Many of the problems for Sections 2.1-2.4 require the determination of principal stresses...

Many of the problems for Sections 2.1-2.4 require the determination of principal stresses and maximum shear stress, as well as the location of the planes on which they act. These quantities are required input in design and failure criteria for structural and mechanical systems.

The stress array (Eq. 2.5) relative to axes (x, y, z) is given by

where the stress components are in [MPa].

a. Determine the stress invariants of T.

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b. Consider a rotation of the (x, y) axes by 45° counterclockwise in the (x, y) plane to form axes (X, Y). Let the Z axis and the z axis coincide. Calculate the stress components relative to axes (X, Y, Z). Note that the direction cosines between axes (x, y, z) and (X, Y, Z) are given by Table 2.2, where the rotation of 45° determines l1, m1, n1,....

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c. With the results of part (b), determine the stress invariants relative to axes (X, Y, Z) and show that they are the same as the invariants of part (a).

 (2.5)

TABLE 2.2 Direction Cosines