CEGE0040 Structural Dynamics

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

If the sway natural frequency of in-plane vibrations of the system is 1.5 Hz, deduce the value of k.

[8 marks]

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]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Normalised frequency f/fn

FIGURE Q1b. Dynamic magnification factor against normalised frequency

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]

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]

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.

p(t) m

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

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]

Determine the natural frequencies and the corresponding mode shapes. Sketch the mode shapes.

[9 marks]

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]

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]

Question 3

The three-storey shear frame shown in Figure Q3(a) has the structural characteristics as indicated in the figure, with �� = 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 motion1

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]:

0.8856	−0.5155	0.2738
[Φ] = [0.9667	0.2137	−0.5094].
1.0000	1.0000	1.0000

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]

Explain the key limitations of modal combination rules.

[3 marks]

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]

Estimate the inter-storey drift and the moment demands on the ground floor and first-floor columns.

[5 marks]

Question 4

(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]

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]

A single degree of freedom system is made of a mass ��₀ = 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.