DSpace at VNU: A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates

Comput Mech (2010) 46:679–701 DOI 10.1007/s00466-010-0509-x ORIGINAL PAPER A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates H Nguyen-Xuan · T Rabczuk · N Nguyen-Thanh · T Nguyen-Thoi · S Bordas Received: February 2010 / Accepted: June 2010 / Published online: 23 June 2010 © Springer-Verlag 2010 Abstract In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation A so-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed Several numerical examples are used to illustrate the accuracy and effectiveness of the present method Keywords Plate bending · Transverse shear locking · Finite element method · Node-based smoothed H Nguyen-Xuan (B) · T Nguyen-Thoi Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam e-mail: nxhung@hcmuns.edu.vn URL: http://www.math.hcmuns.edu.vn/∼nxhung H Nguyen-Xuan · T Nguyen-Thoi Division of Computational Mechanics, Faculty of Civil Engineering, Ton Duc Thang University, 98 Ngo Tat To, Binh Thanh District, Ho Chi Minh City, Vietnam T Rabczuk · N Nguyen-Thanh Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, Marienstr 15, 99423 Weimar, Germany S Bordas School of Engineering, Institute of Theoretical, Applied and Computational Mechanics, Cardiff University, Wales, UK finite element · Discrete shear gap (DSG) · Stabilization technique Introduction Static, free vibration and buckling analyses of plate structures play an increasing important role in engineering applications A large amount of research work on plates can be found in the literature reviews [1–5] The analytical solution approaches are restricted to plates with simple shapes Effective numerical methods such as finite difference techniques, splinestrip element methods, the finite element method (FEM), and meshfree methods, etc., have been devised to analyze and simulate the behavior of plates Among these numerical approaches, the FEM is still so far the most popular and reliable tool During the last three decades, lower-order Mindlin– Reissner plate finite elements have often been preferred due to their simplicity and efficiency They require only C -continuity for the deflection and the normal rotations However, these low-order plate elements in the thin plate limit often suffer from shear locking phenomenon due to incorrect transverse forces under bending Therefore, many formulations have been developed to overcome the shear locking phenomenon and to increase accuracy and stability of numerical methods such as mixed formulation/hybrid elements [6–9], stabilization methods [10,11], the enhanced assumed strain (EAS) methods [12,13], the assumed natural strain (ANS) methods [14–17], etc Recently, the discrete shear gap (DSG) method [18] which can be considered as an alternative form of the ANS was proposed The DSG is somewhat similar to the ANS methods in the aspect of modifying the course of certain strains within the element, but is different in the aspect of lack of collocation points The DSG 123 680 method is therefore independent of the order and form of the element On an other front of development of finite element technology, Liu et al have combined the strain smoothing technique [19] used in meshfree methods [20–27] into the finite element method using quadrilateral elements to formulate a cell/element-based smoothed finite element method (SFEM or CS-FEM) [28–30] for 2D solids It is well known that loworder elements for solid problems certain inherent drawbacks such as overestimation of the stiffness matrix and locking problems Therefore applying the strain smoothing technique on smoothing domains to these standard FEM models aims to soften the stiffness formulation, and hence can improve significantly the accuracy of solutions in both displacement and stress In CS-FEM, the smoothing domains are created based on elements, and each element can be subdivided into or several quadrilateral smoothing domains The theoretical aspects of CS-FEM were fully studied in [29,31,32] The SFEM has also been extended to general n-sided polygonal elements (nSFEM) [33], dynamic analysis [34,35], plate and shell analysis [36,37], kinematic limit analysis [38] and coupled to partition of unity enrichment [39] A general framework for this strain smoothing technique in FEM can be found in [40,41] In the effort to overcome shortcomings of low-order elements, Liu et al have then extended the concept of smoothing domains to formulate a family of smoothed FEM (S-FEM) models with different applications such as the node-based S-FEM (NS-FEM) [42,43], edge-based S-FEM (ES-FEM) [44–49], face-based S-FEM (FS-FEM) [50,51] Similar to the standard FEM, these S-FEM models also use a mesh of elements In these S-FEM models, the discrete weak form is evaluated using smoothed strains over smoothing domains instead of using compatible strains over the elements as in the traditional FEM The smoothed strains are computed by integrating the weighted (smoothed) compatible strains The smoothing domains are created based on the features of the element mesh such as nodes [42], or edges [44] or faces [50] These smoothing domains are linear independent and hence stability and convergence of the S-FEM models are ensured They cover parts of adjacent elements, and therefore the number of supporting nodes in smoothing domains is larger than that in elements This leads to bandwidth of the stiffness matrix in S-FEM models increased and the computational cost is hence higher than those in the FEM However, due to contribution of more supporting nodes in the smoothing domains, S-FEM often produces the solution that is much more accurate than that of the FEM Therefore in general, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM models are more efficient than the counterpart FEM models [43,46,47,51] It can be argued that these S-FEM models have the features of 123 Comput Mech (2010) 46:679–701 both models: meshfree and FEM The element mesh is still used but the smoothed strains bring the non-local information from the neighboring elements A general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models was given in [32] Among these S-FEM models, the NS-FEM [42,43] shows some interesting properties in the elastic solid mechanics such as: 1) it can provide an upper bound to the strain energy; 2) it can avoid volumetric locking without any modification on integration terms; 3) super-accurate and super-convergent properties of stress solutions are gained; and 4) the stress at nodes can be computed directly from the displacement