Measurement of Properties
2.4 Rupture Strength
2.4.1 Introduction Basic DefinitionsBasic Definitions
● Stress (σ) is force P per unit of area, expressed as
σ P/A (2.11)
System Equivalent units for stress
English 13.9 psi 2000 psf 2 ksf 1 tsf 1 bar Metric 1 kg/cm2 10 T/m2(≈1 tsf)
SI 100 kN/m2 100 kPa 0.1 MPa (≈1 tsf)
● Strain (ε) is change in length per unit of length caused by stress. It can occur as compressive or tensile strain. Compressive and tensile strain are expressed as
ε ∆L/L (2.12)
● Shear is the displacement of adjacent elements along a plane or curved surface.
● Shear strain (ξ) is the angle of displacement between elements during displace-ment.
● Shear stress (τ) is the stress causing shear.
● Shear strength (S or s) is a characteristic value at which a material fails in rupture or shear under an applied force.
● Dilatancy is the tendency of the volume to increase under increasing shear or stress difference.
Strength of Geologic Materials Components: Friction and Cohesion
Friction is a resisting force between two surfaces as illustrated in Figure 2.18. It is often the only source of strength in geologic materials and is a direct function of the normal force.
Cohesion results from a bonding between the surfaces of particles. It is caused by electro-chemical forces and is independent of normal forces.
Influencing Factors
Strength is not a constant value for a given material, but rather is dependent upon many factors, including material properties, magnitude and direction of the applied force and the rate of application, drainage conditions in the mass, and the magnitude of the confin-ing pressure.
Stress Conditions In Situ Importance
A major factor in strength problems is the existence of stress conditions in the ground, pri-marily because normal stresses on potential failure surfaces result from overburden pres-sures.
Geostatic Stresses
Overburden pressures, consisting of both vertical and lateral stresses, exist on an element in the ground as a result of the weight of the overlying materials. Stress conditions for
N
w
N f
P
T
T = N tan Tmax = Nf = N tan
FIGURE 2.18
Frictional force f resisting shearing force T [P force applied in increments until slip occurs; N normal force component including block weight W; T shearing stress component; f frictional resistance; α
angle of obliquity [resultant of N and T]; φ friction angle, or αmaxat slip; Smax maximum shearing resistance Tmax].
level ground are illustrated in Figure 2.19; sloping ground results in more complex condi-tions. (Changes in geostatic stresses are invoked by surface foundation loads, surface and subsurface excavations, lowering of the groundwater level, and natural phenomena such as erosion and deposition). Vertical earth pressures from overburden weight alone are found by summing the weights from the various strata as follows:
σv冱z
0
γnZn (2.13)
Coefficient of lateral earth pressure “at-rest” K0is the ratio of the lateral to vertical stress in a natural deposit that has not been subject to lateral strain, the values for which vary sub-stantially with material types and properties (see Section 2.4.2). It is expressed as
K0σh/σv (2.14)
or
σh K0σv (2.15)
For an elastic solid
K0ν/1ν (2.16)
In the above expressions, γ is the material unit weight (γn above groundwater level, γb below groundwater level) and νis the Poisson’s ratio (see Section 2.5.1).
Total and Effective Stresses
The total stress on the soil element in Figure 2.19 at depth z is
σvγtZ (2.17)
If the static water table is at the surface, however, and the soil to depth z is saturated, there is pressure on the water in the pores because of a piezoelectric head hwand the unit weight of water γw. This is termed the neutral stress (acting equally in all directions), or the pore-water pressure uwor u and is given as
uσwhw (2.18)
The effective stress σ v, or actual intergranular stresses between soil particles, results from a reduction caused by the neutral stress and is equal to the total stress minus the pore-water pressure, or
σv σv u (2.19)
∞
1
2
3 σv
σh
K0z
σh
σv
z z1
z2
z3
FIGURE 2.19
The geostatic stress condition and “at-rest” earth pressures.
or
σv γbz (2.20)
where γbγtγwis the effective or submerged soil weight.
