THEORY EXAMINATION (SEM–VI) 2016-17 FINITE ELEMENT METHOD

FINITE ELEMENT METHOD (NME012)

SECTION-WISE SOLVED ANSWERS

SECTION – A

(Short Answer Questions | 2 Marks Each)

(a) Merits and Demerits of Finite Element Method

Merits:
FEM can analyze complex geometries, irregular boundaries, and varying material properties. It provides accurate numerical solutions and is applicable to real-life engineering problems.

Demerits:
FEM requires large computational effort, skilled users, and proper meshing. Results depend heavily on element quality and boundary condition accuracy.

(b) Gauss Quadrature Approach

Gauss quadrature is a numerical integration technique used in FEM to evaluate integrals efficiently. In one-dimensional integration, weighted points (Gauss points) are used within the limits −1 to +1. In two-dimensional integration, the method is extended using product rules.

(c) Principle of Minimum Potential Energy

This principle states that among all possible displacement configurations satisfying boundary conditions, the actual displacement minimizes the total potential energy of the system.

(d) Displacement Function & Geometric Variance

A displacement function expresses displacement in terms of nodal values using shape functions.
Geometric variance refers to the change in element geometry due to deformation.

(e) FEM vs Classical Method

Classical methods solve differential equations directly and are suitable only for simple geometries. FEM converts the domain into elements and solves algebraic equations, making it suitable for complex problems.

(f) Element Stiffness Matrix for 2D Beam Element

The stiffness matrix for a 2D beam element includes axial, bending, and rotational degrees of freedom and is derived using beam theory and shape functions.

(g) Strain Components Calculation

Given:
u = (−x² + 2y² + 6xy) × 10⁻⁴                              v = (3x + 6y − y²) × 10⁻⁴

Strain relations:                                                  εxx = ∂u/∂x
εyy = ∂v/∂y                                                        εxy = ½(∂u/∂y + ∂v/∂x)

At (x=1, y=0):                                                     εxx = (−2x + 6y) × 10⁻⁴ = −2 × 10⁻⁴
εyy = (6 − 2y) × 10⁻⁴ = 6 × 10⁻⁴                        εxy = ½(6x + 3) × 10⁻⁴ = 4.5 × 10⁻⁴

(h) Axisymmetric Elements

Axisymmetric elements are used when geometry and loading are symmetric about an axis. Formulation includes radial and axial stresses.
Applications: pressure vessels, pipes, cylinders.
 

(i) Galerkin Approach

Galerkin approach is a weighted residual method where weight functions are chosen same as trial functions. It converts differential equations into algebraic equations.
 

(j) Hermitian Interpolation Function

Hermitian functions include both displacement and slope continuity. They are commonly used in beam and plate elements for better accuracy.
 

SECTION – B

(Descriptive Questions | 10 Marks Each)

(a) Local, Global & Natural Coordinates

Global coordinates define the structure in space.
Local coordinates define element orientation.
Natural coordinates simplify shape function formulation using normalized values (−1 to +1).

Natural coordinates improve numerical integration efficiency.

(b) Uniform Bar Problem

The bar is divided into elements, stiffness matrices are formed, assembled, boundary conditions applied, and nodal displacements solved. Support reactions are obtained using equilibrium equations.

(c) Fixed Beam Deflection Problem

The beam is divided into two elements. Element stiffness matrices are formulated using beam theory. After assembly and applying boundary conditions, deflection at the center is calculated and compared with classical beam theory.

(d)

(i) Shape Functions for 4-Node Rectangular Element
Derived using Lagrange interpolation, ensuring unity at one node and zero at others.

(ii) Jacobian Derivation
Jacobian relates global and natural coordinates and is derived from partial derivatives of shape functions.

(e) FEM Formulation Steps                                       Discretization

Selection of element type                                        Derivation of shape functions

Element stiffness matrix formulation                       Assembly

Boundary condition application                               Solution of equations

Advantages: flexibility, accuracy                             Limitations: computational cost, modeling effort

(f) Beam with Distributed Load

The equivalent nodal loads are calculated, stiffness matrices assembled, and mid-point deflection determined using displacement equations.

(g)

(i) Convergence & Compatibility
Convergence ensures solution approaches exact result as mesh is refined.
Compatibility ensures displacement continuity between elements.

(ii) Conforming vs Non-Conforming Elements
Conforming elements satisfy compatibility; non-conforming do not.

(h) Short Notes

(i) Shape Functions:
Interpolate displacement within an element.

(ii) One-Dimensional Conduction with Convection:
Heat transfer involving conduction inside body and convection at boundaries.

(iii) Jacobian:
Used to transform integrals from global to natural coordinates.

(iv) Boundary Conditions:
Essential for unique FEM solutions.

SECTION – C

(Long Answer Questions | 15 Marks Each)

3.

(i) Constant Strain Triangle (CST)
CST element assumes constant strain within the element. Isoparametric formulation uses natural coordinates to represent geometry and displacement.

(ii) Weighted Residual Method
Residuals are minimized by choosing appropriate weight functions. In single trial function approach, the same function approximates solution across the domain.

4. Truss Problem

Stiffness matrices for members AB and BC are derived using axial deformation theory. After assembly and boundary conditions, nodal displacements are computed.
Stress in each bar is calculated using stress-strain relations.

5.

(i) Total Potential Energy Approach
Finite element equations are derived by minimizing total potential energy. A 1D bar example is used.

(ii) Stress-Strain Relationship
σ = Dε, where D is material matrix.
Strain-displacement relations are derived using displacement gradients.

(iii) Stress and Strain Components
Normal and shear components are explained with diagrams showing x-y axes.