Threshold Load Factor for Liquefaction Triggering Evaluations

The $V_s$-based chart developed by Andrus and Stokoe (2000) becomes less conservative when a constant cyclic shear strain, ${\gamma }_c$, is used for liquefaction triggering (Dobry et al. 2015), where $V_s$ = shear wave velocity. Specifically, natural silty sands in the Imperial Valley of California, heavily preshaken by earthquakes, have a triggering resistance for a given normalized $V_s$ significantly higher than uncompacted clean and silty sand fills. A higher ${\gamma }_c$ is required for triggering in the natural sands than in the fills (0.1–0.2% versus 0.03%). Dobry et al. (2015) reached this conclusion within the usual liquefaction chart format, plotting $\mathrm{CSR}={\tau }_c/{\sigma }_{v0}^{\prime }=0.65(a_{\text{max}}/g)({\sigma }_{v0}/{\sigma }_{v0}^{\prime })r_d$ versus normalized shear wave velocity, $V_{s1}=V_s{(100/{\sigma }_{v0}^{\prime })}^{0.25}$ (${\sigma }_{v0}^{\prime }$ in kPa). The next logical step would be to plot ${\gamma }_c$ directly versus $V_{s1}$. However, using ${\gamma }_c$ is not practical for several reasons. The authors propose instead use of the threshold load factor (TLF)where ${\tau }_c={\sigma }_{v0}^{\prime }$ CSR; and ${\tau }_{tv}$ = volumetric threshold shear stress needed to start buildup of excess pore pressures. TLF measures how much bigger the earthquake demand, ${\tau }_c$, is compared to the shear stress ${\tau }_{tv}$ needed for the development of any excess pore pressure. The denominator in Eq. (1), ${\tau }_{tv}$, is directly related to the volumetric threshold shear strain, ${\gamma }_{tv}$, which is notably constant and equal to about $0.01\%=1\times {10}^{-4}\mathrm{m}/\mathrm{m}$ in most clean and silty sands of interest. That is, ${\tau }_{tv}=G{\gamma }_{tv}=G_{\text{max}}{(G/G_{\text{max}})}_{tv}{\gamma }_{tv}$, where ${(G/G_{\text{max}})}_{tv}\approx 0.75$ is a representative value of the modulus reduction curve in sands for a shear strain, ${\gamma }_c={\gamma }_{tv}=1\times {10}^{-4}\mathrm{m}/\mathrm{m}$. Also, $G_{\text{max}}=\rho V_s^2$, where $\rho$ = saturated soil density. In first approximation, ${\tau }_{tv}=0.75\times {10}^{-4}\rho V_s^2$. Finally

As ${\tau }_c=G{\gamma }_c=G_{\text{max}}{(G/G_{\text{max}})}_c{\gamma }_c$, then ${\tau }_c=\rho V_s^2{(G/G_{\text{max}})}_c{\gamma }_c$; this expression combined with Eqs. (1) and (2) gives $\mathrm{TLF}=\mathrm{13,333}{(G/G_{\text{max}})}_c{\gamma }_c$, which is a one-to-one relationship between TLF and ${\gamma }_c$. This confirms that TLF and ${\gamma }_c$ are two sides of the same coin. Fig. 1 shows the plot of TLF versus $V_{s1}$ for an earthquake magnitude, $M_w=7.5$, for the same case histories of natural sands in the Imperial Valley already plotted as CSR versus $V_{s1}$ by Dobry et al. (2015). All TLF were calculated with Eq. (2) using the same parameters listed by Dobry et al. (2015). Fig. 1 is consistent with the assumption that $\mathrm{TLF}={(\mathrm{TLF})}_l=3.0$ needed for triggering in the Imperial Valley soils for $M_w=7.5$ is independent of $V_{s1}$. The dashed line corresponds to ${(\mathrm{TLF})}_l=2.0$, computed in a similar way as used for the uncompacted fills. As expected, ${(\mathrm{TLF})}_l$ in the Imperial Valley is significantly greater than for uncompacted fills.