Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements

Transcription

1 Plate bending approximation: thin (Kirhhoff) plates and C1 ontinuity requirements 4.1 Introdution The subjet of bending of plates and indeed its extension to shells was one of the first to whih the finite element method was applied in the early 1960s. At that time the various diffiulties that were to be enountered were not fully appreiated and for this reason the topi remains one in whih researh is ative to the present day. Although the subjet is of diret interest only to applied mehaniians and strutural engineers there is muh that has more general appliability, and many of the proedures whih we shall introdue an be diretly translated to other fields of appliation. Plates and shells are but a partiular form of a threedimensional solid, the treatment of whih presents no theoretial diffiulties, at least in the ase of elastiity. However, the thikness of suh strutures (denoted throughout this and later hapters as t) is very small when ompared with other dimensions, and omplete threedimensional numerial treatment is not only ostly but in addition often leads to serious numerial illonditioning problems. To ease the solution, even long before numerial approahes beame possible, several lassial assumptions regarding the behaviour of suh strutures were introdued. Clearly, suh assumptions result in a series of approximations. Thus numerial treatment will, in general, onern itself with the approximation to an already approximate theory (or mathematial model), the validity of whih is restrited. On oasion we shall point out the shortomings of the original assumptions, and indeed modify these as neessary or onvenient. This an be done simply beause now we are granted more freedom than that whih existed in the preomputer era. The thin plate theory is based on the assumptions formalized by Kirhhoff in 1850, and indeed his name is often assoiated with this theory, though an early version was presented by Sophie Germain in A relaxation of the assumptions was made by Reissner in 1945 and in a slightly different manner by Mindlin6 in These modified theories extend the field of appliation of the theory to ririk p1rrte.v and we shall assoiate this name with the ReissnerMindlin postulates. It turns out that the thik plate theory is simpler to implement in the finite element method. though in the early days of analytial treatment it presented more diffiulties. As it is more onvenient to introdue first the thik plate theory and by imposition of

2 I 12 Plate bending approximation additional assumptions to limit it to thin plate theory we shall follow this path in the present hapter. However, when disussing numerial solutions we shall reverse the proess and follow the historial proedure of dealing with the thin plate situations first in this hapter. The extension to thik plates and to what turns out always to be a mixed formulation will be the subjet of Chapter 5. In the thin plate theory it is possible to represent the state of deformation by one quantity IY, the lateral displaement of the middle plane of the plate. Clearly, suh a formulation is irreduible. The ahievement of this irreduible form introdues seond derivatives of w in the strain definition and ontinuity onditions between elements have now to be imposed not only on this quantity but also on its derivatives (C, ontinuity). This is to ensure that the plate remains ontinuous and does not kink.* Thus at nodes on element interfaes it will always be neessary to use both the value of M, and its slopes (first derivatives of w) to impose ontinuity. Determination of suitable shape funtions is now muh more omplex than those needed for C, ontinuity. Indeed, as omplete slope ontinuity is required on the interfaes between various elements, the mathematial and omputational diffiulties often rise disproportionately fast. It is, however, relatively simple to obtain shape funtions whih, while preserving ontinuity of w, may violate its slope ontinuity between elements, though normally not at the node where suh ontinuity is imposed.+ If suh hosen funtions satisfy the path test (see Chapter 10, Volume 1) then onvergene will still be found. The first part of this hapter will be onerned with suh nononforming or inompatible shape funtions. In later parts new funtions will be introdued by whih ontinuity an be restored. The solution with suh onforming shape funtions will now give bounds to the energy of the orret solution, but, on many oasions, will yield inferior auray to that ahieved with nononforming elements. Thus, for pratial usage the methods of the first part of the hapter are often reommended. The shape funtions for retangular elements are the simplest to form for thin plates and will be introdued first. Shape funtions for triangular and quadrilateral elements are more omplex and will be introdued later for solutions of plates of arbitrary shape or, for that matter, for dealing with shell problems where suh elements are essential. The problem of thin plates is assoiated with fourthorder diflerentiul equations leading to a potential energy funtion whih ontains seond derivatives of the unknown funtion. It is harateristi of a large lass of physial problems and, although the hapter onentrates on the strutural problem, the reader will find that the proedures developed also will be equally appliable to any problem whih is of fourth order. The diffiulty of imposing C, ontinuity on the shape funtions has resulted in many alternative approahes to the problems in whih this diffiulty is sidestepped. Several possibilities exist. Two of the most important are:, 1. independent interpolation of rotations 8 and displaement ~ 3 imposing ontinuity as a speial onstraint, often applied at disrete points only; * If kinking ours the seond derivative or urvature beomes infinite and squares of infinite terms our in the energy exprssion. + Later w show that even slope disontinuity at the node may b used.

3 The plate problem: thik and thin formulations the introdution of lagrangian variables or indeed other variables to avoid the neessity of C, ontinuity. Both approahes fall into the lass of mixed formulations and we shall disuss these briefly at the end of the hapter. However, a fuller statement of mixed approahes will be made in the next hapter where both thik and thin approximations will be dealt with simultaneously. 4.2 The plate problem: thik and thin formulations Governing equations The mehanis of plate ation is perhaps best illustrated in one dimension, as shown in Fig Here we onsider the problem of ylindrial bending of plates.* In this problem the plate is assumed to have infinite extent in one diretion (here assumed the y diretion) and to be loaded and supported by onditions independent of y. In this ase we may analyse a strip of unit width subjeted to some stress resultants M,, P,, and S,, whih denote xdiretion bending moment, axial fore and transverse Fig. 4.1 Displaements and stress resultants for a typial beam.

