3. Stress Distribution in Soil 55
3.5 Additional Stress in Ground Base
The increase in pressure is triggered by constructing the structure after the earth is excavated, so it is the contact pressure of the underside of the foundation after subtracting the original overburden pressure.
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 74
74 Soil Mechanics
Fig. 3.13. Stress state of the soil due to a vertical concentrated load.
σx = 3F 2π ·
x2z
R5 +1−2υ 3
1
R(R+z) − (2R+z)x2 (R+z)2R3 − z
R3
, (3.23c) τxy = 3F
2π · xyz
R5 +1−2υ
3 ·(2R+z)xy (R+z)2R3
, (3.23d)
τzy= 3F 2π ·yz2
R5, (3.23e)
τzx= 3F 2π ·xz2
R5 . (3.23f)
The above equations are the well-known Boussinesq solutions.
They are basic formulae for solving the stress increment in the ground. In soil mechanics (as against in elasticity), a uniform sign convention has been used in that the following are considered as positive: compressive stress, reduction in length or volume, and displacement in the positive coordinate direction.
In soil mechanics, the vertical (or normal) stress componentσzon the horizontal plane is of special importance, as it is the major cause of the compressive deformation of the ground soil. Therefore, the calculation of the additional stress and the analyses of its distribution pattern are discussed below.
In the light of the geometrical relationship R2 = r2 + z2 in Fig. 3.13, Eq. (3.23a) can be rewritten as given in the
following form:
σz = 3F 2π · z3
R5 = 3F
2π·z2 · 1
1 +r
z
25
2
=α· F
z2, (3.24) where
α= 3
2π · 1
1 +r
z
252
is the coefficient of the additional stress beneath the underside of the foundation due to a vertical concentrated load. It is a function ofr/z and it can be read off in Table 3.2.
It can be seen from Eq. (3.24) that the following three conclusions can be obtained:
(1) On the concentrated load line (r = 0, α= 2π3 , σz = 2π3 · zp2), the additional stress decreases with increasing depthz, as shown in Fig. 3.14.
(2) At a certain distance r away from the concentrated load line, the additional stress σz is zero at the ground surface, and it increases gradually with increasing depth. However,σz decreases with increasing depth, as shown in Fig. 3.14.
(3) On a horizontal plane at a certain depth z, the additional stress decreases with increasingr, as shown in Fig. 3.14.
3.5.2 Additional stress beneath the corners of the underside of a rectangular foundation due to a vertical uniform load
When a vertical uniform load (hereby referred to as the compressive stress, ditto) is applied to the underside of a rectangular foundation, the additional stress under the corners of the foundation can be calculated by integrating the basic Eq. (3.24) with respect to the whole rectangular area, as shown in Fig. 3.15. If the vertical uniform load intensity on the underside of a foundation is p, then the acting force dp on the infinitesimal area dxdy is pdxdy and it can be
September3,202017:14SoilMechanics-9inx6inb3878-ch03page76
76SoilMechanics
Table 3.2. The coefficient of the additional stressαdue to a vertical concentrated load.
r/z α r/z α r/z α r/z α r/z α r/z α r/z α
0.00 0.4775 0.30 0.3849 0.60 0.2214 0.90 0.1083 1.20 0.0513 1.50 0.0251 2.00 0.0085 0.02 0.4770 0.32 0.3742 0.62 0.2117 0.92 0.1031 1.22 0.0489 1.54 0.0225 2.10 0.0070 0.04 0.4756 0.34 0.3632 0.64 0.2024 0.94 0.0981 1.24 0.0466 1.58 0.0209 2.20 0.0058 0.06 0.4732 0.36 0.3521 0.66 0.1934 0.96 0.0933 1.26 0.0443 1.60 0.0200 2.40 0.0040 0.08 0.4699 0.38 0.3408 0.68 0.1846 0.98 0.0887 1.28 0.0422 1.64 0.0183 2.60 0.0029 0.10 0.4657 0.40 0.3294 0.70 0.1762 1.00 0.0844 1.30 0.0402 1.68 0.0167 2.80 0.0021 0.12 0.4607 0.42 0.3181 0.72 0.1681 1.02 0.0803 1.32 0.0384 1.70 0.0160 3.00 0.0015 0.14 0.4548 0.44 0.3068 0.74 0.1603 1.04 0.0764 1.34 0.0365 1.74 0.0147 3.50 0.0007 0.16 0.4482 0.46 0.2955 0.76 0.1527 1.06 0.0727 1.36 0.0348 1.78 0.0135 4.00 0.0004 0.18 0.4409 0.48 0.2843 0.78 0.1455 1.08 0.0691 1.38 0.0332 1.80 0.0129 4.50 0.0002 0.20 0.4329 0.50 0.2733 0.80 0.1386 1.10 0.0658 1.40 0.0317 1.84 0.0119 5.00 0.0001 0.22 0.4242 0.52 0.2625 0.82 0.1320 1.12 0.0628 1.42 0.0302 1.88 0.0109
0.24 0.4151 0.54 0.2518 0.84 0.1257 1.14 0.0595 1.44 0.0288 1.90 0.0106 0.26 0.4054 0.56 0.2414 0.86 0.1196 1.16 0.0567 1.46 0.0275 1.94 0.0097 0.28 0.3954 0.58 0.2313 0.88 0.1138 1.18 0.0539 1.48 0.0263 1.98 0.0089
Fig. 3.14. Distribution of stress increment due to a point load.
