In this research, the smoothed finite element methods (S-FEM) based on the edge-based (ES) and node-based (NS) approaches are combined to develop for the 3-node triangular plate element which uses the mixed interpolation of tensorial components (MITC3) technique to remove the shear-locking phenomenon. This approach is based on the βFEM in which the parameter β is used to tune the contribution ratio of the edge-based and node-based smoothed domains.
Journal of Science and Technology in Civil Engineering NUCE 2019 13 (3): 45–57 STATIC ANALYSIS OF REISSNER-MINDLIN PLATES USING ES+NS-MITC3 ELEMENTS Chau Dinh Thanha,∗, Ho Thi Conb , Le Phuong Binha a Faculty of Civil Engineering, HCMC University of Technology and Education, Vo Van Ngan street, Thu Duc district, Ho Chi Minh city, Vietnam b Center for Vocational and Regular Education, Chau Phu district, An Giang province, Vietnam Article history: Received 09/08/2019, Revised 29/08/2019, Accepted 30/08/2019 Abstract In this research, the smoothed finite element methods (S-FEM) based on the edge-based (ES) and node-based (NS) approaches are combined to develop for the 3-node triangular plate element which uses the mixed interpolation of tensorial components (MITC3) technique to remove the shear-locking phenomenon This approach is based on the βFEM in which the parameter β is used to tune the contribution ratio of the edge-based and node-based smoothed domains The strain fields of the proposed ES+NS-MITC3 element are smoothed on a part of the edge-based domains and the other on the node-based domains which are respectively defined by elements sharing common edges and common nodes The ES+NS-MITC3 element passes the patch test and is employed to statically analyze some benchmark Reissner-Mindlin plates, including square and rhombus ones Numerical results show that, in both thin and thick plates the ES+NS-MITC3 element can give results better than similar elements using the ES-FEM or NS-FEM only Keywords: Reissner-Mindlin plates; MITC3; ES-FEM; NS-FEM https://doi.org/10.31814/stce.nuce2019-13(3)-05 c 2019 National University of Civil Engineering Introduction Plate is one of the most popular structures in construction, shipbuilding, automotive or aerospace industries due to its advantages of load-carrying capacity and aesthetics Instead of using analytical approaches [1–3], to determine the behaviors of complex plate structures the finite element methods (FEM) are widely employed Then many plate finite elements have been developed, especially triangular elements based on the thick plate theory of Reissner-Mindlin which includes the transverse shear strains [1] One of the simplest triangular elements is the 3-node triangular element because it uses the C -type displacement approximation and is most efficient to discretize arbitrary plate geometries However, the original C0-type elements always exists non-zero transverse shear strains and leads to underestimate the deflection, or the shear-locking phenomenon, of the thin plates which ignore the transverse shear strains according to the Kirchhoff-Love plate theory To make the C -type elements be used for analysis of both thin and thick plates, various techniques have been suggested and successfully applied to alleviate the shear locking The Mindlin-type node (MIN3) [4], the discrete shear gap (DSG3) [5], or the mixed interpolation of tensorial components (MITC3) [6] techniques are some of efficient approaches to attenuate the shear locking in the 3-node triangular element Especially, the ∗ Corresponding author E-mail address: chdthanh@hcmute.edu.vn (Thanh, C D.) 45 Thanh, C D., et al / Journal of Science and Technology in Civil Engineering MITC3 approach satisfies the requirement of spatial isotropy, meaning that the element stiffness matrices are independent of the sequence of node numbering Consequently, the plate elements MIN3, DSG3 or MITC3 can be used to analyze both thin and thick Reissner-Mindlin plates The strain fields of the 3-node triangular elements are constant on element domains because of C -type displacement approximation To reduce much difference in strain fields between the elements, the smoothed finite element methods (SFEM) have been proposed [7] According to the SFEM, the strain fields can be averaged over smoothed domains defined by adjacent elements having common edges or common nodes, namely the edge-based smoothed (ES) or the node-based smoothed (NS) methods respectively Although the cell-based smoothed (CS) method is the other type of the SFEM, it is identical to the FEM when applied for the isotropic 3-node triangular elements The ES- and NS-FEM have been successfully developed for the DSG3 and MITC3 plate elements [8–11] Numerical results show that the ES-FEM usually brings overly stiff effects on the behaviors of the discretized model In contrast, the NS-FEM causes overly soft behaviors in comparison with analytical solutions To narrow the gap in results provided by the SFEM and the analytical solutions, the hybrid SFEM or βFEM has been suggested [12, 13] by reconstructing a new smoothed strain fields which includes the ES- and NS-strain fields In this approach, a scale factor β ∈ [0,1] is used to tune the contribution ratio of ES- and NS-domains into the hybrid smoothed strain fields The βFEM for the DSG3 plate element has demonstrated the superior performance when analyzing statics and vibration of the Reissner-Mindlin plates [12, 13] Therefore, the βFEM will be developed for the MITC3 triangular plate element in this work The proposed plate element, called ES+NS-MITC3 element, will be studied the accuracy and efficiency in the static analysis of the Reissner-Mindlin plates The paper is organized as follows In the next section, the finite element formulae of the MITC3, ES-MITC3, NS-MITC3 elements are briefly reviews and then the