- Email: [email protected]
Nonlinear analysis of reinforced concrete
Computers 81 Slruetvres Vol. 12. pp. 571J79 @ Pergamon Press Ltd., 1980. Printed in Great Britain
NONLINEAR ANALYSIS OF REINFORCED CONCRETE A. ARNESEN,S. I. SORENSEN and P. G. BERGAN The Norwegian
Institute of Technology, The University
(Received 4 October 1979; receiued for publicafion
of Trondheim,
Norway
29 January 1980)
Abstract-This paper discusses the development of two different computer programs for nonlinear analysis of reinforced concrete structures. The first program handles plane stress problems. Flow theory of plasticity is used in the modelling of concrete and reinforcement General four-noded quadrilateral elements with selective sampling of strain are used in the discretization. The second program is developed for analysis of plates and shells. Endochronic theory is used in the constitutive law for concrete whereas an overlay model is utilized for the reinforcement Geometric nonlinearities are accounted for through updating of coordinates for the triangular shell elements. Several examples of applications of the two programs are given. The plane stress programis used for analysis of a beam and two different corbels, while the shell program has been applied to a square plate and a shell with geometric nonlinearities.
1. INTRODUCTION
Nonlinear analysis of reinforced concrete structures has become increasingly important over the last decade. Thorough investigation of capacity and safety aspects of important concrete structures require that the entire structural response up to collapse is established. Today, it is possible to simulate such behaviour by means of advanced computational models and the use of modern computers [ I-31. The three major ingredients in developing a successful nonlinear analysis program can be listed as follows: (i) A realistic material model (ii) An efficient discretization technique (iii) An efficient and reliable solution algorithm. The problem of developing a good material model for reinforced concrete is probably the most difficult. Although this material has been profusely tested and its phenomenological behaviour is well understood, it has proved difficult to cast this knowledge into mathematical forms that can be used in computations. Classical elasticity and plasticity theory can be sufficiently accurate for many applications; this is particularly true when the concrete structure fails in cracking of concrete and yielding of reinforcement bars. Plasticity theory for concrete is sufficiently accurate for uniaxial stress states and two-dimensional states of stress with proportional loading. But for complex loading histories and threedimensional states of stress it has been shown that these theories are inadequate, and more comprehensive formulations must be resorted to. Socalled internal variable theories are very powerful in describing materials in which several rheological phenomena (like gradual deterioration) take place simultaneously. One such material model is the endochronic theory, which has been adapted to concrete behaviour by Bazant[4-6]. The finite element method is the only logical choice for spatial discretization of complex structural systems. The basic equilibrium and incremental equation for this method are easily obtained using the principle of virtual work. Experience has shown that it is advantageous to use relatively low order of interpolation in the finite elements for concrete structures, this is due to abrubt changes of deformations and material behaviour (concentrated yield and failure zones). Further, the order of
integration to obtain the stiffness matrices should represent the same order of approximation as the interpolation functions. There is no point in using nearly exact integration for polynomials that themselves are approximations. Another important aspect of the interpolation is to watch out for spurious self-straining that may result in artificial cracking of concrete. It will be shown that this problem may readily be handled with selective integration. A major requirement to the solution algorithm is that it should account for the history of deformations without great deviations from the true equilibrium path. Standard incremental-iterative schemes satisfy this requirement[7]. However, it is necessary that such algorithms also be adapted to the specific form of the material law. For example, it has been necessary to develop special solution algorithms for the endochronic model[8_lO]. In general it may be stated that practical problems require practical insight and understanding in order to develop efficient methods of analysis. This is also true in developing complex computer programs for nonlinear analysis of concrete structures. This paper discusses the development of two different computer programs for reinforced concrete structures. The first program handles plane stress problems and uses flow theory of plasticity. The second program is based on endochronic theory for concrete and can be used for analysis of slabs and thin shells of arbitrary shapes. This program also accounts for geometric nonlinearities which may play an important role for such structures. 2. MATERIAL MODELS FOR REINFORCED CONCRETE
2. I Concrete in compression 2.1.1 Plasticity model. A relatively crude plasticity model may be used when the failure mode is dominated by concrete cracking and yielding of steel. The program for plane stress problems is based on the flow theory of plasticity to model concrete in compression. The material is assumed to exhibit linear hardening beyond the elastic proportionality limit, uY, see Fig. I(a). The von Mises ellipse is used to define the elastic limit, the flow function and the failure envelope in biaxial states of stress; this is shown in Fig. l(b). 571
A. ARNESEN et al.
512
dh = L(& gii,uiii)d[
Fig. 1. Assumed concrete behaviour in one- and two-dimensional states of stress.
The incremental constitutive relation has the linear form Au = D” AE
(1)
where Au and AE contain the incremental stress and strain components, respectively, and D” is the material matrix which reflects elastic or plastic material behaviour depending on the stress level. The concrete is assumed to be crushed and lose its strength momentaneously when the failure envelope (in stress or strain space) is exceeded. This implies that DP and u are set equal to zero, when the strength envelope (solid line) in Fig. I(b) has been crossed. The sudden loss of strength as well as the elliptic shape of the failure envelope are only crude approximations of the true behaviour. 2.1.2 Endochronic model. In the program for the plate and shell analysis the behaviour of concrete is modelled using the endochronic theory. This theory was originally promoted by Valanis[ 1I], but has later been extended and adapted to concrete behaviour by Bazant [4,6]. The conceptual basis for the formulation is the accumulation of damage done to the material through a scalar parameter & called “intrinsic time measure”. This parameter expresses the distortion that the material has suffered, and is found by accumulating the deviatoric strain increments dl= (i de, deij)“*.
(2)
A sum should be carried out over i and j. The incremental constitutive equations of the endochronic theory are
(6)
F and L depend on other functions which are fit to experimentally observed effects like strain hardening and strain softening, sensitivity to hydrostatic pressure, hysteretic behaviour and degradation of elastic moduli. Bazant built certain material properties into these functions in such a way that the material parameter necessary to supply for practical applications is the uniaxial compressive cylinder strength f:. This model is characterized by a continuous relationship between stresses and strains within the entire range of application, and no specific criterion for loading and unloading is uncorporated. The degradation of the shear and bulk moduli is assumed to depend on the scalar parameter h through K=K,,(l-0.25$),
G=G,(I-0.25$).
(7)
At disruptive failure of the concrete G and K are reduced with only 25% of their initial values. 2.1.3 Discussion of endochronic formulations and flow theories of plasticity. The capability of the endochronic formulation in describing the complex behaviour of concrete has been verified by Bazant [b61. However, it is also clear that inelastic behaviour can to a large extent be described by general flow theories of plasticity. Comparisons between endochronic and more classical plasticity theories have been discussed in Refs. [6, 12,131.The concept for “inelastic stiffness locus” was introduced [6] as a basis for direct comparisons between the different theories. This is a locus (surface) in the strain space on which all points produce the same amount of incremental inelastic strain, de;, measured from a given initial state. The form of the locus gives information about the inelastic properties of the model. The further away a point on this locus lies from the reference state the stiffer is the response in that direction. Three different stiffness loci are shown in Fig. 2. The loci appear in the deviatoric strain space as: a straight line for the incremental plasticity associated with a von Mises type of yield surface, a circle for the endochronic formulation, and a quadratic curve as shown in Fig. 2(c) for the vertex hardening model of Rudnicki and Rice[l4]. It is seen from these loci that incremental plasticity produce no inelastic strain for tangential loading (loading to the side), while the endochronic formulation and the vertex hardening model produce such strain. The circular form of the locus indicates that the same inelastic response is produced in any load direction for the endochronic
(3) dc=de’+de*
I =-du+dA 3K
(4)
G and K are the current shear and bulk moduli, respectively, and eii is the deviatoric and E the volumetric strain components; the corresponding stress quantities are sij and u. Superscript e and p indicate elastic and inelastic, respectively. Inelastic strains are governed by the “intrinsic time scale” z, and the inelastic dilatancy (volume change) is determined by A. Both state variables z and A are highly history dependent and are tied to 5 through dr = F( 5, lii, ui/) dl
(5)
Fig. 2. Inelastic stiffness locus for (a) classical incremental plasticity, (b) endochronic theory, and (c) Rudnicki-Rice vertex hardenmg model.
