Finite Element Method Questions and Answers – One Dimensional Problems – Galerkin Approach

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This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “One Dimensional Problems – Galerkin Approach”.

1. Galerkin technique is also called as _____________
a) Variational functional approach
b) Direct approach
c) Weighted residual technique
d) Variational technique

Explanation: The equivalent of applying the variation of parameters to a function space, by converting the equation into weak formulation. Galerkin’s method provide powerful numerical solution to differential equations and modal analysis. The Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method.

2. In the equation, $$\int_{L} \sigma^T \epsilon(\phi)Adx -\int_{L} \phi^T f Adx -\int_{L}\phi^Tdx – \sum_{i}\phi_i P_i=0$$ First term represents _______
a) External virtual work
b) Virtual work
c) Internal virtual work
d) Total virtual work

Explanation: In the given equation first term represents internal virtual work. Virtual work means the work done by the virtual displacements. The principle of virtual work is equivalent to the conditions for static equilibrium of a rigid body expressed in terms of total forces and torques. The virtual work done by internal forces is called internal virtual work.

3. Considering element connectivity, for example for element ψ=[ψ1, ψ2]n for element n, then the variational form is ______________
a) ψT(KQ–F)=0
b) ψ(KQ-F)=0
c) ψ(KQ)=F
d) ψ(F)=0

Explanation: Element connectivity means Assemble the element equations. To find the global equation system for the whole solution region we must assemble all the element equations. For formulation of a variational form for a system of differential equations. First method treats each equation independently as a scalar equation, while the other method views the total system as a vector equation with a vector function as a unknown.

4. Write the element stiffness matrix for a beam element.
a) K=$$\frac{2EI}{l}$$
b) K=$$\frac{2EI}{l}\begin{bmatrix}2 & 1 \\ 1 & 2 \end{bmatrix}$$
c) K=$$\frac{2E}{l}\begin{bmatrix}2 \\ 1 \end{bmatrix}$$
d) K=$$\frac{2E}{l}\begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix}$$

Explanation: Element stiffness matrix means it is a matrix method that makes use of the members stiffness relations for computing member forces and displacements in the structures.

5. Element connectivities are used for _____
a) Traction force
b) Assembling
c) Stiffness matrix
d) Virtual work

Explanation: Element connectivity means “Assemble the element equations. To find the global equation system for the whole solution region we must assemble all the element equations. In other words we must combine local element equations for all the elements used for discretization.

6. Virtual displacement field is _____________
a) K=$$\frac{EA}{l}$$
b) F=ma
c) f(x)=y
d) ф=ф(x)

Explanation: Virtual work is defined as work done by a real force acting through a virtual displacement. Virtual displacement is an assumed infinitesimal change of system coordinates occurring while time is held constant.

7. Virtual strain is ____________
a) ε(ф)=$$\frac{dx}{d\phi}$$
b) ε(ф)=$$\frac{d\phi}{dx}$$
c) ε(ф)=$$\frac{dx}{d\varepsilon}$$
d) ф(ε)=$$\frac{d\varepsilon}{d\phi}$$

Explanation: Virtual work is defined as the work done by a real force acting through a virtual displacement. A virtual displacement is any displacement is any displacement consistent with the constraints of the structure.

8. To solve a galerkin method of approach equation must be in ___________
a) Equation
b) Vector equation
c) Matrix equation
d) Differential equation

Explanation: Galerkin method of approach is also called as weighted residual technique. This method of approach can be used for irregular geometry with a regular pattern of nodes. The solution function is substituted in a differential equation, this differential equation will not be satisfied and will give a residue.

9. By the Galerkin approach equation can be written as __________
a) {P}-{K}{Δ}=0
b) {K}-{P}{Δ}=0
c) {Δ}-{p}{K}=0
d) Undefined

Explanation: Galerkin’s method of weighted residuals, the most common method of calculating the global stiffness matrix in fem. This requires the boundary element for solving integral equations.

10. In basic equation Lu=f, L is a ____________
a) Matrix function
b) Differential operator
c) Degrees of freedom
d) No. of elements