Finite Element Applications
Finite Element Applications
(GEEN1125)
RESIT COURSEWORK
Level:7
Credit:15
Department:Engineering Science
Faculty:Engineering and Science
Lecturer:Dr. Michael I. Okereke (MIO)
Email (MIO):m.i.okereke@gre.ac.ukOffice (MIO):Room P160, Pembroke South
Instructions
Attempt all 4 questions.
All questions must be answered compulsorily.
Each question is worth 25 marks
You should start each questions answer on a fresh page.
Question 1:
The Finite Element Method is one of the methods required in solving structural analysis problems. Another promising approach is the Virtual Work Principle. Briefly discuss the Virtual Work Principle and how different this is to the Finite Element Method.
[5 marks]
The length scales required for analysis of an engineering structure from nanoscale to structural scales is given in Figure Q1A. Identify the individual components at each length scale for the trainer shoes structure shown in Figure Q1B.
[5 marks]
Figure Q1A: Typical length scales used in computational materials science. The different scales are shown in order on increasing dimensions.
Figure Q1B: A Trainer Shoes
During material model development of the FE process, visualizing the 3D plots of functions is essential in understanding the spatial variation of material parameters, say a and b with c. Take the range of values of a and b to vary from -3 to +3 using incremental steps of 0.1. Using MATLAB, construct a 3D plot of the equations: and .
Ensure that both graphs are plotted on the same xyz-axis. Document your solution procedure within a MATLAB M-file.
[10 marks]
To prescribe a kinematic constraint to a typical finite element model, Multi-Freedom Constraint (MFC) equations are required. Write the ABAQUS-style linear constraint equation for the following equations:
[5 marks]
Question 2
A flat fillet bar is designed as an axially loaded member subjected to a simple tensile testing load. The bar dimensions given in Figure Q2(A). The bar is made from steel with properties: Youngs Modulus, E = 210GPa, and Poisson ratio, v = 0.3.
Generate the virtual domains of the fillet bar in an appropriate virtual domain creating software of your choice.
[6 marks]
Assume the bar behaves in an elastic manner, run simulations of the tensile deformation of the bar based on mesh discretization of the virtual domain using a hexahedral element shape (swept meshing with adequate mesh density).
Show contour plots of the von Mises stress (S Mises), in-plane shear strain (E12) and magnitude of displacement (U Magnitude).
[8 marks]
In order to verify the numerical solutions, the analytical relationships between the maximum effective (von Mises) stress of the bar and the average stress are defined as:
where K=stress concentration factor for the fillet bar, P = axial pressure load, h=height of the smaller cross-section, t = thickness of bar.
Use the chart given in Figure Q2(B) to determine the value of K.
[6 marks]
Determine the percentage error between the numerically derived maximum von Mises stress, and the analytically determined maximum stresses for the structure. Take the percentage error expression to be:
where = analytically determined maximum tress for the fillet bar and =numerically determined maximum stress.
[5 marks]
Figure Q2(A) A diagram of a fillet bar subjected to load, P
(Not drawn to scale).
Figure Q2 (B) Stress concentration chart for axially loaded fillet bars.
Question 3
An inverted T-support structure is made of cylindrical steel trusses of diameter, d. The tent is designed to support a distributed load of 6.5 kN/m at the under-side of the truss-structure, as shown in Figure Q3.
Take the Youngs modulus of steel, Esteel = 210 GPa. As a design specification, none of the nodes in the structure should sustain a maximum displacement greater than 20 mm i.e., mm, where u = displacement.
Using the MATFESolverTM finite element solver, determine the diameter, of the steel bars that meet the nodal displacement design specification. Show some of the different iterations that led to the required answer.
[12 marks]
Show the deformed profile that corresponds to the required diameter. Use a scaling factor of 50.
[3 marks]
Identify the nodes that experience the largest displacement.
[3 marks]
To reduce the localized deformation at some of the nodes, it was recommended that the roller supports are replaced with fixed supports. With the change implemented, re-run the simulation, and based on the predicted new load the structure can support, calculate the percentage improvement in the load bearing capacity of the structure? Using this equation:
where Fold = 6500kN/m and Fnew = new load that corresponds to umax.
[7 marks]
Figure Q3: An inverted T-support structure with uniformly distributed load.
Question 6
An 80x80 m2 glass fibre reinforced polypropylene matrix unidirectional (UD) composite material is to be analysed using finite element methods. The properties of the fibre and matrix are given in Table Q4. The volume fraction of the fibre is 35%. The polypropylene matrix shows and elastoplastic response while the fibre shows an elastic response. The numerical exercise aims to determine the effective properties i.e., E11, E22 and G12 by using ABAQUS. Consider the diameter of the fibre to be 15 m.
Table Q4: Table of elastoplastic properties of the UD composite
Material Modulus, E (GPa) Poisson Ratio, Yield Stress, y [MPa] Plastic Strain
Glass Fibre 73 0.20 - Polypropylene Matrix 1.38 0.40 40 0.0
42 0.05
45 0.20
47 0.35
Using ABAQUS create the constituent parts and the virtual domain for the composite shown in Figure Q4.
[10 marks]
You are required to carry out simulations of (a) uniaxial deformations in X-axis, (b) uniaxial deformation along Y-axis and (c) in-plane shear deformation on XY plane for a given two-dimensional representative volume element (2DRVE) shown in Figure Q4 circles as fibres, square box as matrix and insert coordinates as x- and y-coordinates positions of the fibres (circles). In all cases, you must apply a 10% strain on the model.
You must use PBC2DGen software to impose periodic boundary conditions on the virtual domain and run simulations for the three load cases.
Ensure you use an adequate mesh to ensure convergent solution. Obtain the stresses-strain plots for all three simulations. Combine the plot of X and Y- axis together for comparison for all three cases.
[10 marks]
Determine the effective properties of the composite for all three load cases. Do you believe these values are realistic? How can you check the results to ensure they are representative of the effective properties of the UD composite?
[5 marks]
Figure Q4: 2D RVE of a particulate composite with different inclusion types(C) Copyright 2022: Dr. Michael I. Okereke.
Email: m.i.okereke@gre.ac.uk ;
Phone: +44(0)1634 88 3580.
YouTube: www.youtube.com/c/michaelokereke
