# Finite Element Method Questions and Answers – Plane Elasticity – Finite Element Model

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This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Plane Elasticity – Finite Element Model”.

1. If only the first derivatives of ux and uy appear in the weak forms, then their interpolation must be at least bilinear.
a) True
b) False

Explanation: An examination of the weak form, 0=∫Ωehe$$[\frac{\partial w_1}{\partial x}(c_{11}\frac{\partial u_x}{\partial x}+c_{12}\frac{\partial u_y}{\partial y}) + c_{66}\frac{\partial w_1}{\partial y} (\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x})$$+ρw1ux]dxdy-∫Ωehew1fxdxdy-∮dhew1txds reveals that only first derivatives of ux and uy with respect to x and y appear respectively. Therefore, ux and uy must be approximated by the Lagrange family of interpolation functions, and at least a bilinear (i.e., linear both in x and y) interpolation is required.

2. In FEM, if two independent variables are components of the same vector, then they can be approximated by two different types of interpolations.
a) True
b) False

Explanation: ux and uy are the primary variables in the expanded weak forms of the plane elasticity problems. Although ux and uy are independent of each other, they are the components of the displacement vector. Therefore, both components should be approximated using the same type and degree of interpolation.

3. What can one conclude about the displacement components ux and uy in the finite element model of the plane elasticity equations?
a) They are primary variables and must be carried as primary nodal degrees of freedom
b) They are secondary variables and must be carried as primary nodal degrees of freedom
c) They are primary variables and must be carried as secondary nodal degrees of freedom
d) They are secondary variables and must be carried as secondary nodal degrees of freedom

Explanation: An examination of the weak form, 0=∫Ωehe$$[\frac{\partial w_1}{\partial x}(c_{11}\frac{\partial u_x}{\partial x}+c_{12}\frac{\partial u_y}{\partial y}) + c_{66}\frac{\partial w_1}{\partial y} (\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x})$$+ρw1ux]dxdy-∫Ωehew1fxdxdy-∮dhew1txds reveals the following:
(i) ux and uy are the primary variables, which must be carried as the primary nodal degrees of freedom.
(ii) only first derivatives of ux and uywith respect to x and y, respectively, appear.

4. Which interpolation functions must be used for the primary variables in weak forms of plane elasticity equations?
a) Hermite family interpolation function
b) Lagrange family interpolation function
c) Hierarchical interpolation function

Explanation: An examination of expanded weak forms of the plane elasticity problems reveals that the variables ux and uy are the primary variables, and only first derivatives of ux and uy with respect to x and y appear, respectively. Therefore, ux and uy must be approximated by the Lagrange family of interpolation functions, and at least bilinear (i.e., linear both in x and y) interpolation is required.

5. What are the simplest elements that fit for the finite element model of the plane elasticity equations?
c) Linear triangular and linear quadrilateral elements

Explanation: Because the first-derivatives of the primary variables, ux and uy with respect to x and y, respectively, appear in the expanded weak forms of the plane elasticity problems, they must be approximated by the Lagrange family of interpolation functions, with at least a bilinear interpolation. The simplest elements that satisfy those requirements are the linear triangular and linear quadrilateral elements.

6. What is the correct statement regarding the shape function S of a linear triangular element?
a) The first derivatives of S and hence, all the strains are element-wise constant
b) The first derivatives of S are linear, but the strains are element-wise constant
c) The first derivatives of S and hence the strains are linear functions
d) The first derivatives of S are element-wise constant, but the strains are linear

Explanation: A linear triangular element (number of nodes=3) has two degrees of freedom, ux and uy per node and a total of six nodal displacements per element. Since the shape functions are linear, their first derivatives are element-wise constant, and hence, all the strains computed for the linear triangular element are element-wise constant.

7. What is the other name of a linear triangular element for plane elasticity problems?
a) Constant-strain triangle
b) Linear-strain triangle
d) Variable-strain triangle

Explanation: In a linear triangular element (number of nodes=3), there are two degrees of freedom, ux and uy per node and a total of six nodal displacements per element. Since the first derivatives of shape functions for a triangular element are element-wise constant, all the strains computed for the linear triangular element are element-wise constant. Therefore, the linear triangular element for a plane elasticity problem is known as the constant-strain triangular (CST) element.

8. What is the function on $$\frac{\partial s}{\partial n}$$ if S (n, e) denotes the shape function of a linear quadrilateral element?
a) Linear in e and constant in n
b) Constant in e and Linear in n
c) Linear in both e and n
d) Constant in both e and n

Explanation: For a quadrilateral element, the first derivatives of the shape function are not constant. $$\frac{\partial s}{\partial n}$$ is linear in e and constant in n whereas $$\frac{\partial s}{\partial e}$$ is linear in n and constant in e. Also, the first derivatives of shape functions for a triangular element are constant element wise, and hence all the strains computed are constant element wise.

9. For a linear triangular 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

Explanation: 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 triangular element has three nodes, thus n=3.
Order of B is 3×2*3
=3×6.

10. 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

Explanation: 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.

11. Which option is not correct with respect to the orders of the matrices in the following finite element model of plane elastic equations?
Me$$\ddot{\Delta}^e$$+KeΔe=Fe+Qe
a) Mass matrix M has order 2n x 2n
b) Stiffness matrix K has order 2n x n
c) The element load vector F has order 2n x 1
d) The vector of internal forces Q has order 2n x 1

Explanation: For the given vector form of finite element model of plane elastic equations, the element mass matrix M and stiffness matrix K are of the order 2n x 2n, the element load vector F and the vector of internal forces Q are of the order 2n x 1, where n is the number of nodes in a Lagrange finite element.

12. Which form of a periodic solution is sought for the natural vibration study of plane elastic bodies?
a) {Δ}={Δ0}eiωt
b) {Δ}={Δ0}e-iω
c) {Δ}={Δ0}e-iωt
d) {Δ}={Δ0}e-iω/t

Explanation: For natural vibration study of plane elastic bodies, we seek a periodic solution of the form {Δ}={Δ0}e-iωt, where Δ denotes the displacements, ω is the frequency of natural vibration and i=$$\sqrt{-1}$$. With this, the finite element models of plane elastic problems reduce to an eigen value problem (-ω2Me+Ke)$$\Delta_0^e$$=Qe.

13. Which option is correct for the first derivative of the shape function S in the study of plane elasticity problems?
a) It is element-wise constant for triangular element whereas not a constant for quadrilateral element
b) It is element-wise constant for both the triangular element as well as a quadrilateral element
c) It is not a constant for triangular element whereas element-wise constant for quadrilateral element
d) It is a combination of a linear function and constant for both the triangular element and a quadrilateral element

Explanation: In the study of plane elasticity problems, the first derivatives of the shape function S for a triangular element are element-wise constant, whereas, for a quadrilateral element, they are not constant. Notably, $$\frac{\partial s (n, e)}{\partial n}$$ is linear in e and constant in n whereas $$\frac{\partial s (n, e)}{\partial e}$$ is linear in n and constant in e. 