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Seismic Engineering
Seismic Response Spectrum (Sa/Sv/Sd) Simulator
The response spectrum is the core tool of earthquake-resistant design: for a structure with period T and damping zeta, it gives the maximum acceleration Sa(T), pseudo-velocity Sv(T) and pseudo-displacement Sd(T) experienced during an earthquake. Change site class, return period and code shape (Eurocode 8 / IBC / Japanese) to see the design base shear and required ductility move in real time.
Parameters
Peak Ground Acceleration PGA
g
Bedrock peak ground acceleration of the design event
Peak Ground Velocity PGV
cm/s
Site class
Soil amplification and spectrum shape
Return period T_R
yr
475 yr = design EQ, 2475 yr = MCE
Structural period T
s
1st-mode period; ~1 s for a 10-story RC building
Damping ratio ζ
RC=0.05, pier=0.02, isolation=0.20-0.30
Code standard
Reference spectrum shape
Results
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Site PGA (g)
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Sa (T) (g)
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Pseudo-velocity Sv (cm/s)
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Pseudo-displacement Sd (cm)
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Design base shear (g)
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Return-period adj. Sa (g)
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Response spectrum + SDOF building response
Top: T vs Sa response spectrum with the current T marker. Bottom: SDOF pendulum-building shaken by the ground motion at the current T and zeta. Colour indicates Sa level (green to orange to red).
a_g = PGA in g, eta = damping correction, S = soil factor; shape is the period-dependent factor over short / medium / long bands. Sv and Sd are pseudo-spectra.
Eurocode 8 Type 1 spectrum. T_B / T_C / T_D depend on site class; zeta is the damping ratio. Short period = constant acceleration, medium = constant velocity, long = constant displacement.
Seismic Response Spectrum — Design Use of Sa/Sv/Sd
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Every earthquake-engineering textbook talks about the "response spectrum". What does that curve actually represent? It is not the same as Mercalli intensity, right?
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Right, those are very different things. Intensity (MMI or JMA) is a felt scale - what people and buildings experience. The response spectrum is far more engineering-oriented: take an infinite family of pendulums with different periods T and damping zeta, hit them all with the same ground motion, and plot each one's peak response against its period. The horizontal axis is the building's natural period; the vertical axis is Sa, the peak acceleration. For a 10-story RC building with T = 1 s, you read Sa straight off the curve, multiply by floor weights, and you immediately have the design base shear.
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OK, the default case (PGA 0.3 g, site B rock, T = 1.0 s) gives me Sa = 0.375 g. What does that number mean in practice?
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The ground itself peaks at 0.3 g. Multiply by the site amplification (S = 1.0 for class B), feed it into the Eurocode 8 shape, and a structure with T = 1.0 s gets shaken at 0.375 g. So the building's roof acceleration is 25% higher than the ground - because T = 1.0 s falls in the medium-period band (T_C = 0.5 s to T_D = 2.0 s) where pseudo-velocity is nearly constant. Drag T down to 0.3 s with the slider and Sa jumps to 0.75 g on the constant-acceleration plateau.
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When I switch site class from B to E (soft soil) Sa shoots up by ~1.7x. Does the soil really matter that much?
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It really does - it is fair to say the soil shapes the spectrum. Soft sediments are a natural filter that amplifies the bedrock motion 1.5 to 3x and pushes the spectral peak toward longer periods. The textbook case is the 1985 Mexico City earthquake: 400 km from the epicentre, only 8-18 story buildings on the old lake-bed clay collapsed, because the soil's dominant period of about 2 s resonated with their fundamental periods. The same buildings on hard rock would have been fine. If you don't take Vs30 or SPT-N seriously, your design force can be off by a factor of two.
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Damping must matter too, then. I've heard isolated buildings use zeta = 0.20-0.30.
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Look at eta = sqrt(0.10 / (0.05 + zeta)): at the standard zeta = 0.05 you get eta = 1.0; push zeta to 0.20 and eta drops to about 0.63, knocking 37% off Sa. So lead-rubber bearings and viscous dampers cut the force transmitted into the superstructure by adding damping. Isolation also lengthens the period to 3-4 s, moving the building into the constant-displacement tail where Sa is roughly 1/4 of the fixed-base value. The "longer period plus higher damping" double benefit is exactly why base isolation works.
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And changing the return period from 475 yr to 2475 yr bumps Sa by ~1.7x. Which one do designers actually use?
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Both - that is the heart of performance-based earthquake engineering. Ordinary buildings are designed elastically for the "repairable damage" level at T_R = 475 yr (10% in 50 yr), and checked against collapse at the MCE level T_R = 2475 yr (2% in 50 yr). Critical facilities like nuclear plants or hospitals go all the way to the SSE at ~10,000 yr. Because Sa scales as roughly T_R^(1/3), making T_R five times longer multiplies Sa by ~1.7. This tool uses returnFactor = (T_R/475)^0.33 to let you slide through that range and feel the trade-off.
Frequently asked questions
A response spectrum plots the maximum response of an infinite family of single-degree-of-freedom (SDOF) oscillators with period T and damping ratio zeta when the same ground motion is applied. Sa is the maximum acceleration (inertia force when multiplied by mass = base shear), Sv is the maximum relative velocity and Sd is the maximum relative displacement. They are related approximately by Sv = Sa * T/(2*pi) and Sd = Sa * T^2/(4*pi^2) (pseudo-spectra), and design practice uses the Sa curve to read both lateral force and drift demand.
Relative to site class A (hard rock), soft soils at site class E amplify the PGA by roughly 1.7x and shift the spectral peak toward longer periods. Eurocode 8 captures this through soil factor S; this tool uses S = 0.8/1.0/1.2/1.4/1.7 for A/B/C/D/E. In practice the class is estimated from Vs30 or SPT-N, and tall RC buildings or base-isolated structures on soft soil require especially careful checking of the long-period part of the spectrum.