solution without using any post-processing In this paper, we exploit several interesting properties of the NS-FEM for analyzing plates The NS-FEM has been already extended to perform adaptive analysis [52], linear elastostatics and vibration 2D solid problems [53] Also, alpha finite element methods (αFEM) have been recently proposed have been recently proposed as an alternative to the NS-FEM and shown to significantly improve the results obtained by conventional and smoothed FEM techniques, at the cost of the introduction of a problem-dependent parameter α [54–56] This paper presents a formulation of the node-based smoothed finite element method (NS-FEM) for Reissner– Mindlin plates using only three-node triangular meshes which are easily generated for complicated domains The evaluation of the discrete weak form is performed by using a strain smoothing technique over smoothing domains associated with nodes of elements Transverse shear locking can be avoided through the discrete shear gap (DSG) method The stability and accuracy of NS-FEM formulation is further improved by a stabilization technique to give a so-called node-based smoothed finite element method with a stabilized discrete shear gap method (NS-DSG) Several numerical examples are presented to demonstrate the reliability and effectiveness of the present method The layout of the paper is as follows Next section describes the weak form of governing equations and the formulation of 3-node plate element In Sect 3, a formulation of NS-FEM with the stabilized discrete shear technique is introduced Section recalls some techniques relevant to the present approach Section presents and discusses numerical results Finally, we close our paper with some concluding remarks The formulation of 3-node plate element 2.1 Discrete governing equations We consider a domain Ω ⊂ R2 occupied by the reference middle surface of a plate Let w and β = (βx , β y )T be the transverse displacement and the rotations about the y and x Comput Mech (2010) 46:679–701 681 Fig a 3-Node triangular element; b Local coordinates axes, see Fig 1, respectively We assume that the material is homogeneous and isotropic with Young’s modulus E and Poisson’s ratio ν, the governing differential equations of the static and dynamic Mindlin–Reissner plates can be expressed in the following form [57] For the free vibration analysis of a Mindlin–Reissner plate model, a weak form may be derived from the dynamic form of the principle of virtual work under the assumptions of first order shear-deformation plate theory [58]: find ω ∈ R+ and = (w, β) ∈ V such that ρt ω β = in Ω 12 kt∇ · γ + ρtω2 w = p in Ω (2) ∀(v, η) ∈ V0 , a(β, η) + kt (∇w − β, ∇v − η) = ω2 ρt (w, v) + ρt (β, η) 12 w = w0 , β = β on Γ = ∂Ω (3) ∇ · Db κ(β) + ktγ + (1) where t is the plate thickness, ρ is the mass density, ω is the natural frequency, p = p(x, y) is the transverse loading per unit area, k = μE/2(1 + ν), μ = 5/6 is the shear correction factor and Db is the tensor of bending modulus given by ⎤ ⎡ ν Et b ⎣ν ⎦ D = (4) 12 − ν 0 1−ν [∇β + (∇β)T ], γ = ∇w − β (5) (7) wv dΩ, (β, η) = Ω β · η dΩ, Ω κ(β) : D : κ(η) dΩ a(β, η) = {(∇βx )T σˆ ∇ηx + (∇β y )T σˆ ∇η y } dΩ Ω (6) with B denotes a set of the essential boundary conditions and the L inner products are given as (w, v) = b where σˆ = σ 0x σ 0x y σ 0x y σ 0y (11) Let us assume that the bounded domain Ω is discretized into Ne Ne finite elements such that Ω ≈ e=1 Ω e and Ω i ∩ Ω j = ∅ , i = j The finite element solution of the static problem of a low-order1 element model for the Mindlin–Reissner plate is to find (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , η)+kt (∇w h −β h , ∇v−η) = ( p, v) Ω The weak form of the static equilibrium equations is: find (w, β) ∈ V such that ∀(v, η)∈V0 , a(β, η)+kt (∇w − β, ∇v − η)=( p, v) (10) (∇w)T σˆ ∇v dΩ, b1 (w, v) = b2 (β, η) = V0 = {(v, η) : v ∈ H (Ω), η ∈ H (Ω)2 : v = 0, η = on ∂Ω} ∀(v, η) ∈ V0 , a(β, η) + kt (∇w − β, ∇v − η) = λcr tb1 (w, v) + t b2 (β, η) 12 Ω where ∇ = (∂/∂ x, ∂/∂ y)T is the gradient vector Let V and V0 be defined as V = {(w, β) : w ∈ H (Ω), β ∈ H (Ω)2 } ∩ B In the case of in-plane buckling analyses and assuming prebuckling stresses σˆ , nonlinear strains appear and the weak form can be reformulated as [58]: find ω ∈ R+ and = (w, β) ∈ V s.t where The bending κ and shear strains γ are defined as κ= (9) (8) (12) In our case a three-node triangular linear finite element 123 682 Comput Mech (2010) 46:679–701 where the finite element spaces, V h and V0h , are defined by V h = {(w h , β h ) ∈ H (Ω) × H (Ω)2 , w h |Ω e ∈ P1 (Ω e ), β h |Ω e ∈ P1 (Ω e )2 } ∩ B V0h (13) = {(v , η ) ∈ H (Ω) × H (Ω) : v = 0, h h 1 h ηh = on ∂Ω} (14) where P1 (Ω e ) stands for the set of polynomials of degree for each variable The finite element problem of the free vibration modes is to find the natural frequency ωh ∈ R+ and = (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , η) + kt (∇w h − β h , ∇v − η) (15) = (ωh )2 ρt (w h , v) + ρt (β h , η) 12 and for the buckling analysis is to find the critical buckling h ∈ R+ and = (w h , β h ) ∈ V h load λcr ∀(v, η) ∈ V0h , a(β h , η) + kt (∇w h − β h , ∇v − η) h tb1 (w h , v) + t b2 (β h , η) = λcr 12 (16) Since only linear triangular elements are used to obtain stiffness matrices, the finite Reissner–Mindlin plate-bending element approximation is simply interpolated using the linear basis functions for both deflection and rotations without any additional variables (C -continuity for the transverse displacement and the normal rotations) Hence, the bending and geometric strains are constant and unchanged from the standard finite elements while the transverse shear strains contain linear interpolated functions It is known that these low-order plate elements in the thin plate limit often suffer from shear locking In order to avoid shear locking, the discrete shear gaps (DSG) [18] which were proposed to reform the shear strains are adopted As a result of using the threenode triangular elements, the shear strains γ DSG become then constant Table Summary elements 123 Fig Triangular elements and smoothing domains associated with nodes 2.2 Brief on the DSG3 element In the linear triangular DSG3 element [18], the finite element approximation (w h , β h ) is simply interpolated using the linear basis functions for both deflection and rotations without any additional variables The bending strains used in the standard finite elements are unchanged while the transverse shear strains are reformulated by the interpolated shear gaps The shear strains can be