Solid Mechanics / Strength of Materials taught at the undergraduate level formulates
problems (e.g., pressure vessels, bending, torsion) based on approximate engineering
theories involving assumptions related to geometry and loading.

Thus the theory
developed therein has limited applicability.

The present Advanced Solid Mechanics
course mostly uses an elasticity approach to formulate these problems for more general
geometry and loading, thus expanding the applicability of theories developed herein to
a wider range of problems.

A rigorous and systematic approach is adopted.

This begins
with the derivation of basic field equations (Topics 2-4) which are then applied to
formulate various two - and three - dimensional problems without invoking
assumptions of engineering theories.

The 2-D problems of thick-walled cylinders,
curved beams, deep beams, plate with hole, etc. (Topic 5), and 3-D problems of bending
and torsion of bars of arbitrary cross-section (Topics 6, 7) are studied by this approach.

Exact analytical solutions are obtained for certain simple geometries. Numerical
methods are introduced for complex geometries.

Using rational assumptions, the
problems of bending and torsion of thin walled cross-sections, and shear center
location, are formulated and solved.

Comparisons with results of engineering theories
are made wherever possible, so that the range of validity of the engineering theories is
well understood.

Lastly the engineering theory of curved beams and beams on elastic
foundation is studied (Topic 8). Example problems are solved throughout the course.

Contents:

Analysis of stress and strain, field equations (equilibrium, straindisplacement,
constitutive, compatibility), boundary conditions; boundary value
problems of elasticity;

Two dimensional problems in rectangular and polar coordinates;
Torsion of general cross-section shafts; Bending of general cross-section beams, shear
center; Curved beams; Beams on elastic foundation.

Sl. No.

Topic

No. of Hours

1

Introduction:

Motivation for course; Review of elementary solid mechanics.

01

2

Analysis of Stress:

Surface forces and traction/stress vector, body forces and moments;

Components of stress matrix and its relation to stress vector;

Normal and shearing stresses on a plane; Stress transformations and stress tensor, introduction to tensors;

Principal stresses and axes; Maximum shearing stress; Equilibrium equations; Boundary conditions.

04

3

Analysis of Deformation and Strain:

Deformation map, displacement gradient, straining of line element and strain components as measure of deformation;

Strain-displacement relations, infinitesimal strain and linearization, physical interpretation of normal and shear strain components;

Infinitesimal rotation vector and relative displacement; Straining of arbitrary line element, strain transformation and strain tensor, principal strains and axes;

Analogies with stress tensor; Volumetric strain and cubical dilation; Strain Compatibility equations.

05

4

Constitutive Relations, Boundary Value Problems:

Generalized Hooke's law, 3-D stress-strain relation for linear elastic Isotropic solid;

Compatibility equations in terms of stress; Types of boundary value problems (BVPs) - displacement and stress formulations, Saint Venant's principle.

02

5

Two Dimensional Elasticity in Cartesian and Polar Coordinates:

Plane stress, plane strain; Formulation of BVP using Airy stress function, inverse and semi-inverse methods of solution;

Problems in rectangular coordinates - polynomial solutions, determination of displacements, Fourier series solutions;

Problems in polar coordinates - transformation of field equations in polar coordinates, axisymmetric problems, non-axisymmetric problems, stress concentrations; Use of symmetry in solving 2-D problems.

09

6

End Torsion of Bars (prismatic, general cross-section):

Review of torsion of circular sections; Formulation of BVP using Prandtl stress function and Saint Venant's semi-inverse method (Warping function method), membrane analogy;

Solutions for solid cross-section bars; Torsion of thin-walled open-section and closed-section (multi-celled) members;

Formulation for torsion of multi-celled thick-walled cross-sections; Finite difference method.

08

7

Bending of Beams (prismatic, general cross-section):

Preliminaries - sign conventions, area moments of inertia, their transformation, principal inertias;

Pure bending of beam with terminal couples; Bending of beam with end shear - BVP formulation, examples, shear center and its determination;

One-dimensional shear flow in open thin-walled beams and shear center problem solving.

09

8

Specific topics:

Bending of Curved Beams

(Prismatic, symmetric sectioned) - Assumption, derivation of basic results (kinematics, stresses), obtaining maximum stresses, determining deflections using energy methods.

Beams on Elastic Foundation

Basic problem of infinite beam with point load, various modifications of basic problem and application of superposition for solving them.

04

Solid Mechanics / Strength of Materials (first course at UG level).

L. S. Srinath, Advanced Mechanics of Solids, 2^{nd} ed., Tata McGraw Hill, 2003.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3^{rd }ed., McGraw Hill, 1970.

A. P. Boresi, R. J. Schmidt, Advanced Mechanics of Materials, 6^{th} ed., Wiley, 2003.

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