Practice

A structure has a natural frequency of 2.5 Hz. What is its natural period, in seconds?

A beam deflects 8 mm under a slowly applied load. The same load is dropped onto it instead. Ignoring damping, what is the peak deflection in mm?

Estimate the natural period, in seconds, of a 14-storey building using the rule T_n ≈ 0.1N.

Which statement best defines a dynamic load?

A machine runs at 1 200 rpm. What is its forcing frequency in Hz?

A floor has a natural frequency of 5.0 Hz and 2% damping. If a harmonic force is applied at exactly that frequency, by what factor is the response amplified compared with the same force applied slowly?

Which of these loadings can produce true resonance?

A tank of mass 12 000 kg sits on a support structure with a lateral stiffness of 4.8 MN/m. What is its natural frequency in Hz?

A structure has a natural period of 0.5 s. A load is applied over 0.1 s. Give the ratio t_r/T_n, and use it to say which treatment applies.

A floor deflects 6 mm under its own weight. Estimate its natural frequency in Hz using the deflection shortcut.

A pump forces at 8 Hz. A floor has a natural frequency of 25 Hz. What is the appropriate treatment?

Which of these is the clearest sign that a load must be treated as dynamic rather than static?

Two identical loads are applied to two identical structures. One is applied over 5 natural periods, the other over 0.05 natural periods. Which produces the larger response?

A mass of 25 tonnes is supported on a structure of lateral stiffness 10 MN/m. What is the natural period in seconds?

A beam deflects 5 mm under a load applied slowly. The same load is applied suddenly to a structure with 5% damping. What is the peak deflection in mm? Use DLF = 1 + exp(−ζπ/√(1−ζ²)).

An office floor is uncomfortably lively. Which change raises its natural frequency most effectively?

A machine runs at 900 rpm on a floor whose natural frequency is 12 Hz. What is the frequency ratio β?

A six-storey building is modelled as a shear building. How many degrees of freedom does the model have?

A three-dimensional building model uses rigid diaphragms and includes floor torsion. How many degrees of freedom does an eight-storey model have?

What does a rigid diaphragm assumption remove from a model?

A twelve-storey building is modelled with a rigid diaphragm at every floor. How many dynamic degrees of freedom does the model have?

The same twelve-storey building is reduced to a plane shear model in one direction. How many degrees of freedom now?

A three-dimensional frame model has 40 nodes, of which 4 are fully fixed at the base. Taking six degrees of freedom per free node, how many degrees of freedom does the model have?

A reduced model has been built by lumping mass at floor levels. Which question can it NOT answer?

A twelve-storey building is modelled in three dimensions with rigid diaphragms and torsion included. How many modes will a complete modal analysis return?

A reduced model reports a first natural frequency LOWER than the full model it was reduced from. What does this tell you?

A mass of 4 000 kg is on a spring of stiffness 1.6 MN/m with a damping coefficient of 8 000 N·s/m. What is the damping ratio?

For the same system (m = 4 000 kg, k = 1.6 MN/m, c = 8 000 N·s/m), the mass is at x = 20 mm moving at 0.4 m/s with an acceleration of −7.6 m/s². What applied force F(t) is acting, in kN?

In free vibration, at the instant the mass passes through its equilibrium position, which force is zero?

A water tank of 45 tonnes sits on a frame whose measured period is 0.62 s. What lateral stiffness does that imply, in MN/m?

A single-storey frame has four columns, each contributing 12EI/h³. Take E = 205 GPa, I = 1.2 × 10⁻⁴ m⁴ and h = 3.6 m. What is the total lateral stiffness, in MN/m?

In the equation mü + cu̇ + ku = p(t), which term carries the energy that leaves the system permanently?

A 15 tonne mass sits on columns of total lateral stiffness 6.0 MN/m with 4% damping. What is the damping coefficient c, in kN·s/m?

Why does gravity not appear in the equation of motion for a vertically hanging mass?

A system has ω_n = 25 rad/s and ζ = 0.08. What is the damped circular frequency, in rad/s?

A structure with 3% damping is displaced and released. After how many cycles has the amplitude fallen to 10% of its starting value?

A mass of 6 000 kg on a spring of 2.4 MN/m has a damping coefficient of 12 000 N·s/m. What is the damping ratio?