Fig. 3.15. Underside of a rectangular foundation subjected to vertical uniform load.
considered as a concentrated point load. Therefore, the additional stress at a depth z under the foundation corner O induced by this point load is given by
dσz = 3p
2π · 1
1 +r
z
25/2 ·dxdy
z2 . (3.25)
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 78
78 Soil Mechanics
Substitutingr2 =x2+y2into the above equation and integrating it with respect to the whole area of the underside of a foundation, we see that the additional stress induced by the vertical uniform load at depth z under the corner O of the underside of the rectangular foundation will be obtained, as expressed by
σz= b
o
l
o
3p
2π · z3dxdy (
x2+y2+z2)5
= p 2π
mn
√1 +m2+n2 · 1
m2+n2 + 1 1 +n2
+ arctan
m n√
1 +m2+n2
=αcp, (3.26)
whereαc is the coefficient of the additional stress under the cornerO of the underside of a rectangular foundation due to a vertical uniform load. It is a function of m(=l/b) and n(=z/b), namely,
αc =f(m, n) = p 2π
mn
√1 +m2+n2 · 1
m2+n2 + 1 1 +n2
+ arctan
m n√
1 +m2+n2
and it can be read off in Table 3.2, wherelis the length of the longer side of the underside of the foundation and b is the width of the shorter side of the underside of the foundation.
For any point inside and outside the range of underside of the foundation, the additional stress can be calculated by using Eq. (3.26) and the principle of superposition.
3.5.2.1 The point is in the side of the foundation
Consider a vertical uniform load p acting on the underside of a rectangular foundationbhfc, as shown in Fig. 3.16(a). In order to find the additional stressσz at any depthzunder pointM, we can draw one auxiliary line eM parallel to the side of the foundation. Point M is the common corner of two rectangles hbMe(I) andeMfc(II).
Therefore, the additional stress at any depth z under M is the
(a) (b) (c) (d)
Fig. 3.16. Point that is considered for calculation inside or outside of the underside of the foundation.
sum of the additional stress on the above two new undersides of the foundation, namely,
σz = (αcI+αcII)p.
3.5.2.2 The point is inside of the foundation
Consider a vertical uniform load p acting on the underside of a rectangular foundation abcd, as shown in Fig. 3.16(b). In order to find the additional stress σz at any depth z under point M, we can draw two auxiliary lines eg and hf parallel to the longer side and the shorter side of the foundation, respectively. Point M is the common corner of four rectangles bhMe(I), eMfc(II), hagM(III), and Mgdf(IV). Therefore, the additional stress at any depth z under M is the sum of the additional stress on the above four new undersides of the foundation, namely,
σz = (αcI+αcII+αcIII+αcIV)p.
3.5.2.3 The point is outside of the side of the foundation
Consider a vertical uniform load p acting on the underside of a rectangular foundation abcd, as shown in Fig. 3.16(c). In order to find the additional stress σz at any depth z under point M, we can draw two auxiliary lines eM and hf parallel to the longer side and the shorter side of the foundation, respectively.
Point M is the common corner of four rectangles bhMe(I), eMfc(II), hagM(III), and Mgdf(IV). Therefore, the additional stress at any depth z under M is the sum of the additional
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 80
80 Soil Mechanics
stress on the above four new undersides of the foundation, namely,
σz = (αcI+αcII−αcIII−αcIV)p.
