Microsoft Word - ETASR_V13_N2_pp10466-10470 Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10466-10470 10466 www.etasr.com Tran & Vo Nguyen: Computation of Limit Loads for Bending Plates Computation of Limit Loads for Bending Plates Trung-Dung Tran Faculty of Civil Engineering, Ho Chi Minh City Open University, Vietnam dung.ttrung@ou.edu.vn Phu-Huan Vo Nguyen Infrastructural Technique Department, Faculty of Civil Engineering, Ho Chi Minh City Open University, Vietnam huan.vnp@ou.edu.vn (corresponding author) Received: 8 January 2023 | Revised: 2 February2 2023 | Accepted: 17 February 2023 ABSTRACT The purpose of this paper is to present a method for calculating the upper bound limit loads of plate bending using a conforming Hsieh-Clough-Tocher (HCT) element. These limit loads can be obtained from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic limit theorem. When combining this theorem with the approximated displacement field, the limit analysis turns into an optimization problem and can be effectively solved by Second-Order Cone Programming (SOCP). Several benchmark plate problems such as square, rectangular, and L-shape plates are investigated to illustrate the effectiveness of the proposed solution. The results of the proposed method show good agreement with the results of previous studies. The maximum error is only 2.91% for the fully clamped rectangular plate problem. Keywords-plate bending; limit analysis; second-order cone programming; plates; shakedown I. INTRODUCTION One of the most significant issues in structure engineering is the determination of the safety load factors. Limit analysis is a well-known efficient approach for resolving the issue. These limit loads can be calculated by using the lower- or upper- bound theorems of limit analysis. However, it is very difficult to apply these two theorems to tackle general problems when using analytical methods [1]. Therefore, the current research is mainly concentrated on computational methods. Authors in [2, 3] carried out experimental tests in order to understand the punching shear behavior of concrete slabs reinforced by steel collars and hybrid fibers. Author in [4] studied the limit state of local instability of steel column structures under axial compression using the energy method combined with the finite element method using fiber beam-column elements. Authors in [5] employed the shaft-grouted method to improve the bearing capacity of barrette piles for high-rise buildings in Ho Chi Minh City, Vietnam. Authors in [6] developed a new effective finite element modeling for predicting the ultimate strength of concrete-filled steel tube columns under axial compression. Authors in [7, 8] studied steel frames under static and earthquake loadings with limit load analysis methods. Various discretization techniques have been developed to treat the limit problems such as finite elements [9-12], meshfree methods [13- 17], and isogeometric analysis [18]. Limit analysis of plate bending has been extensively investigated, analytically and numerically [19, 20]. In [19], limit loads of plate bending were studied in both Kirchhoff and Mindlin plate models based on the upper bound theorem and the finite element method. A dual algorithm combined with DKQ elements for limit and shakedown analyses of plate bending was proposed in [21]. One of the most efficient methods to conduct plate bending limit analysis was developed by Canh [12, 17], where SOCP was utilized in conjunction with conforming the HCT element or the meshless Element‐ Free Galerkin (EFG) method. Based on this approach, the aim of this research is to provide a solution to determine the upper bound load factors of plate bending with the combination of the HCT element and SOCP. However, the limit loads will be determined from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic theorem of limit analysis as in [12]. The advantage of this approach is that it can be easily extended to the shakedown problem. In addition, having calculated the stress components, it is possible to determine the failure mode of structures such as alternating plasticity for non-shakedown problems [22]. II. LIMIT ANALYSIS FORMULATION OF PLATE BENDING BASED ON THE SHAKEDOWN APPROACH Consider a rigid-perfectly plastic plate that has a closed boundary with plane area A. Let A and A denote static and kinematic boundaries respectively. Neglecting the shear deformation, the kinematic relations of thin plates can be described as: Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10466-10470 10467 www.etasr.com Tran & Vo Nguyen: Computation of Limit Loads for Bending Plates 2 x yy 2x xy       ɺ ɺ ɺ ɺ ɺ (1) where ɺ denotes the curvature rate vector, ɺ is the transversal