Solving Stiffness Matrix: Multiplication & Addition of Matrices

In summary, the conversation discusses attempts to solve a stiffness matrix for a system with known forces and displacements. The matrix is partitioned into two parts, but the equation cannot be solved. The goal is to find the magnitude of forces and displacements for all nodes in the system.
  • #1
tomcenjerrym
37
0
I spent almost 2 days to solve the following stiffness matrix

[tex]\left(\begin{array}{c}f1\\f2\\f3\\f4\\f5\end{arr ay}\right)=\left(\begin{array}{ccccc}20&-20&0&0&0\\-20&40&-20&0&0\\0&-20&40&-20&0\\0&0&-20&40&-20\\0&0&0&-20&20\end{array}\right)\left(\begin{array}{c}0\\d2\\d3\\d4\\0\end{arr ay}\right)?[/tex]

I have tried to partition the matrix like the following

FIRST

[tex]\left(\begin{array}{c}f1\\f2\\f3\end{array}\right)=\left(\begin{array}{ccc}20&-20&0\\-20&40&-20\\0&-20&40\end{array}\right)\left(\begin{array}{c}0\\d2\\d3\end{array}\right)+\left(\begin{array}{cc}0&0\\0&0\\-20&0\end{array}\right)\left(\begin{array}{c}d4\\0\end{array}\right)[/tex]

SECOND

[tex]\left(\begin{array}{c}f4\\f5\end{array}\right)=\left(\begin{array}{ccc}0&0&-20\\0&0&0\end{array}\right)\left(\begin{array}{c}0\\d2\\d3\end{array}\right)+\left(\begin{array}{cc}40&-20\\-20&20\end{array}\right)\left(\begin{array}{c}d4\\0\end{array}\right)[/tex]

but still cannot solve for the equation.

The partitioned matrix produced UNDETERMINED Multiplication and Addition of 2 matrices.

Can anyone help me on this, please...
 
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  • #2
What exactly are you trying to obtain? Are you trying to find the d-s?
 
  • #3
All of forces fx and displacements dx magnitude. Please look at my attached image for the system. Note that the magnitude of f3 = 10 kN/m and the displacement of node 1 and 5 is 0 also the magnitude of k1 is = k2 = k3 = k4 = 20 kN/m.
 

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1. What is a stiffness matrix?

A stiffness matrix is a square matrix that represents the relationship between the external forces acting on a structure and the corresponding displacements of that structure. It is a fundamental tool used in structural analysis and is essential in solving problems related to elasticity and mechanics.

2. How do you multiply two stiffness matrices?

To multiply two stiffness matrices, we use the standard matrix multiplication method. This involves multiplying the corresponding elements in each row of the first matrix by the corresponding elements in each column of the second matrix, and then adding the products to get the resulting element in the new matrix. This process is repeated for each element in the resulting matrix.

3. Can you add two stiffness matrices together?

Yes, it is possible to add two stiffness matrices together. This is done by simply adding the corresponding elements in each matrix to get the resulting element in the new matrix. However, the two matrices must have the same dimensions for this operation to be possible.

4. What are some applications of solving stiffness matrices?

Solving stiffness matrices is used in various fields such as civil engineering, mechanical engineering, and aerospace engineering. It is used to analyze the behavior of structures under different loads, determine the stability and strength of structures, and design structures that can withstand external forces.

5. Is there a specific order in which you should perform matrix operations for stiffness matrices?

Yes, when solving stiffness matrices, it is important to follow the order of operations for matrix multiplication and addition, as with any other mathematical operation. This means multiplying matrices first and then adding the resulting matrices together. Performing operations in a different order can lead to incorrect results.

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