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Structural engineers need to calculate how beams bend, and they do so by \ using principles of structural mechanics and calculus. In this module, you \ will investigate some of the ways that engineers use calculus to ensure that \ the structures they design are both safe and functional. \ \>", "Text", PageWidth->PaperWidth], Cell["\<\ Before you begin this module, we recommend that you refer to \"Maximums, \ Minimums, and Inflection Points,\" a JAVA applet included in this supplement. \ This applet allows you to explore the relationship between the shape of the \ graph of a function and the values of its first and second derivatives.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", PageWidth->PaperWidth, CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \ on the right cell bracket and double click.", CellFrame->True, Background->None], "\nINITIALIZATION CELLS\n\tWhen asked if you want to \". . . automatically \ evaluate all the initialization cells in the \tnotebook . . . ,\" respond by \ pressing the \"Yes\" button.\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. 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In contrast to beams, cables and hangars support loads by \ stretching, columns and arches by compressing, and shafts and rods by \ twisting. 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It is shown as the ", StyleBox["x", FontSlant->"Italic"], "-axis in the figure below. When the loads are applied, the beam bends, \ resulting in a vertical deflection of the long axis. The deflected axis is \ called the elastic curve, and it is denoted by ", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). The independent variable ", StyleBox["x ", FontSlant->"Italic"], "measures the position along the axis of the unloaded/undeflected beam, and \ ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") measures the vertical deflection of the point at position ", StyleBox["x", FontSlant->"Italic"], " along the beam. Knowing the deflections, ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "), the structural engineer can calculate the maximum deflections of the \ beam to ensure that they meet the requirements and specifications for the \ design. For example, deflections that exceed the maximum allowable deflection \ may result in damage to windows, wall partitions, ceiling panels, and so on. \ Therefore, design specifications usually place limits on the maximum \ deflections." }], "Text", PageWidth->PaperWidth], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0007Ml0@0005P000000000ooooojX7002Y0P00 00000000000C=P00:1<00215CDH00040O7L00;L1000500000000000000000000?`/00>T>003;0000 3@4000X0000@0000000000000009000040000:X7002g0P004P0000`000010000600000`000000002 DP0004`100010000a?ooo`0000000000000009010000000014004U@0J@1]06D0L`0P04h0I@1g0200 DP1_06d0H@1^0000000300000000000000000000000004=:DU=DDP0000000000I000004000010040 0P020000001B0d`4K`SZ2P00DP=<16l8jPY_2>X:@00N03l0AP1X0FP10`03000:@00?0040C51DH5002Y0P0010000:@5b@7T1Gl1i0EM0Z@5Y`8U0000300000L0080X0000 30000080000U0000300000P0080U0000300000D0080[000060000=l6003Q0@00ZPL00882000<0000 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For statically indeterminate beams, we have to start with ", Cell[BoxData[ \(TraditionalForm\`\(u\^\((4)\)\)(x)\)]], ", the fourth derivative of the elastic curve. In addition, the support \ conditions can usually be specified. For example, the beam in the first \ figure above rests on two columns, which prevent the ends of the beam from \ displacing up or down. Mathematically, these conditions are ", StyleBox["u", FontSlant->"Italic"], "(0) = 0 and ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["L", FontSlant->"Italic"], ") = 0, where ", StyleBox["L", FontSlant->"Italic"], " is the length of the beam between the supports. Knowing ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(x)\)]], ", the engineer can integrate this function twice and apply appropriate \ support conditions to determine ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "), the equation of the elastic curve. If we start with ", Cell[BoxData[ \(TraditionalForm\`\(u\^\((4)\)\)(x)\)]], ", we need to integrate four times and apply the support conditions, and we \ will need four conditions instead of two.\n\nIn mathematics, the support \ conditions are often called boundary conditions or initial conditions. The \ combination of a differential equation and boundary conditions is called a \ boundary value problem; the combination of a differential equation and \ initial conditions is called an initial value problem. \n\nStructural \ engineers are interested in the concavity of a beam's elastic curve. Wherever \ the elastic curve is concave up, the beam is said to be in positive bending \ and, wherever it is concave down, the beam is in negative bending. For \ positive bending, the material above the long axis of the beam is compressed \ while the material below the long axis is stretched. The situation is \ reversed for negative bending. \n\nIn reinforced concrete beams, concrete is \ used to resist compression while the steel reinforcement bars or \"rebars\" \ are used to resist stretching or tension. Consequently, the structural \ engineer must design reinforced concrete beams with the rebar on the bottom \ for positive bending and on the top for negative bending. In some cases, a \ beam can have positive bending over part of its length and negative bending \ over the rest. In such a situation, the rebar is switched from the top to the \ bottom or vice versa at the inflection points of the elastic curve.