Soil behaviour in shear
Chapter 5
Soil behaviour in shear
Learning outcomes
After working through the material in this chapter, you should be able to:
1 Understand how soil may be modelled as a continuum, and how its mechanical behaviour (strength and stiffness) may be adequately described using elastic and plastic material (constitutive) models (Section 5.1–5.3);
2 Understand the method of operation of standard laboratory testing apparatus and derive strength and stiffness properties of soil from these tests for use in subsequent geotechnical analyses (Section 5.4);
3 Appreciate the different strength characteristics of coarse and fine-grained soils and derive material parameters to model these (Sections 5.5–5.6 and Section 5.8);
4 Understand the critical state concept and its important role in coupling strength and volumetric behaviour in soil (Section 5.7);
5 Use simple empirical correlations to estimate strength and stiffness properties of soil based on the results of index tests (see Chapter 1) and appreciate how these may be used to support the results from laboratory tests (Section 5.9).
element of soil are shown in Figure 5.1(a), the stresses being positive in magnitude as shown; the stresses vary across the element. The rates of change of the normal stresses in the x- and z-direc- tions are ∂σx/∂x and ∂σz/∂z respectively; the rates of change of the shear stresses are ∂τzx/∂x and
∂τxz/∂z. Every such element within a soil mass must be in static equilibrium. By equating moments about the centre point of the element, and neglecting higher-order differentials, it is apparent that τxz = τzx. By equating forces in the x- and z-directions the following equations are obtained:
∂
∂ ∂
∂ sx tzx
x + z =0 (5.1a)
∂
∂
∂
∂
txxz + szz - =g 0 (5.1b)
These are the equations of equilibrium in two dimensions in terms of total stresses; for dry soils, the body force (or unit weight) γ = γdry, while for saturated soil, γ = γsat. Equation 5.1 can also be written in terms of effective stress. From Terzaghi’s Principle (Equation 3.1) the effective body forces will be 0 and γ′ = γ–γw in the x- and z-directions respectively. Furthermore, if seepage is taking place with hydraulic gradients of ix and iz in the x- and z-directions, respectively, then there will be additional body forces due to seepage (see Section 3.6) of ixγw and izγw in the x- and z-directions, i.e.:
∂
∂
∂
∂
¢ + ¢ - =
sxx tzzx ix wg 0 (5.2a)
∂
∂ ∂
′ + ′ − ′ +∂
( )
=txz sz g z wg
x z i 0 (5.2b)
The effective stress components are shown in Figure 5.1(b).
Due to the applied loading, points within the soil mass will be displaced relative to the axes and to one another, as shown in Figure 5.2. If the components of displacement in the x- and Figure 5.1 Two-dimensional state of stress in an element of soil: (a) total stresses, (b)
effective stresses.
Soil behaviour in shear
z-directions are denoted by u and w, respectively, then the normal strains in these directions (εx
and εz, respectively) are given by ex = ∂ ez =
∂
∂
∂ u x
w , z and the shear strain by
gxz = ∂ +
∂
∂
∂ u z
w x
These strains are not independent; they must be compatible with each other if the soil mass as a whole is to remain continuous. This requirement leads to the following relationship, known as the equation of compatibility in two dimensions:
∂
∂
∂
∂
∂
∂ ∂
2 x 2
z xz
e e g
z2 + x2 - x z =0 (5.3)
The rigorous solution of a particular problem requires that the equations of equilibrium and com- patibility are satisfied for the given boundary conditions (i.e. applied loads and known displace- ment conditions) at all points within a soil mass; an appropriate stress–strain relationship is also required to link the two equations. Equations 5.1–5.3, being independent of material properties, can be applied to soil with any stress–strain relationship (also termed a constitutive model). In general, soils are non-homogeneous, exhibit anisotropy (i.e. have different values of a given prop- erty in different directions) and have non-linear stress–strain relationships which are dependent on stress history (see Section 4.2) and the particular stress path followed. This can make solution difficult.
In analysis, therefore, an appropriate idealisation of the stress–strain relationship is employed to simplify computation. One such idealisation is shown by the dotted lines in Figure 5.3(a), lin- early elastic behaviour (i.e. Hooke’s Law) being assumed between O and Y′ (the assumed yield Figure 5.2 Two-dimensional induced state of strain in an element of soil, due to
stresses shown in Figure 5.1.
point) followed by unrestricted plastic strain (or flow) Y′P at constant stress. This idealisation, which is shown separately in Figure 5.3(b), is known as the elastic–perfectly plastic model of mate- rial behaviour. If only the collapse condition (soil failure) in a practical problem is of interest, then the elastic phase can be omitted and the rigid–perfectly plastic model, shown in Figure 5.3(c), may be used. A third idealisation is the elastic–strain hardening plastic model, shown in Figure 5.3(d) by OY′P (solid line), in which plastic strain beyond the yield point necessitates further stress increase, i.e. the soil hardens or strengthens as it strains. If unloading and reloading were to take place subsequent to yielding in the strain hardening model, as shown by the dotted line Y″U in Figure 5.3(d), there would be a new yield point Y″ at a stress level higher than that at the first yield point Y′. An increase in yield stress is a characteristic of strain hardening. No such increase takes place in the case of perfectly plastic (i.e. non-hardening) behaviour, the stress at Y″ being equal to that at Y′ as shown in Figure 5.3(b) and (c). A further idealisation is the elastic–strain softening plastic model, represented by OY′P′ (chain-dotted line) in Figure 5.3(d), in which plastic strain beyond the yield point is accompanied by stress decrease or softening of the material.
In plasticity theory (Hill, 1950; Calladine, 2000) the characteristics of yielding, hardening and flow are considered; these are described by a yield function, a hardening law and a flow rule, Figure 5.3 (a) Typical stress–strain relationship for soil, (b) elastic–perfectly plastic
model, (c) rigid–perfectly plastic model, and (d) strain hardening and strain softening elastic–plastic models.
Soil behaviour in shear respectively. The yield function is written in terms of stress components or principal stresses, and defines the yield point as a function of current effective stresses and stress history. The Mohr–
Coulomb criterion, which will be described in Section 5.3, is one possible (simple) yield function if perfectly plastic behaviour is assumed. The hardening law represents the relationship between the increase in yield stress and the corresponding plastic strain components, i.e. defining the gra- dient of Y′P in Figure 5.3(d). The flow rule specifies the relative (i.e. not absolute) magnitudes of the plastic strain components during yielding under a particular state of stress. The remainder of this book will consider simple elastic–perfectly plastic material models for soil, as shown in Figure 5.3(b), in which elastic behaviour is isotropic (Section 5.2) and plastic behaviour is defined by the Mohr–Coulomb criterion (Section 5.3).