Higher-order linked interpolation in triangular thick plate finite elements. Dragan Ribarić and Gordan Jelenić

Transcription

1 Hgher-order lnked nterpolaton n trangular thck plate fnte elements Dragan Rbarć and Gordan Jelenć Unverst of Reka, Facult of Cvl Engneerng, R. Matečć, 5000 Reka, Republc of Croata

2 Abstract In ths ork e emplo the so-called lnked nterpolaton concept alread tested on beam and quadrlateral plate fnte elements n the desgn of hgher-order trangular plate fnte elements. Startng from the analog beteen the Tmoshenko beam theor and the Mndln plate theor a faml of trangular lnked-nterpolaton plate fnte elements of arbtrar order are desgned from the underlng faml of lnked-nterpolaton beam elements. The derved elements pass the standard patch tests as the convergence benchmarks and possess the potental to pass hgher-order patch tests of an order drectl related to the order of the element. The elements are tested on the standard set of test eamples, and numercall assessed aganst the reference results from lterature usng varous mesh denstes and orders of nterpolaton. Ke ords: Mndln plate theor, trangular plate fnte elements, hgher-order lnked nterpolaton. Introducton Ong to ts close relatonshp th the Tmoshenko theor of thck beams, the dea of lnkng the dsplacement feld to the rotatons of the cross-sectons has been often studed and thoroughl nvestgated and eploted n fnte-element applcatons of the Mndln moderatel thck plate theor [-6]. It has been found out that the dea on ts on cannot elmnate the problem of shear lockng even though ths result ma be acheved for the Tmoshenko beam elements [7,7-]. Dfferent mprovements have been proposed b dfferent authors, hch nvolve adusted materal parameters [4] or the assumed or enhanced stran concepts [,4,5,5-8] or are based on the use of med and hbrd approaches [,,6,9,,]. In ths paper e buld on the deas gven n [9] here e have re-vsted ths classc topc and, remanng frml n the frameork of the standard dsplacement-based desgn technque, derved a faml of quadrlateral thck plate elements b etendng hgher-order lnked nterpolaton functons developed for the Tmoshenko beams. Here e appl the methodolog to the popular class of

3 trangular elements and contrast t to an alternatve methodolog of devsng hgher-order lnked nterpolaton for ths class of elements [8]. In Secton e present the faml of nterpolaton functons for the Tmoshenko beam elements hch provde the eact soluton for arbtrar polnomal loadngs [8]. Even though the Mndln theor of thck plates ma be regarded as a D generalsaton of the Tmoshenko theor of thck beams, the dfferental equatons of equlbrum for thck plates cannot be solved n terms of a fnte number of parameters and so, n contrast to beams, there does not est an eact fnte-element nterpolaton. Nonetheless, n [,,0] such nterpolaton has been used to formulate three-node trangular and fournode quadrlateral thck plate elements, hle n [,6] a s-node trangular and an eght-node quadrlateral elements have been proposed. A faml of trangular elements desgned n ths a has been proposed n [8]. In Secton e outlne the Mndln plate theor and contnue b consderng a trangular three-node element, for hch the constant shear stran condton mposed on the element edges s knon to lead to an nterpolaton for the dsplacement feld hch s dependent not onl on the nodal dsplacements, but also on the nodal rotatons around the n-plane normal drectons to the element edges. The same result ma be obtaned b generalsng the lnked nterpolaton for beams [8] to D stuatons. Ths approach enables a straghtforard generalsaton of the lnked-nterpolaton beam concept to hgherorder trangular plate elements leadng to addtonal nternal degrees of freedom hch do not est n the beam elements. A smlar goal ma be acheved follong the approach presented n [8] here a faml of dsplacement-based lnked-nterpolaton trangular elements has been derved b prescrbng the order of the shear dstrbuton over the element hch, n contrast, does not nvolve nternal degrees of freedom. Generalsng ether of these deas to arbtrar curvlnear trangular shapes (on hgher order elements) s non-trval and specal care needs to be taken for such elements to satsf the standard patch tests. In Secton 4 e compare the to approaches and n Secton 5 summarse the fnte-element results. Fnall, In Secton 6 e conduct numercal tests and n Secton 7 dra the conclusons.

4 . Soluton of the Tmoshenko beam problem for polnomal loadng usng lnked nterpolaton of approprate order In contrast to the Bernoull beam theor, n the Tmoshenko beam theor the cross secton of a beam remans planar after the deformaton, but not necessarl orthogonal to the beam centrodal as. Ths departure from orthogonalt s the shear angle d γ ' d Fg. : Intal and deformed confguraton of a moderatel thck beam here s the lateral dsplacement of the beam shon n Fg., the dash (') ndcates a dfferentaton th respect to the co-ordnate, and s the rotaton of a cross secton. Let, the cross-sectonal stress-couple and shear stress resultants M and S be lnearl dependent on curvature (change of cross-sectonal rotaton) and shear angle va M EI ' and S GA s γ, here EI and GA s are the bendng and shear stffness respectvel. As the equlbrum equatons are M ' S and S' q, here q s the dstrbuted loadng per unt of length of the beam, ths results n the follong dfferental equatons:

