Category: Finite Element Objective Questions

Finite Element Method Multiple Choice Questions on “Eigen Value and Time Dependent Problems – 1”.

1. The simultaneous linear equations used in FEM for solution of static problems are KX=F, the methods available for solving these equations are divided into two types: direct and iterative.
a) True
b) False
Answer: a
Clarification: When FEM is used for solution of static problems, we deal with a set of simultaneous Linear Equations of the form KX=F, where K is stiffness Matrix, X is displacement matrix and F is load vector. The order of matrix K is very large and the methods available for solving the equation are divided into two types: direct and iterative. Direct methods are used for equations without any round of error and iterative methods are used for the equations which start with an initial approximation.

2. For the following equations, what is the value of x₂ using Gaussian elimination method?
x₁-x₂+3x₃=10——– (i)
5x₂-5x₃=-5———— (ii)
-7x₃=-28 —————- (iii)
a) 1
b) 2
c) 3
d) 4
Answer: c
Clarification: From the 3^rd equation it’s seen that x₃=(frac{-28}{-7})=4. Using x₃ in 2^nd equation we get
5*x₂-5*4=-5
5*x₂=15
x₂=3

3. For the following equations, what is the value of x₁ using Gaussian elimination method?
x₁-x₂+3x₃=10——– (i)
5x₂-5x₃=-5———— (ii)
-7x₃=-28—————- (iii)
a) 1
b) 2
c) 3
d) 4
Answer: a
Clarification: From the 3^rd equation it’s seen that x₃=(frac{-28}{-7})=4. Using x₃ in 2^nd equation we get
5*x₂-5*4=-5
5*x₂=15
x₂=3
Using x₃, x₂ in 1^st equation
x₁-3+3*4=10
x₁-3+12=10
x₁=1

4. Which option is not correct about direct methods for solving system of linear equations?
a) In the absence of errors it yields exact solution
b) Errors arising from round off and truncation may give useless results
c) Gaussian elimination method is an example
d) Starts with an initial approximation
Answer: d
Clarification: The methods used for solving a system of linear equations are classified as: direct and iterative. Direct methods are those, which, in the absence of round-off and other errors, yield an exact solution in a finite number of elementary arithmetic operations. Indeed the errors arising from round-off and truncation may lead to extremely poor or even useless results. The fundamental method used for direct solutions is Gaussian elimination.

5. Which option is not correct about iterative methods for solving system of linear equations?
a) Convergence yields a good approximate solution
b) Insensitive to the growth of round-off errors
c) Gaussian elimination method is an example
d) Starts with an initial approximation
Answer: c
Clarification: The methods available for solving a system of linear equations can be divided into two types: direct and iterative. Iterative methods are those, which start with an initial approximation. When the process converges, we can expect to get a good approximate solution. The main advantages of iterative methods are the simplicity and uniformity of the operations to be performed, which make them well suited for use on computers and their relative insensitivity to growth of round-off errors.

6. In structural mechanics, which option is not correct about linear analysis?
a) Displacements are infinitesimally small
b) Material is linearly elastic
c) Externally applied loads are a function of time
d) Applied loads are not a function of time
Answer: c
Clarification: In a linear analysis, we assume that the displacements of a finite element assemblage are infinitesimally small and the material is linearly elastic. In addition, we also assume that a nature of boundary conditions remains unchanged during application of loads on Finite element assemblage. Loads are constant with respect to time.

7. A generalized Eigen value problem [K- ω²M]X=0 has a non-zero solution for X. What can be the value of determinant of the matrix [K- ω²M]?
a) Any integer
b) 0
c) +1
d) Positive integer
Answer: b
Clarification: A generalized Eigen value problem is represented by homogeneous matrix equation, [K- ω²M]X=0. From matrix equations methods, the equation has a non-zero solution for X if the determinant of the matrix [K-ω²M] equals to zero.

