Stress Distribution Solutions
This spreadsheet contains solutions for stresses within an infinte elastic half-space when
subject to various surface loadings. All solutions use the Boussinesq solution. Two sources
are used for the solutions:
Poulos & Davis (1974) Elastic Solutions for Soil and Rock Mechanics, Wiley, New York
Coduto, Yeung, and Kitch (2011) Geotechnical Engineering: Principles and Practices,
2nd Ed., Printice Hall, New York
Stresses due to point load–Boussinesq’s Solution
Coduto, Yeung & Kitch (2011) Equations 9.23 through 9.30
© William A Kitch 2010
Ver 1.2 5/14/2010 Title:
P = 350.00 force z-Incr
x = 1 0.2
y = 0
n = 0.3
r = 1
z
s
x
s
y
s
z
t
zx
t
xy
t
zy R
0.00 0.0000 0.0000 #N/A 0 0 0 1.0000
0.20 12.3890 13.7102 1.2120 6.060188 0 0 1.0198
0.40 32.1173 6.8727 7.3798 18.44949 0 0 1.0770
0.60 35.6672 2.3886 16.7346 27.89104 0 0 1.1662
0.80 30.4517 -0.1249 24.8410 31.0513 0 0 1.2806
1.00 23.0155 -1.3516 29.5416 29.54163 0 0 1.4142
1.20 16.3989 -1.8509 31.0512 25.87601 0 0 1.5620
x
y
xf
yf
P
r
Stresses due to line load–Boussinesq’s Solution
Coduto, Yeung & Kitch (2011) Equation 9.31
© William A Kitch 2010
Ver 1.0 4/14/2010 Title:
P/b = 10.00 force/unit length z-Incr
x = 0 0.5
z
s
x
0.00 #N/A
0.50 12.73
1.00 6.37
1.50 4.24
2.00 3.18
2.50 2.55
3.00 2.12
3.50 1.82
4.00 1.59
x
y
xf
P/b
Stresses due to line load–Boussinesq’s Solution
Poulos & Davis (1974) pg 26
© William A Kitch 2010
Ver 1.2 5/14/2010 Title:
P/b = 1.00 force/unit length z-Incr
x = 0 0.5
n = 0.35
z
s
x
s
y
s
z
t
xz R
0.00 #N/A #N/A #N/A #N/A 0.00
0.50 0.00 0.45 1.27 0.00 0.50
1.00 0.00 0.22 0.64 0.00 1.00
1.50 0.00 0.15 0.42 0.00 1.50
2.00 0.00 0.11 0.32 0.00 2.00
2.50 0.00 0.09 0.25 0.00 2.50
3.00 0.00 0.07 0.21 0.00 3.00
3.50 0.00 0.06 0.18 0.00 3.50
4.00 0.00 0.06 0.16 0.00 4.00
x
y
xf
P/b
Stresses under center of a circularly loaded area-Boussinesq’s Solution
Poulos & Davis (1974) pg 43
© William A Kitch 2010
Ver 1.0 5/14/2010 Title:
q = 105.00 force/unit area z-Incr
b = 10.5 0.5
n = 0.4
z
s
z
s
r
sq
0.00 105.000 94.500 94.500
0.50 104.989 87.514 87.514
1.00 104.911 80.608 80.608
1.50 104.703 73.860 73.860
2.00 104.312 67.338 67.338
2.50 103.695 61.104 61.104
P/b
B
b
Stresses under infinite strip load–Boussinesq’s Solution
Poulos & Davis (1974) pg 36ff
© William A Kitch 2007
Ver 1.2 5/24/2009 Title:
q = 1 force/unit area
z-Incr
b = 1 0.05
x = 3
n = 0.35
for B = 2b
z z/b z/B
s
z
s
x
s
y
t
zx
d a
0.00 0.00 0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.05 0.05 0.03 0.0000 0.0080 0.0028 0.0001 1.5458 0.0125
0.10 0.10 0.05 0.0000 0.0159 0.0056 0.0006 1.5208 0.0250
0.95 0.95 0.48 0.0151 0.1187 0.0469 0.0416 1.1273 0.2103
1.00 1.00 0.50 0.0172 0.1220 0.0487 0.0449 1.1071 0.2187
1.05 1.05 0.53 0.0193 0.1250 0.0505 0.0483 1.0873 0.2267
