# Theory of Elasticity and Plasticity

Note : Attempt all questions.

All questions carry equal marks.

Assume missing data, if any, suitably.

1 . Attempt any two parts of the following :

(a)  Explain about plane stress and plane strain problems. Give two examples also.

(b) Derive the compatibility equation in terms of stress for a plane stress problem. Is this equation valid for plane strain also ?

(c)  The general displacement field in a body in certain coordinates is given as:-

u = 0.015 x2y + 0.03

v = 0.005y2 + 0.03 xz

w = 0.003 z2 + O.OOlyz + 0.005

Find all  the strains for the point (1,0,2)

2. Attempt any two parts of the following :

(a) Derive the expression for circumferential stress in a curved beam with large initial armature and subjected to pure binding. State clearly the assumptions and its limitations.

(b)  A circular plate with a circular hole is simply supported around its edge and subjected to linearly varying distributed load. Derive the expressions for maximum stress.

(c)  A narrow, simply supported beam of rectangular cross-section is subjected to a uniformly distributed load. Determine the stress distribution in the beam.

3. Attempt any one part of the following :

(a)  Determine the distribution of stress is a circular cylindrical shell having the ends supported by the diagraphs. The shell has been filled with oil of density P such that P(Q) = 10′ Pa cos Q Where a = radius

(b)   Derive the expressions for the stress resultants and displacements for the case of a cylindrical shell with a uniform pressure.

4.   Attempt any one part of the following:

(a)  Derive an expression for strain energy per unit volume for a two-dimensional linearly elastic body for plane stress or plane strain in terms of Airy’s stress function.

(b)  How do you determine the stress distribution due to cracks? Explain with a suitable example.

5. Attempt any one part of the following :

(a)  Derive an expression for strain energy per unit volume for a two dimensional linearly elastic body for plane stress or plane strain in terms of Airy’s stress functions.

(b)  How do you determine the stress distribution due to cracks? Explain with a suitable example.