The Delft Sand, Clay & Rock Cutting Model, 2019a

Basic Soil Mechanics.
2.8. The Mohr Circle.
In the derivation of the Mohr circle the vertical stress σv and the horizontal stress σh are assumed to be the principal
stresses, but in reality these stresses could have any orientation. It should be noted here that the Mohr circle
approach is valid for the stress situation in a point in the soil. Now consider an infinitesimal element of soil under
plane strain conditions as is shown in Figure 2-44. On the element a vertical stress σv and a horizontal stress σh are
acting. On the horizontal and vertical planes the shear stresses are assumed to be zero. Now the question is, what
would the normal stress σ and shear stress τ be on a plane with an angle α with the horizontal direction? To solve
this problem, the horizontal and vertical equilibriums of forces will be derived. Equilibriums of stresses do not
exist. One should consider that the surfaces of the triangle drawn in Figure 2-44 are not equal. If the surface (or
length) of the surface under the angle α is considered to be 1, then the surface (or length) of the horizontal side is
cos(α) and the vertical side sin(α). The stresses have to be multiplied with their surface in order to get forces and
forces are required for the equilibriums of forces, see Figure 2-45. The derivation of the Mohr circle is also an
exercise for the derivation of many equations in this book where equilibriums of forces and moments are applied.
Figure 2-44: The stresses on a soil element.
Figure 2-45: The forces on a soil element.
Since an equilibrium of stresses does not exist, only an equilibrium of forces exists, the forces on the soil element
have to be known, or the ratio of the forces has to be known.
These forces are, assuming the length of the side under an angle α is 1:
h h v v
F sin and F cos
(2-45)
And:
F and F (2-46)
n
s
The equilibrium of forces in the horizontal direction:
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