JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year FINITE ELEMENT METHODS Previous Question Paper (September - 2021 - R18)

The total potential energy Π of an elastic body is defined as the sum of the internal strain energy U and the potential energy of the external loads V. For an elastic body occupying a volume V with surface area S, the expression is derived as follows:

1. Internal Strain Energy (U): The strain energy stored in the body due to deformation is given by the integral of the strain energy density over the volume:

where σ is the stress vector and ε is the strain vector.

2. Potential Energy of External Loads (V): This accounts for the work done by body forces, surface traction forces, and point loads. The potential energy is the negative of the work done by these forces:

a) Body Forces (f): Forces acting throughout the volume (e.g., gravity), defined by force per unit volume f:

b) Traction Forces (T): Surface forces acting on the boundary S[t], defined by force per unit area T:

c) Point Forces (P): Concentrated loads acting at specific points i:

3. Total Potential Energy (Π): Combining these terms, the total potential energy functional is:

In this expression, u represents the displacement field. The principle of minimum potential energy states that for a stable equilibrium configuration, the total potential energy Π is stationary with respect to small admissible variations in displacement, leading to the governing equations of the finite element method.

To determine the displacements and support reactions for a uniform bar, we apply the finite element method using the stiffness matrix approach. For a bar element of length L, cross-sectional area A, and Young's modulus E, the element stiffness matrix is given by:

Assuming the bar is discretized into elements based on the geometry provided in the figure, we assemble the global stiffness matrix K by summing the element matrices at the corresponding nodes. The global equilibrium equation is expressed as:

Where {u} is the vector of nodal displacements and {F} is the vector of nodal forces. Given the load P = 300 kN, we apply the boundary conditions by setting the displacement at the fixed support to zero (u = 0). This allows us to partition the matrix equation to solve for the unknown displacements at the free or intermediate nodes.

1. Assemble the global stiffness matrix [K] for the entire bar system.
2. Apply the boundary conditions by removing the rows and columns corresponding to the fixed nodes.
3. Solve the resulting reduced system of linear equations for the unknown nodal displacements {u}.
4. Calculate the internal element forces using the relation F = (AE/L)(u_j - u_i).
5. Determine the support reactions by substituting the calculated displacements back into the original global equilibrium equation for the constrained nodes, R = Σ(K_ij * u_j) - F_i.

Final support reactions are obtained by summing the forces at the supports, ensuring that the total external load P is balanced by the reaction forces to satisfy static equilibrium.