Nonlinear analysis of reinforcedconcrete
model. Thus no special unloading property is exhibited. This is contrary to plasticity formulations which give pure elastic response for unloading. Recently, refined versions of the endochronic theory in which special unloading properties are included, have been suggested[l3,15,161. Another important distinction between classical plasticity theories and the endochronic formulation is that the latter does not possess a simple linear incremental form but depends on the quadratic expression in eqn (2). This has consequences for the solution algorithm which for end~hronic formulations must be of the “initial load” type. However, it is possible to linearize the theory by fixing the direction of the strain increments. It can thus be shown that such linearized versions of the endochronic theory can be made identical to the incremental plasticity associated with the von Mises type of yield surface[6]. 2.2 opening and closing of cracks 2.2. I Plusticity model. The tension cracking process in this model is governed by the state of stress. Cracks are assumed to open perpendicular to the highest principal tensile stress direction when the failure envelope in Fig. l(a) has been reached. The material is assumed to behave linearly in tension up to the onset of cracking. At this point there are of course no shear stresses to be transferred across the crack. By further straining, however, shear strains may occur parallel to the crack. This raises the question of whether aggregate interlocking is capable of transferring such shear stress over the crack. Shear transfer is taken into account by assuming that a “cracked” shear m~ulus is retained through a factor 0 I (Y5 I times the elastic shear modulus. This factor a is made dependent upon average crack widths computed in the program. Secondary cracks are allowed to form when the other principal stress component exceeds the failure criterion. 2.2.2 ~~doch~o~ic model. In this model a crack is assumed to open when a principal strain or stress component exceeds a prescribed allowable. tensife strain or the uniaxial tensile strength. The crack is formed normal to the principal strain/stress direction. The combined criterion is necessary in case of cyclic loading, since a situation of tensile stress at compressive strain may occur when unloading from a compressive state, see Fig. 3(b). Criteria for closing and reopening of cracks must also be included. Figure 3 demonstrates the effects of the criteria used in the present work. The problem of shear transfer across open cracks is treated in the same way as in the plasticity model, but with a constant value a < 1. Secondary cracking may take place at an already cracked point according to the same criterion as for primary cracking.
(o ) Firs+ boding
in tension
( b ) Firs,
leading
in compression
Fig. 3. Criteriafor opening,closing and reopening of cracks.
573
The concepts for opening and closing of cracks are more detailed described in Refs. [&lO]. 2.3 The reinforcement 2.3.1 Representation in plasticity model. The ~haviour of the rei~orcement steel is approximate by a uniaxial stress-strain relationship, A plasticity formulation is used, assuming linear, isotropic strain hardening after initial yielding. Unloading follows the initial modulus of elasticity. The stress-strain curve is assumed to be the same in tension and compression. 2.3.3 Representation in e~dochronic model. In this model the behaviour of the rei~orcement steel is approximated by a plasticity approach based on the socalled overlay techniqueHO, 171. This means that the material is assumed to be composed of several strands or “layers”; each layer behaves elastic-ideally plastic, but have different yield levels. Combination of several layers gives the possibility to model arbitrary nonline~ strain hardening as well as unloadi~ behaviour with a Bauschinger-type kinematic effect. 3.A PROGRAM FOR PLANESTRESSPROBLEMS
3.1 The finite element model The concrete is modelled by quadrilateral, isoparametric finite elements, based on the assumption of linear inter~lation functions in terms of displacements. This quadrilateral has four corner nodes with two translational degrees of freedom each. It is well known that this element is far too stiff in in-plane bending modes due to spurious shear strains. It was suggested by Doherty et al.[lS] that the in-plane bending performance can be improved by selective intention: the stain energy associated with stretching is computed using regular 2 x 2 Gaussian integration, whereas the shear strain energy is obtained from a one point integration using the centroid. There are no spurious shear strain at the centroid. The same concept is used here for the nonlinear concrete element, the shear strain is sampled at the centroid but used in accum~ation of strain history at the four Gaussian points. The reinforcement bars are modelled by simple twonoded bar elements with linear displacement interpolation. Compatibility between concrete and reinforcement bars is assured at common nodal points. 3.2 Solution of the ~onlineur equations The elasto-plastic material model and the finite element discretization are used with the principle of virtual work in order to establish the total equilibrium equations and the incremental equilibrium equations. The solution of these nonlinear equations is based on a standard Euler-Cauchy incrementation combing with NewtonRaphson iteration. After a load increment has been applied, the internal forces are not in equilibrium with the external, applied loads. This is due to cracking or crushing of concrete, or yielding of concrete or steel. These unbalanced forces are computed, and applied as a load correction, and this process is repeated iteratively. The incremental (~ngential) stiffness matrix serves gradient for the iterative process, and considerable saving is obtained by keeping the same gradient for several iteration cycles (“Quasi-Newton”). The iteration is terminated when the displacement corrections become stdhciently small measured in terms of a modified displacement norm[ 191, or when a prescribed maximum number of cycles has been reached.
574
A. ARNESEN
It could be expected that the crude material model with instantaneous loss of strength at crushing and cracking would result in considerable problems with poor convergence. However, the experience from many applications of the program has not shown any special numerical difficulties. 4. A PROGRAM FOR THIN, FREE-FORM SHELL
Large displacement analysis The socalled “updated Lagrangian description” is used to define the motion of the body[20]. This formulation uses an auxiliary coordinate system that moves rigidly along with the shell elements, so-called “co-rotational coordinates”[21]. Local reference frames are attached to each point of the shell surface and move in harmony with the deformations. The rotational degrees of freedom of the shell elements are referred to this moving surface system [22]. The updated coordinate formulation is efficient in dealing with finite rotations and allow for use of small strain expressions in the incremental analysis. The Love-Kirchoff assumptions[23] for plates and shells are adopted.
4.1
4.2 The finite element model A flat triangular element consisting of the conforming plate bending element of Irons and Razzaque [24,25], and the constant strain membrane element is used for the plate and shell analysis. The shell element has two rotational and three translational degrees of freedom per corner node. In elastic analysis this plate bending element has identically the same stiffness as the nine degrees of freedom hybrid element of Allwood and Corners [26] and Allman[271. The shell element properties are integrated numerically. Simple integration based on the three midside points is carried out over the middle surface of the element; the integration through the thickness is performed using a trapezoidal rule that is adapted to the location of the reinforcement bars. Discrete reinforcement bars are transformed to equivalent “plate layers” parallel to the middle surface. 4.3 Solution of the nonlinear system The general constitutive eqns (3) and (4) are adapted to plane stress analysis, and can be written in matrix notation as [8-IO] Au t Au” = CAr
(8)
where C is the elastic constitutive matrix based on the current G and K. The sum (Ao.+ Aup) is the elastic
et al.
stress increment which consists of a fictitious inelastic stress AaP and the effective stress increment (giving real forces) Au. Determination of the effective stress increment requires calculation of the inelastic stress increment A# which depends on the endochronic state variables z and A. However, these variables are unknown for the current increment and should thus be determined through iterations. Figure 4 illustrates two versions of this procedure: points I, 2, 3, etc. are the iteration points. In the first version the C matrix is updated for every cycle in accordance with the changes of the state variables, see Fig. 4(a). In the second version C is kept constant during these iterations, see Fig. 4(b). The iterations are carried out until Au”, z and h have converged. The second version is cheaper in use and is recommended because C changes very little in practice. The equilibrium equations for the total structure are derived from the principle of virtual work. The stiffness in the incremental force-displacement relation consists of a secant stiffness based on C for the materials; however, the geometric stiffness is also added to account for the change in geometry. Standard load incrementation followed by equilibrium iterations are used. It should be noted that for every iteration cycle a series of iterations for the inelastic stress and the endochronic state variable is carried out. The equilibrium iterations are also affected by the cracking of concrete and yielding of reinforcement bars. They are terminated when the artificial inelastic stresses have vanished and balance between the applied forces and the effective inelastic stresses has been established. A convergence criterion based on the modified Euclidean norm of displacements controls the equilibrium iterations[l9]. The iterations are terminated after a prescribed number of iterations if convergence has not been attained. 5. APPLICATIONS Bresler-Scordelis beam The simply supported beam tested by Bresler and Scordelis[28] has become a classical case for trying out computer programs for reinforced concrete. This beam is shown in Fig. 5, and it is analyzed using the program for plane stress problems. The finite element idealization is also indicated in the figure. During the experiments it was observed that the beam failed by a rapid diagonal tension failure mechanism for a load of 258.1kN. The experimental midspan loaddeflection curve is shown by the solid line in Fig. 6. The horizontal plateau corresponds to the failure load level. Load-deflection curves from four different analyses are
5.1
Strain
o ) Updoting matrix.
b ) The conrtitutive cO”*tMt.
of the conrtitutive
Fig. 4. Iterative
procedures
for solution
matrix
is kept
of eqn (8).
