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ECMM108
UNIVERSITY OF EXETER
COLLEGE OF ENGINEERING, MATHEMATICS
AND PHYSICAL SCIENCES
ENGINEERING
May 2022
Advanced Structural Engineering
Module Convenor: M. K. Wadee
Duration: TWO HOURS + 30 minutes upload time
Answer THREE out of FOUR questions
This is an OPEN BOOK examination.
ECMM108 1 CONTINUE
Question 1 (20 marks)
The column of length L shown in Figure Q1 is made up of two sections. The upper
part has bending stiffness EI whereas the lower part has bending stiffness 1.5EI. The
ends of the column are simply-supported as indicated and it is acted upon by an axial
force P.
P
EI
1.5EI
L/2
L/2
Figure Q1: Simply-supported column.
(a) Using the Rayleigh method and a quadratic polynomial function for the deflection,
derive the total potential energy of the structure as it begins to buckle. (12 marks)
(b) Find an approximation to the value of the critical load, P
C. (4 marks)
(c) Comment on your results and explain why the above polynomial for deflection is not
the best type to choose for this problem and therefore suggest another polynomial
which would provide a better approximation to the critical load. (4 marks)
ECMM108 2 CONTINUE
Question 2 (20 marks)
Figure Q2(a) shows a tower made of an aluminium tube length L, modelled as a single_xfffe_degree-of-freedom structure with its mass, m, concentrated at its free end.
tt1
P(t)
P(t)
P0
m
L
(a)
(b)
Figure Q2: A single-degree-of-freedom model of a tower.
(a) Calculate the structure’s stiffness against lateral displacement of its unsupported
end and hence determine its undamped natural period, T, given that m = 400 kg,
E = 70 kN mm 2
, I = 2 × 107 mm4
, L = 4 m. (8 marks)
(b) The structure is acted upon by a short-duration impulse of the form shown in Fig ure Q2(b) where P0 = 2 kN and t1 = 0.8 s. Estimate the maximum deflection of
the free end and the maximum bending moment induced at the fixed end of the
cantilever. (8 marks)
(c) If the aluminium tube were replaced with a steel tube with identical physical dimen sions but three times the Young’s modulus, what would the maximum values above
(4 marks)
ECMM108 3 CONTINUE
Question 3 (20 marks)
Use two families of beam strips, one set each along the x and y directions, to evaluate
the maximum design moments for the homogeneous and isotropic slab in Figure Q3.
The slab is simply-supported on three edges AB, BC and CD, and carries a uniformly
distributed load of magnitude 10 kN m 2
as illustrated in the figure.
(a) Briefly describe the load routes adopted for the slab. Indicate the loads taken by
the beam strips along the x- and y-directions and how the strips interact. (5 marks)
(b) Plot reaction forces along the boundaries taking care to convert any twist moments
into equivalent shear forces. (3 marks)
(c) Provide shear and bending moment diagrams for one beam strip along the x direction that has the maximum bending moments. (4 marks)
(d) Provide shear and bending moment diagrams for one beam strip along the y direction that has the maximum bending moments. (8 marks)
A
B C
D
8 m
10 kN/m^2
Figure Q3: A structural slab.
ECMM108 4 CONTINUE 2 m2 m
Question 4 (20 marks)
(a) State the assumption in Mindlin–Reissner plate bending theory that allows for
shear deformation. Explain using relevant equations how shear forces are related
to the corresponding shear strains for a plate element based on this theory.
(3 marks)
(b) Use two families of beam strips, one set each along x and y directions, to eval uate the maximum design moments for the homogeneous and isotropic slab in
Figure Q4. The slab is simply-supported on two edges AB and DC, and carries a
uniformly distributed load of magnitude 8 kN m 2 as illustrated in the figure.
i. Briefly describe the load routes adopted for the slab. Indicate the loads taken
by the beam strips along the x- and y-directions and how the strips interact.
(5 marks)
ii. Plot reaction forces along the boundaries taking care to convert any twist mo ments into equivalent shear forces. (3 marks)
iii. Derive the equations for shear and bending moments for one beam strip along
the x-direction that has the maximum bending moments. Also give qualitative
sketches of the shear and bending moment diagrams. (9 marks)
x
BA
C D
2 m
u.d.l. applied here
6 m
y
Figure Q4: Partially loaded plate.
END OF QUESTION PAPER
ECMM108 5
4 m