\n" }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Part I: A Cantilever Beam", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["A Description of the Beam and its Supports", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@00024@0@0005P000000000oooooj05003F0`00 00000000002n9`0061/00215CDH00040424007@1000400000000000000000000?`/00>T>003;0000 3@4000X0000@0000000000000009000040000:85003G0`004P0000`000010000600000`000000002 DP0004`100010000a?ooo`0000000000000009010000000014004U@0J@1]06D0L`0P04h0I@1g0200 DP1_06d0H@1^0000000300000000000000000000000004=:DU=DDP0000000000I000004000010040 0P020000001B0d`4K`SZ2P00DP=<16l8jPY_2>X:@00N03l0AP1X0FP10`03000:@00?0040C51D06P0BH0000L00000`0000000004000010000000000U0000 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7@0001/0000D00003`0002L0000K00006`0002l0001D0000E00001h400350@00>0@00?H100010000 Aj;Q@4NRhD0N1000a@400040001<0000100008P200350@00IPD00?L1001@0000800001/0000R0000 30000?oooolU0000300000P0080U0000300000D0080[000060000000050000000000@00005000 \>"], "Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{430, 293.438}, ImageMargins->{{8, 0}, {0, 4}}, ImageRegion->{{0, 1}, {0, 1}}], "\n" }], "Graphics", PageWidth->PaperWidth, GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{430, 293.438}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "A cantilever beam has one end rigidly fixed in a column or wall and the \ other end free, as shown in the figure above. The second derivative of the \ elastic curve for the beam shown above is\n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(u\^\[DoublePrime]\)(x)\), "=", RowBox[{ FractionBox[\(w\ L\^2\), RowBox[{"2", StyleBox["EI", FontSlant->"Italic"]}]], "[", \(2 \((x\/L)\) - \((x\/L)\)\^2 - 1\), "]"}]}], TraditionalForm]]], "\n\t\t\nwhere ", StyleBox["w", FontSlant->"Italic"], " is the load on the beam (in pounds per inch of length or newtons per \ meter of length), ", StyleBox["L", FontSlant->"Italic"], " is the length of the beam, ", StyleBox["E", FontSlant->"Italic"], " is a material constant known as Young's modulus, and ", StyleBox["I", FontSlant->"Italic"], " is a constant determined by the geometry of the cross section of the \ beam. For the support at the wall, the initial conditions are ", StyleBox["u", FontSlant->"Italic"], "(0) = 0 and ", Cell[BoxData[ \(TraditionalForm\`\(\(\(u\^\[Prime]\)(0)\)\(=\)\(0\)\(\ \)\)\)]], "(i.e., the beam cannot move up or down, and it cannot rotate at the \ support). These are called initial conditions because the value of the \ function and the derivative are both specified at ", Cell[BoxData[ \(TraditionalForm\`x = 0\)]], "." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["The Elastic Curve and Maximum Deflections", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "Let's determine the equation for the elastic curve and calculate the \ maximum deflection of the beam. \n\nSince ", StyleBox["E", FontSlant->"Italic"], " and ", StyleBox["I", FontSlant->"Italic"], " are reserved symbols in ", StyleBox["Mathematica", FontSlant->"Italic"], ", we will use ", StyleBox["e ", FontSlant->"Italic"], "and", StyleBox[" i", FontSlant->"Italic"], " to represent these quantities." }], "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(\(Clear[u, w, L, e, i, slope, momemt, shear];\)\), "\[IndentingNewLine]", \(\(u\^\[DoublePrime]\)[ x_] = \(\(w\ L\^2\)\/\(2\ e\ i\)\) \((2 \((x\/L)\) - \((x\/L)\)\^2 - 1)\)\)}], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Integrate ", Cell[BoxData[ \(TraditionalForm\`\(\(\(u\^\[DoublePrime]\)(x)\)\(\ \)\)\)]], "to find the slope of the elastic curve. (Don't forget to add the constant \ of integration.)" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[Prime]\)[ x_] = \[Integral]\(u\^\[DoublePrime]\)[x] \[DifferentialD]x + c1\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Integrate ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(x)\)]], " to find ", StyleBox["u", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "), the elastic curve." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = \[Integral]\(u\^\[Prime]\)[x] \[DifferentialD]x + c2\)], "Input",\ PageWidth->PaperWidth], Cell["\<\ Apply the initial conditions at the support to determine the constant of \ integration.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(ics = {u[0] \[Equal] 0, \(u\^\[Prime]\)[0] \[Equal] 0}\)], "Input", PageWidth->PaperWidth], Cell["\<\ In this case, the solution of the initial condition equations is obvious; \ however, in general we would need to solve a system of more complicated \ linear equations to determine c1 and c2. We include that step next for \ completeness of the process.