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The effect of finite element mesh size in nonlinear analysis of reinforced concrete structures
Computers
Printedin
& .Wuctures Great Britain.
Vol.36.
No.
5. pp. 807-815,
0045-7949/w s3.00 + 0.00 Pergamon Press plc
1990
THE EFFECT OF FINITE ELEMENT MESH SIZE IN NONLINEAR ANALYSIS OF REINFORCED CONCRETE STRUCTURES C.-K. CHOI and H.-G. KWAK Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Seoul 130-650, Korea (Received
1 August 1989)
Abstract-A
new approach is developed to the nonlinear analysis of reinforced concrete (R/C) structures subjected to a monotonically increasing load, i.e. from zero up to the ultimate load. Tensile cracking and the nonlinear stress-strain relationship for concrete and reinforcement are taken into account in the analysis. Concrete is assumed to be elastic in the tension region and elasto-hardening plastic in the compression region. The reinforcing bar is considered as a linear strain hardening material. The tension stiffening effect of concrete between cracks is also considered. The effect of finite element mesh sizes on the analysis of the behavior of the structure, which is the most significant among the effects such as load step, and integration orders, etc. is investigated and a new criterion to reduce the numerical error associated with the mesh sizes is developed. This newly developed criterion is based on the fracture energy concept and can be easily implemented into a numerical analysis procedure. In particular, this approach can be used effectively with relatively large finite element mesh sizes. The proposed criterion is tested by comparing the analytical results from this study with those of experimental studies and other previous numerical studies.
INTRODUCTION
Reinforced concrete (R/C) structures are commonly designed to satisfy two criteria in terms of the serviceability and the safety. In order to ensure the serviceability requirement, it is necessary to predict accurately the cracking and deflections of R/C structures at working loads. To assess the safety of structures against failure, an accurate estimation of the ultimate load is essential. Although experimental studies can provide good information about members tested, they are timeconsuming and costly, if not impossible. Therefore, it is desirable to develop a more reliable numerical analysis model which can substitute for experiment. The development of an analytical method for R/C structures, however, is complicated because of the following features: (1) they are composites of two different materials, i.e. concrete and steel; (2) the nonlinear behavior of concrete due to the nonlinear constitutive relations, tension cracking, biaxial stiffening, and strain softening; (3) bond slip and aggregate interlocking between concrete and reinforcing bars, etc. Two basically different approaches have been used to analyze R/C slabs and beams by the finite element method: (1) the modified stiffness approach; and (2) the layer approach. The former is based on an overall moment-curvature relationship reflecting the CM
36,s-z
various stages of material behavior. Jofriet and McNeice [I] used a bilinear moment-curvature relationship based on an empirically determined effective moment of inertia of the cracked slab section in order to allow for the beneficial effect of tension stiffening. Some other researchers [2,3] used a momentcurvature relationship based on the theoretically determined yield line to consider the nonlinear variation of material properties through the depth and proved that the yield line theory is an excellent tool not only for predicting the ultimate strength but also for formulating the full load-deflection behavior of R/C structures. The layer approach is based on the idealized stress-strain relationship for concrete and reinforcement in which a finite element is divided into imaginary concrete and steel layers. This approach has been broadly used by many previous investigators [4-81. Lin and Scordelis [S] utilized layered triangular finite elements in R/C shell analysis taking the coupling of the membrane and bending actions into account. Gilbert and Warner [7] analyzed the behavior of R/C slabs based on the variation of the slope of the descending branch of the stress-strain relationship of concrete. They were among the first to point out that the numerical analysis results of concrete structures are greatly dependent on the finite element mesh sizes used in modeling and are so affected by the tension stiffening effect in concrete. In order to account for the tension stiffening effect, and to improve the realism of the post-cracking 807
C.-K. CHOIand H.-G. KWAK
808
representations of the behavior of R/C, some investigators have artificially increased the stiffness of steel by modifying its stress-strain relationship [7], or have modified the stress-strain curve of concrete in tension by considering the descending branch [4,5,7]. The overall nonlinear response of R/C slabs and beams subjected to bending is significantly affected by the tension stiffening [5,7,9] and the analysis results are also dependent on the finite element mesh sizes [9, lo] when the analysis is carried out by using the smeared crack model. The determination of the descending branch, however, relies more on the type of problems under consideration and the experience of the analyzer rather than on the rational approach. As yet, many of the previously proposed analysis schemes have not been fully verified. In this paper, a new energy criterion which automatically adjusts the slope of the descending branch of the stress-strain curve of concrete in accordance with the finite element mesh sizes used in modeling is proposed. By assuming a new microcrack distribution in an element and adjusting the slope of the descending branch of concrete as the mesh sizes change, the numerical error associated with mesh sizes can be reduced. Moreover, this proposed model can be used for relatively large finite element mesh sizes with reasonable accuracy.
u
1
-I% I
Fig. 2. Idealized uniaxial stress-strain relationship for steel.
referenced to the local axes oriented in the reinforcing bar direction and normal to it, can be written as
in which ES, is the first modulus of steel. When steel has been yielded, E,, is replaced by the second modulus EJ2. The yield criterion used in this study is the von Mises formula. Concrete
As shown
MATERIAL PROPERTY ASSUMPTIONS In order to formulate the constitutive
relationships in a layer of R/C element, the following simplified assumptions have been made:
(1) the element is divided into imaginary concrete
e
I
in Fig. 3, up to the peak
stress
feq,concrete is assumed to be linear elastic in compression-tension and to remain in the biaxial tension region. Then, the stress drops linearly with increasing equivalent uniaxial strain. In the biaxial compression region, the elasto-hardening plastic model is usually adopted. One such yield criterion can be expressed by [l l]
and steel layers (see Fig. 1);
(2) the Mindlin hypothesis of plate bending and the
F = (0, + a2)*/(02 + 3.65a,) - Af = 0,
Timoshenko beam theory are adopted for the slabs and beams, respectively; (3) the reinforcing steel is assumed to carry the uniaxial stress only; (4) a perfect bond between steel and concrete is assumed.
in which 6, and a2 are the principal stresses; f;, is the uniaxial compressive strength; and A is the parameter which represents the plastic flow from the initial yield surface (A = 0.6) to the ultimate loading surface (A = 1.0). In addition, the associated flow rule is
Steel The reinforcing steel is assumed to be a linear strain hardening material whose yield stress is cY (Fig. 2). The stress-strain relationships which are
Fig. 1. Layer system.
Fig. 3. Biaxial strength envelops of concrete.
