## Finite Element Analysis [MCQ’s]

#### Module 01

1. Axisymmetry implies that points lying on the z- axis remains _____ fixed.
a) Tangentially
b) Spherically
d) Circularly
Explanation: 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
Explanation: 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 rj 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
Explanation: 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
Explanation: 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
Explanation: 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
Explanation: 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
Explanation: 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
Explanation: 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
Explanation: 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
Explanation: 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.

11. If the structure is divided into discrete areas or volumes then it is called an _______
a) Structure
b) Element
c) Matrix
d) Boundaries
Explanation: An element is a basic building block of finite element analysis. An element is a mathematical relation that defines how the degrees of freedom of node relate to next. The structure is divided into discrete areas or volumes known as elements.

12. In finite element modeling nodal points are connected by unique ________
a) Surface
b) Shape
c) Eigen values
d) Matrix
Explanation: A node is a co-ordinate location in a space where the degrees of freedom can be defined. A node may be limited in calculated motions for a variety of reasons. Element boundaries are defined when nodal points are connected by unique polynomial curve or surface.

13. In finite element modeling every element connects to _______
a) 4 nodes
b) 3 nodes
c) 2 nodes
d) Infinite no of nodes
Explanation: In finite element modeling, each element connects to 2 nodes. Better approximations are obtained by increasing the number of elements. It is convenient to define a node at each location where the point load is applied.

14. In one dimensional problem, each node has _________ degrees of freedom.
a) 2 degrees of freedom
b) 3 degrees of freedom
c) No degrees of freedom
d) 1 degree of freedom
Explanation: A degrees of freedom may be defined as, the number of parameters of system that may vary independently. It is the number of parameters that determines the state of a physical system. In one dimensional problem, every node is permitted to displace only in the direction. Thus each node has only one degree of freedom.

15. Which relations are used in one dimensional finite element modeling?
a) Stress-strain relation
b) Strain-displacement relation
c) Total potential energy
d) Total potential energy; Stress-strain relation; Strain-displacement relation.
Explanation: The basic procedure for a one dimensional problem depends upon total potential energy, stress-strain relation and strain-displacement relation are used in developing the finite element modeling.

16. One dimensional element is the linear segments which are used to model ________
a) Bars and trusses
b) Plates and beams
c) Structures
d) Solids
Explanation: In finite element method elements are grouped as one dimensional, two dimensional and three dimensional elements. One dimensional element is the linesegment which is used to model bars and trusses.

17. Discretization includes __________ numbering.
a) Element and node
b) Only nodal
c) Only elemental
d) Either nodal or elemental
Explanation: The process of dividing a body into equivalent number of finite elements associated with nodes is called discretization. Discretization includes both node and element numbering, in this model every element connects two nodes.

a) Body force
b) Traction force
d) Body force, Traction force & Point load
Explanation: The loading on an element includes body force; traction force & point load. Body force is distributed force acting on every elemental volume. Traction force is a distributed load along the surface of a body.

19. Global nodes corresponds to _______
a) Entire body
b) On surface
c) On interface
d) On element
Explanation: Global coordinate system corresponds to the entire body. It is used to define nodes in the entire body.

20. Local node number corresponds to ______________
a) Entire body
b) On element
c) On interface
d) On surface
Explanation: Local coordinate system corresponds to particular element in the body. The numbering is done to that particular element neglecting the entire body.

21. How is Assembly of stiffness matrix symbolically denoted?
a) K={k}e
b) K←∑eKe
c) K←∑Ke
d) Undefined
Explanation: The stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to differential equation.

22. What is the Strain energy equation?

Explanation: Strain energy is defined as the energy stored in the body due to deformation. The strain energy per unit volume is known as strain energy density and the area under stress-strain curve towards the point of deformation. When the applied force is released, the system returns to its original shape.

23. What is the actual equation of stiffness matrix?

Explanation: Stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. The stiffness matrix is an inherent property of the structure. A stiffness matrix is a positive definite.

24. From where does the global load vector F is assembled?
a) Element force vectors only
c) Both element force vectors and point loads
d) Undefined
Explanation: Global load vector is assembling of all local load variables. This global load vector is get from assembling of both element force vectors and point loads.

25. For an element as given below, what will be the 1ST element stiffness matrix?

Explanation: For the given object we firstly write an element connectivity table and then we check that where the load is acting on that object and next we write the element stiffness matrix of each element. For this object first element stiffness matrix is as given.

26. Principal of minimum potential energy follows directly from the principal of ________
a) Elastic energy
b) Virtual work energy
c) Kinetic energy
d) Potential energy
Explanation: The total potential energy of an elastic body is defined as sum of total strain energy and the work potential energy. Therefore the principal of minimum potential energy follows directly the principal of virtual work energy.

27. The points at where kinetic energy increases dramatically then those points are called _______
a) Stable equilibrium points
b) Unstable equilibrium points
c) Equilibrium points
d) Unique points
Explanation: If an external force acts to give the particles of the system some small initial velocity and kinetic energy will developed in that body then the point where kinetic energy decreased that point is Stable equilibrium point and the point where the kinetic energy dramatically increased then the point is called Unstable equilibrium points.