8. In FEM, the forced vibrations equation after Finite Element discretization of a structure can be expressed as which option?
a) Mẍ+Kẋ=F
b) Mẍ+Kẋ=0
c) Mẍ+Kx=F
d) Mẍ+Kx=0
Answer: c
Clarification: The forced vibrations equation after Finite Element discretization of a structure can be expressed as Mẍ+Kx=F where M and K are the mass and stiffness matrices of the structure, F is the external load vector; x and ẍ are the displacement and acceleration vectors. In the forced vibration equation the force vector is non-zero.

9. The free vibrations equation after Finite Element discretization of a structure is expressed as Mẍ+Kx=0. Which option is not correct about the free vibration case?
a) Displacements are harmonic
b) x=Xe^iωt where X is amplitude
c) [K-ω²M]X=0
d) KX=Mω²
Answer: d
Clarification: In a free vibration analysis, the external load vector is zero and the displacements, x are harmonic x=Xe^iωt where X is amplitude, on substituting x in governing equation we get [K-ω²M]X=0 or KX=Mω²X.

10. The generalized Eigen value problem [K-ω²M]X=0 has a non-zero solution for X. What is the value of natural frequency, ω if K=(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix}),
M=(begin{pmatrix}9&9&9\9&9&9\9&9&9end{pmatrix})?
a) 3
b) 1/9
c) 9
d) 1/3
Answer: d
Clarification: The generalized Eigen value problem [K-ω²M]=0 has a non-zero solution for X if the determinant of the matrix [K-ω²M] equals zero,
K=ω²M
(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix})=ω²*(begin{pmatrix}9&9&9\9&9&9\9&9&9end{pmatrix})
Equating corresponding elements, we get 9*ω²=1
ω²=1/9
Natural frequency, ω =1/3.

11. Which option is not correct about free vibration analysis problem KX= λMX, where X represents the amplitude of displacement x?
a) The displacements are harmonic
b) X represent mode shapes or Eigen vectors
c) λ represent Eigen value
d) ω represents Eigen value
Answer: d
Clarification: In a free vibration analysis KX= λMX, the external load vector is zero and the displacements are harmonic x=Xe^iωt where X represents the amplitude of displacement x called Eigen vectors and λ= ω² represent Eigen value.

12. After Finite Element discretization of a structure, which option expresses the free vibrations equation?
a) Mẍ+Kẋ=F
b) Mẍ+Kẋ=0
c) Mẍ+Kx=F
d) Mẍ+Kx=0
Answer: d
Clarification: After Finite Element discretization of a structure, the free vibrations equation can be expressed as Mẍ+Kx=0 where M and K are the mass and stiffness matrices of the structure; x and ẍ are the displacement and acceleration vectors respectively. In a free vibration analysis, the external load vector is zero.

13. For the eigenvalue problem of the form A(u) = λB(u), which option is not correct about the parameters used in the equation below?
(-frac{d^2x}{dx^2}=lambda u)
a) A=(frac{d^2x}{dx^2})
b) B=1
c) B=0
d) λ is called eigenvalue
Answer: c
Clarification: For the eigenvalue problem of the form A(u) = λB(u), A and B denote linear differential operators, has nontrivial solutions u. The values of λ are called eigenvalues and the associated functions U are called Eigen functions. For example, the given equation has A=(frac{d^2x}{dx^2}) and B=1.

14. The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X. What is the value of λ if K=(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix}), M=(begin{pmatrix}4&4&4\4&4&4\4&4&4end{pmatrix})?
a) 1
b) (frac{1}{2})
c) 4
d) (frac{1}{4})
Answer: d
Clarification: The generalized Eigen value problem [K- λM]X=0 has a non-zero solution for X if the determinant of the matrix [K- λM] equals zero or K=λM
(begin{pmatrix}1&1&1\1&1&1\1&1&1end{pmatrix})=λ(begin{pmatrix}4&4&4\4&4&4\4&4&4end{pmatrix})
Equating corresponding elements, we get 1=4*λ
λ=(frac{1}{4}).

15. For the following eigenvalue equation to represent a heat transfer problem, a=kA and C=ρcA.
(-frac{d}{dx}(afrac{dU}{dx}))=λCU
a) True
b) False
Answer: a
Clarification: For the given eigenvalue equation the quantities a and C depend on the physics of problem. For a heat transfer problem, a=kA and C=ρcA where k is thermal conductivity, A is cross-sectional area and c is specific heat.

