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Page 1: 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.

Page 2: Summary of Earth Pressure Formulas

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

Page 3: Summary of Earth Pressure Formulas

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)

Page 4: Summary of Earth Pressure Formulas

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

Page 5: Summary of Earth Pressure Formulas

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

Page 6: Summary of Earth Pressure Formulas

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:

Page 7: Summary of Earth Pressure Formulas

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/

Page 8: Summary of Earth Pressure Formulas

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

Page 9: Summary of Earth Pressure Formulas

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

Page 10: Summary of Earth Pressure Formulas

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

Page 11: Summary of Earth Pressure Formulas

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)

Page 12: Summary of Earth Pressure Formulas

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

Page 13: Summary of Earth Pressure Formulas

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:

Page 14: Summary of Earth Pressure Formulas

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.

Page 15: Summary of Earth Pressure Formulas