(Editors: H. A. Mang, F. G. Rammerstorfer, and
J. Eberhardsteiner), Vienna University of Technology,
Austria, http://wccm.tuwien.ac.at.

[2] Kabele P., Kalousková M.: “Multiscale Stochastic Si-
mulation of Building Demolition.” In Proceedings of
the Sixth World Congress on Computational Mechanics
(WCCM VI), to appear.

[3] Shames I. H., Dym C. L.: “Energy and Finite Element
Methods in Structural Mechanics.” Taylor & Francis
1991.

Doc. Ing. Petr Kabele, Ph.D.
phone: 1420224354485
e-mail: petr.kabele@fsv.cvut.cz

Department of Structural Mechanics

Czech Technical University in Prague
Faculty of Civil Engineering
Thákurova 7
166 29 Prague 6, Czech Republic

168 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No.5–6/2004

0

5

10

15

20

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

P
(k

N
)

u (m)

Proposed model

FEM

Fig. 5: Post-buckling load-displacement curves of a reinforce-
ment bar

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.01 0.02 0.03 0.04 0.05 0.06

S
tr

a
in

(-
)

u (m)

Proposed model

FEM - compression side

FEM - tension side

Fig: 6: Extreme strains vs. end displacement at x � 0 and� � � d 2;
absolute values of logarithmic strains are plotted as FEM
results