(Solved): solve this step by step and explain the main steps An axial strain gauge and a transverse strain gau ...

An axial strain gauge and a transverse strain gauge are mounted to the top surface of a steel beam that experiences a uniaxial stress \( 15.3 \mathrm{MPa} \). The gauges are connected as shown to arms 1 and 2 of a Wheatstone bridge. With a purely axial load applied. If \( \delta E 0=250 \mathrm{mV} \) and \( E_{i}=10 \mathrm{~V} \), determine: a- the bridge constant for the measurement system. b- estimate the average gauge factor of the strain gauges. For this material, Poisson's ratio is \( 0.3 \) and the modulus of elasticity is 207 MPa. \( \frac{\delta E_{0}}{E_{1}}=\frac{G F}{4}\left(\varepsilon_{1}-E_{2}+\varepsilon_{4}-\varepsilon_{j}\right) \) In general \[ \frac{\delta E_{0}}{E_{i}}=\frac{G F}{4}\left(\varepsilon_{1}-\varepsilon_{2}+\varepsilon_{4}-\varepsilon_{3}\right) \] which in this case, for one axial and one transverse gauge yields \[ \begin{aligned} \frac{\delta E_{0}}{E_{i}} &=\frac{G F}{4}\left(\varepsilon_{\max }-0.3 \varepsilon_{\max }\right) \\ &=\frac{G F}{4}\left(0.7 \varepsilon_{\max }\right) \end{aligned} \] for a single gauge sensing the maximum strain \[ \begin{array}{l} \frac{\delta E_{0}}{E_{i}}=\frac{G F}{4} \delta_{\max } \\ \text { and } \\ \kappa_{y}=0.7 \end{array} \] To find the average value of GF \[ \begin{array}{l} \frac{\delta E_{0}}{E_{i}}=\frac{G F}{4} \varepsilon_{\max } \kappa_{B} \\ \frac{250 \times 10^{-6}}{10}=\frac{G F}{4} \varepsilon_{\max } \kappa_{B} \end{array} \] \[ G F=\frac{4 \times 250 \times 10^{-6}}{10(0.7 \times 0.0739)}=1.932 \]