The CST (Constant Strain Triangle) An insidious survivor from the infancy of FEA by R.P. Prukl, MFT Computing (Pty) Ltd P.O. Box 221, Rivonia 2128 (This paper was delivered at the FEMSA 92 Symposium, 11 th Symposium on Finite Element Methods in South Africa, 15-17 January 1992, organised by FRD/UCT Center for Research in Computational and Applied Mechanics, University of Cape Town) The CST was the first element, which was developed for finite element analysis (FEA) and thirty years ago it served its purpose well. In the meantime more accurate elements have been created and these should be used to replace the CST. A simply supported beam loaded vertically with a uniformly distributed load and modeled with eight CST elements gives zero stress where it should have its maximum. This is a 100% error and shows that such elements are useless and very dangerous in the hands of inexperienced users. Some finite element programs, which are being marketed worldwide, have still only the CST in their element library for plane stress and three-dimensional shells. Over the past seven years, the author has discussed the problems associated with the CST with a number of lecturers who claim to be finite element experts. Some of these erudite gentlemen still consider three-noded plane stress elements to be better than four-noded ones! In many lecture notes, textbooks and pamphlets on FEA only triangular elements are shown. Furthermore, some automatic mesh generators can create only triangular elements, while others create meshes with unnecessary triangular elements. In considering that, in the future, under-qualified people will be using FEA, this paper is intended to create an awareness of the problems associated with the CST and eventually to have these elements removed from element libraries for the sake of safer structures and thus saving time, money and probably human lives. 1. THE SHOCK A concrete beam, 8m long, 2m high and 0,2m thick is loaded with a vertical u.d.l. of 100 kn/m at the top (see fig. 1). The material properties are: Young s modulus 3,0 x 10 7 kpa and Poisson s ratio v = 0,2. The horizontal stresses at the centre of the span can easily be calculated manually using the bending theory. They are 6000 kpa (compression) at the top and +6000 kpa (tension) at the bottom. Let us model this two-dimensional problem with finite elements, using eight three-noded plane stress elements with a mesh layout according to fig. 2a. The results of the finite element analysis look strange (see table 1). At the bottom centre of the beam we only get +1347 kpa compared to the theoretical value of +6000 and at the top centre, where we should have the maximum compression of 6000 kpa we get zero stress. The error, which has been defined as the maximum of the errors for the two stress columns, is, therefore, 100%. HATCH Technical Training Finite Element Analysis htt-fea.the_cst1-8.doc Page 1 of 10 05/07/04 / 08:45 PM
Let us use four-noded plane stress elements with a mesh according to fig. 2c. Now the results look much better. We are getting values of +5250 kpa at the bottom and 5250 kpa at the top, an error of 13%. Various other finite element models were analysed. The numerical results can be found in Table 1 and the plots for the Sxx stresses for line numbers 2 to 9 are displayed in figs. 3a-3h. In the past seven years the author has asked several hundred people to estimate the stresses for the two above mentioned finite element meshes. Some of these people had no finite element background at all; others were experienced users of FEA and a number of them were even university lecturers, teaching FEA to students. The author first asked for an error estimate for the mesh with the four-noded elements. Some people had such faith in FEA that they first did not understand the question and then said that there would be no error at all in such a simple problem. When they were told the error was 13%, they were surprised. When they were asked for an error estimate for the mesh with the three-noded elements, almost everybody, including the academics, estimated lower errors than before (typically 3%). When confronted with the facts they could firstly not believe it and then they were truly shocked. The author first discovered the misbehaviour of the three-noded triangular plane stress elements about eleven years ago although he had already five years of experience with FEA and had analysed and designed in that time major structures using the finite element method. 2. THE EXPLANATION (2.1) Consider a 3-noded plane stress element in the xy-plane with node points 1, 2 and 3. The x-deflections are u 1, u 2, u 3 and the y-deflections v 1, v 2 and v 3, six values together. The displacement function then has the following form i to describe the behaviour of the element): (using six constants u = 1 + 2 x + 3 y v = 4 + 5 x + 6 y The direct strains can then be calculated by differentiation u ε x = = α 2 x v ε y = = α 6 y What does this mean? The strains in such an element are constants. We know, however, that in a beam as in fig. 1 we have compression at the top and tension at the bottom. Our single element is, therefore, not capable of modelling bending behaviour of a beam, it cannot model anything at all. HATCH Technical Training Finite Element Analysis htt-fea.the_cst1-8.doc Page 2 of 10 05/07/04 / 08:45 PM
