Finite Element Method Questions and Answers – One Dimensional Problems – Co-ordinates and Shape Functions

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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

Answer: b
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.
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2. In q=[q1,q2]T is defined as __________
a) Element displacement vector
b) Element vector
c) Displacement vector
d) Shape function vector
View Answer

Answer: a
Explanation: Once the shape functions are defined, the linear displacement field within in the element can be written in terms of nodal displacements q1 and q2 and matrix notation as q=[q1,q2]. 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

Answer: c
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.
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4. Isoparametric formula is ______________
a) x=N1x1+N2x2
b) x=N2x1+N1x2
c) x=N1x1-N2x2
d) x=N2x1-N1x2
View Answer

Answer: a
Explanation: From nodal displacement equation we can write that isoparametric equation as
x=N1x1+N2x2
Here both displacement u and co-ordinate x are interpolated within the element using shape functions N1 and N2. 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

Answer: b
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.
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6. Deformation at the end of elements are called _____________
a) Load
b) Displacement functions
c) Co-ordinates
d) Nodes
View Answer

Answer: d
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.
Find the Shape Function of the element. u= N1u1(e)+N2u2(e). Here N1 & N2 are
a) N1=1-x/le&N2=x/le
b) N1=x/le&N2=1-x/le
c) N1=0 & N2=x
d) N1=x & N2=0
View Answer

Answer: a
Explanation:
Find the Shape Function of the element.

1			             2 --- local variables
I			             j --- global variables
u1(e)	                             u2(e)
x1=0			     x2=0
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Then matrix notation form is
u=\(\begin{bmatrix} 1 & x \end{bmatrix}\begin{Bmatrix}c_1 \\ c_2 \end{Bmatrix}\)
u1(e)=c1+c2(0)=c1
u2(e)= c1+c2(le)
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
N1=1-x/le& N2=x/le.

8. In shape functions, first derivatives must be _______ within an element.
a) Infinite
b) Finite
c) Natural
d) Integer
View Answer

Answer: b
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.
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9. In shape functions, _________ must be continuous across the element boundary.
a) Derivatives
b) Nodes
c) Displacement
d) Shape function
View Answer

Answer: c
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

Answer: c
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

Answer: a
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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