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7. STRESS ANALYSIS AND STRESS PATHS
7.1 THE MOHR CIRCLE
The discussions in Chapters 2 and 5 were largely concerned with vertical stresses. A more detailed examination of soil behaviour requires a knowledge of stresses in other directions and two or three dimensional analyses become necessary. For the graphical representation of the state of stress on a soil element a very convenient and widely used method is by means of the Mohr circle. In the treatment that follows the stresses in two dimensions only will be considered.
Fig. 7.1(a) shows the normal stresses σy and σx and shear stresses τxy acting on an element of soil. The normal stress σ and shear stress τ acting on any plane inclined at θ to the plane on which σy acts are shown in Fig. 7.1(b). The stresses σ and τ may be expressed in terms of the angle θ and the other stresses indicated in Fig. 7.1(b). If a, b and c represent the sides of the triangle then, for force equilibrium in the direction of σ:
σ a = σx b sin θ +τxy b cos θ + σy c cos θ + τxy c sin θ
σ = σx sin^2 θ + τxy sin θ cos θ + σy cos^2 θ + τxy cos θ sin θ
σ = σx sin^2 θ + σy cos^2 θ + τxy sin 2 θ (7.1)
Similarly if forces are resolved in the direction of τ
τ a = σy c sin θ - τxy c cos θ - σx b cos θ + τxy b sin θ
τ = σy sin θ cos θ - τxy cos^2 θ - σx sin θ cos θ + τxy sin^2 θ
= σy^2 -^ σx sin 2 θ - τxy cos 2 θ (7.2)
Equation (7.1) can be further expressed as follows
σ - (σx^ + 2 σy) = (σy^2 -^ σx)cos 2 θ + τxy sin 2 θ
This equation can be combined with equation (7.2) to give
Fig. 7.1 Stress at a Point
Fig. 7.2 The Mohr Circle
Fig. 7.
(a) Determine the inclination of the planes on which the maximum shear stresses act.
(b) Determine the inclination of the planes on which the following condition is satisfied τ = ± σ tan 45˚
(c) On how many planes are shear stresses having a magnitude of 5kN/m^2 acting?
In Fig. 7.3(b) the Mohr circle has been drawn, A and B representing the major and minor principal stresses respectively. By drawing line A OP parallel to the major principal plane the origin of planes OP may be located.
(a) The maximum shear stress may be calculated from equation (7.3) or it may simply be read from the Mohr circle.
Clearly τ max = σ^1 2 -^ σ^3
= 15 kN/m^2
The points of maximum shear stress are represented by C and D. Therefore the planes on which these stresses act are parallel to lines OP C and OP D respectively. As shown on the figure these planes are inclined at 45_ to the principal planes. This will always be the case regardless of the inclination of the principal planes.
(b) The lines representing the relationship
τ = ± σ tan 45˚
have been drawn in Fig. 7.3(b). Since the circle touches neither of these lines there are no planes on which the relationship holds.
(c) The points on the circle representing a shear stress of 5kN/m^2 are E,F, G and H so there are four planes on which this shear stress acts. These planes are parallel to the lines OP E, OP F, OP G and OP H respectively.
7.2 STRESS PATHS
When the stresses acting at a point undergo changes, these changes may be conveniently represented on a plot of shear stress against normal stress. Such a situation is illustrated in Fig.
Fig. 7.4 Stress Paths
Fig. 7.5 Examples of Stress Paths
This demonstrates that q is the same regardless of whether total stresses or effective stresses are being considered. In other words the shear stress is unaffected by pore pressure (this point was also made in section 2.5) Since q is equal to the radius of the Mohr circle this means that the total and effective Mohr circles must always have the same size.
If the mean normal stress is expressed in terms of effective stresses
p' = σ'^1 + 2 σ'^3
= (σ^1 - u) + ( 2 σ^3 - u)
= σ^1 +^ σ 23 - 2u
= p - u
This shows (in agreement with the principle of effective stress) that the difference between the total and effective mean normal stresses is equal to the pore pressure. This means that there is not one stress path to consider but two - a total stress path and an effective stress path (see Lambe and Whitman, 1979). Lambe (1967) and Lambe and Marr (1979) have described the use of the stress path method in solving stress-strain problems in soil mechanics.
Some examples of stress paths are shown in Fig. 7.5. Fig. 7.5(a) shows a number of stress paths that start on the p axis (σ 1 = σ 3 ), the stress paths going in different directions depending on the relative changes to σ 1 and σ 3. Fig. 7.5(b) shows stress paths for loading under conditions of constant stress ratio (σ 3 /σ 1 ) from an initial zero state of stress. With this type of loading
(q/p) = (1 - K) / (1+ K) (7.6)
where K = σ 3 /σ 1
The line marked K = 1 corresponds to isotropic compression for which the principal stresses (σ 1 and σ 3 ) are maintained equal during the loading. The line marked K = Ko corresponds to compression under conditions of no lateral strain, as discussed in Chapter 2.
Fig. 7.
Fig. 7.7 Undrained Loading of a Soil
p'f = 140 - 60 = 80kN/m^2
qf = 220 - 60 2 = 80kN/m^2
This enables the total and effective stress paths to be completed by joining the intermediate and final points as illustrated in Fig. 7.6.
7.3. PORE PRESSURE PARAMETERS
When a soil sample is sealed in a testing apparatus so that water is prevented from moving into or out of the soil, pore pressures develop in the sample when it is subjected to external stress changes. The application of external stresses under these conditions is referred to as undrained loading, since water is unable to drain from the sample.
If water is allowed to drain from the sample and no pore pressure changes are allowed to develop the application of external stresses is referred to as drained loading. It is seen that drained loading involves the process of consolidation. In other words the sample is being consolidated under the externally applied stresses in a drained test.