28. We can obtain same assembly procedure by Stiffness matrix method and _______
a) Potential energy method
b) Rayleigh method
c) Galerkin approach
d) Vector method
Explanation: Galerkin method provides powerful numerical solution to differential equations and modal analysis. Assembling procedure is same for both stiffness matrix method and galerkin approach method in Finite element modeling.

29. By element stiffness matrix we can get relation of members in an object in _____
a) Different matrices
b) One matrix
c) Identity matrix
d) Singular matrix
Explanation: Element stiffness matrix method is that make use of the members of stiffness relations for computing member forces and displacement in structures. So by this element stiffness matrix method we can get relation of members in an object in one matrix.

30. What is the Global stiffness method called?
a) Multiple matrix
b) Direct stiffness matrix
c) Unique matrix
d) Vector matrix
Explanation: Global stiffness matrix method makes use of the members stiffness relations for computing member forces and displacements in structures. Hence Global stiffness matrix or Direct stiffness matrix or Element stiffness matrix can be called as one.

a) Scale out technique
b) Scale up technique
c) Building technique
d) Shrinking technique
Explanation: When the workload increases on the system, the machine scales up by adding more RAM, CPU and storage spaces.

#### Module 02

1. From solid mechanics, what is the correct displacement(u) boundary condition for the following plane stress problem of the beam?

Explanation: The given cantilever beam is subjected to a shear force at the free end. The other end is supported by roller and hinge support.

2. From solid mechanics, which traction(t) boundary condition is not correct for the following beam of thickness h?

Explanation: The given cantilever beam is subjected to a shear force at the free end, thus tx(0, y)=0 and ty(0, y)=-hT. The other end is supported by both roller and hinge support.

3. In Finite Element Analysis of the beam, which primary variable does not belong to the following mesh?

a) U9=0
b) U19=0
c) U10=0
d) U20=0
Explanation: The given cantilever beam is subjected to a shear force at the free end. The other end is supported by roller and hinge support. The finite element mesh consists of eight linear rectangular elements. The node 1, 2, 3… represents the DOF (1, 2), (3, 4), (5, 6)… respectively. Since the translation along x is constrained, U9=U19 =U29=0. Because of the hinge at node 10, U20=0. The roller support doesn’t restrain vertical movement, thus U10≠0.

4. What is the total size of the assembled stiffness matrix of a plane elastic structure such that its finite element mesh has eight nodes and two degrees of freedom at each node?
a) 16×16
b) 8×8
c) 2×2
d) 4×4
Explanation: The size of the assembled stiffness matrix is equal to the total DOF of a structure. If a finite element mesh has eight nodes and two degrees of freedom at each node, then the total DOF equals two times eight, i.e., sixteen. Thus the order of the assembled stiffness matrix is 16×16.

5. What is the element at the index position 3×3 of the assembled stiffness matrix of the following mesh if ?

a) 9
b) 11
c) 13
d) 4

6. In the Finite Element Method, if two different values of the same degree of freedom are specified at a point, then such point is called as a singular point.
a) True
b) False
Explanation: The points at which both displacement and force degrees of freedom are known or when two different values of the same degree of freedom are specified are called as singular points. In problems with multiple DOF, we are required to decide as to which degree of freedom is known when singular points are encountered.

7. For time-dependent problems in FEA, which variables must be specified for each component of the displacement field problems?
a) The initial displacement and velocity
b) The initial displacement only
c) The final velocity
d) The initial displacement and final velocity
Explanation: Concerning the specification of the displacements (the primary degrees of freedom) and forces (the secondary degrees of freedom) in a finite element mesh, in general, only one of the quantities of each of the pairs (ux, tx) and (uy, ty) is known at a nodal point in the mesh. For time-dependent problems, the initial displacement and velocity must be specified for each component of the displacement field.

8. What is the magnitude of the force at node 22 if the moment M is replaced by an equivalent distributed force at x=acm?

a) 2Mb
b) Always zero
c) Mb
d) Mb
Explanation: To calculate the magnitude, assume that the force causing the moment is linear with y. At node 11, the beam is pushed towards negative x; thus, the effective force at 11 is negative. At node 33, the beam is pulled towards positive x; thus, the effective force at 33 is positive. As node 22 is located at the center, it is neither pushed nor pulled; thus, the effective force at node 22 is always zero.

#### Module 03

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
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=[q1,q2]T is defined as __________
a) Element displacement vector
b) Element vector
c) Displacement vector
d) Shape function vector
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
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=N1x1+N2x2
b) x=N2x1+N1x2
c) x=N1x1-N2x2
d) x=N2x1-N1x2
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=__1__[-1 1] is an ___________
x2x1
a) Strain matrix
b) Element-strain displacement matrix
c) Displacement matrix
d) Elemental matrix
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 _____________
b) Displacement functions
c) Co-ordinates
d) Nodes
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= 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
Explanation:

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

12. Continuum is discretized into_______ elements.
a) Infinite
b) Finite
c) Unique
d) Equal
Explanation: 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.

