Moments and Moment Equilibrium

Moments

In the Forces section, the equilibrium of forces was discussed. If forces are equal and opposite, then the net effect will be zero and no acceleration will occur. However, there are instances when this doesn’t always hold true.

Imagine a rig bar, with a pivot at the middle. If masses are placed on the bar equally spaced each side of the pivot and with equal mass, then the pivot will remain in balance. The force equilibrium is shown in the diagram below:

Figure 1: Equal masses at equal spacing, no rotation and forces in equilibrium.

Now, if we move one of the masses to be twice as far from the pivot as it started, we can still draw the force equilibrium diagram (and the vertical reaction force will be the same), but we all know that the bar will start to rotate.

Unequal masses, rotation occurs
Figure 2: Masses unequally spaced, leading to rotation of the bar, however as both masses are the same the vertical forces are still in equilibrium.

There is clearly something additional going on in the system other than simply the equilibrium of vertical forces, clearly the position of forces from the pivot has a bearing. This introduces us to the concept of “Moments”.

A moment is very simply defined as:

M = Fz

Where F is the applied force, z is the distance to the point of rotation, measured at right angles to the direction of force and M is the moment that results A moment usually applies along an axis, with the axis of the moment being perpendicular to both the direction of the force and also to the orientation of the lever arm

Moments help to explain a number of things we observe in everyday life. If you carry a heavy object, you will know that it is easier to carry close to you than an arm’s length. The force (F) of the load is the same, but when the object is further away from you the lever arm (z) is greater and therefore your body has to resist a greater moment (M).
In the metric system, the typical units for Force is the newton, N, the typical units for lever arm are metres, m, this gives the units of a moment equal to Nm. 1Nm is equal to 1 newton, placed at a lever arm of 1m. In Structural Engineering practise it is more usual to use kNm, 1kNm = 1000Nm or 1000 newtons placed at a lever arm of 1m.
Bending moment sign conventions
Figure 3: Sign convention used to calculate moments

By breaking down the two previous examples we can see how the moments applied change:

Bending moment comparisons
Figure 4: Comparison of moments applied in Figure 1 and Figure 2.

Moments can also be used to understand the action of levers, such as a crowbar. By positioning a pivot close to a load and applying a small load to the lever far away from the pivot a much greater load will result at the other end

Archimedes once said, “Give me a fulcrum (pivot) and lever long enough, I can move the world”. By understanding the principles of moments it is possible to understand his meaning, however one thing he forgot to mention was that his lever and pivot would have to be pretty strong!
Bending moments and levers
Figure 5: Using moments it is possibe to see how a small force with a long lever arm can produce a big force.

If a force is out of balance, it gives rise to an acceleration. If a moment is out of balance then it gives rise to an angular acceleration, i.e. it rotates. For a structural system to be in complete balance, both the forces and moments must be in balance.

Again, Structural and Civil Engineers aim for a system to be in equilibrium, with no out of balance forces or moments. Just as you don’t want a building or structure to accelerate away (forces), you also don’t want it to fall over (moments).

Mechanical Engineers in particular often desire moments to be out of balance, using the out of balance moment to give rise to a rotation, or vice versa

You may have heard the phrase “Torque”, particularly with relation to cars. A “Torque” is a moment that usually applies around the axis of a shaft or wheel

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