515
Nonlinear analysis of reinforced concrete P =249.2kN
P=
38kN
‘----
f 0
I 2
/
I 4
4 I
Midrpon
Fig. 6. Load-deflection
Fig. 5. Beam tested by Bresler and Scordelis[28].
shown in the same figure. The curves numbered 1,2 and 3 refer to analyses carried out in the present study, using the plasticity model. Curve 4 is taken from an earlier study by Siirensen[9], based on endochronic inelasticity for concrete in compression. Table 1 shows different assumptions made with respect to shear retention after cracking, and numerical integration. In the cases of curves 3 and 4, some shear stiffness is retained after secondary cracking and these curves do not reflect a sudden limit load. In the cases 1 and 2, the concrete is assumed to lose its entire strength when cracks open in two directions. In these cases a relatively sudden diagonal tension failure is reflected by the computed loaddeflection curve, which agrees with the experimental curve. Curve 1 is obtained by a model where the shear strains are sampled at the element centroids whereas standard 2 x 2 Gaussian integration is used otherwise. In this case the computed ultimate load is found at approx. 92% of the experimental value. Crack patterns from experiment and analysis are shown in Fig. 5. 5.2 Corbels Symmetric corbel. The symmetrical corbel shown in Fig. 7(a) was tested by Kriz and Raths[29]. The test specimen was reinforced by a single layer of reinforcement bars. Initial yielding of reinforcement occurred at 266.7 kN in the test specimen, and it failed in vertical cracking and crushing at the base at a load level of 426.8 kN. The finite element model which is used in the analysis is shown in Fig. 7(b). Based on the good experience from the beam, selective sampling of shear strain is also used in this case. Experimental and computed crack patterns, and zones where the concrete is crushed in compression are shown in Fig. 7. The numerical model cannot express discrete Table
I. Approximations
Material model
Shear retention factor,a
1
Plasticity
Variable
7
_ 11_
_ 11_
3 4
- I(_
Endochronic
_(V_
Constant =O.5
( 28 )
Endochmnic I 6
(9 ) I 8
deflection,
mm
curves at midspan
cracks, otherwise the agreement between the two crack patterns is good. Experimental and numerical results are shown in Fig. 8. The onset of yielding of reinforcement bars in the experiment is indicated by P,._, while the corresponding value for the analysis is P,... The ultimate load in the experiment is marked by P,.,,, and the corresponding limit value is marked P,,., for the analysis. The latter value is approx. 10% lower than the experimental value. 5.2.2 Corbel tested at NEZ. The corbel shown in Fig. 9(a) was tested by Jonsson and Svare[30]. This corbel has a more extensive reinforcement in that it contains stirrups, as well as column reinforcement. Figure 9(b) shows the finite element model used in the analysis. The test specimen failed at a load level of 450 kN by diagonal tension in a “shear type” failure. The selective sampling of shear strains is also here used in the finite element analysis. Experimental and computed crack patterns and crushed zones at failure load are shown in Fig. 9, and agree quite well. Figure IO shows the load-deflection curve obtained in the analysis. The calculated ultimate load is approx. 84% of the experimental load. The relatively large discrepancy between computed and experimental failure load for this corbel may possibly be due to triaxial compression introduced by the stirrups. This added strength can only be accounted for by using a full three-dimensional analysis. 5.3 Simply supported square slab Figure II shows the dimensions and reinforcement of a simply supported square slab tested by Taylor et al.[31]. A uniform pressure is applied to the slab in the vertical direction. One eigth of the slab is analyzed applying appropriate boundary conditions along the lines of symmetry. The element mesh is shown in Fig. 12. Nine integration points are used through the thickness. The load is increased in steps of AP = 6.25 kN (load over entire plate) after initial cracking has occurred.
used in different beam
Curve
Experiment
analyses of Bresler-Scordelis
SeCOnilary
0
of ShPar
1 (CentrxJid)
516
A. ARNESEN et al.
Moteriol
pammeters:
( a ) Test specimen
( b )
Finite
element
model
Fig. 7. Corbel tested by Kriz and Raths[29].
In Fig. 12 the midpoint deflection is plotted against the total load applied to the slab. The importance of the geometric stiffening is clearly demonstrated in that the failure load is significantly higher than the limit load of yield line theory. A separate analysis in which the geometric nonlinearities are neglected shows the same effect. -
500
a
400
2Z
MO
++=u,exp= 426,*kN
-
One analysis with linear material properties of the concrete in compression is carried out in order to test the influence of using nonlinear stress-strain curve for the concrete. Only small difference is observed for the linear and nonlinear case. This demonstrates that the important factors for failure are cracking of concrete, yielding of reinforcement bars, and geometric nonlinearities,
z T
------%;78TikN
-___L5e_lp_=~Y!- --~-----
400
a
P urn
-
= 380 kN
$
0
,2
,4
.6
.8
‘,O
I.2
I,4
Deflection,
Fig. 8. Load-deflection
curve. Kriz-Raths
0
I,6 4
Corbel.
,4
,6
Fig. IO. Load-deflection
Material Concrete :
Porameterr:
dy = 34,0
N/mm’
Steel E, = 2,1*105
f c = 40.0
N/mm2
E,,=
f,
N/mm2
=
5,O
E, = 3,5.104 -
( 0 ) Test specimen
,2
(mm )
460
N/mm*
E, = 0,0035
(b
) Finite
Fig. 9. Corbel tested at NBI [30].
element
N/,,mz
N/mm’
model
,8
I,2
I,4
I,6
Deflection,
I,0
aA
(mm)
curve-NBI
Corbel.
Nonlinear analysis of reinforced concrete Waterid
577
date:
f; = 35 M’,” d
aI=
Concrete:
+&!!Tm
E, = 3 ’ lo MPa cc,=1,20. 10-4
Reinforcement: fy= 375
MPa
E, = 2, I . IO5 MPa ES”= 2 . IO’ 5 I
Fig.
0 = 4,76
MPo
mm
Cover to the bottom layer 4,7b mm
I
I I. One quarter of simply supported square slab. 0,2
whereas the assumed behaviour of concrete in compression is only of minor importance for this example. The in-plane displacement at the midside of the reference plane is shown in Fig. 13. The cracking leads to stretching of the midplane while the large deflection effects tend to force this point back towards the center of the plate. 5.4 Cylindrical shell with edge beam Figure I4 shows the dimensions of a cylindrical barrel shell with edge beams tested experimentally by van Riel et aL[32]. The shell is simply supported in the circumferential direction at the curved edges. It is subjected to uniformly distributed load in the vertical diretion. Due to the double symmetry of geometry and loading only one quarter of the shell is idealized by finite elements, see Fig. IS. The edge beam is modelled by elements lying in the horizontal plane because the shell elements are much more accurate in out-of-plane bending than in in-plane bending. The computational model is modified so that the curved shell surface is assumed to be attached to the midline of the edge beams, see Fig. IS. In this way the overall stiffness of the shell is somewhat underestimated. In Fig. 15 the central defiection of the edge beam is plotted against the total load applied to the shell. Two analyses are carried out: one which includes the geometric nonlinearities and one which does not. The failure load for the geometrically linear analysis is close to the ultimate load calculated according to a simple beam-type rupture analysis reported in Ref. [32]. Good agreement with the experimental curve is obtained for the geometrically nonlinear analysis up to a
75
0
-0,2
-0,b
-0,4
Displacement,
load level of P = 38 kN. Above this load an increased stiffness is observed for the finite element solution. This is probably due to stiffening of the shell surface along the edge beam. The experimental failure of the shell showed large inward bending of the edge beam; this is not the case in the analysis. It is likely that the edge beam has become too stiff in the finite element model for inward bending and that this has restricted the edge movement excessively. For this reason buckling tendencies in the circumferential direction are restrained. Failure of the shell takes place when yielding of the reinforcement in the edge beam has progressed more than half through the beam height. Large deflections then prevail over the shell and give yielding of the reinforcement in the shell body.
