Module 3. DYNAMIC SOIL PROPERTIES (Lectures 10 to 16)

Module 3 DYNAMIC SOIL PROPERTIES (Lectures 10 to 16) Lecture 16 Topics 3.9 DAMPING RATIO 3.10 CYCLIC NONLINEAR MODELS 3.11 ADVANCED CONSTITUTIVE MODELS 3.12 STRENGTH OF CYCLICALLY LOADED SOILS 3.12.1 Definitions of failure 3.12.2 Cyclic strength 3.12.3 Monotonic strength 3.9 DAMPING RATIO Theoretically, no hysteretic dissipation of energy takes place at strain below the linear cyclic threshold shear strain. Experimental evidence, however, shows that some energy is dissipated even at very low strain levels (the mechanism is not well understood), so the damping ratio is never zero. above the threshold strain, the breadth of the hysteresis loops exhibited by a cyclically loaded soil increase with increasing cyclic strain amplitude, which indicates that the damping ratio increases with increasing strain amplitude. Damping ratios of highly plastic soils are lower than those of low plasticity soils at the same cyclic strain amplitude (figure 3.50) The damping behavior of gravel is very similar to that of sand (Seed et al., 1984). Dept. of Civil Engg. Indian Institute of Technology, Kanpur 1

Figure 3.50: Variation of damping ratio of fine-grained soil with cyclic shear strain amplitude and plasticity index. (After Vucetic and Dobry 1991) Damping behavior is also influenced by effective confining pressure, particularly for soils of low plasticity, Ishibashi and Zhang (1993) developed an empirical expression for the damping ratio of plastic and no plastic soils. Using (equation 3.52) to compute the modulus reduction factor,, the damping ratio is given by [ ( ) ] (3.54) The influence of various environmental and loading conditions on the damping ratio of normally consolidated and moderately over consolidated soils is described in (table 3.9). Table 3.8 Effect of Environmental and Loading Conditions on Modulus Ratio (at a Given Strain Level) of Normally Consolidated and Moderately Over consolidated Soils (from Dobry and Vucetic 1987). Increasing Factor Confining pressure, Increases with ; effect decreases with increasing PI Void ratio, Increases with Geologic age, May increase with Cementation, May increase with Over consolidation ratio, OCR Not affected Plasticity index, PI Increases with PI Cyclic strain, Decreases with Strain rate, G increases with but / probably not affected if G and Dept. of Civil Engg. Indian Institute of Technology, Kanpur 2

Number of loading cycles, N are measured at same Decreases after N cycles of large ( measured before N cycles) for clays; for sands, can increase (under drained conditions) decrease (under undrained conditions) 3.10 CYCLIC NONLINEAR MODELS The nonlinear stress-strain behavior of soils can be represented more accurately by cyclic non-linear models that follow the actual stress-strain path during cyclic loading. Such models are able to represent the shear strength of the soil, and with an appropriate pore pressure generation model, changes in effective stress during undrained cyclic loading. A variety of cyclic nonlinear models have been developed; all are characterized by (1) a backbone curve and (2) a series of rules that govern unloading-reloading behavior, stiffness degradation, and other effects. The simplest of these models have relatively simple backbone curves and only a few basic rules. The applicability of cyclic nonlinear models, however, is generally restricted to a fairly narrow, albeit important range of initial conditions and stress paths. The performance of cyclic nonlinear models can be illustrated by a very simple example in which the shape of the backbone curve is described by. The shape of any backbone curve is tied to two parameters, the initial (low-strain) stiffness and the (high-strain) shear strength of the soil. For the simple example, the backbone function,, can be described by a hyperbola (3.55) Table 3.9 Effects of Environmental and Loading Conditions on Damping Ratio of Normally and Moderately Overconsolidated Soils (from Dobry and Vucetic 1987). Increasing Factor Confining pressure, Decreases with ; effect decreases with increasing PI Void ratio, Decreases with Geologic age, Decrease with Cementation, May decrease with Overconsolidation ratio, OCR Not affected Plasticity index, PI Decreases with PI Cyclic strain, Increases with Strain rate, Stays constant or may increase with Number of loading cycles, N Not significant for moderate and N The shape of the hyperbola backbone curve is illustrated in (figure 3.51). Other expressions (e.g., the RAmberg-Osgood model (Ramberg and Osgood, 1943) can also be used to describe the backbone curve. Alternatively, backbone curves can be Dept. of Civil Engg. Indian Institute of Technology, Kanpur 3