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(soln = Flatten[Solve[ics, {c1, c2}]]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ ButtonBox[ ButtonBox[ RowBox[{ StyleBox["\[MathematicaIcon]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0.792981, 0.777356, 0.144533], FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["About", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.500008]], StyleBox[" ", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Mathematica", FontSize->14, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008]]}], ButtonStyle->"Hyperlink"], ButtonData:>"h1", ButtonStyle->"Hyperlink"]], "Input", PageWidth->PaperWidth, Evaluatable->False, CellTags->"h1b"], Cell["\<\ Replace c1 and c2 in the elastic curve function with the values found in the \ previous step.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = u[x] /. soln\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ ButtonBox[ ButtonBox[ RowBox[{ StyleBox["\[MathematicaIcon]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0.792981, 0.777356, 0.144533], FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["About", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.500008]], StyleBox[" ", FontSize->14, FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Mathematica", FontSize->14, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008]]}], ButtonStyle->"Hyperlink"], ButtonData:>"h2", ButtonStyle->"Hyperlink"]], "Input", PageWidth->PaperWidth, Evaluatable->False, CellTags->"h2b"], Cell[TextData[{ "The maximum deflection will occur at critical points, or at one of the two \ ends of the beam. To find the critical values, look for places where ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(x)\)]], ", the slope of the elastic curve, is either 0 or is undefined. There are \ no values of ", StyleBox["x", FontSlant->"Italic"], " where ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(x)\)]], " is undefined." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(Solve[\(u\^\[Prime]\)[x] \[Equal] 0, x]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Since ", StyleBox["x", FontSlant->"Italic"], " = 0 is not in the interior of the domain 0\[LessEqual]", StyleBox["x\[LessEqual] L ", FontSlant->"Italic"], "and the other roots are complex, there are no critical values of ", StyleBox["x", FontSlant->"Italic"], ". Therefore, the maximum deflection will occur at ", StyleBox["x", FontSlant->"Italic"], " = 0 or at ", StyleBox["x ", FontSlant->"Italic"], "=", StyleBox[" L", FontSlant->"Italic"], ". " }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[0]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(u[L]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "The maximum deflection is at the free end of the beam and is ", Cell[BoxData[ FormBox[ RowBox[{"-", FormBox[\(\(w\ L\^4\)\/\(8 EI\)\), "TraditionalForm"]}], TraditionalForm]]], ". \n\nNow let's put in some numbers and plot the elastic curve and its \ derivative. Let ", Cell[BoxData[ \(TraditionalForm\`L\ = \ 120\)]], " in., ", Cell[BoxData[ \(TraditionalForm\`\(\(w\)\(\ \)\(=\)\)\)]], " 200 ", Cell[BoxData[ \(TraditionalForm\`lb\/in\)]], ", ", Cell[BoxData[ FormBox[ RowBox[{"E", " ", "=", " ", RowBox[{"29", RowBox[{"(", FormBox[\(10\^6\), "TraditionalForm"], ")"}]}]}], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`lb\/in\^2\)]], ", and ", Cell[BoxData[ FormBox[ RowBox[{"I", " ", "=", " ", RowBox[{"500", " ", FormBox[\(in\^4\), "TraditionalForm"]}]}], TraditionalForm]]], "." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(L = 120; w = 200; e = 29*10^6; i = 500;\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[u[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[\(u\^\[Prime]\)[x], {x, 0, L}, PlotRange \[Rule] All, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell["We can also calculate the maximum deflection. ", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[L] // N\)], "Input", PageWidth->PaperWidth], Cell["\<\ The free end of the beam will deflect down 0.358 inches when the specified \ load is applied. \ \>", "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Will the Beam Break?", "Subsection", PageWidth->PaperWidth], Cell["\<\ The engineer is also interested in how much force is required to break the \ beam. There are two ways a beam can break. The first is due to excessive \ bending, and this type of failure is depicted for positive bending in the \ next figure. 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Usually, we are interested in the quantity ", StyleBox["EI", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(x)\)]], ", which is a force quantity called the bending moment. Bending moments \ have units of force times distance (e.g., pound-feet). A plot of the bending \ moment shows that the cantilever beam, with the type of load considered here, \ would fail at the support if overloaded." