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The performance of Mindlin plate finite strips in geometrically nonlinear analysis
Lw5-‘94946 13 Qll l .@I C 1986 Pergamon Press Ltd.
THE PERFORMANCE OF MINDLIN PLATE FINITE STRIPS IN GEOMETRICALLY NONLINEAR ANALYSIS Depanment
D. J. DAWE and Z. G. AZIZIAN of Civil Engineering, The University of Birmingham, P.O. Box 363, Birmingham B15 XT. United Kingdom (Received
19 February
1985)
Abstract-The finite strip method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates within the context of Sfindlin plate theory wherein the effects of transverse shear deformation are included. The analysis is of the Lagrangian type with the nonlinearity introduced by the inclusion of certain nonlinear terms in the strain-displacement equations. Following on from a related earlier investigation which dealt with a particular finite strip model, the performance of a range of different models is investigated. Linear, quadratic, cubic, and quartic polynomial interpolation is used in the different models in representing the variation of the five relevant displacement type quantities across a strip: also, both analytical (exact) and numerical (reduced, selective) schemes of integration are used in the crosswise direction in evaluating the stiffness properties of the various models. The ends of the finite strips are simply supported for out-of-plane behaviour and immovable for inplane behaviour. Detailed results are presented of the application of seven types of finite strip model to a range of plate problems, all involving uniformly loaded, square plates but with thin or moderately thick geometry and with simply supported or clamped longitudinal edges.
I. INTRODUCTIOS
In a recent paper[l] the authors described a finite strip method for the nonlinear analysis of laterally loaded, rectangular, isotropic plates. The procedure was based on the use of Mindlin plate theory (MPT) which incorporates the through-the-thickness shear deformation effects which are ignored in the classical plate theory (UT). The MPT finite strip method was described initially in quite general terms but then specialised to particular end conditions and to a particular level of crosswise polynomial interpolation for each of the five reference quantities-namely the three components of displacement and the two rotations-of the problem[ 11. The particular level of interpolation used by way of example was in fact cubic and presented numerical results demonstrated the efficiency of the derived procedures for both thin and moderately thick isotropic plates. Description of past finite strip plate analysis work has been discussed quite fully in the Introduction of Ref. [l], both for linear analysis, when based on CPT and on MPT, and for nonlinear analysis, when based on CPT alone. Such description will not therefore be repeated here, and allusion will be made when required only to that past work which is directly pertinent to the theme of the present paper. The aim of the present paper is to examine the numerical performance of various members of a family of MPT finite strip models when used in the solution of geometrically nonlinear plate problems: the various members of the family may be generated by changing the order of the adopted crosswise polynomial interpolation or by changing the scheme
of integration used in evaluating the strip stiffness properties. In the simpler problem of linear analysis there does exist a reasonable body of knowledge on the relative performance of different types of MPT finite strip model (as will be discussed in the following section) but there is an absence of such information in the realm of nonlinear analysis. Particularly in view of the great demands made on computer resources by nonlinear analyses it is important to look in some detail at the relative efflciency of MPT strip models and this is what wil be pursued here. BASIC ANALYSIS DETAILS
2.
As has already been pointed out, the development of a quite general formulation for the geometrically nonlinear analysis of laterally loaded plates using MPT finite strips is described by the authors in detail elsewhere[l]. Since the present study is a direct extension of this earlier work it is only necessary here to recall very limited details of the basic analysis with some modifications introduced to suit the present purpose. A typical finite strip is illustrated in Fig. 1 and forms part of a rectangular plate of dimensions A x I3 in plan which is subjected to a lateral loading p,,,. At any general point in the plate the displacement components in the X, y, and z directions, namely U, 5, and W, are assumed to be, within the philosophy of MPT, 7?(X,y, 5(x, qx,
z)
=
Lb,
y)
+
zd&,
Y),
y. z)
=
4x,
y)
+
zJly(x,
YL
y, z)
=
H’(X y).
(1)
D. J. DAWE
and Z. G. AZIZIA?;
(6)
+; (g)’($)I]dr
d>
(7)
and (8) Also Fig. 1. A finite strip.
On the right-hand side of this equation appear five funiamental displacement quantities at the middle surface; namely II, u, and W, the components of translation, and & and ti,,, the rotations about the y and .r axes. The nonlinear analysis is of the Lagrangian type in which deformation is measured with respect to the initial configuration of the plate and the nonlinearity appears through the inclusion of certain squares and products of displacement derivatives in the strain-displacement equations. By rearrangement of the equations of Ref. [l] the potential energy of an individual finite strip can be expressed n,=upmfUpb+u::+U~fup4+Vp,
(2)
where
u,m= ;J_b~2,,Ac[(g)’ +($ + 2~:;
+ l3 (;
+ ;)‘I
drdy,
(3)
Eh c = (1 - vz) *
Eh’ D = I?(1 - u’)
p = (1 - v) 2
E G=p 2(1 + V) ’
(9)
where E and v are Young’s modulus and Poisson’s ratio of the isotropic material and K’ is the shear correction factor. The quantities L$’ to C/g, defined in eqns (3)(7), are contributions to the total strain energy of the finite strip. All of UT, U,“, and UF are quadratic functions of the displacements, and they represent distinct contributions from membrane, bending, and transverse shearing actions, respectively. (Ii is a cubic function of the displacements which incorporates the effect of nonlinear bending-stretching coupling. r/g is a quartic function of the displacements arising from bending action. The quantity VP defined in eqn (8) is the potential energy of the lateral loading acting on the strip. It is noted that the bending strain energy U,” [eqn (4)] is proportional to the cube of plate thickness whereas all other strain energy contributions are proportional to the thickness. In the MPT finite strip approach assumptions are made for the variations of each of the fundamental displacement quantities U. U, W, 1,, and JI, over the middle surface. Leaving aside, for the moment, consideration of the detailed specification of these assumed variations, it is possible (see Ref. [I]) to eventually re-express the strip potential energy leqn (31 as IIp = ;id,T(K; + K,b i + bd;K,,d,
K:)d,
+ &d;K-,d,
- d,TP,.
(IO)
Here d, is the column matrix of strip degrees of freedom and P, is the corresponding column matrix of forces: the subscript n relates to the order of crosswise polynomial interpolation adopted in specifying the displacement field. The five stiffness
Performance of .Llindlin plate finite strips
3
matrices K:, Iii. Ii:. Ki,, and Kz, are developed as in the earlier work. Thus, for every strip model. from the five strain energy contributions cl;, upb, L$‘. Ub. and Ui;l. respectively. The coefficients of U,(X) = V,(X) = W;(X) = CpJ.r) = KT. KR. and Kz are constants whilst those of K,, (13) and K?,, are linear and quadratic functions. respec@Jx) = tively. of the fundamental displacement quantities. Addition of the potential energies of all strips gives the potential energy of an assemblage of strips These functions are suitable for end conditions (at and the usual energy minimisation procedure yields I = 0 and x = A) defined as the set of nonlinear plate equations
sin% , co,?
--
u
where an overbar indicates a complete-plate quantity. Equation (11) is a modified form of eqn (44) of Ref. [l] with k of that reference replaced here by the three component contributions Em, Eb, and ??
3, THE FIVITE STRIP MODELS In Ref. [l], the variations over the strip middle surface of the five fundamental quantities were expressed initially in very general form as summations of r products of longitudinal (x-direction) series functions and crosswise (y-direction) polynomial functions, as
u = i
uicdgY(Y),
v =
5 i=l
Vi(X)gY(Y)t
‘V =
i wi(X)g?(Y)t i= I
i=l
4J.r =
i i=
I
(12)
@.ri(x)gF(Y).