5 EI ''' q, GA s ( '' ') q, th the follong closed-form soluton for a polnomal loadng q of order n-4 [8]: n n n n L n I, I N ( ), () n here L s the beam length, and are the values of the dsplacements and the rotatons at the n nodes equdstantl spaced beteen the beam ends, I are the standard Lagrangan polnomals of order n-, and N for and L N n otherse. In the natural co-ordnate sstem L n n L th the dsplacement soluton reads I I L. n. Overve of the Mndln plate theor and a faml of trangular lnked-nterpolaton elements The Mndln plate theor s closel related to the Tmoshenko beam theor and ma be regarded as ts generalsaton to to-dmensonal problems. The plate s assumed to be of a unform thckness h th a md-surface lng n the horzontal co-ordnate plane and a dstrbuted loadng q assumed to act on the plate md-surface n the drecton perpendcular to t. The changes of the angles hch the vertcal fbres close th the md-surface are the shear angles: γ z 0 Γ e γ z 0 () hle the curvatures (the fbre s changes of rotatons) are

6 κ L κ κ κ 0 0 () here s the rotaton vector th components and around the respectve horzontal global coordnate aes, s the transverse dsplacement feld, Γ s the shear stran vector and κ s the curvature vector, s a gradent on the dsplacement feld and L s a dfferental operator on the rotaton feld (see e.g. []). Let us consder a lnear elastc materal th L D κ D M b b Eh M M M ν ν ν ν ) ( (4) ) ( 0 0 kgh S S z z e D DΓ S s s γ γ, here M, M, and M are the bendng and tstng moments around the respectve co-ordnate aes, S and S are the shear-stress resultants, E and G are the Young and shear modul, hle ν and k are Posson s coeffcent and the shear correcton factor usuall set to 5/6. The dfferental equatons of equlbrum are S M M, S M M, q S S. (5) Substtutng (4) n (5), results n the dfferental equatons hch no cannot be solved n terms of a fnte number of parameters as before. Stll, e shall attempt to etend the results from Secton n

7 order to derve more accurate Mndln plate elements. To do so, e shall need the functonal of the total potental energ: T T T T Π(,, ) ( M κ) da ( S Γ) da Πet ( κ Dκ b ) da ( Γ DsΓ) da Π et, (6) here the last term descrbes the potental energ of the dstrbuted and boundar loadng.. Lnked nterpolaton for a three-node trangular plate element We shall frst appl the result gven n () to a trangular element th three nodal ponts at the element vertces as n [8,,5,4,0,] (see Fg. ). The dsplacements and rotatons are epressed n the so-called area coordnates hch, for an nteror pont, make the rato of the respectve nteror area to the area of the hole trangle -- as shon n Fg.. Fg. : Three-node trangular plate element and ts area coordnates of an nteror pont Because n to dmensons an pont s unquel defned b onl to coordnates, the three coordnates,, and are not ndependent of each other and for an pont thn the doman the are related b the epresson:

8 . The area coordnates of an pont thn the doman are transformed nto the Cartesan coordnates as and vce versa ( ) ( ) A A A A a b b a a b ( ) ( ) A A b a b a a b ( ) ( ) A A b a a b a b, here k a and k b are the drected sde-length proectons along the coordnate aes and the ndces,, and k denotng the trangle vertces are cclc permutatons of, and. The area ( ) ( ) [ ] k k a b A denotes the area of the nteror trangle hose one verte s at the pont (,) hle the other to are the vertces and k, hle ( ) b a b a A s the area of the hole trangle (Fg.). Note that the area co-ordnates,, are n fact the standard lnear Lagrangan shape functons over a trangular doman.

9 Fg. : Trangular element s sde and ts rotaton degrees of freedom If an trangle sde k of length s k (here s k a k b k ) s taken as a beam element (Fg. ), the epressons for the dsplacement and the rotaton around the n-plane normal n can be derved n lnked form (): sk n n ( ) bk ( ) a sk ( ) [( ) cosα ( ) snα ] [ ] k, k k, n n n hle k 0. Such nterpolaton provdes constant moments and constant shear along the element sde. At each nodal pont (,,) of an element there est three degrees of freedom (dsplacement, and rotatons, and n the global coordnate drectons). The lnked nterpolaton for the dsplacement and rotaton feld over the hole trangular doman ma be no proposed as [( ) b ( ) a ] [( ) b ( ) a ] [ ( ) b ( ) a ] (7)

10 (8). (9) Here, are the rotaton components at the element vertces (see Fg. 4). Terms n brackets are rotatonal proectons of respectve rotaton components to the normal on each element sde tmes the sde length. Therefore the nterpolaton s soparametrc for the rotatons, hle for the dsplacement functon t ncludes an addtonal lnkng part schematcall presented n Fg. 5. Fg. 4: Three-node trangular plate element and ts nodal rotatons. The nodal dsplacements are perpendcular to the element plane

11 [( ) b ( ) a ] Fg. 5: Lnkng part of the shape functon on sde - of the element The lnked nterpolaton as emploed n a to-node Tmoshenko beam element can eactl reproduce the quadratc dsplacement functon, and the same should be epected for the D nterpolaton consdered here. The fnte element developed on ths bass ll be named T-U, denotng the threenode element th the second-order dsplacement dstrbuton.. Lnked nterpolaton for a s-node trangular plate element A s-node lnked-nterpolaton trangular element (Fg. 6) ma be defned correspondngl.

