7. STRESS ANALYSIS AND STRESS PATHS
7-17. STRESS ANALYSIS AND STRESS PATHS7.1THE MOHR CIRCLEThe discussions in Chapters 2 and 5 were largely concerned with vertical stresses. Amore detailed examination of soil behaviour requires a knowledge of stresses in other directionsand two or three dimensional analyses become necessary. For the graphical representation of thestate of stress on a soil element a very convenient and widely used method is by means of theMohr 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 anelement of soil. The normal stress σ and shear stress τ acting on any plane inclined at θ to theplane on which σy acts are shown in Fig. 7.1(b). The stresses σ and τ may be expressed in termsof the angle θ and the other stresses indicated in Fig. 7.1(b). If a, b and c represent the sides of thetriangle then, for force equilibrium in the direction of σ:σa σx b sin θ τxy b cos θ σy c cos θ τxy c sin θσ σx sin2 θ τxy sin θ cos θ σy cos2 θ τxy cos θ sin θσ σx sin2 θ σy cos2 θ τxy sin 2 θ(7.1)Similarly if forces are resolved in the direction of ττa σy c sin θ - τxy c cos θ - σx b cos θ τxy b sin θτ σy sin θ cos θ - τxy cos2 θ - σx sin θ cos θ τxy sin2 θ σy - σxsin 2 θ - τxy cos 2 θ2Equation (7.1) can be further expressed as followsσ -(σx σy)(σy - σx) cos 2 θ τxy sin 2 θ22This equation can be combined with equation (7.2) to give(7.2)
7-2Fig. 7.1 Stress at a PointFig. 7.2 The Mohr Circle
7-322 σx σ y σ σ y 2 σ τ 2 x τ xy2 2 (7.3)This is the equation of a circle with a centre atσ σx σy,2τ 0and a radius of (σy - σx)2 τxy 2 1/22 This circle known as the Mohr circle is represented in Fig. 7.2. In this diagram point Arepresents the stresses on the σy plane and point B represents the stresses on the σx plane. Theshear stresses are considered as negative if they give a couple in the clockwise direction. Ingeomechanics usage the normal stresses are positive when compressive.Point C represents the stresses τ and σ on the θ plane. The location of point C may befound by rotating a radius by an angle equal to 2θ in an anticlockwise direction from the radiusthrough point A.Alternatively point C may be found by means of the point OP known as the “origin ofplanes” or the “pole”. This point is defined as follows: if any line OP X is drawn through theorigin of planes and intersects the other side of the Mohr circle at point X then point X representsthe stresses on the plane parallel to OP X. In other words line OP A in Fig. 7.2 is parallel to theplane on which the stress σy acts and line OP B is parallel to the plane on which the stress σx acts.To find the point on the circle representing the stresses on the θ plane, line OP C is drawn parallelto that plane to yield point C. Both of the constructions just described for the location of point Cmay be verified by means of equations (7.1) and (7.2).From Fig. 7.2 the major and minor principal stresses σ1 and σ3 and the inclinations ofthe planes on which they act may also be determined. A more detailed treatment of the Mohrcircle may be found in most books on the mechanics of solids.EXAMPLEMajor and minor principal stresses of 45kN/m2 and 15kN/m2 respectively act on anelement of soil where the principal planes are inclined as illustrated in Fig. 7.3(a).
7-4Fig. 7.3(a)Determine the inclination of the planes on which the maximum shear stressesact.
7-5(b)Determine the inclination of the planes on which the following condition issatisfiedτ σ tan 45 (c)On how many planes are shear stresses having a magnitude of 5kN/m2 acting?In Fig. 7.3(b) the Mohr circle has been drawn, A and B representing the major and minorprincipal stresses respectively. By drawing line A OP parallel to the major principal plane theorigin of planes OP may be located.(a)The maximum shear stress may be calculated from equation (7.3) or it may simply beread from the Mohr circle.Clearly τ max σ1 - σ3215 kN/m2The points of maximum shear stress are represented by C and D. Therefore the planeson which these stresses act are parallel to lines OP C and OP D respectively. As shownon the figure these planes are inclined at 45 to the principal planes. This will always bethe case regardless of the inclination of the principal planes.(b)The lines representing the relationshipτ σ tan 45 have been drawn in Fig. 7.3(b). Since the circle touches neither of these lines there areno planes on which the relationship holds.(c)The points on the circle representing a shear stress of 5kN/m2 are E,F, G and H so thereare four planes on which this shear stress acts. These planes are parallel to the lines OPE, OP F, OP G and OP H respectively.7.2STRESS PATHSWhen the stresses acting at a point undergo changes, these changes may be convenientlyrepresented on a plot of shear stress against normal stress. Such a situation is illustrated in Fig.
