Statics

Lesson 13

3D Statics, Part I

🟦 13.1 The wonderful world of 3D!

🟦 13.2 Condensing a 3D problem to 2D

🟦 13.3 Roadmap of the Statics Toolbox for 3D problems

🟦 13.4 Vector symbols and notation

🟦 13.5 Review: points vs. vectors

🟦 13.6 Review: position vectors

🟦 13.7 Review: magnitude of a vector

🟦 13.8 Review: unit vectors

🟦 13.9 The bounding cuboid for 3D vectors

🟦 13.10 Review: how to compute the dot product of two vectors

🟦 13.11 Review: how to project vectors with the dot product

🟦 13.12 Review: how to compute the cross-product of two vectors

➜ Practice Problems

🟦 13.1 The wonderful world of 3D!

While all Statics problems represent the reality of the 3D world, many can be fully analyzed in 2D.

Some Statics problems require a 3D approach, though, and our approach depends on the complexity of the problem statement.

If the problem is relatively simple, we may decide to conduct three planar two-dimensional analyses (the xy plane, xz plane, and yz plane).

❏ A simple 3D problem

For intermediate problems, we may decide to use vector notation.

It's for that reason that this lesson is principally a review of 3D vector operations.

❏ An intermediate 3D problem

When a problem is extremely complex, you would use software (finite element analysis) for the analysis. It would be too time-consuming to do these types of calculations by hand.

We would use finite element analysis, an approximate method that falls under the umbrella of so-called numerical methods.

These types of problems are not in the scope of Statics.

❏ A complex 3D problem

Image source: S. Reynolds, photograph, playground equipment, Amsterdam, Netherlands (designer unknown)

🟦 13.2 Condensing a 3D problem to 2D

A three-dimensional structure that is symmetric (with respect to its geometry and loading) can be condensed to a 2D problem.

For instance, let's say that you are trying to tip a sibling's chair by exerting force at B and C on the chair with your two hands.

This is depicted as System I.

But does the chair tip or remain in static equilibrium?

❏ Some 3D problems can be modeled in 2D

You recall principles of static equivalency and put the forces at D instead (System II). This is not fully equivalent, but it is statically equivalent.

Now that the loads lie in a single xy plane, we can project the geometry with respect to x. This creates equivalent system III. It's a 2D planar statics problem; you can use scalar approaches to solve it (sum moments about E to determine whether or not the chair tips).

The key takeaway is that you don't need to use the power of vector notation on every 3D statics problem. Use 2D scalar notation whenever you can.

🟦 13.3 Roadmap of the Statics Toolbox for 3D problems

The infographic below provides an overview for our transition from 2D problem-solving to 3D problem-solving.

The concepts are the same. The tools are a bit different. We'll go through this material in detail one concept at a time; this is just a quick-reference.

Solution of six unknowns

In 3D, we can use the six E.o.E. to solve up to six unknowns. If you're familiar with using linear algebra (matrix methods) to solve a system of linear equations, you're welcome to do that. The problems introduced in this text may all be solved using simple algebraic methods (substitution and elimination). If you'd like to learn how to solve a system of linear equations using a matrix approach, you are welcome to do so: check out this link.

🟦 13.4 Vector symbols and notation

In our 2D problems, we had the luxury of being a little lax in our vector notation. For instance, we generally didn't go to the trouble of putting a little arrow on top of our force vectors. This is OK, because everyone who studies Statics knows that force is a vector.

In 3D problems, we don't want to mix up vectors and magnitudes, so it's a good idea to be a little more formal with our vector symbols and notation. For example, it's helpful to put little arrows on top of symbols for vectors when we are writing by hand. When typing (e.g. this website), we make the vector boldface instead of drawing the arrow on top.

When we want to write the magnitude of a vector, we use single or double vertical bars (either is OK):

| F | = ||F|| = the magnitude of the vector F

Types of vectors:

we use i, j, and k for unit vectors in the x-, y-, and z-directions
we use u for other unit vectors
we use r for position (with 2 subscripts)
we use F for force (with 2 subscripts)
we use M for moment (with 2 subscripts)

How to use subscripts:

Subscripts can be used with several types of vectors, but the ones that need 2 subscripts are r and u. Let's take the position vector rAB for example. It can be read as the change in position "from A to B." Note that rAB is equal to (-1)rBA. You reverse the signs of all three components of rAB if you want to write rBA in your calculation.