(2)
Nonlinear analysis of reinforced concrete structures
809
make the determination of an effective shear modulus more complicated. Therefore. it is assumed that the cracked shear modulus is a constant of fixed value after cracking, i.e. 1 = 0.4 in eqn (4). Then, the cracked stiffness is assumed as follows: CRACKED
0 0
0 CRUSHED
Fig. 4. Failure surface of concrete. assumed to govern the post-yielding stress-strain relationships for concrete [5]. When the biaxial stresses exceed the Kupfer failure envelope, concrete is assumed to exist in the strain softening region in which an elastic orthotropic model is used to express the stress-strain relationships [12]. In order to check the failure state of concrete, i.e. the crushing and the cracking, it is necessary to adopt a failure surface expressed in terms of the strain components. In this study, a failure surface (see Fig. 4) is established analogously to the yield surface of eqn (2) in terms of strains and is expressed by
where 6, Fig. 5) is Finally, have lost
(3)
and c2 are principal strains, and 6,” (see the ultimate strain of concrete. when concrete is crushed, it is assumed to all its stiffness.
Cracked st@ness of concrete
When any of the principal tensile strains exceed Q (Fig. 5), cracks will develop in a direction perpendicular to that of principal stress. The stress normal to the crack may be zero and the shear modulus must be reduced by cracking. The effect of dowel action and the aggregate interlocking, however, tend to t7
Fig. 5. Idealized uniaxial stress-strain concrete.
relationship
El 62 9 (4)
0 I(l-v)G 2
Y12
I[}
in which 1 and 2 are the parallel and the perpendicular direction to the cracks, respectively; G is the shear modulus for untracked concrete; and 1 is a cracked shear constant.
I
C = (L, +E~)~/(Q + 3.656,) - ecu= 0,
0
for
TENSION STIFFENING EFFECT
Cracking in concrete will develop and propagate in the direction normal to that of the major principal stress starting from the section where a crack is first originated. Even after cracking, however, concrete is still partially capable of resisting tensile forces due to the bond between concrete and reinforcement. This effect can be adequately accounted for by increasing the average stiffness of the element which has relatively large dimensions when compared with the size of the cracked section. Increase of the tensile stiffness of concrete can be accomplished by using a stressstrain relationship which includes a descending branch in the tension region [4, $71. On the other hand, in order to predict more accurate deflections of the structure, it is necessary to use a cracking model in the nonlinear analysis since the cracks in concrete are the major source of material nonlinearity. As shown in Fig. 6, however, the fracture behavior of concrete is quite different from that of metals, mainly due to the fact that the fracture of concrete is preceded by microcracking instead of yielding. Thus, the stress controlled smeared cracking model first used by Rashid [ 131has found wide use in the numerical analysis of R/C structures. This model represents cracked concrete as an elastic orthotropic material with reduced moduli normal to the crack plane. Although this approach is simple to apply and widely accepted, it nevertheless has a drawback, i.e. the dependency of finite element mesh sizes (lo]. A lot of research efforts have been devoted to the
(a) Concrete
(b) Steel Fig. 6. Relative sizes of fracture zone and yielding zone.
C.-K. CHOIand H.-G. KWAK
810
structures. The procedures for derivation of the model are described in the following sections. Microcrack distributions
In the first place, a formula for the distribution of microcracks in an element is developed with the general exponential function (see Fig. 7) which can well express the characteristic of assumed microcracks concentrated at the crack tip when the gage length (i.e. finite element mesh size) is fairly large,
Lb*& Fig. 7. Assumed distribution of microcracks
in an element.
removal of this drawback, based on the fracture mechanics concept in particular. Bazant and Oh [14] introduced the crack band theory in the analysis of a plain concrete panel, which may be one of the simplest types of fictitious crack models. The two basic assumptions in the model are that the width of the fracture zone has a certain value w, which is proportional to three times the maximum aggregate size (approximately 1 in.), and that the strain is uniform within the band. Thus, the final equation derived for determining E,,can be expressed as follows:
This model can be successfully applied to the R/C problems when the finite element mesh size used is relatively small. Their analytical results, however, are significantly different from the experimental data as the mesh size is increased since this model assumes a uniform distribution of microcracks in the entire region of a relatively large finite element (see Fig. 8a), while the actual microcracks are concentrated in the relatively small cracked region in the element (see Fig. 8~). Therefore, it is not proper to apply eqn (5) directly to the numerical analysis of R/C structures modeled with relatively large finite element mesh sizes. A PROPOSED MODEL In this study,
the development
which can be applied mesh
size is
of a new criterion
to a fairly large finite element
used in the nonlinear
analysis of R/C
f(x) = a eBr,
in which a and /I are constants to be determined. Substituting the boundary conditions, i.e. f(0) = 1.0 and f(b/2) = 3/b, into eqn (6), the following equation can be obtained:
f(x)
=
e-2/b
I”$,
0
(7)
in which b is the element width. The functionf(x) in eqn (7) portrayed in Figs 7 and 8 expresses: (1) that the distribution of microcracks has a symmetric characteristic as shown in Fig. 7; and (2) that at the end of the finite element mesh, the typical size of microcracks is 3/b. The second feature indicates that the uniform distribution of microcracks is assumed when the element width is less than 3 in., that is, three times the maximum aggregate size [14]. Fracture energy
The fracture energy is expressed by the stressstrain relationships of concrete as
j”(x) dx .2, wheref, is the tensile strength of the concrete, cois the strain at the end of strain softening (Fig. 5), at which the microcracks coalesce into a continuous crack, and G,is the fracture energy consumed in formulating and opening a crack of unit length per unit thickness. If
J
0
(6)
0
s
Fig. 8. Distribution function of microcracks according to the finite element mesh size. (a) Relatively small mesh. (b) Medium mesh. (c) Relatively large mesh.
811
Nonlinear analysis of reinforced concrete structures G, andJ; are known from measu~ments, be calculated as follows:
then 6 may
Gr
4”
Based on the Mindlin hypothesis, the ~spIacement field can be expressed in the following matrix form:
(9)
By using eqn (9), co can be recalculated whenever the finite element mesh size is changed. The proposed criterion in eqn (9) becomes identical with eqn (5) when f(x) = 1.O, i.e. the element mesh size is less or equal to 3 in. When the finite element mesh size is greater than 3 in., as is the case in most practical situations, it is assumed that the microcracks are distributed in an element in accordance with the function f(x) in eqn (7). In this case, the calculated value of =cObecomes different from that calculated by eqn (5) and will better approximate the reality of the microcracks in R/C structures. Thus, eqn (9) can be used in the nonlinear numerical analysis of R/C structures with a wide range of finite element mesh sizes.
in which n is the number of nodes and N, is the shape function. The relationship between strains and displacements can be written as
(z-4.3
0
(z - 0 4,)
00
0
0 (2 - 0dm)1
dN
0
-ax
0
0
2
0
ax
--dN ay
or FINITE ELEM~~
ID~LIZA~ON
A typical finite element is divided into the imaginary concrete and steel layers as shown in Fig. 9. The displacement field in the element is assumed to be continuous and there are no gaps between layers. Each layer may have different material properties, but these properties are assumed to be constant through the thickness of each layer. Therefore, the volume integration involving the different material properties can be expressed as
(13) in which d,, d,,,, and d,,. represent the arbitrary reference depths (see Fig. 9) and are introduced to prevent the undesired development of the in-plane forces. They can be calculated by using the condition of j B, dz = j cy dz = l T.~,, dz = 0, where z is the depth from the neutral axis (see Fig. 9). Moreover, using the simplified assumption that the shear modulus G is constant over the depth, dnry can be assumed to be d/2, in which d is the thickness of the element. Also, the relationship between the transverse shear strains and displacement components can be expressed as
aiv
ax where [D,],, [D,],, n, and n, mean the material matrix of the ith steel layer, the material matrix of the jth concrete layer, the number of steel layers and the number of concrete layers, respectively.
