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Based on the first-order shear deformation theory (FSDT) and the moving least-squares approximation, a new meshless model to study the geometric nonlinear problem of ribbed rectangular plates is presented. Considering the plate and the ribs separately, the displacement field, the stress, and strain of the plate and the ribs are obtained according to the moving least-squares approximation, the von Karman large deflection theory, and the FSDT. The ribs are attached to the plate by considering the displacement compatible condition along the connections between the ribs and the plate. The virtual strain energy formulation of the plate and the ribs is derived separately, and the nonlinear equilibrium equation of the entire ribbed plate is given by the virtual work principle. In the new meshless model for ribbed plates, there is no limitation to the rib position; for example, the ribs need not to be placed along the mesh lines of the plate as they need to be in FEM, and the change of rib positions will not lead to remeshing of the plate. The proposed model is compared with the FEM models from pieces of literature and ANSYS in several numerical examples, which proves the accuracy of the model.

Ribbed plate has been widely used in engineering, such as bridges, ship hulls, and aviation, and it is a popular structure with obvious advantages. The ribs make the structure stiffer and allow it to achieve larger bearing capacity than flat plate with roughly the same weight. However, the ribs also bring difficulties to analysis, and the calculation of ribbed plate is more complicated than that of flat plates. Based on the fact that the ribs of many ribbed plates are attached to the plate with uniform spacing and close to one another, and that ribbed plates show different elastic characteristics in the two perpendicular directions, early researchers transformed the ribs to an addition layer to the plate and used the orthotropic model to approximate the ribbed plates [

Not many nonlinear analyses of ribbed plates can be found in pieces of literature, and most of them were based on the FEM [

In MLS [_{
x},

Therefore, (

The meshless model of a ribbed plate, shown in Figure

Meshless model of a ribbed plate.

Based on the FSDT [

Along the axis of the

Section of plate and

For a node

Point

With this new transformation equation (

According to the FSDT and the von Karman theory, the strains of an isotropic plate are

The stress is

The virtual work equation of the ribbed plate is
_{1} is the sum of all of the loading vectors,

This paper used the Newton-Raphson method to solve nonlinear equilibrium equation (

Take linear solution

Substituting

Employ (

Obtain the incremental displacements by

and the improved solution

Return to step

Due to a lack of Kronecker delta properties in the shape functions given in (

To show the convergence of the proposed model, and the influences of the support size of the nodes and the order of the basic functions, a clamped square plate subjected to a uniformly distributed pressure of 100 Pa is studied. The width of the plate is 1.8 m, and the thickness is 0.018 m. The Young’s modulus of the plate is

The nonlinear deflection of the central point of the plate that is obtained by the proposed model under different support sizes (which are denoted by scaling factors,

Nonlinear deflection of the flat plate under different _{
c}.

In this paper, rectangular support is employed, and thus the scaling factors _{
c}) need a larger support size to make the solution converge.

Secondly, we vary the meshless scheme and obtain the variations of the nonlinear central deflection under certain completeness order of the basic functions

Variation of nonlinear deflection of the flat plate,

Variation of linear deflection of the flat plate,

Variation of linear deflection of the flat plate,

From the studies, we find that when the order of basic functions

A rectangular plate clamped at two opposite sides and with one rib (Figure

Rectangular plate with one rib.

The two other sides of the plate are free. Both the plate and the rib are made of the same material, with Young’s modulus

Central deflection of the rectangular plate with one rib under different loads.

Load (MPa) |
Koko and Olson [ |
Present results (mm) | Relative errors |
---|---|---|---|

0.2 | 5.526 | 5.31654 | 3.8% |

0.3 | 7.172 | 6.9253 | 3.4% |

0.4 | 8.631 | 8.47243 | 1.8% |

0.5 | 9.868 | 9.82917 | 0.4% |

0.6 | 11.053 | 11.0406 | 0.1% |

0.7 | 11.974 | 12.1373 | −1.4% |

0.8 | 12.961 | 13.1409 | −1.4% |

A simply supported square plate with two ribs is studied. The ribs are located at

Square plate with two cross ribs.

Young’s modulus and Poisson’s ratio are

Load-central deflection curve of the simply supported square plate with two ribs.

Load-stress curve of the clamped square plate with two ribs.

3D finite element model of a square plate with two ribs.

If the thickness of the plate is increased to 0.1 m and the plate is clamped, the load-central deflection curves of the plate are shown in Figure

Load-central deflection curve of the clamped square plate with two ribs.

A square plate with one rib located at

Square plate with one rib.

The plate is under a uniformly distributed load

3D finite element model of a square plate with one rib.

Firstly, the present results (Figure

Nonuniform distribution of the 81 plate nodes.

Uniform distribution of the 9 × 9 plate nodes.

Central deflection of the clamped plate with one rib (81 nodes).

Secondly, the present results (Figure

Nonuniform distribution of the 121 plate nodes.

Uniform distribution of the 11

Central deflection of the clamped plate with one rib (121 nodes).

Finally, the present results (Figure

Nonuniform distribution of the 169 plate nodes.

Uniform distribution of the 13 × 13 plate nodes.

Central deflection of the clamped plate with one rib (169 nodes).

In Figures

This paper presents a meshless model, which is based on the FSDT and the MLS approximation to study the geometric nonlinear behaviors of ribbed plate structures. Considering a ribbed plate as a composite structure of plate and ribs, and starting from the large deflection theory of von Karman, the nonlinear behaviors of the plate and the ribs were studied, respectively. Then, employing the meshless advantages of the proposed model, the nonlinear governing equations of the plate and the ribs were superposed with a new equation derived for the nodal parameter transformation of the plate and the ribs, and the geometric nonlinear equilibrium equation of the entire structure is established. The advantages of the proposed model are that the ribs can be placed anywhere on a plate and any changes of their positions will not lead to the remeshing of the plate, which enhances computational efficiency in solving the optimization of rib layout under the consideration of nonlinear deformation. And the proposed model do not rely on mesh; therefore, mesh disorder due to the large deformation of problem domain is avoided. The present results are compared with those from three-dimensional FEM analysis and pieces of literature. Good agreement can be observed, which proves the accuracy of the proposed meshless model.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work that is described in this paper has been supported by the grants awarded by the National Natural Science Foundation of China (Projects nos. 11102044, 51168003, and 51168005) and the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Structural Safety (Project no. 2012ZDX07).