Are units important?

They are very important! It's best to put units at the end of the vector, like this: rAB = <5, 3, 1> m.

It is also acceptable to do this: rAB = <5m, 3m, 1m>.

Aside from unit vectors, all other vectors require units.

🟦 13.5 Review: points vs. vectors

We use parentheses to designate points. For example, A: (1, 2, 3) meters means that Point A lies at coordinates x=1m, y=2m, and z=3m.

We use chevrons for vector components. If Vector B is a force, and it's designated as <4, 5, 6> kips, it means that the force components are 4k in the x-direction, 5k in the y-direction, and 6k in the z-direction.

In order to fully communicate everything there is to know about a force vector, you will need to specify both the point of application (using coordinates in parentheses) as well as the vector itself (using components in chevrons).

🟦 13.6 Review: position vectors

Position vectors are used to measure the distance between two points. We typically use the symbol r for this idea. The r stands for radius, because if you can create a sphere using the coordinates of the first point and the position vector that defines the second point.

Author's Note: This flipbook is in progress and will be updated soon.

In this image, rectangles C, D, and E all measure 3 feet by 5 feet.

The position vector from A to B is <-5,-3,-3> feet.

In other words, to get from A to B, you have to move in the negative x-direction, negative y-direction, and negative z-direction.

Practice the position vector from G to A on your own. The answer is <5,-2,-3> feet.

Also practice the position vector from B to G. This one is <0,5,6> feet.

🟦 13.7 Review: magnitude of a vector

Consider the force vector <3,4,5> Newtons.

It's depicted as head-to-tail addition in the interactive model.

It shows that a vector in 3D space can always be visualized in terms of two right triangles.

The first triangle lies in the xy plane.

You see Fx=3N and Fy=4N, but not Fz (as it's coming directly towards you. You can calculate the hypotenuse of this triangle using the Pythagorean Theorem. We can call the hypotenuse Fxy. It's equal to 5N.

The second right triangle has a base of Fxy=5N and a height of Fz=5N.

Its hypotenuse is equal in length to the magnitude of the original vector. That's sqrt(5^2 + 5^2) = 7.071 N. Again, this is a magnitude.

We would need to multiply that magnitude by the proper unit vector to express it as the original <3,4,5> Newton force vector.

❏ Interactive: the two-triangle approach

There is a 3D version of the Pythagorean Theorem shown here. The sum of the squares of the three components is equal to the square of the resultant vector.

❏ Key equations

🟦 13.8 Review: unit vectors

Unit vectors have a length of one (or unity). They are used to communicate the direction of a vector. They do not have units.

When we want to use a unit vector in the x-direction, y-direction, or z-direction, we use i, j, and k.

These are pronounced as "i-hat," "j-hat," and "k-hat."

When typed, they are bold and italicized.

When handwritten, they wear pointed hats.

Sometimes we want a unit vector that defines some other direction in space. Use u for this type of unit vector. Also, use two subscripts to indicate the tail-to-head direction of the unit vector.

Important: note that the order of the subscripts matter.

Example problem 1:

Points A and B lie at an inclination of 28 degrees, as shown.

You need to write the unit vector that defines the inclination of the line of action that is parallel to line AB. You want to go from A to B.

If an angle is given, the components of the unit vector can be expressed in terms of sine and cosine.

Remember that you can use the unit circle and Pythagorean Theorem to check your work. (cos^2 + sin^2 = 1^2).

Example problem 2:

Points A and B lie 5m apart, as shown.

As before, you want to the unit vector that defines the line of action from A to B.

Since we know the dimensions, use ratios. You shouldn't calculate the angle when ratios are provided.

In this example, you'd take sqrt(0.8^2 + 0.6^2) = 1 to verify that you have written a unit vector.

Example problem 3:

In 3D space, point A lies at <2, 0, 3> inches and point B lies at <0, 6, 0> inches.

We compute the length AB using the 3D Pythagorean Theorem.

Then, the unit vector is written by dividing the position vector by the magnitude of the length.

Remember: since all unit vectors equal unity by definition, you can always check your unit vector by using the 3D version of the Pythagorean Theorem. Make sure it has a length of 1.

🟦 13.9 The bounding cuboid for 3D vectors

This flipbook provides a review of:

position vectors
vector components and resultants
unit vectors

In 2D, we visualize a bounding box for vector components in a plane.

