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In this paper, a gradient stable node-based smoothed discrete shear gap method (GS-DSG) using 3-node triangular elements is presented for Reissner–Mindlin plates in elastic-static, free vibration, and buckling analyses fields. By applying the smoothed Galerkin weak form, the discretized system equations are obtained. In order to carry out the smoothing operation and numerical integration, the smoothing domain associated with each node is defined. The modified smoothed strain with gradient information is derived from the Hu–Washizu three-field variational principle, resulting in the stabilization terms in the system equations. The stabilized discrete shear gap method is also applied to avoid transverse shear-locking problem. Several numerical examples are provided to illustrate the accuracy and effectiveness. The results demonstrate that the presented method is free of shear locking and can overcome the temporal instability issues, simultaneously obtaining excellent solutions.

Thin-walled structures (shells) render a majority of engineering structures, and as one special case of shells, the plate has been widely used in mechanical, civil, marine, aerospace, and other engineering science fields. The analyses of plate structures in elastic-static, free vibration, and buckling fields stand for the key three aspects in their engineering application. There exist two first-order plate theories, namely, the Kirchhoff plate theory and the Reissner–Mindlin one. Kirchhoff theory is usually applied to thin structures with negligible shear strain, and the C^{1}-continuous shape function is required. In view of its simplicity and efficiency, the lower-order Reissner–Mindlin plate which considered the shear effects is appealing in practical and only requires the

Up to now, the finite element method (FEM) still holds its place as the most widely used numerical tool to simulate different behaviors of plates [

NS-FEM can be regarded as a modified model of FEM. It has very attractive properties and can be easily applied to tetrahedral or triangular elements without any modification of formulas and procedures. NS-FEM wins the favor recently for its prominent inherent properties [

In this work, a gradient stable node-based smoothed discrete shear gap method (GS-DSG) using 3-node triangular elements is formulated for elastic-static, free vibration, and buckling analyses of Reissner–Mindlin plates. In order to overcome the temporal instability problem encountered in the nodal integration process, the smoothed Galerkin weak form is applied by using the strain smoothing technique with gradient information, which is derived from the Hu–Washizu three-field variation principle. The stabilized discrete shear gap method is also incorporated into the presented method to avoid the transverse shear locking and improve the accuracy of the present formulation. The numerical examples presented herein demonstrate that the present method is both free of shear locking and temporal instability. It also achieves high accuracy compared with the exact solutions and other existing methods in the literature.

The outline of this paper is as follows.

Based on the assumption of the first-order shear-deformation plate theory, the displacements in the Cartesian coordinate system can be expressed as follows:

Positive directions of the deflection and two rotations.

The relevant strain vector

By applying the principle of virtual work, the weak form can be stated as follows:

Based on the assumption of the first-order shear-deformation plate theory, the weak form for the free vibration analysis of the Reissner–Mindlin plate can be derived from the dynamic form of energy principle, i.e.,

For the buckling analysis, when the plate is subjected to in-plane prebuckling stresses

Equation (

The bounded domain

Substituting equation (

From equations (

Substituting equations (

In equation (

For free vibration, we have

For the buckling analysis, we have

According to the introduction in

Based on the Hu–Washizu three-field variational principle, Duan [

Assume that

Clearly, if the interpolated strain

Then, a form containing only independent variables can be obtained as simple as the classical one:

Deriving by reference [

Different from reference [

In the nodal integration scheme, node

Substitution of equations (

By solving equation (

The equation for

To simplify the calculation, the equivalent circle domain can be assumed for the subdomain

It should be noted that the concept of equivalent circles is only introduced to simplify the calculation of these integrals, and the actual smooth region is still a polygon composed of elements and nodes.

In this part, the nodal integration formulation will be introduced. We can discretize the problem domain

Triangular elements and smoothing cells associated with nodes.

From equations (

The discrete shear gap method is adopted here to eliminate the shear locking. In each triangular element, the nodes are denoted anticlockwise as

From equations (

To improve significantly the accuracy of approximate solutions and to stabilize shear force oscillations presenting the triangular element, a stabilization technique [

We now seek for a weak form solution of the generalized displacement field

Substituting equations (

The summation in equation (

For free vibration, we have

For the buckling analysis, we have

The nodal geometrical stiffness matrix

In this section, static, free vibration, and buckling analyses of square, T-shape, elliptical, and rectangular plates are considered. In addition, the present method is compared with other three methods, the FEM-DSG, NS-FEM, and NS-DSG methods. To examine the numerical error precisely, the displacement error norm is defined as

In the following example, material parameters’ Young's modulus is expressed as

Consider the model of a simply supported square plate subjected to a uniform load

A square plate with a simply supported boundary condition.

