Finite Element Displacement Analysis of Plate Bending Based on Rectangular. Elements. Norges byggforslmingsinstitutt

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1 REPRINT 132 Finite Element Displacement Analysis of Plate Bending Based on Rectangular. Elements By Civil Engineer HARALD ~N5TEEN Norwegian Building Research Institute Norges byggforslmingsinstitutt NORWEGIAN BUILDING RESEARCH INSTITUTE ~ NUl OSLO 1966

2 University of Newcastle upon Tyne Department of Civil Engineering WORKING SESSION No.4. PAPER No. 14. INTERNATIONAL SYMPOSIUM The use ojelectronic Digital Computers in Structural Engineering FINITE ELEMENT DISPLACEMENT ANALYSIS OF PLATE BENDING BASED ON RECTANGULAR ELEMENTS Harald Hansteen 1 SYNOPSIS This paper describes a finite element procedure which ensures complete continuity of displacements between adjacent elements. The reliability of the procedure is illustrated by numerical solutions given at the end of the paper. INTRODUCTION In plate bending problems few of the cases met with in practice can be solved by analytically exact methods. Very often one has to rely on approximate procedures of variable accuracy. It is natural, therefore, that investigators have shown great interest in examining the possibilities offered by the finite element method, and as a result a number of papers on the subject have been published in the recent years. The major part of these works is concerned with plate elements of rectangular form. Numerical results reported are yet too few to give reason for any conclusions concerning the accuracy and reliability of the approach in general. However, encouraging results have been obtained in particular cases, Zienkiewicz (8). Some investigators have wanted to utilize the advantage of triangular elements in representing plates of arbitrary shape. Numerical examples carried through by this procedure have not proved.to be 1 Civil engineer, Norwegian Building Research Institute, Oslo, Norway. A",.,.'~. ~ _.1.

3 as accurate as for rectangular elements. The results show a converging character as the element size decreases; however, the values towards which they converge greatly depend. on the displacement patterns selected for the elements and have generally little resemblance to the correct values, Clough (4). Melosh (7) hap shown that a necessary condition for the finite element displacement method to converge towards the correct solution is that the selected displacement patterns must be continuous in the internal of the elements, and maintain continuity with displacements of adjacent elements. This condition has not been fully accomplished by the procedures published so far, all of which fail to establish continuity in normal slopes along the lines connecting the nodal points. Clough (4) claims this circumstance to be the basic reason for the lack of accuracy of his results. One should anticipate a similar tendency to oceur for the procedures applying rectangular elements. When such a tendency is not as yet found, the reason is probably that in the limited number of test examples examined the discrepancy in normal slope between the elements has been of little significanoe. ANALYSIS OF ELEMENT STIFFNESS IN BENDING The fundamental idea of the finite element method is to represent the actual structure by a finite number of individual elements, interconnected at a finite number of nodal points. The stiffness of the idealized structure is obtained by adding the stiffnesses possessed by the individual elements. To obtain a good computational model it is therefore of great importance to be careful when deciding the stiffness properties of the elements. To be able.to derive the element stiffnesses, one has to assume a finite number of displacement modes [degrees of freedom] for the eiement. The final dispiacement pattern of the element Will be l:ljnited to a linear I combination of these displacement modes. These :ljnposed restpictlods to the deformation of the element is equivalent to the introduction of constraints, and it is fairly obvious; therefore, that the computed stiffnes~~~ w1l~.be greater than the true ones, provided that the selected displacement patterns satisfy the Melosh conditions referenced above. Further it should be emphasized that the Melosh conditions are necessary, but not sufficient, to ensure convergency of the results. Another important condition must be that the selected displace~nts should be able to describe all the essential modes of the true solution in all points of the