Finite Element Method Multiple Choice Questions on “One Dimensional Problems – Potential Energy Approach”.

1. Continuum is discretized into_______ elements.
a) Infinite
b) Finite
c) Unique
d) Equal
Answer: b
Clarification: The continuum is a physical body structure, system or a solid being analyzed and finite elements are smaller bodies of equivalent system when given body is sub divided into an equivalent system.

2. U_e=(frac{1}{2}int) σ^TεA dx is a _____________
a) Potential equation
b) Element strain energy
c) Load
d) Element equation
Answer: b
Clarification: The given equation is Element strain energy equation. The strain energy is the elastic energy stored in a deformed structure. It is computed by integrating the strain energy density over the entire volume of the structure.

3. Which is the correct option for the following equation?
K^e=(frac{E_eA_e}{l_e}begin{bmatrix}
1 & -1 \ -1 & 1 end{bmatrix})
a) Load vector
b) Energy matrix
c) Node matrix
d) Element stiffness matrix
Answer: d
Clarification: The given matrix is element stiffness matrix. A stiffness matrix represents the system of linear equations that must be solved in order to as certain an approximate solution to the differential equation. The stiffness matrix is a inherent property of a structure. Stiffness matrix is positive definite. K_e is linearly proportional to the product E_eA_e and inversely proportional to length l_e.

4. Body force vector f^e = _____________
a) (frac{A_el_ef}{2}begin{Bmatrix}1 \ 1 end{Bmatrix})
b) (frac{A_el_e}{2}begin{Bmatrix}1 \ 1 end{Bmatrix})
c) A_el_e(begin{Bmatrix}1 \ 1 end{Bmatrix})
d) A_el_ef (begin{Bmatrix}1 \ 1 end{Bmatrix})
Answer: a
Clarification: A Body force is a force that acts throughout the volume of the body. Forces due to gravity, electric and magnetic fields are examples of body forces.

5. Between wheel and ground how much of traction force is required?
a) High traction force
b) Low traction force
c) Infinite traction force
d) No traction force
Answer: a
Clarification: Traction or tractive force is the force used to generate motion between a body and a tangential surface, through the use of dry friction, through the use of shear force of the surface. In the design of wheeled or tracked vehicles, high traction between wheel and ground should be more desirable.

6. Element traction force is given by ___
a) T^e=Tl_e(begin{Bmatrix}1 \ 1 end{Bmatrix})
b) T^e=Tl_e
c) T^e=(frac{Tl_e}{2}begin{Bmatrix}1 \ 1 end{Bmatrix})
d) Undefined
Answer: c
Clarification: Traction or tractive force, is the force used to generate motion between a body and a tangential surface, through the use of dry friction, through the use of shear force of the surface.

7. ∏ = (frac{1}{2}) Q^TKQ-Q^T F In this equation F is defined as _________
a) Global displacement vector
b) Global load vector
c) Global stiffness matrix
d) Local displacement vector
Answer: b
Clarification: Global load vector is assembly of all local load vectors. This load vector is obtained by due to given load. In the given equation F is defined as global load vector.

8. What are the basic unknowns on stiffness matrix method?
a) Nodal displacements
b) Vector displacements
c) Load displacements
d) Stress displacements
Answer: a
Clarification: Stiffness matrix represents systems of linear equations that must be solved in order to as certain an approximate solution to the differential equation. In stiffness matrix nodal displacements are treated as basic unknowns for the solution of indeterminate structures. The external loads and the internal member forces must be in equilibrium at the nodal points.

9. Write the element stiffness for a truss element.
a) K=(frac{A}{l})
b) K=(frac{AE}{l})
c) K=(frac{E}{l})
d) K=AE
Answer: b
Clarification: Truss is a structure that consists of only two force members only. Where the members are organized so that the assemblage as a whole behaves as a single object.

