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TRANSPORT and ROAD RESEARCH LABORATORY
Department of the Environment
TRRL SUPPLEMENTARY REPORT 51 UC
SOME APPROXIMATIONS IN THE NON-LINEAR ANALYSIS OF RECTANGULAR PLATES USING FINITE ELEMENTS
by
M A Crisfield BSc PhD
Any views expressed in this Report are not necessarily those of the Department of the Environment
Bridge Design Division Structures Department
Transport and Road Research Laboratory Crowthorne, Berkshire
1974
Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on I st April 1996.
This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.
CONTENTS
Abstract
1. Introduction
1.1 Poor in-plane elements
1.2 Non-conformity
1.3 Incompatible orders of accuracy
1.4 Nodal averaging at boundary points
1.5 Continuous and discontinuous stresses
2. Reduced integration
2.1 In-plane functions
2.1.1 Shape functions
2.1.2 Full strain matrix
2.1.3 Reissner's variational principle
2.1.4 Examples
2.2 Out-of-plane functions
2.2.1 Curvatures
2.2.2 Slopes
2.2.3 Integration mesh
2.2.4 Examples
3. Nodal averaged versus Gauss point stresses
3.1 Boundary stresses
3.2 Continuous and discontinuous stresses
3 .2 .10den ' s method
3.2.2 Out-of-balance forces
3.2.3 Tests for in-plane elements
3.2.4 Tests for out-of-plane elements
3.2.5 Some conclusions
3.3 Gauss point stresses
4. Non-conformity
4.1 Examples relating to the iterative correction
5. Proposed approach
5.1 Examples
6. Conclusions
7. Acknowledgements
8. References
9. Notation
9.1 Subscripts
9.2 Vectors
9.3 Matrices
10. Appendix 1 - Ilyushin's yield criterion
10.1 Kirchoff assumptions
10.2 Equivalent strain
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10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
i0.13
10.14
Stress-strain laws
Stress resultants
Stress resultant-strain laws
Quadratic stress intensities
Quadratic stress intensity-strain intensity laws
Equivalent strain ratios
Ideal plastic behaviour
The integrals J 1, J2 and J3
Non-dimensional stress intensities
Full yield criterion
Approximate yield criterion
Limits
10.14.1 In-plane only
10.14.2 Bending only
10.14.3 One dimensional case
10.14.4 Discussion
11. Appendix 2 - Full and reduced 'strain' matrices for bending element
11.1 Displacement function
11.2 Slope matrix
11.3 Curvature matrix
11.4 Reduced curvature matrix
11.5 Reduced slope matrix
12. Appendix 3 -- Element tangent stiffness matrix from Reference 1
12.1 Sub-matrices
12.2 Corrigendum to Reference 1
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© CROWN COPYRIGHT 1974 Extracts f rom the tex t may be reproduced, except for commercial
purposes, pro vided the source is acknowledged.
SOME APPROXIMATIONS IN THE NON-LINEAR ANALYSIS OF RECTANGULAR PLATES USING FINITE ELEMENTS
ABSTRACT
In earlier work the formulation of the large deflection behaviour of elasto- plastic plates has been described. The approximations that were made are examined in detail and alternative methods have been developed and tested. In some cases, the aim of these changes is a more economic solution; in others, it is a more exact analysis.
1. INTRODUCTION
Finite element models for the analysis of the large-deflection elasto-plastic behaviour of imperfect steel plates have recently been developed. Since such behaviour involves the complex interaction of material and geometric non-linearities, many simplifying assumptions are required to ensure that the computation is economic. In the earlier work 1, simple shape functions are used for the rectangular elements. These functions are bilinear for the in-plane displacements and of a restricted quartic form for the out-of-plane deformations. The latter shape functions lead to a non-conforming bending element 2. The significant simplifying assumptions used in the analysis of Reference 1 are:
(a) Ideal elastic - perfectly plastic behaviour (with no strain hardening)
aw aw (b) Only moderately large deflections so that ax , ay ~ 1.
(c) The use, in some cases, of an approximate yield criterion given by llyushin 3 (See Appendix 1)...:- This yield criterion is a function of the six generalized stress resultants N x, Ny, Nxy, Mx; My and Mxy , and pre-supposes a sudden plastification of the plate section.
(d) The assumption, for the formation of the 'geometric stiffness matrix' , that the in-plane stress resultants vary in a bilinear manner through the averaged nodal values.
(e) The assumption that the constituent terms of the elasto-plastic modular matrices vary bilinearly between the modal values, the latter being calculated from the averaged nodal 'stresses'.
(0 The use of a lower order (four-by-four) Gaussian integration scheme for the computation of the element tangent stiffness matrix than that strictly required (even allowing for assumptions (d) and (e)):.
(g) The assumption of bilinear variations of the total stress resultants (M) and (N) (through the averaged nodal values) when calculating the internal force vector and hence the out-of-balance load vector.
In the light of these approximations and also in view of the chosen shape functions, the following criticisms could be made of-the approach adopted in Reference I.
* The notation used throughout the report is given in Section 9.
1.1 Poor in-plane elements
The bilinear in-plane element is known to give particularly poor results for the analysis of in-plane bending problems unless a large number of elements are employed.
1.2 Non-eonformiW
The use of a non-conforming bending element throws doubts on the convergence with increasing fineness of mesh. Some of the work dissipated between the elements is not accounted for in the formulation.
1.3 Incompatible orders of accuracy
The use of lower order (reduced) numerical integration (assumption ( f ) ) makes it somewhat illogical to retain all the higher order terms in the 'strain matrices' relating the 'strains' (including curvatures and slopes) to the nodal deflections. This point is particularly pertinent with regard to the high order slope matrix:
The retention of the higlaer order terms is also illogical in view of assumptions (d) and (g).
1.4 Nodal averaging at boundary points
While the nodal averaging process leads to a satisfactory representation of the true stresses at the internal nodes, the method is less satisfactory at the external boundaries.
1.5 Continuous and discontinuous stresses
The derivation of the tangent stiffness matrix for the structure is based on discontinuous strains (as obtained by differentiation of the shape functions) while the internal load vector is derived from an assumed continuous stress field which passes through the averaged nodal values. This inconsistency means that, before iteration, small out-of-balance loads will exist even for linear problems.
These points are examined below and, where necessary, alternative approaches are investigated and modifications proposed. Some of the issues considered are illustrated by linear examples. The improvements, however, will also lead to better non-linear solutions.
2. REDUCED INTEGRATION
2.1 In-plane functions
The poor performance of the bilinear in-l~lane shape functions, when applied to in-plane bending problems, has been discussed in Section 1.1. Taig & Kerr'+,proposed that the element be improved by artificially modifying the shear modulus. This procedure is similar to reduced numerical integration 5, 6 of the shear energy which is designed to eliminate the 'parasitic' shear strains set up when the element is subject to in-plane bending (Figure 1)*. These parasitic shear strains do not exist at the centroid of the element so that, if numerical integration is used, a one point (centroid) integration scheme (for the shear strain energy) will by-pass this unwanted energy.
In practice, instead of using reduced numerical integration, it is more convenient to modify the strain matrix which relates the strains to the nodal displacements. The process may then be seen to be the equivalent of a hybr id element model 7, 8 based on Reissner's variational principle 9-
* 1"his Figure is from Reference 5.
2
2.1.1 Shape functions The bilinear displacement functions are:
where a i = ¼ (1 + ~'i~') (1 + 7/i 77)
• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . .
and ~'i and r/i take the respective nodal values (Figure 2) for i = 1,4
(2.1)
i.e. ~'i =* - - 1 , - - 1 , 1, 1
r/i =~ --1, 1 , -1 , 1
(u} and (v} are vectors containing the nodal displacements.
2.1.2 Full strain matrix Differentiation of equation (2.1) gives
{o}-- 3'xy
f ~)u
a__Ev = ay
au+av i)y ax
= [ H I { u}v ................................................. (2.2)
where [HI is the full strain matrix given by:
[HI = 0
1 T
{oil} and { %~} are given by:
afi = ¼ fi (1 + r/i r/)
ar/i = ¼ r/i (1 + ~'i ~')
0
1 T .................................................. (2.3)
The strains{g} can be Seen to vary linearly with x and y. In particular the shear strain 7xy has a bilinear distribution.
