Dynamic Analysis of Structures Questions and Answers

Answer: C

Correct answer: build up amplitude without limit

At resonance the magnification is \( 1/2\xi \), which grows without bound as \( \xi \to 0 \). With no damping the forcing adds energy every cycle and nothing removes it, so the amplitude grows linearly with time.

Real structures always have some damping, so the response reaches a finite steady state - but the mathematics shows why damping is what limits resonant response, and why a small change in it matters so much near a natural frequency.

Answer: D

Correct answer: a relatively high damping ratio

Damping is the mechanism that removes energy from the system, so the rate of decay measures it directly - which is what a logarithmic decrement calculation exploits.

The other options mistake the cause. Low damping gives slow decay, the opposite of what is described; mass and stiffness together fix the period and the number of cycles per second, but not how quickly the motion dies. A structure with a long period simply oscillates slowly, however fast it decays.

Answer: A

Correct answer: they dissipate less energy internally and at their connections

Damping in a real structure comes largely from internal friction, from micro-cracking, and from movement at joints and interfaces with non-structural elements.

Reinforced concrete offers all of those in abundance; welded and bolted steel offers far less. Hence the usual assumptions of 2 to 3 per cent for steel and 5 per cent for concrete. The consequence is a higher spectral demand on the steel structure, since the resonant magnification is inversely proportional to the damping.

Answer: B

Nearly all the mass sits at the top of the staging and the structure has one obvious sway direction, so one degree of freedom captures the behaviour. Tall or irregular buildings need many degrees of freedom to be represented properly.

Answer: C

Correct answer: the distribution of mass over the structure

The mass matrix records where the mass is, and it is assembled from the physical mass at or around each degree of freedom.

The lumped formulation places the tributary mass at each node, giving a diagonal matrix; a consistent formulation distributes it using the same shape functions as the stiffness, giving off-diagonal terms. Natural frequencies and damping ratios are results of the analysis or assumptions imposed on it - neither is an input to the mass matrix.

Answer: D

Correct answer: Maxwell's reciprocal theorem

Maxwell's theorem states that the displacement at \( i \) due to a unit force at \( j \) equals the displacement at \( j \) due to a unit force at \( i \), for a linear elastic system.

Written in matrix terms that is \( k_{ij} = k_{ji} \) - symmetry. The property is not incidental: it halves the storage required and permits solution methods that assume symmetry, which is why the eigenvalue problem for structures is so much cheaper than the general case.

Answer: A

Correct answer: 5 × 5

The stiffness matrix is square with order equal to the number of degrees of freedom, and five masses with one lateral degree of freedom each gives five.

The mass matrix has the same order, and the eigenvalue problem yields five natural frequencies and five mode shapes. Allowing two horizontal directions and a rotation per floor would give fifteen degrees of freedom and a 15 × 15 matrix - the count follows from what the model permits to move, not from the structure itself.

Answer: B

Correct answer: at the floor levels

Most of a building's mass is in its floor slabs and the loading they carry, and the floors act as rigid diaphragms tying each level together.

Lumping the tributary mass of each storey at its floor level is therefore both convenient and physically reasonable, and it reduces a continuous structure to a handful of degrees of freedom. The columns' own mass is small and is usually apportioned to the floors above and below rather than modelled separately.

Answer: C

Correct answer: 10

The number of cycles is the elapsed time divided by the period:

\[ \frac{10}{1.0} = 10 \]

The period is the time for one complete cycle, so this is a matter of definition rather than of dynamics. It is worth noting how slow that is: a one-second building sways at 1 Hz, which is visible and perceptible to occupants, and is why serviceability under wind is checked for tall buildings as well as strength under earthquake.

Answer: D

Correct answer: exponentially with time

The free vibration solution contains the factor \( e^{-\xi\omega_n t} \), so the envelope decays exponentially.

The consequence is that the ratio of successive peaks is constant, which is exactly what makes the logarithmic decrement a well-defined quantity: taking the logarithm of a constant ratio gives the same value for every pair of peaks, so several cycles can be averaged. Linear decay would give a constant difference instead, and no such measurement would be possible.