- Email: [email protected]
Nonlinear analysis for concrete frame structures using the finite element method
Comparers & Stnrcfure~ Vol. 48, No. I, pp. 73-79,
0045-7949/93 56.00 + 0.00
1993
~1993PergamonPnssLtd
Printed in Great Britain.
NONLINEAR ANALYSIS FOR CONCRETE FRAME STRUCTURES USING THE FINITE ELEMENT METHOD C. H.
SUN, M. A. BRADFORD and R. 1.
GILBERT
Lkpartment of Structural Engineering, School of Civil Engineering, The University of New South Wales, Kensington, NSW 2033, Australia (Received 21 April 1992) Abstract-A
new finite element layered model used for beam-column elements in concrete frame structures is developed. Using an automatic incremental/iteration process, the model can account for both geometric and material nonlinearity in concrete frame structures. The model is compared with independent experimental results, and a stable and reliable prediction is obtained. The finite element method forms a basis for the efficient analysis of large reinforced concrete frames.
a special large displacement stiffness matrix which is derived to deal with the change of geometry according to the initial undeformed structure rather than the deformed structure. The auto~ti~ incremen~l-iterative solution technique presented by Bergan and Mollested 171 was selected to model the nonlinear behaviour of the concrete frames. This method is applicable for both geometric and material nonlinear problems and is computationally efficient particularly when the loading is less than its ultimate value. At this stage, no attempt is made in the formulation to investigate the behaviour beyond the limit point, since a complete and accurate pre-failure analysis is of principal interest. In accordance with the above-mentioned algorithms, a new computer package for the nonlinear analysis of concrete frame structures has been developed. A self-adjusting loading process forms the basis of the numerical analysis. Each loading stage achieves ~uilib~~ of the total force system by an iterative process. The program is shown to exhibit stable convergence, and good agreement with independent test results.
using
1. INTRODUmION
A significant amount of effort has been devoted to developing methods of nonlinear analysis of concrete frame structures over the last two decades or so. Despite using different computer-o~ented strategies, the main goal has been to model the geometric and material nonlinear behaviour as closely as possible. Two basic types of algorithm, the finite element method and the direct stiffness method, have been chosen to solve nonlinear concrete problems. The finite element method has been used by Kang and Scordelis [I] and by Chajes and Churchill [2], while the direct stiffness method has been used by Lazaro and Richards [3], Bakoss and Corderoy [4] and Wang et al. [5]. The finite element method directly uses stress and strain as the variant, while the direct stiffness method is based on internal forces. The finite element procedure has become the more popular method for the solution of nonlinear problems, and is adopted in this paper. For nonlinear problems that contain geometric and material nonhnearities, stress and strain are the key parameters which lead to a change of all other outcomes, such as stiffness and actions. Because of this, the finite element method is more ‘direct’ than the so-called direct stiffness method when carrying out a nonlinear analysis. A full-range nonlinear analysis of reinforced or prestressed concrete frame structures using the finite element method is presented. A layered model [6l is developed, which involves a basic element type which allows for the nonhomogeneous arrangement of material in any part of the structure and for the changing of material properties through the thickness of the element during loading. The model can also account for longitudinal changes of material properties and the nonprismatic nature of the beam-column element through numerical integration. Geometric nonlinearity is included
2. FORMULATION
In the following formulation, geometrical and material nonlinea~ty are considered simultaneousiy and the same solution technique is adopted to cover both. A layered model for the beam-column element, as shown in Fig. 1, is chosen to model the variation of the material properties throughout the depth of the section. By use of this model, cracking of the concrete can easily be modelled by a simple adjustment to the stress-strain relationship of the cracked concrete. The following theoretical derivation of the nonlinear eq~Iib~um equations follows that presented by Zienkiewicz 181.In Fig. 1, the displacement model 73
c. H.
c
SUN et al.
Sec. A-A
I
1
Fig. 1. Layer model for the beam-column
is expressed in terms of the generalized coordinates q)r as follows: (U)=(U,,+.., u(x) = a, + azx
(la)
u(x) = GJ+ a,x + ff*x2 + &X3.