Y
1
Experiment
[31]
B
Geometrically
nonlinear
Lc-d
Geometrically
linear
onalysir
analysis
25
I
I
20
30
336klm
Fig. 14. Dimensions of the shell and edge beams.
50
I
VA (mm)
Fig. 13. In-plane displacement at midside point.
-
10
-0,8
40
I
I
I
I
50
60
70
80
Midpoint
deflection
Fig. 12. Central deflection of the slab.
(mm)
578
A.
ARNESEN et al.
Element mesh
_
GeometricoIly linear
-
Geometrically
analyrir
nonlinear
analyrir
Reinforcement:
Real edge beam. 5I
10 I
I 15
Finite element ideolizotion of the edge beam. 20I
I 25
I 30
I
35
I
I
40
45
Deflection
Fig. 15. Load-deflection
6. CONCLUSIONS
The plane stress program is efficient and inexpensive in use. A main reason for developing this program was to find out if good results could be obtained in spite of a crude material model. The results clearly demonstrate that the assumed constitutive law for untracked concrete is of only minor importance for normal structures which fail in cracking and yielding of reinforcement. This conclusion applies to the entire load-deflection curve including the failure load. But some of the results also indicate that a more refined model should be used in case of triaxial stresses (confinement from reinforcement) and crushing of concrete. The studies performed with the plate and shell program also show good agreement with experimental results for the load-deflection curves as well as for the failure load. This indicates that both the geometric and the material nonlinearities are well represented in the program. The endochronic theory can be implemented in a finite element formulation which also considers cracking and yielding of reinforcement bars. Great efforts have been necessary in order to develop an efficient and reliable solution algorithm. The examples studied with the two programs demonstrate that very extensive and accurate information about the nonlinear behaviour of reinforced concrete structures can be obtained using modern computer simulation. Acknowledgement-The authors would like to thank Prof. R. Lenschow for useful discussions. REFERENCES I. A. C. Scordelis and W. Schnobrich, Finite element analysis of reinforced concrete structures. Seminar on Finite Element Analysis of Reinforced Concrete Structures, pp. 61-334. Politecnico di Milan0 (1978).
2. W. Schnobrich, Behavior of reinforced concrete predicted by finite element method. Proc. of the 2nd Nat. Symp. on Computerized
Structural
Analysis
and
Design.
Washington University, Washington, D.C. (1976).
George
ot midrpan
(mm )
curve for the edge beam.
3. P. G. Bergan and 1. Holand, Nonlinear finite element analysis of concrete structures. Jnt. Conf. on finite ElementsI .in.. Nonlinear Mechanics. ISD, University of Stuttgart (1978); Comput. Math. Appl. Mech. Engng 1711443-467 (1979). 4. Z. P. Bazant and P. D. Bhat, Endochronic theory of inelasticity and failure of concrete. ASCE J. Engng Mech. Div.
102,701-722 (1976). 5. Z. P. Bazant and P. D. Bhat, Prediction of hysteresis of reinforced concrete members. ASCE J. Struct. Mech. Div. 103, 153-167(1977). 6 Z. P. Bazant, On endochronic inelasticity and incremental plasticity. Structural Engineering Rep. No. 76-121259. Northwestern University, Evanston, Illinois (1976). 7 P. G. Bergan, G. Horrigmoe, B. Krikeland and T. H. Soreide, Solution techniques for nonlinear finite element problems. Int. J. Numerical Methods in Engng 12, l677-16% (1978). 8. S. I. Sorensen, A. Arnesen and P. G. Bergan, Nonlinear finite element analysis of reinforced concrete using endochronic theory. Finite Elements in Nonlinear Mechanics (Edited by P. G. Bergan et al.). TAPIR, Trondheim(1978). 9. S. I. Sorensen, Endochronic theory in nonlinear finite eiement analysis of reinforced concrete. Rep. No. 78-l. Division of Structural Mechanics. The Norwegian lnstitute of Technology, The University of Trondheim (1978). 10. A. Arnesen, Analysis of reinforced concrete shells considering material and geometric nonlinearities. Rep. No. 79-l. Division of Structural Mechanics. The Norwegian Institute of Technology, The University of Trondheim (1979). Il. K. C. Valanis, A theory of viscoplasticity without a yield surface. Archiwum Mechaniki Stosowanej 3, 517-551 (1971). 12. G. Wempner and J. Aberson, A formulation of inelasticity from thermal and mechanical concepts. Int. J Solids Structures 12, 705-721(1976). 13. Z. P. Bazant, Endochronic inelasticity and incremental plasticity. Int. J. So/ids Structures 14, (1978). 14. J. W. Rudnickiand J. R. Rice, Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371-394(1975). 15. K. C. Valanis, Some recent developments in the endochronic theory of plasticity-the concept of internal barriers. In Constitutiue Equations of Plasticity, AMD-Vol. 21, pp. 15-32. Am. Sot. of Mechanical Engineers, New York (presented at ASME Water Annual Meeting, New York, Dec. 1976).
579
Nonlinear analysis of reinforced concrete 16. K. C. Valanis and H. E. Read, A theory of plasticity for hysteretic materials-I: Shear response. C’omput. Structures 8.503510 (1978). 17. 0. C. Zienkiewicz, G. C. Navak and D. R. J. Owen. Composite and overlay models in numerical analysis of elastoplastic continua. Int. Symp. on Foundations of Plasticity. Warsaw (1972). 18. W. P. Doherty, E. L. Wilson and R. L. Taylor. Stress analysis of ax&symmetric solids using higher order quadrilateral finite elements. Structural Engineeting Laboratory Rep. No. SESM 69-3. Universitv of California. Berkelev (1%9). 19. P. G. Bergan and R. W. Clough, Convergence criteria for iterative processes. AIAA J. 10, 1107-l 108 (1972). 20. K. J. Bathe, E. Ramm and E. L. Wilson, Finite element formulation for large deformation dynamic analysis. ht. 1. Numer.
Methods
in Engng
9.353-386
(1975).
21. T. Belytschko and B. J. Hsieh, Non-linear transient finite element analyses with convected coordinates, Int. J. Numer. Methods
in En,ena 1. 255-271
(1973).
22. T. Soreide, Collapse behaviour of stiffened plates using alternative finite element formulations, Rep. No. 77-3. Division of Structural Mechanics, The Norwegian Institute of Technology, Trondheim (1977). 23. L. H. Donnell. Beams, Plates and Shells. McGraw-Hill, New York (1976). 24. B. M. Irons and A. Razzaque, Shape function formulations for elements other than displacement models. In Variational
Methods in Engineering, Proceedings presented at University of Southampton, Sept. 1972 (Edited by C. Brebbia and H. Tottenham). University Press (1972). 25. A. Razzaaue. Program for triangular bending elements with derivative’ smoothing. Int. J. Nimer. Methods in Engng 6. 332-343 ( 1973). 26. R. J. Allwood and G. M. M. Corners, A polygonal finite element for plate bending problems using the assumed stress approach. Int. J. Numer. Methods in Engng 1, 135-149 (l%9). 27. D. J. Allmann, Triangular finite elements for plate bending with constant and linearly varying bending moments. Proc. WTAM Symp. on High Speed Computing tures. University of Liege (1970).
of Elastic
Struc-
28. B. Bresler and A. C. Scordelis, Shear strength of reinforced concrete beams-series II. SESM Reo. No. 64-2. Universitv of California, Berkeley (1964). 29. L. B. Kriz and C. H. Raths. Connections in precast concrete structures-strength of corbels. PC1 1. 10, 16-61 (1965). 30. E. Jonsson and T. I. Svare. Tests of concrete corbels. NBI Rep. (1976) (in Norwegian). 31. R. Taylor, D. R. H. Maher and B. Hayes, Effect of the arrangement of reinforcement on the behaviour of reinforced concrete slabs. Magazine of Concrete Res. 98, (l%6). 32. A. C. van Riel. W. J. Beranek and A. L. Bouma. Tests on shell roof models of reinforced mortar, Proc. of the 2nd Symp. on Concrete She// Roof Construction. Oslo (1957).
