A beam is 7,5 m long. It is supported at the left end and has another support 1,5 m from the right end. From the left end, a uniformly distributed load of 60 N/m rests on the beam over a length of 3 m. Another distributed load of 20 N/m rests 3 m from the left end of the beam over a length of 4,5 m. Another support is positioned 1,5 m from the right-side end of the beam. Perform the following tasks in your groups and present them on the board to the class: a) Draw the loading diagram. b) Calculate bending moments
To calculate the bending moments of the beam, we first need to draw the loading diagram.
a) Loading Diagram:
We have two uniformly distributed loads on the beam:
- 60 N/m over a length of 3 m, starting from the left end.
- 20 N/m over a length of 4.5 m, starting 3 m from the left end.
The distance of the first support from the left end is 0 m, and the distance of the second support from the right end is 1.5 m.
The loading diagram will look like this:
```
<----60 N/m-----> <---------20 N/m-------->
(0 m) |–––[==== 3 m ====]–|–––[=======4.5 m =======]–––––| (7.5 m)
^ ^ ^
| | |
Support Support Right End
```
b) Calculating Bending Moments:
To calculate the bending moments at different points along the beam, we need to analyze each section of the beam separately.
1. Section between the left end and the first support (0 m to 1.5 m):
No distributed loads act on this section, only the reaction force from the first support. Thus, the bending moment is zero.
2. Section between the first and second supports (1.5 m to 6 m):
This section has the first distributed load of 60 N/m (from 0 m to 3 m). The load is uniformly distributed, so the load per unit length is constant.
To calculate the bending moment at any point in this section, we can consider the reaction force (R1) due to the first support acting at 1.5 m. We can integrate the load per unit length (60 N/m) over the length to find the total load (force) acting on this section.
The bending moment (M) at any point (x) between 1.5 m and 6 m can be calculated as:
M(x) = R1 * (x - 1.5) - (60 N/m) * (x - 1.5) * (x - 1.5) / 2
3. Section between the second support and the right end (6 m to 7.5 m):
This section has two distributed loads - the first one ends at 6 m, and the second one starts at 6 m and goes up to 7.5 m. Similar to section 2, we can calculate the bending moment at any point by considering the reaction force (R2) due to the second support acting at 6 m and integrating the load per unit length (20 N/m) over the length.
The bending moment (M) at any point (x) between 6 m and 7.5 m can be calculated as:
M(x) = R2 * (x - 6) - (20 N/m) * (x - 6) * (x - 6) / 2
These equations will give the bending moments at different points along the beam. By substituting the corresponding values of x and solving these equations, we can find the bending moments at desired locations.