(lb)
The elements are assumed to sustain only axial and flexural deformation, with shear deformation being disregarded. The axial strain E(X) at level y,for the ith layer in the cross-section in Fig. 1 can be expressed as
Wx)
c(x) = - dx
d2v +Yis+z
I
0
dv z
2
*
(2)
In eqn (2), the first two terms represent the linear effect, while the squared term represents the nonlinear effect. By substituting eqn (I) into (2), the variation of the strain de(x) becomes de(x) = (8){du) = (CR), + W,)fdaL
(3)
(5a)
0%
(5b)
(C) = (0, 0, 0, 1,2x, 3x2).
(5c)
In the above relationships, (B), represents the linear relationship between strain and displacement while (R), depends nonlinearly on the displacement. For the layered model, the constitutive relationship between the range of stress and the change of strain is expressed as da = E, dc,
references present the empirical relationship between stress and strain in concrete from which E, can be derived. For any nonlinear system, ~~lib~urn between the external loads and internal forces should be maintained. By denoting (Rf as the vector of external loads according to the generalized displacement {a), and {f} as the vector of forces induced by member or surface loads, then
@)rcr dV - (ff. IY
(R)=
where V is the volume of the beam-column The variation of {R} then becomes {dR) =
(dR)‘a I
V
dV +
(@Tda
(7) element.
dV.
(8)
s Y
By using eqns (3) and (6), the relations~p of the variations of stress and those of displacement is expressed as
(4)
where
= (~f’
element.
da = E,(B)(da}.
(9)
Since (B), is the independent of displacement, the variation of <8) with respect to the displacements is only related to (R},, so that (da)
= (dB),.
WV
Hence by substituting eqns (9) and (10) into (8) for da and (da), eqn (8) becomes (dB>,Ta dV + [Rl(da),
fdR) = f
(11)
V
where
(6)
where E, is the tangent modulus at a certain state of stress or strain. Many standard texts and
=[K&+[Kb.
(13)
15
Nonlinear analysis of concrete frame structures
By substituting eqn (3) into (12) for (B), the matrices [Kk, and [K],, can be expressed as (14)
In accordance with standard tin&e element transformations, transforming the vector generalized coordinates {a} into the nodal point degrees of freedom {q}, shown in Fig. 2 results in the following relationships
{qh= L+I
(20)
so that
(21) where [Kb represents the usual small displacement stiffness matrix, while [K1, is the large displacement stiffness matrix and contains terms which are both linear and quadratic in {a}. It can be seen from the above equations that the large displacement effect could be included by calculating the matrix [K], according to the generalized displacement {u} at the present state, rather than adjusting the geometry of the deformed structure in the computation of the stiffness. By use of eqn (5b), the first term in (11) can be expressed as
[A]-’ =
1
0
0
0
0
0
I-’
0
0
1-l
0
0
0
1
0
0
0
0
0
0
1
0
0
-21-’
0
31-l
-I-’
I-’
0
-21-3
1-i
0
-31-l
.O
21-3
0
(22) and
s
(dS),Ta dV = [%{da}
(16)
V
in which WI, =
(C)%(C)
dV.
sV
(17)
The matrix [K], is known as the geometric stiffness matrix, and depends on the stress level. Therefore, by using eqns (13) and (16) (11) can be rewritten as
W >= ([40 + Kl, + [KL)(da1 =
Wl,{daL
By using eqn (21) and the principle of contragradient [9] the relationship between the forces {Q ), in the (q}, coordinate system and the forces {R} in the {a} coordinate system can be written as
VQ>,= kWTW) so that the nonlinear becomes
(24)
tangent stiffness relationship
(18) VQI, = Ml-%Wl-‘&I,.
(19)
where [K], is the tangent stiffness matrix in the generalized coordinate system. Using eqn (19) allows the incremental/iterative method to be implemented to solve the problem incorporating both geometric and material nonlinearities.
(25)
Equation (25) is according to local coordinates, and can be transformed into global coordinates by the relation
@QI, = [~~Al-~Kl,[Al-‘[T3(dq}~ = [WdqL
(27)
where the transformation cos0
VI=
Fig. 2. Global and local coordinates.
(26)
matrix [7J is
sin 6
0
0
0
0
-sine
cos6
0
0
0
0
0
0
1
0
0
0
0
0
0
cos0
sin6
0
0
0
0
-sin 8
cos tl
0
0
0
0
0
0
1 (28)
76
C.