expressed as a reduced operator Rh : H (Ω e ) → Γ h (Ω e ), where Γ h (Ω e ) is defined as: Γ h (Ω e ) = γ h |Ω e = (J−1 )T γˆ h , γˆ h = [γ1h γ2h ]T (17) The shear strain can be rewritten in the incorporation of reduction operator as γ DSG (w h , β h ) = ∇w h − Rh β h = (J−1 )T γˆ h MITC4 Four node mixed interpolation of tensorial component [14] MIN3 Three node Mindlin [15] DSG3 Discrete shear gap triangle element [18] (18) ES-DSG3 Edge-based smoothed discrete shear gap triangular element [46] Q4BL Quadrilateral bubble linked [69] DKMQ Discrete Kirchhoff Mindlin quadrilateral [70] ANS4 Four node assumed natural strain [71] ANS9 Nine node Assumed natural strain [72] RPIM Radial point interpolation method [73] Pb-2 Ritz Two-dimensional polynomial function Rayleigh–Ritz method [74] Comput Mech (2010) 46:679–701 683 Δwγ11 = Δwγ31 = Δwγ12 = Δwγ22 = 1 Δwγ21 = w2 −w1 + a(βx1 +βx2 )+ b(β y1 +β y2 ) (20) 2 1 Δwγ32 = w3 − w1 + d(βx1 + βx3 ) + c(β y1 + β y3 ) 2 Fig Patch test of the element (E = 100,000; ν = 0.25; t = 0.01) Table Patch test Methods w5 θx5 θ y5 m x5 m x5 m x y5 MIN3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033 DSG3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033 ES-DSG3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033 with a = x2 − x1 , b = y2 − y1 , c = y3 − y1 , d = x3 − x1 and (wi , βxi , β yi ), i = 1, 2, are the degree of freedoms at node i of the element In order to further improve the accuracy of approximate solutions and to stabilize shear force oscillations appearing in the triangular element, the stabilization technique [59] can be used here The idea for the stabilization of the original DSG3 element was also introduced in [60] With this remedy, the DSG3 element problem to the static problem is to find (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , ηh ) + kt (γ DSG (w h , β h ), γ DSG (v, η)) = ( p, v) (21) NS-DSG3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033 where x,ξ y,ξ is the Jacobian matrix of the bilinear where J = x,η y,η mapping from the local triangular element Ωˆ into the physical triangular element Ω e (see Fig 1b) and γˆ h contains the derivatives of the shape functions (N1 = − ξ − η, N2 = ξ, N3 = η) that are only constant (cf Bletzinger al [18] for more detail) (γ DSG (w h , β h ), γ DSG (v, η)) = (∇w h −Rh β h , ∇v−Rh η) Ne = e=1 Ne = e=1 ⎡ γ1h ⎤ γˆ h = ⎣ γ h ⎦ = N2,ξ Δwγ21 + N3,η Δwγ31 N2,ξ Δwγ22 + N3,η Δwγ32 (19) where Δwγi ς (i = 1, 2, 3) are the discrete shear gaps at the triangular element nodes related to the ς -local coordinate axis (ς = 1, 2) that are reported as t2 t2 (∇w h − Rh β h , ∇v − Rh η) L (Ω e ) + αh 2e t2 (∇w h − Rh β h ).(∇v − Rh η)Ae t + αh 2e (22) where h e is the longest length of the edges of the element, α is a positive constant and Ae is the area of the triangular element It is evident that the original DSG3 element is recovered when α = For free vibration and buckling problems, the second terms ((∇w h − β h , ∇v − η)) in the left hand side of Eq (15) and Eq (16) are replaced by the terms in Eq (22) In what follows, we utilize these constant strains to establish a formulation of a node-based smoothed triangular Fig Square plate model: a simply supported plate; b full clamped plate (a) (b) 123 684 Comput Mech (2010) 46:679–701 1.1 Normalized central moment Normalized deflection w 1.05 0.95 0.9 0.85 Exact solu MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 0.8 0.75 0.7 10 15 20 25 1.05 0.95 0.9 Exact solu MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 0.85 0.8 0.75 30 10 Number of elements per edge 15 25 30 (a) 1.5 1.5 0.5 −0.5 −1 MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 −1.5 −2 0.6 0.8 1.2 1.4 1.6 10 log (Relative error of central moment)[%]) 10 log (Relative error of central deflection)[%]) (a) −2.5 0.4 20 Number of elements per edge 0.5 −0.5 −1 MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 −1.5 −2 0.4 1.8 0.6 0.8 log10(h) 1.2 1.4 1.6 1.8 log10(h) (b) (b) Fig Simply supported plate (t/L = 0.01): a normalized central deflection; b relative error under log–log scale Fig Simply supported plate (t/L = 0.01): a normalized central moment; b relative error under log–log scale element with the stabilized discrete shear gap technique (NS-DSG3) for Reissner–Mindlin plates ated by connecting sequentially the mid-edge-points to the centroids of the surrounding triangular elements of node k as shown in Fig ¯ and the average shear strains The average curvatures κ, γ¯ over the cell Ω (k) are defined by A formulation of NS-FEM with stabilized discrete shear technique In the NS-FEM [42,43], the domain discretization is the same as that of the standard FEM using Ne triangular elements, but the integration required in the weak form of the FEM is now performed based on the nodes, and strain smoothing technique [19] is utilized In such a nodal integration process, the problem domain Ω is again divided into a set Nn Ω (k) and of smoothing domains Ω (k) such as Ω ≈ k=1 j) (i) ( Ω ∩ Ω = ∅, i = j in which Nn is the total number of nodes of the problem domain For triangular elements, the smoothing domain Ω (k) associated with the node k is cre- 123 κ¯ k = (k) A Ω (k) γ¯ k (w h , β h ) = = A(k) Nek i=1 κ dΩ = (k) A A(k) Nek i=1 e A κi i (23) γ DSG (w h , β h ) dΩ Ω (k) e DSG h h A γ (w , β ) i i (24) Comput Mech (2010) 46:679–701 685 1.04 1.1 1.02 Normalized central deflection w C Normalized strain energy 0.98 0.96 0.94 Exact sol MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 0.92 0.9 0.88 10 15 20 25 30 Exact sol MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 0.9 0.8 0.7 0.6 0.5 35 10 15 log10(Relative error of central deflection)[%]) MITC4 (r = 1.12) MIN3 (r = 1.42) DSG3 (r = 1.3) ES−DSG3 (r = 1.6) NS−DSG3 (r=0.99) 1.6 1.4 log10(Error in strain energy) 30 1.8 1.2 0.8 0.6 0.4 0.2 −0.2 0.4 1.5 0.5 −0.5 −1 −2 −2.5 0.4 0.6 0.8 1.2 MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 −1.5 0.6 0.8 1.4 10 Fig Simply supported plate (t/L = 0.01): a strain energy; b convergence rate ¯ ∇β ¯ x, The average gradients related to geometric strains ∇w, (k) ¯ ∇β y over the smoothing domain Ω are given by = (k) A i=1 Nek i=1 e ¯ xk = A ∇wi , ∇β i A(k) e h A ∇β yi i 1.6 1.8 Fig Convergence in the deflection of clamped plate (t/L = 0.001): a normalized central deflection; b relative error under log–log scale A (k) = Ω (k) Nek 1.4 (b) (b) A(k) 1.2 log10(h) log (h) ¯ yk ∇β 25 (a) (a) ¯ k= ∇w 20 Number of elements per edge Number of elements per edge Nek i=1 e h A β , i xi (25) where A(k) is the area of the smoothing domain Ω (k) and is