Advanced Mechanics of Materials L1 3184.pptx

lecture

1. 1. Stress Function The solution of two dimensional problems in elasticity requires integration of the differential equations of equilibrium together with the compatibility equation and the boundary conditions. For no body forces These equations are the same for both plane stress and plane strain problems.

2. The equations of equilibrium are identically satisfied by a new function ( x,y ), called the stress function, introduced by G.B. Airy, related to stresses as: Substituting the above expressions into the compatibility equation Biharmonic Equation Where

3. For a more general case of a body forces and when the force components X and Y are given by In which V is the potential function. The equilibrium equations become These equations can be satisfied by taking Substituting into the compatibility equation for plane stress distribution.

4. An analogous equation can be obtained for the case of plane strain. Note: When the body force is simply the weight, the potential V is (- . g.y ). Example: Show that the given function gives the stresses correctly on all boundaries except at the end x=l. x y p l h h 1

5. Example: For the given cantilever show that is the proper stress function. Determine the constants A and B so that the shear stress is zero on the top and bottom face while the resultant vertical force on the free surface is P y x p l h h 1

6. 2. Strain Energy due to the work of elastic deformation dx dy o x y (a) dx dy o x y (b)  x  F x F o (c) Fig.(1) Basis for determining strain energy induced by applied stresses Figures (a) and (b) above, show a normal and simple shear stress acting on stress elements of dimensions dx , dy , dz. In part (a), the stress  x produces a strain  x and the work done is, Where, and

7. Therefore, Thus, the work done per unit volume, Similarly, in figure (b), Thus,

8. In the most general case, and with the use of superposition, the work or strain energy per unit volume is expressed by, Where principal direction are involved,