Thereafter, in producing numerical solutions a particular selection of longitudinal functions was made to satisfy a given set of strip end conditions, and a particular level of crosswise polynomial interpolation was adopted. This level was cubic, for each of the five fundamental quantities, and exact, analytical integration was used in calculating strip properties. In the numerical applications of the present study the same longitudinal functions are employed
=
i’
=
,,’
=
bI,
=
0
6, free,
(14)
and correspond to simple supports for bending/ shearing behaviour and to immovable edges for membrane behaviour. In the crosswise direction, for any particular strip model the form of the g(y) polynomial functions is assumed to be the same for each of the five fundamental quantities and for all i, but the order of polynomial used, that is n, does vary from one strip model to another, of course. The degrees of freedom of any particular strip model in the C” type continuity problem are the values of u, V, bv, b,, and I&.for all values of i, located at (n + 1) reference lines equispaced across the strip model. In evaluating the properties of the finite strip models some markedly different schemes of integration in the crosswise direction are employed here, but before detailing the various strip models we refer briefly in the next paragraph to the concept of under integration as developed in finite element analysis. The use of deliberate under integration in evaluating the stiffness properties of isoparametric plate and shell finite elements, based on a shear deformation theory such as MFT, is now common practice. The earliest work on this topic was by Zienkiewicz, Taylor, and Too[Z] and by Pawsey and Clough[3] who detailed the dramatic improvement obtained in using under integration in connection with the isoparametric shell elements of Ahmad et a1.[4]. When using “full” Gaussian numerical integration these elements are efficient performers for thick plates and shells but are excessively overstiff for thin geometry: reduction in the order of numerical integration leads to very considerable improvement in efficiency for thin geometry, whether such reduction is “selective” or “reduced”[3, 41. Regarding the terminology used here, “full” integration (p x p points over the element middle surface) means an order of integration sufficient to calculate the stiffness matrix exactly for a rectangular or parallelogrammic-shaped element: “selective” integration implies the use of (p - 1) x (p - 1) integration points in evaluating the transverse shear stiffness contribution and p x p points in evaluating all other contributions; and “reduced” integration implies the use of (p - 1) x (p - 1) points in evaluating all stiffness contributions. Following the initial work mentioned above there has been considerable investigation of the use of the under in-
D. J.
4
D.AWE and Z.
tegration philosophy in MPT-based isoparametric finite elements of the serendipity. Lagrangian. and combined serendipity./Lagrangian types: amongst such investigations are those described in Refs. [S91. Certain difficulties can arise for particular element types. Thus, in serendipity elements “locking” behaviour occurs. manifested by the generation of overstiff numerical results for very thin plates (due to the predominance of the transverse shear contribution for such plate geometry), whilst in Lagrangian elements there can be difficulties associated with the production of spurious zero-energy modes. The “heterosis” element[7] of the combined SerendipityiLagrangian type appears to avoid both kinds of difficulty. The finite element studies referred to thus far are concerned with linear analysis but it is noted that Pica, Wood, and Hinton[ IO] have investigated geometrically nonlinear plate behaviour using 4-, 8-. and 9-noded elements of the serendipity, Lagrangian, and heterosis types, with reduced and selective numerical integration schemes. The numerical applications are restricted to plates of thin geometry and in this context, whilst the performance of the elements is generally good, no clearly superior element emerges. Returning to the subject topic of nonlinear MPT finite strip analysis the basic details of the strip models utilised in the present study are recorded in Table I: these details relate to the orders of crosswise polynomial interpolation used in the displacement fields and to the manner of integration of
Table I. Details of Strip
Order of crosswise interpolation n
Model
G. AZIZI.LV
terms in the crosswise direction. (In the longitudinal direction analytical integration is used in all cases.) Half the models listed in Table 1 are based upon the use of analytical (that is exact) integration of all stiffness contributions, and the order of polynomial interpolation ranges from linear (n = 1) to quartic (n = J). In fact the linear, analytical model LA, though listed in the Table for completeness, will not be considered later in the paper when numerical results are discussed since this model yields hopelessly overstiff results in most circumstances. The remaining three analytically integrated models, namely QA, C4, and FA, have been tested earlier in linear analyses. This was first done by Dawe[ 1l] in a study of the vibration of thin and moderately thick plates with simply supported ends where, additionally, quintic interpolation was also employed. Numerical applications showed clearly that increasing the order of crosswise interpolation leads to significant improvements in accuracy of solution for a given number of total degrees of freedom in the problem. They also showed that the accuracy of the higher-order models is not greatly dependent upon the relative thickness of the plate under consideration, whilst the accuracy of the quadratic QA model is, and shows a marked decline with decrease in thickness. Further details of the use of the FA model in linear eigenvalue problems (for plates with a variety of end conditions) are given by Dawe and Roufaeil[lZ, 131, whilst more recently Onate and Suarez have produced numerical results for the linear static analysis of simply supported plates when
the
MPT finite strip models No. of reference lines
Crosswise integration scheme
(Linear,
LA analytical)
Analytical.
(Linear.
LS selective)
Selective. tS I point for K 2 points for%ther
(Linear,
LR reduced)
Reduced. I point for
QA (Quadratic,
analytical)
(Quadratic,QS
selective)
(guadratic,
QRreduced) CA
(Cubic,
analytical)
(Quartic analytical)
FA [Fourth-order],
K
allK
Analytical.
2
3
Selective. 2 points 3 points
for Kts for 08~
for
2
3
Reduced. 2 points
3
4
Analytical.
4
5
Analytical
all
C
5
Performance
of Xlindlin plate finite strips
using the LA. QA. and CA models[l4]. Of course, the CA model (shown in Fig. I) is also the model used by the authors in their earlier study of geometrically nonlinear static behaviour[ 11 which, so far as is known, is the only application to nonlinear analysis of any of the strip models detailed in Table I. The four numerically integrated strip models listed in Table 1, namely LS, LR. QS, and QR, are based on the use of either linear or quadratic polynomial interpolation across the strip, each in conjunction with either selective or reduced Gaussian numerical integration. The quadratic strip model QR was introduced by Mawenya and Davies[ 151 in the solution of linear static problems and its use was extended to vibration and stability applications by Benson and Hinton[l6]. The linear strip model LS was first described by Hinton and Zienkiewicz[l7] in the context of static analysis. The more-recent work of Onate and Suarez[ 141, mentioned earlier with regard to the analytically integrated models, contains detailed comparative discussion of the use of strip models LS, LR, QS, and QR (and cubic models) in the linear static analysis of uniformly loaded plates with one pair of ends simply supported. On the theoretical level, the singularity of the stiffness contribution Kf of each strip model is examined, using the simple rule suggested in Ref. [51. This rule predicts that singularity of K: does not exist, and hence that the problem of locking behaviour for very thin plates will be present, for the linear strip with full Gaussian integration but not for the equivalent strip with selective or reduced integration (models LS and LR). It also predicts that locking behaviour will be present for some specific boundary conditions (clamped edges) when using the quadratic strip with full integration but not with reduced or selective integration (models QS and QR). Presented numerical results1141 confirm these predictions and show that the linear, quadratic, and cubic MPT strips perform well for thick and thin geometry when their properties are based on the use of selective or reduced integration but that only the cubic strip is completely reliable when full integration (four integration points for the cubic strip) is used. Onate and Suarez[l4] recommend the LR strip model as the “best value” MPT strip of those studied in their work for reasons of its excellent performance, for thick and thin geometry, and its inherent simplicity which corresponds to the use of only one integration point. They point out that the LR strip model has one possible spurious zero energy mode but that this mode is not propagated if an assembly of two or more strips is considered, which would be the case in practical analysis using the linear model. Of course, it must be borne in mind that all past studies referred to above (except that by the present authors in Ref. [l]) relate to linear, small-deflection analysis in which the only strain energy contribu-
5
tions present are those of bending (I/i) and transverse shearing (c/F) actions: consequently only Kf: and KY are then present on the right-hand side of eqn (10). In moving in the present study to consideration of nonlinear analysis, the extra strain energy contributions UT, Vi, and c’z are introduced and the proportional balance of energy terms is changed radically, so that conclusions that apply for linear analysis may no longer apply. It is with such matters, amongst others, that the present study is concerned, of course. 4. NLXERICAL APPLICATIONS The problems detailed here are basically the same as those considered in the earlier, related publication[ 11 where only the CA strip model was considered. Thus the plates are square, of side length A, and are subjected to a uniformly distributed lateral loading. All plate edges are immovable in the plane of a plate whilst for out-of-plane behaviour either all four edges are simply supported (the SSSS plate) or one pair of opposite edges (at .r = 0 and x = A) is simply supported and the other pair is clamped (the SCSC plate). It is noted that the boundary conditions at a simply supported edge are defined in eqn (14) for an edge x = constant (with QI and JIYinterchanged for an edge y = constant) whilst the definition of a clamped edge is that all of U, V, w, JIX,and ti, are zero at the edge. Two relative thicknesses are considered, corresponding to hlA = 0.01 (thin plate) and h/A = 0.05 (moderately thick plate), and the values of I? and v are taken to be 3 and 0.3, respectively. As in Ref. [l], the presented results are referred to values of a nondimensional loading factor Q = qA4/Dh, where q is the intensity of the uniform lateral loading. All of the problems have a symmetry in the longitudinal direction such that, with reference to eqns (13), only the terms corresponding to i = 2, 4, 6, 8 . . for Ui(x) and to i = 1, 3, 5, 7 . . . for Vi(x), &i(x), @xi(X), and @y{(X)make a contribution and hence need be employed. There also exists a symmetry in the crosswise direction whbh is incorporated in the usual way by modelhng only half the plate with finite strips. In what follows results of the application of the MPT finite strip models are presented for displacement and force quantities, both in tabular form for point values and in diagrammatic form showing distributions across or along a plate. For all the strip models the displacement quantities are calculated directly at the reference lines, of course, and at other intermediate points as desired, and no difftculty arises in their representation. With regard to the force quantities for the analytically integrated strip models QA, CA, and FA, these are calculated, as are the displacement quantities, at reference lines and at any convenient intermediate points. For the numerically integrated strip models LR, LS, QR, and QS calculation of force quantities is made