computed by dΩ = Nek Aie (26) i=1 in which Aie is the area of the ith element attached to node k and Nek is the number of elements associated with node k illustrated in Fig With the above definitions, a solution of the NS-FEM with the stabilized discrete shear gap technique using three—node triangular elements (NS-DSG3) is now established The NSDSG3 solution to the static problem is to find (w h , β h ) ∈ V h such that 123 686 Comput Mech (2010) 46:679–701 1.4 1.3 Normalized central moment 1.2 1.1 0.9 0.8 Normalized strain energy 1.05 Exact sol MITC4 DSG3 MIN3 ES−DSG3 NS−DSG3 0.95 0.9 Exact solu MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 0.7 0.85 0.6 0.5 0.4 0.8 10 15 20 25 10 15 (a) 1.5 log10(Error in energy norm) log10(Relative error of central moment)[%]) MITC4 (r = 1.0) MIN3 (r = 1.1) DSG3 (r = 0.9) ES−DSG3 (r = 1.03) NS−DSG3 (r = 0.97) 1.8 0.5 −0.5 −1 MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3 −1.5 −2 0.8 1.2 1.4 1.6 1.6 1.4 1.2 0.8 1.8 log (h) 0.4 10 0.6 (b) 1.2 1.4 10 ∀(v, η) ∈ V0h , a(β ¯ h , ηh )+kt (γ¯ (w h , β h ), γ¯ (v, η)) = ( p, v) (27) The NS-DSG3 solution of the free vibration modes is to find the natural frequency ωh ∈ R+ and = (w h , β h ) ∈ V h such that ¯ h , η) + kt (γ¯ (w h , β h ), γ¯ (v, η)) ∀(v, η) ∈ V0h , a(β = (ωh )2 ρt (w h , v) + ρt (β h , η) 12 0.8 log (h) Fig Clamped plate (t/L = 0.001): a Normalized central moment; b relative error under log–log scale (b) Fig 10 Clamped plate model (t/L = 0.001): a strain energy; b convergence rate ∀(v, η) ∈ V0h , a(β ¯ h , η) + kt (γ¯ (w h , β h ), γ¯ (v, η)) h t b¯1 (w h , v) + t b¯2 (β h , η) = λcr 12 Nn a(β ¯ , η) = (28) (29) where a(·, ¯ ·) is a smoothed bilinear form given by h and for the buckling analysis is to find the critical buckling h ∈ R+ and = (w h , β h ) ∈ V h such that load λcr 123 30 2 0.6 25 (a) Number of elements per edge −2.5 0.4 20 Number of elements per edge 30 κ¯ k : Db : η¯ k A(k) (30) k=1 ¯ ∇β ¯ x, and the geometric terms related to the gradients (∇w, (k) ¯ ∇β y ) over the smoothing domain Ω are given by Comput Mech (2010) 46:679–701 687 c Central deflection (100w D/pL4) 0.425 0.42 0.415 0.41 0.405 10 100 1000 10000 100000 1000000 Ratio L/t (a) Central displacements wc/(pL /1000D) 0.8 Exact DSG3 MIN3 NS−DSG3 Morley sol Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 Central deflection (100wcD/pL4) 0.155 15 20 25 30 Number of element per edge Exact DSG3 MIN3 NS−DSG3 0.15 10 Fig 13 Convergence of central deflection for skew Morley’s plate 0.145 Nn (γ¯ (w , β ), γ¯ (v, η)) = h 0.14 h k=1 × γ¯ k (w h , β h ) · γ¯ k (v, η)A(k) (32) 0.135 0.13 0.125 10 100 1000 10000 100000 1000000 Ratio L/t (b) Fig 11 Performance of NS-DSG3 with varying L/t ratios: a Simply supported plate; b Clamped plate Fig 12 A simply supported skew Morley’s model b¯1 (w h , v) = t2 t + αh 2k Nn Nn b2 (β , η) = h Relationship to similar techniques ¯ k )T σˆ ∇v ¯ k A(k) , (∇w k=1 (31) h T h T ¯ xk ¯ yk ¯ xk+(∇β ¯ yk (∇β ) σˆ ∇η ) σˆ ∇η √ in which h k = A(k) is considered as the character length of the smoothing domain Ω (k) The necessity of stabilization for lower order plate elements in bending was shown in [59,60] Stabilization significantly improves the accuracy in the case of very thin plates and distorted meshes and to reduce oscillations of transverse shear forces It is found from numerical experiments that the stabilization parameter α fixed at 0.1 can produce the reasonable accuracy for all cases tested The stiffness matrix of NS-DSG3 becomes too flexible, if α is chosen too large On the contrary, the accuracy of the solution will reduce due to the oscillation of shear forces, if α is chosen too small So far, how to obtain an optimal value of parameter α is an open question A(k) k=1 and modified shear terms are now obtained by performing the smoothing operation via the smoothing domain Ω (k) : The present method can be considered as an alternative form of nodally integrated techniques in finite element formulations [61–68] The crucial idea of these methods is to formulate a nodal deformation gradient via a weighted average of the surrounding element values The major contribution to the nodal-integral method has been pointed out by Bonet and Burton [61] In their approach, the node-based formulation is applied to the volumetric component of the strain energy in order to eliminate volumetric locking of the 123 688 Comput Mech (2010) 46:679–701 10 Morley sol Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3 2.3 Central max principal moment (Mmax/qL /100) 2.4 2.2 2.1 4.4 x 10 Reference DKMQ MITC4 Q4BL DSG3 ES−DSG3 NS−DSG3 4.2 Strain energy 1.9 1.8 1.7 3.8 3.6 3.4 1.6 3.2 1.5 10 15 20 25 30 Number of elements per edge (a) 10 20 30 40 50 60 Number of elements per edge Central principal moment (Mmin/qL /100) Fig 15 Strain energy of a simply supported skew Morley’s plate Morley sol Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3 1.7 1.6 1.5 1.4 1.3 1.2 1.1 0.9 0.8 0.7 10 15 20 25 30 Number of elements per edge (b) Fig 14 Skew Morley’s plate: a central max principle moment; b central principle moment standard tetrahedral element Subsequently, Dohrmann et al [62] proposed a nodally averaged formulation for entire components of the strain energy (i.e including deviatoric component) This approach is simple while the method possesses very interesting properties such as 1) the upper bound property in strain energy; 2) free of volumetric locking; 3) superaccurate and super-convergent properties of stress solutions; 4) the stress at nodes computed directly from the displacement solution without using any post-processing Further improvements on the stability condition of the nodally averaged formulation have also been devised in Bonet et al 123 [63], Puso and Solberg [65] and Gee et al [66] Recently, a weighted-residual method that is used to weakly impose both the equilibrium equation and the kinematic equation was also introduced to create the average nodal strain formulation [67,68] for a variety of solid and plate elements The NS-FEM approach originates from the computation of smoothed strains via the smoothing domains associated with nodes of elements In the NS-FEM, the way to create smoothing domains is similar to nodal backgrounds in Dohrmann et al [62] Furthermore, the NS-FEM works for arbitrary n-sided polygonal elements When only linear triangular or tetrahedral elements are used, the NS-FEM produces the same results as the method proposed by Dohrmann et al [62] The extension of the nodally integrated techniques to Reissner–Mindlin plate elements has been