Plane Trusses Finite Elements 2 - Assembled Matrix

In summary, the global matrix is assembled by taking the element freedom tables and multiplying each element by the corresponding global number.
  • #1
bugatti79
794
1
Folks,

I am having difficulty understanding how this global matrix is assembled with the naming convention used as shown in attached.

The numbers in the corners such as 1(1,2) etc in figure 4.6.3 (b) denote the global and element numbers respectively.

Can anyone shed light on how this assembled...? Thanks
 

Attachments

  • Assembled Matrix.jpg
    Assembled Matrix.jpg
    11.9 KB · Views: 443
Engineering news on Phys.org
  • #2
bugatti79 said:
Folks,

I am having difficulty understanding how this global matrix is assembled with the naming convention used as shown in attached.

The numbers in the corners such as 1(1,2) etc in figure 4.6.3 (b) denote the global and element numbers respectively.

Can anyone shed light on how this assembled...? Thanks

Maybe if I write out the matrix attached for easier read...the blanks indicate symmetry.

##\begin{bmatrix}
k^1_{11}+k^3_{11} &k^1_{12}+k^3_{12} &k^1_{13} &k^1_{14} &k^3_{13} &k^3_{14} \\
&k^1_{22}+k^3_{22} &k^1_{23} &k^1_{24} &k^3_{23} &k^3_{24} \\
& & k^1_{33}+k^2_{11} &k^1_{34}+k^2_{12} &k^2_{13} &k^2_{14} \\
& & &k^1_{44}+k^2_{22} &k^2_{23} &k^2_{24} \\
& & & & k^2_{33}+k^3_{33} &k^2_{24}+k^3_{34} \\
& & & & & k^2_{44}+k^3_{44}
\end{bmatrix}##

The above is the matrix I am trying to understand how it was assembled based on the attached picture..

The numbers in the corners such as 1(1,2) etc in figure 4.6.3 (b) denote the global and element numbers respectively.

There are 2 displacement degrees of freedom (horizontal and vertical) at each node of the element...thanks
 

Attachments

  • DSC_0892.jpg
    DSC_0892.jpg
    30.2 KB · Views: 420
  • #3
bugatti79 said:
Maybe if I write out the matrix attached for easier read...the blanks indicate symmetry.

##\begin{bmatrix}
k^1_{11}+k^3_{11} &k^1_{12}+k^3_{12} &k^1_{13} &k^1_{14} &k^3_{13} &k^3_{14} \\
&k^1_{22}+k^3_{22} &k^1_{23} &k^1_{24} &k^3_{23} &k^3_{24} \\
& & k^1_{33}+k^2_{11} &k^1_{34}+k^2_{12} &k^2_{13} &k^2_{14} \\
& & &k^1_{44}+k^2_{22} &k^2_{23} &k^2_{24} \\
& & & & k^2_{33}+k^3_{33} &k^2_{24}+k^3_{34} \\
& & & & & k^2_{44}+k^3_{44}
\end{bmatrix}##

The above is the matrix I am trying to understand how it was assembled based on the attached picture..

The numbers in the corners such as 1(1,2) etc in figure 4.6.3 (b) denote the global and element numbers respectively.

There are 2 displacement degrees of freedom (horizontal and vertical) at each node of the element...thanks

After some searching online I have a found an easy way of assembling the global matrix for this problem.

If we focus on element 3 which has global nodes 1 and 3. we can create the element freedom table 'EFT' for this element by the following

2 dof's times global number 1 minus 1=1
2 dof's times the global number 1 =2

2 dof's times global number 3 minus 1=5
2 dof's times the global number 3 =6

( I am interested to know what the above technique is based on)

Thus the EFT is {1,2,5,6}. Similarly for the other 2 elements.

Then one combines the EFT for each element into the global matrx (2 dof's times number elements 3= 6 gives a 6 matrix.)
 

What is a plane truss?

A plane truss is a structural system composed of interconnected bars joined together at their ends to form a stable and rigid framework. It is commonly used in engineering and architecture to support loads and resist forces, such as in bridges and roofs.

What are finite elements in relation to plane trusses?

Finite elements are small, simplified portions of a larger structure that are used to approximate the behavior of the entire structure. In the case of plane trusses, finite elements are used to model the behavior of individual bars and joints within the truss.

What is an assembled matrix in the context of plane trusses?

An assembled matrix is a mathematical representation of the stiffness and flexibility of a plane truss. It is created by combining the stiffness and flexibility matrices of each individual element in the truss, and is used to solve for the displacements and forces within the truss.

How is the assembled matrix used in analyzing plane trusses?

The assembled matrix is used in conjunction with external loads and boundary conditions to solve for the displacements and forces within a plane truss. By applying the principles of equilibrium and compatibility, the assembled matrix can determine the internal forces and deformations of the truss.

What are the limitations of using assembled matrices in plane truss analysis?

While assembled matrices are a powerful tool for analyzing plane trusses, they have some limitations. They assume that the truss is made up of rigid and perfectly connected elements, and do not account for factors such as material properties, geometric imperfections, and non-uniform loading. Therefore, assembled matrices should be used in conjunction with other analysis methods to ensure accurate results.

Similar threads

Replies
4
Views
1K
Replies
2
Views
1K
  • Mechanical Engineering
Replies
3
Views
9K
  • Mechanical Engineering
Replies
2
Views
3K
  • Mechanical Engineering
Replies
2
Views
2K
  • Mechanical Engineering
Replies
7
Views
3K
Replies
3
Views
978
  • Mechanical Engineering
Replies
1
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
7
Views
620
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
1K
Back
Top