Shear modulus

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

Calculating Speed Earthquake

Strain

Damping Area of hysteresis loop ratio

4p OAB

Higher strain Higher damping Lower modulus

Strain

Damping Area of hysteresis loop ratio

4p OAB

Higher strain Higher damping Lower modulus

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

Shear Modulus SoilShear Modulus Soil

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.

Table 5.3 Typical modulus of elasticity values for soils and rocks

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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