development of the ES+NS-MITC3 element is presented The numerical performance of the ES+NS-MITC3 element is evaluated through the static analyses of some benchmark plate problems in Section In the last section, significant conclusions about the proposed element are withdrawn Finite element formulation of ES+NS-MITC3 based on the Reissner-Mindlin plate theory 2.1 MITC3 plate element Consider a bending plate with the mid-surface of area A as shown in Fig The plate is subjected to loadings q normal to the mid-surface According to the Reissner-Mindlin thick plate theory, the translational displacements u, v, w related to the x-, y-, z-directions are determined by [1] u (x, y, z) = zβ x (x, y) ; v (x, y, z) = zβy (x, y) ; w (x, y, z) = w0 (x, y) (1) where w0 , β x , βy are respectively the deflection and the rotations of the mid-surface about y- and x-axis with positive directions as shown in Fig The mid-surface is discretized by the 3-node triangular elements The displacements of the midsurface are approximated by [14] w0 = Ni wi ; i=1 βx = Ni θyi ; i=1 46 βy = − Ni θ xi i=1 (2) u ( x, y, zu)( x=, yz, b byy((xx, y, )y; )w; (w z )x=( zxb, xy()x;, yv) ;( x v ,( xy,,yz, z))== zzb x, (yx , z,)y=, zw)0 (=x, w y )0 ( x, y ) (1) (1) here w0,here bx, wb0,y bare and rotations the mid-surface are respectivelythe the deflection deflection and the the rotations of the of mid-surface x, by respectively about y- about and x-axis with positive directions asshown shown in Fig y- and x-axis with directions Fig in1 Thanh, C.positive D., et al / Journal of as Science andin Technology Civil Engineering of loadings andthe thepositive positive Figure 3-node triangular plateelementplate Figure 1.Figure Definition of loadings and positive Figure 3-node The 3-node triangular Figure1.1.Definition Definition of loadings and the Figure The The triangular plate and its displacements of the mid-surface of the positive directions of the nodal displacements displacements of mid-surface the mid-surfaceof of the the element and its positive directionsdirections of displacements of the element and its positive of Reissner-Mindlin plate Reissner-Mindlin plate the nodal displacements Reissner-Mindlin plate the nodal displacements The mid-surfacerespectively isdiscretized discretized by 3-node triangular elements The The mid-surface by the the 3-node elements The in which wi , θ xi , θyi areis the deflection and rotations oftriangular node i with the positive directions displacements of the mid-surface are approximated by [12] displacements the are approximated by [12] defined inof Fig 2; mid-surface and the shape functions are 3 3 33 (2) N w ; bx = N ; by (x =-å NNiqiq (2) w0 = åwN qNiq;1yi b å = y= iw i ; bi x i = å Nå − x y xi)xi+ (y2 − y3 ) x + (x3 − x2 ) y 1i =yi y3å i =1 ii==11 i =1 i =1 i =1 2Ae which , yi qxi,are qyi are (x3the ) x +rotations (x1 of N2respectively = ythe x1 y3 ) + (y3and − yand − xnode (3) in whichin w , iq respectively deflection nodethei with the i, qxiw − deflection 1rotations ) y ofi with 2Ae directions defined in Fig.2;2;and and the functions are are positive positive directions defined in Fig theshape shape functions (x1 y2 − x2 y1 ) + (y1 − y2 ) x + (x2 − x1 ) y N3 = 2A é(x2 x y3 x3 y+2 )(+y( ye- y ) x + ( x3 - x2 x ) 2y ùû) y ûù N1 = N1 ëé=( 2xA yë 3 y2 ) 2 y33) x + ( x e A e where xi , yi are the nodal coordinates of node i as shown in Fig 2; and Ae is the area of the element (3) 1N Eqs (3) between éë(-x3and =( x y(1) x)1 y+3 )the +y( yrelationships é - y1 ) x + ( x1 - x3 ) y ù ùû strains and the nodal displacements are N =From xy11 y-3(2), û ythe ( 3 - y1 ) x + ( x1 - x3 ) ë A e 2A determined e = é( x y - x y ) + ( y - y ) x + ( x - x ) y ù ∂β /∂x N = N 3 éë(εxx1 y2ë -1 x2 + ( x2 3-1 x Ni,x wi 0)û y ùû y1 ) + ( yx1 - y2 ) x Ae Ae ∂β ∂y −Ni,y εy = z θ xi = z =z Bbi dei (4) y i, yγi are the nodal Aθe is the area where x coordinates of node i as shown in Fig 2; and of i=1 i=1 −N N ∂β + ∂β ∂x /∂y i,x in Fig i,y xy nodal coordinates where xi, yi are the 2; yiand Ae is the area of x y of node i as shown the element Bbi dei the element From (1) Eqs and (1) and the relationships relationships between the strains and nodalthe nodal the and strains From Eqs (2),(2),the between the wi 3 γ β + ∂w N N /∂x xz determined x i,x i displacements are B d (5) = = displacements are determined θ = si ei xi γyz βy + ∂w0 /∂y Ni,x −Ni i=1 i=1 θ yi ì xü ì é0 ¶b x ¶x ü ü N i , x ù ì wùi üì ü é0 ì e x ü ï eì N B ï ï¶b x ¶x ï úsiïi , x ï wi d ê (4) ï ï í ẹ ï Beibi d ei3 ¶b ¶y ï ý = z å êê0 - N i , y ú íq xiúýï= z å (4) y ý = zí ¶b y ¶yy = zå N q = z B d í e y ý =ïz í ý í ý å i , y xi bi ei =1 ê i =1 ï ï¶b B ¶yof ï i i=strains ïq úï ê ú g + ¶ b ¶ x N N i = where the gradients the in-plane are given by ùg ù ợ xyùỵảb ợảy x+biảb ảy x ù ỵ ù i,x i , y ỷ ợ yiỳỵù ở"## ờ0 # -$### N N! % y i,x i , y ỷ ợq yi ỵ ợ xy ỵ ợ x ỵ ở"## ! # % d B $### b − c c dei B0bi 0 −b a − d d −a (6) ; Bb2 = ; Bb3 = Bb1 = 2Ae 2Ae 2Ae c−b d−a −c −d b a bi ei in which a = x2 ˘x1 , b = y2 ˘y1 , c = y3 ˘y1 , d =3x3 ˘x1 With the approximation of the transverse3shear strains given by Eq (5), there is always existence of the transverse shear strains in analyzed plates In other words, the pure 3-node triangular element cannot be used for analysis of thin plates in which there are not the transverse shear strains according 47 Thanh, C D., et al / Journal of Science and Technology in Civil Engineering to the Kirchhoff-Love thin plate theory To be employed for both thin and thick plates, from the mixed interpolation of tensorial components approach the transverse shear strains in Eq (5) are reinterpolated to be linear variations corresponding to the three edge directions of the element but be constant on the edges The interpolations of the transverse shear strains connect the displacement approximations at tying points