Zj ~ rf-----t-
NEUNRAL Axts(dn)
L
EFFEXTWE STEEL LAYER CONCRETE LAYER Fig.
I
aN -ay
9. A layered section.
dN
dv
-N
0
0
-N
or
Substituting eqns (13) and (15) into eqn (10) and rearranging the material matrix, the element stiffness matrix can be formulated.
812
C-K.
CHol and H.-G. KWAK
in which [D,,] and [D,] are the bending and shear parts of the material matrix, respectively, Once the nodal displacement parameters are calculated, the strain components of each layer can be calculated by using eqns (13) and (IS),
where
+
$ $R(z,+i - 4, I2- (2,-
d”, Yb,
,=I
(17)
a,
1% I, = (8,)=
fx:=J=l c (z,,
1
I -
Z,)G,
n, fy;= c (z,+, - z,)r,., 1=1
After the equivalent nodal forces are known, unbalanced nodal forces can be computed by and the stress components are calculated based on the stress-strain relationship and used to check the failure state of the material.
(23)
the
These unbalanced nodal forces are applied to obtain the increment in solution and recalculated iteratively to satisfy the convergence tolerance. NUMERICAL SOLUTION PROCEDURE
where {rrsj, and {cs>, denote the stress and strain at the mid-depth of the ith steel layer, and (cr,1, and (cc), denote those of the jth concrete layer. Since the calculated stresses do not generally coincide with the true stresses in a nonlinear problem, the unbalanced nodal forces and the equivalent nodal forces must be calculated. The equivalent nodal forces are statically equivalent to the stress field obtained at each iteration and can be computed as follows:
+ r =
s
W,IT{~JdV
voi
r
[B,,]’
J
The finite element used in this study is a quadratic plate element developed by Choi and Kim [15]. This element is established by the combined use of reduced integration and addition of nonconforming displacement modes and gives very good results in linear elastic analysis. The application of this element has been extended to nonlinear analysis in this study. To obtain the load-deflection relationship for the structures, loads are monotonically increased until the structure fails. Load increments are gradually reduced in magnitude after the first detection of the cracking in concrete. The iteration method used is the combined constant-variable stiffness method, i.e. the stiffness matrix is only updated at every loading step. The essential steps in the solution procedure for a typical load increment are given in the flow diagram in Fig. 10.
[d,,lr{cr,)
J
dz - dA
NUMERICAL EXAMPLES
Example I: reinforced concrete slab
A corner-supported two-way slab was tested to investigate the validity of proposed model for the tension stiffening effect. This slab was 36in. square and 1.75 in. thick with an isotropic mesh of
Nonlinear analysis of reinforced concrete structures
813
1
r
and analyzr the Applya load Incmment trwture to obtoln tha nodd dlaplacmm
I Add tha lncrunantd dbpkmrnntm to the total dlaplocmant~
I Cdculate the rtrsw and etraln In the lag u&g tha dd rnotarld propm-tie~
1.2
.-
I
Lh and .Saxddh[S
--.-
Chuk the calculated stren and mtrdn agalnmt the glwn yield crltwla
hahur
wid Dandn[3]
,6
I
0
among the applied total nodal forcea
4
6
12
16
24
20
32
28
DEFLECTION AT NODE 2 IN INCHES Fig. 12. Load-deflection
I”“,
//
36 x10*
relationship at node 2 in Fig. 11.
3.0
3
*.
2 E
2.4
z a
1.1
% o -1
1.2
.e
Fig. IO. Flow diagram for solution procedure.
0 0
4
8
12
DEFLECllON
4
16
AT NODE
20
24
za
2 IN INCHES
32
B
X10*
Solutim A : Smin Softening NOT Considacd(9 clemnts)
k
Solution B
: Both Strain Softening and the Element Size Effecrt Considered (9 elements)
-ic
Solution
P
Solution D : Bxp&mt[l]
I
-----
2
36 in
3in
Fig. I 1. Corner-supported
C : Both Strain Softening and the Element Size Effect? Considctul (4 elements)
two-way slab [I].
0.85% reinforcing steel and subjected to a concentrated load at the center (see Fig. 11). The assumed material properties were as follows: the Poisson ratio v = 0.167, the tensile strength f, = 5,& and the fracture energy G,= 0.5 lb/in.; the number of Table 1. Material properties
Soludon B : Elcmnt Sii
Effect in Case C NOT Cmsidered
Fig. 13. Load-deflection relationships at node 2 with different finite element mesh size.
concrete layers n, = 8. Other basic material properties are summarized in Table 1. This slab has also been analyzed previously by a number of investigators [ 1,3,6,7] to check their studies. Figure 11 shows the finite element grid used and Figs 12 and 13 show the profile of load-deflection relationships at node 2 as the applied load increases. The deflection at node 2 is obtained approximately by interpolating the deflections at adjacent nodes. As shown in Fig. 12, the numerical results obtained by using the fracture energy criterion proposed in this study are in good agreement with the experimental used
in numerical examples
E, (ksi)
E, (ksi)
f; (ksi)
f,
(ksi)
P (%)
d (in.)
Slab
Jofriet and McNeice [l]
415
29.0
5.50
40.0
0.85
1.31
Beam
TIMA T3MA J-4
3.86 3.94 3.80
28.2 28.2 29.5
4.60 4.80 4.82
46.0 41.0 44.9
0.62 3.22 0.99
10.72 10.35 18.00
814
C.-K.
CHOI
and H.-G.
curve. However, the deflection obtained by neglecting tension stiffening in the previous studies is significantly underestimated [63. In the analysis of the same slab, Jofriet and McNeice [l] used the modified stiffness approach and treated both steel and concrete as elastic materials; Lin and Scordelis included the elastoplastic behavior for steel and concrete, and accounted for the coupling effect between membrane and bending actions; Bashur and Darwin [3] used the analytical moment-curvature relationship based on the yield line theory; Hand et al. [6] considered both the shear effect and in-plane deformation. Results from these previous studies are plotted in Fig. 12 for comparison with the results from the present study. Also, in order to test the effects of finite element mesh sizes on the load-deflection relationships, the problem was solved by two different meshes, i.e. nine-element and four-element meshes; the results are shown in Fig. 13. This figure shows that the use of the proposed criterion [eqn (9)] which considers the tension stiffening effect virtually eliminated entire numerical errors which appear when the relatively large finite element mesh size is used (solutions B and C). The two curves in Fig. 13, i.e. solutions E and D, show that the numerical results are dependent on the finite element mesh sizes used in the analysis and that the load-deflection curve obtained by considering the tension stiffening effect in plate bending problems is much closer to the experimental curve (solution D) than to the conventional solution (solution E). Example 2: reinforced concrete beams
Three simply supported R/C beams were investigated in this example, TlMA and T3MA, which were tested by Gaston et al. [16], and J-4, which was tested by Burns and Siess [17]. TlMA and J-4 are under-reinforced, while T3MA has a balanced reinforcement ratio. The geometries of the beams tested are shown in Figs 14 and 15 and the material properties are summarized in Table 1.
