FEA 23: 2-D Triangular Elements
Summary
TLDRThis video script offers a comprehensive guide to developing a 2D linear triangular element in structural analysis. It covers defining displacement and strain fields, setting up element geometry and degrees of freedom, and establishing shape functions. The script delves into deriving the B matrix and D matrix for stress-strain relationships, culminating in the stiffness matrix formulation. It also addresses handling distributed forces, distinguishing between body and surface forces, and calculating elemental force vectors, providing a foundational understanding of finite element analysis.
Takeaways
- đ The video focuses on the development of a 2D triangular linear element, specifically a right triangle with a 90-degree corner.
- đ It emphasizes the importance of defining the displacement field, strain, and stress fields for the element.
- đ The element's geometry and degrees of freedom are defined, which are crucial for relating back to the displacement field vector.
- đ A shape function matrix is established to define how the displacement field varies within the element.
- đ The relationship between the strain vector and the displacement field vector U is explored, leading to the partial derivative matrix operator.
- đ The D matrix, which contains material properties, is derived from the relationship between stress and strain.
- 𧟠Matrix multiplication of the partial derivative matrix and the shape function matrix results in the B matrix.
- đ The stiffness matrix K of the element is calculated using B and D matrices.
- đȘ The element force vector is discussed, including how to account for distributed forces acting on the element.
- đ The concept of transformation from local to global coordinate systems is introduced but will be detailed in later videos.
- đ The script uses a specialized right triangle element to simplify the development process and focus on the methodology.
Q & A
What is the focus of the short video introduced in the transcript?
-The short video focuses on introducing 2D elements, specifically the three-noded triangular linear element, and discusses the process of defining a new element in finite element analysis.
What are the initial decisions one needs to make when defining a new element?
-When defining a new element, one needs to decide on the displacement field of interest, the element geometry, the degrees of freedom, and how the displacement field varies within the element.
What is a degree of freedom vector in the context of this video?
-A degree of freedom vector relates back to the displacement field vector and consists of six terms for a three-noded triangle element, with each node having two degrees of freedom for horizontal and vertical displacement.
Why is the shape function matrix important in the development of an element?
-The shape function matrix is crucial as it defines how the displacement field varies within the element and how that variation relates back to the degrees of freedom.
What does the partial derivative matrix operator represent in the context of this video?
-The partial derivative matrix operator represents the relationship between the strain vector and the displacement field vector U.
What is the D matrix and how is it derived?
-The D matrix represents the relationship between stress and strain, incorporating material properties. It is derived from the stress-strain relationship for a given material under plane stress or plane strain conditions.
Why is the B matrix significant in finding the stiffness matrix of an element?
-The B matrix is significant because it is the product of the partial derivative matrix operator and the shape function matrix, which is then used to calculate the stiffness matrix K.
How is the stiffness matrix K for a new element calculated?
-The stiffness matrix K is calculated by integrating the product of B transpose and D matrices over the volume of the element.
What is the significance of the element force vector in the context of this video?
-The element force vector represents the forces acting on the element, which when added to the nodal forces, gives the global force vector.
Why is the right triangle element chosen for the development process in the video?
-The right triangle element is chosen because it simplifies the development process by limiting the complexity of the math involved in defining the position of the three points, allowing the focus to be on the development process itself.
How does the video script differentiate between body forces and surface forces?
-Body forces are forces that act throughout the volume of the element, while surface forces act along the edges or surface of the element. The script uses an example of a uniformly distributed downward acting body force and a surface traction force at a 45-degree angle to illustrate this.
Outlines
đ Introduction to 2D Elements and Displacement Field
The script begins by introducing 2D elements, specifically focusing on a triangular linear element. It emphasizes the need to understand the displacement field, strain, and stress fields. The process involves defining element geometry, degrees of freedom, and the relationship between displacement and strain. The script also mentions the importance of establishing shape functions, partial derivative matrices, and material properties through the D Matrix. The goal is to find the stiffness matrix (k) and the elemental force vector, with a note on the importance of transformation from local to global coordinate systems.