located at the mid-edges The assumed transverse shear strains have been designed by Lee and Bathe [6] for the continuum mechanics based 3-node triangular shell finite elements, namely MITC3 technique to remove the shear locking As a result, the transverse shear strains in Eq (5) can be rewritten as MITC3 γ xz MITC3 γyz = B MITC3 dei si (7) i=1 in which by using one Gaussian quadrature point located at the centroid of the element, B MITC3 si have been derived by Chau-Dinh et al [10] in the explicit formulation, which only depends on nodal coordinates, as follows 2Ae = 2Ae = 2Ae MITC3 B s1 = MITC3 B s2 MITC3 B s3 (b − c) (b + c)/6 b−c Ae + (d − a) (b + c)/6 d − a −Ae − (b − c) (a + d)/6 − (d − a) (a + d)/6 c −bc/2 + c (b + c)/6 ac/2 − d (b + c)/6 −d bd/2 − c (a + d)/6 −ad/2 + d (a + d)/6 −b −bc/2 − b (b + c)/6 −bd/2 + a (b + c)/6 a −ac/2 + b (a + d)/6 ad/2 − a (a + d)/6 (8) The constitutive relations between the stresses and the strains in the isotropic linear plates give εx 0 σx v v Ez E εy = v v 0 σ Bbi dei (9) = y 2 τ − v 0 (1 − ν)/2 γ − v 0 (1 − ν)/2 i=1 xy xy τ xz τyz = E 2(1 − v) MITC3 γ xz MITC3 γyz = E 2(1 − v) B MITC3 dei si (10) i=1 with the Young’s modulus E and the Poisson’s ratio v The total potential energy of the plate subjected to the normal loadings q is expressed in matrix notation as [14] h/2 h/2 σx kh2 MITC3 MITC3 τ xz σy dzdA+ ε x εy γ xy γ xz γyz dzdA− wqdA = Π= τyz 2 h + αhe τ xy A −h/2 A A −h/2 (11) where the shear correction k is 5/6; the stabilized factor α is 0.1; and he is the maximum length of the element’s edges [15] Using Eqs (4), (7), (9), and (10), the total potential energy is written by Ne Ne Ne T T T MITC3 T MITC3 Π= de Bb Db Bb dA de + de Bs DsBs dA de − dTe NqdA = 2 e=1 e=1 e=1 Ae Ae Ae (12) 48 Tạp chí Khoa học Cơng nghệ Xây dựng NUCE 2019 Tạp chí Khoa học Cơng nghệ Xây dựng NUCE 2019 Thanh, C D., et al / Journal of Science and Technology in Civil Engineering T MITC MITC k = ò ΒTTb Db Bb dA + ò ( ΒMITC )T D s ΒMITC dA sMITC3 s MITC3 k ee = Db1b BbBdb2 A +Bòb3 Β D Β dAB MITC3 B MITC3 ]; de = [dT dT dT ]T ; N(16) ( ) s s s in which BbòAe=Βb[B ]; B = [B = Ae s e1 e2 e3 s1 s2 s3 Ae Ae [N1 0 N2 T 0 N3 0]TMITC and T MITC = Β D B A + Β s 33 )T Ds ΒMITC A = ΒTbb DbbBbb Aee + ((Β sMITC Ds Β ss Aee ) v Eh3 assembled Db = D v and with D = and F is the global load vector from the element load vectors from the element and F is the global load vector and assembled 12 − v2load vectors 0 (1 − v)/2 = ffee = NqdA òò NqdA Ae Ae Ds = kEh3 h2 + αh2e 2(1 + v) 0 (16) (13) (17) (17) (14) 2.2 ES-MITC3 ES-MITC3 plate plate element element 2.2 Differentiating Π in Eq (12) with respect to de and equating each term to zero to minimize Π, the In the theequilibrium edge-based smoothed FEM [5], strain fields fields are are averaged averagedon ondomains domainsofof discretized equations are obtains as[5], follows In edge-based smoothed FEM strain two adjacent adjacent elements elements Particularly, Particularly, the the edge-based edge-based smoothed smootheddomains domainsare aredefined definedby by two Kd = F (15) straight lines lines which which connect connect the the edge's edge's nodes nodes with with the the centroids centroids of of two two elements elements straight where d is the nodal displacements of the plate; K is the global stiffness matrix and assembled from sharing this this edge edge as as shown shown in Fig Therefore, the ES-MITC3 plate element [8] is the sharing the element stiffness matrices in Fig Therefore, the ES-MITC3 plate element [8] is the MITC3 one one in in which which the the strain strain fields fields given given by by Eqs Eqs (9) (9) and and (10) (10) are are smoothed smoothedasas MITC3 T MITC3 T MITC3 k = B D B dA + B D B dA b b s e follows s s b follows Ae Ae ! ì ee!x ü ü ìee x üü ! ì ì ! x x ü 11 B MITC3 ìtt üT D s B MITC3 Ae ïï ïï = BTììD ï !! ï ï ttxz ï bxzBüb A=e + sìí xzxzüý d b ! ! e = e d A ; ! í ý í ý í ý ! ! e = e d A ; = dAA s y ò í yy ý í t! ý ò ò ! í y ý ! ítt ý A A ! ! t ! yz yz A A ỵ yzthe Ak ï ïisg!thẹ ï ïg vector kk A kk AA ợợ yz ỵỵ ỵỵ element load vectors ùù k kk ợ and Fù xy ỵ global load ợg xy ợg xyxy ỵỵ and assembled from ợ þ ỵ ! fe = NqdA here Akk is the edge-based edge-based smoothed smoothed domain domain ofedge edge"k" "k" of (16) (18) (18) (17) Ae Using the relationships relationships between between the the strain strain fields fields and and nodal nodaldisplacements displacementsgiven given 2.2 ES-MITC3 plate element (10), the the smoothed smoothed strains strains fields fieldsin inEq Eq.(18) (18)can canbe beexpressed expressed by Eqs (9) and (10), Figure Edge-based smoothed Edge-based smoothed domains Figure 3.3.Edge-based smoothed domainsdomains for a plate discretized by 3-node elements for a plate discretized discretized by by 3-node 3-node elements elements Figure smoothed domains Figure 4.Node-based Node-based smoothed domains Figure 4.4 Node-based smoothed domains for a plate discretized by 3-node elements for foraaplate platediscretized discretizedby by3-node 3-nodeelements elements ! ü e x edge-based 11 vv 00 [7],ùùstrainN!! fields are éésmoothed Inìthe FEM averaged on domains of two adjacent eleEz ï !x ï êê úú 11 N ææ AAee 33 ee öö Ez (19) (19) ! e yy ý = v B ments.íParticularly, the edge-based smoothed domains are defined ! v B å 22 ỳỳ A ồỗỗ bibiddeieiữữby straight lines which connect the ê v ! e = i = è ø v A k è $### i =1 k e =sharing ïnodes ï with theêëêcentroids % edge’sï this edge g!xy ï 0 1of-two n elements ú "### %øas shown in Fig Therefore, the !!$### ợ ỵ 0 ((1 -n )) 2ỷỳỷ "### ợg xy ỵ BB dd ES-MITC3 plate element [10] is the MITC3 one in which the strain fields given by Eqs (9) and (10) !! !! are smoothed ì ü as follows E 11 NN ỉỉ A ưư ìtt xz Aee 3 B MITC E (20) xz ü = MITC3, 3,ee (20) ! ! í ý å å ! sisi d eiei÷ B d ỗ ớt!yz ý = 2(1 - ồ ỗ ữ v ) A x x e = i = è ø t kk e =1 è i = v ) A ỵ ứ "### # $#### % ợ yz ỵ þ 2(1 - 1 τ τ "### xz xz # !