Finite Element Analysis - Author J.N Reddy Book

In summary, there was a discussion about understanding a concept in the book "An Introduction to the Finite Element Method" by J.N. Reddy. The concept involved assembling stiffness matrices for a system of linear elements. The conversation included suggestions to write the concept in matrix form, with an example provided for two elements. There was also a suggestion to use a graphical representation for better understanding. The conversation ended with appreciation for the clarification provided by AlephZero's notation and the use of paper and pens for visualizing the concept.
  • #1
bugatti79
794
1
Folks,

Is there anyone out there familiar with 'An introduction to the Finite Element Method' by J.N. Reddy?

I am struggling to decipher what is happening on page 129 as shown in the attachment. If some-one is willing to help I will reply with a more specific query on that page. Thanks
 

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  • #2
I think this would be easier to understand if you write it in matrix form.
Suppose you have two linear elements joined end to end, so there are 3 nodes.
If the stiffness matrices of the two elements are
$$\begin{bmatrix}
k^1_{11} & k^1_{12} \\
k^1_{21} & k^1_{22}
\end{bmatrix}$$
and $$\begin{bmatrix}
k^2_{11} & k^2_{12} \\
k^2_{21} & k^2_{22}
\end{bmatrix}$$
The assembled stiffness matrix is
$$\begin{bmatrix}
k^1_{11} & k^1_{12} & 0 \\
k^1_{21} & k^1_{22}+k^2_{11} & k^2_{12} \\
0 & k^2_{21} & k^2_{22}
\end{bmatrix}$$
And similarly for the right hand side vectors.

You probably first met this idea in a dynamics course, setting up the equations of motion for mass-and-spring systems.
 
  • #3
AlephZero said:
I think this would be easier to understand if you write it in matrix form.
Suppose you have two linear elements joined end to end, so there are 3 nodes.
If the stiffness matrices of the two elements are
$$\begin{bmatrix}
k^1_{11} & k^1_{12} \\
k^1_{21} & k^1_{22}
\end{bmatrix}$$
and $$\begin{bmatrix}
k^2_{11} & k^2_{12} \\
k^2_{21} & k^2_{22}
\end{bmatrix}$$
The assembled stiffness matrix is
$$\begin{bmatrix}
k^1_{11} & k^1_{12} & 0 \\
k^1_{21} & k^1_{22}+k^2_{11} & k^2_{12} \\
0 & k^2_{21} & k^2_{22}
\end{bmatrix}$$
And similarly for the right hand side vectors.

You probably first met this idea in a dynamics course, setting up the equations of motion for mass-and-spring systems.

Thanks.Actually further down the page the matrix form is shown (see attached). However, I don't see how the assembled matrix you have shown for 2 linear elements can be derived 'explicitly' from the matrix attached. Ie, the second row of attached contains ##k_{11}^3## which does not exist for a system of 2 elements...? Of course we know it does not exist hence we can simply not write it in but...
 

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  • #4
I think the matrix in the book is slightly misleading. There should also be a vertical dotted line showing that some columns are missing, lile
$$\begin{bmatrix}
\color{red}{ k^1_{11}} & \color{red}{k^1_{12}} & & \vdots \\
\color{red}{k^1_{21}} & \color{red}{k^1_{22} + k^2_{11}} & \color{blue}{k^2_{12}} & \vdots \\
& \color{blue}{k^2_{21}} & \color{blue}{k^2_{22} + k^3_{11}} & \vdots \\
\cdots & \cdots & \cdots & \ddots & \cdots & \cdots \\
& & & \vdots & k^{N-1}_{22} + k^N_{11} & \color{red}{K^N_{12}} \\
& & & \vdots & \color{red}{k^N_{21}} & \color{red}{K^N_{22}}
\end{bmatrix}$$

When N = 1, you just have the first and last rows and columns forming a 2x2 matrix.

When N = 2, you have the first second and last rows and columns forming a 3x3 matrix, i.e. the matrix entries shown in red.

When N = 3, you have the entries shown in red and blue.
 
Last edited:
  • #5
In two elements K11^3 = 0 as there is no U4 either. Its just the notations that are generally written for more than two elements
 
  • #6
As a mechanical engineer I found it much easier to lay out the matrices as AlephZero has done (i.e. graphically), the reason the author of the book lays them out in equation or linear algebra form is basically that's what you would need if you were writing your own FEA software.

Great posts AlephZero! That's some impressive application of TEX!
 
  • #7
AlephZero said:
I think the matrix in the book is slightly misleading. There should also be a vertical dotted line showing that some columns are missing, lile
$$\begin{bmatrix}
\color{red}{ k^1_{11}} & \color{red}{k^1_{12}} & & \vdots \\
\color{red}{k^1_{21}} & \color{red}{k^1_{22} + k^2_{11}} & \color{blue}{k^2_{12}} & \vdots \\
& \color{blue}{k^2_{21}} & \color{blue}{k^2_{22} + k^3_{11}} & \vdots \\
\cdots & \cdots & \cdots & \ddots & \cdots & \cdots \\
& & & \vdots & k^{N-1}_{22} + k^N_{11} & \color{red}{K^N_{12}} \\
& & & \vdots & \color{red}{k^N_{21}} & \color{red}{K^N_{22}}
\end{bmatrix}$$

When N = 1, you just have the first and last rows and columns forming a 2x2 matrix.

When N = 2, you have the first second and last rows and columns forming a 3x3 matrix, i.e. the matrix entries shown in red.

When N = 3, you have the entries shown in red and blue.

THanks to all and particularly AlephZero. His matrix notation greatly clarifies things for me.

Appreciated.
Ed
 
  • #8
Mech_Engineer said:
As a mechanical engineer I found it much easier to lay out the matrices as AlephZero has done (i.e. graphically),

Actually, there's a better technology than TeX for doing this. It's called "some big sheets of paper and a pack of colored pens." :smile:

(But Mech_Engineer probably knew that already.)
 

What is Finite Element Analysis?

Finite Element Analysis (FEA) is a computerized method used to predict how a product reacts to real-world forces, vibration, heat, fluid flow, and other physical effects. It is a numerical method for solving problems that involve complicated geometries, loads, and material properties.

Who is the author of the book "Finite Element Analysis"?

The author of the book "Finite Element Analysis" is J.N Reddy, an Indian American mechanical engineer and researcher. He is a distinguished professor and holder of the Oscar S. Wyatt Jr. Endowed Chair in Mechanical Engineering at Texas A&M University.

What topics does the book cover?

The book covers a wide range of topics related to Finite Element Analysis, including mathematical foundations, element and stiffness matrices, assembly of elements, applications to structural mechanics and heat transfer, and advanced topics such as dynamic analysis and fluid-structure interaction.

Is the book suitable for beginners?

No, the book is not suitable for beginners. It is intended for graduate students and researchers in engineering and applied mathematics who have a basic understanding of engineering mechanics and numerical analysis.

How is the book different from other books on Finite Element Analysis?

The book stands out for its comprehensive coverage of the mathematical foundations of Finite Element Analysis, its clear and concise explanations, and its numerous examples and exercises. It also includes advanced topics such as fluid-structure interaction and nonlinear analysis, making it a valuable reference for researchers and professionals in the field.

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