NONLINEAR ANALYSIS OF REINFORCED CONCRETE A. ARNESEN,S. I. SORENSEN and P. G. BERGAN The Norwegian
Institute of Technology, The University
(Received 4 October 1979; receiued for publicafion
of Trondheim,
Norway
29 January 1980)
Abstract-This paper discusses the development of two different computer programs for nonlinear analysis of reinforced concrete structures. The first program handles plane stress problems. Flow theory of plasticity is used in the modelling of concrete and reinforcement General four-noded quadrilateral elements with selective sampling of strain are used in the discretization. The second program is developed for analysis of plates and shells. Endochronic theory is used in the constitutive law for concrete whereas an overlay model is utilized for the reinforcement Geometric nonlinearities are accounted for through updating of coordinates for the triangular shell elements. Several examples of applications of the two programs are given. The plane stress programis used for analysis of a beam and two different corbels, while the shell program has been applied to a square plate and a shell with geometric nonlinearities.
1. INTRODUCTION
Nonlinear analysis of reinforced concrete structures has become increasingly important over the last decade. Thorough investigation of capacity and safety aspects of important concrete structures require that the entire structural response up to collapse is established. Today, it is possible to simulate such behaviour by means of advanced computational models and the use of modern computers [ I-31. The three major ingredients in developing a successful nonlinear analysis program can be listed as follows: (i) A realistic material model (ii) An efficient discretization technique (iii) An efficient and reliable solution algorithm. The problem of developing a good material model for reinforced concrete is probably the most difficult. Although this material has been profusely tested and its phenomenological behaviour is well understood, it has proved difficult to cast this knowledge into mathematical forms that can be used in computations. Classical elasticity and plasticity theory can be sufficiently accurate for many applications; this is particularly true when the concrete structure fails in cracking of concrete and yielding of reinforcement bars. Plasticity theory for concrete is sufficiently accurate for uniaxial stress states and two-dimensional states of stress with proportional loading. But for complex loading histories and threedimensional states of stress it has been shown that these theories are inadequate, and more comprehensive formulations must be resorted to. Socalled internal variable theories are very powerful in describing materials in which several rheological phenomena (like gradual deterioration) take place simultaneously. One such material model is the endochronic theory, which has been adapted to concrete behaviour by Bazant[4-6]. The finite element method is the only logical choice for spatial discretization of complex structural systems. The basic equilibrium and incremental equation for this method are easily obtained using the principle of virtual work. Experience has shown that it is advantageous to use relatively low order of interpolation in the finite elements for concrete structures, this is due to abrubt changes of deformations and material behaviour (concentrated yield and failure zones). Further, the order of
integration to obtain the stiffness matrices should represent the same order of approximation as the interpolation functions. There is no point in using nearly exact integration for polynomials that themselves are approximations. Another important aspect of the interpolation is to watch out for spurious self-straining that may result in artificial cracking of concrete. It will be shown that this problem may readily be handled with selective integration. A major requirement to the solution algorithm is that it should account for the history of deformations without great deviations from the true equilibrium path. Standard incremental-iterative schemes satisfy this requirement[7]. However, it is necessary that such algorithms also be adapted to the specific form of the material law. For example, it has been necessary to develop special solution algorithms for the endochronic model[8_lO]. In general it may be stated that practical problems require practical insight and understanding in order to develop efficient methods of analysis. This is also true in developing complex computer programs for nonlinear analysis of concrete structures. This paper discusses the development of two different computer programs for reinforced concrete structures. The first program handles plane stress problems and uses flow theory of plasticity. The second program is based on endochronic theory for concrete and can be used for analysis of slabs and thin shells of arbitrary shapes. This program also accounts for geometric nonlinearities which may play an important role for such structures. 2. MATERIAL MODELS FOR REINFORCED CONCRETE
2. I Concrete in compression 2.1.1 Plasticity model. A relatively crude plasticity model may be used when the failure mode is dominated by concrete cracking and yielding of steel. The program for plane stress problems is based on the flow theory of plasticity to model concrete in compression. The material is assumed to exhibit linear hardening beyond the elastic proportionality limit, uY, see Fig. I(a). The von Mises ellipse is used to define the elastic limit, the flow function and the failure envelope in biaxial states of stress; this is shown in Fig. l(b). 571
A. ARNESEN et al.
512
dh = L(& gii,uiii)d[
Fig. 1. Assumed concrete behaviour in one- and two-dimensional states of stress.
The incremental constitutive relation has the linear form Au = D” AE
(1)
where Au and AE contain the incremental stress and strain components, respectively, and D” is the material matrix which reflects elastic or plastic material behaviour depending on the stress level. The concrete is assumed to be crushed and lose its strength momentaneously when the failure envelope (in stress or strain space) is exceeded. This implies that DP and u are set equal to zero, when the strength envelope (solid line) in Fig. I(b) has been crossed. The sudden loss of strength as well as the elliptic shape of the failure envelope are only crude approximations of the true behaviour. 2.1.2 Endochronic model. In the program for the plate and shell analysis the behaviour of concrete is modelled using the endochronic theory. This theory was originally promoted by Valanis[ 1I], but has later been extended and adapted to concrete behaviour by Bazant [4,6]. The conceptual basis for the formulation is the accumulation of damage done to the material through a scalar parameter & called “intrinsic time measure”. This parameter expresses the distortion that the material has suffered, and is found by accumulating the deviatoric strain increments dl= (i de, deij)“*.
(2)
A sum should be carried out over i and j. The incremental constitutive equations of the endochronic theory are
(6)
F and L depend on other functions which are fit to experimentally observed effects like strain hardening and strain softening, sensitivity to hydrostatic pressure, hysteretic behaviour and degradation of elastic moduli. Bazant built certain material properties into these functions in such a way that the material parameter necessary to supply for practical applications is the uniaxial compressive cylinder strength f:. This model is characterized by a continuous relationship between stresses and strains within the entire range of application, and no specific criterion for loading and unloading is uncorporated. The degradation of the shear and bulk moduli is assumed to depend on the scalar parameter h through K=K,,(l-0.25$),
G=G,(I-0.25$).
(7)
At disruptive failure of the concrete G and K are reduced with only 25% of their initial values. 2.1.3 Discussion of endochronic formulations and flow theories of plasticity. The capability of the endochronic formulation in describing the complex behaviour of concrete has been verified by Bazant [b61. However, it is also clear that inelastic behaviour can to a large extent be described by general flow theories of plasticity. Comparisons between endochronic and more classical plasticity theories have been discussed in Refs. [6, 12,131.The concept for “inelastic stiffness locus” was introduced [6] as a basis for direct comparisons between the different theories. This is a locus (surface) in the strain space on which all points produce the same amount of incremental inelastic strain, de;, measured from a given initial state. The form of the locus gives information about the inelastic properties of the model. The further away a point on this locus lies from the reference state the stiffer is the response in that direction. Three different stiffness loci are shown in Fig. 2. The loci appear in the deviatoric strain space as: a straight line for the incremental plasticity associated with a von Mises type of yield surface, a circle for the endochronic formulation, and a quadratic curve as shown in Fig. 2(c) for the vertex hardening model of Rudnicki and Rice[l4]. It is seen from these loci that incremental plasticity produce no inelastic strain for tangential loading (loading to the side), while the endochronic formulation and the vertex hardening model produce such strain. The circular form of the locus indicates that the same inelastic response is produced in any load direction for the endochronic
(3) dc=de’+de*
I =-du+dA 3K
(4)
G and K are the current shear and bulk moduli, respectively, and eii is the deviatoric and E the volumetric strain components; the corresponding stress quantities are sij and u. Superscript e and p indicate elastic and inelastic, respectively. Inelastic strains are governed by the “intrinsic time scale” z, and the inelastic dilatancy (volume change) is determined by A. Both state variables z and A are highly history dependent and are tied to 5 through dr = F( 5, lii, ui/) dl
(5)
Fig. 2. Inelastic stiffness locus for (a) classical incremental plasticity, (b) endochronic theory, and (c) Rudnicki-Rice vertex hardenmg model.