D. J. D~WE and Z. G. AZIZIAY
6
specifically at the Gauss points: where values are required at other points these are obtained by suitable extrapolation from the Gauss-point values. We begin the presentation of results by examining the performance of seven strip models in calculating the lateral deflection M’,.at a plate centre and the v-direction stress u,, occurring on the lower surface of a plate. again at its centre of plan: the u,~ stress combines contributions from the central bending moment lMycand extensional force NY,, of course. Numerical details of the convergence of the calculated values of the two specified quantities with increase in the value of a parameter N are given in Tables 2-5 for four plate problems. The parameter N is the number of degrees of freedom per contributing series term in a symmetric halfplate after applying the appropriate boundary conditions at the plate longitudinal edge and the longitudinal centre line: the total number of degrees of freedom in a probiem is thus N x no where n,, is the number of terms used in the longitudinal series of each reference quantity. For the results given in Tables 2-5 attention is concentrated on convergence with respect to the crosswise modelling and the value of no is fixed at four. The values of N quoted in the tables correspond to the use of 2, 3, 4, 6, 8, 10, and 12 strips for models LR and LS; of I, 2, 3, 4, 5, and 6 strips for models QR, QS, and QA; of 1, 2, 3, and 4 strips for model CA; and of 1, 2, and 3 strips for model FA. The four plate problems to which Tables 2-5
refer are for the thin and moderately thick geometries of both the SSSS and SCSC plates at the higher load levels considered in the earlier investigation[l], i.e. Q = 6400 for h/A = 0.01 and Q = 100 for h/A = 0.05. For these problems there are comparative approximate solutions available[ l] which are based on a Rayleigh-Ritz analysis, though within the confines of classical plate theory: these “R-R CPT” results are recorded in Tables 2-5. It can be seen in Tables 2-5 that convergence with increase in N of values of central deflection calculated using the MPT finite strip approach is orderly and rapid for all types of strip model. The central deflection values reached by each model at the largest values of N used (58 or 59) are very close to one another in any of the four documented problems, the greatest differences being less than 1% even for the difficult problem of the thin SCSC plate. (Incidentally it is noted that no bound conditions apply to the tabulated values of central deflection even for the analytically integrated strip models, since the loading is a distributed one.) Normally results of good accuracy are obtained for central deflection using far lower values of N than the greatest values detailed here. The point-stress values given in Tables 2-5 for the finite strip method also reveal good rates of convergence to common levels of stress for any particular problem. In comparing the values to which the MPT finite strip method results are converging with the R-R
Table 2. Values of central deflection and stress for thin SSSS square plate: no = 4, Q = 6400
I
ISI
I
Strip Model LRI
LS
1
QR
1
QS
1
QA
1
CA
I
‘?A
1
Performance
of Mindlin
plate finite strips
Table 3. Values of central deflection and stress for moderately Q = 100
thick SSSS square plate: no = 4.
Table 4. Values of central deflection and stress for thin SCSC square plate: no = 4, Q = 6400
7
D. .I.
8 Table 5. Vaiucj of
central
dztlsction
D.AW and Z. G. .~ZIZIA~
and S~KSS
for moderately
thick SCSC
square
plate:
no = 1.
Q = 100
1
13
I .Sl72
1.8168
-
CPT results it is apparent that there is little difference between the predictions of the two procedures for the thin plates. This is as expected since transverse shear effects have very little influence for thin, isotropic plates under uniform static loading. For the moderately thick plates there is some noticeable transverse shear effect on the central deflection value, demonstrated by the increased flexibility of the shear-deformable plate model, although the effect is still not terribly large (something over 3%) even for the SCSC plate. Comparison of the values of deflection and stress exhibited in Tables 2-5 for the various MPT finite strip models suggests the following points. (a) For both the linearly and quadratically interpolated models, when numerical integration is employed there is very little difference in the results whether reduced or selective integration is used, i.e. the LRand LS models perform very similarly one to another, as do the QR and QS models. (b) On the basis of a common value of N the performance of the LR/LS models is only slightly less good than that of the QWQS models. (c) The QWQS models are somewhat superior to the analytically integrated QA model. (Also, the LWLS models are very markedly superior to the analytically integrated LA model, though
?.ji?6
-
03
details of the performance of the latter model are not given here). (d) The higher-order, analytically integrated models CA and FA are accurate for stress and very accurate for deflection when two strips are employed in the half-plate, and sometimes give adequate performance when employing only a single strip. It should be noted that use of the value of the parameter N in Tables 2-5 as a basis for assessing the relative efficiency of the various strip models is convenient but it by no means tells the whole story as regards the computational effort involved in obtaining a solution of given accuracy when using different types and numbers of strip models. It is not possible to be very specific about relative corn: putational effort here, since this is both machine and program dependent, and the numerical results presented have been obtained using more than one computer. However, obviously one significant time-consuming stage is that of solution for displacements and here it is known that the size of the half-bandwidth of the structure stiffness matrix is of great importance, with solution time being dependent on the half-bandwidth to the power 2 at the very least. Since, for a given number of degrees of freedom the sizes of the half-bandwidths associated with assemblies of linear, quadratic, cubic, and
9
Performance of hlindlin plate tinitr strips quartic finite strips. respectively. are in the ratio 2/ 3i4iS. it is clear that there are very considerable economies associated with the use of low-order strips. Continuing with the examination of the efftciency of the crosswise modelling. Figs. 2-S illustrtate the distributions of M’,dV,. and NY across one half of each of the four plates discussed above. The distributions shown are along the x = A/2 centre line and results predicted by the use of various numbers of each of the seven strip models are presented. For comparison purposes the distributions predicted by the R-R CRT approach are also shown in the figures: these distributions are themselves not exact. of course (as evidenced by the waviness of
(0)
LR
the N, and M, distributions for the SCSC thin plate in Fig. I), but they do provide a very useful measure of comparison. The results shown in Figs. 2-j serve to verify the proper convergence of each strip model and provide further evidence of the statements made above concerning the relative performance of the various models. It is interesting to note the high accuracy of the force type distributions obtained for the numerically integrated strip models when Gauss point values alone are used, and to observe the manner in which the calculated distributions of moment and membrane force vary across the full width of strips for the analytically integrated models QA. CA. and F.4. Thus far, all the results presented correspond to
(d)
(b) LS
-
R - R CPT solution A
(e)
0.4
(f)
CA
(9)
OS
1 strip
1
FA
Fig. 2. Thin SSSS square plate with Q = 6400: distributions of w, I&. and NYat x = A/2 for no = 4 (R-R CPT central values are w,/A = 0.023890, 6My,AzlEh4 = 11.535, and N.&‘/Eh’ = 18.315).
D. J. D.AWEand Z. G. Azrzr-is
10
(a)
LR
lb)
Cd) OS
LS
-
(e)
QA
(f)
CA
R-R
CPTsaktlian
(P) FA
Fig. 3. Moderately thick SSSS square plate with Q = 100: dist~b~tioRs of W, .\fY, and N,. at x = A/ 2 for no = 4 (R-R CPT central values are w,iA = 0.017335, 6M,,A’iEh’ = t.2043, and Ny,A21Eh’ = 0.36650).
the use of a particufar number (no), namely 4, of terms in the longitudinal series expressions for the reference quantities. The manner of convergence of displacement and stress quantities with the number of longitudinal series terms used is also of interest, and Table 6 and Fig. 6 are included to give an idea of such convergence in the present circumstance where the applied loading is a uniformly distributed one. Table 6 gives details of point values and demonstrates that the rate of convergence with no does not vary greatly with the type or the number of strips used: results are presented, somewhat arbitrarily, for two types of problem and for the use of particular numbers of strips having the four differert levels of crosswise interpolation considered
in this paper. The point values considered are the same as those considered earlier in Tables 2-5 and the values obtained with no = 1 do not differ greatly from the corresponding values with no = 4. Figure 6 shows the manner in which the value of no affects the predicted dist~butions of W, N,, and M, along the longitudinal centre line for the same two types of plate problem detailed in Table 6. The strip method distributions shown in Fig. 6 were actually drawn using the results obtained for an assembly of eight LR strips, but it is noted that there is no discernible difference in a graphical presentation from distributions obtained for alternative strip assemblies having around the same N value, such as from an assembly of two FA strips for instance.
Performance
of Mndlin plate finite strips
II
lb) LS
i’---.,J (1)
CA
(9)
FA
Fig. 4. Thin SCSC square plate with Q = 6400: distributions of W, My, and NY at x = A/2 for no = 4 (R-R CPT central values are w,/A = 0.022728. 6My,A2/EhJ = 13.350, and N,,A’IEh = 16.835).