13. Ue=1/2 σT εA dx is a _____________
a) Potential equation
b) Element strain energy
d) Element equation
Explanation: 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.

14. Which is the correct option for the following equation?

b) Energy matrix
c) Node matrix
d) Element stiffness matrix
Explanation: 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. Ke is linearly proportional to the product EeAe and inversely proportional to length le.

15. Body force vector fe = _____________

Explanation: 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.

16. 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
Explanation: 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.

17. ∏ = 12 QTKQ-QT F In this equation F is defined as _________
a) Global displacement vector
c) Global stiffness matrix
d) Local displacement vector
Explanation: 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
d) Stress displacements
Explanation: 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=Al
b) K=AEl
c) K=El
d) K=AE
Explanation: 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.

20. 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
Explanation: 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.

1. Dimension of global stiffness matrix is _______
a) N X N, where N is no of nodes
b) M X N, where M is no of rows and N is no of columns
c) Linear
d) Eliminated
Explanation: A global stiffness matrix is a method that makes use of members stiffness relation for computing member forces and displacements in structures. The dimension of global stiffness matrix K is N X N where N is no of nodes.

22. Each node has only _______
a) Two degrees of freedom
b) One degree of freedom
c) Six degrees of freedom
d) Three degrees of freedom
Explanation: Degrees of freedom of a node tells that the number of ways in which a system can allowed to moves. In a stiffness matrix each node can have one degree of freedom.

23. Global stiffness K is a______ matrix.
a) Identity matrix
b) Upper triangular matrix
c) Lower triangular matrix
d) Banded matrix
Explanation: A banded matrix is a sparse matrix whose non zero entities are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. A global stiffness matrix K is a banded matrix. That is, all the elements outside the band are zero.

24. The dimension of Kbanded is _____ (Here NBW is half bandwidth)
a) [N X NBW ]
b) [NBW X N]
c) [N X N]
d) [NBW X NBW]
Explanation: K can be compactly represented in banded form. As Kbanded is of dimension [N X NBW] where NBW is the half band width.

25. In many one-dimensional problems, the banded matrix has only two columns. Here NBW=____
a) 6
b) 3
c) 7
d) 2
Explanation: NBW means half bandwidth. Many of the One- dimensional problems banded matrix has only 2 columns then NBW=2. We know that
NBW=max(Difference between dof numbers connecting an element)+1

26. Stiffness matrix represents a system of ________
a) Programming equations
b) Iterative equations
c) Linear equations
d) Program CG SOLVING equations
Explanation: Stiffness is amount of force required to cause the unit displacement same concept is applied for stiffness matrix. The stiffness matrix represents a system of linear equations that must be solved in order to ascertain an approximate solution to differential equation.

27. Stiffness matrix is _____
a) Non symmetric and square
b) Symmetric and square
c) Non symmetric and rectangular
d) Symmetric and rectangular
Explanation: Stiffness matrix is a inherent property of the structure. The property of a stiffness matrix, as the stiffness matrix is square and symmetric.

28. In stiffness matrix, all the _____ elements are positive.
a) Linear
b) Zigzag
c) Diagonal
d) Rectangular
Explanation: Stiffness matrix represents system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. The stiffness matrix is an inherent property of a structure. In stiffness matrix all the diagonal elements are positive.

29. The size of global stiffness matrix will be equal to the total ______ of the structure.
a) Nodes
b) Degrees of freedom
c) Elements
d) Structure
Explanation: For a global stiffness matrix, a structural system is an assemblage of number of elements. These elements are interconnected to form the whole structure. The size of global stiffness matrix will be equal to the total degrees of freedom of the structure.

30. Element stiffness is obtained with respect to its ___
a) Degrees of freedom
b) Nodes
c) Axes
d) Elements
Explanation: A stiffness matrix represents system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. Element stiffness is obtained with respect to its axes.

31. Types of Boundary conditions are ______
a) Potential- Energy approach
b) Penalty approach
c) Elimination approach
d) Both penalty approach and elimination approach
Explanation: Boundary condition means a condition which a quantity that varies through out a given space or enclosure must be fulfill at every point on the boundary of that space. In fem, Boundary conditions are basically two types they are Penalty approach and elimination approach.

32. Potential energy, π = _________
a) 1/2[QTKQ-QTF]
b) QKQ-QF
c) 1/2[KQ-QF]
d) 1/2[QF]
Explanation: Minimum potential energy theorem states that “Of all possible displacements that satisfy the boundary conditions of a structural system, those corresponding to equilibrium configurations make the total potential energy assume a minimum value.”
Potential energy π=1/2[QTKQ-QTF]

33. Equilibrium conditions are obtained by minimizing ______
a) Kinetic energy
b) Force
c) Potential energy
Explanation: According to minimum potential energy theorem, that equilibrium configurations make the total potential energy assumed to be a minimum value. Therefore, Equilibrium conditions are obtained by minimizing Potential energy.

