Download CIVL 332 Structural Dynamics Final Exam (Open

CIVL 332
Structural Dynamics
Final Exam (Open-Book)
11 December 2010
16:30 – 19:30
Rm 2465
INSTRUCTIONS
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Print your name and your student ID number on the bottom of this page.
Write down your ideas and approaches clearly. You are responsible to explain
your concepts to me in a clear and understandable fashion.
Do not talk nor communicate to other people during the course of the exam.
Cheating or dishonesty in any form during the exam will automatically render you
an F grade for this course.
Violation of the above instructions will be penalized accordingly.
NAME :
STUDENT ID :
Problem
1
2
3
4
5
Total
Maximum Score
20
20
25
20
15
100
1
1. Multiple-choice and short questions:
(1) The followings are some descriptions about the structural damping. Which
ones are correct statements?
(a) The damping ratios for RC tall buildings can be assumed to be between 30
to 50 %.
(b) The damping ratios for steel tall buildings can be assumed to be between 1
to 2 %.
(c) The damping ratios for RC bridges can be assumed to be between 20 to 30
%.
(d) The damping ratios for long-span cable-supported bridges can be assumed
to be between 0.5 to 2 %.
(e) The damping ratios of a structure can be obtained from mathematical
modelling.
(2) Assume that you are asked to describe/comment the dynamic
characteristics/behaviour of a 100-story SRC tall building in Hong Kong.
Which ones are correct descriptions?
(a) Based on a rule-of-thumb formula, its first natural period is about 10
seconds.
(b) The natural frequencies measured at the 100th story would be completely
different from those measured at the 50th story as their heights are different.
(c) The building would not vibrate at all under wind since the engineer
designed it very heavy, bulky and safe.
(d) The building’s dynamic characteristics (frequencies and dampings) cannot
be modified at all once it has been built.
(e) Its damping ratios can be bigger than 1 (or 100%) since the building is 100
stories high.
2
(3) The D-V-A plots for the EW component of the Kobe earthquake that occurred
on the 17th January 1995 are shown in the following figure (the damping ratio
is assumed to be 5%). Estimate the peak ground acceleration and the peak
ground displacement for this earthquake. Explain your answer.
0.4
D, m
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
Tn, sec
2
2.5
3
3
V, m/sec
2.5
2
1.5
1
0.5
0
2.5
A, g
2
1.5
1
0.5
0
(4) Explain the differences between static analysis and dynamic analysis. Write an
example on how the dynamic analysis can benefit the design or analysis of a
civil engineering project.
3
2. You are asked to analyze a tall building subjected to the EW component of the 1995
Kobe earthquake using the generalized coordinate approach. The properties of
building are: mass/length m = 2.64 × 10 5 kg / m ; rigidity EI = 1.95 × 1013 N ⋅ m 2 ;
height L = 100m ; and damping ratios=5%.
(a) Select a proper shape function for the building from the following list of
functions. Explain your choices.
x
x2 x
⎛ πx ⎞
(i) φ ( x ) = ;
(ii) φ (x ) = 2 − ;
(iii) φ ( x ) = sin⎜ ⎟ ;
L
L L
⎝ 2L ⎠
⎛ πx ⎞
⎛ πx ⎞
⎛ πx ⎞
(iv) φ ( x ) = cos⎜ ⎟ ; (v) φ ( x ) = 1 − sin⎜ ⎟ ; (vi) φ (x ) = 1 − cos⎜ ⎟ ;
⎝ 2L ⎠
⎝ 2L ⎠
⎝ 2L ⎠
(b) Using the selected shape function in (a), determine the natural frequency of the
building.
(c) Using the selected shape function in (a), establish the equation of motion for the
building.
(d) Use the D-V-A plots shown in the previous problem and determine the
maximum relative displacement and absolute acceleration at the top of building.
(e) Determine the maximum shear force and the maximum overturing moment at
the base of building.
L = 100m
m = 2.64 ×105 kg / m
EI = 1.95 × 1013 N ⋅ m 2
x
4
3. It is decided to model the building in the 2nd problem using one 2-node 4-DOF
beam element.
(a) Write the mass and stiffness matrices of the building.
(b) Determine its natural frequencies and mode shapes.
(c) Assuming that the damping ratios are all equal to 5%, determine the damping
matrix.
(d) Write the equations of motion for the building.
(e) Estimate the maximum relative displacement at the top of building using the
first mode response only.
(f) Compare the results with those obtained in the 2nd problem and comment on the
accuracy of the two results.
5
4. The governing equation of motion for a cable under a constant tensile force N as
shown can be obtained as,
N
∂2y
∂x2
+ p( x , t ) = m
∂2y
∂t2
(a) Write natural frequencies and mode shapes for a cable of length L with both
ends fixed.
(b) The first natural frequency for one of the Kap Shui Mun Bridge cables is
measured to be 0.5 Hz. The cable length is 200 m and its mass per unit length
m = 100 kg / m , calculate the axial tensile force in the cable.
(c) To reduce the wind-induced cable vibration, a damper is installed between the
cable and the deck as shown. Assuming that the damper’s force can be modeled
as f = − y& c × 10 6 N ( y& c is the velocity of the cable at the damper’s location in
m/s), estimate the increase of 1st mode damping ratio due to the installation of
this damper.
displacement
y(x,t)
N
Cable:
Length = L
Tensile force = N
loading Mass/length = m
p(x,t)
axial force
N
x
0m
20
2m
y& c
f = − y& c MN
damper
6
x
5. A car with a mass of m is driving through a highway bridge at a constant speed of u
as shown. The bridge comes with a span of 2L and is clamped to two rigid piers.
The bridge is stiffened by a column of length H which is rigidly-connected to the
bridge deck at mid-span. The mass/length and the rigidity for the bridge deck are
denoted as EIB and mB, respectively, and those for the column are denoted as EIC
and mC, respectively. You are asked to present a method to analyze the dynamic
response of the bridge. The following deliverables should be included in the method:
(a) How the bridge is modeled;
(b) How the equations governing the dynamic behavior of the bridge are established
(need to provide the estimation of mass, damping and stiffness parameters);
(c) How the loading acting on the bridge is modeled;
(d) How the maximum deformation and internal forces of the bridge deck and the
maximum reaction forces at the supports can be obtained;
(e) What level of accuracy you expect from your method.
Mass: m
mB, EIB
Clamped
Clamped
L
mC
EIC H
Clamped
7
L
Rigid pier
Rigid pier
Speed: u