Uniformly loaded rhombic orthotropic plates supported at corners

Shanmugam, N. E.; Huang, Rongfeng; Yu, C.H.; Lee, S.L.

doi:10.1016/0045-7949(88)90148-4

Cited by 10 publications

(1 citation statement)

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“…Subsequently, Pan (1961) proved that the results of Lee and Ballesteros (1960) were in reasonable agreement with those of Galerkin (1953). Shanmugam and his associates used the polynomial deflection function to solve the cornersupported plates with rhombic (Shanmugam et al, 1988) and triangular shapes (Shanmugam et al, 1989). The vibration of corner-supported thick Mindlin plate was investigated by Kitipornchai et al (1994), who employed a hybrid numerical approach combining the Rayleigh-Ritz method and the Lagrange multiplier in order to impose zero lateral deflection constraints at plate corners.…”

Section: Introductionmentioning

confidence: 99%

Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

Lim

Yao

Cui

2008

The IES Journal Part A: Civil & Structural Engineering

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This work presents the first ever analytical solutions for bending of a rectangular, thin plate supported only at its four corners. This breakthrough analysis employs a new symplectic elasticity approach that extends beyond the limitation of the classical plate bending methods such as Timoshenko's method, Navier method, Levy method and the polynomial approximation analysis of Lee and Ballesteros (Int J Mech Sci 1960;2:206). The classical methods are, in fact, special cases of this symplectic approach in the real eigenvalue regime for wavenumber with at least one pair of opposite sides of plates simply supported. For plate problems that do not fall into this category, the classical methods fail to yield any analytical solutions, but the symplectic approach does because in these cases the plate bending problems enter the complex eigenvalue regime for wavenumber. Another distinctive feature of this new approach is its necessity to pose an eigenvalue problem even for plate bending. In short, this innovative approach establishes the relationship between eigenvalue problem and bending. The novelty of this approach lies in the use of the Hamiltonian principle in a symplectic geometry space to derive a Hamiltonian system and a full state vector. The free boundaries with corner supports are dealt with using the variational principle. Analytical bending modes are then derived by expansion of eigenfunctions. The solutions are compared with other known (approximate) results and numerical finite element solutions but some of the results are not in agreement. Because the analytical bending moment and shear force solutions thus derived fulfill all natural and geometric boundary conditions, it leaves ample room for authentication of the benchmarks in the future. In addition, the twisting moment at the corners satisfies the condition for static bending equilibrium, in which the finite element solutions fail.

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