![[ME 323 Lec 5 - Stress and strain (generalized concepts)#^7e92b9]] ![[ME 323 Lec 5 - Stress and strain (generalized concepts)#^cfb52e]] ![[ME 323 Lec 5 - Stress and strain (generalized concepts)#^5e2215]] $ e=u(L)-u(0)=\int_{0}^L \left( \frac{P}{AE}+\alpha\Delta T \right)dx $ Assuming $\alpha,P,E,A,\Delta T$ are constant, $ e=\frac{PL}{EA}+\alpha\Delta TL $ If elongation is constrained, thermal stresses make up for it: $ e=\frac{\sigma_{x}L}{EA}+\alpha\Delta T=0 $ $ \boxed{ \sigma_{x}=-\alpha\Delta TEA } $ $ \Delta T>0\to\mp\sigma_{x} $ $ \Delta T<0\to \pm\sigma_{x} $