# Cyclic Stress Approach In Earthquake Engineering

In the cyclic stress approach, both the loading imposed on the soil by the earthquake and the resistance of the soil to liquefaction are characterized in terms of cyclic shear stresses. By characterizing both loading and resistance in common terms, they can be directly compared to determine the potential for liquefaction. For this approach, estimation of two variables is needed in order to evaluate the potential for liquefaction. These two variables are the seismic (or loading) demand on a soil layer expressed in terms of the Cyclic Stress Ratio (CSR) induced by an earthquake (or other loading) and the capacity of the soil to resist liquefaction expressed in terms of the Cyclic Resistance Ratio (CRR), which is simply the value of the CSR required to cause liquefaction.

Characterization of Loading: Loading is characterized in terms of the cyclic stress ratio, CSR, which is defined as the ratio of the equivalent cyclic shear stress, τcyc, to the initial vertical effective stress σ_v^'. CSR=τ_cyc/(σ_v^' )Seed and Idris (1971) performed numerous tests to determine the value of τcyc in terms of other variables that can be readily obtained. Based on their testing, the equivalent cyclic shear stress is assumed to be equal to 65% of the peak cyclic shear stress. Seed and Idris arrived at this value by comparing rates of porewater pressure generation caused by transient earthquake shear stress histories with rates caused by uniform harmonic shear stress histories. Seed and Idris stated, “the factor was intended to allow an adequate comparison of a transient shear stress history from an earthquake of a magnitude M with that of N cycles of harmonic motion of amplitude 0.65τ_max, where N is an equivalent number of cycles of harmonic motion.” Utilizing graphs showing Neq at 0.65τ_max versus Earthquake Magnitude M, shear stress versus time, and depth versus the stress reduction factor, rd, Idris and Seed showed that the maximum shear stress can be expressed as: τ_max=a_max/g σ_v r_dwhere a_max is the peak ground surface acceleration, g is the acceleration of gravity, σ_v the total vertical stress, and rd the value of the stress reduction factor at the depth of interest. These values can all be readily obtained from either past events or present day in-situ testing. The CSR can consequently be expressed as:CSR=0.65 a_max/g σ_v/(σ_v^' ) r_d Note that in this procedure, the earthquake-induced loading was characterized by a level of uniform cyclic shear stress that is applied for an equivalent number of cycles. This is to allow a comparison between earthquake-induced loading with laboratory-determined resistance since this is estimated from tests in which the cyclic shear stresses have uniform amplitude.

Characterization of Resistance: As detailed previously, the liquefaction resistance of an element of soil depends on how close its initial state is to the state corresponding to “failure” and on the nature required to move it from the initial state to the failure state. Flow liquefaction and cyclic mobility are both loaded differently and fail differently. However, when the cyclic stress approach was developed, the distinction between flow liquefaction and cyclic mobility was not well defined. Rather, liquefaction in general was the goal and consequently this approach does not have separate cases for each type of liquefaction. Characterization of liquefaction resistance can be determined based on results either from lab tests or in-situ tests and observations of liquefaction behavior in past earthquakes. A couple of laboratory tests were significant in developing an equation for the CRR. Figure 18 shows that the number of loading cycles required to produce liquefaction failure, NL, decreases with increasing shear stress amplitude and with decreasing density. In figure 18a, the loose sand reaches initial liquefaction after 9 cycles while in figure 18b the dense sand, with a much greater amplitude loading, does not even reach initial liquefaction after 16 cycles. Liquefaction failure may occur in a loose specimen in just a few cycles, but for a dense specimen it might take thousands of cycles. Correlations between density, cyclic stress amplitude, and the number of cycles to liquefaction failure can be expressed graphically by laboratory cyclic strength curves, such as those shown in figure 19.

