Question

A state of stress that occurs at a point on the free surface of a solid body is: σx = 60 MPa; σy =0 MPa; τxy =30 MPa a. Evaluate the two principal normal stresses and the one principal shear stress that can be found by coordinate system rotations in the x‐y plane, and show the coordinate system rotations to these principal stresses with sketches of the elements for both principal normal stresses and principal shear stresses, correctly oriented relative to x & y axes. b. Determine the maximum normal stress and the maximum shear stress at this point. Indicate which planes upon which these maxima are acting. That is, be sure the resulting maxima are labeled with the correct and appropriate subscripts.

Answer 1

To determine the principal stresses and maximum stresses in the x-y plane, use the given stress components and solve the characteristic equation. Then use the transformation equations to find the orientations of the planes where these stresses act.

Step-by-step explanation:

The student's question involves finding the principal normal stresses and principal shear stress at a point on a body subjected to a state of stress, and subsequently determining the maximum normal and shear stresses. Using the given stress components σx, σy, and τxy, the principal stresses and their orientations can be calculated via coordinate transformations or Mohr's Circle method.

To find the principal stresses, one must solve for the roots of the characteristic equation obtained from setting the determinant of the stress tensor minus the identity tensor multiplied by an unknown λ equal to zero. The principal shear stress can be calculated as the maximum shear stress occurring at a 45-degree angle with respect to the principal stresses.

The maximum normal stress is the largest principal stress, and the maximum shear stress is half the difference between the maximum and minimum principal normal stresses. The planes on which these stresses act are oriented at specific angles with respect to the x and y axes, which are calculated using the transformation equations for stress.