Rankine theory

Active and passive soil pressures

This theory, which considers the soil to be in a state of plastic equilibrium, makes the assumptions that the soil is homogeneous, isotropic and has internal friction. The pressure exerted by soil against the wall is referred to as active pressure. The resistance offered by the soil to an object pushing against it is referred to as "passive pressure". Rankine's theory is applicable to incompressible soils. The equation for cohesionless active earth pressure is expressed as:

${\displaystyle P_a=K_{a}wh}$

where:

${\displaystyle K_a=\frac{\cos \beta -{\left({\cos }^2\beta -{\cos }^2\varphi \right)}^{1/2}}{\cos \beta +{\left({\cos }^2\beta -{\cos }^2\varphi \right)}^{1/2}}\ast \cos \beta }$

and:

K_a = Coefficient of active pressure

w = weight density of soil

h = depth of the section (below top soil) where the pressure is being evaluated.

β = angle that the top surface of soil makes with the horizontal.

φ = angle of internal friction of soil.

The expression for passive pressure is:

${\displaystyle P_p=K_{p}wh}$

where:

${\displaystyle K_p=\frac{\cos \beta +{\left({\cos }^2\beta -{\cos }^2\varphi \right)}^{1/2}}{\cos \beta -{\left({\cos }^2\beta -{\cos }^2\varphi \right)}^{1/2}}\ast \cos \beta }$

Or in the case of β=0, then the two coefficients are inversely proportional, such that:

${\displaystyle K_p=\frac{1}{K_a}}$