고체역학 (Gere) — Analysis of Stress and Strain: Problem 7.2-21

7.2-21 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 \(\theta = 0^\circ\). From the figure for problem 7.2-21, the given parameters are: \(\theta = 36^\circ\) \(\sigma_{x1} = -90 \text{ MPa}\) (compression, as arrows point into the face) \(\sigma_{y1} = 15 \text{ MPa}\) (tension, as arrows point out of the face) \(\tau_{x1y1} = 23 \text{ MPa}\) (positive direction, pointing towards the top-right corner on the \(x_1\) and \(y_1\) faces)