Compurers & .Wtct~~es. Vol 13, pp. 223-227. 1981 0045-7949/~71~10223-0%02.

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Printed in Great Britain All rights KCSW& Copyri@t @ 1981 Pcquncm PIES Ltd

NONLINEAR ANALYSIS OF REINFORCED

CONCRETE FRAMES

CHENGTZU THOMASHsti
Department of Civil and Environmental Engineering, New Jersey Jnstitute of Technology, Newark,
NJ 07102, U.S.A.

Department of Civil Engineering and Applied Mechanics, h4cGill University, Montreal, Quebec, Canads
Canada H3A 2K6
and
C.S.SuNNySEA§
Carolina Dewatering Corporation, Irvington, New York, U.S.A.
(Received25 April 1980)

Abstrwt- A computer analysis is described for the elastic-perfectly plastic analysis of reinforced concrete
planar frames. This computer program rquires less computer time and memory space and is intended as
practical analysis and design uses. The computer program is capable of complete analysis of reinforced
concrete frames from zero load until failure under any system of static gravity and lateral loads. The
program can cater for any given geometry and end conditions. The analysis of reinforced concrete frames
uses a computer program as a subroutine to calculate the moment-curvature characteristics under a
constant load applied at the section ccntroid. A reinforced concrete frame tested by Cranston and Cracknell
[l] are analyzed using the computer program developed and the results are compared with the experimental
results. It is noted that the txoposed nonlinear analysis satisfactorily reproduced the behavior of reinforced
concrete plane frames from Zero loads until failure.

INTRODUCTION establishing the redistribution of bending moments in

Many rigorous and wmputationally com’plex methods the frame. The post-elastic behavior of the wncrete
[2-9] for predicting the non-linear behavior of rein- frames for both of these aspects is dependent on the
forced concrete frames have been developed over the shape of the moment-curvature and the moment-
last two decades. Various ones of these methods have rotation curves for the frame members.
been devised to include some of the following aspects :
(i) the nonlinear concrete stress strain relationship ;
(ii) the varying cross section due to nonuniform crack- Load-moment-curvature and inelastic rotation
ing and inelastic behavior ; (iii) the efiect of axial load ; It is well known that the plastic hinges formed in a
(iv) the effects of creep and shrinkage ; (v) the post steel frame have a very large rotation capacity. How-
yielding behavior of sections; (vi) the residual effects ever, the rotation capacity of hinging regions in a
of overloading. Most of the effort in this area has been structural wncrete member is dependent on several
concentrated on developing suitable computer pro- parameters, e.g. material properties, membergeometry,
grams to provide information on strength, ductility applied loads, etc. Recent limit design research in
and elastic and plastic behavior including deflections structural wncrete has led to the definition of the hinge
at specific load levels. These analyses were based on length; the rotation capacity of the hinge can then be
the matrix flexibility, matrix displacement and finite calculated from the wnstitutive relationships for the
element methods and assumed that a structure behaves materials and the section geometry. Hsu and Mirza [7l
linearly under small increments of load or deforma- have developed a computer program to determine the
tion. Many of these computer programs require a great moment curvature characteristics of a reinforced wn-
deal of computer time and memory space and are Crete section subjected to a constant axial load applied
therefore intended as a research tool. at the section centroid. The inelastic rotation capacity
This paper presents a more practical and time- of the hinge can be calculated using the following
saving computer analysis for reinforced wncrete plane equations (Mattock [l l] and Corley [12]) :
frames. A computer program developed by Wang
[ 13-141 for an elastic-perfectly plastic analysis of steel (la)
plane frames was modified to acwunt for the behavior
of reinforced concrete materials and structures. where f+,,=the inelastic rotation of the hinge ; d=the
dfective depth ; 4y = the curvature at yield ; M, =the
PRESENTMETHOD bending moment at yield ;+. = the curvature at ultimate
General load and M, = the ultimate bending moment.
In any inelastic analysis, it is necessary to consider For an under-reinforced section, the moment-
the rotation capacity of the hinging regions before curvature curve can be approximated by an elastic-

223
224 CHENGTZIJTHOMASHsu et al.