$#### dA; % = dA (18) = εy B! d εy B d τ τ yz yz γ Ak A k γ xy xy Ak Ak 66 49 ee k k bk k b e e k k sk k s Thanh, C D., et al / Journal of Science and Technology in Civil Engineering where Ak is the edge-based smoothed domain of edge “k” Using the relationships between the strain fields and nodal displacements given by Eqs (9) and (10), the smoothed strains fields in Eq (18) can be expressed ε Ne v x A Ez e e v B d = (19) ε y bi ei − v γ A e=1 i=1 (1 k 0 − ν)/2 xy k B b dk τ xz τyz E = 2(1 − v) A k Ae e=1 Ne i=1 B MITC3,e dei si (20) k B s dk in which Ak is the area of edge-based smoothed domain “k”; N e = for edge “k” on the boundary k k and N e = for the others; Bb , B s are respectively the gradient matrices of the in-plane and transverse shear smoothed strains; and dk is the nodal displacements related to the smoothed domain “k” Substituting Eqs (19) and (20) into the total potential energy in Eq (11) and following the standard FEM procedure, the equilibrium equations of the plate discretized by the ES-MITC3 elements are rewritten as Kd = F (21) where K is the smoothed global stiffness matrix and assembled from the edge-based smoothed stiffness matrices k T k T k k kk = Bb Db Bb Ak + B s D s B s Ak (22) 2.3 NS-MITC3 plate element According to the node-based smoothed FEM [7], strain fields are averaged on domains of elements sharing nodes These smoothed domains are defined by straight lines connecting the edges’ midpoints with the centroids of node-sharing elements as demonstrated in Fig As a result, the strain fields in Eqs (9) and (10) are smoothed on the node-based smoothed domains as follows [11] ε εx x 1 τ xz τ xz ε = dA; = dA (23) ε y y τyz τyz γ γ Al A l xy xy Al Al where Al is the smoothed domain of node “l” Substituting the strain – nodal displacement relations in Eqs (9) and (10) into the Eq (23), the node-based smoothed strains are rewritten εx Ne v A Ez e e v B d = (24) ε ei y bi − v2 0 (1 − ν)/2 Al e=1 i=1 γ xy l Bb dl 50 tandard FEM using procedure, the discretized equations the plate simulated Similarly, the expressions of the equilibrium nodal smoothed strains of in Eqs (24) and 25)the forNS-MITC3 the strain energy in the potential energy in Eq (11) and following the by elements cantotal be obtained tandard ! FEM procedure, the discretized equilibrium equations of the plate simulated Thanh, C D., et al / Journal of Science and Technology in Civil Engineering Kd = F by the NS-MITC3 elements can be obtained ! Ne (26 Ae E 1matrix τ xz stiffness MITC3,e here !K is the smoothed global and assembled from the (25) node-based = B d ei (26) Kd = F τyz 2(1 − v) A e=1 i=1 si l moothed stiffness matrices ! l here K is the smoothed global stiffness matrix and Bassembled from the node-based l sd ! !l T !l ! !l T !l ! moothed k l =stiffness ( Bb ) Dmatrices b Bb Al + ( B s ) Ds B s Al in which Al is the area of node-based smoothed domain “l”; N e and dl are respectively number of (27 l l ! !l T !l ! !l T !l ! elements andBthe nodal displacements belonging to the smoothed domain “l”; and Bb , B s are the (27) k = B D A + B D B A l b b b l s s s l 2.4 ES+NS-MITC3 element gradient matricesplate of the in-plane and transverse shear smoothed strains, respectively ( ) ( ) Similarly, using the expressions of the nodal smoothed strains in Eqs (24) and (25) for the strain 2.4 ES+NS-MITC3 plate In the ofelement combining ES-following and the NS-FEM, theprocedure, strain thefields energyapproach in the total potential energy in Eq the (11) and standard FEM dis- of the cretized element equilibrium equations of the plate simulated by theportion NS-MITC3of elements edge-based can be obtained MITC3 plate now smoothed In the approach of are combining the ES- on and aNS-FEM, thethe strain fields of smoothed the Kd = (26) 5(b) To domains and element the otherare of now the node-based smoothed as edge-based illustrated in Fig MITC3 plate smoothed on a Fportionones of the smoothed domains the otherdomains of the node-based onesNS-ones, as illustrated Fig 5(b).edge To ed i build theand smoothed including smoothed the ES- and eachinelement's where K is the smoothed global stiffness matrix and assembled from the node-based smoothed stiffness matrices build theinto smoothed domains theT with ES- and each element's edge ed is divided segments as including in Fig 5(b) the NS-ones, ratio l l l T l Bb Alratio + Bs divided into segments as in Fig k5(b) l = Bwith b Dbthe ed L ed L1ed = Led ; Led = bLed = (1 - b ) L ES+NS-MITC3 element ed 2; Ledplate L1ed =2.4 Led = b = (1 - b ) L (27) D s B s Al (28) (28 In the approach of combining the ES- and NS-FEM, the strain fields of the MITC3 plate element ed ed ed here Led are b Ỵ [0,1] is a scale factordomains used and to the tune the contribution of now+ smoothed of the edge-based smoothed other of the node-based =L L +edLon3 a; portion ed ed1 ed here L =smoothed ; b Ỵ [0,1] a scale factor used todomains tune the contribution L1 + Lones L3illustrated in Fig.is5(b) To build the smoothed including the ES- and of NS-the + as node-based smoothed domains in the ES+NS-domains It means that if b = 0, ones, each element’s edge ed is divided into segments as in Fig 5(b) with the ratio the the node-based smoothed domains in the ES+NS-domains It means that if b = 0, the ES+NS-domains become the edES-domains, and if b ed= 1, the ES+NS-domains are Led (1 −=β)1, ES+NS-domains become the ES-domains, ; L2edif= b L the ES+NS-domains (28)are L1 = L3ed = β and purely NS-ones.This Thisapproach approach is also called the bFEM [10,11] purely NS-ones is also called the b FEM [10,11] Figure (a) (a) (a) (b) (b) (b) Figure (a) Edge and node-based smoothed domains for a plate discretized by 3-node elements; (a) and node-based smoothed domains forsmoothed a plateareas