Derivation of compatibility equation are 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 ϵx twice w.r.t. y
………………(b)
Differentiating ϵy twice w.r.t. x
……………….(c)
Differentiating ϒxy 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 ϒyz w.r.t. x ;
…………..(i)
Differentiating ϒzx w.r.t. y ;
………….......(ii)
Differentiating ϒxy 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
Δ4ɸ = 0
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;

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; 𝛥4
ɸ = 0, Airy’ Stress Function
Where;