Show that the function " .xy - —xyh 215 an Airy's stress function and
hence show that it represents the stress distribution in a cantilever beam loaded at
the free end with a load ‏.م‎ The width of beam is '0' and depth is ‏'ط'‎

h
Assume —Oar=+—
2
5. 3. State and prove uniqueness theorem in theory of elasticity 3

‎State generalized Hooke's law with clear explanation to the reduction in elastic 6‏ .م
‎constants for different cases, Hence write down the stress-strain relations for a‏
‎three - dimensional orthotropic and transversely isotropic body.‏

‎6 a. Derive Betrami-Michell's equations for a three - dimensional stress state 7 0. State and
explain Saint Venant's principle 2
PART C

‎7: Show that the stress concentration around a hole in 8 plate of infinite 12 dimension
under uni-axial tension is 3. Plot the variation of stresses around the hole.

‎8. a. Derive the compatibility equation in polar co-ordinate system for two-8
dimensional stress state

‎b. Show that by making a small hole at the centre in a solid rotating disc with 4 radius
'b', the circumferential stress will increase twice.

‎9. a. Discuss Prandtl's membrane analogy on torsion 6 b. Determine the maximum
torque that can be applied onthe section shown in 6 the figure below if the allowable
shear stress is 300 N/mm2 . What is the angle
of twist per unit length of the shaft under the above torque. Determine the
shear stress in various parts of the section. The wall thickness is uniform and

‎has a value of 15mm. The modulus of rigidity G —2 x 104 N/mm2

‎450mm