Example 1: Analyze the beam shown in the figure that is subjected to a distributed load and a point force, use the matrix method of stiffness.
EI = Constant
![](https://hebmerma.com/wp-content/uploads/2021/09/E1.png)
Solution:
a) Define the degrees of freedom and orientation of the element.
Note:
- The beam is not subjected to axial load, therefore two degrees of freedom are defined at each node (shear and moment).
- First we define the free degrees of freedom (green color).
- Second, we define the restricted degrees of freedom (red color).
- Finally, we define the orientation of the element.
![](https://hebmerma.com/wp-content/uploads/2021/09/E1-Grados-de-Libertad.png)
b) Assemble the stiffness matrix of each element.
![](https://hebmerma.com/wp-content/uploads/2021/09/image-2.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-3.png)
c) Assemble the stiffness matrix of the beam.
![](https://hebmerma.com/wp-content/uploads/2021/09/image-4.png)
d) Partition of the sub matrix K₁₁.
![](https://hebmerma.com/wp-content/uploads/2021/09/image-6.png)
e) Load vector analysis.
Element 1
![](https://hebmerma.com/wp-content/uploads/2021/09/E1-Vector-Cargas-1-1024x290.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-8.png)
Element 2
![](https://hebmerma.com/wp-content/uploads/2021/09/E1-Vector-Cargas-2.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-9.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-10.png)
f) Displacement vector calculation. |DD| = |K ₁₁|⁻¹ * |Cc|
![](https://hebmerma.com/wp-content/uploads/2021/09/image-11.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-12.png)
g) Calculation of reactions. |CD| = |DD| * |K₂₁|
![](https://hebmerma.com/wp-content/uploads/2021/09/image-13.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-14.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/E1-Reacciones-1024x313.png)
![](https://hebmerma.com/wp-content/uploads/2021/09/image-5-1024x576.png)
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