Table 6. Convergence
of central deflection and stress with no Strip
Plate Problem
“a
I z z 1 .” 5
,-E
xf;
II
LR
Quantity (4 WC/A
(2
strips)
yodel CA (2
strips)
PA (I
strip)
3.5560
3.5362
3.5340
3.5321
2
3.5110
3.4927
3.4898
3.4885
3
3.5164
3.4981
3.4952
3.4939
4
3.5155
3.4972
3.4943
3.4930
1
200
scrips)
QR
2.6058
2.6459
2.5996
2.5302
-G Ec u
2
2.5750
2.6283
2.5724
2.4996
2: ;z
3
2.5834
2.6346
2.5806
2.5096
r;;
4
2.6050
2.6324
2.5774
2.5064
=YC
u a-
l
2.4308
2.3865
2.3586
2.3098
m s
2
2.3123
2.2722
2.2430
2.2007
3
2.3274
2.2882
2.2587
2.2149
4
2.3242
2.2846
2.2554
2.2121
: m gz c
c II
I
100
A2/Eh2
WC/A
32.208
31.347
31.340
14.357
;c
2
cycA2/Eh’
31 .I70
30.938
30.475
15.553
.z L: c
3
31.475
31.357
30.938
15.830
4
31.295
31 .089
30.703
15.548
D. 1.
12
(01
LR
Ib)
DAWE
and
Z. G.
AZIZI.AN
(d)
LS
OS
R-R
CPT solution
1 strip 2 strips 1
6slrips Bstrips J
fr)
QA
(f)
CA
(q1 fA
Fig. 5. Moderately thick SCSC square plate with Q = NO: distributions 6M,,A’iEh’ 2 for no = 4 (R-R CPT central values are w,/A = 0.0093355,
of W, h&, and iy, at .r = A/ = 1.7378, and Ny,A21Eh3
= 0.10840).
Finally, we show the manner of variation with increasing load level of central deflection and stress for the moderately thick SSSS plate in Fig. 7 and for the thin SCSC plate in Fig. 8. These figures relate to the use of the two strip models at the opposite ends of the spectrum so far as sophistication is concerned, namely the linear LR model and the quartic FA model. Again, the R-R CPT predictions are shown for comparison purposes and it is seen that the finite strip results converge rapidly toward these R-R CPT predictions. (Fully converged MPT finite strip vaIues will differ somewhat from the RR CPT values but not by amounts which are noticeable in these particular graphical presentations.) 5. CONCtt’SrO3-S
All the MPT finite-strip models used in this detailed study of nonlinear plate behaviour demon-
strate convergence to proper levels of displacement and stress for piates of thin and moderately thick geometry, as evidenced by tabulated point values and by graphical distributions. The high-order, analytically integrated cubic and quartic models CA and FA are very accurate in predicting values of deflection, with the use of very few strips, and on a degree-of-freedom basis are probably the most accurate and consistent performers in this respect. These models also represent the dist~b~ions of force type quantities we11 in general, though they do not have an advantage in this respect over the other models considered. In the present nonlinear analysis an important practical disadvantage of the CA and FA models is their relative expense due to the large ham-bandwidth of the structure stiffness matrices. Of the three quadratic models the analytically integrated model QA is the least impressive per-
Performance
of Mindiin plate finite strips
13 W
!2 h
0 _-:-0123L56
-
CPT
R-R “,
.
“, = 2
I
l-lo L 3
0
“* = L
=
solution
1
d
MPT Frmtc stnp 50iut10ns
-R-R
I
Fig. 6. Longitudinal distributions of W. &I,,, and N,: (a) Moderately thick SSSS square plate with Q = 100; (b) Thin SCSC square plate with Q = 6400. [R-R CPT central values are H+,./.+= 0.017335, &M,,A’i Eh’ = 2.2043, and N,,A’/E113 = 0.36650 for case (a), and w,iA = 0.022728, 61~~~‘~~/~~~= 11.063. and N,,.A’iElr3 = 17.383 for case (b).]
CPT solution
d
1 strip
.
2 strips
x
3 strips
0
4 stnps
1 MPT F,n,te strtp solutions
Fig. 7. Moderately thick SSSS square plate: variations of central deflection and stress with load: (a) LR model; (bf FA model.
20
h I-O
3
~ ‘0
0
Y
10“
1
2
3
0
-
R-
R
CPT
A
1 strtp
.
2 strips
II
3 strtps
0
4 strips
(I
6 strips
l
6 strips
x
4
5
6
IO“
solution
MPT strip
Fmste s0lut10ns
I
Fig. 8, Thin SCSC square plate: variations of central deflection and stress with load: (a) LR model: (b) FA model.
I4
D. J. D.AWEand Z. G. AZIZI.\~
former in numerical terms. particularly so for the plates having clamped edges. The performance of the numerically integrated modeis QR and QS is very good and is similar. Of these two models it would therefore appear to make sense to use the QR model with its simpler and cheaper scheme of integration. The efficiency of the numerically integrated linear models LR and LS is remarkably good in the considered applications. (The analytically integrated linear model has not been discussed in detail but its performance is so poor that it is simply not a viable model.) A particularly impressive feature of the LR and LS models, considering their simplicity, is their accurate representation of stress type distributions when using few strips and when Gauss point values are plotted. The LR and LS models perform very similarly to each other and on a degree-of-freedom basis give results almost as good as do the QR and QS models: however, the quadratic models are significantly more expensive than are the linear models. Finally, of the numericaily integrated models considered in this study the linear LR model must be recommended in view of its simplicity, with just one integrating point used for the evaluation of all strain energy cont~butions, and of its economy that results both from this iow-level integration and from the small half-bandwidth of the structure stiffness matrices. This judgement appears justified on the evidence available but perhaps should be qualified with a slight note of caution untii a wider range of numerical tests is made. The alternative, analytically integrated, cubic and quartic models have much to recommend them as accurate and consistent performers which are based rigorously on the principle of minimum potential energy. However, in the nonlinear realm they are much more demanding of computing power than is the numerically integrated linear model, at least when using the solution procedures adopted in the present study. rlcknowledgmenl-The authors are grateful to the Science and Engineering Research Council for providing a grant to support the work described in this paper. They are also pleased to acknowledge the use of facilities at the Computer Centre of 3i~ingham University and at the L’niversity of Manchester Regional Computer Centre in obtammg the numerical results reported here.
REFERESCES 1. 2. G. Azizian and D. J. Dawe. Geometrically
nonlinear analysis of rectangular Mindlin plates using the finite strip method. Co~rrprrr. Srnrcr. 21, 113-436 (19%). 0. C. Ztenkiewicz. R. Taylor. andJ. hf. Too. Reduced integration techniques in general analysis ofplates and shells. In?. J. ~VMXT.Merh. Engng 3.275-290 (1971). S. E. Pawsey and R. W’. Ciough. improved numerical integration of thick shell tinite elements. IN. J. Numer. Mrlh. Engng 3. 575-586 (1971). S. Ahmad. 8. Sf. Irons, and 0. C. Zienkiewicz, Analysis of thick and thin shel! structures by curved finite elements. lnr, J. .V~xr. .?ferh. Engng 2, 419-431 (1970). 5. 0. C. Zienkievvicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Frnnklin fnsr. 302, 443-461 (1976). I‘. J. R. Hughes, R. L. Taylor. and W. ~anoknuku~-
9.
10.
Il. 12.
13.
14.
chai, A simple and efficient element for plate bending, Inr. J. Nztmer. Merh. Engng 11, 1529-1543 (1977). T. J. R. Hughes and Xl. Cohen. The heterosis finite element for plate bending. Comprtr. Srnrct. 9, 445450 (1978). E. D. L. Pugh. E. Hinton. and 0. C. Zienkiewicz. A study of quadrilateral plate-bending elements with “reduced” integration. Inr. J. Numer. Mefh. Engng 12, 1059-1079 (1978). E. Hinton and S. Bicanic, A comparison of Lagrangian and serendipity Mindlin piate elements for free vibration analysis. Compact. Srrrccr. 10, 483-493 (1979). i\. Pica, R. D. Wood, and E. Hinton, Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Comprrr. Strucr. 11, ?03215 T1980). D. J. Dawe, Finite strip models for vibration of Mtlindho elates. J. Sofrnd Vihr. $9. 441-452 (19%). 0. i. Roufaeil and D. J. Da&e. Vibration analysis of rectangular Mindlin plates by the finite strip method. Corn&r. Srruct. 12, -833-842 (1980). D. J. Dawe and 0. L. Roufaeil. Buckling of rectangular Mindlin plates. fompuf. ‘Slrurr. fs, 461-171 (1982). E. Onate and B. Suarez. A comparison of the linear, quadratic, and cubic .Ciindlin strip elements for the analysis of thick and thin plates. Comp~r. Srrttcr. 17, 427-439
(1983).
IS. A. S. Mawenya and J. D. Davies. Finite strip analysis of plate bending including transverse shear effects. Build.
Sci. 9. 175-180
(1974).
16 P. R. Benson and E. Hinton. A thick finite strip solution for static, free vibration and stability problems. fnr. J. ;Xurner. Xerh. Engng IO, 665-678 (1976). 17. E. Hinton and 0. C. Zienkiewicz, A note on a simple thick finite strip. Inr. J. il’~mer. :Merlr. Engng 11,905906 (1977).
