4.1.1 Mindlin plates: Bending theory and variational formulation

Transcription

1 Chaper 4 soropic fla shell elemens n his chaper, fia shell elemens are formulaed hrough he assembly of membrane and plae elemens. The exac soluion of a shell approximaed by fia faces compared o he exac soluion of a ruly curved shell may reveal considerable differences in he disribuion of bending momens, shearing forces, ec. However, for 'simple' elemens he discreizaion error is approximaely of he same order and excellen resuls can be obained wih he fia shell approximaion [39]. Apar from being easy o define geomerically, fia shell elemens will always converge o he correc deep shell soluion in he limi of mesh refinemen [46J. 4.1 Plae formulaion For he plae componen of he fia shell elemen he shear deformable formulaion of Mindlin is employed. The final formulaion is modified o include he assumed srain inerpolaion of Bahe and Dvorkin [47] Mindlin plaes: Bending heory and variaional formulaion n his secion he reamens of Hinon and Huang [48] and Papadopoulos and Taylor [49] are followed closely, albei wih differen noaions. However, he same may be found in he sandard works of, for insance Hughes [38], Zienkiewicz and Taylor [39] and Bahe [50]. The simples plae formulaion which accouns for he effec of shear deformaion, is presened. The ransverse shear is assumed consan hroughou he hickness. T he assumpions of he firs order Mindlin heory are (4.1) 1From now on, he drilling degree of freedom 1/), inroduced in Chaper 2, is denoed 1/)3 for reasons of clariy. 43

2 CHAPTER 4. SOTROPC FLAT SHELL ELEMENTS 44 (4.2) U, U2 and 'll3 are he displacemens componens in he X l ) X2 and.t3 direcions respecively, U3 is he laeral displacemen and 'l/j [ and 'l/j2 are he normal roaions in he Xl3 and X23 planes respecively (See Figure 4.1). The elemen is assumed o be fia, wih hickness. Flaness of he plae is no a necessary assumpion, bu merely simplifies he required noaion and implemenaion. The elemen area is denoed D. / 1/ / r ~-~--~~ Figure 4.1: Four-node shell elemen (4.1) is obviously inconsisen wih hree-dimensional elasiciy. However, he ransverse normal sress may be negleced for plaes he hickness is small compared wih he oher dimensions. Moreover, when a linear or consan hrough-he-hickness displacemen assumpion is made, as is cusomary in he shear-deformable plae heories, limied locking occurs due o he Poisson effec, when (J3 3 is resrained. (4.2) implies ha sraigh normals o he reference surface, X3 = 0, remain sraigh, bu do no necessarily remain normal o he plae afer deformaion (Figure 4.2). Also, he ransverse displacemen 'U3 is consan hrough he hickness. The displacemen field assumed in (4.2) yields in-plane srains of he form X3'l/J2,2 1'12 X3 ('l/j,2 + 'l/j2,d (4.3) (4.4)

3 CHAPTER 4. SOTROPC FLAT SHELL ELEMEl TS 45 <ij; ~ fu l 'A c ssumed delormalol True deformaion Normal on neural axis afer deformaion Figure 4.2: Mindlin heory T he ransverse shear srains are obained as 1 13 = 'U3,1 + 'lh 123 = 'U3,2 + <ij1 ( 4.5) For plane sress and linear isoropic elasiciy he forgoing srain field defines he in-plane sresses as E CTll :-2 [Ell + VE22] (4.6) 1 - v -1- E - v- h 2 + VEn ] (4.7) 2 CT21 = G,12 (4.8) E is Young's modulus and v is Poisson's raio. Similarly, he ou-of-plane sresses are given by CT13 CT31 = G'13 (4.9) (/23 CT32 = G,'23 (4.10) G = E (4.11) 2(1 + v)

