Measurement of Properties
2.4 Rupture Strength
2.4.2 Shear Strength Relationships Basic ConceptsBasic Concepts
Shear strength may be given in several forms, depending on various factors, including the drained strength, the undrained strength, the peak strength, the residual or ultimate strength, and strength under dynamic loadings. In addition, strength is the major factor in determining active and passive Earth pressures.
Under an applied force, a specimen will strain until rupture occurs at some peak stress;
in some materials, as strain continues, the resistance reduces until a constant minimum value is reached, termed the ultimate or residual strength.
Factors Affecting Strength
Material type: Some materials exhibit only a frictional component of resistance φ; others exhibit φas well as cohesion c. In soft clays, at the end of construction, it is normally the undrained strength s that governs.
Confining pressure: In materials with φ acting, the strength increases as the confining pressure increases.
A(σn,τ)
σ1−σ3
τmax = 2
σnf
2c
Normal stress
σ3
σ1
σ1
σ3
+
Shearing stress τ
c
FIGURE. 2.22
The Mohr diagram relating τ, σnf, σ3, and σ1(θcrθat failure; σnf normal stress at failure).
Undrained or drained conditions: It relates to the ability of a material to drain under applied stress and determines whether total or effective stresses act.
Loading direction: Forces applied parallel to weakness planes, as represented by stratifi-cation in soils, foliation planes, or joints in rock masses, will result in lower strengths than if the force is applied perpendicularly. Compressive strengths are much higher than ten-sile strengths.
Displacement and normal stress: In some cohesive materials, such as overconsolidated fis-sured clays and clay shales, the strain at which the peak stress occurs depends on the nor-mal stress level and the magnitude of the peak strength varies with the magnitude of normal stress (Peck, 1969) as shown in Figure 2.24. As strain continues the ultimate strength prevails. These concepts are particularly important in problems with slope stability.
Angle of Internal Friction φ
The stresses acting on a confined specimen of dry cohesionless soil, either in the ground or in a triaxial testing device (see Section 2.4.4), are illustrated in Figure 2.25. p is the applied stress, σ3is the confining pressure, and σ1 p σ3(in testing p is termed the deviator stress σd). As stress p is gradually increased, it is resisted by the frictional forces acting between
F
Some field conditions involving failure by rupture: (a) foundations; (b) retaining structures; (c) slopes;
(d) ground anchors; (e) wall anchors; (f) embankments.
grains until some characteristic stress level is reached at which resistance is exceeded and rupture occurs. This stress is termed the peak strength.
If σ3and σ1(for the peak strength) are plotted to define a Mohr’s circle at failure, a line drawn tangentially to the circle passes through the origin at an angle φas shown in Figure 2.26 and Figure 2.27. Other specimens of the same material loaded to failure, but at different confining pressures, will have Mohr’s circles tangent to the line defined by φ. The circle tan-gent line represents the limits of stability and is termed Mohr’s envelope. It defines shear strength in terms of the friction angle φ, and the normal or total stresses, and it expressed as
sτmaxσntan φ (2.23)
(In actuality, the envelope line will not be straight, but will curve downward slightly at higher confining pressures.)
Total vs. Effective Stresses (φvs. φ )
In a fully saturated cohesionless soil, the φvalue will vary with the drainage conditions prevailing during failure. In undrained conditions, total stresses prevail and in drained conditions, only effective stresses act. The friction angle is expressed as φ .
Total Stresses Saturated Soils
If no drainage is permitted from a fully saturated soil as load is applied, the stress at fail-ure is carried partially by the pore water and partially by the soil particles which thus
σn = 5 kg/cm2 σn = 3 kg/cm2
σn = 1 kg/cm2 σn = 0.5 kg/cm2
Peak strengths
∆L
FIGURE 2.24
Peak strength vs. displacement and normal stress.
(From Peck, R. B., Proceedings of ASCE, Stability and Performance of Slopes Embankments, Berkeley, CA, 1969, pp. 437–451. With permission.)
