University FEA Assignment: ENEM20004, Short Answer Questions, T2/2019

This document presents a comprehensive solution to an FEA assignment, addressing a range of fundamental concepts in finite element analysis. The solution begins by outlining the basic approach to FEA within an engineering context, followed by definitions and examples of degrees of freedom. It then delves into the distinctions between static, dynamic, transient, linear, and nonlinear analyses, providing illustrative examples for each. The assignment also explores axisymmetry, its significance in modeling, and its governing conditions. Furthermore, the solution covers different element types in FEA, the role of isoparametric elements, and the process of discretization, including its impact on solution accuracy. Other topics addressed include aspect ratio, boundary conditions, and the factors affecting vibrational analysis. The solution also includes the determination of strains, definitions of nodes, and explanations of body forces, traction forces, and point loads. Shape functions, total potential energy, and the minimum potential energy principle are defined, along with the difference between structural and non-structural problems. The required material data for linear elastic FEA is also provided. Finally, the solution discusses methods to reduce peak stress, mesh sensitivity, convergence studies, and their interrelation.

Running head: FINITE ELEMENT ANALYSIS
FINITE ELEMENT ANALYSIS
Name of student
Name of university
Author’s note:

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FINITE ELEMENT ANALYSIS
Question 1:
Describe the basic approach to finite element analysis in the engineering context.
Finite element analysis or FET is a process of analysing how a product performs when it is
exposed to various forces which the product might encounter during application in real-
worlds. These forces includes vibration, heat, fluid flow, and other physical effects. This
provides a comprehensive idea of durability and quality of the product along with details
regarding the performance of the product when it is considered for application (Kurowski
2017). However, it is important to note that this process is not executed manually and it is
completely automated for which computerised methods are considered.
In order to execute the finite element analysis, the approach is to breakdown a real object in
different numbers of finite elements which ranges from thousands to hundreds of thousands
for example little cubes. Now in this context, different mathematical equations are considered
for predicting the behaviour of each of the elements which is then combined and analysed by
a computer for predicting the behaviour in reference to the actual object considered in this
analysis
Question 2:
Define degrees of freedom (DOF) using 3 typical examples.
Degree of freedom refers to the minimum number of variables that is required to describe the
position as well as configuration of dynamic system in the space (Filippi, Carrera and
Zenkour 2015)

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FINITE ELEMENT ANALYSIS
If a rigid body is kept in space, then there is 3 transitional as well as 3 rotational motion is
possible. Therefore, the degree of freedom is 6 in this context
If a rigid body is kept in plane, then there is 2 transitional and 1 rotational motion is possible.
Therefore, the degree of freedom is 3 in this context
Question 3:
Define and briefly discuss the following with appropriate examples:
(a) Static analysis: static analysis is the process of analysing the behaviour of a physical
structure when it is subjected to a load which is applied slowly on the structure without
significant acceleration in reference to the natural frequency of the structure (Filippi, Carrera
and Zenkour 2015)
(b) Dynamic analysis: dynamic analysis refers to the process of analysing the behaviour of a
physical structure when it is subjected to a load which is applied suddenly on the structure
with significant acceleration in reference to the natural frequency of the structure (Kurowski
2017)
(c) Transient analysis: transient analysis refers to the process of analysing the behaviour of a
physical structure due to some applied load where the load varies with time (Filippi, Carrera
and Zenkour 2015)
(d) Linear analysis: linear analysis is referred to structural analysis where there is a linear
relationship between the applied force and the displacement. Therefore, in this type of
analysis, it is possible to determine the displacement of the structure through some linear
equation if the applied force is known and defined properly for the analysis (Komzsik 2016)

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FINITE ELEMENT ANALYSIS
(e) Nonlinear analysis: non-linear analysis is referred to structural analysis where there is no
linear relationship defined between the applied force and the displacement. Therefore, in this
type of analysis, it requires some complex non-linear equations to determine the displacement
of the structure because it is not possible to determine this through some linear equation even
if the applied force is known and defined properly for the analysis (Labiod et al. 2017)
Question 4:
(a) What is axisymmetry?
Axisymmetry is referred to a different type of rotational symmetry and in this symmetry,
rotation of a one or two-dimensional shape is considered which include a 360 degrees
rotation about a central axis (Wan et al. 2016)
(b) How is it significant in a modelling context?
The main application of Axisymmetry in modelling context is that it reduces 3D structure
problems into 2D structure problem (Wan et al. 2016). The benefit of 2D approximation is
that it reduces computational time due to less number of elements. Along with that it also
increase accuracy in the mesh generation and convergence error is reduced as well.
(c) What are the governing conditions for axisymmetry?
In axisymmetry, it is not only enough for the structure to be symmetric about the axis, the
boundary conditions and other related parameters of the structure need to be symmetric as
well about the axis as well

