FEA 24: 2-D Rectangular Elements
Summary
TLDRThis video explains the development of a 2D rectangular finite element with four nodes, using a bilinear displacement field. It covers defining the element's displacement, stress, and strain fields, and constructing its shape function matrix and stiffness matrix. The video explores how strain and displacement are related, deriving the B matrix, and emphasizes the difference between the rectangular and triangular elements. It concludes by discussing the issue of shear locking in pure bending scenarios and how it affects the accuracy of the model.
Takeaways
- đ The video explains the development of a 2D rectangular element with four nodes and a bilinear displacement field.
- đ Defining a new element requires specifying the displacement, stress, and strain field vectors, along with element geometry and degrees of freedom.
- đą Shape function matrices help relate strain and displacement, leading to the B matrix and stiffness matrix, but transformations are not covered in this video.
- 𧟠The displacement field is expanded with four coefficients for each polynomial, making it bilinear â linear in both x and y directions.
- âïž The degree of freedom vector has eight terms in total (two per node), representing displacements in both x and y directions for each node.
- 𧩠Matrix manipulations simplify the resolution of unknown coefficients, transforming them into expressions dependent on degrees of freedom.
- đșïž Shape functions are derived for each degree of freedom and arranged into a shape function matrix for efficient computation.
- đ The B matrix is derived from partial derivatives of the shape function matrix, resulting in a linear variation of strain in the bilinear rectangular element.
- đ The B matrix can be used to calculate the strain at specific points within the element, such as the center and the node locations.
- â The video highlights a phenomenon called 'shear locking,' where unexpected shear strain appears during pure bending, leading to less deflection in finite element analysis compared to reality.
Q & A
What is a 2D rectangular element in the context of this video?
-A 2D rectangular element is a finite element with four nodes, each with two degrees of freedom. It has a bilinear displacement field inside, meaning the displacement is linear in both the x and y directions.
What are the key components needed to define a new element?
-To define a new element, we need to specify the displacement field, the stress and strain field vectors, the element geometry, the degree of freedom vector, and the number of nodes. These help form the shape function matrix and the B matrix, leading to the stiffness matrix and the element force vector.
What is meant by a 'bilinear displacement field'?
-A bilinear displacement field refers to a displacement that is linear in both x and y directions. Even though the displacement may include a quadratic term like XY, it is still linear in terms of each individual direction.
Why is the X'Y' coordinate system used in defining the element geometry?
-The X'Y' coordinate system is chosen for balance, with the origin placed at the center of the element to make it easier to balance displacements and deformations symmetrically from left to right and top to bottom.
How are the shape functions derived for the rectangular element?
-The shape functions are derived by evaluating the displacement field at each node of the element and resolving the unknown coefficients of the polynomial displacement expression in terms of the degrees of freedom. The displacement field is then expressed as a combination of shape functions multiplied by the degrees of freedom.
What is the B matrix, and how is it computed?
-The B matrix, or the strain-displacement matrix, is obtained by multiplying the partial derivative matrix operator by the shape function matrix. It relates the strain to the nodal displacements in the element.
How does the B matrix differ between triangular and rectangular elements?
-For the rectangular element, the B matrix depends on x and y, leading to a linear variation of strain within the element. In contrast, for a triangular element, the B matrix is constant, resulting in constant strain throughout the element.
Why is evaluating the B matrix at different positions within the element important?
-Evaluating the B matrix at different positions, such as the center or the nodes, allows us to calculate the strain at those specific points. Since strain varies within the element, this helps in understanding the strain distribution.
What happens when the rectangular element is subjected to pure bending?
-When the rectangular element undergoes pure bending, one side gets compressed while the other side is stretched, turning the rectangle into a trapezoid. The displacement field corresponds to this bending, and the strain can be calculated using the B matrix.
What is 'shear locking,' and how does it occur in this context?
-Shear locking occurs when unexpected shear strain appears in pure bending scenarios, where no shear forces are applied. This happens because the bilinear element introduces shear strain, which absorbs some bending energy, resulting in less deflection than expected.