10. Formula for global stiffness matrix is ____________
a) No. of nodes*Degrees of freedom per node
b) No. of nodes
c) Degrees of freedom per node
d) No. of elements
Answer: a
Clarification: Generally global stiffness matrix is used to complex systems. Stiffness matrix method is used for structures such as simply supported, fixed beams and portal frames. Size of stiffness matrix is defined as:
Size of global stiffness matrix=No. of nodes*Degrees of freedom per node.

Finite Element Method Multiple Choice Questions on “Axis Symmetric Problem Modelling and Boundary Conditions”.

1. Axisymmetry implies that points lying on the z- axis remains _____ fixed.
a) Tangentially
b) Spherically
c) Radially
d) Circularly
Answer: c
Clarification: Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. Surface element may refer to an infinitesimal portion of a 2D surface, as used in a surface integral in a 3D space.

2. Modeling of a cylinder of infinite length subjected to external pressure. The length dimensions are assumed to be _____
a) Finite
b) Non uniform
c) Perpendicular
d) Constant
Answer: d
Clarification: The traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology.

3. Press fit of a ring of length L and internal radius r_j onto a rigid shaft of radius r₁+δ is considered. When symmetry is assumed about the mid plane, this plane is restrained in the _____
a) X direction
b) Y direction
c) Z direction
d) Undefined
Answer: c
Clarification: A drive shaft, driveshaft, driving shaft, propeller shaft (prop shaft), or Cardan shaft is a mechanical component for transmitting torque and rotation, usually used to connect other components of a drive train that cannot be connected directly because of distance or the need to allow for relative movement between them.

4. The condition that nodes at the internal radius have to displace radially by δ , a large stiffness C is added to the _____
a) Co-ordinates
b) Length
c) Diagonal locations
d) Radius
Answer: c
Clarification: A shaft is a rotating machine element, usually circular in cross section, which is used to transmit power from one part to another, or from a machine which produces power to a machine which absorbs power. The various members such as pulleys and gears are mounted on it.

5. Press fit on elastic shaft, may define pairs of nodes on the contacting boundary, each pair consisting of one node on the _____ and one on the ______
a) Shaft and couple
b) Sleeve and shaft
c) Shaft and sleeve
d) Sleeve and couple
Answer: b
Clarification: A shaft is a rotating machine element, usually circular in cross section, which is used to transmit power from one part to another, or from a machine which produces power to a machine which absorbs power. A flexible shaft or an elastic shaft is a device for transmitting rotary motion between two objects which are not fixed relative to one another.

6. For a Belleville spring the load is applied on _____
a) Shaft
b) Hole
c) Periphery of the circle
d) Coupling
Answer: c
Clarification: The Belleville spring, also called the Belleville washer, is a conical disk spring. The load is applied on the periphery of the circle and supported at the bottom.

7. On Belleville spring the load is applied in ______
a) X direction
b) Z direction
c) Y direction
d) Axial direction
Answer: d
Clarification: A Belleville washer, also known as a coned-disc spring, [1] conical spring washer, [2] disc spring, Belleville spring or cupped spring washer, is a conical shell which can be loaded along its axis either statically or dynamically. A Belleville washer is a type of spring shaped like a washer. It is the frusto-conical shape that gives the washer a spring characteristic.

8. In the Belleville spring, the load-deflection curve is _____
a) Linear
b) Curved
c) Non linear
d) Parabolic
Answer: c
Clarification: A Belleville washer, also known as a coned-disc spring, [1] conical spring washer, [2] disc spring, Belleville spring or cupped spring washer, is a conical shell which can be loaded along its axis either statically or dynamically.

9. A steel sleeve inserted into a rigid insulated wall. The sleeve fits snugly, and then the temperature is raised by _____
a) Uniform
b) Non uniform
c) σ
d) ΔT
Answer: d
Clarification: A sleeve is a tube of material that is put into a cylindrical bore, for example to reduce the diameter of the bore or to line it with a different material. Sometimes there is a metal sleeve in the bore to give it more strength. The pistons run directly in the bores without using cast iron sleeves.

