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UNIVERSITY OF SOUTHAMPTON CENV2031W1
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SEMESTER 1 EXAMINATIONS 2019-20
TITLE: Structural Analysis
DURATION: 120 MINS
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This paper contains Four Questions
Please use separate answer books for Sections A, B and C.
Answer ALL questions on this paper.
Each Question carries 25% of the total marks for the exam paper
and you should aim to spend about 30 minutes on it.
An outline marking scheme is shown in brackets to the right of each
question.
An Engineering Data Book by Calvert and Farrar is provided
Note that a formula sheet is provided at the end of this paper
Only University approved calculators may be used.
A foreign language direct ‘Word to Word’ translation dictionary (paper
version ONLY) is permitted, provided it contains no notes, additions
or annotations.
Copyright 2019 University of Southampton Page 2 of 10
SECTION A (Answer ALL Questions)
A.1
(i) The 40m-high dam of Figure A.1.1 has a triangular cross section and retains 30m of water, whose pressure at the base of
the dam is 300kPa. The passive earth pressure from the soil in
front of the dam is idealised as a triangular distributed load with a
maximum value 400kPa, as shown. The dam weighs 10000kN per
meter length of the dam.
Calculate the horizontal force that must develop at the base of the
dam to stop it from sliding.
Assuming the vertical pressure from the foundation is linearly
distributed, calculate its values at points A and B.
Assuming the friction angle at the foundation is = 30°, calculate
the factor of safety of the dam against sliding.
[20 marks]
(ii) For the cantilever beam shown in Figure A.1.2, calculate the
reaction moment at point A using the Principle of Virtual Work,
sketching clearly the mechanism you use.
[5 marks]
[Total 25 marks]
Figure A.1.1: A dam with triangular cross-section.
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Figure A.1.2: A cantilever beam.
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SECTION B (Answer ALL Questions)
(i) The cross-section of a rectangular steel member shown in
Figure B.1.1 (all dimensions in millimetres) is subjected to a
combined bending and axial loading. If the yield stress of steel
is 250 N/mm2 and a factor of safety 1.6 is required against yield
in either tension or compression, determine:
(a) the maximum bending moment about the z-z axis that may
be applied if the axial load is 1200 kN in tension.
[8 marks]
(b) the minimum compression axial load required to prevent
tensile stress occurring anywhere in the section when the
bending moment applied about the z-z axis is 150 kN/m.
[7 marks]
(Hint: Area and second moment of area of the cross section are
2.528×104 mm2 and 4.76×108 mm4
, respectively)
(ii) Using appropriate sketches explain why open cross-sections
are much less efficient in carrying torsion compared to closed
cross sections. [4 marks]
(iii) The thin-walled member with the cross section shown in Figure
B.1.3 is subjected to torsional moment T = 100 Nm. If shear
modulus (G) of the material is 70 KN/mm2
, calculate the maximum
shear stress. [6 marks]
[Total 25 marks]
Figure B.1.1: A hollow steel
cross section
Figure B.1.3: A thin-walled
steel cross section
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SECTION C (Answer ALL Questions)
C.1
(i) State two advantages of the use of moment-distribution
method for the structural analysis of portal frames compared to
the use of slope-deflection method for the same analysis.
[4 marks]
(ii) (a) To apply the moment distribution method to solve the
moments at Joints A, B, C and D of the frame shown in
Figure C.1.2, determine the fixed-end moments at each
joint. [2 marks]
(b) Determine the moment distribution factors at joints B and
C. [2 marks]
(c) Determine the moments at A, B, C and D after balancing
the initial out-of-balance fixed end moments. [7 marks]
(iii) Draw the influence line for the reaction at A of the beam
system shown in Figure C.1.3. Assume that the unit load
moves from left to right along span AB. (Hint: Beam AB is
supported on the lower beam at D using a roller support.)
[5 marks]
(iv) A steel column (E = 210 GPa and yield stress = 210 MPa) of
length L = 4 m has a solid square cross section of unknown
dimension b. The support conditions of the column are “fixed
support” at one end and “pinned support” at other end.
Determine b (in millimetres) that will ensures simultaneous
yielding and Euler buckling of the column. [5 marks]
[Total 25 marks]
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Figure C.1.2: Portal frame
Figure C.1.3: Beam AB is supported on the lower beam at D
using a roller support
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C.2
(i)
(a) State one advantage of the use of force method of
structural analysis compared to the use of the matrix
method of structural analysis
[2 marks]
(b) State one limitation of the use of force method of
structural analysis method compared to the use of matrix
method of structural analysis. [2 mark]

(ii) Figure C.2.2 shows a two-span beam where the plastic
moment capacity of span AC is twice as that of span CB.
Using the lower-bound theory of plastic analysis show that
plastic moment capacity of 275 KNm in span CB is a safe
design for the given system of applied loads. (Hint: you should
check the moment capacities at A, C and at midspan of AB
and BC) [8 marks]

(iii) Let the stress state at a given point in a structure is
represented by three principal stress components -200 MPa,
-100 MPa and 0 MPa. Using the knowledge of Von-Mises
yield criterion determine whether the material has yielded or
not. The uniaxial tensile yield stress of the material is 250
MPa. [4 marks]
(iv) What role does equilibrium play in the theory of upper bound
plastic analysis of structures [2 marks]
(v) Using the upper bound theory of plastic analysis determine the
ultimate load W (in terms of Mp and length of the beam, L) of
the beam shown in Figure C.2.5. [7 marks]
[Total 25 marks]
Copyright 2019 University of Southampton Page 8 of 10
Figure C.2.2: Two-span continuous beam. Strong beam
continues 1 m beyond support C
Figure C.2.5: Single span beam
END OF PAPER (Formula sheet overleaf)
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Useful Formulae
Section B
Second moment of area:
Rectangle : =

3
12
Circle : =

4
64

′ = + 2
Shear
Torsion of a circular section:
For an open thin-walled cross section made up of rectangles

z
b I
SA y
0
‘
=
T
J
=
t
r
=
Gq
L
Copyright 2019 University of Southampton Page 10 of 10
Section C
Member stiffness matrix of a pin-jointed truss member inclined by
an angle (anti- clockwise) to the horizontal is given by:

where c = cos and s = sin and A and L are the cross sectional
area and the length of the member and E is the Young’s modulus.
Slope-deflections equations (with standard notations)

Von-Mises yield criterion
( ) ( ) ( )
1/ 2 2 2 2
2
1
I III III II II I y + + =
Rankine Buckling formula

Where P is the failure load and other terms have the respective
usual meanings.
Southwell equation

=
1
( )
( )
0
P
P
x
x
cr

=
2 2
2 2
2 2
2 2
s sc sc s
cs c cs c
s sc s cs
cs cs c c
L
K
AE
END OF FORMULA SHEET
2
1

+

=
r
k
L
A
P
e
s
E
k
s
2

=

( A B
) AB
F
AB M
L
E M + + = 2 3
2 I
( B A
) BA
F
BA M
L
E M + + = 2 3
2 I