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yahoo.com Faculty of Civil Engineering, Hanoi This paper develops a new finite element method (FEM)-based upper bound Architectural University, Nguyen Trai algorithm for limit and shakedown analysis of hardening structures by a direct Street, Thanh Xuân District, Hanoi, Vietnam plasticity method. The hardening model is a simple two-surface model of plasticity Full list of author information is with a fixed bounding surface. The initial yield surface can translate inside the available at the end of the article bounding surface, and it is bounded by one of the two equivalent conditions: (1) it always stays inside the bounding surface or (2) its centre cannot move outside the back-stress surface. The algorithm gives an effective tool to analyze the problems with a very high number of degree of freedom. Our numerical results are very close to the analytical solutions and numerical solutions in literature. Keywords: Ratchetting; Kinematic hardening; Two-surface plasticity; Shakedown; FEM Background Shakedown analysis for hardening structures has been investigated by many researchers. Among hardening models, the isotropic hardening law is generally not reasonable in situa- tions where structures are subjected to cyclic loading because it does not account for the Bauschinger effect and rejects the possibility of incremental plasticity. The unbounded kine- matic hardening model has already been introduced theoretically by Melan [1] and later by Prager [2]. Applications of this model have been investigated by Maier [3] and Ponter [4]. The unbounded kinematic hardening model cannot estimate the plastic collapse and also in- cremental plasticity but only low-cycle fatigue, while low-cycle fatigue limit with the ki- nematical hardening model seems not to be essentially different from the perfectly plastic model, cf. Gokhfeld and Cherniavsky [5] and Stein and Huang [6]. Introducing a bounding surface in Melan-Prager's model, a two-surface model of plasticity with a fixed bounding surface is achieved which appears to be most basic, suitable and simple for shakedown analysis. Application of bounded kinematic harden- ing model was introduced theoretically and numerically by Weichert and Groß-Weege [7] who used the generalized standard material model (GSM). They used Airy's stress function to satisfy the equilibrium conditions in the interior of the structures fulfilled. Shakedown theorems for bounded linear and nonlinear kinematic hardening have been proposed by Bodovillé and de Saxcé [8], Pham [9,10] and Nguyen [11]. Numerical investigations for bounded kinematic hardening using basic reduction technique have been introduced by Staat and Heitzer [12,13] and Stein and Zhang [14]. By the lower bound approach, it permits to avoid the nondifferentiability of the © 2014 Phạm and Staat; 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 cited. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 2 of 13 http://www.apjcen.com/content/1/1/4 objective function, which must be regularized via internal dissipation energy and there is no incompressibility constraint in nonlinear programming problem, but this ap- proach suffers from nonlinear inequality constraints. A company of lower bound algorithm is the upper bound algorithm, which is based on Koiter theorem. For perfectly plastic structures, the upper bound algorithm has been estab- lished by Yan and Nguyen Dang [15,16] and Yan et al. [17]. The major numerical obstacle in this approach is the singular property of plastic dissipation function. Dealing with this diffi- p p p p 2 culty, the researchers replaced the original dissipation function D ε_ by D ε_ þ ε ,where ij ij 0 ε is a very small number. This technique is also used in our algorithm. By using the static approach and the criterion of the mean, Nguyen Dang and König [18] showed that the shakedown solution can be obtained by a maximization or a minimization problem. The