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(moment[x_] = \(e*i*\(u\^\[DoublePrime]\)[x] // N\) // Simplify\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[moment[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "If a cantilever beam is made of reinforced concrete, steel rebars are \ placed in the top of the beam along its entire length because ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(x)\)]], " is always negative (i.e., negative bending). Consequently, the material \ in the lower portion of the beam is compressed while the material in the \ upper portion is in tension. Can you think of a cantilever beam where the \ load pushes up from underneath causing positive bending?" }], "Text", PageWidth->PaperWidth], Cell["\<\ The second way a beam can fracture is by shearing off, as shown in the next \ figure. The failure shown in the figure is for positive shear. In cases of \ negative shear failure, the picture is inverted.\ \>", "Text", PageWidth->PaperWidth], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0000FX0@0005P000000000000000l3001=0@00 00000000002L5@00=@T00215CDH00040Z0D00280000300000000000000000000?`/00>T>003;0000 3@4000X0000@0000000000000009000040000103001>0@004P0000`000010000DP0004`100010000 a?ooo`0000000000000009010000000014004U@0J@1]06D0L`0P04h0I@1g0200DP1_06d0H@1^0000 0000000000000000000300000000000000000000000004=:DU=DDP008@010000I000004000010040 0P020000001B0d`4K`SZ2P00DP=<16l8jPY_2>X:@00N03l0AP1X0FP10`03000:@00?0040C51D"], "Graphics", PageWidth->PaperWidth, ImageSize->{327, 139}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The third derivative of the elastic curve measures the tendency of a beam \ to shear at any location along its length. Usually we are interested in ", StyleBox["EIu'''(x)", FontSlant->"Italic"], ", which is called the shear force, and it has units of force (e.g., \ pounds). A plot of the shear force shows that the cantilever, subject to the \ load considered here, is most likely to shear off at the support. " }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(shear[x_] = \(e*i*\(\(u\^\[DoublePrime]\)\^\[Prime]\)[x] // N\) // Simplify\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[shear[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell["\<\ Structural engineers must check the shear force and bending moments in beams \ to ensure the beams are strong enough to support their expected loads.\ \>", "Text", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part II: A Beam with Fixed Supports", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["A Description of the Beam and its Supports", "Subsection", PageWidth->PaperWidth], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0002=P0@0005P0003ooooo00000><6003E0`00 00000000002G<0004A/00215CDH00040H2<007h1000500000000000000000000?`/00>T>003;0000 3@4000X0000@0000000000000009000040000><6003F0`004P0000`000010000600000`000000002 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4@0000l0000N0000500005@0001D0000k00001d000060@00H00000400017X^50Aj;Q@>`0000M0000 0@0004`000040000=P0001d0000U0@00_0000500000P00006`0005@0001`0000=P0006D0003g0000 Z00000400017X^50Aj;Q@3H0001U00001P0004`000040000=P0001d0000U0@00_00005P0001306l0 K01e06d0KP0X00007P000140000N0000;`0001h0001D0000E0000?P0001U00004P400:P000010000 Aj;Q@4NRhD3h0000I@000040001<0000100003H0000M00009@400;`0001@0000800001/0000R0000 30000?oooolU000030000040081=0000K00009T500000000]@H00=D3002I1@00000001d1003F0`00 8@3`0000000000000020?`00000000000020?`000000000000000000000000000000000000000000 00000000000U0000300000P0080U0000300000D0080[000060000:/5000;0000h`H00"], "Graphics", PageWidth->PaperWidth, ImageSize->{478, 265.75}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The beam shown above has fixed supports at both ends. In this case, we can \ use the principles of mechanics to determine the fourth derivative of the \ elastic curve. It is \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(u\^\((4)\)\)(x)\), "=", RowBox[{"-", FractionBox["w", StyleBox["EI", FontSlant->"Italic"]]}]}], TraditionalForm]]], ",\t\t\n\nand the boundary conditions are ", StyleBox["u", FontSlant->"Italic"], "(0) = 0, ", Cell[BoxData[ \(TraditionalForm\`\(\(u(L) = 0\)\(,\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(0) = 0\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(L) = 0\)]], ". The first two conditions specify that the beam cannot be displaced up or \ down at the two ends, and the second two conditions specify that it cannot \ rotate at these locations (i.e., the slope of the elastic curve must be 0). \ Zero displacement and zero rotation characterize a fixed support. These \ conditions are called boundary conditions rather than initial conditions \ because they specify conditions on the unknown function ", Cell[BoxData[ \(TraditionalForm\`u(x)\)]], " and its derivatives at more than one place in the domain, that is, at ", Cell[BoxData[ \(TraditionalForm\`x = 0\)]], " and at ", Cell[BoxData[ \(TraditionalForm\`x = L\)]], ".\n\nNow let's perform the following tasks." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Find Elastic Curve, Slopes, Moments, and Shears", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "1. For generic load ", StyleBox["w", FontSlant->"Italic"], ", length ", StyleBox["L", FontSlant->"Italic"], ", material stiffness ", StyleBox["E", FontSlant->"Italic"], ", and cross-section parameter ", StyleBox["I", FontSlant->"Italic"], ", ", "determine the functions for the elastic curve, the slope of the elastic \ curve, the bending moment, and the shear force. To do this, integrate ", Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox["w", StyleBox["EI", FontSlant->"Italic"]]}], TraditionalForm]]], " four times, adding a new constant of integration each time. Then apply \ all four boundary conditions to obtain a system of four linear equations in \ four unknowns (the four constants of integration), and solve the system. ", StyleBox["Mathematica", FontSlant->"Italic"], " Help will show you how to solve a system of equations using the ", StyleBox["Solve[ ]", FontWeight->"Bold"], " command.\n\nFirst, integrate four times, adding a new constant of \ integration each time.\n" }], "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(\(Clear[u, L, e, i, w, shear, moment, c1, c2, c3, c4];\)\), "\n", \(\(\(u\^\[DoublePrime]\)\^\[DoublePrime]\)[ x_] = \(-w\)/\((e*i)\)\)}], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(\(u\^\[Prime]\)\^\[DoublePrime]\)[ x_] = \[Integral]\(\(u\^\[DoublePrime]\)\^\[DoublePrime]\)[ x] \[DifferentialD]x + c1\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[DoublePrime]\)[ x_] = \[Integral]\(\(u\^\[Prime]\)\^\[DoublePrime]\)[ x] \[DifferentialD]x + c2\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[Prime]\)[ x_] = \[Integral]\(u\^\[DoublePrime]\)[x] \[DifferentialD]x + c3\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = \[Integral]\(u\^\[Prime]\)[x] \[DifferentialD]x + c4\)], "Input",\ PageWidth->PaperWidth], Cell["\<\ Apply the boundary conditions, and solve for the constants of integration.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(eqns = {u[0] \[Equal] 0, u[L] \[Equal] 0, \(u\^\[Prime]\)[0] \[Equal] 0, \(u\^\[Prime]\)[L] \[Equal] 0}\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(soln = Flatten[Solve[eqns, {c1, c2, c3, c4}]]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Substitute the solutions back into ", StyleBox["u[x] ", FontWeight->"Bold"], "to determine the equation of the elastic curve for the fixed beam, and \ determine expressions for the slope, bending moment, and shear force." }], "Text", PageWidth->PaperWidth], Cell["a) Elastic Curve:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = u[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell["b) Slope of Elastic Curve:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[Prime]\)[x] // Simplify\)], "Input", PageWidth->PaperWidth], Cell["c) Bending Moment:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(moment[x_] = e*i*\(u\^\[DoublePrime]\)[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell["d) Shear Force:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(shear[x_] = e*i*\(u'''\)[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "2. Given ", Cell[BoxData[ FormBox[ RowBox[{\(L\ = \ 240\ \(\(in\)\(.\)\)\), ",", " ", RowBox[{"w", " ", "=", " ", RowBox[{"200", " ", FormBox[\(lb\/in\), "TraditionalForm"]}]}], ",", " ", RowBox[{"E", " ", "=", " ", RowBox[{"29", RowBox[{"(", FormBox[\(10\^6\), "TraditionalForm"], ")"}], " ", FormBox[\(lb\/in\^2\), "TraditionalForm"]}]}], ",", " ", RowBox[{\(and\ I\), " ", "=", " ", RowBox[{"500", " ", FormBox[\(in\^4\), "TraditionalForm"]}]}]}], TraditionalForm]]], ", plot graphs of the functions found in part (1).\n" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(L = 240; w = 200; e = 29*10^6; i = 500;\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[u[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[\(u\^\[Prime]\)[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[moment[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[shear[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Determine the Extreme Values", "Subsection", PageWidth->PaperWidth], Cell["\<\ 3. Determine the absolute maximum and minimum values of the functions found \ in part (1), and specify where they occur.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ From the graph above, it is evident that the maximum and minimum shear forces \ occur at the two supports. Therefore, the extreme values are as follows:\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(shear[0]\), "\[IndentingNewLine]", \(shear[L]\)}], "Input", PageWidth->PaperWidth], Cell["\<\ Since the shear force is the derivative of the bending moment function, \ critical values of the bending moment function occur wherever the shear force \ is 0 or undefined. There are no places where the shear force is undefined.