4 1 14 Plate bending approximation shear fore, respetively. For rosssetions that are originally normal to the middle plane of the plate we an use the approximation that at some distane from points of support or onentrated loads plane setions will remain plane during the deformation proess. The postulate that setions normal to the middle plane remain plane during deformation is thus the first and most important assumption of the theory of plates (and indeed shells). To this is added the seond assumption. This simply observes that the diret stresses in the normal diretion, z, are small, that is, of the order of applied lateral load intensities, q, and hene diret strains in that diretion an be negleted. This inonsisteny in approximation is ompensated for by assuming plane stress onditions in eah lamina. With these two assumptions it is easy to see that the total state of deformation an be desribed by displaements uo and wo of the middle surfae (z = 0) and a rotation e\ of the normal. Thus the loal displaements in the diretions of the x and z axes are taken as U(X,Z) = uo(x) ze.,(x) and w(x,z) = wo(x) (4.1) Immediately the strains in the x and z diretions are available as du duo 80, E, z = Z ax ax ax E, = 0 du dw dwo i, = + = e, + dz ax dx For the ylindrial bending problem a state of linear elasti, plane stress for eah lamina yields the stressstrain relations The stress resultants are obtained as and where B is the inplane plate stiffness and D the bending stiffness omputed from with u Poisson s ratio, E and G diret and shear elasti moduli, respetively.* A onstant IC has been added here to aount for the Fdt that the shar stresss are not onstant aross the setion. A value of K = 5/6 is xat for a rtangular, homogeneous setion and orresponds to a paraboli shear stress distribution.

5 The plate problem: thik and thin formulations 1 15 Three equations of equilibrium omplete the basi formulation. These equilibrium equations may be omputed diretly from a differential element of the plate or by integration of the loal equilibrium equations. Using the latter approah and assuming zero body and inertial fores we have for the axial resultant ap, =o ax where the shear stress on the top and bottom of the plate are assumed to be zero. Similarly, the shear resultant follows from as, +q,=o ax where the transverse loading q, arises from the resultant of the normal tration on the top and/or bottom surfaes. Finally, the moment equilibrium is dedued from a 112 zo, dz + r,, dz = 0 (4.7) In the elasti ase of a plate it is easy to see that the inplane displaements and fores, uo and P,, deouple from the other terms and the problem of lateral deformations an be dealt with separately. We shall thus only onsider bending in the present hapter, returning to the ombined problem, harateristi of shell behaviour, in later hapters. Equations (4.1)(4.7) are typial for thik plates, and the thin plate theory adds an additional assumption. This simply neglets the shear deformation and puts G = x. Equation (4.3) thus beomes This thin plate assumption is equivalent to stating that the normals to the middle plane remain normal to it during deformation and is the same as the wellknown BernoulliEuler assumption for thin beams. The thin, onstrained theory is very

6 116 Plate bending approximation Fig. 4.2 Support (end) onditions for a plate or a beam. Note: the onventionally illustrated simple support leads to infinite displaement reality is different. widely used in pratie and proves adequate for a large number of strutural problems, though, of ourse, should not be taken literally as the true behaviour near supports or where loal load ation is important and is three dimensional. In Fig. 4.2 we illustrate some of the boundary onditions imposed on plates (and beams) and immediately note that the diagrammati representations of simple support as a knife edge would lead to infinite displaements and stresses. Of ourse, if a rigid braket is added in the manner shown this will alter the behaviour to that whih we shall generally assume. The onedimensional problem of plates and the introdution of thik and thin assumptions translate diretly to the general theory of plates. In Fig. 4.3 we illustrate the extensions neessary and write, in plae of Eq. (4.1) (assuming uo and vo to be zero) u = dy(x,y),u= zo,.(x,y) M. = wo(x,y) (4.9) where we note that displaement parameters are now funtions of.y and y.

7 The plate problem: thik and thin formulations 11 7.( d 0 dx &.Y e?}i 0 $ rq d d Lay ax, {;;}=m (4.10)

8 118 Plate bending approximation We note that now in addition to normal bending moments M, and My, now defined by expression (4.3) for the x and y diretions, respetively, a twisting moment arises defined by (4.12) Introduing appropriate onstitutive relations, all moment omponents an be related to displaement derivatives. For isotropi elastiity we an thus write, in plae of Eq. (4.3), where, assuming plane stress behaviour in eah layer, D=D[i 1; 0 ] 0 0 (1 v)/2 (4.13) (4.14) where u is Poisson s ratio and D is defined by the seond of Eqs (4.4). Further, the shear fore resultants are (4.15) For isotropi elastiity (though here we deliberately have not related G to E and v to allow for possibly different shear rigidities) a = ngti (4.16) where I is a 2 x 2 identity matrix. Of ourse, the onstitutive relations an be simply generalized to anisotropi or inhomogeneous behaviour suh as an be manifested if several layers of materials are assembled to form a omposite. The only apparent differene is the struture of the D and a matries, whih an always be found by simple integration. The governing equations of thik and thin plate behaviour are ompleted by writing the equilibrium relations. Again omitting the inplane behaviour we have, in plae of Eq. (4.6), and, in plae of Eq. (4.7), (4.17) (4.18)

