🟦 13.1 The wonderful world of 3D!
While all Statics problems represent the reality of 3D, many can be analyzed in 2D.
Any three-dimensional structure that is symmetric (in terms of both geometry and loading) can be condensed to a 2D problem.
When the geometry requires a 3D approach, such as the whimsical childrens' play structure shown in the photo, we have a few analysis options to pursue.
If the problem is relatively simple, we may decide to conduct three planar two-dimensional analyses (the xy plane, xz plane, and yz plane). For simple problems, this approach is recommended.
If the problem is more complex, we may apply the mathematical rigor of vector notation. We will learn to apply these fundamentals by hand.
Rest assured that when you are confronted with any degree of complexity (e.g. the playground in the photograph), you will not be doing these calculations by hand. That would waste time, and the likelihood of errors is high. Instead, you would use software (finite element analysis) in your calculations.
The problems we solve in this class do not require linear algebra or matrix solution methods. Instead, we will use simple algebra (substitution and elimination methods to solve systems of equations). If you'd like to learn more about solving a system of linear equations, check out this link.
A structure that merits a 3D approach
🟦 13.2 Common symbols for vectors in 3D analysis
uAB = a unit vector (meaning that it has a unit length) that defines the inclination of the line of action from Point A to Point B
rAB = a position vector that goes from Point A to Point B
F = a force vector (perhaps a load vector or a reaction vector)
M = a moment vector (yes, moment is a vector, that will be explained a little later)
🟦 13.2 Vector components and resultants (the bounding cuboid)
we will do this in class on 7/26/2023 and then I will paste the flipbook here after class (please remind me if I forget)
🟦 13.3 Project vectors by using the dot product
Peruse this flipbook to see how we can use a dot product to project a vector component onto a line (or an axis).
In this example problem, a component of force vector F is projected to the line (axis) OA, through the dot product operation.
🟦 12.x Using unit vectors (i, j, k) in vector notation
Put in a 2D planar problem, like problem 2 on the midterm exam, and express in 2D notation and in 3D notation.