THE PERFORMANCE OF MINDLIN PLATE FINITE STRIPS IN GEOMETRICALLY NONLINEAR ANALYSIS Depanment
D. J. DAWE and Z. G. AZIZIAN of Civil Engineering, The University of Birmingham, P.O. Box 363, Birmingham B15 XT. United Kingdom (Received
19 February
1985)
Abstract-The finite strip method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates within the context of Sfindlin plate theory wherein the effects of transverse shear deformation are included. The analysis is of the Lagrangian type with the nonlinearity introduced by the inclusion of certain nonlinear terms in the strain-displacement equations. Following on from a related earlier investigation which dealt with a particular finite strip model, the performance of a range of different models is investigated. Linear, quadratic, cubic, and quartic polynomial interpolation is used in the different models in representing the variation of the five relevant displacement type quantities across a strip: also, both analytical (exact) and numerical (reduced, selective) schemes of integration are used in the crosswise direction in evaluating the stiffness properties of the various models. The ends of the finite strips are simply supported for out-of-plane behaviour and immovable for inplane behaviour. Detailed results are presented of the application of seven types of finite strip model to a range of plate problems, all involving uniformly loaded, square plates but with thin or moderately thick geometry and with simply supported or clamped longitudinal edges.
I. INTRODUCTIOS
In a recent paper[l] the authors described a finite strip method for the nonlinear analysis of laterally loaded, rectangular, isotropic plates. The procedure was based on the use of Mindlin plate theory (MPT) which incorporates the through-the-thickness shear deformation effects which are ignored in the classical plate theory (UT). The MPT finite strip method was described initially in quite general terms but then specialised to particular end conditions and to a particular level of crosswise polynomial interpolation for each of the five reference quantities-namely the three components of displacement and the two rotations-of the problem[ 11. The particular level of interpolation used by way of example was in fact cubic and presented numerical results demonstrated the efficiency of the derived procedures for both thin and moderately thick isotropic plates. Description of past finite strip plate analysis work has been discussed quite fully in the Introduction of Ref. [l], both for linear analysis, when based on CPT and on MPT, and for nonlinear analysis, when based on CPT alone. Such description will not therefore be repeated here, and allusion will be made when required only to that past work which is directly pertinent to the theme of the present paper. The aim of the present paper is to examine the numerical performance of various members of a family of MPT finite strip models when used in the solution of geometrically nonlinear plate problems: the various members of the family may be generated by changing the order of the adopted crosswise polynomial interpolation or by changing the scheme
of integration used in evaluating the strip stiffness properties. In the simpler problem of linear analysis there does exist a reasonable body of knowledge on the relative performance of different types of MPT finite strip model (as will be discussed in the following section) but there is an absence of such information in the realm of nonlinear analysis. Particularly in view of the great demands made on computer resources by nonlinear analyses it is important to look in some detail at the relative efflciency of MPT strip models and this is what wil be pursued here. BASIC ANALYSIS DETAILS
2.
As has already been pointed out, the development of a quite general formulation for the geometrically nonlinear analysis of laterally loaded plates using MPT finite strips is described by the authors in detail elsewhere[l]. Since the present study is a direct extension of this earlier work it is only necessary here to recall very limited details of the basic analysis with some modifications introduced to suit the present purpose. A typical finite strip is illustrated in Fig. 1 and forms part of a rectangular plate of dimensions A x I3 in plan which is subjected to a lateral loading p,,,. At any general point in the plate the displacement components in the X, y, and z directions, namely U, 5, and W, are assumed to be, within the philosophy of MPT, 7?(X,y, 5(x, qx,
z)
=
Lb,
y)
+
zd&,
Y),
y. z)
=
4x,
y)
+
zJly(x,
YL
y, z)
=
H’(X y).
(1)
D. J. DAWE
and Z. G. AZIZIA?;
(6)
+; (g)’($)I]dr
d>
(7)
and (8) Also Fig. 1. A finite strip.
On the right-hand side of this equation appear five funiamental displacement quantities at the middle surface; namely II, u, and W, the components of translation, and & and ti,,, the rotations about the y and .r axes. The nonlinear analysis is of the Lagrangian type in which deformation is measured with respect to the initial configuration of the plate and the nonlinearity appears through the inclusion of certain squares and products of displacement derivatives in the strain-displacement equations. By rearrangement of the equations of Ref. [l] the potential energy of an individual finite strip can be expressed n,=upmfUpb+u::+U~fup4+Vp,
(2)
where
u,m= ;J_b~2,,Ac[(g)’ +($ + 2~:;
+ l3 (;
+ ;)‘I
drdy,
(3)
Eh c = (1 - vz) *
Eh’ D = I?(1 - u’)
p = (1 - v) 2
E G=p 2(1 + V) ’
(9)
where E and v are Young’s modulus and Poisson’s ratio of the isotropic material and K’ is the shear correction factor. The quantities L$’ to C/g, defined in eqns (3)(7), are contributions to the total strain energy of the finite strip. All of UT, U,“, and UF are quadratic functions of the displacements, and they represent distinct contributions from membrane, bending, and transverse shearing actions, respectively. (Ii is a cubic function of the displacements which incorporates the effect of nonlinear bending-stretching coupling. r/g is a quartic function of the displacements arising from bending action. The quantity VP defined in eqn (8) is the potential energy of the lateral loading acting on the strip. It is noted that the bending strain energy U,” [eqn (4)] is proportional to the cube of plate thickness whereas all other strain energy contributions are proportional to the thickness. In the MPT finite strip approach assumptions are made for the variations of each of the fundamental displacement quantities U. U, W, 1,, and JI, over the middle surface. Leaving aside, for the moment, consideration of the detailed specification of these assumed variations, it is possible (see Ref. [I]) to eventually re-express the strip potential energy leqn (31 as IIp = ;id,T(K; + K,b i + bd;K,,d,
K:)d,
+ &d;K-,d,
- d,TP,.
(IO)
Here d, is the column matrix of strip degrees of freedom and P, is the corresponding column matrix of forces: the subscript n relates to the order of crosswise polynomial interpolation adopted in specifying the displacement field. The five stiffness
Performance of .Llindlin plate finite strips
3
matrices K:, Iii. Ii:. Ki,, and Kz, are developed as in the earlier work. Thus, for every strip model. from the five strain energy contributions cl;, upb, L$‘. Ub. and Ui;l. respectively. The coefficients of U,(X) = V,(X) = W;(X) = CpJ.r) = KT. KR. and Kz are constants whilst those of K,, (13) and K?,, are linear and quadratic functions. respec@Jx) = tively. of the fundamental displacement quantities. Addition of the potential energies of all strips gives the potential energy of an assemblage of strips These functions are suitable for end conditions (at and the usual energy minimisation procedure yields I = 0 and x = A) defined as the set of nonlinear plate equations
sin% , co,?
--
u
where an overbar indicates a complete-plate quantity. Equation (11) is a modified form of eqn (44) of Ref. [l] with k of that reference replaced here by the three component contributions Em, Eb, and ??
3, THE FIVITE STRIP MODELS In Ref. [l], the variations over the strip middle surface of the five fundamental quantities were expressed initially in very general form as summations of r products of longitudinal (x-direction) series functions and crosswise (y-direction) polynomial functions, as
u = i
uicdgY(Y),
v =
5 i=l
Vi(X)gY(Y)t
‘V =
i wi(X)g?(Y)t i= I
i=l
4J.r =
i i=
I
(12)
@.ri(x)gF(Y).
Thereafter, in producing numerical solutions a particular selection of longitudinal functions was made to satisfy a given set of strip end conditions, and a particular level of crosswise polynomial interpolation was adopted. This level was cubic, for each of the five fundamental quantities, and exact, analytical integration was used in calculating strip properties. In the numerical applications of the present study the same longitudinal functions are employed
=
i’
=
,,’
=
bI,
=
0
6, free,
(14)
and correspond to simple supports for bending/ shearing behaviour and to immovable edges for membrane behaviour. In the crosswise direction, for any particular strip model the form of the g(y) polynomial functions is assumed to be the same for each of the five fundamental quantities and for all i, but the order of polynomial used, that is n, does vary from one strip model to another, of course. The degrees of freedom of any particular strip model in the C” type continuity problem are the values of u, V, bv, b,, and I&.for all values of i, located at (n + 1) reference lines equispaced across the strip model. In evaluating the properties of the finite strip models some markedly different schemes of integration in the crosswise direction are employed here, but before detailing the various strip models we refer briefly in the next paragraph to the concept of under integration as developed in finite element analysis. The use of deliberate under integration in evaluating the stiffness properties of isoparametric plate and shell finite elements, based on a shear deformation theory such as MFT, is now common practice. The earliest work on this topic was by Zienkiewicz, Taylor, and Too[Z] and by Pawsey and Clough[3] who detailed the dramatic improvement obtained in using under integration in connection with the isoparametric shell elements of Ahmad et a1.[4]. When using “full” Gaussian numerical integration these elements are efficient performers for thick plates and shells but are excessively overstiff for thin geometry: reduction in the order of numerical integration leads to very considerable improvement in efficiency for thin geometry, whether such reduction is “selective” or “reduced”[3, 41. Regarding the terminology used here, “full” integration (p x p points over the element middle surface) means an order of integration sufficient to calculate the stiffness matrix exactly for a rectangular or parallelogrammic-shaped element: “selective” integration implies the use of (p - 1) x (p - 1) integration points in evaluating the transverse shear stiffness contribution and p x p points in evaluating all other contributions; and “reduced” integration implies the use of (p - 1) x (p - 1) points in evaluating all stiffness contributions. Following the initial work mentioned above there has been considerable investigation of the use of the under in-
D. J.
4
D.AWE and Z.
