Integral nonlocal stress gradient elasticity of functionally graded porous Timoshenko nanobeam with symmetrical or anti‐symmetrical condition

Abstract

Utilization of symmetrical or anti‐symmetrical condition could improve the calculation efficiency. In this paper, a mathematical formulation is proposed to deal with the symmetrical or anti‐symmetrical condition in an integral nonlocal stress gradient model (INSGM), which is transformed equivalently into differential form with constitutive boundary condition as well as constitutive symmetrical or anti‐symmetrical condition. Unlike general constitutive boundary conditions, an integral item is introduced to constitutive symmetrical and anti‐symmetrical conditions, and they are opposite to each other. Based on INSGM with symmetrical or anti‐symmetrical conditions, static bending of simply‐supported (SS) and clamped‐clamped (CC) functionally graded porous Timoshenko nanobeams is investigated for symmetrical loads, including uniformly distributed load (UDL) and middle point force, as well as anti‐symmetrical loads, including anti‐symmetrical UDL and middle point moment. The exact solutions are deduced and expressed in explicit form for different boundary and loading conditions. Calculation shows that, under UDL, bending deflections of half Timoshenko nanobeams based on current model agree well with those for whole Timoshenko nanobeams based on general INSGM for both SS and CC boundary conditions. Numerical study is performed to show the effectiveness of current model.