9 The plate problem: thik and thin formulations 1 19 Equations (4.13)(4.18) are the basis from whih the solution of both thik and thin plates an start. For thik plates any (or all) of the independent variables an be approximated independently, leading to a mixed formulation whih we shall disuss in Chapter 5 and also briefly in Se of this hapter. For thin plates in whih the shear deformations are suppressed Eq. (4.15) is rewritten as and the straindisplaement relations (4.10) beome E = ZLVW = vwe=o (4.19) Z< I \ a2 U' a2 Mi ay2 a2 U' 2 ax ay ZK (4.20) where K is the matrix of hanges in urvature of the plate. Using the above form for the thin plate, both irreduible and mixed forms an now be written. In partiular, it is an easy matter to eliminate M, S and 6 and leave only w as the variable. Applying the operator VT to expression (4.17), inserting Eqs (4.13) and (4.17) and finally replaing 6 by the use of Eq. (4.19) gives a salar equation where, using Eq. (4.20), (LV)*DLV~ = o (4.21) a2 a2 I' [ax2 ay2 axay (LV)=,, 2 In the ase of isotropy with onstant bending stiffness D this beomes the wellknown biharmoni equation of plate flexure a2 (4.22) The boundary onditions The boundary onditions whih have to be imposed on the problem (see Figs 4.2 and 4.4) inlude the following lassial onditions. 1. Fixed boundary, where displaements on restrained parts of the boundary are given speified values.* These onditions are expressed as 14' ~ 8, = 8, ~ and 8,v = Note that in thin plates the speifiation of IC along s automatially speifis 0, by Eq. (4.19). but this is not the ase in thik plates where the quantities are independently presribed.

10 120 Plate bending approximation Fig. 4.4 Boundary tration and onjugate displaement. Note: the simply supported ondition requiring M,, = 0, 0, = 0 and w = 0 is idential at a orner node to speifying 0, = 0, = 0, that is, a lamped support. This leads to a paradox if a urved boundary (a) is modelled as a polygon (b). Here n and s are diretions normal and tangential to the boundary urve of the middle surfae. A lamped edge is a speial ase with zero values assigned. 2. Tration boundary, where stress resultants M,,, M,,s and S,, (onjugate to the displaements e,,, Os and w) are given presribed values. A free edge is a speial ase with zero values assigned. 3. Mixed boundary onditions, where both tration and displaements an be speified. Typial here is the simply supported edge (see Fig. 4.2). For this, learly, M,, = 0 and u = 0, but it is less lear whether M,,, or 8,y needs to be given. Speifiation of M,,, = 0 is physially a more aeptable ondition and does not lead to diffiulties. This should always be adopted for thik plates. In thin plates 8, is automatially speified from w and we shall find ertain diffiulties, and indeed anomalies, assoiated with this ass~mption.~ For instane, in Fig. 4.4 we see how a speifiation of 8,s = 0 at orner nodes impliit in thin plates formally leads to the presription of all boundary parameters, whih is idential to boundary onditions of a lamped plate for this point The irreduible, thin plate approximation The thin plate approximation when ast in terms of a single variable w is learly irreduible and is in fat typial of a displaement formulation. The equations (4.17) and (4.18) an be written together as (LV)~M q = o (4.23) and the onstitutive relation (4.13) an be reast by using Eq. (4.19) as M = D LV H, (4.24)

11 The plate problem: thik and thin formulations 121 The derivation of the finite element equations an be obtained either from a weak form of Eq. (4.23) obtained by weighting with an arbitrary funtion (say = Ni) and integration by parts (done twie) or, more diretly, by appliation of the virtual work equivalene. Using the latter approah we may write the internal virtual work for the plate as (4.25) where R denotes the area of the plate referene (middle) surfae and D is the plate stiffness, whih for isotropy is given by Eq. (4.14). Similarly the external work is given by2 HI, = 1 Swq dr + SO,, M,, dr + 1 SO, I@,, dr + Sr, 6w S,, dr (4.26) r,, r, where M,,, M,,, S,, are speified values and r,,, ri and Ts are parts of the boundary where eah omponent is speified. For thin plates with straight edges Eq. (4.19) gives immediately = dw/& and thus the last two terms above may be ombined as 1 SO, M,,, SW (SI,%) dr + SW, R, (4.27) dr + SWS,, dr = sr. Ti r, 1 where Ri are onentrated fores arising at loations where orners exist (see Fig 4.2).2 Substituting into Eqs (4.25) and (4.26) the disretization w=na (4.28) where a are appropriate parameters, we an obtain for a linear ase standard displaement approximation equations with and Ka=f (4.29) (4.30) f=/ NTqdo+fh (4.31) 12 where fb is the boundary ontribution to be disussed later and with M = DBa (4.32) B = (LV)N (4.33) It is of interest, and indeed important to note, that when trations are presribed to nonzero values the fore term fh inludes all presribed values of M,,, M,,,y and S,, irrespetive of whether the thik or thin formulation is used. The reader an verify that this term is (4.34a)

12 122 Plate bending approximation where A?,,, A?,,,? and S,, are presribed values and for thin plates [though, of ourse, relation (4.34a) is valid for thik plates also]: dn dn N and N,T = dn ds (4.34b) The reader will reognize in the above the wellknown ingredients of a displaement formulation (see Chapter 2 of Volume 1, and Chapter 1 of this volume) and the proedures are almost automati one N is hosen Continuity requirement for shape funtions (C, ontinuity) In Setions we will be onerned with the above formulation [starting from Eqs (4.24) and (4.26)], and the presene of the seond derivatives indiates quite learly that we shall need C1 ontinuity of the shape funtions for the irreduible, thin plate, formulation. This ontinuity is diffiult to ahieve and reasons for this are given below. To ensure the ontinuity of both w and its normal slope aross an interfae we must have both w and dw/dn uniquely defined by values of nodal parameters along suh an interfae. Consider Fig. 4.5 depiting the side 12 of a retangular element. The normal diretion n is in fat that of y and we desire w and dw/dy to be uniquely determined by values of w, dwldx, dw/dy at the nodes lying along this line. Following the priniples expounded in Chapter 8 of Volume 1, we would write along side 12, and w = A, + A*x + A3y +... (4.35) (4.36) with a number of onstants in eah expression just suffiient to determine a unique solution for the nodal parameters assoiated with the line. Thus, for instane, if only two nodes are present a ubi variation of w should be permissible noting that dw/dx and w are speified at eah node. Similarly, only a linear, or twoterm, variation of dw/dy would be permissible. Fig. 4.5 Continuity requirement for normal slopes.

