# Concentrated and Distributed loads in regards to moments

1. A load is any externally applied force or moment.

2. A load which is spread over a very small area of the beam is called a concentrated load.

3. A load which is spread over a significant area of the beam is called a distributed load.

I'm able to understand concentrated and distributed loads, when these loads are forces,

but I'm unable to make sense of the definitions 2 and 3 when it comes to moments.

What actually is a concentrated moment and a distributed moment?

• If you draw the bending moment diagram for concentrated and distributed moment, and the shear forces diagram for the concentrated and distributed loads you will see the parallels. If you are looking for a real life example that you can related, that is a bit more difficult, because uniform distributed moments are not common in my experience.
– NMech
Commented Jan 4, 2022 at 9:03
• A gear held to a shaft with a key which has a large thickness due to the load will apply a distributed load to the shaft over the length of the key. But during analysis that will be taken as a concentrated load. Commented Jan 4, 2022 at 9:22

I don't know of any example of distributed bending moment, but I think it may be useful concept for understanding shear deformation component caused by transverse force. Just imagine typical situation where a beam with length $$L$$ is fixed at one end and loaded by transverse force $$F$$ at the other and what happens when we add constant distributed moment $$m$$ using force couples:

Total bending moment at distance $$x$$ from the left will be: $$M_b(x) = F\cdot (L-x) - m\cdot (L-x)$$ This naturaly brings up the question what happens when $$m=F$$? In such case the bending moment will be $$0$$ along the whole length, but the beam will still be deformed by shear as you can clearly see from the free body diagram of any beam element along the axis:

This kind of deformation represents the difference between Timoshenko and Euler-Bernoulli beam formulations:

• Euler-Bernoulli: $$\delta_{EB} = \frac{F}{EI}\cdot \left(L\frac{x^2}{2}-\frac{x^3}{6}\right)$$
• Timoshenko: $$\delta_{T} = \frac{F}{EI}\cdot \left(L\frac{x^2}{2}-\frac{x^3}{6}\right)+\frac{F}{\kappa AG}\cdot x$$
• the difference: $$\delta_{T}-\delta_{EB} = \frac{F}{\kappa AG}\cdot x$$

A concentrated moment and uniformly distributed moment are exactly the same as concentrated load and uniform load but the direction of the applied load. The former (moment) is a rotational force about the member's longitudinal axis, and the latter is a linear force acting on one of the transverse axes, or both. Note the rotational force is usually caused by an offset linear force or a couple about the centroid of the member cross-section. In the sketches below, F & w represent the concentrated and uniformly distributed linear forces; M & m represent the resulting rotational forces due to the offset of F & w respectively.

Note, the rotational force, M, is called "twisting force" and "torque/torsion" as well.

One example of continuous moment applied on a shaft is the torque applied by many threads on a fabric weaving machine shaft pulling the threads.

We can compare this to a simply supported beam with a uniformly distributed load.

The torque hundreds of threads impart on a unit length of the shaft is approximately the uniformly distributed moment. The rotation(angle of twist), $$\ \theta=max$$ at the center, and zero at supports like the moment in a beam.

Torsion stress is maximum at supports $$\ \tau= \frac{1}{2} T_{distributed}*L$$

And it is zero at center, similar to shear stress in the beam.

## Edit

A good industrial example of distributed moment on a rod is cold rolling sheet metal.

the friction on the rollers is close to uniform torque.

Typically, a continually varying moment is due to a distributed force (i.e. the moment is the integral of the shear force). A concentrated moment will cause a step change in this moment, while distributed forces will cause gradual changes in moment across the length of the distributed force. In practice I haven't seen an example of a "distributed moment" such as you've drawn. If it does exist, it's rare.