Slab-beam stiffness matrix assembly

[StructMnguy]

Hi everyone,

I have a topic on my mind, Has someone ever seen the assembly matrix of a beam and a slab? Does Anyone know if there is a book that shows matrix algorithms used in structural software?
I'm curious about that.

Sorry for my writing but English is my second language.


Thank you,

StructMnguy

 

[ANE91]

Matrix Analysis of Structures by Sennett
Elementary Structural Analysis by Norris et al.
Structural Analysis by Kassimali has 1 chapter on it
Structural Analysis in Theory and Practice by Williams

 

[StructMnguy]

@ANE91 Thanks for the help!

Do you know exactly how a shell element and a bar element are assembled in the stiffness matrix?

 

[ANE91]

Bar element is the first thing you learn — 2x2 stiffness matrix. I don’t do plates.

 

[StructMnguy]

Yes @ANE91 , I know the first thing you learn is a 2x2 stiffness matrix.

But I'm looking for an example of the connection between the shell and bar stiffness matrix.

Thanks BTW.

StructMnguy

 

[Celt83]

The elements stiffnesses are summed based on degree of freedom, at this stage there is nothing unique about either element.

Matrix Analysis of Structures by Kassimali has pseudo code for up to 3D bar structures first edition of the book can be found relatively inexpensive online and covers the topic sufficiently.

A First Course in the Finite Element Method by Logan any edition of the book has a section that addresses plate elements.

 

[StructMnguy]

@Celt83 Thank you.

But have you ever seen a book or an example that covers the connection between a plate and a bar element?

BTW thanks for the books, I will check them.

StructMnguy.

 

[IDS]

You might find my attached conference paper of interest, relating to analysis of bridge decks. It doesn't go into details of the FEA matrix, but does look at alternative ways of modelling the interaction between beams and slabs, and compares results from alternative approaches.

 

[Celt83]

@Celt83 Thank you.

But have you ever seen a book or an example that covers the connection between a plate and a bar element?

BTW thanks for the books, I will check them.

StructMnguy.

Ah you are looking more for how to model the intersection between these elements not the summation of the stiffness matrices.

Concept and Applications of Finite Element Analysis by Cook, Malkus, Plesha, and Witt
Modern Structural Analysis by MacLeod
Finite-Element Design of Concrete Structures by Rombach
Finite Element Modeling in Engineering Practice by Spyrakos

My understanding is for compatibility you generally need to engage more than one node at the interface between bar and shell elements this can be achieved by extending the bar element across a number of common nodes or by using rigid links to connect to additional shell nodes vertically to approximate the member depth.

 

[StructMnguy]

Thank you, @IDS . I'm checking the paper.

StructMnguy.

 

[StructMnguy]

Yes, @Celt83. I'm looking for the summation of stiffness matrices.

But how to model is a plus. My goal is replicate shell-bar connection in a model and using the stiffness matrices.

StructMnguy.

 

[Celt83]

Yes, @Celt83. I'm looking for the summation of stiffness matrices.

But how to model is a plus. My goal is replicate shell-bar connection in a model and using the stiffness matrices.

StructMnguy.

When it comes to forming the global stiffness matrix there is nothing unique between a bar element and a shell element. At nodes that have a shell and beam attached you just sum the stiffness components for each degree of freedom.

The majority of the complexity occurs in the initial modeling and ensuring compatibility at the intersection of the two element types.