Bending moments are very important aspects of mechanics. Mechanical engineers are generally dealing with these kinds of problems on a theoretical basis. Definitions of correct forces and moments are very important to obtain the required results.
Drawing bending moment diagrams for different beam sections is also a very important part of the strength of material problems. Here, we explain how to draw bending moment diagrams effectively for all problems.
Bending diagrams are represented with a straight horizontal line right below the beam. This line represents zero. Beneath the line, the bending moment is negative, above the line bending moment is positive. To understand the sign conventions of bending moment and shear forces, consider this article.
How To Draw Bending Moment Diagrams?
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First of all, we need to understand this fact; Bending moment in beam sections is the result of the shear forces acting on the beam. If we know the shear force situation and application areas, we can make comments about possible bending moment diagrams.
The moment is actually is the multiplication of the distance of a force where the moment is calculated, with the force itself. With the increasing distance(x), the bending moment is also increasing. So;
dM(x)/dx = F(Shear Force)
According to the derivation calculus, when the shear force is zero or changes sign from maximum to minimum, the bending moment will have the maximum or minimum on this point.
Some Graphing Techniques From Math
- If the shear force application is singular on a beam, the shear force diagram will be constant just as shown below. So, the bending moment diagram will be inclined linear.
- If the shear force application is a uniformly distributed load on a beam, from left(where the beam is considered as cantilevered) to right, the shear force will decrease as inclined linear. So, the bending moment will decrease as a second-order curve.
So, you can understand the decrement and increment relation between shear force and bending moment diagrams.
A Bending Diagram Drawing Method
At the free side of the cantilevered beam, the bending moment will be zero. So, you need to make your calculations from the free side of the beam.
For Singular Shear Forces
Say that the length of the beam is ‘l’. And say the distance from the free side of the beam is ‘x’. This ‘x’ must be at both sides of the shear force application to see the complete situation of bending moment.
First, you need to calculate the left side bending moment situation of the singular force, by placing ‘x’ to the left side of the singular shear force. Then calculate the bending moment at that point, from the left side of the point.
Then place the ‘x’ on the right side, make your calculation again.
For Uniformly Distributed Shear Forces
If the force is uniformly distributed on a random side of the beam, you need to place your ‘x’ on the place that that distributed load is not applied(left side of distributed load). Then you need to place your ‘x’ on the place that distributed load is applied.
If the right side of the distributed force is empty, place your ‘x’ on this side.
For each placement, make your bending moment calculations according to these places by considering the left side(free side actually).
For Shear Forces That Distributed As Inclined Linear Curve
In this case, you must have the second or third order of the equation of shear force. If you derive this equation with ‘x’, you will have the equation of bending moment.
Actually, we want to obtain the graphs of equations in bending moment diagrams. The mathematical methods above for obtaining the equations.
For Complex Problems
You can deal with complex problems such as different shear force forms are acting on beams. In this case, you need to make separate calculations for each force as they are only acting forces on beams. Then you need to add these bending moment diagrams to obtain the complete system’s bending moment diagram.
If you understand the general logic of obtaining bending moment and shear force diagrams, you will able to solve all the problems related to this topic.
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