The first peak of a decay trace is 12.0 mm and the fifth is 5.4 mm. What is the logarithmic decrement?

For that trace, what is the damping ratio as a percentage?

A structure has 2% damping. Over how many cycles must peaks be separated for the amplitude ratio to reach 2.0?

A decay trace shows peaks falling by a constant AMOUNT each cycle — 10 mm, 8 mm, 6 mm, 4 mm — rather than by a constant ratio. What does this indicate?

Two identical structures are tested. One shows 12 visible cycles before the motion dies away, the other 3. What differs?

A structure has m = 20 000 kg and k = 8.0 MN/m. What is the critical damping coefficient, in kN·s/m?

Peaks in a decay trace are 25.0 mm and 9.2 mm, eight cycles apart. What is the damping ratio as a percentage?

What does critical damping mean physically?

A structure with 5% damping is set vibrating. What fraction of its initial ENERGY remains after 3 cycles? Give the answer as a percentage.

A structure with 2.5% damping is forced at exactly its natural frequency. What is the dynamic magnification factor?

A machine forces at 12 Hz on a floor with a natural frequency of 15 Hz and 4% damping. What is the magnification factor?

For the same system (β = 0.8, ζ = 0.04), what is the phase lag in degrees?

At what frequency ratio β does the magnification factor peak, for a system with 10% damping?

At resonance, which force balances the applied force?

What is the magnification factor exactly at resonance for a system with 4% damping?

A machine runs at 80% of a floor's natural frequency. With 5% damping, what is the magnification factor?

At what frequency ratio does the true maximum of the magnification curve occur, for a lightly damped system?

A structure has 1.5% damping. What is its resonant magnification factor?

A 5 kN harmonic force acts at β = 1.4 on a structure of stiffness 50 MN/m with 5% damping. What is the steady-state amplitude in mm?

A machine is to be isolated from a floor. At what frequency ratio does isolation begin?

How many cycles does a resonant response take to reach 63% of its steady-state amplitude, at 2% damping?

A structure of mass 30 000 kg is struck by an impulse of 45 kN·s. What velocity does it acquire, in m/s?

That structure has a natural frequency of 12 rad/s. Ignoring damping, what is its peak displacement in mm?

A unit impulse response h(t) has an initial slope of 4 × 10⁻⁵ m per N·s². What is the mass of the structure, in kg?

A rectangular pulse lasts 0.06 s on a structure with a natural period of 0.4 s. Using the undamped shock spectrum, what is the dynamic load factor?

For the same structure, what pulse duration in seconds would give the maximum dynamic load factor of 2?

A triangular pulse has a peak of 80 kN and lasts 0.025 s. What is its impulse, in kN·s?

For a pulse with t_d/T_n = 0.2, when does the peak response occur?

In a convolution, an impulse arrives at τ = 1.2 s and the response is being evaluated at t = 3.0 s. At what elapsed time is the impulse response function evaluated, in seconds?

A convolution is evaluated over 6 000 time steps. Roughly how many multiply–add operations does that require, expressed in millions?

Which single assumption does the convolution integral depend on most fundamentally?

Why does the Duhamel integral apply only to linear systems?

A structure of mass 18 000 kg and natural frequency 9 rad/s is struck by an impulse of 24 kN·s. Ignoring damping, what is the peak displacement in mm?

A rectangular pulse acts for 0.09 s on a structure whose natural period is 0.5 s. What is the undamped dynamic load factor?

Two pulses have the same area but different shapes, and both are very short compared with the natural period. What can be said about the responses?

A half-sine pulse of peak 120 kN lasts 0.04 s. What is its impulse in kN·s? The area under a half sine is 2F₀t_d/π.

Why is the convolution integral rarely used to compute earthquake response in practice?

A structure of stiffness 3.5 MN/m is displaced 40 mm and released from rest. What is its total mechanical energy, in J?

That structure has a mass of 14 000 kg. What is its maximum velocity, in m/s?

A vibration decays to 30% of its original amplitude. What percentage of the original energy remains?

A viscous damper with c = 8 kN·s/m operates at 10 rad/s with an amplitude of 20 mm. How much energy does it dissipate per cycle, in J?

A measured hysteresis loop encloses 450 J at an amplitude of 25 mm. The stiffness is 4.0 MN/m. What is the equivalent viscous damping ratio?

