Experimental Lab. Principles of Superposition

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Piers Williamson
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1 Experimental Lab Principles of Superposition Objective: The objective of this lab is to demonstrate and validate the principle of superposition using both an experimental lab and theory. For this lab you and your team members will be required to submit a joint lab report. Theoretical Background: The principle states: If the structural behavior is linearly elastic the forces acting on a structure may be separated or divided in any convenient fashion and the structure analyzed for the separate cases. The final results can then be obtained by adding algebraically the individual results. Many of the analytical techniques used in structural analysis are based on this principle. The following illustration shows how the principle of superposition can be applied to a cantilever beam. M o1 w M o w o C Δ c = R o1 + Δ c1 R o P M o2 Δ c2 From this illustration the following expressions are valid: R o2 P R R R o o1 o2 M M M o o1 o2 c c1 c2 x The magnitude of the vertical deflection at the point of interest will be affected by: a.) the magnitude of the load, P or w b.) the span length, L c.) the moment of inertia of the cross-section, I d.) the modulus of elasticity of the material, E The ability of a beam to resist bending is called the beam stiffness, which is characterized by the following equation: K EI Beam tables can be found in most structural analysis textbooks with numerous support and loading conditions, along with formulas for calculating reactions, shear, moment and deflections at any location along the beam.

2 Lab Instructions 1. Measure the dimensions of the beam: L, h, b e, b w, and h f and calculate I, the moment of inertia and EI, the beam stiffness. Record the dimensions on the appropriate charts. The beam material is Aluminum with a Young s modulus that equals 10,600 ksi. 2. Determine the loading of each axle of the semi-truck by individually placing each wheel set on the center of the scale. Measure the distances between each wheel set. 3. Adjust the dial gauge to be placed exactly at midspan (L/2). 4. Zero out the dial gauge (or record the current reading on the dial gauge). 5. Load the specimen according to Series A. Place the truck along the beam in order to produce the maximum deflection (Hint: Use the resultant force). 6. Record the measured deflection in the attached table. 7. Prepare and submit a joint lab report as instructed by the TA. Lab Report Write-up Requirements: Item 1. Major Report Sections: Cover Page, Table of Contents, Summary, Introduction, Procedure, Results and Discussion, Conclusions, Appendices. Item 2. Show all Measured Values and Required Calculations. This includes measured bar properties, measured deflection readings for each load test, calculated moment of inertia (I), calculated stiffness (EI), calculated theoretical deflections ( v T ) using both the resultant force method and superposition method, and calculated % error from theoretical. Item 3. Compare and discuss the concept of superposition from your experimental measurements. The measured vs. the theoretical and the theoretical (resultant) vs. theoretical (superposition). How do these compare and does the principle of superposition satisfy your measured values within a reasonable level of error. Item 4. Discuss possible variables in your experimental tests which could affect your measured error between experiment and theory. Item 5. Cite two examples or applications as a minimum where you could use the principle of superposition to make your life easier as a structural engineer (Hint one was done in your lab test.) Item 6. In the Appendix, sample hand calculations must be included. Lab Report is due in 2 lab periods after you perform the lab.

3 b w =0.25" H=0.25" h f =0.125" b e =5" Figure 1 Beam Cross-section Series A: Simply Supported F 3 F 2 F 1 d 3 d 2 d 1 L Figure 2 Series A: Test Set-Up Weight Station Setup: Begin by checking the force on each axle of the truck using a simulated highway system weigh station. The weigh station in this lab includes a platform and a scale installed in the middle. Each group will place the truck with only one axle force loading on the scale at a time to evaluate the force for each axle. At least three measurements should be taken for each axle. The average values should be noted in the lab data sheet attached below.

4 Lab Data Sheet Principles of Superposition Table 1: Axle Weight Specimen Data (U.S. Customary Units) Axle F 1 F 2 F 3 Weight, lbf Table 2: Material and Beam Properties Specimen Data (U.S. Customary Units) 4 Material E (ksi) b e (in ) b w (in) h (in ) h f (in) L (in) I ( in ) Aluminum 10, EI 2 ( kip in ) Table 3: Measured Deflection Values at Midspan Load Distance from support vm in vt in Test to trailer axle set Measured Theoretical Pass 1 * Pass 2 * Pass 3 (optional) Pass 4 (optional) * Pass 1 = truck going one direction; Pass 2 = truck going other direction. v M v T in v M v T T % Error

5 Helpful Beam Tables to Compute Theoretical Deflections:

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200 Notes for Strength of Materials, ET 00 Steel Six Easy Steps Steel beam design is about selecting the lightest steel beam that will support the load without exceeding the bending strength or shear strength