3.5.2.4 The point is outside of the corner of the foundation
Consider a vertical uniform load p acting on the underside of a rectangular foundation abcd, as shown in Fig. 3.16(d). In order to find the additional stress σz at any depthz under pointM, we can draw two auxiliary lines eM and hM parallel to the longer side and the shorter side of the foundation, respectively. Point M is the common corner of four rectangles bhMe(I), eMfc(II), hagM(III), and Mfdg(IV). Therefore, the additional stress at any depth z under M is the sum of the additional stress on the above four new undersides of the foundation, namely,
σz = (αcI−αcII−αcIII+αcIV)p,
where lis the longer side and bis the shorter side in Table 3.3.
Example 3.3. There are two adjacent foundationsA andB, their sizes, positions, and the additional stress distributions are all shown in Fig. 3.17. Considering the effect of the adjacent foundationB, try to find the additional stress at a depth z of 2 m under the center point O of foundationA.
Solution.
• Step 1: The additional stress on the center pointO due to the vertical uniform load of the foundationA.
In order to obtain the additional stress under point O, the underside of foundationAis divided into four rectangles of equal area of 1 m×1 m. The sum of the additional stress of the four rectangles is the same as the additional stress on the center point O of the foundationA, namely,
σz = 4αcpA.
With l/b = 1/1 = 1 and z/b = 2/1 = 2, and from Table 3.3, we can get αc = 0.0840. Therefore, the additional stress under
Table 3.3. Additional stress coefficient αc under the corner of the underside of rectangular foundation due to a vertical load.
σz=αcp
l/b
z/b 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 10.0
0.0 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.250 0.2500 0.2500 0.2500 0.2 0.2486 0.2489 0.2490 0.2491 0.2491 0.2491 0.2492 0.2492 0.2492 0.2492 0.2492 0.4 0.2401 0.2420 0.2429 0.2434 0.2437 0.2439 0.2442 0.2443 0.2443 0.2443 0.2443 0.6 0.2229 0.2275 0.2300 0.2315 0.2324 0.2329 0.2339 0.2341 0.2342 0.2342 0.2342 0.8 0.1999 0.2075 0.2120 0.2147 0.2165 0.2176 0.2196 0.2200 0.2202 0.2202 0.2202 1.0 0.1752 0.1851 0.1911 0.1955 0.1981 0.1999 0.2034 0.2042 0.2044 0.2045 0.2046 1.2 0.1516 0.1626 0.1705 0.1758 0.1793 0.1818 0.1870 0.1882 0.1885 0.1887 0.1888 1.4 0.1308 01423 0.1508 0.1569 0.1613 0.1644 0.1712 0.1730 0.1735 0.1738 0.1740 1.6 0.1123 0.1241 0.1329 0.1396 0.1445 0.1482 0.1567 0.1590 0.1598 0.1601 0.1604 1.8 0.0969 0.1083 0.1172 0.1241 0.1294 0.1334 0.1434 0.1463 0.1474 0.1478 0.1482 2.0 0.0840 0.0947 0.1034 0.1103 0.1158 0.1202 0.1314 0.1350 0.1363 0.1368 0.1374 2.2 0.0732 0.0832 0.0917 0.0984 0.1039 0.1084 0.1205 0.1248 0.1264 0.1271 0.1277 2.4 0.0642 0.0734 0.0813 0.0879 0.0934 0.0979 0.1108 0.1156 0.1175 0.1184 0.1192 2.6 0.0566 0.0651 0.0725 0.0788 0.0842 0.0887 0.1020 0.1073 0.1095 0.1106 0.1116 2.8 0.0502 0.0580 0.0649 0.0709 0.0761 0.0805 0.0942 0.0999 0.1024 0.1036 0.1048 3.0 0.0447 0.0519 0.0583 0.0640 0.0690 0.0732 0.0870 0.0931 0.0959 0.0973 0.0987 3.2 0.0401 0.0467 0.0526 0.0580 0.0627 0.0668 0.0806 0.0870 0.0900 0.096 0.0933 3.4 0.0361 0.0421 0.0477 0.0527 0.0571 0.0611 0.0747 0.0814 0.847 0.0864 0.0882 3.6 0.0326 0.0382 0.0433 0.0480 0.0523 0.0561 0.0694 0.0763 0.0799 0.0816 0.0837 3.8 0.0296 0.0348 0.0395 0.0439 0.0479 0.0516 0.0646 0.0717 0.0753 0.0773 0.0796 4.0 0.0270 0.0318 0.0362 0.0404 0.0441 0.0474 0.0603 0.0674 0.0712 0.0733 0.0758 4.2 0.0247 0.0291 0.0333 0.0371 0.0407 0.0439 0.0563 0.0634 0.0674 0.0696 0.0724 4.4 0.0227 0.0268 0.0306 0.0343 0.0376 0.0407 0.0527 0.0597 0.0639 0.0662 0.0692 4.6 0.0209 0.0247 0.0283 0.0317 0.0348 0.0378 0.0493 0.0564 0.0606 0.0630 0.0663 4.8 0.0193 0.0229 0.0262 0.0294 0.0324 0.0352 0.0463 0.0533 0.0576 0.0601 0.0635 5.0 0.0179 0.0212 0.0243 0.0274 0.0302 0.0328 0.0435 0.0504 0.0547 0.0573 0.0610 6.0 0.0127 0.0151 0.0174 0.0196 0.02218 0.0238 0.0325 0.0388 0.0431 0.0460 0.0506 7.0 0.0094 0.0112 0.0130 0.0147 0.0164 0.0180 0.0251 0.0306 0.0346 0.0376 0.0428 8.0 0.0073 0.0087 0.0101 0.0114 0.0127 0.0140 0.0198 0.0246 0.0283 0.0311 0.0367 9.0 0.0058 0.0069 0.0080 0.0091 0.0102 0.0112 0.0161 0.0202 0.0235 0.0262 0.0319 10.0 0.0047 0.0056 0.0065 0.0074 0.0083 0.0092 0.0132 0.0167 0.0198 0.0222 0.0280
