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Active earth pressure – the Mazindrani theory Active earth pressure is given by the following formula: where: σ z - vertical geostatic stress K a - 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: σ a - 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.
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### Transcript of Summary of Earth Pressure Formulas

Active earth pressure the Mazindrani theory

Active earth pressure is given by the following formula:

where:z-vertical geostatic stress

Ka-coefficient of active earth pressure due to Rankin

-slope inclination

-weight of soil

z-assumed depth

-coefficient of active earth pressure due to Mazindrani

where:-slope inclination

-angle of internal friction of soil

C-cohesion of soil

Assuming cohesionless soils (c = 0) and horizontal ground surface ( = 0) yields the Rankin solution, for which the active earth pressure is provided by:

and the coefficient of active earth pressure becomes:

where:-angle of internal friction of soil

Horizontal and vertical components of the active earth pressure become:

where:a-active earth pressure

-angle of friction structure - soil

-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): 110112.Active earth pressure - the Coulomb theory

Active earth pressure is given by the following formula:

where:z-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 pressureKais given by:

The coefficient of active earth pressureKacis given by:

for:

for:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

Horizontal and vertical components of the active earth pressure become:

where:a-active earth pressure

-angle of friction structure - soil

-back face inclination of the structure

Active earth pressure - the Mller-Breslau theory

Active earth pressure is given by the following formula:

where:z-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 pressureKais given by:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

The coefficient of active earth pressureKacis given by:

for:

for:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

Horizontal and vertical components of the active earth pressure become:

where:a-active earth pressure

-angle of friction structure - soil

-back face inclination of the structure

Literature:Mller-Breslau's Erddruck auf Stutzmauern,Stuttgart: Alfred Kroner-Verlag, 1906 (German)Active earth pressure - the Caqouot theory

Active earth pressure is given by the following formula:

where:z-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 pressureKa:

where:Ka-coefficient of active earth pressure due to Caquot

KaCoulomb-coefficient of active earth pressure due to Coulomb

-conversion coefficient see further

where:-slope inclination behind the structure

-angle of internal friction of soil

-angle of friction structure - soil

The coefficient of active earth pressureKacis given by:

for:

for:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination behind the structure

-back face inclination of the structure

Horizontal and vertical components of the active earth pressure become:

where:a-active earth pressure

-angle of friction structure - soil

-back face inclination of the structure

Active earth pressure - the Absi theory

Active earth pressure is given by the following formula:

where:z-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 pressureKafrom a database, built upon the values published in the book: Krisel, Absi: Active and passive earth Pressure Tables, 3rd Ed. A.A. Balkema, 1990 ISBN 90 6191886 3.

The coefficient of active earth pressureKacis given by:

for:

for:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

Horizontal and vertical components of the active earth pressure become:

where:a-active earth pressure

-angle of friction structure - soil

-back face inclination of the structure

Literature:Krisel, Absi: Active and Passive Earth Pressure Tables, 3rd ed., Balkema, 1990 ISBN 90 6191886 3Passive earth pressure - the Rankin and Mazindrani theory

Passive earth pressure follows from the following formula:

where:z-vertical geostatic stress

Kp-coefficient of passive earth pressure due to Rankin

-slope inclination

-weight of soil

z-assumed depth

-coefficient of passive earth pressure due to Mazindrani

The coefficient of passive earth pressureKpis given by:

where:-slope inclination

-angle of internal friction of soil

c-cohesion of soil

If there is no friction ( = 0) between the structure and cohesionless soils (c = 0), the ground surface is horizontal ( = 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:-angle of internal friction of soil

Passive earth pressurepby Rankin for cohesionless soils is given:

where:-unit weight of soil

z-assumed depth

Kp-coefficient of passive earth pressure due to Rankin

Literature:Passive earth pressure - the Coulomb theory

Passive earth pressure follows from the following formula:

where:z-effective vertical geostatic stress

Kp-coefficient of passive earth pressure due to Coulomb

c-cohesion of soil

The coefficient of passive earth pressureKpis given by:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

The verticalpvand horizontalphcomponents of passive earth pressureare given by:

where:-angle of friction structure - soil

-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 Krisel theory

Passive earth pressure follows from the following formula:

where:Kp-coefficient of passive earth pressure for = -, see thetable

-reduction coefficientfor|| < , see thetable

c-cohesion of soil

z-vertical geostatic stress

The verticalpvand horizontalphcomponents of passive earth pressureare given by:

where:-angle of friction structure - soil

-back face inclination of the structure

Coefficient of passive earth pressure Kp

Coefficient of passive earth pressureKpfor =-

[][]Kpwhen

051015202530354045

101,171,411,53

151,301,701,922,08

201,712,082,422,712,92

252,142,812,983,884,224,43

-30302,783,424,185,015,988,947,40

353,754,735,877,218,7810,8012,5013,80

405,318,878,7711,0013,7017,2024,8025,4028,40

458,0510,7014,2018,4023,8090,6038.9049,1060,7069,10

101,361,581,70

151,681,972,202,38

202,132,522,923,223,51

252,783,343,994,805,295,57

-20303,784,818,588,817,849,129,77

355,388,898,2810,1012,2014,8017,4019,00

408,0710,4012,0018,5020,0025,5038,5037,8042,20

4513,217,5022,9029,8038,3048,9082,3078,8097,30111,04

101,521,721,83.

