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nguyenthoitrung@tdt.edu.vn Division of Computational Background: The paper presents a numerical procedure for kinematic limit analysis Mathematics and Engineering of Mindlin plate governed by von Mises criterion. (CME), Institute for Computational Science (INCOS), Ton Duc Thang Methods: The cell-based smoothed three-node Mindlin plate element (CS-MIN3) is University, Hochiminh City, Viet combined with a second-order cone optimization programming (SOCP) to Nam Department of Mechanics, Faculty determine the upper bound limit load of the Mindlin plates. In the CS-MIN3, each of Mathematics and Computer triangular element will be divided into three sub-triangles, and in each sub-triangle, Science, VNUHCM University of the gradient matrices of MIN3 is used to compute the strain rates. Then the gradient Science, Hochiminh City, Viet Nam Full list of author information is smoothing technique on whole the triangular element is used to smooth the strain available at the end of the article rates on these three sub-triangles. The limit analysis problem of Mindlin plates is formulated by minimizing the dissipation power subjected to a set of constraints of boundary conditions and unitary external work. For Mindlin plates, the dissipation power is computed on both the middle plane and thickness of the plate. This minimization problem then can be transformed into a form suitable for the optimum solution using the SOCP. Results and Conclusions: The numerical results of some benchmark problems show that the proposal procedure can provide the reliable upper bound collapse multipliers for both thick and thin plates. Keywords: Limit analysis; Upper bound; Mindlin plates; Cell-based smoothed three- node Mindlin plate element (CS-MIN3); Smoothed finite element methods (S-FEM); Second-order cone programming (SOCP) Background Limit analysis is a branch of plasticity analysis and plays an important role in deter- mining the limit loads of a structure. The fundamental theorems of limit analysis ig- nore the evolutive elastoplastic computations but focus to determine the upper or lower bound loads which cause the plastic collapse of structures. Using analytical methods and different yield criteria such as the maximum principal stress criterion, Tresca criterion, and von Mises criterion, many scholars derived the analytical solutions for the limit loads of plates. Some systematic and comprehensive summaries can be found in the monographs of Hodge [1], Save and Massonnet [2], Zyczkowski [3], Xu and Liu [4], Lubliner [5], Yu et al. [6], etc. Using numerical methods, some early works for the limit loads of plates can be mentioned such as © 2014 Trung et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 2 of 26 http://www.apjcen.com/content/1/1/6 those by Hodge and Belytschko [7] and Nguyen [8]. However, due to the lack of effi- cient optimization algorithms and the limit of the computing power, the numerical limit analysis of plates seems to be ignored for a certain times. Recently, the interest of scientists in numerical limit analysis [9-15] has been resurged, principally thanks to the rapid development of efficient optimization algorithms and the continuous improvement in computer facilities. Current research is focusing on develop- ing numerical limit analysis tools which are efficient and robust for the practice usage of engineers. In the numerical limit analysis, once the stress or displacement/velocity fields are approximated and the bound theorems are applied, the limit analysis becomes a prob- lem of optimization involving either linear programing (LP) or nonlinear programming (NLP) which can be solved respectively by the available LP or NLP algorithms [16-23]. For the LP algorithms, some significant contributions have been published such as the active set LP algorithm by Sloan [16], the bespoke interior-point algorithm for LP by Andersen and Christiansen [17], and the commercial LP code XA by Pastor et al. [18]. For the NLP algorithms, some recently important contributions can be mentioned such as the algorithm based on feasible directions by Zouain et al. [10] or by Lyamin and Sloan [19], the algorithm based on the interior-point method by Andersen et al. [20] or by Krabbenhoft and Damkilde [21], and the general-purpose NLP codes CONOPT and MINOS by Tin-Loi and Ngo [22]. Recently, one of the most efficient NLP algorithms based on the primal-dual interior-point method was proposed by Andersen et al. [23]. The algorithm can be applied to von Mises-type yield functions and can handle problems with any nonlinear yield functions. The algorithm is implemented in second-order cone programming (SOCP) [24] of the commercial software MOSEK [25] and has been applied for the limit loads of some limit analysis problems [26,27]. Using such LP and NLP algorithms for the numerical limit analyses of plate struc- tures, many significant researches have been published. For the Kirchhoff plates, we can list the works by Christiansen and Larsen [28], Turco and Caracciolo [29], Corradi and Vena [30], Corradi and Panzeri [31], Tran et al. [32], Le et al. [33-35], and Zhou et al. [36]. For the Mindlin plates, we can list the works by Capsoni and Corradi [37] and Capsoni and Vicente da Silva [38]. In comparison, it is seen that many studies in the literature are concerned with the limit analysis of Kirchhoff plates, while the litera- ture related to those of Mindlin plates is somehow still limited. This paper hence aims to further contribute a numerical limit analysis of Mindlin plates by using a Mindlin plate element proposed recently together with the SOCP. In the other front of the development of numerical methods, Liu and Nguyen Thoi [39] have integrated the strain smoothing technique [40] into the finite element method (FEM) to create a series of smoothed FEMs (S-FEMs) such as cell/element-based smoothed FEM (CS-FEM) [41-43], node-based smoothed FEM (NS-FEM) [44-46], edge-based smoothed FEM (ES-FEM) [47,48], face-based smoothed FEM (FS-FEM) [49], and a group of alpha-FEM [50-53]. Each of these smoothed FEMs has different properties and has been used to produce desired solutions for a wide class of bench- mark and practical mechanics problems. Several theoretical aspects of the S-FEM models have been provided in [54,55]. The S-FEM models have also been further inves- tigated and applied to various problems such as plates and shells [56-68], piezoelectri- city [69,70], fracture mechanics [71], visco-elastoplasticity [72-74], limit and shakedown analysis for solids [75-77], and some other applications [78,79]. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 3 of 26 http://www.apjcen.com/content/1/1/6 Among these S-FEM models, the CS-FEM [39,41] shows some interesting properties in solid mechanics problems. Extending the idea of the CS-FEM to plate structures, Nguyen-Thoi et al. [80] have recently formulated a cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of isotropic Mindlin plates by incorporating the CS-FEM with the original MIN3 element [81]. In the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub- triangle, the MIN3 is used to compute the strains. Then, the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The numerical results showed that the CS-MIN3 is free of shear locking and achieves high ac- curacy compared to the exact solutions and other existing elements in the literature. In this paper, the CS-MIN3 is further extended to the kinematic limit analysis of Mindlin plates governed by the von Mises criterion. The CS-MIN3 is combined with a SOCP to determine the limit load of the plates. The limit analysis problem of Mindlin plates is formulated by minimizing the dissipation power subjected to a set of constraints of boundary conditions and unitary external work. For Mindlin plates, the dissipation power is computed on both the middle plane and the thickness of the plate. This minimization problem can then be transformed into a form suitable for the optimum so- lution using the SOCP. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. Methods Limit analysis of Mindlin plates-kinematic formulation We now consider a rigid-perfectly plastic plate identified by its middle plane Ω, the thickness h, and the boundary Γ = Γ ∪ Γ where Γ is the boundary subjected to a sur- u t t face traction λt and Γ is the constrained boundary. Let w be the transverse displace- β β ment (deflection) and β ¼ be the vector of rotations, in which β and β are x y x y the rotations of the middle plane around the y-axis and x-axis, respectively, with the positive directions defined as shown in Figure 1. The unknown vector of three independent field variables at any point in the problem domain of the Mindlin plates can be written as w β β u ¼ ð1Þ x y The curvature of the deflected plate κ and the shear strains γ are defined, respectively, as 2 3 xz 4 5 κ ¼ y ¼L β ; γ ¼ ¼ ∇w þ β ð2Þ yz xy Figure 1 Mindlin plate and positive directions of the displacement w and two rotations β and β . x y Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 4 of 26 http://www.apjcen.com/content/1/1/6 where ∇ ¼½ ∂=∂x ∂=∂y and L is a differential operator matrix defined by 2 3 = 0 ∂x 6 7 6 7 0 = L ¼ ð3Þ d ∂y 4 5 ∂ ∂ = = ∂y ∂x Because the material in the limit analysis is assumed to be rigid-perfectly plastic, the bending stress σ and transverse shear τ are confined within the convex domain Φ(σ, τ) ≤ 0, where Φ(σ, τ) is the yield function. If von Mises's criterion is adopted, we have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T T ΦσðÞ ; τ ¼ðÞ σ P σ þ τ P τ −σ ≤ 0 ð4Þ b s 0 where σ is the yield stress and 2 3 2 −10 1 30 4 5 P ¼ −120 P ¼ ð5Þ b s 00 6 Deformations cannot occur as long as Φ(σ, τ) < 0, while plastic flow may develop when Φ(σ, τ) = 0. In this case, strain rates ε_ and γ_ obey the normality flow rule as ∂Φ ∂Φ _ _ _ ε_ ¼ λ ; γ_ ¼ λ ; λ ≥ 0 ð6Þ ∂σ ∂τ Equation 6 might impose restrictions on strain rates, by confining them within a con- vex domain ΨðÞ ε_ ; γ_ , the sub-space spanned by the outward normals to the yield surface. LetðÞ σ ~; τ ~ represent the admissible stresses contained within the convex yield surface ^ ^ andðÞ σ ; τ represent the stress point on the limit surface associated to any given strain rateðÞ ε_ ; γ_ through the plasticity condition, then the plastic dissipation power dðÞ ε_ ; γ_ (per unit volume) is defined by Hill's maximum principle as T T T T dðÞ ε_ ; γ_ ¼ max σ ~ ε_ þ τ ~ γ_ ¼ σ ^ ε_ þ τ ^ γ_ ð7Þ ΦðÞ σ ~;ε ~ ≤0 The plastic dissipation power is a uniquely defined function of strain rates, and its ex- plicit expression is available for a number of yield criteria [5]. If von Mises's criterion is adopted, one has [37] pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^ T T dðÞ ε_ ; γ_ ¼ σ ε_ Γ ε_ þ γ_ Γ γ_ ð8Þ 0 b s with 2 3 42 0 1 1 −1 −1 4 5 Γ ¼ P ¼ 24 0 Γ ¼ P ¼ ð9Þ b s b s 3 3 00 1 Using the relation ε = zκ between the membrane strain ε with the curvature κ,Equation8 can be rewritten as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T T ^ 2 dðÞ κ_ ; γ_ ¼ σ z c þ c c ¼ κ_ Γ κ_ ; c ¼ γ_ Γ γ_ ð10Þ 0 b s b b s s Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 5 of 26 http://www.apjcen.com/content/1/1/6 The internal dissipation power for the two-dimensional plate domain Ω with the thickness h is now expressed as Z Z Z Z h=2 h=2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^ 2 DðÞ κ_ ; γ_ ¼ d dz dΩ ¼ σ z c þ c dz dΩ ð11Þ 0 b s Ω −h=2 Ω −h=2 The limit analysis of the Mindlin plate considers a rigid-perfectly plastic plate sub- jected to body forces λb ¼½ λp 00 on its middle plane Ω and to surface tractions λt on the boundary Γ . The constrained boundary Γ is fixed. Loads now are defined as t u basic values b and t , affected by a load multiplier λ. Then, the kinematic theorem of limit analysis states that the limit value λ (collapse multiplier) of λ is the optimal value of the minimization problem [37] _ _ λ ¼ min DðÞ κ; γ ð12Þ w _ ;θ subject to