ECMM107 Mechanics of Materials 2021
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ECMM107
ENGINEERING
January 2021
Mechanics of Materials
Question 1 (25 marks)
(a) Using appropriate diagrams, explain what the term complementary shear stress in elastic materials. State any assumptions that you make. (5 marks)
(b) A stress tensor at a particular point in an elastic body is given by
\
( ) ,
measured in 102 Nmm—2 and where m and n are unknown constants. An inde- pendent experiment has determined that the principal stresses are −500 Nmm—2 , 100 Nmm—2 and 300 Nmm—2 . Find the values of m and n. (20 marks)
Question 2 (25 marks)
(a) Deine a state of plane stress. Give an example where this condition arises. (5 marks)
(b) The triangular cantilever shown in Figure Q2 is acted upon by a loading to be deter- mined. It has the following stress function associated with it:
= Cr2 [(α − a) + sin 2a − cos2 a tan α] ,
where α and C are constants. Show that satisies the biharmonic equation ∇4 = 0. (10 marks)
(c) Determine the radial and tangential components of direct stress, ar and ae and shear stress τre and hence determine the loading on the horizontal and sloping edges.
y
α r
a
x
Figure Q2: Triangular cantilever (10 marks)
Question 3 (25 marks)
(a) Your company designs and manufactures work vehicles such as refuse collection vehicles (bin lorries) and road sweepers. As part of a programme of weight loss for a road sweeper vehicle, a prototype composite chassis has been itted into one vehicle. This is the irst time your company has used composites in load bearing roles, and even though they have undertaken extensive inite element analysis dur- ing the design phase, so they are not conident they have got their design absolutely correct.
You have been asked to undertake an experimental stress analysis of the chassis in the vehicle during testing. The objective of this analysis is to measure real patterns of stresses in the composite chassis, compared to those predicted during design, under realistic loading conditions.
Describe how you would undertake an experimental stress analysis of the compos- ite chassis in the prototype vehicle, indicating the key steps in your process. (10 marks)
(b) Set out your arguments as to why the technique or techniques you have speciied are appropriate for this case. Explain how the results of your investigation could be used to inform the redesign of the chassis if required. (7 marks)
(c) Describe the principles of operation of the electronic resistance strain gauge. In- clude the governing equations relating the measured physical parameters, material properties, and strain.
Include diagrams and equations in your answer as appropriate. (8 marks)
Question 4 (25 marks)
Various metal alloys can be used to make electronic resistance strain gauges. Most are described as being ‘compensated’ for a speciic temperature (or temperature range).
(a) Deine the term ‘compensated, and explain why different alloys have different com- pensation temperatures. (15 marks)
(b) Specify a gauge strain alloy and its compensation temperature, and give an example situation where it could be used successfully in an experimental stress analysis. (5 marks)
(c) Give an example situation where the strain gauge alloy speciied in your answer to Question 4(b) could NOT be used successfully in an experimental stress analysis. (5 marks)
Question 5 (25 marks)
A plate with two side cracks, made of 2014-T651 aluminium, has dimensions b = 50 mm, t = 5 mm, and large h as speciied in the Figure Q5. A tensile force of P = 50 kN is applied to the plate. The material has the following properties Kk = 24 MPam1/2 and a0 = 390 MPa.
Figure Q5: A plate with two side cracks each of length a.
(a) What is the stress intensity factor when the crack length is 5 mm? (10 marks)
(b) Determine the critical crack length for fracture under the given loading condition. (5 marks)
(c) For a particular design scenario involving this plate, the minimum safety factor of 4.0 is required against fracture. What are the largest crack lengths allowed and what is the safety factor on crack length in this case? (10 marks)
Question 6 (25 marks)
Engineering stress-strain data from a tension test on 7075-T651 aluminium are given in Table Q6. The diameter before testing was 10 mm, and after fracture at a stress of 571 MPa, the minimum diameter in the necked region was 8 mm.
a, MPa |
ε, % |
a, MPa |
ε, % |
0 |
0.000 |
587 |
5.980 |
168 |
0.248 |
593 |
8.020 |
326 |
0.474 |
599 |
9.520 |
415 |
0.605 |
605 |
10.970 |
505 |
0.797 |
600 |
12.500 |
572 |
1.209 |
591 |
13.900 |
563 |
2.300 |
571 |
15.330 |
577 |
4.020 |
|
|
Table Q6: Tensile test data of 7075-T651 aluminium.
These data points have been plotted on the graph below as Engineering Stress (MPa) against Engineering Strain (%) in Figure Q6
500
400 300 200 stre1SS(MPa)00
5.000 10.000 strain(%) |
Figure Q6: Stress-strain graph.
(a) Reproduce a sketch of the above plotted stress-strain curve and label elastic de- formation, yielding, strain hardening and necking phases on this sketch. Annotate points on sketch with the stress and strain values at which different phases transi- tion. (8 marks)
(b) Determine the following parameters, elastic modulus, 0.2% offset yield strength, ultimate strength, percent elongation, and percent reduction in the cross-sectional area at fracture. (10 marks)
(c) What is the plastic strain after the fracture? What is the new length of the specimen if the initial length was 12 mm? Sketch return path after failure on a sketch. (7 marks)
2023-01-10