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 82
82 Soil Mechanics
Fig. 3.17. Diagram for Example 3.3.
point O is
σz = 4αcpA= 4×0.0840×200 = 67.2 kPa.
• Step 2: The additional stress on the center pointO due to the vertical uniform load of the foundationB.
According to the figure, the point O lies outside of the foundationBand the additional stress is calculated as shown in Fig. 3.18, that is
σz = (αcI−αcII−αcIII+αcIV)pB, where
rectangle I: l = 4 m, b = 4 m; with l/b = 1, z/b = 0.5, from Table 3.3, we can getαc = 0.2315;
rectangle II: l = 4 m, b = 2 m; with l/b = 2, z/b = 1, from Table 3.3, we can getαc = 0.1999;
rectangle III: l = 4 m, b = 1 m; with l/b = 4, z/b = 2, from Table 3.3, we can getαc = 0.1350;
rectangle IV: l = 2 m, b= 1 m; with l/b = 2, z/b = 2, from Table 3.3, we can getαc = 0.1202.
σz = (αcI−αcII−αcIII+αcIV)pB = (0.2315−0.1999−0.1350 + 0.1202)×300 = 5.28 kPa.
• Step 3:The additional stress on the center point O at the 2 m depth of the foundationA
σz= 67.2 + 5.28 = 72.48 kPa.
Fig. 3.18. The additional stress on the center pointOof the foundationA.
3.5.3 Additional stress under the corners of the underside of a rectangular foundation due to a vertical triangular load
When a triangularly distributed load (i.e. the triangularly distributed compressive pressure, ditto) is applied to the underside of a rectan- gular foundation, the additional stress under the corner where the load intensity is zero can also be calculated by integrating Eq. (3.24) with respect to the whole loading area. If the maximum intensity of the triangular load on the underside of the rectangular foundation is pt, then the acting force dF on the infinitesimal area dxdy equals
ptx
b dxdy, which can be considered as a concentrated point load, as shown in Fig. 3.19. Therefore, the additional stress at an arbitrary depth z under the corner O induced by this point load can be calculated as given by
dσz = 3pt
2πb· 1
1 +r
z
25/2 ·xdxdy
z2 , (3.27)
Substituting r2 = x2+y2 into the above equation and integrating it with respect the whole area of the underside of the foundation, the additional stress induced by the vertical uniform load at depthz under the corner O of the rectangular foundation will be obtained,
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 84
84 Soil Mechanics
Fig. 3.19. Rectangular foundation subjected to a triangularly distributed load.
as expressed by
σz =αtcpt, (3.28)
αtc= mn 2π
1
√m2+n2 − n2 (1 +n2)√
1 +m2+n2
, (3.29) whereαtcis the coefficient of the additional stress under the corner of the underside of a rectangular foundation due to a vertical triangular load. It is a function of m(= l/b) and n(= z/b) and it can be read off in Table 3.4, whereb is the length of one side of the underside of the foundation in the loading variation direction and l is the length of another side.
For the additional stress at any point inside and outside the range of the underside of the foundation, it can also be calculated using the principle of superposition. Two points should, however, be noted: the calculation point should be on the vertical line under the point where the triangular load intensity is zero; b represents the side length of the underside of the rectangular foundation in the loading variation direction.
