Simple Finite Element noob question

Question

Simple Finite Element noob question

RudiBR (Structural)

(OP)

7 Dec 20 16:04

Hi,

I am new to FEM theory, and I have a small issue with a very simple example.

It's a square in x/y as shown below. "Bottom" nodes, (0) and (3), are fixed.
There is a 10000 kN down force at node (1) and a horizontal force of 11000 kN at node (1).

CODE -->

y                                                          
            ^                                                          
            |                                                          
            |           Fy                                            
            |           |                                             
                        |                                             
                beam 1  |                                             
          (2)           v (1)                                            
             +----------+--->Fx                                       
             |          |                                             
             |          |                                             
     beam 2  |          |   beam 0                                          
             |          |                                             
             |          |                                             
             +----------+               ------> x                              
          (3)^  beam 3  ^(0)

Simple 2D mesh.
Nodes (0) and (3) are fixed.
All sides equal length and k values.
length = 1000 mm
k = 10000 kN/mm^2

Here are the stiffness matrices for each beam one at the time, and finally the sum.
As the last step, the rows/columns representing the fixed nodes are removed.
However, I end up with an unsolvable matrix.

What am I doing wrong?

CODE -->

beam 0 stiffness matrix											
											
		0		1		2		3			
		0	1	2	3	0	1	2	3	F [N]	
0	0	0	0	0	0					0	x
	1	0	10000	0	-10000					0	y
1	2	0	0	0	0					-11000	x
	3	0	-10000	0	10000					-10000	y
2	4										x
	5										y
3	6										x
	7										y


beam 1 stiffness matrix											
											
		0		1		2		3			
		0	1	2	3	0	1	2	3	F [N]	
0	0										x
	1										y
1	2			10000	0	-10000	0			-11000	x
	3			0	0	0	0			0	y
2	4			-10000	0	10000	0			0	x
	5			0	0	0	0			0	y
3	6										x
	7										y


beam 2 stiffness matrix											
											
		0		1		2		3			
		0	1	2	3	0	1	2	3	F [N]	
0	0										x
	1										y
1	2										x
	3										y
2	4					0	0	0	0	0	x
	5					0	10000	0	-10000	0	y
3	6					0	0	0	0	0	x
	7					0	-10000	0	10000	0	y


beam 3 stiffness matrix											
											
		0		1		2		3			
		0	1	2	3	0	1	2	3	F [N]	
0	0	10000	0					-10000	0	0	x
	1	0	0					0	0	0	y
1	2										x
	3										y
2	4										x
	5										y
3	6	-10000	0					10000	0	0	x
	7	0	0					0	0	0	y


Main (summed) stiffness matrix											
											
		0		1		2		3			
		0	1	2	3	0	1	2	3	F [N]	
0	0	10000	0	0	0	0	0	-10000	0	0	x
	1	0	10000	0	-10000	0	0	0	0	0	y
1	2	0	0	10000	0	-10000	0	0	0	-22000	x
	3	0	-10000	0	10000	0	0	0	0	-10000	y
2	4	0	0	-10000	0	10000	0	0	0	0	x
	5	0	0	0	0	0	10000	0	-10000	0	y
3	6	-10000	0	0	0	0	0	10000	0	0	x
	7	0	0	0	0	0	-10000	0	10000	0	y


Constrained (fixed) nodes removed stiffness matrix

		1		2		
		2	3	0	1	F [N]
1	2	10000	0	-10000	0	-22000
	3	0	10000	0	0	-10000
2	4	-10000	0	10000	0	0
	5	0	0	0	10000	0

Mustaine3 (Mechanical) · Answer 1 · 2020-12-07 16:26:42

RE: Simple Finite Element noob question

Mustaine3 (Mechanical) 7 Dec 20 16:26

Beams or Trusses? There is a difference.

GregLocock (Automotive) · Answer 2 · 2020-12-07 19:51:14

You've overconstrained beam 3, which doesn't need to exist.

Cheers

Greg Locock

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IDS (Civil/Environmental) · Answer 3 · 2020-12-07 20:26:30

Your matrices only include the axial stiffness of the members, with no moment restraint, so your structure is unstable. You need a diagonal member to make it stable.

The link below works through setting up a 2D frame analysis on a spreadsheet starting with a single beam.:

https://newtonexcelbach.com/2009/01/31/frame-analy...

There is a link to the next post in the series at the bottom of each post.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

rb1957 (Aerospace) · Answer 4 · 2020-12-07 21:48:17

you say "stiffness = ..." ... axial stiffness ? (you'd think so) ... but your frame requires bending stiffness to work ... you need to specify a diagonal member (say 1-3).

looking at beam "0" ("odd" that number !)
node "0" is fixed in x (zero deformation) but has deformation in y (finite k)
node "1" is also fixed in x, deformable in y
... that seems very "odd" ...

is this homework ? (doesn't look like "real" work)

another day in paradise, or is paradise one day closer ?

RudiBR (Structural) · Answer 5 · 2020-12-08 00:36:36

Thanks for you reply. Yes - off course. As Doug Jenkins write, I need at least one diagonal, or the rectangular structure will collapse into a parallelogram.
And no, it’s not “homework”. I am implementing some basic FEM in software and I am in the process of adding some basic unit tests, where this came up.
With one or two diagonals, I get a solvable stiffness matrix.
... and I guess this is trusses? Sorry for the confusion.

rb1957 (Aerospace) · Answer 6 · 2020-12-08 13:13:53

you need to understand the difference between axial members ... RODs carry axial load only, BEAMs carry bending (and lateral loads) as well.

node "0" should not deflect under X or Y loads.
node "1" should only deflect under Y loads (as far as beam "0" is concerned, and deflect under X loads as far as beam "1" is concerned.

There are plenty of texts out there on this.

another day in paradise, or is paradise one day closer ?

Mohanlal0488 (Structural) · Answer 7 · 2020-12-11 09:05:15

Have a look at the below book, and you will be a master in no time!!!

I bought a copy and it has been one of the best FEA books I have.

https://www.amazon.com/Introduction-Finite-Element...

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Simple Finite Element noob question