đ Developing the Shape Function Matrix
This section delves into the development of the shape function matrix for a 2D right triangle element. It discusses how to define the displacement field in two dimensions, focusing on horizontal and vertical translations. The script explains the process of determining coefficients for the shape functions by using degrees of freedom at specific nodes. It then rearranges terms to define the shape functions for each node and presents the shape function matrix in matrix form, which is crucial for calculating the B Matrix and understanding displacement variations within the element.
đ Deriving the B Matrix and Constant Strain in Linear Triangles
The script continues with the derivation of the B Matrix, which relates strain to the degree of freedom vector. It discusses the partial derivative matrix operator and how it acts on the shape function matrix to yield the B Matrix. The result is a constant strain across the element due to the linear nature of the function, which is a characteristic of linear triangle elements. This section also highlights that the entire element will experience the same strain, regardless of its shape, due to the linearity of the element.
đ Stress-Strain Relationship and Stiffness Matrix Assembly
This part of the script reviews the stress-strain relationship in a 2D plane stress state and how it leads to the formulation of the D Matrix. It then describes the assembly of the stiffness matrix (K) by integrating the product of B transpose and D over the volume of the element. The script provides a detailed example of calculating the stiffness matrix for a right triangle element, highlighting the simplifications that occur due to the element's geometry. The section concludes with an explanation of how to handle distributed forces, differentiating between body forces and surface forces, and providing an example of how these forces are integrated into the element model.
𧟠Handling Distributed Forces in the Element Model
The final paragraph focuses on how to account for distributed forces, such as body forces and surface tractions, in the element model. It provides an example of applying a uniform body force and a surface traction at a 45-degree angle on the right triangle element. The script explains the process of evaluating shape functions at the edges and surface of interest, and how these are used to calculate the elemental force vector. The section concludes with an expression that represents the force vector for the given distributed forces, demonstrating the method's applicability to various load scenarios.
Mindmap
Keywords
đĄDisplacement field
đĄStrain
đĄStress
đĄDegrees of freedom
đĄShape function matrix
đĄPartial derivative matrix operator
đĄD Matrix
đĄStiffness matrix (K)
đĄElement force vector
đĄTransformation
đĄRight triangle element
Highlights
Introduction to 2D elements, specifically focusing on the triangular linear element.
The necessity of defining displacement, strain, and stress fields for element analysis.
Importance of deciding element geometry and degrees of freedom related to the displacement field vector.
Explanation of the shape function matrix and its role in defining displacement field variation.
The relationship between strain vector and displacement field vector U, leading to the partial derivative matrix operator.
Stress-strain relationship resulting in the D Matrix, which contains material properties.
Matrix multiplication of the partial derivative matrix operator and shape function matrix to obtain matrix B.
Construction of the stiffness matrix K using matrices B and D for the new element.
Discussion on obtaining the element force vector and its contribution to the global force vector.
Introduction to the concept of transformation between local and global coordinate systems.
Choice of a right triangle element with a 90-degree corner for simplified development process.
Defining the displacement field in two dimensions for translation of nodes in both horizontal and vertical directions.
Development of the shape functions for a linear right triangle element with three degrees of freedom in each direction.
Calculation of coefficients for the displacement field equation using node degrees of freedom.
Derivation of the shape function matrix for both X and Y displacements.
Explanation of the partial derivative matrix operator relating strain to displacement.
Construction of the B Matrix by multiplying the partial derivative matrix operator with the shape function matrix.
Development of the stiffness matrix K through the integral of B transpose DB over the element's volume.
Handling of distributed forces, including body forces and surface forces, and their impact on the element.
Example of applying body force and surface traction to the linear right triangle element.
Integration of shape functions and forces to determine the elemental force vector.
