(Solved): i. Show that for a 2-D state of stress, the conjugate stresses are equal. ii. Figure Q1 represents t ...

i. Show that for a 2-D state of stress, the conjugate stresses are equal. ii. Figure Q1 represents the state of stress at a point in a material. By the use of Mohr's circle, determine the magnitude of the principal stresses and the maximum shearing stress. i. For a 3-D state of stress expressed by the compendium A, derive expressions for the first, second and third invariants. ii. State two properties of the compendium. Find

and

and determine the invariants of compendium B .

A vertical rod of cross.sectional area A is secured firmly at its top and has a collar of inextensible mass attached firmly to its bottom as shown in figure Q3. If a mass of magnitude, M , is dropped onto the collar from a distance, h above it, determine the maximum values of stress and deflection that will occur owing to this impact. Neglect energy and other losses, and assume that the mass is in the form of an annular ring. Determine the maximum stress if the mass, M , is applied at effective height of zero.