Rankine theory

Rankine theory

Rankine's Theory says that failure will occur when the maximum principal stress at any point reaches a value equal to the tensile stress in a simple tension specimen at failure. This theory does not take into account the effect of the other two principal stresses. Rankine's theory is satisfactory for brittle materials, and not applicable to ductile materials. This theory is also called the Maximum Stress 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:

: P_a = K_a w h

where:

: K_a = \frac{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}*\cos\beta

and:

: Ka = 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:

: P_p = K_p w h

where:

: K_p = \frac{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}*\cos\beta : Or in the case of β=0, then the two coefficients are inversely proportional, such that: : K_p = \frac{1}{K_a}

References

References

1. Rankine, W. (1857) On the stability of loose earth. Philosophical Transactions of the Royal Society of London, Vol. 147.