(2.2) Three-noded elements for other applications than plane stress and strain are quite acceptable, e.g., plate bending and heat transfer. (2.3) The results for configuration 2c (for four-noded elements) from programmes COSMOS *** (incompatible elements), ALGOR and ANSYS are identical with the FESDEC results. It is confusing for the novice user that COSMOS ** (full integration) is inferior to COSMOS * (reduced integration) and that the best of all, COSMOS *** is an incompatible element. It should be noted that one row of 6- and 8-noded elements (fig. 2e and 2f) performs equally well as two rows of four-noded elements (fig. 2g). The results from the boundary programme BEASY are excellent and one should also consider that for plane stress and plane strain problems no meshes are required in boundary element programmes. The whole interior is defined by the boundary only. 3. THE REMEDY (3.1) Use elements with four nodes. We then have eight constants to describe the behaviour of the element: u = 1 + 2 x + 3 y + 4 xy v = 5 + 6 x + 7 y + 8 xy The direct strains are then as follows: u ε x = = α 2 + α 4 y x v ε y = = α7 + α y 8 X The strain in the x-direction is now a linear function of its y-value. This is much better than for the triangular element. (3.2) Use triangular elements which also include the rotational degree of freedom about the z-axis normal to the xy-plane. The idea was implemented by D. Neille [1] in his programme PST. The error for a mesh configuration 2a is reduced from 100% (CST) to 56%. Using 2 layers of elements according to 2h the error is 27% compared with 72% for the CST. (3.3) Many CST elements are required to give satisfactory results (with say an error of less than 10%). These elements should, therefore, only be permitted as fillers between four-noded elements (6- and 8-noded elements are preferred). Finite element programmes that have CST elements only should be prohibited by a Code of Practice for Finite Elements. HATCH Technical Training Finite Element Analysis htt-fea.the_cst1-8.doc Page 3 of 10 05/07/04 / 08:45 PM
It should be noted that programmes like MICROSAFE have four-noded elements, which gave an error of 65% for configuration 2c. This is due to the fact that these programmes do not have real four-noded elements. They subdivide the fournoded elements into four triangles. Automatic mesh generators should be improved to avoid the creation of meshes with triangles only or meshes with unnecessary triangular elements. Triangular elements for plane stress and shells should be banned from textbooks and courses and should only be mentioned as bad examples. 4. LIST OF FEA AND BEA PROGRAMS ALGOR from ALGOR Inc., Pennsylvania, USA ANSYS from Swanson Analysis Systems Inc., USA BEASY from Computational Mechanics, Southampton, UK COSMOS from SRAC, California, USA FESDEC from ETA, Hinckley, UK MICROSAFE from Microstress Corporation, Washington, USA PSQ and PST from Dr. D. Neille, Sandton, RSA 5. REFERENCES [1] D Neille, The Finite Element for Infinite Applications, NCL Stewart Scott, Consulting Engineers 6. ACKNOWLEDGEMENTS The author would like to thank the following persons for carrying out test runs: Mr Jose Calixto da Silva (ALGOR runs) Mr Mark Kilfoil (ANSYS run) Mr Jerzy Miszczyk and Mr Richard Harfield (BEASY runs) as well as Mr Adrian Peirson for editing the manuscript. HATCH Technical Training Finite Element Analysis htt-fea.the_cst1-8.doc Page 4 of 10 05/07/04 / 08:45 PM
Sxx stress (kpa) Line No. Program Name Program Type Fig. Mid Span Top Mid Span Bottom Error % 1 Bending Theory - - 6000 + 6000 0 2 FESDEC FEA 2a 0 + 1347 100 3 FESDEC FEA 2b - 426 + 424 93 4 FESDEC FEA 2c - 5250 + 5250 13 5 FESDEC FEA 2d - 3940 + 6772 34 6 FESDEC FEA 2e - 5962 + 5978 1 7 FESDEC FEA 2f - 6022 + 6275 5 8 FESDEC FEA 2g - 5802 + 5795 3 9 FESDEC FEA 2h - 1683 + 3289 72 10 COSMOS FEA 2a 0 + 1347 100 11 COSMOS * FEA 2c - 5323 + 5177 14 12 COSMOS ** FEA 2c - 3787 + 3713 38 13 COSMOS *** FEA 2c - 5250 + 5250 13 14 ANSYS FEA 2c - 5250 + 5250 13 15 ALGOR FEA 2a 0 + 1347 100 16 ALGOR FEA 2c - 5250 + 5250 13 17 PST FEA 2a - 2660 + 3999 56 18 PST FEA 2h - 4371 + 4976 27 19 PSQ FEA 2c - 5791 + 4709 22 20 PSQ FEA 2d - 5647 + 4230 30 21 PSQ FEA 2g - 5957 + 5598 8 22 MICROSAFE FEA 2c - 2094 + 2106 65 23 BEASY BEA 2i - 5750 + 5750 4 Table 1 HATCH Technical Training Finite Element Analysis htt-fea.the_cst1-8.doc Page 5 of 10 05/07/04 / 08:45 PM
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