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# Simple Finite Element noob question

## Simple Finite Element noob question

(OP)
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 

### RE: Simple Finite Element noob question

Beams or Trusses? There is a difference.

### RE: Simple Finite Element noob question

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/

### RE: Simple Finite Element noob question

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)

### RE: Simple Finite Element noob question

(OP)
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.

### RE: Simple Finite Element noob question

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.

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