To determine the components of the support reactions at the fixed support A of a cantilevered beam, we need to analyze the beam using static equilibrium conditions. A cantilever beam is one that is fixed at one end and free at the other, and the reactions at the fixed support can be broken down into vertical and horizontal forces as well as a moment.
First, we need to identify the loads applied to the beam and their locations. For instance, if there is a point load or distributed load on the beam, those will affect the reactions at support A. We will denote the reactions at support A as follows:
- Vertical Reaction Force (Ay)
- Horizontal Reaction Force (Ax)
- Moment Reaction (Ma)
Next, we set up the equilibrium equations based on the following conditions:
- Sum of vertical forces (∑Fy = 0)
- Sum of horizontal forces (∑Fx = 0)
- Sum of moments about point A (∑Ma = 0)
Starting with the vertical forces, we set up the equation:
∑Fy = 0: Ay + Σ (Downward Loads) = 0
This implies that Ay will balance out the total downward loads applied to the beam.
For the horizontal forces, if there are any loads acting horizontally, we write:
∑Fx = 0: Ax + Σ (Rightward Loads) – Σ (Leftward Loads) = 0
From this, we can find Ax.
Lastly, the moment about point A will include all external moments from loads applied to the beam:
∑Ma = 0: Ma – Σ (Moment Due to Loads about A) = 0
This will allow us to solve for the moment reaction at support A.
By systematically applying these equations, we can compute the values of Ax, Ay, and Ma at the fixed support A. This process ensures that the beam remains in static equilibrium, which is fundamental in structural analysis.