Summary of Earth Pressure Formulas
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Transcript of Summary of Earth Pressure Formulas
Active earth pressure – the Mazindrani theoryActive earth pressure is given by the following formula:
where: σz - vertical geostatic stress
Ka - coefficient of active earth pressure due to Rankin
β - slope inclination
γ - weight of soil
z - assumed depth
- coefficient of active earth pressure due to Mazindrani
where: Β - slope inclination
Φ - angle of internal friction of soil
C - cohesion of soil
Assuming cohesionless soils (c = 0) and horizontal ground surface (β = 0) yields the Rankin solution,
for which the active earth pressure is provided by:
and the coefficient of active earth pressure becomes:
where: Φ - angle of internal friction of soil
Horizontal and vertical components of the active earth pressure become:
where: σa - active earth pressure
Δ - angle of friction structure - soil
Α - back face inclination of the structure
Literature:
Mazindrani, Z.H., and Ganjali, M.H. 1997. Lateral earth pressure problem of cohesive backfill with
inclined surface. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(2): 110–
112.
Active earth pressure - the Coulomb theoryActive earth pressure is given by the following formula:
where: σz - vertical geostatic stress
cef - effective cohesion of soil
Ka - coefficient of active earth pressure
Kac - coefficient of active earth pressure due to cohesion
The coefficient of active earth pressure Ka is given by:
The coefficient of active earth pressure Kac is given by:
for:
for:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
Horizontal and vertical components of the active earth pressure become:
where: σa - active earth pressure
δ - angle of friction structure - soil
α - back face inclination of the structure
Active earth pressure - the Müller-Breslau theoryActive earth pressure is given by the following formula:
where: σz - vertical geostatic stress
cef - effective cohesion of soil
Ka - coefficient of active earth pressure
Kac - coefficient of active earth pressure due to cohesion
The coefficient of active earth pressure Ka is given by:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
The coefficient of active earth pressure Kac is given by:
for:
for:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
Horizontal and vertical components of the active earth pressure become:
where: σa - active earth pressure
δ - angle of friction structure - soil
α - back face inclination of the structure
Literature:
Müller-Breslau's Erddruck auf Stutzmauern,Stuttgart: Alfred Kroner-Verlag, 1906 (German)
Active earth pressure - the Caqouot theoryActive earth pressure is given by the following formula:
where: σz - vertical geostatic stress
cef - effective cohesion of soil
Ka - coefficient of active earth pressure
Kac - coefficient of active earth pressure due to cohesion
The following analytical solution (Boussinesque, Caqouot) is implemented to compute the coefficient of active
earth pressure Ka:
where: Ka - coefficient of active earth pressure due to Caquot
KaCoulomb - coefficient of active earth pressure due to Coulomb
ρ - conversion coefficient – see further
where: β - slope inclination behind the structure
φ - angle of internal friction of soil
δ - angle of friction structure - soil
The coefficient of active earth pressure Kac is given by:
for:
for:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination behind the structure
α - back face inclination of the structure
Horizontal and vertical components of the active earth pressure become:
where: σa - active earth pressure
δ - angle of friction structure - soil
α - back face inclination of the structure
Active earth pressure - the Absi theoryActive earth pressure is given by the following formula:
where: σz - vertical geostatic stress
cef - effective cohesion of soil
Ka - coefficient of active earth pressure
Kac - coefficient of active earth pressure due to cohesion
The program takes values of the coefficient of active earth pressure Ka from a database, built upon the values
published in the book: Kérisel, Absi: Active and passive earth Pressure Tables, 3rd Ed. A.A. Balkema, 1990 ISBN
90 6191886 3.