The Eurocode 8 damping correction is eta = sqrt(0.10 / (0.05 + zeta)). With the standard zeta = 0.05, eta = 1.0; with zeta = 0.02, eta is about 1.20 (spectrum 20% higher); with zeta = 0.10, eta is about 0.82; with zeta = 0.20, eta is about 0.63. Typical values are zeta = 0.02 for chimneys and bridge piers, 0.05 for RC buildings, 0.10 for dams and 0.20-0.30 for base-isolated buildings. The 30-40% Sa reduction at high zeta is the essence of base-isolated and damped designs.
Return period T_R is the average interval between earthquakes of a given size. The standard design earthquake is T_R = 475 yr (10% exceedance in 50 yr); for safety checks T_R = 2475 yr (2% in 50 yr, the MCE) is also used. Sa scales roughly as T_R^(1/3), so going from 475 to 2475 yr increases Sa by about 1.74x, and 50 to 475 yr by 0.46x. This tool applies returnFactor = (T_R/475)^0.33 to produce the return-period adjusted Sa.
Real-world applications
Code base-shear calculations: Japan's Building Standard Law, the US IBC (ASCE 7), Eurocode 8 and China's GB 50011 all start from a response spectrum. The fundamental period T_1 is estimated from equivalent mass and stiffness, Sa(T_1) is read off the design spectrum, and floor weights are multiplied to give the story shears. A response modification factor R (Ds in Japan) accounts for inelastic action to give the final design base shear.
Base isolation and supplemental damping: Lead-rubber or friction-pendulum isolators stretch the period to 3-4 s and lift damping to 0.20-0.30. On the spectrum this is a "move right and drop down" trajectory, cutting the Sa transmitted to the superstructure by a factor of 3-5. Drag T from 0.5 s up to 3.5 s and zeta from 0.05 to 0.25 in this tool to feel that dramatic reduction.
Performance-based design (PBEE) tiers: FEMA P-58 and Japan's limit-state method use spectra at multiple return periods - Service Level (T_R = 43 yr), Design Basis Earthquake (475 yr), Maximum Considered Earthquake (2475 yr) - and check different objectives (continued operation, repairable damage, life safety) at each level. The return-period slider here lets you simulate that multi-level check.
Pairing with hazard maps: USGS, Japan's J-SHIS and the National Annexes to Eurocode 8 publish site-specific PGA, Sa(0.2 s) and Sa(1.0 s). Feeding those values into the PGA input here gives a quick site-specific response spectrum. Detailed design moves on to nonlinear time-history analysis, but a spectral check like this is more than enough at concept stage.
Common misconceptions and pitfalls
The biggest trap is treating the code spectrum as if it were a real ground motion. A design spectrum is the statistical envelope of many recorded earthquakes; the spectrum of any single record (e.g. 1995 Hyogoken-Nanbu JMA Kobe NS, 2011 Tohoku K-NET Sendai, 2016 Kumamoto KMMH16) shows sharp local peaks at its dominant periods that can exceed the code curve in a narrow band. For important structures you have to run nonlinear time-history analyses with several real records and look at the scatter.
Second, do not confuse the pseudo spectrum with the true spectrum. This tool computes Sv = Sa * T/(2*pi) and Sd = Sa * T^2/(4*pi^2), the pseudo-spectra. At zeta = 0.05 they match the true peak velocity and displacement very well, but at zeta > 0.20 (isolated buildings) or zeta < 0.005 (very long-period structures), Duhamel-integrated true spectra are noticeably different. Design practice absorbs the gap inside a 5-10% safety margin, but it matters in research or precise evaluation.
Third, never assume linear SDOF response equals real-building response. The spectrum method assumes an elastic SDOF; it does not include inelasticity, multiple modes, torsion or P-Delta effects. Response modification factors R = 2-8 approximate the first, and modal combinations (SRSS, CQC) the second, but pilotis buildings, eccentric structures and supertalls can violate spectrum-method assumptions significantly. Events such as the 2007 Niigata-Chuetsu-oki record (1-2.6 g at Kashiwazaki-Kariwa), the long-period 2011 Tohoku motion that swung Osaka WTC by 2.7 m, and the 2024 Noto near-fault pulse all drove updates to code methods because they exposed phenomena the response-spectrum approach alone cannot capture.
How to Use
Enter site Peak Ground Acceleration (PGA) in g-units (typical range 0.1–0.8g for seismic zones) and peak ground velocity in cm/s.
Specify structural period T in seconds (e.g., T=0.5s for stiff frame, T=2.5s for flexible building) and damping ratio (typically 5% for reinforced concrete).
Set design return period (e.g., 475 years for Design Basis Earthquake) to adjust spectral acceleration; simulator computes Sa, Sv, Sd, and equivalent base shear coefficient.
Worked Example
For a 6-story RC office building on firm soil: PGA=0.35g, PGV=25 cm/s, structural period T=0.8s, damping=5%. The simulator returns Sa(0.8s)=0.72g, Sv=22.5 cm/s, Sd=5.8 cm. Design base shear = 0.72g × total mass. If return period increases from 475 to 2475 years, Sa amplifies to 0.91g, raising code-required shear by 26%.
Practical Notes
Use USGS seismic hazard maps or local building code spectra to calibrate PGA input; Japan (0.6g) and California (0.4g+) differ significantly.
Higher damping (10–15%) is valid for damped systems or tuned mass dampers; reduces Sa by 20–40% compared to 5% baseline.
Return-period adjustment reflects risk tolerance: critical facilities (hospitals, data centers) often design for 2475-year events; ordinary buildings use 475-year.