12 Fg. 6: S-node trangular plate element and ts geometr. Agan, f an trangle sde k s taken as a beam element, the epressons for the dsplacement and the rotaton around the n-plane normal n can be derved n the lnked form (), th node located at the mddle of the sde as n Fg. 7: Fg. 7: Trangular element s sde and ts rotaton degrees of freedom ( ) ( ) ( ) ( ) n n n k s,,, 4

13 ( ) ( ) ( ) ( ) ( ) [ ] k k a b,, 4 ( ) ( ),,, 4 n n n n, snce k 0. Such nterpolaton ma descrbe a lnear moment and shear change along the element sde. The lnked nterpolaton for the dsplacement feld over the hole trangle doman ma be no gven as ( ) ( ) ( ) * ( ) ( ) ( ) [ ] 4 4 a b ( ) ( ) ( ) [ ] 5 5 a b ( ) ( ) ( ) [ ] 6 6 a b (0) hle the nterpolaton for the rotatons takes the standard Lagrangan form ( ) ( ) ( ) () ( ) ( ) ( ) () here, are the nodal rotaton components at the element vertces and mdponts. As before, the dsplacement and rotaton felds are nterpolated usng the same nterpolaton functons, but the dsplacement feld has an addtonal lnkng part epressed n terms of the rotatonal components on each element sde.

14 The rotatons n () and () have a full quadratc polnomal form, but the dsplacement feld does not have a full cubc polnomal form snce epresson (0) msses the 0 th tem n Pascal s trangle th the functon that has zero values along all the element sdes hch cannot be assocated th an nodal degree of freedom. To provde the full cubc epanson e need to epand the result from (0) th an ndependent bubble degree of freedom b.e. () * b The fnte element developed on ths bass ll be named T6-U, denotng the s-node element th the thrd-order dsplacement dstrbuton. The same nterpolaton for the dsplacements has been appled to the med-tpe s-node trangular plate element n [6].. Lnked nterpolaton for a ten-node trangular plate element A ten-node lnked-nterpolaton trangular element (Fg. 8) follos analogousl from the lnked nterpolaton for the four-node Tmoshenko beam element. Fg. 8: Ten-node trangular plate element and ts geometr.

15 An trangle sde can be taken as a beam element and epressons for and n can be epressed n the lnked form (). Completed over the hole trangle, the nterpolatons for the dsplacement and the rotatons follo as ( )( ) ( ) ( ) 5 4 * 9 9 ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) [ ] a b ( )( ) ( ) ( ) [ ] a b ( )( ) ( ) ( ) [ ] a b (4) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) (5) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) (6)

16 here, are the nodal rotaton components at the element vertces and the md-sde ponts. The rotatons are epressed as a full cubc polnomal, but the dsplacement feld does not have a full quartc polnomal forms. To etend epresson (4) to a full quartc form (th all 5 tems n Pascal s trangle), to more parameters are needed and the are related th the functons that have zero values along an element sde and at the central pont (node th nde 0). Those parameters are some nternal bubbles so fnall the dsplacement feld ma be completed as: * ( ) b ( ) b (7) The thrd term that appears to be mssng n epresson (7) to complete the cclc trangle smmetr, namel, ( ) b s actuall lnearl dependent on the to other added terms and the 0 th term n (4). The fnte element developed on the bass of ths nterpolaton ll be named T0-U4, denotng the ten-node element th the fourth-order dsplacement dstrbuton. 4. Comparson th Lu-Rggs faml of purel dsplacement-based trangular elements If arbtrar drecton s crossng the trangle element s chosen (Fg. 9), the shear stran can be epressed n terms of the shear strans along the drectons of the co-ordnate aes and as γ γ cosα γ snα, (8) s here α s the angle beteen the s-drecton and the -as.

17 In the lnked nterpolaton formulaton presented n ths ork, the epresson for the shear along an element sde s a polnomal hch s to orders loer than the dsplacement nterpolaton polnomal. Ths s also vald for an drecton parallel to an element sde. Lu and Rggs [8] have derved the faml of trangle elements lkese based purel on dsplacement nterpolatons, hch eventuall, turn out to be of the lnked tpe n the sense that the dsplacement dstrbuton also depends on the nodal rotatons. In contrast to the approach presented here, hoever, the requrement that the authors set s that the shear stran along arbtrar drecton s, and not onl those parallel to the element sdes, should satsf the above condton (Fg. 9), thus mposng the p-th dervatve of (8) to be zero for arbtrar α, hle the dsplacement nterpolaton s of the order p and the nterpolaton for the rotatons s of the order p. γ γ α Fg. 9: Dfference n the shear stran condton beteen the present formulaton and the Lu-Rggs formulaton [8].

18 4. Lu-Rggs nterpolaton for a three node trangular plate element Lu and Rggs [8] and Tessler and Hughes [4] before them, have derved nterpolaton functons for the trangular element named MIN th three nodes and nne degrees of freedom (the same degrees as the T-U element presented n.) from the condton that the shear stran must be constant along an drecton thn the element (Tessler and Hughes have addtonall ntroduced a shear-relaaton factor n order to mprove the element performance). Ther nterpolatons for the dsplacement and the rotaton felds are: N L M, N and N (9) here the nterpolaton functons are N, L ( b b ) k k and M ( a a ) k k (0) th, and k agan beng the cclc permutatons of, and. It can be verfed that the rgd bod condtons are satsfed because N, L 0 and M 0, () and t can be also verfed b drect calculaton that the Lu Rggs nterpolaton s the same as the lnked nterpolaton gven n.. Therefore, for the trangular element th three nodes, n the present formulaton the shear stran s also constant n an drecton and not onl along the drectons parallel to the sdes of the trangle. 4. Lu-Rggs nterpolaton for a s node trangular plate element MIN6