2.1.3 Reissner's variational principle It will now be shown that Reissner's variational principle may be used to derive an element identical to that obtained by reduced integration of the shear strain energy. Reissner's functional is
T f = f A t ( { o } T ( g ) _ ½ ( o } T [ C ] - I ( o } ) d A _ ( U} ( ~ } ......................... (2.4)
{U} { : t ( ) aXy] where are the nodal forces corresponding to the nodal displacements , o are the stresses, Cry T X
and [C] is the modular matrix which for an isotropic plate is given by:
[c ] I 1 v 0 1 = - - u 1 0 l-v2 0 0 1--v
2
...................................................... (2 .5)
3
f 1 Equation (2.2) provides a relationship between the strains and the nodal displacements. The stresses ! o~ are assumed to be related to 'nodal' values by:
Ox -- } o, -- ( ° , )
rxy = r o
where ~fOx'lf, ~fOy~ and r 0 are the 'nodal' stresses k ~ k 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 . 6 )
i.e. bilinear distributions are assumed for o x and o. while the shear stress rxy is assumed to be constant within Y
the element. Substituting equations (2.2) and (2.6) into equation (2.4), and applying variation (at the element level) to equation (2 .4)wi th respect of the 'nodal' stresses ( O x ) , ( O y ) and w o, leads to a relationship between the 'nodal' stresses and the nodal displacements. ox} (:)
Oy = [GI
T O
Further substftution into equation (2.6) gives the implicit relationship,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 . 7 )
E 1 T v T ] Ox- E v T 1 T
Oy = l - v : (a (o~-) { u ) + ~ (otr/) ( v ) )
E 1 1 T
( 2 . 8 )
The resulting distribution of stress is, for o x and o3i, identical to the conventional explicit relationship
{ : } ....................................................... { o ) = ~c]
which would be used in a Potential Energy formulation. For the shear stress Wxy, however, the implicit compa- tibility relationship is that obtained by replacing the last row of the 'full' strain matrix [HI with a constant term that coincides with the original bilinear function at the centroid.
I n } If, now, functional (2.4) were to be varied with respect to the nodal displacements and equation (2.7)
v used to relate the 'nodal' stresses to the nodal displacements, a conventional set of element stiffness equations would be obtained,
( U } = [Ke] ( : } ....................................................... (2.10)
The element stiffness matrix [Ke] is identical to that given by the conventional total potential energy formulation if the last orw of the 'full' strain matrix [H] were to be altered to that given by equation (2.8). Hence
[Kel = fA [ ~ ] T [C] [HI d a ...................................................... (2.11)
where [H] is the modified (or 'reduced') strain matrix.
2.1.4 Examples In this section some examples are given that show the improved displacements and stresses that accrue from the use of the 'reduced' element. Figure 3 shows a cantilever analysed under three separate loadings. These loadings are:
4
(i) a point load at the tip
(ii) a pure moment at the tip
and (iii) a uniformly distributed line load.
The errors in the predicted tip deflection are shown in Table 1. The "exact' solutions are taken to be those given by beam theory with corrections made for shear deformation.
TABLE 1
Percentage error in tip deflection for a cantilever
LOADING TYPE
ELEMENT TYPE
MESH
Point Load
Full
-42.1
-39.1
Reduced
- 1 0 . 0
- 5 . 3
Pure Moment
Full
-41 .3
- 3 8 . 9
Reduced
-8 .9
-3 .5
Full
--39.4
--37.4
- 1 4 . 5
U.D.L.
Reduced
- 8 . 3
- 3 . 6
- 2 . 5
For each loading case, the 'reduced element' solutions are significantly superior to those given by the 'full element ' . The 'reduced element' , when used with mesh 1, gives an error of 8.9 per cent for the tip deflection of a cantilever when subject to a pure moment. The computed result is, however, an exact 'plane strain' solution (with Ey = O) i.e. the answer is a factor of 1 lower than that predicted by beam theory (which has Oy = 0). The predicted bending
1 - u 2
stresses are exact but vertical stresses Oy are computed as ua x instead of zero. As the mesh is refined, the computed results tend towards the correct 'plane stress' solution.
The improved stresses given by the 'reduced element' are clearly evident from Figure 4 which shows the calculated shear force and bending moment diagrams for the cantilever subject to a point load at its tip. The shear forces given by the 'reduced element' are exact while those given by the 'full element ' are very poor. It should be noted, however, that these very poor shear forces are considerably improved by nodal averaging at the interior nodes. Unfortunately, the large error at the cantilever root cannot be improved in this manner.
2.2 Out-of-plane functions
2.2.1 Curvatures Double differentiation of the restricted cubic shape functions gives curvatures X x and ×y that vary linearly with x and y (Appendix 2). The distribution of twisting curvature ×xy is, however, quadratic. The desirability of reducing the order of the terms in the strain matrices' has been discussed in Section 1.3. In the present context, this means reducing the quadratic distribution of Xxy to a linear distribution. This ' reduct ion ' may be simply achieved by numerically integrating the twisting strain energy with a two-by-two Gaussian integration scheme, instead of the three-by-three scheme that is strictly required. It is, however, again more convenient to modify the 'strain' (curvature) matrix at source.
Reissner's variational principle can once more be used for this purpose. Bilinear distributions are assumed for the moments M x, My and Mxy , whilst the curvatures are derived by double differentiation of the restricted quartic shape functions. Variations with respect to the moment variables are applied to Reissner's functional at the element level. This leads to an implicit relationship between the moment variables and the nodal displacements. For the moments M x and My, this implicit distribution is identical to the explicit relationship
I )fWe} .............................................
5
that is the basis of the total potential energy model. [F] is the curvature matrix relating the curvatures to the nodal displacements and [D] is the modular bending matrix while (We) is the vector of element nodal displace- ments. The relationship, given by Reissner's variational principle between the twisting moment and the nodal displacements is the same as that obtained if the quadratic distribution of equation (2.1 2) is replaced by a linear distribution that coincides with the quadratic distribution at the four two-point Gauss stations. If the IF] matrix is so modified, and the resulting 'reduced strain' matrix used in a potential energy forrrrulation, the derived stiffness matrix is identical to that given by the hybrid model. The reduced curvature (×xy)R is related to the
full curvature (Xxy)F by:
4 . .(Xxy) R = ¾ Z j=l
1 ( + (Xxy)Fj ............................................. (2.13)
~ a n d 7/. are the non-dimensional co-ordinates of the nodes, and (×x ")F: takes the values of the full function at e fou~ consecutive Gauss stations (Figure 2). In effect all the quadrYatiJc terms ~2 and ,12 are replaced by the
constantS3.
2 .2 .2 S l o p e s The slope matrix [B],
where
{s} = aw = [B] w e
•
............................................. (2.14)
is of order as high as cubic. (Appendix 2). To make this matrix more consistent with the assumed bilinear variation of in-plane stress resultant, these terms should be 'reduced' to be linear with x and y. This is once more achieved by replacing the original function with a bilinear distribution that coincides with the former at the four (two-point) Gauss stations. In the present case (see Appendix 2) the effect is to replace the full cubic terms ~3 and 77 3 by the linear terms ~'/3 and r~/3 respectively. The quadratic terms are again replaced by the
constant 3 a-.
2 . 2 . 3 I n t e g r a t i o n m e s h The'reduced strain' matrices allow the use of lower order numerical integration for the tangent stiffness matrices. Gaussian integration is used so that a polynomial of degree 2n-1 may be exactly integrated using n integration stations. The required number of integration stations is shown in the
following table.
TABLE 2
Required integration mesh for the tangent stiffness matrix
Type of problem
In-plane only - elastic or elasto-plastic
Bending only - elastic or elasto-plastic
Linear buckling (eigenvalue) with constant stress (elastic)
Large-deflection elasto-plastic
Type of element
Full
2 x 2
3 x 3
4 x 4
7 x 7
( 4 x 4 )
Reduced
2 x 2
2 x 2
2 x 2
3 x 3
It is assumed in the above Table that the elasto-plastic formulation is based on a bilinear distribution of the consti tuent terms of the tangential modular matrices 1 . The 'full element' approach for the large-deflection elasto- plastic problems relates to the method given in Reference 1. The tangent stiffness matrix is given in Appendix 3.
6
For economy, a four-by-four integration mesh was used in place of the seven-by-seven mesh.
Z 2 . 4 Examples While the modifications made to the strain matrix of the in-plane element were designed specifically to improve its accuracy, those made to the slope and curvature matrices for the bending element were designed to reduce the complexity of the integration. Tests were therefore Conducted on simple examples to investigate the effect of these latter changes on the accuracy of the solutions.
Figures 5 and 6 show some results from thesmall-deflection bending analysis of square plates in which both the full and the reduced elements were employed. (The chain dotted line 'reduced with iterative correction' in Fig. 6 refers to a further modification which will be discussed in Section 4). It can be seen that, for the plates supported along the sides (Figure 5), the reduced element gives slightly worse solutions for some of the loadings. The departure from the full element solutions is, however, small and vanishes as the mesh is refined. (No results have been plotted for the simply supported plate subject to a uniformly distributed load (U.D.L.), since the difference between the two solutions was negligible in this case). For the plate supported only at the corners (Figure 6), the "reduced element' slightly improves the solution. In all cases, the solutions furnished by the reduced element appear to be converging with increasing fineness of mesh. This point was amplified by checking the element against the patch test 10. (Further details on this test are given in Section 4).
To investigate the effect of using the reduced slope matrix, a linear buckling (eigenvalue) problem was analysed. Such a solution can be obtained by finding the minimum value of X that satisfies the equation:
where ;k is the amplification factor on the in-plane stresses that is required for neutral equilibrium (instability). [Ko] is the ordinary small deflection stiffness matrix and [Kg] is the 'geometric stiffness matrix' which is given
by ~
[Kg] = fA [BIT [~1 [B] dA ............................................. (2.16)
where [~] is a matrix holding the basic values of the in-plane stress resultants I . In the current work, X is found, by monitoring the magnitude and sign of the determinant of the stiffness matrix [K]. Provided the first infinity is not passed 11, the critical load lies between loads which give positive and negative values for the determinant. For a simply supported square plate subject to constant shear 12 : .
