[SOLVED] Computer Modelling Techniques MM3CMT or MMME3026 and AERO3009 Python

Department of Mechanical. Materials and Manufacturing Engineering, Faculty of Science and Engineering

LEVEL: 3

MODULE: Computer Modelling Techniques (MM3CMT or MMME3026 and AERO3009)

ASSIGNMENT: Computer Modelling Techniques – Coursework-Part III: FEA-2D Modelling

ISSUE DATE: 26^th November 2024

SUBMISSION DATE: 4 pm, Friday 15^th December 2024

Computer Modelling Techniques Coursework – Part III: FEA-2D

Modelling

Note: The whole coursework (consisting of parts I, II and III) is equivalent to 30% of the final mark of the module assessment. This coursework, part III- FEA-2D modelling will contribute 9% to the final mark.

Question 1. [Total: 50 marks]

To be completed individually

A long square cross-section steel beam, with a width of w m for each side, contains a uniformly distributed internal heat source of G kW/m³. The beam is subjected to a constant temperature of 25°C on its outer surfaces and the steel material that the beam is made from has a constant thermal conductivity, k, of 50 W/m.K. Assume the heat flux density conducted through the steel beam surface heat insulator to be q=40 kW/m².

Figure Q1. A steel beam with a square cross-section and uniform. internal heat source.

Each student is allocated a different value for the beam width, w, and internal heat source, G, which can be found in “MMME3026 AERO3009 CW Part III  Student Data 2024- 2025.xlsx” on MMME3026/AERO3009 Moodle.

Using the symmetries about the centre of the square cross-section, the problem can be reduced and resolved using only a single right-angle triangle, as shown in Figure Q1. Solve the problem using this element by following the steps given in sub-questions below.

(a) The shape functions for this triangular element can be given as:

 Find the kinematic matrix, [B], given:

[20 Marks]

(b) Find the element stiffness matrix [K], where:

[10 Marks]

(c) Find the forcing vector {F}, where:

[10 Marks]

(d) Due to the symmetry of beam, the square cross-section is modelled by a single element and its stiffness matrix is given by that of the element. Construct the structural stiffness equation.

[5 Marks] 

(e) Solve the stiffness equation in order to calculate the temperature at the central node.

[5 Marks]

Question 2. [Total: 50 marks]

To be completed in groups

A thin notched plate, with dimensions shown in Figure Q2, is tested under uniaxial tension with a load of σ0 MPa. This plate is made from a steel material with Young’s modulus E = 210 GPa and Poisson’s ratio ν = 0.25. Unit thickness is assumed. Respond to the sub questions below in order to resolve the displacements and stresses in this 2D continuum using the ANSYS Mechanical APDL software.

Each group is allocated a different value for the load, σ₀, which can be found in “MMME3026 AERO3009 CW Part II Student Data 2024-2025.xlsx” on MMME3026/AERO3009 Moodle.

When working in groups, if you have any problems with your groupmates (ie. they are not contributing to the coursework or not responding to your emails) you must contact the lecturer well before the submission due date.

Figure Q2. A notched plate beam with a square cross-section and uniform. internal heat source.

(a) First, identify any possible simplification of the model via symmetry, and re-draw the simplified geometry. Include details of the load application and boundary conditions. You may need to add a constraint at a single node to resist possible rigid body motion in the x-direction.

[4 marks]

(b) Note what type of element you will use for the model, and why.

[2 marks]

(c) Before reporting any results, confirm the validity of your mesh by performing a mesh convergence study for the y-displacement at point B:

· Report the results of this study using a graph with “y-displacement at B” for the dependent axis, against the “number of elements” in the mesh, for the independent axis.

· Plot the results for at least 4 different meshes.

· Also write a sentence or two explaining how many elements are needed for a suitable mesh… in this case, the desired accuracy is within 0.1% for the y-displacement results (you can calculate this accuracy by comparing results from different meshes to see if the change in y-displacement is less than 0.1% when the number of elements is increased).

(Note: the number of elements in your mesh is displayed in the “Output Window” when you mesh the geometry. It is recommended that you start from a very coarse mesh to best demonstrate this convergence behaviour.)

[20 marks]

(d) Now, with greater confidence in the quality of the mesh, please report the following values as predicted by the FE model:

· x- and y-displacements at points A, B and C

· Maximum normal stress in the x-direction, including the location of this stress

· Maximum normal stress in the y-direction, including the location of this stress

(Note: you may wish to use images to show the location of these stresses, which can be saved through “PlotCrtls > Capture Image…”)

[16 marks]

(e) Lastly, use a “path” to graph the change in y stresses between points D and E. This should have y stress as the dependent variable against the distance from D on the independent axis.

[8 marks]