Outlines
đ Introduction to Bilinear 2D Rectangular Elements
The video begins with an explanation of the development process for a 2D rectangular element with four nodes, featuring a bilinear displacement field. To define this element, it is essential to determine the displacement field, stress and strain fields, element geometry, and the degree of freedom vector. Once these aspects are defined, the shape function matrix can be derived. This leads to the relationship between strain and displacement, with the partial derivative matrix operator playing a key role. The stress-strain relationship is captured in the D matrix, while the B matrix (strain nodal displacement matrix) helps obtain the stiffness matrix and the element force vector. Transformation aspects are mentioned but not covered in this video.
𧩠Defining the Geometry and Degrees of Freedom
The geometry of the bilinear rectangular element is defined with local X' and Y' axes, and the nodes are placed symmetrically. The width of the element is 2b, and the height is 2h. Each node has two degrees of freedom (X and Y directions), resulting in a total of eight degrees of freedom. The displacement field for this element can be expressed using polynomials with four coefficients, allowing a bilinear behavior in both X and Y directions. This distinction makes the element more complex than a triangular element, incorporating a quadratic term but remaining linear in both X and Y.
đą Solving for Displacement in the X Direction
The displacement field in the X direction is evaluated at each node, producing a series of expressions based on the polynomial coefficients. Due to the placement of the origin in the middle of the element, each expression involves all four polynomial terms, making the resolution more complex. Using matrix manipulation simplifies this process, enabling the determination of unknown coefficients in terms of the degrees of freedom. This matrix approach resolves the displacement field, allowing it to be expressed in terms of the degrees of freedom.
đ Shape Function Matrix and the B Matrix
The displacement field is rewritten in terms of shape functions for each degree of freedom, resulting in a shape function matrix. This matrix has eight columns and two rows, corresponding to the eight degrees of freedom in the element. Once the shape functions are established, the B matrix can be calculated. The B matrix, or strain-displacement matrix, is derived by applying the partial derivative matrix operator to the shape function matrix. Due to the bilinear nature of the element, the B matrix depends on the variables X and Y, leading to a linear variation of strain within the element.
đ§ Strain Evaluation and Complications
Evaluating strain using the B matrix involves more complexity than the constant strain seen in a triangular element. The B matrix, when multiplied by the D matrix and its transpose, leads to quadratic terms inside the stiffness matrix integrals, making the computation more challenging. While this video does not cover the stiffness matrix computation for this specialized element, it mentions that future videos will address transformations needed for more versatile elements. Additionally, the B matrix allows for strain evaluation at specific points inside the element, such as at the center or at a node.
đ Strain Evaluation at the Element Center and Node 1
Strain is evaluated at two locations: the element center and node 1. At the center, the B matrix results in no strain, as expected for the neutral axis in pure bending scenarios. When evaluated at node 1, the B matrix yields both normal strain and shear strain, the latter of which is unexpected in pure bending. This discrepancy is a result of 'shear locking,' where the element erroneously absorbs bending energy as shear strain, leading to less deflection than expected. The video highlights the importance of addressing such issues in future discussions on element transformation.
Mindmap
Keywords
đĄBilinear displacement field
đĄShape function matrix
đĄStrain-displacement matrix (B matrix)
đĄStiffness matrix
đĄDegree of freedom (DOF)
đĄElement geometry
đĄStress-strain relationship (D matrix)
đĄPure bending
đĄShear locking
đĄTransformation
Highlights
Introduction to the development of a 2D rectangular element with four nodes and a bilinear displacement field.
Explanation of the key choices: displacement field, stress-strain vectors, element geometry, degree of freedom vector, and number of nodes.
Description of the shape function matrix and the derivation of the partial derivative matrix operator.
Introduction to the stress-strain relationship captured in the D matrix.
Multiplying the partial derivative matrix by the shape function matrix to derive the B matrix (strain nodal displacement matrix).
Geometry of the rectangular element with local axis X'Y' and centered coordinates for clarity.
Definition of the bilinear displacement field with linear variation in both X and Y directions.
Derivation of displacement equations using matrix notation to solve for the coefficients in terms of degrees of freedom.
Explanation of shape functions for each node and how they relate to displacement in both X and Y directions.
Formation of the B matrix by taking partial derivatives of the shape function matrix, with B matrix depending on X and Y coordinates.
Notable difference from the linear triangle element: bilinear rectangle has a linear variation of strain, resulting in a more complex B matrix.
Introduction of quadratic terms in the stiffness matrix integral, making the calculation more complex.
Evaluation of strain using the B matrix at specific positions, like the element center and node 1.
