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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

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

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Best answer
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.

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