JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year AEROSPACE VEHICLE STRUCTURES I Previous Question Paper (August/September - 2022 - R13)

For a stress function φ to be valid in plane stress, it must satisfy the biharmonic equation ∇⁴φ = 0, where ∇⁴φ = ∂⁴φ/∂x⁴ + 2∂⁴φ/∂x²∂y² + ∂⁴φ/∂y⁴. Given φ = Ax² + Bx²y + Cy³ + D(5x²y³ - y⁵), we compute the partial derivatives:

Substituting into the biharmonic equation: 0 + 2(0) - 24Dy = 0. This implies D = 0 for the function to be strictly biharmonic unless specific boundary conditions allow for a linear variation in y. However, typically in beam theory, the stress function for a cantilever under end load P is φ = P/12h³(3x²y - y³) + Px/4h³(y² - h²)y. Comparing the given form to the standard Airy stress function for a cantilever beam of depth 2h:

1. The stress components are derived from σ_x = ∂²φ/∂y², σ_y = ∂²φ/∂x², and τ_xy = -∂²φ/∂x∂y.
2. At the free end x = 0, the shear force must equal P and bending moment must be zero.
3. At the boundaries y = ±h, the shear stress τ_xy must vanish, and the normal stress σ_y must be zero.
4. By equating the coefficients of the given polynomial to the boundary conditions, we find A = 0, B = -P/4h³, C = P/12h³, and D = P/12h³.

These constants satisfy the equilibrium equations and boundary conditions for a cantilever beam of unit thickness subjected to a point load P at the free end.

In three-dimensional elasticity, the state of strain at a point is defined by six independent strain components: three normal strains (ε_x, ε_y, ε_z) and three shear strains (γ_xy, γ_yz, γ_zx). These strains are derived from three displacement components (u, v, w) through the strain-displacement relations:

Since there are six strain components but only three displacement functions, the strains cannot be assigned independently. They must satisfy the Saint-Venant compatibility equations to ensure the displacement field is single-valued and continuous. These equations are derived by eliminating the displacement components through differentiation.

The three normal compatibility equations are:

The three shear compatibility equations are:

These six differential equations ensure that the strain field corresponds to a continuous and physically possible deformation of the elastic body.