Equivalent-linear method¶

RSL III implements the equivalent-linear method of Idriss-Seed (1968, 1970), the industry standard for 1D site response analysis in the frequency domain.

Basic idea¶

Real soil is non-linear: G and ξ depend on γ_max reached during the earthquake. A truly non-linear analysis solves the equation of motion step-by-step in the time domain, with complex hysteretic constitutive laws.

The equivalent-linear approach is a practical compromise:

Solve a linear analysis in the frequency domain (simple and fast)
Estimate γ_max in each layer
Update G and ξ to the "secant" corresponding to γ_eff = 0.65 · γ_max
Re-solve the linear analysis with the new G and ξ
Iterate until convergence

The γ_eff = 65% of γ_max is an empirical approximation to "represent" the average effect of non-linearity over the entire cyclic history.

Calculation scheme¶

Step 1 — Discretisation¶

The stratigraphic column is discretised into N sub-layers of thickness Δh. Criterion to avoid aliasing at high frequencies:

\[ \Delta h \leq \frac{V_s^{\min}}{4 \cdot f_{\max}} \]

with f_max = 25 Hz (default). For Vs minimum = 100 m/s → Δh ≤ 1 m.

RSL III computes Δh automatically; typically discretises into 20-50 sub-layers.

Step 2 — Transfer function¶

For vertical plane S waves in horizontally stratified columns, the transfer function H(ω) between surface and bedrock is computed with the reflection/transmission coefficients method (Haskell-Thomson, i.e. the propagator matrix method of Aki-Richards):

\[ H(\omega) = \frac{u_{\text{surface}}(\omega)}{u_{\text{bedrock}}(\omega)} \]

where each layer interface generates coefficients r (reflection) and t (transmission) as a function of acoustic impedances Z = ρ · Vs.

For N layers the transformation becomes a product of 2×2 matrices (one per layer). RSL III computes it at ~2000-4000 frequencies of the accelerogram Fourier transform.

Step 3 — Output Fourier spectrum¶

\[ F_{\text{out}}(\omega) = H(\omega) \cdot F_{\text{in}}(\omega) \]

The inverse Fourier transform gives the surface accelerogram in the time domain.

Step 4 — γ_max estimate¶

In each sub-layer, given the computed maximum shear τ_max:

\[ \gamma_{\max} = \frac{\tau_{\max}}{G_{\text{secant}}} \]

And hence:

\[ \gamma_{\text{eff}} = 0{,}65 \cdot \gamma_{\max} \]

Step 5 — Update G and ξ¶

For each sub-layer, read the degradation curves at γ_eff:

G_new = G_max · (G/G_max)(γ_eff)
ξ_new = ξ(γ_eff)

Step 6 — Convergence¶

Compare γ_eff(iter) with γ_eff(iter-1). If the maximum variation across all sub-layers is < tolerance (default 0.5%), convergence reached. Otherwise go back to step 2 with the new G and ξ.

Typically converges in 4-8 iterations for standard accelerograms.

Method limitations¶

1. Average equivalence¶

γ_eff = 65% of γ_max is a proxy for the average effect. For accelerograms with a single, very isolated peak + low tail (e.g. near-impulsive events) it overestimates non-linearity. For accelerograms with many similar-amplitude cycles (long tail) the estimate is good.

2. Strong-motion non-linearity¶

For strains γ > 1% the equivalent-linear method becomes inaccurate: heavily strained clays have asymmetric hysteretic behaviour that the secant model does not capture. For extreme shaking (unstable slopes in the Friuli / L'Aquila zone) use non-linear SRA in the time domain.

3. 1D vertical¶

Assumes vertical propagation of S waves. Does not capture:

2D / 3D topography effects (ridges, valleys)
Sedimentary basin effects (lateral reflections)
Surface waves (Rayleigh, Love)

For sites with significant topography or marked basin geometry, pair with 2D analysis using Geostru RSL 2D, our dedicated software for two-dimensional site response.

4. Horizontal bedrock¶

The bedrock-soil interface is assumed horizontal. For strongly inclined bedrock (e.g. slope accumulations) the 1D representation is inadequate.

When to use RSL III (and when not)¶

✅ Suitable:

Seismic microzonation Level 3 studies (ICMS)
NTC spectrum vs site spectrum verification for standard design
Horizontally stratified granular/cohesive masses with known Vs
Moderate intensity seismic events (a_max ≤ 0.4 g, γ_max < 1%)

❌ Not suitable (or requires integration with other tools):

Studies on slopes or ridges (need 2D topographic FA)
Real-time liquefaction analysis (RSL estimates γ_max but not pore-pressure build-up — use LiquiTer for liquefaction)
Highly non-linear events (a_max > 0.6 g, γ > 2%)
Deep sedimentary basins with 2D/3D effects

References¶

Idriss I.M., Seed H.B. (1968) — Seismic response of horizontal soil layers. JSMFD, ASCE
Schnabel P.B., Lysmer J., Seed H.B. (1972) — SHAKE: A computer program for earthquake response analysis of horizontally layered sites. EERC Report, UC Berkeley
Bardet J.P., Tobita T. (2001) — NERA / EERA: Equivalent-linear earthquake site response analyses
Kramer S.L. (1996) — Geotechnical Earthquake Engineering. Prentice Hall
ICMS 2008/2018 — Indirizzi e Criteri per la Microzonazione Sismica, Italian Department of Civil Protection