Transcripts
this short video introduces 2D elements
and focuses on initially the three noted
triangular linear element
so let's review what we need to do to
find a new element we need to make some
decisions and we also need to Define
some vectors and matrices so first off
you want to decide what are what is the
displacement field that you're
interested in what does that Vector look
like also the strain and stress fields
are important then you can Define your
element geometry and the degrees of
freedom that are going to be relating
back to that displacement field Vector
that will give us a degree of Freedom
Vector then we need to establish a shape
function Matrix that defines how the
displacement field varies within the
element and how that variation relates
back to the degrees of freedom we also
look at the relationship between the
strain vector and the displacement field
Vector U that gives us this partial
derivative Matrix operator
next we need to look at the relationship
between stress and strain that's going
to give us our D Matrix which has our
material properties in it we also need
to define a matrix that's the product of
the partial derivative Matrix operator
and the shape function Matrix so this is
just a matrix multiplication that will
give us B and once we have B and D we
can go and find our stiffness Matrix k
for the new element
finally we want the element Force Vector
if we have a distributed force acting on
the element we need to Define how we get
an elemental Force Vector which then
gets added to the nodal forces to give
us a global Force vector
and one of the last things that we need
to discuss is transformation how are we
going to take an element from its local
coordinate system and apply it to
whatever shape it becomes in the global
system we will defer transformation for
a few videos but the rest of these we'll
focus on today
so let's get started making some
decisions and defining a 2d right
triangle element this is a specialized
element it is a three noted triangle but
it's going to have a 90 degree Corner in
it and the reason that I choose this
I'll explain at the end of this slide so
uh start out we have to Define what our
displacement field is this is in two
dimensions and I'm interested in
translation of the nodes of my right
triangle element in both the horizontal
and the vertical direction or the X in
the y direction
I am also interested in the strains in
the element so in this case I'm
interested in the strain in the X
Direction the normal strain X Direction
the normal strain in the y direction and
then also the shear strain that links
those two directions I don't have any of
the Z component strains because this is
a 2d element I'm developing similarly
I've got the stress Vector that has the
same components as the strain Vector did
so this is my element geometry it's
going to be a right triangle it's going
to have a base of b and a height of H
those are variables so it can be any
shape right triangle but it needs to
have that 90 degree corner for the one
that I'm developing I'm going to add my
local coordinate axes so an X Prime and
a y Prime axis in the locations you'd
expect and then I'm going to Define my
node numbers so 1 is going to be right
if the origin of my coordinate system 2
is at the end of the horizontal leg and
three is at the end of the vertical leg
now this gives me my degree of Freedom
Vector I have six terms in it because I
have three nodes and each one has two
degrees of freedom see how this is all
related back I choose the geometry and
the nodes but then I also defined my U
Vector which gives me the directions
that I'm interested in now why did I
choose to do a right triangle element
well most texts certainly look at a more
complicated element they don't restrict
it to a right triangle so you can have
any angles any sizes which is a very
useful element but the problem is you
get so caught up in the math of trying
to Define any position for all your
three points that you lose the actual
development process so I'm focusing on
this simpler element which is not as
general you won't even see it used in
most Fe codes however it gives me the
power to focus in on the process all the
way through so that's why I'm going
through that in this video
so continuing our development of this
linear right triangle element we want to
define the shape functions we have three
degrees of freedom in each Direction now
because it's 2D element we need to talk
about directions so we have d1x d2x and
d3x in the X direction that means that
for the polynomial for U I can have
three unknown coefficients in it so I
can write U of X Y is equal to a0 plus
a1x plus a2y so by doing One X term and
one y term I've balanced this variation
field so my field is not
um is not biased towards the X direction
or the y direction in terms of the
variation I'm allowing for strain or
displacement inside the element
so now I want to determine what those
coefficients are so I start out with
each one of my degrees of freedom so d1x
is going to be equal to whatever U is at
0 0 and in this case that's a not d2x is
going to be U evaluated at B comma 0
that's node two and that is an a0 plus
a1b and d3x is U evaluated at zero comma
H so that's a0 plus a2h I can take these
three equations and solve them for my
three A's so a0 becomes d1x A1 is 1 over
B times d2x minus d1x and A2 is 1 over H
times d3x minus d1x so now I don't need
the A's because I'm just going to
substitute them in but it's an important
intermediate step to calculate them so I