13 The plate problem: thik and thin formulations 123 Note, however, that a similar exerise ould be performed along the side plaed in the y diretion preserving ontinuity of dw/dx along this. Along side 12 we thus have dw/ay, depending on nodal parameters of line 12 only, and along side 13 we have dw/dx, depending on nodal parameters of line 13 only. Differentiating the first with respet to x, on line 12 we have d2wj/dx8y, depending on nodal parameters of line 12 only, and similarly, on line 13 we have, d2w/aydx, depending on nodal parameters of line 13 only. At the ommon point, 1, an inonsisteny arises immediately as we annot automatially have there the neessary identity for ontinuous funtions d2w a2w ~ axay ayax (4.37) for arbitrary values of the parameters at nodes 2 and 3. It is thus impossible to speify simple polynomial expressions for shape funtions ensuring full ompatibility when only w and its slopes are presribed at orner nodes.' Thus if any funtions satisfying the ompatibility are found with the three nodal variables, they must be suh that at orner nodes these funtions are not ontinuously differentiable and the rossderivative is not unique. Some suh funtions are disussed in the seond part of this hapter.'0'6 The above proof has been given for a retangular element. Clearly, the arguments an be extended for any two arbitrary diretions of interfae at the orner node 1. A way out of this diffiulty appears to be obvious. We ould speify the rossderivative as one of the nodal parameters. This, for an assembly of retangular elements, is onvenient and indeed permissible. Simple funtions of that type have been suggested by Bogner et ai.'7 and used with some suess. Unfortunately, the extension to nodes at whih a number of element interfaes meet with different angles (Fig. 4.6) is not, in general, permissible. Here, the ontinuity of rossderivatives in several sets of orthogonal diretions implies, in fat, a speifiation of all seond derivatives at a node. This, however, violates physial requirements if the plate stiffness varies abruptly from element to element, for then equality of moments normal to the interfaes annot be maintained. However, this proess has been used with some suess in homogeneous plate situations'8p25 although Smith and Dunan" omment adversely on the effet of imposing suh exessive ontinuities on several orders of higher derivatives. Fig. 4.6 Nodes where elements meet in arbitrary diretions.

14 124 Plate bending approximation The diffiulties of finding ompatible displaement funtions have led to many attempts at ignoring the omplete slope ontinuity while still ontinuing with the other neessary riteria. Proeeding perhaps from a naive but intuitive idea that the imposition of slope ontinuity at nodes only must, in the limit, lead to a omplete slope ontinuity, several suessful, nononforming, elements have been developed.,2640 The onvergene of suh elements is not obvious but an be proved either by appliation of the path test or by omparison with finite differene algorithms. We have disussed the importane of the path test extensively in Chapter 11 of Volume 1 and additional details are available in referenes In plate problems the importane of the path test in both design and testing of elements is paramount and this test should never be omitted. In the first part of this hapter, dealing with nononforming elements, we shall repeatedly make use of it. Indeed, we shall show how some of the most suessful elements urrently used have developed via this analytial interpretati~n.~~~~ Nononforming shape funtions 4.3 Retangular element with orner nodes (12 degrees of freedom) Shape funtions Consider a retangular element of a plate ijkf oiniding with the xy plane as shown in Fig At eah node, n, displaements a, are introdued. These have three omponents: the first a displaement in the z diretion, w,, the seond a rotation about the x axis, (d,y), and the third a rotation about the y axis (6,,),,.* The nodal displaement vetors are defined below as ai. The element displaement will, as usual, be given by a listing of the nodal displaements, now totalling twelve: (4.38) A polynomial expression is onveniently used to define the shape funtions in terms of the 12 parameters. Certain terms must be omitted from a omplete fourthorder * Note that we have hanged here the onvention from that of Fig. 4.3 in this hapter. This allows transformations needed for shells to be arried out in an easier manner. However, when manipulating the equations of Chapter 5 we shall return to the orginal definitions of Fig Similar diffiulties are disussed by Hughes, and a simple transformation is as follows: 6=Te where T= [ : :I

15 Retangular element with orner nodes (1 2 degrees of freedom) 125 Fig. 4.7 A retangular plate element. polynomial. Writing 2 2 u' = (Irl + a2.x + a3y + (Y4x + asxy f a(jy + Q7X3 + a8x2y + a9xy2 + aloy3 + ~ u~,x~y + (ri2,~y3 = Pa (4.39) has ertain advantages. In partiular, along any x onstant or y onstant line, the displaement w will vary as a ubi. The element boundaries or interfaes are omposed of suh lines. As a ubi is uniquely defined by four onstants, the two end values of slopes and the two displaements at the ends will therefore define the displaements along the boundaries uniquely. As suh end values are ommon to adjaent elements ontinuity of w will be imposed along any interfae. It will be observed that the gradient of w normal to any of the boundaries also varies along it in a ubi way. (Consider, for instane, values of the normal dw/dy along a line on whih x is onstant.) As on suh lines only two values of the normal slope are defined, the ubi is not speified uniquely and, in general, a disontinuity of normal slope will our. The funtion is thus 'nononforming'. The onstants al to ai2 an be evaluated by writing down the 12 simultaneous equations linking the values of w and its slopes at the nodes when the oordinates take their appropriate values. For instane, (E), (E), = e,, = a3 + asx, +. ' ' M', = a1 + a2x, + a3y1 + '. ' = e,, = a2 asy,... Listing all 12 equations, we an write, in matrix form, ae = Ca (4.40)

