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Project 1 for CAE Design from Norton...

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Design Project 1 Main Bridge Beam

Carson Britt Sizemore EGMN 420-001 Spring 2020

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Table of Contents

Summary … 3 Recommendations for Future Designs … 4 Introduction … 4 Design Basis … 5 Design Methodology … 5 Design Results and Discussion … 5 Conclusions … 5 References … 6 Appendix … 7

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Summary The objective of the following report is to communicate with engineering management a design for the main supporting beam of a beam bridge. The following table contains the Beam’s design specifications.

FOS Base (meters) Height (meters) Material

5.56 2 4 Solid American White Oak

Table 1 – Beam Specifications

An AutoCAD drafting of the bridge beam design is included below. Assume simply supported on the left end by pin and on the right end by roller.

ISO

TOP

SIDE 10.00 (m)

4.00 (m)

FRONT

CARSON MATERIAL: WHITE SIZEMORE AMERICAN OAK PROJECT DENSITY: EGMN 770 kg/m^3 ONE 420

2.00 (m)

Figure 1 – Beam Drafting – Made in AutoCAD

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Recommendations for Future Designs While the decision to make the beam out of solid American white oak provided the strength necessary for the desired factor of safety of 5.56, this bridge could be redesigned with a material of a higher UTS such as S355 steel to achieve the same FOS while simultaneously taking up less physical space. A bridge of this design disrupts the environment around it less and could allow for larger objects to pass underneath.

Introduction In this project we were given the task of designing a 10m long main beam for a bridge that can support an evenly distributed load of 450,000N. The bridge must be simply supported at either end. The base, height, FOS, and material were decided by the designer in such a way that was most optimal for the project. This report was created for an engineering management in your organization in order to help them decide whether to continue with the bridge design. The calculations and analysis are important because if the bridge design passes the company can increase funds towards the project.

Design Basis The basis for the design was a bridge that was optimally strong, met all the project guidelines, used a minimum amount of resources, and was also as simple as possible. For the material I decided to go with solid American white oak for its abundance in America, medium to low cost, and ultimate tensile strength (UTS) of 5.50 MPa (matweb.com). Plus, I just like wood.

Design Methodology The first idea I had when designing this beam was that I wanted to use oak wood. I wasn’t quite sure what the base and height required for such a bridge would be, so I guessed, and 4

my initial design was 1 (m) * 1 (m) * 10 (m), and a desired factor of safety of 6. Some hand calculations, including free body diagrams, reaction forces, and shear force and bending moment diagrams were done on paper in order to assist with the MATLAB calculations. These hand calculations can be found as Fig. 2 in Appendix A.1. After running these calculations in MATLAB, the calculated ultimate tensile strength was too small for the desired FOS. After the initial design parameters failed, I lowered the minimum FOS to 5, which still falls in the recommended range of 5-7 for structural bridges (engineeringtoolbox.com) and increased the base dimension to 5 (m) and the height 3 (m). The idea here was that a thicker and larger bridge would be stronger, and the estimation was correct with the design passing the FOS comparison test in MATLAB, the code of which can be found in Appendix A.2.

Design Results & Conclusions After this initial design calculation was made, a second MATLAB code was written to find the most optimal values of b and h possible. A contour plot was created (which can be found in Appendix B) that plotted (base vs height) vs FOS, and with a minimum FOS value of 5 in mind, the contour plot was analyzed, and based on the shape of the plot given, a b and h value of 2 and 4 (m) respectively were chosen, to give this bridge design an FOS of 5.56. These b and h calculations, as well as their combined UTS and FOS values can be found in the MATLAB code in Appendix B.

Conclusions After picking my desired material, American white oak, and my desired (minimum) factor of safety (FOS), 5, stress calculations were done for different combinations of base, b, and height, h, for a bridge of length, l, of 10 meters, simply supported on either end. A contour plot was created that graphed these different base and height values vs their 5

combined factor of safety. That contour plot can be found as Fig. 3 in Appendix B. After an analysis of the shape and values of this contour plot, the values b = 2m and h = 4m were chosen, which provide a factor of safety of 5.56. These values were chosen because they met the FOS requirement while also providing the smallest volume possible, thus reducing material cost. This design exceeds the project’s strength requirement by 565%.

References Works Cited American White Oak Wood, www.matweb.com/search/datasheet_print.aspx? matguid=44cdf6b01d004baaa7e9510575891dc3&n=1.

“Density of Various Wood Species.” Engineering ToolBox, www.engineeringtoolbox.com/wood-density-d_40.html.

“Factors of Safety.” Engineering ToolBox, www.engineeringtoolbox.com/factors-safety-fosd_1624.html.

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Appendix A.1

Figure 2 – Hand Calculations

Appendix A.1 %Sizemore, Carson B.