discretized 3-node (b) Edge Definition of the ES- (line hatching) and the NS(dot hatching) of a triangularby element Figure (a) Edge and node-based smoothed domains for a plate discretized by 3-node elements; (b) Definition of the ES- (line hatching) and the NS- (dot hatching) 51 elements; (b) Definition of the ES(line hatching) and the NS- (dot hatching) smoothed areas of a triangular element smoothed areas of a triangular element ed ed ed From the middle segments L2 and the end segments L1 and L3 , the ES-domains and Thanh, C D., et al / Journal of Science and Technology in Civil Engineering where Led = L1ed + L2ed + L3ed ; β ∈ [0, 1] is a scale factor used to tune the contribution of the node-based smoothed domains in the ES+NS-domains It means that if β = 0, the ES+NS-domains become the ES-domains, and if β = 1, the ES+NS-domains are purely NS-ones This approach is also called the βFEM [12, 13] From the middle segments L2ed and the end segments L1ed and L3ed , the ES-domains and NSdomains are respectively constructed for elements having common edges and nodes to have the smoothed areas of Aˆ k = β2 Ak ; A˜ l = − β2 Al (29) Consequently, the strain fields in Eqs (9) and (10) averaged on the ES+NS-domains are determined by εˆ x εˆ y γˆ xy ε˜ x ε˜ y γ˜ xy = Aˆ k = A˜ l Aˆ k A˜ l εx εy γ xy εx εy γ xy dA; τˆ xz τˆ yz dA; τ˜ xz τ˜ yz Aˆ k = A˜ l = τ xz τyz dA (30) τ xz τyz dA (31) Aˆ k A˜ l Using Eq (29) and substituting Eqs (9), (10) into Eqs (30), (31), the relationships between ESand NS-strain fields and the nodal displacements in the ES+NS-MITC3 plate element can be derived εˆ x εˆ y γˆ xy Ez = − v2 v v 0 (1 − ν)/2 β2 A Ne k e=1 Ae β2 3 i=1 Bebi dei (32) k Bb dk τˆ xz τˆ yz E = 2(1 − v) β2 A k Ne e=1 Ae β2 i=1 B MITC3,e dei si (33) k B s dk ε˜ x ε˜ y γ˜ xy Ez = − v2 v v 0 (1 − ν)/2 − β2 Al − β2 Ae e=1 Ne i=1 Bebi dei (34) l B b dl τ˜ xz τ˜ yz E = 2(1 − v) − β2 A l − β2 Ae e=1 Ne i=1 B MITC3,e dei si (35) l B s dl And then, from the total potential energy expressed in the smoothed strain fields given in Eqs (32)–(35), the equilibrium equations of the plates discretized by ES+NS-MITC3 elements can be written KES +NS d = F (36) 52 ! k" l = Bbl ( ) T ! ! " + Bl DbBbl A l s ( ) T ! ! " = (1 - b ) k Ds Bls A l l Numerical examples Thanh, C D., et al.In / Journal of Science the and Technology Civil Engineering this section, accuracyin and convergence of the ES+NS-MITC via the global patch stiffness test and some plate where K is the edge- be and evaluated node-based smoothed matrix and benchmark assembled from the problem smoothed stiffness matricesprovided by the ES+NS-MITC3 element are compared with similar kin k T k k like ES-DSG3 [6],k TMITC3 [4], ES-MITC3 [8] and NS-MITC3 [ kˆ k = Bb Db Bb Aˆ k + B s D s B s Aˆ k = β2 kk (37) examples, we choose the scale factor b = 0.6 To compare with r l T l l T and moments atl Athe are normalized by ˜kdeflection ˜ ˜ = plate = B D B A + B DB − βcenter k (38) ES +NS l b b b l wc = wc Numerical examples s s s l l 100 D 10 ; Mc = Mc qL qL In this section, the accuracy and convergence of the ES+NS-MITC3 element will be evaluated 3.1 Patch test via the patch test and some benchmark plate problems The results provided by the ES+NS-MITC3 Consider patch test a 0.01 [8], m-thick rectangular plate with the element are compared with similar kinds ofaelements likebe ES-DSG3 MITC3 [6], ES-MITC3 [10] and NS-MITC3 [11] In all m the´examples, the scale factorEβ == 10 0.6.7 kN/m To compare 0.24 0.12 m, we thechoose Young's modulus andwith the Poisson's references, the deflection and moments at the plate center are normalized by The plate is discretized by 3-node triangular elements as in Fig 100D of ¯the plate 10 deflection + y2 ) / 200 m w¯ c = equation wc ; Mc = Mc 2w = (1 + x + 2y + x + xy (39) qL can reproduce qL the deflection and moments at node MITC3 element Table It means that the ES+NS-MITC3 plate element passes the patch 3.1 Patch test Consider a patch test be a 0.01 m-thick rectangular plate with the dimension of 0.24 m × 0.12 m, the Young’s modulus E = 107 kN/m2 and the Poisson’s ratio v = 0.25 The plate is discretized by 3-node triangular elements as in Fig [8] With the deflection equation of the plate w = (1 + x + 2y + x2 + xy + y2 )/200 m, the ES+NSMITC3 element can reproduce the deflection and moments at node as shown in Table It means that the ES+NS-MITC3 plate elementFigure passes theNodal coordinates Figure Nodal coordinates of elements for the patc of elements discretized discretized for the patch test patch test Table Deflection and moments of the patch test at node Table Deflection and moments of the patch test at node Methods w5 (×10−2 m) θ x5Methods (×10−2 rad.) ES+NS-MITC3+ Exact solution 0.6422 0.6422 1.1300 1.1300 ES+NS-MITC3+ Exact solution w5 θy5 (×10−2 rad.) -2 (´10 −0.6400 −0.6400 qx5 M x5 (kNm/m) qy5 My5 (kNm/m) Mx5 M xy5 (kNm/m) -2 m) (´10 (´10-2 rad.) −0.0111rad.) −0.0111 0.6422 0.6422 3.2 Simply supported plate under uniform distributed loading −0.0111 1.1300 1.1300 −0.0111 -0.6400 -0.6400 (kNm/m) −0.0033 −0.0033 My5 (kNm/ -0.0111 -0.01 -0.0111 -0.01 3.2 Simply loading A square plate of the length L and supported the thicknessplate h is under simply uniform supporteddistributed on the boundary and subjected to the uniform loading q = kN/m as illustrated in Fig The material properties are E A square plate of the length L and the thickness h is simply sup = 1092000 kN/m2 and v = 0.3 The plate is modelled by × N × N triangular elements in which N is and subjected to the uniform loading q = kN/m2 as illustr number of elements on eachboundary edge thin plate with The accuracy and convergence of the ES+NS-MITC3 are studied for the The material properties areelement E = 1092000 kN/m and n = 0.3 The plate the ratio h/L = 0.001 and the thick one with h/L = 0.1, and the meshes of N = 4, 8, 12, and 