Question 1: Concepts of Solid Mechanics

Objective: Apply basic concepts of solid mechanics, e.g. displacement field, strain and stress tensors. The displacement ???? in a second-order (quadratic) element is calculated from the FEA of static analysis of a linear elastic, isotropic, homogeneous solid as following

u = 2.0 * 10^-4(x₁ + 2x₂ - x₃)e₁ + 1.0 * 10^-4x₂x₃e₂ - 1.5 * 10^-4(x₁ - x₃)²e₃

(Note: the displacements and the coordinates have the same length unit, m)

At point P(4, 2, 2), determine:

• The displacement vector;
• The strain tensor;
• The stress tensor, given E = 200 GPa and ν = 0.3; and
• If the yield strength of this material is 550 MPa, is the material safe at point P according to von Mises yield criteria?

Question 2: FEA of a 1-D bar system

Objectives:

To understand bar element and know how to use it to analyse a 1-D bar system

A system consisting of three bars is shown in the following figure and all the parameters of the members are given in the figure as well.

With the minimum number of linear elements, determine:

• the element stiffness matrices for each part;
• the assembled global stiffness matrices;
• the boundary conditions in the system;
• the condensed matrix equations for the unknown displacements after applying the boundary conditions;
• the displacement at each interface for two neighbouring parts;
• the reaction forces applied at the left end of the bar;
• the internal forces in each part; and
• the internal stress in each part.

Question 3: Structural Analysis of a planar truss

Objectives:

1 To understand truss element and know how to use FEA to analyse a truss structure. A planar truss structure is shown in the following figure and all the parameters of the members are given in the figure as well:

Determine:

• the transformed element stiffness matrices for each truss member;
• the assembled global stiffness matrix;
• the boundary conditions at each node;
• the condensed matrix equations for the unknown displacements after applying the boundary conditions;
• the displacements at each node when the load is applied statically;
• the internal forces in each truss member; and
• the internal stress in each truss member.

Question 4: FEA of Beam Structure

Objectives:

To understand beam element and know how to use it to analyse a 1-D beam system subjected to combined loadings.

Use the minimum number of Euler-Bernoulli beam elements, which only one beam element is used for one part made of a material, to analyse the beam structure as depicted in the image below subjected to combined loading.

(a) Determine:

• the element stiffness matrices for each part;
• the assembled global stiffness matrices;
• the boundary conditions in the system;
• the condensed matrix equations for the unknown displacements after applying boundary conditions;
• the displacement at each interface for two neighbouring parts;
• the reaction forces and moments; and
• the maximum stress in each part.

(b) Use Abaqus to run a static analysis of this problem.