What fraction of the stored energy does a structure with 4% damping lose per cycle? Give the answer as a percentage, using the small-damping relationship.

Why is the energy dissipated by a viscous damper proportional to frequency?

Over one complete cycle of steady harmonic response, what is the net work done by the inertia force?

A structure of mass 25 000 kg is moving at 0.8 m/s as it passes through its equilibrium position in free vibration. What is its total mechanical energy, in kJ?

A damper with c = 15 kN·s/m operates at 8 rad/s with amplitude 30 mm. What is the energy dissipated per cycle, in J?

In a computed time-history analysis, the cumulative damping energy is seen to DECREASE at some steps. What does this indicate?

A loop encloses 220 J at an amplitude of 18 mm, on a system of stiffness 6.0 MN/m. What is the equivalent viscous damping ratio, as a percentage?

A linear time-history analysis closes its energy balance to within 0.0001%. What does this establish?

A structure has m = 2 000 kg, k = 500 kN/m and no damping. Using γ = ½, β = ¼ and Δt = 0.02 s, what is the effective stiffness k̂, in MN/m?

A structure of mass 4 000 kg and stiffness 900 kN/m is at rest when a force of 18 kN is suddenly applied. What is the initial acceleration, in m/s²?

What does the Newmark parameter β = ¼ correspond to physically?

A structure has a natural period of 0.8 s. What is the largest time step, in seconds, for which the central-difference method is stable?

Using constant average acceleration at Δt/T = 0.15, what is the approximate period elongation, as a percentage?

A building has T₁ = 1.2 s and its highest significant mode has T₆ = 0.09 s. Using a T/20 rule on the mode that governs, what time step should be used, in seconds?

Using the general Newmark stability condition Δt/T ≤ 1/(π√2·√(γ − 2β)), what is the limit for γ = 0.5 and β = 1/6?

Which failure mode is more dangerous in practice?

An explicit central-difference run of a finely meshed model needs an extremely small time step. Which period governs it?

A structure has m = 5 000 kg, c = 20 kN·s/m and k = 2 MN/m. Using γ = ½, β = ¼ and Δt = 0.01 s, what is the effective stiffness k̂, in MN/m?

At Δt/T = 0.05, what is the approximate period elongation for constant average acceleration, as a percentage?

Why does the central-difference method have a stability limit when constant average acceleration does not?

A record is sampled at 0.01 s. An analysis needs Δt = 0.002 s. By what integer factor must the record be sub-sampled?

An analyst reports: 'Δt = T₁/100, which is well within normal practice, so the analysis is converged.' What is wrong with this statement?

A two-storey shear building has k₁ = 80 MN/m and k₂ = 50 MN/m. What is the entry K[0][0], in MN/m?

For the same building, what is the off-diagonal entry K[0][1], in MN/m?

A 2DOF system has m₁ = m₂ = 1 000 kg and k₁ = k₂ = 100 kN/m. What is the first natural frequency, in rad/s?

For that system, what is the mode-1 shape ratio φ₂/φ₁?

A 2DOF system has mode shapes φ₁ = {1, 1.6} and φ₂ = {1, −0.9}, with m₁ = 4 000 kg and m₂ = 2 500 kg. Compute φ₁ᵀMφ₂ to check orthogonality. The answer should be near zero — give it in kg.

Why can only the RATIO of a mode shape's components be determined, and not their absolute values?

In the second mode of a uniform two-storey shear frame, how do the two floors move?

A two-storey shear building has k₁ = 120 MN/m and k₂ = 70 MN/m. What is K[0][0], in MN/m?

A 2DOF system has m₁ = m₂ = 2 000 kg and k₁ = k₂ = 300 kN/m. What is the second natural frequency, in rad/s?

A modal analysis of a two-storey building returns a first mode shape of {1, −0.8}. What does this indicate?

Mode shapes φ₁ = {1, 2} and φ₂ = {1, −0.5} are claimed for a system with m₁ = 5 000 kg and m₂ = 2 500 kg. Compute φ₁ᵀMφ₂, in kg, to test orthogonality.

What is orthogonality of mode shapes actually FOR?

A five-storey shear building has all storey stiffnesses equal to 400 MN/m. What is K[2][2] — the second floor's diagonal entry — in MN/m?