When a horizontal uniform load ph is applied to the underside of a rectangular foundation (Fig. 3.20), the additional stress at an
arbitrary depth zunder the corner of the foundation is given as
σz =±αl·ph, (3.30a)
αl = m 2π
1
√m2+n2 − n2 (1 +n2)√
1 +m2+n2
, (3.30b)
Table 3.4. Additional stress coefficientαtc under the corner of the underside of a rectangular foundation due to a vertical triangular load.
l/b
z/b 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2 0.0223 0.0280 0.0296 0.0301 0.0304 0.0305 0.0305 0.0306 0.4 0.0269 0.0420 0.0487 0.0517 0.0531 0.0539 0.0543 0.0545 0.6 0.0259 0.0448 0.0560 0.0621 0.0654 0.0673 0.0684 0.0690 0.8 0.0232 0.0421 0.0553 0.0637 0.0688 0.0720 0.0739 0.0751 1.0 0.0201 0.0375 0.0508 0.0602 0.0666 0.0708 0.0735 0.0753 1.2 0.0171 0.0324 0.0450 0.0546 0.0615 0.0664 0.0698 0.0721 1.4 0.0145 0.0278 0.0392 0.0483 0.0554 0.0606 0.0644 0.0672 1.6 0.0123 0.0238 0.0339 0.0424 0.0492 0.0545 0.0586 0.0616 1.8 0.0105 0.0204 0.0294 0.0371 0.0435 0.0487 0.0528 0.0560 2.0 0.0090 0.0176 0.0255 0.0324 0.0384 0.0434 0.0474 0.0507 2.5 0.0063 0.0125 0.0183 0.0236 0.0284 0.0326 0.0362 0.0393 3.0 0.0046 0.0092 0.0135 0.0176 0.0214 0.0249 0.0280 0.0307 5.0 0.0018 0.0036 0.0054 0.0071 0.0088 0.0104 0.0120 0.0135 7.0 0.0009 0.0019 0.0028 0.0038 0.0047 0.0056 0.0064 0.0073 10.0 0.0005 0.0009 0.0014 0.0019 0.0023 0.0028 0.0033 0.0037 (Continued)
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch03 page 86
86 Soil Mechanics
Table 3.4. (Continued) l/b
z/b 1.8 2.0 3.0 4.0 6.0 8.0 10.0
0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2 0.0306 0.0306 0.0306 0.0306 0.0306 0.0306 0.0306 0.4 0.0546 0.0547 0.0548 0.0549 0.0549 0.0549 0.0549 0.6 0.0694 0.0696 0.0701 0.0702 0.0702 0.0702 0.0702 0.8 0.0759 0.0764 0.0773 0.0776 0.0776 0.0776 0.0776 1.0 0.0766 0.0774 0.0790 0.0794 0.0795 0.0796 0.0796 1.2 0.0738 0.0749 0.0774 0.0779 0.0782 0.0783 0.0783 1.4 0.0692 0.0707 0.0739 0.0748 0.0752 0.0752 0.0753 1.6 0.0639 0.0656 0.0697 0.0708 0.0714 0.0715 0.0715 1.8 0.0585 0.0604 0.0652 0.0666 0.0673 0.0675 0.0675 2.0 0.0533 0.0553 0.0607 0.0624 0.0634 0.0636 0.0636 2.5 0.0419 0.0440 0.0504 0.0529 0.0543 0.0547 0.0548 3.0 0.0331 0.0352 0.0419 0.0449 0.0469 0.0474 0.0476 5.0 0.0148 0.0161 0.0214 0.0248 0.0283 0.0296 0.0301 7.0 0.0081 0.0089 0.0124 0.0152 0.01860 0.0204 0.0212 10.0 0.0041 0.0046 0.0066 0.0084 0.0111 0.0128 0.0139
whereαl is the coefficient of the additional stress under the corner of the underside of a rectangular foundation due to a horizontal uniform load. It is a function of m(=l/b) andn(=z/b) and it can be read off in Table 3.5, wherebis the length of one side of the underside of the foundation in the direction parallel to the horizontal load direction and l is the length of another side.
In the above equation, the positive sign “+” is taken at the calculation point under the stopping end of the horizontal uniform load (under point 2) and the negative sign “–” is taken at the calculation point under the starting end of the horizontal uniform load (under point 1).
The additional stress at any point inside and outside the range of the underside of the foundation can also be calculated using the principle of superposition.
Fig. 3.20. Rectangular foundation subjected to a horizontal uniform load.