This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “One Dimensional Problems – Co-ordinates and Shape Functions”.

1. Natural or intrinsic coordinate system is used to define ___________

a) Co-ordinates

b) Shape functions

c) Displacement functions

d) Both shape functions and co-ordinate functions

View Answer

Explanation: Natural coordinate system is another way of representing direction. It is based on the relative motion of the object. We use this system of coordinates in defining shape functions, which are used in interpolating the displacement field.

2. In q=[q_{1},q_{2}]^{T} is defined as __________

a) Element displacement vector

b) Element vector

c) Displacement vector

d) Shape function vector

View Answer

Explanation: Once the shape functions are defined, the linear displacement field within in the element can be written in terms of nodal displacements q

_{1}and q

_{2}and matrix notation as q=[q

_{1},q

_{2}]. Here q is referred as element displacement function.

3. Shape function is just a ___________

a) Displacement function

b) Equation

c) Interpolation function

d) Matrix function

View Answer

Explanation: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Low order polynomials are typically chosen as shape functions. Interpolation within the shape functions is achieved through shape functions.

4. Isoparametric formula is ______________

a) x=N_{1}x_{1}+N_{2}x_{2}

b) x=N_{2}x_{1}+N_{1}x_{2}

c) x=N_{1}x_{1}-N_{2}x_{2}

d) x=N_{2}x_{1}-N_{1}x_{2}

View Answer

Explanation: From nodal displacement equation we can write that isoparametric equation as

x=N

_{1}x

_{1}+N

_{2}x

_{2}

Here both displacement u and co-ordinate x are interpolated within the element using shape functions N

_{1}and N

_{2}. This is called isoparametric formulation in literature.

5. B=\(\frac{1}{x_2-x_1}\)[-1 1] is an ___________

a) Strain matrix

b) Element-strain displacement matrix

c) Displacement matrix

d) Elemental matrix

View Answer

Explanation: ε=

*Bq*

Here B is element strain displacement matrix. Use of linear shape functions results in a constant B matrix. Hence, in a constant strain within the element. The stress from Hooke’s law is

σ=

*EBq*.

6. Deformation at the end of elements are called _____________

a) Load

b) Displacement functions

c) Co-ordinates

d) Nodes

View Answer

Explanation: Nodes are the points where displacement, reaction force, deformation etc.., can be calculated. Corner of each element is called a node. A node is a co-ordinate location in space where degrees of freedom are defined.

7. Write the shape function of the given element.

u= N_{1}u_{1}^{(e)}+N_{2}u_{2}^{(e)}. Here N_{1} & N_{2} are

a) N_{1}=1-x/l_{e}&N_{2}=x/l_{e}

b) N_{1}=x/l_{e}&N_{2}=1-x/l_{e}

c) N_{1}=0 & N_{2}=x

d) N_{1}=x & N_{2}=0

View Answer

Explanation:

1 2 --- local variables I j --- global variables u_{1}^{(e)}u_{2}^{(e)}x_{1}=0 x_{2}=0

Then matrix notation form is

u=\(\begin{bmatrix} 1 & x \end{bmatrix}\begin{Bmatrix}c_1 \\ c_2 \end{Bmatrix}\)

u_{1}^{(e)}=c_{1}+c_{2}(0)=c_{1}

u_{2}^{(e)}= c_{1}+c_{2}(l_{e})

In matrix equation

\(\begin{Bmatrix}u_1 \\ u_2 \end{Bmatrix} = \begin{bmatrix}

1 & 0 \\ 1 & l_e \end{bmatrix} \begin{Bmatrix}c_1 \\ c_2 \end{Bmatrix}\)

By solving we get

N_{1}=1-x/l_{e}& N_{2}=x/l_{e}.

8. In shape functions, first derivatives must be _______ within an element.

a) Infinite

b) Finite

c) Natural

d) Integer

View Answer

Explanation: In general shape functions need to satisfy that, first derivatives must be finite within element. Shape functions are interpolation functions. First derivatives are finite within element because for easy calculations.

9. In shape functions, _________ must be continuous across the element boundary.

a) Derivatives

b) Nodes

c) Displacement

d) Shape function

View Answer

Explanation: Shape functions are interpolation functions. In general shape functions need to satisfy that, displacements must be continuous across the element boundary.

10. Stresses due to rigid body motion are _______________

a) Zero

b) Considered

c) Not considered

d) Infinite

View Answer

Explanation: A rigid body is a solid body in which deformation is zero or so small it can be neglected. A rigid body is usually considered as a continuous distribution of mass. By rigid body deformation is neglected so stresses are not considered.

11. The expressions u=Nq; ε=*Bq;* σ=*EBq* relate ____________

a) Displacement, Strain and Stress

b) Strain and stress

c) Strain and displacement

d) Stress and displacement

View Answer

Explanation: Stress is defined as force per unit area. Strain is defined as the amount of deformation in the direction of applied force. Displacement is the difference between the final and initial position of a point. The given expressions show the relationship between stress, strain and displacement of a body.

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