THEORY EXAMINATION (SEM–VIII) 2016-17 FINIT ELEMENT METHOD

SECTION – A

Explain the following (Short Answer Type)

A stiffness matrix represents the relationship between nodal forces and displacements in a finite element system. It defines how a structure resists deformation.

Finite Element Analysis (FEA) is a numerical technique used to solve complex engineering problems by dividing structures into smaller elements.

The main advantage of FEM is its ability to analyze complex shapes, boundary conditions, and material properties accurately.

The limitation of FEM is that it requires high computational effort and accurate meshing for reliable results.

Galerkin method is a weighted residual method used to obtain approximate solutions by minimizing error in differential equations.

Convergence in FEM means that as the number of elements increases, the numerical solution approaches the exact solution.

A global matrix is formed by assembling stiffness matrices of all elements into one system equation.

The stiffness matrix for a beam element depends on material properties, length, and cross-sectional area.

Stages in finite element formulation include discretization, element selection, assembling matrices, applying boundary conditions, and solving equations.

The stiffness matrix for a truss element relates axial forces to nodal displacements in the structure.

Questions (Section A)

What is stiffness matrix?

Define finite element analysis.

Write advantages of FEM.

What is convergence in FEM?

What is global matrix?

SECTION – B

Attempt any five (Long Answer Type)

Shape functions describe displacement variation within an element. Linear rectangular elements use local coordinate systems to define these functions.

Errors in FEM occur due to discretization, numerical approximation, and modeling assumptions. These errors reduce as mesh density increases.

In displacement approximation problems, finite element methods use interpolation functions to estimate unknown values.

Kirchhoff theory assumes thin plate bending with no transverse shear deformation, while Mindlin theory considers shear effects for thick plates.

Isoparametric formulation uses the same shape functions for geometry and displacement representation, simplifying analysis.

Jacobian transformation converts local element coordinates into global coordinates during finite element calculations.

Bar element modeling helps determine displacement and stress distribution in structural components.

Temperature distribution problems can be solved using one-dimensional finite element analysis.

Questions (Section B)

What are shape functions?

What are FEM errors?

Explain Kirchhoff and Mindlin theories.

What is isoparametric formulation?

What is Jacobian transformation?

SECTION – C

Attempt any two (Detailed Explanation)

In truss analysis using FEM, global stiffness matrices are formed by assembling element matrices. Displacements and reaction forces are calculated by solving system equations.

Thermal conduction problems can be solved using FEM by developing global matrices based on thermal resistance and heat transfer coefficients.

Rayleigh-Ritz method is an energy-based approach used to approximate displacement and stress in structural elements under distributed loads.

Questions (Section C)

How is global stiffness matrix formed?

Explain FEM in thermal conduction problems.

What is Rayleigh-Ritz method?