A rectangular element as shown in the figure is used to represent a pure bending problem. Due to the bending moment \(M\), the element is deformed as shown in the figure with displacement \(\{\mathbf{q}\}=\left\{u_{1}, v_{1}, u_{2}, v_{2}, u_{3}, v_{3}, u_{4}, v_{4}ight\}^{\mathrm{T}}=\{-1,0,1,0,-1,0,1,0\}^{\mathrm{T}}\).

a. Write the mathematical expressions of strain component \(\varepsilon_{x x}, \varepsilon_{y y}\), and \(\gamma_{x y}\), as functions of \(x\) and \(y\).

b. Does the element satisfy pure bending condition? Explain your answer.

c. If two CST elements are used by connecting nodes 1-2-4 and 4-2-3, what will be \(\varepsilon_{x x}\) along line A-B?

M y 4(0,2) (3,2) (0,0) 2 (3,0) M D X A U1 B U2