34. In elimination approach, which elements are eliminated from a matrix ____
a) Force
c) Rows and columns
d) Undefined
Explanation: By elimination approach method we can construct a global stiffness matrix by load and force acting on the structure or an element. Then reduced stiffness matrix can be obtained by eliminating no of rows and columns of a global stiffness matrix of an element.

35. In elimination approach method, extract the displacement vector q from the Q vector. By using ___
a) Potential energy
c) Force
d) Element connectivity
Explanation: By elimination approach method we can construct a global stiffness matrix by load and force acting on the structure or an element. Then we extract the displacement vector q from the Q vector. By using Element connectivity, and determine the element stresses.

36. Penalty approach method is easy to implement in a ______
a) Stiffness matrix
b) Iterative equations
c) Computer program
d) Cg solving
Explanation: Penalty approach is the second approach for handling boundary conditions. This method is used to derive boundary conditions. This approach is easy to implement in a computer program and retains it simplicity even when considering general boundary conditions.

37. If Q1=a1 then a1is _________
a) Displacement
b) Symmetric
c) Non symmetric
d) Specified displacement
Explanation: In penalty approach method a1 is known as specified displacement of 1. This is used to model the boundary conditions.

38. The first step of penalty approach is, adding a number C to the diagonal elements of the stiffness matrix. Here C is a __________
a) Large number
b) Positive number
c) Real number
d) Zero
Explanation: Penalty approach is one of the method to derive boundary conditions of an element or a structure. The first step is adding a large number C to the diagonal elements of the stiffness matrix. Here C is a large number.

39. In penalty approach evaluate _______ at each support.
b) Degrees of freedom
c) Force vector
d) Reaction force
Explanation: By penalty approach we can derive boundary conditions of an element or a structure. The first step of this approach is to add a large number to the diagonal elements. Second step is to extract element displacement vector. Third step is to evaluate reaction force at each point.

40. For modeling of inclined roller or rigid connections, the method used is ___
a) Elimination approach
b) Multiple constraints
c) Penalty approach
d) Minimum potential energy theorem
Explanation: Multiple constraints is one of the method for boundary conditions it is generally used in problems for modeling inclined rollers or rigid connections.

41. With temperature effect which will vary linearly?
b) Potential energy
d) Kinematic energy
Explanation: Temperature is a variant which varies from one point to another point. It has adverse effects on different structures. By temperature effect Vertical stress load vary linearly.

42. α means ____
a) Co-efficient of thermal expansion
b) Co-efficient of linear expansion
c) Thermal expansion
d) Thermal effect
Explanation: The co-efficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in size per degree change in temperature at constant pressure. It is denoted by symbol α.

43. In temperature effect, initial strain, ε0= ____
a) α ΔT
b) α+ΔT
c) α-ΔT
Explanation: Strain is relative change in shape or size of an object due to externally applied forces. Temperature is a variant which varies from one point to another point. In temperature effect of FEM, Initial strain ε0=α ΔT.

44. In a structure, a crack is formed as a result of ______
a) Thermal expansion
b) Thermo couple
c) Thermal strain
d) Thermal stress
Explanation: Thermal stress is caused by differences in temperature or by differences in thermal expansion. A crack formed as a result of Thermal stress produced by rapid cooling from a high temperature.

#### Module 04

1. In 2D elements. Discretization can be done. The points where triangular elements meet are called ____
a) Displacement
b) Nodes
c) Vector displacements
d) Co-ordinates
Explanation: The two dimensional region is divided into straight sided triangles, which shows as typical triangulation. The points where the corners of the triangles meet are called nodes.

2. Each triangle formed by three nodes and three sides is called a ______
a) Node
b) Force matrix
c) Displacement vector
d) Element
Explanation: An element is a basic building block of finite element analysis. An element is a mathematical relation that defines how the degrees of freedom of a node relate to next. In discretization of 2D element each triangle is called element.

3. The finite element method is used to solve the problem ______
a) Uniformly
b) Vigorously
c) Approximately
d) Identically
Explanation: The finite element method is a numerical method for solving problems of engineering and mathematical physics. Typical problems areas of interest include structure analysis, heat transfer, fluid flow, mass transport and electromagnetic potential etc..,. The method yields approximate values of the unknowns at discrete number of points.

4. In two dimensional modeling each node has ____ degrees of freedom.
a) One
b) Infinity
c) Finite
d) Two
Explanation: In two dimensional problem, each node is permitted to displace in the two directions x and y. Thus each node has two degrees of freedom.

5. For a triangular element,element displacement vector can be denoted as ___
a) q=[q1,q2,q3]T
b) q=[q1,q2]T
c) q=[q1,q2,……q6]T
Explanation: The displacement components of a local node is represented in x and y directions, respectively. For that we denote element displacement vector as
q=[q1,q2,……q6]T.