1.10 1.10 0.55 0.0216 0.1277 0.0522 0.0515 1.0680 0.2345
1.15 1.15 0.58 0.0240 0.1300 0.0539 0.0548 1.0490 0.2419
1.20 1.20 0.60 0.0264 0.1321 0.0555 0.0580 1.0304 0.2490
1.25 1.25 0.63 0.0290 0.1338 0.0570 0.0611 1.0122 0.2557
1.30 1.30 0.65 0.0316 0.1353 0.0584 0.0641 0.9944 0.2621
1.35 1.35 0.68 0.0342 0.1365 0.0598 0.0671 0.9770 0.2683
1.40 1.40 0.70 0.0369 0.1375 0.0611 0.0699 0.9601 0.2741
1.45 1.45 0.73 0.0397 0.1383 0.0623 0.0727 0.9435 0.2795
1.50 1.50 0.75 0.0425 0.1388 0.0634 0.0753 0.9273 0.2847
1.55 1.55 0.78 0.0453 0.1391 0.0645 0.0779 0.9115 0.2896
1.60 1.60 0.80 0.0481 0.1392 0.0656 0.0803 0.8961 0.2942
1.65 1.65 0.83 0.0510 0.1391 0.0665 0.0826 0.8810 0.2986
2.50 2.50 1.25 0.0952 0.1196 0.0752 0.1047 0.6747 0.3375
2.55 2.55 1.28 0.0974 0.1179 0.0753 0.1051 0.6651 0.3382
2.60 2.60 1.30 0.0994 0.1162 0.0755 0.1054 0.6557 0.3387
2.65 2.65 1.33 0.1015 0.1145 0.0756 0.1057 0.6465 0.3392
2.70 2.70 1.35 0.1034 0.1127 0.0756 0.1059 0.6375 0.3395
2.75 2.75 1.38 0.1053 0.1109 0.0757 0.1060 0.6288 0.3397
2.80 2.80 1.40 0.1072 0.1092 0.0757 0.1061 0.6202 0.3398
2.85 2.85 1.43 0.1089 0.1074 0.0757 0.1061 0.6119 0.3398
2.90 2.90 1.45 0.1106 0.1056 0.0757 0.1060 0.6037 0.3397
2.95 2.95 1.48 0.1123 0.1039 0.0757 0.1059 0.5958 0.3396
3.00 3.00 1.50 0.1139 0.1021 0.0756 0.1058 0.5880 0.3393
2b
q
Boussinesq for corner of rectagular area
Coduto, Yeung & Kitch (2011) Equations 9.34 & 9.35
Title:
© William A Kitch 2010
Ver 1.1 1/9/2007 q = 3200 z-Incr
R1 = b2 + l2 + z2
l = 2.5 0.2
R2 = b2+l2+2z2
b = 2.5
R3 = b2 * l2
b/
l
= 1R4 = b * l * z
R5 = b2 * l2 / z2
z
s
zIqR1 R2 R3 R4 R5
0.00 800.000 0.250
0.50 795.436 0.249 12.75 13.00 39.06 3.13 156.2500
1.00 768.318 0.240 13.50 14.50 39.06 6.25 39.0625
1.50 713.250 0.223 14.75 17.00 39.06 9.38 17.3611
2.00 639.777 0.200 16.50 20.50 39.06 12.50 9.7656
2.50 560.709 0.175 18.75 25.00 39.06 15.63 6.2500
3.00 485.155 0.152 21.50 30.50 39.06 18.75 4.3403
14.00 46.260 0.014 208.50 404.50 39.06 87.50 0.1993
14.50 43.273 0.014 222.75 433.00 39.06 90.63 0.1858
15.00 40.562 0.013 237.50 462.50 39.06 93.75 0.1736
15.50 38.094 0.012 252.75 493.00 39.06 96.88 0.1626
16.00 35.842 0.011 268.50 524.50 39.06 100.00 0.1526
16.50 33.782 0.011 284.75 557.00 39.06 103.13 0.1435
17.00 31.892 0.010 301.50 590.50 39.06 106.25 0.1352
17.50 30.155 0.009 318.75 625.00 39.06 109.38 0.1276
18.00 28.555 0.009 336.50 660.50 39.06 112.50 0.1206
18.50 27.077 0.008 354.75 697.00 39.06 115.63 0.1141
19.00 25.710 0.008 373.50 734.50 39.06 118.75 0.1082
19.50 24.443 0.008 392.75 773.00 39.06 121.88 0.1027
20.00 23.267 0.007 412.50 812.50 39.06 125.00 0.0977
20.50 22.173 0.007 432.75 853.00 39.06 128.13 0.0930
21.00 21.154 0.007 453.50 894.50 39.06 131.25 0.0886
21.50 20.203 0.006 474.75 937.00 39.06 134.38 0.0845
