Chapter 2, Problem 27E

Determine shape functions of a bar element shown in the figure by assuming the following form of displacement: $u\left(x\right)=a_{1}x+a_2{x}^2$; that is, obtain $N_1\left(x\right)$ and $N_2\left(x\right)$ such that $u\left(x\right)=N_1\left(x\right)u_1+N_2\left(x\right)u_2$. Calculate axial strain ${\epsilon }_{xx}=\frac{du}{dx}$ when $u_1=u_2=1$ (rigid body motion). Explain why strain is not zero under the rigid-body motion.