9.34 rr2E (~_)2 " (2.17) rcrit = 1 2 ( 1 - v 2)
The percentage errors given by the persent method for two different meshes are:
Mesh 4 x 4 6 x 6
% Error in rcrit -10 .7 - 7 . 0
Working to three significant figures, there was no difference between the solutions obtained using the full slope matrix and those derived by means of the reduced slope matrix.
3. NODAL AVERAGED VERSUS GAUSS POINT STRESSES
The points raised in Sections 1.4 and 1.5 are considered here and an alternative to nodal averaging is investigated.
3.1 Boundary stresses
The first criticism of the nodal averaging spproach is that it gives unsatisfactory stresses at the boundaries. (Section 1.4). This is particularly true of the in-plane shear stresses when the full bilinear in-plane element is employed. However, the examples considered in Section 2.1.4. show that this situation is radically improved if
7
the reduced strain matrix is used for the in-plane element. A further improvement could be achieved by calculating and storing 'stresses' at mid-side positions for boundary elements and at the centroid for corner
elements (Figure 7).
3.2 Continuous and discontinuous stresses
3 . 2 . 1 0 d e n ' s m e t h o d A 'consistent' continuous stress field based on nodal values has been derived by Oden et a113 using the theory of conjugate approximations 14. A 'fundamental matrix '13 is formed for the whole structure and this matrix must subsequently be inverted. In structural terms the same results could be obtained
as follows:
(1) Derive the displacements in the usual manner (from a potential energy formulation).
(2) Set up a Reissner functional 9 in which both stresses and displacements are variables with the stresses assumed to vary continuously through the unknown nodal values.
(3) Apply variations to this functional with respect to the stress unknowns so that, after inversion of a structure flexibility matrix (similar to Oden's fundamental matrix), a relationship is obtained at the structural level between the stress variables and the nodal displacements.
(4) Obtain the stresses by substituting the nodal displacements obtained in step (1) into the stress- displacement relationship found in step (3).
This process is very time-consuming and from the results given in Reference 14 appears to give solutions that are very similar to the nodal averaged values - particularly if the mid-element stresses are used for the boundary
elements.
3 . 2 . 2 O u t - o f - b a l a n c e f o r c e s An inconstistency has been noted in Section 1.5 between the tangent stiffness matrix which is based on discontinuous strains (obtained from the shape functions), and the internal load vector ~-P}which is derived in Reference 1 from a continuous stress field passing through the averaged nodal stresses. This inconsistency means that, before iteration, small out-of-balance loads will be present even for linear problems.
The following questions are therefore posed:
(1) If this out-of-balance load vector is applied in an iterative manner to the original stiffness matrix using
a modified Newton-Raphson approach,
{ d - .............................................
will the results converge so that {Ap} ~ 07
(The term { P} - { P } in equation (3.1) is the out-of-balance load vector-)
(2) To what values will the results converge?
The process will certainly converge for very fine meshes since in the limit there will be no difference between the discontinuous and the continuous stress fields. For coarse meshes however, there does not appear to be a theoretical answer to these questions and consequently resort will be made to test examples.
3.2.3 Tests for in-plane elements The first set of tests relates to linear elastic in-plane analysis using both the full bilinear element and the "reduced shear' element (Section 2.1). The results of the tests which were applied to the cantilever of Figure 3 are shown in Figures 8 to 10. Maximum and average nodal errors have been plotted
in Figures 9 and 10.
The error at node i is deffmed as
8
° c ) i - (°a)i ] ............................................. (3.2) e i = (Oa) ~
Subscript c refers to the computed value while subscript a refers to the exact (beam theory) value and (Oa) r is the beam theory stress at the cantilever root. The average error is merely the sum of the errors e i at the nodes divided by the number of nodes.
For all the cases analysed, the iterative process appears not only to converge but also to improve the solutions. This improvement is most marked for the 'full element' particularly where the mid-element values are used for the stresses in the boundary elements. The results that are shown relate to a cantilever subject to a point load at its tip. A similar pattern of behaviour was observed for other loadings which included a pure moment at the tip and a uniformly distributed load.
3.2.4 Tests for out-of-plane elements The convergence of the iterative procedure (equation 3.1) was checked for linear small-deflection bending problems by analysing a range of square plates using both the full and reduced curvature matrices (Section 2.2.1.). Both simply supported and fully clamped boundary conditions were applied and the plates were subject to uniformly distributed loads and central point loads respectively. The mesh divisions were all uniform and ranged from a four-by-four to a fourteen-by-fourteen mesh for the whole plate. Iterations were applied until the maximum deflection change, (from(Ap} Equation ( 3 . 1 ) ) was within some specified percentage (convergence criterion) of the maximum total deflection (from ( p )
where { p } = [ K ] - I { P ) + 1=, "~" {Ap}
and n is the number iterations). The following iable summarises the results.
TABLE 3
Changes in deflection
CONVERGENCE CRITERION < 1 %
No. of iterations
n
Max total change
(fr°m i~i {AP} )
< 0 . 1 %
No of iterations
n MESH
COARSE (up to 6 x 6) 1-2 < (+) 6% Did not always converge
FINE (8 x 8 to 14 x 14) 1-2 < (+) 2% 1--3 < 2%
Max total change
(from i~i {AP} )
It can be seen that for practical purposes (i.e., with a convergence criterion of less than one per cent change as used in the non-linear work of Reference 1), convergence occurred in all cases. For a medium mesh (say 8 x 8) there was little total change from the initial solutions (between one and two per cent). For the coarse meshes, however, absolute convergence did not always occur. A typical result for a four-by-four mesh would be:
Iteration number, i 1 2 3 4
Percentage change 5.0 0.25 0.24 0.23
with convergence either taking place very slowly (and to a patently false solution) or else not at all. For practical purposes, this lack of absolute convergence would not be significant in a non-linear problem since the iterative changes due to the non-linearities would predominate. The small change (of the order of 0.25 per cent) caused by the nodal averaging process at the second iteration would be insignificant so that the next load increment would be applied instead of proceeding with the iteration. The lack of convergence for the coarse meshes reflects the difference between the original discontinuous moment field and the assumed continuous field through the averaged
9
nodal moments. For freer meshes this difference vanishes and the iterative process converges.
3 . 2 . 5 S o m e e o n e l u s i o n s In Section 3.2, the question was raised 'Does the iterative process always converge for linear problems?' In the light of the results given in Section 3.2.4 it is clear that whilst for practical purposes convergence is achieved, in absolute terms no guarantee can be given for convergence when coarse meshes are employed. It is therefore advisable to look for some alternative method for storing the stresses and calculating the internal load vector. This will be discussed below.
3.3 Gauss point stresses
An alternative to the use of nodal averaged stresses is to use (and store) the stresses at the Gauss points. This method is theoretically attractive since it allows for a discontinuous distribution of stress, thereby enabling the internal load vector to be calculated in a manner that is consistent with the derivation of the stiffness matrix. The approach is often used in small deflection elasto-plastic problems 15 in which the stresses, stored at the Gaussian integration stations are used at these points to calculate the elasto-plastic modular matrices. For large- deflection elasto-plastic problems, however, where the strain energy contains high order terms in x and y, the method would lead to a significant increase in both storage and computer time. In the present case, even with the reduced strain matrices of Section 2, a three-by-three Gaussian integration scheme is required to form the tangent stiffness matrix. With Ilyushin's yield criterion, six stress resultants must be stored at each integration station. If von Mises' yield criterion is adopted in conjunction with a volume integral approach, the problem is amplified by integration through the depth. For a plate divided into N x N elements (with (N + 1) x (N + 1) nodes), the storage space required for the stresses would be:
Nodal averaged Gauss point
6(N 2 + 2N + 1 .) 54N 2
if Ilyushin's yield criterion were used.
The increase in storage would be accompanied by an increase in time as the yield criterion would have to be tested and the elasto-plastic modular matrices calculated at all the Gauss points. If a volume integral approach was used with five integration stations through the depth of the plate then the required storage space would be:
Nodal averaged Gauss point
1 5 ( N 2 + 2N +1) 135N 2
It is apparent that some form of compromise is required between the two approaches. If the constituent terms of the tangent stiffness matrix are inspected (Appendix 3), it is clear that for an elastic element subject to large-deflection behaviour, all the terms may be integrated using two-by-two Gaussian integration with the
exception of the term
fA [BIT [ T s I T [C] [TS] [B] dA
which strictly requires three-by-three integration if the reduced strain matrices are employed. This expression contains the most highly non-linear terms and it would seem reasonable to integrate it in an approximate manner using two-by-two integration along wlth the other terms. It would therefore be logical to store the stresses at the four two-point Gauss stations thus implying a bilinear distribution of 'stress'. Internal load vectors~P~ calculated on this basis would be entirely consistent with the linear stiffness matrices if the reduced curvature matrix of Section 2.2. l was used for the bending element. With this scheme there would be no out-of-balance forces for
linear problems.