In calculations, therefore, above the groundwater level, the effective soil weight is the total weight γt, and below the groundwater level (or any other water surface), the effective soil weight is the submerged soil weight γb.
Prestress in Soil Formations
General: Soils compress naturally under the weight of overlying materials or some other applied load, resulting in strength increase over values inherent as deposited, or shortly thereafter. Three categories of prestress are defined according to the degree of compression (termed consolidation) that has occurred.
Normally consolidated (NC): The soil element has never been subjected to pressures greater than the existing overburden pressures.
Overconsolidated (OC): The soil element has at some time in its history been subjected to pressures in excess of existing overburden, such as resulting from glacial ice loads, removal of material by erosion, desiccation, or lowering of the groundwater level.
Underconsolidated (UC): The soil element exists at a degree of pressure less than existing overburden pressures. This case can result from hydrostatic pressures reducing overbur-den load as illustrated in Figure 2.20. Such soils are normally relatively weak. Weakening of strata can also occur due to removal of a cementing agent or other mineral constituents by solution.
Principal Stresses and the Mohr Diagram Importance
Fundamental to the strength aspects of geologic materials are the concepts of principal stresses and the Mohr diagram on which their relationships may be illustrated.
Hydrostatic head removes some effective overburden pressure; load on soil can be less than existing overburden load (under consolidated)
Sand Cementing agent dissolved
leaving loose structure Cementing agent in granular soils Solution of cementing agent
Rainfall
Hydrostatic uplift
(approximate)∆H
Clay
FIGURE 2.20
Soil profile weakening processes.
Principal Stresses
Stresses acting on any plane passed through a point consist of a normal stress σ (com-pression or tension) and a shearing stress τ. (Soil mechanics problems are normally con-cerned with compressive stresses.) On one particular plane, the normal stress will be the maximum possible value and the shearing stress will be equal to zero. On one plane per-pendicular to this plane, the normal stress will be the minimum possible value, with shear stress also equal to zero. On a second plane perpendicular to this plane, the normal stress will have an intermediate value and the shearing stress will also be zero. These planes are termed the principal planes.
The principal stresses are the stresses acting perpendicular to the principal planes including the maximum (major) principal stress σ1, the minimum (minor) principal stress σ3, and the intermediate principal stress σ2. The relationship between principal stresses and the normal stress and the shear stress acting on a random plane through a point is shown in Figure 2.21. The intermediate principal stress is the plane of the paper and, in soil mechanics problems, is normally considered to be equal to σ3.
The Mohr Diagram
To attain equilibrium, the sum of the forces given in Figure 2.21 should be zero. Therefore, σnand τcan be expressed in terms of the principal stresses and the angle θas
σn [ (σ1σ2 )/2 (σ1σ3 )/2 ] cos 2θ (2.21)
τ [( σ1σ3 )/2] sin 2θ (2.22)
If points are plotted to represent coordinates of normal and shearing stresses acting on a particular plane for all values of θgiven in equations 2.21 and 2.22, their loci form a circle which intersects the abscissa at coordinates equal to the major (σ1) and minor (σ3) principal stresses. The circle is referred to as the Mohr diagram, or Mohr’s circle, given in Figure 2.22.
Applications of Strength Values Stability Analysis
The values for strength are used in stability analyses; the discussion is beyond the scope of this book, except for evaluations of slopes. In general terms, stability is based on plastic equi-librium or a condition of maximum shear strength with failure by rupture imminent. When the imposed stresses cause the shear strength to be exceeded, rupture occurs in the mass along one or more failure surfaces. Analyses are normally based on the limit equilibrium
σn
Stresses on a random plane through a point (σ2is the plane of the paper).
approach, i.e., a limiting value that can be reached when the forces acting to cause failure are in balance with the forces acting to resist failure. Resistance to failure is provided by the shear strength mobilized along the failure surface.
Typical Problems
Some field conditions involving failure by rupture are illustrated in Figure 2.23, showing the relationships between the force acting to cause failure, the strength acting along the failure surface, and the principal and normal stresses.
2.4.2 Shear Strength Relationships