Nonlinear analysis of reinforcedconcrete
model. Thus no special unloading property is exhibited. This is contrary to plasticity formulations which give pure elastic response for unloading. Recently, refined versions of the endochronic theory in which special unloading properties are included, have been suggested[l3,15,161. Another important distinction between classical plasticity theories and the endochronic formulation is that the latter does not possess a simple linear incremental form but depends on the quadratic expression in eqn (2). This has consequences for the solution algorithm which for end~hronic formulations must be of the “initial load” type. However, it is possible to linearize the theory by fixing the direction of the strain increments. It can thus be shown that such linearized versions of the endochronic theory can be made identical to the incremental plasticity associated with the von Mises type of yield surface[6]. 2.2 opening and closing of cracks 2.2. I Plusticity model. The tension cracking process in this model is governed by the state of stress. Cracks are assumed to open perpendicular to the highest principal tensile stress direction when the failure envelope in Fig. l(a) has been reached. The material is assumed to behave linearly in tension up to the onset of cracking. At this point there are of course no shear stresses to be transferred across the crack. By further straining, however, shear strains may occur parallel to the crack. This raises the question of whether aggregate interlocking is capable of transferring such shear stress over the crack. Shear transfer is taken into account by assuming that a “cracked” shear m~ulus is retained through a factor 0 I (Y5 I times the elastic shear modulus. This factor a is made dependent upon average crack widths computed in the program. Secondary cracks are allowed to form when the other principal stress component exceeds the failure criterion. 2.2.2 ~~doch~o~ic model. In this model a crack is assumed to open when a principal strain or stress component exceeds a prescribed allowable. tensife strain or the uniaxial tensile strength. The crack is formed normal to the principal strain/stress direction. The combined criterion is necessary in case of cyclic loading, since a situation of tensile stress at compressive strain may occur when unloading from a compressive state, see Fig. 3(b). Criteria for closing and reopening of cracks must also be included. Figure 3 demonstrates the effects of the criteria used in the present work. The problem of shear transfer across open cracks is treated in the same way as in the plasticity model, but with a constant value a < 1. Secondary cracking may take place at an already cracked point according to the same criterion as for primary cracking.
(o ) Firs+ boding
in tension
( b ) Firs,
leading
in compression
Fig. 3. Criteriafor opening,closing and reopening of cracks.
573
The concepts for opening and closing of cracks are more detailed described in Refs. [&lO]. 2.3 The reinforcement 2.3.1 Representation in plasticity model. The ~haviour of the rei~orcement steel is approximate by a uniaxial stress-strain relationship, A plasticity formulation is used, assuming linear, isotropic strain hardening after initial yielding. Unloading follows the initial modulus of elasticity. The stress-strain curve is assumed to be the same in tension and compression. 2.3.3 Representation in e~dochronic model. In this model the behaviour of the rei~orcement steel is approximated by a plasticity approach based on the socalled overlay techniqueHO, 171. This means that the material is assumed to be composed of several strands or “layers”; each layer behaves elastic-ideally plastic, but have different yield levels. Combination of several layers gives the possibility to model arbitrary nonline~ strain hardening as well as unloadi~ behaviour with a Bauschinger-type kinematic effect. 3.A PROGRAM FOR PLANESTRESSPROBLEMS
3.1 The finite element model The concrete is modelled by quadrilateral, isoparametric finite elements, based on the assumption of linear inter~lation functions in terms of displacements. This quadrilateral has four corner nodes with two translational degrees of freedom each. It is well known that this element is far too stiff in in-plane bending modes due to spurious shear strains. It was suggested by Doherty et al.[lS] that the in-plane bending performance can be improved by selective intention: the stain energy associated with stretching is computed using regular 2 x 2 Gaussian integration, whereas the shear strain energy is obtained from a one point integration using the centroid. There are no spurious shear strain at the centroid. The same concept is used here for the nonlinear concrete element, the shear strain is sampled at the centroid but used in accum~ation of strain history at the four Gaussian points. The reinforcement bars are modelled by simple twonoded bar elements with linear displacement interpolation. Compatibility between concrete and reinforcement bars is assured at common nodal points. 3.2 Solution of the ~onlineur equations The elasto-plastic material model and the finite element discretization are used with the principle of virtual work in order to establish the total equilibrium equations and the incremental equilibrium equations. The solution of these nonlinear equations is based on a standard Euler-Cauchy incrementation combing with NewtonRaphson iteration. After a load increment has been applied, the internal forces are not in equilibrium with the external, applied loads. This is due to cracking or crushing of concrete, or yielding of concrete or steel. These unbalanced forces are computed, and applied as a load correction, and this process is repeated iteratively. The incremental (~ngential) stiffness matrix serves gradient for the iterative process, and considerable saving is obtained by keeping the same gradient for several iteration cycles (“Quasi-Newton”). The iteration is terminated when the displacement corrections become stdhciently small measured in terms of a modified displacement norm[ 191, or when a prescribed maximum number of cycles has been reached.
574
A. ARNESEN
It could be expected that the crude material model with instantaneous loss of strength at crushing and cracking would result in considerable problems with poor convergence. However, the experience from many applications of the program has not shown any special numerical difficulties. 4. A PROGRAM FOR THIN, FREE-FORM SHELL
Large displacement analysis The socalled “updated Lagrangian description” is used to define the motion of the body[20]. This formulation uses an auxiliary coordinate system that moves rigidly along with the shell elements, so-called “co-rotational coordinates”[21]. Local reference frames are attached to each point of the shell surface and move in harmony with the deformations. The rotational degrees of freedom of the shell elements are referred to this moving surface system [22]. The updated coordinate formulation is efficient in dealing with finite rotations and allow for use of small strain expressions in the incremental analysis. The Love-Kirchoff assumptions[23] for plates and shells are adopted.
4.1
4.2 The finite element model A flat triangular element consisting of the conforming plate bending element of Irons and Razzaque [24,25], and the constant strain membrane element is used for the plate and shell analysis. The shell element has two rotational and three translational degrees of freedom per corner node. In elastic analysis this plate bending element has identically the same stiffness as the nine degrees of freedom hybrid element of Allwood and Corners [26] and Allman[271. The shell element properties are integrated numerically. Simple integration based on the three midside points is carried out over the middle surface of the element; the integration through the thickness is performed using a trapezoidal rule that is adapted to the location of the reinforcement bars. Discrete reinforcement bars are transformed to equivalent “plate layers” parallel to the middle surface. 4.3 Solution of the nonlinear system The general constitutive eqns (3) and (4) are adapted to plane stress analysis, and can be written in matrix notation as [8-IO] Au t Au” = CAr
(8)
where C is the elastic constitutive matrix based on the current G and K. The sum (Ao.+ Aup) is the elastic
et al.
stress increment which consists of a fictitious inelastic stress AaP and the effective stress increment (giving real forces) Au. Determination of the effective stress increment requires calculation of the inelastic stress increment A# which depends on the endochronic state variables z and A. However, these variables are unknown for the current increment and should thus be determined through iterations. Figure 4 illustrates two versions of this procedure: points I, 2, 3, etc. are the iteration points. In the first version the C matrix is updated for every cycle in accordance with the changes of the state variables, see Fig. 4(a). In the second version C is kept constant during these iterations, see Fig. 4(b). The iterations are carried out until Au”, z and h have converged. The second version is cheaper in use and is recommended because C changes very little in practice. The equilibrium equations for the total structure are derived from the principle of virtual work. The stiffness in the incremental force-displacement relation consists of a secant stiffness based on C for the materials; however, the geometric stiffness is also added to account for the change in geometry. Standard load incrementation followed by equilibrium iterations are used. It should be noted that for every iteration cycle a series of iterations for the inelastic stress and the endochronic state variable is carried out. The equilibrium iterations are also affected by the cracking of concrete and yielding of reinforcement bars. They are terminated when the artificial inelastic stresses have vanished and balance between the applied forces and the effective inelastic stresses has been established. A convergence criterion based on the modified Euclidean norm of displacements controls the equilibrium iterations[l9]. The iterations are terminated after a prescribed number of iterations if convergence has not been attained. 5. APPLICATIONS Bresler-Scordelis beam The simply supported beam tested by Bresler and Scordelis[28] has become a classical case for trying out computer programs for reinforced concrete. This beam is shown in Fig. 5, and it is analyzed using the program for plane stress problems. The finite element idealization is also indicated in the figure. During the experiments it was observed that the beam failed by a rapid diagonal tension failure mechanism for a load of 258.1kN. The experimental midspan loaddeflection curve is shown by the solid line in Fig. 6. The horizontal plateau corresponds to the failure load level. Load-deflection curves from four different analyses are
5.1
Strain
o ) Updoting matrix.
b ) The conrtitutive cO”*tMt.
of the conrtitutive
Fig. 4. Iterative
procedures
for solution
matrix
is kept
of eqn (8).
