(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1177367, 43971]*) (*NotebookOutlinePosition[ 1179999, 44050]*) (* CellTagsIndexPosition[ 1179578, 44035]*) (*WindowFrame->Normal*) Notebook[{ Cell["MechanicsExplorers`Beam`", "Chapter", PageWidth->WindowWidth, TextAlignment->Center, TextJustification->0], Cell["\<\ A User-friendly Exploration Program for the Bending of Beams \ \>", "Text", CellDingbat->None, TextAlignment->Center, TextJustification->0, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["Author's Address", "SubsubsectionIcon"], Cell[TextData[{ StyleBox[ "Stephan Kaufmann\nInstitute of Mechanics\nSwiss Federal Institute of \ Technology, ETH Z\[UDoubleDot]rich\nCH-8092 Z\[UDoubleDot]rich, Switzerland\n\ Email: ", "Text"], ButtonBox["kaufmann@ifm.mavt.ethz.ch", ButtonData:>{ URL[ "mailto:kaufmann@ifm.mavt.ethz.ch"], None}, ButtonStyle->"Hyperlink"], StyleBox["\nURL: ", "Text"], ButtonBox["http://www.ifm.ethz.ch/~kaufmann/\n", ButtonData:>{ URL[ "http://www.ifm.ethz.ch/~kaufmann/"], None}, ButtonStyle->"Hyperlink"], "\[Copyright] 1997 by Stephan Kaufmann\nVersion 1.0, March 1997" }], "Text", CellDingbat->None, TextAlignment->Center, TextJustification->0, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]], Cell[CellGroupData[{ Cell["Requirements", "SubsubsectionIcon"], Cell[TextData[{ "This package is written for ", StyleBox["Mathematica", FontSlant->"Italic"], " Version 3.0 or higher. It does not work with earlier versions." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Installation", "SubsubsectionIcon", CellTags->"Installation"], Cell[TextData[{ "Copy the file ", StyleBox["BeamPalette.nb", FontFamily->"Courier"], " (or a link to it) into the directory <", StyleBox["Mathematica", FontSlant->"Italic"], " installation directory>", StyleBox["/Configuration/FrontEnd/Palettes", FontFamily->"Courier"], "." }], "Text", CellDingbat->"\[FilledSmallCircle]"], Cell[TextData[{ "Copy the directory ", StyleBox["MechanicsExplorers", FontFamily->"Courier"], " with its all contents into the directory <", StyleBox["Mathematica", FontSlant->"Italic"], " installation directory>", StyleBox["/AddOns/Autoload", FontFamily->"Courier"], ".\nYou can also use the ", StyleBox["AddOns/Applications", FontFamily->"Courier"], " directory. In this case, you have to ", ButtonBox["manually load the package", ButtonData:>"Load package", ButtonStyle->"Hyperlink"], " before using its functions." }], "Text", CellDingbat->"\[FilledSmallCircle]"], Cell[TextData[{ "Rebuild your help index (use the command ", StyleBox["Rebuild Help Index", FontWeight->"Bold"], " in the menu ", StyleBox["Help", FontWeight->"Bold"], ")." }], "Text", CellDingbat->"\[FilledSmallCircle]"], Cell[TextData[{ "On multi-user systems, a personal installation can be accomplished by \ copying the file ", StyleBox["Beam.m", FontFamily->"Courier"], " to the directory "Italic"], " directory>", StyleBox["/Kernel", FontFamily->"Courier"], " and the file ", StyleBox["BeamPalette.nb", FontFamily->"Courier"], " to the directory "Italic"], " directory>", StyleBox["/Configuration/FrontEnd/Palettes", FontFamily->"Courier"], ", where "Italic"], " directory> usually is ", StyleBox["~/.Mathematica/3.0", FontFamily->"Courier"], ". In this case, the documentation will not be available through the \ Function Browser." }], "Text", CellDingbat->"\[FilledSmallCircle]"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Theoretical Information", "SubsectionIcon"]], "SubsectionIcon"], Cell[CellGroupData[{ Cell["Assumptions", "SubsubsectionIcon"], Cell["\<\ This package uses the Euler-Bernoulli theory for (small) \ deflections of thin elastic straight beams.\ \>", "Text"], Cell[TextData[{ "The coordinate system is chosen such that the ", StyleBox["x", FontSlant->"Italic"], "-axis lies parallel to the beam and points to the right, the ", StyleBox["y", FontSlant->"Italic"], "-axis points vertically downwards, and the ", StyleBox["z", FontSlant->"Italic"], "-axis points backwards into the screen." }], "Text", CellTags->"Coordinates"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .21905 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.069161 0.907029 0.186621 0.907029 [ [.94898 .15034 -4.5 -7 ] [.94898 .15034 4.5 7 ] [.10544 .03243 -4.5 -7.5 ] [.10544 .03243 4.5 7.5 ] [.02381 .18662 -4.5 -7 ] [.02381 .18662 4.5 7 ] [ 0 0 0 0 ] [ 1 .21905 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords arrl 0. eq { 0 0 Mdot } { [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 0 0 ] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke } ifelse grestore } def % End of user PostScript 0 g .5 Mabswid [ ] 0 setdash .06916 .18662 m .97619 .18662 L s % Start of user PostScript 0 0 1. 