FEA 24: 2-D Rectangular Elements

Summary

This 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

• 📐 The video explains the development of a 2D rectangular element with four nodes and a bilinear displacement field.
• 📝 Defining a new element requires specifying the displacement, stress, and strain field vectors, along with element geometry and degrees of freedom.
• 🔢 Shape function matrices help relate strain and displacement, leading to the B matrix and stiffness matrix, but transformations are not covered in this video.
• 🧮 The displacement field is expanded with four coefficients for each polynomial, making it bilinear — linear in both x and y directions.
• ⚙️ The degree of freedom vector has eight terms in total (two per node), representing displacements in both x and y directions for each node.
• 🧩 Matrix manipulations simplify the resolution of unknown coefficients, transforming them into expressions dependent on degrees of freedom.
• 🗺️ Shape functions are derived for each degree of freedom and arranged into a shape function matrix for efficient computation.
• 📊 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.
• 🔍 The B matrix can be used to calculate the strain at specific points within the element, such as the center and the node locations.
• ❗ 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

00:00

📐 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.

05:01

🧩 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.

10:04

🔢 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.

📊 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.

🔧 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.

📐 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.

Keywords

💡Bilinear displacement field

A bilinear displacement field describes the relationship between the displacements within an element, where the displacement is linear in both the x and y directions. In the video, the rectangular element has four nodes, allowing for a more complex displacement field than triangular elements. This term is key to understanding how the element responds to forces.

💡Shape function matrix

The shape function matrix is a mathematical tool used to express the displacement field in terms of nodal displacements. In the video, it is used to simplify the process of calculating displacements across the rectangular element. The shape function matrix forms the basis for determining strain and stress fields.

💡Strain-displacement matrix (B matrix)

The B matrix, or strain-displacement matrix, links nodal displacements to strain in an element. It is derived by applying the partial derivative matrix operator to the shape function matrix. In the video, the B matrix is used to evaluate how strain varies inside the rectangular element at different locations.

💡Stiffness matrix

The stiffness matrix relates forces and displacements within an element, helping determine how an element resists deformation under loads. In the video, the development of the stiffness matrix for a bilinear rectangular element is mentioned, though its full calculation is postponed due to the complexity of the element’s geometry.

💡Degree of freedom (DOF)

Degrees of freedom refer to the independent movements allowed at each node of the element, such as displacements in the x and y directions. In the video, the rectangular element has two degrees of freedom per node, resulting in a total of eight degrees of freedom for the four-node element. This concept is crucial for formulating the displacement field.

💡Element geometry

Element geometry defines the physical shape and size of the element. In the video, the rectangular element has a width of 2b and a height of 2h, and the local coordinate system is placed at the element’s center. The geometry of an element affects how displacements and strains are calculated.

💡Stress-strain relationship (D matrix)

The D matrix represents the stress-strain relationship in an element, often derived from material properties such as elasticity. In the video, the D matrix is introduced as part of the process for calculating stress based on strain within the rectangular element. This matrix plays a key role in determining the internal forces within the element.

💡Pure bending

Pure bending refers to a deformation mode where an element bends without experiencing shear forces. In the video, pure bending is used as an example of a displacement field that causes the element to deform into a trapezoid. Understanding pure bending helps explain how the rectangular element behaves under specific load conditions.

💡Shear locking

Shear locking is an issue where an element experiences artificial shear strain, leading to an overly stiff response in finite element analysis (FEA). In the video, this phenomenon occurs when the rectangular element undergoes pure bending, resulting in unexpected shear strain and a reduced deflection compared to reality. This highlights a limitation of the element's formulation.

💡Transformation

Transformation refers to the mathematical process of changing the coordinate system or shape of an element. In the video, transformation is mentioned as an important concept for future videos, where it will be necessary to apply transformations to make elements more versatile in FEA models. Transformation allows an element to be adapted to different shapes and coordinate systems.

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