Summary of Earth Pressure Formulas
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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
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
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.