yield criterion of the mean was further applied in practical computations by displacement method and equilibrium finite element by Nguyen Dang and Palgen [19]. A very efficient primal-dual algorithm, which can derive lower and upper bound sim- ultaneously of shakedown limit load factor for complicated structures, has been intro- duced by Vu, Yan and Nguyen Dang [20-22] and Vu [23]. In these works, dual relationship between upper bound and lower bound for shakedown analysis of perfectly plastic structures has been proven. Theoretically speaking, primal-dual algorithm helps to find a very accurate solution of shakedown analysis problem. While using the finite element method (FEM) for limit and shakedown analysis, the stress method can be used, but this method is restricted since for certain structures, it is very difficult to find appropriate stress function, so the displacement method is pre- ferred to make the numerical approach as general as possible. For the structures with hardening material, it is difficult to prove the relationship between upper bound and lower bound because of the complication of the objective function. Fur- thermore, in the static approach, it is difficult to present alternating limit and ratcheting limit separately. In this paper, we have presented a FEM-based upper bound algorithm for shake- down analysis of bounded kinematic hardening structures with von Mises yield criterion. By the direct plasticity methods, shakedown analysis is a nonlinear programming problem. The present algorithm can deal with complicated realistic structures which are modelled by 3D, 20-node elements with huge number of degree of freedom. Two numerical examples are in- cluded to validate the algorithm and to study the influence of hardening effect. Methods Bounded kinematic hardening model For kinematic hardening model, the initial yield surface can translate in the multi-axial stress space, without changing its shape and size. If the translation is unlimited, or in other words, the ultimate strength of material σ is infinite, we have unbounded model (Figure 1). This model is inadequate to predict the plastic collapse (both incremental and instantaneous) of structure. It can only describe the alternating plasticity mode. The initial yield surface for von Mises material is defined as below F½ σ −σ ¼ 0: ð1Þ The subsequent surface is defined as F½ σ−π −σ ¼ 0 ð2Þ where π is the back stress. If hardening is unbounded, π is infinite. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 3 of 13 http://www.apjcen.com/content/1/1/4 Figure 1 Unbounded kinematic hardening model. For more realistic material, yield stress σ must be bounded by ultimate strength σ . y u A simple two-surface model is used to model the bounded hardening. The subsequent yield surface may or may not touch the fixed bounding surface; see Figure 2. This is satisfied by one of the two following conditions: 1. Centre of subsequent yield surface cannot move outside the back-stress surface. This is expressed by F½ π ≤ σ −σ : ð3Þ u y 2. Subsequent yield surface always stays inside bounding surface. This is expressed by F½ σ ≤σ : ð4Þ Figure 2 A simple two-surface plasticity with fixed bounding surface. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 4 of 13 http://www.apjcen.com/content/1/1/4 In the preceding conditions, Equations 3 and 4, equalities occur when the subsequent surface touches bounding surface. We have proven that bounding conditions (3) and (4) are exactly equivalent. See detail in the study of Pham and Staat [24]. Shakedown formulation based on Koiter's theorem Problem establishment Upper bound solution of shakedown load multiplier is the solution of a constrained nonlinear programming problem Z Z p p α ¼ min D ðÞ ε dV dt ðÞ a sd ε_ 0 V p p > Δε ¼ ε_ dt ðÞ b > 0 > p trðÞ ε_ ¼0cðÞ ð5Þ p 1 Δε ¼ ∇ðÞ Δu þ ∇ðÞ Δu in V ðÞ d s:t: : Δu ¼ 0 on ∂V ðÞ e Z Z E p σ ðÞ x; t : ε_ dVdt ¼1gðÞ 0 V p p where total plastic energy dissipation D ðÞ ε_ in