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Mechanics And Mathematics of
FEA
This is the first part of the ‘Introduction to
FEA’, which describes basic theory of mechanics and various analysis
types. Click here to go to the second
page. Together with the web page outlining ‘Ten Common Mistakes in
FEA’ (click here to see it), these
three pages are intended to help engineers not trained in the field of FEA,
better understand the process. It is not software specific but could help to
avoid some common problems and bad practice.
The types of, and applications for analysis considered in these
pages are those typically required of FEA in mechanical engineering: namely
various forms of static, dynamic and thermal analysis.
We are able to advise on best practice in developing your analysis
strategies. Contact us if you are unsure about
the best way to formulate a particular analysis problem.
Additional Sources of Information
Dermot Monaghan's site has more extensive information. Click on
the 'FE Information' menu: www.dermotmonaghan.com
There are many books on mechanics; a comprehensive book on the
subject is:
Mechanics of Materials, by J.M. Gere & Stephen P.
Timoshenko
Follow this link to view this book at Amazon:
Mechanics
of Materials (Mechanics of)
A more basic text that we have used extensively is:
Mechanics of Materials: An Introduction to the Mechanics
of Elastic and Plastic Deformation of Solids and Structural Components, by E.J.
Hearn
Follow this link to view this book at Amazon:
Mechanics
of Materials: An Introduction...
The amazon website sells
these titles as well as many others on the subject.
Evaluating a Component and Its Application -
Structural Considerations
Static Equilibrium
For static and quasi-static problems, the body to be analysed
should be in st atic equilibrium. This is an important concept to understand:
analysis should include all the forces that contribute to the state of
equilibrium; omission of any one force will invalidate the analysis. Drawing a
free body diagram is often helpful in determining whether all the forces acting
on the component or system have been considered.
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Tension, Compression,
Torsion and Bending
Judgements must be made as to what static loads are significant
based on their magnitude relative to the bending, torsional or tensile strength
of the component. Often, the relevant simple beam, torsion or tension equations
can be applied in preliminary calculations, to determine the most critical
loads for subsequent FE analysis.
Some examples of these important
structural equations are given below:
Beams
An understanding of beam behaviour can be of benefit in very
many static situations. The beam bending stress formula is given by:
σ = M.y / I
Where σ is the direct stress, M is the moment,
y is the distance from the neutral axis and I is the second moment of area of
the cross section. This equation is derived from a consideration of pure
bending (where the radius of curvature of the beam is constant along its
length) and ignores the effect of warping. This is acceptable providing shear
stresses are small compared with bending stresses, which they are in most
situations. Bending stresses are greatest at the outer fibres of the beam,
furthest from the neutral axis; shear stresses are greatest in the middle of
the beam.
If the depth of a beam section is doubled, I increases by 8 and y
by 2, so the bending stress is reduced to 2/8, or ¼ of its original
value.
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Membrane and Bending Stiffness of
Plates
The stiffness (i.e. load/deflection) of a thin sheet when loaded
in its plane will be many orders greater than its stiffness normal to the
direction of the plane; thus a small lateral load could be more important than
a larger in plane one.
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Torsion in Open and Closed Channels
A thin walled open channel section will have large deformations
under torsion or bending compared to an axial load and the concept of the shear
centre is an important one to understand in such cases.
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Shear Failure
A shaft in torsion is less likely to fail under the same peak
torsional shear stress than the same material in a direct shear case. Features
such as notches and changes of radius can have less of an effect as stress
concentrations in torsional rather than in direct shear. Also, a shaft can
sustain greater strain energy without failing by brittle fracture, than say a
shaft or rivet sustaining the same calculated direct shear stress over its
cross section.
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Buckling
Axial tensile loads can often be ignored in slender beams due to
the relatively small stresses they will cause in comparison to the bending
stresses. Conversely, a compressive axial load can cause buckling that
cannot be assessed using static analysis and requires special consideration.
Parts that are prone to buckling are often termed 'slender', describing their
shape. Beams and panels of thin section can buckle but, importantly, an
ordinary linear static analysis will not show this behaviour. This is discussed
further in the next section.
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Static Finite Element
Analysis
Mathematical and historical
background
The essence of the FE method is that the mesh of finite elements
filling the component allows the most complex of shapes and load patterns to be
represented by a lengthy set of equations. These equations are usually linear.
This means the independent variables (deflections) or their derivatives appear
once in each term. The maths of FEA is essentially centred on the manipulation
and solution of matrix equations, which will is now described in more detail
below. As an alternative, a
non-mathematical description of the maths of FEA is given here. For a basic
static analysis, mathematics is used to formulate the governing differential
equations of the problem and the geometry into a set of matrix equations. These
equations are of the form:
{F} = [K] {X}
where each term above represents a matrix.
The {F} matrix is only 1 column wide (and is usually called a vector)
and is a numerical representation of the loads on the model.