For elasto-plastic elements it is proposed that the stresses again be stored at the four two-point Gauss stations, the elasto-pla.stic n~odular matrices being calculated at these points. The constituent terms of these modular matrices, ( [~] , [DI and [cd] in Appendix 3) will be assumed to have a bilinear distribution which now passes through the values at the two-by-two Gauss stations instead of through the nodes as assumed in
10
Reference 1. Inspection of the constituent terms of the tangent stiffness matrix (Appendix 3) shows that a three- by-three Gaussian integration scheme will be adequate for the elasto-plastic elements.
The proposed method is therefore as follows:
(1) Store all stresses (or stress resultants) at the four two-by-two Gauss stations and monitor these points for plasticity.
(2) If the element is completely elastic, use two-by-two integration for the tangent stiffness matrix.
(3) If any one of the two-by-two Gauss stations is plastic, use three-by-three Gaussian integration. The "F *
values of the in-plane stress resultant matrix [N] and the modular matrices [C], [I~] and [cd] (at these points) will then be obtained by bilinear interpolation from the values at the two-by-two Gauss stations.
The storage required for the stresses using this procedure is shown below:
Nodal averaged Present procedure
Area approach (Ilyushin) 6(N 2 + 2N + 1) 24N 2
Volume approach (yon Mises with 5 stations thro' the depth) 15(N2 + 2N + 1) 60N 2
It would seem that the proposed method is probably viable for the area approach but despite certain theoretical limitations, the nodal averaged procedure is, on economic grounds, more appropriate for the volume approach.
4. NON-CONFORMITY
The adopted restricted quartic shape functions lead to a non-conformity of inter-element slope. This non-cbnformity raises doubts concerning the satisfaction of both the compatibility and the equilibrium e~uations. Full compatibility can, however, be ensured in the limit (as the mesh is refined) by means of the patch test 10. This test is applied to a small patch of elements which is given boundary displacements consistent with some arbitrary state of constant strain (curvature). The free central node should then deform so as to satisfy this strain exactly. This test, for non- conforming elements, is the logical extension of the constant strain criterion for conforming elements. It is argued that in the limit, as the mesh is refined, there will be no stress (moment) variation over the patch. For convergence to be possible, discontinuity effects must vanish in these limiting conditions so that the elements conform and the constant strain is exactly satisfied. Fortunately the present non-conforming bending element satisfies the patch test both with the full curvature matrix and the reduced matrix of Section 2.2.1.
A method to improve the implicit equilibrium satisfaction of non-conforming elements is given in a paper which is in preparation. A brief summary of the approach is presented here. If a virtual work formulation is applied to the small-deflection bending problem, the following equilibrium equation is given:
T Ps = Z J" [F] T M d A + Z Ws fs (50nl + 50n2) MnavdS (4.1) elements boundaries
where Mna v is the average normal moment across an inter-element boundary while ( M } are the moments inside the element. If the displacements and moments are calculated using the conventional non-conforming approach, these moments may be subsequently substituted into equation (4.1) so that an internal load vector( P~f may be derived. This internal load vector will not coincide with the original external load vector, since the initial non-conforming formulation does not include the work dissipated across the inter-element boundaries. An out-of-balance load vector may therefore be derived which can be applied in an interative manner to equation (3.1) until convergence is achieved. Since this iterative cycle is a fundamental part of the present non-linear analysis, the addition of the line integral terms in equation (4.1) will add very little extra computer time.
11
4.1 Examples relating to the iterative correction
The effect of this iterative correction on small-deflection bending problems was investigated by re-analysing the examples of Section 2.2.4. In every case, convergence occurred very rapidly - a change of less than 0.1 per cent in either the maximum or Euclidean norm deflection being achieved at the second iteration. All significant change, therefore, took place on the first iteration with the second only serving as a check. (This second iteration would be unnecessary if, instead of using the deflection change as a convergence criterion, the out-of-balance forces were used). The results of the convergence tests (with increasing fineness of mesh) are shown in Figures 6 and 11. It is clear that the iterative correction significantly improves all the solutions.
5. PROPOSED APPROACH
The'proposed method of analysis includes both the reduced strain matrices of Section 2 and the use of the two-by- . two Gauss stations for storing the stresses (Section 3.3). The iterative correction of Section 4 is also included to
improve the implicit equilibrium satisfaction given by the non-conforming bending elements.
5.1 Examples
Two examples are given which illustrate the application of the present approach to the non-linear problems. Both involve the uniaxial compression of simply supported imperfect plates. In each case, comparisons are made with the solutions previously given in Reference 1 as well as with theoretical results due to other workers.
The first plate analysed was elastic and the results are given in Figure 12. The classical solutions are due to Yamaki 16 who used a double trigonometric series with four coefficients to solve Marguerre's 17 fundamental equations. Results are given for two different boundary conditions relating to the unloaded edges. In one case the edge is free to pull in; in the other the edges are restrained to remain straight so that there is no resultant transverse edge load. For the finite element solutions, the latter boundary condition is achieved by adding a rigid bar element to the unloaded edge of the plate. An incremental loading process was adopted. To avoid a build-up of errors, modified Newton-Raphson iterations were applied after each load increment until there was less than one per cent change in the total central deflection. 1
The second example includes elasto-plastic behaviour and involves a plate previously analysed by Moxham 18 using a Ritz procedure. (Comparisons with experimental results 19 and finite element solutions obtained using a volume integral approach are given in Reference 1). The finite element solutions shown in Figure 13 are based on Ilyushin's yield criterion and, as anticipated, the predicted collapse loads are higher than those given by Moxham since the latter solutions allow for the progressive growth of plasticity from the plate surfaces. It should be noted, however, that Moxham's series method predicts a critical buckling load for the plate which is some nine per cent less than the true solution while the finite element result is only lower by 1.7 per cent 1.
The present finite element method gives slightly higher collapse loads than those given in Reference 1. This difference is mainly due to the current use of Gauss station stresses which, in some respects, are less satisfactory than nodal stresses for this plate. The dimension of the present plate are critical in the sense that the squash load and the critical buckling loads coincide. For such a plate, collapse follows very rapidly after the first attainment of "full section yield' (as defined by Ilyushin's yield criterion) 1 . 'Yield' first occurs in the centre of the plate 1 and is instantly monitored by the nodal averaging process which has 'stress points' at the centre. In the present approach, however, there are no 'stress points ' at the centre and further load must be applied until the first
Gauss point yields.
Residual (welding) stresses are included in the analysis by means of an assumed rectangular stress-block distribution 20. This approach was followed strictly by Moxham and also in the present approach. In Reference 1, however, the tensile stresses at the edge o f the plate were assumed to take a triangular form with the net residual forces remaining the same. This triangular distribution was dictated by the nodal averaging assumptions
which require a continuous stress distribution.
12
Despite the critical nature of the example, the differences between the two finite element solutions are small. The average computing time per increment (with two modified Newton-Raphson interations) was, however, reduced from 70 sees. (Central Processor Unit time) to 53 seconds. For this example, therefore, the savings due to the reduced integration order outweigh the losses due to the larger number of 'stress points'.
6. CONCLUSIONS
Finite element methods for the large-deflection analysis of imperfect elasto-plastic plates have been presented in earlier work. In the present report, these methods have been modified so that the 'area approach' based on Ilyushin's yield criterion is both more economic and theoretically better established. The modifications will therefore be adopted for the 'area approach'. For the 'volume approach' (based on von Mises yield criterion), a nodal averaging method is retained and reduced strain functions employed. It is considered that in this case the adoption of the Gauss stress-points would lead to solutions that are prohibitively uneconomic.
7. ACKNOWLEDGEMENTS
The work described in this Report was carried out in the Bridge Design Division (Head of Division Dr G P Tilly) of the Structures Department of TRRL. The author would like to thank Dr L C P Yam (Head of Structural Analysis Section) for his help and encouragement.
2.
~3.
4.
8. REFERENCES
CRISFIELD, M A. Large-deflection elasto-plastic buckling analysis of plates finite elements. Transport and Road Research Laboratory Report LR 593, Crowthorne, Berks, 1973.
ZIENKIEWICZ, O C and Y K CHEUNG. The finite element method for the analysis of elastic isotropic and orthotropic slabs. Proc. Instn. Cir. Engrs. 28, 1964, pp 471-488.
ILYUSHIN, A A Plasticite, Edititions Eyrolles, Paris, 1956.
TAIG, I C and KERR, R I. Some problems in the discrete element representation o f aircraft structures. Matrix methods of structural analysis. Edited by B Fraeijs de Veubeke, Pergamon Press, 1964, pp 267-316.
5. ZIENKIEWICZ, O C, TAYLOR, R L and TOO, J M. Reduced integration techniques in general analysis of plates and shells. Int. Journal for num. meth. in Engng, Vol. 3, 1971, pp 275-290.
6. PAWSEY, S F and R W CLOUGH. Improved numerical integration of thick shell finite elements. Int. Journal for num. meth. in Engng., Vol. 3, 1971, pp 575-586.
7. PLAN, T H H and P TONG. Basis of finite element methods for solid continua. Int. Journal num. meth. in Engng., 1, 1, 1969, pp 3-28.
.
.
