JNTU B.Tech III FINITE ELEMENT METHODS Previous Question Paper (August - 2022 - R18)

To determine the stress in each material of a composite block subjected to an axial load, we utilize the principles of compatibility and equilibrium. Let the two materials be denoted by subscripts 1 and 2, with cross-sectional areas A₁ and A₂, and elastic moduli E₁ and E₂.

The total load P is shared between the materials according to the equilibrium equation:

Assuming the block is constrained such that both materials undergo the same deformation (strain compatibility), the strain ε is equal in both materials:

From the compatibility condition, we express the stress in one material in terms of the other:

Substituting this into the equilibrium equation:

Solving for σ₂:

Once σ₂ is calculated, σ₁ is found using the ratio of the elastic moduli. For the given load P = 385 kN, ensure that all units are consistent (e.g., converting kN to N and areas to mm²) to obtain the stress values in MPa (N/mm²).

To find the temperature distribution in a stepped fin using two finite elements, we model the fin as a one-dimensional heat conduction problem with convection. The governing differential equation for a fin is given by:

For a linear element of length L, the element stiffness matrix [k] and load vector {f} are derived using the Galerkin approach or variational principle. The element stiffness matrix is:

The global system is assembled by connecting two elements. Let the nodes be 1, 2, and 3. Element 1 spans nodes 1-2 with area A1, and element 2 spans nodes 2-3 with area A2. The global stiffness matrix [K] is the summation of element matrices [k1] and [k2] at the shared node 2.

1. Define the nodal temperatures T1, T2, and T3.
2. Apply the boundary conditions: at the base (node 1), the temperature is usually fixed (T1 = Tb), and at the tip (node 3), a convective boundary condition is applied using the heat balance equation: -kA(dT/dx) = hA(T3 - T∞).
3. Assemble the global matrix equation [K]{T} = {F}.
4. Substitute the known values for thermal conductivity k, convection coefficient h, perimeter P, and areas A1, A2.
5. Solve the resulting system of linear algebraic equations for the unknown temperatures T2 and T3.

The temperature distribution within each element is then given by the linear interpolation function: T(x) = N1(x)T_i + N2(x)T_j, where N1 and N2 are shape functions.