Question 7.8: A 45° strain rosette (also called a rectangular rosette) con...
A 45° strain rosette (also called a rectangular rosette) consists of three electrical-resistance strain gages arranged to measure strains in two perpendicular directions and also at a 45° angle between them, as shown in Fig. 7-37a. The rosette is bonded to the surface of the structure before it is loaded. Gages A, B, and C measure the normal strains ϵ_{a}, ϵ_{b}, and ϵ_{c} in the directions of lines Oa, Ob, and Oc, respectively.
Explain how to obtain the strains ϵ_{x_{1}}, ϵ_{y_{1}}, and \gamma_{x_{1}y_{1}} associated with an element oriented at an angle θ to the xy axes (Fig. 7-37b).
At the surface of the stressed object, the material is in plane stress. Since the strain-transformation equations (Eqs. 7-71a and 7-71b) apply to plane stress as well as to plane strain, we can use those equations to determine the strains in any desired direction.
\frac{\gamma_{x_{1}y_{1}}}{2} =-\frac{\epsilon_{x}-\epsilon_{y}}{2}\sin 2_\ \theta+\frac{\gamma_{xy}}{2}\cos 2_\ \theta (7-71b)
Strains associated with the xy axes. We begin by determining the strains associated with the xy axes. Because gages A and C are aligned with the x and y axes, respectively, they give the strains ϵ_{x} and ϵ_{y} directly:
ϵ_{x} = ϵ_{a} ϵ_{y} = ϵ_{c} (7-77a,b)
To obtain the shear strain \gamma_{xy}, we use the transformation equation for normal strains (Eq. 7-71a):
\epsilon_{x_{1}}=\frac{\epsilon_{x}+\epsilon_{y}}{2}+\frac{\epsilon_{x}-\epsilon_{y}}{2}\cos 2_\ \theta +\frac{\gamma_{xy}}{2}\sin 2_\ \theta (7-71a)
For an angle θ = 45° , we know that \epsilon_{x_{1}} = ϵ_{b} (Fig. 7-37a); therefore, the preceding equation gives
\epsilon_{b}=\frac{\epsilon_{a}+\epsilon_{c}}{2}+\frac{\epsilon_{a}-\epsilon_{c}}{2}(\cos 90^{\circ})+\frac{\gamma_{xy}}{2}(\sin90^{\circ})Solving for \gamma_{xy}, we get
\gamma_{xy}=2\epsilon_{b}-\epsilon_{a}-\epsilon_{c} (7-78)
Thus, the strains \epsilon_{x}, \epsilon_{y}, and \gamma_{xy} are easily determined from the given strain-gage readings.
Strains associated with the x_{1}y_{1} axes. Knowing the strains \epsilon_{x}, \epsilon_{y}, and \gamma_{xy}, we can calculate the strains for an element oriented at any angle θ (Fig. 7-37b) from the strain-transformation equations (Eqs. 7-71a and 7-71b) or from Mohr’s circle. We can also calculate the principal strains and the maximum shear strains from Eqs. (7-74) and (7-75), respectively.
\epsilon_{1,2}=\frac{\epsilon_{x}+\epsilon_{y}}{2}\pm \sqrt{\left(\frac{\epsilon_{x}-\epsilon_{y}}{2}\right)^{2}+\left(\frac{\gamma_{xy}}{2}\right)^{2}} (7-74)
\frac{\gamma_{\max}}{2}=\sqrt{\left(\frac{\epsilon_{x}-\epsilon_{y}}{2}\right)^{2}+\left(\frac{\gamma_{xy}}{2}\right)^{2}} (7-75)