10. In a Belleville spring, load-deflection characteristics and stress distribution can be obtained by dividing the area into ____
a) Surfaces
b) Nodes
c) Elements
d) Loads
Answer: c
Clarification: A Belleville washer, also known as a coned-disc spring, [1] conical spring washer, [2] disc spring, Belleville spring or cupped spring washer, is a conical shell which can be loaded along its axis either statically or dynamically.

Finite Element Method Questions on “Eigen Value and Time Dependent Problems – 2”.

1. Suppose the following eigenvalue equation represents a bar problem, then the value of the parameters a and c₀ should be EA and ρA, respectively.
(-frac{d}{dx}(afrac{dU}{dx}))=λc₀ U
a) True
b) False
Answer: a
Clarification: For the given eigenvalue equation, the values of the parameters a and c₀ depends upon the physical properties and phenomena involved in the problem. For a bar problem, a=EA and c₀=ρA, where E is Young’s modulus, A is cross-sectional area and m is the mass density. For a heat transfer problem, a=kA and c₀=ρcA.

2. A plane wall of length L units and Cross-section area A units was initially maintained at a temperature of T units. It is subjected to an ambient temperature of T_∞ units at one surface. If the heat transfer coefficient at the surface of the wall is assumed to be h units, then what is the temperature gradient developed at the surface?
a)(T_∞-T)(frac{h}{k})
b)(T_∞-T)(frac{1}{L} )
c) T_∞-T
d) T-T_∞
Answer: a
Clarification: Let T_x be temperature gradient developed at the surface. If the heat transfer coefficient at the surfaces of a wall is assumed to be h units, then the heat interaction at the surfaces of the wall is evaluated by Equating the conduction heat transfer to the convection heat transfer, i.e.,
kAT_x = hA(T-T_∞)
T_x=(T_∞-T)(frac{h}{k}).

3. A plane wall of thermal conductivity of 45(frac{W}{mK}) was initially maintained at a temperature of 35°C. It is subjected to an ambient temperature of 45°C at one surface. If the heat transfer coefficient at the surface of the wall is 9(frac{W}{m^2K}), then what is the temperature gradient developed at the surface?
a) 1
b) 2
c) 3
d) 4
Answer: b
Clarification: Let T_x be temperature gradient developed at the surface. If the heat transfer coefficient at the surface of a wall is is 9(frac{W}{m^2K}) then the heat interaction at the surface of the wall is evaluated by equating the conduction heat transfer to the convection heat transfer, i.e.,
45T_x = 9(35-45)
T_x = (frac{9}{-45}) (35-45)
=(frac{1}{-5}) (-10)
=2.

4. A plane wall was maintained initially at a temperature of T units. It is subjected to an ambient temperature of T_∞ units at one surface. If the heat transfer coefficient at the surfaces of the wall is assumed to be infinite, then what is the new temperature at the wall?
a) T
b) T_∞
c) T_∞-T
d) T-T_∞
Answer: b
Clarification: Let X be the unknown new temperature at the wall surface. If the heat transfer coefficient at the surfaces of a wall is assumed to be h, then the heat interaction at the surfaces of the wall is evaluated by equating the conduction heat transfer to the convection heat transfer, i.e.,
kAT_x = hA(X – T_∞)
(frac{-kT_x}{h}) = X – T_∞
Given h = ∞
(frac{-kT_x}{infty}) = X – T_∞
0 = X – T_∞
X = T_∞.

5. A plane wall was maintained initially at a temperature of 35°C. It is subjected to an ambient temperature of 45°C at one surface. If the heat transfer coefficient at the surfaces of the wall is assumed to be infinite, then what is the new temperature at the wall surface?
a) 35°C
b) 45°C
c) 40°C
d) 50°C
Answer: b
Clarification: Let X be the unknown new temperature at the wall surface. If the heat transfer coefficient at the surfaces of a wall is assumed to be h, then the heat interaction at the surfaces of the wall is evaluated by equating the conduction heat transfer to the convection heat transfer, i.e.,
kAT_x=hA(X-T_∞)
(frac{-kT_x}{h})=X-T_∞
Given h = ∞
(frac{-kT_x}{infty})=X-T_∞
0=X-T_∞
X=T_∞, given T_∞=45°C
X=45°C.

