1
MATH35021: SOLUTION SHEET VII1
1.) It is natural to work in spherical polar coordinates. The geometry and the boundary
conditions are independent of and so we pose a solution of the form
u = u(r)er .
It follows that curl u = 0 and so the NavierLam equat
MATH35021
Two hours
UNIVERSITY OF MANCHESTER
ELASTICITY
25 January, 2011
14:00 16:00
Answer ALL questions in Section A and ALL questions in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH35021
SECTIO
MATH30271
TWO hours
UNIVERSITY OF MANCHESTER
ELASTICITY
16 January 2007
14.00 V 16.00
Answer ALL questions in Section A and ALL question in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 6 PTO. MATH30271
S_EQ
MATH35021
Two hours
THE UNIVERSITY OF MANCHESTER
ELASTICITY
20 January, 2012
14:00 16:00
Answer ALL questions in Section A and ALL questions in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH35021
SE
3271
Two Hours
UNIVERSITY OF MANCHESTER
ELASTICITY
2005
Answer all four questions in SECTION A (40 marks in total)
and
all of the two questions in SECTION B (20 marks each).
The total for the paper is 80 marks. A further 20 marks are available from course
MATH35021
Two hours
UNIVERSITY OF MANCHESTER
ELASTICITY
Friday 18th January, 2008
14:00 16:00
Answer ALL questions in Section A and ALL question in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH3502
1
MATH35021: Comments on Exam 2013/14
Most students attempted every question on the exam and the majority did reasonably well. There
were few algebraic and calculus problems, but there was confusion about formulating boundary conditions
(particularly in q
1
MATH35021: Comments on Exam 2012/13
The majority of students attempted every question, but it is clear that this was a dicult exam.
General problems were lack of condence with vector calculus and partial dierential equations; not
being able to formulate
MATH35021
Two hours
UNIVERSITY OF MANCHESTER
ELASTICITY
Wednesday 2lst January, 2009
09.45 11.45
Answer ALL questions in Section A and ALL question in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 5 P.T.O
1
MATH35021: Comments on Exam 2011/12
Overall, the vast majority of students attempted every question. The general diculties were problems
with index notation; not understanding how to compute the deformed position from the undeformed
position and displac
1
MATH35021: Comments on Exam 2010/11
Overall, the vast majority of students attempted every question. The general diculties were a lack
of condence in dierentiation and solving standard ODEs, index notation and the derivation/imposition
of boundary condi
MATH35021
Two hours
UNIVERSITY OF MANCHESTER
ELASTICITY
21 January, 2010
09:45 11:45
Answer ALL questions in Section A and ALL questions in Section B.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH35021
SECTIO
3271
Two Hours
UNIVERSITY OF MANCHESTER
ELASTICITY
2003
Examinationtime not specied
Answer all three questions in SECTION A (30 marks in total)
and
all of the two questions in SECTION B (25 marks each).
The total for the paper is 80 marks. A further 20 ma
1
MATH35021: SOLUTION SHEET1 VI
1.) a) The boundary conditions are:
Top: n = e3 ; air pressure is zero, p = 0:
t = 0.
Curved sides: n = (n1 , n2 , 0); the water pressure increases linearly with
depth p = gx3 :
t = gx3 n.
Bottom: n = e3 ; the water pressur
1
MATH35021: EXAMPLE SHEET1 IX
y
B
s
A
x
Figure 1: A rigid body in plane strain.
1.) From the lecture notes, if is the Airy stress function then on the boundary curve
tx (s) =
d
ds
y
and ty (s) =
,
d
ds
x
,
where s is the arc length which parameterises t
1
MATH35021: SOLUTION SHEET V1
1.) The principal axes of stress are the eigenvectors of the stress tensor, ij , i.e. the
vectors, vi such that
ij vj = vi ,
(1)
where is a constant, in fact it is the principal stress.
For a homogeneous, isotropic, linear e
1
MATH35021: SOLUTION SHEET I1
1.) Which one of these equations in index notation are valid? Remember the summation
convention!
a) c = ai bi (OK, this is the dot product c = a b)
b) c = aij bi (Wrong, the free index j doesnt appear on LHS)
c) ci = aij bi
1
MATH35021: SOLUTION SHEET II1
1 1
, see Example Sheet I. Using the formula for
1 1
the normal strain, en = eij ni nj , we have
1.) a) The strain tensor is eij =
1 1
1 1
3/5 4/5
en1 =
3/5
4/5
49
.
25
=
Now, n1 n2 = 0 and so the angle between the undeform
MATH35021
Two hours
THE UNIVERSITY OF MANCHESTER
ELASTICITY
15 January 2013
09:45 11:45
Answer ALL SIX questions.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH35021
1. A two-dimensional elastic body has an un
MATH35021
Two hours
THE UNIVERSITY OF MANCHESTER
ELASTICITY
13 January 2014
09:45 11:45
Answer ALL FIVE questions.
Electronic calculators may be used, provided that they cannot store text.
1 of 5
P.T.O.
MATH35021
1. A two-dimensional elastic body has an u
1
MATH35021: SOLUTION SHEET III1
(1)
(2)
1.) i.) The two displacement elds ui and ui corrrespond to the same strain eld eij .
(1)
(2)
If we let u = ui ui then the corresponding strain eld is
i
e =
ij
=
1
1 (1)
(2)
(1)
(2)
ui,j + u =
ui,j ui,j + uj,i uj,i
1
MATH35021: Comments on Exam 2009/10
A1 Apart from a few arithmetic errors, nearly everybody got (i) and (ii), but part (iii) proved to be a
problem. The majority got to the correct eigenproblem:
2 (x2 1)
0
0
x2
v = v,
but did not know (or realise) that