6. In two dimensional analysis, stresses and strains are related as ___
a) σ=Dε
b) σ=ε
d) ε=Dσ
Explanation: When a material is loaded with force, it produces stress. Which then cause material to deform. Strain is response of a system t an applied stress.

7. In two dimensional modeling, body force is denoted as ___
a) f=[fx,fy]T
b) σ=Dε
c) q=lq
d) f=[2|i-j|+1]
Explanation: A body force is a force that acts throughout the volume of the body. Body forces contrast with contact forces or the classical definition of surface forces which are exerted to the surface of the object. Body force is denoted as
f=[fx,fy]T.

8. The information of array of size and number of elements and nodes per element can be seen in ___
a) Column height
b) Element connectivity table
c) Matrix form
d) Undefined
Explanation: An element connectivity table specifies global node number corresponding to the local node element. Element connectivity is the nodal information for the individual element with details how to fit together to form the complete original system.

9. In two dimensional modeling, traction force is denoted as ____
a) Row vector
b) T=[Tx,Ty]T
c) f=[fx,fy]T
d) σ=Dε
Explanation: Traction or tractive force is the force used to generate motion between body and a tangential surface, through the use of dry friction, through the use of hear force. Tractive force is defined as
T=[Tx,Ty]T

10. In two dimensional modeling, elemental volume is given by ____
a) dV=tdA
b) dV=dA
c) f=[fx,fy]T
d) Trussky program
Explanation: In mathematics, a volume element provides a means for integrating a function with respect to volume in various co-ordinate systems such as spherical co-ordinates and cylindrical co-ordinates. Then elemental volume is given by
dV=tdA.

11. Finite element method is used for computing _____ and _____
a) Stress and strain
b) Nodes and displacement
c) Nodes and elements
d) Displacement and strain
Explanation: The finite element method is a numerical method for solving problems of engineering and mathematical physics. To solve the problem it subdivides a larger problem into smaller, simpler parts that are called finite elements.

12. In deformation of the body, the symmetry of ______ and symmetry of ____ can be used effectively.
a) Stress and strain
b) Nodes and displacement
c) Geometry and strain
Explanation: Deformation changes in an object’s shape or form due to the application of a force or forces. Deformation proportional to the stress applied within the elastic limits of the material.

13. For a circular pipe under internal or external pressure, by symmetry all points move _____
b) Linearly
c) Circularly
d) Along the pipe
Explanation: The boundary conditions require that points along x and n are constrained normal to the two lines respectively. If a circular pipe under internal or external pressure, by symmetry all the points move radially.

14. Boundary conditions can be easily considered by using _______
a) Rayleigh method
b) Penalty approach method
c) Galerkin approach
d) Potential energy approach
Explanation: In computation of Finite element analysis problem defined under initial or boundary conditions. For implementation of boundary conditions we need a staggered grid.

15. When dividing an area into triangles, avoid large _____
a) Dimensions
c) Aspect ratios
d) Boundary conditions
Explanation: Aspect ratio is defined as ratio of maximum to minimum characteristics dimensions. For this reason we can avoid large aspect ratios when dividing an area into triangles.

16. In dividing the elements a good practice may be to choose corner angles in the range of ____
a) 30-120°
b) 90-180°
c) 25-75°
d) 45-180°
Explanation: The best elements are those that approach an equilateral triangular configuration. Such configurations are usually not possible. A good practice is to choose corner angle in the range of 30-120°.

7. Stresses can be change widely at ____
a) Large circular sections
b) Notches and fillets
c) Corners
d) Crystals
Explanation: In a structure geometrical notches, such as holes cannot be avoided. The notches are causing in a homogeneous stress distribution, as notches fillets are also a cause for in homogenous stress distribution.

18. The Constant strain triangle can give____ stresses on elements.
a) Linear
b) Constant
c) Uniform
d) Parallel
Explanation: The constant strain triangle or cst is a type of element used in finite element analysis which is used to provide an approximate solution in a 2D domain to the exact solution of a given differential equation. By this we get constant stresses on elements.

19. The _____ can be obtained even with coarser meshes by plotting and extrapolating.
a) Minimum stresses
b) Minimum strain
c) Maximum stresses
d) Maximum strain
Explanation: Coarse mesh is more accurate in getting values. The smaller elements will better represent the distribution. Better estimates of maximum stress may be obtained even with coarser meshes. Coarse meshes are recommended for initial trails.

20. Coarser meshes are recommended for _____
b) Notches and fillets
c) Crystals
d) Initial trails
Explanation: The smaller elements will better represent the distribution. Coarse mesh is more accurate in getting values. Better estimates of maximum stress may obtained even with the coarse meshes.

21. Give an example of orthotropic material?
a) Topaz
b) Aluminum
c) Barium
d) Sodium
Explanation: Orthotropic materials have material properties that differ along three mutually orthogonal two fold axis of rotational symmetry. They are a subset of anisotropic materials, because their properties change when measured from different directions.

