ECMM107 Mechanics of Materials 2018
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ECMM107
ENGINEERING
Mechanics of Materials
January 2018
Question 1 (25 marks)
(a) In the theory of elasticity, what is meant when a material is described as
homogeneous and isotropic? (3 marks)
(b) What is a principal strain and what is a principal axis? How do the axes orientate with respect to each other in a three-dimensional homogeneous and isotropic material? (7 marks)
(c) A state of strain (measured in millistrain or 10—3 ) in a material is given by the
following tensor
1(3) 1(0)
1 −2) .
Find the principal strains and their directions. (15 marks)
Question 2 (25 marks)
(a) Deine a state of plane stress. Give an example where this condition arises. (3 marks)
(b) In polar coordinates, if a stress function is independent of a, i.e. = (T) only, show that the biharmonic equation, ∇4 = 0, simpliies to 2
d4 2 d3 1 d2 1 d (10 marks)
dT4 T dT3 T2 dT2 T3 dT
(c) Show that the stress function
(T) = Aln T + CT2
satisies the above biharmonic equation (A and C are constants). (7 marks)
(d) Determine the radial and tangential components of direct stress, ar and ae respectively, and identify any singularities in the stress ield. Why is the shear stress zero? (5 marks)
Question 3 (25 marks)
A new hybrid electric vehicle is being designed by your company. A prototype must be put through a series of tests to make sure the design is sound before it is passed over to the Production Team who will work out how to manufacture it.
Whilst being tested around a track a load bearing beam located internally underneath the passenger door buckles in bending and then fails catastrophically. The beam is made from carbon ibre composite. Finite element analysis conducted on the design did not indicate any problem with this component.
(a) i. Set out how you would undertake an experimental stress analysis of this
problem, thereby identifying the cause of failure. Indicate the important practical steps in the technique or techniques you plan to use. (8 marks)
ii. Another similar vehicle is available for you to use in your investigation.
Set out arguments as to why the technique or techniques you have selected are appropriate for this case. Explain how the results of your investigation could be used to inform the redesign of the vehicle. (9 marks)
(b) Choose one of the experimental techniques you selected in 3(a) and describe its principles of operation, including the governing equations relating the measured physical parameter and strain or stress.
Include diagrams and equations in your answer as appropriate. (8 marks)
Question 4 (25 marks)
(a) It is possible to locate and orient strain gauges onto components so they are sensitive to some loads but insensitive to others. This is advantageous because it allows measurement of important loads without more costly sensors and electronic equipment, or any uncertainty.
One such example is to locate two linear gauges on the top surface of a cantilever and two gauges on the lower surface. These are connected into a ‘full bridge’ Wheatstone circuit. When placed in combined axial and lexural loads, this coniguration is sensitive only to lexural loads.
Describe another such example coniguration of strain gauges which are sensitive to load of one type but insensitive to load of another type. Include a sketch showing the location of the gauges on the component, and their connectivity within a Wheatstone bridge circuit. Describe how this coniguration functions. (10 marks)
(b) Explain how the Digital Image correlation method detects strains on a surface. Include diagrams in your answer as appropriate. (15 marks)
Question 5 (25 marks)
(a) A central crack, with a length of 50 mm, develops in an aluminium plate, of width
100 mm. If the plate fails at the applied stress value of 200 MPa,
i. What is the stress intensity factor for this plate (KI )? (4 marks)
ii. Using the value from the part (i) for the stress intensity (assuming KI = KIc ), if E = 70 GPa for the assigned plate, estimate its strain energy release rate (Gc). (3 marks)
iii. How much should the applied load change (as a percentage) to produce fracture for the same crack length in a two symmetrical edge-crack (refer to Table Q5(a)) where each has a length of 25 mm? (5 marks)
Type of crack |
Stress intensity |
Centre-crack, length 2a, in an ininite plate |
KI = aapp (Ta)1/2 |
Centre-crack, length 2a, in a plate of width W |
KI = aapp [W tan(Ta/W)]1/2 |
Central penny-shaped crack, radius a, in ininite body |
KI = 2aapp (a/T)1/2 |
Edge crack, length a, in a semi-ininite plate |
KI = 1.12aapp (Ta)1/2 |
Two symmetrical edge cracks, each of length a, in a plate of total width W |
KI = aapp W1/2[tan(Ta/W) + 0.1 sin(2Ta/W)]1/2 |
Table Q5(a): Table of stress intensity for different crack types.
(b) If the maximum applied stress for the same plate in part (a) is 200 MPa, and the minimum applied is 120 MPa,
i. Find the maximum shear stress using Tresca method. (3 marks)
ii. What is the von Mises failure stress? (3 marks)
iii. Draw the Tresca and von Mises stress envelopes in 2D. (4 marks)
iv. Comment on the differences between the two stress envelopes in part (iii).
(3 marks)
Question 6 (25 marks)
(a) For one steel component, which has length 50 mm in section, the stress generated during the heat transfer in a surface law is 150 MPa. The KIc , which has been found experimentally, is 40 MPam1/2, and the proof stress is 600 MPa. If the maximum allowable size of surface defect is 1.4 mm,
i. How large is the tolerable defect size if a/(2c) = 0.2? (5 marks)
ii. If the generated stress approaches the proof strength of the material, what will be the critical length of crack? Use the data from Table Q6(a). (5 marks)
Use
( )cr = .
a/ay |
Q for a/(2c) values | |
||||
|
0.10 |
0.20 |
0.25 |
0.30 |
0.40 |
1.0 |
0.88 |
1.07 |
1.21 |
1.38 |
1.76 |
0.9 |
0.91 |
1.12 |
1.24 |
1.41 |
1.79 |
0.8 |
0.95 |
1.15 |
1.27 |
1.45 |
1.83 |
0.7 |
0.98 |
1.17 |
1.31 |
1.48 |
1.87 |
0.6 |
1.02 |
1.22 |
1.35 |
1.52 |
1.90 |
<0.6 |
1.10 |
1.29 |
1.43 |
1.60 |
1.98 |
Table Q6(a): Table of critical value of a/Q for different a/ay values.
(b) For a specimen tested in a laboratory, the measured values are illustrated in Table Q6(b)
|
Stress (MPa) |
Strain |
Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 Point 7 |
0 2 4 6 8 10 12 |
0 0.2 0.4 1.1 4.5 5.7 6.2 |
Table Q6(b): Table of experimental stress–strain data.
i. Draw the stress-strain curve based on interpolating the points. (5 marks)
ii. Show the three main regions of elastic, yielding and plastic deformations on the graph. (3 marks)
iii. Based on this graph, calculate the Young’s modulus, E . (1 mark)
iv. At which point, does the strain hardening start? (1 mark)
v. Show the Considere criterion on the graph to estimate the necking location. (5 marks)
2023-01-10