The [K]
matrix is 'square'; i.e. it has as many rows as columns and, for a 3-D solid
model, is a numerical representation of the stiffnesses in three directions
throughout the solid.
The {X} matrix is a single column vector of
displacements and most of its values are unknown before the solution is carried
out.
Once all the x's within {X} are found, differentiation of {X} is
required to obtain the strains, which can be multiplied by a matrix of material
properties to get the stresses. The stiffness matrix as it stands is singular;
i.e. no unique solution exists. This is in reality the same as having no, or
insufficient constraints to prevent rigid body motion, which represents a
meaningless or trivial solution. The imposition of adequate restraints
effectively eliminates some rows and columns from the stiffness matrix and this
provides a non-singular matrix, i.e., one that has a meaningful physical
solution.
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Non Linearity and
Buckling in FEA
If the component is a thin plate or 'slender' column (as defined
by the Euler buckling equation) and is loaded in compression, there is a risk
of buckling. This cannot be investigated by a static finite element analysis
which would give a safe compressive stress with no indication of the components
susceptibility to failure through buckling.
There are two approaches
to assessment of buckling by finite elements.
1. Although buckling is
a non-linear phenomenon, an estimate of the buckling load can be obtained by
hand calculation, using the Euler buckling formula. Using finite elements, a
simple buckling mode shape can be determined by extracting the first Eigen
vector of the system equations.
2. A more complicated approach
involves non-linear analysis and is suitable for determining local as well as
global buckling effects. This requires the splitting of the load application
into several contiguous steps such that a small portion of the load is applied
in each. The model deforms by a relatively small amount that is treated
linearly, under the action of the portion of the load in the step. The model
shape is then updated with the displacements from this completed step for the
next step and small portion of load. In general, the large buckling
displacements cause the load vs. displacement gradient to change (even though
the material stiffness remains constant) from one load step to the next which
is what defines it as a non-linear problem.
In fact, all types of non
linear analysis adopt similar procedures to these described in 2 above. Two
important effects can occur in non-linear static analysis: (i) the geometry of
the structure changes; (ii) the load application point moves. At least one of
these effects is important in the common types of non-linear analysis; called
respectively ‘large displacements’ and ‘follower
forces’. Facilities exist to activate either or both of them within most
solvers.
There can be a tendency for some more difficult buckling
analyses to become numerically unstable, even though the model is a physically
valid one. This is often due to a negative force displacement gradient
occurring at some point in the solution, often called 'snap through'. Special
solver algorithms can be deployed to ensure that the solution attained is a
valid one.
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Quasi-static Analysis
This refers to static analysis as described previously, but where
some or all of these forces are induced as a result of some physical motion of
the body, such as cornering or braking of a land vehicle, stable flight of an
aircraft etc. The component or system must again be in equilibrium under the
action of all the forces, including those that are motion induced.
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Dynamic Forces
A quasi-static analysis is not to be confused with a dynamic
analysis and it is necessary to understand in what circumstances the respective
analyses are applicable. Generally, (but see below) where the forces are not
changing rapidly relative to the natural frequencies of the system, then a
quasi-static analysis is valid.
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Impulsive Loading
When a structure is struck by a sudden force, this can produce
stresses up to twice as large as the values obtained for the same load applied
statically. Shock waves (i.e. stress and strain waves) travel through the
structure and can be reflected back from a boundary to the point of application
of the load. Superposition of these waves could result in a doubling up of the
stresses where two waves positively reinforce each other. This magnification
factor of two will occur when the duration of the pulse (t) is equal to or
greater than the natural period (T) of the structure. The maximum stress is
twice the static value but reduces, as the impulse becomes shorter. If the
impulse becomes longer, it will tend to approach a static load and therefore
the stresses will be the same as for the static case again.
The shock
behaviour of the structure must be included for any such analyses, implying
material density and stiffness need to be specified.
Resonance is a
well-known engineering phenomenon and implies an increase in peak deflection of
a structure due to a periodic excitation load, (or displacement) in comparison
with the deflection induced by the same load applied statically. The impulsive
stresses described ignore the effect of resonance which may be important. If
so, a vibration analysis would be required, as described in the next section.
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Vibration (Dynamic)
Analysis
This refers to the analysis of a component, under the action of a
time fluctuating load, which has at least some frequency components that are
close to or higher than the first natural frequency of the system and may
therefore excite a resonant response, in which case, it is not possible to
determine the extent of the response without undertaking a dynamic analysis.
Many loads are harmonic, i.e. they have several frequency components, and may
even be described only by a statistical distribution of frequency vs. load or
energy (PSD) and are no longer perceivable as a set of discrete frequencies.
Typically, when considering vibration, a modal analysis is carried out,
possibly followed by a response analysis, requiring the specification of
damping for the structure.