19 Lu and Rggs n [8] have net derved a faml of elements based on upgradng the crtera for the shear stran along an arbtrar drecton over the element. The second member of the faml s the socalled MIN6 element th s nodal ponts. The nterpolatons are derved to provde lnear shear n an drecton crossng the element. Interpolatons n MIN6 for the dsplacement and rotatons are agan: N L M, N and N for,, 6 () here the actual nterpolaton functons are N ( ), N 4 for,, (a) L k k, L 4 b b (b) ( ) ( b b ) and M ( ) ( a a ) k k, M 4 a a (c) It should be stressed that n the epressons for L and M gven here a tpographc error n the Lu- Rggs orgnal [8] s corrected to satsf the rgd bod condtons: N, L 0 and M 0. (4) The element based on nterpolaton () denoted as MIN6 - has been coded n the fnte-element programme envronment FEAP [7] along th the earler elements T-U, T6-U and T0-U4. In contrast to MIN (thout shear relaaton), hch corresponds eactl to the T-U presented n., the MIN6 element s dfferent from the T6-U presented n., hch has an addtonal bubble degree of freedom. It can be verfed b drect calculaton that the Lu Rggs nterpolaton for MIN6 should concde th the T6-U nterpolaton gven n. f the bubble degree of freedom ere constraned to

20 [( b b ) ( a a ) ( b b ) ( a a ) ( b b ) ( a a ) ] b [( b b ) ( a a ) ( b b ) ( a a ) ( b b ) ( a a ) ] (5) It should be made clear that the shear stran dstrbuton of a certan order along an arbtrar drecton s the basc underlng condton from hch the MINn faml of elements has been derved, hle the shear stran dstrbuton of a certan order along a drecton parallel to the element sdes s a consequence, rather than the orgn of the faml of elements presented n Secton. The presented methodolog generates the lnked nterpolaton from the underlng nterpolaton functons developed for the beam elements and ma be consstentl and straght-forardl appled to trangular plate elements of arbtrar order. In contrast, the MINn methodolog requres a smbolc manpulaton of algebrac epressons hch get progressvel more complcated as the order of the element s rased. 5 Fnte element stffness matr and load vector The earler nterpolatons ma be rtten n matr form as I N N b, (6), b I,, (7) here I s a matr of all nterpolaton functons concernng the nodal dsplacement parameters th the dmenson N nd, here n( n ) / N nd and n s the number of nodes per element sde. Also, s the vector of nodal dsplacement parameters th the dmenson N nd : T L n, N s the matr of all lnked nterpolaton functons th the dmenson N nd and, s the vector of T nodal rotatons n global coordnate drectons th the dmenson N nd :,, L n,. n

21 Further, N b s the matr of bubble nterpolaton functons gven n () or (7) th the dmenson T N b and b s the bubble parameter vector th the dmenson N b n-: b b, Lb, n, onl for n. I s agan the matr of all nterpolaton functons concernng rotatonal parameters descrbed n (8,9), (,) or (5,6) and has the dmenson N nd. The formaton of the element stffness matr and the eternal load vector for the nterpolaton functons defned n ths a follos the standard fnte-element procedure descrbed n tet-books e.g. [-4]. A functonal of the total energ of the sstem s gven n (6) and from the statonart condton for the total potental energ of an element, a sstem of algebrac equatons s derved: K K K S S Sb K K B K T S K Sb S K K K T Sb T Sb Sbb f f b fb,, here vectors f, f and f b are the terms due to eternal loadng. The submatrces n the stffness matr follo K B T ( LI ) Db( LI )da A K K K K K K S Sbb S S Sb Sb T ( I ) Ds( I ) da A T ( Nb ) Ds( Nb ) da A T ( ei N ) Ds( ei N ) A T ( ei N ) Ds( I ) da A T ( Nb ) Ds( I ) da A T ( Nb ) Ds( ei N )da A da here L and are the dfferental operators from () and () actng on the nterpolaton functons n (7) and (6) respectvel, hle e s a transformaton matr gven n ().

22 6. Test eamples In all the eamples the results for the elements T-U, T6-U and T0-U4 are compared to the medtpe element of Aurccho and Talor denoted as T-LIM [0] and ntegrated n FEAP (A Fnte Element Analss Program) b the same authors, or to the TBL element []. Also, comparson s made to the MIN6 element [8] and the hbrd-tpe element 9β Q4 [9] as ell as the lnkednterpolaton quadrlateral elements Q4-U, Q9-U and Q6-U4 [9]. 6.. Patch test and egen-analss of the stffness matr Consstenc of the developed elements s tested for the constant stran condtons on the patch eample th ten elements, coverng a rectangular doman of a plate as shon n Fg. 0. The dsplacements and rotatons for the four nternal nodes thn the patch are checked for the specfc dsplacements and rotatons gven at the four eternal nodes [4, 5, 5, 6]. The plate propertes are E0 5, ν0.5, k5/6, hle to dfferent thcknesses correspondng to a thck and a thn plate etremes are consdered: h.0 and h 0.0. To stran-stress states are analsed [6]: Constant bendng state Dsplacements and rotatons are epressed respectvel b ( ) /, ( ) /, ( ) /. The eact dsplacements and rotatons at the nternal nodes and the eact strans and stress resultants at ever ntegraton pont are epected. The moments are constant M M -. h³, M -. h³ and the shear forces vansh (S S 0). Constant shear state