KWAK
Fwbonmd
schnobrlch[4]
Epahmtd[l7] I'MStudy
0'
0
I
I
I
I
.4
CENTER
.8
I
I
_
1.2
DEFLECTION
Fig. 15. Load-deflection
relationships of J-4.
The properties not mentioned in Table 1 were assumed to be the same as those used in example I in this study. The number of finite elements used was nine, the number of concrete layers in an element was eight and the Timoshenko beam theory was used. The load-deflection curves obtained are shown in Figs 14 and 15 along with the previous results by Bashur and Darwin [3] and Fariborz and Schnobrich [4]. Those authors took the tension stiffening effect into account by determining the slope of the strain softening region according to the Hillerborg model [ 181. These figures also show that the numerical results obtained in this study are in good agreement with the experimental curves [16, 171 and with those of previous researchers [3,4]. In Table 2, it is shown that the differences between the numerical results are hardly noticeable. It may suggest that if the correct t,, is obtained by the proposed criterion in this study [eqn (9)], the numerical results are not affected by the variation of the tensile strength of concrete.
CONCLUSIONS
0
.5
CENTER
1
1.5
2
Boshur and Omnln[3]
-----
Expwtmmtol[l6]
-
lhlsStudy
25
DEFLECTlON(in)
Fig. 14. Load+Ieflection relationships of TIMA and T3MA.
The behavior of R/C structures in the post-cracking stage is significantly influenced by the tension stiffening effect between adjacent cracks in the concrete. If this effect is ignored, the calculated deflections will be much larger than the experimental data. Therefore, this should not be neglected. In this paper, a new criterion to determine co (the strain at the end of strain softening) is proposed [see eqn (9)] which reduces the numerical errors associated with the tension stiffening effect. Adjusting the slope of the strain softening region in accordance with the proposed criterion, the calculated deflections closely agreed with the experimental data and the
Nonlinear analysis of reinforced concrete structures
815
Table 2. The load-deflection relationship of J-4 with the variation of tensile strength of concrete OSP (kips)
Experiment (in.)
a = 5.5
a = 6.3
a = 6.7
a = 7.5
a = 9.0
2.0 4.0 6.0 8.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0
0.0088 0.0176 0.0525 0.0949 0.1390 0.1630 0.1780 0.2000 0.2200 0.2410 0.2590 0.2760 0.5410 1.0860
0.0112 0.0227 0.0384 0.0719 0.1118 0.1294 0.1476 0.1642 0.1819 0.1976 0.2132 0.2289 0.7024 -
0.0112 0.0227 0.0361 0.0709 0.1112 0.1296 0.1473 0.1636 0.1822 0.1976 0.2128 0.2287 0.7010
0.0112 0.0227 0.0355 0.0723 0.1106 0.1301 0.1474 0.1632 0.1822 0.1976 0.2126 0.2280 0.7010 -
0.0112 0.0227 0.0346 0.0669 0.1104 0.1301 0.1470 0.1623 0.1824 0.1974 0.2124 0.2274 0.7031 -
0.0112 0.0227 0.0341 0.0636 0.1101 0.1241 0.1471 0.1614 0.1823 0.1981 0.2125 0.2273 0.7098 -
Note: f, = aJf<, the number of finite elements used is nine and the values given are the deflection (in.) at the center. numerical error caused by the larger mesh sizes was hardly noticeable, thus making it possible to model
the structure with relatively large finite element mesh sizes. REFERENCES
I. J. C. Jofriet and G. M. McNeice, Finite element analysis of RC slabs. J. struct. Div., AXE 97, 785-806 (1971).
2. A. Vebo and A. Ghah, Moment-curvature relation of reinforced concrete slabs. J. struct. Div., AXE 103, 515-531 (1977). 3. F. K. Bashur and D. Darwin, Nonlinear model for reinforced concrete slabs. J. struct. Div.. ASCE 104. 157-170 (1978). 4. B. J. Fariborz and W. C. Schnobrich, Nonlinear finite element analysis of reinforced concrete under short term monotonic loading. Structural Research Series No. 530, University of Illinois at Urbana, IL (1986). 5. C. S. Lin and A. C. Scordelis, Nonlinear analysis of RC shells of general form. J. struct. Div., ASCE 101, 523-538 (I 975). 6. F. R. Hand, D. A. Pecknold and W. C. Schnobrich,
Nonlinear layered analysis of RC plates and shells. J. struct. Div., ASCE 99, 1491-1505 (1973). 7. R. I. Gilbert and R. F. Warner, Tension stiffening in reinforced concrete slabs. J. struct. Div., ASCE 104, 1885-1900 (1978). 8. T. E. Wamba, Nonlinear analysis of reinforced concrete beams by finite element method. Ph.D. dissertation, Department of Civil Engineering, State University of New York at Buffalo, NY (1980).
9. L. D. Leibengood, D. Darwin and R. H. Dodds, Parameters affecting FE analysis of concrete structures. J. sfrucf. Engng Div., AXE 112, 326-341 (1986). 10. Z. P. Bazant and L. Cedolin. Fracture mechanics of reinforced concrete. J. Engng‘Med.Div., ASCE 106, 1287-1306 (1980). 11. H. Kufer, H. K. Hilsolort and H. Rusch, Behavior of concrete under biaxial stresses. J. Am. Concr. hr. 66, 656-666 (1969). 12. D. Darwin and D. A. W. Pecknold, Analysis of cyclic loading of plane R/C structures. Comput. Struct. 7, 137-147 (1977). 13. Y. R. Rashid, Analysis of prestressed concrete pressure vessels. Nucl. Engng Des. 7, 334-344 (1968). 14. Z. P. Bazant and B. H. Oh. Crack band theorv for fracture of concrete. Mate;. Struct. 16, 155-17; (1983). 15. C. K. Choi and S. H. Kim, Reduced
integration, nonconforming modes and their coupling in thin plate elements. Comput. Struct. 29, 52-62 (1988). 16. J. R. Gaston, C. P. Siess and N. M. Newmark, A layered finite element non-linear analysis of reinforced concrete plates and shells. Civil Engineering Studies, SRS No. 389, University of Illinois at Urbana, IL (1972). 17. N. H. Burns and C. P. Siess, Load-deformation characteristics of beamcolumn connections in reinforced concrete. Civil Engineering Studies, SRS No. 234, University of Illinois, Urbana, IL (1962). 18. A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of crack formation and growth in concrete by means of fracture mechanics and finite element. Cement Concr. Res. 6, 773-782 (1976).