constructed from modulus reduction curves. Figure 3.51: Hyperbolic backbone curve asymptotic to The quantities may be measured directly, computed, or obtained by empirical correlation. For the example model, the response of the soil to cyclic loading is governed by the following four rules: 1. For initial loading, the stress-strain curve follows the backbone curve. 2. If a stress reversal occurs at a point defined by ( the stress-strain curve follows a path given by ( ) In other words, the unloading and reloading curves have the same shape as the backbone curve (with the origin shifted to the loading reversal point) but is enlarged by a factor of 2. These first two rules, which describe Masing behavior (Masing, 1926), are not sufficient to describe soil response under general cyclic loading. As a result, additional rules are needed. 3. If the unloading or reloading curve exceeds the maximum past strain and intersects the backbone curve, it follows the backbone curve until the next stress reversal. 4. If an unloading or reloading curve crosses an unloading or reloading curve from the previous cycle, the stress-strain curve follows that of the previous cycle. Models that follow these four rules are often called extended Masing models. An example of the extended Masing model is shown in (figure 3.52). Cyclic loading begins at point A, and the stress-strain curve during initial loading (from A to B) follows the backbone curve as required by rule 1. At point B, the loading is reversed and the unloading portion of the stress-strain curve moves away from B along the path required by rule 2. Note that the initial unloading modulus is equal to. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 4

The unloading path intersects the backbone curve at point C, and according to rule 3, continues along the backbone curve until the next loading reversal at point D. the reloading curve then moves away from D as required by rule 2, and the process is repeated for the remainder of the applied loading. Although this model is very simple and is expressed only in terms of effective stresses, it inherently incorporates the hysteretic nature of damping and the strain-dependence of the shear modulus and damping ratio. Other unloading-reloading models are available (e.g., Iwan, 1967; Finn et al., 1977; Vucetic, 1990); the Cundall-Pyke, 1979) is particularly straight forward and easily implemented into ground response analyses. To avoid spurious response at very low strain levels, some cyclic nonlinear models require the addition of a small amount of low strain damping. Note that the cyclic nonlinear model does not require the shear strain to be zero when the shear stress is zero. The ability to represent the development of permanent strains is one of the most important advantages of cyclic nonlinear models over equivalent linear models. Figure 3.52: Extended Massing rules: (a) variation of shear stress with time; (b) resulting stress-strain behavior (backbone curve indicated by dashed line) This simple example model does not, however, allow for the determination of shearinduced volumetric strains that can lead to hardening under drained conditions or to pore pressure development with attendant stiffness degradation under undrained conditions. Such factors are accounted for in the majority of the cyclic nonlinear models commonly used in geotechnical earthquake engineering practice (e.g., Finn et al., 1977; Pyke, 1979, 1985). The ability to compute changes in pore pressure, hence also changes in effective stress, represent another significant advantages of cyclic nonlinear models over equivalent linear models. As pore pressures increase, effective stresses decrease, and consequently the values of decrease. Since the shape and position of the backbone curve depends on the backbone curve degrades with increasing pore pressure. As with actual soils, the stiffness in a stress-strain model depends not only on the cyclic strain amplitude, as implied by the equivalent linear model, but also on the stress history of the soil. ADVANCED CONSTITUTIVE MODELS The most accurate and general methods for representation of soil behavior are based on advanced constitutive models that use basic principles of mechanics to describe observed soil behavior for (a) general initial stress conditions, (b) a wide variety of Dept. of Civil Engg. Indian Institute of Technology, Kanpur 5

stress paths, (c) rotating principal stress axes, (d) cyclic or monotonic loading, (e) high or low strain rates, and (f) drained or undrained conditions. Such models generally require a yield surface that describes the limiting stress conditions for which elastic behavior is observed, a hardening law that describes changes in the size and shape of the yield surface as plastic deformation occurs, and a flow rule that relates increments of plastic strain to increment of stress. The Cam- Clay (Roscoe and Schofield, 1963) and modified Cam-Clay (Roscoe and Burland, 1968) models were among the first of this type. Improvement in the prediction of shear strains have resulted from the use of multiple nested yield loci within the yield surface (Mroz, 1967; Pervost 1977) and the development of bounding surface models (Dafalias and Popov, 1979) which incorporate a smooth transition from elastic to plastic behavior. Detailed treatment of such advanced constitutive models is beyond the scope of this book. The interested reader can refer to a number of sources, including Desai and Siriwardane (1984), Defalias and Hermann (1982), Wroth and Houlsley (1985), Lade (1988), and Wood (1991). 3.11 STRENGTH OF CYCLICALLY LOADED SOILS The effect of cyclic loading on the limiting strength of soils is considerable importance in geotechnical earthquake engineering. Problems of slope stability, foundation performance, and retaining wall behavior, amount others, are strongly influenced by the strength that the soil can mobilize at large strains. 3.12.1 Definitions of failure The shear strength of an element of soil is typically defined as the shear stress mobilized at the point of failure, but failure can be defined in many different ways. In the field, failure is usually associated with deformations that exceed some serviceability limit. Since deformation results from the integration of strains over some volume of soil, the point of failure of an element of soil is often defined in terms of a limiting strain. Consider an element of soil in drained equilibrium under anisotropic stress conditions in a cyclic direct simple shear test (point A in figure 3.53). The application of a cyclic shear stress,, produces (under stress-controlled conditions) a cyclic shear strain,, but also an increase in the average strain,. The average shear strain increases with increasing numbers of loading cycles. Clearly the strength of the soil during cyclic loading could be defined in terms of limiting values of or or of some combination of the two. The available strength of the sol under monotonic loading (after the cyclic loading has ended) may also be of interest. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 6