23 Dsplacements and rotatons are epressed respectvel b h ( ) ( ) ν, 8 6, ( 6 4 ). The eact dsplacements and rotatons at the nternal nodes and the eact strans and stress resultants at ever ntegraton pont are epected agan. The shear forces are constant S h³ and S h³ n ever Gauss pont and the moments are lnearl dstrbuted accordng to Eh M [ ( 6 8ν ) ( 6 ν )], ( ν ) M Eh ( ν ) [ ( 8 6ν ) ( 6ν )] and Eh ν M 6 ( ν ) ( ). Fg. 0: Element patch for consstenc assessment of three-node elements The three node trangle element T-U s tested on the patch gven n Fg. 0. For the gven values for the dsplacements and rotatons at the eternal nodes calculated from the above data, the dsplacements

24 and rotatons at the nternal nodes as ell as the bendng and torsonal moments and the shear forces at the ntegraton ponts are calculated and found out to correspond eactl to the analtcal results gven above for the constant bendng test, but not for the constant shear test. It should be noted that the constant shear test performed here s related to a lnear change n curvature (thrd-order clndrcal bendng) and n fact b defnton requres an element to enable cubc dstrbuton of the dsplacement feld and the quadratc dstrbuton of the rotaton felds, for hch the analsed element s not desgned. Ths test should not be mstaken for the constant shear test th no curvature as a consequence of a sutable choce of dstrbuted moment loadngs [9], hch the analsed element also passes. Fg. : Element patch for consstenc assessment of s-node elements The s node trangle element T6-U s tested on the smlar patch eample (the mesh s gven n Fg. ). Agan, onl the dsplacements and rotatons at the boundar nodes are gven (8 dsplacements and 6 rotatons), hle all the nternal nodal dsplacements and rotatons are to be calculated b the fnteelement soluton procedure. In fact the are calculated eactl for both the constant bendng and the constant shear test. The moments and shear forces at the ntegraton ponts are also eact.

25 Fg. : Element patch for consstenc assessment of ten-node elements The same patch tests are also successfull performed b the ten-node elements T0-U4, here 48 parameters for the degrees of freedom are prescrbed and 08 others are checked (Fg. ). The results of the patch tests for all three proposed elements are gven n Table for the dsplacement at node th co-ordnates (0.04, 0.0). The results are not altered f an of the nternal nodes changes ts poston n the mesh (for eample node n Fg. 0) for T-U element s constant curvature test or for T6-U and T0-U4 elements n both tests. The results of the patch tests are not senstve to mesh dstorton. Furthermore, the quartc nterpolaton for the dsplacement feld ould enable the ten-node element T0-U4 to eactl reproduce even the clndrcal bendng of the fourth order. Table : The patch test results for the proposed elements (dsplacement at pont : ) Elements Patch test for constant curvature Patch test for constant shear h.0 h0.0 result h.0 h0.0 result T-U () pass fal

26 T-U(a) pass fal T-U(b) pass fal T6-U pass pass T0-U pass pass Analtcal soluton For stablt assessment of the elements, a patch test must be obvousl satsfed, but the egenanalss on the sngle element should be also checked out [,0]. Several cases of span-to-thckness rato are consdered: L/h 0, L/h000 and L/h (see Fg. ). In egen-analss the bendng stffness s kept constant b scalng the Young modulus proportonall to (/h)³. The elements have alas the correct number of zero egenvalues that correspond to rgd bod modes. The T-U element has three egen-values that are assocated th shear, hch eperence consderable groth as the element thckness s reduced ndcatng a propenst of the element to lock. The other three egen-values are bendng dependent and the reman constant. The results are gven n Table.. In T6-U and T0-U4 elements there also est the grong shear-related egen-values, but there s also an ncreased number of the egen-values hch reman constant (see Tables. and.). It ll be shon n Sectons that, n spte of the grong egen-values, these elements consderabl reduce or completel elmnate the lockng effect. Fg. : Sngle element egen-analss s performed over the rght angle trangle geometr

27 Table.: Rght angle trangle element T-U. Egenvalues of the element stffness matr L/h E E0.8644E0.584E E E E E E E E05.85E05.589E E-0.060E-0 -.9E-.5980E E E E09.85E09.589E E-0.060E E E E-07 Table.: Rght angle trangle element T6-U. Egenvalues of the element stffness matr L/h E E0.4600E0.80E0.759E0 9.85E E E00 4.0E00.45E00.0E E-0.0E-0.90E E E E-6.570E E E05.67E05.747E E E E E E00.59E00.7E E-0.6E-0.0E E E- -.8E- 6.0E E E09.67E09.747E E E E E E00.59E00.7E E-0.6E-0.0E E-0.58E E E-08 Table.: Rght angle trangle element T0-U4. Egenvalues of the element stffness matr L/h E E E E0 4.5E0.887E0.449E0.99E0.589E E E E E E E E E00.769E00.475E00.786E E-0.707E-0.49E-0.50E-0.58E E E E E E E E E E E05.748E05.059E05.7E E E04 7.0E E E E04.050E E E00.98E00.79E E E E-0.880E-0.809E-0.684E E E E E-.076E E E E E E09.748E09.059E09.7E E E08 7.0E E E08.607E E E E00.984E00.79E E E E-0.88E-0.809E-0.684E E E E E E Clamped square plate In ths eample a square plate th clamped edges s consdered. Onl one quarter of the plate s modeled th smmetrc boundar condtons mposed on the smmetr lnes. To ratos of span

