BEAMBEND How Much Does a Beam Bend? BRIEF ABSTRACT This problem shows an application of 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. The calculus skill you will be applying is integration. GENERAL INFORMATION 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. STATEMENT OF PROBLEM 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. Read this to become a structural engineer. 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. Here is a sketch. 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. 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. 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. 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. Here is a simple problem worked out. Let's take a cantilever beam with uniform distributed load (constant) w=W. Find the deflection curve w(x). Step 1 -- Calculate boundary conditions V0 and M0. In[2]:= V0 = Integrate[ W, {u,0,L}] Out[2]= L W In[3]:= M0 = Integrate[ u W, {u,0,L} ] Out[3]= 2 L W ---- 2 Step 2 -- Calculate y''' curve In[4]:= yppp[x_] = V0 / EI - Integrate[ W/EI, {u,0,x} ] Out[4]= L W W x --- - --- EI EI Step 3 -- Calculate y'' curve In[5]:= ypp[x_] = M0 / EI + Integrate[ yppp[u], {u,0,x}] Out[5]= 2 2 L W L W x W x ---- + ----- - ---- 2 EI EI 2 EI Step 4 -- Calculate y' curve In[6]:= yp[x_] = 0 + Integrate[ ypp[u], {u,0,x} ] Out[6]= 2 2 3 L W x L W x W x ------ + ------ - ---- 2 EI 2 EI 6 EI Step 5 -- Calculate y curve In[7]:= y[x_] = 0 + Integrate[ yp[u], {u,0,x} ] Out[7]= 2 2 3 4 L W x L W x W x ------- + ------ - ----- 4 EI 6 EI 24 EI Here is the tip deflection. In[8]:= y[L] Out[8]= 4 3 L W ------ 8 EI Here is your problem. 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. KEYWORDS Beam, bending, deflection, shear, moment, derivative, integral, piecewise functions, dummy variable of integration. TEACHER NOTES ISSUES RELATED TO THE PROBLEM 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. Prerequisites A knowledge of integration is essential to carrying out the manipulations needed to work the problem. Time allotment - time management 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. Expectations 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. Future payoffs 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. Extensions 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. References and Sources Mechanics of Materials, by Ferdinand P. Beer and Russell Johnston, Jr. McGraw-Hill, 1992, ISBN 0-07-004340-X POSSIBLE SOLUTION In[9]:= EI = 10. ^ 10; Step 1 -- Calculate shear and moment at the wall. In[10]:= w1[x_] = 0; w2[x_] = 100; w3[x_] = 0; w4[x_] = (x-100) * 100 / 50; In[14]:= 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}] Out[14]= 5500 In[15]:= 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 Out[15]= 468333. Step 2 --Calculate the shear (y''') in the four segments In[16]:= 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}]; Step 3 -- Calculate the moment (y'') in the four segments In[20]:= 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}]; Step 4 -- Calculate the slope (y') in the four segments In[24]:= 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}]; Step 5 -- Calculate the deflection in four segments In[28]:= 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}]; Step 6 -- Produce a plot of the solution In[32]:= Y[x_] := Which[ 0 <= x < 30, y1[x], 30 <= x < 60, y2[x], 60 <= x < 100, y3[x], 100 <= x <= 150, y4[x]] In[33]:= Plot[ Y[x], {x,0,150}] Out[33]= -Graphics- 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. In[34]:= Y[150] Out[34]= 0.776667 In[35]:= Y'[150] Out[35]= 0.0114954 ISSUES IN SOLUTION 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.