H. SUNet al.
Equation (27) is the basic nonlinear equation incorporating both geometric and material nonlinearities, and forms the basis for the incremental loading process in the nonlinear analysis. 3. COMPUTER
PROGRAMMING
The nonlinear analysis is complicated and tedious, but can be undertaken readily using modern highspeed digital computers. For the formulation here, a desktop workstation was used and produced rapid solutions for the comparison analysis presented in the next section. Since the incremental/iterative solution technique is an efficient means with which to trace the nonlinear behaviour, a linear structural analysis program, similar to those presented in standard finite element theory texts, could be applied and modified directly. However, it was found that stress and strain were much more convenient to use as the basic input rather than internal forces such as axial force and bending moment in the frame analysis. This is because a stress-strain-based approach is better suited to handling the nonlinearities. The layer model can trace the variation of material properties through the depth of the cross-section of the beam-column element. However, the variation of geometric and material properties along the x-axis, can also be handled as shown in Fig. 1. In eqns (7) or (15), for example, the method of Gaussian integration [IO] is used to account for the variation of properties along the member. This also allows nonprismatic members with multiple layers of reinforcement to be treated. Despite the existing range of software that could be applied to handle the current algorithms, extensive revisions of these programs needed to be made. For large problems, it is suggested that the ‘dynamic storage method’ could have a great deal of memory and make the revision or expansion of existing programs much easier and reliable. As large numbers of degrees of freedom are inevitably encountered in realistic frame analyses, it is appropriate to use the ‘variable band solution’ technique to save CPU time and memory. If core memory is not able to accommodate the problem, then an out-of-core arrangement could be considered. The procedure for the nonlinear analysis used here is as follows: 1. Set up the tangent stiffness matrix [RI, at the present loading state. This matrix changes gradually at each loading stage. 2. Solve the nonlinear equation eqn (27) for the given incremental loading {AQ}:, and the incremental displacement {q}: is added to the present deformed structure. 3. Use eqn (7) to calculate the internal force vector {R) of the structure at the present stage. Comparing this with the external loading, which is assumed
constant during the deformation step, the unbalanced forces are determined. 4. Using the unbalanced forces and current tangent stiffness matrix [RI,, the equilibrium equation (27) is re-solved. Steps 24 are repeated until the Euclidian norm of the unbalanced forces is sufficiently small. 5. For the next incremental loading {AQ}:“, the whole procedure from steps 1 to 4 is restarted until all incremental loadings have been applied. The automatic incremental/iterative technique converges well while the loading is under the ultimate load. The technique theoretically could pass the limit point without difficulty for a ductile concrete frame structure. A method such as that in [7] would have to be deployed to handle brittle failure. 4. COMPARISON
WITH TEST RESULTS
In 1965, Read [1 l] carried out a series of tests on full-size single-storey reinforced concrete portal frames as shown in Fig. 3. Since flyash was used in the concrete mix, the resulting strength of the concrete was relatively high. Tests on the materials were carried out after the tests were completed, and included lengths of reinforcement and cylinders cut from the concrete. The frame model used for the theoretical analysis is shown in Fig. 4. The tapered beam-column element as shown in Fig. 1 was designed to suit the tapered columns for this portal frame, thereby avoiding a much finer mesh subdivision which would have been symm. :
s3 -
s2 -
Sl II
I
i --iiFig. 3. Shape and size of portal frame (Read [I I]).
Nonlinear analysis of concrete frame structures
r-l 4 - lQ”
0 . .s2’ . . 4-1- 1 #, 8 4 -
2 _ 1”92 _1## a L
,p
381
381
1
4 - $0 4 - 1”
1
6290
_I
6290
4 - ,$,, t-
1 2 - 1” 4 - ,i”
Fig. 4. Analytical model for the portal frame.
