can be designed to resist earth pressure. The active state occurs when a retained soil mass is allowed to relax or deform laterally and outward (away from the soil mass) to the point of mobilizing its available full shear resistance (or engaging its shear strength) in trying to resist lateral deformation. That is, the soil is at the point of incipient failure by shearing due to unloading in the lateral direction. It is the minimum theoretical lateral pressure that a given soil mass will exert on a retaining wall that will move or rotate away from the soil until the soil active state is reached (not necessarily the actual in-service lateral pressure on walls that do not move when subjected to soil lateral pressures higher than the active pressure). The passive state occurs when a soil mass is externally forced laterally and inward (towards the soil mass) to the point of mobilizing its available full shear resistance in trying to resist further lateral deformation. That is, the soil mass is at the point of incipient failure by shearing due to loading in the lateral direction. It is the maximum lateral resistance that a given soil mass can offer to a retaining wall that is being pushed towards the soil mass. That is, the soil is at the point of incipient failure by shearing, but this time due to loading in the lateral direction. Thus active pressure and passive resistance define the minimum lateral pressure and the maximum lateral resistance possible from a given mass of soil.
Coulomb's earth pressure coefficients Coulomb (1776) a number of potential failure surfaces must be analysed to identify the critical failure surface (i.e. the surface that produces the maximum or minimum thrust on the wall). Coulombs main assumption is that the failure surface is planar. In simpler terms, Coulomb's theory imagines a wedge of soil behind the wall that is just on the verge of sliding. By comparing the forces trying to push the wedge forward with those resisting movement, the maximum or minimum pressure on the wall can be estimated. Mayniel (1808) later extended Coulomb's equations to account for wall friction, denoted by \delta. Müller-Breslau (1906) further generalized Mayniel's equations for a non-horizontal backfill and a non-vertical soil-wall interface (represented by an angle \theta from the vertical). The symbols K_a and K_p represent dimensionless "earth pressure coefficients": K_a for active pressure (when the wall moves away from the soil and the soil expands slightly) and K_p for passive pressure (when the wall moves into the soil and the soil is compressed). These coefficients scale the weight of the soil to give the actual horizontal pressure on the wall. : K_a = \frac{ \cos ^2 \left( \phi' - \theta \right)}{\cos ^2 \theta \cos \left( \delta + \theta \right) \left( 1 + \sqrt{ \frac{ \sin \left( \delta + \phi' \right) \sin \left( \phi' - \beta \right)}{\cos \left( \delta + \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2} : K_p = \frac{ \cos ^2 \left( \phi' + \theta \right)}{\cos ^2 \theta \cos \left( \delta - \theta \right) \left( 1 - \sqrt{ \frac{ \sin \left( \delta + \phi' \right) \sin \left( \phi' + \beta \right)}{\cos \left( \delta - \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2} Instead of evaluating the above equations or using commercial software applications for this, books of tables for the most common cases can be used. Generally, instead of K_a, the horizontal part K_{ah} is tabulated. It is the same as K_a times \cos(\delta + \theta). [Note that under certain conditions, the equation for K_p "blows up". For example, if \theta = \beta = 0 and \phi' = \delta = 45 ^ \circ, then K_p \to \infty.] The actual earth pressure force E_a is the sum of the part E_{ag} due to the weight of the earth, a part E_{ap} due to extra loads such as traffic, minus a part E_{ac} due to any cohesion present. E_{ag} is the integral of the pressure over the height of the wall, which equates to K_a times the specific gravity of the earth, times one half the wall height squared. In the case of a uniform pressure loading on a terrace above a retaining wall, E_{ap} equates to this pressure times K_a times the height of the wall. This applies if the terrace is horizontal or the wall vertical. Otherwise, E_{ap} must be multiplied by \frac{\cos \theta \cdot \cos \beta}{\cos(\theta - \beta)}. E_{ac} is generally assumed to be zero unless a value of cohesion can be maintained permanently. E_{ag} acts on the wall's surface at one third of its height from the bottom and at an angle \delta relative to a right angle at the wall. E_{ap} acts at the same angle, but at one half the height.
