This simple pair of K’NEX models can demonstrate how trusses greatly increase the strength/stiffness of structures without adding much weight. The trusses are built out of K’NEX pieces and topped with construction paper to produce a roadway. Text books are used to load the trusses to demonstrate its strength to weight ratio. This demonstration should take 15 minutes.

The basic equations required for analysis are the equations of equilibrium and trigonometry. However, no real numerical analysis is required beyond measuring the weight. The model is used primarily for developing a feel for the importance of trusses as a structure, especially when considering the structure weight to load carried ratio, and that the weight of the truss can be initially ignored during the design (add weight of truss members back in during the final design checks).

Item | Quantity | Description/Clarification |
---|---|---|

K’NEX Pieces | Lots! | These pieces are connected into a number of triangle shapes to form 2D and then 3D trusses with supports. The connection of the 2D trusses forms a rudimentary “road” for the bridge. Also, a so-called “road” without trusses is made to demonstrate how flexible a bridge can be without trusses (long members overly dramatized for effect). All of the K’NEX pieces shown are part to the Big Ball Factory set. Individual pieces can be purchased to build your own kit. For the bridges in Figures 1, 2, and 3, you will need: Bridge without trusses – 7 yellow (3.25 inches) straight pieces, 12 blue (2.125 inches) straight pieces, 28 purple connectors (2 each forming a joint); Bridge with trusses – 17 yellow (3.25 inches) straight pieces, 30 blue (2.125 inches) straight pieces, 2 red (5.125 inches) straight pieces, 4 white (1.25 inches) straight pieces, 20 purple connectors (2 each forming a joint), 4 green connectors, 14 yellow connectors (2 each forming 4 supports), 2 red connectors. |

Load (textbooks) | Any | Test how many textbooks is safe for the K’NEX truss model so as to not actually break the connections, but make sure the number of books is effective in making a point about the strength of the bridge. |

Construction Paper | 2 | Two pieces of construction paper make the roads for the two bridges. |

Tape | 1 | Tape is used to attach the paper to the two bridges. |

Supports | 2 | Supports are needed to hold the ends of the bridges off of the desk while they are being loaded. Anything can be used for this: more textbooks for example. |

Scale | 1 | A scale is used to weigh the truss and the load it is able to carry successfully. No scale is really required. A student can provide a qualitative answer by placing the truss in one hand and the load carried in the other hand. |

**Before Class:** Construct the bridges out of K’NEX and load test the truss bridge to make sure you don’t overload it and break it during class (normally only connections come apart, but occasionally, the textbooks can cause a connector to occasionally break a small piece off), especially if teaching multiple sections of the same lesson.

**In Class:** Show the class the bridge without trusses. Place it in the hands of a student and ask how much it weighs (qualitatively). Place it on the supports and place a light object on it and show how it begins to deflect under the load. Ask the student to compare the weight of the structure and the load placed on it (qualitatively). Show the students that it is bending and ask why it isn’t very strong. Generally long slender members acting as simply supported members are not efficient in supporting loads over long spans. How can we make the structure stronger – greater depth to the members (Draw on their experience of building a tree house using 2×4 members. The member is stronger when loaded perpendicular to the greatest depth – use a ruler to demonstrate)? How can we get greater depth without too much more weight? If and when you can elicit a statement about trusses from the class, bring out the other bridge.

Ask the same student how much it weighs compared to the first non-truss bridge (not much more…). Set the bridge with trusses on the supports and place the same small object on it. Lead the class to see how much more load it can support. Begin to load textbooks on the bridge (can use some of the students’ texts).

Weight the bridge, then the textbooks. There should be quite a significant difference in ratio between the weights. A light truss structure can support a load that is many times greater than the weight of the actual structural members over long spans.