4 plate. If this is not achieved the finite element s.olution will converge towards values for the displacements which are smaller than the true values. In Fig. 1a are shown the four rectangular elements interconnected at nodal point number k. The outer edges of the elements are supposed to be completely fixed, and the object is to study the stiffness properties of the elements due to generalized unit displacements of point k. The general displacement pattern of the built-in plate is composed by four essentially different modes of displacements, as described in Table 1. To the right in the Table are given the corresponding values of the nodal displacements: Displ. at node k. Displacement patterns w w w k k,x w k,y k,xy. 1. Positive displ. of all four elements Positive displ. of elements I and IV, negative displ. of elements II and III Positive displ. of elements I and II. negative displ. of elements III and IV Positive displ. of elements I and III, negative displ. of elements II and IV Table 1 Here w is the normal displacement of node k k and the comma denotes partial differentation. Fig. lb shows element III provided with, an internal coordinate system (x,y,z), z positive downwards, and with internal numbering of the nodes from 1 to 4 [internal node 1 corresponds to the global numbering k]. The side lengths of the element are a and b, and a new set of dimensionless coordinates (~,~) which may substitute coordinates (x,y) are defined as follows: 5 = x/a, "l = y/b (1 )

5 The simplest polynomial representation of the displacements satisfying both the Melosh conditions and the displacement patterns of Table 1, will for node 1 be: The notation V (5,1) is adopted to describe the continuous normal displacements corresponding to displacements at node 1. The equation may be 1 written on matrix form, yielding (3) where IA 1 is a row vector and"" 1 a column vector as follows: (4).., 1 = { wi' aw 1,x' bw 1,y' abw 1,xy} (5) Similarily one can find the displacement patterns corresponding to unit displacements at nodes 2-4. Hence the total displacement of an element is expressed by: v(s,') = [A 1,A 2,Ay "'4 ] "'1 = A(S'7) 'W ""2 1>01 3 '''''4 (6) The "'i matrices are easily obtained from Eq. (5) by substituting the actual nodal point number. The IA i matrices will be A = 2 [ 4<t+~)2(1+S)(~7)2(1+'l), +2(~+~)2(1+S)(~7)2(~+'l), = [ 4(~+5)2(1+5)(~+?)2(1+7), 2(i+s)2( 1+s)(i+'(')2( ~'1), = [ 4(~5)2(1+5)(~+7)2(1+'Z), 2( ~ )2(1+:5)(~+'7)2(i+?), 2( ~+5h~+5) (i+')2( 1+?) +(~+~l(i+5)(i+'l)2(t+?) ] 2( ~+5)2(i+s)a+?l( 1+?) (~+S)2(~5)(t+?)2(i+?) ] +2(~5)2( t+s) (~+?)2(1+'() +(~S)2(~+5)(~+?)2(~?) ] (7a-c)

6 Differentiating Eq. (6) the internal element curvatures,~, may be expressed qy the generalized displacements at the nodes, as follows: (8) Introducing now the elastic characteristics Df the finite element material defined by the stress-strain matrix ID, the internal moments,m, in the element are expressed by where [) ~ Et 3 " 0 12(1'l-\?) " (1+,,) (10) E ~ [~odulus of elasticity " ~ t Pcisson's ratio thickness of the plate The elastic energy stdred in an element upon deformation, Wi' should equalize the external work, W, performed by the load on the element. e This load consists of the reactive nodal forces from adjacent elements, S, and distributed and concentrated loads at the surface of the plate. For brevity Dnly distributed surface loads, q(s'~)' are considered in the derivation below; however, there are no di~ficulties in taking into account also concentrated forces or moments arbitrarily situated on the elements. The expressions for work then are: Wi ~ is )(.t M dv ~ -kw t [J Bt[)1B dv]w ~ ~ "...h~... (11) Vol Vol We ~ t \,}S + -k lre~t q d A ~ t"hs + {ret't q d A] ~ ~...t[s +5] (12) Equating the two expressions one obtains (13 )

