Dynamic Analysis of Structures Questions and Answers

Answer: A

Correct answer: conditionally stable, with a limit on the time step

The central difference method is explicit, and explicit schemes are conditionally stable: the step must satisfy \( \Delta t \le T_{min}/\pi \), where \( T_{min} \) is the shortest period in the model.

The limit is set by the highest mode, which for a fine mesh may be of no interest whatever - so a model refined for accuracy forces very small steps. That is the trade against implicit methods such as Newmark's, which are unconditionally stable but require an equation solution at every step.

Answer: B

Correct answer: 0.0064 s

Apply the stability condition:

\[ \Delta t \le \frac{T_{min}}{\pi} = \frac{0.02}{3.1416} = 0.00637\ \mathrm{s} \]

Note that the shortest period governs, not the fundamental - so refining a mesh, which introduces ever higher modes, tightens the limit even though those modes contribute nothing to the response. That is the characteristic difficulty of explicit integration and the reason implicit schemes are preferred for structural earthquake analysis.

Answer: C

Instability announces itself as oscillations that double in amplitude every few steps, which is at least easy to recognise. The dangerous case is a stable but inaccurate solution from an over-large step in an unconditionally stable scheme.

Answer: D

Correct answer: accuracy rather than stability

The two requirements are distinct. An unconditionally stable scheme has no stability limit at all, yet a large step still gives a poor answer - period elongation and amplitude decay in the computed response.

A step of about \( T/10 \) resolves the shortest period of interest well enough to keep those errors small. Where a scheme is conditionally stable, the stability limit is usually the tighter of the two and governs instead; the accuracy requirement remains either way.

Answer: A

Correct answer: one or more ground acceleration records

Time-history analysis computes the response at every instant, so it needs the excitation at every instant - a digitised accelerogram, not a summary of it.

A response spectrum, a peak ground acceleration or a zone factor are all condensations of ground motion into a single number or curve, and none can be integrated. Codes generally require several records, scaled and matched to the site's spectrum, because the response to a single record is sensitive to its particular sequence of pulses.

Answer: B

Pushing the structure gradually traces a capacity curve of base shear against roof displacement, showing where hinges form and in what order. It is a static nonlinear procedure and says nothing about the timing of demands.

Answer: C

Correct answer: base shear against roof displacement

The capacity curve records the structure's resistance as it is pushed progressively further: base shear on one axis and roof displacement on the other.

Its shape tells the story - an initial linear portion, a knee as members begin to yield, a plateau at the mechanism, and eventual degradation. That is what makes pushover analysis a nonlinear method: the curve shows not only the strength but the displacement over which it is sustained, which is the ductility available.

Answer: D

Correct answer: 1, 2 and 3

All three are correct. Mode shapes are orthogonal with respect to both the mass and stiffness matrices, which is what uncouples the equations of motion and makes modal superposition possible.

Their magnitude is arbitrary, since the eigenvalue problem is homogeneous, so a normalisation must be chosen - unit modal mass, or unit value at the roof. And the number of modes equals the number of degrees of freedom, the eigenvalue problem being of that order.

Answer: A

Correct answer: 1, 2 and 3

All three are correct and the first two describe complementary experiments. The logarithmic decrement is read from a free vibration decay, needing only a means of displacing the structure; the half-power bandwidth is read from the resonance peak of a forced vibration test, needing a shaker.

Rayleigh damping is constructed as a combination of the mass and stiffness matrices precisely so that it is orthogonal to the mode shapes, which keeps the modal equations uncoupled - a mathematical convenience rather than a physical description.

Answer: B

Correct answer: 1-Q, 2-P, 3-R

Each method answers a different question. Eigenvalue analysis solves \( (K-\omega^2M)\phi = 0 \) and yields periods and mode shapes - properties of the structure alone, with no loading involved.

Pushover analysis produces a capacity curve of base shear against roof displacement, showing strength and ductility. Time-history analysis integrates the equation of motion against a record and produces the response at every instant. Structure alone, capacity, and response to a particular event.