0 MAarrowhead1 % End of user PostScript .06916 .18662 m .06916 .00522 L s % Start of user PostScript 0 0 0 -0.2 MAarrowhead1 % End of user PostScript newpath .06916 .18662 .02721 0 365.73 arc s .04992 .16738 m .0884 .20586 L s .04992 .20586 m .0884 .16738 L s 3 Mabswid .09637 .18662 m .88549 .18662 L s gsave .94898 .15034 -65.5 -11 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto /Times-Italic findfont 12.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (x) show 1.000000 setlinewidth grestore gsave .10544 .03243 -65.5 -11.5 Mabsadd m 1 1 Mabs scale currentpoint translate 0 23 translate 1 -1 scale 63.000000 15.000000 moveto /Times-Italic findfont 12.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (y) show 1.000000 setlinewidth grestore gsave .02381 .18662 -65.5 -11 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto /Times-Italic findfont 12.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (z) show 1.000000 setlinewidth grestore 0 0 m 1 0 L 1 .21905 L 0 .21905 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Text", ImageSize->{276.688, 60.1875}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg_[jn[Zj^WYjNSXj0<3 0d92@WinO[fm_Oclo5D00:X00?l00=oOgmkNg]gMgMcLg=_Kfm[Jf]WIfMSHf0L71dI6AX61PL30`000 EED0EJX0EOl0ELo?clk>c/g=cLcCXV9RLS8b000oeD0ojX0ool0ojn_[jj^[Zf][Jb/[:^[ ZjZZZZVYZJRXZ1SXf=SHbCTe=CDacXj>STi>CPh>2l_;fi^KZVYZNSXj02Z oeFZojZZoonZobl_;bh^;Rd];B`/;2/[:bXZ:RTY:BPX:33Pd=3@`<30/; 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Only forces in the ", StyleBox["y", FontSlant->"Italic"], "-direction and moments in the ", StyleBox["z", FontSlant->"Italic"], "-direction are considered. Torsion and elongation are neglected." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The Algorithm", "SubsubsectionIcon"], Cell[TextData[{ "The basic function of this package is ", Cell[BoxData[ StyleBox["SolveBeam", FontWeight->"Plain"]], "Input"], ". It calculates the shear force ", Cell[BoxData[ FormBox[ SubscriptBox["S", "y"], TraditionalForm]]], ", the bending moment ", Cell[BoxData[ FormBox[ SubscriptBox["M", "z"], TraditionalForm]]], ", and the deflection ", StyleBox["d", FontSlant->"Italic"], " (as well as its slope ", StyleBox["s", FontSlant->"Italic"], ") for beams with given loads (discrete forces, distributed forces, and \ discrete moments), supports (simple or fixed), and possibly hinges. 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E is Young's modulus, ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["I", FontSlant->"Plain"], "z"], TraditionalForm]]], " the main moment of inertia about the ", StyleBox["z", FontSlant->"Italic"], "-axis." }], "Text"], Cell[TextData[{ "The solution is calculated by integrating the following well known \ differential relations ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["f", "z"]}], TraditionalForm]]], " is the distributed force in ", StyleBox["z", FontSlant->"Italic"], "-direction):" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{ SubscriptBox["S", "y"], "(", "x", ")"}]}], RowBox[{"\[PartialD]", "x"}], MultilineFunction->None], "==", RowBox[{"-", RowBox[{ SubscriptBox["f", "z"], "(", "x", ")"}]}]}], TraditionalForm]]]], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{ SubscriptBox["M", "z"], "(", "x", ")"}]}], RowBox[{"\[PartialD]", "x"}], MultilineFunction->None], "==", RowBox[{"-", RowBox[{ SubscriptBox["S", "y"], "(", "x", ")"}]}]}], TraditionalForm]]]], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[PartialD]", "2"], RowBox[{"d", "(", "x", ")"}]}], RowBox[{"\[PartialD]", SuperscriptBox["x", "2"]}], MultilineFunction->None], "==", FractionBox[ RowBox[{"M", "(", "x", ")"}], SubscriptBox["EI", "z"]]}], TraditionalForm]]]], "Text"], Cell[TextData[{ "Discrete forces and moments (either given externally or in the supports) \ are added in the form of ", Cell[BoxData[ StyleBox["UnitStep", FontWeight->"Plain"]], "Input"], " functions. An unknown force has to be added in a simple support, an \ unknown force and an unknown moment in a fixed support." }], "Text"], Cell["\<\ The unknown forces and moments, as well as the constants of \ integration, can be determined by taking into account the following \ relations.