Printedin
& .Wuctures Great Britain.
Vol.36.
No.
5. pp. 807-815,
0045-7949/w s3.00 + 0.00 Pergamon Press plc
1990
THE EFFECT OF FINITE ELEMENT MESH SIZE IN NONLINEAR ANALYSIS OF REINFORCED CONCRETE STRUCTURES C.-K. CHOI and H.-G. KWAK Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Seoul 130-650, Korea (Received
1 August 1989)
Abstract-A
new approach is developed to the nonlinear analysis of reinforced concrete (R/C) structures subjected to a monotonically increasing load, i.e. from zero up to the ultimate load. Tensile cracking and the nonlinear stress-strain relationship for concrete and reinforcement are taken into account in the analysis. Concrete is assumed to be elastic in the tension region and elasto-hardening plastic in the compression region. The reinforcing bar is considered as a linear strain hardening material. The tension stiffening effect of concrete between cracks is also considered. The effect of finite element mesh sizes on the analysis of the behavior of the structure, which is the most significant among the effects such as load step, and integration orders, etc. is investigated and a new criterion to reduce the numerical error associated with the mesh sizes is developed. This newly developed criterion is based on the fracture energy concept and can be easily implemented into a numerical analysis procedure. In particular, this approach can be used effectively with relatively large finite element mesh sizes. The proposed criterion is tested by comparing the analytical results from this study with those of experimental studies and other previous numerical studies.
INTRODUCTION
Reinforced concrete (R/C) structures are commonly designed to satisfy two criteria in terms of the serviceability and the safety. In order to ensure the serviceability requirement, it is necessary to predict accurately the cracking and deflections of R/C structures at working loads. To assess the safety of structures against failure, an accurate estimation of the ultimate load is essential. Although experimental studies can provide good information about members tested, they are timeconsuming and costly, if not impossible. Therefore, it is desirable to develop a more reliable numerical analysis model which can substitute for experiment. The development of an analytical method for R/C structures, however, is complicated because of the following features: (1) they are composites of two different materials, i.e. concrete and steel; (2) the nonlinear behavior of concrete due to the nonlinear constitutive relations, tension cracking, biaxial stiffening, and strain softening; (3) bond slip and aggregate interlocking between concrete and reinforcing bars, etc. Two basically different approaches have been used to analyze R/C slabs and beams by the finite element method: (1) the modified stiffness approach; and (2) the layer approach. The former is based on an overall moment-curvature relationship reflecting the CM
36,s-z
various stages of material behavior. Jofriet and McNeice [I] used a bilinear moment-curvature relationship based on an empirically determined effective moment of inertia of the cracked slab section in order to allow for the beneficial effect of tension stiffening. Some other researchers [2,3] used a momentcurvature relationship based on the theoretically determined yield line to consider the nonlinear variation of material properties through the depth and proved that the yield line theory is an excellent tool not only for predicting the ultimate strength but also for formulating the full load-deflection behavior of R/C structures. The layer approach is based on the idealized stress-strain relationship for concrete and reinforcement in which a finite element is divided into imaginary concrete and steel layers. This approach has been broadly used by many previous investigators [4-81. Lin and Scordelis [S] utilized layered triangular finite elements in R/C shell analysis taking the coupling of the membrane and bending actions into account. Gilbert and Warner [7] analyzed the behavior of R/C slabs based on the variation of the slope of the descending branch of the stress-strain relationship of concrete. They were among the first to point out that the numerical analysis results of concrete structures are greatly dependent on the finite element mesh sizes used in modeling and are so affected by the tension stiffening effect in concrete. In order to account for the tension stiffening effect, and to improve the realism of the post-cracking 807
C.-K. CHOIand H.-G. KWAK
808
representations of the behavior of R/C, some investigators have artificially increased the stiffness of steel by modifying its stress-strain relationship [7], or have modified the stress-strain curve of concrete in tension by considering the descending branch [4,5,7]. The overall nonlinear response of R/C slabs and beams subjected to bending is significantly affected by the tension stiffening [5,7,9] and the analysis results are also dependent on the finite element mesh sizes [9, lo] when the analysis is carried out by using the smeared crack model. The determination of the descending branch, however, relies more on the type of problems under consideration and the experience of the analyzer rather than on the rational approach. As yet, many of the previously proposed analysis schemes have not been fully verified. In this paper, a new energy criterion which automatically adjusts the slope of the descending branch of the stress-strain curve of concrete in accordance with the finite element mesh sizes used in modeling is proposed. By assuming a new microcrack distribution in an element and adjusting the slope of the descending branch of concrete as the mesh sizes change, the numerical error associated with mesh sizes can be reduced. Moreover, this proposed model can be used for relatively large finite element mesh sizes with reasonable accuracy.
u
1
-I% I
Fig. 2. Idealized uniaxial stress-strain relationship for steel.
referenced to the local axes oriented in the reinforcing bar direction and normal to it, can be written as
in which ES, is the first modulus of steel. When steel has been yielded, E,, is replaced by the second modulus EJ2. The yield criterion used in this study is the von Mises formula. Concrete
As shown
MATERIAL PROPERTY ASSUMPTIONS In order to formulate the constitutive
relationships in a layer of R/C element, the following simplified assumptions have been made:
(1) the element is divided into imaginary concrete
e
I
in Fig. 3, up to the peak
stress
feq,concrete is assumed to be linear elastic in compression-tension and to remain in the biaxial tension region. Then, the stress drops linearly with increasing equivalent uniaxial strain. In the biaxial compression region, the elasto-hardening plastic model is usually adopted. One such yield criterion can be expressed by [l l]
and steel layers (see Fig. 1);
(2) the Mindlin hypothesis of plate bending and the
F = (0, + a2)*/(02 + 3.65a,) - Af = 0,
Timoshenko beam theory are adopted for the slabs and beams, respectively; (3) the reinforcing steel is assumed to carry the uniaxial stress only; (4) a perfect bond between steel and concrete is assumed.
in which 6, and a2 are the principal stresses; f;, is the uniaxial compressive strength; and A is the parameter which represents the plastic flow from the initial yield surface (A = 0.6) to the ultimate loading surface (A = 1.0). In addition, the associated flow rule is
Steel The reinforcing steel is assumed to be a linear strain hardening material whose yield stress is cY (Fig. 2). The stress-strain relationships which are
Fig. 1. Layer system.
Fig. 3. Biaxial strength envelops of concrete.
(2)
Nonlinear analysis of reinforced concrete structures
809
make the determination of an effective shear modulus more complicated. Therefore. it is assumed that the cracked shear modulus is a constant of fixed value after cracking, i.e. 1 = 0.4 in eqn (4). Then, the cracked stiffness is assumed as follows: CRACKED
0 0
0 CRUSHED
Fig. 4. Failure surface of concrete. assumed to govern the post-yielding stress-strain relationships for concrete [5]. When the biaxial stresses exceed the Kupfer failure envelope, concrete is assumed to exist in the strain softening region in which an elastic orthotropic model is used to express the stress-strain relationships [12]. In order to check the failure state of concrete, i.e. the crushing and the cracking, it is necessary to adopt a failure surface expressed in terms of the strain components. In this study, a failure surface (see Fig. 4) is established analogously to the yield surface of eqn (2) in terms of strains and is expressed by
where 6, Fig. 5) is Finally, have lost
(3)
and c2 are principal strains, and 6,” (see the ultimate strain of concrete. when concrete is crushed, it is assumed to all its stiffness.
