Question

In Classical Plate Theory, what is the correct comment for the polynomial w=a1+a2x+a3y+a4xy+a5x^2+a6y^2+a7(x^2y+xy^2)+a8x^3+a9y^3?

(a) It is used in the approximation of deflection for a rectangular element

(b) It is a complete third order polynomial

(c) It’s an incomplete third order polynomial

(d) w varies as a quadratic along any line x=constant

This question was posed to me during an online interview.

I'd like to ask this question from Classical Plate Model in portion Bending of Elastic Plates of Finite Element Method

Answer 1

Right option is (c) It’s an incomplete third order polynomial

Explanation: Several C1 rectangular and triangular plate bending elements with (w, \(\frac{dw}{dx}\),\(\frac{dw}{dy}\)) or with (w,\(\frac{dw}{dx}\),\(\frac{dw}{dy}\),\(\frac{d^2w}{dy^2}\)) as the degrees of freedom at each node exists in the literature. A triangular element with three nodes, with (w, \(\frac{dw}{dx}\),\(\frac{dw}{dy}\)) at each node, requires the 9-term (n=9) polynomial approximation of w, i.e., w=a1+a2x+a3y+a4xy+a5x^2+a6y^2+a7(x^2y+xy^2)+a8x^3+a9y^3. This is an incomplete third-order polynomial because x^2y and y^2x do not vary independently. We note from the equation that w varies as a cubic along with any line x=constant or y=constant.