A uniform beam has a weight of 500N(acting at the centre of the beam). It also carries a point load of 1,2kN at the centre of the left half of the beam, and a point load of 820N at distance of one eighth of the beam's lenght from the right hand side. The support are at the two ends. Calculate the magnitude of the forces at the two supports
draw the structure.
sum vertical forces and set them to zero. Call the support forces upward Fl and Fr
Now, sum moments about either end, and set to zero.
You should have two equations, two unknowns.
To calculate the magnitude of the forces at the two supports, we need to find the reactions at each end of the beam.
Let's denote the length of the beam as L. The weight of the beam, acting at its center, is 500 N. The point load at the center of the left half of the beam is 1.2 kN (or 1200 N), and the point load at a distance of one eighth of the beam's length from the right-hand side is 820 N.
Step 1: Calculate the total weight acting on the beam.
The total weight acting on the beam is the weight of the beam itself (500 N) plus the point loads at the center of the left half (1200 N) and the load at a distance of one eighth of the beam's length from the right-hand side (820 N).
Total weight = 500 N + 1200 N + 820 N = 2520 N
Step 2: Calculate the reaction forces at the supports.
Since the beam is in equilibrium, the sum of the vertical forces acting on it must be zero. This means that the reactions at the supports must balance out the total weight and point loads.
Let's denote the reaction force at the left support as R1 and the reaction force at the right support as R2.
Considering the vertical equilibrium, we can write the equation:
R1 + R2 = Total weight
Step 3: Calculate the position of the center of gravity of the beam.
The position of the center of gravity (CG) of the beam will help us determine the distribution of the total weight and point loads between the supports. The position of the CG can be calculated as follows:
Position of CG = (Distance from the left support to the CG * Total weight) / Beam length
In this case, the distance from the left support to the CG is half the beam length (L/2).
Position of CG = (L/2 * Total weight) / L = Total weight / 2
Step 4: Determine the distribution of weight and point loads between the supports.
Since the CG is at the center of the beam, the weight and point loads are symmetrically distributed between the supports.
The weight on each support will be half of the total weight.
Weight on each support = Total weight / 2
Step 5: Calculate the reaction forces at the supports.
Using the equation from Step 2, we can substitute the value of the total weight and solve for R1 and R2.
R1 + R2 = 2520 N
Since the weight is symmetrically distributed, R1 = R2 = Weight on each support.
So, 2R1 = 2520 N
R1 = R2 = 2520 N / 2 = 1260 N
Therefore, the magnitude of the forces at the two supports (R1 and R2) is 1260 N each.
To calculate the magnitude of the forces at the two supports, we need to analyze the forces acting on the beam and apply the principle of equilibrium.
First, let's draw a free body diagram of the beam and label the forces acting on it.
At the left support:
- There is a horizontal force (reaction) acting to the right, which we'll call R1.
- There is a vertical force (reaction) acting upward, which we'll call V1.
At the right support:
- There is a horizontal force (reaction) acting to the left, which we'll call R2.
- There is a vertical force (reaction) acting upward, which we'll call V2.
Now let's consider the forces acting on the beam:
1. The weight of the beam (500N) acts vertically downward at the center of the beam.
2. The point load of 1,2kN (1200N) acts vertically downward at the center of the left half of the beam. Since the beam is uniform, we can assume this point load acts at the center of the beam.
3. The point load of 820N acts vertically downward at a distance of one-eight (1/8) of the beam's length from the right side.
To find the forces at the supports, we can consider the vertical and horizontal forces separately and apply the principle of equilibrium.
Vertical Equilibrium:
The sum of the vertical forces must be zero since the beam is in equilibrium vertically.
V1 + V2 - 500N - 1200N - 820N = 0
Horizontal Equilibrium:
The sum of the horizontal forces must be zero since the beam is in equilibrium horizontally.
R1 + R2 = 0
Now we have two equations:
V1 + V2 = 500N + 1200N + 820N
R1 + R2 = 0
Solving these equations will give us the magnitudes of the forces at the supports.