28 versus thckness are analsed, L/h0 representng a relatvel thck plate and L/h 000 representng ts thn counterpart. The loadng on the plate s unforml dstrbuted of magntude q. The plate materal propertes are E 0.9 and ν 0.. The numercal results for the mesh pattern n Fg. 4 are gven n Tables. and 4. and compared to the elements presented n [,0] based on the med approach. The dmensonless results * / (ql 4 /00D) and M*M / (ql²/00), here DEh³/((-ν²)) and L s the plate span, gven n these tables are related to the central dsplacement of the plate and the bendng moment at the ntegraton pont nearest to the centre of the plate. The number of elements per mesh n these tables s gven for one quarter of the structure as shon n Fg. 4 for the 44 mesh consstng of elements. C L C L Fg. 4: A quarter of the square plate th clamped boundar condtons under unform load (- element mesh) Clearl, all the ne elements converge toards the same soluton as the elements from the lterature, and the hgher-order elements ehbt an epected faster convergence rate. Stll, the loest-order element T-U s somehat nferor to TBL, hch s not surprsng knong that that element s actuall based on the lnked nterpolaton as n T-U on top of hch addtonal mprovements are made. The mnute dfferences beteen the results of T6-U and MIN6 are attrbuted to the slght

29 dfference n these elements as eplaned n Secton 4.. Table.: Clamped square plate: dsplacement and moment at the centre, L/h 0 Element T-U T6-U T0-U4 mesh * M* * M* * M* Ref. sol. [7] Element TBL [ ] MIN6 mesh * M* * M* Ref. sol. [7] Comparng the present trangular lnked-nterpolaton elements to ther quadrlateral counterparts [9] shos that for the same number of the degrees of freedom the latter, converge a lttle faster (Table.). Table.: Clamped square plate: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 0

30 Element 9βQ4 [9] Q4-U Q9-U Q6-U4 mesh * M* * M* * M* * M* Ref. sol. [7] Convergence of the central dsplacement for the thck-plate case s presented n Fg. 5, th respect to the number of degrees of freedom (n logarthmc scale). Best convergence th respect to the number of degrees of freedom can be observed n elements th hgher-order lnked nterpolaton and t ma be concluded that for the thck clamped plate the present elements converge compettvel for a comparable number of degrees of freedom.

31 Fg. 5: Convergence of the transverse dsplacement at the centre for L/h0 The M moment dstrbuton along the as s computed at the Gauss ponts closest to the as and, begnnng from the centre pont of the plate, shon n Fg. 6. These results are the same as the results for the moment M along the drecton. For T-U element, the moment s constant across the element, ong to ts dependence on the dervatves of rotatons (8) and (9) n both drectons. For elements T6-U and T0-U4, the moment dstrbuton s accordngl lnear or quadratc, respectvel, and the results converge toards the eact dstrbuton fast. Smlar observatons ma be made for the dstrbuton of the shear-stress resultants. C Fg. 6: Moment M dstrbuton along element s Gauss ponts closest to the as on the 44 regular mesh for the clamped plate th L/h0 For the thn plate case shon n Table 4., the elements T-U and T6-U suffer from some shear lockng hen the meshes are coarse, but as epected the converge to the correct result. The hgherorder elements ehbt an epected faster convergence rate. As for the case of the thck plate, the loest-order element T-U s stll somehat nferor to TBL, hle T6-U s margnall better than MIN6. Lkese, comparng the trangular lnked-nterpolaton

32 elements to ther quadrlateral counterparts [9] shos that for the same number of the degrees of freedom the latter, converge a lttle faster (Table 4.). Table 4.: Clamped square plate: dsplacement and moment at the centre, L/h 000. Element T-U T6-U T0-U4 mesh * M* * M* * M* Ref. sol. [] Element TBL [ ] MIN6 mesh * M* * M* Ref. sol. []

33 Table 4.: Clamped square plate: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 000 Element 9βQ4 [9] Q4-U Q9-U Q6-U4 mesh * M* * M* * M* * M* Ref. sol. [] If the same eample ere run th a dfferent orentaton of the trangular elements (th the longest element sde orthogonal to the dagonal passng through the centre of the plate) the results ould turn out to be slghtl orse even though for the hgher-order elements the trend gets reversed as the mesh s refned. Ths s shon n Table 4. for the thn plate case. Table 4.: Clamped square plate: dsplacement and moment at the centre usng opposte orentaton of trangular elements n the mesh, L/h 000. Element T-U T6-U T0-U4 mesh * M* * M* * M* Ref. sol. []

34 The dfferences n the results gven for the to orentatons (Tables 4. and 4.) drop belo. % for the T-U dsplacement th a 66 mesh alread. 6. Smpl supported square plate In ths eample the square plate as before s consdered, but ths tme th the smpl supported edges of the tpe SS (dsplacements and rotatons around the normal to the edge set to zero) as shon n Fg.7. The same elements as before are tested and the results are gven n Tables 5. and 6. for the thck and the thn plate, respectvel, compared agan to the elements presented n [,0]. C L C L Fg. 7: A quarter of the square plate th smpl supported boundar condtons (SS), under unform load (-element mesh) The dmensonless results * / (ql 4 /00D) and M*M / (ql²/00) gven n these tables are related to the central dsplacement of the plate and the bendng moment at the ntegraton pont nearest to the centre of the plate. The number of elements per mesh n these tables relates to one quarter of the plate. Table 5.: Smpl supported square plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre, L/h 0. Element T-U T6-U T0-U4

