Due: Mid-Semester Break Week - Sunday (23:59pm 2 October 2022)

Assignment 2

Due: Mid-Semester Break Week - Sunday (23:59pm 2 October 2022)

Q1 Boundary conditions (10 marks)

A rigidly clamped beam with rectangular cross-section of side length 2a is subjected to a set of external loads, as illustrated in Fig. 1, where P is a transverse shear force on the right end surface along the negative direction of z-axis and M is a bending moment on plane EFGH about the x-axis. Describe the boundary conditions of the surfaces ABCD and EFGH.

Figure 1

Q2 Thick cylindrical tube (10 marks)

As shown in Fig. 2, a very long tube A (inner tube made of aluminium) has been thermally shrink-fitted into another very long tube B (outer tube made of mild steel). The final dimensions of the fitted tubes are measured to be RAi=25 mm, RAo=50 mm, RBi=50 mm, and RBo=75 mm.

After fitting, it was found (via the strain gauges as shown) that the two-tube system generates a circumferential stress with a magnitude of =160 MPa on the outer surface of the outer tube B. If the material properties are aluminium EAl=70 GPa, Al=0.3, Yielding strength [Al]=250 MPa; and mild steel ESt=210 GPa, St=0.3, Yielding strength [St]=350 MPa.

(a) Determine the radial interfacial pressure p between the inner and outer tubes.

(b) Please use von Mise criterion to determine whether or not the two-tube system is free of yielding.

Figure 2

Q3 1D finite element method (10 marks)

The horizontal bar is connected by two linear springs. The system is fully clamped at ends A and D, as illustrated in Fig. 3(a).

Figure 3(a)

Discretise the system into two spring elements and one bar element. Write each elemental equilibrium equation; and compile the global equilibrium equation.

Calculate the displacements at B (uB) and C (uC), and the reaction forces at A (FA) and D (FD).

Calculate the displacement at O (uO), which is the midpoint of the bar element, by using the corresponding shape functions.

Plot the distributions of displacement (u(x)) and strain (x) in the system.

If a stopper is placed to the left of node B with a distance (gap) of as shown in Fig. 3(b), calculate the nodal reaction forces at A, B, and D, as well as the nodal displacements at A, B, C and D.

Figure 3(b)

Q4 FEA Mini-project (50 marks)

We can use FE modelling to independently compare variations in geometry and/or loading.

Consider a thin plate with a thickness of t=0.1 m made of aluminium (please find material properties yourself from a reputable source, the reference needs to be listed), which has a central through-hole of radius a. The structure is subject to a uniform tensile stress of =10 MPa, as shown in Fig. 2. Write up a 6-page report for the following case studies.

Figure 2

Case 1 (base case):

For a=0.2 m:

Use appropriate symmetry and boundary conditions to create a finite element model and determine the stress and strain states at points A and B.

Graph the stress xx along line AE and determine the stress concentration factor k.

Case 2:

Investigate the effect of changes in radius a on the stress at point A using the finite element model. Test at least 4 different radii. Verify the numerical results against the analytical solution.

Case 3:

Here, the top (FD) and bottom (GH) edges are fixed, and a uniform temperature increase of T=200 C is applied to the whole plate.

How do xx along AE and yy along BC change relative to Case 1?

Case 4:

Consider a thick prismatic block with a cross section illustrated in Fig. 2.

Compare the stress and strain states at A and B relative to Case 1.