16 126 Plate bending approximation where C is a 12 x 12 matrix depending on nodal oordinates, and a is a vetor of the 12 unknown onstants. Inverting we have a = ' ae (4.41) This inversion an be arried out by the omputer or, if an expliit expression for the stiffness, et., is desired, it an be performed algebraially. This was in fat done by Zienkiewiz and Cheung.26 It is now possible to write the expression for the displaement within the element in a standard form as where u E w = Nae = PCplae p = (1, x, Y, x21 XYl Y2, x3, X2Y, XY2, Y3, x3y1 XY3) The form of the B is obtained diretly from Eqs (4.28) and (4.33). We thus have +2~4 +6~7x i +~QSY + { 6~ 1 1 XY LVW = +2r6 +2agx +6QlOY + 6Ql2XY 2 $ 2~5 +~Q*x +4agy + 6 ~ ~ ~12~ ~ We an write the above as in whih LVw = Qa = QC'ae = Bae and thus B = QC' (4.42) (4.43) O x 2 y 0 06xy x 6y 0 6x;] (4.44) x 4y 0 6x2 6y2 It is of interest to remark now that the displaement funtion hosen does in fat permit a state of onstant strain (urvature) to exist and therefore satisfies one of the riteria of onvergene stated in Volume I.* An expliit form of the shape funtion N was derived by Me10sh~~ and an be written simply in terms of normalized oordinates. Thus, we an write for any node NT = $(I + Eo)(l + 770) E2 r12 brli(1 v2) al;(l E2) with normalized oordinates defined as: x x,. <= where to = Ei U Y v= Y,. h where 70 = (4.45) * If a7 to yl2 are zero, then the 'strain' defined by seond derivatives is onstant. By Eq. (4.40), the orresponding ae an be found. As there is a unique orrespondene between ae and a suh a state is therefore unique. All this presumes that C' does in fat exist. The algebrai inversion shows that the matrix C is never singular.

17 Retangular element with orner nodes (12 degrees of freedom) 127 This form avoids the expliit inversion of C; however, for simpliity we pursue the diret use of polynomials to dedue the stiffness and load matries Stiff ness and load matries Standard proedures an now be followed, and it is almost superfluous to reount the details. The stiffness matrix relating the nodal fores (given by lateral fore and two moments at eah node) to the orresponding nodal displaement is Ke = J BTDBdxdy or, substituting Eq. (4.43) into this expression, Ke = C R E (4.46) (4.47) The terms not ontaining x and y have now been moved from the operation of integrating. The term within the integration sign an be multiplied out and integrated expliitly without diffiulty if D is onstant. The external fores at nodes arising from distributed loading an be assigned by inspetion, alloating speifi areas as ontributing to any node. However, it is more logial and aurate to use one again the standard expression (4.31) for suh an alloation. The ontribution of these fores to eah of the nodes is f, = { I ti,, =J } h u NTgdxdy (4.48) or, by Eq. (4.42), fq,> fj=ctj h a PTqdxdy (4.49) The integral is again evaluated simply. It will now be noted that, in general, all three omponents of external fore at any node will have nonzero values. This is a result that the simple alloation of external loads would have missed. The nodal load vetor for uniform loading q is given by The vetor of nodal plate fores due to initial strains and initial stresses an be found in a similar way. It is neessary to remark in this onnetion that initial strains, suh as may be due to a temperature rise, is seldom onfined in its effets on urvatures. Usually, diret (inplane) strains in the plate are introdued additionally, and the omplete problem an be solved only by onsideration of the plane stress problem as well as that of bending.

18 128 Plate bending approximation 4.4 Quadrilateral and parallelogram elements The retangular element developed in the preeding setion passes the path test4' and is always onvergent. However, it annot be easily generalized into a quadrilateral shape. Transformation of oordinates of the type desribed in Chapter 9 of Volume 1 an be performed but unfortunately now it will be found that the onstant urvature riterion is violated. As expeted, suh elements behave badly but by arguments given in Chapter 9 of Volume 1 onvergene may still our providing the path test is passed in the urvilinear oordinates. Henshell et a[.40 studied the performane of suh an element (and also some of a higher order) and onluded that reasonable auray is attainable. Their paper gives all the details of transformations required for an isoparametri mapping and the resulting need for numerial integration. Only for the ase of a parallelogram is it possible to ahieve states of onstant urvature exlusively using funtions of < and 11 and the path test is satisfied. For a parallelogram the loal oordinates an be related to the global ones by the expliit expression (Fig. 4.8) x y ot0 E= a y sa q=b (4.51) and all expressions for the stiffness and loads an therefore also be derived diretly. Suh an element is suggested in the disussion in referene 26, and the stiffness matries have been worked out by Dawe.** A somewhat different set of shape funtions was suggested by Argyri~.~~ Fig. 4.8 Parallelogram element and skew oordinates. 4.5 Triangular element with orner nodes (9 degrees of freedom) At first sight, it would seem that one again a simple polynomial expansion ould be used in a manner idential to that of the previous setion. As only nine independent