Example of pure bending displacement field, resulting in strain distribution and demonstrating the neutral axis for pure bending.
Unexpected shear strain found during pure bending analysis, leading to a phenomenon called shear locking.
Transcripts
this video goes through the development
of a 2d rectangular element one that has
four nodes so it's going to have a
bilinear displacement field inside of it
as we'll see let's start out with a
quick reminder of the things that we
need to define in order to be able to
say that we have defined a new element
we have to make choices about what the
displacement field is that we're
interested in and what the stress and
the strain field vectors look like then
we need to define the element geometry
and the degree of freedom vector
corresponding to the displacement field
and the number of nodes once we have
those we can get the shape function
matrix and we go back and remind
ourselves about how strain and
displacement are related and that gives
us the partial derivative matrix
operator we have the stress-strain
relationship captured in the d matrix
and we can then multiply the partial
derivative matrix operator by the shape
function matrix to give us the b matrix
or the strain nodal displacement matrix
that allows us to get to the stiffness
matrix and then finally the element
force vector one of the last things we
need to consider is transformation which
will not be covered in this video all
right let's work through the development
of a bilinear rectangular element you're
going to see it is a rectangle I'll
explain bilinear in a couple of slides
when we look at the shape functions so
we need to initially define what our
displacement field looks like and what
our strain and our stress fields look
like and this is the same as what we did
for the triangular element in the last
video so now we look at the element
geometry I've got a rectangle defined
here and it's got a local axis X prime Y
prime axis and then the nodes I've
chosen the X prime Y prime axis in the
middle this is not the only way I could
have defined this element obviously I
could have centered it down on the node
1 here but this makes it a little bit
more clear how I'm balancing things left
to right and top to bottom so this is
the more preferred elemental coordinate
system to be using for the rectangular
element so you can see that my width of
my element is 2b and my height is 2
times H
and I've got the node numbers there the
degrees of freedom each one of these
nodes I'm going to want to allow them to
move in both the X in the y direction so
I have two degrees of freedom per node
and that gives me my degree of freedom
vector which has four terms for each
direction a total of eight terms so
again here's my element remember I've
got now four degrees of freedoms in each
direction so that means in my polynomial
I can have a slightly longer one than
the triangle hat I can have four
coefficients for each polynomial so what
I'm going to do is a 0 plus a 1 X plus a
2 wide same thing we had for the
triangle element but now I've got
another term and I'm going to do a 3 X Y
now X Y is really the first quadratic
term which makes this element higher
degree than purely linear but it is
still linear in X and it's linear in Y
so that's why we call it a bilinear
rectangle it's linear in both of those
even though we have a first quadratic
term okay now I want to resolve these
aids in terms of the degrees of freedom
D so I start out with I go through each
one of my degrees of freedom in the
x-direction now everything I'm doing
here for the X displacement would be
completely analogous for the Y
displacement which we call V so I'm
going to be able to use the same shape
functions for the two directions I don't
need to go through the development again
so D 1 X is going to be the displacement
in the X direction at node 1 which means
evaluating the function U at negative B
comma negative H which gives us the
expression shown here D 2 X is
evaluating the displacement field at
become a negative H which gives us this
expression D 3 X is evaluating it at B
comma H which gives us this expression
and then D 4 X is evaluating the
displacement field at negative B comma H
and we get this expression so this is a
bit messy we haven't because we didn't
put the origin down at node one every
single one of these expressions has all
four of my A's so to resolve this it
actually makes a lot of sense to use
some of the tools we've been developing
for matrix manipulations and write this
out as a matrix equation when it's in
this form now you can go through and
find the inverse of that matrix and then
that allows you to solve for the a terms
you don't have to solve it that way but
it's a straightforward way to do it and
it allows you then to determine what
each one of those formerly unknown
coefficients were we're now defining
them in terms of the degrees of freedom
so I plug those into my expression for
you remember I'm not done yet I've just
eliminated the a terms now I have my
displacement field defined in terms of
the DS but what I want to do is gather
all the terms for each degree of freedom
so I'm going to have something x d1 X
plus something x D 2 X and so on that's
something in each case is going to be
the shape function here's that
expression again and when I rearrange
the terms I end up with something x d1 X
and that something happens to be 1 over
4 B H times B minus x times H minus y