Fig. 5. Reinforcement for the portal frame.
necessary if uniform elements had been used. Since Gaussian integration could account for the variation of material properties and stress in the longitudinal direction of each element, accuracy could still be maintained by meshing the elements relatively coarsely. Two different cases, pinned based with vertical loads and fixed based with vertical loads, were chosen to test the accuracy of the program. The same distribution of reinforcement used in the tests was adopted for the computer model, and this is shown approximately in Fig. 5. The material properties used for the concrete and steel were ff = 40 MPa and f, = 240 MPa for the pinned base portal and ff= 45 MPa and h,, = 240 MPa for the fixed based portal frame. The stress-strain curves for the concrete
are shown in Fig. 6, and the constitutive model adopted is given in the Appendix. The reinforcement was assumed to be elastic-perfectly plastic, with a Young’s modulus of 207 x 10r MPa. The theoretical and experimental results are compared in Figs 7 and 8. Also shown in these figures are the results of the mechanism method proposed by Read [ll]. In the early part of the loading history, good agreement is obtained with the test results by accounting for tension stiffening effects. When loading approaches the final stages, the nonlinearities produced from the concrete constitutive relationship and yielding of the steel lead to a better prediction of the bchaviour of the concrete frame than the mechanism approach proposed by Read. The numerical solutions
I
10. t
--E c
Fig. 6. Stress-strain law for concrete.
15fc'
C. H. SUNet
78
al.
kN
0
Experiment
A
Mechanism method
-
Layer model
Central vertical deflection Fig. 7. Pinned-base vertical loads.
were found to be quite rapid on a desktop workstation, and produced a good prediction of the tests. 5. CONCLUSIONS A finite element model of a beam-column element that incorporates both material and geometric non-
linearities has been developed. Material nonlinearity is accounted for by a layer model, while geometric nonlinearity is introduced by the use of a large strain formulation. Gaussian integration enables the layer model to account for continuity of material and geometric properties in the longitudinal direction, rather than referring to values at the centre point of
kN 0
A
0
Experiment
A
Mechanism method
-
Layer model
Central vertical deflection Fig. 8. Fixed-base vertical loads.
19
Nonlinear analysis of concrete frame structures
each element as was done by Kang and Scordelis [l] and Wang et al. [5]. Because of this, the efficiency and accuracy of the algorithms increase dramatically. The method was used to analyse two reinforced concrete frames that were tested independently, and good agreement between theory and test were obtained. The computer program thus provides a very powerful means for analysing ductile concrete frames and can be extended readily to cover prestressed concrete and composite concrete-steel frameworks.
7. P. G. Bergan and E. Mollested, Static and dynamic solution strategies in non-linear analysis. In Numerical Metho& for Non-linear Problems, Vol. 2 (Edited by C. Taylor et al.), pp. 3-17. Pineridge Press, Swansea (1984). The Finite Element Method. 8. 0. C. Zienkiewia, McGraw-Hill (1977). 9. A. S. Hall and R. W. Woodhead, Frame Analysis, 2nd Edn. University of New South Wales Press (1973). 10. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ. 11. J. B. Read, Tests to destruction of full-size portal frames. Technical report TRA/390, Cement and Concrete Association, London (1965).
Acknowledgement-The authors are appreciative of the assistance given by Dr P. W. Kneen of the University of New South Wales for implementing the programs on a Sun workstation in the School of Civil Engineering’s workstation laboratory.
APPENDIX:
CONCRETE CONSTITUTIVE RELATIONSHIPS
REFERENCES
1. Y. J. Kang and A. C. Scordelis, Non-linear analysis of presttessed concrete frames. J. Struct. Div., ASCE 106, 445-462 (1980). 2. A. Chajes and J. E. Churchill, Non-linear frame analysis by finite element methods. J. Struct. Engng, AXE 113, 1221-1223 (1987). 3. A. L. Lazaro and R. Richards Jr, Full-range analysis of concrete frames. J. Srruct. Div., ASCE 99, 1761-1783 (1973). 4. S. L. Bakoss and H. J. B. Corderoy, Analysis of non-linear behaviour of reinforced concrete frames. Conference on Computers and Engineering, Institution of Engineers, Australia, pp. 8489 (1983). 5. K. W. Wang, M. F. Yeo and R. F. Warner, Analysis of non-linear concrete structures by deformation control. First National Structural Engineering Conference, Melbourne, pp. 181-185 (1987). 6. R. I. Gilbert, Time-dependent behaviour of structural concrete slabs. Ph.D. thesis, University of New South Wales (1979).
(I =
0,
L < 6”
u = E,c,
0 < 6 < t, cr < c < 106,,
o=a0+a,c,+a,c3+a,62, where E, = 0.0025,
e, = 0.005
EC=25
x5 c, = c C,
60
6,
=
f, 4
,
a, = 1.28,
f,
=
0.6,k
a, = -0.311
a, = 0.0324, a, = -0.00141.