P
σ3
σ3 σ3
σn
σ1
τ
σ1
Tan =τ/σn
FIGURE 2.25
Stresses on a specimen in a confined state.
develop some intergranular stresses. The peak “undrained” strength parameters mobi-lized depend on the soil type:
● Cohesionless soils — the envelope passes through the origin (Figure 2.26 and Figure 2.27) and strength is expressed in terms of φas in equation 2.23.
● Soils with cohesion — the envelope intercepts the shear stress ordinate at a value taken as cohesion c, similar to the case shown in Figure 2.28, and the strength is expressed by the Mohr–Coulomb failure law:
s c σntan φ (2.24)
Shearing strength s
Effective stress
P3 P3
P1 P1
udf udf
Total stress
d
FIGURE 2.26
Mohr’s envelopes for total and effective stresses from CU triaxial tests on loose saturated sands.
Shearing strength s
Effective stress
P1 P1 P1
P3
udf udf
Total stress
d
Normal stress P FIGURE 2.27
Mohr’s envelopes for total and effective stresses from CU triaxial tests on dense saturated sands.
Shear stress τ
c
Confining, applied and normal stresses
τmax = c + σntan
σ FIGURE 2.28
Drained triaxial test on soil with cohesion.
● Normally to slightly overconsolidated clays have no frictional component (φ 0) and the strength is expressed as the undrained strength su, as will be discussed later.
Applicable Field Conditions
● Initial phases of slope excavations or retaining wall construction in slowly drain-ing soils, where time is too short to permit drainage. (Results are conservative if drainage occurs.)
● Sudden drawdown conditions in the upstream slope of a dam embankment, or when flood waters recede rapidly from stream banks and the soils remain satu-rated.
● Rapid placement of embankments or loading of storage tanks over soft clays, or high live loads applied to slow-draining soils.
Effective Stresses Saturated Soils
If the soil is permitted to drain as load is applied, the stress is initially carried partly by the pore water, but as drainage occurs the load is transferred to the soil grains, and the effec-tive stresses (intergranular stresses) are mobilized. The effeceffec-tive stress is equal to the total normal stress σnminus the pore-water pressure u, and the peak “drained” shear strength is expressed as
s (σn u)tan φ σn tan φ (2.25) As shown in Figures 2.26 and 2.27, φvalues based on effective stresses are often higher than those based on total stresses as long as u is positive. In dense sands, u is negative and the undrained strength is higher, i.e., φis higher. If pore pressures are measured during undrained loadings, effective stresses can be computed.
For soils with a cohesion intercept (Figure 2.28), the drained strength is expressed by the Coulomb–Terzaghi equation as
s c (σn u)tan φ c σn tan φ (2.26) Applicable Field Conditions
Drained strength prevails in the field under relatively slow loading conditions during which pore pressures can dissipate. Such conditions generally exist in:
● Most foundations, except for cases involving rapid load application.
● Natural slopes, except for the sudden drawdown case.
● Cut slopes, embankments, and retaining structures some time after construction completion.
Partially Saturated Soils
In partially saturated soils, strength is controlled by effective stresses, but the effective stress concept cannot be applied directly because of pressures in the air or gas in the par-tially saturated voids. Strength should be estimated from tests performed to duplicate in situ conditions as closely as possible in terms of percent saturation, total stress, and pres-sure on the liquid phase (Lambe and Whitman, 1969).
Apparent cohesion results from capillary forces in partially saturated fine-grained granu-lar soils, such as fine sands and silts, and provides a temporary strength which is lost upon saturation or drying. The apparent cohesion has been expressed in terms of the depth D to the water table (Lambe and Whitman, 1969) as
ca Dγwtan φ (2.27)
Pore-Pressure Parameters Definition
Pore-pressure parameters express the portion of a stress increment carried by the pore fluid in terms of the ratio of the pore-pressure increment (∆u) to the total stress increment (∆σ). As indicated in Lambe and Whitman (1969) and Bishop and Henkel (1962), the parameters are
C∆u/∆σ1 (2.28)
for loading in the odeometer (one-dimensional compression),
B∆u/∆σ (2.29)
for isotopic loading (three-dimensional compression),
A (∆u∆σ3)/(∆σ1∆σ3) (2.30)
for triaxial loading and
A∆u/∆σ1 (2.31)
for the normal undrained test where σ3 0.