For that same building, what is K[4][4] — the top floor's diagonal entry — in MN/m?

A bar element has ρAL = 900 kg. What is the leading diagonal entry of its consistent mass matrix, in kg?

For that same bar, what is the diagonal entry of the LUMPED mass matrix, in kg?

A fixed–free bar is 5 m long with E/ρ = 2.675 × 10⁷ m²/s². What is the exact first axial circular frequency, in rad/s?

What do the off-diagonal terms of a consistent mass matrix represent physically?

A rectangular floor is 24 m by 15 m with a mass of 400 tonnes. What is its mass moment of inertia about the vertical axis through its centre of mass, in kg·m²?

For that floor, what is the radius of gyration in metres?

A three-dimensional model has 9 storeys with rigid diaphragms including torsion. How many modes should a complete modal analysis return?

What happens to a model's torsional mode if the diaphragm rotational inertia is left at zero?

A rigid-diaphragm model is being used for a floor with a large stair opening and a long narrow plan. What is the risk?

A six-storey shear building has all storey stiffnesses of 550 MN/m. What is the sum of the diagonal entries of its stiffness matrix, in MN/m?

A bar element has ρAL = 1 200 kg. What is the off-diagonal entry of its consistent mass matrix, in kg?

A square floor is 20 m by 20 m with a mass of 600 tonnes. What is its mass moment of inertia, in kg·m²?

A modal analysis of an eight-storey rigid-diaphragm model returns 26 modes. What does this indicate?

Why do lumped and consistent mass models bracket the exact natural frequency?

A model has 8 degrees of freedom. How many natural frequencies does it have?

A uniform three-storey shear building has m = 500 tonnes per floor and k = 750 MN/m per storey. What is its first natural frequency, in rad/s? Use the ratio ω₁ = 0.4450√(k/m).

For that building, what is the third natural frequency in rad/s? Use ω₃ = 1.8019√(k/m).

Why do real eigensolvers not find the roots of det(K − ω²M) = 0 directly?

A mode shape is {0.30, 0.65, 1.00} on a building with 500 tonnes per floor. What is its modal mass, in tonnes?

A mass-normalised mode has a natural frequency of 24 rad/s. What is its modal stiffness φᵀKφ?

Two mode shapes are φ₁ = {0.5, 1.0} and φ₂ = {1.0, −0.4} on a system with m₁ = 300 t and m₂ = 375 t. Compute φ₁ᵀMφ₂ in tonnes to test orthogonality.

Which single property of M and K does the orthogonality derivation actually require?

A mode shape is reported with a maximum ordinate of 1.0 in one analysis and 0.043 in another, for the same building. What has changed?

A uniform four-storey shear building has a first natural frequency of 12 rad/s. Using the uniform-shear-building ratios (0.3473, 1.0000, 1.5321, 1.8794), what is its fourth natural frequency in rad/s?

A mode shape {0.25, 0.55, 0.80, 1.00} is on a building with 350 tonnes per floor. What is its modal mass, in tonnes?

A modal analysis of a symmetric square building returns modes 1 and 2 at exactly the same frequency. Is this an error?

A mass-normalised mode has a period of 0.8 s. What is its modal stiffness φᵀKφ?

Which of these quantities is UNCHANGED when a set of mode shapes is renormalised?

A mode has shape {0.4, 0.7, 1.0} on a building with 300 tonnes per floor. A horizontal force of 150 kN is applied at the roof only. What is the generalised force φᵀp, in kN?

A mode has modal mass 500 tonnes and natural frequency 20 rad/s. What is its modal stiffness, in MN/m?

A mode with modal mass 800 tonnes and frequency 15 rad/s is assigned 4% damping. What is its modal damping coefficient Cₙ, in kN·s/m?

Which matrix is NOT automatically diagonalised by the mode shapes?

A mode has L = φᵀMι = 1 800 tonnes and modal mass M = 1 500 tonnes. What is its participation factor?

For that same mode, what is the effective modal mass, in tonnes?

Rayleigh damping is anchored at 5% on ω = 4 rad/s and ω = 40 rad/s. What damping does it deliver at ω = 12 rad/s? Use a₀ = 2ζω₁ω₂/(ω₁+ω₂) and a₁ = 2ζ/(ω₁+ω₂).

Cumulative effective mass reaches 92% with six modes. What has this established?