6. In thermodynamics, the following equation represents a diffusion process. If k is thermal conductivity, p is density, and c is the specific heat at constant pressure, then what is α?
(frac{partial^2 T}{partial x^2} = frac{1}{alpha} frac{partial T}{partial t})
a) (frac{k}{pc})
b) (frac{pc}{k})
c) (frac{c}{kp})
d) (frac{c}{k})
Answer: a
Clarification: The term α is called diffusion coefficient, and it is equal to (frac{k}{pc}). This equation governs one-dimensional temperature distribution in a plain wall. A one-dimensional problem is solved using bar elements with one degree of freedom at each node.

7. The governing equation of an unsteady one-dimensional heat transfer problem is given below. It has a solution u(x,t) = U(x)exp(λt). What is λ appropriately called?
(frac{-partial}{partial x} (a frac{partial u}{partial x}) + b frac{partial u}{partial t}) + cu = 0 for 0a) Natural frequency
b) Eigenvalue
c) Thermal diffusivity
d) Thermal flux
Answer: b
Clarification: The governing equation of an unsteady one-dimensional heat transfer problem is a parabolic equation. Hence, its solution is given by u(x,t) = U(x)exp(λt), where u represents temperature along a direction x at any time t, U(x) is the corresponding mode shape, and λ is the eigenvalue of the equation. The solution is periodic.

8. The unsteady natural axial oscillations of a bar are periodic, and they are determined by assuming a solution u(x, t) = U(x) e^-iwt. Which option is not correct about the solution equation?
a) w denotes the natural frequency
b) w² denotes eigenvalue
c) U(x) denotes mode shape
d) u(x, t) denotes transverse displacements
Answer: d
Clarification: The unsteady natural axial oscillations of a bar are periodic. They are measured by assuming a solution u(x, t) = U(x) e^-iwt, where w is natural frequency, w² is an eigenvalue, U(x) is mode shape, and u is instantaneous axial displacement. The problem is solved in FEM by employing bar elements and appropriate shape functions.

9. In matrix algebra, which option is not correct about an eigenvalue problem of the type Ax = Lx?
a) It has a discrete solution
b) It has solution only if A non-singular
c) x is called eigenvector
d) L is called eigenvalue
Answer: b
Clarification: An eigenvalue problem of the type Ax = Lx looks as if it should have a continuous solution, but instead, it has discrete ones. The problem is to find the numbers denoted by L, called eigenvalues, and their matching vectors denoted by x, called eigenvectors. It may have a solution irrespective of whether the matrix A is singular or not.

10. The dynamic equation of motion of a structure contains M, C and K as mass, damping and stiffness matrices of the structure, respectively. If F is an external load vector, then which option is correct about the equation?
a) M(ddot{x}) + K(dot{x}) + Cx = F
b) M(ddot{x}) is time-dependent
c) All the forces are time-independent
d) The equation is of 3^rd order
Answer: b
Clarification: The dynamic equation of motion of a structure is a 2^nd order equation. It is written as M(ddot{x}) + C(dot{x}) + Kx = F, where M, C and K are the mass, damping and stiffness matrices of structure, respectively. All the forces in the equation are time-dependent. M(ddot{x}) is inertia force, Kx is spring force, and C(ddot{x}) is damping force.

11. In matrix algebra, a matrix K equals (begin{pmatrix} 1&0&0 \ 0&1&0 \ 0&0&3 end{pmatrix}). What is the value of a, if K⁷ = (begin{pmatrix} c&0&0\ 0&b&0 \ 0&0&a end{pmatrix})?
a) 2187
b) 729
c) 6561
d) 5⁷
Answer: a
Clarification: Since K is a diagonal matrix, its higher powers are obtained by raising its diagonal elements to the same power. If K=(begin{pmatrix} 1&0&0 \ 0&1&0 \ 0&0&3 end{pmatrix}) then K⁷=(begin{pmatrix} 1^7&0&0 \ 0&1^7&0 \ 0&0&3^7end{pmatrix}). Equating the corresponding elements of (begin{pmatrix} c&0&0\ 0&b&0 \ 0&0&a end{pmatrix}) and (begin{pmatrix} 1^7&0&0 \ 0&1^7&0 \ 0&0&3^7end{pmatrix}) we get
a=3⁷
a=2187.

