For soils the stress-strain behaviour of most interest in earthquakes is that involving shear, and, except for competent rock, engineering soils behave in a markedly non-linear fashion in the stress range of interest.
For small strains the shear modulus of a soil can be taken as the mean slope of the stress-strain curve. At large strains the stress-strain curve becomes markedly non-linear so that the shear modulus is far from constant but is dependent on the magnitude of the shear strain (Figure 5.1).
There are various field and laboratory methods available for finding the shear modulus G of soils. Field tests may be used for finding the shear-wave velocity, vs, and calculating the maximum shear modulus from the relationship
where p is the mass density of the soil. Typical values of vs and p are given in Tables 5.1 and 5.2, respectively.
Stress
Stress
Lower strain Lower damping Higher modulus
Strain
Stress
Strain
Strain
Damping Area of hysteresis loop ratio
4p OAB
Higher strain Higher damping Lower modulus
Strain
Damping Area of hysteresis loop ratio
4p OAB
Figure 5.1 Illustration defining the effect of shear strain on damping and shear modulus of soils. (Reprinted from Seed and Idriss (1969), Influence of soil conditions on ground motions during earthquakes. J Soil Mech and Found Divn 95(SM1): 99-137, by permission of the American Society of Civil Engineers)
Table 5.1 Mean shear-wave velocities (m/s) for the top 30 m |
of ground (mainly from Borcherdt, | |
1994) | ||
General description |
Mean shear-wave velocity | |
Minimum |
Average Maximum | |
Firm and Hard Rocks | ||
Hard Rocks |
1400 |
1620 |
(e.g. metamorphic rocks with very widely spaced fractures) | ||
Firm to Hard Rocks* |
700 |
1050 1400 |
(e.g. granites, igneous rocks, conglomerates, sandstones, | ||
and shales with close to widely spaced fractures) | ||
Gravelly Soils and Soft to Firm Rocks |
375 |
540 700 |
(e.g. soft igneous sedimentary rocks, sandstones, and shales, | ||
gravels, and soils with >20% gravel) | ||
Stiff Clays and Sandy Soils |
200 |
290 375 |
(e.g. loose to v. dense sands, silt loams and sandy clays, | ||
and medium stiff to hard clays and silty clays (N > 5 | ||
blows/ft)) | ||
Soft Soils | ||
Non-special Study Soft Soils |
100 |
150 200 |
(e.g. loose submerged fills and very soft (N < 5 blows/ft) | ||
clays and silty clays <37 m (120 ft) thick) | ||
Very Soft Soils |
50? |
75? 100 |
(e.g. loose saturated sand, marshland, recent reclamation) |
*The NGA project (Section 4.6.4) has adopted vs = 1000 m/s as the threshold for engineering rock.
*The NGA project (Section 4.6.4) has adopted vs = 1000 m/s as the threshold for engineering rock.
Table 5.2 Typical mass densities of basic soil types
Soil type Mass density p(Mg/m3)*
Poorly graded soil Well-graded soil
Table 5.2 Typical mass densities of basic soil types
Range |
Typical value |
Range |
Typical value | |
Loose sand |
1.70-1.90 |
1.75 |
1.75-2.00 |
1.85 |
Dense sand |
1.90-2.10 |
2.00 |
2.00-2.20 |
2.10 |
Soft clay |
1.60-1.90 |
1.75 |
1.60-1.90 |
1.75 |
Stiff clay |
1.90-2.25 |
2.07 |
1.90-2.25 |
2.07 |
Silty soils |
1.60-2.00 |
1.75 |
1.60-2.00 |
1.75 |
Gravelly soils |
1.90-2.25 |
2.07 |
2.00-2.30 |
2.15 |
*Values are representative of moist sands and gravels and saturated silts and clays.
The idea that the stress-strain behaviour of a soil can be modelled as a linear elastic material is a very considerable idealization. First, the stiffness of a soil is dependent on the effective stresses. It is generally agreed that the small-strain stiffness is proportional to the square root of the mean principal stress. For example, Seed et al. (1986) proposed the following relation for the small-strain shear modulus of normally consolidated
Figure 5.2 Influence of mean effective confining pressure (kPa) on modulus reduction curves for (a) non-plastic (PI = 0) soil, and (b) plastic (PI = 50) soil. (Reprinted from Ishibashi (1992). Discussion to: Effect of soil plasticity on cyclic response, by M Vucetic and R Dobry, J Geotech Eng 118: 830-832, by permission of the American Society of Civil Engineers)
sands:
where Gmax denotes the small-strain shear modulus (the maximum value that it may take for a given material and effective stress), o'm is the mean principal effective stress (kPa) and (N1)60 is a corrected N value.
Laboratory methods generally measure G more directly from stress-strain tests. It is clear from Figure 5.1 that the level of strain at which G is measured must be known. This is further illustrated in Figure 5.2 which shows how G also varies with confining pressure and plasticity index (PI). In a study of normally consolidated and moderately overconsolidated soils, Dobry and Vucetic (1987) found that G/ Gmax depends also upon other factors, i.e. void ratio, number of cycles of loading, and sometimes geologic age and cementation.
The difficulties involved in finding a reliable shear modulus model for any given project are compounded by the fact that there is no simple linear relationship between laboratory and field tests (Tani, 1995; Yasuda et al., 1994). The latter found that the ratio of G from laboratory tests to G from field tests decreases markedly with increasing shear stiffness.
Shear strains developed during earthquakes may increase from about 10-3% in small earthquakes to 10-1% for large motions, and the maximum strain in each cycle will be different. For earthquake design purposes a value of two-thirds G measured at the maximum strain developed may be used. Alternatively, an appropriate value of G can be calculated from the relationship
where E is Young's modulus and v is Poisson's ratio. In the absence of any more specific data, low strain values of E may be taken from Table 5.3. Values of Poisson's ratio from Table 5.4 may be used in the above formula.
Soil type |
E(MPa/m2) |
E/cu |
Soft clay |
up to 15 |
300 |
Firm, stiff clay |
10-50 |
300 |
Very stiff, hard clay |
25-200 |
300 |
Silty sand |
7-70 | |
Loose sand |
15-50 | |
Dense sand |
50-120 | |
Dense sand and gravel |
90-200 | |
Sandstone |
up to 50,000 |
400 |
Chalk |
5,000-20,000 |
2000 |
Limestone |
25,000-100,000 |
600 |
Basalt |
15,000-100,000 |
600 |
Note that the values of E vary greatly for each soil type depending on the chemical and physical condition of the soil in question. Hence the above wide ranges of E value provide only vague guidance prior to test results being available. The ratio E/cu may be helpful, if the undrained shear strength cu is known, although the value of this ratio also varies for a given soil type.
Table 5.4 Typical values of Poisson's ratio for soils
Soil type Poisson's ratio, v
Clean sands and gravels 0.33
Stiff clay 0.40
Soft clay 0.45
A value of 0.4 will be adequate for most practical purposes.
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