0045-7949/93 56.00 + 0.00
1993
~1993PergamonPnssLtd
Printed in Great Britain.
NONLINEAR ANALYSIS FOR CONCRETE FRAME STRUCTURES USING THE FINITE ELEMENT METHOD C. H.
SUN, M. A. BRADFORD and R. 1.
GILBERT
Lkpartment of Structural Engineering, School of Civil Engineering, The University of New South Wales, Kensington, NSW 2033, Australia (Received 21 April 1992) Abstract-A
new finite element layered model used for beam-column elements in concrete frame structures is developed. Using an automatic incremental/iteration process, the model can account for both geometric and material nonlinearity in concrete frame structures. The model is compared with independent experimental results, and a stable and reliable prediction is obtained. The finite element method forms a basis for the efficient analysis of large reinforced concrete frames.
a special large displacement stiffness matrix which is derived to deal with the change of geometry according to the initial undeformed structure rather than the deformed structure. The auto~ti~ incremen~l-iterative solution technique presented by Bergan and Mollested 171 was selected to model the nonlinear behaviour of the concrete frames. This method is applicable for both geometric and material nonlinear problems and is computationally efficient particularly when the loading is less than its ultimate value. At this stage, no attempt is made in the formulation to investigate the behaviour beyond the limit point, since a complete and accurate pre-failure analysis is of principal interest. In accordance with the above-mentioned algorithms, a new computer package for the nonlinear analysis of concrete frame structures has been developed. A self-adjusting loading process forms the basis of the numerical analysis. Each loading stage achieves ~uilib~~ of the total force system by an iterative process. The program is shown to exhibit stable convergence, and good agreement with independent test results.
using
1. INTRODUmION
A significant amount of effort has been devoted to developing methods of nonlinear analysis of concrete frame structures over the last two decades or so. Despite using different computer-o~ented strategies, the main goal has been to model the geometric and material nonlinear behaviour as closely as possible. Two basic types of algorithm, the finite element method and the direct stiffness method, have been chosen to solve nonlinear concrete problems. The finite element method has been used by Kang and Scordelis [I] and by Chajes and Churchill [2], while the direct stiffness method has been used by Lazaro and Richards [3], Bakoss and Corderoy [4] and Wang et al. [5]. The finite element method directly uses stress and strain as the variant, while the direct stiffness method is based on internal forces. The finite element procedure has become the more popular method for the solution of nonlinear problems, and is adopted in this paper. For nonlinear problems that contain geometric and material nonhnearities, stress and strain are the key parameters which lead to a change of all other outcomes, such as stiffness and actions. Because of this, the finite element method is more ‘direct’ than the so-called direct stiffness method when carrying out a nonlinear analysis. A full-range nonlinear analysis of reinforced or prestressed concrete frame structures using the finite element method is presented. A layered model [6l is developed, which involves a basic element type which allows for the nonhomogeneous arrangement of material in any part of the structure and for the changing of material properties through the thickness of the element during loading. The model can also account for longitudinal changes of material properties and the nonprismatic nature of the beam-column element through numerical integration. Geometric nonlinearity is included
2. FORMULATION
In the following formulation, geometrical and material nonlinea~ty are considered simultaneousiy and the same solution technique is adopted to cover both. A layered model for the beam-column element, as shown in Fig. 1, is chosen to model the variation of the material properties throughout the depth of the section. By use of this model, cracking of the concrete can easily be modelled by a simple adjustment to the stress-strain relationship of the cracked concrete. The following theoretical derivation of the nonlinear eq~Iib~um equations follows that presented by Zienkiewicz 181.In Fig. 1, the displacement model 73
c. H.
c
SUN et al.
Sec. A-A
I
1
Fig. 1. Layer model for the beam-column
is expressed in terms of the generalized coordinates q)r as follows: (U)=(U,,+.., u(x) = a, + azx
(la)
u(x) = GJ+ a,x + ff*x2 + &X3.
