Question 4.SP.7: For the beam of Sample Prob. 4.5, determine the residual str...
For the beam of Sample Prob. 4.5, determine the residual stresses and the permanent radius of curvature after the 10,230-kip \cdot in. couple M has been removed.
Loading. In Sample Prob. 4.5 a couple of moment M = 10,230 kip \cdot in. was applied and the stresses shown in Fig. 1 were obtained.
Elastic Unloading. The beam is unloaded by the application of a couple of moment M = -10,230 kip\cdot in. (which is equal and opposite to the couple originally applied). During this unloading, the action of the beam is fully elastic; recalling from Sample Prob. 4.5 that I = 1524 in^{4} , we compute the maximum stress
\sigma_{m}^{\prime}=\frac{M c}{I}=\frac{(10,230 kip \cdot in .)(8 in .)}{1524 in ^{4}}=53.70 ksi
The stresses caused by the unloading are shown in Fig. 2.
Residual Stresses. We superpose the stresses due to the loading (Fig. 1) and to the unloading (Fig. 2) and obtain the residual stresses in the beam (Fig. 3).
Permanent Radius of Curvature. At y = 7 in. the residual stress is \sigma = -3.01 ksi. Since no plastic deformation occurred at this point, Hooke’s law can be used and we have \epsilon_{x} = \sigma/E. Recalling Eq. (4.8), we write
\epsilon_{x}=-\frac{y}{\rho} Eq. (4.8)
\rho=-\frac{y}{\epsilon_{x}}=-\frac{y E}{\sigma}=-\frac{(7 in .)\left(29 \times 10^{6} psi \right)}{-3.0 1 ksi }=+67,400 in \rho = 5620 ft
We note that the residual stress is tensile on the upper face of the beam and compressive on the lower face, even though the beam is concave upward.
