Learning Objectives

1. To review the theoretical fundamentals of the Finite Element Method.

2. To apply the finite element method to solve some simple 1D and 2D problems manually.

3. Demonstrate the ability to understand and apply the fundamentals of Finite Element Analysis.

4. To get familiar with the design, setup and solution of basic FEA models.

Question 1: Application of 2D Solid Elements

Simplification of complex geometry to perform an efficient analysis of a mechanical or structural problem is a critical step the Computer Aided Engineering analyst must make. In particular, simplifying complex 3D problems into 2D can be beneficial, reducing setup, solve times and allowing for rapid design optimisation.

For this task you are required to identify a series of physical problems which lend themselves to being simplified to the use of a 2D solid element.

You are required to:

a) Describe and discuss the differences of the 2D Solid Element types:
o Plane Stress;
o Plane Strain; and
o Axisymmetric.

b) Provide 6 realistic case studies, in total, in which the simplification from 3D to 2D solid elements above can be made:
o 2x plane stress case studies;
o 2x plane strain case studies; and
o 2x axisymmetric case studies.
You must describe and explain why the simplification is suitable for each for them. Include schematics and support mathematically if able to do so. This should be conducted clearly, with clearly defined boundary conditions, etc.

c) Setup an ANSYS FE model for one of the case studies discussed above for each element type (3 case studies in total):
o 1x plane stress;
o 1x plane strain; and
o 1x axisymmetric.

Present the FE setup (element selection, mesh, boundary conditions, etc.), results and a detailed discussion of the suitability for the 2D assumption in a clear and structured report. Describe and discuss the critical results for the particular analysis.

Question 2: Bar Element

Considering the composite member below, discretize the problem using the minimum number of bar elements required to solve it. The elastic properties and geometry of sections 1 and 3 are identical: Young's modulus E₁ = E₃ = 210 GPa and cross-section area A₁ = A₃ = 0.1 m². For the middle section we have E₂ = 170 GPa and A₂ = 0.05 m². The length L is 0.5 m

Determine:

a) the nodal displacements at the interfaces of the sections;

b) the reaction force at the fixed end; and

c) the axial stresses and strains in each section.

Question 3: Truss Structure

Consider the three membered planar truss-structure shown in the figure below. All members of the truss have identical square cross-sectional area A = 0.15 m², and Young's Modulus E = 210 GPa. The hinged joints of the truss structure allow free rotation of the members about the z-axis.

For L = 1 m, β = 60° and an applied force F = 20 kN, determine:

a) the horizontal and vertical displacements at node 3;

b) the reaction forces at each support; and

c) the stress and strain of each member.