22. Unidirectional fiber- reinforced composites also exhibit _______ behavior.
a) Isotropic
b) Orthotropic
c) Material
d) Unidirection composite
Explanation: Orthotropic materials have material properties that differ along three mutually orthogonal two fold axis of rotational symmetry. They are a subset of anisotropic materials, because their properties change when measured from different directions.

23. Orthotropic planes have ____ mutually perpendicular planes of elastic symmetry.
a) One
b) Two
c) Three
d) Four
Explanation: Orthotropic materials have material properties that differ along three mutually orthogonal two fold axis of rotational symmetry. Wood may also consider to be orthotropic. These materials have three mutually perpendicular planes.

24. The principal material axes that are normal to the _______
a) Co-ordinates
b) Number of nodes
c) Principal axes
d) Plane of symmetry
Explanation: Orthotropic materials are a subset of anisotropic; their properties depend upon the direction in which they are measured. Orthotropic materials have three planes of symmetry. That is normal to principal material axes.

25. v12 indicates that the poisson’s ratio that characterizes the decrease in ______ during tension applied in ______
a) 2- direction and 1- direction
b) 2- direction and 3- direction
c) 1- direction and 2- direction
d) 2- direction and 4- direction
Explanation: Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Tensile deformation is considered positive and compressive deformation is considered negative.

26. Unidirectional composites are stacked at different fiber orientations to form a ______
a) Laminate
b) Orthotropic material
c) Isotropic material
d) Anisotropic material
Explanation: A unidirectional (UD) fabric is one in which the majority of fibers run in one direction only. A laminate is a tough material that is made by sticking together two or more layers of a particular substance.

27. When an orthotropic plate is loaded parallel to its material axes, it results only _____
a) Shear strains
b) Normal strains
c) Parallel strains
d) Uniform strains
Explanation: Orthotropic materials are a subset of anisotropic; their properties depend upon the direction in which they are measured. When an orthotropic plate is loaded parallel to its material axes, it results normal strains.

28. When the stresses are determined in an orthotropic material, then they are used to determine ____
a) Strains
b) Deformation
c) Factor of safety
Explanation: Factors of safety (FoS), is also known as safety factor (SF), is a term describing the load carrying capacity of a system beyond the expected or actual loads. Essentially, the factor of safety is how much stronger the system is than it needs to be for an intended load. After determining the stresses in orthotropic materials by using an appropriate failure theory we can find factor of safety.

29. Stress- strain law defined as ______
a) σ=D(ε-ε0)
b) σ=D
c) σ=Dε
d) σ=Dε0
Explanation: The relationship between the stress and strain that a particular material displays is known as that particular material’s stress–strain curve. It is unique for each material and is found by recording the amount of deformation (strain) at distinct intervals of tensile or compressive loading (stress).

30. E1 value of Balsa wood is ___
a) 0.125*106psi
b) 12.04*106psi
c) 23.06*106psi
d) 7.50*106psi
Explanation: A material’s property (or material property) is an intensive, often quantitative, property of some material. Quantitative properties may be used as a metric by which the benefits of one material versus another can be assessed, thereby aiding in materials selection.

31. What is a shape function?
a) Interpolation function
b) Displacement function
c) Iterative function
d) Both interpolation and displacement function
Explanation: The shape function is a function which interpolates the solution between discrete values obtained at the mesh nodes. Lower order polynomials are chosen as shape functions. Shape function is a displacement function as well as interpolation function.

32. Quadratic shape functions give much more _______
a) Precision
b) Accuracy
c) Both Precision and accuracy
d) Identity
Explanation: The shape function is function which interpolates the solution between discrete values obtained at the mesh nodes. The unknown displacement field was interpolated by linear shape functions within each element. Use of quadratic interpolation leads to more accurate results.

33. Strain displacement relation ______
a) ε=du/dx
b) ε=du/dϵ
c) x=dϵ/du
d) Cannot be determined
Explanation: The relationship is that connects the displacement fields with the strain is called strain – displacement relationship. If strain is ε then strain – displacement relation is
ε=du/dx

4. The _____ and ______ can vary linearly.
b) Precision and accuracy
c) Strain and stress
d) Distance and displacement
Explanation: Strain is defined as a geometrical measure of deformation representing the relative displacement between particles in a material body. Stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other. In quadratic shape functions strain and stress can vary linearly.

35. By Hooke’s law, stress is ______
a) σ=Bq
b) σ=EB
c) B=σq
d) σ=EBq
Explanation: Hooke’s law states that the strain in a solid is proportional to the applied stress within the elastic limit of that solid.

36. Nodal displacement as _____
a) u=Nq
b) N=uq
c) q=Nu
d) Program SOLVING
Explanation: Nodes will have nodal displacements or degrees of freedom which may include translations, rotations and for special applications, higher order derivatives of displacements.

7. At the condition, at , N1=1 at ξ=-1 which yields c=1/2. Then these shape functions are called ____
a) Computer functions
b) Programming functions
c) Galerkin function
d) Lagrange shape functions
Explanation: The lagrange shape function sum to unity everywhere. At the given condition the shape functions are named as Lagrange shape functions.

