🌍 Earthquake Response Spectrum

The Eurocode 8 elastic and design response spectrum, the natural period of a building and the seismic base shear – the first numbers of every earthquake calculation. Everything is calculated in your browser.

Ground shaking

Reference ground acceleration a_gR [g]475-year map, rock …or your own value [g] Importance class Ground type

Structure

Natural period T [s] Building height H [m] above the foundation, max. 40 m Structural system Lateral stiffness k [kN/m] Mass m [t] all floors together Number of storeys Behaviour factor q 1.5 = no ductility Damping ξ [%]

The behaviour factor q tells how much the structure may yield and still stay standing. Eurocode 8 gives it by material and ductility class – for a concrete frame of medium ductility around 3.9, for unreinforced masonry 1.5. Use 1.5 if you are not sure.

Results

Response spectrum – acceleration of the structure [g] against its period [s]

Elastic S_e(T)Design S_d(T)Your structure

What the spectrum says

An earthquake shakes the ground back and forth. A structure does not simply follow – it swings with its own rhythm, its natural period T. The response spectrum answers one question: how strongly does a structure with period T shake during this earthquake? Very stiff structures (T close to 0) move with the ground. Structures with periods between T_B and T_C – typical low and mid-rise buildings – resonate and get about 2.5 times the ground acceleration. Tall, flexible buildings (long periods) feel less acceleration, but move more.

Soft soil makes the shaking stronger (factor S) and longer (bigger T_C). That is why the same earthquake can be very different for two houses in the same town.

How it is calculated

EN 1998-1, horizontal spectrum Type 1 with the recommended values (check the national annex for your country). The design ground acceleration is a_g = γ_I · a_gR.

Ground	S	T_B [s]	T_C [s]	T_D [s]
A	1.00	0.15	0.4	2.0
B	1.20	0.15	0.5	2.0
C	1.15	0.20	0.6	2.0
D	1.35	0.20	0.8	2.0
E	1.40	0.15	0.5	2.0

Elastic spectrum (η = √(10 / (5 + ξ)) ≥ 0.55, η = 1 for 5 % damping)
0 ≤ T ≤ TB:   Se = ag·S·[1 + T/TB·(2.5·η − 1)]
TB ≤ T ≤ TC:  Se = ag·S·2.5·η
TC ≤ T ≤ TD:  Se = ag·S·2.5·η·TC/T
TD ≤ T ≤ 4 s: Se = ag·S·2.5·η·TC·TD/T²

Design spectrum (divided by q, never below β·ag with β = 0.2)
0 ≤ T ≤ TB:   Sd = ag·S·[2/3 + T/TB·(2.5/q − 2/3)]
TB ≤ T ≤ TC:  Sd = ag·S·2.5/q
TC ≤ T ≤ TD:  Sd = max(ag·S·2.5/q·TC/T, β·ag)
TD ≤ T:       Sd = max(ag·S·2.5/q·TC·TD/T², β·ag)

Period: T = 2π·√(m/k) for one mass on a spring, or the Eurocode estimate T₁ = C_t·H^3/4 for buildings up to 40 m. Base shear (lateral force method): F_b = S_d(T₁) · m · λ, with λ = 0.85 when T₁ ≤ 2·T_C and the building has more than two storeys, otherwise λ = 1.0. The displacement shown is the elastic displacement of a single mass, S_e·(T/2π)².

The background – resonance, damping, ductility and a worked example – is in my note Structural Dynamics and Earthquakes. The official Slovenian map is published by ARSO, and more data and programs are in my Structural Dynamics & Earthquakes resources.

For learning and quick checks only. No torsion, no vertical component, no combination with other loads, no detailing rules. Real structures must be designed by a licensed engineer.