10.
CONNOR, J. Mixed models for plates. Cahp 5 of Finite Element Techniques in Struct. Mech. Edited by H Tottenham and C Brebbia, Southampton University Press, 1970.
REISSNER, E. On a variational theorem in Elasticity J Math. Phys, 1950, pp 90-95.
IRONS, B M and A RAZZAQUE. Shape functions for elements other than displacement models. Variational methods in Engng., Vol. 1, Edited by C Brebbia and H Tottenham, Southampton University Press, 1973, pp 4159-4172.
11. GREGORY, M. Elastic instability pub. E & F N Spon Ltd., London, 1967.
* Sometimes spelt ILIOUCHINE 13
12. BULSON, P S. The stability of flat plates, pub. Chatto and Windus, London 1970.
13. ODEN, J T and BRAUCHLI, H J. On the calculation of consistent stress distributions in finite element applications. Int. Journal num. meth. in Engng., 3, 1971, pp 317-325.
14. ODEN, J T. Finite elements of non-linear continua, pub. McGraw-Hill, 1972.
15. NAYAK, G C and O C ZIENKIEWlCZ. Elasto-plastic stress analysis. A generalisation for various constitutive relationships including strain softening. Int. Journal num. meth. in Engng. Vol. 5, 1972,
pp 113-135.
16. YAMAKI, N. Post-buckling behaviour of rectangular plates with small initial curvature loaded in edge compression. Journal of Applied Mech., 26, 3 Sept. 1959, pp 407-414.
17. MARGUERRE, K. Zur Theorie der gekrummten platten formanderung Proc. Fifth Int. Congress of App. Mech., Cambridge, 1938, pp 93-101.
18. MOXHAM, K E. Theoretical prediction of the strength of welded steel plates in compression. Cambridge University Report No. CUED/C - struct/TR2, 1971.
19. MOXHAM, K E. Buckling tests on individual welded steel plates in compression. Cambridge University Report No. CUED/C - struct/TR3, 1971.
20. DWIGHT, J B and MOXHAM, K E. Welded steel plates in compression. The Structural Engineer, Vol. 43,
No. 2, Feb 1969, pp 49-66.
21. HENCKY, H. Zur Theorie plastrischer Deformation und der mier durch im Material hervorgerufenen Nachspannungen Z. ang. Math. Mech., 4, 1924, p 323.
a
A
b
D
e
e i
eil
el2
eio
e s
exx, eyy, exy
E
9. NOTATION
length of plate or half length of element
area
Width of plate or half width of element
flexural rigidity
error
equivalent strain
value o f e i at z = --hi2
value of e i at z = h/2
minimum value of e i
yield strain
strains at depth z
Young's modulus
Section 10 only
14
f
h
H 1, H 2, HI2
J1, J2' J3
M 1, M 2, M12
Mx, My, Mxy
m
m 1 , m 2, m12
n
N
PE, PX, PE×
PS' PH' PSH
S x, Sy, Sxy
S 1, S 2, S12
S
t
T 1 , T 2, T12
Ts
t
t 1 , t 2, t12
U~ V~ W
X, y , Z
Xx, Yy, Xy, Yx
Z o
3'
6 o
functional
depth of plate
linear combinations of M 1' M2' M 12
integrals
hending moments per unit width t Section 10 only
bending moments per unit width
uniaxial yield moment
non-dimensional value of M s
non-dimensional values of M 1' M2' M 12
number of nodes
i Section 10 only
number of elements in x or y direction
quadratic equivalent strain intensities
quadratic equivalent stress intensities
devioteric stresses
linear combinations of T 1 , T 2, T12
Section 10 only
surface
plate thickness
in-plane stress resultants
uniaxial yield force
non-dimensional value of T s
non-dimensional forms of T 1 , T 2, T 12
Section 10 only
deflections in x, y and z directions
coordinates
stresses Section 10 only
value of z at which e i = eio Section 10 only
shear strain
increment or virtual displacement such as 8w or ~50
initial imperfection at centre of buckle measured from initial medium plane
15
E
%
O.
%
o R
oi
%
O n
Onl
On2
ro
v
x
9.1 Subscripts
a
av
c
cn t
c
i
n
o
s
9.2 Vectors
16
direct strain
uniaxial yield strain
direct stress
uniaxial yield stress
residual compression in centre of plate non-dimensionalised w.r.t, o o
equivalent stress ~ Section 10 only
J uniaxial yield stress
slope a__ww where n is outward directed normal an
rotation O n on one side of boundary
rotation O n on the other side
shear stress
'nodal' shear stress
Poisson's ratio
non-dimensional forms of x, y
amplification factor
curvature
exact
average
computed or central
critical
refers to the dement
refers to the node number (or else in-plane)
normal
out-of-plane or value at yield
refers to the structure
bending moments
{P)
{ut {v t {w} {ot (o t (o4 {o}
{ °x~ ' (°Y)
9.3 Matrices
[BI
[B]
[Cl
[C], [D] & [cd]
[D]
[F]
[HI
[I~]
[K]
[KE]
[Kg]
[K o ]
[K e ]
generalised nodal forces - increments (AP)
generalised internal nodal forces
generalised nodal displacements - increments { Ap}
• i0w0w t slopes with s = , ax ay
in-plane nodal deflections
in-plane nodal forces
generalised out-of-plane nodal displacements
generalised out-of-plane nodal forces
in-plane shape functions
differentials w.r.t. ~" and r/respectively of (c~)
linear in_plane strains with ~ )T = 13UxU av ' ay
'nodal' stresses
au + a_y_v} 3y ax
slope matrix
'reduced' slope matrix
in-plane modular matrix
elasto-plastic modular matrices
out-of-plane modular matrix
curvature matrix
'reduced' curvature matrix
in-plane linear strain matrix
'reduced' in-plane linear strain matrix
stiffness matrix
tangent stiffness matrix
'geometric' stiffness matrix
linear stiffness matrix
element stiffness matrix
17
[Kii], [Kool, [Kio]
[TS]
sub-matrices of [KE]
matrix of in-plane stress resultants
total slope matrix
- aw XX O
O ay
aw aw ay ax
I N x
Nxy Nxy 1 Ny
18
10. APPENDIX 1
llyushin's yield criterion
A French translation (from the original Russian) of Ilyushin's treatise on plasticity is given in Reference 2. Unfortunately, no English translation appears to be available and, further, Reference 2 is now out-of-print. Since the present work (including Reference 1) relies heavily on Uyushin's yield criterion, this Appendix contains a summarised English translation of the relevant sections of Reference 2. The original notation is generally used but no attempt has been made to produce an exact translation. Instead a precis is p resen ted along with some further discussion.
10.1 Kirchoff assumptions
If plane sections remain plane,
exx = E 1 -- zXl
eyy = E 2 -- z× 2 ~, ............................................. (10.1 )
exy 2 = E l 2 - zXl2 _,
where exx, e and e x (the 'engineering' shear strain) are the strains at depth z below the centre of the plate YY Y while El , E~ and E12 are the values of these strains at z = 0. X~, X2 and X~ 2 are the curvatures.
10.2 Equivalent strain
The equivalent strain e i is defined as:
e i =~-- / e x x 2 +eyy 2 + exxeyy + ¼ e x y 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( l O . 2 )
_ 2 / ( 10 .3 ) or ei - 4~ PE - 2z PE× + z2 P× .............................................
where P e P× and PE× are quadratic 'equivalent strain intensities' given by"
PE = Et2 + El E2 + E22 + E122
PX = Xl2 + Xi X2 + X22 + Xl2 z ~ .......................................... (10.4)
PEx = EIX l + ½EIx2 + ½E2XI + E2X2 +E~a2X~
10.3 Stress-strain laws
i.e.
The adopted stress-strain laws are based on deformation theory as given by Hencky 21.
S x = X x - ½ Y y ei ( E l - z × 1)
_ °i Sy = y y - ½ X x - ~ ( E 2 - z × 2)
2 °i Sxy = Xy - 5 ei ( E I 2 - z x 12 )
(10.5)
X x, Y and X (= Yx) are the stresses while S x, S and S x are their deviatoric components, e i is the equivalent y - ~ y Y
strain as given by equation (10.2) while o i is an equivalent stress given by:
°i = / Xx 2 + yy2 _ Xx y y + 3Xy2 ............................................. (10.6)
19
10.4 Stress resultants
The in-plane stress resultants are T~, T2 and T12- They are obtained by integration of X x, Yy and Xy
through the thickness of the plate, h.
