Statics

The equations of equilibrium (E.o.E.) continue to be a key concept in our course. 

A body is in a state of static equilibrium when all E.o.E. are satisfied.

In 3D, we have six equations of equilibrium to satisfy. A body is only in static equilibrium when all six equations are satisfied. 

With six equations, we can also solve up to six unknowns in a 3D problem.

While studying abroad in Rome, Italy, you look out the window one day and notice a steel beam cantilevering out from the exterior wall. The beam is "built-in" to the brick.

The beam supports a pulley, and someone seems to be hoisting a mysterious purple prism from the ground below.

You estimate that the purple prism weighs about 80 Newtons.

That means that tension in the cable is also 80 Newtons.

The two downward forces (the prism and the pull force from the person on the ground) sum to 160 N (neglecting friction in the pulley itself).

You remember how cantilever beams work from Lesson 8 (Beams). The support (in this case, the brick wall) provides a vertical reaction and a reaction moment.

You draw a side view of the structure in the xy plane. 

Technically speaking, we could say that you drew an xy elevation or that you projected the geometry with respect to z.

You compute the moment at the support as –160 N∙m (or 160 N∙m ↻).

Before, we simply called this MB. Now, we'll need a second subscript. We can write MB,z (the moment at B with respect to rotation about z).

Let's also draw a projection of the yz plane. This elevation helps us visualize the moment about x.

That works out to:

MB,x = 160N*90mm = –14.4 N∙m

In 3D space, a moment vector has three components. For this problem, the tendency to rotate about x is –14.4 N∙m; the tendency to rotate about y is zero; and the tendency to rotate about z is –160 N∙m.

We could therefore express the moment reaction vector as <–14.4, 0, –160> N∙m.

The curly arrow symbol for moment is useful in 2D projections, when the moment is about the axis that is perpendicular to the screen.

For other views -- and in 3D illustrations, it's best to use a double-arrow vector instead.

Moment has always been a vector, because it has both a magnitude and direction. 

The "direction" of a moment vector is just a bit more abstract than the direction of a force vector. It's a reference to the axis about which there is a tendency to rotate.

We use the right-hand rule to help us convert double-arrow notation to curly-arrow notation and vice versa.

Moments have x, y, and z components in the same way that forces have components.

For the moment components, please use the double-headed vector.

We can illustrate the x, y, and z components of a moment using the idea of the bounding cuboid.

This image depicts forces (single vectors) and moments in 3D space.

When the double-arrow points in the +x direction, we say it's a positive moment about x.

When the double-arrow points in the -y direction, we say it's a negative moment about y.

In a 3D problem, we say "moment at a point" when we want to calculate the three moment components (Mx, My, and Mz).

Here, we see the moment about node A.

Remember, in both 2D and 3D problems, we use the word "moment" for three different concepts:

(1)  An applied moment is a load or input (information that is given to you in the problem statement).

(2) A reacting moment is the reaction that is needed for static equilibrium (what you want to solve).

(3) A moment summation is an operation (i.e. "the sum of the moments about an axis equals zero"). It's the equation of equilibrium.

Let's say that you want to turn this crank by applying a force. 

Unfortunately, the crank (of undetermined purpose) is in a cramped space in your basement utility room. There are a bunch of HVAC ducts in the way. Instead of being able to push in the y-direction, your push force is at a weird angle, as shown.