FEM

Assignment 1: Introduction to FEM – Unit 1

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 List three major applications of FEM in civil engineering and explain why FEM is preferred over other numerical methods for those cases. CO3 L2, L4 5
2 Derive the stiffness relation for a linear spring element using Hooke’s law and matrix formulation. Include a neat diagram. CO1 L3 5
3 Derive the stiffness matrix for a 1D bar element of length $L$, cross-sectional area $A$, and modulus of elasticity $E$. State assumptions clearly. CO1 L3 5
4 Using the Minimum Potential Energy Principle, derive the FEM equation for a spring element. CO1, CO2 L3 5
5 Explain the Direct Stiffness Method with the steps: (a) numbering, (b) connectivity, (c) formation of local stiffness matrices, (d) assembly. Draw a schematic. CO1, CO4 L2, L3 5
6 For the spring system shown below with $k_1=50 \ \text{lb/in}$ and $k_2=25 \ \text{lb/in}$, determine the force $F_3$ required to displace node 2 by $0.75$ in. Also compute displacement of node 3. CO1, CO3 L3, L4 5
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7 In the spring assembly of the figure shown below, forces $F_2$ and $F_4$ are to be applied such that the resultant force in element 2 is zero and node 4 displaces an amount of 1 in. Determine:
(a) the required values of forces $F_2$ and $F_4$,
(b) the displacement of node 2, and
(c) the reaction force at node 1.

Given spring constants: $k_1 = 30 \, \text{lb/in.},\; k_2,\; k_3$.
CO1, CO3 L3, L4 5
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8 A steel rod as shown in figure below is modeled by two bar elements of length $0.5 \ \text{m}$ each, $E=207 \ \text{GPa}$, $A=500 \ \text{mm}^2$, subjected to a $12 \ \text{kN}$ compressive load. Determine the nodal displacements and the axial stress in each element. What other concerns should be examined? CO1, CO3 L3, L4 5
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9 In the two-member truss shown in below, let:
• $\theta_1 = 45^\circ$, $\theta_2 = 15^\circ$
• $F_5 = 5000 \, \text{lb}$, $F_6 = 3000 \, \text{lb}$

(a) Using only static force equilibrium equations, solve for the force in each member as well as the reaction force components.
(b) Assuming each member has axial stiffness $k = 52000 \, \text{lb/in.}$, compute the axial deflection of each member.
(c) Using the results of part (b), calculate the X and Y displacements of node 3.
CO1, CO3 L3, L4 5
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10 The two-element truss in figure below is subjected to external loading as shown. Determine:
• The displacement components of node 3,
• The reaction force components at nodes 1 and 2,
• The element displacements, stresses, and forces.

Material properties: $E_1 = E_2 = 10 \times 10^6 \ \text{lb/in}^2$, $A_1 = A_2 = 1.5 \ \text{in}^2$.
CO1, CO3 L3, L4 5
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Course Outcomes (COs)

CO # Course Outcomes
CO1 Develop stiffness matrix and load vector for a given structural element
CO2 Formulate finite element equation for a given problem
CO3 Select suitable element type and solution form in the Finite element analysis
CO4 Illustrate the use of FEA tools/software in obtaining the structural response of a given problem

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