19 Triangular element with orner nodes (9 degrees of freedom) 129 p (4, L2, L3) area P23. et, L, = area 123 a! b!! I, L; L,b Lg da = 2A (a + b + + 2)! A = area Fig. 4.9 Area oordinates. movements are imposed, only nine terms of the expansion are permissible. Here an immediate diffiulty arises as the full ubi expansion ontains 10 terms [Eq. (4.39) with all = a12 = 01 and any omission has to be made arbitrarily. To retain a ertain symmetry of appearane all 10 terms ould be retained and two oeffiients made equal (for example "8 = yg) to limit the number of unknowns to nine. Several suh possibilities have been investigated but a further, muh more serious, problem arises. The matrix orresponding to C of Eq. (4.40) beomes singular for ertain orientations of the triangle sides. This happens, for instane, when two sides of the triangle are parallel to the x and y axes respetively. An 'obvious' alternative is to add a entral node to the formulation and eliminate this by stati ondensation. This would allow a omplete ubi to be used, but again it was found that an element derived on this basis does not onverge to orret answers. Diffiulties of asymmetry an be avoided by the use of area oordinates desribed in Se. 8.8 of Volume 1. These are indeed nearly always a natural hoie for triangles, see (Fig. 4.9) Shape funtions As before we shall use polynomial expansion terms, and it is worth remarking that these are given in area oordinates in an unusual form. For instane, gives the three terms of a omplete linear polynomial and a, L1 + "2 L2 + "3 L3 (4.52) "I L: + "2L: + "3L: + "4LIL2 + "5L2L3 + "fjl3li (4.53) gives all six terms of a quadrati (ontaining within it the linear terms).* The 10 terms of a ubi expression are similarly formed by the produts of all possible ubi * However, it is also possible to write a omplete quadrati as ql, +a2l2 +CU3L3+CL4L,L?+NSL2L)+N6L3LI and so on, for higher orders. This has the advantage of expliitly stating all retained terms of polynomials of lower order.

20 130 Plate bending approximation Fig Some basi funtions in area oordinate polynomials. ombinations, that is, L?, L:, L:, LtL2, Lh, L%3, GLI, L:Lh L:L2, L1L2L3 (4.54) For a 9 degreeoffreedom element any of the above terms an be used in a suitable ombination, remembering, however, that only nine independent funtions are needed and that onstant urvature states have to be obtained. Figure 4.10 shows some funtions that are of importane. The first [Fig. 4.10(a)] gives one of three funtions representing a simple, unstrained rotation of the plate. Obviously, these must be available to produe the rigid body modes. Further, funtions of the type LtLz, of whih there are six in the ubi expression, will be found to take up a form similar (though not idential) to Fig. 4.10(b). The ubi funtion L, L, L3 is shown in Fig 4. IO(), illustrating that this is a purely internal (bubble) mode with zero values and slopes at all three orner nodes (though slopes are not zero along edges). This funtion ould thus be useful for a nodeless or internal variable but will not, in isolation, be used as it annot be presribed in terms of orner variables. It an, however, be added to any other basi shape in any proportion, as indiated in Fig. 4.10(b).

21 Triangular element with orner nodes (9 degrees of freedom) 131 The funtions of the seond kind are of speial interest. They have zero values of MI at all orners and indeed always have zero slope in the diretion of one side. A linear ombination of two of these (for example L: L1 and L: L3) are apable of providing any desired slopes in the x and y diretions at one node while maintaining all other nodal slopes at zero. For an element envisaged with 9 degrees of freedom we must ensure that all six quadrati terms are present. In addition we selet three of the ubi terms. The quadrati terms ensure that a onstant urvature, neessary for path test satisfation, is possible. Thus, the polynomials we onsider are and we write the interpolation as w=pa (4.55) where a are parameters to be expressed in terms of nodal values. The nine nodal values are denoted as Upon noting that (4.56) where 2A = bi2 b2~i bi = YJ Yk, = xk XI with i, j, k a yli permutation of indies (see Chapter 9 of Volume l), we now determine the shape funtion by a suitable inversion [see Se , Eq. (4.42)], and write for node i 1 3LF 2L; (4.57) { NY = L: (b, LI, bk Ll) + i (6, bk) Ll L2 L3 L? Lk k Lj) f i (CJ k) L1 L2 L3 Here the term Ll L2L3 is added to permit onstant urvature states. The omputation of stiffness and load matries an again follow the standard patterns, and integration of expressions (4.30) and (4.31) an be done exatly using the general integrals given in Fig However, numerial quadrature is generally used and proves equally effiient (see Chapter 9 of Volume 1). The stiffness matrix requires omputation of seond derivatives of shape funtions and these may be

22 132 Plate bending approximation onveniently obtained from in whih Ni denotes any of the shape funtions given in Eq. (4.57). The element just derived is one first developed in referene 11. Although it satisfies the onstant strain riterion (being able to produe onstant urvature states) it unfortunately does not pass the path test for arbitrary mesh onfigurations. Indeed, this was pointed out in the original referene (whih also was the one in whih the path test was mentioned for the first time). However, the path test is fully satisfied with this element for meshes of triangles reated by three sets of equally spaed straight lines. In general, the performane of the element, despite this shortoming, made the element quite popular in pratial appliation^.^' It is possible to amend the element shape funtions so that the resulting element passes the path test in all onfigurations. An early approah was presented by Kikuhi and Ando" by replaing boundary integral terms in the virtual work statement of Eq. (4.26) by (4.59) in whih, re is the boundary of eah element e, M,(w) is the normal moment omputed from seond derivatives of the w interpolation, and s is the tangent diretion along the element boundaries. The interpolations given by Eq. (4.57) are Co onforming and have slopes whih math those of adjaent elements at nodes. To orret the slope inompatibility between nodes, a simple interpolation is introdued along eah element boundary segment as where s' is 0 at node j and 1 at node k, and mi and ni are diretion osines with respet to the x and y axes, respetively. The above modifiation requires boundary integrals in addition to the usual area integrals; however, the final result is one whih passes the path test. Bergen44'46.47 and Samuelss~n~~ also show a way of produing elements whih pass the path test, but a suessful modifiation useful for general appliation with elasti and inelasti material behaviour is one derived by Spe~ht.~~ This modifiation uses three fourthorder terms in plae of the three ubi terms of the equation preeding