proposed very recently in [68] Numerical results show some advantages of the nodally integrated formulation compared to the standard FEM However, it was proved in [68] that these formulations can not fulfill sufficiently, in general, constant bending strain patch test with an arbitrary mesh While only static plate problems were studied in [68], we dealt with static, free vibration and buckling solutions of Reissner–Mindlin plates using the NS-FEM Moreover, we show numerically the fulfilment of the constant bending strain patch test Although in several cases the numerical results using the NS-FEM are slightly less accurate than those using the ES-FEM,2 its performance is much better than several other existing This method for analysis of plates has already been investigated in [46] Comput Mech (2010) 46:679–701 689 Fig 16 Supported and clamped plate Table A non-dimensional frequency parameter of a SSSS plate (a/b = 1) t/a Method DSG3 (%) 0.005 0.1 Table A non-dimensional frequency parameter of a CCCC plate (a/b = 1) Mode t/a 0.1 NS-DSG3 (%) Exact [76] 4.443 4.5131 (1.58) 4.4641 (0.48) 4.4509 (0.18) 7.1502 (1.78) 7.0870 (0.88) 7.0441 (0.27) 7.025 7.3169 (4.15) 7.1193 (1.34) 7.0572 (0.46) 7.025 9.3628 (5.36) 9.0582 (1.94) 8.9519 (0.74) 8.886 10.3772 (4.45) 10.1444 (2.11) 9.9944 (0.60) 9.935 10.3772 (4.45) 10.1489 (2.15) 9.9954 (0.61) 9.935 4.3943 (0.56) 4.3846 (0.33) 4.3725 (0.06) 4.37 6.8227 (1.23) 6.7922 (0.77) 6.7516 (0.17) 6.74 6.8587 (1.76) 6.8196 (1.18) 6.7627 (0.34) 6.74 8.5447 (2.33) 8.4744 (1.49) 8.3801 (0.36) 8.35 9.4557 (2.56) 9.3666 (1.60) 9.2278 (0.08) 9.22 9.4616 (2.62) 9.3698 (1.62) 9.2285 (0.09) 9.22 Mode Method DSG3 (%) 0.005 ES-DSG3 (%) ES-DSG3 (%) NS-DSG3 (%) Exact [76] 6.1786 (3.00) 6.0355 (0.61) 5.9693 (−0.50) 5.999 8.8759 (3.60) 8.6535 (1.00) 8.5085 (−0.70) 8.568 9.0680 (5.83) 8.7081 (1.64) 8.5269 (−0.48) 8.568 11.2452 (8.05) 10.6584 (2.42) 10.3880 (−0.18) 10.407 12.2182 (6.50) 11.7430 (2.36) 11.3913 (−0.70) 11.472 12.2992 (6.97) 11.7720 (2.38) 11.4215 (−0.67) 11.498 5.7616 (0.90) 5.7250 (0.26) 5.6746 (−0.62) 5.71 7.9935 (1.44) 7.9211 (0.52) 7.8158 (−0.81) 7.88 8.0525 (2.19) 7.9627 (1.05) 7.8313 (−0.62) 7.88 9.5772 (2.65) 9.4499 (1.29) 9.2686 (−0.66) 9.33 10.4153 (2.82) 10.2631 (1.31) 10.0222 (−1.06) 10.13 10.4697 (2.85) 10.3126 (1.30) 10.0683 (−1.09) 10.18 methods In addition, nodally integrated techniques for Reissner–Mindlin plate formulations found in the literature are very limited This may be due to the various difficulties generated from plate models Therefore, the motivation of this work is to complement the nascent body of litera- ture on nodally integrated formulations for static, free vibration and buckling analyses of Reissner–Mindlin plates Further developments of the present technique for plates with complicated behaviors or shell problems will be investigated in forthcoming papers 123 690 Comput Mech (2010) 46:679–701 1.9 1.7 Normalized frequencies Numerical examples Exact DSG3 (mode1) DSG3 (mode2) DSG3 (mode3) DSG3 (mode4) DSG3 (mode5) NS−DSG3 (mode1) NS−DSG3 (mode2) NS−DSG3 (mode3) NS−DSG3 (mode4) NS−DSG3 (mode5) 1.8 1.6 1.5 1.4 In what follows, the present element (NS-DSG3) is compared to several published elements from the literature, summarized in Table Static, free vibration and buckling and analyses of square, rectangular, circular and triangular plates are considered 1.3 5.1 Static analysis 1.2 5.1.1 Patch test 1.1 0.9 10 12 14 16 Number of elements per edge (a) 1.6 Normalized frequencies √ Table A non-dimensional frequency parameter = ωa ρt/D of a square plate (t/a = 0.005) with various boundary conditions Exact DSG3 (mode1) DSG3 (mode2) DSG3 (mode3) DSG3 (mode4) DSG3 (mode5) NS−DSG3 (mode1) NS−DSG3 (mode2) NS−DSG3 (mode3) NS−DSG3 (mode4) NS−DSG3 (mode5) 1.5 1.4 1.3 The patch test is introduced to examine the convergence of finite elements It is checked if the element is able to reproduce a constant distribution of all quantities for arbitrary meshes A rectangular plate is modeled by several Plate type SSSF 1.2 1.1 SFSF 0.9 10 12 14 16 Number of elements per edge CCCF (b) Fig 17 Convergence of normalized frequency a/b = 1; t/a = 0.005: a SSSS plate; b CCCC plate h/ exact with CFCF Fig 18 The circular plates and a typical mesh 123 Mode Methods DSG3 ES-DSG3 NS-DSG3 Exact [76] 11.7720 11.6831 11.6481 11.685 28.3759 27.8382 27.7495 27.756 41.9628 41.4312 41.0324 41.197 61.5092 59.6720 59.0864 59.066 9.6673 9.6425 9.6224 9.631 16.3522 16.1239 16.0636 16.135 37.6792 36.9054 36.8072 36.726 39.5026 39.2167 38.9116 38.945 24.2848 23.8947 23.6016 24.020 41.7698 40.1998 39.9046 40.039 65.0068 63.5127 61.8530 63.493 80.9461 77.8776 77.1318 76.761 22.3437 22.1715 21.8768 22.272 27.1814 26.4259 26.1366 26.529 45.8829 43.9273 43.6459 43.664 62.5225 62.9466 62.9090 64.466 Comput Mech (2010) 46:679–701 Table The parameterized natural frequencies √ = ωa ρt/D of a clamped circular plate with t/(2R) = 0.01 Table The parameterized natural frequencies √ = ωa ρt/D of a clamped circular plate with t/(2R) = 0.1 691 Mode Methods DSG3 ES-DSG3 NS-DSG3 ANS4 [71] ANS9 [72] Exact [77] 10.2941 10.2402 10.2580 10.2572 10.2129 10.2158 21.6504 21.3966 21.4620 21.4981 21.2311 21.2600 21.6599 21.4096 21.4800 21.4981 21.2311 21.2600 35.9885 35.3012 35.4611 35.3941 34.7816 34.8800 35.9981 35.3277 35.5009 35.5173 34.7915 34.8800 41.1864 40.3671 40.6001 40.8975 39.6766 39.7710 53.4374 52.0138 52.3402 52.2054 50.8348 51.0400 53.5173 52.1013 52.4428 52.2054 50.8348 51.0400 64.2317 62.3053 62.8261 63.2397 60.6761 60.8200 10 64.4073 62.4665 62.9265 63.2397 60.6761 60.8200 11 74.2254 71.6554 72.2458 71.7426 69.3028 69.6659 12 74.3270 71.7269 72.3162 72.0375 69.3379 69.6659 13 91.4366 87.7019 88.5316 88.1498 84.2999 84.5800 14 91.5328 87.7861 88.6825 89.3007 84.3835 84.5800 Mode Methods DSG3 ES-DSG3 NS-DSG3 ANS4 [71] Exact [77] 9.3012 9.2527 9.2789 9.2605 9.240 18.0038 17.8372 17.9195 17.9469 17.834 18.0098 17.8428 17.9366 17.9469 17.834 27.6010 27.2344 27.4301 27.0345 27.214 27.6082 27.2391 27.4531 27.6566 27.214 30.9865 30.5173 30.7906 30.3221 30.211 37.9464 37.2817 37.6719 37.2579 37.109 37.9817 37.3128 37.7152 37.2579 37.109 43.9528 43.0626 43.6325 43.2702 42.409 10 44.0324 43.1328 43.6664 43.2702 42.409 11 48.9624 47.8823 48.5592 47.7074 47.340 12 48.9793 47.8976 48.5887 47.8028 47.340 13 57.2487 55.7747 56.7283 56.0625 54.557 14 57.2776 55.8052 56.7876 57.1311 54.557 triangular elements as shown in Fig The boundary deflection is assumed to be w(x, y) = (1+x +2y+x +x y+y )/2 It is seen from Table that the NS-DSG3 element passes the constant bending