(lb)
The elements are assumed to sustain only axial and flexural deformation, with shear deformation being disregarded. The axial strain E(X) at level y,for the ith layer in the cross-section in Fig. 1 can be expressed as
Wx)
c(x) = - dx
d2v +Yis+z
I
0
dv z
2
*
(2)
In eqn (2), the first two terms represent the linear effect, while the squared term represents the nonlinear effect. By substituting eqn (I) into (2), the variation of the strain de(x) becomes de(x) = (8){du) = (CR), + W,)fdaL
(3)
(5a)
0%
(5b)
(C) = (0, 0, 0, 1,2x, 3x2).
(5c)
In the above relationships, (B), represents the linear relationship between strain and displacement while (R), depends nonlinearly on the displacement. For the layered model, the constitutive relationship between the range of stress and the change of strain is expressed as da = E, dc,
references present the empirical relationship between stress and strain in concrete from which E, can be derived. For any nonlinear system, ~~lib~urn between the external loads and internal forces should be maintained. By denoting (Rf as the vector of external loads according to the generalized displacement {a), and {f} as the vector of forces induced by member or surface loads, then
@)rcr dV - (ff. IY
(R)=
where V is the volume of the beam-column The variation of {R} then becomes {dR) =
(dR)‘a I
V
dV +
(@Tda
(7) element.
dV.
(8)
s Y
By using eqns (3) and (6), the relations~p of the variations of stress and those of displacement is expressed as
(4)
where
= (~f’
element.
da = E,(B)(da}.
(9)
Since (B), is the independent of displacement, the variation of <8) with respect to the displacements is only related to (R},, so that (da)
= (dB),.
WV
Hence by substituting eqns (9) and (10) into (8) for da and (da), eqn (8) becomes (dB>,Ta dV + [Rl(da),
fdR) = f
(11)
V
where
(6)
where E, is the tangent modulus at a certain state of stress or strain. Many standard texts and
=[K&+[Kb.
(13)
15
Nonlinear analysis of concrete frame structures
By substituting eqn (3) into (12) for (B), the matrices [Kk, and [K],, can be expressed as (14)
In accordance with standard tin&e element transformations, transforming the vector generalized coordinates {a} into the nodal point degrees of freedom {q}, shown in Fig. 2 results in the following relationships
{qh= L+I
(20)
so that
(21) where [Kb represents the usual small displacement stiffness matrix, while [K1, is the large displacement stiffness matrix and contains terms which are both linear and quadratic in {a}. It can be seen from the above equations that the large displacement effect could be included by calculating the matrix [K], according to the generalized displacement {u} at the present state, rather than adjusting the geometry of the deformed structure in the computation of the stiffness. By use of eqn (5b), the first term in (11) can be expressed as
[A]-’ =
1
0
0
0
0
0
I-’
0
0
1-l
0
0
0
1
0
0
0
0
0
0
1
0
0
-21-’
0
31-l
-I-’
I-’
0
-21-3
1-i
0
-31-l
.O
21-3
0
(22) and
s
(dS),Ta dV = [%{da}
(16)
V
in which WI, =
(C)%(C)
dV.
sV
(17)
The matrix [K], is known as the geometric stiffness matrix, and depends on the stress level. Therefore, by using eqns (13) and (16) (11) can be rewritten as
W >= ([40 + Kl, + [KL)(da1 =
Wl,{daL
By using eqn (21) and the principle of contragradient [9] the relationship between the forces {Q ), in the (q}, coordinate system and the forces {R} in the {a} coordinate system can be written as
VQ>,= kWTW) so that the nonlinear becomes
(24)
tangent stiffness relationship
(18) VQI, = Ml-%Wl-‘&I,.
(19)
where [K], is the tangent stiffness matrix in the generalized coordinate system. Using eqn (19) allows the incremental/iterative method to be implemented to solve the problem incorporating both geometric and material nonlinearities.
(25)
Equation (25) is according to local coordinates, and can be transformed into global coordinates by the relation
@QI, = [~~Al-~Kl,[Al-‘[T3(dq}~ = [WdqL
(27)
where the transformation cos0
VI=
Fig. 2. Global and local coordinates.
(26)
matrix [7J is
sin 6
0
0
0
0
-sine
cos6
0
0
0
0
0
0
1
0
0
0
0
0
0
cos0
sin6
0
0
0
0
-sin 8
cos tl
0
0
0
0
0
0
1 (28)
76
C.
