## 🟦 15.1 3D connections

Review the three main types of 2D connections and the types of reactions they can develop for static equilibrium.

When a connection prohibits translation, it develops a force reaction.

When a connection prohibits rotation, it develops a moment connection.

To review:

The 2D pin-roller connection prevents translation in one direction. It develops one force reaction, perpendicular to the surface.

The 2D pinned connection prevents translation in that plane. It develops a force at an angle of inclination (or, as more commonly expressed, x- and y- components).

The 2D fixed connection prevents translation in the plane and prevents rotation about the axis that is perpendicular to the plane. It therefore can develop two force reactions (x- and y- components) and a moment about z.

### ❏ 2D connections and reactions

In 3D, we look at a connection assembly and go through the following logic tests:

Does the connection assembly prevent translation in a given direction? If so, it has the potential to develop a reaction force in that direction.

Does the connection assembly prevent rotation about a given axis? If so, it has the potential to develop a reaction moment about that axis.

This image illustrates three common connections drawn in 3D.

The first is a roller that develops Fy.

The second can be called a 3D pin connection or a ball-and-socket joint. Translation in x, y, and z is constrained, but the member is not constrained from rotation. Therefore three force reactions can develop.

The third is a 3D fixed connection. They are powerful: you can support the weight of any body with one 3D fixed connection! They develop all 6 potential reaction forces and reaction moments.

### ❏ 3D connections and reactions

### ❏ a 3D connection in Amsterdam

### ❏ a 3D connection in Rome

### ❏ 3D connections at Heathrow

## 🟦 15.2 Axial (or "two-force") members in 3D

In 2D, we learned that the geometry of an axial member can help us solve for unknown forces or force components. This is because the angles (alpha and beta) in the geometry diagram are congruent to the angles in the component force diagram.

The same logic holds true for 3D.

An axial member in 3D space has the same ratios between the lengths and force components.

### Example problem 1:

Cable AB carries a tension force of 100 kN. Solve Tx, Ty, and Tz. You know the geometry: Lx = 2m; Ly = 3m; and Lz = 6m.

Solution:

LAB = sqrt(2^2 + 3^2 + 6^2) = 7m

Fx = (2/7)*100 kN

Fy = (3/7)*100 kN

Fz = (6/7)*100 kN

### Example problem 2:

You deduce that the z-direction force in cable AB is 100 pounds. You also know the geometry: Lx = 2 feet; Ly = 3 feet; and Lz = 6 feet. What is the tension in cable AB?

Solution:

LAB = sqrt(2^2 + 3^2 + 6^2) = 7m

T = 100#*(7 feet / 6 feet) = 117#

## 🟦 15.3 Concurrent force problems in 3D

This flipbook solves a 3D truss. All forces are concurrent at D.

When assessing nodal equilibrium in a 3D concurrent force problem, you may find it helpful to draw the point as a cube instead of a circle or sphere. A depiction of this style of FBD is shown below. Remember to use the geometry given to break the forces into their x-, y-, and z-components.

## 🟦 15.4 Computing the center of weight of mass in 3D

This sculpture is comprised of three identical cuboids.

Each measures 1" x 1" x 10" and weighs three pounds.

We want to determine the center of weight.

Set up an equivalent system with a 9# vector in space.

Then, we can solve for x bar, y bar, and z bar (from the origin) by summing moments.

Summer 2024: I'm out of time to build out a solution for this concept, so please take a stab at this one on your own, and we will regroup in class. The procedure is nearly identical to the one we learned for centroids, except that in 3D you have to do 3 moment summations. Here's the first one to get you started.

System I = the three weights

System II = the singular 9# weight

The sum of the moments about the y-axis in System I equals the sum of the moments about the y-axis in System II (a static equivalency equation).

3#(5" + 9.5" + 9.5") = 9# * xbar

## 🟦 15.5 Summing moments about an axis

Summer 2024: This is what we did in class today. Pick an axis (AB, AC, BC, CD, whatever), define a positive direction, and sum moments about it to solve unknowns.

## 🟦 15.6 How to project moment to a given axis

Summer 2024: We can talk about this in class as well. Basically, you can dot a moment vector with a unit vector to project it on an axis of choice.

## ➜ Practice Problems

Problems 1 and 2:

Problems 2.7 and 2.8 at: http://mechanicsmap.psu.edu/homework_problems/Chapter2_Problems.pdf. These are 3D concurrent force problems.

Problems 3 and 4:

Problems 3.14 and 3.15 at http://mechanicsmap.psu.edu/homework_problems/Chapter3_Problems.pdf.

Problem 5:

A box (W = 30 kips) is hung by three cables from three pin supports. The three supports are all at the same elevation and form an equilateral triangle (12' by 12' by 12'). Node D lies 10 feet below the ceiling.

How much tension force is carried through each of the three cables? Here is an interactive model that may help you think through the geometry.

That's it for Summer 2024!

not for Summer 2024.

3D fluids problem - opening swinging gates and such. animations with environmentally-friendly designs for fish.