u_ w _ ; θ ¼ 0 on Γ ðÞ a T T c ¼ κ_ Γ κ_ ; c ¼ γ_ Γ γ_ ðÞ b b b s s ð13Þ _ _ _ _ _ where κ ¼ L β and γ ¼ ∇w þ β Z Z T T W ðÞ u_ ¼ b u_ dΩ þ t u_ dΓ ¼1cðÞ ext Ω Γ Equations 13(a) and 13(b) express the compatibility of the constrained boundary and the strain rate with a velocity field u_ , respectively, and Equation 13(c) denotes the power of basic loads, which is normalized to unity. Note that in Equation 11, the dissipation power DðÞ κ_ ; γ_ is a positively homogeneous function of degree 1 in the strain rates and not differentiable at strain rate zero. Equations 13(a) to 13(c) hence bring the computation of the collapse multiplier to the search of the minimum of a convex but not everywhere differentiable functional. The functional minimized in Equation 12 is only differentiable in the region Ω where plastic flow develops, but not so in the remaining portion Ω of the plate, which keeps rigid in the mechanism, and hence, the minimum does not correspond to a stationary point. Brief on kinematic formulation of CS-MIN3 for Mindlin plates Kinematic formulation of the MIN3 for Mindlin plates In the original MIN3 [81], the rotations are assumed to be linear through the rotational degrees of freedom (DOFs) at three nodes of the elements, and the deflection is initially assumed to be quadratic through the deflection DOFs at six nodes (three nodes of the elements and three mid-edge points). Then, by enforcing continuous shear constraints at every element edge, the deflection DOFs at three mid-edge points can be removed and the deflection is now approximated only by vertex DOFs at three nodes of the ele- ments. The MIN3 element can hence overcome shear locking and produces convergent solutions. In this paper, we just brief on the kinematic formulation of the MIN3 which is necessary for the kinematic formulation of the CS-MIN3. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 6 of 26 http://www.apjcen.com/content/1/1/6 Using a mesh of three-node triangular elements, the approximation of displacement hi _ _ flow u_ ¼ w _ β β for an element Ω shown in Figure 2 can be written as x y 2 3 N ðÞ x 00 3 3 X X 4 5 _ _ u_ ¼ 0 N ðÞ x 0 d ¼ N ðÞ x d ð14Þ I el I el I¼1 I¼1 00 N ðÞ x |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} N ðÞ x hi _ _ _ where d ¼ w _ β β , I = 1, 2, 3, is the flow vector of the nodal degrees of freedom eI I xI yI of u associated to node I and N (x), I = 1, 2, 3, are linear shape functions at node I. The curvature rates of the deflection flow in an element are then defined by κ_ ¼ Bd ð15Þ _ _ _ where d ¼ d d d is the vector of nodal displacement flow of the element e1 e2 e3 and B contains the constants which are derived from the derivatives of the shape func- tions as 2 3 00 N ;x 4 5 0 N 0 B ¼ ð16Þ ;y 0 N N ;x ;y in which N and N are the matrices of derivatives of the shape functions in the x-direction ,x ,y and y-direction, respectively. The shear strain rates of the deflection flow in an element are then defined by γ_ ¼ Sd ð17Þ Figure 2 Three-node triangular plate element in the MIN3. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 7 of 26 http://www.apjcen.com/content/1/1/6 where N L þNH ;x ;x ;x S ¼ ð18Þ N L H þ N ;y ;y ;y in which L , L , H ,and H are the matrices of derivatives of the shape functions in ,x ,y ,x ,y the x-direction and y-direction, respectively, and L ¼½ L L L and H ¼ 1 2 3 ½ H H H are the vectors of shape functions, with L and H , I=1,2,3,given by 1 2 3 I I 1 1 1 L ¼ðÞ b N N −b N N ; L ¼ðÞ b N N −b N N ; L ¼ðÞ b N N −b N N 1 3 1 2 2 3 1 2 1 2 3 3 1 2 3 2 3 1 1 2 3 2 2 2 ð19Þ 1 1 1 H ¼ðÞ a N N −a N N ; H ¼ðÞ a N N −a N N ; H ¼ðÞ a N N −a N N 1 2 3 1 3 1 2 2 3 1 2 1 2 3 3 1 2 3 2 3 1 2 2 2 ð20Þ in which a and b (i = 1 ÷ 3) are the geometric distances as shown in Figure 2. i i Kinematic formulation of CS-MIN3 In the CS-MIN3 [80], the domain discretization is the same as that of the MIN3 using N nodes and N triangular elements. However, in the formulation of the CS-MIN3, each n e triangular element Ω is further divided into three sub-triangles Δ , Δ ,and Δ by con- e 1 2 3 necting the central point O of the element to three field nodes as shown in Figure 3. In the CS-MIN3, we assume that the vector of displacement flow d at the central eO _ _ _ point O is the simple average of three vectors of displacement flow d , d , and d of e1 e2 e3 three field nodes as _ _ _ _ d ¼ d þ d þ d ð21Þ eO e1 e2 e3 On the first sub-triangle Δ (triangle O-1-2), the linear approximation u_ ¼ hi _ _ w _ β β is constructed by ex ey Δ Δ Δ Δ Δ Δ 1 1 1 1 1 1 _ _ _ _ u_ ¼ N ðÞ x d þ N ðÞ x d þ N ðÞ x d ¼ N ðÞ x d ð22Þ eO e1 e2 e 1 2 3 _ _ _ where d ¼ is the vector of displacement flow of nodal degrees of d d d eO e1 e2 Δ Δ Δ Δ 1 1 1 1 freedom of the sub-triangle Δ and N ¼ is the vector containing N N N 1 2 3 the linear shape functions at nodes O, 1, 2 of the sub-triangle Δ . central point sub-triangle Figure 3 Three sub-triangles (Δ , Δ , and Δ ) created from the triangle 1-2-3 in the CS-MIN3. 1 2 3 Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 8 of 26 http://www.apjcen.com/content/1/1/6 Δ Δ 1 1 The curvature rates of the deflection flow κ_ and the altered shear strain rates γ_ in the sub-triangle Δ are then obtained by 2 3 eO Δ Δ Δ 1 Δ Δ Δ 1 1 1 1 1 4 5 _ κ_ ¼ ¼b d ð23Þ b b b d e1 1 2 3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} e2 2 3 eO Δ Δ Δ 1 Δ Δ Δ 1 1 1 1 1 4 5 _ γ_ ¼ ¼s d ð24Þ s s s d e1 1 2 3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} e2 1 Δ where b and s are respectively computed similarly as the matrices B and S of the MIN3 in Equations 16 and 18 but with two following changes: (1) the coordinates of three-node x ¼½ x y , i = 1, 2, 3, are replaced by x , x , and x , respectively, and (2) i i O 1 2 the area A is replaced by the area A of sub-triangle Δ . e Δ 1 Substituting d in Equation 21 into Equations 23 and 24, and then rearranging, we eO obtain 2 3 e1 1 1 1 Δ Δ 1 Δ Δ Δ Δ Δ 1 1 1 1 1 1 4 5 _ κ_ ¼ ¼ B d ð25Þ b þ b b þ b b d e2 1 2 1 3 1 3 3 3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} e3 2 3 e1 1 1 1 Δ Δ Δ Δ Δ Δ Δ 1 1 1 1 1 1 4 5 1 _ γ_ ¼ s þ s s þ s s ¼ S d ð26Þ d e e2 1 2 1 3 1 3 3 3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} e3 Similarly, by using cyclic permutation, we easily obtain the curvature rates of the de- Δ Δ Δ j j j j flection flow κ_ , the shear strains γ_ , and matrices B and S , j = 2, 3, for the second sub-triangle Δ (triangle O-2-3) and third sub-triangle Δ (triangle O-3-1), respectively. 