7 where j i3 t(s,?)[jib (S.,,/) dv Vol j At(5''7)q(~,?)dA Area (14 ) (15 ) ~ is the 16 by 16 stiffness matrix of an element and 5 a 16 by 1 matrix representing the external load on the plate. is has generally nouzero elements and may be interpreted as generallized loadings [both vertical forces and moments] acting in the nodal points. [See also Zienkiewicz (8)]. If the size of the elements is small compared to the dimensions of the plate the substitution of distributed loads by concentrated vertical forces at the nodes is a reasonably good approximation. When using relatively large elements, however, one should anticipate improvement of the accuracy by doing the more detailed analysis shown above. As the latter procedure completely corresponds to the actual loads on the plate, andas the elements ors are usually easy to calculate, there should be little reason not to prefer this procedure in all cases. The stiffness of the assembled structure is obtained from Eq. (13) by adding the stiffnesses of the individual elements meeting in each nodal point. The final equation relating generalized nodal loads, ~ to the corresponding nodal displacements, r, may be written. R ; D<.. (16 ) ~ being the stiffness matrix of the assembled structure. After inserting the actual boundary conditions in Eq. (16), th~ nodal displacements are' found by solving the system of simultaneous equations. With the nodal displacements known the plate moments may be computed from Eq. (9).. EXAMPLES OF NUMERICAL SOLUTIONS The numerical results shown below are intended to illustrate the accuracy of the procedure. The moment values repo~ted are the mean values of the moments at each nodal point. In the cases where distributed loads occur the corresponding nodal loads have been computed from Eq. (15).

8 Example l: The first example is the standard problem of a simply sup-' ported quadratic plate og uniform thickness shown in Fig. 2. Two loading cases are considered: ~ase 1: A uniform load allover the plate. case 2: A concentrated load in the middle of the plate. To study the convergence of the solution four different finite elemen~ meshes of successively reduced sizes have been employed as shown in the figure. The accuracy of the various solutions is exemplified by comparing the values obtained for the central point of the plate to the exact values. The results are given in the table at the bottom of Fig. 2. Example 2: Fig. 3 shows the same plate as in example 1, this time partially loaded by a uniform load over a square 1/8 by 1/8 of the side lengths. The applied finite element mesh, as well as the position of the loaded area -~; [shaded in the figure], are shown in the figure. In the accompaning table are given the computed and the exact displacements and moments at the nodal. points of the section A-A. Example 3: The third example is a uniformly loaded plate completely fixed at three edges' and free at the fourth edge. The results are cumpared to a nearly exact solution of the problem proposed by Hellan (6). Hellan has given the displacement values with three significant digits. These digits are verified by the finite element solution. The necessary data of the example and the moments in significant sections of the plate are shown in Fig. 4..,, 1 CONCLUSIONS The numerical examples cover some of the loading cases and'ed!e conditions that frequently oocur in practical problems. It ia intere!ting to note that the degree of accuracy obtained for the displacements and the moments in the various caaes do not depend on the type of the problem. This indicates that equally accurate results may be expected for other types of problems. In example 1 are tested tbe convergence properties of the procedure. That the results do co;"';"~e toward the correct solution is clearly demonstrated by the resultfr. Another interesting feature is that even the