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%CAE Design Project 1 %Beam Calculations clear; clc; %GIVEN: %Length of the beam (m) l = 10; %Uniformly distributed load (N) dl = 450000; %Rectangular cross-section. %Simply Supported. %BEAM PARAMETERS: %Base of beam cross section (m) b = 5; %Height of beam cross section (m) h = 3; %Factor of safety (Ultimate load/Actual load) FOS = 5; %Material: American White Oak. %PROPERTIES: %Density (kg/m^3) (from engineeringtoolbox.com) d = 770; %Ultimate Tensile Strength (Pa) (from engineeringtoolbox.com)

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uts = 5.50*10^6; %Acceleration downward due to gravity (m/s^2) g = 9.81; %BEAM CALCULATIONS: %Volume of beam (m^3) v = b*h*l; %Distance from Neutral Axis to Distributed Load(m) c = h/2; %Mass of Beam (kg) m = v*d; %Weight of the beam (N) w = m*g; %Reaction force A (forces acting upwards are neg.) (N) RA = - (dl + w) / (2) ; %Reaction Force B (N) RB = RA ; %Sum of forces acting down on the beam (N) Ey = dl + w; %Maximum Shear Force (N) sf = RA + w*dl; %Maximum Bending Moment (N*m) bm = (w + dl)*(l / 2); %Moment of inertia (m^4)

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i = (b * (h^3)) / 12; %Max tensile stress (Pa) omax = (bm * c) / i; %Allowable tensile stress (Pa) oallow = (omax) * (FOS); %STRENGTH TEST: if oallow < uts fprintf 'a good strong bridge' else fprintf 'a dinky lame bridge' end %{ WHEN RUN: __________COMMAND WINDOW__________ |a good strong bridge>>

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%} %THEREFORE: THE DESIGN PASSES! Yippee! Funding!

Appendix B %Sizemore, Carson B. %CAE Design Project 1 %OTIMIZATION AND CONTOUR PLOT clear; 10

clc; %Finding the smallest max tensile stress value possiblefor an array %of squarevalues for b and h. %GIVEN: %Length of the beam (m) l = 10; %Uniformly distributed load (N) dl = 450000; %Rectangular cross-section. %Simply Supported. %BEAM PARAMETERS: %Base of beam cross section (m) [using an array of values from 0-5] b = [0.5,1,1.5,2,2.5,3,3.5,4,4.5,5]; %Height of beam cross section (m) [using an array of values from 0-5] h = [0.5;1;1.5;2;2.5;3;3.5;4;4.5;5]; %Factor of safety (Ultimate load./Actual load) FOS = [0 1 2 3 4 5 6 7 8 9 10]; %Material: American White Oak. %PROPERTIES: %Density (kg./m.^3) (from engineeringtoolbox.com) d = 770; %Ultimate Tensile Strength (Pa) (from engineeringtoolbox.com) uts = 5.50.*10.^6;

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%Acceleration downward due to gravity (m./s.^2) g = 9.81; %BEAM CALCULATIONS: %Volume of beam (m.^3) v = b.*h.*l; %Distance from Neutral Axis to Distributed Load(m) c = h./2; %Mass of Beam (kg) m = v .* d; %Weight of the beam (N) w = m .* g; %Reaction force A (forces acting upwards are neg.) (N) RA = -1.* (dl + w) ./ (2) ; %Reaction Force B (N) RB = RA ; %Sum of forces acting down on the beam (N) Ey = dl + w; %Maximum Shear Force (N) sf = RA + w.*dl; %Maximum Bending Moment (N.*m) bm = (w + dl).*(l ./ 2); %Moment of inertia (m.^4) i = (b .* (h.^3)) ./ 12;

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%Max tensile stress (Pa) omax = (bm .* c) ./ i; %{ MAX TENSILE STRESSES omax = 1.0e+08 * Columns 1 through 6 1.1253 0.5853 0.4053 0.3153 0.2613 0.2253 0.2927 0.1577 0.1127 0.0902 0.0767 0.0677 0.1351 0.0751 0.0551 0.0451 0.0391 0.0351 0.0788 0.0451 0.0338 0.0282 0.0248 0.0226 0.0523 0.0307 0.0235 0.0199 0.0177 0.0163 0.0376 0.0226 0.0176 0.0151 0.0136 0.0126 0.0285 0.0175 0.0138 0.0120 0.0109 0.0101 0.0225 0.0141 0.0113 0.0099 0.0090 0.0085

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0.0184 0.0117 0.0095 0.0084 0.0077 0.0073 0.0153 0.0099 0.0081 0.0072 0.0067 0.0063 Columns 7 through 10 0.1996 0.1803 0.1653 0.1533 0.0612 0.0564 0.0527 0.0497 0.0323 0.0301 0.0284 0.0271 0.0210 0.0198 0.0188 0.0181 0.0152 0.0145 0.0139 0.0134 0.0118 0.0113 0.0109 0.0106 0.0096 0.0092 0.0089 0.0087 0.0081 0.0078 0.0075 0.0074 0.0069 0.0067 0.0065 0.0064 0.0061 0.0059 0.0057 0.0056 %} %CONTOUR PLOT %Factor of safety SF = uts ./ omax; %Creating meshgrid workspace [b,h] = meshgrid (b, h); %Creating contour plot contourf (b,h,SF, 25)

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colorbar %Based on contour plot, the design will be according to the following %dimensions: %{ b=2 h=4 FOS=5.565 %}

Fig. 3 – (b v h) v FOS Contour Plot Made in MATLAB

“On my honor, I have neither given nor received aid on this assignment, and I pledge that I am in compliance with the VCU Honor System.”

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-CARSON BRITT SIZEMORE

2.5.2020

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