Answer the following descriptive questions.

a. What are nonzero stress and strain components for plane stress problems?

b. What are nonzero stress and strain components for plane strain problems?

c. What are nonzero stress and strain components for axisymmetric problems?

d. When would a 3node triangular element be invalid?

e. If only nonzero displacements of a 3node triangular element are \(u_{3}\) and \(v_{3}\), what would be the displacement along the edge of nodes 1 and 2?

f. How do the strains, \(\varepsilon_{x x}, \varepsilon_{y y}\), and \(\gamma_{y x}\), vary within a 3node triangular element?

g. How do the strains, \(\varepsilon_{x x}, \varepsilon_{y y}\), and \(\gamma_{x y}\), vary within a 4node rectangular element?

h. If a gravitational force is applied in the ycoordinate direction for a 3node triangular element, what would the equivalent nodal forces be? Assume that the total gravitational force is \(ho h A g\), where \(ho\) is density, \(h\) is thickness, \(A\) is area, and \(g\) is gravitational acceleration.

i. For a fournode rectangular plane element, we use \(u(x, y)=a_{0}+a_{1} x+a_{2} y+a_{3} x y\) as the form of solution. What would be the problem if we use \(u(x, y)=a_{0} x^{2}+a_{1} x y+a_{2} y^{2}\) \(+a_{3} x^{2} y^{2}\) instead?

j. For a rectangular element, plot the shape function \(N_{1}(x, y)\).

k. Define interelement displacement compatibility.