Figure 3.53: Definition of average and cyclic shear stress and shear strain (After Goulois et al., 1985) 3.12.2 Cyclic strength The levels of both cycles and permanent deformations are of interest in a number of geotechnical earthquake engineering problems. They are also important in the design of foundations for marine structures subjected to wave loading, and much of the current state of knowledge of cyclic strength has come from research in that area. The cyclic strength of an element of soil depends on the relationship between the average shear stress,, and the cyclic shear stress,. When the average shear stress is low, unidirectional strains will accumulate slowly, so the average shear strain will also be low. The amplitude of the cyclic strain, however, may become large if the cyclic shear stress is large. If, on the other hand, the average shear stress is high (relative to the static shear strength, ), substantial unidirectional strains can develop even when the cyclic shear stress is small. For the case of, no unidirectional strain will develop, so failure must be defined in terms of the cyclic shear strain,. When failure is defined in terms of a specific level of cyclic shear strain (often 3 percent), the cyclic strength ratio, defined as, decreases with increasing numbers of cycles as shown in (figure 3.54). Dept. of Civil Engg. Indian Institute of Technology, Kanpur 7

Figure 3.54: Variation of cyclic strength ratio with number of cycles for different soils. (After Lee and Focht, 1976) For cases in which is greater than zero, both, will depend on and (Seed and Chan, 1966). Investigations of the cyclic response of marine clays (e.g., Meimon and Hicher, 1980; Goulois et al. 1985; Anderson et al., 1988) have shown that depends predominantly on and the number of cycles, and depends predominantly on and the numbers of cycles (figure 3.55). Figure 3.55: variation of average shear strain with average shear stress, cyclic shear stress, and number of cycles in cyclic direct simple shear tests on plastic drammen clay. (After Goulois et al., 1985.) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 8

3.12.3 Monotonic strength Evaluation of the static stability of slopes and retaining walls and the capacity of foundations after earthquake shaking has ended is another important problem in geotechnical earthquake engineering. Such problems require evaluation of the available shear strength of the soil after the earthquake has ended. This post earthquake strength must reflect any effects of cyclic loading imposed by the earthquake. The ultimate (residual, high-strain) undrained shear strength of a saturated soil is controlled by its void ratio and structure. Barring any change in soil structure, a saturated soil at a particular void ratio will mobilize a specific undrained strength, with little influence of the history of stresses and strains by which that strength is arrived at. For such soil conditions, the undrained strength after cyclic loading will be equal to the undrained strength before undrained loading (at the same strain rate). Since cyclic loading induces positive excess pore pressures, the effective stress in an element of soil sheared monotonically after being subjected to cyclic loading will be lower than that in an identical element that is sheared monotonically without prior cyclic loading. Consequently, the element that had been cycled would be expected to exhibit more delative behavior but to have a lower stiffness in the early stages of monotonic undrained loading than the element that had not been cycled. Changes in monotonic strength can be caused by disturbance of the soil structure during cyclic loading. The extent to which the structure of the soil is disturbed is influenced by the relationship between the cyclic strain amplitude and the strain at which failure occurs under monotonic loading conditions (Thiers and Seed, 1969). Substantial structural disturbance can modify the stress-strain behavior and reduce the monotonic shear strength. The six triaxial specimens shown in (figure 3.56) had similar void ratios (except specimen 6, which had a somewhat higher void ratio than the rest) at the end of consolidation. Specimen 1 was sheared monotonically immediately after consolidation, but specimens 2 to 6 were first subjected to varying levels of cyclic loading. Since the void ratios were nearly the same, the specimens would therefore be expected to have similar monotonic strengths. As shown by the stress-strain curves and stress paths, they behaved largely as would be expected. After being subjected to different levels of cyclic strain, their ultimate (large strain) strength was similar (except specimen 6, which was lower than the others). Differences in the ultimate strength can be explained by small differences in the void ratios and also by differences in the extent of structural disturbance induced by the cyclic loading. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 9

Figure 3.56: Effect of cyclic loading on subsequent monotonic undrained loading behavior of traixial specimens of slightly plastic silt: (a) stress-strain behavior; (b) effective stress path behavior. (After Castro and of Christian, 1976) Thiers and Seed (1969) found that the ultimate strength of three clays decreased by less than 10% when the cyclic strain amplitude was less than one-half of the failure strain from monotonic tests. At higher cyclic strain amplitudes, the reduction in strength was more dramatic, as illustrated in (figure 3.57). Similar results have been obtained by others (e.g., Koutsoftas, 1978; Byrne et al., 1984). Figure 3.57: Effect of peak cyclic strain on monotonic strength after cyclic loading. ( Thiers and Seed, (1969)) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 10