Cyclic Stress Ratio for Liquefaction

Loose saturated sand shaken hard enough stops being a solid. Each cycle of shear tries to compact the grains, the water in the pores cannot escape fast enough, so it takes up the load instead — pore pressure climbs, effective stress falls, and when effective stress reaches zero the grains are no longer touching under load and the deposit behaves as a heavy liquid. Buildings settle or tip over intact, buried tanks and pipes float upward, and the ground itself can flow on a slope of a degree or two. Niigata and Anchorage, both in 1964, are where the profession learned this properly.

H. Bolton Seed and Izzat Idriss's 1971 simplified procedure turned that into an engineering check, and the cyclic stress ratio is its demand half: $\mathrm{CSR} = 0.65\,(a_{max}/g)(\sigma_v/\sigma'_v)\,r_d$. Read it left to right. The 0.65 converts an irregular earthquake record into an equivalent number of uniform cycles at 65% of the peak stress — a judgement call from 1971 that has survived because it works, not a derived constant. The $a_{max}/g$ term is the horizontal inertia of the soil column above, from Newton's second law. The $\sigma_v/\sigma'_v$ ratio normalises the shear stress by the effective stress holding the grains together, because it is that ratio, not the absolute stress, that decides whether the sand loses contact. And $r_d$ accounts for the column being deformable rather than rigid.

That last term deserves its own paragraph, because it is empirical and often treated as though it were not. $r_d$ is 1 at the ground surface by definition — there is no column above to deform — and falls with depth: roughly 0.9 at 10 m and 0.6 or so at 25 m. The original linear approximation $r_d \approx 1 - 0.00765z$ with $z$ in metres is still widely used in the top 9 m, and later formulations by Idriss and by Cetin and Seed add a magnitude dependence and considerable scatter. It is an input on this page rather than a buried constant precisely because burying somebody's fitted curve inside the arithmetic is how a site-specific correlation quietly becomes an invisible assumption.

Finally, CSR is only half the assessment. It is the demand; the capacity is the cyclic resistance ratio, CRR, obtained from an SPT blow count, a CPT tip resistance or a shear-wave velocity through case-history correlations, corrected for overburden and fines content and scaled for earthquake magnitude — because a magnitude 7.5 delivers about fifteen equivalent cycles while a magnitude 6 delivers about five. Liquefaction is predicted when CSR exceeds CRR, and the factor of safety is their ratio. Three things also fall outside this arithmetic entirely and must be judged separately: whether the soil is susceptible at all (clean loose sands and non-plastic silts are, plastic clays essentially are not), whether the deposit is saturated, and what the CONSEQUENCE of liquefaction would be — a metre of settlement under a warehouse is a repair, whereas the same metre under a bridge pier or a tailings dam is not.