Pore-pressure parameter A is the most significant in practice. Values depend on soil type, state of stress, strain magnitude, and time. Typical values are given in Table 2.17 for con-ditions at failure but important projects always require measurement by testing.
High values occur in soft or loose soils. Negative values indicate negative pore pres-sures, which occur in dense sands and heavily preconsolidated clays as the result of vol-ume increase during shear (dilatancy). Pore pressures are most responsive to applied
TABLE 2.17
Typical Values of Pore-Pressure Parameter A at Failurea
Soil Type Parameter A
Sensitive clay 1.5–2.5
Normally consolidated clay 0.7–1.3
Overconsolidated clay 0.3–0.7
Heavily overconsolidated clay 0.5–0.0
Very loose fine sand 2.0–3.0
Medium fine sand 0.0
Dense fine sand 0.3
Loess 0.2
a From Lambe, T.W., Proc. ASCE, J. Soil Mech. Found. Eng. Div., 88, 19–47, 1962.
stress in sensitive clays and loose fine sands that are subject to liquefaction. In soft or loose soils, higher shear strain magnitudes result in higher values for parameter A.
Applications
Pore-pressure parameter A is used to estimate the magnitude of initial excess pore pres-sure produced at a given point in the subsoil by a change in the total stress system. With piezometers used to measure in situ pore pressures, the validity of the estimates can be verified and the loading rate during surcharging or embankment construction can be con-trolled to avoid failure caused by exceeding the undrained shear strength.
Undrained Shear Strength su Concepts
During undrained loading in soft to firm saturated clays, the applied stress is carried partly by the soil skeleton and partly by the pore water. Increasing the confining pressure does not increase the diameter of Mohr’s circle, since the pore pressure increases as much as the confining pressure. The undrained strength, therefore, is independent of an increase in normal stress (φ 0), and as shown in Figure 2.29, is given by the expression
su ½ σd (2.32)
where σd (σ1σ3) is the applied or deviator stress. In the figure, Ucrepresents the uncon-fined compressive strength where σ3 0. As shown, su 1/2Uc.
For NC clays, su falls within a limited fraction of the effective overburden pressure (σv or p ), usually in the range su/p 0.16 to 0.4 (based on field vane and K0
triaxial tests). Therefore, if su 0.5p the clay may be considered as overconsolidated (see Section 2.5.2).
Factors Affecting suValues
Time rate of loading: For soft clays, su, measured by vane shear tests, has been found to be greater than the field strength mobilized under an embankment loading, and the differ-ence increases with the plasticity of the clay (Casagrande, 1959; Bjerrum, 1969, 1972). The difference is attributed to the variation in loading time rates; laboratory tests are per-formed at much higher strain rates than those that occur during the placement of an embankment in the field, which may take several months. A correction factor, therefore, is often applied to test values for su(see Section 2.4.4).
τ
σ3 σ1
σ Uc
Su = C
FIGURE 2.29
Undrained tests on saturated soil with cohesion.
Direction of loading: Soil anisotropy may cause shear strength measured in a horizontal direction to be significantly higher than when measured in a vertical direction (see In Situ Vane Shear Test, in Section 2.4.4).
Residual or Ultimate Shear Strength sr, or φφr Concepts
The lowest strength that a cohesive material can attain in the confined state is termed the ultimate strength. In many cohesive materials, if strain continues past the peak strength under continued stress, the strength will decrease until some minimum value is reached.