Table

Numerical results of normalized central deflection

Mesh | FEM-DSG | NS-FEM | NS-DSG | GS-DSG |
---|---|---|---|---|

8 × 8 | 0.005442 | 0.005678 | 0.005567 | 0.005519 |

16 × 16 | 0.005516 | 0.005585 | 0.005554 | 0.005540 |

24 × 24 | 0.005532 | 0.005563 | 0.005550 | 0.005544 |

32 × 32 | 0.005538 | 0.005555 | 0.005549 | 0.005545 |

Reference | 0.005545 |

Central deflection.

Convergence properties.

In this section, a T-shaped plate with clamped edges and subjected to two kinds of loads, i.e., uniform and concentrated loads, is analyzed to further examine the efficiency of the present method. The geometric parameters are shown in Figure

Geometry illustration of a T-plate.

Figure

Domain discretization using triangular.

Deflections along the line OA with different thickness (uniformly distributed load): (a)

Deflections along the line OA with different thickness (concentrated load): (a)

In this section, numerical examples of free vibration for various plates are given. The nondimensional frequency parameter is normalized by

Square plates of length

First, the SSSS plate corresponding to length-to-width ratios

(a) Supported plate, (b) clamped plate, and (c, d) triangular meshes.

A nondimensional frequency parameter

Method | Mode sequence number | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

