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4.2 Stress At A Point - 3D

4.2.1 Stresses on Rotated Planes – 3D

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, σ^ij, can be determined from the stress components in the reference state, σij, 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.

σ^=aσaT

In matrix form:

[σ^11σ^12σ^13σ^21σ^22σ^23σ^31σ^32σ^33]=[a11a12a13a21a22a23a31a32a33][σ11σ12σ13σ21σ22σ23σ31σ32σ33][a11a21a31a12a22a32a13a23a33] 2000px-Stress_transformation_3D.svg.png

Figure 2.1.1. Stresses in a reference coordinate system (axes x1, x2, x3), and rotated coordinate system (axes x1, x2, x3). https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Stress_transformation_3D.svg/2000px-Stress_transformation_3D.svg.png

Try it yourself

Note: The a matrix must be orthogonal (i.e., the axes in the rotated coordinate system must form 90 degree angles with each other)

σ
a
σ^
2 0 0
0 1 0
0 0 1

Directly computing components of the a matrix is not always convenient. If you prefer, you may specify the rotation angles about the x1, x2, and x3 axes (α, β, and γ, respectively) below to set the a matrix.

αdeg
βdeg
γdeg