f hI2 i.e. Tl = --h/2 Xx dz ............................................. (10.7)
with similar expressions for Yy and Xy. The bending moments are obtained by expressions of the form:
h/2 (10.8) -- z X x dz ............................................. M, f h [2
Instead o f working directly with the stress resultants T l, T2, Tt 2, Mr, M2 and M l 2, it is computationally more convenient to use the following linear combinations:
Sl = Tl -- ½ T2 ~ ............................................. (10.9)
$2 = T2 -- ~ Tl
3 Sl2 = -~ Tl2
Hi = MI - ½ M2
H2 = M2 -- ½ Ml
and
• 3 MI2 H12 = ~
10.5 Stress resultant-strain laws t ............................................. (10.10)
and
where
Integration of equation (10.5)gives:
Sl = Jl E1 -- J2XI
$2 = Jl E2 2_ j2×2
S12 ---- J l El2 -- J2XI2
HI = J2 ~1 - J3X1
H2 = J2 ~ 1 - - J3X2
H12 = J2 ~12 - J3X12
) )
............................................. (10.1 1)
............................................. (10.12)
h/2 0 i h[2 h/2 0 i J t = f h / 2 ( ~ - i ) d z ; J 2 = fh [2 ( e - ~ ) zdz; J3 = [h]2 ( ~ - i ) z2dz .......................................... (10.13)
10.6 Quadratic stress intensities
The following quadratic stress intensities are defined:
2 0
= 2 +S 1 S2 +522 +S122 = ~ ( T I 2 +T22 - -Tl T2 +3T122) Ps S t
2 +M2 2 M! M2 +3M12 2) P H = H I 2 +Hi H2 +H2 2 +HI2 2 =¾(M=
P s H = S I Hi +$2 H2 +1,6S1 H2 +½82 HI +S12 Ht2
= 3A(Tt M1 +T2 M2 - ½ J ' t M2 - ½ T 2 Mt +3Tt2 M12)
10.7 Quadratic stress intensity-strain intensity laws
....................................... (10.14)
Substituting equations (10.11) and (10.12) into equation (10.14) gives:
PS =Jr2 PE-- 2J i J2 PE×+J22 P×
PH =J22 PE-- 2J2 J3 PEX+ J32 PX
PSH = JI J2 PE - (JIJ3 + J22) PEX + J2 J3 PX
10.8 Equivalent strain ratios
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 0 . 1 5 )
The equivalent strains at the top surface (eil at z = -h /2 ) and the bot tom surface (ei2 at z = h/2) are
h 2 =A J + h % + P× eil ~-
2 ~ h 2 ei2 = ~ PE- -hPE x + ~ PX
obtained from equation (10.3) as:
) Also from equation (10.3), the minimum (absolute)value of e i is
eio =4- 5 x d~p× PE P× - PE× 2
at z = z o = PEX
PX
............................................. (10.16)
............................................. (10.17)
Two fundamental equivalent sfrain ratios are defined as
h - ei2 / eil
tt = e i__9_o eil
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( l O . 1 8 )
Using equation (10.3), the three quadratic strain intensities PE, P- and P=. are expressed in terms of the two ratios ;~ and/a and the equivalent strain on the top surface, eil . The relationships are:
2 PEX = 3eil A 1 A
8h
PX = 3ei12 Ai 2 4h 2
k 2 (41u2 + A 2 ) PE = eil
............................................. (10.19)
21
where AtA == (v/1--/'t21--X2A, + /X2- - t t2 ) t ............................................. (10.20)
111 equation (10.20) and subsequently throughout this Appendix the top sign occurs when
-h /2 ~< z o ~< h/2
This condition is referred to as 'bending dominant'. The lower ( - ve in the current case) sign occurs when
z o > h / 2 o r z o < - h / 2
This condition is referred to as 'in-plane dominant'.
10.9 Ideal plastic behaviour
It is assumed that the magnitude of the equivalent strain e i is throughout the section very much greater than the limit of proportionality, e s.
i . e . e i >> e s
and hence the equivalent stress o i (equation (10.6) ) is everywhere equal in magnitude to the uniaxial yield stress o s which is assumed to be constant (no strain hardening).
10.10 The integrals J 1, J2, and J3
With the aid of equation (10.3) and the ideal plastic assumptions given above, the integrals J1, J2 and Ja (equation (10.13)) may be evaluated as:
JI - h x ~bo s eil A~
h 2 A ~ O s h 2 J 2 = + tp . 0 s
2Ai2 eil e i lAl 2
h 3 J3 - (X--/"t2 ~k)°s + ~ °s + Aha~q o 6A1 ell 4Ala eil Al3 eil s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( l O . 2 1 )
where A~ and A have been related to the fundamental ratios ~ and/a in equation (10.20) and ~0, ~ and X are so related by:
~o = ~.-- 1 t qj= ] log e ( l + J l - . 2 ) + l°ge ( X + J • 2 - , 2 . 1 I /1 /~
10.11 Non-dimensional stress intensities
............................................. (10.22)
It is convenient to non-dimensionalise the quadratic stress intensities PS' PH and PSH (equation (10.14) ) by introducing the uniaxial yield force,
T s = h o s
2 2
and the uniaxial yield moment
h 2 M s = ~ - °s
The in-plane stress resultants Tl, T2, T12 (equation 00.7) ) and the bending moments MI, M2, Ml 2 (equation (10.8)) may therefore be non-dimensionalised as t i, t2, t 12 and m i, m2, m 12 respectively where, for instance T }
tl -- --k T s (10.23)
Mi and m ~ - Ms
The non-dimensional forms of PS' PH and PSH are Qt' Qm and Qtm where:
_ 4Ps Qt - - 2 3T s
4P H Qm- 2 3M s
4PsH Qtm - 3TsM S
- t l 2 + t 2 2 - t , t2 + 3 t 1 2 2 "~
2 + m 2 2 2 - ml - m , m2 +3ml2
= tt m~ - - ½ t t m 2 - - ½ t2 ml + t2 m2 + 3t~ 2 m~ 2
................................. 00.24)
10.12 Full yield criterion
If equations (10.19) and (10.21) are substituted into equation (10.15), the quadratic stress intensities PS' PH and PSH are found to be dependent on the thickness of the plate h, the yield stress o s and the two strain intensity ratios X and/a (equations (10.18) ). The dependence on o s and h can be eliminated by replacing PS' PH and PSH by the non-dimensional forms Qt' Qm and Qtm (equations (10.24)) so that
I ~ 2 ~b 2 + ~02) Qt = ~,2
=4 (/22 022 +A2)~ 2 +(4//2 + A2) ~02 +2/22 A~0~ --2/22 ~ X + 2 A t p X + X 2) .. . . . . . . . . . . . . . . . . (10. 251) Qm AI4
_ 2 a ~p2 /a2 Q t m - ~ (/a2 A~2 + A + ~ ~k + ~oX) "
In the three dimensional space of the variables Qt' Qm and Qtm equations (10.251 ) represent the parametric form of a yield surface
F (Qt' Qm' Qtm) = 0 .............................................. (1'0.25)
10.13 Approximate yield criteria
The yield surface (equation (10.25)) intersects the plane Qtm = 0 (in the positive sector of Qt, Qm' Qtm space).at
= /a2 4 I Qt 1-/~2 l°ge 2 (1 + ~ l - / a 2) ) Qm = ( ~ _ l°ge ( 1 + J / a l - . ) _ l _ , f T . 7 ~ 2 ) u 2 1 2 ............................................. (10.26)
2 3
The curve given by equations (10.26) is shown in Figure 14 along with the straight line (in Qt' Qm space)
Qt + Qm = 1 ............................................. (10.27)
It is clear that equation (10.27) is a good approximation to equation (10.26).
Equation (10.25) can be shown to be symmetric with respect to the plane Qtm = 0 and the Schwartz inequality may be used to show that Qtm is limited by
Qtm < Qt Qm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 o . 2 8 )
It can further be shown that the maximum value of Qtm is
I I ) Qtm 3 4"3 ............................................. (10.29)
which occurs at Qt = 1/3, Qm = 4/9
These latter equations are strictly only valid for the 'bending dominant' condition (See Section 10.8). For this condition, a good approximation to the yield surface of equation (10.25) is given by two planes passing through the straight line (10.27) and through the points (10.29). Such an approximate yield surface is given by:
Qt + Qm + ~-3 Qtm = 1 ............................. ................ (10.30)
10.14 Limits
Three particular cases are investigated. Two of these were looked at by Ilyushin; the third was not.
10.14.1 In-plane only If there is no bending,
XI =X2 =X1 2 =0
so that eil = ei2 = eio
a n d X = la = 1
Substituting into equation (10.251 ) gives:
Qt = Qm = 0
and Q t = t l 2 +t22 - t s t2 +3t122 = 1 ............................................. (10.31)
which is the yon Mises' yield criterion. The same function is given by the approximate yield criterion of equation (10.30) (and also by equation (10.27)).
10.14.2 Bending only If there are no in-plane strains,
eil = ei2
eio =tz= 0
and X= 1
24
Substituting into equation (10.251 ) gives:
Qt = Qtm = 0
Qm=m~ 2 +m22 - m ~ m2 +3m~22 = 1
............................................. (10.32)
The same criterion is given by the approximate yield surface of equation (10.30) (as well as that of equation (10.27)).
10.14.3 One dimensional case (uniaxial) . *
L e t t = t j , t 2 = t t 2 = O
and m = m l , m 2 m12 = 0
Then/a - 0 (equation (10.18)) and from equations (10.251 )
Qt = t2 = - I + X - ( 1 - )~2
) Qm = m2 - 16)~ 2 (1 + ),)*
Qtm = Qt Qm
so that the yield surface (equivalent to equations (10.251 ) ) is
t 2 + l m l =1
for bending dominant. For in-plane dominant,
............................................. (10.34)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( l O . 3 5 )
or
Qt = 1 while Qm = Qtm = 0
t=_+l ............................................. (10.36)
These equations are shown graphically in Figure 15.