**Additional Application: **Before loading the truss bridge, ask a student how much it weighs by placing the bridge into his/her hand. Then ask how much load the bridge might be able to support? Once the students see how much stronger the bridge with trusses is as compared to the one without trusses, call one of the students up to the front of the class. Give the student the two bridges to hold and ask if there is much significant difference in weight. Then take away the bridge without trusses and begin to stack the textbooks that had been loaded on the second bridge onto the student’s hand until all are there of he/she cannot hold any more (drama and fun!). Then bring out a scale to show numerically how different the weight of the bridge is when compared to the weight of the load.

More elaborate truss structures/supports that represent real bridges can be developed for this same demonstration. This bridge is an approximate example of the Falls Bridge over the Schuylkill River Philadelphia, PA (Figure 7, left).

Building a 2D K’NEX section (see Figure 7, right) of a 3D K’NEX truss allows for a discussion of actual analysis of 3D structures. The 3D bridge structure is mentally taken apart into two 2D sections with one-half of the deck load (for this case) placed on each half. The engineer then draws a Free Body diagram (FBD) of the 2D section and completes the analysis to determine the loads in each member. This K’NEX model matches the planned in class truss analysis problem using both the Method of Joints and Method of Sections.

A discussion can ensue concerning other portions/assumptions of the definition of a truss: a structure composed of slender members joined together at end points by frictionless pins; loads only applied at joints; all truss members are 2-force members; and member weight is negligible. Slender members is obvious, but showing connections using K’NEX pieces leads to discussion/review of a previous topic – concurrent force systems and frictionless pins of years gone by. The fact that truss members are 2-force members will be shown during analysis using Method of Joints and Sections. The load (books) not totally going through the joints leads to a possible discussion that truss members are not always only 2-force members (if the class has been prompted to know the difference between the current theory and what real structures experience and adjustments to analysis at later times). Usually, the deck will carry the load to the joints. Member weight is generally negligible – readily apparent by the initial demonstration.

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This is a demonstration of the basic principles underlying the behavior of rotating bodies. A cylinder “race” is used to show that the closer the mass of an object is concentrated to an axis of rotation, the faster it will spin because it has a lower moment of inertia, which is a measure of a body’s resistance to rotation. The video below provides a brief synopsis of the demonstration.

The mass moment of inertia (pg. 1296 of the McGraw-Hill Vector Mechanics for Engineers—Dynamics text) is a rigid body’s resistance to rotation and is a measure of the distribution of mass of a rigid body relative to a given axis of rotation. In its most general form, the mass moment of inertia is given by:

where:

“I” is the mass moment of inertia

“dm” is a differential element of mass of the rigid body

“r” is the perpendicular distance from the axis of rotation to a differential element of mass

“B” represents the rigid body

For a solid cylinder and a hollow cylinder, the equations for the mass moment of inertia about the axis of interest in our demonstration reduce to those in figure 1 (http://hyperphysics.phy-astr.gsu.edu). “M” represents the mass of the rigid body and “R” represents the radius of the solid cylinder, and “a” and “b” represent the inner and outer radii of the hollow cylinder.

These materials can be easily manufactured. Additionally they (or similar materials) can be obtained from businesses that specialize in building teaching aids, such as Arbor Scientific.

Item | Quantity | Description/Clarification |
---|---|---|

Inclined Plane | 1 | An inclined plane able to support both rolling cylinders simultaneously (Figure 2) |

“Steel Wheel” | 1 | A steel cylinder of approximately 3” length and 1” diameter with a wooden core which weighs approximately the same as “Rolling Timber” (Figure 3) |

“Rolling Timber” | 1 | A wooden cylinder of approximately 3” length and 1” diameter with a steel core which weighs approximately the same as “Steel Wheel” (Figure 3) |

12″ Ruler | 1 | Used to start the race |

*Before Class:* Obtain materials. Measure and calculate the basic parameters (mass, diameters). Practice demonstration.

*In Class:* First establish the scenario by introducing and describing “the players” in your best race announcer’s voice. Then, without any analysis, ask the students to guess which “player” will win the competition. Ensure you record this on the board. Build up to the start of the race and stop just short of letting them go. Ask students to consider if weight might be a factor affecting the outcome, then provided the class with the weight of each player and see if their guesses change (record on the board). Again, build up to the start and stop short to discuss the concept of mass being a resistance to translation and moment of inertia as resistance to rotation. Have students help you calculate the moment of inertia for each of the players (figure 4) and take a final tally of the bets. Finally, let the race happen!