0.005 | FEM-DSG | 4.8067 | 8.5038 | 9.2631 | 12.3353 | 16.6573 | 17.9400 |

4.5190 | 7.3529 | 7.5123 | 9.7058 | 11.2071 | 11.2798 | ||

4.4545 | 7.0943 | 7.1344 | 9.0748 | 10.2148 | 10.2180 | ||

4.4440 | 7.0499 | 7.0680 | 8.9591 | 10.0481 | 10.0521 | ||

NS-FEM | 3.8523 | 4.6180 | 4.6696 | 5.4803 | 5.5805 | 5.7068 | |

4.2873 | 4.7464 | 4.9528 | 5.4243 | 5.9702 | 6.2474 | ||

4.3931 | 4.8182 | 4.8238 | 5.2862 | 5.8463 | 6.1054 | ||

4.4156 | 4.7659 | 4.7660 | 5.4864 | 6.2891 | 6.2921 | ||

NS-DSG | 4.1952 | 6.2489 | 6.3781 | 7.4032 | 7.5154 | 8.2944 | |

4.3191 | 6.7445 | 6.7573 | 8.4370 | 9.3131 | 9.3758 | ||

4.3853 | 6.9234 | 6.9248 | 8.7009 | 9.7366 | 9.7462 | ||

4.4053 | 6.9658 | 6.9667 | 8.7753 | 9.8308 | 9.8348 | ||

GS-DSG | 4.2893 | 6.7888 | 6.9313 | 8.3816 | 9.5817 | 9.6035 | |

4.3637 | 6.8888 | 6.8966 | 8.6707 | 9.6991 | 9.7403 | ||

4.4101 | 6.9687 | 6.9729 | 8.7846 | 9.8260 | 9.8393 | ||

4.4228 | 6.9918 | 6.9945 | 8.8249 | 9.8738 | 9.8807 | ||

Reference [ | 4.443 | 7.025 | 7.025 | 8.886 | 9.935 | 9.935 | |

0.1 | FEM-DSG | 4.6700 | 7.9713 | 8.5811 | 10.3261 | 10.6011 | 10.6017 |

4.3964 | 6.9971 | 7.1328 | 8.9615 | 9.7532 | 10.1849 | ||

4.3200 | 6.7565 | 6.7874 | 8.4201 | 9.4119 | 9.4185 | ||

4.3012 | 6.7086 | 6.7216 | 8.3093 | 9.2733 | 9.2794 | ||

NS-FEM | 3.7366 | 4.4102 | 4.4397 | 5.1403 | 5.2270 | 5.3701 | |

4.1438 | 4.4389 | 4.5862 | 5.0152 | 5.4576 | 5.6616 | ||

4.2327 | 4.4534 | 4.4553 | 4.8923 | 5.3434 | 5.5359 | ||

4.2534 | 4.4057 | 4.4058 | 5.0280 | 5.6971 | 5.6991 | ||

NS-DSG | 4.1026 | 5.9418 | 6.0366 | 6.7521 | 6.7843 | 7.1540 | |

4.2175 | 6.4611 | 6.4728 | 7.3821 | 7.6795 | 7.7982 | ||

4.2675 | 6.6158 | 6.6158 | 7.6205 | 7.8446 | 7.8451 | ||

4.2759 | 6.6439 | 6.6458 | 7.7018 | 7.8603 | 7.8604 | ||

GS-DSG | 4.1863 | 6.4643 | 6.5416 | 7.7482 | 8.5051 | 8.6428 | |

4.2546 | 6.5859 | 6.5877 | 8.0866 | 8.9413 | 8.9637 | ||

4.2816 | 6.6463 | 6.6487 | 8.1817 | 9.0967 | 9.0979 | ||

4.2830 | 6.6576 | 6.6593 | 8.2002 | 9.1314 | 9.1365 | ||

Reference [ | 4.37 | 6.74 | 6.74 | 8.35 | 9.22 | 9.22 |

First 6 modes of the square plate (SSSS) solved by different methods with

Modes | FEM-DSG | NS-FEM | NS-DSG | GS-DSG |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 |

The second problem is that the mesh of CCCC square plate as shown in Figure

A nondimensional frequency parameter

Method | Mode sequence number | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

0.005 | FEM-DSG | 8.6743 | 14.6071 | 18.4781 | 22.1215 | 28.3891 | 29.1794 |

6.3839 | 9.4693 | 9.7732 | 12.4103 | 13.8892 | 14.2365 | ||

6.0740 | 8.7422 | 8.8097 | 10.8089 | 11.9473 | 11.9823 | ||

6.0284 | 8.6355 | 8.6554 | 10.5653 | 11.6629 | 11.6897 | ||

NS-FEM | 4.8662 | 27.2994 | 27.4030 | 30.5485 | 31.2468 | 33.7182 | |

5.5113 | 5.6369 | 5.6908 | 6.1573 | 6.7345 | 6.8578 | ||

5.1463 | 5.1567 | 5.6597 | 5.8989 | 6.2153 | 6.5428 | ||

4.9637 | 4.9639 | 5.7411 | 5.9520 | 6.5102 | 6.5153 | ||

NS-DSG | 6.2399 | 7.7833 | 14.5143 | 16.8787 | 18.5504 | 19.4584 | |

8.8488 | 8.0945 | 8.2608 | 9.9613 | 10.4157 | 10.8271 | ||

5.9356 | 8.3981 | 8.4178 | 10.1743 | 11.1144 | 11.1576 | ||

5.9660 | 8.4810 | 8.4879 | 10.2770 | 11.2890 | 11.3201 | ||

GS-DSG | 7.1391 | 10.4788 | 14.7362 | 17.0639 | 18.8060 | 19.5371 | |

5.9902 | 8.4671 | 8.6174 | 10.6220 | 11.2868 | 11.6486 | ||

5.9694 | 8.4856 | 8.5019 | 10.3112 | 11.3058 | 11.3398 | ||

5.9807 | 8.5191 | 8.5249 | 10.3363 | 11.3724 | 11.4008 | ||

Reference [ | 5.999 | 8.568 | 8.568 | 10.407 | 11.472 | 11.498 | |

0.1 | FEM-DSG | 7.0255 | 10.2284 | 11.3539 | 11.3850 | 11.8340 | 12.8015 |

5.9576 | 8.4011 | 8.6489 | 10.3774 | 11.2084 | 11.3691 | ||

5.7609 | 7.9968 | 8.0525 | 9.5838 | 10.4278 | 10.4803 | ||

5.7274 | 7.9271 | 7.9514 | 9.4369 | 10.2517 | 10.2995 | ||

NS-FEM | 4.6154 | 6.3432 | 6.7867 | 6.8568 | 7.0858 | 7.6405 | |

4.8119 | 4.9749 | 5.3796 | 5.4743 | 5.9144 | 5.9380 | ||

4.5501 | 4.5572 | 5.1111 | 5.5108 | 5.6206 | 5.7290 | ||

4.4494 | 4.4494 | 5.2545 | 5.6658 | 5.7524 | 5.7533 | ||

NS-DSG | 5.7131 | 7.0630 | 7.8426 | 8.2222 | 8.3801 | 8.4698 | |

5.5540 | 7.4852 | 7.5508 | 7.7638 | 7.9153 | 8.2500 | ||

5.6544 | 7.7558 | 7.7675 | 7.9294 | 7.9310 | 8.0354 | ||

5.6797 | 7.8190 | 7.8242 | 7.9132 | 7.9133 | 7.9705 | ||

GS-DSG | 6.1305 | 8.3987 | 8.9986 | 10.2059 | 10.3095 | 10.3511 | |

5.6505 | 7.7377 | 7.7683 | 9.2385 | 9.8542 | 9.8966 | ||

5.6787 | 7.8130 | 7.8225 | 9.2477 | 9.9985 | 10.0421 | ||

5.6902 | 7.8433 | 7.8479 | 9.2831 | 10.0575 | 10.1036 | ||

Reference [ | 5.71 | 7.88 | 7.88 | 9.33 | 10.13 | 10.18 |

First 6 modes of the square plate (CCCC) solved by different methods with

Modes | FEM-DSG | NS-FEM | NS-DSG | GS-DSG |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 |