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FINITE ELEMENT ANALYSIS
Question 5:
(a) What are the different types of elements one would encounter in finite element analysis?
Give appropriate examples for the application of 3 element types.
There are three different type of elements considered in FEA which are 1D, 2D and 3D
elements (Deshmukh et al. 2017)
1D elements are applied to model tower, bridges or buildings. These elements are applied for
linear elastic structural analysis
2D elements are applied for plane stress or plane strain analysis
3D elements are applied for modelling structures to actual structures as close as possible
(b) Discuss the significance of Isoparametric elements.
Isoparametric elements are considered for identifying area without applying integration and
this makes it easier to calculate areas of different shapes, especially elements where
boundaries are curvy which makes the calculation of area complex (Deshmukh et al. 2017)
Question 6:
What is discretization? What role does it play in finite element analysis?
The process where the element is equivalently divided into number of finite elements is
known as discretization

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FINITE ELEMENT ANALYSIS
Discretization allows to analyse easily and effectively as it divides the elements into
equivalent number of finite elements, therefore increasing the accuracy of the finite element
analysis (Stein and Olavi Rüter 2018)
Question 7:
What is meant by aspect ratio of an element? Discuss its significance in relation to solution
accuracy.
Aspect ratio is referred to the process of measuring how and to what extent elements differ
from their perfect shapes - for example equilateral triangles or squares while analysing in 2D
and regular tetrahedrons or tetrahedron while analysing in 3D (Rao 2017)
Aspect ratio provides a quantitative measures for identifying the quality of an element in
FEA. If aspect ratio of an element is 1 then it is in perfect shape and increase in aspect ratio
means the mesh is not perfect which affects structural simulation in FEA
Question 8:
Discuss the role of boundary conditions in finite element analysis. The sketch on the right
shows a propped cantilever beam under a uniformly distributed load. Briefly explain what
idealizations of reality may have been introduced in arriving at this model?
The boundary conditions are important for finding solution of boundary value problems in
FEA. Boundary value problems are considered for modelling different phenomena and

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FINITE ELEMENT ANALYSIS
applications in FEA (Auricchio et al. 2017). Therefore, boundary conditions are important in
FEA
Idealizations:
Loads are distributed throughout the beam evenly
There is no variation in the magnitude of the load
Question 9:
An FE analyst performed a vibrational analysis on a rotating shaft and noted that the
percentage error between the FEA and analytical solution increased with increasing mode
shape number. List the possible factors to which you would ascribe this observation.
Boundary conditions are not properly defined according to the conditions of the experiment
The model might not has non-structural mass
Question 10:
A displacement field in the +ve x direction is given as 𝑢 = 3𝑥2 + 14𝑦2 − 8𝑥𝑦.
Determine the strains in the x and y directions.
Strain in x direction:
6x-8y
Strain in y direction:
28y-8x

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FINITE ELEMENT ANALYSIS
Question 11:
What is a node? During discretization, what are the places where you need to place a node?
The node is referred to as coordinates defined in space which provide context for defining
degree of freedom (Brenner and Carstensen 2017)
Concentrated load acting point
Cross-section changing point
Different material interjections point
Sudden change in point load
Question 12:
You may come across body forces, traction forces and point loads in assigning loads. Define
each of these concepts with examples.
A body force is defined as the force that exerts force not to any specific context, but to the
overall volume of the body of the structure. Gravity is one such force that exerts force to the
overall volume of the body of the structure. Other body forces include forces due to electric
and magnetic field (Cazzani, Malagù and Turco 2016)
In order to produce motion for a body on a tangential surface, it is required to apply proper
force on that body. This force is known as traction force. One of the examples of traction
forces is the force that is applied against dry fiction for generating motion and this force is
referred to as the shear force that is associated with the surface (Cazzani, Malagù and Turco
2016)