plug them back into my U equation and
now I have the one shown here without
any A's in it so now I just have B and H
which are my dimensions and then my X
and Y which are the the variables that
make this a function
and now I need to rearrange terms to
gather everything multiplied by D1 X d2x
and d3x and when I do that this is the
expression I get now this defines my
shape functions everything multiplied by
d1x is N1 everything multiplied by T2 X
is N2 and everything multiplied by d3x
is N3 now to put it into Matrix form I
want to think about the fact that I want
to take these degrees I want to end up
with this equation here U of x y equals
the shape functions multiplied by the
degrees of freedom but I want to do this
for both the X displacement that's U and
also the Y device displacement which is
V and the V is going to look the same as
U is here it's just going to have the
d1y d2y and d3y
so when I put that all together I get
this relationship for my shape function
Matrix this will give me a u of X Y that
looks like this and a v of X Y that
looks very similar just with a Y
subscript for each of my degrees of
freedom terms
so now that we have the shape function
Matrix we can move forward to get the B
Matrix and of course on the way we need
to define the partial derivative Matrix
operator in fact we're not going to
Define it we're just going to plop it
out here because I've discussed this in
the strength material review video
earlier in this series so remember that
we've got a relationship between strain
and displacement that's the partial
derivative Matrix operator but what I
really want to do is Define strain in
terms of the degree of Freedom Vector D
that's where the B Matrix comes into
play so remember my strain Matrix I
decided it looked like this
we know from again the review of
strength materials that the strain in
the X direction is dudx in the Y
directions dvdy those are the normal
strains and then the shear strain is d u
d y plus DV DX so written in a matrix
operator form I now have my displacement
field Vector U which has the terms UV
and it's pre-multiplied by this partial
derivative Matrix operator which I call
the parcel symbol is equal to the The
Matrix shown here
so now to get B I take the partial
derivative Matrix operator and I have it
act on the shape function Matrix now
when you do this for every new element
you need to make sure this
multiplication works and if you've set
up your matrices properly it will always
work
so let's continue with that I'm going to
multiply these matrices together when I
do that I end up with the Matrix in this
form where I've used some initial
notation to try to minimize the amount
of space I'm taking up so the comma X or
comma y actually represents a partial
derivative with respect to that variable
so N1 comma X is dn1 DX
now I know what my shape functions are
so I can actually calculate all of these
partial derivatives here are the shape
functions I previously developed
so the partial derivatives with respect
to X are straightforward and similarly
with respect to Y there's a
straightforward I can take all of these
partial derivatives plug them into that
b Matrix and I get this Matrix now
notice what happened here I lost all the
x's and y's which we would expect
because it was a linear function and I'm
taking the derivative of linear
functions I'm going to get constant
terms but recall that strain is
determined by multiplying the B Matrix
times the degree of Freedom Vector that
means because the B is constant and the
degree of Freedom Vector never depends
on position I have no variation of
strain in this element the entire
element will have exactly the same
strain this is what you get with a
linear triangle this is what would
happen even if this was a any triangle
any triangle shape the right triangle
makes it look a little simpler but it
would always be constant
okay let's continue our development
looking at the stress strain
relationship this is just a quick review
because we did this in uh the strength
of materials review video earlier in the
series so Sigma X looks like this Sigma
Y and Tau XY so that's my stress strain
relationship in a 2d plane stress State
plane strain would be somewhat different
this is plain stress specifically so now
what I want to do is I want to write my
stress Vector is equal to some Matrix D
multiplied by my strain vector and
remember that my stress Vector has three
terms and my strain Vector has three
terms so D is going to be a three by
three Matrix and you can see that if I
take those three equations at the top
and put them into the Matrix form this
is the Matrix D I end up with and this
again was developed in more detail in an
earlier video
okay so now we have all the pieces to
put together our stiffness Matrix K is
equal to the integral over the volume of
B transpose DB
and here we go plug in the B's that we
found and the D and in addition I'm
going to convert my volume integral into
an integral over the thickness of the
element
um inside of the integral over the cross
section or the the surface of the
element rather the d a
so here the integral from 0 to T of DZ
is just going to be T the thickness of
my element so that will remain as a
constant here assuming that I have
constant thickness throughout the
element and then the integral of d a
remember nothing else in here if you
look depends on X or Y so I can just
resolve that integral right away so the
integral over d a I'm sorry the integral
of the cross-sectional area is BH over
2.
