JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III FINITE ELEMENT TECHNIQUES Previous Question Paper (September - 2021 - R15)

The potential energy approach is based on the principle of minimum potential energy, which states that for a stable conservative system, the displacement configuration that satisfies the boundary conditions and equilibrium equations minimizes the total potential energy.

The total potential energy Π is defined as the sum of the internal strain energy (U) and the potential energy of external loads (V):

Substituting the displacement relation {ε} = [B]{u}, where [B] is the strain-displacement matrix:

Defining the stiffness matrix [K] = ∫ [B]^T [D] [B] dV, the expression becomes Π = 0.5 {u}^T [K] {u} - {u}^T {F}. To minimize Π, we take the partial derivative with respect to the nodal displacement vector {u} and set it to zero:

The salient parts include the Stiffness Matrix [K], which represents the structural resistance; the Displacement Vector {u}, representing the unknown nodal degrees of freedom; and the Force Vector {F}, representing the applied external nodal loads.

The finite element method follows a systematic procedure to transform a continuous problem into a discrete one:

1. Discretization: The domain is divided into a finite number of elements connected at nodes. The geometry and topology are defined by the element type and mesh density.
2. Selection of Interpolation Functions: Shape functions are chosen to represent the variation of the field variable (e.g., displacement) within each element, usually expressed as polynomials.
3. Element Stiffness Matrix and Load Vector: Using the constitutive laws and strain-displacement relations, the element stiffness matrix [k] and force vector {f} are derived, typically via the principle of minimum potential energy.
4. Assembly: Individual element matrices are assembled into a global stiffness matrix [K] and global force vector {F} by enforcing displacement compatibility at shared nodes.
5. Application of Boundary Conditions: Known displacements or forces are applied to the global system of equations to remove singularities and ensure a unique solution.
6. Solution of Equations: The resulting system of linear algebraic equations [K]{u} = {F} is solved for nodal unknowns using methods like Gaussian elimination or Cholesky decomposition.
7. Post-processing: Strains and stresses are calculated from the nodal displacements using the strain-displacement and constitutive matrices.