Exact microstructure-dependent Timoshenko beam element

• General solution for couple-stress Timoshenko beam equations is derived.
• Any distributed load expressible as a Maclaurin series can be modeled.
• The general solution is presented in terms of finite element degrees of freedom.
• Exact microstructure-dependent beam finite element is developed.
• Examples with applications in micron systems and sandwich structures are presented.

In this study, we develop an exact microstructure-dependent Timoshenko beam finite element. First, a Timoshenko beam model based on the modified couple-stress theory is reviewed briefly. The general closed-form solution to the equilibrium equations of the beam is derived. The two-dimensional in-plane stress distributions along the beam in terms of load resultants are obtained. In addition, the general solution can be used to model any distributed load which can be expressed as a Maclaurin series. Then, the solution is presented in terms of discrete finite element (FE) degrees of freedom. This representation yields the exact shape functions for the beam element. Finally, the exact beam finite element equations are obtained by writing the generalized forces at the beam ends using the FE degrees of freedom. Three calculation examples which have applications in micron systems and sandwich structures are presented.