Thereafter, strength remains constant with increasing strain as shown in Figure 2.30.
Residual strength is a strength lower than peak strength remaining after failure has occurred. It has become generally accepted that the residual strength is the lowest strength that can be obtained during shear. In a normally consolidated clay the remolded undrained shear strength is considered to be equal to the ultimate or residual strength sr.
The envelope for the ultimate drained strength passes through the origin as a straight line on the Mohr diagram and has no cohesion intercept (even in cohesive materials) as shown in Figure 2.31 (Lambe and Whitman, 1969). The drained ultimate shear strength is expressed as
sσn tan φr (2.33)
Other Factors
Natural slopes: The residual strength, rather than the peak strength, often applies. An approximate relationship between φrand plasticity index for rock gouge materials is given in Figure 2.32.
Peak strength− overconsolidated clay
Peak strength−normally consolidated clay
Ultimate strength
Displacement
FIGURE 2.30
Peak and ultimate strength vs. displacement.
σ
r
r τmax = σntan τ
FIGURE 2.31
Mohr’s envelope for ultimate drained strength in clay.
Sensitivity St: The ratio of the natural peak strength to the ultimate undrained strength when a sample is completely remolded at unaltered water content is referred to as the soil sensitivity, expressed as
St su/sr (2.34)
A clay classification based on Stis given in Table 2.18. Soils termed as “quick” have high sensitivities and are extremely sensitive to vibrations and other forms of disturbance. They can quickly lose their strength, change to a fluid, and flow, even on very flat slopes.
Thixotropy refers to the regain in strength occurring in remolded soils because of a “reha-bilitation of the molecular structure of the adsorbed layers” (Lambe and Whitman, 1969).
Dynamic Shear Concepts
Under dynamic or cyclic loading a soil specimen subject to shear initially undergoes defor-mations that are partially irreversible, irrespective of the strain amplitude; hence, stress–strain curves in loading and unloading do not coincide. If strain amplitude is small, the difference between successive reloading curves tends to disappear after a few loading
20 Drained angle of shearing resisitance r (deg)
FIGURE 2.32
Approximate relationship between the drained angle of residual shearing resistance and plasticity index for rock gouge material. (From Patton, F. D. and Hendron Jr., A. J., Proceedings of the 2nd International Congress, International Association Engineering Geology, Sao Paulo, 1974, p. V-GR. 1. Kanji, M. A., M.S. Thesis, Department of Geology, University of Illinois, Urbana, 1970. With permission.)
TABLE 2.18
Clay Classification by Sensitivity Sta
Sensitivity su/srb Classification
2 Insensitive
a From Skempton, A.W., Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, Switzerland, Vol. I, 1953, pp. 57–61.
b su peak undrained strength, st remolded strength.
cycles of a similar amplitude and the stress–strain curve becomes a closed loop that can be defined by two parameters as shown in Figure 2.33:
● Shear modulus (Section 2.5.2) is defined by the average slope and has been found to decrease markedly with increasing strain amplitude.
● Internal damping refers to energy dissipation and is defined by the enclosed areas as shown in the figure (damping ratio λ).
In clay soils, if strain amplitude is large there is a significant reduction in the undrained strength.
In cohesionless soils, pore pressure increases almost linearly with the number of cycles until failure occurs. The simple shear device used for dynamic testing in the laboratory is shown on Figure 2.34. Liquefaction occurs when pore pressures totally relieve effective stresses.
Field Occurrence
Wave forces against offshore structures cause low-frequency cyclic loads. Seismic waves from earthquakes cause low- to high-frequency cyclic loads.
At-Rest, Passive, and Active Stress States At-Rest Conditions (K0)
The coefficient of lateral at-rest earth pressure has been defined as the ratio of lateral to vertical stress, K0σh/σv(Equation 2.15).