515
Nonlinear analysis of reinforced concrete P =249.2kN
P=
38kN
‘----
f 0
I 2
/
I 4
4 I
Midrpon
Fig. 6. Load-deflection
Fig. 5. Beam tested by Bresler and Scordelis[28].
shown in the same figure. The curves numbered 1,2 and 3 refer to analyses carried out in the present study, using the plasticity model. Curve 4 is taken from an earlier study by Siirensen[9], based on endochronic inelasticity for concrete in compression. Table 1 shows different assumptions made with respect to shear retention after cracking, and numerical integration. In the cases of curves 3 and 4, some shear stiffness is retained after secondary cracking and these curves do not reflect a sudden limit load. In the cases 1 and 2, the concrete is assumed to lose its entire strength when cracks open in two directions. In these cases a relatively sudden diagonal tension failure is reflected by the computed loaddeflection curve, which agrees with the experimental curve. Curve 1 is obtained by a model where the shear strains are sampled at the element centroids whereas standard 2 x 2 Gaussian integration is used otherwise. In this case the computed ultimate load is found at approx. 92% of the experimental value. Crack patterns from experiment and analysis are shown in Fig. 5. 5.2 Corbels Symmetric corbel. The symmetrical corbel shown in Fig. 7(a) was tested by Kriz and Raths[29]. The test specimen was reinforced by a single layer of reinforcement bars. Initial yielding of reinforcement occurred at 266.7 kN in the test specimen, and it failed in vertical cracking and crushing at the base at a load level of 426.8 kN. The finite element model which is used in the analysis is shown in Fig. 7(b). Based on the good experience from the beam, selective sampling of shear strain is also used in this case. Experimental and computed crack patterns, and zones where the concrete is crushed in compression are shown in Fig. 7. The numerical model cannot express discrete Table
I. Approximations
Material model
Shear retention factor,a
1
Plasticity
Variable
7
_ 11_
_ 11_
3 4
- I(_
Endochronic
_(V_
Constant =O.5
( 28 )
Endochmnic I 6
(9 ) I 8
deflection,
mm
curves at midspan
cracks, otherwise the agreement between the two crack patterns is good. Experimental and numerical results are shown in Fig. 8. The onset of yielding of reinforcement bars in the experiment is indicated by P,._, while the corresponding value for the analysis is P,... The ultimate load in the experiment is marked by P,.,,, and the corresponding limit value is marked P,,., for the analysis. The latter value is approx. 10% lower than the experimental value. 5.2.2 Corbel tested at NEZ. The corbel shown in Fig. 9(a) was tested by Jonsson and Svare[30]. This corbel has a more extensive reinforcement in that it contains stirrups, as well as column reinforcement. Figure 9(b) shows the finite element model used in the analysis. The test specimen failed at a load level of 450 kN by diagonal tension in a “shear type” failure. The selective sampling of shear strains is also here used in the finite element analysis. Experimental and computed crack patterns and crushed zones at failure load are shown in Fig. 9, and agree quite well. Figure IO shows the load-deflection curve obtained in the analysis. The calculated ultimate load is approx. 84% of the experimental load. The relatively large discrepancy between computed and experimental failure load for this corbel may possibly be due to triaxial compression introduced by the stirrups. This added strength can only be accounted for by using a full three-dimensional analysis. 5.3 Simply supported square slab Figure II shows the dimensions and reinforcement of a simply supported square slab tested by Taylor et al.[31]. A uniform pressure is applied to the slab in the vertical direction. One eigth of the slab is analyzed applying appropriate boundary conditions along the lines of symmetry. The element mesh is shown in Fig. 12. Nine integration points are used through the thickness. The load is increased in steps of AP = 6.25 kN (load over entire plate) after initial cracking has occurred.
used in different beam
Curve
Experiment
analyses of Bresler-Scordelis
SeCOnilary
0
of ShPar
1 (CentrxJid)
516
A. ARNESEN et al.
Moteriol
pammeters:
( a ) Test specimen
( b )
Finite
element
model
Fig. 7. Corbel tested by Kriz and Raths[29].
In Fig. 12 the midpoint deflection is plotted against the total load applied to the slab. The importance of the geometric stiffening is clearly demonstrated in that the failure load is significantly higher than the limit load of yield line theory. A separate analysis in which the geometric nonlinearities are neglected shows the same effect. -
500
a
400
2Z
MO
++=u,exp= 426,*kN
-
One analysis with linear material properties of the concrete in compression is carried out in order to test the influence of using nonlinear stress-strain curve for the concrete. Only small difference is observed for the linear and nonlinear case. This demonstrates that the important factors for failure are cracking of concrete, yielding of reinforcement bars, and geometric nonlinearities,
z T
------%;78TikN
-___L5e_lp_=~Y!- --~-----
400
a
P urn
-
= 380 kN
$
0
,2
,4
.6
.8
‘,O
I.2
I,4
Deflection,
Fig. 8. Load-deflection
curve. Kriz-Raths
0
I,6 4
Corbel.
,4
,6
Fig. IO. Load-deflection
Material Concrete :
Porameterr:
dy = 34,0
N/mm’
Steel E, = 2,1*105
f c = 40.0
N/mm2
E,,=
f,
N/mm2
=
5,O
E, = 3,5.104 -
( 0 ) Test specimen
,2
(mm )
460
N/mm*
E, = 0,0035
(b
) Finite
Fig. 9. Corbel tested at NBI [30].
element
N/,,mz
N/mm’
model
,8
I,2
I,4
I,6
Deflection,
I,0
aA
(mm)
curve-NBI
Corbel.
Nonlinear analysis of reinforced concrete Waterid
577
date:
f; = 35 M’,” d
aI=
Concrete:
+&!!Tm
E, = 3 ’ lo MPa cc,=1,20. 10-4
Reinforcement: fy= 375
MPa
E, = 2, I . IO5 MPa ES”= 2 . IO’ 5 I
Fig.
0 = 4,76
MPo
mm
Cover to the bottom layer 4,7b mm
I
I I. One quarter of simply supported square slab. 0,2
whereas the assumed behaviour of concrete in compression is only of minor importance for this example. The in-plane displacement at the midside of the reference plane is shown in Fig. 13. The cracking leads to stretching of the midplane while the large deflection effects tend to force this point back towards the center of the plate. 5.4 Cylindrical shell with edge beam Figure I4 shows the dimensions of a cylindrical barrel shell with edge beams tested experimentally by van Riel et aL[32]. The shell is simply supported in the circumferential direction at the curved edges. It is subjected to uniformly distributed load in the vertical diretion. Due to the double symmetry of geometry and loading only one quarter of the shell is idealized by finite elements, see Fig. IS. The edge beam is modelled by elements lying in the horizontal plane because the shell elements are much more accurate in out-of-plane bending than in in-plane bending. The computational model is modified so that the curved shell surface is assumed to be attached to the midline of the edge beams, see Fig. IS. In this way the overall stiffness of the shell is somewhat underestimated. In Fig. 15 the central defiection of the edge beam is plotted against the total load applied to the shell. Two analyses are carried out: one which includes the geometric nonlinearities and one which does not. The failure load for the geometrically linear analysis is close to the ultimate load calculated according to a simple beam-type rupture analysis reported in Ref. [32]. Good agreement with the experimental curve is obtained for the geometrically nonlinear analysis up to a
75
0
-0,2
-0,b
-0,4
Displacement,
load level of P = 38 kN. Above this load an increased stiffness is observed for the finite element solution. This is probably due to stiffening of the shell surface along the edge beam. The experimental failure of the shell showed large inward bending of the edge beam; this is not the case in the analysis. It is likely that the edge beam has become too stiff in the finite element model for inward bending and that this has restricted the edge movement excessively. For this reason buckling tendencies in the circumferential direction are restrained. Failure of the shell takes place when yielding of the reinforcement in the edge beam has progressed more than half through the beam height. Large deflections then prevail over the shell and give yielding of the reinforcement in the shell body.
