Gere & Goodno — Analysis of Stress and Strain: Problem 7.2-22

7.2-22 Solve the preceding problem for the element shown in the figure. [Preceding problem 7.2-21 text]: An element in plane stress from the frame of a racing car is oriented at a known angle \(\theta\) (see figure). On this inclined element, the normal and shear stresses have the magnitudes and directions shown in the figure. Determine the normal and shear stresses acting on an element whose sides are parallel to the \(xy\) axes, that is, determine \(\sigma_x\), \(\sigma_y\), and \(\tau_{xy}\). Show the results on a sketch of an element oriented at \(\theta = 0^\circ\). [Data from Figure 7.2-22]: Element orientation: \(\theta = 50^\circ\) (angle between the \(x\)-axis and the \(x'\)-axis). Stresses on the inclined element: - Normal stress on the face with normal at \(\theta + 180^\circ\) (the \(-x'\) face): \(27 \text{ MPa}\) (compression, arrow points toward the face). Thus, \(\sigma_{x'} = -27 \text{ MPa}\). - Normal stress on the face with normal at \(\theta + 90^\circ\) (the \(+y'\) face): \(18 \text{ MPa}\) (tension, arrow points away from the face). Thus, \(\sigma_{y'} = 18 \text{ MPa}\). - Shear stress on the \(+y'\) face: \(55 \text{ MPa}\), pointing toward the \(-x'\) corner. According to the sign convention, a shear stress on the \(+y'\) face is positive if it points in the \(+x'\) direction. Since it points in the \(-x'\) direction, \(\tau_{x'y'} = -55 \text{ MPa}\).