the structure is as follows: T T rﬃﬃ rﬃﬃ Z Z Z Z Z 2 2 p p p p _ _ D ðÞ ε dVdt ¼ σ kk ε dVdt þ σ −σ kk Δε dV ð6Þ y u y 3 3 0 V 0 V V The first term in the right hand side of Equation 6 is plastic energy dissipation of per- fect plasticity material, and the second term is hardening effect. Evidently, if σ = σ ,we u y have ideal plastic material. Constraint (5b) is the definition of plastic strain accumulation. The plastic strain rate p p ε_ may not necessarily be compatible, but Δε must be compatible. This is expressed by constraints (5d) and (5e). Constraint (5c) is the incompressibility condition, and (5g) is the normalized condition. Problem discretization Based on FEM, whole structure V is discretized into n finite elements with NG = n × n Gaussian points, where n is number of Gaussian points in each element. If e g g the load domain L is convex, it is sufficient to check if shakedown will happen at all vertices of L. So the load domain can be discretized into finite number of load combinations P , k =1, …, m,and m is total number of vertices of L.Bythese dis- cretizations, the shakedown analysis is reduced to checking shakedown conditions at all Gaussian points and all load vertices m, instead of checking for whole Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 5 of 13 http://www.apjcen.com/content/1/1/4 structure V andentireloaddomain L. Then, numerical form of Equation 5 is as follows: () sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m NG NG m m XX X X X pﬃﬃ pﬃﬃ blkh 2 T 2 2 2 T 2 2 2 2 α ¼ min σ w ε Dε þ w ε þ σ −σ w ε D ε þ w ε ðÞ a y ik u y ik 3 i i 0 3 i ik i 0 sd ik ε_ k¼1 i¼1 i¼1 k¼1 k¼1 s:t: : ε ¼ B u ∀i ¼ 1; NG ðÞ b > ik i k¼1 D ε ¼ 0 ∀i ¼ 1; NG ∀k ¼ 1; m ðÞ c M ik m NG XX > T E : w ε σ ¼1dðÞ ik ik i¼1 k¼1 ð7Þ blkh where α denotes the shakedown multiplier in bounded linearly kinematic hardening. sd ε is the strain vector corresponding to load vertex k at Gaussian point i ik ik ik ik ik ik ik ε ¼ ε ε ε 2ε 2ε 2ε ik 11 22 33 12 23 13 ð8Þ ik ik ik ik ik ik ¼ ε ε ε γ γ γ : 11 22 33 12 23 13 σ is the fictitious elastic stress vector corresponding to load vertex k at Gaussian ik point i, u is the nodal displacement vector, B is the deformation matrix and ε is the i 0 small number to avoid singularity. D and D are square matrices, expressed in Equation 9: 2 3 1 110 00 6 7 1 110 00 6 7 6 7 1 1 1 1 110 00 D ¼ Diag ; D ¼ : ð9Þ 11 1 M 6 7 0 000 00 2 2 2 6 7 4 5 0 000 00 0 000 00 For the sake of simplicity, we define some new plastic strain e , fictitious elastic ik stress t , deformation matrix B , respectively as ik i 1 1 = = 1= E 2 2 ^ 2 e ¼ w D ε ; t ¼ D σ ; B ¼ w D B : ð10Þ ik i ik ik i i i ik Then Equation (7) becomes () sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m NG NG m m XX X X X pﬃﬃ blkh 2 T 2 2 α ¼ σ min e e þ ε þ a e e þ ε ðÞ a y e_ ik ik sd 3 ik ik ik k¼1 i¼1 i¼1 k¼1 k¼1 > ^ > e ¼ B u ∀i ¼ 1; NG ðÞ b ik i k¼1 ð11Þ s:t: : D e ¼ 0 ∀i ¼ 1; NG ∀k ¼ 1; m ðÞ c M ik > 3 m NG >XX > T e t ¼1dðÞ : ik ik k¼1 i¼1 where a ¼ σ −σ =σ : ð12Þ u y y to solve problem (11), using penalty function method for constraints (11b) and (11c), combined with Lagrange multiplier method for constraint (11d). Penalty Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 6 of 13 http://www.apjcen.com/content/1/1/4 function F and Lagrange function F are expressed in Equations 13 and 14, P PL respectively. ( sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NG m m m X X X X T 2 2 F ¼ e e þ ε þ a e e þ ε P ik ik ik ik i¼1 k¼1 k¼1 k¼1 ! !) m m m X X X c c ^ ^ þ e D e þ e −B u e −B u M ik ik i ik i ik 2 2 k¼1 k¼1 k¼1 s:t: : ð13Þ m NG XX e t ¼ 1 ik ik k¼1 i¼1 ( sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NG m m m X X X X 2 T 2 F ¼ a e e þ ε þ e e þ ε PL ik ik ik ik i¼1 k¼1 k¼1 k¼1 ! !) ! m m m m NG X X X XX c c T T ^ ^ þ e D e þ e −B u e −B u þ α e t −1 M ik ik i ik i ik ik ik 2 2 i¼1 k¼1 k¼1 k¼1 k¼1 ð14Þ Algorithm 0 0 Step 1: Choose starting point: displacement and strain vectors u and e such