\ \>", "Text"], Cell[TextData[{ "In a simple support at ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ":" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", "(", SubscriptBox["x", "0"], ")"}], "==", "0"}], TraditionalForm]]]], "Text"], Cell[TextData[{ "In a fixed support at ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ":" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", "(", SubscriptBox["x", "0"], ")"}], "==", "0"}], TraditionalForm]]]], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{"d", "(", "x", ")"}]}], RowBox[{"\[PartialD]", "x"}], MultilineFunction->None], "/.", RowBox[{"x", "\[Rule]", RowBox[{ SubscriptBox["x", "0"], "==", "0"}]}]}], TraditionalForm]]]], "Text"], Cell[TextData[{ "In a hinge at ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], ":" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["M", "z"], "(", SubscriptBox["x", "0"], ")"}], "==", "0"}], TraditionalForm]]]], "Text"], Cell[TextData[{ "Furthermore, the limits of ", Cell[BoxData[ FormBox[ SubscriptBox["S", "y"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["M", "z"], ","}], TraditionalForm]]], " as ", StyleBox["x", FontSlant->"Italic"], " approaches the end points of the beam from outside of the beam, have to \ vanish." }], "Text"], Cell["\<\ This works for statically determinate and statically indeterminate \ systems.\ \>", "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Using the Package", "SubsectionIcon"], Cell[TextData[{ "The functions of this package work best if the positions and values of \ supports and loads are dimensionless numerical quantities. As far as \ possible, the solutions can also be derived for symbolic values. But, in this \ case, the results tend to be complicated, and the plotting functions will not \ work. If you want to use units, please read the ", ButtonBox["remark below", ButtonData:>"Remark about Units", ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "All functions defined in the package can be used from the ", StyleBox["BeamPalette", FontWeight->"Bold"], " which appears in the ", StyleBox["File", FontWeight->"Bold"], " > ", StyleBox["Palette", FontWeight->"Bold"], " menu after the ", ButtonBox["installation", ButtonData:>"Installation", ButtonStyle->"Hyperlink"], ". There are three levels of simplicity for the interaction with the \ package:" }], "Text"], Cell[CellGroupData[{ Cell["Simple", "SubsubsectionIcon"], Cell[TextData[{ "You are guided through the use of the functions, either by dialogs, or by \ text which is pasted into your notebook. The corresponding switch (", StyleBox["Input by Dialogs", FontWeight->"Bold"], " or ", StyleBox["Input by Evaluating Expressions", FontWeight->"Bold"], ") can be set from the palette. All calculations are done with hidden work \ objects." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Medium", "SubsubsectionIcon"], Cell["\<\ The medium level also uses hidden work objects for the definitions \ and calculations. But here, the palettes only paste templates into the \ notebook. You are expected to supply the parameters and to evaluate the \ expressions.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Advanced", "SubsubsectionIcon"], Cell[TextData[{ "The advanced level basically uses the same functions as above, but you are \ requested to give a name for the beam and for the solution. Instead of \ constructing a beam step by step, you can directly define the end points, \ supports, and loads in a ", StyleBox["Beam", FontFamily->"Courier"], " object." }], "Text"], Cell[TextData[{ "In the advanced mode, the solution has to be calculated explicitly with \ the function ", StyleBox["SolveBeam", FontFamily->"Courier"], ". 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