A birch wood log has the following elastic constants (FPS, 1999) relative to orthotropic a...

A birch wood log has the following elastic constants (FPS, 1999) relative to orthotropic axes (x, y, z):

Ex = 15,290 MPa, Ey = 1195 MPa, Ez =765 MPa

Gxy = 1130 MPa, Gxz = 1040 MPa, Gyz = 260 MPa

νxy = 0.426, νxz = 0.451, νyz = 0.697

where the x axis is longitudinal to the grain, the y axis is radial in the tree, and the z axis is tangent to the growth rings of the tree. The unit of stress is [MPa]. At a point in a birch log, the components of stress are σxx = 7 MPa, σyy = 2.1 MPa, σzz = −2.8 MPa, σxy = 1.4 MPa, and σxz = σyz = 0.

a. Determine the orientation of the principal axes of stress.

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b. Determine the strain components.

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c. Determine the orientation of the principal axes of strain.