12. In matrix algebra, what is the eigenvalue of the matrix (begin{pmatrix} 1&1&1 \ 1&1&1 \ 1&1&1 end{pmatrix})?
a) 1
b) 2
c) 3
d) 4
Answer: c
Clarification: The eigenvalue, L of a matrix is equal to the root (factor) of the equation |K-LI|=0.
Let the given matrix be denoted by K then K-LI = (begin{pmatrix} 1&1&1 \ 1&1&1 \ 1&1&1 end{pmatrix} – L begin{pmatrix}1&0&0 \ 0&1&0 \0&0&1 end{pmatrix})
= (begin{pmatrix} 1-L&1&1 \ 1&1-L&1 \ 1&1&1-L end{pmatrix})
|K-LI| = (begin{vmatrix} 1-L&1&1 \ 1&1-L&1 \ 1&1&1-L end{vmatrix})
= (1-L)((1-L)²-1)-1(-L)+1(L)
= (1-L)(L²-2L) + 2L
= -L³ + 3L²
= -L² (L-3).
Given -L² (L-3) = 0
On simplification L = 0, 0 and 3.

13. In matrix algebra, what is the value of a-b if the eigenvector of (begin{pmatrix}1&1&1 \ 1&1&1 \ 1&1&1 end{pmatrix}) corresponding to eigenvalue three is (begin{pmatrix}a \ b \ a end{pmatrix})?
a) 0
b) 1
c) 2
d) 3
Answer: a
Clarification: If X is an eigenvector corresponding to an eigenvalue L of a matrix K, then KX=LX. The eigenvector of (begin{pmatrix}1&1&1 \ 1&1&1 \ 1&1&1 end{pmatrix}) corresponding to eigenvalue three is (begin{pmatrix}1 \ 1 \ 1 end{pmatrix}). Equating the corresponding elements of (begin{pmatrix}a \ b \ a end{pmatrix}) and (begin{pmatrix}1 \ 1 \ 1 end{pmatrix})
a=b=1
a-b=0.

14. From the Euler-Bernoulli beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the stiffness matrix?
a) (frac{12EI}{h^3})
b) (frac{12EA}{h^3})
c) (frac{12EA}{h})
d) (frac{12AI}{h^3})
Answer: a
Clarification: In the formulation of the Euler-Bernoulli beam theory, there are two degrees of freedom at a point, w and (frac{dw}{dx}). Typically, the finite element model of this theory uses cubic polynomial. The first element of the stiffness matrix is (frac{12EI}{h^3}), where E is Young’s modulus, I is the area moment of inertia and h is the length of the element.

15. From the Timoshenko beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the mass matrix?
a) (frac{rho A}{3})
b) (frac{rho A}{6})
c) 0
d) (frac{rho I}{3})
Answer: a
Clarification: Using the Timoshenko beam theory applied to natural vibrations, mode shape is approximated using the cubic Hermite polynomials (psi_i^e) and (psi_j^e). The first element of a mass matrix is (M_{ij}^{11} = int_{x_a}^{x_b} rho A psi_i^e psi_j^e) dx, where x is the length of the element. For the 1^st element, using appropriate values of (psi_i^e) and (psi_j^e), the term (M_{ij}^{11}) reduces to (frac{rho A}{3}), where ρ is the density of the beam material, and A is the cross-section area of the beam.