2 3 Now, by applying the cell-based strain smoothing operation in the CS-FEM [39,41], Δ Δ j j the bending and shear strain rates κ_ and γ_ , j = 1, 2, 3, are used to create the _ _ ~ ~ smoothed bending and smoothed shear strain rates κ and γ , respectively, on the tri- angular element Ω , such as Z Z h Δ κ ~ ¼ κ_ Φ ðÞ x dΩ ¼ κ_ Φ ðÞ x dΩ ð27Þ e e e j¼1 Ω Δ e j Z Z h Δ _ j ~ _ _ γ ¼ γ Φ ðÞ x dΩ ¼ γ Φ ðÞ x dΩ ð28Þ e e j¼1 Ω Δ e j where Φ (x) is a given smoothing function that satisfies the unity property Φ ðÞ x dΩ ¼ 1: Using the following constant smoothing function 1=A x∈Ω e e Φ ðÞ x ¼ ð29Þ 0 x∉Ω where A is the area of the triangular element, the smoothed bending strain rate κ ~ and e e the smoothed shear strain rate γ ~ in Equations 27 and 28 become 3 3 X X 1 1 Δ Δ j j _ _ κ ~ ¼ A κ_ ; γ ~ ¼ A γ_ ð30Þ e Δ Δ j e j A A e e j¼1 j¼1 Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 9 of 26 http://www.apjcen.com/content/1/1/6 Δ Δ j j Substituting κ_ and γ_ , j = 1, 2, 3, into Equation 30, the smoothed bending strain _ _ rate κ ~ and the smoothed shear strain rate γ ~ are expressed by _ _ _ _ ~ ~ κ ~ ¼ Β d ; γ ~ ¼ S d ð31Þ e e e e e ~ ~ where Β and S are the smoothed bending and shear strain gradient matrices given by e e 3 3 X X 1 1 Δ Δ j ~ j Β ¼ A Β ; S ¼ A S ð32Þ e Δ e Δ j j A A e e j¼1 j¼1 Discretization of kinematic formulation by CS-MIN3 In Equation 11, when c becomes small at the thin plate limit, the last term is very nearly singular and numerical integration is preferable. To avoid inaccuracies associated with the point z = 0, twice the integral over half thickness is considered, and Equation 11 can be rewritten as [37] rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z Z h = 2 n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X 2 2 DðÞ κ_ ; γ_ ¼ 2 σ z c þ c d z dΩ ¼ m W 1 þ ζ c þ W c dΩ 0 b s 0 b s g g g Ω 0 g¼1 ð33Þ where ζ =4z/h − 1 and ζ and W are the usual Gauss integration point coordinates and g g weights, respectively; n is the number of Gauss integration points; m = σ h /4 is the G 0 0 plastic moment of resistance per unit width of the plate of thickness h. By discretizing the domain Ω into n triangular plate elements such that Ω ¼ ∪ Ω e e e¼1 and Ω ∩ Ω = ∅, i ≠ j, and using the kinematic formulation of the CS-MIN3 as pre- i j sented in the ‘Brief on kinematic formulation of CS-MIN3 for Mindlin plates’ section, the plastic dissipation in Equation 33 is expressed as rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n n e G XX CS‐DSG3 2 2 D ¼ A m W 1 þ ζ ~c þ W ~c e 0 g b s ðÞ κ_ ;γ_ g g 2 e¼1 g¼1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð34Þ n n e G 2 2 ¼ A m WðÞ 1 þ ζ ~c þ W ~c i 0 i b s i i 2 i¼1 hi hi T T T T _ _ _ _ _ _ _ _ _ _ where ~c ¼ κ ~ Γ κ ~ and ~c ¼ γ ~ Γ γ ~ in which κ ~ ¼ κ ~ γ ~ and γ ~ ¼ γ ~ κ ~ γ ~ b b e s s e y e e e x xy e yz xz are computed at the Gauss points by Equation 31. ~ ~ Combining Equations 9 and 31, c and c are now expressed explicitly as b s hi 2 1 1 1 2 2 2 2 2 2 _ _ _ _ _ _ _ ~c ¼ κ ~ þ κ ~ þ κ ~ þ κ ~ þ γ ~ ; ~c ¼ γ ~ þ γ ~ ð35Þ b x y x y s xy xz yz 3 3 3 3 The plastic dissipation in Equation 34 is hence expressed as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n n e G CS‐DSG3 2 2 2 2 2 2 D ¼ A m z þ z þ z þ z þ z þ z ð36Þ i 0 ðÞ κ_ ;γ_ i1 i2 i3 i4 i5 i6 i¼1 Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 10 of 26 http://www.apjcen.com/content/1/1/6 where pﬃﬃﬃﬃﬃﬃﬃﬃ _ _ ~ ~ z ¼ 2=3WðÞ 1 þ ζ κ þ κ > i1 i i x y pﬃﬃﬃﬃﬃﬃﬃﬃ z ¼ 2=3WðÞ 1 þ ζ κ > i2 i x > pﬃﬃﬃﬃﬃﬃﬃﬃ z ¼ 2=3WðÞ 1 þ ζ κ ~ i3 i y pﬃﬃﬃﬃﬃﬃﬃﬃ ð37Þ z ¼ 2=3WðÞ 1 þ ζ γ ~ > i4 i i xy pﬃﬃﬃﬃﬃﬃﬃﬃ z ¼ 2=3ðÞ 4=h W γ > i5 i xz > pﬃﬃﬃﬃﬃﬃﬃﬃ z ¼ 2=3ðÞ 4=h W γ ~ i6 i yz Equation 37 can be rewritten in the matrix form as H y −z ¼ 0; i ¼ 1; 2; …; n n ð38Þ i e G i i where 2 3 ðÞ 1 þ ζ ðÞ 1 þ ζ 00 0 i i 6 7 ðÞ 1 þ ζ 00 0 0 rﬃﬃﬃ i 6 7 6 7 2 01ðÞ þ ζ 00 0 T i 6 7 H ¼ W ð39Þ 6 7 00 ðÞ1 þ ζ 00 6 7 000 ðÞ4=h 0 000 0 ðÞ4=h and hi _ _ _ _ ~ ~ ~ ~ y ¼ κ ~ κ γ γ γ at the ith Gauss point ð40Þ i x xy xz yz z ¼½ z z z z z z ð41Þ i i1 i2 i3 i4 i5 i6 Note that using Equations 31 and 32, the vector y in Equation 40 can be rewritten in the form of the discrete element displacement flow vector d y ¼ ¼ G d ð42Þ i i γ_ Figure 4 Square plate models and their discretizations using triangular elements. (a) Clamped plate. (b) Simply supported plate. (c) Four forms of discretization of the plate using triangular elements. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 11 of 26 http://www.apjcen.com/content/1/1/6 Figure 5 Convergence of collapse multipliers of the clamped square plate versus the number of Gauss points. where 2 3 ~ ~ 0 B B i11 i12 6 7 ~ ~ 0 B B i21 i22 6 7 6 7 ~ ~ G ¼ ð43Þ i 0 B B i31 i32 4 5 ~ ~ S S 0 i11 i12 ~ ~ S 0 S i21 i22 ~ ~ and B and S are the components extracted, respectively, from the matrices Β and ixx ixx e S in Equation 32. Similarly, the external energy W ðÞ u_ in Equation 13(c) and the boundary condition ext of displacement flow Equation 13(a) can be combined and rewritten in the matrix form of the discrete system displacement flow vector d as [33] A d ¼ b ð44Þ eq eq Figure 6 Convergence of collapse multipliers of the supported square plate versus the number of Gauss points. