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# Craig's Soil Mechanics 7th Edition

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was shown that at small strains the shear stress in the soil at the wall of the expanding cavity is given by � ¼ "c dp d"c ð5:8Þ and at larger strains by � ¼ dp d½lnð�V=VÞ� ð5:9Þ both the circumferential and volumetric strains being defined with respect to the reference state. Equations 5.8 and 5.9 can be used to derive the entire stress–strain curve for the soil. An analysis for the interpretation of pressuremeter tests in sands has been given by Hughes et al. [9]. The analysis enables values for the angle of shearing resistance (�0) and the angle of dilation ( ) to be determined. A comprehensive review of the use of pressuremeters, including examples of test results and their application in design, has been given by Mair and Wood [10]. 5.2 STRESSES FROM ELASTIC THEORY The stresses within a semi-infinite, homogeneous, isotropic mass, with a linear stress– strain relationship, due to a point load on the surface, were determined by Boussinesq in 1885. The vertical, radial, circumferential and shear stresses at a depth z and a horizontal distance r from the point of application of the load were given. The stresses due to surface loads distributed over a particular area can be obtained by integration from the point load solutions. The stresses at a point due to more than one surface load are obtained by superposition. In practice, loads are not usually applied directly on the surface but the results for surface loading can be applied conservatively in problems concerning loads at a shallow depth. A range of solutions, suitable for determining the stresses below foundations, is given in the following sections. Negative values of loading can be used if the stresses due to excavation are required or in problems in which the principle of superposition is used. The stresses due to surface loading act in addition to the in-situ stresses due to the self-weight of the soil. Point load Referring to Figure 5.5(a), the stresses at X due to a point load Q on the surface are as follows: �z ¼ 3Q 2 z2 1 1þ ðr=zÞ2 ( )5=2 ð5:10Þ 144 Stresses and displacements Figure 5.5 (a) Stresses due to point load and (b) variation of vertical stress due to point load. �r ¼ Q 2 3r2z ðr2 þ z2Þ5=2 � 1� 2v r2 þ z2 þ zðr2 þ z2Þ1=2 ( ) ð5:11Þ �� ¼ � Q 2 ð1� 2vÞ z ðr2 þ z2Þ3=2 � 1 r2 þ z2 þ zðr2 þ z2Þ1=2 ( ) ð5:12Þ �rz ¼ 3Q 2 rz2 ðr2 þ z2Þ5=2 ( ) ð5:13Þ It should be noted that when v ¼ 0:5 the second term in Equation 5.11 vanishes and Equation 5.12 gives �� ¼ 0. Equation 5.10 is used most frequently in practice and can be written in terms of an influence factor Ip, where Ip ¼ 3 2 1 1þ ðr=zÞ2 ( )5=2 Then, �z ¼ Q z2 Ip Values of Ip in terms of r/z are given in Table 5.1. The form of the variation of �z with z and r is illustrated in Figure 5.5(b). The left-hand side of the figure shows the variation of �z with z on the vertical through the point of application of the load Q (i.e. for r ¼ 0); the right-hand side of the figure shows the variation of �z with r for three different values of z. It should be noted that the expression for �z (Equation 5.10) is independent of elastic modulus (E ) and Poisson’s ratio (�). Table 5.1 Influence factors for vertical stress due to point load r/z Ip r/z Ip r/z Ip 0.00 0.478 0.80 0.139 1.60 0.020 0.10 0.466 0.90 0.108 1.70 0.016 0.20 0.433 1.00 0.084 1.80 0.013 0.30 0.385 1.10 0.066 1.90 0.011 0.40 0.329 1.20 0.051 2.00 0.009 0.50 0.273 1.30 0.040 2.20 0.006 0.60 0.221 1.40 0.032 2.40 0.004 0.70 0.176 1.50 0.025 2.60 0.003 146 Stresses and displacements Line load Referring to Figure 5.6(a), the stresses at point X due to a line load ofQ per unit length on the surface are as follows: �z ¼ 2Q z3 ðx2 þ z2Þ2 ð5:14Þ �x ¼ 