that the normalized condition (11d) is satisfied: NG m XX T 0 t e ¼ 1 ð15Þ ik ik i¼1 k¼1 Step 2: Calculate du, de ,(α + dα) from current values of u, e ik −1 −1 ~ ~ ~ ~ > du ¼ −u þ S f þðÞ α þ dα S f 1 2 ! ! m m X X −1 −1 −1 −1 −1 −1 −1 −1 ~ ~ ~ ~ ~ ~ > de ¼ M N Q M −M β þðÞ α þ dα M N Q M −M β ik ik ik : 1 2 ik i ik ik ik i ik ik k¼1 k¼1 ð16Þ where NG ~ ^ ~ ^ S ¼ B E B ð17Þ i i i¼1 ! sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NG m NG m m qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m X X X X X X X T T −1 −1 T ~ T 2 2 ^ ~ ^ ~ f ¼ B E e − B Q M a e e_ e þ ε þ e e e þ ε 1 i ik ik ik ik ik i i i ik ik ik i¼1 k¼1 i¼1 k¼1 k¼1 k¼1 k¼1 NG m X X T −1 −1 ^ ~ − B Q M cb D e ik M ik i i ik i¼1 k¼1 ð18Þ NG m X X T −1 −1 ~ ^ ~ f ¼ − B Q M t b ð19Þ 2 ik ik i i ik i¼1 k¼1 −1 −1 ~ ~ E ¼ I −cQ b M ð20Þ i i ik i ik k¼1 Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 7 of 13 http://www.apjcen.com/content/1/1/4 −1 Q ¼ I þ M N ð21Þ i ik i ik k¼1 ! sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m X X M ≈ e e þ ε I þ cb D ð22Þ ik ik ik ik M ik k¼1 k¼1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N ≈ a e e þ ε þ cb I ð23Þ ik ik ik ik ik sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X X T T 2 2 b ¼ e e þ ε e e þ ε ð24Þ ik ik ik ik ik k¼1 k¼1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m X X X T T 2 2 β ¼ a e e e þ ε þ e e e þ ε þ cD e b ik ik ik ik M ik ik 1 ik ik k¼1 k¼1 k¼1 ^ ^ þ c e −B u b −cb B du ð25Þ ik i ik ik i 1 k¼1 β ¼ t −cB du ð26Þ ik i 2 NG m NG m XX XX T T 1− t e þðÞ de t ðÞ de ik ik ik ik 1 ik 1 i¼1 i¼1 k¼1 k¼1 ðÞ α þ dα¼ ¼ − ð27Þ NG m NG m XX XX T T t ðÞ de t ðÞ de ik ik ik 2 ik 2 i¼1 k¼1 i¼1 k¼1 Step 3: Perform a line search to find λ such that λ ¼ FðÞ u þ λdu; e þ λde → min ð28Þ u P Update displacement u, plastic strain e ik u ¼ u þ λ du ðÞ a ð29Þ e ¼ e þ λ de ðÞ b ik ik u ik Step 4: Check convergence criteria: if they are all satisfied, then stop; otherwise go to step 2. Figure 3 Continuous beam. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 8 of 13 http://www.apjcen.com/content/1/1/4 Figure 4 Load domain for example 4.1. Results and discussions Two examples are reported. To compare the results on shakedown limit for perfectly plastic materials with other researches, we choose σ = σ . To investigate the effect of u y bounded hardening, we choose σ < σ <2σ . When σ ≥ 2σ , we have unbounded kine- y u y u y matic hardening model. Continuous beam The continuous steel beam is described in Figure 3 subjected to uniform distributed loads: p and P vary independently in the domain: p ∈ [1.2, 2], p ∈ [0, 1]. The load 1 2 1 2 domain is described in Figure 4. 5 2 The material mechanical properties are Young's modulus, E = 1.8 ⋅ 10 N/mm ; yield stress, σ =100 N/mm ; ultimate strength, σ =1.35σ and Poisson's ratio, ν =0.3. By the y u y symmetryofthe problem,onlyhalfof the structure is discretized into 589 elements, 8-node quadrangle, Figure 5. The structure is considered as a plane stress problem. Numerical limit and shakedown analysis for this structure made of perfectly plastic material were presented in Garcea et al. [25] and Tran et al. [26]. Table 1 shows the results of limit and shakedown analysis. Present results are close to others in literature. Interaction diagram of shakedown load multiplier is plotted in Figure 6. In this struc- ture, when p is not very large, the structure fails in ratcheting mode, and benefit of hardening is quite clear. Cylindrical pipe under complex loading This closed-end pipe is investigated for perfectly plastic material in Vu [23] using primal-dual shakedown algorithm. The structure is subjected to bending M and Figure 5 FEM mesh. