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# Mechanics of Deformable Structures: Part 2

## Overview

In this course: (1) you will learn to model the multi-axial stress-strain response of isotropic linear elastic material due to combined loads (axial, torsional, bending); (2) you will learn to obtain objective measures of the severity of the loading conditions to prevent failure; (3) you will learn to use energy methods to efficiently predict the structural response of statically determinate and statically indeterminate structures.

This course will give you a foundation to predict and prevent structural failure and will introduce you to energy methods, which form one basis for numerical techniques (like the Finite Element Method) to solve complex mechanics problems

This is the third course in a 3-part series. In this series you will learn how mechanical engineers can use analytical methods and “back of the envelope” calculations to predict structural behavior. The three courses in the series are:

Part 1 – 2.01x: Elements of Structures. (Elastic response of Structural Elements: bars, shafts, beams).

Part 2 – 2.02.1x Mechanics of Deformable Structures: Part 1. (Assemblages of Elastic, Elastic-Plastic, and Viscoelastic Structural Elements).

Part 3 – 2.02.2x Mechanics of Deformable Structures: Part 2. (Multi-axial Loading and Deformation. Energy Methods).

These courses are based on the first subject in solid mechanics for MIT Mechanical Engineering students. Join them and learn to rely on the notions of equilibrium, geometric compatibility, and constitutive material response to ensure that your structures will perform their specified mechanical function without failing.

## Syllabus

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)

### Taught by

David Parks and Simona Socrate

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