*Observations:* The student should observe that the outcome is a factor of the moment of inertia and not the weight (given the overall dimensions and the weight of the players are approximately the same.

Hype up the event by dramatizing the race! Consider using racetrack videos and noises which can easily be found on the internet. Ask students to guess which will win once the scenario is set up and before the principles are discussed. Build suspense by working your way to the “gun shot” starting the race, then backing off the start to analyze another aspect. If you’re using Power Point, incrementally build your slides to build suspense!

Furthermore, cylinders of different materials and diameters might be considered (figure 5).

]]>Understanding the concept of particle equilibrium is critical for success in the mechanics curriculum. The equilibrium demonstrator described here helps students to grasp this concept as well and provide a means of visualizing vector components in two or three dimensions

The primary principle relevant in this example is Newton’s first law of motion, which states that if the vector sum of all forces acting on an object is zero then the object must either be at rest or moving with constant velocity. Since in this case we have specified that the object will be specifically at rest, we can look at the problem from the other direction and say that since the object is static, the vector sum of all the forces acting upon the object must be zero. This is shown using the following equation. Since we are talking specifically about particles in this case, moments acting on the object are ignored.

This equation can be further separated into its vector components as shown in the following equations.

Item | Quantity | Description/Clarification |
---|---|---|

Frame | 1 per group | The frame can be made out of any thin material such as cardboard, wood, plastic, or metal. My example is laser cut from clear acrylic using a template (DXF, SLDDRW). |

Rubber band | 3 per frame | Any rubber bands should work as long as they are not too long for the frame. |

S-hooks | 6 per frame | S-hooks attach the rubber bands to a weight and to the frame. |

Weight | 1 per frame | You can provide a weight or students can use their own objects, car keys work well. If students are using their own weights then a scale will be necessary to quantify the weight. |

Students are provided with the two pieces of the frame and sufficient materials to complete the activity. Worksheets can also be supplied to guide students through the activity. See sample 2D worksheet and sample 3D worksheet. These worksheets include blanks for students to fill out with the necessary values and a pre-defined calibration for converting rubber band elongation into Newtons of force which is specific to the rubber bands selected.

To complete the activity, students must first hang their weight from holes in the frame using the rubber bands. They they measure the start and end points of the rubber band to determine the orientation of each rubber band in the XYZ coordinate system. These vectors allow the student to predict the force in each rubber band using the equilibrium equations. This is considered the theoretical force.

Next the students measure the elongation of each rubber band convert that into force using the provided calibration. This is considered the experimental force.

Finally students can calculate a “performance metric” which provides a measure of how well their experimental and theoretical forces matched as shown on the worksheets linked above.

An example of a completed worksheet is shown here.

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Thrill your students with Tales of Torque using the versatile Tower of Torque! In it’s simplest form, the device allows students to predict and then measure the capacity of a simple bolt, modeled as a rod in torsion.

For cylindrical rods in torsion, the maximum shear stress experienced is seen at the outer surface, and for materials in the elastic range the shear stress is computed using the expression

where the shear stress is equal to the applied torque *T* times the distance from the neutral axis to the outer surface of the rod, *c*, divided by the polar moment of inertia, *J*. The students should observe that the bolt undergoes permanent deformations when the measured torque on the bolt is computed to cause a shear stress at or somewhat above the published yield stress in shear for the material out of which the bolt is made. This demonstration ignores the fact that the sample is under a combined load rather than a pure torsional load in this demonstration. Don’t bring that up, but be ready to discuss it.

