Homework answers / question archive / Problem 1 What is the Finite Element Method (FEM)? And what are the 10 steps in FEM? Also, discuss all possible ways to evaluate FEM results

Problem 1 What is the Finite Element Method (FEM)? And what are the 10 steps in FEM? Also, discuss all possible ways to evaluate FEM results

Mechanical Engineering

Share With

---

Problem 1

What is the Finite Element Method (FEM)? And what are the 10 steps in FEM? Also, discuss all possible ways to evaluate FEM results.

Problem 2 (30 points) Consider the two-member arch-truss structure shown in Figure 1. Take span S = 8, height H = 3, elastic modulus E = 1000, cross section areas A⁽²⁾ = 4 and downward crown force P = 12. Using FEM to carry out the following steps:

(a) Is the problem symmetric with respect to the central axis? Discuss and justify your answer.

(b) Show element equation in local and global coordinates, assemble the global stiffness equations.

(c) Apply the boundary conditions and solve the reduced system for the crown displacements ux2, uy2. (d) What is the actual direction of ux2 (to the left or right)? And why? Discuss and justify your answer for ux2 from engineering sense.

(e) Compute the reaction forces and stresses.

 

Problem 3 (20 points) Show the shape functions N^e(x) of linear and quadratic 1D elements that span from x = 0 to x = L. After that, determine the corresponding geometry matrix Be (x) = d/dxN^e. Verify that the summation of all shape functions per element equal to 1 and the summation of all geometry matrix components equal to 0• i.e.

∑_i=1^{# of nodes} N^e(x) = 1, ∑^#of nodes B_e(x)=0

 

Explain why the above conditions always have to be satisfied from engineering sense.

 

Problem 4 (30 points) Given an elastic bar of length i = 4 m with constant cross-sectional area A = 0.1 m² and a piecewise constant Young's modulus as shown in Figure

2. The bar is constrained at x = 4 m, a prescribed traction t= 500 Nm^-2 acts at x = 0 m in the positive x-direction, and a body force b(x) = 100x NM-2 distributed over the entire bar.

a) Model the bar with two linear elements: derive element stiffness matrix, force matrix, assemble the global equations, and solve for the unknown displacements using the boundary conditions.

b) Model the bar with one quadratic element: derive element stiffness matrix, force matrix, and solve for the unknown displacements using the boundary conditions (Hint: in element integrals, you can integrate separately for [0.1/2] and [1/2.1] as they have different E's, and sum them up).

c) Evaluate the strains and stresses over the entire bar (as functions of x) using quadratic and linear elements. Compare the results. Which one is better (from engineering sense)? In term of design of finite element meshes, what kind of a recommendation can you make based on the results of this problem? Choose higher-order elements with less mesh (one quadratic element) or lower-order elements but with more meshes (two linear elements)?

d) Check whether the equilibrium conditions (force balance) and traction boundary conditions (stress at x = 0) are satisfied for the two element types. Discuss your answer.

E=10⁵ N/m²     E₂= 10⁸  N/m²

                                                x

 

X=0                                        x=1/2                                    x=1