22.00 19.314 0.006 496.50 980.50 39.06 137.50 0.0807
22.50 18.482 0.006 518.75 1025.00 39.06 140.63 0.0772
23.00 17.703 0.006 541.50 1070.50 39.06 143.75 0.0738
23.50 16.971 0.005 564.75 1117.00 39.06 146.88 0.0707
24.00 16.284 0.005 588.50 1164.50 39.06 150.00 0.0678
24.50 15.637 0.005 612.75 1213.00 39.06 153.13 0.0651
25.00 15.028 0.005 637.50 1262.50 39.06 156.25 0.0625
25.50 14.454 0.005 662.75 1313.00 39.06 159.38 0.0601
26.00 13.912 0.004 688.50 1364.50 39.06 162.50 0.0578
26.50 13.399 0.004 714.75 1417.00 39.06 165.63 0.0556
27.00 12.915 0.004 741.50 1470.50 39.06 168.75 0.0536
27.50 12.456 0.004 768.75 1525.00 39.06 171.88 0.0517
28.00 12.020 0.004 796.50 1580.50 39.06 175.00 0.0498
28.50 11.608 0.004 824.75 1637.00 39.06 178.13 0.0481
29.00 11.216 0.004 853.50 1694.50 39.06 181.25 0.0464
29.50 10.843 0.003 882.75 1753.00 39.06 184.38 0.0449
30.00 10.489 0.003 912.50 1812.50 39.06 187.50 0.0434
30.50 10.152 0.003 942.75 1873.00 39.06 190.63 0.0420
31.00 9.830 0.003 973.50 1934.50 39.06 193.75 0.0406
y
b
l
3.50 417.610 0.131 24.75 37.00 39.06 21.88 3.1888
4.00 359.394 0.112 28.50 44.50 39.06 25.00 2.4414
4.50 310.163 0.097 32.75 53.00 39.06 28.13 1.9290
5.00 268.886 0.084 37.50 62.50 39.06 31.25 1.5625
5.50 234.358 0.073 42.75 73.00 39.06 34.38 1.2913
6.00 205.435 0.064 48.50 84.50 39.06 37.50 1.0851
6.50 181.122 0.057 54.75 97.00 39.06 40.63 0.9246
10.00 86.466 0.027 112.50 212.50 39.06 62.50 0.3906
10.50 79.124 0.025 122.75 233.00 39.06 65.63 0.3543
11.00 72.654 0.023 133.50 254.50 39.06 68.75 0.3228
11.50 66.926 0.021 144.75 277.00 39.06 71.88 0.2954
12.00 61.835 0.019 156.50 300.50 39.06 75.00 0.2713
12.50 57.291 0.018 168.75 325.00 39.06 78.13 0.2500
13.00 53.220 0.017 181.50 350.50 39.06 81.25 0.2311
13.50 49.561 0.015 194.75 377.00 39.06 84.38 0.2143
37.50 6.741 0.002 1418.75 2825.00 39.06 234.38 0.0278
38.00 6.566 0.002 1456.50 2900.50 39.06 237.50 0.0271
38.50 6.397 0.002 1494.75 2977.00 39.06 240.63 0.0264
39.00 6.236 0.002 1533.50 3054.50 39.06 243.75 0.0257
39.50 6.080 0.002 1572.75 3133.00 39.06 246.88 0.0250
40.00 5.930 0.002 1612.50 3212.50 39.06 250.00 0.0244
40.50 5.785 0.002 1652.75 3293.00 39.06 253.13 0.0238
41.00 5.646 0.002 1693.50 3374.50 39.06 256.25 0.0232
41.50 5.511 0.002 1734.75 3457.00 39.06 259.38 0.0227
42.00 5.382 0.002 1776.50 3540.50 39.06 262.50 0.0221
42.50 5.256 0.002 1818.75 3625.00 39.06 265.63 0.0216
43.00 5.136 0.002 1861.50 3710.50 39.06 268.75 0.0211
43.50 5.019 0.002 1904.75 3797.00 39.06 271.88 0.0206
Boussinesq for corner of rectagular area
Poulos & Davis (1974) pg 54 [Holl’s solution, 1940]
Title:
© William A Kitch 2010
Ver 1.1 1/9/2007 q = 100 z-Incr R1 = sqrt(l^2+z^2)
l = 5 0.1 R2 = sqrt(b^2+z^2)
b = 2R3 = sqrt(l^2+b^2+z^2)
b/
l
= 0.4
z/l z
s
x
s
y
s
z
t
xz
t
yz
t
xy R1 R2 R3
