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ECMM108P 1 TURN OVER
ECMM108P
UNIVERSITY OF EXETER
COLLEGE OF ENGINEERING,
MATHEMATICS AND PHYSICAL SCIENCES
ENGINEERING
Advanced Structural Engineering
Module Convenor: M. K. Wadee
Duration: TWO HOURS
May 2018
Answer any THREE questions out of FOUR
Materials to be supplied on request:
Graphpaper
Approved calculators are permitted.
This is an OPEN BOOK examination.
Question 1 (20 marks) The pin-ended steel column of length L shown in Figure Q1 is
made up of two sections of equal length welded at their interface. One section has
bending stiffness EI while the other has the value 2EI as indicated in Figure Q1. The
column is loaded by an axial force P acting at its top end.
PSfrag replacements
EI
2EI
L/2
L/2
P
x
Figure Q1: Axially-loaded column
(a) Using Rayleigh’s method and by approximating the deflection with a
single-degree-of-freedom (SDOF) mode comprising a half sine wave of the form
y = Q sin
πx
L
,
where Q is a constant, find an expression for the total potential energy, V , of the
structure. (10 marks)
(b) Use your answer for (a) to find the load carrying capacity (buckling load) of the
column. (5 marks)
(c) Explain why, if the deflection were to be assumed to be in the form of a quadratic
polynomial, that the buckling load found is likely to be a poorer approximation than
using the sinusoidal form above. Which type of polynomial would be a better
approximant (5 marks)
ECMM108 2 CONTINUE
Question 2 (20 marks)
(a) A weight W falls freely from a height h onto the midpoint of an undeflected,
prismatic and massless beam of length L, where
h =
L
384
,
whereupon it adheres permanently to the beam (see Figure Q2(a)). Calculate the
maximum bending moment at this point if the static deflection due to the weight W,
denoted ust, is
ust =
L
480
.
You may assume that the structure is undamped. (14 marks)
(b) An industrial building has been instrumented to monitor its structural behaviour and
performance. When a crane load is accidentally dropped on it, it undergoes
damped oscillations as shown in Figure Q2(b). Estimate the damping ratio from
these data and also the undamped natural circular frequency of the structure.
(6 marks)
PSfrag replacements
W
L
h
ust
(a) Beam subject to dropped weight
Time (s)
-20
-15
-10
-5
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4
PSfrag replacements
W
L
h
ust
(b) Damped vibration data
Figure Q2
ECMM108 3 TURN OVER Displacement (mm)
Question 3 (20 marks)
(a) State the assumption in Mindlin-Reissner plate bending theory that allows for
shear deformation. Using relevant equations, show how shear forces are related
to the corresponding shear strains for a plate element based on this theory.
(3 marks)
(b) Explain briefly why Mindlin–Reissner theory is not applicable to structures
undergoing large deformations. State the other modes of deformation that have
to be considered in such scenarios. (2 marks)
(c) The floor slab given in Figure Q3 is to be analysed using Mindlin-Reissner plate
elements. A uniformly distributed load of magnitude q kN mm 2
is applied over
the shaded portion of the slab. Edges AD and BC have simple supports. Edges
AB and DC have fixed supports that prevent rotation about the edge and vertical
translation.
A B
C D
x
6 m
3 m
3 m
y
2 m 2 m 2 m
Figure Q3
i. Define displacement boundary conditions that need to be prescribed along
the four supported edges. (2 marks)
ii. State the known force boundary conditions with respect to Mx, My, Mxy, Qx
and Qy. (3 marks)
iii. How can symmetry be used to model only a symmetric portion of the floor
slab given in Figure Q3 Clearly state the boundary conditions that need to
be prescribed when modelling only a portion of the slab. Also state the
known force boundary conditions along the planes of symmetry. (5 marks)
iv. Label on a diagram drawn in your answer book where you expect to see
maximum sagging and hogging bending moments for Mx and My. Also give
a qualitative sketch of the distribution of the reactions along edge AB.
(5 marks)
ECMM108 4 CONTINUE
Question 4 (20 marks)
(a) Explain why yield-line techniques are referred to as unsafe for evaluating collapse
loads. (2 marks)
(b) What are the assumptions made on deformations and the mode of failure when
using yield line techniques to evaluate collapse loads (2 marks)
(c) Figure Q4 shows the yield line pattern used to compute the yield bending
moment for a slab. Hogging and sagging yield lines are shown using dashed and
thick solid lines respectively. The slab is simply supported on edges AD and BC.
It is fixed against rotation and vertical translation along edge AB and free along
edge DC. A uniformly distributed load of magnitude 5 kN m 2
acts on the shaded
region indicated in Figure Q4. Complete the following tasks towards calculating
the yield bending moment for the slab based on the given yield line pattern.
i. Parameterize the yield line pattern and illustrate this using a diagram.
(2 marks)
ii. Write down the equations for internal and external work. (8 marks)
iii. Derive the equation for the yield bending moment by relating the equations
obtained for internal and external work. (4 marks)
iv. Describe how the yield bending moment can be obtained from the equation
derived in part iii through the use of optimization. (2 marks)
A B
C D
x
8 m
3 m
3 m
y
1 m 3 m 1 m
Figure Q4
END OF QUESTION PAPER
ECMM108 5