What exactly does classical damping buy you?

A mode has L = 2 400 tonnes and modal mass 2 000 tonnes, on a building of total mass 3 000 tonnes. What percentage of the total mass does this mode mobilise?

A mode with modal mass 600 tonnes and frequency 25 rad/s is given 3% damping. What is its modal damping coefficient, in kN·s/m?

Rayleigh damping is anchored at 4% on ω = 5 rad/s and ω = 30 rad/s. What damping does it deliver at ω = 60 rad/s?

Why do design codes accumulate effective modal mass rather than participation factors?

Which situation makes modal superposition invalid rather than merely inaccurate?

A structure of mass 40 tonnes experiences a peak ground acceleration of 3.2 m/s². What is the peak effective earthquake force, in kN?

A structure has a relative acceleration of −2.1 m/s² at an instant when the ground acceleration is 1.4 m/s². What is its absolute acceleration, in m/s²?

Ground motion is harmonic at 20 mm amplitude and 2 Hz. A structure with a natural frequency of 8 Hz and 5% damping sits on it. What is the relative displacement amplitude, in mm? Use u = β²R_d U_g.

Why does the base-excitation equation contain relative displacement on the left and absolute acceleration on the right?

A record has a peak ground acceleration of 0.28g. What is that in m/s²?

A very stiff structure of 900 tonnes sits on ground with a PGA of 0.28g. What horizontal force does it attract, in MN?

Which displacement strains the structure during an earthquake?

A 12-storey building has a total seismic mass of 9 600 tonnes. At an instant the ground acceleration is 2.5 m/s². What is the total effective earthquake force, in MN?

A structure's relative displacement peaks at 95 mm, and the record's peak ground displacement is 110 mm. What is the peak absolute displacement?

Two records have the same peak ground acceleration. One lasts 8 s and the other 40 s. For which type of structure does the difference matter most?

Ground motion is harmonic at 25 mm and 1 Hz. A structure has a natural frequency of 0.5 Hz and 5% damping. What is its relative displacement amplitude, in mm?

An oscillator with a period of 0.5 s has a spectral displacement of 42 mm. What is its pseudo-acceleration, in m/s²?

A structure of mass 5 000 tonnes has a spectral acceleration of 3.4 m/s². What is the peak elastic base shear, in MN?

For that structure, the base shear coefficient V/W is what value?

Why is pseudo-acceleration called 'pseudo'?

Modal base shears are 3.2 MN, 0.9 MN and 0.4 MN. What is the SRSS combination, in MN?

Two modes have frequencies of 5.0 and 5.5 rad/s, both with 5% damping. What is the CQC correlation coefficient ρ between them?

Responses of 100 and 60 units act in two orthogonal directions. What is the 100/30 directional combination?

When does CQC reduce to SRSS?

Why is the pseudo-acceleration, rather than the true peak acceleration, tabulated in design spectra?

An oscillator of period 0.8 s has a spectral displacement of 65 mm. What is its pseudo-acceleration, in m/s²?

Modal base shears are 6.0, 2.5, 1.2 and 0.8 MN. What is the SRSS combination, in MN?

A response-spectrum analysis reports a base shear coefficient V/W of 2.8. What should be concluded?

Two modes have a frequency ratio r = 0.90 and 5% damping each. What is the CQC correlation coefficient?

What information does a response spectrum deliberately discard?

A building has T₁ = 1.5 s and its highest significant mode has T₈ = 0.075 s. Using Δt ≤ T/20 on the mode that governs, what time step should be used, in seconds?

A 40 s record is analysed at Δt = 0.002 s. How many time steps does the analysis take?

A building of seismic mass 6 500 tonnes has a peak base shear of 11.2 MN. What is the base shear coefficient V/W?

Floors at 24.0 m and 20.6 m have peak displacements of 132 mm and 118 mm. What inter-storey drift would subtracting the peaks give, as a percentage?

Which response quantity converges LAST as the time step is refined and modes are added?

What does a time-history analysis give that a response spectrum cannot?

A structure has T₁ = 2.0 s and its highest retained mode has T₁₀ = 0.11 s. Using Δt ≤ T/20 on the governing mode, what step should be used, in seconds?

A building of 11 000 tonnes has a peak base shear of 19.4 MN. What is V/W?