patch test within machine precision 5.1.2 Square plates Consider the model of a square plate (length L, thickness t) with simply supported and clamped boundary conditions, respectively, subjected to a uniform load p = as shown in Fig The material parameters are: Young’s modulus E = 1,092,000 and Poisson’s ratio ν = 0.3 Due to symmetry, only the below left quadrant of the plate is modeled and uniform meshes N × N with N = 2, 4, 8, 16, 32 are employed For a simply supported plate, Fig illustrates the convergence of the normalized deflection and the relative error on a log–log plot The central moments are depicted in Fig The 123 692 Comput Mech (2010) 46:679–701 Fig 19 A triangular cantilever plates and mesh of it strain energy together with its convergence rate for a relation t/L = 0.01 are shown in Fig It is observed that the NSDSG3 element reveals higher accuracy the original DSG3 It is seen that all the results of the NS-DSG3 converge to the exact value from above More details concerning upper bound solutions provided by NS-FEMs are given in [42] For the convergence of the deflection, the MITC4 element is the most effective For the convergence of moment and energy, the ES-DSG3 element is superior Based on the above results, it can be concluded that the NS-DSG3 gives relatively good results compared with the MIN3, MITC4 and ES-DSG3 elements For a clamped plate, numerical results are displayed in Figs 8, and 10 It is seen that the NS-DSG3 produces upper bound solutions to the exact value and shows high reliability compared to the other elements For the deflection, the NS-DS3 convergence is slower than the other elements with fine meshes For the central moment, the NS-DSG3 element produces very reasonable results compared with the MITC4 and ES-DSG3 models Figure 10 plots the convergence in strain energy and its energy error for a relation t/L = 0.001 It is again seen that the present element can produce an upper bound in strain energy and its result shows good agreement with those from MITC4 and ES-DSG3 for this case 123 Now we illustrate the performance of the NS-DSG3 when the plate becomes very thin Theoretically, it is well known that shear effect will reduce when the ratio of length– thickness (L/t) increases Hence, solutions of Reissner– Mindlin theory will approach solutions of Kirchhoff theory Figure 11 plots the central deflection with respect to the L/t ratio It is found that the NS-DSG3 is locking-free in the thin plate limit In addition, the results of NS-DSG3 are more accurate than those of the original DSG3 element 5.1.3 Skew plate subjected to a uniform load Let us consider a rhombic plate subjected to a uniform load p = as shown in Fig 12 This plate was originally studied by Morley [75] Geometry and material parameters are length L = 100, thickness t = 0.1, Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3 The values of the deflection, principle moments and strain energy derived from the NS-DSG3 in comparison with those of other methods are shown in Figs 13, 14 and 15, respectively It is observed that the NS-DSG3 can produce upper bound solutions to the exact value Although the NS-DSG3 is less accurate than the ES-DSG3, it shows very good performance compared to the other elements in the literature Comput Mech (2010) 46:679–701 Table The parameterized natural frequencies = ωa (ρt/D)1/2 /π of triangular platess with t/b = 0.001 693 ϕo Mode Methods DSG3 0◦ 15◦ 30◦ 45◦ 60◦ ES-DSG3 NS-DSG3 ANS4 [71] Rayleigh–Ritz [79] 0.6252 0.6242 0.6235 0.624 2.3890 2.3789 2.3737 2.379 0.625 2.377 3.3404 3.3159 3.3040 3.317 3.310 5.7589 5.7124 5.6828 5.724 5.689 7.8723 7.7919 7.7312 7.794 7.743 10.3026 10.1547 10.0674 10.200 0.5855 0.5840 0.5835 0.583 0.586 2.1926 2.1833 2.1783 2.181 2.182 3.4528 3.4163 3.4024 3.413 3.412 5.3481 5.3020 5.2777 5.303 5.279 7.3996 7.3112 7.2517 7.289 7.263 10.2498 10.0779 9.9925 10.095 0.5798 0.5766 0.5745 0.575 2.1880 2.1778 2.1720 2.174 2.178 3.7157 3.6539 3.6287 3.638 3.657 5.5983 5.5361 5.5095 5.534 5.518 7.109 – – 0.578 7.2814 7.1628 7.0954 7.139 10.7753 10.5108 10.3948 10.477 0.6006 0.5923 0.5855 0.588 0.593 2.3564 2.3359 2.3243 2.324 2.335 4.2795 4.1699 4.1289 4.126 4.222 6.5930 6.4424 6.3662 6.381 6.487 7.8615 7.6658 7.5973 7.614 7.609 11.7850 11.3496 11.1757 11.224 0.6497 0.6261 0.6298 0.613 2.7022 2.6101 2.5709 2.564 2.618 5.6491 5.4283 5.3683 5.353 5.521 – – 0.636 8.3505 7.7333 7.5127 7.460 8.254 10.7757 10.3756 10.2769 10.306 10.395 14.6003 13.3296 12.9519 12.942 – 5.2 Free vibration of plates In this section, we examine the accuracy and efficiency of the NS-DSG3 element for analyzing natural frequencies of plates The plate may have free (F), simply (S) supported or clamped (C) edges A non-dimensional frequency parameter is often used for the presentation of the results for regular meshes 5.2.1 Square plates Let’s consider square plates of length a, width b and thickness t as shown in Fig 16 The material parameters are Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3 and the density mass ρ = 8,000 For comparison, the plate is modeled with uniform meshes of 16 elements per side A nondimensional frequency parameter = (ω2 ρa t/D)1/4 is used, where D = Et /(12(1 − ν )) is the flexural rigidity of the plate Thin and thick SSSS plates corresponding to length-to-width ratios, a/b = and thickness-to-length t/a = 0.005 (and t/a = 0.1) are considered in this problem The geometry of the plate and the typical mesh are shown in Fig 16 The convergence of the first six modes corresponding to meshes using 16 × 16 rectangular elements is presented in Tables and The relative error percentages compared with the exact results are given in parentheses The NS-DSG3 123 694 Comput Mech (2010) 46:679–701 0.65 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4 0.64 0.63 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4 5.5 Mode Mode 0.62 0.61 0.6 4.5 0.59 3.5 0.58 10 20 30 40 50 60 10 20 angle 30 40 50 60 50 60 angle 8.5 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4 2.7 7.5 Mode 2.6 Mode 2.5 2.4 6.5 2.3 2.2 5.5 2.1 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4 10 20 30 40 50 60 10 20 30 40 angle angle Fig 20 Convergence of the frequencies of triangular plates with t/a = 0.001 shows good agreement with the exact results [76] It is seen that computed frequencies of the NS-DSG3 element are more accurate than those of the DSG3 element, and slightly more accurate than those of ES-DSG3 element The convergence of computed frequencies of SSSS and CCCC plates is also displayed in Fig 17 It is clear that the