Theoretical Foundation Engineering
Featuring an easy-to-understand format where detailed mathematical treatment is kept to a minimum, this in-depth book provides numerous charts, drawings, and illustrations to illuminate the problems and theory.
TABLE OF CONTENTS Theoretical Foundation Engineering
Appendix B: Properties of Logarithmic Spirals and Logarithmic Spiral Sectors
Overview
In the analysis of lateral earth pressure, bearing capacity of shallow foundations, and stability of slopes, the entire or a part of the failure surface in the soil is assumed to be the arc of a logarithmic spiral. Following are some useful properties of logarithmic spirals and logarithmic spiral sectors.
The general equation of a logarithmic spiral (Fig. B.1) can be given as
where
| ? | = | angle, in radians |
| ? | = | soil friction angle |
Or
Table B.l gives the variations of r/ r with ? and ?.
| ? | r/ r o | ||||
|---|---|---|---|---|---|
| ? (deg) | |||||
| (deg) | 20 | 25 | 30 | 35 | 40 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 10 | 1.066 | 1.085 | 1.106 | 1.130 | 1.158 |
| 20 | 1.135 | 1.127 | 1.223 | 1.277 | 1.340 |
| 30 | 1.210 | 1.277 | 1.353 | 1.443 | 1.552 |
| 40 | 1.289 | 1.385 | 1.496 | 1.63 | 1.796 |
| 50 | 1.375 | 1.502 | 1.655 | 1.842 | 2.080 |
| 60 | 1.463 | 1.63 | 1.831 | 2.082 | 2.408 |
| 70 | 1.560 | 1.768 | 2.025 | 2.352 | 2.786 |
| 80 | 1.662 | 1.917 | 2.239 | 2.658 | 3.227 |
| 90 | 1.771 | 2.080 | 2.477 | 3.004 | 3.736 |
| 100 | 1.887 | 2.257 | 2.739 | 3.394 | 4.325 |
| 110 | 2.011 | 2.448 | 3.029 | 3.836 | 5.008 |
| 120 | 2.143 | 2.655 | 3.351 | 4.334 | 5.797 |
| 110 | 2.284 | 2.881 | 3.706 | 4.897 | 6.712 |
| 140 | 2.434 | 3.125 | 4.099 | 5.534 | 7.770 |
| 150 | 2.593 | 3.390 | 4.534 | 6.253 | 8.996 |
| 160 | 2.763 | 3.677 | 5.014 | 7.066 | 10.415 |
| 170 | 2.944 | 3.989 | 5.546 | 7.985 | 12.057 |
| 180 | 3.138 | 4.327 | 6.134 | 9.023 | 13.959 |
Another property of a log spiral is that, at any given point, the radial line makes an...
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