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 12 of 26 http://www.apjcen.com/content/1/1/6 Table 1 Convergence of collapse multipliers of clamped square plate subjected to uniform pressure versus various DOFs Boundary Method Degrees of freedom Reference condition solution [38] with 75 146 243 363 507 867 DOFs Clamped MIN3 67.2856 25.8273 17.7147 14.8945 13.5918 12.314 CS-MIN3 29.0137 16.9008 14.2239 13.1852 12.6364 Supported MIN3 14.169 8.597 7.1727 6.6701 6.4414 6.289 CS-MIN3 9.4622 7.1383 6.5829 6.3688 6.265 Basic load p = M /L . Combining Equations 36, 38, 42, and 44, the minimization problem (12) associated with the CS-MIN3 now becomes the problem of finding the optimal value λ such that n n e G λ ¼ min A m kk z ð45Þ i 0 i i¼1 subjected to the constraints H G d −z ¼ 0; i ¼ 1; 2; …; n n ðÞ a i i i e G ð46Þ A d ¼ b ðÞ b eq eq The minimization problem (45) is a convex programming problem in which the ob- jective function is a positively homogeneous function of degree 1 in the variables z (or in the strain rates) and is not differentiable at any points in the rigid domain which do not undergo plastic flow (‖z ‖ = 0). The minimization problem (45) is also categorized into the group of the problems of minimizing a sum of Euclidean norms which has a natural dual maximization formulation [23]. For solving this group of problem, one of the well-known approaches used is to replace qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the terms ‖z ‖ in the objective by the differentiable quantity kk z þ μ ,where μ is a i i fixed positive number. This method is robust but converges slowly as μ→ 0 because some of the norms in the objective function have zero as their optimal value [23]. Figure 7 Convergence of collapse multipliers of the clamped square plate subjected to uniform pressure versus various DOFs. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 13 of 26 http://www.apjcen.com/content/1/1/6 Figure 8 Convergence of collapse multipliers of the supported square plate subjected to uniform pressure versus various DOFs. Recently, Andersen et al. [24] recently employed the aspect of duality of the problem to propose a primal-dual interior-point method for solving a homogeneous self-dual model of conic quadratic programming. In this method, the terms ‖z ‖ are also replaced qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ by kk z þ μ , but the quantity μ is treated as an extra variable, whose value is deter- mined by duality estimates. Using this method, the minimization problem (45) is now solved rapidly and accurately even if there are a large number of variables and many norms ‖z ‖ are zero at a solution point. Also, the primal-dual interior-point method is recently integrated in an available general software (e.g., MOSEK [25]) which special- izes second-order cone programming (SOCP) problems [24]. The limit analysis prob- lem can hence be solved efficiently using such software. The minimization problem (45) is hence rewritten in the form of a standard SOCP problem by introducing auxiliary variables t , i =1,2, …, n × n , such that i e G n n e G λ ¼ min A m t ð47Þ i 0 i i¼1 Figure 9 Patterns of displacement and plastic energy dissipation of the clamped square plate at collapse by CS-MIN3. (a) Displacement. (b) Plastic energy dissipation. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 14 of 26 http://www.apjcen.com/content/1/1/6 Figure 10 Patterns of displacement and plastic energy dissipation of the supported square plate at collapse by CS-MIN3. (a) Displacement. (b) Supported plate. subjected to the constraints > H G d −z ¼ 0 ðÞ a i i i > i A d ¼ b ðÞ b eq eq ð48Þ t ≥kk z ðÞ c > i i i ¼ 1; 2; …; n n e G where Equation 48(c) represents quadratic cone constraints. With the form of standard SOCP problem, the minimization problem (47) for finding the collapse multipliers of the Mindlin plates can now be solved efficiently by using the software MOSEK. Note that the formulation of the minimization problem of the plastic dissipation power in the form of standard SOCP problem was also presented in [33,35]; however, the form of standard SOCP problem in these references is only for Kirchhoff plates. Also, note that in the kinematic limit analysis of plates, the ability to obtain the strict upper bound depends not only on the efficient solution of the arising optimization problem but also on the effectiveness of the elements employed. It is required that the flow rule needs hold throughout each element. For the C1-continuous elements, this requirement can be satisfied naturally. However, for the C0-continuous elements, it can Figure 11 Convergence of collapse multipliers of the clamped square plate subjected to uniform pressure for various (L/t). Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 15 of 26 http://www.apjcen.com/content/1/1/6 Figure 12 A rectangular plate and four forms of discretization. (a) A simply supported rectangular plate. (b) Four discretizations of a quarter of plate using triangular elements. be violated due to the appearance of plastic hinge lines on boundaries of elements. In order to overcome this violation, the internal work dissipated in resulting hinge lines on boundaries of elements should be taken into account as done by Hodge and Belytschko [7] and Makrodimopoulos and Martin [82]. In this paper, the CS-MIN3 uses only three-node triangular plate elements and hence belongs to the C0-continuous ele- ments. However, for the sake of simplicity of using the CS-MIN3 in the limit analysis of plates, we can ignore considering the internal work dissipated in resulting hinge lines on boundaries of elements. It is therefore no longer possible to guarantee that the solu- tion obtained from the minimization problem (47) is a strict upper bound on the col- lapse multiplier. However, using the smoothed strain rates which are constant over elements, the flow rule only needs to be enforced at any point in each element, and it is guaranteed to be satisfied almost everywhere in the problem domain. Therefore, the computed collapse load obtained using the proposed method can still be reasonably considered as an upper bound on the actual value. Results and discussion The number performance of the proposed limit analysis will now be tested by examining a number of benchmark uniformly loaded