Item | Quantity | Description/Clarification |
---|---|---|

A Tower of Torque | 1 | A steel base plate with a 1-inch-square steel “mast” projecting from the base. The top of the mast is capable of receiving a real or simulated bolt. Complete plans are here. |

Torque Wrench | 1 | The torque wrench should have a large dial, and the failure torque for the chosen sample should be roughly half the total range of the wrench. It only helps a little if the wrench has a maximum value indicator (a follower). |

Torsion Samples | Various | Samples consist of 1/2″ aluminum or other hex stock turned down to a quarter of an inch in the shank. The samples should be tested thoroughly and a proposed configuration , though certainly not the only one possible, can be seen on page two of Sample Detail Drawings. |

*Building the Demonstrator: *The demonstrator is actually fairly straightforward, and measurements/etc can be seen in the plans*. *The device basically consists of a steel baseplate normal to which a 1″ square steel mast is attached by welding. Near the peak of the mast, at a recommended distance of 30″ above the base, a square hole is milled to accept the torsion samples. Don’t worry too much about making the milled hole exactly square, as the sample will not make contact with the corners of the square since it is a hex. If you like, you can leave the milled hole closed at the back to facilitate keeping the samples positioned right.

*The Samples:* Details of the samples we use can be found at Sample Detail Drawings. The key property of the samples is that the shafts must be turned down to the point that they can be readily failed in torsion using something like half the total available capacity of the wrench you choose. Practice, practice, practice! Sample strength varies even within the same hex rod, so make sure that the sample will do what you want it to do when the money is down. We also mark the samples with a line along the shaft (use a Sharpie to do this easily) so that the permanent set in the samples due to yielding can be directly observed by students.

*Before Class:* Secure the base of the demonstrator to a desk or workbench somewhere in the classroom where it will be highly visible to all the students. To save time and raise curiosity, mount the sample in the demonstrator prior to the beginning of class. Check out the direction of the torque wrench to make sure that you rotate it the right way when the moment comes, as not all torque wrenches are bi-directional. Make sure to test the whole apparatus at least once prior to use. There’s nothing worse than demonstrating that theory has no bearing on reality…!

*In Class: * If you’d like, you can view a complete video of a run-through of the demonstration by clicking on the demonstrations at the top of this page. Basically, the procedure can be broken into the following steps:

- If time is plentiful, ask the students what they think about the demonstrator, and see if they can guess what the demonstration is.
- Have the students compute the minimum applied torque required to cause yielding of the sample. Dig into the fact that the measured torque will likely be higher than the minimum computed value and why. Consider discussing 95% confidence limits if time is available.
- Slowly apply torque to the sample, having a student call out values as they increase. Try to observe the onset of yielding, and show the students the deformed line on the bolt indicating permanent (non-elastic) deformation.
- After observing yielding, continue to twist the bolt all the way until failure.
- Show/pass around the failed part, and discuss what they just saw.

Be careful not to try to measure the angle of twist using the torque wrench handle, as the angle through which the handle moves is larger than the that of the bolt (there’s a spring in the wrench for measuring the torque!). Further, holding the point of contact of the bolt and wrench steady with your hand helps lower the effects of the combined load on the sample.

*Observations*: Students should observe that the basic equation for torsion allows for a good estimate of the actual torque necessary to fail a bolt with a wrench. They should also observe that permanent deformation can take place in thin parts through the application of very little torque.

More fun and drama can be added to this class by adding the “Helicopter Mechanics” scenario. Basically, you describe a scenario in which a helicopter maintenance person feels it is necessary to get an aluminum bolt “good and tight”. Then, while turning the sample with the wrench at or over the predicted failure torque, say to the class “It must really be getting tight and strong now !”. They will realize that tightening the bolt further will really just weaken it, causing the fiery death of the very pilot the mechanics was trying to protect. Lessons on that little extra in this case includes ‘Knowledge is power’.

]]>This is a simple demonstration of the basic principals underlying the behavior of thin-walled pressure vessels (TWPVs). A balloon or balloons are used to show that hoop stresses are twice as high as longitudinal stresses in cylindrical pressure vessels.

