Timoshenko beam theory

About: Timoshenko beam theory is a research topic. Over the lifetime, 9426 publications have been published within this topic receiving 200570 citations.

Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading

Abstract: An exact, closed-form solution is obtained for the nonlinear static responses of beams made of functionally graded materials (FGM) subjected to a uniform in-plane thermal loading. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear first-order shear deformation beam theory and the physical neutral surface concept. The three equations are reduced to a single nonlinear fourth-order integral–differential equation governing the transverse deformations. For a fixed–fixed FGM beam, the equation and the corresponding boundary conditions lead to a differential eigenvalue problem, while for a hinged–hinged FGM beam, an eigenvalue problem does not arise due to the inhomogeneous boundary conditions, which result in quite different behavior between clamped and simply supported FGM beams. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal post-buckling or bending deformation is obtained as a function of the applied thermal load. The exact solutions explicitly describe the nonlinear equilibrium paths of the deformed beam and thus are able to provide insight into deformation problems. To show the influence of the material gradients, transverse shear deformation, in-plane loading, and boundary conditions, numerical examples are given based on exact solutions, and some properties of the post-buckling and bending responses of FGM beams are discussed. The exact solutions obtained herein can serve as benchmarks to verify and improve various approximate theories and numerical methods.

A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory

Abstract: The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation.

Mode-II Interlaminar Fracture of Composites

Abstract: This chapter will present fracture mechanics and experimental mechanics approaches used to characterize mode-II interlaminar fracture of composites. This chapter is organized into four major sections: background, analytical, numerical, and experimental results. The first section documents from an historical viewpoint, the various mode-II specimen geometries that have been proposed prior to the emergence of the end notch flexure (ENF) specimen as the most frequently used test method to characterize mode-II interlaminar fracture toughness. The second section focuses on analytical approaches to model the ENF specimen that includes shear deformation beam theory, shear deformation plate theory where the crack-tip singularity is introduced as a surface traction and a higher-order beam theory. Specimen design procedures to maintain linear elastic response and to minimize geometric non-linearities and friction between crack surfaces are presented. The third section reviews numerical results. Compliance, strain energy release rates, delamination offset from the mid-plane and frictional effects are investigated. Results are compared to analytical solutions. The last section reviews experimental mechanics techniques used to characterize the static mode-II interlaminar fracture toughness of composites.