perfectly plastic relationship without any serious error. and end conditions (fixed and/or pmned end con-
The plastic rotation capacity is then given by ditions).
The details of computer program can be found in the
Ref. [lo]. The flow diagram is shown in Fig. 1. The
program is coded in FORTRAN IV and can be run
either on the RAX or O/S systems of IBM 370/t% or
The following points must be considered in develop- IBM 360/75.
mg a computer program for an elasti*perfectly plastic
analysis of reinforced concrete frames :
(a) The inelastic rotation capacity of the hinging Example and dismssmn oj’ remits
region in a reinforced concrete section is dependent The specimen analysed using the present computer
on the section geometry, material properties and program is Frame FP4 tested by Cranston and Crack-
applied axial load and can be limited in some cases. nell [l] (Figs. 2 and 3). It must be noted-that the value
The axial loads applied at the members are based on of flexurai rigidity (En used in the main computer
the values obtained from the linear structural analysis program is the slope of the moment-c~ature curve
for plane frames. (under a constant axial force at the section centroid).
(b) The appearance of cracks in a structural concrete Moment-curvature curves for typical sections are
member gives rise to a varying flexural rigidity (EI) shown in Figs. 4 and 5. The location of formation of
along its length. Flexural rigidity (EI) used in the hinges and the final collapse mechanism are shown in
present method is on the basis of the &xural rigidity Fig. 6; these have been obtained from the computer
after concrete cracking. The zero flexural rigidity is analysis results. The. failure mechanism obtained
assumed after yielding of the tension steels. experimentally shows excellent agreement with the
(c) The descending branch of the moment-curvature present analysis results (Fig. 7).
curve at a section (or the moment-rotation curve at a The experimental and the computed loaddeforma-
hinge) provides the section with added ductility tion curves are shown in Fia. 8. The commuted ultimate
although its bending strength decreases. strengths of the frame is l;wer than thi experimental
(d) The strength of a reinforced concrete section in
positive and negative bending is dependent on the
quantity and the arrangement of tension and com-
pression steels. The bending strength of a reinforced
concrete section can therefore, be ~~~n~y differ-
ent under reversal of applied foads. Similarly, the
ductility of a reinforeed concrete section is dependent

r
on the reinforcement details.
(e) The provision of stirrups in a reinforced concrete
beam does not only prevent shear failures, but also
increases the ductility of the concrete. This increased
compressive strain capacity significantIy improves the
rotation capacity of the hinging region although it does
not add to the strength of the section

A computer program was developed by the writers

[lo] for elastic-perfectly plastic analysis of reinforced
concrete plane frames, based on a modification of
Wang’s program [13, 141 which was developed as a
general purpose program (matrix displacement formu-
lation) for limit analysis of steel plane fiames {see
Appendix). Wange 113, 141 used the conventional
mechanism approach to handle the collapse stage in
steel plane frames. The present program was based on
an assumed rotational capacity of the “plastic hinges”
and the formation of a “m~~nism” at the ultimate
load stage
The reinforced concrete frame was divided into
several small elements. For each element, the section
geometry, material properties and inelastic hinge
rotation are known through the input data and sub-
programs. The computer program developed is capable
of complete analysis of reinforced concrete plane
frames from zero load until failure under any system
of loads. The computer output gives the complete
load-deformation behavior, the location and the
sequence of the formation of “plastic hinges” until a
“collapse mechanism” is formed. The program can
cater for any given geometry (structural layout, Fig. 1. Flow diagram: elastic-perfectly plastic analysis of
member length, section geometry) material properties reinforced concrete plane frames.
 Nonlinear analysis of reinforced concrete frames 225

Fig. 2. Dimensions and loading offrame FP4 (after Cranston

I:,, , , , , , J
l
and Cracknell [ 11). z
C”“Wvml
3
I IV’ ,I/l(rl
s s I

Fig. 5. Moment-curvature curves.

Fig. 6. Process of mechanism formed by the present analysis.

200.

,-P*SSOO Ibs

Fig. 4. Moment-curvature curves.

value because the moment-curvature relationship used CONCLUSIONS

in analysis is a conservative elasticplastic idealization A computer-aided limit analysis and design of
of the moment-curvature results obtained from the reinforced concrete frames was developed to predict
present computer program. However, both horizontal the failure mechanism and the load-cldltction curves.
and the vertical deflections at the maximum load agree The computer program basically follows the limit
well with the test results showing that the curvature analysis of steel plane frames, and has been outlined
formulation used is satisfactory. Moreover, there is for incorporating the effects of rotational capacity of
excellent agreement between the experimental and the the plastic hinges by modifying a general purpose
computed A, - As curves. computer program developed by Wang 113,141.
226 CHENG-TZUTHOMAS Hsuet al.