#### Module 05

1. In solid mechanics, what does linearized elasticity deal with?
a) Small deformations in linear elastic solids
b) Large deformations in linear elastic solids
c) Large deformations in non-Hookean solids
d) Small deformations in non-Hookean solids
Explanation: The part of solid mechanics that deals with stress and deformation of solid continua is called Elasticity. Linearized elasticity is concerned with small deformations (i.e., strains and displacements that are very small compared to unity) in linear elastic solids or Hookean solids (i.e., obey Hooke’s law).

2. For plane elasticity problems in three dimensions, which option is not responsible for making the solutions independent of one of the dimensions?
a) Geometry
b) Boundary conditions
d) Material
Explanation: Elasticity is the part of solid mechanics that deals with stress and deformation of solid continua. There is a class of problems in elasticity whose solution (i.e., displacements and stresses) is not dependent on one of the coordinates because of their geometry, boundary conditions, and externally applied loads. Such problems are called plane elasticity problems.

3. For a plane strain problem, which strain value is correct if the problem is characterized by the displacement field ux=ux(x,y), uy=uy(x,y) and uz=0?
a) εxy=0
b) εxz=0
c) εyz≠0
d) εxz≠0
Explanation: The plane strain problems are characterized by the displacement field ux=ux(x,y), uy=uy(x,y) and uz=0, where (ux, uy, uz) denote the components of this displacement vector u in the (x, y, z) coordinate system.

4. For an orthotropic material, if E and v represent Young’s modulus and the poisons ratio, respectively, then what is the value of v12 if E1=200 Gpa, E2=160 Gpa and v21=0.25?
a) 0.3125
b) 0.05
c) 0.2125
d) 0.3
Explanation: For an orthotropic material, E1 and E2 are the principal (Young’s) moduli in the x and y directions, respectively. The poisons ratio and Young’s moduli are related by the equation
v12=v21 E1/E2.
v12=0.25*200/160
=0.25*1.25
=0.3125.

5. Under plane stress condition in the XYZ Cartesian system, which stress value is correct if a problem is characterized by the stress field σxxxx(x,y), σyyyy(x,y) and σzz=0?
a) σxy=0
b) σyx≠0
c) σzx≠0
d) σyz≠0
Explanation: A state of plane stress in XYZ Cartesian system is defined as one in which the following stress field exists:
σxzyzzz=0, σxx(x,y), σxyxy(x,y) and σyyyy(x,y).
Thus, σxx , σxy and σyy are non-zero stresses. Such a problem in three dimensions can be dealt with as a two-dimensional (plane) problem.

6. For theplane stress problem in XYZ Cartesian system, σxxxx(x,y), σyyyy(x,y) and σzz=0, which option is correct regarding the associated strain field?
a) εxx=0
b) εyx=0
c) εzx=0
d) εyy=0
Explanation: The strain field associated with the given stress field has the form ε=Sσ, where the matrix S is a symmetric matrix, and it is called elastic compliances matrix. In the XYZ Cartesian system, all the strain components except εyz and εzx are non-zero. Thus, εxx≠0, εyy≠0, εzz≠0, εxy≠0, where as εyz=0 and εzx=0.

7. For any two cases of plane elasticity problems, if the constitutive equations are different, then their final equations of motion are also different.
a) True
b) False
Explanation: The equations of motion for plane elasticity problems are given by D*σ+f=ρü in the vector form, where f denotes body force vector, σ is the stress vector, u is displacement vector, D is a matrix of the differential operator, and ρ is the density. Note that the equations of motion of plane stress and plane strain cases differ from each other only on account of the difference in their constitutive equations.

8. In solid mechanics, which option is not a characteristic of a plane stress problem in the XYZ Cartesian system?
a) One dimension is very small compared to the other two dimensions
b) All external loads are coplanar
c) Strain along any one direction is zero
d) Stress along any one direction is zero
Explanation: An example of a plane stress problem is provided by a plate in the XYZ Cartesian system that is thin along the Z-axis. It is acted upon by external loads lying in the xy plane (or parallel to it) that are independent of the Z coordinate. Thus, stresses and strains are observed in all directions except that the stress is zero along the Z-axis.

9. In solid mechanics, what is the correct vector form of the equations of motion for a plane elasticity problem?
a) D*σ+f=ρü
b) D*σ+f=ρu̇
c) D2*σ+f=ρü
d) D*σ+f=ρu
Explanation: For plane elasticity problems, the equations of motion are one of the governing equations. The vector form of equations of motion is D*σ+f=ρü, where f denotes body force vector, σ is the stress vector, u is the displacement vector, D is a matrix of differential operator and ρ is the density.