H. SUNet al.
Equation (27) is the basic nonlinear equation incorporating both geometric and material nonlinearities, and forms the basis for the incremental loading process in the nonlinear analysis. 3. COMPUTER
PROGRAMMING
The nonlinear analysis is complicated and tedious, but can be undertaken readily using modern highspeed digital computers. For the formulation here, a desktop workstation was used and produced rapid solutions for the comparison analysis presented in the next section. Since the incremental/iterative solution technique is an efficient means with which to trace the nonlinear behaviour, a linear structural analysis program, similar to those presented in standard finite element theory texts, could be applied and modified directly. However, it was found that stress and strain were much more convenient to use as the basic input rather than internal forces such as axial force and bending moment in the frame analysis. This is because a stress-strain-based approach is better suited to handling the nonlinearities. The layer model can trace the variation of material properties through the depth of the cross-section of the beam-column element. However, the variation of geometric and material properties along the x-axis, can also be handled as shown in Fig. 1. In eqns (7) or (15), for example, the method of Gaussian integration [IO] is used to account for the variation of properties along the member. This also allows nonprismatic members with multiple layers of reinforcement to be treated. Despite the existing range of software that could be applied to handle the current algorithms, extensive revisions of these programs needed to be made. For large problems, it is suggested that the ‘dynamic storage method’ could have a great deal of memory and make the revision or expansion of existing programs much easier and reliable. As large numbers of degrees of freedom are inevitably encountered in realistic frame analyses, it is appropriate to use the ‘variable band solution’ technique to save CPU time and memory. If core memory is not able to accommodate the problem, then an out-of-core arrangement could be considered. The procedure for the nonlinear analysis used here is as follows: 1. Set up the tangent stiffness matrix [RI, at the present loading state. This matrix changes gradually at each loading stage. 2. Solve the nonlinear equation eqn (27) for the given incremental loading {AQ}:, and the incremental displacement {q}: is added to the present deformed structure. 3. Use eqn (7) to calculate the internal force vector {R) of the structure at the present stage. Comparing this with the external loading, which is assumed
constant during the deformation step, the unbalanced forces are determined. 4. Using the unbalanced forces and current tangent stiffness matrix [RI,, the equilibrium equation (27) is re-solved. Steps 24 are repeated until the Euclidian norm of the unbalanced forces is sufficiently small. 5. For the next incremental loading {AQ}:“, the whole procedure from steps 1 to 4 is restarted until all incremental loadings have been applied. The automatic incremental/iterative technique converges well while the loading is under the ultimate load. The technique theoretically could pass the limit point without difficulty for a ductile concrete frame structure. A method such as that in [7] would have to be deployed to handle brittle failure. 4. COMPARISON
WITH TEST RESULTS
In 1965, Read [1 l] carried out a series of tests on full-size single-storey reinforced concrete portal frames as shown in Fig. 3. Since flyash was used in the concrete mix, the resulting strength of the concrete was relatively high. Tests on the materials were carried out after the tests were completed, and included lengths of reinforcement and cylinders cut from the concrete. The frame model used for the theoretical analysis is shown in Fig. 4. The tapered beam-column element as shown in Fig. 1 was designed to suit the tapered columns for this portal frame, thereby avoiding a much finer mesh subdivision which would have been symm. :
s3 -
s2 -
Sl II
I
i --iiFig. 3. Shape and size of portal frame (Read [I I]).
Nonlinear analysis of concrete frame structures
r-l 4 - lQ”
0 . .s2’ . . 4-1- 1 #, 8 4 -
2 _ 1”92 _1## a L
,p
381
381
1
4 - $0 4 - 1”
1
6290
_I
6290
4 - ,$,, t-
1 2 - 1” 4 - ,i”
Fig. 4. Analytical model for the portal frame.
Fig. 5. Reinforcement for the portal frame.