8. B. L. Gunmn. F N. Radand R. W. Furlong, A general

non-linear analysis of concrete structures and com-
parison with frame tests. Comput. Structures 7,257-265
(1977)
9. J. S. Ford. D. C. Chang and J. E. Breen. Experimental
and analytical modeling of unbraced multi-panel con-
crete frames. Part 2. Final Report. RC RC Project # 3 1.
Civtl Engineering Structures Research Laboratory. The
Univ of Texas at Austm. Texas, 39 pp. (1978)
10. C T Hsu. M. S Mirza and A. A. Mufti, An elastic-
plasttc, computer analysis of steel and reinforced con-
crete plane frames. Structural Concrete Series 72-5.
Dept. of Civil Engineering and Applied Mechanics.
McGill Umv.. Montreal, Quebec (19721.
. PlOSllC hlnp*s ‘0m.d I” 1.11 cy Cranlton a cracnnell (I] Il. A. H. Mattock, Rotational capacity of hingmg regions
x PIO~1IC hl”.q., rormrd I” pr**.nt comput.r onal”*m
m reinforced concrete beams. Proc. Int. Symp. FIexural
Fig. 7. Modes of failure for Frame FP4 ,Wechanics and Remforced Concrete, Miami. Florida,
ASCE-AC1 SP-1’. pp. 143-182 (1964)
12. W. G. Corley. Rotatton capacity of reinforced concrete
beams. Proc. .4SCE Structl. Div. 92(ST4). 121-146
1 I 1966).
13 C. K. Wang, General computer program for limit
analysts. J. Struct. Div. ASCE 89(ST6), 101-l 18 (1963).
14. C. K. Wang, Matrix Methods of Structural Analysrs.
2nd Edn. International Textbook, Pennsylvania (1970).
15 R. K. Livesley, Automatic design of structural frames.
Q. J. Mech. Appl. Math. 9(3), 257-278 (1956).
16 A. Jennings and K. Majid, An elastic-plastic analysis
by computer for framed structures loaded up to col-
lapse. The Structural Engr 43(12). 407-412 (1965).

Fig. 8. Load-deflection curves for Frame FP4. Formulations for elastic-perfectly plastic analysis of plane
frames
The program can be used to examine the strength and The stiffness matnx of a member S relates the joint
ductility of reinforced concrete plane frames and the moments F and the corresponding rotations e as follows :
resulting factor of safety against failure.
F&e.
Acknowledgements-The writers are grateful to the Canada Equation (2a) can be written explicitly for a member 0 as
Emergency Measures Organization for the financial assist-
ance to the continuing program of study of the strength and F, =S,,e, +S,,e, 12b)
behavior of three-dimensional reinforced and prestressed
concrete structures. The financial assistance of the National F, =Slie, +Sj,e,. 12:)
Research Council of Canada and New Jersey Institute of It must be noted that eqns (2) account only for flexural
Technology towards the computer costs of this program is deformations in the member 0 while the axial and sheer
gratefully acknowledged. deformations are neglected. This follows an earlier formula-
tion by Wang [13, 141.A more complete formulation for an
REFERENCES elasticplastic material has been developed by Livesley [ 15]
and Jennings and Majid [16].
1. W. B. Cranston and J. A. Cracknell, Tests on reinforced Using the results from basic slope deflection equation,
concrete frames 2: Portal frames with fixed feet. the member stiffness matrix is given by.
Cement and Concrete Association TRA/420, London,
England (1969).
2. J. Ferry-Borges and E. R. Olivetra, Non-linear analysis
of reinforced concrete structures. Pub. LABS,!? 23, 51- where E=the modulus of elastictty: I=the moment of
69 (1963). inertia and L-the length of the member 0.
3. W. B. Cranston, A computer method for inelastic If a simple or a plastic hinge is introduced at the /th end
analysis of plane frames. Cement and Concrete Associa- of the member ii, the stiffness matrix S gets modified as
tion, TRA/386, London, England (1965).
4. J. M. Becker, Inelastic analysis of reinforced concrete (4)
frames. M.Sc. Diss., Cornell University, Ithaca. New
York (1967). Similarly if a simple or plastic hinge IS mtroduced at the
5. R. G. Drysdale, Prediction of the behavior of concrete ith end of the member ij, then S gets modified as
frames. IABSE Symp.. Design of Concrete Structures
for Creep, Shrinkage and Temperature Changes. (5)
Madrid (1970)
6. H. A. Franklin, Non-linear analysts of reinforced For a typtcal member g in bendmg
concrete frames and panels. Ph.D. Diss., Univ. of
California, Berkeley, Calif. (1970). F=Se
7 C. T. Hsu, Behavior of structural concrete subjected to or
biaxial Bexure and axial compression. Ph.D. DIS-
sertation. McGill Univ.. Montreal. Quebec (1974). e=S -I F =DF (61
 Nonlinear analysis of reinforced concrete frames 227

where D is the flexibility matrix for the member @and is where A’ =B and r denotes transport of a matrix and A is
given by the statics matrix relates the externally applied forces P to
internal end moments f: as follows :

P=AF. (11)

The external rotations X,, X,, and the deflection X, of If there are no hinges either at i or at j, then continuity
end j relative to end i are related to the internal rotations requires that the internal end rotations c 8s caused by the
ei and e, by the equation end moments F, be equal to those caused by the external
joint rotations or ~spI8~ments. Therefore it follows from
e=BX (8) eqns (6) and (lo), that

I0 1
where B=the deformation matrix DF=A’X (12)
-l/L Pa) whence
= 0 1 -l/L
[
F=D -‘A’X=SA’X (13)

W
and
which is familiar equation from the displacement method of
rigid fr8me at&y&.
If there is a hinge at any member end, then the hinge
rotation H is given by the angle from the direction of the
member as required by the external joint rotations or dis-
placements to that caused by the end moments. Therefore
Equation (8) can be rewritten as
H=DF-A’X. (14)
c=A’X (10)