For the current one-dimensional case, the approximate yield surface (equation ( 1 0 . 3 0 ) ) gives:
t 2 + m 2 +,/~3 Itm I = 1 ............................................. (10.37)
This approximate curve is compared with the exact curves (equations (10.35) and (10 .36)) in Figure 15. Also shown is the curve
t 2 + m 2 = 1 ............................................. (10.38)
which is the one-dimensional form of equation (10.27).
10.14.4 Discussion Both the approximate yield surfaces (equations (10.27) and (10.30)) satisfy exactly (within the limits of the ideal plastic assumptions) the bending only and the in-plane only tests. They also both give approximate solutions for the one-dimensional limit. The approximate yield criterion given in equation (10.30) (that adopted in Reference 1) gives the closer overall agreement with the 'exact ' solution. However as t ~ -+ 1 and m ~ +- 1, equation (10.27) is more accurate. This is because Qtm ~ 0 in these cases and equation (10.27) was derived for the condition Qtm = O.
* This limit was not considered by Ilyushin.
25
The approach adopted in the earlier work 1 is to use equation (10.30) unless Qtm is small in which case equation (10.27) is used. Equation (10.27)has the numerical advantage that it posesses no discontinuities in its partial derivatives as Qtm ~ 0. The movement from one yield surface to the other when Qtm is small can be accomplished by means of a normal shift.
11. APPENDIX 2
Full and 'reduced' strain matrices for bending element
11.1 Displacement function
{w} w--lb T'{w } a { a ~ }
where the element topography is as shown in Figure 2. (W~, {A~
' (1 + ~'i ~') (1 + •i •) (2 + ~'i ~" (1 -- ~'i ~') + Hi ~? (1 - r/i r/) ) / Wi= g
A i = - - ~- ~'i (1 +S'i ~')(1 +H i r/)(1 _~-2)
1 S i = - g 7/i ( l + ~ ' i ~ ' ) ( 1+77 i n ) ( 1 - n 2)
11.2 Slope matrix
(We} where ( s ) V = aw
( ) T { { T { } T a w w e -- w } a
and [B] is the slope matrix given by:
[BI
where W~-i = - ~- ~'i (1 + 77 i r/) (3~ "2 + 7/2 - r/i 7 / - 3)
= ~ (1+7/ i r / ) (3~ 2 +2~" i ~ ' - 1) A~- i ~-
B~" i - ~ ~'i r/i (1 + r/i r/) (1 - r / 2 )
W~i is as W~- i but with ~ replaced by r /and vice versa
AT/i is as B~i but with ~ replaced by r /and vice versa
BT? i is as A~- i but with ~ replaced by r /and vice versa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 1 . 1 )
and ( B ) are vectors of shape functions given by:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 1 . 2 )
............................................. (1 1.4)
............................................. (11 .5)
............................................. (11.6)
............................................. (1 1.7)
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 1 . 8 )
2 6
11.3 Curvature matrix
( q = [F] {We)
where{x) T=(Xx Xy X x y ) = { 3=W3x 2
and [F] is the curvature matrix given by:
[F] =
02w Oy 2
1
2
232w } 3xOy
where
{ A~-~- B T)
W~'~'i = -¾ ~'i ~" (1 + r/i r/)
A~.~. i = ¼ (1+ 7/i r/) (3~" + ~'i)
B~-~i = 0
W~-r~i = -~ ' i r/i (3~2 + 3r/2 - 4)
A~rt i = ~ rti (3~ 2 + 2~'i~ - 1)
B~-r/i = 8 l - ~i (3r/2 + 2r/it/- 1)
Wrtrt i is as W~-~. i but with ~" replaced by r/and vice versa
A~/r/i is as B~.~- i but with ~" replaced by r/and vice versa
Br/r/i is as A~.~i but with ~" replaced by 77 and vice versa
11.4 Reduced curvature matrix
The last line of equation (11.9) is replaced by
()T = [ 2 ( ff~n ×xy ab
where ~/~'~i = ¼ ~i Hi
~'r/i = ¼ ~'i r/i ~"
B ~ni = ¼ ~i 7/i r~
so that equation (1 1.9) is replaced by
{q= {We) where [F-] is the reduced curvature matrix.
}
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1 1.9)
.............................................. (11.10)
............................................. (11.11)
(11.12)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 1 . 1 3 )
............................................. (11.14)
.............................................. (11.15)
27
11.5 Reduced slope matrix
The vectors W , A ~-} ( ~-}, (B~-}, {Wr/}, (At/) and (Br/} (equations (l l.8) are replacedby reduced
vectors {W~-), { A @ , (g~-), (~/r/) , {~'r /)and [g~)
~'i (4 r/. r/+ 3) where W~'i = [-2 t
A~'i = ¼ ~'i ~" (1 + r/i r/)
B~'i = - ~ ~'ir/i(l +r/it/)
Wr/i is as ~/~'i but with ~" replaced by 7/and vice versa
~'r/i is as B~'i but with ~" replaced by 77 and vice versa
Br/i is as Afi but with ~" replaced by 7/and vice versa
( l l . 1 6 )
12. APPENDIX 3
Element tangent stiffness matrix from Reference 1
In Reference 1, an expression is derived for the tangent stiffness matrix of a large-deflection elasto-plastic element. The element stiffness equinions are given by
(AP~" = [KE] {Ap) ............................................. (12.1)
(} where [KE] is the tangent stiffness matrix, AP is the vector of generalised nodal forces such that
AU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and (Ap) is the corresponding vector of generalised nodal displacements.
12.1 Sub-matrices f
[KE] may be divided into sub-matrices as follows:
• [ K E ] = [Kii] -[Ki0l [Kio ]T [Koo]
where [Kiil = fA [HIT [~1 [H] dA
[K°°] =fA ([FIT [I~] [F] + [BI T [TS] T [~] [TS] [B]
+ + [BI T [N] [B] + [FI T [cd] T [TS] [B]
+ [BIT [TS] [cd] [F]) dA
[Ki°] =fA ([HIT [~] [TS] [BI + [H] T [cd] [F])dA
The notation used in equation (12.4) is given in Section 9.
............................................. (12.3)
............................................. (12.4)
2 8
12.2 Corrigendum to Reference 1
A minus sign has been ommitted in the expression for the elasto-plastic modular matrix [cd] in equation (10.13) on page 27 of Reference 1. The correct expression is:
I c d l - - t* [El [NMI [El ............................................. (12.5) 12 (m + n)
2 9
\ \
\
f < \
\
f
~_. ~ i I -- , -- -4- -4 - ~ - ~ -
\ ~ I I / I -- ~----~-- -- 4-- ---I--- ---/- - ÷ - - /
\ / \ /
f
----7 /
/ / Constrained
mode
b /
/ /
True mode
Fig. 1. PARASITIC SHEAR STRESSES IN LINEAR ELENEtiT SUBJECT TO PURE IN-PLANE BEHDIHG
2b
J, 0 2
r
0
2 •
Y(?)
1 3
Two -point Nodes
2a
Gauss stations
~i = - 1 , - 1 , 1 , 1 [
"~i = -1, 1 ,-1,1
4
x(~)
)3
a5 y= b~
Fig. 2. RECTAHGULAR ELENENT
Y
=I - 5 d
" = 0 - 5
I d
MESH 1
MESH 2
M ESH 3
Fig. 3. CANT[LEVER TEST EXAMPLE
A B
~_ 1.0
:E i~'~v 0.75 p
0.50 o o.25
0 A
. ~ ~ , Reduced element I
- - _ _
-- "Full-element, . . . . , ~ - - ~ ~ . ~ 1 ~'~'-~" ~-~1
B
10
F- U x i.I
E ~g u L
L O ¢-
8
G
4
2
0
-2
-4
-6
- 8
"Full" element
k I~_ 'Reduced' element
_ \ I \ - - - -~-- ,1" \- I r - _A \ \ '1 \ \ 1 "\~
B
Fig. A. SHEAR FORCE ,AND BENDING MOMENT DIAGRAMS FOR CANTILEVER WITH POINT LOAD (MESH 2)
R e d u c e d e l e m e n t
Ful l e l e m e n t
8 L
¢-
0 ° - -
*~ 12
6 LOAD AT C E N T R E
4
2
0 I I I I 4 x 4 G x 6 8x l5 10x10 12x12
Mesh s ize
14x14
13
8 -
.E
o
c 4 -
o
' - 0 4 x 4
o L
n
-CLAMPED WITH .POINT LOAD AT CENTRE
I I I I 6 x 6 B x 8 10x10 1 2 x 1 2
Mesh s ize 14>414
12
B
6
4
2 -
0 I I I
4 x 4 6 x 6 8 x 8 10x10 Mesh s ize
I 12x12 14x14
Fig. 5. CONVERGENCE TESTS FOR SOUARE PLATE : CENTRAL OEFLECTIOtl
F u l l e l e m e n t
. . . . R e d u c e d e l e m e n t
R e d u c e d e l e m e n t w i t h i t e r ' a t i v e c o r r e c t i o n
M e s h s i z e
2 x 2 4 x 4 6 x 6 8 × 8 10 x l O 12 XI2
0 I I I I ___ !
- 1 0 - . - - - - - - - o " - - - " ° " - - ' - " °
-20
g -4o /
~ - 5 0
a. - 6 0
- 7 0
Fig. 6. CONVERGENCE TEST FOR SQUARE PLATE SIMPLV SUPPORTED AT CORNERS AND SUBJECT TO U.O.L. : CURliER TWISTING MOMENT
0
0
0
0
0
0
Stress points
0
0
0
0
0
Fig. 7. STRESS POINTS TO IMPROVE ACCURACY FOR BOUROARV ELEMENTS
Internal load vector based on continuous stress field through averaged nodal values
. . . . As above with the exception that mid-side stresses are used for boundary elements (see Fig.7)
t - O ,i.i L)
.c_ { . 0 ( - L.
w-
(3 { .
n
0 Or
-51
-15
-20
-25
-30
-35
-401
Number of i t e r a t i o n s
. . . . + . . . .
