With respect to the coordinate system \(x y z\), the state of stress at a point \(P\) in a solid is:

a. \(\mathbf{m}^{1}, \mathbf{m}^{2}\) and \(\mathbf{m}^{3}\) are three mutually perpendicular vectors such that \(\mathbf{m}^{1}\) makes \(45^{\circ}\) with both the \(x\) and \(y\) axis and \(\mathbf{m}^{3}\) is aligned with the zaxis. Compute the normal stresses on planes normal to \(\mathbf{m}^{1}, \mathbf{m}^{2}\), and \(\mathbf{m}^{3}\).

b. Compute two components of shear stress on the plane normal to \(\mathbf{m}^{1}\) in the directions \(\mathbf{m}^{2}\) and \(\mathbf{m}^{3}\).

c. Is the vector \(\mathbf{n}=\{0,1,1\}^{\mathrm{T}}\) a principal direction of stress? Explain. What is the normal stress in the direction \(\mathbf{n}\) ?

d. Draw an infinitesimal cube with faces normal to \(\mathbf{m}^{1}, \mathbf{m}^{2}\), and \(\mathbf{m}^{3}\) and display the stresses on the positive faces of this cube.

e. Express the state of stress at the point \(P\) with respect to the \(x^{\prime} y^{\prime} z^{\prime}\) coordinates system that is aligned with the vectors \(\mathbf{m}^{1}, \mathbf{m}^{2}\) and \(\mathbf{m}^{3}\) ?

f. What are the principal stress and principal directions of stress at the point \(P\) with respect to the \(x^{\prime} y^{\prime} z^{\prime}\) coordinate system? Explain.

g. Compute the maximum shear stress at point \(P\). Which plane(s) does this maximum shear stress act on?

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x -20 0 0 b [6] 0 50 0 MPa. 0 0 50 N P m m 2 m y