tegration philosophy in MPT-based isoparametric finite elements of the serendipity. Lagrangian. and combined serendipity./Lagrangian types: amongst such investigations are those described in Refs. [S91. Certain difficulties can arise for particular element types. Thus, in serendipity elements “locking” behaviour occurs. manifested by the generation of overstiff numerical results for very thin plates (due to the predominance of the transverse shear contribution for such plate geometry), whilst in Lagrangian elements there can be difficulties associated with the production of spurious zero-energy modes. The “heterosis” element[7] of the combined SerendipityiLagrangian type appears to avoid both kinds of difficulty. The finite element studies referred to thus far are concerned with linear analysis but it is noted that Pica, Wood, and Hinton[ IO] have investigated geometrically nonlinear plate behaviour using 4-, 8-. and 9-noded elements of the serendipity, Lagrangian, and heterosis types, with reduced and selective numerical integration schemes. The numerical applications are restricted to plates of thin geometry and in this context, whilst the performance of the elements is generally good, no clearly superior element emerges. Returning to the subject topic of nonlinear MPT finite strip analysis the basic details of the strip models utilised in the present study are recorded in Table I: these details relate to the orders of crosswise polynomial interpolation used in the displacement fields and to the manner of integration of
Table I. Details of Strip
Order of crosswise interpolation n
Model
G. AZIZI.LV
terms in the crosswise direction. (In the longitudinal direction analytical integration is used in all cases.) Half the models listed in Table 1 are based upon the use of analytical (that is exact) integration of all stiffness contributions, and the order of polynomial interpolation ranges from linear (n = 1) to quartic (n = J). In fact the linear, analytical model LA, though listed in the Table for completeness, will not be considered later in the paper when numerical results are discussed since this model yields hopelessly overstiff results in most circumstances. The remaining three analytically integrated models, namely QA, C4, and FA, have been tested earlier in linear analyses. This was first done by Dawe[ 1l] in a study of the vibration of thin and moderately thick plates with simply supported ends where, additionally, quintic interpolation was also employed. Numerical applications showed clearly that increasing the order of crosswise interpolation leads to significant improvements in accuracy of solution for a given number of total degrees of freedom in the problem. They also showed that the accuracy of the higher-order models is not greatly dependent upon the relative thickness of the plate under consideration, whilst the accuracy of the quadratic QA model is, and shows a marked decline with decrease in thickness. Further details of the use of the FA model in linear eigenvalue problems (for plates with a variety of end conditions) are given by Dawe and Roufaeil[lZ, 131, whilst more recently Onate and Suarez have produced numerical results for the linear static analysis of simply supported plates when
the
MPT finite strip models No. of reference lines
Crosswise integration scheme
(Linear,
LA analytical)
Analytical.
(Linear.
LS selective)
Selective. tS I point for K 2 points for%ther
(Linear,
LR reduced)
Reduced. I point for
QA (Quadratic,
analytical)
(Quadratic,QS
selective)
(guadratic,
QRreduced) CA
(Cubic,
analytical)
(Quartic analytical)
FA [Fourth-order],
K
allK
Analytical.
2
3
Selective. 2 points 3 points
for Kts for 08~
for
2
3
Reduced. 2 points
3
4
Analytical.
4
5
Analytical
all
C
5
Performance
of Xlindlin plate finite strips
using the LA. QA. and CA models[l4]. Of course, the CA model (shown in Fig. I) is also the model used by the authors in their earlier study of geometrically nonlinear static behaviour[ 11 which, so far as is known, is the only application to nonlinear analysis of any of the strip models detailed in Table I. The four numerically integrated strip models listed in Table 1, namely LS, LR. QS, and QR, are based on the use of either linear or quadratic polynomial interpolation across the strip, each in conjunction with either selective or reduced Gaussian numerical integration. The quadratic strip model QR was introduced by Mawenya and Davies[ 151 in the solution of linear static problems and its use was extended to vibration and stability applications by Benson and Hinton[l6]. The linear strip model LS was first described by Hinton and Zienkiewicz[l7] in the context of static analysis. The more-recent work of Onate and Suarez[ 141, mentioned earlier with regard to the analytically integrated models, contains detailed comparative discussion of the use of strip models LS, LR, QS, and QR (and cubic models) in the linear static analysis of uniformly loaded plates with one pair of ends simply supported. On the theoretical level, the singularity of the stiffness contribution Kf of each strip model is examined, using the simple rule suggested in Ref. [51. This rule predicts that singularity of K: does not exist, and hence that the problem of locking behaviour for very thin plates will be present, for the linear strip with full Gaussian integration but not for the equivalent strip with selective or reduced integration (models LS and LR). It also predicts that locking behaviour will be present for some specific boundary conditions (clamped edges) when using the quadratic strip with full integration but not with reduced or selective integration (models QS and QR). Presented numerical results1141 confirm these predictions and show that the linear, quadratic, and cubic MPT strips perform well for thick and thin geometry when their properties are based on the use of selective or reduced integration but that only the cubic strip is completely reliable when full integration (four integration points for the cubic strip) is used. Onate and Suarez[l4] recommend the LR strip model as the “best value” MPT strip of those studied in their work for reasons of its excellent performance, for thick and thin geometry, and its inherent simplicity which corresponds to the use of only one integration point. They point out that the LR strip model has one possible spurious zero energy mode but that this mode is not propagated if an assembly of two or more strips is considered, which would be the case in practical analysis using the linear model. Of course, it must be borne in mind that all past studies referred to above (except that by the present authors in Ref. [l]) relate to linear, small-deflection analysis in which the only strain energy contribu-
5
tions present are those of bending (I/i) and transverse shearing (c/F) actions: consequently only Kf: and KY are then present on the right-hand side of eqn (10). In moving in the present study to consideration of nonlinear analysis, the extra strain energy contributions UT, Vi, and c’z are introduced and the proportional balance of energy terms is changed radically, so that conclusions that apply for linear analysis may no longer apply. It is with such matters, amongst others, that the present study is concerned, of course. 4. NLXERICAL APPLICATIONS The problems detailed here are basically the same as those considered in the earlier, related publication[ 11 where only the CA strip model was considered. Thus the plates are square, of side length A, and are subjected to a uniformly distributed lateral loading. All plate edges are immovable in the plane of a plate whilst for out-of-plane behaviour either all four edges are simply supported (the SSSS plate) or one pair of opposite edges (at .r = 0 and x = A) is simply supported and the other pair is clamped (the SCSC plate). It is noted that the boundary conditions at a simply supported edge are defined in eqn (14) for an edge x = constant (with QI and JIYinterchanged for an edge y = constant) whilst the definition of a clamped edge is that all of U, V, w, JIX,and ti, are zero at the edge. Two relative thicknesses are considered, corresponding to hlA = 0.01 (thin plate) and h/A = 0.05 (moderately thick plate), and the values of I? and v are taken to be 3 and 0.3, respectively. As in Ref. [l], the presented results are referred to values of a nondimensional loading factor Q = qA4/Dh, where q is the intensity of the uniform lateral loading. All of the problems have a symmetry in the longitudinal direction such that, with reference to eqns (13), only the terms corresponding to i = 2, 4, 6, 8 . . for Ui(x) and to i = 1, 3, 5, 7 . . . for Vi(x), &i(x), @xi(X), and @y{(X)make a contribution and hence need be employed. There also exists a symmetry in the crosswise direction whbh is incorporated in the usual way by modelhng only half the plate with finite strips. In what follows results of the application of the MPT finite strip models are presented for displacement and force quantities, both in tabular form for point values and in diagrammatic form showing distributions across or along a plate. For all the strip models the displacement quantities are calculated directly at the reference lines, of course, and at other intermediate points as desired, and no difftculty arises in their representation. With regard to the force quantities for the analytically integrated strip models QA, CA, and FA, these are calculated, as are the displacement quantities, at reference lines and at any convenient intermediate points. For the numerically integrated strip models LR, LS, QR, and QS calculation of force quantities is made
D. J. D~WE and Z. G. AZIZIAY
6