The stress paths drawn in Fig. 7.6 are clearly for an undrained test since pore pressure changes have developed during the loading as a result of the applied stresses. These pore pressure changes may be calculated from the changes in the major and minor principal stresses by means of an equation that was developed by Skempton. (1954).
In developing this equation a three dimensional state of stress is considered in which σ 2 is always equal to σ 3 (axially symmetric). Three stages of loading are considered in this development and these are illustrated in Fig. 7.7.
Stage I
Initially the soil is consolidated under an all around stress of σ'c. In other words drained loading is applied and at the end of this loading the pore pressure is zero and external stresses of σ'c exist in all three coordinate directions.
The stress path representing this consolidation stage is given by OP in Fig. 7.8. Since σ'c is an all around stress the value of q remains at zero throughout the loading. Since the pore pressure is zero, line OP represents both the total and the effective stress paths.
- Fig. 7.8 Stress Paths for the Loading in Fig. 7.
∆σ' 2 = ∆σ' 3 = 0 - ∆ud = - ∆ud
If the soil is assumed to behave according to elastic theory, in which the volume change of the soil skeleton is governed by the mean principal effective stress change, then using the same symbols as defined in stage II.
∆V = Cs V^13 (∆σ' 1 + 2 ∆σ' 3 )
= Cs V
3 (∆σ^1 -^ ∆σ^3 - 3^ ∆ud)
and this volume change must equal the volume change of the void space.
∆Vv = - Cv n V ∆ud
= ∆V
This leads to the expression
∆ud = (^) 1 + n C Csv^ -1^.^13. (∆σ 1 - ∆σ 3 )
= B.^13. (∆σ 1 - ∆σ 3 )
In order to remove the dependence upon elastic theory and to make the expression more
generally applicable, Skempton replaced the
3 by the pore pressure parameter A
∆ud = B. A. (∆σ 1 - ∆σ 3 ) (7.8)
so the total pore pressure change _u throughout all stages of loading is
∆u = ∆ua + ∆ud
= B [∆ σ 3 + A ( ∆ σ 1 −∆ σ 3 )] (7.9)
The A parameter varies with stress and soil type but commonly it lies within the range + 1 to -1/2. At failure (discussed in Ch. 8) typical ranges of values for the A parameter are given in Table 7.1.
At the end of consolidation the effective and total stresses are equal to 60kN/m^2. This stage is represented by point R in Fig. 7.9. At the end of loading the change in the major principal stress is ∆σ 1 = 170 - 60 = 110kN/m^2
The pore pressure change may be calculated from equation (7.9)
∆u = B [∆ σ 3 + A ( ∆ σ 1 −∆ σ 3 )] (9.9)
= 1.0 [ 0 + 0.5 (110 - 0)]
= 55kN/m^2
Since the pore pressure was zero at the beginning of the undrained loading stage the pore pressure at the end of loading is equal to 55kN/m^2.
In Fig. 7.9 the total stress Mohr circle at the end of loading has been drawn with
σ 3 = 60kN/m^2 and σ 1 = 170kN/m^2
Clearly the total stress path is represented by line RQ where Q is located at the top of the circle. Alternatively the stress path may have been drawn after calculating the coordinates of point Q
p = σ^1 + 2 σ^3 = 170 + 60 2 = 115kN/m^2
q = σ^1 2 -^ σ^3 = 170 - 60 2 = 55kN/m^2
For the end point P of the effective stress path the coordinates are:
q = 55kN/m^2
p' = p - u
= 115 - 55 = 60kN/m^2 This data enables the effective stress path RP to be drawn.
7.4 PRINCIPAL STRESS PLOT
There are many ways of graphically representing changes in the state of stress on a soil specimen, apart from the q - p diagram discussed in section 7.2. Another way of representing the changes in state of stress is by means of a principal stress plot. That is successive values of σ 1 are plotted against σ 3 during loading to identify the total stress path as illustrated in Fig. 7.10. For the effective stress path σ' 1 is plotted against σ' 3. The stress paths RT and QS plotted on Fig. 7. correspond to the total and effective stress paths respectively that are plotted on Fig. 7.8.
EXAMPLE
(a) On a principal stress plot describe the direction in which lines representing ∆q = 0 should be drawn.
(b) On a principal stress plot describe the direction in which lines representing ∆p = 0 should be drawn.
(c) Loading of a soil sample commences from an initial stress state represented by q = 10kPa and p = 50kPa. The loading is such that ∆q = 2∆p and the loading continues until p = 80kPa. Plot the stress path on a principal stress plot and determine the values of σ 1 and σ 3 at the end of loading.
(a) The lines are parallel to the diagonal line σ 1 = σ 3 , sometimes called the space diagonal.
(b) The lines are perpendicular to the space diagonal.
(c) q = (σ 1 - σ 3 ) /
p = (σ 1 + σ 3 ) /
∴ σ 1 = p + q = 60kPa initially
and σ 3 = p - q = 40kPa initially
Fig.7.
The stress path which is a straight line is plotted in Fig. 7.11 and the final values of σ 1 and σ 3 are given in the table above.
REFERENCES
Lambe, T.W., (1967), “Stress Path Method”, Jnl. Soil Mech. & Found. Division, ASCE, Vol. 93, No. SM6, pp 309-331.
Lambe, T.W. and Marr, W.A., (1979), “Stress Path Method: Second Edition”, Jnl. Geot. Eng. Division, ASCE, Vol. 105, No. GT6, pp 727-738.
Lambe, T.W. and Whitman, R.V., (1979), “Soil Mechanics SI Version”, John Wiley & Sons, 553p.
Skempton, A.W., (1954), “The Pore Pressure Coefficients A and B”, Geotechnique, Vol. 4, No. 4, pp 143-147.