velocity, and the operator 2 is expressed by: 2 2 2 2 2 2 2 T x yx y             Let x yy 2 T x xym m m   m and p be the vector of bending moments and the transversal load, respectively. The equilibrium relation can be defined as:  2 0 T p  m (2) In this study, the von Mises yield condition is adopted, and the internal dissipation power of the plate bending (or the dissipation rate) can be described as: ( )κ κ κQp T p D m=ɺ ɺ ɺ (3) where: 4 2 0 1 2 4 0 3 0 0 1          Q (4) and 2 4p ym h is the plastic limit moment per unit length of a plate section in which h and y denote the plate thickness and the yield stress, respectively. Next, consider a structure simultaneously submitted to NL linearly independently varying loads  ˆ 1, 2,...,kP k NL  which form a convex polyhedral load domain. According to Koiter’s theorem [23], by applying mathematical programming theory, an upper bound of the shakedown load factor  can be rewritten in the form of non-linear programming as follows (the superscript p of dissipation rate function is skipped for simplicity):   1 min NL k A k D dA    ɺκ   2 1 1 . 0 ˆ, 1, NL k k u NL T e k k A k s t on A x P dA                     κ m ɺ ɺ ɺ ɺ κ κ (5) where m e is the fictitious elastic moment vector of the point x on the domain A. Note that, when NL = 1 the problem (5) reduces that of limit analysis. Then, the formulation for the upper bound limit load is determined as follows:   2 min . 0 1, P A u eT A D dA s t on A dA                 κ κ m κ ɺ ɺ ɺ ɺ ɺ (6) Compared to the kinematic theorem of limit analysis as presented in [12], the difference of (6) is that the work rate of the applied load constraint is replaced by the external work rate constraint. In general, problem (6) is complicated because it has to calculate the stress components. However, as mentioned above, it can be advantageous to develop the shakedown problem. In addition, the failure mode of alternating plasticity can be determined easily when the structure is subjected to repeated loads. III. HCT-BASED KINEMATIC FORMULATION The conforming HCT element has been detailed in [12, 24]. The key idea of the HCT element is to divide an original triangular element into three sub-elements as shown in Figure 1. Fig. 1. HCT element. The number of the total degrees of freedom of the element is 12 including 3 degrees at each corner node and 3 degrees at three mid-side nodes. In terms of area coordinates  1 2 3, ,    , at each sub-triangle, we can express the displacement expansion  k w as follows:             0 , 1, 2, 3kk ke ew k    L L F q (7) where qe is the element displacement matrix ανδ the partitions    ke L and    0 k L denote the interpolation functions, respectively. F is the matrix determined from compatible conditions. Then, the curvatures can be defined by: ( ) ( ) ( ) ( ) , 0, , ( ) ( ) ( ) ( ) ( ) , 0, , ( ) ( ) ( ) ( ) , 0, , ) k k k k xx e xx xx e xx k k k k k yy e yy yy e e yy e k k k k xy e xy xy e xy                                    L L F B κ L L F q B q L L F B (8) and we can obtain the plastic dissipation of the plate as: Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10466-10470 10468 www.etasr.com Tran & Vo Nguyen: Computation of Limit Loads for Bending Plates     3 1 κ κ      Qɺ ɺ ne NG T p j j j j D m (9) where ne and NG are the total number of elements and Gaussian points, respectively. The upper-bound limit analysis based on the shakedown approach of plate bending is now described as:         3 1 3 1 min . 1 0 κ κ κ κ                       Q B q m q ɺ ɺ ɺ ɺ ɺ ɺ ne NG T p j j j j j j ne NG e T i j j j u m s t on A (10) By formulating the objective function to a form comprising a sum of norms, the optimization problem (10) can be re- expressed as a standard SOCP as follows [25]:   3 1 κ     J ɺ ne NG T p j j j D m (11) where J is defined as: 2 0 0 1 1 3 0 3 0 0 1            J (12) It can be seen that J is also the Cholesky factor of Q. Finally, by establishing supplementary variables x1, x2,…, xne3NG, problem (10) can be stated as follows:     3 1 3 1 min 1 . 