Application of Galerkin iterative technique to nonlinear bending response of three-directional functionally graded slender beams subjected to hygro-thermal loads

Abstract

In the view of some progressive devices such as shuttles in the aerospace field, the vessels in the marine engineering and reaction equipments in the nuclear field are exposed to super-high thermal or hygro-thermal loads simultaneously varied along the three directions in the operation, we propose the three-directional functional graded materials (3D FGMs) to carry out the composition of slender beams to meet the requirements of the hygro-thermal distribution in the multi-directions and improve the endural ability so that catastrophic failure is refrained due to large deflections. Built up on von Kármán geometric nonlinearity, one takes advantages of the principle of minimum total potential energy to derive the nonlinear governing equation of the structures subjected to hygro-thermal loadings of the uniform, linear and sinusoidal distributions. A novel analytical solution for the nonlinear bending response is presented based on the Galerkin iterative technique, which overcomes the shortcoming of finding difficultly the closed-form solution to antisymmetric nonlinear structures such as multi-directional FGMs. The generalized differential quadrature method (GDQM) is performed for validating the effectiveness of the presented methodology. Numerical simulations are implemented to reveal the effects of 3D FGMs indexes, mechanical and hygro-thermal loadings on the nonlinear bending response. Results exhibit the nonlinear behavior is profoundly dependent on the 3D FGMs and hygro-thermal properties, which is applied for optimizing the 3D FGMs structures in multiple physical fields.