10. For plane elasticity problems, which type of boundary condition is represented by the equation tx≡σxxnxxyny, where tx is surface traction force and n is direction cosine?
a) Essential boundary condition
b) Natural boundary condition
c) Both Essential and natural boundary conditions
d) Dirichlet boundary condition
Explanation: For plane elasticity problems, the boundary conditions are one of the governing equations. There are two types of boundary conditions, namely, essential boundary conditions and natural boundary conditions. The equation tx≡σxxnxxyny represents natural boundary condition or Neumann boundary condition.

11. Finite element method uses the concept of _____
a) Nodes and elements
b) Nodal displacement
c) Shape functions
d) Assembling
Explanation: The finite element method is a numerical method for solving problems of engineering and mathematical physics. Finite element method uses the concept of shape functions in systematically developing the interpolations.

12. For constant strain elements the shape functions are ____
a) Spherical
c) Polynomial
d) Linear
Explanation: The constant strain triangle element is a type of element used in finite element analysis which is used to provide an approximate solution in a 2D domain to the exact solution of a given differential equation. For CST shape functions are linear over the elements.

13. Linear combination of these shape functions represents a ______
a) Square surface
b) Linear surface
c) Plane surface
d) Combinational surface
Explanation: A constant strain element is used to provide an approximate solution to the 2D domain to the exact solution of the given differential equation. The shape function is a function which interpolates the solution between the discrete values obtained at the mesh nodes.

14. In particular, N1+N2+N3 represent a plane at a height of one at nodes ______
a) One
b) Two
c) Three
d) One, two and three
Explanation: Any linear combination of these shape functions also represents a plane surface. In particular, N1+N2+N3 represents a plane height of one at nodes one, two, and, three and thus it is parallel to the triangle 123.

15. If N3 is dependent shape function, It is represented as ____
a) N3
b) N3=1-ξ
c) N3=1-η
d) N3=1-ξ-η
Explanation: The shape function is a function which interpolates the solution between the discrete values obtained at the mesh nodes. N1, N2, N3 are not linearly independent only one of two of these are independent.

16. In two dimensional problems x-, y- co-ordinates are mapped onto ____
a) x-, y- co-ordinates
b) x-, ξ – co-ordinates
c) η-, y- co-ordinates
d) ξ-η-Co-ordinates
Explanation: The similarity with one dimensional element should be noted ; in one dimensional problem the x- co-ordinates were mapped onto ξ- co-ordinates and the shape functions were defined as functions of ξ. In the two dimensional elements the x-, y-, co-ordinates are mapped onto ξ-,,η – co-ordinates.

17. The shape functions are physically represented by _____
a) Triangular co-ordinates
b) ξ-,η-Co-ordinates
c) Area co-ordinates
d) Surface co-ordinates
Explanation: The shape function is a function which interpolates the solution between discrete values obtained at the mesh nodes. Therefore appropriate functions have to be used and as already mentioned; low order typical polynomials are used in shape functions. The shape functions are physically represented by area co-ordinates.

18. A1 is the first area and N1 is its shape function then shape function N1= ___
a) A1/A
b) A-A1
c) A1+A
d) A1
Explanation: The shape functions are physically represented by area co-ordinates. A point in a triangle divides into three areas. The shape functions are precisely represented as
N1=A1/A .

19. The equation u=Nq is a _____ representation.
a) Nodal
b) Isoparametric
c) Uniparametric
d) Co-ordinate
Explanation: The isoparametric representation of finite elements is defined as element geometry and displacements are represented by same set of shape functions.

10. For plane stress or plane strain, the element stiffness matrix can be obtained by taking _____
a) Shape functions, N
b) Material property matrix, D
c) Iso parametric representation, u
d) Degrees of freedom, DoF
Explanation: The material property matrix is represented as ratio of stress to strain that is σ=Dε . Therefore by this relation element stiffness matrix can be obtained by material property matrix.

11. In a constant strain triangle, element body force is given as ____

Explanation: A body force is a force which acts through the volume of the body. Body forces contrast with the contact forces or the classical definition of the surface forces which are exerted to the surface of the body.

22. Traction force term represented as ___
a) ∫uT Tl
b) ∫uTT
c) ∫uT
d) uTTl
Explanation: Traction force or tractive force are used to generate a motion between a body and a tangential surface, through the use of dry friction, through the use of shear force of the surface is also commonly used.

23. In the equation KQ=F, K is called as ____
a) Stiffness matrix
b) Modified stiffness matrix
c) Singular stiffness matrix
d) Uniform stiffness matrix
Explanation: The stiffness matrix represents system of linear equations that must be solved in order to ascertain an approximate solution to differential equation. The stiffness and force modifications are made to account for the boundary conditions.

24. Principal stresses and their directions are calculated by using ____
a) Galerkin approach
b) Rayleigh method
c) Potential energy method
d) Mohr’s circle method
Explanation: Mohr’s circle is two dimensional graphical representation of the transformation law. While considering longitudinal stresses and vertical stresses in a horizontal beam during bending.

25. I the distribution of the change in temperature ΔT, the strain due to this change is ____
a) Constant strain
b) Stress
c) Initial strain
d) Uniform strain