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xcrit = Solve[shear[x] \[Equal] 0, x]\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the values of the bending moment at the critical values and the ends of \ the beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(moment[0]\), "\[IndentingNewLine]", \(moment[xcrit[\([1, 1, 2]\)]]\), "\[IndentingNewLine]", \(moment[L]\)}], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Extreme values of the slope occur where ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["EI", FontSlant->"Italic"], "u"}]], "\[DoublePrime]"], "(", "x", ")"}], TraditionalForm]]], ", the bending moment, is 0 or undefined, or at the ends of the beam. Note \ that the bending moment is 0 at the inflection points of the elastic curve \ where ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(x) = 0\)]], ". There are no places where the bending moment function is undefined." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xip = Solve[\(u\^\[DoublePrime]\)[x] \[Equal] 0, x] // N\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the values of the slope at the critical points and the ends of the \ beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(\(u\^\[Prime]\)[0]\), "\[IndentingNewLine]", \(\(u\^\[Prime]\)[xip[\([1, 1, 2]\)]]\), "\[IndentingNewLine]", \(\(u\^\[Prime]\)[xip[\([2, 1, 2]\)]]\), "\[IndentingNewLine]", \(\(u\^\[Prime]\)[L]\)}], "Input", PageWidth->PaperWidth], Cell["\<\ In this case, the maximum and minimum slopes occur at the two inflection \ points. The maximum and minimum deflections will occur where the slope is 0 or \ undefined, or at the ends of the beam. There are no places where the slope is \ undefined.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xcritdisp = Solve[\(u\^\[Prime]\)[x] \[Equal] 0, x]\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the displacement at the critical value and the ends of the beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(u[0]\), "\[IndentingNewLine]", \(u[120] // N\), "\[IndentingNewLine]", \(u[240]\)}], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Interpret the Results", "Subsection", PageWidth->PaperWidth], Cell["\<\ 4. Specify where the beam would be most likely to fail in bending and where \ it would be most likely to fail in shear.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ The beam would be most likely to fail in positive bending at the center of \ the span, and it would most likely fail in negative bending at either of the \ supports. The shear force is maximum and minimum at the two supports, and so \ it would likely fail in shear at either one.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ 5. If the beam were made of reinforced concrete, specify where you would put \ the tension steel.\ \>", "Text", PageWidth->PaperWidth], Cell[TextData[{ "The rebars would go in the upper portion of the beam between ", StyleBox["x", FontSlant->"Italic"], " = 0 and ", StyleBox["x", FontSlant->"Italic"], " = 50.75 inches and between ", StyleBox["x", FontSlant->"Italic"], " = 189.25 inches and ", StyleBox["x", FontSlant->"Italic"], " = 240 inches because, in these portions of the span, the beam is in \ negative bending. The rebars would go in the lower portion of the beam \ between ", StyleBox["x", FontSlant->"Italic"], " = 50.75 inches and ", StyleBox["x", FontSlant->"Italic"], " = 189.25 inches because this portion of the beam is in positive bending. \ (Note: The American Concrete Institute's design code requires that the top \ and bottom steel bars extend a specified distance beyond the inflection \ points to ensure that the bars are adequately anchored in the concrete so \ that they can adequately support tension forces.)" }], "Text", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: A Propped Cantilever Beam", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["A Description of the Beam and its Supports", "Subsection", PageWidth->PaperWidth], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@00028d0@0005P00000000000000?D5003E0`00 00000000000?:P004A/00215CDH00040=28007X1000500000000000000000000?`/00>T>003;0000 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In this case, we can use the \ principles of mechanics to determine the fourth derivative of the elastic \ curve. It is \n\n", Cell[BoxData[ FormBox[ RowBox[{\(\(u\^\((4)\)\)(x)\), "=", RowBox[{"-", FractionBox["w", StyleBox["EI", FontSlant->"Italic"]]}]}], TraditionalForm]]], ",\n\t\t\t\t\nand the boundary conditions are ", StyleBox["u", FontSlant->"Italic"], "(0) = 0, ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[Prime]\)(0) = 0, \ u(L) = 0, \)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(L) = 0\)]], ". The first two conditions indicate that the beam cannot move up or down \ at the left support and it cannot rotate there. The second two conditions \ indicate that the beam cannot move up or down at the right support, and the \ bending moment (i.e. ", StyleBox["EI", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(y\^\[DoublePrime]\)(x)\)]], ") is 0 there. \n\nSee if you can accomplish the following tasks." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Find the Elastic Curve, Slopes, Moments, and Shears", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "1. For generic load ", StyleBox["w", FontSlant->"Italic"], ", length ", StyleBox["L", FontSlant->"Italic"], ", material stiffness ", StyleBox["E", FontSlant->"Italic"], ", and cross-section parameter ", StyleBox["I", FontSlant->"Italic"], ", determine the functions for the elastic curve, the slope of the elastic \ curve, the bending moment, and the shear force. Hint: Integrate ", Cell[BoxData[ \(TraditionalForm\`\(-\(w\/EI\)\)\)]], " four times, adding a new constant of integration each time. Then apply \ all four boundary conditions to obtain a system of four equations in four \ unknowns (the four constants of integration), and solve the system. ", StyleBox["Mathematica", FontSlant->"Italic"], " Help will show you how to solve a system of equations using the ", StyleBox["Solve[ ]", FontWeight->"Bold"], " command. To help you, we copied some of the commands from Part II in the \ next cell. You need to change the items that are in red.\n" }], "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(\(Clear[u, L, e, i, w, shear, moment, c1, c2, c3, c4];\)\), "\n", \(\(\(u\^\[DoublePrime]\)\^\[DoublePrime]\)[ x_] = \(-w\)/\((e*i)\)\)}], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(\(u\^\[Prime]\)\^\[DoublePrime]\)[ x_] = \[Integral]\(\(u\^\[DoublePrime]\)\^\[DoublePrime]\)[ x] \[DifferentialD]x + c1\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[DoublePrime]\)[ x_] = \[Integral]\(\(u\^\[Prime]\)\^\[DoublePrime]\)[ x] \[DifferentialD]x + c2\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[Prime]\)[ x_] = \[Integral]\(u\^\[DoublePrime]\)[x] \[DifferentialD]x + c3\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = \[Integral]\(u\^\[Prime]\)[x] \[DifferentialD]x + c4\)], "Input",\ PageWidth->PaperWidth], Cell["\<\ Apply the boundary conditions, and solve for the constants of integration.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ RowBox[{"bondaryconditions", "=", RowBox[{"{", StyleBox[\(u[0] \[Equal] 0, u[L] \[Equal] 0, \(u\^\[Prime]\)[0] \[Equal] 0, \(u\^\[Prime]\)[L] \[Equal] 0\), FontColor->RGBColor[1, 0, 0]], "}"}]}]], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(soln = Flatten[Solve[bondaryconditions, {c1, c2, c3, c4}]]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Substitute the solutions back into ", StyleBox["u[x] ", FontWeight->"Bold"], "to determine the equation of the elastic curve for the fixed beam, and \ determine expressions for the slope, bending moment, and shear force." }], "Text", PageWidth->PaperWidth], Cell["a) Elastic Curve:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(u[x_] = u[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell["b) Slope of Elastic Curve:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(u\^\[Prime]\)[x] // Simplify\)], "Input", PageWidth->PaperWidth], Cell["c) Bending Moment:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(moment[x_] = e*i*\(u\^\[DoublePrime]\)[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell["d) Shear Force:", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(shear[x_] = e*i*\(u'''\)[x] /. soln // Simplify\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "2. Given ", Cell[BoxData[ FormBox[ RowBox[{\(L\ = \ 240\ \(\(in\)\(.\)\)\), ",", " ", RowBox[{"w", " ", "=", " ", RowBox[{"200", " ", FormBox[\(lb\/in\), "TraditionalForm"]}]}], ",", " ", RowBox[{"E", " ", "=", " ", RowBox[{"29", RowBox[{"(", FormBox[\(10\^6\), "TraditionalForm"], ")"}], " ", FormBox[\(lb\/in\^2\), "TraditionalForm"]}]}], ",", " ", RowBox[{\(and\ I\), " ", "=", " ", RowBox[{"500", " ", FormBox[\(in\^4\), "TraditionalForm"]}]}]}], TraditionalForm]]], ", plot graphs of the functions found in part (1).\n" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(L = 240. ; w = 200. ; e = 29. *10^6; i = 500. ;\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[u[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[\(u\^\[Prime]\)[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[moment[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}, PlotRange \[Rule] All];\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(Plot[shear[x], {x, 0, L}, AxesLabel \[Rule] {"\", "\"}];\)\)], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Determine the Extreme Values", "Subsection", PageWidth->PaperWidth], Cell["\<\ 3. Determine the absolute maximum and minimum values of the functions found \ in part (1), and specify where they occur.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ From the graph above, it is evident that the maximum and minimum shear forces \ occur at the two supports. Therefore, the extreme values are as follows:\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(shear[0]\), "\[IndentingNewLine]", \(shear[L]\)}], "Input", PageWidth->PaperWidth], Cell["\<\ Since the shear force is the derivative of the bending moment function, \ critical values of the bending moment function occur wherever the shear force \ is 0 or undefined. There are no places where the shear force is undefined.