6 Nodes, 4 Elements, Size of Stiffness Matrix?

In summary, the size of a stiffness matrix for a system is determined by the number of nodes and the degrees of freedom per node. It is calculated by multiplying the number of nodes by the degrees of freedom per node. For example, in a system with 6 nodes and 2 degrees of freedom per node, the stiffness matrix would be 12x12. This is considered one of the more challenging aspects of FEM.
  • #1
ThurmanMurman
12
0
Howdy,

If I have the following configuration of nodes:

2---4--6
| \ | /|
| \ | / |
| \| / |
| \/ |
1---3--5

What should the dimensions of my FEM stiffness be?

Thanks
 
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  • #2
Alright my attempt at formatting got messed up once I posted. Nodes 2-4-6 are equally spaced across the top, and nodes 1-3-5 sit below 2-4-6 on the bottom. 2-4-6 are respectively connected to 1-3-5 vertically, and 2 is connected to 3 diagonally, and 3 is connected to 6 diagonally.
 
  • #3
The matrix size is (number of nodes) times (number of degrees of freedom per node).

The number of elements and how they connect is irrelevant.
 
  • #4
So is the (number of nodes) = (number of rows) and the (DOFs per node) = (number of columns)?
 
  • #5
Just to check my understanding, I think that my arrangement of nodes (4 total) with 2 DOFs each (8 total) leaves me with a 4X8 stiffness matrix. Is that correct?
 
  • #6
Stiffness matrices are square and symmetric. So your stiffness matrix will be 8x8.
 
  • #7
So is there a (nodes,DOFs) equation that states the size of a stiffness matrix for a system?
 
  • #8
ThurmanMurman said:
So is the (number of nodes) = (number of rows) and the (DOFs per node) = (number of columns)?

No, the number of rows and columns are both equal to (number of nodes) x (DOFs per node) .

Sorry if that wasn't clear - I took it as "obvious" that stiffness matrices are square.
 
  • #9
So for my 6 node example, the stiffness would be sized (number of nodes = 6) x (DOFs per node = 2) = 12 x 12?
 
  • #10
Yes.
 
  • #11
So that's the most difficult part of FEM, right?
 

1. What is the significance of 6 nodes and 4 elements in a stiffness matrix?

The number of nodes and elements in a stiffness matrix determines the level of detail and accuracy in a finite element analysis. Having 6 nodes and 4 elements suggests a relatively simple and coarse mesh, which may be appropriate for certain applications.

2. How is the size of the stiffness matrix determined?

The size of the stiffness matrix is determined by the number of degrees of freedom in the system. In a 6-node, 4-element system, there are 6 degrees of freedom (3 translational and 3 rotational) at each node, resulting in a 24x24 stiffness matrix.

3. Can a 6-node, 4-element system accurately represent a complex structure?

It depends on the complexity of the structure and the level of accuracy required. In general, a larger number of nodes and elements will result in a more accurate representation of a structure. However, for simpler structures, a 6-node, 4-element system may be sufficient.

4. How does the stiffness matrix affect the accuracy of a finite element analysis?

The stiffness matrix plays a crucial role in the accuracy of a finite element analysis. It contains the stiffness coefficients for each element, which are used to calculate the displacements, stresses, and strains in the structure. A larger stiffness matrix with more nodes and elements can result in a more accurate analysis.

5. Is it possible to modify the size of the stiffness matrix in a finite element analysis?

Yes, the size of the stiffness matrix can be modified by changing the number of nodes and elements in the mesh. A finer mesh with more nodes and elements will result in a larger stiffness matrix, while a coarser mesh with fewer nodes and elements will result in a smaller stiffness matrix.

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