9 results obtained for the very coarse mesh n = 1 are well within the limits of accuracy needed in practical computations. One should, however, be careful not to draw any conclusions from one simple example. Further investigations are needed to clarify the connection between the size of the finite elements and the accuracy of the results. The results of Examples 2 and 3 are close to the exact values. The greatest errors occur at points where the gradient of the curvatures are. steep, as in point 3 of Example 2. The maximum values seem to be very accurate. An exception is the moment value in point 1 of Example 3. However, in this particular point Hellan (6) points out that the reference value is less reliable. Note. - During the preparation of this paper it came to the author's knowledge that a report, that in parts may develop along similar lines as the present work, have been made by Butlin (3). As this report has not been available for the author he has not been able to check the points of similarity. REFERENCES 1. Argyris, J. H. "Matrix Displacement Analysis of Anisotropic Shells by Triangular Elements" Journal of the Royal AerCftau.tlcal Society, Vol. 69, Nov Argyris, J. H. "The Trondheim Lectures on the Matrix Theory of Structures". John Wiley & Sons, Ltd., London ButUn, G. A. "On the Finite Element Technique in Plate Bending Analysis; a Derivation of a Basic Stiffness 'Matrix" Internal re P9rt at Engineering Dep., Churchill College, Cambridge. 4. Clough, R. W. "The Finite Element Method in Strutural Mechanics" in "Stress Analysis." John Wiley & Sons, Ltd., London Clough, R. W. and Tocher, J. L. "Analysis of Thin Arch Dama by the Finite Element Method': Int. Symp. on the Theory of Arch Dams, Southampton Univ., Pergamon Press, Hellan, K. "The Rectangular Plate with One Free Edge", Trondhelm Melosh, R. J. "Basis for the Derivation of Matrices for the Direct Stiffness Method", J." Am. Aero. Astro., 1, 1631, 1963.

10 Displacements and ldoments at the central point. case I common n = exact factor w qa /Et x10 ' M 4,68 4,66 4,73 4,75 4,79 qa 2 xl0+ 2. case II w Pa 2 /Et 3 x10+4 Fig. 2. y, I.., I I, a X Simply support~d ~dg~s Loads Cas~ I: uniform load 9 Ca$~lI: conc~ntrat~d load P in c~ntral point v =0,3 Analysis of this ljuart~r M~sh SIZ~S

11 Dis~lacements and moments along section A - A r. /.. common Point factor.w, f.e.m exact ~7 qa /Eexl0! f.e.m. 0,27 1,08 2,70 2,62 0,94 MJli.. exact 0,26 1,19 2,70 2,63 0,94 My foe.m. 0,75 1,85 2,70 2,48 1,18 exact 0,73 1,83 2,70 2,49 1, qa xl0 f.e.m. ='finite element method Fig: 3. Simply supported edges :j--+--t-e~~::t- ~!!~i form load 9 A-.4-~--+-~ ~-+--rh-t---t-- 'X V :: 0,3.It 1 A II ---A

12 8. Zienkiewicz, O. C. "Finite Element Procedures in the Solution of Plate and Shell Problems"in Stress Analysis". John Wiley & Sons Ltd., London, Zienkiewicz, O. C. and Cheung, Y. K. "Finite Element Method of Analysis of Arch Dam Shells and Comparison with Finite Difference Procedure,s". Intern. Symp. Press, on the Theory of Arch Dams, Southampton Univ., Pergamon

13 41><, I - -- I~....:.: ~ ~ ~!sf ~.

14 Moments. common Point factor Mx f.e.m. + 6,89 + 5,18 + 3,25 + 1,20 Hellan (6 ) + 7,14 + 5,18 + 3,22 + 1,15 Point My f.e.m. 0 1,20 + 3,20 + 4,64 + 5,15 Hellan (6) + 1,13 + 3,17 + 4,64 + 5,16 Point Mx 0 0,09 1,16 1,98 2,56 My f.e.ro. + 5}15 + 1,29 0,40 0,71 Mx 0,11 1,17 1,98 2,54 Hellan (6) My + 5, ,35 0,68 f.e.ro. = finite element method qa x Fig ~ I 8 9 ~f---_~a.. 1 Upp~r ~dg~ fre~. oth~r edg~s fixed Uniform load CJ v x

CRITERIA FOR SELECTION OF FEM MODELS.

CRITERIA FOR SELECTION OF FEM MODELS. Prof. P. C.Vasani,Applied Mechanics Department, L. D. College of Engineering,Ahmedabad- 380015 Ph.(079) 7486320 [R] E-mail:pcv-im@eth.net 1. Criteria for Convergence.