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FINITE ELEMENT ANALYSIS
Sometime force is applied on a single and specific point of a structure and this force is known
as point load. This is also referred to as concentrated load. For example, when a nail is hit by
a hammer, it applies force on a specific point and therefore, it is referred to as point load
(Scherneck 2016)
Question 13:
Define a shape function? Why are polynomial shape functions more commonly used in FEA?
The shape function refers to the function that is considered for interpolating the solution
while considering the discrete values which are associated with the mesh nodes (Lee 2018).
Polynomial shape functions more commonly used in FEA because its formulation is easy
along and computerization of equations associated with the finite element is easy as well.
Along with that it is also possible to increase the accuracy of the solution through increasing
the order of the polynomial.
Question 14:
Define the following: (a) Total Potential Energy; (b) Minimum Potential Energy Principle
The total potential energy associated with an elastic body, is defined as the summation
of total strain energy (U) and the work potential (WP) (Reddy 2017).
MEP provides that if all the displacements associate with a body satisfies the compatibility
conditions along with boundary conditions, the displacement which satisfies the equilibrium
condition consist of minimum PE (Langhaar 2016).
Question 15:
Distinguish between a structural and non-structural problem.

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FINITE ELEMENT ANALYSIS
In structural problem, displacement is obtained for each nodal points. These displacement
solutions obtained in this process are considered for calculating stress and strain for each
element (Ranzi and Gilbert 2018).
In structural problem, temperature or fluid pressure is obtained for each nodal points. These
values obtained in this process are considered for heat flow or fluid flow for each element.
Question 16:
What material data is required for a typical linear elastic FEA? What additional data would
you require if you are considering body loads?
The following material data are required (Ranzi and Gilbert 2018):
 Young's modulus of elasticity and yield strength.
 Bulk and shear modulus.
 Poisson's ratio.
 Density
Question 17:
A steel cantilever beam is to be modelled with a fixed downward displacement at the free
end. The initial analysis has given a certain value of the peak stress which is beyond the
material’s yield strength. What would you do to lower this peak stress from a geometry
perspective? What would the peak stress be if the beam was made of aluminium?
In order to reduce peak stress, following measures will be taken (Ranzi and Gilbert 2018):
 Considering rounded corners instead of sharp corner

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FINITE ELEMENT ANALYSIS
 Sanding and surface polishing
 Reducing stiffness associated with straight segments that bears load
If the beam was made of aluminium, then the peak stress would be lower compared to the
steel
Question 18:
What is mesh sensitivity? What is a convergence study? How are the two related?
Mesh sensitivity refers to the number of potential singular points of the mesh
Convergence study is the process of identifying the extent to which mesh should be refined to
ensure that the simulation process is accurate and provide results which have low chance of
error (Ranzi and Gilbert 2018)
If the sensitivity of the mesh is low then it is required to refine the mesh to higher extent to
ensure that the simulation is accurate, even though it is not ensured
Question 19:
Discuss some of the common sources of errors in FEA.
There are four type of errors in FEA and those errors are (Kurowski 2017):
Discretization error: the reason for this type of error is issue in transformation of physical
problem into model associated with finite element analysis. This issue is significant in 2D

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FINITE ELEMENT ANALYSIS
and 3D problems and the strategy for reducing these type of errors is to increase the number
of element into an extent that is sufficient in this context.
Formulation error: the reason for this type of errors is application of some element that does
not have the capacity of describing behaviour associated with the physical problem. This type
of error is also important to consider for finite element analysis or FET.
Numerical error: the reason for this type of error is due to truncation error or round-off errors.
When there is small error due to round-off in reference to a one specific element, it affects the
overall system.
Human error: the reason for this type of error is ineffective data collection and thus providing
wrong input into the system.
Question 20:
Why is verification an important aspect in FEA?
In finite element analysis, it is important to consider that this process is just an approximation
and it not robust as well. If there is any error, due to modelling, ineffective method of data
input, or some error in the boundary condition, there will be significant error in the result.
One important issue in this context is that, this errors not always significant and they are not
easy to identify, however these errors has the capability to affect the performance of the
product significantly (Natarajan et al. 2015). Therefore, in order to ensure that these errors
are identified effectively and efficiently so that these errors do affect the quality and
productivity of the product, verification is considered. Therefore, verification is considered as
an important aspect in FEA.