2Q x2z ðx2 þ z2Þ2 ð5:15Þ �xz ¼ 2Q xz2 ðx2 þ z2Þ2 ð5:16Þ Figure 5.6 (a) Stresses due to line load and (b) lateral pressure due to line load. Stresses from elastic theory 147 Equation 5.15 can be used to estimate the lateral pressure on an earth-retaining structure due to a line load on the surface of the backfill. In terms of the dimensions given in Figure 5.6(b), Equation 5.15 becomes �x ¼ 2Q h m2n ðm2 þ n2Þ2 However, the structure will tend to interfere with the lateral strain due to the load Q and to obtain the lateral pressure on a relatively rigid structure a second load Q must be imagined at an equal distance on the other side of the structure. Then, the lateral pressure is given by px ¼ 4Q h m2n ðm2 þ n2Þ2 ð5:17Þ The total thrust on the structure is given by Px ¼ Z 1 0 pxh dn ¼ 2Q 1 m2 þ 1 ð5:18Þ Strip area carrying uniform pressure The stresses at point X due to a uniform pressure q on a strip area of width B and infinite length are given in terms of the angles and � defined in Figure 5.7(a). �z ¼ q f þ sin cosð þ 2�Þg ð5:19Þ �x ¼ q f � sin cosð þ 2�Þg ð5:20Þ �xz ¼ q fsin sinð þ 2�Þg ð5:21Þ Contours of equal vertical stress in the vicinity of a strip area carrying a uniform pressure are plotted in Figure 5.8(a). The zone lying inside the vertical stress contour of value 0.2q is described as the bulb of pressure. Strip area carrying linearly increasing pressure The stresses at point X due to pressure increasing linearly from zero to q on a strip area of width B are given in terms of the angles and � and the lengths R1 and R2, as defined in Figure 5.7(b). 148 Stresses and displacements Figure 5.7 Stresses due to (a) uniform pressure and (b) linearly increasing pressure, on strip area. Figure 5.8 Contours of equal vertical stress: (a) under strip area and (b) under square area. �z ¼ q x B � 1 2 sin 2� � � ð5:22Þ �x ¼ q x B � z B ln R21 R22 þ 1 2 sin 2� � � ð5:23Þ �xz ¼ q 2 1þ cos 2� � 2 z B � ð5:24Þ Circular area carrying uniform pressure The vertical stress at depth z under the centre of a circular area of diameter D ¼ 2R carrying a uniform pressure q is given by �z ¼ q 1� 1 1þ ðR=zÞ2 ( )3=224 3 5 ¼ qIc ð5:25Þ Values of the influence factor Ic in terms of D/z are given in Figure 5.9. The radial and circumferential stresses under the centre are equal and are given by �r ¼ �� ¼ q 2 ð1þ 2vÞ � 2ð1þ vÞ f1þ ðR=zÞ2g1=2 þ 1 f1þ ðR=zÞ2g3=2 " # ð5:26Þ Figure 5.9 Vertical stress under the centre of a circular area carrying a uniform pressure. 150 Stresses and displacements Rectangular area carrying uniform pressure A solution has been obtained for the vertical stress at depth z under a corner of a rectangular area of dimensions mz and nz (Figure 5.10) carrying a uniform pressure q. The solution can be written in the form �z ¼ qIr Values of the influence factor Ir in terms of m and n are given in the chart, due to Fadum [5], shown in Figure 5.10. The factors m and n are interchangeable. The chart can also be used for a strip area, considered as a rectangular area of infinite length. Superposition enables any area based on rectangles to be dealt with and enables the vertical stress under any point within or outside the area to be obtained. Figure 5.10 Vertical stress under a corner of a rectangular area carrying a uniform pressure. (Reproduced from R.E. Fadum (1948) Proceedings of the 2nd International Conference of SMFE, Rotterdam, Vol. 3, by permission of Professor Fadum.) Stresses from elastic theory 151 Contours of equal vertical stress in the vicinity of a square area carrying a uniform pressure are plotted in Figure 5.8(b). Influence factors for �x and �y (which depend on v) are given in Ref. [16]. Influence chart for vertical stress Newmark [12] constructed an influence chart, based on the Boussinesq solution, enabling the vertical stress to be determined