23 Triangular element of the simplest form (6 degrees of freedom) 133 Eq. (4.55). The partiular form of these is so designed that the path test riterion whih we shall disuss in detail later in Se. 4.7 is identially satisfied. We onsider now the nine polynomial funtions given by where 2 lk f pi=i: and li is the length of the triangle side opposite node i.' The modified interpolation for w is taken as w=pa (4.62) (4.63) and, on identifiation of nodal values and inversion, the shape funtions an be written expliitly in terms of the omponents of the vetor P defined by Eq. (4.61) as 1 p1+3 + pk+3 + { 2 (p~+6 Pk+6) NF = bl (Pkf6 pk+ 3) bk PI +6 (4.64) l(pk+6 pk+3) kpr+6 where i, j, k are the yli permutations of 1, 2, 3. One again, stiffness and load matries an be determined either expliitly or using numerial quadrature. The element derived above passes all the path tests and performs e~ellently.~' Indeed, if the quadrature is arried out in a 'redued' manner using three quadrature points (see Volume 1, Table 9.2 of Se ) then the element is one of the best triangles with 9 degrees of freedom that is urrently available, as we shall show in the setion dealing with numerial omparisons. 4.6 Triangular element of the simplest form (6 degrees of freedom) If onformity at nodes (C, ontinuity) is to be abandoned, it is possible to introdue even simpler elements than those already desribed by reduing the element interonnetions. A very simple element of this type was first proposed by M~rley.~' In this element, illustrated in Fig , the interonnetions require ontinuity of the displaement M? at the triangle verties and of normal slopes at the element midsides. * The onstants k, are geometri parameters ourring in the expression for normal derivatives. Thus on side /, the normal derivative is given by

24 134 Plate bending approximation Fig The simplest nononforming triangle, from M~rley,~' with 6 degrees of freedom. With 6 degrees of freedom the expansion an be limited to quadrati terms alone, whih one an write as W= [LI, L2, L3, L1L2, L2L3, L3LiIa (4.65) Identifiation of nodal variables and inversion leads to the following shape funtions: for orner nodes bibk C;Ck bjbj C;C~ N; = L; L;( 1 L;) L,( 1 Lj) Lk( 1 L3) (4.66) bf + ; bi + i and for 'normal gradient' nodes 2A Ni+3 = J iml;(l LJ (4.67) where the symbols are idential to those used in Eq. (4.56) and i,j, k are a yli permutation of 1,2,3. Establishment of stiffness and load matries follows the standard pattern and we find that one again the element passes fully all the path tests required. This simple element performs reasonably, as we shall show later, though its auray is, of ourse, less than that of the preeding ones. It is of interest to remark that the moment field desribed by the element satisfies exatly interelement equilibrium onditions on the normal moment M,, as the reader an verify. Indeed, originally this element was derived as an equilibrating one using the omplementary energy priniple,52 and for this reason it always gives an upper bound on the strain energy of flexure. This is the simplest possible element as it simply represents the minimum requirements of a onstant moment field. An expliit form of stiffness routines for this element is given by Wood.3' 4.7 The path test an analytial requirement The path test in its different forms (disussed fully in Chapters 10 and 1 1 of Volume 1) is generally applied numerially to test the final form of an element. However, the basi requirements for its satisfation by shape funtions that violate ompatibility an be foreast aurately if ertain onditions are satisfied in the hoie of suh funtions. These onditions follow from the requirement that for onstant strain states the virtual work done by internal fores ating at the disontinuity must be zero. Thus if the

25 The path test an analytial requirement 135 trations ating on an element interfae of a plate are (see Fig. 4.4) M,,, M,,, and S,, (4.68) and if the orresponding mismath of virtual displaements are AQ, = A (E), AQ, = A (E) and Awl (4.69) then ideally we would like the integral given below to be zero, as indiated, at least for the onstant stress states: j MI, MI, dr + 1 Mn.7 AQ,s dr + 6, S,, Aw dr = 0 (4.70) r,. r', The last term will always be zero identially for onstant M,, M,,, M.rF fields as then S, = S,. = 0 [in the absene of applied ouples, see Eq. (4.1 S)] and we an ensure the satisfation of the remaining onditions if jr,, AQ,, dt = 0 and J, ao,s dr = o (4.71) is satisfied for eah straight side re of the element. For elements joining at verties where dw/dn is presribed, these integrals will be identially zero only if antisymmetri ubi terms arise in the departure from linearity and a quadrati variation of normal gradients is absent, as shown in Fig. 4.12(a). This is the motivation for the rather speial form of shape funtion basis hosen to desribe the inompatible triangle in Eq. (4.61), and here the first ondition of Eq. (4.71) is automatially satisfied. The satisfation of the seond ondition of Eq. (4.71) is always ensured if the funtion w and its first derivatives are presribed at the orner nodes. For the purely quadrati triangle of Se. 4.6 the situation is even simpler. Here the gradients an only be linear, and if their value is presribed at the element midside as shown in Fig. 4.1 l(b) the integral is identially zero. The same arguments apparently fail when the retangular element with the funtion basis given in Eq. (4.42) is examined. However, the reader an verify by diret Fig Continuity ondition for satisfation of path test [j(aw/an) ds = 01, variation of awjan along side (a) Definition by orner nodes (linear omponent ompatible), (b) definition by one entral node (onstant omponent ompatible)

26 136 Plate bending approximation Fig A square plate with lamped edges; uniform load 9; square elements. Table 4.1 Computed entral defletion of a square plate for several meshes (retangular elements) Mesh Total Simply supported plate Clamped plate number of nodes ff* Pi a* Pi 2x I x x x Series (Timoshenko) * wmdx = 0yL4/D for uniformly distributed load 4. I wmdl = i3pl2/d for entral onentrated load P. Note: Subdivision of whole plate given for mesh. Table 4.2 Corner supported square plate Method Mesh Point I Point 2 II M, It M, Finite element 2x x x Marus IO Ballesteros and Lee Multiplier ql4/d ql2 ql4/d ql2 Note: point I, entre of side; point 2, entre of plate.