The basic derivation for TWPV behavior states that for a cylindrical pressure vessel

where the hoop stress is equal to the pressure times the inner radius divided by the wall thickness. The longitudinal stress is exactly half the hoop value. Further, according to Hooke’s Law, the strain will follow the stress, so observed deformations are indicators of the magnitude of stress.

Item | Quantity | Description/Clarification |
---|---|---|

Party Balloons | Lots! | Both round and cylindrical. Spheres should be 10″ minimum diameter when inflated, and the cylindrical balloons should be about 5 times as long as they are around. Choose really big balloons for big lecture halls. |

Inflation Device | 1 | We use a Super Soaker filled with air rather than water. Either way, the device needs to be able to fill the balloon rapidly to avoid “dead air”. |

Marker/Pen | 1 | You’ll have to experiment with this somewhat to get a clear, dark mark without having the carrier of the ink weaken the balloon. |

*Before Class:* Prepare the balloons by drawing a square stress block midway down the length to negate the effects of the ends. Be careful drawing the square; ensure the sides are equal with 90^{o} corners and the lines are very bold. If the lines are not dark enough, they will be very hard to see after the balloon is inflated because they fade when they stretch. Regular ball-point ink pens work well for drawing the stress block. Definitely test this out a few times before going into the classroom to make sure the marks and the inflation device won’t fail you when it counts.

*In Class: * We typically have fun with the demonstration and play up the inherent party nature of balloons. Several pre-marked, deflated balloons of each type are displayed at the start of class. We show the students the square stress blocks on the deflated balloon and ask the students what shape the square will be after the balloon is inflated for each balloon type. Will the sides be the same length? The corners still 90^{o}? The balloons are then inflated and the behavior is observed.

*Observations*: The students should observe that on a spherical balloon, the stress block remains square, with 90^{o} corners. This demonstrates that the magnitude of the stress in both the longitudinal (along the length of the balloon) direction and the hoop (around the circumference) direction are the same. On the cylindrical balloon, the students should observe that the initially square stress block has deformed into a rectangle, with the longitudinal deformation being half as great as in the hoop direction. This demonstrates that the magnitude of the hoop stress is twice that of the longitudinal stress. For both balloons, the persistence of 90^{o} corners demonstrates that there is no shear stress acting on the stress block. The lack of any shear deformation shows that the stress blocks are oriented along the principal stress directions in both cases.

*In Class: * We typically have fun with the demonstration and play up the inherent party nature of balloons. Several pre-marked, deflated balloons of each type are displayed at the start of class. We show the students the square stress blocks on the deflated balloon and ask the students what shape the square will be after the balloon is inflated for each balloon type. Will the sides be the same length? The corners still 90^{o}? The balloons are then inflated and the behavior is observed.

*Observations*: The students should observe that on a spherical balloon, the stress block remains square, with 90^{o} corners. This demonstrates that the magnitude of the stress in both the longitudinal (along the length of the balloon) direction and the hoop (around the circumference) direction are the same. On the cylindrical balloon, the students should observe that the initially square stress block has deformed into a rectangle, with the longitudinal deformation being half as great as in the hoop direction. This demonstrates that the magnitude of the hoop stress is twice that of the longitudinal stress. For both balloons, the persistence of 90^{o} corners demonstrates that there is no shear stress acting on the stress block. The lack of any shear deformation shows that the stress blocks are oriented along the principal stress directions in both cases.

We sometimes ask student volunteers to blow up the balloons, having purchased balloons that were difficult to inflate. The students have always succeeded in blowing up the balloons, although usually only after multiple attempts. Do not choose students who smoke or have lung ailments for this part of the demonstration, as they may injure themselves trying to inflate the balloons. Next, we ask the students if they’d like to see us blow up the balloon. The students are usually quite eager to see what shade of red the instructor’s face will turn during the attempt, but are surprised when the instructor produces a hidden Super Soaker water gun fully charged with air and easily inflates the balloon. Two embedded lessons within this demonstration are the spherical pressure vessels on the Super Soaker (why did they choose spheres?) and the fact that pre-yielding the latex balloons by stretching significantly weakens them and allows for easy inflation.

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