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 9 of 13 http://www.apjcen.com/content/1/1/4 Table 1 Comparison of plastic limit collapse and shakedown results Limit Shakedown p ∈½ 1:2;2 Author [p , p ] = [2.0, 0.0] [p , p ]=[0.0, 1.0] [p , p ]=[1.2, 1.0] [p , p ] = [2.0, 1.0] 1 2 1 2 1 2 1 2 p ∈½ 0;1 Garcea et al. [25] 3.280 8.718 5.467 3.280 3.244 Tran et al. [26] 3.402 9.192 5.720 3.388 3.377 Present (perfectly 3.300 8.744 5.500 3.300 3.264 plastic) Present (kin. 4.455 11.804 7.425 4.455 4.406 hardening) torsion M moments, internal pressure p and axial tension T. Material properties are 5 2 2 Young's modulus, E = 2.1 ⋅ 10 N/mm ; yield stress, σ = 160 N/mm ; ultimate strength, σ = 1.25σ and Poisson's ratio, ν = 0.3. Using 20-node 3D elements to model whole u y structure with the dimensions: length L = 2, 700 mm, mean radius r = 300 mm and thickness h = 60 mm, see Figure 7. The analytical solutions of plastic collapse limit for cylindrical pipe under complex loading can be cited from Vu [23]. Pure bending capacity: 2 6 M ¼ 4σ hr þ ¼ 3647:52⋅10 Nmm: ð30Þ b lim y Pure torsion capacity: 2 6 pﬃﬃﬃ M ¼ πr hσ ¼ 3134:24⋅10 Nmm: ð31Þ t lim y Pure tension capacity: T ¼ 2πrhσ ¼ 18095573:6N: ð32Þ lim y Pure internal pressure capacity: p ¼ σ ¼ 32 N=mm ð33Þ lim Figure 6 Interaction diagram for shakedown bounds of continuous beam. The results are not normalized. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 10 of 13 http://www.apjcen.com/content/1/1/4 Figure 7 FEM mesh of cylindrical pipe. and the normalized load multiplier when bending, internal pressure and tension are combined is as follows: 2 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4−3n n −2n π 6 φ x 7 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m ¼ cos4 5; ð34Þ 2 2 4−3n where m ¼ M=M b lim n ¼ p=p : ð35Þ lim n ¼ T =T x lim If the axial tension force comes from only internal pressure on closed ends, then n = n /2, and formula (35) can be rewritten as x φ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4−3n m ¼ ð36Þ FE analysis is fulfilled for structure subjected to combined internal pressure p and bending M . Results are presented in Table 2, normalized by pure bending capacity in Table 2 Limit and shakedown load multipliers of cylindrical pipe subjected to internal pressure and bending Load combination Elastic Limit factor Shakedown Shakedown Shakedown factor factor (perfectly factor (perfectly factor (bounded (unbounded plastic) plastic) hardening) hardening) 0.0p_1.0 M 0.7338 1.0012 0.7338 0.7338 0.7338 0.2p_1.0 M 0.7228 0.9870 0.7297 0.7310 0.7304 0.4p_1.0 M 0.7011 0.9478 0.7228 0.7236 0.7231 0.6p_1.0 M 0.6570 0.8914 0.7131 0.7132 0.7134 0.8p_1.0 M 0.6023 0.8267 0.7011 0.7013 0.7014 1.0p_1.0 M 0.5509 0.7608 0.6667 0.6855 0.6853 1.0p_0.8 M 0.6168 0.8540 0.7696 0.8128 0.8127 1.0p_0.6 M 0.6921 0.9556 0.8906 0.9894 0.9894 1.0p_0.4 M 0.7727 1.0546 1.0179 1.2318 1.2346 1.0p_0.2 M 0.8486 1.1306 1.1204 1.3874 1.5506 1.0p_0.0 M 0.9019 1.1589 1.1586 1.4482 1.8091 Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 11 of 13 http://www.apjcen.com/content/1/1/4 Figure 8 Interaction diagram for limit bounds. Comparison between analytical and numerical solutions. formula (28) and pure internal pressure in formula (33). Limit analysis is implemented for σ /σ = 1.0 to be compared to formula (36), and interaction diagram is plotted in u y Figure 8. Shakedown analysis with and without hardening effect is implemented for the load domain: p ∈ [0, 1]; M ∈ [−1, 1]. Interaction diagram is plotted in Figure 9. Figure 8 shows that the present results of limit analysis for σ /σ = 1 are close to ana- u y lytical solutions. Figure 9 shows that the hardening effect is clear if the applied moment is less than 0.5M .If σ ≥ 2σ , bounded hardening model becomes unbounded, and b lim u y shakedown limit of structure cannot exceed two times of elastic limit. Conclusions The paper developed a new upper bound algorithm for shakedown analysis of elastic plastic-bounded linearly kinematic hardening structures. This is an efficient tool for prac- tical computation, especially for complicated structures subject to mechanical loads. Figure 9 Interaction diagram for elastic and shakedown bounds, normalized by pure plastic collapse limits, M and p b lim lim. Phạm and Staat