or point-loaded plate problems for which nu- merical solutions have been published in the literature. For all the examples considered, the following were assumed: yield stress σ = 250 MPa and yield moment M = σ t /4. p p p Square plates We now consider a square plate subjected to a uniform out-of-plane pressure loading (with basic load p = M /L ) with two different boundary conditions: (1) clamped sup- ports on all edges as shown in Figure 4a and (2) simply supported supports on all edges as shown in Figure 4b. For this problem, the full plate is considered and the upper Table 2 Convergence of collapse multipliers of clamped rectangular plate subjected to uniform pressure versus various DOFs Method Degrees of freedom Reference solution [33] with 135 273 360 510 1350 DOFs MIN3 62.2853 39.4181 35.883 33.0316 29.88 CS-MIN3 42.2025 33.4743 31.9999 30.7917 Basic load p = M /(LH). p Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 16 of 26 http://www.apjcen.com/content/1/1/6 Figure 13 Convergence of collapse multipliers of the supported rectangular plate subjected to uniform pressure versus DOFs. bound reference solution using quadrilateral elements with 867 degrees of freedom (DOFs) can be found in [37,38]. Figure 4c illustrates four forms of discretization of the plate using triangular elements. In this example, firstly, the thickness of the plate is chosen such that the ratio L/t = 10 and the plate is discretized by the mesh 12 × 12 with 507 DOFs. We first consider the effect of the collapse multipliers when the number of Gauss points along the half of the thickness of the plate is changed from 1 point to 7 points. Figures 5 and 6 show the convergence of the collapse multipliers versus the different numbers of Gauss points for both cases of boundary conditions by MIN3 and CS-MIN3. The results show that both solutions of the CS-MIN3 and MIN3 converge to the upper bound reference solu- tion [38] when the number of Gauss points increases, but those of the CS-MIN3 are more accurate than those of the MIN3. This hence implies that the CS-MIN3 can pro- vide the reliable upper bound collapse multipliers for the Mindlin plates when a suit- able number of Gauss points is used along the half of the thickness of the plate. In these analyses, it is seen that the usage of 6 Gauss points is the most suitable for both cases of boundary conditions and hence will be recommended for default employing in the CS-MIN3 (and also in the MIN3) for all numerical examples in this paper. Next, the convergence of the collapse multipliers versus various degrees of freedom of the system is considered. The numbers of degrees of freedom of the system are now changed from 75 (corresponding to the mesh 4 × 4) to 507 (corresponding to the mesh Figure 14 Patterns of displacement and plastic energy dissipation of the supported rectangular plate at collapse by CS-MIN3. (a) Displacement. (b) Plastic energy dissipation. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 17 of 26 http://www.apjcen.com/content/1/1/6 Figure 15 A rhombic plate and four forms of discretization. (a) A rhombic plate. (b) Four discretizations of a quarter of plate using triangular elements. 12 × 12). The results for both cases of boundary conditions by the CS-MIN3 and MIN3 are listed in Table 1 and plotted in Figures 7 and 8, respectively. The results show that both solutions of the CS-MIN3 and MIN3 converge to the reference solutions when the number of degrees of freedom increases. In addition, Figures 9 and 10 show the patterns of displacement and plastic energy dissipation at collapse for both cases of boundary conditions by using the CS-MIN3. It can be observed that the forms of the yield lines are clearly identified reasonably from these dissipation patterns. These re- sults hence imply that the CS-MIN3 can provide the reliable upper bound collapse multipliers for the Mindlin plates when a suitable number of degrees of freedom is used. Note that in the above analyses, the results of the CS-MIN3 are more accurate than those of the MIN3, especially in the coarse meshes. This hence implies that the cell-based strain smoothing technique in the CS-MIN3 is very necessary to improve the accuracy of the MIN3 in the limit analysis of Mindlin plates. Last, the analysis for the solutions of the thin plate by the CS-MIN3 and MIN3 is performed for the clamped square plate by changing the slenderness ratios (L/t) from 5 to 6,250 with the mesh 12 × 12. Convergence of the collapse multipliers of the clamped square plate versus various slenderness ratios (L/t) by the CS-MIN3 and MIN3 is plot- ted in Figure 11. The Kirchhoff reference results can be found in [33,37]. As expected, Table 3 Convergence of collapse multipliers of clamped rhombic plate subjected to uniform pressure versus various DOFs Skewness Method Degrees of freedom Reference angle α solution [38] with 75 146 243 363 507 867 DOFs 30 CS-MIN3 20.5975 13.109 11.0884 10.2647 9.8309 9.852 MIN3 36.1594 21.4943 16.6904 14.0769 12.4709 45 CS-MIN3 23.8074 14.2608 12.3219 11.4793 11.0211 10.847 MIN3 46.298 24.0417 17.946 14.7981 13.0246 60 CS-MIN3 26.561 15.3363 13.2884 12.3919 11.8979 11.641 MIN3 55.5217 25.6668 18.3164 15.0692 13.3652 75 CS-MIN3 28.4641 16.2358 13.9159 12.9635 12.438 12.143 MIN3 62.9974 26.2878 18.2042 15.0939 13.5819 90 CS-MIN3 29.0137 16.9008 14.2239 13.1852 12.6364 12.314 MIN3 67.2856 25.8273 17.7147 14.8945 13.5918 Basic load p = M /R . p Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 18 of 26 http://www.apjcen.com/content/1/1/6 Figure 16 Convergence of collapse multipliers of clamped rhombic plate for two cases. (a) α = 30°. (b) α = 60°. the solutions of the CS-MIN3 converge to the reference Kirchhoff solution [33] when the slenderness ratio is increased to the limit of the thin plate. This hence shows that the CS-MIN3 is free of shear locking in the limit analysis of thin plates. Also, note that the convergence of solutions of MIN3 is much higher than the expected value when the slenderness