specifically at the Gauss points: where values are required at other points these are obtained by suitable extrapolation from the Gauss-point values. We begin the presentation of results by examining the performance of seven strip models in calculating the lateral deflection M’,.at a plate centre and the v-direction stress u,, occurring on the lower surface of a plate. again at its centre of plan: the u,~ stress combines contributions from the central bending moment lMycand extensional force NY,, of course. Numerical details of the convergence of the calculated values of the two specified quantities with increase in the value of a parameter N are given in Tables 2-5 for four plate problems. The parameter N is the number of degrees of freedom per contributing series term in a symmetric halfplate after applying the appropriate boundary conditions at the plate longitudinal edge and the longitudinal centre line: the total number of degrees of freedom in a probiem is thus N x no where n,, is the number of terms used in the longitudinal series of each reference quantity. For the results given in Tables 2-5 attention is concentrated on convergence with respect to the crosswise modelling and the value of no is fixed at four. The values of N quoted in the tables correspond to the use of 2, 3, 4, 6, 8, 10, and 12 strips for models LR and LS; of I, 2, 3, 4, 5, and 6 strips for models QR, QS, and QA; of 1, 2, 3, and 4 strips for model CA; and of 1, 2, and 3 strips for model FA. The four plate problems to which Tables 2-5
refer are for the thin and moderately thick geometries of both the SSSS and SCSC plates at the higher load levels considered in the earlier investigation[l], i.e. Q = 6400 for h/A = 0.01 and Q = 100 for h/A = 0.05. For these problems there are comparative approximate solutions available[ l] which are based on a Rayleigh-Ritz analysis, though within the confines of classical plate theory: these “R-R CPT” results are recorded in Tables 2-5. It can be seen in Tables 2-5 that convergence with increase in N of values of central deflection calculated using the MPT finite strip approach is orderly and rapid for all types of strip model. The central deflection values reached by each model at the largest values of N used (58 or 59) are very close to one another in any of the four documented problems, the greatest differences being less than 1% even for the difficult problem of the thin SCSC plate. (Incidentally it is noted that no bound conditions apply to the tabulated values of central deflection even for the analytically integrated strip models, since the loading is a distributed one.) Normally results of good accuracy are obtained for central deflection using far lower values of N than the greatest values detailed here. The point-stress values given in Tables 2-5 for the finite strip method also reveal good rates of convergence to common levels of stress for any particular problem. In comparing the values to which the MPT finite strip method results are converging with the R-R
Table 2. Values of central deflection and stress for thin SSSS square plate: no = 4, Q = 6400
I
ISI
I
Strip Model LRI
LS
1
QR
1
QS
1
QA
1
CA
I
‘?A
1
Performance
of Mindlin
plate finite strips
Table 3. Values of central deflection and stress for moderately Q = 100
thick SSSS square plate: no = 4.
Table 4. Values of central deflection and stress for thin SCSC square plate: no = 4, Q = 6400
7
D. .I.
8 Table 5. Vaiucj of
central
dztlsction
D.AW and Z. G. .~ZIZIA~
and S~KSS
for moderately
thick SCSC
square
plate:
no = 1.
Q = 100
1
13
I .Sl72
1.8168
-
CPT results it is apparent that there is little difference between the predictions of the two procedures for the thin plates. This is as expected since transverse shear effects have very little influence for thin, isotropic plates under uniform static loading. For the moderately thick plates there is some noticeable transverse shear effect on the central deflection value, demonstrated by the increased flexibility of the shear-deformable plate model, although the effect is still not terribly large (something over 3%) even for the SCSC plate. Comparison of the values of deflection and stress exhibited in Tables 2-5 for the various MPT finite strip models suggests the following points. (a) For both the linearly and quadratically interpolated models, when numerical integration is employed there is very little difference in the results whether reduced or selective integration is used, i.e. the LRand LS models perform very similarly one to another, as do the QR and QS models. (b) On the basis of a common value of N the performance of the LR/LS models is only slightly less good than that of the QWQS models. (c) The QWQS models are somewhat superior to the analytically integrated QA model. (Also, the LWLS models are very markedly superior to the analytically integrated LA model, though
?.ji?6
-
03
details of the performance of the latter model are not given here). (d) The higher-order, analytically integrated models CA and FA are accurate for stress and very accurate for deflection when two strips are employed in the half-plate, and sometimes give adequate performance when employing only a single strip. It should be noted that use of the value of the parameter N in Tables 2-5 as a basis for assessing the relative efficiency of the various strip models is convenient but it by no means tells the whole story as regards the computational effort involved in obtaining a solution of given accuracy when using different types and numbers of strip models. It is not possible to be very specific about relative corn: putational effort here, since this is both machine and program dependent, and the numerical results presented have been obtained using more than one computer. However, obviously one significant time-consuming stage is that of solution for displacements and here it is known that the size of the half-bandwidth of the structure stiffness matrix is of great importance, with solution time being dependent on the half-bandwidth to the power 2 at the very least. Since, for a given number of degrees of freedom the sizes of the half-bandwidths associated with assemblies of linear, quadratic, cubic, and
9
Performance of hlindlin plate tinitr strips quartic finite strips. respectively. are in the ratio 2/ 3i4iS. it is clear that there are very considerable economies associated with the use of low-order strips. Continuing with the examination of the efftciency of the crosswise modelling. Figs. 2-S illustrtate the distributions of M’,dV,. and NY across one half of each of the four plates discussed above. The distributions shown are along the x = A/2 centre line and results predicted by the use of various numbers of each of the seven strip models are presented. For comparison purposes the distributions predicted by the R-R CRT approach are also shown in the figures: these distributions are themselves not exact. of course (as evidenced by the waviness of
(0)
LR
the N, and M, distributions for the SCSC thin plate in Fig. I), but they do provide a very useful measure of comparison. The results shown in Figs. 2-j serve to verify the proper convergence of each strip model and provide further evidence of the statements made above concerning the relative performance of the various models. It is interesting to note the high accuracy of the force type distributions obtained for the numerically integrated strip models when Gauss point values alone are used, and to observe the manner in which the calculated distributions of moment and membrane force vary across the full width of strips for the analytically integrated models QA. CA. and F.4. Thus far, all the results presented correspond to
(d)
(b) LS
-
R - R CPT solution A
(e)
0.4
(f)
CA
(9)
OS
1 strip
1
FA
Fig. 2. Thin SSSS square plate with Q = 6400: distributions of w, I&. and NYat x = A/2 for no = 4 (R-R CPT central values are w,/A = 0.023890, 6My,AzlEh4 = 11.535, and N.&‘/Eh’ = 18.315).
D. J. D.AWEand Z. G. Azrzr-is
10
(a)
LR
lb)
Cd) OS
LS
-
(e)
QA
(f)
CA
R-R
CPTsaktlian
(P) FA
Fig. 3. Moderately thick SSSS square plate with Q = 100: dist~b~tioRs of W, .\fY, and N,. at x = A/ 2 for no = 4 (R-R CPT central values are w,iA = 0.017335, 6M,,A’iEh’ = t.2043, and Ny,A21Eh’ = 0.36650).
the use of a particufar number (no), namely 4, of terms in the longitudinal series expressions for the reference quantities. The manner of convergence of displacement and stress quantities with the number of longitudinal series terms used is also of interest, and Table 6 and Fig. 6 are included to give an idea of such convergence in the present circumstance where the applied loading is a uniformly distributed one. Table 6 gives details of point values and demonstrates that the rate of convergence with no does not vary greatly with the type or the number of strips used: results are presented, somewhat arbitrarily, for two types of problem and for the use of particular numbers of strips having the four differert levels of crosswise interpolation considered
in this paper. The point values considered are the same as those considered earlier in Tables 2-5 and the values obtained with no = 1 do not differ greatly from the corresponding values with no = 4. Figure 6 shows the manner in which the value of no affects the predicted dist~butions of W, N,, and M, along the longitudinal centre line for the same two types of plate problem detailed in Table 6. The strip method distributions shown in Fig. 6 were actually drawn using the results obtained for an assembly of eight LR strips, but it is noted that there is no discernible difference in a graphical presentation from distributions obtained for alternative strip assemblies having around the same N value, such as from an assembly of two FA strips for instance.
Performance
of Mndlin plate finite strips
II
lb) LS
i’---.,J (1)
CA
(9)
FA
Fig. 4. Thin SCSC square plate with Q = 6400: distributions of W, My, and NY at x = A/2 for no = 4 (R-R CPT central values are w,/A = 0.022728. 6My,A2/EhJ = 13.350, and N,,A’IEh = 16.835).