0.00 0.00 25.0000 25.0000 25.0000 15.9155 15.9155 15.9156 5.00 2.00 5.39
0.10 0.50 20.2139 17.3345 24.8415 15.3820 14.9710 19.6634 5.02 2.06 5.41
0.20 1.00 15.9072 11.2132 23.9539 14.0117 12.7006 22.8176 5.10 2.24 5.48
0.30 1.50 12.3270 7.0613 22.2943 12.2622 10.1195 25.1621 5.22 2.50 5.59
0.40 2.00 9.4881 4.4725 20.2359 10.4896 7.8508 26.7997 5.39 2.83 5.74
0.50 2.50 0.6331 0.6340 0.6418 0.7654 2.2546 16.1434 35.25 41.50 100.00
0.60 3.00 0.5270 0.5282 0.5358 0.6753 2.0909 16.1524 38.00 47.00 100.00
0.70 3.50 0.4513 0.4527 0.4598 0.5927 1.9257 16.1633 41.25 53.50 100.00
0.80 4.00 0.3947 0.3961 0.4027 0.5193 1.7650 16.1810 45.00 61.00 100.00
0.90 4.50 0.3507 0.3521 0.3581 0.4553 1.6124 16.2080 49.25 69.50 100.00
2.40 12.00 0.1320 0.1324 0.1335 0.0989 0.4588 17.3239 173.00 317.00 100.00
2.50 12.50 0.1267 0.1272 0.1281 0.0918 0.4285 17.4136 185.25 341.50 100.00
2.60 13.00 0.1219 0.1223 0.1231 0.0854 0.4009 17.5033 198.00 367.00 100.00
2.70 13.50 0.1174 0.1178 0.1185 0.0796 0.3758 17.5930 211.25 393.50 100.00
2.80 14.00 0.1132 0.1136 0.1142 0.0744 0.3528 17.6826 225.00 421.00 100.00
2.90 14.50 0.1094 0.1096 0.1103 0.0696 0.3318 17.7721 239.25 449.50 100.00
3.00 15.00 0.1057 0.1060 0.1066 0.0653 0.3125 17.8613 254.00 479.00 100.00
3.10 15.50 0.1023 0.1026 0.1031 0.0614 0.2948 17.9504 269.25 509.50 100.00
3.20 16.00 0.0992 0.0994 0.0999 0.0578 0.2785 18.0392 285.00 541.00 100.00
3.30 16.50 0.0962 0.0964 0.0968 0.0545 0.2635 18.1277 301.25 573.50 100.00
3.40 17.00 0.0934 0.0935 0.0940 0.0515 0.2496 18.2160 318.00 607.00 100.00
3.50 17.50 0.0907 0.0909 0.0913 0.0488 0.2368 18.3041 335.25 641.50 100.00
3.60 18.00 0.0882 0.0884 0.0887 0.0462 0.2249 18.3919 353.00 677.00 100.00
3.70 18.50 0.0858 0.0860 0.0863 0.0438 0.2138 18.4794 371.25 713.50 100.00
3.80 19.00 0.0836 0.0837 0.0840 0.0416 0.2035 18.5667 390.00 751.00 100.00
3.90 19.50 0.0814 0.0816 0.0819 0.0396 0.1940 18.6538 409.25 789.50 100.00
4.00 20.00 0.0794 0.0795 0.0798 0.0377 0.1850 18.7406 429.00 829.00 100.00
5.90 29.50 0.0539 0.0539 0.0540 0.0176 0.0883 20.3538 899.25 1769.50 100.00
6.00 30.00 0.0530 0.0530 0.0531 0.0171 0.0854 20.4372 929.00 1829.00 100.00
6.10 30.50 0.0521 0.0522 0.0522 0.0165 0.0828 20.5206 959.25 1889.50 100.00
6.20 31.00 0.0513 0.0513 0.0514 0.0160 0.0802 20.6038 990.00 1951.00 100.00
6.30 31.50 0.0505 0.0505 0.0506 0.0155 0.0777 20.6870 1021.25 2013.50 100.00
6.40 32.00 0.0497 0.0497 0.0498 0.0150 0.0754 20.7700 1053.00 2077.00 100.00
6.50 32.50 0.0489 0.0490 0.0490 0.0146 0.0731 20.8529 1085.25 2141.50 100.00
6.60 33.00 0.0482 0.0482 0.0483 0.0141 0.0710 20.9358 1118.00 2207.00 100.00
6.70 33.50 0.0475 0.0475 0.0476 0.0137 0.0689 21.0186 1151.25 2273.50 100.00
6.80 34.00 0.0468 0.0468 0.0469 0.0133 0.0670 21.1013 1185.00 2341.00 100.00
For u = 0.5 ?
x
y
b
l
R3