UNIVERSITY OF BRISTOL
FACULTY OF ENGINEERING
Second Year Examination for the Degrees of
Bachelor and Master of Engineering
EXAMPLE PAPER 1
for
May/June 2015
3 hours
Materials 1
MENG11100
This paper contains 4 sections. You must answer 1 question from each section.
Use separate answer books for each question
Section A contains one question worth 30 marks
Section B contains 2 questions worth 20 marks. Answer only 1 question.
Section C contains one question worth 30 marks
Section D contains 2 questions worth 20 marks. Answer only 1 question.
The maximum for this paper is 100 marks
A formula sheet is provided
Calculators to have Faculty of Engineering Seal of Approval.
TURN OVER ONLY WHEN TOLD TO START WRITING
SECTION A
The section contains one question. This question is compulsory. Use a separate
answer booklet for this question.
Q1
(a) Question
(X marks)
2
SECTION B
You must answer 1 question from this section. Use a separate answer booklet for
this question.
Q2
(a) Question
(X marks)
Q3
(a) Question
(X marks)
3
turn over ...
SECTION C
The section contains one question. This question is compulsory. Use a separate
answer booklet for this question.
Q4
The stress tensor is σij = 0, except σ12 = σ21 = −100MPa.
(a) Write the stress tensor as a 3 × 3 matrix.
(1 marks)
(b) Illustrate the stress tensor on an elementary cube of material.
(2 marks)
(c) Calculate the stress invariants.
(2 marks)
Remember that if all elements in a column or in a row of a matrix are zeros,
then the determinant is zero.
(d) Find the principal stresses.
(4 marks)
(e) Write the stress tensor in the principal coordinate system as a 3 × 3 matrix.
(1 marks)
(f) Find the principal directions using the eigenvalue/eigenvector problem.
(10 marks)
(g) Write down the rotation tensor as a 3 × 3 matrix.
(2 marks)
The rotation tensor is constructed from the principal vectors, written as rowvectors:
(h) Prove that the new coordinate system is right handed by calculating the
determinant of the rotation tensor.
(1 marks)
(i)
Illustrate the principal stress tensor on an elementary cube of material.
(2 marks)
Try to orient the cube according the rotation tensor.
(4 marks)
(j)
Describe the stress state in words
(1 marks)
4
SECTION D
You must answer 1 question from this section. Use a separate answer booklet for
this question.
Q5
The stress tensor is σij = 0, except σ12 = σ21 = 200MPa, σ22 = −200MPa and
σ33 = 800MPa.
(a) Calculate the principal stresses using the Mohr’s circle.
(19 marks)
(b) Calculate the maximum shear stress.
(1 marks)
Q6
The motion of a body is described by:
x1 = X1 + αX1
x2 = X2 + αX2
x3 = X 3
(1)
(2)
(3)
where α is a small constant.
(a) Calculate the displacement vector.
(4 marks)
(b) Calculate the small strain tensor and write it as a 3 × 3 matrix.
(6 marks)
(c) Illustrate the strain tensor on an elementary cube of material.
(6 marks)
(d) Describe the strain state in words.
(4 marks)
5