4 CHAPTER 4. SOTROPC FLAT SHELL ELE"fENTS 46 negraing he in-plane sresses, which vary linearly along he plae hickness, gives sress resulans of.he form nroducing marix noaion, he forgoing are wrien as Mll l: O'U X 3 dx3 (4. 12). 2 M22 1_21 0'22 1:3 dx3 (4. 13) 2 M12 = M21 = / 2 0'12 X 3 dx3 (4.14) -2!vll 1 M= M22 [ M12 (4. 15) and he curvaures as (4. 16) follows ha he momen-curvaure relaion may be expressed as ( 4.17) D - _ b - 12(1-1/ 2 ) 0 0 1;// _ E_ 3 _ [~~ ~ 1 (4.18) Similarly, he ou-of-plane sresses, when inegraed along he hickness, glve ransverse shear forces 1_21 0'13 dx3 "2 [ 21 0'23 dx3 2 (4. 19) ( 4.20) which, using marix noaion, resuls in Q= D si (4.21)

5 CHAPTER 4. SOTROPC FLAT SHELL ELEMENTS 47 ( 4.22) ( 4.23) ( 4. 24) Summaion convenion is implied over Xl, X2 and X3 for Lain indices and over Xl and X2 for Greek indices, so ha he local equilibrium equaions may be appropriaely inegraed hrough he hickness o deduce he plae equilibrium equaions SO,O + P o (4.25) p denoes he ransverse surface loading. The firs equaion relaes he bending momens o he shear forces, as he second is a saemen of ransverse force equilibrium. n he limiing case he Kirchhoff hypohesis of zero ransverse shear srains mus hold. Therefore U31 + 1/;2 0 o (4.26) (4.5) imply ha he ransverse shear srain remains consan hrough he elemen hickness. This is inconsisen wih classical heory, he corresponding ransverse shear sress varies quadraically. Also, he ransverse shear srain on he plae surface is required o be zero. Consequenly, a emporary modificaion o he displacemen field is made, namely mposing he consrain (4.27) 1 ~2l (x~ +,BX3)X3 dx3 = 0. 2 and seing 113 = 0 on he plae faces resuls in ( 4.28) [ 5 ( 2 32) 1 Moreover, subsiuing (4.29) ino (4.21) leads o 1 13 = X3-20 (1/;2 + u3,d (4.29 )

6 CHAPTER 4. SOTROPC FLA.T SHELL ELEMENTS 48 (4.30) Therefore, for consisency reasons, a 'shear correcion' erm is inroduced as, 6 k=- 5 (4,31) ino (4,21), which now becomes ( 4.32) (4.33) T he oal plae energy, based on poenial energy for bending and shear, is wrien as ( 4.34) ex is he poenial energy of he applied loads. The hin plae Kirchhoff condiions of (4.26) should be saisfied in he finie elemen inerpolaion Finie elemen inerpolaion The displacemen in he reference surface of he elemen is defined by (4,35 ) w here N (~, TJ) are he isoparameric shape funcions The secional (normal) roaions are inerpolaed as ( 4.36) Nn~, 17)1/Jl N (~, 17 )1/J~ ( 4.37) (4.38)

7 CHA PTER 4. SOTROPC FLAT SHELL ELEMENTS 49 and he ransverse mid-surface displacemens are inerpolaed as U3, 'l/jl and 'l/j~ are he nodal poin values of he variables U3, 'l/j] and 'l/j2 respecively. The curvaure-displacemen relaions are now wrien as ( 4.39) 4 '" = L B bi qi ( 4.40) i=l The elemen curvaure-displacemen marix is given in Appendix A. The unknowns a node z are (4.41) The shear srain-displacemen relaions are wrien as 4 1= L B siqi (4.42) i = l The elemen shear srain-displacemen marix is given in Appendix A Assumed srain inerpolaions The saionary condiion of (4.34) direcly resul s in he plae force-displacemen relaionship K eq = T ( 4.43) wih K C = (K b + K s) ( 4.44) K b n ",T D b'\, dd (4.45) K s n T D S dd ( 4.46) Subscrips band s indicae bending and shear respecively. For elemens wih 4 nodes, he expression for K b is problem free, a leas in erms of locking. The employed inerpolaion field of (4.36) in K S resuls in severe lo cking when full inegraion is used.

