A sixnode finite element as shown in the figure is used for approximating the beam problem.

a. Write the expressions of displacements \(u(x, y)\) and \(v(x, y)\) in terms of polynomials with unknown coefficients. For example, \(u(x, y)=a_{0}+a_{1} x+\cdots\).

b. Can this element represent the pure bending problem accurately? Explain your answer. A bending moment \(M\) is applied to the edge 2-3.

c. Can this element represent a uniformly distributed load problem accurately? The distributed load \(q\) is applied along the edge 4-6-3.

y 4 6 3 5 2 X