Importance of equilibrium and compatibility equations in elasticity

Question

What is the importance and relevance of equilibrium and compatibility equations in elasticity. What problem do they solve?

Answer 1

The equilibrium equations mean every small element of the material is in equilibrium , i.e. the forces balance at every point.

The compatibility equations mean the deformed material is "continuous" everywhere, i.e. it doesn't have any internal holes, cracks, or overlapping regions.

Answer 2

Equilibrium is statics means the member under consideration is not accelerating, moving or turning.

e.g., On a simply supported beam with a length L with a load P at L/2 if the right-hand support provides any value less than P/2 resistance the beam will rotate down.

Compatibility means there is a unique geometrical displacement, rotation, deflection at any point in a structure. e.g., if we cut a beam at a point to study its free body diagram and this section under stresses has rotated an angle A, and a deflection delta, the other part of the beam adjacent to it has an identical angle of rotation and deflection. This property is used to drive additional equations to help solve the stresses and forces on a member.