Asia Pacific Journal on Computational Engineering 2014, 1:4 Page 12 of 13 http://www.apjcen.com/content/1/1/4 The proposed algorithm gives results that are close to the results in literatures. If σ = σ , u y it leads to perfectly plastic material; if σ ≥ 2σ , it leads to unbounded kinematic hardening u y material; otherwise, σ < σ <2σ , we have bounded kinematic hardening material. y u y pp blkh Let α ; α ; and α denote respectively elastic limit, shakedown limit for elastic per- el sd sd fectly plastic and shakedown limit for bounded kinematic hardening material, respect- ively, then: pp blkh pp α ≤ α ≤ α ≤2α : el sd sd sd In the preceding expression, the left equality occurs if the subsequent yield surface translates inside the bounding surface, the middle equality occurs if the subsequent yield surface fixed on the bounding surface and the last equality occurs when yield sur- face translates unboundedly. If the structure shakes down in alternating plasticity mode, then there is no difference between perfectly plastic and kinematic hardening models. Competing interests The authors declare that they have no competing interests. Author details Faculty of Civil Engineering, Hanoi Architectural University, Nguyen Trai Street, Thanh Xuân District, Hanoi, Vietnam. Faculty of Medical Engineering and Technomathematics, Aachen University of Applied Science, Jülich Campus, Heinrich-Mußmann-Str. 1Jülich 52428, Germany. Received: 15 August 2013 Accepted: 30 December 2013 Published: 29 April 2014 References 1. Melan E (1938) Zur Plastizität des räumlichen Kontinuums. Ing-Arch 8:116–126 2. Prager W (1956) A new method of analyzing stress and strain in work hardening plastic solids. J Appl Mech ASME 23:493–496 3. Maier GA (1973) Shakedown matrix theory allowing for work hardening and second-order geometric effects. In: Sawczuk A (ed) Foundations of plasticity. Springer, North-Holland, Amsterdam, pp 417–433 4. Ponter ARS (1975) A general shakedown theorem for elastic plastic bodies with work hardening. In: Proc. SMiRT-3, paper L5/2 5. Gokhfeld DA, Cherniavsky OF (1980) Limit analysis of structures at thermal cycling. 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Nguyen Dang H, Palgen L (1980-81) Shakedown analysis by displacement method and equilibrium finite element. Transac CSME 6(Nr. 1):32–39 20. Vu DK, Yan AM, Nguyen Dang H (2003) A dual form for discretized kinematic formulation in shakedown analysis. Int J Solids Struct 41(1):267–277 21. Vu DK, Yan AM, Nguyen Dang H (2004) A primal-dual algorithm for shakedown analysis of structures. Comp Methods Appl Mech Eng (Elsevier) 193(42–44):4663–4674 22. Yan AM, Vu DK, Nguyen Dang H (2004) Dual in kinematical approaches of limit and shakedown analysis of structures. In: David Y (ed) Complimentarily, duality and symmetry in nonlinear mechanics, vol 6. Gao, Kluwer Academic Publishers, Springer, Netherlands, pp 127–148 23. Vu DK (2001) Dual Limit and Shakedown analysis of structures. PhD Thesis. Université de Liège, Belgium 24. Phạm PT, Staat M (2013) An upper bound algorithm for limit and shakedown analysis of bounded linearly kinematic hardening bodies. In: De Saxcé G et al. (ed) Direct Methods. Springer, Netherlands 25. Garcea G, Armentano G, Petrolo S, Casciaro R (2005) Finite element shakedown of two-dimensional structures. Int J Numer Mech Engng 63:1174–1202 26. Tran TN, Liu GR, Nguyen XH, Nguyen TT (2010) An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int J Numer Engng 82:917–938 doi:10.1186/2196-1166-1-4 Cite this article as: Phạm and Staat: FEM-based shakedown analysis of hardening structures. Asia Pacific Journal on Computational Engineering 2014 1:4. Submit your manuscript to a journal and beneﬁ t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the ﬁ eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com
Asia Pacific Journal on Computational Engineering – Springer Journals
Published: Apr 29, 2014
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