0 , 1, 2,..., 3 κ κ                           m q ρ J ρ ɺ ɺ ɺ ne NG p k k k ne NG e T j j j j T k k k k m x s t on A x k ne NG (13) where ρk kx denotes quadratic cones and the k are supplementary variables. IV. NUMERICAL EXAMPLES To illustrate the effectiveness of the proposed solution, in this section we investigate various benchmark problems that have numerical or analytical solutions. The following parameters are adopted for all examples: length a = 10m and plate thickness h = 0.1m. The first example considers a square plate with side length a, and restraints on all edges that are either simply supports or clamped. The plate is subjected to uniform pressure p, as shown in Figure 2. Due to symmetry, we only modeled the upper-right quarter of the plate as illustrated in Figure 3. The results of the proposed method, normalized with mp/a 2 , are presented in Table I. It can be observed that when the number of the elements increases, the collapse load factors reduce and converge. When compared with previous results, the present results are in good agreement with those obtained numerically in [18, 19, 21], in which the EFG method, a C1 continuity plate element, and the the DKQ element were used, respectively. In addition, Table I also shows that the results of the proposed method are similar to the results obtained analytically in [20, 26]. The yield patterns in terms of plastic dissipation distribution are plotted in Figure 4. It can be seen that they are very consistent with the experiment, for both simply supported and clamped plates. (a) (b) Fig. 2. Square slabs: geometry and loading. (a) Simply supported plate, (b) clamped plate. Fig. 3. Square slab: finite element mesh. TABLE I. LIMIT LOAD FACTOR OF THE SQUARE PLATE (pa2/mp) Method Simply supported Clamped UB LB UB LB Proposed Ne = 50 25.18 - 51.83 - Ne = 200 25.06 - 48.17 - Ne = 800 25.03 - 46.19 - Ne = 1800 25.03 - 45.58 - [19] 25.02 - 45.29 - [17] 25.01 - 45.07 - [21] 25.04 25.04 45.06 45.06 [20] 26.54 24.86 49.25 42.86 [26] 27.71 23.81 52.01 - Ne: number of elements, LB – Lower bound, UB – Upper bound Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10466-10470 10469 www.etasr.com Tran & Vo Nguyen: Computation of Limit Loads for Bending Plates (a) (b) Fig. 4. Square slabs: plastic dissipation distribution. (a) Simply supported plate, (b) clamped plate. Fig. 5. Rectangular slab: geometry and loading. Fig. 6. Rectangular slab: finite element mesh. TABLE II. LIMIT LOAD FACTOR OF THE RECTANGULAR PLATE (pab/mp) Method Simply supported Clamped Proposed 29.90 56.20 [19] 29.88 - [17] 29.88 54.61 (a) (b) Fig. 7. Rectangular slabs: plastic dissipation distribution. (a) Simply supported plate, (b) clamped plate. Next, we consider the rectangular slab of Figure 5 (dimensions ab and b/a = 2). Similar to the previous example, only the upper-right quarter of the plate is modeled by 800 elements (Figure 6). Limit load factors are reported in Table II. When compared with the previously obtained results, the proposed method provides better solutions than in [17, 19] by 0.07% and 2.91% for the case of simply supported and clamped edges, respectively. The plastic dissipation distribution of rectangular slabs is plotted in Figure 7. The last example is an L-shape plate, investigated in the case of uniform load. The geometry and uniform mesh of the L-shape plate are shown in Figure 8 and the limit load factors are shown in Table III. Again, the results of the proposed method show good agreement with previous studies. The maximum error is only 2.59% for the case of clamped edges. The plastic dissipation distribution using 2400 elements is plotted in Figure 9. (a) (b) Fig. 8. L-shape plate. (a) Geometry, (b) Mesh. TABLE III. LIMIT LOAD FACTOR OF THE L-SHAPE PLATE (pa2/mp) Method Simply supported Clamped Proposed 6.20 16.24 [10] 6.29 - [15] 6.17/6.04 15.83/15.69 (a) (b) Fig. 9. L-shape slabs: Plastic dissipation distribution. (a) Simply supported, (b) clamped. V. CONCLUSION This paper presented a numerical solution for the computation of the limit loads of bending plates. Using von Mises yield criterion and the conforming HCT element, the upper bound limit load can be derived from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the upper bound theorem of limit analysis. Based on this approach, the work rate of the applied load constraint in the kinematic theorem of limit analysis is replaced by the external work rate constraint. The formulation based on Koiter’s kinematic shakedown theorem is more complicated than the Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10466-10470 10470 www.etasr.com Tran & Vo Nguyen: Computation of Limit Loads for Bending Plates formulations for the upper bound of the limit load, however, it can be developed to the shakedown problem and the failure mode of alternating plasticity can be determined easily. One benefit of this research is that the problems can be formulated as SOCP, so that they can be solved rapidly, even when many variables are involved. The accuracy and reliability of the proposed solution method were proven by numerical examples. ACKNOWLEDGMENT This research is funded by Ho Chi Minh City Open University under Grant No. E2020.01.1 REFERENCES [1] W. Prager and P. G. Hodge Jr., Theory of Perfectly Plastic Solids, 1st ed. New York, NY, USA: John Wiley & Sons, 1951. [2] A. A. Abdulhussein and M. H. 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