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4.1 Stress At A Point - 2D

4.1.1 Stresses on Rotated Planes

Suppose that we know the stress state of the soil on a particular plane, and we wish to determine the stress state on a different, rotated plane. This problem is equivalent to a transformation of the coordinate system. The components of the Cauchy stress tensor in the rotated coordinate space, ˆσij, can be determined from the stress components in the reference state, ˆσij, using the rotation matrix a with components aij. Note that in soil mechanics, we often represent the shear stress components (i.e., the off-diagonal elements of the stress tensor) as τij instead of σij. Furthermore, the prime in σ usually denotes effective stress, but is used here to denote stresses in a rotated coordinate system.

ˆσ=aσaT In soil mechanics, we often simplify three-dimensional stress conditions to two dimensions. The Mohr Circle is a very common tool used to interpret soil stress states, and it is inherently a two-dimensional representation of stresses. We should therefore consider rotation of stresses in two dimensions. Figure 4.1.1 defines the terms of the rotation tensor, aij.

RotatedStresses2D.png

Figure 4.1.1. Two dimensional coordinate system rotation (note rotation is about the x3 axis by angle γ).

The rotated stress state can therefore be defined as follows: [σ11ˆσ12σ21ˆσ22]=[cos(γ)sin(γ)sin(γ)cos(γ)][σ11σ12σ21σ22][cos(γ)sin(γ)sin(γ)cos(γ)]

Expanding out the matrix algebra results in the following expressions for the rotated stresses: ˆσ11=σ11σ12sin(2γ)sin2(γ)(σ11σ22)ˆσ22=σ22+σ12sin(2γ)+sin2(γ)(σ11σ22)ˆσ12=σ12cos(2γ)+sin(2γ)2(σ11σ22)

Try it yourself

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