27 The path test an analytial requirement 137

28 138 Plate bending approximation algebra that the integrals of Eqs (4.71) are identially satisfied. Thus, for instane, I. a ay awl dx = 0 when y = f b and aw)/ay is taken as zero at the two nodes (i.e. departure from presribed linear variations only is onsidered). The remarks of this setion are verified in numerial tests and lead to an intelligent, a priori, determination of onditions whih make shape funtions onvergent for inompatible elements. 4.8 Numerial examples The various plate bending elements already derived ~ and those to be derived in subsequent setions ~ have been used to solve some lassial plate bending problems. We first give two speifi illustrations and then follow these with a general onvergene study of elements disussed. Fig Castleton railway bridge: general geometry and details of finite element subdivision. (a) Typial atual setion; (b) idealization and meshing.

29 Numerial examples 139 Figure 4.13 shows the defletions and moments in a square plate lamped along its edges and solved by the use of the retangular element derived in Se. 4.3 and a uniform mesh.26 Table 4.1 gives numerial results for a set of similar examples solved with the same element,39 and Table 4.2 presents another square plate with more omplex boundary onditions. Exat results are available here and omparisons are made Figures 4.14 and 4.15 show pratial engineering appliations to more omplex shapes of slab bridges. In both examples the requirements of geometry neessitate the use of a triangular element with that of referene 11 being used here. Further, in both examples, beams reinfore the slab edges and these are simply inorporated in the analysis on the assumption of onentri behaviour. Finally in Fig. 4.16(a)(d) we show the results of a onvergene study of the square plate with simply supported and lamped edge onditions for various triangular and Fig (Continued) Castleton railway bridge general geometry and details of finite element subdivision () moment omponents (tonftftr ) under uniform load of 1501bft with omputer plot of ontours

30 140 Plate bending approximation

31 Numerial examples 141 u F m 0 aju TI aj e e aj u 0 U 5._ 3 aj m Q m 3 2 5: TI aj YL 0 Q Q 3 x VI._ e VI +2._ aj m Q 2 5: m 3 U W TI 8 x : 3 TI 0 Q Q e 2 E x Q CII m w i el. U

32 142 Plate bending approximation

33 Numerial examples 143 W % C E 8. 2 P 8 2 z ru 73 f C._ L a, ol rg C g a e P 8 2 g Y 73 a, Y a, U CL 0 U 5. 3 W m Q ru 3 0 a, Q 2! E 5 U?? h; e s t m Q (0 3 u W 73 m 0 x 0 t 3 73 W Q E, U U A Y e W 3. A ex a, ox u 0 m 7 ei5 L P 4:

34 144 Plate bending approximation Table 4.3 List of elements for omparison of performane in Fig. 4.16: (a) 9 degreeoffreedom triangles; (b) I2 degreeoffreedom retangles; () 16 degreeoffreedom retangle Code Referene Symbol Desription and omment (a) BCIZ I Bazeley et a/. 0 Displaement, nononforming (fails path test) PAT Spe~ht~~ a Displaement, nononforming BCIZ 2 Bazeley et a/. 0 Displaement, onforming (HCT) Clough and Toher DKT Striklin et a/.59 and Dhattm o Disrete Kirhhoff (b) ACM Zienkiewiz and Cheung26 a Displaement, nononforming Q19 Clough and Felippa 0 Displaement, onforming DKQ Batoz and Ben Tohar6 Displaement, onforming () BF Bogner et al. I 0 Displaement onforming retangular elements and two load types. This type of diagram is onventionally used for assessing the behaviour of various elements, and we show on it the performane of the elements already desribed as well as others to whih we shall refer to later. Table 4.3 gives the key to the various element odes whih inlude elements yet to be desribed.55p58 Fig Rate of onvergene in energy norm versus degree of freedom for three elements: the problem of a slightly skewed, simply supported plate (80 ) with uniform mesh subdivision.

35 Singular shape funtions for the simple triangular element 145 The omparison singles out only one displaement and eah plot uses the number of mesh divisions in a quarter of the plate as absissa. It is therefore diffiult to dedue the onvergene rate and the performane of elements with multiple nodes. A more onvenient plot gives the energy norm IIuII, versus the number of degrees of freedom N on a logarithmi sale. We show suh a omparison for some elements in Fig for a problem of a slightly skewed, simply supported plate.7 It is of interest to observe that, owing to the singularity, both high and loworder elements onverge at almost idential rates (though, of ourse, the former give better overall auray). Different rates of onvergene would, of ourse, be obtained if no singularity existed (see Chapter 14 of Volume 1). Conforming shape funtions with nodal singularities 4.9 General remarks It has already been demonstrated in Se. 4.3 that it is impossible to devise a simple polynomial funtion with only three nodal degrees of freedom that will be able to satisfy slope ontinuity requirements at all loations along element boundaries. The alternative of imposing urvature parameters at nodes has the disadvantage, however, of imposing exessive onditions of ontinuity (although we will investigate some of the elements that have been proposed from this lass). Furthermore, it is desirable from many points of view to limit the nodal variables to three quantities only. These, with simple physial interpretation, allow the generalization of plate elements to shells to be easily interpreted also. It is, however, possible to ahieve C, ontinuity by provision of additional shape funtions for whih, in general, seondorder derivatives have nonunique values at nodes. Providing the path test onditions are satisfied, onvergene is again assured. Suh shape funtions will be disussed now in the ontext of triangular and quadrilateral elements. The simple retangular shape will be omitted as it is a speial ase of the quadrilateral Singular shape funtions for the simple triangular element Consider for instane either of the following sets of funtions: or (4.72) (4.73) in whih one again i>,j, k are a yli permutation of 1,2,3. Both have the property that along two sides (ij and ik) of a triangle (Fig. 4.18) their values and the values