The coefficient of active earth pressure Kac is given by:
for:
for:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
Horizontal and vertical components of the active earth pressure become:
where: σa - active earth pressure
δ - angle of friction structure - soil
α - back face inclination of the structure
Literature:
Kérisel, Absi: Active and Passive Earth Pressure Tables, 3rd ed., Balkema, 1990 ISBN 90 6191886 3
Passive earth pressure - the Rankin and Mazindrani theoryPassive earth pressure follows from the following formula:
where: σz - vertical geostatic stress
Kp - coefficient of passive earth pressure due to Rankin
β - slope inclination
γ - weight of soil
z - assumed depth
- coefficient of passive earth pressure due to Mazindrani
The coefficient of passive earth pressure Kp is given by:
where: β - slope inclination
φ - angle of internal friction of soil
c - cohesion of soil
If there is no friction (δ = 0) between the structure and cohesionless soils (c = 0), the ground surface is horizontal
(β = 0) and the resulting slip surafce is also plane with the slope:
the Mazindrani theory then reduces to the Rankin theory. The coefficient of passive earth pressure is then
provided by:
where: φ - angle of internal friction of soil
Passive earth pressure σp by Rankin for cohesionless soils is given:
where: γ - unit weight of soil
z - assumed depth
Kp - coefficient of passive earth pressure due to Rankin
Literature:
Passive earth pressure - the Coulomb theoryPassive earth pressure follows from the following formula:
where: σz - effective vertical geostatic stress
Kp - coefficient of passive earth pressure due to Coulomb
c - cohesion of soil
The coefficient of passive earth pressure Kp is given by:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
The vertical σpv and horizontal σph components of passive earth pressure are given by:
where: δ - angle of friction structure - soil
α - back face inclination of the structure
Literature:
Arnold Verruijt: Soil mechanics, Delft University of Technology, 2001, 2006, http://geo.verruijt.net/
Passive earth pressure - the Caquot – Kérisel theoryPassive earth pressure follows from the following formula:
where: Kp - coefficient of passive earth pressure for δ = -φ, see the table
ψ - reduction coefficient ψ for |δ| < φ, see the table
c - cohesion of soil
σz - vertical geostatic stress
The vertical σpv and horizontal σph components of passive earth pressureare given by:
where: δ - angle of friction structure - soil
α - back face inclination of the structure
Coefficient of passive earth pressure KpCoefficient of passive earth pressure Kp for δ = -φ
α [°] φ [°] Kp when β°
0 5 10 15 20 25 30 35 40 45
10 1,17 1,41 1,53
15 1,30 1,70 1,92 2,08
20 1,71 2,08 2,42 2,71 2,92
25 2,14 2,81 2,98 3,88 4,22 4,43
-30 30 2,78 3,42 4,18 5,01 5,98 8,94 7,40
35 3,75 4,73 5,87 7,21 8,78 10,80 12,50 13,80
40 5,31 8,87 8,77 11,00 13,70 17,20 24,80 25,40 28,40
45 8,05 10,70 14,20 18,40 23,80 90,60 38.90 49,10 60,70 69,10
10 1,36 1,58 1,70
15 1,68 1,97 2,20 2,38
20 2,13 2,52 2,92 3,22 3,51
25 2,78 3,34 3,99 4,80 5,29 5,57
-20 30 3,78 4,81 8,58 8,81 7,84 9,12 9,77
35 5,38 8,89 8,28 10,10 12,20 14,80 17,40 19,00
40 8,07 10,40 12,00 18,50 20,00 25,50 38,50 37,80 42,20
45 13,2 17,50 22,90 29,80 38,30 48,90 82,30 78,80 97,30 111,04
10 1,52 1,72 1,83 .
15 1,95 2,23 2,57 2,88
20 2,57 2,98 3,42 3,75 4,09
25 3,50 4,14 4,90 5,82 8,45 8,81
-10 30 4,98 8,01 7,19 8,51 10,10 11,70 12,80
35 7,47 9,24 11,30 13,80 18,70 20,10 23,70 2ó,00
40 12,0 15,40 19,40 24,10 29,80 37,10 53,20 55,10 61,80
45 21,2 27,90 38,50 47,20 80,80 77,30 908,20 124,00 153,00 178,00
10 1,84 1,81 1,93
15 2,19 2,46 2,73 2,91
20 3,01 3,44 3,91 4,42 4,66
25 4,28 5,02 5,81 8,72 7,71 8,16
0 30 8,42 7,69 9,19 10,80 12,70 14,80 15,90
35 10,2 12,60 15,30 18,80 22,30 28,90 31,70 34,90
40 17,5 22,30 28,00 34,80 42,90 53,30 78,40 79,10 88,70
45 33,5 44,10 57,40 74,10 94,70 120,00 153,00 174,00 240,00 275,00
