Controlled Condition Exam: 3 Hours exam

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2021/22

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TURN OVERCEGE0040

CONTINUED

Page 2 of 7

CEGE0040: Structural Dynamics

MEng/MSc

Course Examinations – 2022

Time allowed: 3 hours

• Answer all 4 questions

• All questions carry equal weight. Where questions are split into sub-categories, the

marks for each part are shown in [ ]

• Include diagrams where they assist in providing an explanation

Question 1

A frame consists of two steel columns (E = 210 GPa) and a top beam that is connected to an

adjacent wall by means of an axial spring of stiffness k (see Figure Q1a). The columns are

3 m high and are inextensible and light so that the mass of the frame, m=1.6 tonne, can be

considered concentrated in the rigid top beam. The second moment of area I = 264 cm4 is

the same for the two columns. All connections are moment-resisting.

FIGURE Q1a

(a) If the sway natural frequency of in-plane vibrations of the system is 1.5 Hz, deduce

the value of k.

[8 marks]

(b) A resonance test was carried out on the structure by applying a sinusoidal lateral

force to the frame over a range of frequencies and measuring the resulting

displacement. The results are plotted in Figure Q1b, which shows the dynamic

magnification factor against the normalised frequency. Using the half-power

bandwidth method, estimate the damping factor for this system.

[4 marks]

3 m

kCEGE0040

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Page 3 of 7

FIGURE Q1b. Dynamic magnification factor against normalised frequency

(c) The system is now subjected to a sinusoidal excitation with frequency f = 1.2 Hz

and force amplitude F0 =1 kN. What is the steady-state amplitude of the response

of the structure to this excitation?

[5 marks]

(d) The distance between the unstressed frame and the fixed wall to its left is 35 mm.

What range of forcing frequencies would cause the frame in its steady-state

response to hit the wall for force amplitudes larger than 1 kN?

[8 marks]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

5

10

15

20

25

30

Normalised frequency f/fn

|U |k/F 0 0CEGE0040

CONTINUED

Page 4 of 7

Question 2

Figure Q2 shows the simplified elevation of a frame structure with a mezzanine floor half

way up the height of the building. The mezzanine slab has the same mass as the roof slab

(top). The structure is supported by columns of stiffness as indicated in the figure. Assume

that the horizontal slabs are rigid and the axial deformations in the columns are negligible.

Ignore any damping.

FIGURE Q2. Schematic elevation of a simple building with mezzanine floor

(a) Write down the governing differential equation of motion for this system for the

case that a lateral force p(t) acts on the roof of the structure as shown in Figure Q2.

[5 marks]

(b) Determine the natural frequencies and the corresponding mode shapes. Sketch the

mode shapes.

[9 marks]

(c) The lateral point load p, applied to the roof of the structure, is ramped up quasi

statically from zero amplitude until the lateral displacement of the roof with respect

to the ground is one unit of length. Explain the significance of the word ‘quasi

static’ in this process and deduce the corresponding displacement of the mezzanine

floor.

[5 marks]

(d) The structure is then released from the final state reached in (c) and allowed to

vibrate freely. Find the expression for the displacements of the two masses in the

subsequent motion.

[6 marks]

k

u2

m

m

u1

k

k

k

p(t) CEGE0040

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Page 5 of 7

Question 3

The three-storey shear frame shown in Figure Q3(a) has the structural characteristics as

indicated in the figure, with 𝐸𝐼 = 20 MNm2 and m0 = 20 tonnes. Figure Q3 shows the

acceleration response spectrum of a single degree of freedom system to the Taiwan 1986

ground motion.

(a) Shear frame

(b) Spectral acceleration

FIGURE Q3

(a) Determine the mass and stiffness matrices for the frame. Calculate their

coefficients numerically.

[4 marks]

The natural periods of this frame are 0.82 s, 0.17 s and 0.12 s and the corresponding mode

shape vectors are given as the columns of matrix [Φ]:

[Φ] = [ 0

.

8856

−

0

.5155 0

.

2738

0

.

9667

0

.

2137

−

0

.5094

1

.

0000

1

.

0000

1

.

0000

].

(b) Calculate the first natural frequency and the two effective modal masses. What

can you deduce in terms of modal contribution to the overall response?

[10 marks]

(c) Explain the key limitations of modal combination rules.

[3 marks]

EI

u2

u1

1.5EI

EI/2

m0

m0

EI

1.5EI

0.25 m0

__

3m

L

u3

3m

4m

EI/2CEGE0040

CONTINUED

Page 6 of 7

(d) Using Figure Q3(b), estimate the total base shear demand on this frame from the

Taiwan 1986 ground motion, assuming 2.5 % modal damping.

[3 marks]

(e) Estimate the inter-storey drift and the moment demands on the ground floor and

first-floor columns.

[5 marks]

Question 4

(a) (i) Describe what numerical damping is and explain its particular significance in the

context of direct integration for structural dynamics.

(ii) Describe the main types of nonlinear analyses in the context of earthquake

engineering and briefly discuss their pros and cons.

[6 marks]

(b) Derive the recurrence relationship that follows from discretising the equation of

motion for a damped single degree of freedom dynamical system using Euler’s

backward scheme. Initialise the first two displacement samples using initial rest

conditions.

[4 marks]

(c) A single degree of freedom system is made of a mass 𝑚0 = 60 tonnes, and the elasto

plastic spring defined by the force-displacement relationship shown in Figure Q4(a).

The mass is subjected to the force defined by the time history shown in Figure Q4(b).

Ignore any viscous damping and assume the system is initially at rest.

(i)

Compare the elastic period to the duration of the impulsive force. How do

you expect the system to respond initially?

[3 marks]

(ii)

Taking a time step

Δ𝑡 =

0.02 s, calculate the displacement of the system for

the first 10 time steps.

[10 marks]

(iii)

Calculate the energy dissipated during this 0.2 s time interval.

[2 marks]CEGE0040

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