Cracked st@ness of concrete
When any of the principal tensile strains exceed Q (Fig. 5), cracks will develop in a direction perpendicular to that of principal stress. The stress normal to the crack may be zero and the shear modulus must be reduced by cracking. The effect of dowel action and the aggregate interlocking, however, tend to t7
Fig. 5. Idealized uniaxial stress-strain concrete.
relationship
El 62 9 (4)
0 I(l-v)G 2
Y12
I[}
in which 1 and 2 are the parallel and the perpendicular direction to the cracks, respectively; G is the shear modulus for untracked concrete; and 1 is a cracked shear constant.
I
C = (L, +E~)~/(Q + 3.656,) - ecu= 0,
0
for
TENSION STIFFENING EFFECT
Cracking in concrete will develop and propagate in the direction normal to that of the major principal stress starting from the section where a crack is first originated. Even after cracking, however, concrete is still partially capable of resisting tensile forces due to the bond between concrete and reinforcement. This effect can be adequately accounted for by increasing the average stiffness of the element which has relatively large dimensions when compared with the size of the cracked section. Increase of the tensile stiffness of concrete can be accomplished by using a stressstrain relationship which includes a descending branch in the tension region [4, $71. On the other hand, in order to predict more accurate deflections of the structure, it is necessary to use a cracking model in the nonlinear analysis since the cracks in concrete are the major source of material nonlinearity. As shown in Fig. 6, however, the fracture behavior of concrete is quite different from that of metals, mainly due to the fact that the fracture of concrete is preceded by microcracking instead of yielding. Thus, the stress controlled smeared cracking model first used by Rashid [ 131has found wide use in the numerical analysis of R/C structures. This model represents cracked concrete as an elastic orthotropic material with reduced moduli normal to the crack plane. Although this approach is simple to apply and widely accepted, it nevertheless has a drawback, i.e. the dependency of finite element mesh sizes (lo]. A lot of research efforts have been devoted to the
(a) Concrete
(b) Steel Fig. 6. Relative sizes of fracture zone and yielding zone.
C.-K. CHOIand H.-G. KWAK
810
structures. The procedures for derivation of the model are described in the following sections. Microcrack distributions
In the first place, a formula for the distribution of microcracks in an element is developed with the general exponential function (see Fig. 7) which can well express the characteristic of assumed microcracks concentrated at the crack tip when the gage length (i.e. finite element mesh size) is fairly large,
Lb*& Fig. 7. Assumed distribution of microcracks
in an element.
removal of this drawback, based on the fracture mechanics concept in particular. Bazant and Oh [14] introduced the crack band theory in the analysis of a plain concrete panel, which may be one of the simplest types of fictitious crack models. The two basic assumptions in the model are that the width of the fracture zone has a certain value w, which is proportional to three times the maximum aggregate size (approximately 1 in.), and that the strain is uniform within the band. Thus, the final equation derived for determining E,,can be expressed as follows:
This model can be successfully applied to the R/C problems when the finite element mesh size used is relatively small. Their analytical results, however, are significantly different from the experimental data as the mesh size is increased since this model assumes a uniform distribution of microcracks in the entire region of a relatively large finite element (see Fig. 8a), while the actual microcracks are concentrated in the relatively small cracked region in the element (see Fig. 8~). Therefore, it is not proper to apply eqn (5) directly to the numerical analysis of R/C structures modeled with relatively large finite element mesh sizes. A PROPOSED MODEL In this study,
the development
which can be applied mesh
size is
of a new criterion
to a fairly large finite element
used in the nonlinear
analysis of R/C
f(x) = a eBr,
in which a and /I are constants to be determined. Substituting the boundary conditions, i.e. f(0) = 1.0 and f(b/2) = 3/b, into eqn (6), the following equation can be obtained:
f(x)
=
e-2/b
I”$,
0
(7)
in which b is the element width. The functionf(x) in eqn (7) portrayed in Figs 7 and 8 expresses: (1) that the distribution of microcracks has a symmetric characteristic as shown in Fig. 7; and (2) that at the end of the finite element mesh, the typical size of microcracks is 3/b. The second feature indicates that the uniform distribution of microcracks is assumed when the element width is less than 3 in., that is, three times the maximum aggregate size [14]. Fracture energy
The fracture energy is expressed by the stressstrain relationships of concrete as
j”(x) dx .2, wheref, is the tensile strength of the concrete, cois the strain at the end of strain softening (Fig. 5), at which the microcracks coalesce into a continuous crack, and G,is the fracture energy consumed in formulating and opening a crack of unit length per unit thickness. If
J
0
(6)
0
s
Fig. 8. Distribution function of microcracks according to the finite element mesh size. (a) Relatively small mesh. (b) Medium mesh. (c) Relatively large mesh.
811
Nonlinear analysis of reinforced concrete structures G, andJ; are known from measu~ments, be calculated as follows:
then 6 may
Gr
4”
Based on the Mindlin hypothesis, the ~spIacement field can be expressed in the following matrix form:
(9)
By using eqn (9), co can be recalculated whenever the finite element mesh size is changed. The proposed criterion in eqn (9) becomes identical with eqn (5) when f(x) = 1.O, i.e. the element mesh size is less or equal to 3 in. When the finite element mesh size is greater than 3 in., as is the case in most practical situations, it is assumed that the microcracks are distributed in an element in accordance with the function f(x) in eqn (7). In this case, the calculated value of =cObecomes different from that calculated by eqn (5) and will better approximate the reality of the microcracks in R/C structures. Thus, eqn (9) can be used in the nonlinear numerical analysis of R/C structures with a wide range of finite element mesh sizes.
in which n is the number of nodes and N, is the shape function. The relationship between strains and displacements can be written as
(z-4.3
0
(z - 0 4,)
00
0
0 (2 - 0dm)1
dN
0
-ax
0
0
2
0
ax
--dN ay
or FINITE ELEM~~
ID~LIZA~ON
A typical finite element is divided into the imaginary concrete and steel layers as shown in Fig. 9. The displacement field in the element is assumed to be continuous and there are no gaps between layers. Each layer may have different material properties, but these properties are assumed to be constant through the thickness of each layer. Therefore, the volume integration involving the different material properties can be expressed as
(13) in which d,, d,,,, and d,,. represent the arbitrary reference depths (see Fig. 9) and are introduced to prevent the undesired development of the in-plane forces. They can be calculated by using the condition of j B, dz = j cy dz = l T.~,, dz = 0, where z is the depth from the neutral axis (see Fig. 9). Moreover, using the simplified assumption that the shear modulus G is constant over the depth, dnry can be assumed to be d/2, in which d is the thickness of the element. Also, the relationship between the transverse shear strains and displacement components can be expressed as
aiv
ax where [D,],, [D,],, n, and n, mean the material matrix of the ith steel layer, the material matrix of the jth concrete layer, the number of steel layers and the number of concrete layers, respectively.