Finite Element Method for Entrance exams,

Finite Element Method Multiple Choice Questions 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
Answer: c
Clarification: 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
Answer: c
Clarification: 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) ψ^T(KQ–F)=0
b) ψ(KQ-F)=0
c) ψ(KQ)=F
d) ψ(F)=0
Answer: a
Clarification: 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})
Answer: b
Clarification: 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
Answer: b
Clarification: 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)
Answer: d
Clarification: 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}{dphi})
b) ε(ф)=(frac{dphi}{dx})
c) ε(ф)=(frac{dx}{dvarepsilon})
d) ф(ε)=(frac{dvarepsilon}{dphi})
Answer: b
Clarification: 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
Answer: d
Clarification: 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
Answer: a
Clarification: 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
Answer: b
Clarification: The method of weighted residual technique uses the weak form of physical problem or the direct differential equation. The basic equation Lu=f in that L is an differential operator. It uses the principle of orthogonality between Residual function and basis function.

Finite Element Method Multiple Choice Questions on “Two Dimensional Isoparametric Elements – Four Node Quadrilateral”.

1. In two dimensional isoparametric elements, we can generate element stiffness matrix by using ____
a) Numerical integration
b) Differential equations
c) Partial derivatives
d) Undefined
Answer: a
Clarification: The term isoparametric is derived from the use of the same shape functions (or interpolation functions) [N] to define the element’s geometric shape as are used to define the displacements within the element.

2. The vector q=[q₁,q₂………q₈]^T of a four noded quadrilateral denotes ____
a) Load vector
b) Transition matrix
c) Element displacement vector
d) Constant matrix
Answer: c
Clarification: A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.

3. For a four noded quadrilateral, we define shape functions on _____
a) X direction
b) Y direction
c) Load vector
d) Master element
Answer: d
Clarification: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements. The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes.

4. The master element is defined in ______
a) Co-ordinates
b) Natural co-ordinates
c) Universal co-ordinates
d) Radius
Answer: b
Clarification: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements.

5. Shape function can be written as _____
a) N_t=(1-ξ)(1-η)
b) N_t=(1-ξ)
c) N_t=(1-η)
d) N_t=(frac{1}{4})(1-ξ)(1-η)
Answer: d
Clarification: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.

6. For a four noded element while implementing a computer program, the compact representation of shape function is ____
a) N_t=(frac{1}{4})(1-ξ)(1-η)
b) N_t=(1-ξ)(1-η)
c) N_t=(frac{1}{4})(1+ξξ_i)(1+ηη_i)
d) Undefined
Answer: c
Clarification: FourNodeQuad is a four-node plane-strain element using bilinear isoparametric formulation. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot’s theory of porous medium. Each element node has 3 degrees-of-freedom (DOF): DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p).

7. For a four noded quadrilateral elements, In u^T=[u.v]^T the displacement elements can be represented as u=N₁q₁+N₂q₃+ N₃q₅+ N₄q₇
v= N₁q₂+N₂q₄+ N₃q₆+ N₄q₈
then the shape function can be represented as _____
a) (N=left[begin{array}{ |c c c c}q_1 & q_5 \ q_2 &q_6\q_3 &q_7\q_4 & q_8end{array}right])
b) (N=begin{bmatrix}q_1 &q_3 &q_5 &q_7 \ q_2 &q_4&q_6&q_8end{bmatrix})
c) (N=begin{bmatrix}q_1 \ q_2end{bmatrix})
d) (N=begin{bmatrix}N_1 & 0 & N_3 & 0&N_5&0&N_7 & 0 \ 0 & N_2 &0 &N_4&0&N_6&0&N_8end{bmatrix})
Answer: d
Clarification: Displacement function in FEM. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution.

8. The stiffness matrix from the quadrilateral element can be derived from _____
a) Uniform energy
b) Strain energy
c) Stress
d) Displacement
Answer: b
Clarification: In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to as certain an approximate solution to the differential equation.

9. For four noded quadrilateral element, the global load vector can be determined by considering the body force term in _____
a) Kinetic energy
b) Potential energy
c) Kinematic energy
d) Temperature
Answer: b
Clarification: A body force that is distributed force per unit volume, a vector, many people probably call up Vector’s definition (from Despicable Me). He says: It’s a mathematical term. A quantity represented by an arrow with both direction and magnitude. … Vector: a quantity with more than one element (more than one piece of information).

10. Shape functions are linear functions along the _____
a) Surfaces
b) Edges
c) Elements
d) Planes
Answer: b
Clarification: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.