Seismic Response Calculator — Period, Drift & Base Shear
Analyze a multi-story building’s earthquake response: natural periods, mode shapes, story-drift ratio, and base shear by Newmark-β time integration.
Earthquake response, modal mass, drift, and base shear
- Ground acc
- Roof disp (m)
- Roof acc
- Base shear (MN)
- Max drift (%)
- Pseudo Sa (g)
- Eff. mass (%)
- Mode 1
- Mode 2
- Mode 3
- Mode 4
A clear path from inputs to results.
Estimate how a multi-story building responds to earthquake ground shaking. The tool models the building as a stack of floor masses connected by story stiffnesses, finds its natural periods and mode shapes, then integrates the full equations of motion in time under a chosen ground motion to report roof drift, floor accelerations, story drift ratios, and base shear.
How to use
- Choose a ground-motion scenario (moderate design motion, near-fault pulse, design sweep, or paste a custom acceleration record in g).
- Set the shaking duration and the integration time step; a smaller step resolves higher-frequency response.
- Set the number of floors, the mass per floor, and the base story stiffness to define the structure.
- Apply the mass taper and stiffness taper to make upper floors lighter and more flexible, as in a real building.
- Set the modal damping ratio; the tool builds a Rayleigh damping matrix tuned to the first two modes.
- Read the fundamental period, peak roof displacement and acceleration, peak story-drift ratio, and peak base shear, and scrub the time cursor to inspect the 2D/3D deflected shape at any instant.
The formula
Reference: Linear MDOF base-excitation equation of motion with Newmark-β average-acceleration integration (β = 1/4, γ = 1/2) and pseudo-spectral ordinates; Chopra, Dynamics of Structures (base excitation, response spectrum); FEMA/NEHRP Recommended Seismic Provisions (modal response-spectrum analysis).
Worked example
Six-story shear building — periods and modal demand
- Floors n = 6; base floor mass 420,000 kg with a 12% mass taper (402,360 kg at the roof); base story stiffness 160 MN/m with a 20% stiffness taper (177.6 MN/m at the base, 160 MN/m at the top); modal damping 5%.
- 1
The tool assembles the tridiagonal shear-building stiffness matrix K from the six story stiffnesses and a diagonal mass matrix M, then solves the generalized eigenproblem by forming A = M^(−1/2) K M^(−1/2) and extracting its eigenvalues. The circular frequencies ω = √λ give natural frequencies f = ω/2π of 0.789, 2.282, 3.651, 4.809, 5.688, and 6.242 Hz. The fundamental mode has ω₁ = 4.959 rad/s, so f₁ = 0.789 Hz and the fundamental period is T₁ = 2π/ω₁ = 1.267 s. The mass-orthonormal mode shapes give effective modal mass ratios of 86.2%, 9.3%, 2.8%, 1.1%, 0.4%, and 0.1%, so the first mode alone carries about 86% of the seismic mass and the first two carry about 96%. For modal demand, if the design spectrum returns a spectral displacement S_d = 30 mm at the fundamental period, the pseudo-acceleration follows from the exact identity S_a = ω₁² S_d = 0.738 m/s² ≈ 0.075 g.
Fundamental period T₁ ≈ 1.27 s (f₁ = 0.789 Hz) with the first mode carrying ~86% of the effective mass; six modal frequencies up to 6.24 Hz. Peak roof displacement, acceleration, story drift, and base shear then follow from Newmark-β time integration of the chosen ground motion.
Interpreting the result
The fundamental period is the single most important dynamic property: it sets where the building sits on a response spectrum and therefore how hard a given earthquake shakes it. A long period (tall, flexible building) generally draws lower acceleration but larger displacement and drift; a short period does the opposite. The effective mass ratios tell you how many modes you must include — here the first mode dominates, so the response is close to a tuned single-degree-of-freedom system. Peak story-drift ratio is the key damage indicator: values approaching or exceeding 2% signal a severe response that would demand design review. Base shear is the total horizontal force the foundation must resist.
Assumptions & limitations
- The building is idealised as a lumped-mass shear building: rigid floor diaphragms, mass concentrated at floor levels, one lateral degree of freedom per floor, and no vertical or torsional response.
- Behaviour is linear elastic throughout; there is no yielding, stiffness degradation, P-delta effect, or soil-structure interaction, so results are a demand estimate rather than a collapse prediction.
- Modal properties come from a generalized eigenanalysis of a realistic (tapered, non-degenerate) building; the response-spectrum relations S_a = ω²S_d and S_v = ωS_d are exact pseudo-spectral definitions.
- Damping is classical Rayleigh damping calibrated to the target ratio at the first two modes, so intermediate modes are damped somewhat less and higher modes somewhat more.
- The built-in scenarios are synthetic analytic ground motions (smooth pulses and bursts), not recorded accelerograms; use the custom-record input to replay a real motion, and remember peak responses depend on the input.
- The Newmark average-acceleration scheme (β = 1/4, γ = 1/2) is unconditionally stable, but the time step must still be small enough to resolve the highest mode of interest.
Frequently asked questions
What is the fundamental period and why does it matter?
The fundamental period T₁ is the time for one full sway of the lowest vibration mode; for the example building it is about 1.27 s (f₁ = 0.789 Hz). It matters because seismic demand is read from a response spectrum at this period — if the ground motion has strong energy near T₁, the response amplifies. It is also the main thing that changes when a building gets taller, heavier, or more flexible.
How are the natural frequencies and mode shapes found?
The tool builds the stiffness matrix K and mass matrix M, forms the symmetric matrix M^(−1/2)KM^(−1/2), and extracts its eigenvalues and eigenvectors. The square roots of the eigenvalues are the circular frequencies ω_i (and f_i = ω_i/2π), and the eigenvectors, scaled to be mass-orthonormal, are the mode shapes.
Why is the peak story-drift ratio emphasised?
Interstory drift ratio — the relative horizontal movement between adjacent floors divided by the story height — correlates far better with structural and non-structural damage than roof displacement alone. Codes cap it (often around 2%), so the tool flags a warning when the peak drift crosses that threshold.
What does the Newmark-β method actually compute?
Newmark-β is a step-by-step time-integration scheme that advances the coupled equations of motion Mü + Cẅ + Ku = −Mrü_g from one instant to the next. With β = 1/4 and γ = 1/2 (average acceleration) it is unconditionally stable and energy-preserving, so it produces the full displacement, velocity, and acceleration history of every floor rather than just a peak estimate.
What is pseudo-spectral acceleration and how does S_a = ω²S_d arise?
For a damped oscillator the peak spring force equals the peak displacement times the stiffness, which per unit mass is ω² times the spectral displacement S_d. That product, S_a = ω²S_d, is the pseudo-acceleration used in design spectra; the pseudo-velocity is S_v = ωS_d. They are near-exact for lightly damped structures and are the quantities tabulated in seismic codes.
Can I analyse a real earthquake record?
Yes. Select the custom scenario and paste a comma-separated acceleration series in g; the tool linearly interpolates it onto the integration time step and runs the same Newmark-β analysis. The built-in scenarios are smooth synthetic motions for exploring behaviour, so a recorded accelerogram gives the most realistic peak demands.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.