ENGR 244 Lab 5

.docx

School

Concordia University *

*We aren’t endorsed by this school

Course

244

Subject

Mechanical Engineering

Date

Dec 6, 2023

Type

docx

Pages

Uploaded by AgentSnail2002

Lab 5: Deflection of Beams Lab Section CI – X Winter 2022 Professor Ahmed Soliman Concordia University Montreal, QC, Canada Tuesday March 8 th , 2022

Table of Contents Nomenclature pg 3 List of Tables pg 3 List of Figures pg 3 Objective pg 4 Introduction pg 4 Procedure pg 4-5 Results pg 5-9 Discussion and Conclusion pg 9-10 References pg 10 Original Data pg 11

Nomenclature ρ = radius of curvature b = width E = modulus of elasticity h = height I = moment of inertia L = length of span P = load x = distance along span y = deflection List of Tables Table 1: Experimental and Theoretical Values of Deflection for Brass Beam at L/2 p. 5 Table 2: Experimental and Theoretical Values of Deflection for Brass Beam at L/4 p. 5 Table 3: Experimental and Theoretical Values of Deflection for Steel Beam at L/2 p. 6 Table 4: Experimental and Theoretical Values of Deflection for Steel Beam at L/4 p. 6 Table 5: Experimental and Theoretical Values of Deflection for Aluminum at L/2 p. 6 Table 6: Experimental and Theoretical Values of Deflection for Aluminum at L/4 p. 6 Table 7: Experimental and Theoretical Values of Deflection for Cantilever Brass at L/2 p. 7 Table 8: Experimental and Theoretical Values of Deflection for Cantilever Brass at L p. 7 Table 9: Experimental and Theoretical Values of Deflection for Cantilever Steel at L/2 p. 8 Table 10: Experimental and Theoretical Values of Deflection for Cantilever Steel at L p. 8 Table 11: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L/2 p. 8 Table 12: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L p. 8 List of Figures Figure 1: Theoretical vs Experimental Deflection of the Beams at L/2 p.7 Figure 2: Theoretical vs Experimental Deflection of the Cantilever Beams at L/2 p.9

Objective: The goal of this experiment is to determine the modulus of elasticity using the relationship between simply supported beams and load deflection[ CITATION Con \l 4105 ]. Introduction: Subjecting a beam to a transverse load will cause deflection in the beam. Different types of loads can induce deflection [CITATION Des22 \l 4105 ], but in this experiment only transverse loading will be tested. The length of the beam, the magnitude of the loads and the positioning of the loads will make a difference in the magnitude of the deflection. Deflection is an important aspect in engineering as many building codes require certain limits on deflections[ CITATION Con \l 4105 ]. These limits can include things such as deflections from earthquakes or winds [ CITATION Mov22 \l 4105 ]. The data collected in this experiment is important as it allows us to calculate the curvature of the beam about the neutral axis through this relation [ CITATION Con \l 4105 ]: 1/ρ = M(x)/EI The equation above is only valid within the elastic range where ρ is the radius of curvature, M(x) is the bending moment at distance x, E is the modulus of elasticity of the assumed material, and I is the moment of inertia of the entire cross-section [ CITATION Con \l 4105 ]. Various beam cross-sections give different advantages when it comes to beam deflection [ CITATION Mov22 \l 4105 ], but for this experiment, a full rectangular cross-section beam is used, so the value of I can be expressed as: I = (1/12)bh 3 Though the radius of curvature, ρ, is related to the deflection of the beam, it does not allow us to calculate it directly. The curvature has the following equation that will allow us to solve for the deflection at any point on the beam: 1/ρ = (d 2 y/dx 2 )/(1 + (dy/dx) 2 ) 3/2 Since the beam is only subjected to deflection in the elastic region, it can be said that the value of the slope, dy/dx, is negligible and therefore our new relation between the curvature and the deflection will be [ CITATION Con \l 4105 ]: 1/ρ = (d 2 y/dx 2 ) Substituting 1/ρ with M(x)/EI, we get: d 2 y/dx 2 = M(x)/EI  (d 2 y/dx 2 )EI = M(x) This equation will allow us to calculate the deflection of the beam at any point once we integrate twice [ CITATION Con \l 4105 ]. Procedure: Point load on a simply supported beam: Make sure to measure and record the cross section of each beam and its span. The span should be 455mm. Place the brass, steel or aluminum sample on the supports of the test machine and ensure that load is at the center of the sample. Place the two deformation gauges at the center and quarter portions of the beam and ensure it measure the vertical

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version