Lab curves such as those in figure 19 aid in tests to determine the CRR value. The CRR value is defined differently for each test. For the cyclic simple shear test, the CRR is the ratio of the cyclic shear stress to the initial vertical effective stress, 〖CRR〗_ss=τ_cyc/(σ_v^' ). For the cyclical triaxial test, the CRR is the ratio of the maximum cyclic shear stress to the initial effective confining pressure, 〖CRR〗_tx=σ_dc/(〖2σ〗_3c^' ). The two laboratory tests are related by the formula, 〖CRR〗_ss=c_r 〖CRR〗_tx, where cr is a tabulated correction factor. Yet the CSR required to produce initial liquefaction in the field (CRR value) is not the same as that required in these laboratory cyclic shear tests. Earthquakes produce shear stresses in different directions, as opposed to the laboratory cyclic simple shear test and cyclic triaxial test. This shaking in multiple directions has been shown experimentally to cause more pore water pressure increases than the unidirectional shaking experienced in the lab. The CSR required to produce initial liquefaction in the field is estimated to be about 10% less than that required in the unidirectional cyclic simple shear test. Therefore the liquefaction resistance of an element of soil in the field is:〖CRR〗_field=τ_cyc/(σ_v^' )=0.9〖CRR〗_ss=0.9c_r 〖CSR〗_tx Another approach to determine the CRR is through in-situ testing and utilizing case histories. Whitman (1971) first proposed that liquefaction case histories be used to characterize liquefaction resistance in terms of measured in-situ test parameters. Previous case histories can be characterized by the combination of a loading parameter, L, and a liquefaction resistance parameter, R, which can be plotted on a graph such as that shown in figure 20 to determine a boundary for whether or not liquefaction was observed. For the cyclic stress approach, the loading parameter is usually the CRR while the resistance parameter is the in-situ test parameters that affect the density and generation of pore pressures.

The in-situ test used most often in the United States is the Standard Penetration Test. The SPT is so useful because the factors that tend to increase liquefaction (density, prior seismic straining, overconsolidation ratio, lateral earth pressures, time under sustained pressure, etc.) also tend to increase the SPT resistance. Seed (1983) corrected the SPT resistance and compared this to the cyclic stress ratio for clean sand (figure 21) and silty sand (figure 22) sites at which liquefaction was both observed and not observed in earthquakes of magnitude M=7.5 in order to determine the minimum cyclic stress ratio at which liquefaction could be expected in a sand of a given SPT resistance. These figures are very powerful because a boundary line is formed between where liquefaction was observed and not observed at particular sites when subjected to the same magnitude earthquake. With very few exceptions, liquefaction was observed to occur only to the left or on top of the boundary line. The boundary line serves as either the minimum value of the CSR required to produce liquefaction (CRR) for a given (N1)60 value or the minimum value of (N1)60 required to resist liquefaction. Liquefaction observed to the left of the boundary line is reasonable because fewer blow counts means lower strength soil just as liquefaction observed above the boundary line makes sense because a higher CRR means that the equivalent cyclic shear stress is greater and hence the soil is more prone to liquefaction.

Another in-situ test, the Cone Penetrometer Test, has an advantage over the SPT in its ability to detect thin seams of loose soil. The tip resistance of the CPT is used as a measure of liquefaction resistance. However, because the data of samples at which CPT resistance was measured and where the occurrence or nonoccurrence of liquefaction was noted is small, there are two ways to generate a graph of CSR versus the normalized cone resistance, qc1. Seed supplemented this lack of data with correlations between CPT and SPT resistances. Using these correlations, the cyclic stress ratio that triggers liquefaction in a clean sand can be determined and is shown in figure 23a. Since the SPT-CPT correlations depend on grain size, CPT-based liquefaction curves have been developed for different grain sizes. These curves, like that of the SPT curves in figures 21 and 22 serve as a boundary separating whether liquefaction can be expected or not. Figure 23a shows a grain size of ≤5% fines. Table 2 shows a correlation factor to account for the grain size. Mitchell and Tseng (1990) produced CPT based liquefaction curves based on lab tests and theoretically derived values of CPT resistance (figure 23b). As in the case for the SPT, anything plotting above and to the left of the given curve experiences liquefaction while below and to the right of the curve experiences no liquefaction. Figure 23: CPT based liquefaction curves: (a) based on correlations with SPT data, (b) based on theoretical/experimental results

A shear Wave Velocity or Dilatometer Index are in-situ tests that can also be used to determine whether or not liquefaction occurs and what the CRR value is. However, these tests are not nearly used as often as the SPT and CPT. SPT resistance is by far the most predominant in-situ test parameter for characterization of liquefaction resistance. The SPT allows a sample to be retrieved for further examination and has the largest case history of any in-situ test available. However, because the CPT is faster, less expensive, and has a continuous record of penetration resistance, the CPT is becoming more popular.