FEA 23: 2-D Triangular Elements

Summary

This 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

• 📐 The video focuses on the development of a 2D triangular linear element, specifically a right triangle with a 90-degree corner.
• 🔍 It emphasizes the importance of defining the displacement field, strain, and stress fields for the element.
• 🌐 The element's geometry and degrees of freedom are defined, which are crucial for relating back to the displacement field vector.
• 📏 A shape function matrix is established to define how the displacement field varies within the element.
• 🔗 The relationship between the strain vector and the displacement field vector U is explored, leading to the partial derivative matrix operator.
• 📚 The D matrix, which contains material properties, is derived from the relationship between stress and strain.
• 🧮 Matrix multiplication of the partial derivative matrix and the shape function matrix results in the B matrix.
• 🔑 The stiffness matrix K of the element is calculated using B and D matrices.
• 💪 The element force vector is discussed, including how to account for distributed forces acting on the element.
• 🔄 The concept of transformation from local to global coordinate systems is introduced but will be detailed in later videos.
• 📉 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

00:00

📐 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.

05:02

🔍 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.

10:02

📉 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.

15:02

🔗 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.

🧮 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.

Keywords

💡Displacement field

The displacement field refers to the vector representation of how points within a body move due to deformation. In the video, the displacement field is described as a vector that includes both horizontal and vertical translations of nodes in a right triangle element. It is crucial for understanding how the element behaves under load.

💡Strain

Strain is a measure of deformation in a body. The video discusses normal strain in the X and Y directions and shear strain, which links these two directions. Strain is important for determining how much a material deforms under stress.

💡Stress

Stress is the internal resistance of a material to deformation. The script mentions normal stress in the X and Y directions and shear stress, which is related to the strain through the stress-strain relationship. Understanding stress is essential for predicting material failure.

💡Degrees of freedom

Degrees of freedom refer to the number of independent values needed to define the configuration of a system. In the context of the video, each node in the triangular element has two degrees of freedom, corresponding to displacements in the X and Y directions.

💡Shape function matrix

The shape function matrix defines how the displacement field varies within an element and relates this variation back to the degrees of freedom. It is crucial for mapping the local behavior of an element to its global behavior.

💡Partial derivative matrix operator

This operator relates the strain vector to the displacement field vector U. It is used to calculate the strain distribution within the element based on the displacement field. The script explains that this operator is essential for constructing the B matrix.

💡D Matrix

The D Matrix, also known as the material property matrix, relates stress to strain and includes the material's elastic properties. It is integral to calculating the stiffness matrix of the element.

💡Stiffness matrix (K)

The stiffness matrix K is derived from the B and D matrices and is used to calculate the element's resistance to deformation. It is a key component in finite element analysis for predicting how structures will behave under load.

💡Element force vector

The element force vector represents the forces acting on an element, which can be due to distributed forces. The video explains how to define this vector, which is then added to the nodal forces to create a global force vector.

💡Transformation

Transformation in the context of the video refers to the process of applying an element from its local coordinate system to the global system. This is important for accurately modeling complex structures where elements may be oriented in various directions.

💡Right triangle element

The right triangle element is a specialized type of triangular element used in the video. It has a 90-degree corner and is simpler than a general triangle, making it easier to demonstrate the finite element development process without getting lost in complex mathematics.

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