16 53 10 Tạp chí Khoa học Công nghệ Xây dựng NUCE 2019 C D., etinal.which / Journal N of Science and Technology in Civil Engineering 2´N´N triangularThanh, elements is number of elements on each edge Figure plate simply simply supported supported on on all all edges edges and and subjected subjectedto touniform uniform Figure 7 Square Square plate distributed loading, and and regularly regularly meshed meshed by by NN == 44 on on each eachplate's plate'sedge edge distributed loading, The and convergence convergence of of the the ES+NS-MITC3 ES+NS-MITC3 element element are are studied studied for for The accuracy accuracy and the the ratio ratio h/L h/L == 0.001 0.001 and and the the thick thick one one with with h/L h/L == 0.1, 0.1, and and the the the thin thin plate plate with with the meshes 8, 12, 12, and and 16 16 meshes of of N N= = 4, 4, 8, Figure Square plate simply supportedat on the all edges subjected to uniform loading, and The normalized deflections plateandcenter provided bydistributed the ES+NS-MITC3 The7.normalized deflections at the plateoncenter provided subjected by the ES+NS-MITC3 Figure Square plate simplymeshed supported edges to uniform regularly by N = onall each plate’sand edge element for the ratio h/L = 0.001 and and h/L h/L == 0.1 0.1 are are demonstrated demonstrated in in Fig Fig 8.8 In In both both distributed loading, and regularly meshed by N = on each plate's edge cases of the plate thickness, theplate convergence curve given by the the ES+NS-MITC3 ES+NS-MITC3 The normalized deflections at the center provided by given the ES+NS-MITC3 element for the convergence curve by ratio h/L = 0.001 andand h/L = 0.1ofare in Fig In both elements cases of the platestudied thickness, The accuracy convergence of the ES+NS-MITC3 element are element lies between those thedemonstrated ES-MITC3 and NS-MITC3 elements Therefore, thefor ES-MITC3 and NS-MITC3 Therefore, the thedeflection convergence curve given by the ES+NS-MITC3 element lies between those of the ES-MITC3 the the ES+NS-MITC3 element approaches to one the analytical analytical solution [14]the the thin plate ofwith ratio h/L = element 0.001 and the thick with h/L solution = 0.1, and approaches to the [14] and NS-MITC3 elements Therefore, the deflection of the ES+NS-MITC3 element approaches to the moreofrapidly ES-MITC3 and NS-MITC3 elements.elements However, the those of ES-MITC3 NS-MITC3 elements However, the meshes N = 4, than 8, 12, andrapidly 16.thethan analytical solution [16] more those of theand ES-MITC3 and NS-MITC3 However, and NS-domains does not improve improve theofresults results ofas theESESNS-domains does not the of theproposed proposed combination combination ofofthe andand NS-domains does not improve the results moments The normalized deflections at the plate center provided by the ES+NS-MITC3 illustrated in as Fig.illustrated moments in Fig element for the ratio h/L = 0.001 and h/L = 0.1 are demonstrated in Fig In both cases of the plate thickness, the convergence curve given by the ES+NS-MITC3 element lies between those of the ES-MITC3 and NS-MITC3 elements Therefore, the deflection of the ES+NS-MITC3 element approaches to the analytical solution [14] more rapidly than those of the ES-MITC3 and NS-MITC3 elements However, the proposed combination of the ES- and NS-domains does not improve the results of moments as illustrated in Fig (a) h/L =Tạp 0.001chí h/L = 0.1 2019 Tạp chí Khoa học Cơng nghệ Xây dựng NUCE (b) (a) h/L = 0.001 (b) (b) h/L h/L==0.1 0.1 Figure Convergence of the normalized deflections at the center of the simply supported plates under uniform distributed loading deflections at Figure Convergence of the normalized normalized deflections at the the center center of ofthe thesimply simplysupported supported plates under uniform uniform distributed distributed loading loading (a) h/L = 0.001 11 11 (b) h/L = 0.1 Figure Convergence of the normalized deflections at the center of the simply supported (a) h/L =plates 0.001 (b) h/L = 0.1 (a) h/L h/L == 0.001 under uniform distributed loading (b) h/L = 0.1 (a) Figure Convergence of the normalized moments at the center of the simply supported plates under uniformmoments distributed loading Figure 9 Convergence Convergence of the normalized Figure at the center of the simply supported 54 distributed loading plates under uniform 3.3 Simply Simply supported supported Morley plate subjected to uniform distributed loading 3.3 Consider the the rhombus Morley plate [15] of the length L = 100 cm and the Consider 3.3 Simply supported Morley plate subjected to uniform distributed loading Consider the rhombus Morley plate [15] of the length L = 100 cm thickness h = cm as shown in Fig 10 The plate is simply supported on all Thanh, C D., et al / Journal of Science and Technology in Civil Engineering and subjected uniform distributed loading q = 0.1 N/cm2 The Young's mod uniform distributed loading 3.3 Simply supported Morley plate subjected 109200 N/cmto and the Poisson's ratio n is 0.3 Consider the rhombus Morley plate [17] of the length L = 100 cm and the thickness h = cm as shown in Fig 10 The plate is simply supported on all the edges and subjected uniform distributed loading q = 0.1 N/cm2 The Young’s modulus E is 109200 N/cm2 and the Poisson’s ratio v is 0.3 The Morley plate is discretized by different meshes of N = 4, 8, 12, and 16, in which N is the number of elements on each edge of the plate (Fig 10) The normalized deflections and(a) moFigure 10 Geometry,Figure uniform loading, and simply supported 10 distributed Geometry, uniform distributed ments at the plate center provided by the proposed loading, and simply supported boundary of the Morley plate with a mesh of Nof=the element and the other reference ones are compared Morley plate with a mesh of N = in Fig 11 and Fig 12, respectively As shown