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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2011/
Module Title: Advanced Theory of Structures
Module Code: CIVL
School: Building and Civil Engineering
Programme Title: Bachelor of Engineering (Honours) in Structural Engineering – Year 4
Programme Code: CSTRU_8_Y
External Examiner(s): Dr M.G. Richardson Mr J. O’Mahony
Internal Examiner(s): Mr J.J. Murphy
Instructions: Answer all four questions.
Duration: 2 Hours
Sitting: Summer 2012
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
Q1. The pin-jointed steel framework shown in Fig. Q1 is attached to pinned supports at A and
D and a roller support at B. The framework is loaded as shown. The cross-sectional area of
each member is 600 mm^2. Use the stiffness matrix method to determine the horizontal and
vertical deflections at C and the vertical deflection at B. Hence determine the resulting
forces in each of the members, the horizontal and vertical reactions at A and D and the
horizontal reaction at B. (25 marks)
E = 210 kN/mm^2
Q2. (a) A rod of length l and flexural rigidity EI is attached to a support at A, which has a torsional
stiffness β. An axial load P is applied to the free end of the rod, as shown in Fig. Q2(a).
Show that the characteristic equation, whose lowest root yields the critical load Pcr is given
by tan (k l ) – β/(EI k ) = 0, where k^2 = P/EI.
Hence determine the value of Pcr for the limiting cases β → ∞ and β → 0.
(9 marks)
(b) Fig. Q2(b) shows a strut, which is attached to a roller support at A and a fixed support at B.
It is subjected to an axial force P at A. The section of the strut between 0 and 0.4 l has
flexural stiffness EI, while the remainder has flexural stiffness 2EI. If an approximation of
the buckled shape of the strut is given by the equation
4 3
3 3 2 l
x l
x l
x y K where x is
measured from A, use the Rayleigh energy method to obtain an upper bound
approximation of the critical buckling load. (12 marks)
(c) The uniform frame shown in Fig. Q2(c) is attached to pinned supports at A, B, C and D
and to fixed supports at E and F. It is subjected to equal vertical loads P at G and H as
shown. Formulate equations in terms of stability functions, which express the conditions of
instability of the frame in its own plane and hence determine the critical value of P.
(9 marks)
Table of Stability Functions
s c sc s(1+c) s(1-c*c) m m 1 1.21 1.728 2.070 1.264 2.618 4.688 - 1.239 - 3.654 0. 1.22 1.735 2.051 1.280 2.626 4.676 - 1.311 - 3.478 0. 1.23 1.742 2.031 1.297 2.633 4.664 - 1.384 - 3.318 0. 2.39 2.428 - 1.258 - 3.463 4.356 3.098 13.825 - 0.356 - 1. 2.4 2.4 33 - 1.301 - 3.370 4.383 3.083 13.472 - 0.352 - 1. 2.41 2.439 - 1.344 - 3.283 4.411 3.067 13.138 - 0.348 - 1. 2.42 2.444 - 1.387 - 3.201 4.439 3.052 12.820 - 0.343 - 1.
FORMULAE
4 4
4 2 2
4 4
4 2
x x y y
^4 w = q/D
Mx = D
2
2 2
2
y
w x
w
My = D
2
2 2
2
dx
w y
w
Mxy = D(1-)
x y
(^2) w
Qx = D
2
3 3
3
x y
w x
w Qy = D
x y
w y
w 2
3 3
3
Vy = Qy + x
M (^) xy
Vx = Qx + y
M (^) xy
D.L.F. = ((1-
2
2
2
-0.
= l / = kM
Transmissibility = D.L.F. x 1 ( 2 )^2
t
p r r
2
2 1
P =
dx dx
dy
dx dx
d y EI
2
2
2
2
(Rayleigh) P =
dx EI
y
dx dx
dy
2
2
(V.S.)
k = sEI/L ; scEI/L - fixed k = s(1-c^2 )EI/L - pinned