For sands and NC clays, K0is normally in the range of 0.4 to 0.5 and is a function of φin accordance with
K0 1.0 sin φ (2.35)
4πAT
Definitions: For G2
Shear modulus, G =G2 G1
1 1
1 G2 Damping ratio, D =
AL = Loop area AT = Triangle area Locus of hysteresis loops tips
Shear stress τ
(1) =
AL (1)
Shear strain ζ
ζ2 FIGURE 2.33
Hysteretic stress-strain relationship from cyclic shear test at different strain amplitudes. (After USAEC, National Technical Information Service
Publication TID-25953, U. S. Department of Commerce, Oak Ridge National Laboratory, Oak Ridge, TN, 1972.)
In clay soils, K0 has been found to be directly related to the amount of prestress or pre-consolidation in the formation, ranging from 0.5 to almost 3. The degree of prestress is given in terms of the overconsolidation ratio (OCR), i.e., the ratio of the maximum past pressure to the existing overburden pressure (see Section 2.5.4).
Passive State of Stress (Kp)
The passive state exists when a force pushes against a soil mass and the mass exerts its maximum resistance to the force. The principal stresses for the passive state are shown in Figure 2.35a; as the soil element is pushed, the vertical stress remains unchanged but the horizontal stress increases (σhσv). As movement continues the shear stress increases from the at-rest condition (1) until σhσv(2), then continues to increase as slip lines (rup-ture planes) form and finally failure occurs at (4). At this point of plastic equilibrium, the passive state has been reached.
The relationship between φand the principal stresses at failure, as shown on the Mohr diagram of Figure 2.35a, may be expressed for a cohesionless granular soil with a hori-zontal ground surface as
σ1/σ3σv/σh (1 sin φ)/(1 sin φ) (2.36)
de = horizontal cyclic displacement amplitude Fe = horizontal cyclic shear force
H = height of specimen A = area of specimen
o = do / H = horizontal cyclic shear strain amplitude τo = Fo / A = horizontal cyclic shear stress amplitude
H
FIGURE 2.34
NGI-type simple shear device at the UCLA Soil Dynamics Laboratory. (From Vucetic, 2004.)
or
σv/σh tan2(45°φ/2) Kp (2.37) where Kpis the coefficient of passive stress.
Active State of Stress
The active state exists when a soil mass is allowed to stretch, for example, when a retain-ing wall tilts and σvσh. In Figure 2.35b, as the mass stretches, σvremains unchanged and σhdecreases (2) until the induced shear stress is sufficient to cause failure (3). At this point of plastic equilibrium, the active state has been reached.
The coefficient of active stress Ka, (the ratio σh/σv) represents a minimum force, expressed for a cohesionless soil with a horizontal ground surface as
Ka tan2(45°φ/2) (2.38)
and
Ka 1/Kp (2.39)
Applications
The at-rest coefficient K0has a number of practical applications. It is used to compute lateral thrusts against earth-retaining structures, where lateral movement is anticipated to be too small to mobilize Ka. It is fundamental to the reconsolidation of triaxial test specimens according to an anisotropic stress path resembling that which occurred in situ (CKoU tests).
It is basic to the computation of settlements in certain situations (Lambe, 1964). It has been used for the analysis of progressive failure in clay slopes (Lo and Lee, 1973), the prediction of pore-water pressure in earth dams (Pells, 1973), and the computation of lateral swelling pressures against friction piles in expansive soils (Kassif and Baker, 1969).
τ τ
Mohr diagrams for the (a) passive and (b) active states of stress.
Coefficients of lateral earth pressure is the term often used to refer to Kpand Ka. When rup-ture occurs along some failure surface, earth pressures are mobilized. The magnitudes of the pressures are a function of the weight of the mass in the failure zone and the strength acting along the failure surface. These pressures (Ppor Pa) are often expressed in terms of the product of the weight of the mass times the active or passive coefficients, as follows:
Pp Kpγz (2.40)
Pa Kaγz (2.41)
Examples of the occurrence of Ppand Pain practice are given in Figure 2.36.
2.4.3 Rock Strength Measurements