Two analyses of the same linear building give peak drifts differing by 0.4% and peak floor accelerations differing by 38%. What should be concluded?

A 25 s record sampled at 0.01 s must be analysed at Δt = 0.002 s. By what integer factor must it be sub-sampled?

Why do design standards require several ground-motion records rather than one?

A bilinear system has k = 12 MN/m and f_y = 480 kN. What is its yield displacement, in millimetres?

The same system reaches a peak displacement of 120 mm. What is the displacement ductility demand?

Using the equal-energy rule, what strength reduction factor R corresponds to a ductility of 4?

For the system in question 1 (k = 12 MN/m, f_y = 480 kN) with α = 0.05 and μ = 3, what is the secant stiffness at peak, in MN/m?

An elastic–perfectly plastic system has f_y = 480 kN and u_y = 40 mm and is cycled to μ = 3. How much energy does one full symmetric cycle dissipate, in kJ?

Which stiffness does a nonlinear solver use inside its Newton iteration?

Why can a response spectrum not be used directly on a structure that yields?

A structure has an elastic base shear demand of 1 460 kN and is designed with a behaviour factor of 4. What is the design base shear, in kN?

An elastic–perfectly plastic system reaches a ductility of 4. What equivalent viscous damping ratio does the energy-equivalence formula ζ_eq = (2/π)(1 − 1/μ) give, as a percentage?

The same formula at a ductility of 2 gives what equivalent damping ratio, as a percentage?

A frame yields at 31.2 mm and is expected to reach a ductility of 4. What peak displacement should the cladding connections be detailed for, in millimetres?

Which rule should be used to relate strength reduction and ductility for a stiff, short-period structure?

Two nonlinear analyses of the same building under different records give peak displacements within 5% of each other but residual displacements differing by a factor of three. What should be concluded?

A bilinear system has k = 8.0 MN/m and f_y = 360 kN. What is its yield displacement, in millimetres?

The same system reaches 158 mm. What is its displacement ductility demand?

Using the equal-energy rule, what strength reduction factor corresponds to a ductility of 3?

A nonlinear time-history analysis reports 210 non-converged steps out of 24 000. What should be done?

A building of total mass 2 400 tonnes sits on isolators of total horizontal stiffness 10.53 MN/m. What is the isolated period, in seconds?

What total isolator stiffness is needed to give a 2 400 tonne building a period of 2.5 s? Answer in MN/m.

A fixed-base building has a period of 0.494 s and its isolated version 3.030 s. What is the period shift ratio?

The isolated base shear is 1.09 MN and the fixed-base value is 7.27 MN. What is the force ratio?

Base isolation reduces the force on a building primarily by:

A directional isolator displacement of 105 mm is combined bidirectionally by SRSS with an equal peak in the orthogonal direction. What is the resultant, in millimetres?

Starting from a 148.5 mm resultant, apply a torsional amplification of 1.15 and a lower-bound bearing factor of 1.20. What displacement results, in millimetres?

Why do isolation systems need a wind-restraint mechanism?

Which building is the best candidate for base isolation?

Why does isolator damping mainly help the displacement rather than the force?

What total isolator stiffness gives a 1 800 tonne building a period of 3.2 s? Answer in MN/m.

A fixed-base period of 0.52 s becomes 3.2 s when isolated. What is the period shift ratio?

An analysis gives a directional isolator displacement of 130 mm. Take the bidirectional resultant of two equal peaks, then apply torsional amplification 1.15 and a lower-bound bearing factor 1.20. What displacement results, in millimetres?

For which of these should base isolation NOT be used?

A linear damper has C = 900 kN·s/m. What force does it deliver at a relative velocity of 0.35 m/s, in kN?

A damper with C = 900 kN·(s/m)^0.4 and α = 0.4 is driven at 0.35 m/s. What force does it deliver, in kN?

The same α = 0.4 damper is driven at 0.70 m/s. What force does it deliver, in kN?

A damper is braced at 38° to the horizontal. What fraction of a horizontal damper's effectiveness does it retain?

A linear damper with C = 2.0 MN·s/m is cycled at 30 mm amplitude at ω = 4.19 rad/s. How much energy does it dissipate per cycle, in kJ?

Why are seismic viscous dampers usually made with a velocity exponent below 1?

