Hi, and welcome to module thirteen of an Introduction to Engineering Mechanics.
Today, we're going to express the 2D and 3D static equilibrium equations. We're
going to recognize and apply the principle of transmissibility and we're going to
explain the relationships between sums of moments. So let's start off by looking at
the static equilibrium equations. In vector form for static equilibrium, we
have to have a balance of forces and we have to have a balance of moments about a
point p, where p is any point that we choose, and so, let's look at this first
of all in, in Dd. So we have to have a balance of forces in two orthogonal
directions in the plane, so I've got an x direction here and a y direction. So I, I
don't have any acceleration in the x direction, I don't have any acceleration
in the y direction, and so, that's a balance of forces that keeps it in static
equilibrium there. And I also don't have any angular acceleration about any point
and this point can be on or off the body. And so, it could be a point p down here,
at the origin or up here, but there is no angular acceleration either, and so,
that's for 2D in, in a plane. we have three independent equations in that case.
When we go to 3D we have six independent equations, because now, we not only have
to have a balance of forces in the x and y direction, but we also have to have a
balance of forces in a z direction. And, instead of a moment around the, the normal
to the plane of the moment, moment about the z-axis in this case, we also have to
have a balance of moments about the, the x-axis and about the y-axis for any point,
again, on the body or off the body. And so, those are, are our static equilibrium
equations. Let's next look at the principle of transmissibility. And so,
what the principle of transmissibility says is that for a body like this or if we
consider this a rigid body, if I have a force and I push on the body here, that
force, if it's the same magnitude, and I pull on the other side in the same line of
action, it, it externally, it doesn't matter. Okay?
That force can be transmitted along the line of absolute force and works anywhere.
Now, internally, if the body were to be able to deform, then it would make a
difference, but that's, that's for a future course in mechanics and materials.
In this course, we're only dealing with rigid bodies. let's next look at the
relationship between sums of moments. If a body is in static equilibrium, again, the
sum of the forces vectorially has to be equal to zero and the sum of the moments
about any point p has to be equal to zero. And I said that we could pick point p
maybe down here at the origin or another place, but if, if we have static
equilibrium, then the moments about any point, any arbitrary point a we choose on
or off the body has to also be equal zero. And so, here is an example. I've got a
have a cantilever beam here, and, if I have some of the forces equal to zero, and
sum of the moments about p equals zero, then the sum of the moments about point A,
or about point B as it's shown, or about point C, all have to be equal to zero. So
we may sum moments about any point to enforce static equilibrium and we'll do
that in example problems in future modules, so, what that leads to is, is for
the static equilibrium equations, in 2D. Again, we said, some of the force in x,
and the y, and moments about a point equals zero in fact we may even have two
orthogonal directions, which would be on a slope. So we may have a parallel,
direction the slope and a perpendicular direction to the slope and a, and a moment
again about any point, force and static equilibrium, or since there's three
independent equations, we may have the sum of the forces in just one direction, which
may be along an arbitrary line equal to zero and we could sum moments about two
points, maybe a point here and a point here. again, or we can sum moments about
three points and have three independent equations and, and we'll use all of these
cases in, in, in problems that we solve. But, just realize that the sum of the
forces, the sum of the moments have to be equal to zero vectorially and we can get
three independent equations in 2D. And so I look forward to working with you in
future modules to use these equations to solve some real world problems.
Dr. Wayne Whiteman, PESenior Academic Professional
