# RETRACTED ARTICLE: A limit analysis of Mindlin plates using the cell-based smoothed triangular element CS-MIN3 and second-order cone programming (SOCP)

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## Abstract

### Background

The paper presents a numerical procedure for kinematic limit analysis of Mindlin plate governed by von Mises criterion.

### Methods

The cell-based smoothed three-node Mindlin plate element (CS-MIN3) is combined with a second-order cone optimization programming (SOCP) to determine the upper bound limit load of the Mindlin plates. In the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the gradient matrices of MIN3 is used to compute the strain rates. Then 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 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 determining the limit loads of a structure. The fundamental theorems of limit analysis ignore 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 those by Hodge and Belytschko [7] and Nguyen [8]. However, due to the lack of efficient 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, 10, 11, 12, 13, 14, 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 developing 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 problem of optimization involving either linear programing (LP) or nonlinear programming (NLP) which can be solved respectively by the available LP or NLP algorithms [16, 17, 18, 19, 20, 21, 22, 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 structures, 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, 34, 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 literature 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, 42, 43], node-based smoothed FEM (NS-FEM) [44, 45, 46], edge-based smoothed FEM (ES-FEM) [47, 48], face-based smoothed FEM (FS-FEM) [49], and a group of alpha-FEM [50, 51, 52, 53]. Each of these smoothed FEMs has different properties and has been used to produce desired solutions for a wide class of benchmark 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 investigated and applied to various problems such as plates and shells [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], piezoelectricity [69, 70], fracture mechanics [71], visco-elastoplasticity [72, 73, 74], limit and shakedown analysis for solids [75, 76, 77], and some other applications [78, 79].

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 accuracy 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 solution 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

*h*, and the boundary Γ = Γ

_{ u }∪ Γ

_{ t }where Γ

_{ t }is the boundary subjected to a surface traction Open image in new window and Γ

_{ u }is the constrained boundary. Let

*w*be the transverse displacement (deflection) and Open image in new window be the vector of rotations, in which

*β*

_{ x }and

*β*

_{ y }are the rotations of the middle plane around the

*y-*axis and

*x*-axis, respectively, with the positive directions defined as shown in Figure 1.

**κ**and the shear strains

**γ**are defined, respectively, as

**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

**Φ**(

**σ**,

**τ**) < 0, while plastic flow may develop when

**Φ**(

**σ**,

**τ**) = 0. In this case, strain rates Open image in new window and Open image in new window obey the normality flow rule as

Equation 6 might impose restrictions on strain rates, by confining them within a convex domain Open image in new window , the sub-space spanned by the outward normals to the yield surface.

**ϵ**=

*z*

**κ**between the membrane strain

**ϵ**with the curvature

**κ**, Equation 8 can be rewritten as

*h*is now expressed as

_{ t }. The constrained boundary Γ

_{ u }is fixed. Loads now are defined as basic values

**b**and Open image in new window , 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]

Equations 13(a) and 13(b) express the compatibility of the constrained boundary and the strain rate with a velocity field Open image in new window , respectively, and Equation 13(c) denotes the power of basic loads, which is normalized to unity.

Note that in Equation 11, the dissipation power Open image in new window 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_{ p } where plastic flow develops, but not so in the remaining portion Ω_{ r } 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 elements. 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.

*Ω*

_{ e }shown in Figure 2 can be written as

*I*= 1, 2, 3, is the flow vector of the nodal degrees of freedom of Open image in new window associated to node

*I*and

*N*

_{ I }(

**x**),

*I*= 1, 2, 3, are linear shape functions at node

*I*.

**B**contains the constants which are derived from the derivatives of the shape functions as

in which **N**_{,x} and **N**_{,y} are the matrices of derivatives of the shape functions in the *x*-direction and *y*-direction, respectively.