in plate is discretized by different meshes of N = 4, 8, 12, a The Morley these figures, the results of the ES+NS-MITC3 element average values of those given the(Fig ES- 10) The n which N is the number are of elements on each edge of the by plate Tạp chí Khoa học Cơng nghệ Xây dựng NUCE 2019 MITC3 and NS-MITC3 elements The deflection of the ES+NS-MITC3 element well converge the Tạp chí deflections Khoa học Công Xây dựng NUCE andnghệ moments at the plate2019 center provided by thetoproposed eleme reference solution [17] However, the accuracy and convergence of the moment given by the ES+NSother reference ones are compared in Fig 11 and Fig 12 respectively As MITC3 are not good due to the bad results provide by the NS-MITC3 element In this case, we can these figures, results of reduce the ES+NS-MITC3 value tune the scale factor β to be nearly equal 1.0 tothe dramatically the overly softelement behaviorare of average the node-based smoothed approach node-based smoothed approach given by the ES-MITC3 and NS-MITC3 elements The deflection of the node-based smoothed approach MITC3 element well converge to the reference solution [15] However, the and convergence of the moment given by the ES+NS-MITC3 are not good bad results provide by the NS-MITC3 element In this case, we can tune factor b to be nearly equal 1.0 to dramatically reduce the overly soft behav 12 Figure 11 Convergence of the normalized deflections at the center ofof thethe Morley plate Figure 11 Convergence normalized Figure 12 Convergence of the normalized moments at the center of the Morley normalized Figure 12 Convergence of theplate normalized deflections at the center of the Morley plate moments at the center of the Morley Morley plate plate 3.4 Clamped circular plate under uniform distributed loading 3.4 Clamped circular plate Give a circular plate with theunder radiusuniform R = mdistributed clamped onloading its circumference and subjected to uniformGive distributed loading q =with kN/m as shown Fig 13(a) The plate thickness h with theand ratio a circular plate the radius R= 5m clamped on its circumference circumference and h/R = 0.02 and h/R = 0.2 are studied The isotropic homogeneous material of the plate has E = subjected to uniform distributed loading q = kN/m22 as shown Fig 13(a) The The plate plate 1092000 kN/m2 , v = 0.3 thickness h with the ratio ofh/R 0.02is meshed and h/Rby=6, 0.2 areorstudied The asisotropic isotropic Due to symmetry, a quarter the =plate 24, 54 96 elements shown in Fig 13(b) The deflections moments at the plate center solved ES+NS-MITC3 and other homogeneous material and of the plate has E= 1092000 kN/m 2by , nthe = 0.3 reference elements are respectively demonstrated in Fig 14 and Fig 15 Numerical results show that the hybrid model of the ES+NS-MITC3 element can reduce the overly soft behaviors of the NSMITC3 element and the overly stiff behaviors of the ES-MITC3 to rapidly approach the reference solutions [1] for both thin (h/R = 0.02) and thick (h/R = 0.2) plates 55 Give a circular plate with the radius R = m clamped on its circumference and a circular plate with radius2 Ras=shown m clamped on its The circumference and subjected to uniformGive distributed loading q =the kN/m Fig 13(a) plate subjected to uniform distributed loading q = kN/m as shown Fig 13(a) The plate thickness h with the ratio h/R = 0.02 and h/R = 0.2 are studied The isotropic thickness h with the ratio h/R = 0.02 and 2h/R = 0.2 are studied The isotropic homogeneous material ofThanh, the plate EJournal = 1092000 kN/m , n = 0.3 C D., ethas al of Science and1092000 Technology in Civil Engineering homogeneous material of/ the plate has E = kN/m , n = 0.3 Tạpchí chíKhoa Khoahọc họcCơng Cơngnghệ nghệXây Xâydựng dựngNUCE NUCE2019 2019 Tạp Tạp Tạpchí chíKhoa Khoahọc họcCơng Côngnghệ nghệXây Xâydựng dựngNUCE NUCE2019 2019 both thin thin (h/R (h/R == 0.02) 0.02) and and thick thick (h/R (h/R==0.2) 0.2)plates plates both (a) (a) (b) (b) (a) (b)circular plate; Figure 13 (a) Geometry and loading of the clamped both (h/R = 0.02) and thick (h/R = 0.2) plates Figure 13 (a) Geometry and loading of the clamped circular plate boththin thin (h/R = 0.02) and thick (h/R = 0.2) plates (b) A quarter of the plate discretized by 24 triangular elements and symmetric boundaries Figure (a) Geometry anddiscretized loading of clamped circular (b) 13 A quarter of the plate bythe 24 triangular elementsplate and symmetric boundaries (b) A quarter of the plate discretized by 24 triangular elements and symmetric boundaries Due to symmetry, a quarter of the plate is meshed by 6, 24, 54 or 96 elements as shown inaFig 13(b) deflections and moments theorplate center solved Due to symmetry, quarter of The the plate is meshed by 6, 24,at54 96 elements as by the ES+NS-MITC3 and otherand reference elements respectively demonstrated shown in Fig 13(b) The deflections moments at theareplate center solved by thein Fig 14 Numerical results show that the hybrid model of the ES+NS-MITC3 ES+NS-MITC3 and andFig other15.reference elements are respectively demonstrated in Fig 14 element can reduce the overly soft behaviors of the NS-MITC3 element and and Fig 15 Numerical results show that the hybrid model of the ES+NS-MITC3the overly stiff behaviors of the ES-MITC3 to rapidly approach the reference solutions [1] for element can reduce the overly soft behaviors of the NS-MITC3 element and the overly (a) h/R h/R == 0.02 0.02 (b) h/R==0.2 0.2 stiff behaviors of the (a) ES-MITC3 to rapidly approach13the reference (b) solutions [1] for h/R h/R = 0.02 at the center of the clamped circular (b) plate h/R = 0.2 Figure 14 14 (a) (a)(a)Deflections Deflections correspondingtoto Figure center of the clamped circular plate (a)h/R h/R==0.02 0.02at the 13 (b)h/R h/R==corresponding 0.2 (a) (b) 0.2 Figure 14 Deflections the centerof the24, clamped circular plate corresponding to differentatmeshes meshes 6, 54and and 96elements elements different ofof6, 24, 54 96 different meshes ofofof 6,the 24, and 96 elements Figure14 14.