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xcrit = Solve[shear[x] \[Equal] 0, x]\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the values of the bending moment at the critical values and the ends of \ the beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{\(moment[0]\), "\[IndentingNewLine]", StyleBox[\(moment[xcrit[\([1, 1, 2]\)]]\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(moment[ L]\)}], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Extreme values of the slope occur where ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["EI", FontSlant->"Italic"], "u"}]], "\[DoublePrime]"], "(", "x", ")"}], TraditionalForm]]], ", the bending moment, is 0 or undefined, or at the ends of the beam. Note \ that the bending moment is 0 at the inflection points of the elastic curve \ where ", Cell[BoxData[ \(TraditionalForm\`\(u\^\[DoublePrime]\)(x) = 0\)]], ". There are no places where the bending moment function is undefined." }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xip = Solve[\(u\^\[DoublePrime]\)[x] \[Equal] 0, x] // N\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the values of the slope at the critical points and the ends of the \ beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{\(\(u\^\[Prime]\)[ 0]\), "\[IndentingNewLine]", \(\(u\^\[Prime]\)[ xip[\([1, 1, 2]\)]]\), "\[IndentingNewLine]", StyleBox[\(\(u\^\[Prime]\)[xip[\([2, 1, 2]\)]]\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(\(u\^\[Prime]\)[ L]\)}], "Input", PageWidth->PaperWidth], Cell["\<\ The maximum and minimum deflections will occur where the slope is 0 or \ undefined, or at the ends of the beam. There are no places where the slope is \ undefined.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(xcritdisp = Solve[\(u\^\[Prime]\)[x] \[Equal] 0, x]\)], "Input", PageWidth->PaperWidth], Cell["\<\ Check the displacement at the critical value and the ends of the beam.\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[{\(u[0]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"u", "[", StyleBox[\(xcritdisp[\([2, 1, 2]\)]\), FontColor->RGBColor[1, 0, 0]], "]"}], "//", "N"}], "\[IndentingNewLine]", \(u[240]\)}], "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Interpret the Results", "Subsection", PageWidth->PaperWidth], Cell["\<\ 4. Specify where the beam would be most likely to fail in bending and where \ it would be most likely to fail in shear. 5. If the beam were made of reinforced concrete, specify where you would put \ the tension steel. \ \>", "Text", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: A Beam on Simple Supports", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["A Description of the Beam and its Supports", "Subsection", PageWidth->PaperWidth], Cell["\<\ If a beam simply rests on a support without being rigidly fixed to it, the \ support is said to be a simple support. A simple support prevents vertical \ movement of the beam at the support location, but it does not prevent \ rotation. If the simple support is at the end of a beam, then the bending \ moment is 0 at that location. The support can be a bearing wall, a column, or \ a bridge pier. For example, the support at the right end of the propped \ cantilever beam in the preceding problem is a simple support. 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the following tasks." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Find the Elastic Curve, Slopes, Moments, and Shears", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "1. For generic load ", StyleBox["w", FontSlant->"Italic"], ", length ", StyleBox["L", FontSlant->"Italic"], ", material stiffness ", StyleBox["E", FontSlant->"Italic"], ", and cross section parameter ", StyleBox["I", FontSlant->"Italic"], ", determine the functions for the elastic curve, the slope of the elastic \ curve, the bending moment, and the shear force? Hint: Integrate ", Cell[BoxData[ \(TraditionalForm\`\(-\(w\/EI\)\)\)]], " four times, adding a new constant of integration each time. Then apply \ all four boundary conditions to obtain a system of four equations in four \ unknowns (the four constants of integration), and solve the system. ", StyleBox["Mathematica", FontSlant->"Italic"], " Help will show you how to solve a system of equations using the ", StyleBox["Solve[ ]", FontWeight->"Bold"], " command. This time we leave you on your own, but you can copy and paste \ some of the commands that we used in the preceding sections.\n\n2. Given ", Cell[BoxData[ FormBox[ RowBox[{\(L\ = \ 240\ \(\(in\)\(.\)\)\), ",", " ", RowBox[{"w", " ", "=", " ", RowBox[{"200", " ", FormBox[\(lb\/in\), "TraditionalForm"]}]}], ",", " ", RowBox[{"E", " ", "=", " ", RowBox[{"29", RowBox[{"(", FormBox[\(10\^6\), "TraditionalForm"], ")"}], " ", FormBox[\(lb\/in\^2\), "TraditionalForm"]}]}], ",", " ", RowBox[{\(and\ I\), " ", "=", " ", RowBox[{"500", " ", FormBox[\(in\^4\), "TraditionalForm"]}]}]}], TraditionalForm]]], ", plot graphs of the functions found in part (1)." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Determine the Extreme Values", "Subsection", PageWidth->PaperWidth], Cell["\<\ 3. Determine the absolute maximum and minimum values of the functions found \ in part (1), and specify where they occur.\ \>", "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["Interpret the Results", "Subsection", PageWidth->PaperWidth], Cell["\<\ 4. 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