Compatibility equation and Airy's stress function of theory of elasticity

simple brief explanation and derivation is done

1. THEORY OF ELASTICITY Topic : 1) Compatibility Equation 2) Saint-Venant’s Equation 3) Airy’s Stress Function Name: - Jayant Chaudhary MTech (Structures)

2. THEORY OF ELASTICITY Compatibility Equations: It is a fundamental problem of the theory of elasticity to determine the state of stress in a body submitted to the action of given forces. In a two-dimensional problem, it is necessary to solve the differential equations of equilibrium and the solution must be such as to satisfy the boundary conditions. In 2D problems we consider three strain components namely, ……………. (a)

3. …………(a) Differentiating ϵ x twice w.r.t. y ………………(b) Differentiating ϵ y twice w.r.t. x ……………….(c) Differentiating ϒ xy w.r.t. x and then w.r.t y ; ……………( d)

4. Adding (b) and (c); ……………………(e) Equating (d) and (e); Similarly, …………………. (f) Equation (f) is called the condition of compatibility.

5. SAINT’S VENANTS EQUATION If the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed.

6. We have ; Differentiating ϒ yz w.r.t. x ; …………..( i ) Differentiating ϒ zx w.r.t. y ; ………….......(ii) Differentiating ϒ xy w.r.t. z ; …………......(iii) ( i ) + (ii) - (iii)

7. So, ……………………(iv) Now again, differentiating w.r.t. z ;

8. These equations are called Saint- Venant equation of Compatibility. These are also known as Continuity Equation.

9. Airy Stress Function Formulation   Where, ɸ = Stress Function   This relation is called the biharmonic equation and its solutions are known as biharmonic functions . To get the complete solution of an elastic problem in addition to the various equation such as equilibrium equation, compatibility equation, boundary condition, etc. require a clear approach to solve the problem. Normally, stress function approach is followed for solving the elastic problem which is known as Airy stress function. The plane problem of elasticity can be reduced to a single equation in terms of the Airy stress function . This function is to be determined in the two-dimensional region bounded by the boundary.

10. From Hook’s Law and by considering plane stress, we have ; Substituting in Compatibility equation i.e.; We get; ………….(a)

11. Also we have, equilibrium equation for 2D; Let the body force per unit width be zero; Differentiating ( i ) twice w.r.t. X; Differentiating (ii) w.r.t. y; Adding (b) and (c);

12. Now, substituting in equation (a) i.e. We get,

13. Equation (e) is the equilibrium equation in terms of plane stress by neglecting the body forces. Let assume; …………………(f) Now, substituting eqn (f) in (e)

14. Which implies; , Airy’ Stress Function Where;