Remember, a vibration analysis will
consider any effects where the structure is likely to resonate in response to
the load. This will not happen if the frequency of the load is significantly
lower than the first natural frequency of the structure.
The above
discussion extends to 'impulsive' loads such as a hammer blow, which is also
harmonic. As already described this can be catered for in a static analysis by
factoring the applied impulse by two. This is true however, providing that no
resonances are set up in the system as a result of dynamic excitation of one of
the harmonic components of the load.
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Mathematical Background - Modal
Analysis
The form of and solution to the matrix equations in a dynamic
solution are different to that described for static analysis as a mass and a
damping matrix is involved. The natural frequencies (usually called normal
modes) of the structure are required for all types of dynamic analysis and
these are related to the Eigenvalues and Eigenvectors of the matrix equations.
They do not involve any assessment of damping (realistic damping values do not
change the values of the natural frequencies obtained from a modal analysis by
a significant amount).
Often, the normal modes are used for
subsequent response analyses of the same structure subject to a forcing
function and also with the inclusion of damping.
For an undamped
system the matrix equations, from Newton 's Second Law, are of the form:
[M]{X} II + [K}{X} = [0]
Where the
displacement vector {X} is differentiated twice on the LHS of the equation ( II
indicating double differentiation) to produce the acceleration vector. This is
an Eigenvalue problem that has as many solutions as there are degrees of
freedom. The solutions are obtained by assuming the displacement vector is time
dependent and has simple harmonic form thus:
{X} = {X
0}.Sinωt
So that:
{X} II = - w 2 .{X 0}.Sinωt
The vector {X} 0 is one of the solutions, termed an
Eigenvector (of peak displacements over time), and represents the shape (called
the mode shape) the item would assume if excited by a forcing frequency of
ω Hz. Substituting:
[K]{X} – ω 2.[M]{X} = [0]
or:
{X} [ [K] – ω 2.[M]
] = [0]
ω 2 is the corresponding
Eigenvalue (square of the natural frequency) to the Eigenvector. For the i th
natural frequency, the solution can be written as:
[K]{X} i = w i 2 [M]{X}
i
Each one of these solutions represents a mode shape and
corresponding natural frequency.
The response of the structure to a
specific forcing frequency would comprise some contribution from up to all the
Eigenvalues and Eigenvectors. A modal analysis will only determine (within a
small error due to the presence of damping) the nearness of any mode of
interest to the excitation force. Generally, the nearer the two, the more
likely is resonance to occur and the more likely the chance of vibration
induced failure.
Damping is not specified in the above equations. The
modes (or mode shapes) are so called since they describe the nodal
displacements, relative to other nodes only; no absolute displacements can be
obtained from this type of solution. Despite this, modal analysis is often the
only type of vibration assessment carried out on many structures to determine a
ceiling on excitation frequencies.
Actual displacements and thus the
likelihood of failure from vibration can be obtained from subsequent response
analyses, but they will depend on damping values which must be specified. These
values are often difficult to obtain accurately.
Modal frequencies
can be derived from first principles for simple assemblies of beam structures
and lumped mass and spring systems. These may sometimes be used to help verify
a FE model.
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Damped Forced Vibration
This may be called dynamic frequency analysis, where the
frequencies are constant over time.
To obtain a clearer understanding
of whether an excitation frequency will cause resonance and possible failure,
then a full dynamic analysis will have to be undertaken. This uses the
information form the modal analysis and calculates the absolute vibration
displacements through the structure, under the action of the forcing load. The
problem with this kind of analysis is that it needs an assessment of damping
which is difficult to determine accurately.
For values of excitation
and forcing frequencies that are close together, implying resonance, increasing
amounts of damping will reduce the amplitude of response that occurs. Typical
values of damping parameters for a range of structure types are available from
various sources and these should be consulted if no specific data are
available.
The requirements of such an analysis would be that the response of
the structure over the frequency range of interest should be sufficiently low
as to not cause high stresses. Since the value of damping used determines the
dynamic deflections, it is critical to the accuracy of these stresses.
There are several different approaches to undertaking this sort of
analysis, often dependant on the way in which the load is described. These
approaches are not considered further here.
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Thermal Analysis - Stress
without Strain & Strain without Stress
It is only temperature gradients that cause stress in an
unconstrained structure; a uniform temperature rise will not. (Strain without
stress.) If there is no change in temperature through an unconstrained
structure, there will not be any thermal stress.
Conversely, a fully
constrained structure will have stresses induced by a uniform temperature rise
(stress without strain). These ideas have often help understand some of the
simpler thermal problems.
Differential thermal expansion may cause
thermal strain and consequent stresses to be set up. If this is thought to be
an area of concern then any analysis will require appropriate thermal boundary
conditions to be considered. Deciding on meaningful appropriate thermal
boundary conditions requires a good understanding of basic heat transfer. Often
semi-empirical data is used, but this is expensive to obtain.
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