• Hooke’s law for isotropic linear elastic materials and homogeneous problems in linear elasticity. Pressure vessels. Superposition of loading conditions.
• Traction on a face. Stress transformation. Principal stress components. Stress and strain invariants. Tresca and Mises yield criteria.
• Elastic strain energy. Castigliano methods. Potential energy formulations. Approximate solutions and the Rayleigh Ritz method

Unit 0: Review of Prerequisites. Integration of field variables. Introduction to MATLAB. Review of 2.01x: structural elements in axial loading, torsion, bending. Review of 2.02.1x: equilibrium and compatibility in 2D elastic assemblages.

Unit 1: Multi-axial stress and strain. The (nominal) stress tensor, the (small) strain tensor. Hooke’s law for linear isotropic elastic materials. Plane stress. Pressure vessels: components of stress and strain in cylindrical coordinates. Stress and strain states from superposition of loading conditions and kinematic constraints: applications to homogeneous states, pressure vessels, bars, shafts, beams.

Unit 2: Failure theories. Cauchy Result: traction vector. Stress and strain transformation and principal components of stress and strain. Principal directions and invariants. Design limits on multiaxial stress: design against fracture for brittle materials and design against plastic yielding for ductile materials (Tresca, Mises).

Quiz 1 (on Units 1 and 2)

Unit 3: Elastic Strain Energy and Castigliano’s theorems. Elastic strain energy; complementary energy. Castigliano’s second theorem to solve for kinematic degrees of freedom. Applications to assemblages of structural elements in axial loading, torsion and bending.

Unit 4: Minimum Potential Energy methods. Total potential energy of loaded structure. Equilibrium conditions. Applications to statically indeterminate trusses. Approximate solutions. Trial functions and the Rayleigh Ritz method. Applications to structural assemblages with bars, beams and shafts.

Quiz 2 (on Units 3 and 4)