and then you have similar functions in
front of D to X D 3x and D 4x and again
you'll have the same functions if we did
this all for the V or Y direction
so rewriting this in matrix form that my
displacement fields vector U is equal to
the shape function matrix shown so it
has eight columns and two rows and it's
going to be multiplied by my degree of
freedom vector which has eight terms
where now each one of the shape
functions is defined here you can see
that written in this form all that I'm
doing is switching pluses and minuses so
you can kind of think about that pattern
here you can obviously multiply it out
if you want to but this this is a nice
convenient way to look at it once I have
the shape functions I can go and get the
B matrix reminded up at the top there is
this is our partial derivative matrix
operator the B matrix is the product of
that partial derivative matrix operator
acting on the shape function matrix so
the B matrix looks like this again I'm
using indicial notation
so the comma X or comma Y represents a
partial derivative with respect to X or
Y remember that I do know those shape
functions I just developed them so when
I take these derivatives I end up with
this matrix here for B now one thing to
note here unlike the linear triangle
that we just developed the three noted
one because I've got that bilinear term
the B matrix actually depends on x and y
that means I have a linear variation of
strain in the bilinear rectangle element
whereas in the triangle I had a constant
strain throughout the whole thing also
this introduces a complication because
I'm going to have B matrix transpose
times the D matrix times the B matrix
when I do that matrix multiplication my
kit my integral inside or the integrand
in the K integral will have quadratic
terms in it so it's going to be a little
bit messy to solve that integral now I'm
not going to find the stiffness matrix
for this bilinear rectangle element
because this element is pretty specialty
it requires this shape and what we
really want is a more an element that we
can transform into different shapes and
so I will discuss what we do in a later
video when I look at transformation but
I'm going to jump forward in the other
use of the B matrix that we just found
is that we can evaluate the strain
inside the element so I'm going to use
the B matrix to evaluate the strain at
particular locations inside the element
first of all what we want to do is take
the b matrix and evaluate it at a given
position because remember strain varies
throughout the element I need to choose
where to evaluate this matrix so I'm
going to choose two locations if we look
at the element center that means we're
evaluating it where x and y are both
equal to 0 and that gives me this matrix
now of course it's a constant because
I'm evaluating it at a specific point I
haven't changed the B matrix to a
constant matrix I've just evaluated it
somewhere I'm also going to evaluate at
node 1 which is the lower left corner
and that gives me this matrix so now
let's use these to find the strain
okay so we're going to look at a
particular displacement field and see
what happens I'm going to look at pure
bending so when pure bending happens
I've got this rectangle and it's going
to get squeezed on one side and
stretched on the other that's how this
can represent bending so it turns into a
trapezoid so for example if I say that
the amount that each corner is squeezed
in at the top is equal to the same
amount that they're squeezed out at the
bottom that would be the pure bending
case so that gives me actually my
displacement field there are no changes
in the Y direction but in the X terms
I've got minus a 4 D 1 X a plus a 4 D 2
X and so on so that gives me my degree
of freedom vector this would be after
the solution so I found a K I've
inverted the K I've solved for this
displacement field so now the post
process will use that D to go to the
strain so back to the B matrix that we
evaluated at the element Center if I
take this B matrix and I multiply it by
the D matrix there I get I'm sorry the D
vector the degree of freedom vector I
get the strain and that multiplication
looks like what's shown here and it ends
up with zero strain at the middle that's
what we would expect there is no strain
on the neutral axis for pure beam
bending and that's what's predicted by
the theory and the F he predicts that
that's great so let's take a look at
what happens when we consider node 1 I
evaluate B at node 1 and I got this
matrix now I take this B matrix multiply
it by the D vector above and evaluate
the strain from there so when I do the
matrix multiplication I get this
expression which simplifies to a over B
H times H 0 B so I've got a term in the
epsilon X location and I've got a term
in the gamma XY location epsilon X ends
up being able to a over B which is what
we would expect that is the bending
strain along that bottom edge and it's
positive because we're stretching the
bottom edge but I also picked up a gamma
XY term which is
a / H that's unexpected in pure bending
we don't have shear strain shear strain
only gets introduced by the shear force
and I don't have one here this is just
pure bending so as a result what I've
had what what happens is some of the
bending energy gets absorbed in this
unexpected shear strain and so we get
less deflection of the beam if it's
modeled by this element then we would
get in reality this is something that's
called shear locking
5.0 / 5 (0 votes)