ratio is increased to the limit of the thin plate. This hence implies that the cell-based strain smoothing technique in the CS-MIN3 is very necessary to improve the instable behavior of the MIN3 in the limit analysis of thin plates. Rectangular plate We now consider a rectangular plate with simply supported supports on all edges and subjected to a uniform out-of-plane pressure loading (with basic load p = M /(L. H)) as shown in Figure 12a. For this problem, the full plate is considered and the upper bound reference solution using a meshfree method with 1,350 DOFs can be found in [33]. Figure 12b illustrates four forms of discretization using uniform meshes of triangular elements. The convergence of collapse multipliers with respect to the number of degrees of freedom is considered by choosing the thickness of the plate t = 0.01 m, the width L = 2 m, and the ratio L/H = 2. The results by the CS-MIN3 and MIN3 are listed in Table 2 and plotted in Figure 13. In addition, Figure 14 shows the patterns of displacement and plastic energy dissipation at collapse by the CS-MIN3. It is seen that the obtained com- ments from the square plates related to the convergence and accuracy of the CS-MIN3 in the limit analysis of Mindlin plates are confirmed for the rectangular plates. Figure 17 Patterns of plastic energy dissipation rate of clamped rhombic plate at collapse by CS-MIN3 for two cases. (a) α = 30°. (b) α =60°. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 19 of 26 http://www.apjcen.com/content/1/1/6 Figure 18 Collapse multipliers of clamped rhombic plate versus various skewness angles α by the CS-MIN3 and MIN3. Rhombic plate We now consider a clamped rhombic plate with the radius R = 0.5 m and the thickness t = 0.02 m subjected to a uniform out-of-plane pressure loading (with basic load p = M /R ) as shown in Figure 15a. For this problem, the full plate is considered and the upper bound reference solutions using quadrilateral elements with 867 degrees of free- dom can be found in [38]. Figure 15b illustrates four forms of discretization using uni- form meshes of triangular elements. The convergence of collapse multipliers of the clamped rhombic plate with respect to the number of degrees of freedom and various skewness angles α is listed in Table 3 and plotted in Figure 16 for two cases of α = 30° and α = 60°. In addition, the patterns Figure 19 Discretization by 294 triangular elements of the upper right quadrant of the clamped circular plate. The plate was subjected to a uniform out-of-plane pressure loading. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 20 of 26 http://www.apjcen.com/content/1/1/6 Figure 20 Convergence of collapse multipliers of clamped circular plate subjected to uniform pressure versus various DOFs. of the plastic energy dissipation at collapse by the CS-MIN3 for two cases of α = 30° and α = 60° are shown in Figure 17. Again, it is seen that the comments obtained from two previous examples related to the convergence and accuracy of the CS-MIN3 in the limit analysis of Mindlin plates are confirmed for the rhombic plates. In addition, an analysis of collapse multipliers of the clamped rhombic plate with re- spect to various skewness angles α by the CS-MIN3 and MIN3 is illustrated in Figure 18. It is observed that the results of the CS-MIN3 are very close to those of ref- erence solutions, especially for small skewness angles α. These results hence imply that the CS-MIN3 can provide the reliable solutions in the limit analysis of skew Mindlin plates. Circular plate We now consider a clamped circular plate (radius R = 1m and thickness of plate t = 0.01 m) subjected to a uniform out-of-plane pressure loading (with basic load p = M / R ). Due to its symmetry, only the upper right quadrant of the plate is discretized by Figure 21 Patterns of displacement and plastic energy dissipation at collapse of clamped circular plate by CS-MIN3. The plate was subjected to uniform pressure. (a) Displacement. (b) Plastic energy dissipation. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 21 of 26 http://www.apjcen.com/content/1/1/6 Table 4 Convergence of collapse multipliers of clamped circular plate versus various slenderness ratios (2R/t) by CS-MIN3 and MIN3 2R/t Methods Reference solution [38] with MIN3 CS-MIN3 1,041 DOFs 2 10.768 4.174 4.740 4 14.201 7.840 8.778 8 15.389 11.058 11.893 10 15.547 11.694 12.378 20 15.771 12.699 12.990 40 15.851 12.986 13.126 80 15.909 13.065 13.160 100 15.927 13.077 13.165 500 16.872 13.275 13.231 294 triangular elements (507 DOFs) as shown in Figure 19. The upper bound reference solutions using quadrilateral elements with 1,041 DOFs can be found in [38]. The convergence of collapse multipliers with respect to the number of degrees of freedom is plotted in Figure 20, and the patterns of displacement and the plastic energy dissipation at collapse by the CS-MIN3 are shown Figure 21. Again, it is seen that the comments obtained from three previous examples related to the convergence and ac- curacy of the CS-MIN3 in the limit analysis of Mindlin plates are confirmed for the cir- cular plates. Next, due to the availability of the thin plate reference solutions [38] (collapse multi- plier = 13.231), we hence perform a convergent analysis of the collapse multipliers with respect to various slenderness ratios (2R/t) by the CS-MIN3 and MIN3. The results are listed in Table 4 and plotted in Figure 22. As expected, the solutions of the CS-MIN3 again converge to the reference solutions when the slenderness ratio is increased to the limit of the thin plate. This hence confirms again that the CS-MIN3 is free of shear locking in the limit analysis of thin plates. Also, note that the convergence of solutions of MIN3 is much higher than the expected value when the slenderness ratio is Figure 22 Convergence of collapse multipliers of clamped circular plate subjected to a uniform pressure versus various (2R/t). Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 22 of 26 http://www.apjcen.com/content/1/1/6 Figure 23 Equilateral triangle plate and discretization. (a) Equilateral triangle plate. (b) A discretization using triangular elements of the equilateral triangle plate. increased to the limit of the thin plate. This hence confirms again that the cell-based strain smoothing technique in the CS-MIN3 is very necessary to improve the instable behavior of the MIN3 in the limit analysis of thin plates. Equilateral triangle plate We now consider an equilateral triangle plate as shown in Figure 23a with the assigned radius R = 1m and thickness t = 0.04m. The plate is clamped on the boundary and sub- jected to a uniform out-of-plane pressure loading (with basic load p = M /R ). For this problem, the full plate is considered and the upper bound reference solutions can be found in [38]. Figure 23b illustrates a discretization using uniform meshes of triangular elements. The convergence of collapse multipliers with respect to the number of degrees is listed in Table 5 and plotted in Figure 24, and the patterns of displacement and the plastic energy dissipation at collapse by the CS-MIN3 are shown in Figure 25. Again, it is seen that the obtained comments from four previous examples related to the conver- gence and accuracy of the CS-MIN3 in the limit analysis of Mindlin plates are con- firmed for the equilateral triangle plate. Conclusions The paper presents a numerical procedure for the kinematic limit analysis of thick plates governed by the von Mises criterion. The cell-based smoothed three-node Mind- lin plate element (CS-MIN3) is combined with a second-order cone optimization pro- gramming (SOCP) to determine the upper bound limit load of the Mindlin plates. In the CS-MIN3, each triangular element is divided into three sub-triangles, and in each sub-triangle, the gradient matrices of MIN3 are used to compute the strain rates. Then, Table 5 Convergence of collapse multipliers of clamped equilateral triangle plate subjected to uniform pressure versus various DOFs Method Degrees of freedom Reference solution [83] 162 273 360 459 MIN3 15.8781 12.1345 11.1476 10.4487 9.61 CS-MIN3 11.5344 10.6225 10.1818 9.7984 Basic load p = M /R . p Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 23 of 26 http://www.apjcen.com/content/1/1/6 Figure 24 Convergence of collapse multipliers of clamped equilateral triangle plate subjected to uniform pressure versus various DOFs. the gradient smoothing technique on whole the triangular element is used to smooth the strain rates on these three sub-triangles. The limit analysis problem of Mindlin plates is formulated by minimizing the dissipation power subjected to a set of con- straints of boundary conditions and unitary external work. For Mindlin plates, the dis- sipation power is computed on both the middle plane and the thickness of the plate. This minimization problem can then be transformed into a form suitable for optimum solution using the SOCP. Through the formulation and numerical examples, some con- cluding remarks can be drawn as follows: 1. The CS-MIN3 uses only three-node triangular elements that are much easily gener- ated automatically for arbitrary complex geometrical domains. 2. The CS-MIN3 can provide reliable upper bound collapse multipliers for both thick and thin plates. 3. The solutions of the CS-MIN3 converge from the upper bound, and the CS-MIN3 is free of shear locking in the limit analysis of thin plates. Figure 25 Patterns of displacement and plastic energy dissipation at collapse of clamped circular plate by CS-MIN3. The plate was subjected to a uniform pressure. (a) Displacement. (b) Plastic energy dissipation. Trung et al. Asia Pacific Journal on Computational Engineering 2014, 1:6 Page 24 of 26 http://www.apjcen.com/content/1/1/6 4. Compared to the MIN3, the CS-MIN3 is more accurate in the limit analysis of thick plates and more stable in the limit analysis of thin plates. 5. The forms of the yield lines by the CS-MIN3 are identified reasonably from the dis- sipation patterns. In addition, the extension of the present CS-MIN3 for the limit analysis of flat shells using triangular elements is very promising. Competing interests The authors declare that they have no competing interests. Authors' contributions NTT proposed the main idea of extending the CS-MIN3 to the limit analysis of Mindlin plates, carriout out the theory model, numerical discretization model, and revising the manuscript. PVP carried out the numerical results and wrote the first draft of the manuscript. LVC carried out the source code of the limit analysis for Kirchhoff plates. All authors read and approved the final manuscript. Acknowledgements This work was supported by Vietnam National Foundation for Science & Technology Development (NAFOSTED), Ministry of Science & Technology, under the basic research program (Project No.: 107.02-2012.05). Author details Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Hochiminh City, Viet Nam. Department of Mechanics, Faculty of Mathematics and Computer Science, VNUHCM University of Science, Hochiminh City, Viet Nam. Department of Civil Engineering, VNUHCM International University, Hochiminh City, Viet Nam. Received: 9 October 2013 Accepted: 4 December 2013 Published: 29 April 2014 References 1. Hodge PG (1963) Limit analysis of rotationally symmetric plates and shells. Prentice-Hall, Englewood Cliffs 2. Save MA, Massonnet CE (1972) Plastic analysis and design of plates, shells and disks. North-Holland, Amsterdam 3. Zyczkowski M (1981) Combined loadings in the theory of plasticity. Polish Scientific, PWN and Nijhoff, Wasarwa, Poland 4. Xu BY, Liu XS (1985) Plastic limit analysis of structures. China Architecture & Building Press, Beijing 5. Lubliner J (1990) Plasticity theory. Macmillan, New York 6. 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Asia Pacific Journal on Computational Engineering – Springer Journals
Published: Apr 29, 2014
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