NS-DSG3 element outperforms the DSG3 element In addition, the reliability of the NS-DSG3 element is also shown for the four sets of various boundary conditions in this example: SSSF, SFSF, CCCF, and CFCF The first four lowest frequencies are listed in Table 5.2.2 Circular plates In this example, a circular plate with a clamped boundary is studied as shown in Fig 18 The problem parameters are 123 Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3, radius R = and mass density ρ = 8000 The plate is discretized with 848 triangular elements with 460 nodes Two thickness-span ratios t/(2R) = 0.01 and 0.1 are considered Table summarizes the frequencies of circular plate with the thickness-span ratio t/(2R) = 0.01 derived from the NS-DSG3 in comparison with other elements The results of the NS-DSG3 element are closer to the analytical solutions [77,78] compared to those of the DSG3 element It also is a good competitor to the ES-DSG3 element and quadrilateral plate elements such as the Assumed Natural Strain solutions (ANS4) with 432 quadrilateral elements [71] and the higher order Assumed Natural Strain solutions (ANS9) [72] In case of the thickness-span ratio t/(2R) = 0.1, as shown in Table 7, reasonable results are obtained for the NS-DSG3 element Comput Mech (2010) 46:679–701 Table The parameterized natural frequencies = ωa (ρt/D)1/2 /π of a triangular plates with t/b = 0.2 695 ϕ◦ 0◦ 15◦ 30◦ 45◦ 60◦ Mode Methods DSG3 ES-DSG3 NS-DSG3 ANS4 [71] Rayleigh–Ritz [79] 0.5830 0.5823 0.5816 0.582 0.582 1.9101 1.9040 1.8980 1.915 1.900 2.4176 2.4083 2.3989 2.428 2.408 3.9772 3.9559 3.9337 3.984 3.936 5.0265 4.9954 4.9599 5.018 – 5.9521 5.8994 5.8454 5.944 – 0.5449 0.5441 0.5433 0.545 0.544 1.7803 1.7749 1.7693 1.764 1.771 2.3959 2.3854 2.3752 2.420 2.386 3.6668 3.6467 3.6266 3.608 3.628 4.8504 4.8208 4.7868 4.820 – 5.6057 5.5385 5.4760 5.431 – 0.5339 0.5328 0.5316 0.532 0.533 1.7815 1.7754 1.7693 1.773 1.772 2.4356 2.4206 2.4077 2.437 2.419 3.6085 3.5842 3.5631 3.591 3.565 4.7829 4.7444 4.7043 4.765 – 5.4532 5.3377 5.2481 5.323 – 0.5412 0.5391 0.5371 0.541 0.540 1.8977 1.8882 1.8800 1.884 1.885 2.5304 2.5004 2.4820 2.518 2.489 3.7518 3.7035 3.6730 3.748 3.674 4.8188 4.6800 4.5794 4.740 – 5.4304 5.2256 5.1299 5.292 – 0.5634 0.5588 0.5556 0.559 0.559 2.0837 2.0623 2.0496 2.095 2.059 2.5355 2.4356 2.4114 2.483 2.396 4.0862 3.8009 3.7229 3.910 3.590 4.6612 4.3393 4.2779 4.517 – 5.9782 5.5835 5.4814 5.763 – 5.2.3 Triangular plates Let us consider cantilever (CFF) triangular plates with various shape geometries, see Fig 19 The material parameters are Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3 and mass density ρ = 8,000 A non-dimensional frequency parameter = ωa (ρt/D)1/2 /π of triangular square plates with aspect ratio t/a = 0.001 and 0.2 is calculated The mesh of 744 triangular elements with 423 nodes is used to analyze the convergence for various skew angles such as ϕ ◦ = 0◦ , 15◦ , 30◦ , 45◦ , 60◦ The first four modes of the thin triangular plate (t/a = 0.001) are shown in Table and Fig 20 The NS-DSG3 element is also compared to the ANS4 and ES-DSG3 elements and two other well-known numerical methods such as the Rayleigh–Ritz method [79] and the pb-2 Ritz method [74] The frequencies of the NS-DSG3 are often bounded by these reference models Note that our method only uses three primitive DOFs at each vertex node without adding any additional DOFs For a thick plate, the results are shown in Table The first eight mode shapes of cantilever triangular square plates are illustrated in Fig 21 It is clear that the NS-DSG3 is stable 5.3 Buckling of plates In the following examples, the buckling load factor is defined as K = λcr b2 /(π D) where b is the edge width of the plate, 123 696 Comput Mech (2010) 46:679–701 The results of the NS-DSG3 are more accurate than those of the DSG3 The NS-DSG3 produces the best result for the SSSS plate case, but it is slightly less accurate than the ES-DSG3 and some other methods for the CCCC plate, see Table 11 Therefore the accuracy of the NS-DSG3 seems to be a problem-dependent Next, we consider the buckling load factors of SSSS, CCCC, CFCF plates with thickness-to-width ratios t/b = 0.05; 0.1 The results given in Table 12 using NS-DSG3 compare well with several other methods We also consider simply supported plates with various thickness-to-width ratios, t/b = 0.05; 0.1; 0.2 and lengthto-width ratios, a/b = 0.5; 1.0; 1.5; 2.0; 2.5 The buckling factors for a 16 × 16 mesh are described in Fig 24 and Table 13 The axial buckling modes of simply-supported rectangular plates with thickness-to-width ratios t/b = 0.01 and various length-to-width ratios, a/b = 1.0; 1.5; 2.0; 2.5 are shown in Fig 25 It is clear that the results of the NS-DSG3 match well those of ES-DSG3 and Pb-2 Ritz models 5.3.2 Simply supported rectangular plates subjected to biaxial compression Fig 21 The first eight mode shapes of the triangular square plate with t/a=0.001 λcr the critical buckling load The material parameters are assumed: Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3 5.3.1 Simply supported rectangular plates subjected to uniaxial compression Let us first consider a plate with length a, width b and thickness t subjected to uniaxial compression SSSS and CCCC boundary conditions are assumed The geometry and a typical mesh are shown in Fig 22a Table 10 gives the convergence of the buckling load factor corresponding to meshes with 4×4, 8×8, 12×12 and 16×16 rectangular elements Figure 23 plots the convergence of the normalized buckling load K h /K exact of the square plate with thickness ratio t/b = 0.01, where K h is the numerical buckling load and K exact is the analytical buckling load [80] 123 Consider the square plate subjected to biaxial compression shown in Fig 22b Table 14 gives the shear buckling factor of the square plate subjected to biaxial compression with three essential boundary conditions (SSSS, CCCC, SCSC) using × 16 × 16 triangular elements The relative error percentages compared with the analytical solutions [80] are given in a parentheses The results of the NS-DSG3 element are more accurate than those of the DSG3 and slightly less accurate than those of the ES-DSG3 5.3.3 Simply supported rectangular plates subjected to in-plane pure shear The final example is the simply supported plate subjected to in-plane shear shown in Fig 22c The shear buckling load factors K of this plate are calculated using a 16×16 mesh The shear buckling factors with thickness-to-width ratio, t/b = 0.001 and various length-to-width ratios, a/b = 1.0; 