Question

For a linear quadrilateral element, what is the order of matrix B in the strain-displacement relation ε=BD, where D denotes the displacement matrix?

(a) 6×3

(b) 3×6

(c) 3×8

(d) 8×3

I had been asked this question in an interview for internship.

My question is based upon Plane Elasticity in chapter Plane Elasticity of Finite Element Method

Answer 1

Correct choice is (c) 3×8

The best I can explain: For plane elasticity problems the strain-displacement relation is given by ε=BD, where ε=[εxxεyyεxy]^T, B=\(\begin{pmatrix}\frac{\partial \psi_1}{\partial x}&0&\frac{\partial \psi_2}{\partial y}&0&…&\frac{\partial \psi_n}{\partial y}&0\\0&\frac{\partial \psi_1}{\partial y}&0&\frac{\partial \psi_2}{\partial y}&…&0&\frac{\partial \psi_n}{\partial y}\\\frac{\partial \psi_1}{\partial y}&\frac{\partial \psi_1}{\partial x}&\frac{\partial \psi_2}{\partial y}&\frac{\partial \psi_2}{\partial x}&…&\frac{\partial \psi_n}{\partial y}&\frac{\partial \psi_n}{\partial x}\end{pmatrix}\) and D=\(\begin{bmatrix}u_x^1&u_y^1&u_x^2&u_y^2&u_x^3&u_y^3\end{bmatrix}\)^T. The order of matrix B is 3x2n, where n is the number of nodes in the element. A linear quadrilateral element has four nodes, thus n=4.

Order of B is 3×2*4

=3×8.