Statics E14/Mitiguy Spring 2009, Lab 2 Page 1/4 Name: ___________________

Note: Please read this handout before your lab. Turn in your pre-lab at the start of the lab section.

This lab explores two concepts in Engineering Statics:

1) Statically Indeterminate Systems – Difficulties encountered when solving for reaction forces. 2) Modeling Assumptions – The effect of modeling assumptions on our ability to solve for reaction

forces. The advantage of including flexibility into a model, an introduction to stiffness and a motivation of Mechanics of Materials [ME80].

In this lab, you will work with spring scales, rubber bands, and a small weight.

Statics E14/Mitiguy Spring 2009, Lab 2 Page 2/4 Pre-lab: Optional Reading: Determinate, Indeterminate, and Undersconstrained Systems – Section 8.4. Pre-Lab Questions: The following shows a Free Body Diagram [FBD] of a rigid uniform-density horizontal beam (S) supported by three evenly-spaced vertical cables. The purpose of this exercise is to find the tension in each cable. Using your physical intuition, guess the magnitude of the force in each cable (no analysis, just guess!) Result: Fa = _______ lbf Fb = _______ lbf Fc = _______ lbf Find FS, the resultant of all contact and distance forces on the beam (S), and set FS = 0. Result: FS = ___ nx + ______________________________________ ny = 0 Form the dot product of FS = 0 with nx, and ny. Decide if each equation is useful (circle one). Result:

FS . nx = _____ = 0 FS . ny = ____________________________ = 0 Useful / Not Useful Useful / Not Useful

Find the moment MS/A of all forces on the beam (S) about point A (left end of S), and set MS/A = 0. Next, form the dot product of MS/A = 0 with nz and decide if this equation is useful (circle one). Result: MS/A = ___________________________________________ nz = 0 MS/A . nz = __________________________________________ = 0 Useful / Not Useful There are three scalar unknowns (Fa, Fb Fc) and only two independent scalar linear equations. Using only this information, is there a unique solution for Fa, Fb, Fc? Yes / No One way to solve for Fa, Fb, and Fc is to guess Fb and solve for Fa and Fc. Use the following guesses for Fb and solve for the corresponding values of Fa and Fc:

Guess for Fb Solution for Fa and Fc (lbf) Fb = 3.3 lbf Fa = Fc = Fb = 0 lbf Fa = Fc = Fb = Your Favorite Number: _____ lbf Fa = Fc =

Do any of the above solutions agree with your previously guessed physical intuition? Yes / No Does this method seem like an effective and rigorous method to find the forces? Yes / No

FA = Fa ny

nx ny FC = Fc ny FB = Fb ny

Fw = -10 lbf ny

Statics E14/Mitiguy Spring 2009, Lab 2 Page 3/4 In-Lab Procedure: Checking the accuracy of the spring scales. Take a weight and attach a rubber band to provide a single hanging point if there isn’t one already. Check to ensure each spring scale (i.e., SS1, SS2, SS3) reads zero when no force is applied. Hang your weight from SS1 and measure its weight. Repeat for SS2, SS3. Record the maximum difference in readings between your spring scales in the table below> Result:

SS1 SS2 SS3 Max. Difference Force: (N)

Part 1: Flexible 1-D redundant support of a weight. Attach two rubber bands to the weight. Hook one of the rubber bands to SS1 and the other to SS2 as shown to the right. 1.1) Ensure the rubber bands are the same length. Hold the tops of SS1 and SS2 at the same height, and record the spring forces in the table below. 1.2) Lengthen the rubber band attached to SS1. Repeat step 1 above. 1.3) Further lengthen the rubber band attached to SS1 and repeat as above. Results:

SS1 SS2 SS1 + SS2 1.1) 1.2) 1.3)

Lengthening the rubber band attached to SS1 Increases / Decreases the force in SS1. The sum of SS1 and SS2 should remain constant True / False Part 2: Inextensible 1-D redundant support of a weight. Replace the rubber bands with a string with two loops. Hook one loop to SS1 and the other to SS2 as shown to the right. 2.1) Ensure the length of string connecting SS1 to the weight is equal to the length of string connecting SS2 to the weight. Hold the tops of SS1 and SS2 at the same height, and record the spring forces in the table below. 2.2) Lengthen the string attached to SS1. Repeat step 1 above. 2.3) Further lengthen the string attached to SS1 and repeat as above. Results:

SS1 SS2 SS1 + SS2 2.1) 2.2) 2.3)

Lengthening the string attached to SS1 Increases / Decreases the force in SS1. The sum of SS1 and SS2 should remain constant True / False

Statics E14/Mitiguy Spring 2009, Lab 2 Page 4/4 Part 3: Flexible 2-D support of a weight. On a piece of white paper, draw a vertical line. Next, draw a line to the left that intersects the vertical line at point P and creates an angle θleft less than 60°. Draw a second line to the right of the vertical line that also intersects the vertical line at point P and creates an angle θright (also less than 60°). Using a protractor, measure θleft and θright. Tape your paper to the wall. Result:

θleft = _____° θright = _____° Align SS1 and SS2 to support the weight as shown to the right (use rubber bands). Record the force measured in SS1 and SS2. Result:

SS1 = _____ N SS2 = _____N The solution for the force in each spring scale is unique. True/False This system is statically determinate. True/False. Post-Lab Use vectors and FS = 0 (i.e, statics for the weight) to solve for SS1 and SS2 (please show and attach work). Find the % difference with your measured values. Result: SS1 = _________ N SS2 = __________ N Difference SS1 = _________ % Difference SS2 = __________ % Part 4: 2-D redundant support of a weight. Attach SS3 to the weight and align it to the vertical line as shown to the right. 4.1) Record the forces in SS1, SS2, and SS3 4.2) Raise SS3 and record the forces in SS1, SS2, and SS3. 4.3) Lower SS3 and record the forces in SS1, SS2, and SS3. Results:

SS1 (N) SS2 (N) SS3 (N) 4.1) 4.2) 4.3)

Post-Lab Find FS, the resultant of all forces on the weight, and set FS = 0. (attach work!). Form the dot products of FS = 0 with nx, and ny. Write these equations in the following matrix form. Results:

There are __________ scalar equations and __________ scalar unknowns. This set of equations represents a statically determinate / indeterminate system. Bonus: Use Least-squares to solve the previous system of equations. Result:

SS1 = _____ N SS2 = _____N SS3 = _____N