Y
1
Experiment
[31]
B
Geometrically
nonlinear
Lc-d
Geometrically
linear
onalysir
analysis
25
I
I
20
30
336klm
Fig. 14. Dimensions of the shell and edge beams.
50
I
VA (mm)
Fig. 13. In-plane displacement at midside point.
-
10
-0,8
40
I
I
I
I
50
60
70
80
Midpoint
deflection
Fig. 12. Central deflection of the slab.
(mm)
578
A.
ARNESEN et al.
Element mesh
_
GeometricoIly linear
-
Geometrically
analyrir
nonlinear
analyrir
Reinforcement:
Real edge beam. 5I
10 I
I 15
Finite element ideolizotion of the edge beam. 20I
I 25
I 30
I
35
I
I
40
45
Deflection
Fig. 15. Load-deflection
6. CONCLUSIONS
The plane stress program is efficient and inexpensive in use. A main reason for developing this program was to find out if good results could be obtained in spite of a crude material model. The results clearly demonstrate that the assumed constitutive law for untracked concrete is of only minor importance for normal structures which fail in cracking and yielding of reinforcement. This conclusion applies to the entire load-deflection curve including the failure load. But some of the results also indicate that a more refined model should be used in case of triaxial stresses (confinement from reinforcement) and crushing of concrete. The studies performed with the plate and shell program also show good agreement with experimental results for the load-deflection curves as well as for the failure load. This indicates that both the geometric and the material nonlinearities are well represented in the program. The endochronic theory can be implemented in a finite element formulation which also considers cracking and yielding of reinforcement bars. Great efforts have been necessary in order to develop an efficient and reliable solution algorithm. The examples studied with the two programs demonstrate that very extensive and accurate information about the nonlinear behaviour of reinforced concrete structures can be obtained using modern computer simulation. Acknowledgement-The authors would like to thank Prof. R. Lenschow for useful discussions. REFERENCES I. A. C. Scordelis and W. Schnobrich, Finite element analysis of reinforced concrete structures. Seminar on Finite Element Analysis of Reinforced Concrete Structures, pp. 61-334. Politecnico di Milan0 (1978).
2. W. Schnobrich, Behavior of reinforced concrete predicted by finite element method. Proc. of the 2nd Nat. Symp. on Computerized
Structural
Analysis
and
Design.
Washington University, Washington, D.C. (1976).
George
ot midrpan
(mm )
curve for the edge beam.
3. P. G. Bergan and 1. Holand, Nonlinear finite element analysis of concrete structures. Jnt. Conf. on finite ElementsI .in.. Nonlinear Mechanics. ISD, University of Stuttgart (1978); Comput. Math. Appl. Mech. Engng 1711443-467 (1979). 4. Z. P. Bazant and P. D. Bhat, Endochronic theory of inelasticity and failure of concrete. ASCE J. Engng Mech. Div.
102,701-722 (1976). 5. Z. P. Bazant and P. D. Bhat, Prediction of hysteresis of reinforced concrete members. ASCE J. Struct. Mech. Div. 103, 153-167(1977). 6 Z. P. Bazant, On endochronic inelasticity and incremental plasticity. Structural Engineering Rep. No. 76-121259. Northwestern University, Evanston, Illinois (1976). 7 P. G. Bergan, G. Horrigmoe, B. Krikeland and T. H. Soreide, Solution techniques for nonlinear finite element problems. Int. J. Numerical Methods in Engng 12, l677-16% (1978). 8. S. I. Sorensen, A. Arnesen and P. G. Bergan, Nonlinear finite element analysis of reinforced concrete using endochronic theory. Finite Elements in Nonlinear Mechanics (Edited by P. G. Bergan et al.). TAPIR, Trondheim(1978). 9. S. I. Sorensen, Endochronic theory in nonlinear finite eiement analysis of reinforced concrete. Rep. No. 78-l. Division of Structural Mechanics. The Norwegian lnstitute of Technology, The University of Trondheim (1978). 10. A. Arnesen, Analysis of reinforced concrete shells considering material and geometric nonlinearities. Rep. No. 79-l. Division of Structural Mechanics. The Norwegian Institute of Technology, The University of Trondheim (1979). Il. K. C. Valanis, A theory of viscoplasticity without a yield surface. Archiwum Mechaniki Stosowanej 3, 517-551 (1971). 12. G. Wempner and J. Aberson, A formulation of inelasticity from thermal and mechanical concepts. Int. J Solids Structures 12, 705-721(1976). 13. Z. P. Bazant, Endochronic inelasticity and incremental plasticity. Int. J. So/ids Structures 14, (1978). 14. J. W. Rudnickiand J. R. Rice, Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371-394(1975). 15. K. C. Valanis, Some recent developments in the endochronic theory of plasticity-the concept of internal barriers. In Constitutiue Equations of Plasticity, AMD-Vol. 21, pp. 15-32. Am. Sot. of Mechanical Engineers, New York (presented at ASME Water Annual Meeting, New York, Dec. 1976).
579
Nonlinear analysis of reinforced concrete 16. K. C. Valanis and H. E. Read, A theory of plasticity for hysteretic materials-I: Shear response. C’omput. Structures 8.503510 (1978). 17. 0. C. Zienkiewicz, G. C. Navak and D. R. J. Owen. Composite and overlay models in numerical analysis of elastoplastic continua. Int. Symp. on Foundations of Plasticity. Warsaw (1972). 18. W. P. Doherty, E. L. Wilson and R. L. Taylor. Stress analysis of ax&symmetric solids using higher order quadrilateral finite elements. Structural Engineeting Laboratory Rep. No. SESM 69-3. Universitv of California. Berkelev (1%9). 19. P. G. Bergan and R. W. Clough, Convergence criteria for iterative processes. AIAA J. 10, 1107-l 108 (1972). 20. K. J. Bathe, E. Ramm and E. L. Wilson, Finite element formulation for large deformation dynamic analysis. ht. 1. Numer.
Methods
in Engng
9.353-386
(1975).
21. T. Belytschko and B. J. Hsieh, Non-linear transient finite element analyses with convected coordinates, Int. J. Numer. Methods
in En,ena 1. 255-271
(1973).
22. T. Soreide, Collapse behaviour of stiffened plates using alternative finite element formulations, Rep. No. 77-3. Division of Structural Mechanics, The Norwegian Institute of Technology, Trondheim (1977). 23. L. H. Donnell. Beams, Plates and Shells. McGraw-Hill, New York (1976). 24. B. M. Irons and A. Razzaque, Shape function formulations for elements other than displacement models. In Variational
Methods in Engineering, Proceedings presented at University of Southampton, Sept. 1972 (Edited by C. Brebbia and H. Tottenham). University Press (1972). 25. A. Razzaaue. Program for triangular bending elements with derivative’ smoothing. Int. J. Nimer. Methods in Engng 6. 332-343 ( 1973). 26. R. J. Allwood and G. M. M. Corners, A polygonal finite element for plate bending problems using the assumed stress approach. Int. J. Numer. Methods in Engng 1, 135-149 (l%9). 27. D. J. Allmann, Triangular finite elements for plate bending with constant and linearly varying bending moments. Proc. WTAM Symp. on High Speed Computing tures. University of Liege (1970).
of Elastic
Struc-
28. B. Bresler and A. C. Scordelis, Shear strength of reinforced concrete beams-series II. SESM Reo. No. 64-2. Universitv of California, Berkeley (1964). 29. L. B. Kriz and C. H. Raths. Connections in precast concrete structures-strength of corbels. PC1 1. 10, 16-61 (1965). 30. E. Jonsson and T. I. Svare. Tests of concrete corbels. NBI Rep. (1976) (in Norwegian). 31. R. Taylor, D. R. H. Maher and B. Hayes, Effect of the arrangement of reinforcement on the behaviour of reinforced concrete slabs. Magazine of Concrete Res. 98, (l%6). 32. A. C. van Riel. W. J. Beranek and A. L. Bouma. Tests on shell roof models of reinforced mortar, Proc. of the 2nd Symp. on Concrete She// Roof Construction. Oslo (1957).