G) Derive the principal strains E_(1) and E_(2) and the angle (D) for; a. A four-element rectangular rosette (0,45,90,135) b. A tec delta rosette (0,60,120,90)

To derive the principal strains $E_{1}$ and $E_{2}$ and the angle $\theta$ for the given rosettes, we can use the following formulas:<br /><br />$E_{1} = \frac{\sigma_{max} + \sigma_{min}}{2}$<br />$E_{2} = \frac{\sigma_{max} - \sigma_{min}}{2}$<br />$\theta = \frac{1}{2} \arctan\left(\frac{\sigma_{max} - \sigma_{min}}{\sigma_{max} + \sigma_{min}}\right)$<br /><br />where $\sigma_{max}$ and $\sigma_{min}$ are the maximum and minimum principal stresses, respectively.<br /><br />a. For the four-element rectangular rosette $(0,45,90,135)$:<br /><br />Given that the rosette has four elements with angles of $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$, we can assume that the maximum and minimum principal stresses occur at the $0^\circ$ and $90^\circ$ elements, respectively.<br /><br />Let's assume the maximum principal stress $\sigma_{max}$ occurs at the $0^\circ$ element and the minimum principal stress $\sigma_{min}$ occurs at the $90^\circ$ element.<br /><br />Using the formulas mentioned above, we can calculate the principal strains $E_{1}$ and $E_{2}$ and the angle $\theta$ as follows:<br /><br />$E_{1} = \frac{\sigma_{max} + \sigma_{min}}{2}$<br />$E_{2} = \frac{\sigma_{max} - \sigma_{min}}{2}$<br />$\theta = \frac{1}{2} \arctan\left(\frac{\sigma_{max} - \sigma_{min}}{\sigma_{max} + \sigma_{min}}\right)$<br /><br />b. For the tee delta rosette $(0,60,120,90)$:<br /><br />Given that the rosette has four elements with angles of $0^\circ$, $60^\circ$, $120^\circ$, and $90^\circ$, we can assume that the maximum and minimum principal stresses occur at the $60^\circ$ and $120^\circ$ elements, respectively.<br /><br />Let's assume the maximum principal stress $\sigma_{max}$ occurs at the $60^\circ$ element and the minimum principal stress $\sigma_{min}$ occurs at the $120^\circ$ element.<br /><br />Using the formulas mentioned above, we can calculate the principal strains $E_{1}$ and $E_{2}$ and the angle $\theta$ as follows:<br /><br />$E_{1} = \frac{\sigma_{max} + \sigma_{min}}{2}$<br />$E_{2} = \frac{\sigma_{max} - \sigma_{min}}{2}$<br />$\theta = \frac{1}{2} \arctan\left(\frac{\sigma_{max} - \sigma_{min}}{\sigma_{max} + \sigma_{min}}\right)$<br /><br />Note that without specific values for the stresses, we cannot provide numerical values for $E_{1}$, $E_{2}$, and $\theta$.