(a) (a)Deflections Deflections the center the54 clamped circularplate platecorresponding correspondingtoto Figure atatthe center clamped circular differentmeshes meshesofof6,6,24, 24,54 54and and96 96elements elements different (a) h/R = 0.02 (b) h/R = 0.2 (a) h/R h/RMoments 0.02 at the center of the clamped circular plate(b) (b) h/R==0.2 0.2 to (a) == 0.02 h/R Figure 15 corresponding different meshes of 6, 24, 54 and 96 elements Figure 15 15 (a) (a) Moments at the the center centerof ofthe theclamped clampedcircular circular plate Figure at toto (a)Moments h/R==0.02 0.02 (b)plate h/R=corresponding =corresponding 0.2 (a) h/R (b) h/R 0.2 different meshes meshesof of6, 6,24, 24,54 54and and96 96elements elements different Conclusions Figure15 15.(a) (a)Moments Momentsatatthe thecenter centerofofthe theclamped clampedcircular circularplate platecorresponding correspondingtoto Figure 4.The Conclusions Conclusions different meshes 24,54 54and and96 96elements elements ofof6, βFEM, which isdifferent the hybridmeshes approach of6,24, the edge-based and node-based smoothed strains, has been developed for the 3-node triangular MITC3 plate elements The suggested ES+NS-MITC3 Conclusions The bbFEM, FEM, which which isis the the hybrid hybrid approach approach of of the the edge-based edge-based and and node-based node-based The 4.4.Conclusions 56 smoothed strains, has beenisdeveloped developed forapproach the3-node 3-nodetriangular triangular MITC3and plate elements smoothed been for the MITC3 plate elements Thestrains, FEM,has which thehybrid hybrid the edge-based edge-based node-based The bbFEM, which is the approach ofofthe and node-based The suggested ES+NS-MITC3 element passes the test attenuates the The suggested ES+NS-MITC3 elementfor passes the patch patch testand and attenuates theshearshearsmoothed strains, hasbeen beendeveloped developed the3-node 3-node triangular MITC3 plateelements elements smoothed strains, has for the triangular MITC3 plate locking phenomenon The analyses of benchmark problems show locking phenomenon The static staticelement analyses of some some benchmark plate problems show Thesuggested suggested ES+NS-MITC3 passes thepatch patch testand andplate attenuates theshearshearThe ES+NS-MITC3 element passes the test attenuates the Thanh, C D., et al / Journal of Science and Technology in Civil Engineering element passes the patch test and attenuates the shear-locking phenomenon The static analyses of some benchmark plate problems show that the ES+NS-MITC3 element can reduce the overly stiff and soft behaviors of the purely ES-MITC3 and NS-MITC3 elements respectively As a result, the ES+NS-MITC3 element improves the accuracy of the plate deflections and moments as compared with the ES-MITC3 and NS-MITC3 elements, especially in the cases of coarse meshes Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.304 References [1] Timoshenko, S P., Woinowsky-Krieger, S (1959) Theory of plates and shells McGraw-hill [2] Hai, L T., Tu, T M., Huynh, L X (2018) Vibration analysis of porous material plate using the first-order shear deformation theory Journal of Science and Technology in Civil Engineering (STCE) -NUCE, 12 (7):9–19 [3] T, T M., Quoc, T H., Tham, V V (2018) Analytical solutions for the static analysis of laminated composite plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory Journal of Science and Technology in Civil Engineering (STCE) - NUCE, 12(4):40–50 [4] Tessler, A., Hughes, T J R (1985) A three-node Mindlin plate element with improved transverse shear Computer Methods in Applied Mechanics and Engineering, 50(1):71–101 [5] Bletzinger, K.-U., Bischoff, M., Ramm, E (2000) A unified approach for shear-locking-free triangular and rectangular shell finite elements Computers & Structures, 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D (2016) Static and frequency analysis of laminated composite plates using the MITC3 elements having the strains averaged on node-based domains (NS-MITC3) In Proceedings of the National Science Conference on "Composite Materials and Structures: Mechanics, Technology and Application", 613–620 [12] Wu, F., Liu, G R., Li, G Y., Cheng, A G., He, Z C (2014) A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates Computational Mechanics, 54(3):865–890 [13] Wu, F., Zeng, W., Yao, L Y., Hu, M., Chen, Y J., Li, M S (2018) Smoothing Technique Based Beta FEM (β FEM) for Static and Free Vibration Analyses of Reissner–Mindlin Plates International Journal of Computational Methods, page 1845006 [14] Logan, D L (2011) A first course in the finite element method Thomson [15] Lyly, M., Stenberg, R., Vihinen, T (1993) A stable bilinear element for the Reissner–Mindlin plate model Computer Methods in Applied Mechanics and Engineering, 110(3-4):343–357 [16] Taylor, R L., Auricchio, F (1993) Linked interpolation for Reissner-Mindlin plate elements: Part II—A simple triangle International Journal for Numerical Methods in Engineering, 36(18):3057–3066 [17] Morley, L S D (1963) Skew plates and structures Pergamon Press 57 ... mid-surfaceof of the the element and its positive directionsdirections of displacements of the element and its positive of Reissner- Mindlin plate Reissner- Mindlin plate the nodal displacements Reissner- Mindlin. .. efficiency in the static analysis of the Reissner- Mindlin plates The paper is organized as follows In the next section, the finite element formulae of the MITC3, ES-MITC3, NS-MITC3 elements are briefly... NS-MITC3 elements Therefore, the deflection of the ES+NS-MITC3 element approaches to the moreofrapidly ES-MITC3 and NS-MITC3 elements. elements However, the those of ES-MITC3 NS-MITC3 elements