FEM

Assignment 2: Beam Elements – Unit 2

Course Code: MST-102

Course Name: Finite Element Method in Structural Engineering


Submission Instructions

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Assignment Questions

Q. No. Question CO RBT Level Marks
1 Derive the displacement function for a beam flexure element of length \(L\) using a cubic polynomial. Apply boundary conditions at the nodes and express in terms of nodal displacements and rotations. CO1 L3 5
2 Derive the shape functions \(N_1, N_2, N_3, N_4\) for the flexure element in terms of the dimensionless coordinate \(\xi = \frac{x}{L}\). CO1 L3 5
3 Compute the second derivatives of the shape functions and show how the element stiffness coefficients \(k_{mn}\) are obtained using Castigliano’s theorem. CO1, CO2 L3 5
4 Using the dimensionless coordinate \(\xi\), transform the element stiffness matrix integrals and compute the final 4×4 stiffness matrix for a beam element of length \(L\) and flexural rigidity \(EI\). CO1, CO2 L3, L4 5
5 For a simply supported beam of length \(L = 2\,\text{m}\) subjected to a point load \(P = 10\,\text{kN}\) at midspan, model the beam using two flexure elements. Compute:
(a) nodal displacements,
(b) element bending moments,
(c) reactions at supports.
CO1, CO3 L3, L4 5
6 A cantilever beam of length \(L = 3\,\text{m}\) with a uniform distributed load \(w = 5\,\text{kN/m}\) is modeled using two flexure elements. Determine:
(a) the nodal displacements,
(b) nodal rotations,
(c) maximum bending moment in each element.
CO1, CO3 L3, L4 5
7 A beam is subjected to a concentrated moment \(M = 20\,\text{kNm}\) at midspan. Using element load vector, determine the equivalent nodal forces and moments for a two-element discretization. CO2, CO3 L3, L4 5
8 For the beam in Q6, draw the bending moment diagram and indicate the location and value of maximum stress. Compute stress at top and bottom fibers. CO3 L3, L4 5
9 Explain how superposition of nodal forces and assembly of element stiffness matrices lead to the global stiffness matrix for a multi-element beam. Draw a schematic with at least three elements. CO2, CO4 L2, L3 5
10 A beam of length \(L = 4\,\text{m}\) is discretized into two flexure elements. If the first element has a point load at \(x = 1\,\text{m}\) and the second element has a uniform load, determine:
(a) nodal displacements,
(b) nodal rotations,
(c) reactions at the supports.
CO1, CO3 L3, L4 5

Course Outcomes (COs)

CO # Course Outcomes
CO1 Develop stiffness matrix and load vector for a given beam element
CO2 Formulate finite element equations for flexure elements
CO3 Compute nodal displacements, rotations, and stresses in beams
CO4 Illustrate the use of FEA tools/software in obtaining the structural response of beams

RBT Levels

Level L1 L2 L3 L4 L5 L6
Cognitive Process Remember Understand Apply Analyze Evaluate Create

Prepared by:
Dr. Yuvraj Singh
Assistant Professor, Department of Civil Engineering
Guru Nanak Dev Engineering College, Ludhiana