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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2008/
Module Title: Advanced Theory of Structures
Module Code: CIVL
School: Building and Civil Engineering
Programme Title: B Eng (Hons) in Structural Engineering
Programme Code: CSTRU_8_Y
External Examiner(s): Prof. P O’Donoghue Mr. P Anthony Internal Examiner(s): Mr JJ Murphy
Instructions: Answer all four questions
Duration: 2 hours
Sitting: Summer 2009
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 beams AD, BD and DC shown in Fig. Q1 are rigidly connected at D and the structure,
which lies on a horizontal plane is loaded vertically and rigidly supported at A, B and C
and also by a vertical steel tie, which is pinned to a support at E and connected to the
beams at D. In addition to the loading shown, the steel tie is subjected to a temperature
reduction of 20 K.
(a) Use the stiffness matrix method to determine the joint displacements.
(b) Determine the bending moments at A, B, C and D and hence draw the bending moment
diagrams for the beams, noting all significant values. Determine also the torques in
members AD, BD and DC and the tensile force in the tie ED.
Beams: EI = 80000 kNm^2 ; GJ = 25000 kNm^2
Tie (ED): EA = 20000 kN; α = 12 x 10-6^ K-
Q2. (a) Fig Q2(a) shows a three-storey portal frame pinned to supports at A and B. The beams may
be assumed to be infinitely stiff and to have the masses as shown. The columns are of
uniform stiffness with E = 205 kN/mm^2 and I = 12000 cm^4. The mass of the columns may
be neglected. Formulate the stiffness matrix for the frame and calculate the value of one of
the natural frequencies of the frame if it is known that it lies between 5.4 hz and 5.5 hz.
(b) Use qualitative analysis to sketch the bending moment diagrams and deflected shapes for
the beams and frames shown in Fig Q2(b). Indicate also the direction in which the
reactions are acting.
Use Answer Sheet provided. Draw the bending moment diagrams on the tension faces of
the members.
FORMULAS
D
q y
w x y
w x
w
∂
4
4 2 2
4 4
4 2
M (^) x = D (^)
2
2 2
2
y
w x
w
ν M y = D
2
2 2
2
dx
w y
w
ν M xy = D(1-ν)
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-β^2 ) 2 + (2ξβ) 2 ) -0.5^ β = ω l /ω ω = kM
Transmissibility = D.L.F. x 1 +( 2 ξβ)^2
t
p r r
2
2 1
P =
dx dx
dy
dx EI
M
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