Rankine's earth pressure coefficients and Bell's extension for cohesive soils Rankine's theory, developed in 1857, is a stress field solution that predicts active and passive earth pressure. It assumes that the soil is cohesionless, the wall is
non-battered and frictionless whilst the backfill is horizontal. The failure surface on which the soil moves is
planar. The expressions for the active and passive lateral earth pressure coefficients are given below. : K_a = \tan ^2 \left( 45 - \frac{\phi'}{2} \right) = \frac{ 1 - \sin(\phi') }{ 1 + \sin(\phi') } : K_p = \tan ^2 \left( 45 + \frac{\phi'}{2} \right) = \frac{ 1 + \sin(\phi') }{ 1 - \sin(\phi') } :For soils with cohesion, Bell and Kapilla's earth pressure coefficients for dynamic active and passive conditions respectively have been obtained on the same basis as Coulomb's solution. These coefficients are given below: K_{ae} = \frac{\cos^2{(\phi'-\psi-\beta)}}{\cos{\psi}\cos^2{\beta}\cos(\delta+\beta+\psi)\Bigl(1+\sqrt{\frac{\sin{(\phi'+\delta)}\sin{(\phi'-\psi+\alpha)}}{\sqrt{\cos{(\delta+\beta+\phi)}\cos{(\alpha-\beta)}}}}\Bigr)^2} K_{pe} = \frac{\cos^2{(\phi'-\psi+\beta)}}{\cos{\psi}\cos^2{\beta}\cos(\delta-\beta+\psi)\Bigl(1-\sqrt{\frac{\sin{(\phi'+\delta)}\sin{(\phi'-\psi+\alpha)}}{\sqrt{\cos{(\delta-\beta+\phi)}\cos{(\alpha-\beta)}}}}\Bigr)^2}with horizontal components of earth pressure: \sigma_a =K_a \gamma z \cos\beta \sigma_p =K_p \gamma z \cos\beta where, k_h and k_v are the seismic coefficients of horizontal and vertical acceleration respectively, \psi=\arctan{(k_h/(1-k_v))}, \beta is the back face inclination angle of the structure with respect to vertical, \delta is the angle of friction between structure and soil and \alpha is the back slope inclination. The above coefficients are included in numerous seismic design codes worldwide (e.g., EN1998-5, AASHTO), since being suggested as standard methods by Seed and Whitman. The problems with these two solutions are known (e.g., see Anderson]) with the most important one being the square root of negative number for \phi' (the minus sign stands for the active case whilst the plus sign stands for the passive case). The various design codes recognize the problem with these coefficients and they either attempt an interpretation, dictate a modification of these equations, or propose alternatives. In this respect: • Eurocode 8 suggests that k_h=(2/3)PGA for the same reason as above • GEO Report No. 45 of Geotechnical Engineering Office of Hong Kong dictates the use of the trial wedge method when the number under the square root is negative. It is noted that the above empirical corrections on k_h made by AASHTO presented an analytical solution to the problem of earth pressures exerted on a frictionless, non-battered wall by a cohesive-frictional soil with inclined surface. The derived equations are given below for both the active and passive states: K_a = \frac{1}{\cos^2\phi'} \biggl(2\cos^2\beta+2\frac{c'}{\gamma z}\cos\phi'\sin\phi'-\sqrt{4\cos^2\beta\Bigl(\cos^2\beta-\cos^2\phi'\Bigr)+4\biggl(\frac{c'}{\gamma z}\biggl)^2\cos^2\phi'+8\biggl(\frac{c'}{\gamma z}\biggl)\cos^2\beta\sin\phi'\cos\phi'}\biggl)-1 K_p = \frac{1}{\cos^2\phi'} \biggl(2\cos^2\beta+2\frac{c'}{\gamma z}\cos\phi'\sin\phi'+\sqrt{4\cos^2\beta\Bigl(\cos^2\beta-\cos^2\phi'\Bigr)+4\biggl(\frac{c'}{\gamma z}\biggl)^2\cos^2\phi'+8\biggl(\frac{c'}{\gamma z}\biggl)\cos^2\beta\sin\phi'\cos\phi'}\biggl)-1 with horizontal components for the active and passive earth pressure are: \sigma_a =K_a \gamma z \cos\beta \sigma_p =K_p \gamma z \cos\beta ka and kp coefficients for various values of \phi', \beta, and c'/(\gamma z) can be found in tabular form in Mazindrani and Ganjale. gave a different expression for ka. It is noted, however, that both Mazindrani and Ganjale's and Gnanapragasam's expressions lead to identical active earth pressure values. Following either approach for the active earth pressure, the depth of tension crack appears to be the same as in the case of zero backfill inclination (see Bell's extension of Rankine's theory).
Pantelidis' unified approach: the generalized coefficients of earth pressure Pantelidis offered a unified fully analytical continuum mechanics approach (based on Cauchy's first law of motion) for deriving earth pressure coefficients for all soil states, applicable to cohesive-frictional soils and both horizontal and vertical pseudo-static conditions. The following symbols are used: k_h and k_v are the seismic coefficients of horizontal and vertical acceleration respectively c', \phi' and \gamma are the effective cohesion, effective internal friction angle (peak values) and unit weight of soil respectively c_m is the mobilized cohesion of soil (the mobilized shear strength of soil, i.e. the c_mand \phi_m parameters, can be obtained either analytically or through relevant charts; see Pantelidis with the application of a Strength Mobilization Factor (SMF) to c′ and tanφ′. According to this Engineer Manual, an appropriate SMF value allows calculation of greater-than-active earth pressures using Coulomb's active force equation. Assuming an average SMF value equal to 2/3 along Coulomb's failure surface, it has been shown that for purely frictional soils the derived coefficient value of earth pressure matches quite well with the respective one derived from Jaky's K_{0} = 1 - \sin \phi' equation. In the solution proposed by Pantelidis, the SMF factor is the 1/f_m ratio and what has been foreseen by EM1110-2-2502, it can be calculated exactly.
Example #1: For c' =20 kPa, \phi' =30o, γ=18 kN/m3, k_h = k_v =0, and z =2 m, for the state at rest K_0 =0.211, c_m =9.00 kPa and \phi_m =14.57o. Using this ( c_m , \phi_m ) pair of values in place of the ( c' , \phi' ) pair of values in the coefficient of active earth pressure ( K_{ae}) given previously, the latter returns a coefficient of earth pressure equal to 0.211, that is, the coefficient of earth pressure at rest.
Example #2: For c' =0kPa, \phi' =30o, γ=18 kN/m3, k_h =0.3, k_v =0.15, and z =2 m, for the state at rest K_{oe} =0.602, c_m =0 kPa and \phi_m =14.39o. Using this ( c_m , \phi_m ) pair of values in place of the ( c' , \phi' ) pair of values and k_h = k_v =0 in the coefficient of active earth pressure ( K_{ae}) given previously, the latter returns a coefficient of earth pressure equal to 0.602, that is, again the coefficient of earth pressure at rest. == See also ==