In this example, we further studied five different boundary conditions: SSSF, SFSF, CCCF, CFCF, and CFSF. In this case, a 16 × 16 regular mesh is adopted for square plates with different boundary conditions. The square of the nondimensional frequency parameter

A nondimensional frequency parameter

Plate type | Method | Mode sequence number | |||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||

SSSF | FEM-DSG | 11.7066 | 27.9376 | 42.1347 | 60.7093 |

NS-FEM | 11.5084 | 23.1688 | 23.2168 | 26.8704 | |

NS-DSG | 11.3960 | 26.6128 | 40.2935 | 56.4514 | |

GS-DSG | 11.5498 | 27.0919 | 40.7145 | 57.6083 | |

Reference [ | 11.685 | 27.756 | 41.197 | 59.066 | |

SFSF | FEM-DSG | 9.6797 | 16.1112 | 37.0384 | 39.8117 |

NS-FEM | 9.5491 | 15.7288 | 23.1230 | 23.1655 | |

NS-DSG | 9.5757 | 15.2989 | 34.8264 | 38.2995 | |

GS-DSG | 9.5996 | 15.7468 | 35.4943 | 38.5924 | |

Reference [ | 9.631 | 16.135 | 36.726 | 38.945 | |

CCCF | FEM-DSG | 24.3065 | 41.2621 | 65.6722 | 80.5606 |

NS-FEM | 23.4263 | 25.5865 | 25.6584 | 30.8248 | |

NS-DSG | 23.4858 | 39.0341 | 60.8599 | 73.7164 | |

GS-DSG | 23.6981 | 39.4852 | 61.9860 | 74.9583 | |

Reference [ | 24.020 | 40.039 | 63.493 | 76.761 | |

CFCF | FEM-DSG | 22.4444 | 26.9890 | 45.1686 | 63.3315 |

NS-FEM | 21.7548 | 24.7811 | 24.8217 | 25.7548 | |

NS-DSG | 21.7768 | 25.8569 | 42.4465 | 58.8471 | |

GS-DSG | 21.9631 | 26.1317 | 42.9097 | 59.9521 | |

Reference [ | 22.272 | 26.529 | 43.664 | 64.466 | |

CFSF | FEM-DSG | 15.3251 | 20.7977 | 40.5433 | 50.8558 |

NS-FEM | 14.9958 | 20.0826 | 23.9089 | 23.9517 | |

NS-DSG | 15.0267 | 19.8965 | 38.1048 | 48.1404 | |

GS-DSG | 15.1039 | 20.2503 | 38.7071 | 48.7522 | |

Reference [ | 15.285 | 20.673 | 39.882 | 49.500 |

In this section, a simply supported elliptical plate is considered. The geometric parameters of the plate are shown in Figure

Geometric illustration.

To illustrate the benefits of triangular grids, we use an unstructured mesh layout with 446 nodes, as shown in Figure

Domain discretization.

First modes of the elliptical plate solved by different methods.

Modes | FEM-DSG | NS-FEM | NS-DSG | GS-DSG | Reference |
---|---|---|---|---|---|

1 | |||||

2 | |||||

3 | |||||

4 | |||||

5 | |||||

6 | |||||

7 | |||||

8 | |||||

9 | |||||

10 | |||||

11 | |||||

12 |

First 12 parameterized natural frequencies

Modes | FEM-DSG | NS-FEM | NS-DSG | GS-DSG | Reference |
---|---|---|---|---|---|

1 | 6.6093 | 6.3951 | 6.463 | 6.4857 | 6.3595 |

2 | 11.938 | 8.587 | 11.513 | 11.572 | 11.369 |

3 | 19.611 | 9.151 | 18.496 | 18.662 | 18.425 |

4 | 23.984 | 9.579 | 22.042 | 22.388 | 22.203 |

5 | 29.916 | 10.671 | 27.367 | 27.782 | 27.628 |

6 | 33.244 | 10.903 | 29.873 | 30.402 | 30.158 |

7 | 43.224 | 11.273 | 38.241 | 39.089 | 39.012 |

8 | 44.833 | 11.661 | 39.064 | 39.934 | 39.874 |

9 | 54.672 | 13.268 | 45.674 | 47.269 | 47.500 |

10 | 59.026 | 13.963 | 49.719 | 51.119 | 51.421 |

11 | 59.871 | 15.246 | 50.818 | 52.405 | 52.582 |

12 | 69.465 | 15.655 | 56.690 | 58.618 | 58.565 |

Relative errors for the first 12 parameterized natural frequencies of the elliptical plate.