Table 6. Convergence
of central deflection and stress with no Strip
Plate Problem
“a
I z z 1 .” 5
,-E
xf;
II
LR
Quantity (4 WC/A
(2
strips)
yodel CA (2
strips)
PA (I
strip)
3.5560
3.5362
3.5340
3.5321
2
3.5110
3.4927
3.4898
3.4885
3
3.5164
3.4981
3.4952
3.4939
4
3.5155
3.4972
3.4943
3.4930
1
200
scrips)
QR
2.6058
2.6459
2.5996
2.5302
-G Ec u
2
2.5750
2.6283
2.5724
2.4996
2: ;z
3
2.5834
2.6346
2.5806
2.5096
r;;
4
2.6050
2.6324
2.5774
2.5064
=YC
u a-
l
2.4308
2.3865
2.3586
2.3098
m s
2
2.3123
2.2722
2.2430
2.2007
3
2.3274
2.2882
2.2587
2.2149
4
2.3242
2.2846
2.2554
2.2121
: m gz c
c II
I
100
A2/Eh2
WC/A
32.208
31.347
31.340
14.357
;c
2
cycA2/Eh’
31 .I70
30.938
30.475
15.553
.z L: c
3
31.475
31.357
30.938
15.830
4
31.295
31 .089
30.703
15.548
D. 1.
12
(01
LR
Ib)
DAWE
and
Z. G.
AZIZI.AN
(d)
LS
OS
R-R
CPT solution
1 strip 2 strips 1
6slrips Bstrips J
fr)
QA
(f)
CA
(q1 fA
Fig. 5. Moderately thick SCSC square plate with Q = NO: distributions 6M,,A’iEh’ 2 for no = 4 (R-R CPT central values are w,/A = 0.0093355,
of W, h&, and iy, at .r = A/ = 1.7378, and Ny,A21Eh3
= 0.10840).
Finally, we show the manner of variation with increasing load level of central deflection and stress for the moderately thick SSSS plate in Fig. 7 and for the thin SCSC plate in Fig. 8. These figures relate to the use of the two strip models at the opposite ends of the spectrum so far as sophistication is concerned, namely the linear LR model and the quartic FA model. Again, the R-R CPT predictions are shown for comparison purposes and it is seen that the finite strip results converge rapidly toward these R-R CPT predictions. (Fully converged MPT finite strip vaIues will differ somewhat from the RR CPT values but not by amounts which are noticeable in these particular graphical presentations.) 5. CONCtt’SrO3-S
All the MPT finite-strip models used in this detailed study of nonlinear plate behaviour demon-
strate convergence to proper levels of displacement and stress for piates of thin and moderately thick geometry, as evidenced by tabulated point values and by graphical distributions. The high-order, analytically integrated cubic and quartic models CA and FA are very accurate in predicting values of deflection, with the use of very few strips, and on a degree-of-freedom basis are probably the most accurate and consistent performers in this respect. These models also represent the dist~b~ions of force type quantities we11 in general, though they do not have an advantage in this respect over the other models considered. In the present nonlinear analysis an important practical disadvantage of the CA and FA models is their relative expense due to the large ham-bandwidth of the structure stiffness matrices. Of the three quadratic models the analytically integrated model QA is the least impressive per-
Performance
of Mindiin plate finite strips
13 W
!2 h
0 _-:-0123L56
-
CPT
R-R “,
.
“, = 2
I
l-lo L 3
0
“* = L
=
solution
1
d
MPT Frmtc stnp 50iut10ns
-R-R
I
Fig. 6. Longitudinal distributions of W. &I,,, and N,: (a) Moderately thick SSSS square plate with Q = 100; (b) Thin SCSC square plate with Q = 6400. [R-R CPT central values are H+,./.+= 0.017335, &M,,A’i Eh’ = 2.2043, and N,,A’/E113 = 0.36650 for case (a), and w,iA = 0.022728, 61~~~‘~~/~~~= 11.063. and N,,.A’iElr3 = 17.383 for case (b).]
CPT solution
d
1 strip
.
2 strips
x
3 strips
0
4 stnps
1 MPT F,n,te strtp solutions
Fig. 7. Moderately thick SSSS square plate: variations of central deflection and stress with load: (a) LR model; (bf FA model.
20
h I-O
3
~ ‘0
0
Y
10“
1
2
3
0
-
R-
R
CPT
A
1 strtp
.
2 strips
II
3 strtps
0
4 strips
(I
6 strips
l
6 strips
x
4
5
6
IO“
solution
MPT strip
Fmste s0lut10ns
I
Fig. 8, Thin SCSC square plate: variations of central deflection and stress with load: (a) LR model: (b) FA model.
I4
D. J. D.AWEand Z. G. AZIZI.\~
former in numerical terms. particularly so for the plates having clamped edges. The performance of the numerically integrated modeis QR and QS is very good and is similar. Of these two models it would therefore appear to make sense to use the QR model with its simpler and cheaper scheme of integration. The efficiency of the numerically integrated linear models LR and LS is remarkably good in the considered applications. (The analytically integrated linear model has not been discussed in detail but its performance is so poor that it is simply not a viable model.) A particularly impressive feature of the LR and LS models, considering their simplicity, is their accurate representation of stress type distributions when using few strips and when Gauss point values are plotted. The LR and LS models perform very similarly to each other and on a degree-of-freedom basis give results almost as good as do the QR and QS models: however, the quadratic models are significantly more expensive than are the linear models. Finally, of the numericaily integrated models considered in this study the linear LR model must be recommended in view of its simplicity, with just one integrating point used for the evaluation of all strain energy cont~butions, and of its economy that results both from this iow-level integration and from the small half-bandwidth of the structure stiffness matrices. This judgement appears justified on the evidence available but perhaps should be qualified with a slight note of caution untii a wider range of numerical tests is made. The alternative, analytically integrated, cubic and quartic models have much to recommend them as accurate and consistent performers which are based rigorously on the principle of minimum potential energy. However, in the nonlinear realm they are much more demanding of computing power than is the numerically integrated linear model, at least when using the solution procedures adopted in the present study. rlcknowledgmenl-The authors are grateful to the Science and Engineering Research Council for providing a grant to support the work described in this paper. They are also pleased to acknowledge the use of facilities at the Computer Centre of 3i~ingham University and at the L’niversity of Manchester Regional Computer Centre in obtammg the numerical results reported here.
REFERESCES 1. 2. G. Azizian and D. J. Dawe. Geometrically
nonlinear analysis of rectangular Mindlin plates using the finite strip method. Co~rrprrr. Srnrcr. 21, 113-436 (19%). 0. C. Ztenkiewicz. R. Taylor. andJ. hf. Too. Reduced integration techniques in general analysis ofplates and shells. In?. J. ~VMXT.Merh. Engng 3.275-290 (1971). S. E. Pawsey and R. W’. Ciough. improved numerical integration of thick shell tinite elements. IN. J. Numer. Mrlh. Engng 3. 575-586 (1971). S. Ahmad. 8. Sf. Irons, and 0. C. Zienkiewicz, Analysis of thick and thin shel! structures by curved finite elements. lnr, J. .V~xr. .?ferh. Engng 2, 419-431 (1970). 5. 0. C. Zienkievvicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Frnnklin fnsr. 302, 443-461 (1976). I‘. J. R. Hughes, R. L. Taylor. and W. ~anoknuku~-
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10.
Il. 12.
13.
14.
chai, A simple and efficient element for plate bending, Inr. J. Nztmer. Merh. Engng 11, 1529-1543 (1977). T. J. R. Hughes and Xl. Cohen. The heterosis finite element for plate bending. Comprtr. Srnrct. 9, 445450 (1978). E. D. L. Pugh. E. Hinton. and 0. C. Zienkiewicz. A study of quadrilateral plate-bending elements with “reduced” integration. Inr. J. Numer. Mefh. Engng 12, 1059-1079 (1978). E. Hinton and S. Bicanic, A comparison of Lagrangian and serendipity Mindlin piate elements for free vibration analysis. Compact. Srrrccr. 10, 483-493 (1979). i\. Pica, R. D. Wood, and E. Hinton, Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Comprrr. Strucr. 11, ?03215 T1980). D. J. Dawe, Finite strip models for vibration of Mtlindho elates. J. Sofrnd Vihr. $9. 441-452 (19%). 0. i. Roufaeil and D. J. Da&e. Vibration analysis of rectangular Mindlin plates by the finite strip method. Corn&r. Srruct. 12, -833-842 (1980). D. J. Dawe and 0. L. Roufaeil. Buckling of rectangular Mindlin plates. fompuf. ‘Slrurr. fs, 461-171 (1982). E. Onate and B. Suarez. A comparison of the linear, quadratic, and cubic .Ciindlin strip elements for the analysis of thick and thin plates. Comp~r. Srrttcr. 17, 427-439
(1983).
IS. A. S. Mawenya and J. D. Davies. Finite strip analysis of plate bending including transverse shear effects. Build.
Sci. 9. 175-180
(1974).
16 P. R. Benson and E. Hinton. A thick finite strip solution for static, free vibration and stability problems. fnr. J. ;Xurner. Xerh. Engng IO, 665-678 (1976). 17. E. Hinton and 0. C. Zienkiewicz, A note on a simple thick finite strip. Inr. J. il’~mer. :Merlr. Engng 11,905906 (1977).