7-67.4 in which the lower part of the diagram has been omitted for simplicity. The initial major andminor principal stresses are indicated by σli and σ3i respectively. Stress changes of σ1 and σ3have been imposed to give the following final stressesσlf σli σ1σ3f σ3i σ3The initial and final Mohr circles representing these conditions have been drawn in Fig.7.4. To provide a simple graphical representation of the stress changes from the initial to the finalstate use has been made of points at the top of the circles. These points A and B, represent therespective circles and if these points only are plotted the circles could easily be drawn should theybe needed. The loci of the tops of the Mohr circles is the stress path. The straight line AB is onlyone of an infinite number of stress paths which indicates they way in which stresses changebetween the initial and final states. Two other possible stress paths between points A and B havebeen drawn. More information than a knowledge of σ1 and σ3 would be needed regardingintermediate stress changes before the correct stress path could be drawn.When stress paths only are plotted then the axes of the diagram are really particularvalues of the shear stress τ and normal stress σ. These values are commonly referred to as q and pwhereσ1 - σ32q τmax p mean normal stress σ1 σ32(7.4)(7.5)If the maximum shear stress τmax is expressed in terms of effective stresses instead oftotal stressesσ'1 - σ'32 (σ1 - u) - (σ3 - u)2 σ1 - σ3 q2
7-7Fig. 7.4 Stress PathsFig. 7.5 Examples of Stress Paths
7-8This demonstrates that q is the same regardless of whether total stresses or effectivestresses are being considered. In other words the shear stress is unaffected by pore pressure (thispoint was also made in section 2.5) Since q is equal to the radius of the Mohr circle this meansthat the total and effective Mohr circles must always have the same size.If the mean normal stress is expressed in terms of effective stressesp' σ'1 σ'32 (σ1 - u) (σ3 - u)2 σ1 σ3 - 2u2 p-uThis shows (in agreement with the principle of effective stress) that the differencebetween the total and effective mean normal stresses is equal to the pore pressure. This meansthat 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 theuse 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 ofstress paths that start on the p axis (σ1 σ3), the stress paths going in different directionsdepending on the relative changes to σ1 and σ3. Fig. 7.5(b) shows stress paths for loading underconditions of constant stress ratio (σ3/σ1) from an initial zero state of stress. With this type ofloading(q/p) (1 - K) / (1 K)where K σ3/σ1(7.6)The line marked K 1 corresponds to isotropic compression for which the principalstresses (σ1 and σ3) are maintained equal during the loading. The line marked K Kocorresponds to compression under conditions of no lateral strain, as discussed in Chapter 2.
7-9EXAMPLEPlot the total and effective stress paths for the following stress changesσ1kN/m2σ3kN/m2ukN/m2initial state804020intermediate state1406040final state2206060initial statepi σ1 σ32 80 402 60kN/m2p'i pi - ui 60 - 20 40kN/m2qi σ1 - σ32 80 - 402 20kN/m2These calculations enable the initial points (pi, qi) and (p'i , qi) to be plotted as shown inFig. 7.6.intermediate statep 140 602 100kN/m2p' 100 - 40 60kN/m2q 140 - 602 40kN/m2Portion of the total and effective stress paths may now be drawn by joining the initial andintermediate points.final statepf 220 602 140kN/m2
7-10Fig. 7.6Fig. 7.7 Undrained Loading of a Soil
7-11p'f 140 - 60 80kN/m2qf 220 - 602 80kN/m2This enables the total and effective stress paths to be completed by joining theintermediate and final points as illustrated in Fig. 7.6.7.3.PORE PRESSURE PARAMETERSWhen a soil sample is sealed in a testing apparatus so that water is prevented frommoving into or out of the soil, pore pressures develop in the sample when it is subjected toexternal stress changes. The application of external stresses under these conditions is referred toas 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 todevelop the application of external stresses is referred to as drained loading. It is seen thatdrained loading involves the process of consolidation. In other words the sample is beingconsolidated 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 pressurechanges have developed during the loading as a result of the applied stresses. These pore pressurechanges may be calculated from the changes in the major and minor principal stresses by meansof an equation that was developed by Skempton. (1954).In developing this equation a three dimensional state of stress is considered in which σ2is always equal to σ3 (axially symmetric). Three stages of loading are considered in thisdevelopment and these are illustrated in Fig. 7.7.Stage IInitially the soil is consolidated under an all around stress of σ'c. In other words drainedloading is applied and at the end of this loading the pore pressure is zero and external stresses ofσ'c exist in all three coordinate directions.The stress path representing this consolidation stage is given by OP in Fig. 7.8. Since σ'cis an all around stress the value of q remains at zero throughout the loading. Since the porepressure is zero, line OP represents both the total and the effective stress paths.
