Structural Dynamics and Plastic Analysis

Q1 Structural Dynamics

Q1 Structural Dynamics

The three-storey building in Figure Q1(a) is square on plan and has a regular grid of columns at 6m spacing. The columns have Young’s modulus 40 kN/mm2 and are circular in cross-section, with those on the ground storey being 400 mm in diameter and those on higher storeys 300 mm. The columns have pinned connections to the foundations but are fully fixed to the floor and roof beams. The beams can be assumed rigid, the mass of the columns can be neglected and the structure can be assumed to remain in its elastic range. The damping ratio of all modes can be assumed to be 3%.

(a) Considering sway vibrations of the building in one direction, the first three natural frequencies are found to be 2.09 Hz, 6.87 Hz and 10.2 Hz. Find the corresponding mode shapes. (14 marks)

(b) A blast at ground level loads one face of the building, with the relative magnitudes of the forces at each floor level as shown in Figure Q1(a) and the time history of the forces (all applied simultaneously) following a quarter cosine function as shown in Figure Q1(b). Calculate the total horizontal displacement of the roof from the time of the blast. Sketch the component of the roof displacement in each mode, as functions of time. Explain the signs and magnitudes of the components in different modes. This explanation must be typed.

(c) Without detailed calculations of the time of the maximum bending moment, give an upper bound for the maximum bending moment in the ground floor columns. (6 marks)

(d) How long will it be until the acceleration amplitude of the roof in the first mode dies down to below 0.01 m/s2. (4 marks)

(e) Discuss potential mitigation measures against this loading scenario. The answer to this part of the question must be typed (though hand-drawn sketches can be incorporated, if desired).

Q2 Plastic Analysis

The structure in Figure Q2 is subjected to a vertical point load P1, a uniformly distributed vertical load w = P1/L per unit length and a horizontal point load P2, as shown. P1 and P2 can both be assumed to be positive. The left hand support is pinned and the right hand support is fixed. All of the members have plastic moment capacity Mp.

(a) Based on calculations of potentially critical mechanisms, present a diagram summarising the various collapse conditions as functions of P1 and P2, labelling all key points. Indicate the safe loading region and identify which mechanism the structure fails in for which ranges of the ratio P2/P1. (30 marks)

(b) For P2/P1 = 7/8, give the value of P1 at collapse and check that at the collapse state the equilibrium and yield conditions are satisfied. Sketch the corresponding bending moment diagram, labelling all values in terms of P1L. (14 marks)

(c) What further issues and details should be addressed in the full design of the frame? The answer to this part of the question must be typed (though handdrawn sketches can be incorporated, if desired).