151,952,232,572,88

202,572,983,423,754,09

253,504,144,905,828,458,81

-10304,988,017,198,5110,1011,7012,80

357,479,2411,3013,8018,7020,1023,702,00

4012,015,4019,4024,1029,8037,1053,2055,1061,80

4521,227,9038,5047,2080,8077,30908,20124,00153,00178,00

101,841,811,93

152,192,462,732,91

203,013,443,914,424,66

254,285,025,818,727,718,16

0308,427,699,1910,8012,7014,8015,90

3510,212,6015,3018,8022,3028,9031,7034,90

4017,522,3028,0034,8042,9053,3078,4079,1088,70

4533,544,1057,4074,1094,70120,00153,00174,00240,00275,00

101,731,871,98

152,402,652,933,12

203,453,904,404,965,23

10255,175,996,907,959,119,67

308,179,6911,4013,5015,9018,5019,90

3513,816,9020,5024,8029,8035,8042,3046,60

4025,532,2040,4049,9061,7076,40110,00113,00127,00

4552,969,4090,90116,00148,00i88,00239,00303,00375,00431,00

101,781,89 I2,01

152,582,8213,113,30

203,904,384,925,535,83

20256,187,128,179,3910,7011,40

3010,412,3014,4016,9020,0023,2025,00

3518,722,8027,6033,3040,0048,0056,8062,50

4037,246,9058,6072,5089,30111,00158,00164,00185,00

4584,0110,00143,00184,00234,00297,00378,00478,00592,00680,00

Reduction coefficient of passive earth pressure

Reduction coefficientfor|| < []for|| <

51,00,80,60,40,20,0

101,000,9990,9620,9290,8980,864

151,000,9790,9340,8810,8300,775

201,000,9680,9010,8240,7520,678

251,000,9540,8600,7590,6660,574

301,000,9370,8110,6860,5740,467

351,000,9160,7520,6030,4750,362

401,000,8860,6820,5120,3750,262

451,000,8480,6000,4140,2760,174

Passive earth pressure - the Mller Breslau theory

Passive earth pressure follows from the following formula:

where:Kp-coefficient of passive earth pressure

c-cohesion of soil

z-vertical normal total stress

The coefficient of passive earth pressureKpis given by:

where:-angle of internal friction of soil

-angle of friction structure - soil

-slope inclination

-back face inclination of the structure

The verticalpvand horizontalphcomponents of passive earth pressure are given by:

where:-angle of friction structure - soil

-back face inclination of the structure

Literature:Mller-Breslau's Erddruck auf Stutzmauern,Stuttgart: Alfred Kroner-Verlag, 1906 (German)Passive earth pressure - the Absi theory

Passive earth pressure follows from the following formula:

where:Kp-coefficient of passive earth pressure

c-cohesion of soil

z-vertical normal total stress

The program takes values of the coefficientKpfrom a database, built upon the tabulated values published in the book: Krisel, Absi: Active and passive earth Pressure Tables, 3rd Ed. A.A. Balkema, 1990 ISBN 90 6191886 3.

The verticalpvand horizontalphcomponents of passive earth pressureare given by:

where:-angle of friction structure - soil

-back face inclination of the structure

Literature:Krisel, Absi: Active and Passive Earth Pressure Tables, 3rd ed., Balkema, 1990 ISBN 90 6191886 3Passive earth pressure - the Sokolovski theory

Passive 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

z-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 SokolovskiAngles describing the slip surface:

where:-angle of internal friction of soil

p-angle of friction structure - soil

-slope inclination

Provided that < 0the both straight edges of the zoner1andr2numerically overlap and resulting in the plane slip surface developed in the overlapping region. The coefficients of passive earth pressureKpg,Kpp,Kpcthen follow from:

where:-angle of internal friction of soil

p-angle of friction structure - soil

-back face inclination of the structure

Auxiliary variables:ipg,ipp,ipc,gpg,gpp,gpc,tpg,tpp,tpcfor:,,

,,

,,

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.