// /
/
•Mesh Eiement~No.
type
Full
Reduced
1 2
D
V 0
Fig.8. PERCENTAGE ERROR IN TIP DEFLECTION FOR CANTILEVER SUBJECT TO POINT LOAO
i ~ m m
Internal load vec to r based on cont inuous s t ress field through averaged nodal values
As above wi th the excep t ion that mid-s ide s t resses are used for boundary e lements ( see F-ig. 7 )
20 i > o
L" 0 L. L
o "o 0 t -
o L
> <
15
10
%
o E
C 0 L L
-6 "o 0 r-
E
E o
- 5
-10
-15
-20
-25
-30
-35 [
- 4 0
S . o . -
f j / /
/ /
/ I
P I
I I
I I
I I Mesh
Element~No. type
2
Fu II A El
Reduced V 0
-45 T I I I I I 0 1 3 5 7 9
Number of i t e ra t i ons 11
Fig. 9. PERCENTAGE ERRORS IH EXTREME FIBRE BENDING STRESS,~ x FOR CANTILEVER SUBJECT TO POINT LOAD
Internal load vec to r based on continuous s t r ess f ie ld t h r o u g h averaged nodal values
As above w i t h the except ion t ha t mid-s ide stresses o re used for boundary e lements (see Fig. 7)
>
- 300 t.
8
I Io I001
200 X o E
300 2 f._ I]J
-5 4 0 0 "O O C
E "-, 5 0 0 E X O
:Z 6 0 0
7 0 0
~ M e s h Element~No.
type 1 2
Fu II A O
Reduced V O
6 0 0 1 - I I I I I 0 1 3 5 7 9
N u m b e r of i t e ra t ions 11
Fig.lO. PERCENTAGE ERRORS It( SHEAR FORCE FOR CAttT[LEVER SUBJECT TO POll'tT LOAO
8
t -
O
U
m
1 3
E ° ~
°r 2 -
O, 4 x 4
12r
Solution wi th i te ra t ive correction
S.S. WITH POINT LOAD AT C E N T R E
"Initial solut ion
I I I I 6 x 6 8 x 8 10x lO 12x12 14x14
L 0 L 10 I . g
w 8
r
U L
n
E E~
Ot- ~.£
c L A M P E D WITH
So lu t ion w i t h i t e r a t i v e correction
POINT LOAD AT C E N T R E
2 - -
0 4 x4
In i t ia l so lu t ion
I I I I 6 x 6 8 x 8 10x10 12x12
10
8
6
4
2
0 4 x4
~ ~ , 1 S.S W I T H U .D .L . Solut ion w i t h i t e r a t i v e c o r r e c t i o n
I A l n i t i a l so lu t ion
- -
6 x 6 8 x 8 10x10
I
D - De f l ec t ion I
I M - M o m e n t
14×14
12
12x12 14x14
t -
O
10 U
~ 8
-i6 w 4 0 1
r
0
C L A M P E D WITH U.D.L.
Solut ion w i th i t e r a t i v e c o r r e c t i o n
In i t ia l s o l u t i o n
% ~ ,imm=l, ~ ~ ~mmD t l l l m , I m II=~=m=, ,~,m~D m 2 -
0 I I I I 4 x 4 6 x 6 8 x 8 10 ×10 12 x12 14x14
Mesh s ize
Fig. 11. CONVERGENCE TESTS FOR SQUARE PLATE = CENTRAL DEFLECTION (AND MOMENT)
- - - - - Yamaki (ser ies soln5 - F.E. ref.(1)(Sx8 mesh)
F.E. present a p p r o a c h ( S x S m e s h ) v- - f ree ---13--- F.E. present approach (12x12 mesh)
Yamaki (ser ies soln.) _ _ _
+ F.E. ref .(1)(8x8 mesh) ~ (3v I ~ . EE. p resen t app roach (SxSmesh ) / ~ : = 0
F. E. present approach (12 x 12 mesh) J
2
1 .
0 L
.}
2 - 8
2 - 4
i
2 - 0 -
1 " G -
1.2 -
0 - 8 - -
0 " 4
v = f ree Jr ~-~ =0
I I
.---~ I I ! !
I I
~ ~ .
I ~y_ L
v= free or O~c = 0
y ( v )
= : c ( u )
r; li-- I I . - - I I I -- I S
0 I I I I I I I I I I I I I I 0 0 -4 0-8 1-2 1-6 2"0 2-4 2-8
Cen t ra l de f lec t ion ra t io ,Wc/t
Fig. 12. IMPERFECT ELASTIC PLATE UHDER IN-PLANE LOAD : CENTRAL
DEFLECTION ( v = 0-3)
" o
10
(3L <
I i -iI
---S--
v = f ree ~;
I I° I
' I I - I
4
I I
~- V= free
a i ,
l Y(v) =
D I M E N S I O N S A N D P R O P E R T I E S
b/t. = 55, a/b = 0 .875 ~ E= 206 ,20ON/ tom 2, V = 0 . 3 cr o = 2 5 0 N / m m 2 , t = 3 -175mm(1 /8 " )
~c(u)
¢ ._o" I,., i/')
ID 1,_
¢)
I : l f,.,..
> <
1"0
0.9 L
0-~
0-7
0-6
Stress f ree,(4 x 4 mesh) ~ } ~'o = 0 .001 b w i th residual stresses (5x4mesh )
#
(~R = 0 .1225 ~ ~o =0-00117b
0"5
0-4
0.3
0-2
0-1
Present app roach
Reference (1)
Moxham- theore t i ca l
O I r I I I I I I I I 0 0-2 0.4 0-6 0-8 1.0 1-2 1 3 14 1 5
Average stra in ra t i o , E:JE:o
Fig.13 RELATION BETWEEN AVERAGE STRESS AND AVERAGE STRAIN FOR SIMPLY SUPPORTED PLATE IN DIRECT COMPRESSION
Qm
O-B
0 .6
0-4
0.2
0
Equation (10.26)
~ . . . = B Equation (10.27)
\
0 0"2 0 4 0 -6 0"8 1-0
Q t
Fig. l&. RELATIONSHIP BETWEEtl Qt AND Qm FOR ILYUSHIN'S YIELO CRITERIA WHEN Qtm = 0
0"8
0"6 /;y// 0"4
L,:, / o ~ / I I I I 0 ~ - o 8 -o.6 - o 4 -o2
~O° 2
' ' q - o . 4 -
m
%\
I I I I 0-2 0-4 . 0 . 6 0 - 8
I
/ J
t
I t + [ m j = 1 ( b e n d i n g d o m i n a n t , e q n ( 1 0 - 3 5 ) ) t 2 = 1 ( i n -p l ane d o m i n a n t , e q n ( l O - 3 6 ) )
2 2 1 t + m ~ - ~ l t m I = 1 ( equat ion (10-37))
t2~-m 2 = 1 ( equa t i on ( 1 0 - 3 8 ) )
Fig. 15. UHIAXIAL YIELD CRITERIA
(751) Dd209410 200 6/ ' /5 H P L t d S o ' t o n G1915 PRINTED IN ENGLAND
ABSTRACT
Some approximations in the non-linear analysis of rectangular plates using finite elements: M A CRISFIELD, BSc PhD: Department of the Environment, TRRL Supplementary Report 51 UC: Crowthorne 1974 (Transport and Road Research Laboratory). In earlier work the formulation of the large deflection behaviour of elasto-plastic plates has been described. The approximations that were made are examined in detail and alternative methods have been developed and tested. In some cases, the aim of these changes is a more economic
solution; in others, it'is a more exact analysis.
ABSTRACT
Some approximations in the non-linear analysis of rectangular plates using finite elements: M A CRISFIELD, BSc PhD: Department of the Environment, TRRL Supplementary Report 51 UC: Crowthorne 1974 (Transport and Road Research Laboratory). In earlier work the formulation of the large deflection behaviour of elasto-plastic plates has been described. The approximations that were made are examined in detail and alternative methods have been developed and tested. In some cases, the aim of these changes is a more economic
solution; in others, it is a more exact analysis.