In 3D, we can do the same kind of thing. We simply need to use a bounding cuboid instead of a bounding box.

We use the bounding cuboid to visualize the three vector components (x-direction, y-direction, and z-direction).

🟦 13.10 Review: how to compute the dot product of two vectors

Skim this page to review the basics of the dot product. We use the dot product to determine how much of a given vector points in the direction of another vector. This is why when you dot two perpendicular vectors, you receive an answer of zero.

Key take-aways:

For dot-products, the end result is a scalar.
For dot-products, the order of the vectors does NOT matter. (Aٜ·B = B·A). That is, the dot product of two vectors is commutative.

Practice a few problems until you have the ability to compute dot products quickly, either by hand, or by programming your calculator to do it for you.

🟦 13.11 Review: how to project vectors with the dot product

Peruse this flipbook to see how we can use the dot product to project a vector component onto a line (or an axis).

Vector projections give us a way to determine the vector components that are parallel to the ray defined by the unit vector.

In this example problem, a component of force vector F is projected to the line (axis) OA, through the dot product operation.

Here is a model for how to write out these calculations. (It's the same problem.)

🟦 13.12 Review: how to compute the cross-product of two vectors

Skim this overview of the cross-product.

Most people are taught to perform the cross-product with either Method 1 or Method 2 in the image.

Stick with whatever method you were first taught. There's no need to learn the other method.

Key takeaways:

For cross-products, the end result is a vector. It's perpendicular to the plane created by the source vectors.
For cross-products, the order of the vectors is important. AxB ≠ BxA.
If you reverse the order of the operation, then you are reversing the sign of the answer.

Practice until you have the ability to compute cross products quickly. Ideally, use your calculator for this operation. If your calculator doesn't have this functionality pre-built in, program it in yourself.

Stay tuned: we will discover applications for the cross-product in Lesson 14.

➜ Practice Problems

Problem 0.

(a) Watch at least the first half of this video.

(b) Figure out how to do a cross-product with your calculator. Either FIND the program that does a cross-product, or WRITE a program that does a cross-product. Optional: do the same for the dot product.

Problem 1.

Work through Example 2.7.7. Dot Products on https://engineeringstatics.org/dot_products_2D.html. Be sure to complete all five parts (a through e).

Problem 2.

Work through Example 2.8.8. 3D Cross Product at https://engineeringstatics.org/cross-product-math.html.

Problem 3.

Write the unit vector that describes the line of action from A to B.

Rectangles C, D, and E all measure 2m by 4m.

C and D lie in the xy plane.

E is parallel to the xz plane.

Problem 4.

Vector B has been created by rotating vector A 30 degrees about z.

Write out the components of Vector B. Express your answers as fractions and radicals.

Bx = ?

By = ?

Bz = ?

Problem 5.

This is a continuation of the previous problem.

Vector C has been created by rotating Vector B (which lies on b-b) 45 degrees about line d-d (which is perpendicular to line b-b).

What are the components of Vector C? Express your answer in fractions and radicals.

Cx = ?

Cy = ?

Cz = ?

Problem 6.

This Vector C is the same one from the previous problem. What are the vector components in the x'y'z coordinate system? (The x'y' axes are rotated 30 degrees about z compared to the xy axes.)

Cx' =

Cy' =

Cz =

Problem 7.

A force vector (F) has components of 1 kN, 3 kN, and 2 kN, as shown.

Project F into the xy plane.

Fxy = ?

Verify that Fxy + Fz = F

Then, project F into the zy plane.

Fzy = ?

Verify that Fzy + Fx = F

Finally, project F into the xz plane.

Fxz = ?

Verify that Fxz + Fy = F

Problem 8.

This is a 3D FBD of Node E (or particle E or point E).

Your friend has already solved the problem and given you the following magnitudes for the forces. The units are kips.

The force in A is 32/(sqrt 3)
The force in the two forces labeled B is 16/(sqrt 3)
The force in C is 16

Your job is to draw all three 2D projections of the node:

an xy view (in which you do not see any z-direction vectors)
a zy view (in which you do not see any x-direction vectors)
a xz view (in which you do not see any y-direction vectors)

After that, use the equations of equilibrium to determine whether or not your friend's answers are correct. If each 2D projection is in static equilibrium, then the forces given to you must be correct.

This will feel like doing three successive concurrent force problems. Refer back to Lesson 03 if you need a refresher on how to solve concurrent force problems.