2.0; 3.0; 4.0 are listed in Table 15 The NS-DSG3 element agrees well with the exact solution and other numerical models Figure 26 illustrates the convergence of the shear buckling load Figure 27 also presents the shear buckling modes of simply-supported rectangular plates Table 16 gives the shear buckling factor of the square plate subjected to in-plane pure shear with three essential boundary conditions (SSSS, CCCC, SCSC) The NS-DSG3 element works well for these cases Comput Mech (2010) 46:679–701 697 Fig 22 Rectangular plates: a Axial compression, b biaxial compression, c shear in-plane, d regular mesh (a) (b) (c) Plates type Methods Index mesh 4×4 SSSS 12 × 12 16 × 16 DSG3 7.5891 4.8013 4.3200 4.1590 ES-DSG3 4.7023 4.1060 4.0368 4.0170 NS-DSG3 CCCC 8×8 4.1313 4.0741 4.0396 4.0231 DSG3 31.8770 14.7592 11.9823 11.0446 ES-DSG3 14.7104 11.0428 10.3881 10.2106 NS-DSG3 11.4457 11.2947 10.7144 10.4473 Exact SSSS (DSG3) CCCC (DSG3) SSSS (ES−DSG3) CCCC (ES−DSG3) SSSS (NS−DSG3) CCCC (NS−DSG3) Normalized buckling load Kh/Kexact Table 10 The axial buckling load factors K h along the x axis of rectangular plates with length-to-width ratios a/b = and thickness-to-width ratios t/b = 0.01 (d) 2.5 1.5 Conclusions A node-based smoothed finite element method (NS-FEM) with a stabilized discrete shear gap technique using triangular elements (NS-DSG3) has been formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates The method is based on the strain smoothing technique over smoothing domains associated with nodes of finite elements [42] Transverse shear locking is solved with help of the discrete shear gap method and a stability of the NS-DSG3 is 10 12 14 16 mesh index N Fig 23 Normalized buckling load K h /K exact of a square plate with t/b = 0.01 ensured using the stabilization technique The present element uses only three primitive DOFs at each vertex node without additional degrees of freedom The NS-DSG3 is simple to implement into finite element programs using triangular meshes that can be generated with ease for complicated 123 698 Comput Mech (2010) 46:679–701 Table 11 The axial buckling load factors K h along the x axis of rectangular plates with length-to-width ratios a/b = and thickness-to-width ratios t/b = 0.01 Plates type Methods DSG3 ES-DSG3 NS-DSG3 Liew Ansys Tham Timoshenko SSSS 4.1590 4.0170 4.0008 3.9700 4.0634 4.00 4.00 (%) 3.97% 0.4% 0.02% −0.75% 1.85% – – 11.0446 10.2106 10.4473 10.1501 10.1889 10.08 10.07 9.68% 1.4% 3.61% 0.8% 1.18% 0.1% – CCCC (%) Table 12 The axial buckling load factors K h along the x axis of rectangular plates with various length-to-width ratios a/b = and various thickness-to-width ratios Table 13 The axial buckling load factors K h along the x axis of rectangular plates with various length-to-width ratios and various thicknessto-width ratios t/b Plates type a/b t/b Methods Methods DSG3 ES-DSG3 NS-DSG3 Meshfree [58] Ritz [81] DSG3 ES-DSG3 NS-DSG3 RPIM [73] Ritz [81] SSSS 0.05 6.0478 5.9873 6.0381 6.0405 6.0372 0.1 5.3555 5.3064 5.3638 5.3116 5.4777 9.5586 0.2 3.7524 3.7200 3.7811 3.7157 3.9963 3.8005 0.05 3.9786 3.9412 3.9687 3.9293 3.9444 0.1 3.7692 3.7402 3.7870 3.7270 3.7865 0.2 3.1493 3.1263 3.1739 3.1471 3.2637 3.9786 3.9412 3.9536 3.9464 3.9444 0.05 CCCC 9.8284 9.5426 9.6633 9.5819 CFCF 3.8365 3.7654 3.8214 3.8187 SSSS 3.7692 3.7702 3.7563 3.7853 3.7873 CCCC 8.2670 8.2674 8.2753 8.2931 8.2921 CFCF 3.4594 3.4966 3.4652 3.5138 3.5077 0.1 0.5 1.0 1.5 DSG3 (t/b=0.05) DSG3 (t/b=0.1) DSG3 (t/b=0.2) ES−DSG3 (t/b=0.05) ES−DSG3 (t/b=0.1) ES−DSG3 (t/b=0.2) NS−DSG3 (t/b=0.05) NS−DSG3 (t/b=0.1) NS−DSG3 (t/b=0.2) Pb−2−Ritz (t/b=0.05) Pb−2−Ritz (t/b=0.1) Pb−2−Ritz (t/b=0.2) Buckling axial factor K 6.5 5.5 4.5 2.0 2.5 0.05 4.3930 4.2852 4.3230 4.2116 4.2570 0.1 4.0604 3.9844 4.0368 3.8982 4.0250 0.2 3.2014 3.1461 3.1558 3.1032 3.3048 0.05 4.1070 3.9811 4.0323 3.8657 3.9444 0.1 3.8539 3.7711 3.8116 3.6797 3.7865 0.2 3.2023 3.1415 3.2208 3.0783 3.2637 0.05 4.3577 4.1691 4.2523 3.9600 4.0645 0.1 4.0644 3.8924 3.9347 3.7311 3.8683 0.2 3.2393 3.1234 3.1811 3.0306 3.2421 3.5 0.5 1.5 2.5 a/b Fig 24 Convergence of axial buckling load K h of SSSS plate with various length-to-width ratios and various thickness-to-width ratios problem domains Numerical results showed that the NSDSG3 is shear-locking free, stable and is superior to the original DSG3 element Furthermore, the present formulation also exhibits good agreement compared with several published methods in the literature As observed from numerical experiments, it is useful to note that the NS-DSG3 can produce an upper bound solution in the elastic energy for static analyses, while the ES-DSG3 produces results of comparable accuracy, but underestimates the elastic energy Both methods could be used in tandem to 123 (a) (b) (c) (d) Fig 25 Axial buckling modes of simply-supported rectangular plates with thickness-to-width ratios t/b = 0.01 and various length-to-width ratios a/b = 1; 1.5; 2.0; 2.5 Comput Mech (2010) 46:679–701 Table 14 The biaxial buckling load factors K h of rectangular plates with length-to-width ratios a/b = 1, thickness-to-width ratios t/b = 0.01 and various boundary conditions 699 Plates type Methods DSG3 (%) ES-DSG3 (%) Tham [82] Timoshenko [80] SSSS 2.0549 (2.74) 2.0023 (0.11) 2.0138 (0.69) 2.00 2.00 CCCC 5.6419 (6.20) 5.3200 (0.20) 5.3537 (0.82) 5.61 5.31 SCSC 4.0108 (4.72) 3.8332 (0.08) 3.8531 (0.60) 3.83 3.83 Table 15 The shear buckling load factors K h of simply supported rectangular plates with various length-to-width ratios, choose t/b = 0.01 a/b NS-DSG3 (%) Methods DSG3 ES-DSG3 NS-DSG3 Meshfree [58] Exact [83] 1.0 9.5195 9.2830 9.3807 9.3962 9.34 2.0 6.7523 6.4455 6.4599 6.3741 6.34 3.0 6.5129 5.8830 5.7380 5.7232 5.784 4.0 6.3093 5.6732 5.4972 5.4367 5.59 Table 16 The shear buckling load factors K h of rectangular plates with length-to-width ratios a/b = 1, thickness-to-width ratios t/b = 0.01 and various boundary conditions Plates type Methods DSG3 ES-DSG3 NS-DSG3 Tham [82] Timoshenko [80] SSSS 9.5195 9.2830 9.3807 9.40 9.33 CCCC 15.6397 14.6591 15.0281 14.58 14.66 SCSC 13.1652 12.5533 12.7296 12.58 12.58 10 DSG3 ES−DSG3 NS−DSG3 Meshfree Exact 9.5 Buckling shear factor K provide an estimate of the global error in energy for general problems where the exact energy 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methods is to formulate a nodal deformation... 46:679–701 Table The parameterized natural frequencies √ = a ρt/D of a clamped circular plate with t/(2R) = 0.01 Table The parameterized natural frequencies √ = a ρt/D of a clamped circular plate with