10 1,73 1,87 1,98
15 2,40 2,65 2,93 3,12
20 3,45 3,90 4,40 4,96 5,23
10 25 5,17 5,99 6,90 7,95 9,11 9,67
30 8,17 9,69 11,40 13,50 15,90 18,50 19,90
35 13,8 16,90 20,50 24,80 29,80 35,80 42,30 46,60
40 25,5 32,20 40,40 49,90 61,70 76,40 110,00 113,00 127,00
45 52,9 69,40 90,90 116,00 148,00 i88,00 239,00 303,00 375,00 431,00
10 1,78 1,89 I 2,01
15 2,58 2,821 3,11 3,30
20 3,90 4,38 4,92 5,53 5,83
20 25 6,18 7,12 8,17 9,39 10,70 11,40
30 10,4 12,30 14,40 16,90 20,00 23,20 25,00
35 18,7 22,80 27,60 33,30 40,00 48,00 56,80 62,50
40 37,2 46,90 58,60 72,50 89,30 111,00 158,00 164,00 185,00
45 84,0 110,00 143,00 184,00 234,00 297,00 378,00 478,00 592,00 680,00
Reduction coefficient of passive earth pressureReduction coefficient ψ for |δ| < φ
φ[°] ψ for |δ| < φ
5 1,0 0,8 0,6 0,4 0,2 0,0
10 1,00 0,999 0,962 0,929 0,898 0,864
15 1,00 0,979 0,934 0,881 0,830 0,775
20 1,00 0,968 0,901 0,824 0,752 0,678
25 1,00 0,954 0,860 0,759 0,666 0,574
30 1,00 0,937 0,811 0,686 0,574 0,467
35 1,00 0,916 0,752 0,603 0,475 0,362
40 1,00 0,886 0,682 0,512 0,375 0,262
45 1,00 0,848 0,600 0,414 0,276 0,174
Passive earth pressure - the Müller – Breslau theoryPassive earth pressure follows from the following formula:
where: Kp - coefficient of passive earth pressure
c - cohesion of soil
σz - vertical normal total stress
The coefficient of passive earth pressure Kp is given by:
where: φ - angle of internal friction of soil
δ - angle of friction structure - soil
β - slope inclination
α - back face inclination of the structure
The vertical σpv and horizontal σph components of passive earth pressure are given by:
where: δ - angle of friction structure - soil
α - back face inclination of the structure
Literature:
Müller-Breslau's Erddruck auf Stutzmauern,Stuttgart: Alfred Kroner-Verlag, 1906 (German)
Passive earth pressure - the Absi theoryPassive earth pressure follows from the following formula:
where: Kp - coefficient of passive earth pressure
c - cohesion of soil
σz - vertical normal total stress
The program takes values of the coefficient Kp from a database, built upon the tabulated values published in the
book: Kérisel, Absi: Active and passive earth Pressure Tables, 3rd Ed. A.A. Balkema, 1990 ISBN 90 6191886 3.
The vertical σpv and horizontal σph components of passive earth pressureare given by:
where: δ - angle of friction structure - soil
α - back face inclination of the structure
Literature:
Kérisel, Absi: Active and Passive Earth Pressure Tables, 3rd ed., Balkema, 1990 ISBN 90 6191886 3
Passive earth pressure - the Sokolovski theoryPassive earth pressure follows from the following formula:
where: Kpg - passive earth pressure coefficient for cohesionless soils
Kpc - passive earth pressure coefficient due to cohesion
Kpp - passive earth pressure coefficient due to surcharge
σz - vertical normal total stress
Individual expressions for determining the magnitude of passive earth pressure and slip surface are introduced in
the sequel; the meaning of individual variables is evident from Fig.:
Passive eart pressure slip
surface after Sokolovski
Angles describing the slip surface:
where: φ - angle of internal friction of soil
δp - angle of friction structure - soil
β - slope inclination
Slip surface radius vector:
Provided that ω < 0 the both straight edges of the zone r1 and r2 numerically overlap and resulting in the plane slip
surface developed in the overlapping region. The coefficients of passive earth pressure Kpg, Kpp, Kpc then follow
from:
where: φ - angle of internal friction of soil
δp - angle of friction structure - soil
α - back face inclination of the structure
Auxiliary variables: ipg, ipp, ipc, gpg, gpp, gpc, tpg, tpp, tpc
for:, ,
, ,
, ,
where:
For soils with zero value for the angle of internal friction the following expressions are employed to determine the
coefficients of passive earth pressure:
where:
Literature:
Sokolovski, V.V., 1960. Statics of Soil Media,Butterworth, London.