(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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In particular, you will be finding the amount of deflection (bend or sag) in a beam carrying a distributed load. The calculus skill you will be applying is integration. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] FileName: BEAMBEND Full title: How Much Does a Beam Bend? Last Update: 6/3/96 Developer: Jerry Fine Department of Mechanical Engineering Rose-Hulman Institute of Technology Terre Haute, IN 47803 Phone: 812-877-8353. Email: fine@nextwork.rose-hulman.edu FAX: 812-877-3198 Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@rose-hulman.edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] This problem shows an application of the calculus to the analysis of one of the simplest load carrying structures: the beam. In particular, you will be finding the amount of deflection (bend or sag) in a beam carrying a distributed load. Here is the schedule of activities: (1) background reading to understand the basics (2) work through a simple problem with hints (3) work through a more advanced problem using what you have learned. The calculus skill you will be applying is integration. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Read this to become a structural engineer. :[font = subsubsection; inactive; preserveAspect; startGroup] The type of beam we will be studying is the cantilever. One end is firmly fixed to a support so that it can neither rotate or move up or down. The other end of the beam is not supported in any way. A load is distributed along the top surface of the beam. The load may vary and is described by the function w(x), with units of force per length, where x is the spatial coordinate measured along the neutral axis of the beam. (The neutral axis is a reference axis which does not stretch or compress during bending.) A very common load is w=constant. 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFF3F303FC00FC300F03030F0FC0FF033F0F03F03F03C3FC33C300030 F03F3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCFFFFFFFFFFFFFCFCFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFCFCCFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCFFFFFFFFFFFFF CFCFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCFCCFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFF3FFFFFFFFFFFFF03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF033 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF grestore end %%EndFile grestore %End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] To model the behavior of the beam it is important to understand the force and couple (or torque) that the built in support must supply. As the following figure indicates, the force, which is called the shear at the wall, is the integral of w(x) from 0 to L. You can see this by thinking of w(x) dx as the force due to the load's application over an infinitesimal length in the neighborhood of x. The force supplied must equal the sum of all these infinitesimal load contributions. In the case of w=constant, the shear at the wall is just wL. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 474; pictureHeight = 250; endGroup] %! %%Creator: Mathematica %%AspectRatio: 0.527426 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 474 250 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:Pasteboard %%Origin:0 720 %%BoundingBox: 0.0 0.0 474.000 250.000 %%EndComments gsave %%BeginFile:Pasteboard %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:ConvertedTIFF %%CreationDate:Wed Aug 2 18:02:23 1995 %%Origin:0 720 %%BoundingBox: 0.0 0.0 474.000 250.000 %%EndComments %ImageData: 474 250 2 1 0 119 0 "beginimage" 20 dict begin /rows 250 def /cols 474 def /str1 119 string def /readit {currentfile exch readhexstring pop} def /beginimage {{str1 readit} image} def gsave 1.000 1.000 scale cols rows scale cols rows 2 [cols 0 0 rows 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FFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 grestore end %%EndFile grestore %End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect] The couple or torque that the wall must supply to the beam is intended to offset the total torque produced by the load distributed over the beam. As the figure indicates, this torque, called the bending moment at the wall, is the integral of x w(x) from 0 to L. You can see this by thinking of x w(x) as the torque of the infinitesimal load w(x) dx. :[font = subsubsection; inactive; preserveAspect; startGroup] We now turn our attention to answering the question, "How much does the beam bend?" What we really need is a function y(x) which describes the vertical deflection at each point x between 0 and L.We use the neutral axis as a reference as the figure indicates. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 402; pictureHeight = 163; endGroup] %! %%Creator: Mathematica %%AspectRatio: 0.405473 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 402 163 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:Pasteboard %%Origin:0 720 %%BoundingBox: 0.0 0.0 402.000 163.000 %%EndComments gsave %%BeginFile:Pasteboard %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:ConvertedTIFF %%CreationDate:Wed Aug 2 18:02:23 1995 %%Origin:0 720 %%BoundingBox: 0.0 0.0 402.000 164.000 %%EndComments %ImageData: 402 164 2 1 0 101 0 "beginimage" 20 dict begin /rows 164 def /cols 402 def /str1 101 string def /readit {currentfile exch readhexstring 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFF0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 grestore end %%EndFile grestore %End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; endGroup] We wish to relate deflection to loading. There is a simple way to do this provided that the deflections y and slope y' is small. Here is that bending theory in a nutshell: E I y'''' = -w(x) The fourth derivative of the deflection is proportional to the loading at each point x. The constant term E I depends on the material the beam is made from and the shape of the cross section. For the case of the cantilever beam we have the following boundary conditions. E I y'''(0) = V0 EI y''(0) = M0 y'(0) = 0 y(0) = 0 With this information, the beam bending problem is reduced to a problem in integral calculus. Here we can take you through a solution. ;[s] 3:0,0;208,1;213,2;960,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,33,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Here is a simple problem worked out. :[font = subsubsection; inactive; preserveAspect] Let's take a cantilever beam with uniform distributed load (constant) w=W. Find the deflection curve w(x). :[font = subsubsection; inactive; preserveAspect; startGroup] Step 1 -- Calculate boundary conditions V0 and M0. ;[s] 5:0,0;42,1;43,2;49,3;50,4;51,-1; 5:1,10,8,Times,1,12,0,0,0;1,10,8,Times,65,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,10,8,Times,65,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = input; preserveAspect; startGroup] V0 = Integrate[ W, {u,0,L}] :[font = output; output; inactive; preserveAspect; endGroup] L*W ;[o] L W :[font = input; preserveAspect; startGroup] M0 = Integrate[ u W, {u,0,L} ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (L^2*W)/2 ;[o] 2 L W ---- 2 :[font = subsubsection; inactive; preserveAspect; startGroup] Step 2 -- Calculate y''' curve :[font = input; preserveAspect; startGroup] yppp[x_] = V0 / EI - Integrate[ W/EI, {u,0,x} ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (L*W)/EI - (W*x)/EI ;[o] L W W x --- - --- EI EI :[font = subsubsection; inactive; preserveAspect; startGroup] Step 3 -- Calculate y'' curve :[font = input; preserveAspect; startGroup] ypp[x_] = M0 / EI + Integrate[ yppp[u], {u,0,x}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (L^2*W)/(2*EI) + (L*W*x)/EI - (W*x^2)/(2*EI) ;[o] 2 2 L W L W x W x ---- + ----- - ---- 2 EI EI 2 EI :[font = subsubsection; inactive; preserveAspect; startGroup] Step 4 -- Calculate y' curve :[font = input; preserveAspect; startGroup] yp[x_] = 0 + Integrate[ ypp[u], {u,0,x} ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (L^2*W*x)/(2*EI) + (L*W*x^2)/(2*EI) - (W*x^3)/(6*EI) ;[o] 2 2 3 L W x L W x W x ------ + ------ - ---- 2 EI 2 EI 6 EI :[font = subsubsection; inactive; preserveAspect; startGroup] Step 5 -- Calculate y curve :[font = input; preserveAspect; startGroup] y[x_] = 0 + Integrate[ yp[u], {u,0,x} ] :[font = output; output; inactive; preserveAspect; endGroup] (L^2*W*x^2)/(4*EI) + (L*W*x^3)/(6*EI) - (W*x^4)/(24*EI) ;[o] 2 2 3 4 L W x L W x W x ------- + ------ - ----- 4 EI 6 EI 24 EI :[font = text; inactive; preserveAspect] Here is the tip deflection. :[font = input; preserveAspect; startGroup] y[L] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] (3*L^4*W)/(8*EI) ;[o] 4 3 L W ------ 8 EI :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Here is your problem. :[font = text; inactive; preserveAspect] Find the deflection y(x) for the following cantilever beam. HINT: y''', y'' , y' and y are all continuous. You must work the problem in pieces. You should assume that the bending theory applies. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 411; pictureHeight = 180; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 0.437956 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 411 180 0 0 ] ] MathScale % Start of Graphics %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:Pasteboard %%Origin:0 720 %%BoundingBox: 0.0 0.0 411.000 180.000 %%EndComments gsave %%BeginFile:Pasteboard %!PS-Adobe-2.0 EPSF-1.2 %%Creator:Scan %%Title:ConvertedTIFF %%CreationDate:Wed Aug 2 18:02:23 1995 %%Origin:0 720 %%BoundingBox: 0.0 0.0 411.000 181.000 %%EndComments %ImageData: 411 181 2 1 0 103 0 "beginimage" 20 dict begin /rows 181 def /cols 411 def /str1 103 string def /readit {currentfile exch readhexstring pop} def /beginimage {{str1 readit} image} def gsave 1.000 1.000 scale cols rows scale cols rows 2 [cols 0 0 rows 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFCFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFCFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFDFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC grestore end %%EndFile grestore %End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Beam, bending, deflection, shear, moment, derivative, integral, piecewise functions, dummy variable of integration. