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 u_{x} and u_{y} appear in the weak forms, then their interpolation must be at least bilinear.

a) True

b) False

View Answer

Explanation: An examination of the weak form, 0=∫

_{Ωe}h

_{e}\([\frac{\partial w_1}{\partial x}(c_{11}\frac{\partial u_x}{\partial x}+c_{12}\frac{\partial u_y}{\partial y}) + c_{66}\frac{\partial w_1}{\partial y} (\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x})\)+ρw

_{1}u

_{x}]dxdy-∫

_{Ωe}h

_{e}w

_{1}f

_{x}dxdy-∮

_{┌d}h

_{e}w

_{1}t

_{x}ds reveals that only first derivatives of u

_{x}and u

_{y}with respect to x and y appear respectively. Therefore, u

_{x}and u

_{y}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

View Answer

Explanation: u

_{x}and u

_{y}are the primary variables in the expanded weak forms of the plane elasticity problems. Although u

_{x}and u

_{y}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 u_{x} and u_{y} 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

View Answer

Explanation: An examination of the weak form, 0=∫

_{Ωe}h

_{e}\([\frac{\partial w_1}{\partial x}(c_{11}\frac{\partial u_x}{\partial x}+c_{12}\frac{\partial u_y}{\partial y}) + c_{66}\frac{\partial w_1}{\partial y} (\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x})\)+ρw

_{1}u

_{x}]dxdy-∫

_{Ωe}h

_{e}w

_{1}f

_{x}dxdy-∮

_{┌d}h

_{e}w

_{1}t

_{x}ds reveals the following:

(i) u

_{x}and u

_{y}are the primary variables, which must be carried as the primary nodal degrees of freedom.

(ii) only first derivatives of u

_{x}and u

_{y}with 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

d) Quadratic interpolation function

View Answer

Explanation: An examination of expanded weak forms of the plane elasticity problems reveals that the variables u

_{x}and u

_{y}are the primary variables, and only first derivatives of u

_{x}and u

_{y}with respect to x and y appear, respectively. Therefore, u

_{x}and u

_{y}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?

a) Linear triangular and quadratic quadrilateral elements

b) Quadratic triangular and linear quadrilateral elements

c) Linear triangular and linear quadrilateral elements

d) Quadratic triangular and quadratic quadrilateral elements

View Answer

Explanation: Because the first-derivatives of the primary variables, u

_{x}and u

_{y}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

View Answer

Explanation: A linear triangular element (number of nodes=3) has two degrees of freedom, u

_{x}and u

_{y}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

c) Quadratic-strain triangle

d) Variable-strain triangle

View Answer

Explanation: In a linear triangular element (number of nodes=3), there are two degrees of freedom, u

_{x}and u

_{y}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

View Answer

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

View Answer

Explanation: For plane elasticity problems the strain-displacement relation is given by ε=BD, where ε=[ε

_{xx}ε

_{yy}ε

_{xy}]

^{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

View Answer

Explanation: For plane elasticity problems the strain-displacement relation is given by ε=BD, where ε=[ε

_{xx}ε

_{yy}ε

_{xy}]

^{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?

M^{e}\(\ddot{\Delta}^e\)+K^{e}Δ^{e}=F^{e}+Q^{e}

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

View Answer

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}={Δ_{0}}e^{iωt}

b) {Δ}={Δ_{0}}e^{-iω}

c) {Δ}={Δ_{0}}e^{-iωt}

d) {Δ}={Δ_{0}}e^{-iω/t}

View Answer

Explanation: For natural vibration study of plane elastic bodies, we seek a periodic solution of the form {Δ}={Δ

_{0}}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 (-ω

^{2}M

^{e}+K

^{e})\(\Delta_0^e\)=Q

^{e}.

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

View Answer

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.

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