Summary of Earth Pressure Formulas

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Active earth pressure the Mazindrani theory
Active earth pressure is given by the following formula:
where:zvertical geostatic stress
Kacoefficient of active earth pressure due to Rankin
slope inclination
weight of soil
zassumed depth
coefficient of active earth pressure due to Mazindrani
where:slope inclination
angle of internal friction of soil
Ccohesion 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:aactive 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:zvertical geostatic stress
cefeffective cohesion of soil
Kacoefficient of active earth pressure
Kaccoefficient 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:aactive earth pressure
angle of friction structure  soil
back face inclination of the structure
Active earth pressure  the MllerBreslau theory
Active earth pressure is given by the following formula:
where:zvertical geostatic stress
cefeffective cohesion of soil
Kacoefficient of active earth pressure
Kaccoefficient 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:aactive earth pressure
angle of friction structure  soil
back face inclination of the structure
Literature:MllerBreslau's Erddruck auf Stutzmauern,Stuttgart: Alfred KronerVerlag, 1906 (German)Active earth pressure  the Caqouot theory
Active earth pressure is given by the following formula:
where:zvertical geostatic stress
cefeffective cohesion of soil
Kacoefficient of active earth pressure
Kaccoefficient of active earth pressure due to cohesion
The following analytical solution (Boussinesque, Caqouot) is implemented to compute the coefficient of active earth pressureKa:
where:Kacoefficient of active earth pressure due to Caquot
KaCoulombcoefficient 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:aactive 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:zvertical geostatic stress
cefeffective cohesion of soil
Kacoefficient of active earth pressure
Kaccoefficient 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:aactive 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:zvertical geostatic stress
Kpcoefficient of passive earth pressure due to Rankin
slope inclination
weight of soil
zassumed 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
ccohesion 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
zassumed depth
Kpcoefficient of passive earth pressure due to Rankin
Literature:Passive earth pressure  the Coulomb theory
Passive earth pressure follows from the following formula:
where:zeffective vertical geostatic stress
Kpcoefficient of passive earth pressure due to Coulomb
ccohesion 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:Kpcoefficient of passive earth pressure for = , see thetable
reduction coefficientfor < , see thetable
ccohesion of soil
zvertical 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:Kpcoefficient of passive earth pressure
ccohesion of soil
zvertical 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:MllerBreslau's Erddruck auf Stutzmauern,Stuttgart: Alfred KronerVerlag, 1906 (German)Passive earth pressure  the Absi theory
Passive earth pressure follows from the following formula:
where:Kpcoefficient of passive earth pressure
ccohesion of soil
zvertical 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:Kpgpassive earth pressure coefficient for cohesionless soils
Kpcpassive earth pressure coefficient due to cohesion
Kpppassive earth pressure coefficient due to surcharge
zvertical 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
pangle of friction structure  soil
slope inclination
Slip surface radius vector:
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
pangle 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.