and when we multiply those three
matrices together we get this Unholy
mess and this is
simpler than the general three node one
so we can see the B Matrix here or I'm
sorry we can see the K Matrix here you
can see that because it's a right
triangle I've picked up a few zero terms
in it the general three node triangle
will not have any three node terms in
general however it will simplify to this
one when you have a right triangle that
you're analyzing so that is the linear
right triangle stiffness Matrix
okay the last piece of the development
that we're going to go through for this
linear right triangle is how do we
handle distributive forces so
distributed forces for uh stress element
consists of body forces or Surface
forces and in all of my earlier videos
when I was dealing with bars and Beams I
said you can choose either one you can't
do that anymore these are now distinct
note that the NS here this becomes
important we're evaluating it on the
surface of Interest so what do I mean by
Body in a surface well FB is the force
that's acting everywhere inside the
edges of the element so it's throughout
the body of the element whereas FS acts
only at the element edges I call it a
surface because there's thickness
remember in a plane stress element there
is a cross section that would get sliced
in the Z Direction so the surface is the
thickness times whatever the length of
that
um that edge is so FS is acting along
finances let's work through an example
here so here's the the linear right
triangle and I've identified t as the
thickness I'm going to apply a body
Force so throughout the inside of the
element and I'm going to call that W
it's going to have units of
Force divided by length cubed so it
always wants to be a force per volume in
order to be a body term so that means FB
is 0 in the X Direction and minus W in
the y direction that would be my FB for
the integral above I'm going to add
another Force here one that's acting on
the edge or on the surface of the
element this is I'm going to call
attraction I'm going to give it the term
T it's at 45 degrees angry angles
downward and it's going to have units of
force per area or Newtons per meter
squared as it's shown here so because of
the 45 degrees I pick up a 1 over the
square root of 2 for each Direction and
the x is going to be a positive for the
surface force or Surface traction and
the Y is going to be in the negative
Direction so that's my FS term so
putting these in and also showing the
transpose of the shape function matrices
let's first look at that first term for
body Force that's the transpose of my
shape function Matrix and I've got the 0
minus W pretty straightforward there add
in the second term here remember what
I'm doing is I'm evaluating the shape
functions on the surface of Interest so
that means I'm evaluating along the left
edge of the element The Edge where X is
equal to zero
so I have to evaluate each of them at
zero comma y
and then I've also got my T over square
root of 2 times the 1 negative 1 term
now note here that when you evaluate
shape function 2 because it's a linear
function and because we know it has to
be zero at nodes one and three it has to
be zero all along that edge so it's
going to drop out it's going to become
zero
continuing this example when I multiply
my shape function Matrix times my body
Force term I end up with this vector and
then when I do the same thing for my
Surface traction term I get this Vector
again the N2 terms dropped out because
N2 was equal to zero all along the left
edge of the element
so remember these are my shape functions
let's go ahead and plug them in here for
this element we are going to integrate
along the edge of the element the left
Edge that's going to be a
straightforward integration from 0 to H
but the the integration over the surface
of the element is going to mean we're
going to have to follow that
the the angled piece there so I'm going
to evaluate that along the hypotenuse of
the triangle Y is equal to H minus h x
over B
so the first term becomes the shape
function substituted in I've converted
my d a to a dydx where the the DX is on
the outside so that's just 0 to B and
then the d y is on the inside so it goes
from 0 to the hypotenuse which again is
H minus h x over B
and the second term is
just integrating from 0 to H of the
shape functions evaluated along that
left Edge
so just wrapping it up you perform those
integrals and then you do a little bit
of algebra and you end up with this
expression so the left piece here
corresponds to the uniformly distributed
downward acting body force and the right
term is for the at a 45 degree angle
downward to the right traction force and
I think if you did these if you guessed
you would probably end up with these
terms as well but remember I chose very
simple examples to show you the process
you could have now any variation of load
as a body force or as a surface traction
and still be able to capture it using
this method
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