Wind, seismic response and dynamic structural hazards

Define the response coordinate and the input

A dot above a displacement denotes its rate of change with time. Thus ẋ is velocity, measured in m/s, and ẍ is the rate of change of that velocity: acceleration, measured in m/s². These are instantaneous signed quantities along the chosen motion direction. A body can have positive displacement while moving back toward equilibrium, so displacement and velocity need not have the same sign. The equation below relates those quantities; solving its complete time history is a separate task.

For a fixed-base linear oscillator, m ẍ+c ẋ+kx=F(t). Here x is displacement from static equilibrium, m is effective mass, c a viscous damping coefficient, k stiffness and F the applied dynamic force. Undamped natural angular frequency is ωn=√(k/m); frequency in cycles per second is ωn/(2π). Damping ratio ζ=c/(2√(km)) describes this specific linear damping model.

Read the equation as a force account: m ẍ = F(t)−c ẋ−kx. Mass times acceleration has units kg·m/s², or newtons (N). Stiffness k has units N/m, so kx is also a force. The viscous coefficient c has units N·s/m, making c ẋ a force. The spring and damper contributions on the right oppose positive displacement and positive velocity respectively in this model. They can change sign as the motion changes; they are not three additional external loads to add to F.

A sinusoidal applied force can produce a steady oscillation after transients decay. Its displacement amplitude relative to static deflection F 0/k is 1/√((1−r²)²+(2ζr)²), where r is forcing frequency divided by undamped natural frequency. Near r=1, small damping permits large amplification. For damped systems, the exact amplitude peak need not occur exactly at r=1.

Identify why wind and earthquakes are different inputs

Wind produces surface pressures that vary with location and time. Their fluctuating component can excite structural modes; the distribution and temporal content matter as well as an average resultant. A single supplied sinusoidal force is a controlled illustration, not a complete turbulent-wind model or a prediction for a particular building shape.

Earthquake shaking moves the base. For the same simple oscillator with relative displacement x and ground displacement ug, the equation becomes m ẍ+c ẋ+kx=−m üg. Ground acceleration is therefore not interchangeable with the fixed-base force amplitude in a graph without a conversion and defined coordinate. Actual structures can have several translational and torsional modes, and foundations can interact with the ground.

In the base-motion equation, ug is ground displacement in metres, and üg is ground acceleration in m/s²: the second time rate of that displacement. Here x is motion of the mass relative to the moving ground, so absolute displacement is x+ug. Moving the ground-acceleration term from m(ẍ+üg) to the right produces −m üg. This coordinate conversion explains the sign and force units; it does not supply an earthquake record or justify replacing the stated force-response curve by a seismic spectrum.

Connect response to hazards and consequences

An earthquake response spectrum records peak response of a family of specified oscillators to a ground-motion record or another stated seismic demand. It is not the steady harmonic amplification curve shown here. Period, damping, site conditions and the input record influence response. A longer period does not universally imply lower demand for every site, motion or performance measure.

Ductile behaviour can dissipate energy through deformation when the system and details support it. That can involve damage and residual displacement. Collapse avoidance, repairability and rapid functional recovery are distinct outcomes. Attached ceilings, equipment and services can also lose function or create hazards even when the primary frame remains standing; response assessment must define the intended performance boundary.

A supplied oscillator illustrates frequency sensitivity

Assume m=20000 kg and k=800000 N/m. Then ωn=√40≈6.3246 rad/s, frequency is 1.0066 Hz and period 0.9935 s. A 5000 N applied harmonic force has static deflection F 0/k=0.00625 m. At frequency ratio r=1 and ζ=0.05, the amplification factor is 10, giving 62.5 mm steady displacement amplitude.

With the same mass, stiffness and force but ζ=0.20, the factor at r=1 is 2.5 and amplitude 15.625 mm. The two curves compare declared linear models, not proposed changes to a real structure. Their peaks sit slightly below r=1. Neither curve supplies a ground-motion spectrum, yield limit or wind-response qualification.

Try the reasoning

1. If stiffness becomes one-quarter while mass stays fixed, what happens to the undamped period?

   Show the answer and reasoning

   It doubles because T=2π√(m/k). Whether that improves a particular hazard response needs the excitation and performance measure.

2. At r=1, does increasing ζ from 0.05 to 0.20 reduce the stated force-response amplitude by four?

   Show the answer and reasoning

   Yes, for this linear steady-force case: 1/(2ζ) changes from 10 to 2.5. It does not describe every input or frequency.

3. Does a standing frame after an earthquake prove immediate usable service?

   Show the answer and reasoning

   No. Residual movement, damaged nonstructural components and interrupted utilities can prevent occupancy or function.

4. In a separate instantaneous fixed-base case, use m = 20000 kg, k = 800000 N/m, c = 4000 N·s/m, x = +0.010 m, ẋ = +0.020 m/s and F = +9000 N. Find ẍ and explain why this is not a steady displacement amplitude. What are the units of −m üg in the separate base-driven equation?

   Show the answer and reasoning

   The spring term is 800000×0.010 = 8000 N and the damper term is 4000×0.020 = 80 N. Hence m ẍ = 9000−8000−80 = 920 N, so ẍ = +0.046 m/s². It is the instantaneous rate of velocity change for the supplied state and force; no forcing frequency, transient decay or steady sinusoidal response has been established. In the base-driven equation, kg times m/s² is N, and the minus sign reflects the relative-coordinate account rather than a negative mass.

What to carry forward

Name the excitation, coordinate and response measure before using a dynamic curve to discuss structural performance.