Static analysis of linearly elastic bodies

The chapter focuses on fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for the static analysis of linearly elastic bodies in two and three dimensions. The formulation and solution of an elasticity problem requires basic equations—equations of equilibrium, strain-displacement relations (geometric relations), conditions of compatibility, stress-strain relations (constitutive law), and boundary conditions. According to the strain-displacement relations, the six components of strain at a point are completely determined by the three displacement components u , v , and w . Therefore, the strain components cannot be arbitrary functions of x , y , and z . Depending on the geometric configuration of the elastic body in consideration, a two-dimensional elasticity problem usually falls into one of the two categories—plane stress and plane strain. A problem of plane strain is concerned with a long prismatic or cylindrical body. The body is subjected to external loads that are perpendicular to the longitudinal ( z ) axis and do not change along the length. It is assumed that frictionless constraints are imposed at the two ends of the body, which permit x , y deformation, but do not allow the displacement in the z direction.