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect; endGroup] The purpose of this problem is to illustrate a very useful application of the calculus. The issue is the extent of the knowledge that the student must acquire before working the problem. Although a serious attempt has been made to supply this background material, this problem may not be for the average calculus student. It might be assigned as a project to students who are oriented towards mechanical and civil engineering. :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] A knowledge of integration is essential to carrying out the manipulations needed to work the problem. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This is a project which will need some "soak" time. It should be assigned over a period of several days. Even with a computer algebra system, the solution to the problem is quite lengthy. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students will have questions like, "Where does this theory come from?" What is presented here is a simplified special case of Euler bending theory. The references cited should help curious students. The important thing to remember is that the problem is focused more on the calculus than on the mechanics. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] The problem is a good preparation for the study of mechanics of materials. A student who masters the problem should feel comfortable applying calculus in a new, unknown technical area. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] Extensions could be created in which different types of loading or supports were used. For some of these problems it would be possible to search for the points where deflection or slope was at a maximum. :[font = subsection; inactive; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Mechanics of Materials, by Ferdinand P. Beer and Russell Johnston, Jr. McGraw-Hill, 1992, ISBN 0-07-004340-X ;[s] 2:0,0;22,1;110,-1; 2:1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION :[font = input; preserveAspect] EI = 10. ^ 10; :[font = subsection; inactive; preserveAspect; startGroup] Step 1 -- Calculate shear and moment at the wall. :[font = input; preserveAspect] w1[x_] = 0; w2[x_] = 100; w3[x_] = 0; w4[x_] = (x-100) * 100 / 50; :[font = input; preserveAspect; startGroup] V0 = Integrate[ w1[u], {u,0,30} ] + Integrate[ w2[u], {u,30,60}] + Integrate[ w3[u], {u,60,100}] + Integrate[ w4[u], {u,100,150}] :[font = output; output; inactive; preserveAspect; endGroup] 5500 ;[o] 5500 :[font = input; preserveAspect; startGroup] M0 = Integrate[ u w1[u], {u,0,30} ] + Integrate[ u w2[u], {u,30,60}] + Integrate[ u w3[u], {u,60,100}] + Integrate[ u w4[u], {u,100,150}] //N :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 468333.3333333334 ;[o] 468333. :[font = subsection; inactive; preserveAspect; startGroup] Step 2 --Calculate the shear (y''') in the four segments :[font = input; preserveAspect; endGroup] yppp1[x_] = V0/ EI - Integrate[ w1[u]/EI, {u,0,x}]; yppp2[x_] = yppp1[30] - Integrate[ w2[u]/EI, {u,30,x}]; yppp3[x_] = yppp2[60] - Integrate[ w3[u]/EI, {u,60,x}]; yppp4[x_] = yppp3[100] - Integrate[w4[u]/EI, {u,100,x}]; :[font = subsection; inactive; preserveAspect; startGroup] Step 3 -- Calculate the moment (y'') in the four segments :[font = input; preserveAspect; endGroup] ypp1[x_] = M0/ EI + Integrate[yppp1[u], {u,0,x}]; ypp2[x_] = ypp1[30] + Integrate[yppp2[u], {u,30,x}]; ypp3[x_] = ypp2[60] + Integrate[yppp3[u], {u,60,x}]; ypp4[x_] = ypp3[100] + Integrate[yppp4[u], {u,100,x}]; :[font = subsection; inactive; preserveAspect; startGroup] Step 4 -- Calculate the slope (y') in the four segments :[font = input; preserveAspect; endGroup] yp1[x_] = 0 + Integrate[ypp1[u], {u,0,x}]; yp2[x_] = yp1[30] + Integrate[ypp2[u], {u,30,x}]; yp3[x_] = yp2[60] + Integrate[ypp3[u], {u,60,x}]; yp4[x_] = yp3[100] + Integrate[ypp4[u], {u,100,x}]; :[font = subsection; inactive; preserveAspect; startGroup] Step 5 -- Calculate the deflection in four segments :[font = input; preserveAspect; endGroup] y1[x_] = 0 + Integrate[yp1[u], {u,0,x}]; y2[x_] = y1[30] + Integrate[yp2[u], {u,30,x}]; y3[x_] = y2[60] + Integrate[yp3[u], {u,60,x}]; y4[x_] = y3[100] + Integrate[yp4[u], {u,100,x}]; :[font = subsection; inactive; preserveAspect; startGroup] Step 6 -- Produce a plot of the solution :[font = input; preserveAspect] Y[x_] := Which[ 0 <= x < 30, y1[x], 30 <= x < 60, y2[x], 60 <= x < 100, y3[x], 100 <= x <= 150, y4[x]] :[font = input; preserveAspect; startGroup] Plot[ Y[x], {x,0,150}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] Tip deflection is now calculated and seen to be small relative to the length of the beam. The slope at the end is also small. :[font = input; preserveAspect; startGroup] Y[150] :[font = output; output; inactive; preserveAspect; endGroup] 0.7766666666666666 ;[o] 0.776667 :[font = input; preserveAspect; startGroup] Y'[150] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 0.01149541666666667 ;[o] 0.0114954 :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] There will probably be difficulties in two areas: (1) understanding and applying boundary conditions using piecewise calculations (2) handling the dummy variable of integration concept. ^*)