Earthquake design with response spectrum analysis

An earthquake doesn't push on a structure — it moves the ground under it, and the structure's own mass resists being moved. The forces depend on how the structure vibrates. Response spectrum analysis is the standard way codes turn this into design forces. This tutorial explains the method step by step: modes, the design spectrum, the behaviour factor, and combining modes.

#Step 1: how the structure vibrates

Push a building sideways and let go: it sways back and forth at its natural period T (seconds per cycle). A real structure has many ways of vibrating — modes — each with its own period and shape: the first mode sways along one axis, the next along the other, then twisting (torsion), then higher modes with the floors moving in opposite directions.

The periods come from the stiffness K and the mass M by solving Kφ = ω²Mφ, with T = 2π/ω. Stiffer structures have shorter periods; heavier ones, longer. A rough rule for frames: T ≈ 0.1 × number of storeys.

#Where the mass comes from

The mass is the weight that moves with the structure: all the dead load plus a share of the live load, because the full live load is unlikely during an earthquake. Struxt3D lumps the mass at the nodes from D + ψL, with ψ = 0.3 (Eurocode, AS/NZS) or 0.25 (the other codes).

#Mass participation

Each mode mobilises part of the total mass in each direction — its effective modal mass. Codes ask that the modes used add up to at least 90 % of the mass in each direction. Struxt3D computes the 12 lowest modes and lists each mode's period, frequency and mass ratio in X and Y, with running totals.

#Step 2: the design spectrum

A response spectrum gives the peak acceleration S_a a simple oscillator would feel during the design earthquake, for every period T. Codes give it as a smooth curve: rising at very short periods, a flat plateau, then falling as 1/T. Structures with periods on the plateau (stiff low-rise buildings) attract the most force.

Struxt3D builds the spectrum from the inputs your code uses — three families:

Codes	Inputs in Struxt3D	Spectrum
Eurocode, AS/NZS	Ground acceleration ag (AS: Hazard kp·Z) in g, Ground type A–E	EN 1998-1 3.2.2.5, Type 1: soil factor S, TB, TC, TD from Table 3.2; lower bound 0.2ag
AISC/ACI (ASCE 7), KDS, CSA	SDS and SD1 (CSA: S(0.2), S(1.0)) in g	ASCE 7 11.4.6: plateau SDS, then SD1/T, TL = 8 s
NSCP 2015	Seismic zone, Soil profile, and in zone 4 the Seismic source type and Distance to the source	Section 208 (Fig. 208-3): Ca, Cv with near-source factors Na, Nv; floors 0.11Ca and 0.8ZNv/R

#Step 3: the behaviour factor q or R

Designing a building to stay elastic in a strong earthquake would be very expensive. Instead, codes let well-detailed (ductile) structures yield in a controlled way, and divide the elastic spectrum by a behaviour factor — q in Eurocode, R in ASCE 7, KDS and NSCP, R_dR_o in NBCC, μ/S_p in AS 1170.4. The more ductile the system, the larger the factor and the smaller the design forces — but the stricter the detailing rules.

Structural system	EC q	ASCE R	KDS R	CSA RdRo	AS μ/Sp	NSCP R
Ductile moment frame	3.9	8	8	7.5	6	8.5
Ordinary moment frame	1.5	3.5	3.5	2.6	2.6	3.5
Braced frame	4	6	6	3.9	4.5	6.4
Walls or core	3	5	5	5.6	4.5	5.5
Elastic (no ductility)	1	1	1	1	1	1

#Step 4: combine the modes

For each mode, the program reads S_d(T) from the spectrum, scales the mode shape by its participation factor Γ, and gets that mode's displacements and forces. But the modal peaks don't happen at the same instant, so they can't simply be added. Struxt3D combines them by CQC (complete quadratic combination, 5 % damping), which reduces to the square root of the sum of squares when periods are well apart and adds correlation when they are close (as in symmetric buildings). Importantly, every result — including the forces at points along each member — is combined by CQC itself, not derived from combined end forces.

The results have no sign (CQC gives magnitudes), so the combinations apply them both ways. Earthquakes in X and Y act together as ±(E_X + 0.3E_Y) and ±(0.3E_X + E_Y), the 30 % rule of EN 1998-1 and ASCE 7.

#Worked example

A four-storey concrete frame, 20 × 30 m, office, Eurocode, a_g = 0.22 g, ground type C, ductile moment frame (q = 3.9). First mode in X: T₁ = 0.6 s with 80 % mass.

1. Spectrum: ground type C gives S = 1.15, T_B = 0.2 s, T_C = 0.6 s. T₁ is at the end of the plateau: S_d = a_g · S · 2.5 / q = 0.22 × 1.15 × 2.5 / 3.9 = 0.162 g.
2. Seismic weight: per floor D + 0.3L = 6.4 + 0.3 × 3.0 = 7.3 kN/m² × 600 m² = 4380 kN; four floors: 17 520 kN.
3. Base shear from mode 1: V ≈ 0.162 × 0.80 × 17 520 ≈ 2270 kN, about 13 % of the weight. The higher modes add a little through CQC.

With ASCE 7, S_DS = 0.5 g, S_D1 = 0.25 g, R = 8 and T₁ = 0.8 s: T_s = S_D1/S_DS = 0.5 s, so S_a = S_D1/T = 0.31 g, and over R: 0.039 g.

#Try it in Struxt3D

1. Open the Shear wall core or Multi-storey frame template and choose your code in What is it made of?.
2. In How is it held and loaded?, switch Earthquake on ("Response spectrum: modes combined by CQC, X and Y").
3. Choose the Structural system; its q or R is shown beside it. Enter the hazard inputs for your code (a_g and ground type, S_DS/S_D1, or the NSCP zone, soil and source).
4. Run the analysis. In Load case, the Vibration modes group lists each mode with its period and mass in X and Y. Pick Mode 1 and press Animate.
5. In the Full interface, the Modes table gives the periods, frequencies and cumulative mass ratios — check they reach 90 %.
6. Pick Seismic X to see the CQC results, and the seismic combinations for design.

#Summary

• Modes and periods come from stiffness and mass; aim for 90 % mass participation.
• The design spectrum gives acceleration by period; the plateau attracts the most force.
• The behaviour factor rewards ductility — and demands ductile detailing.
• Modal results are combined by CQC and applied ± in both directions with the 30 % rule.