necessary if uniform elements had been used. Since Gaussian integration could account for the variation of material properties and stress in the longitudinal direction of each element, accuracy could still be maintained by meshing the elements relatively coarsely. Two different cases, pinned based with vertical loads and fixed based with vertical loads, were chosen to test the accuracy of the program. The same distribution of reinforcement used in the tests was adopted for the computer model, and this is shown approximately in Fig. 5. The material properties used for the concrete and steel were ff = 40 MPa and f, = 240 MPa for the pinned base portal and ff= 45 MPa and h,, = 240 MPa for the fixed based portal frame. The stress-strain curves for the concrete
are shown in Fig. 6, and the constitutive model adopted is given in the Appendix. The reinforcement was assumed to be elastic-perfectly plastic, with a Young’s modulus of 207 x 10r MPa. The theoretical and experimental results are compared in Figs 7 and 8. Also shown in these figures are the results of the mechanism method proposed by Read [ll]. In the early part of the loading history, good agreement is obtained with the test results by accounting for tension stiffening effects. When loading approaches the final stages, the nonlinearities produced from the concrete constitutive relationship and yielding of the steel lead to a better prediction of the bchaviour of the concrete frame than the mechanism approach proposed by Read. The numerical solutions
I
10. t
--E c
Fig. 6. Stress-strain law for concrete.
15fc'
C. H. SUNet
78
al.
kN
0
Experiment
A
Mechanism method
-
Layer model
Central vertical deflection Fig. 7. Pinned-base vertical loads.
were found to be quite rapid on a desktop workstation, and produced a good prediction of the tests. 5. CONCLUSIONS A finite element model of a beam-column element that incorporates both material and geometric non-
linearities has been developed. Material nonlinearity is accounted for by a layer model, while geometric nonlinearity is introduced by the use of a large strain formulation. Gaussian integration enables the layer model to account for continuity of material and geometric properties in the longitudinal direction, rather than referring to values at the centre point of
kN 0
A
0
Experiment
A
Mechanism method
-
Layer model
Central vertical deflection Fig. 8. Fixed-base vertical loads.
19
Nonlinear analysis of concrete frame structures
each element as was done by Kang and Scordelis [l] and Wang et al. [5]. Because of this, the efficiency and accuracy of the algorithms increase dramatically. The method was used to analyse two reinforced concrete frames that were tested independently, and good agreement between theory and test were obtained. The computer program thus provides a very powerful means for analysing ductile concrete frames and can be extended readily to cover prestressed concrete and composite concrete-steel frameworks.
7. P. G. Bergan and E. Mollested, Static and dynamic solution strategies in non-linear analysis. In Numerical Metho& for Non-linear Problems, Vol. 2 (Edited by C. Taylor et al.), pp. 3-17. Pineridge Press, Swansea (1984). The Finite Element Method. 8. 0. C. Zienkiewia, McGraw-Hill (1977). 9. A. S. Hall and R. W. Woodhead, Frame Analysis, 2nd Edn. University of New South Wales Press (1973). 10. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ. 11. J. B. Read, Tests to destruction of full-size portal frames. Technical report TRA/390, Cement and Concrete Association, London (1965).
Acknowledgement-The authors are appreciative of the assistance given by Dr P. W. Kneen of the University of New South Wales for implementing the programs on a Sun workstation in the School of Civil Engineering’s workstation laboratory.
APPENDIX:
CONCRETE CONSTITUTIVE RELATIONSHIPS
REFERENCES
1. Y. J. Kang and A. C. Scordelis, Non-linear analysis of presttessed concrete frames. J. Struct. Div., ASCE 106, 445-462 (1980). 2. A. Chajes and J. E. Churchill, Non-linear frame analysis by finite element methods. J. Struct. Engng, AXE 113, 1221-1223 (1987). 3. A. L. Lazaro and R. Richards Jr, Full-range analysis of concrete frames. J. Srruct. Div., ASCE 99, 1761-1783 (1973). 4. S. L. Bakoss and H. J. B. Corderoy, Analysis of non-linear behaviour of reinforced concrete frames. Conference on Computers and Engineering, Institution of Engineers, Australia, pp. 8489 (1983). 5. K. W. Wang, M. F. Yeo and R. F. Warner, Analysis of non-linear concrete structures by deformation control. First National Structural Engineering Conference, Melbourne, pp. 181-185 (1987). 6. R. I. Gilbert, Time-dependent behaviour of structural concrete slabs. Ph.D. thesis, University of New South Wales (1979).
(I =
0,
L < 6”
u = E,c,
0 < 6 < t, cr < c < 106,,
o=a0+a,c,+a,c3+a,62, where E, = 0.0025,
e, = 0.005
EC=25
x5 c, = c C,
60
6,
=
f, 4
,
a, = 1.28,
f,
=
0.6,k
a, = -0.311
a, = 0.0324, a, = -0.00141.