Mechanical of Materials

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A project that involves designing, building, and analyzing how an aluminum and brass beam deflects due to certain applied loads.

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Mechanical of Materials

  1. 1. Mechanics of Materials: Beam Deflections<br />Engineered by Group 20 <br />Christopher Webb, Anthony Williams, Carlaton Wong, Jonathan Wong<br />03/03/2010<br />
  2. 2. The Concept<br />Scenario: How much will a beam deflect with an applied load?<br />Calculating the beam’s deflection will reveal the beams curvature relationships<br />This project is designed to exhibit the mechanics of a bending moment on a single cantilever beam. Ultimately leading us to our goal.<br />
  3. 3. Goals & Objectives<br />Experimental Goals<br />How much will aluminum and brass deflect with various applied loads?<br />Determine the Young’s Modulus based on the deflection measured<br />Does our calculated Young’s Modulus equal tabulated values?<br />
  4. 4. Beam Failures<br />
  5. 5. What is Beam Deflection?<br />When loads are applied to a beam, the beam’s axis that was once straight will become curved. The displacements from the initial axis are called bending or flexural deflections. <br />
  6. 6. Apparatus Design & Operation<br /><ul><li>The beam in use will be fixed onto the standing wooden plank by securing it at the top of the apparatus with a c-clamp.
  7. 7. At the end of the beam there will be an attachment to serve as the applied load.
  8. 8. This experiment used strategically filled water bottles for the applied loads.</li></li></ul><li>Apparatus Design & Operation (cont.)<br /><ul><li>The deflection of the beam by the applied load can be recorded at the point of loading.
  9. 9. Cantilever beams are interchangeable allowing for supplemental testing of differing materials & structural make-ups .
  10. 10. Negligible factors:
  11. 11. Load’s tiny distance from free end of member
  12. 12. Small bend in beam after multiple loads</li></li></ul><li>Cost Factors Involved<br />
  13. 13. Our Cantilever Beam<br />
  14. 14. Superposition Table<br />
  15. 15. MatLab Code<br />hold on; grid on<br />plot(x/in_m,v2*100,'g-*',x/in_m,v1*100,'ro-',x/in_m,v1*100+v2*100,'b*-')<br />title('Beam Deflection of Brass w/ 0.3lb Load')<br />xlabel('Beam Length (in)')<br />ylabel('Deflection (cm)')<br />legend('DistributedLoad','ConcentratedLoad','Total Deflection',3)<br />V_max=(q*L+F)/(b*h);%Max shear stress<br />sigma_max=(.5*q*L^2+F*L)*(h*0.5)/I;<br />v_max_tot=(v1(end)+v2(end))*100;<br /> <br /> <br /> <br /> <br /> <br />%Analysis of Aluminum Beam<br />close all;clearall;clc<br />%Conversion Factors<br />in_m=0.0254; %inch to meter <br />lb_N=0.22481;%Pound force to Newton <br />%Material Properties<br />E=69*10^9;%Pa Aluminum<br />%Geometric Properties(in SI units)<br />%Al bar data<br />h=(1/8)*in_m;%Cross section height<br />b=(3/4)*in_m;%Cross sect width<br />L=15*in_m;%Length of Beam (m)<br />I=(b*h^3)/12; %second momemnt of inertia(m^4)<br />%Analysis of Brass beam<br />close all;clearall;clc<br />%Conversion Factors<br />in_m=0.0254; %inch to meter <br />lb_N=0.22481;%Pound force to Newton <br />%Material Properties<br />E=100*10^9;%Pa Brass<br />%Geometric Properties(in SI units)<br />%Brass Bar data<br />h=(.064)*in_m;%Cross section height<br />b=(3/4)*in_m;%Cross sect width<br />L=9*in_m;%Length of Beam (m)<br />I=(b*h^3)/12; %second momemnt of inertia(m^4)<br />%Applied force<br />F=(.3)/lb_N; %(N)<br />%Calculating deflection<br />x=linspace(0,L,50);<br />v1=(F*x.^3/6 - F*L*x.^2/2)/(E*I);<br />%Deflection due to concentrated load<br />q=79.7*10^-3*9.81/(12*.0254);<br />v2=zeros(1,length(x));<br />for i=1:length(x)<br />v2(i)=-q*x(i)^2/(24*E*I)*(6*L^2-4*L*x(i) + x(i)^2);<br />%Deflection due to weight of bar<br />end<br />%Applied force<br />F=(.3)/lb_N;<br />%Calculating deflection<br /> %Concetrated load<br />x=linspace(0,L,50);<br /> v1=(F*x.^3/6 - F*L*x.^2/2)/(E*I);%Deflection in (m)<br /> %Distributed weight<br />q=74*10^-3*9.81/(18.5*.0254);<br /> v2=zeros(1,length(x));<br /> for i=1:length(x)<br /> v2(i)=-q*x(i)^2/(24*E*I)*(6*L^2-4*L*x(i) + x(i)^2);<br /> end<br />hold on;grid on; <br />plot(x/in_m,v1*100,'ro',x/in_m,v2*100,'g*',x/in_m,v1*100+v2*100,'b*')<br />title('Beam Deflection of Aluminum w/ 0.3lb Load')<br />xlabel('Beam Length (in)')<br />ylabel('Deflection (cm)')<br />legend('DistributedLoad','ConcentratedLoad','Total Deflection',3)<br />V_max=(q*L+F)/(b*h)%Max shear stress<br />sigma_max=(.5*q*L^2+F*L)*(h*0.5)/I;<br />v_max_tot=(v1(end)+v2(end))*100;<br />
  16. 16.
  17. 17. Experiment Variables<br /><ul><li> Second Moment of Inertia (I) determined by measuring geometry of the beams.
  18. 18. Calculated Linear Weight Density by weighing the beams on a scale.
  19. 19. Water bottles with a known mass used to calculate forces applied to the beam.</li></li></ul><li>Results – Initial Height<br />
  20. 20. Results – Brass Bar<br />.3 Lbs Force<br />.4 Lbs Force<br />.5 Lbs Force<br />
  21. 21. Results – Aluminum Bar<br />.3 Lbs Force<br />.4 Lbs Force<br />.5 Lbs Force<br />
  22. 22. Experimental Results<br />Aluminum :<br />Expected Young’s Modulus:<br />69 GPa<br />Brass:<br />Expected Young’s Modulus:<br />100 GPa<br />
  23. 23. Expected Results<br />Aluminum:<br />Brass:<br />All measured deflections are higher than calculated. Why?<br />
  24. 24. Max Stress in Beam<br />Using the Flexural formula and observing that the Mmax is located at the origin we can derive an expression for Pmax<br />
  25. 25. Max Stress in Beam<br /><ul><li>Approximate Yield stress:
  26. 26. Aluminum = 35MPa
  27. 27. Brass = 70 MPa</li></li></ul><li>Conclusion<br />With the proper build and successful experimentation of our proposed apparatus, deflections of both beams were recorded.<br />Young’s Modulus was calculated given the measured deflection.<br />Comparing our calculated and measured Young’s Modulus’ error was found to be 10-20% due to possible plastic deformation.<br />
  28. 28. Mechanics of Materials:Beam Deflections<br />Group 20<br />Christopher Webb, Anthony Williams, Carlaton Wong, & Jonathan Wong<br />Thank you!<br />

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