In the following examples, we use the proposed method to study the critical buckling load of rectangular plates with different length-width ratios and different edge loads, as shown in Figure

Rectangular plates: (a) model; (b) axial compression; (c) biaxial compression; (d) shear in-plane.

Firstly, a square plate with thickness

The critical buckling load factor

The buckling load factor

Load | FEM-DSG | NS-FEM | NS-DSG | GS-DSG | Wang ( | Tham [ | Timoshenko [ |
---|---|---|---|---|---|---|---|

UC | 3.98 | 3.91 | 3.88 | 3.98 | 3.93 | 4.00 | 4.00 |

BC | 1.99 | 1.96 | 1.94 | 1.99 | 1.96 | 2.00 | 2.00 |

SP | 9.29 | 8.95 | 9.02 | 9.37 | 9.42 | 9.40 | 9.33 |

The buckling modes of the simply supported square with different types of edge loading: (a) axial compression; (b) biaxial compression; (c) shear in-plane.

In this section, uniaxial compression rectangular plates with different length-to-width ratios and thickness-to-width ratios are considered. Simply supported boundary conditions are assumed. Four types of length-to-width ratios,

The critical buckling load factors solved by different schemes are given in Table

The factor

FEM-DSG | NS-FEM | NS-DSG | GS-DSG | Meshfree [ | Pb-2 Ritz [ | ES-DSG [ | ||
---|---|---|---|---|---|---|---|---|

1.0 | 0.05 | 3.8334 | 3.7167 | 3.7653 | 3.8414 | 3.9293 | 3.9444 | 3.9412 |

0.1 | 3.5012 | 3.3584 | 3.4637 | 3.5184 | 3.7270 | 3.7865 | 3.7402 | |

0.2 | 2.6379 | 2.5662 | 2.6526 | 2.6839 | 3.1471 | 3.2637 | 3.1263 | |

1.5 | 0.05 | 4.1998 | 4.0422 | 4.1579 | 4.2337 | 4.2116 | 4.2570 | 4.2852 |

0.1 | 3.8013 | 3.6308 | 3.7871 | 3.8419 | 3.8982 | 4.0250 | 3.9844 | |

0.2 | 2.7879 | 2.7025 | 2.8176 | 2.8490 | 3.1032 | 3.3048 | 3.1461 | |

2.0 | 0.05 | 3.8866 | 3.7597 | 3.8534 | 3.9088 | 3.8657 | 3.9444 | 3.9811 |

0.1 | 3.5867 | 3.4377 | 3.5717 | 3.6131 | 3.6797 | 3.7865 | 3.7711 | |

0.2 | 2.7672 | 2.6843 | 2.7968 | 2.8143 | 3.0783 | 3.2637 | 3.1415 | |

2.5 | 0.05 | 4.0273 | 3.8801 | 4.0049 | 4.0602 | 3.9600 | 4.0645 | 4.1691 |

0.1 | 3.6885 | 3.5255 | 3.6827 | 3.7252 | 3.7311 | 3.8683 | 3.8924 | |

0.2 | 2.7918 | 2.7019 | 2.8167 | 2.8445 | 3.0306 | 3.2421 | 3.1234 |

Axial buckling modes of simply supported rectangular plates with various length-to-width ratios: (a)

In this work, a GS-DSG method is formulated for Reissner–Mindlin plate problems in elastic-static, free vibration, and buckling analyses using 3-node triangular elements. The stabilization term is added to the discretized system equations by applying the smoothed Galerkin weak form. Through the formulations and numerical examples, the accuracy of the proposed method is demonstrated. Some concluding remarks can be drawn as follows:

Several numerical examples indicate that GS-DSG is temporal stable for both free vibration and buckling problems

In elastic-static analysis, the GS-DSG is free of shear locking and has higher accuracy in the displacement field compared with the FEM-DSG, NS-FEM, and NS-DSG methods

In free vibration and buckling analyses, the GS-DSG effectively eliminates the spurious nonzero energy modes produced by the original NS-FEM and NS-DSG and provides a relatively accurate prediction of natural frequencies compared with other methods.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 11472137 and the Fundamental Research Funds for the Central Universities under Grant nos. 309181A8801 and 30919011204.

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