8 CHAPTER 4. SOTROPC FLAT SHELL ELEMENTS 50 One soluion ha overcomes he locking phenomena, while ensuring ha he final elemen formulaion is rank sufficien, is o incorporae he subsiue assumed srain inerpolaion field of Bahe and Dvorkin [7, 47]. 3 A f. ~ ~-+--f--j r; Figure 4.3: nerpolaion funcions for he ransverse shear srains Depiced in Figure 4.3, he assumed inerpolaion field of Bahe and Dvorkin is wrien as (4.47) ( 4.48) he superscrips A hrough D designae he sampling poins for calculaing he covarian shear srains. The shear srain componens in he Caresian coordinae sysem, E13 and E23, are obained [51] using a ransformaion which in he case of a fia plae elemen reduces o he sandard (2 x 2) Jacobian, J { ~~: } = l [~~ :~ ~~ :~ 1 {!:: } = ~ J ~s ( 4.49) Therefore, he subsiue shear srains Es are expressed as

9 CHAPTER 4. SOTROPC FLAT SHELL ELElvENTS 51 ( 4.50) while he associaed ransverse siffness-displacemen relaionship becomes k s = / B; DsBs dd.!n The elemen siffness-displacemen relaionship now becomes ( 4.51) ( 4.52) vvhich is he final elemen formulaion. The assumed srain inerpolaion saisfies he Kirchoff condiions of zero ransverse shear srains in he hin plae limi, while locking is also adequaely prevened. 4.2 Shell formulaion Elemen formulaion Fla shell elemens are simpler han generally curved shell elemens, boh in erms of formulaion and compuer implemenaion. As he elemen Jacobian marix is consan hrough he hickness, analyical hrough-he-hickness inegraion is easily performed. The elemen force-displacemen relaionship of he 8,B(M) and 9,B(M) membrane families is defined by (2.80), and for he 8,B(D) and 9,B(D) membrane families by (2.84). These relaionships are repeaed here using a differen noaion o disinguish beween he membrane and plae componens. T he wo differen force-displacemen relaionships for he membrane families are rewrien in a universal form o clarify he noaion K mqm = rm (4.53) for he mixed formulaion, and (4.54) for he displacemen formulaion. K m= K + P, ( 4.55) K m denoes he membrane siffness marix, q rn he elemen displacemens and rm he elemen body force vecor. The unknown nodal displacemens qm and he specified consisen nodal loads r m are defined by

10 CHAPTER 4. SO TROPC FLAT SHELL ELEMENTS 52 Tm u; 1P;f [u~ [U{ U~ N~lT (4.56) ( 4.57) 1P; is he in-plane roaion and M~ he in-plane nodal momen. Similarly, he Mindlin plae force-displacemen relaionship (see (4.52)) is rewrien as ( 4.58) The displacemens qp ' and he specified consisen nodal loads T p, are respecively defined by ['u1 1P~ 1P;f [U~ M{ M~f (4.59) ( 4.60) T hrough assembly of he membrane and plae elemens, he fla shell elemen siffness marix K e is obained in a local elemen coordinae sysem as ( 4.61) T he local shell force-displacemen relaionship is given by ( 4.62) he shell nodal displacemens and loads for node i respecively are [u~ u~ u1 1P~ 1P; 1P~] T [U{ U~ U~ M{ M~ j\,{jf ( 4.63) ( 4.64) A general warped configuraion The warp correcion employed in his sudy is he so-called 'rigid link' correcion suggesed by Taylor [6], which is depiced in Figure 4.4. Simple kinemaic nodal relaionships are used o evaluae he warp effec. For elemens wih rue roaional degrees of freedom he roaions abou he local x3-axes in he warped and projeced planes may be aken as equal. Assuming reasonably small warp, he effec of he drilling degree on he ou-of-plane bending roaions is negleced. The srain-displacemen modificaion presened by Taylor is herefore exended by addiion of he final row and column as follows [52]

11 CHAPTER 4. SOTROPC FLAT SHELL ELEMENTS 53 2 Figure 4.4: Warped and projeced quadrilaeral shell elemen - i U 1 -i U 2 -i U 3 ib1 ib~ ib _ Xi Xi i u 1 i U 2 i 'U 3 (4.65) /J '1/;2 1/;3 x~ defin es he warp a each node and bared quaniies (for example ul) ac on he fia projecion. This correcion is much simpler han for insance he correcion presened by Robinson [53].