35 mesh * M* * M* * M* Naver seres Ref. [] Element TBL [ ] MIN6 T-LIM [0 ] mesh * M* * M* * M* Naver seres Ref. [] Table 5.: Smpl supported square plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 0 Element 9βQ4 [9] Q4-U Q9-U Q6-U4 mesh * M* * M* * M* * M*

36 Naver seres Ref. [] Table 6.: Smpl supported square plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre, L/h 000. Element T-U T6-U T0-U4 mesh * M* * M* * M* Naver seres Ref. [] Element TBL [ ] MIN6 T-LIM [0 ] mesh * M* * M* * M* Naver seres

37 Ref. [] For the thck plate case, t can be concluded that elements T6-U and T0-U4 converge consderabl faster than elements based on the med approach and no lockng can be observed on coarse meshes, even for the three-node element T-U. In the thn plate eample, lockng on the coarse meshes can be observed for T-U, but hgher order elements, agan, sho ver good convergence rate. In contrast to the clamped plate problem, the mesh pattern used here has slghtl better convergence than the mesh pattern used n that eample. Table 6.: Smpl supported square plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 000 Element 9βQ4 [9] Q4-U Q9-U Q6-U4 mesh * M* * M* * M* * M* Naver seres Ref. [] 6.4 Smpl supported ske plate In ths eample the rhombc plate s consdered th the smpl supported edges (ths tme, of the socalled soft tpe SS [8]) to test performance of the rhombc elements. The problem geometr and materal propertes are gven n Fg. 8, here an eample of a 88-mesh s shon (8 trangular

38 elements). Fg. 8: A smpl supported (SS) ske plate under unform load The same three elements as before are tested and the results are gven n Tables 7. and 8. for the thck and the thn plate, respectvel. The dmensonless results */(ql 4 /0 4 D), M *M /(ql²/00) and M *M /(ql²/00) are related to the central dsplacement of the plate and the prncpal bendng moments n dagonal drectons at the ntegraton pont nearest to the centre of the plate. The tested eample has to orthogonal aes of smmetr, A-C-B and D-C-E, and onl one trangular quarter ma be taken for analss [,9]. Snce there s a sngulart n the moment feld at the obtuse verte, ths test eample s a dffcult one. Even more, the analtcal soluton [40] reveals that moments n the prncpal drectons near the obtuse verte have opposte sgns. In contrast to the earler eamples, t must be noted that here the ne dsplacement-based elements perform orse than the elements gven n [,0], both for the thck and the thn plate eamples. Tables 7. and 8. no reveal slghtl more pronounced dfferences n the results obtaned usng elements T6-U and MIN6, here the latter are somehat orse, apparentl ong to the absence of

39 the nternal bubble parameter present n T6-U (see Secton 4.). Also, from Tables 7. and 8. t s apparent that for ths test eample the ne trangular faml of lnked-nterpolaton elements s n fact superor to the faml of quadrlateral lnked-nterpolaton elements presented n [9]. Table 7.: Smpl supported ske plate (SS): dsplacement and moment at the centre, L/h 00. Element T-U T6-U T0-U4 mesh * M * M * * M * M * * M * M * Ref. [] Element T-LIM [0 ] MIN6 mesh * M * M * * M * M * Ref. []

40 Table 7.: Smpl supported ske plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 00 Element 9βQ4 [9] mesh * M * M * Ref. [] 0.4 Element Q4-U Q9-U Q6-U4 mesh * M * M * * M * M * * M * M * Ref. [] The mesh pattern chosen for computng the results n Table 7. s the best among the unforml dstrbuted meshes. For eample, meshes b) and c) n Fg. 9 gve less good results for the smlar numbers of degrees of freedom as can be notced n Tables 7. and 7.4.

41 Fg. 9: A smpl supported (SS) ske plate th three dfferent mesh patterns Table 7.: Smpl supported ske plate (SS): dsplacement and moment at the centre, L/h 00. Mesh pattern b). Node D.o.f. T-U mesh * M * M * Ref. [5] 0.4

42 Table 7.4: Smpl supported ske plate (SS): dsplacement and moment at the centre, L/h 00. Mesh pattern c). Element D.o.f. T-U T6-U MIN6 (Lu - Rggs) mesh * M * M * * M * M * * M * M * Ref [8] The dstrbuton of the prncpal moments beteen the obtuse angle at A and the centre-pont C s ver comple ong to the presence of sngulart at A and orth partcular consderaton. The prncpal moment M actng around the n-plane normal to the shorter dagonal converges toards the eact soluton satsfactorl, but for the prncpal moment M actng around the shorter dagonal t s obvous that ths faml of elements fnds t dffcult to follo the eact moment dstrbuton near the sngulart pont. These results are shon n Fg. 0. It should be noted that near the sngulart pont the moments are gettng hgh values of opposte sgns, and there even a small relatve dfference beteen the eact result and the fnte-element soluton n one of the prncpal moments ma strongl nfluence the other prncpal moment snce the are related va Posson s coeffcent (ν 0. n ths case) as shon n (4). Specfcall, even though the fnte-element solutons for the prncpal moment M recognse the monotonous trend of the eact soluton, the fact that, as absolute values, these moments are overestmated makes t dffcult for the element to provde a soluton for M hch ould recognse the change n sgn, slope and curvature evdent n the eact soluton. Of course, there est technques to reduce the error n M hch, as a result, ould also correct ths anomal n M, e.g. the shear correcton factor concept (see e.g. Fg. 4 n [4]), an dea that has not been folloed up