Zj ~ rf-----t-
NEUNRAL Axts(dn)
L
EFFEXTWE STEEL LAYER CONCRETE LAYER Fig.
I
aN -ay
9. A layered section.
dN
dv
-N
0
0
-N
or
Substituting eqns (13) and (15) into eqn (10) and rearranging the material matrix, the element stiffness matrix can be formulated.
812
C-K.
CHol and H.-G. KWAK
in which [D,,] and [D,] are the bending and shear parts of the material matrix, respectively, Once the nodal displacement parameters are calculated, the strain components of each layer can be calculated by using eqns (13) and (IS),
where
+
$ $R(z,+i - 4, I2- (2,-
d”, Yb,
,=I
(17)
a,
1% I, = (8,)=
fx:=J=l c (z,,
1
I -
Z,)G,
n, fy;= c (z,+, - z,)r,., 1=1
After the equivalent nodal forces are known, unbalanced nodal forces can be computed by and the stress components are calculated based on the stress-strain relationship and used to check the failure state of the material.
(23)
the
These unbalanced nodal forces are applied to obtain the increment in solution and recalculated iteratively to satisfy the convergence tolerance. NUMERICAL SOLUTION PROCEDURE
where {rrsj, and {cs>, denote the stress and strain at the mid-depth of the ith steel layer, and (cr,1, and (cc), denote those of the jth concrete layer. Since the calculated stresses do not generally coincide with the true stresses in a nonlinear problem, the unbalanced nodal forces and the equivalent nodal forces must be calculated. The equivalent nodal forces are statically equivalent to the stress field obtained at each iteration and can be computed as follows:
+ r =
s
W,IT{~JdV
voi
r
[B,,]’
J
The finite element used in this study is a quadratic plate element developed by Choi and Kim [15]. This element is established by the combined use of reduced integration and addition of nonconforming displacement modes and gives very good results in linear elastic analysis. The application of this element has been extended to nonlinear analysis in this study. To obtain the load-deflection relationship for the structures, loads are monotonically increased until the structure fails. Load increments are gradually reduced in magnitude after the first detection of the cracking in concrete. The iteration method used is the combined constant-variable stiffness method, i.e. the stiffness matrix is only updated at every loading step. The essential steps in the solution procedure for a typical load increment are given in the flow diagram in Fig. 10.
[d,,lr{cr,)
J
dz - dA
NUMERICAL EXAMPLES
Example I: reinforced concrete slab
A corner-supported two-way slab was tested to investigate the validity of proposed model for the tension stiffening effect. This slab was 36in. square and 1.75 in. thick with an isotropic mesh of
Nonlinear analysis of reinforced concrete structures
813
1
r
and analyzr the Applya load Incmment trwture to obtoln tha nodd dlaplacmm
I Add tha lncrunantd dbpkmrnntm to the total dlaplocmant~
I Cdculate the rtrsw and etraln In the lag u&g tha dd rnotarld propm-tie~
1.2
.-
I
Lh and .Saxddh[S
--.-
Chuk the calculated stren and mtrdn agalnmt the glwn yield crltwla
hahur
wid Dandn[3]
,6
I
0
among the applied total nodal forcea
4
6
12
16
24
20
32
28
DEFLECTION AT NODE 2 IN INCHES Fig. 12. Load-deflection
I”“,
//
36 x10*
relationship at node 2 in Fig. 11.
3.0
3
*.
2 E
2.4
z a
1.1
% o -1
1.2
.e
Fig. IO. Flow diagram for solution procedure.
0 0
4
8
12
DEFLECllON
4
16
AT NODE
20
24
za
2 IN INCHES
32
B
X10*
Solutim A : Smin Softening NOT Considacd(9 clemnts)
k
Solution B
: Both Strain Softening and the Element Size Effecrt Considered (9 elements)
-ic
Solution
P
Solution D : Bxp&mt[l]
I
-----
2
36 in
3in
Fig. I 1. Corner-supported
C : Both Strain Softening and the Element Size Effect? Considctul (4 elements)
two-way slab [I].
0.85% reinforcing steel and subjected to a concentrated load at the center (see Fig. 11). The assumed material properties were as follows: the Poisson ratio v = 0.167, the tensile strength f, = 5,& and the fracture energy G,= 0.5 lb/in.; the number of Table 1. Material properties
Soludon B : Elcmnt Sii
Effect in Case C NOT Cmsidered
Fig. 13. Load-deflection relationships at node 2 with different finite element mesh size.
concrete layers n, = 8. Other basic material properties are summarized in Table 1. This slab has also been analyzed previously by a number of investigators [ 1,3,6,7] to check their studies. Figure 11 shows the finite element grid used and Figs 12 and 13 show the profile of load-deflection relationships at node 2 as the applied load increases. The deflection at node 2 is obtained approximately by interpolating the deflections at adjacent nodes. As shown in Fig. 12, the numerical results obtained by using the fracture energy criterion proposed in this study are in good agreement with the experimental used
in numerical examples
E, (ksi)
E, (ksi)
f; (ksi)
f,
(ksi)
P (%)
d (in.)
Slab
Jofriet and McNeice [l]
415
29.0
5.50
40.0
0.85
1.31
Beam
TIMA T3MA J-4
3.86 3.94 3.80
28.2 28.2 29.5
4.60 4.80 4.82
46.0 41.0 44.9
0.62 3.22 0.99
10.72 10.35 18.00
814
C.-K.
CHOI
and H.-G.
curve. However, the deflection obtained by neglecting tension stiffening in the previous studies is significantly underestimated [63. In the analysis of the same slab, Jofriet and McNeice [l] used the modified stiffness approach and treated both steel and concrete as elastic materials; Lin and Scordelis included the elastoplastic behavior for steel and concrete, and accounted for the coupling effect between membrane and bending actions; Bashur and Darwin [3] used the analytical moment-curvature relationship based on the yield line theory; Hand et al. [6] considered both the shear effect and in-plane deformation. Results from these previous studies are plotted in Fig. 12 for comparison with the results from the present study. Also, in order to test the effects of finite element mesh sizes on the load-deflection relationships, the problem was solved by two different meshes, i.e. nine-element and four-element meshes; the results are shown in Fig. 13. This figure shows that the use of the proposed criterion [eqn (9)] which considers the tension stiffening effect virtually eliminated entire numerical errors which appear when the relatively large finite element mesh size is used (solutions B and C). The two curves in Fig. 13, i.e. solutions E and D, show that the numerical results are dependent on the finite element mesh sizes used in the analysis and that the load-deflection curve obtained by considering the tension stiffening effect in plate bending problems is much closer to the experimental curve (solution D) than to the conventional solution (solution E). Example 2: reinforced concrete beams
Three simply supported R/C beams were investigated in this example, TlMA and T3MA, which were tested by Gaston et al. [16], and J-4, which was tested by Burns and Siess [17]. TlMA and J-4 are under-reinforced, while T3MA has a balanced reinforcement ratio. The geometries of the beams tested are shown in Figs 14 and 15 and the material properties are summarized in Table 1.