7-12Stage IIAfter the consolidation in Stage I the soil is subjected to an externally applied all aroundstress change of σ3 under undrained conditions. During this stage a pore pressure of uadevelops. The total and effective stress paths are represented in Fig. 7.8 by lines PR and PQrespectively. If Cs and Cv and defined as the compressibilities of the soil skeleton and the porefluid respectively then the volume change V of the soil sample is V -Cs V σ'3 -Cs V ( σ3 - ua)where V is the total volume of the soil sample. Since the soil mineral particles are relativelyincompressible the volume change V must be the same as the volume change Vv of the voidspace nV, where n indicates the porosity. Vv -Cv n V ua -Cs V ( σ3 - ua)After collecting terms u a σ 3nCv -1 1 C s B(7.7)This ratio of the pore pressure change to the change in all around stress is known as thepore pressure parameter B. It is clear that B varies from 0 to 1 the zero value applying to acompletely dry soil, the value of unity applying to a completely saturated soil.Stage IIIAs shown in Fig. 7.7, stage III involves the application under undrained conditions of astress in one direction only. This stress, somewhat arbitrarily identified by the symbols ( σ1 σ3), is referred to as the deviator stress. During this loading a pore pressure ud develops. Thetotal and effective stress paths corresponding to this stage are given in Fig.7.8 by lines RT and QSrespectively.The effective stress changes during this stage are σ'1 ( σ1 - σ3) - ud
7-13Fig. 7.8 Stress Paths for the Loading in Fig. 7.7Fig. 7.9
7-14 σ'2 σ'3 0 - ud - udIf the soil is assumed to behave according to elastic theory, in which the volume change of the soilskeleton is governed by the mean principal effective stress change, then using the same symbolsas defined in stage II. V 1Cs V ( σ'1 2 σ'3)3 1Cs V ( σ1 - σ3 - 3 ud)3and this volume change must equal the volume change of the void space. Vv - Cv n V ud VThis leads to the expression ud n Cv -1 1 1 C . 3. ( σ1 - σ3) s 1B . 3 . ( σ1 - σ3)In order to remove the dependence upon elastic theory and to make the expression more1generally applicable, Skempton replaced the 3 by the pore pressure parameter A ud B . A . ( σ1 - σ3)(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 inTable 7.1.
7-15TABLE 7.1Values of Pore Pressure Parameter A at FailureSOILClays of high sensitivityA31 to 142Normally consolidated clays1 to 12Compacted sandy clays13 4 to 4Lightly over consolidated clays10 to 2Compacted clay gravels11- 4 to 4Heavily over consolidated clays1- 2 to 0The pore pressure change u is sometimes referred to as a dependent pore pressure,which means that the magnitude of u depends upon the magnitudes of the stresses applied to thesoil. This contrasts with independent pore pressures which are not dependent upon appliedstresses but are governed by the hydraulic boundary conditions. These independent porepressures have been discussed in Chapter 4.EXAMPLEAn undisturbed sample of saturated clay soil is consolidated to an all around stress of60kN/m2. The following stress changes are then imposed under undrained conditions until thevalue of σ1 reaches 170kN/m2: σ2 σ3 0 and σ1 is positiveThe pore pressure parameters for the soil areB 1.0,A 0.5Sketch the total and effective stress paths for the undrained loading and calculate thepore pressure in the soil at the end of loading.
7-16At the end of consolidation the effective and total stresses are equal to 60kN/m2. Thisstage is represented by point R in Fig. 7.9. At the end of loading the change in the major principalstress is σ1 170 - 60 110kN/m2The pore pressure change may be calculated from equation (7.9) u B [ σ 3 A( σ 1 σ 3 )] 1.0 [0 0.5 (110 - 0)] 55kN/m2(9.9)Since the pore pressure was zero at the beginning of the undrained loading stage the porepressure at the end of loading is equal to 55kN/m2.In Fig. 7.9 the total stress Mohr circle at the end of loading has been drawn withσ3 60kN/m2 and σ1 170kN/m2Clearly the total stress path is represented by line RQ where Q is located at the top of thecircle. Alternatively the stress path may have been drawn after calculating the coordinates ofpoint Qp σ1 σ32 170 602 115kN/m2q σ1 - σ32 170 - 602 55kN/m2For the end point P of the effective stress path the coordinates are:q 55kN/m2p' p-u 115 - 55 60kN/m2This data enables the effective stress path RP to be drawn.
7-177.4PRINCIPAL STRESS PLOTThere are many ways of graphically representing changes in the state of stress on a soilspecimen, apart from the q - p diagram discussed in section 7.2. Another way of representing thechanges in state of stress is by means of a principal stress plot. That is successive values of σ1 areplotted against σ3 during loading to identify the total stress path as illustrated in Fig. 7.10. For theeffective stress path σ'1 is plotted against σ'3. The stress paths RT and QS plotted on Fig. 7.10correspond 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 0should be drawn.(b)On a principal stress plot describe the direction in which lines representing p 0should be drawn.(c)Loading of a soil sample commences from an initial stress state represented by q 10kPaand 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 σ1 andσ3 at the end of loading.(a)The lines are parallel to the diagonal line σ1 σ3, sometimes called the space diagonal.(b)The lines are perpendicular to the space diagonal.(c)q (σ1 - σ3) /2p (σ1 σ3) /2 σ1 p q 60kPa initiallyand σ3 p-q40kPa initially
7-18(kPa) σσ1(kPa) 020306090-3015010 p q(kPa)Fig.7.10 Principal Stress Plot
7-19Fig.7.11The stress path which is a straight line is plotted in Fig. 7.11 and the final values of σ1 and σ3 aregiven in the table above.
7-20REFERENCESLambe, 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.
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 ( σ1 σ3), the stress paths going in different directions depending on the relative changes to σ1 and σ3. Fig. 7.5(b) shows stress paths .
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