**L**

_{,x},

**L**

_{,y},

**H**

_{,x}, and

**H**

_{,y}are the matrices of derivatives of the shape functions in the

*x*-direction and

*y*-direction, respectively, and Open image in new window and Open image in new window are the vectors of shape functions, with

*L*

_{ I }and

*H*

_{ I },

*I*= 1, 2, 3, given by

in which *a*_{ i } and *b*_{ i } (*i* = 1 ÷ 3) are the geometric distances as shown in Figure 2.

#### Kinematic formulation of CS-MIN3

*N*

_{ n }nodes and

*N*

_{ e }triangular elements. However, in the formulation of the CS-MIN3, each triangular element Ω

_{ e }is further divided into three sub-triangles Δ

_{1}, Δ

_{2}, and Δ

_{3}by connecting the central point

*O*of the element to three field nodes as shown in Figure 3.

*O*is the simple average of three vectors of displacement flow Open image in new window , Open image in new window , and Open image in new window of three field nodes as

_{1}(triangle

*O*-1-2), the linear approximation Open image in new window is constructed by

where Open image in new window is the vector of displacement flow of nodal degrees of freedom of the sub-triangle Δ_{1} and Open image in new window is the vector containing the linear shape functions at nodes *O*, 1, 2 of the sub-triangle Δ_{1}.

_{1}are then obtained by

where Open image in new window and Open image in new window 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 Open image in new window , *i* = 1, 2, 3, are replaced by **x**_{ O }, **x**_{1}, and **x**_{2}, respectively, and (2) the area *A*_{ e } is replaced by the area Open image in new window of sub-triangle Δ_{1}.

Similarly, by using cyclic permutation, we easily obtain the curvature rates of the deflection flow Open image in new window , the shear strains Open image in new window , and matrices Open image in new window and Open image in new window , *j* = 2, 3, for the second sub-triangle *Δ*_{2} (triangle *O*-2-3) and third sub-triangle *Δ*_{3} (triangle *O*-3-1), respectively.

*j*= 1, 2, 3, are used to create the

*smoothed*bending and

*smoothed*shear strain rates Open image in new window and Open image in new window , respectively, on the triangular element

*Ω*

_{ e }, such as

_{ e }(

**x**) is a given smoothing function that satisfies the unity property Open image in new window Using the following constant smoothing function

*A*

_{ e }is the area of the triangular element, the smoothed bending strain rate Open image in new window and the smoothed shear strain rate Open image in new window in Equations 27 and 28 become

*j*= 1, 2, 3, into Equation 30, the smoothed bending strain rate Open image in new window and the smoothed shear strain rate Open image in new window are expressed by

### Discretization of kinematic formulation by CS-MIN3

*c*

_{ s }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]

where *ζ* = 4*z*/*h* − 1 and *ζ*_{ g } and *W*_{ g } are the usual Gauss integration point coordinates and weights, respectively; *n*_{ G } is the number of Gauss integration points; *m*_{0} = *σ*_{0}*h*^{2}/4 is the plastic moment of resistance per unit width of the plate of thickness *h*.

*Ω*into

*n*

_{ e }triangular plate elements such that Open image in new window and

*Ω*

_{ i }∩

*Ω*

_{ j }= ∅,

*i*≠

*j*, and using the kinematic formulation of the CS-MIN3 as presented in the ‘Brief on kinematic formulation of CS-MIN3 for Mindlin plates’ section, the plastic dissipation in Equation 33 is expressed as

where Open image in new window and Open image in new window in which Open image in new window and Open image in new window are computed at the Gauss points by Equation 31.

**y**

_{ i }in Equation 40 can be rewritten in the form of the discrete element displacement flow vector Open image in new window

and Open image in new window and Open image in new window are the components extracted, respectively, from the matrices Open image in new window and Open image in new window in Equation 32.

*λ*

^{+}such that

The minimization problem (45) is a convex programming problem in which the objective function is a positively homogeneous function of degree 1 in the variables **z**_{ i } (or in the strain rates) and is not differentiable at any points in the rigid domain which do not undergo plastic flow (||**z**_{ i }|| = 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 the terms ||**z**_{ i }|| in the objective by the differentiable quantity Open image in new window , where *μ* is a 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].