KWAK
Fwbonmd
schnobrlch[4]
Epahmtd[l7] I'MStudy
0'
0
I
I
I
I
.4
CENTER
.8
I
I
_
1.2
DEFLECTION
Fig. 15. Load-deflection
relationships of J-4.
The properties not mentioned in Table 1 were assumed to be the same as those used in example I in this study. The number of finite elements used was nine, the number of concrete layers in an element was eight and the Timoshenko beam theory was used. The load-deflection curves obtained are shown in Figs 14 and 15 along with the previous results by Bashur and Darwin [3] and Fariborz and Schnobrich [4]. Those authors took the tension stiffening effect into account by determining the slope of the strain softening region according to the Hillerborg model [ 181. These figures also show that the numerical results obtained in this study are in good agreement with the experimental curves [16, 171 and with those of previous researchers [3,4]. In Table 2, it is shown that the differences between the numerical results are hardly noticeable. It may suggest that if the correct t,, is obtained by the proposed criterion in this study [eqn (9)], the numerical results are not affected by the variation of the tensile strength of concrete.
CONCLUSIONS
0
.5
CENTER
1
1.5
2
Boshur and Omnln[3]
-----
Expwtmmtol[l6]
-
lhlsStudy
25
DEFLECTlON(in)
Fig. 14. Load+Ieflection relationships of TIMA and T3MA.
The behavior of R/C structures in the post-cracking stage is significantly influenced by the tension stiffening effect between adjacent cracks in the concrete. If this effect is ignored, the calculated deflections will be much larger than the experimental data. Therefore, this should not be neglected. In this paper, a new criterion to determine co (the strain at the end of strain softening) is proposed [see eqn (9)] which reduces the numerical errors associated with the tension stiffening effect. Adjusting the slope of the strain softening region in accordance with the proposed criterion, the calculated deflections closely agreed with the experimental data and the
Nonlinear analysis of reinforced concrete structures
815
Table 2. The load-deflection relationship of J-4 with the variation of tensile strength of concrete OSP (kips)
Experiment (in.)
a = 5.5
a = 6.3
a = 6.7
a = 7.5
a = 9.0
2.0 4.0 6.0 8.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0
0.0088 0.0176 0.0525 0.0949 0.1390 0.1630 0.1780 0.2000 0.2200 0.2410 0.2590 0.2760 0.5410 1.0860
0.0112 0.0227 0.0384 0.0719 0.1118 0.1294 0.1476 0.1642 0.1819 0.1976 0.2132 0.2289 0.7024 -
0.0112 0.0227 0.0361 0.0709 0.1112 0.1296 0.1473 0.1636 0.1822 0.1976 0.2128 0.2287 0.7010
0.0112 0.0227 0.0355 0.0723 0.1106 0.1301 0.1474 0.1632 0.1822 0.1976 0.2126 0.2280 0.7010 -
0.0112 0.0227 0.0346 0.0669 0.1104 0.1301 0.1470 0.1623 0.1824 0.1974 0.2124 0.2274 0.7031 -
0.0112 0.0227 0.0341 0.0636 0.1101 0.1241 0.1471 0.1614 0.1823 0.1981 0.2125 0.2273 0.7098 -
Note: f, = aJf<, the number of finite elements used is nine and the values given are the deflection (in.) at the center. numerical error caused by the larger mesh sizes was hardly noticeable, thus making it possible to model
the structure with relatively large finite element mesh sizes. REFERENCES
I. J. C. Jofriet and G. M. McNeice, Finite element analysis of RC slabs. J. struct. Div., AXE 97, 785-806 (1971).
2. A. Vebo and A. Ghah, Moment-curvature relation of reinforced concrete slabs. J. struct. Div., AXE 103, 515-531 (1977). 3. F. K. Bashur and D. Darwin, Nonlinear model for reinforced concrete slabs. J. struct. Div.. ASCE 104. 157-170 (1978). 4. B. J. Fariborz and W. C. Schnobrich, Nonlinear finite element analysis of reinforced concrete under short term monotonic loading. Structural Research Series No. 530, University of Illinois at Urbana, IL (1986). 5. C. S. Lin and A. C. Scordelis, Nonlinear analysis of RC shells of general form. J. struct. Div., ASCE 101, 523-538 (I 975). 6. F. R. Hand, D. A. Pecknold and W. C. Schnobrich,
Nonlinear layered analysis of RC plates and shells. J. struct. Div., ASCE 99, 1491-1505 (1973). 7. R. I. Gilbert and R. F. Warner, Tension stiffening in reinforced concrete slabs. J. struct. Div., ASCE 104, 1885-1900 (1978). 8. T. E. Wamba, Nonlinear analysis of reinforced concrete beams by finite element method. Ph.D. dissertation, Department of Civil Engineering, State University of New York at Buffalo, NY (1980).
9. L. D. Leibengood, D. Darwin and R. H. Dodds, Parameters affecting FE analysis of concrete structures. J. sfrucf. Engng Div., AXE 112, 326-341 (1986). 10. Z. P. Bazant and L. Cedolin. Fracture mechanics of reinforced concrete. J. Engng‘Med.Div., ASCE 106, 1287-1306 (1980). 11. H. Kufer, H. K. Hilsolort and H. Rusch, Behavior of concrete under biaxial stresses. J. Am. Concr. hr. 66, 656-666 (1969). 12. D. Darwin and D. A. W. Pecknold, Analysis of cyclic loading of plane R/C structures. Comput. Struct. 7, 137-147 (1977). 13. Y. R. Rashid, Analysis of prestressed concrete pressure vessels. Nucl. Engng Des. 7, 334-344 (1968). 14. Z. P. Bazant and B. H. Oh. Crack band theorv for fracture of concrete. Mate;. Struct. 16, 155-17; (1983). 15. C. K. Choi and S. H. Kim, Reduced
integration, nonconforming modes and their coupling in thin plate elements. Comput. Struct. 29, 52-62 (1988). 16. J. R. Gaston, C. P. Siess and N. M. Newmark, A layered finite element non-linear analysis of reinforced concrete plates and shells. Civil Engineering Studies, SRS No. 389, University of Illinois at Urbana, IL (1972). 17. N. H. Burns and C. P. Siess, Load-deformation characteristics of beamcolumn connections in reinforced concrete. Civil Engineering Studies, SRS No. 234, University of Illinois, Urbana, IL (1962). 18. A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of crack formation and growth in concrete by means of fracture mechanics and finite element. Cement Concr. Res. 6, 773-782 (1976).