ECE 2416 STRUCTURAL DYNAMICS FINAL EXAMINATION 2021

 

JOMO KENYATTA UNIVERSITY OF AGRICULTURE AND TECHNOLOGY

UNIVERSITY EXAMINATIONS 2020/2021

FOURTH YEAR SECOND SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE IN CIVIL ENGINEERING

ECE 2416 – THEORY OF STRUCTURES VI (STRUCTURAL DYNAMICS)

DATE: 10 FEBRUARY 2021                                                TIME:  2 HRS

Instructions to candidates

• Attempt any THREE questions
• Scientific non-programmable calculator is allowed during this examination.
• Use of mobile phones is prohibited. As such all phones MUST BE SWITCHED OFF!

QUESTION ONE (20 MARKS)

a)    An undamped SDoF dynamic system having a mass of 20 kg and a spring stiffness of 35 N/mm is given an initial displacement of 10 mm and initial velocity of 100 mm/s. Obtain

                               i)            the natural frequency (2 marks)

                             ii)            the period of vibration (1 mark)

                           iii)            the displacement at 2 sec (4 marks)

                           iv)           a sketch of the motion showing the first three peaks for the undamped and damped vibration. For the damped case assume a damping ratio of 10%. (6 marks)

          (Hint: For solution to Q1(a) above, refer to a related solved problem 

b)     Fig. Q1 shows a single degree of freedom (SDoF) dynamic models. Fig. 1(a) is vibrating along a horizontal axis while Fig. 1(b) is vibrating vertically in the direction of gravity. Demonstrate, starting from basic principles, that the equation of motion expressed with reference to the static equilibrium position of the dynamic system is not affected by gravity forces. All symbols have their usual meaning. (7 marks)

(a) SDOF system vibrating horizontally	(b) SDOF system vibrating vertically
Fig. Q1

QUESTION TWO (20 MARKS)

For the two-storey shear building shown in Fig. Q2,  m₁ = m₂ = m and k₁ = k₂ = k . Calculate and sketch the fundamental and the second mode of vibration for this system.

Fig. Q2

QUESTION THREE (20 MARKS)

Fig Q3 shows a 20 MN RC bridge deck resting on concrete piers with different support conditions as shown. The load-displacement relationship for the piers is shown alongside. The structure’s energy conservation property (Fig A-3a), acceleration response spectrum (Fig. A-3b) and the non-linear response spectrum (Fig. A-3c) is given in Appendix I. You are required to compare the dynamic response for the two structures considering the following parameters:

a)      the natural period (4 marks)

b)      the response displacement using elastic spectrum for hard and soft ground (12 marks)

c)      the response displacement using nonlinear spectrum for soft ground (4 marks)

(i) pin-roller support	(ii) pin – pin support	(iii) Load – displacement relationship

Fig. Q3

QUESTION FOUR (20 MARKS)

a)    A reinforced-concrete tank rests on a 20-m-tall single concrete column. A cable attached to the tank applies a lateral force of 80 kN and pulls the tank horizontally by 5 cm. The cable is suddenly cut, and the resulting free vibration is recorded. At the end of four complete cycles, the time is 2.0 sec, and the amplitude is 2.5 cm. From these data compute the following:

(i)        damping ratio (2 marks)

(ii)       natural period of undamped vibration (1 mark)

(iii)     stiffness (1 mark)

(iv)     weight (2 marks)

(v)       damping coefficient (1 mark)

(vi)    number of cycles required for the displacement amplitude to decrease to 0.5 cm (1 mark)

b)     If the weight of water required to fill the tank of Question 4(a) above is 40,000 kg, determine the natural vibration period and damping ratio of the structure with the tank full (4 marks).

c)    The full water tank described in Q4 (a & b) above is subjected to a dynamic force p(t) as shown in Fig. Q4, caused by an aboveground explosion. Determine the maximum base shear and bending moment at the base of the tower supporting the tank. Sketch the resulting shear force and bending moment diagrams over the height of the tower (8 marks).

Fig. Q4

(Hint: Solution to Q4 above is given here 

QUESTION FIVE (20 MARKS)

a)    A one-story building, idealized as a 4-m-high frame with two columns hinged at the base and a rigid beam, has a natural period of 0.5 sec. Each column is a standard UB steel section with properties for bending about its major axis as I_x = 2772 cm⁴, S = 252 cm³; E = 200,000 MPa. Neglecting damping, determine the maximum displacement at the top of the frame and maximum bending stress in the columns of this frame due to a rectangular pulse force of amplitude 20 kN and duration t_d = 0.2 sec . Show the bending moment distribution along the length of the column, as well as the stress distribution across the cross-section of the steel column (12 marks).

b)    A machine weighing W = 1750.87 kg is mounted on simply supported steel beams as shown in Fig. Q5. A piston that moves up and down in the machine produces a harmonic force of magnitude of 3175.15 kg and frequency of 60 rad/sec. Neglecting the weight of the beam and assuming a 10% critical damping, determine,

                                 i)            the amplitude of the motion of the machine (4 marks)

                               ii)            the force transmitted to the beam supports (2 marks)

                             iii)            the corresponding phase angle (2 marks)

Fig. Q5

(Hint: Solution to Q5b above is given in this link)

 

APPENDIX I

Fig. A-3a. Property of energy conservation (short period)
Fig. A-3b. Acceleration response spectrum
Fig. A-3c. Non-linear response spectrum