43 n ths paper (see [8] for evdence and [4] for eplanaton h ths concept has a dmnshng effect as the nterpolaton order s ncreased). CL CL Fg. 0: Smpl supported ske plate under unform load - a) prncpal moment M dstrbuton beteen ponts A and C b) prncpal moment M dstrbuton beteen ponts A and C

44 Table 8.: Smpl supported ske plate (SS): dsplacement and moment at the centre th regular meshes, L/h 000. Element T-U T6-U T0-U4 mesh * M * M * * M * M * * M * M * Ref. [40] Element TBL[ ] MIN6 mesh * M * M * * M * M * Ref. [40]

45 Table 8.: Smpl supported ske plate (SS) under unforml dstrbuted load: dsplacement and moment at the centre usng a quadrlateral hbrd element [9] and lnked-nterpolaton elements [9], L/h 000. Element 9βQ4 [9] mesh * M * M * Ref. [40] Element Q4-U Q9-U Q6-U4 mesh * M * M * * M * M * * M * M * Ref. [40] Smpl supported crcular plate The crcular plate th the smpl supported edges s analsed net. The element mesh s here rregular and the nfluence of such rregulart s studed on the element faml n consderaton.

46 C L C L C L CL C L C L Fg. : A smpl supported (SS) ¼ of the crcular plate under unform load. Three meshes used n analss n Tables 9 and 0 Addtonall, not onl the verte nodes, but also the sde nodes of the hgher-order elements (s-node T6-U and ten-node T0-U4) are no placed on the crcular boundar. The edge elements are not follong the straght lne rule, so the must behave as curvlnear transformed trangles for hch lnked nterpolaton does not solve the patch test eactl (unless the elements become nfntesmall small). Onl the transverse dsplacements of the nodes on the crcular plate boundar are restraned and the rotatons reman free (SS boundar condton). The results are gven n Tables 9 and 0. for the thck and the thn plate, respectvel. The problem geometr and materal propertes are gven n Fg. (onl one quarter of the plate s analsed), here eamples of the three node element mesh s shon. A comparson th the lnked-nterpolaton quadrlateral elements [9] s gven n Table 0.. Table 9: Smpl supported crcular plate (SS) th unform load on meshes from Fg. : dsplacement and moment at the centre, R/h 5. Element T-U T6-U T0-U4 mesh d.o.f. c * M c * d.o.f. c * M c * d.o.f. c * M c *

47 Ref. [9] Element TBL [ ] mesh * M c * Ref. [9] Table 0.: Smpl supported crcular plate (SS) th unform load on meshes from Fg. : dsplacement and moment at the centre, R/h 50. Element T-U T6-U T0-U4 mesh d.o.f. c * M c * d.o.f. c * M c * d.o.f. c * M c * Ref. [9] Element TBL [ ] mesh * M c * Ref. [9]

48 * 00D c, c q ( R) 4 Eh D, ( ν ) * M 00 M c (at closest Gauss pont) c q ( R) M M M c moment at the central pont Table 0.: Smpl supported crcular plate (SS) th unform load: dsplacement and moment at the centre, R/h 50, usng a quadrlateral hbrd element [9] and quadrlateral lnkednterpolaton elements [9]. Element 9βQ4 [9] Q4-U Q9-U Q6-U4 mesh c * M c * c * M c * c * M c * c * M c * Ref. [9] In Table the results for the smpl supported crcular plate subect to a concentrated load at the center pont are gven. The geometr and materal propertes are dentcal to the problem from Fg.. For the thn plate stuaton (R/h50) the eact soluton for the central dsplacement s knon [9], hle the bendng moment at the centre has a sngulart. Table : Smpl supported crcular plate (SS) th the pont load at the centre. Dsplacement and moment at the centre, R/h 50. Element T-U T6-U T0-U4 mesh d.o.f. c * M c * d.o.f. c * M c * d.o.f. c * M c *

49 Ref. [9] Conclusons Wth: * c 00D c, P 4R * M c M c (at closest Gauss pont) P A dsplacement-based lnked-nterpolaton concept for desgnng trangular Mndln plate fnte elements has been presented and numercall verfed. The dea has been presented n ts general form applcable to trangular elements of an order, but the numercal results have been provded for the frst three elements of the faml,.e. the three-, s- and ten-node plate elements. In all the elements from the faml the leadng desgn prncple s borroed from the underlng faml of Tmoshenko beam elements. Thus the orgnal shear stran condton of a certan order mposed along the beam has been appled to the plate element edges leadng to D generalsaton of the beam-tpe lnked nterpolaton for the dsplacement feld, hereb, both nodal rotatons contrbute to the element outof-plane dsplacements. It has been shon that, for the elements th more than to nodes per sde, addtonal nternal degrees of freedom are requred n order to provde full polnomal epanson of certan order. The number of these degrees of freedom s lnearl grong b one, begnnng th no nternal degrees of freedom for the loest-order three-node member of the faml. The elements developed n ths a gve eact result for clndrcal bendng of a correspondng order, e.g. quadratc dstrbuton of the out-of-plane dsplacements for the three-node elements, cubc for the s-node elements and so on. Hoever, n contrast to beams, the developed elements stll suffer from shear-lockng for ver coarse meshes of the loest-order element tpes. In partcular, the results for the thn clamped square plate have shon that the loest-order trangular lnked-nterpolaton element (T-U) ma suffer from consderable shear lockng and cannot be consdered as relable, hle a certan amount of lockng s also present n T6- U. Performance of these elements has been numercall analsed and t has been found out, that the