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**_{ i }|| are also replaced by Open image in new window , but the quantity *μ* is treated as an extra variable, whose value is determined 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**_{ i }|| 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 specializes second-order cone programming (SOCP) problems [24]. The limit analysis problem can hence be solved efficiently using such software.

*t*

_{ i },

*i*= 1, 2, …,

*n*

_{ e }×

*n*

_{ G }, such that

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 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 elements. 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 solution obtained from the minimization problem (47) is a strict upper bound on the collapse 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 numerical solutions have been published in the literature. For all the examples considered, the following were assumed: yield stress *σ*_{ p } = 250 MPa and yield moment *M*_{ p } = *σ*_{ p }*t*^{2}/4.

### Square plates

*p*=

*M*

_{ p }/

*L*

^{2}) with two different boundary conditions: (1) clamped supports 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 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.

*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 solution [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 provide the reliable upper bound collapse multipliers for the Mindlin plates when a suitable 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.

**Convergence of collapse multipliers of clamped square plate subjected to uniform pressure versus various DOFs**

Boundary condition | Method | Degrees of freedom | Reference solution [38] with 867 DOFs | ||||
---|---|---|---|---|---|---|---|

75 | 146 | 243 | 363 | 507 | |||

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 |

*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 plotted in Figure 11. The Kirchhoff reference results can be found in [33, 37]. As expected, 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

*p*=

*M*

_{ p }/(

*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.

*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 comments 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.

**Convergence of collapse multipliers of clamped rectangular plate subjected to uniform pressure versus various DOFs**

Method | Degrees of freedom | Reference solution [33] with 1350 DOFs | |||
---|---|---|---|---|---|

135 | 273 | 360 | 510 | ||

MIN3 | 62.2853 | 39.4181 | 35.883 | 33.0316 | 29.88 |

CS-MIN3 | 42.2025 | 33.4743 | 31.9999 | 30.7917 |

### Rhombic plate

*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*

_{ p }/

*R*

^{2}) 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 freedom can be found in [38]. Figure 15b illustrates four forms of discretization using uniform meshes of triangular elements.

*α*is listed in Table 3 and plotted in Figure 16 for two cases of

*α*= 30° and

*α*= 60°. In addition, the patterns 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.

**Convergence of collapse multipliers of clamped rhombic plate subjected to uniform pressure versus various DOFs**

Skewness angle | Method | Degrees of freedom | Reference solution [38] with 867 DOFs | ||||
---|---|---|---|---|---|---|---|

75 | 146 | 243 | 363 | 507 | |||

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 |

*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 reference 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

*R*= 1m and thickness of plate

*t*= 0.01 m) subjected to a uniform out-of-plane pressure loading (with basic load

*p*=

*M*

_{ p }/

*R*

^{2}). Due to its symmetry, only the upper right quadrant of the plate is discretized by 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].

*R*/

*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 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.

**Convergence of collapse multipliers of clamped circular plate versus various slenderness ratios** (2*R*/*t*) **by CS-MIN3 and MIN3**

2 t | Methods | Reference solution [38] with 1,041 DOFs | |
---|---|---|---|

MIN3 | CS-MIN3 | ||

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 |

### Equilateral triangle plate

*R*= 1m and thickness

*t*= 0.04m. The plate is clamped on the boundary and subjected to a uniform out-of-plane pressure loading (with basic load

*p*=

*M*

_{ p }/

*R*

^{2}). 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.

**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 |

## Conclusions

- 1.
The CS-MIN3 uses only three-node triangular elements that are much easily generated 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.

- 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 dissipation patterns.

In addition, the extension of the present CS-MIN3 for the limit analysis of flat shells using triangular elements is very promising.

## Notes

### 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).

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