A linear damper with C = 2.0 MN·s/m braced at 30° spans a storey with first-mode relative displacement 0.35. The first mode has ω₁ = 4.19 rad/s and modal mass 900 tonnes. What damping ratio does it add?

What fraction of a horizontal damper's effectiveness is retained by one braced at 60°?

A structure has 2% inherent damping. Six dampers add 5.81%. By what factor does the resonant response fall, taking response at resonance as proportional to 1/ζ?

Why might viscous dampers be preferred to added bracing in the seismic retrofit of an existing building?

Two dampers are quoted with the same coefficient C but different velocity exponents. What can be concluded?

A damper with C = 1 200 kN·(s/m)^0.5 and α = 0.5 is driven at 0.50 m/s. What force does it deliver, in kN?

What fraction of a horizontal damper's effectiveness is retained by one braced at 45°?

A linear damper with C = 1.5 MN·s/m braced at 45° spans a storey with first-mode relative displacement 0.28. The mode has ω = 3.5 rad/s and modal mass 750 tonnes. What damping ratio does it add?

Why is a response-spectrum analysis not valid for a structure with nonlinear viscous dampers?

A tuned mass damper has a mass ratio of 2%. What is Den Hartog's optimum tuning ratio f = ω₂/ω₁?

For the same 2% mass ratio, what is the optimum damping ratio of the damper itself?

A building with a first-mode generalised mass of 1 200 tonnes is fitted with a TMD at a mass ratio of 2%. What is the damper mass, in tonnes?

An undamped structure has a TMD with a 2% mass ratio at Den Hartog's optimum. What peak magnification does the theory predict?

A structure with 1% damping has a peak magnification of 8.71 with an optimally tuned TMD, and 16.8 when the damper is mistuned by 10%. By what factor does the peak response increase?

How does a tuned mass damper reduce a structure's resonant response?

A footbridge has 0.6% damping and needs a sixfold reduction in lateral acceleration. Assuming response scales as 1/ζ, what total damping ratio is required, as a percentage?

What distinguishes a semi-active control device from an active one?

Why is active control common in aerospace and rare in buildings?

A tuned mass damper's own damping ζ₂ is halved from its optimum. What happens to the response curve?

A tuned mass damper has a mass ratio of 3%. What is Den Hartog's optimum tuning ratio?

For the same 3% mass ratio, what is the optimum damper damping ratio?

With a 3% mass ratio at the optimum, what peak magnification does the undamped theory predict?

A structure's safety in the design earthquake would depend on an active control system operating. Is that acceptable?

A floor is 24 m × 18 m with an imposed load of 3.0 kN/m² and a quasi-permanent factor of 0.3. What imposed load contributes to the seismic mass, in kN?

An eight-storey building is 27.2 m tall. What first period does the empirical rule T ≈ 0.075H^0.75 predict, in seconds?

A building has a total seismic weight of 35 900 kN over eight floors of 432 m². What is the seismic weight per unit floor area, in kN/m²?

Which constraint should NOT be applied to a floor with a large central atrium opening?

A ten-storey building 34 m tall reports a first period of 0.42 s. What does the rule T ≈ 0.075H^0.75 predict, in seconds?

An analysis reports a base shear of 21 500 kN on a building of seismic weight 41 000 kN. What is the base shear coefficient?

A reported period is 0.42 s where 1.0 s was expected. By what factor is the model's stiffness in error?

A building of plan area 600 m² and ten floors has a reported total seismic weight of 41 000 kN. What is the seismic weight per unit floor area, in kN/m²?

A response-spectrum analysis gives a base shear coefficient of 1.6. What is the most likely cause?

Cumulative effective mass reaches 92% at mode 8. The analysis must produce floor accelerations for equipment qualification. What should be done?

Which of these checks is independent of the stiffness model, and therefore useful when the period check has already failed?

A twelve-storey building is 41 m tall. What first period does T ≈ 0.075H^0.75 predict, in seconds?

An analysis reports a base shear of 3 900 kN on a building of seismic weight 52 000 kN. What is the base shear coefficient?

A building of plan area 480 m² and twelve floors reports a total seismic weight of 52 000 kN. What is the seismic weight per unit floor area, in kN/m²?

A modal analysis of a six-storey frame reports a first mode at 3.9 s carrying 4% of the mass. What does that indicate?