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"fromtitle": "Main Page",
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"totitle": "Direct Stiffness Method",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\" >Line 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Line 1:</td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div><del class=\"diffchange diffchange-inline\"><strong>MediaWiki has been installed.</strong></del></div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">by Katie Lundgren and Eric Jenkins</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">External edit by Lucas McGill</ins></div></td></tr>\n<tr><td class='diff-marker'>\u00a0</td><td class='diff-context'></td><td class='diff-marker'>\u00a0</td><td class='diff-context'></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div><del class=\"diffchange diffchange-inline\">Consult the [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents User's Guide] for information on using the wiki software.</del></div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">__TOC__</ins></div></td></tr>\n<tr><td class='diff-marker'>\u00a0</td><td class='diff-context'></td><td class='diff-marker'>\u00a0</td><td class='diff-context'></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div>== <del class=\"diffchange diffchange-inline\">Getting started </del>==</div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div>==<ins class=\"diffchange diffchange-inline\">Introduction</ins>==</div></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Configuration_settings Configuration settings list</del>]</div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">FAQ MediaWiki FAQ</del>]</div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[image:StuartburnTrussBridge.jpg|right|thumb|250px|Truss Bridge in Gardenton, Manitoba, Canada]]</ins></div></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">mailman</del>/<del class=\"diffchange diffchange-inline\">listinfo</del>/<del class=\"diffchange diffchange-inline\">mediawiki</del>-<del class=\"diffchange diffchange-inline\">announce MediaWiki release mailing list</del>]</div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">This Wiki is meant to give you just a taste of structural engineering by presenting a crash-course in using the Direct Stiffness Method for structural analysis.\u00a0 Below, you will read about the analysis of simple 2-dimensional truss structures.\u00a0 These same techniques are used in real-world applications, such as truss-bridge design!\u00a0 </ins></div></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/wiki/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Localise MediaWiki for your language</del>]</div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td class='diff-marker'>\u2212</td><td class='diff-deletedline'><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/wiki/Special:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual:Combating_spam Learn how </del>to <del class=\"diffchange diffchange-inline\">combat spam on your wiki]</del></div></td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Structural Analysis===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Structural analysis is the computation of deflections, deformations, and internal forces or stresses on and within static structures.\u00a0 Structural analysis methods allow one to both analyze the strength of existing structures as well as predict the strength of structures in design.\u00a0 Engineers use various standard methods in the design of structures such as bridges, buildings, roadways, and pipelines.\u00a0 But these methods of analysis we will address can also be applied to structures of much smaller or larger scales - everything from the durability of a microchip to the safety of a coal mine.\u00a0 Analysis requires input data, including (but not limited to) structural geometry, support conditions, and structural loads.\u00a0 Output can include reactions, stresses, and displacements that result from the input.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">In any form of structural analysis, simplifications, or 'idealizations', must be made to structures to analyze them mathematically.\u00a0 When designing a bridge, structural engineers cannot predict every wind storm, earthquake, or material defect that may weaken the strength of the structure.\u00a0 So assumptions must be made, and factors of safety must be applied to account for uncertainty in design.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Material Science===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Any material can fail with enough applied stress.\u00a0 However, predicting strength can be difficult, since defects in a material's design are unpredictable and inevitable.\u00a0 In addition to production defects, a material's durability can be affected by outside factors, like weather and age.\u00a0 Aluminum, for example, loses structural strength every time it undergoes stress changes, even if the stresses are magnitudes lower than the material's failing ultimate strength.\u00a0 This is why airplanes must be constantly checked for signs of wear.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The vast majority of structural analysis and design considers only the ''linear-elastic range'' of a material's strength behavior.\u00a0 Linear elasticity means that there is a direct linear relationship between the amount of force applied and deflection, and any deformation caused from stresses in this range are temporary and recoverable.\u00a0 If a force is applied to a steel beam causing it to deflect, then halving the applied force will half the beam's deflection.\u00a0 This linear-elastic property holds true until the applied stress reaches ''yield stress''.\u00a0 Yielding is\u00a0 the point at which the material begins to deform irreversibly and non-linearly.\u00a0 Yielding strength is typically much lower than the material's breaking point, but structural designers generally attempt to keep structures in this region of linear load-deformation response.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Finite Element Method===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The finite element method (FEM) is a simple and efficient method of finding a numerical approximation for a mathematical model of a structure.\u00a0 It involves taking a complex problem and decomposing it into pieces upon each of which a simple approximation of the solution may be constructed, and then putting the local approximate solutions together to obtain a global approximate solution.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">==Direct Stiffness Method==</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The direct stiffness method (DSM) is the most common implementation of the FEM.\u00a0 It is particularly used for computer-automated analysis of complex structures.\u00a0 The method begins with breaking down the model of the complex structure into simpler idealized parts.\u00a0 These material stiffness properties of these parts are then individually evaluated.\u00a0 The behavior of the entire idealized structure is determined by compiling the stiffness equations of the smaller parts into one single matrix equation.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The DSM process can be broken down into three major steps: '''Breakdown''', '''Assembly''', and '''Solution'''.\u00a0 These three major steps are outlined below.\u00a0 In this Wiki, we will focus specifically on 2-dimensional truss structures.\u00a0 Trusses are assumed to only transmit forces in compression and tension, which simplifies the idealization process greatly.\u00a0 In a truss structure, all ends are pinned, meaning these hinges are free to rotate, but they can still be restricted from horizontal and vertical movement.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Breakdown===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">There are three main sub-steps in the Breakdown process: Disconnection, Localization, and Member Formation.\u00a0 First, individual elements in the structure must be identified.\u00a0 The structure can then be broken down into individual elements, separated at the structure's ''nodes'', the points which hold the elements together:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[Image:DSMImage2.png]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Each member can then be analyzed individually.\u00a0 In 2-dimensional truss analysis, each end of the truss can have two degrees of freedom (corresponding to horizontal and vertical displacement).\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Analysis of a truss member is done with respect to its localized coordinate system.\u00a0 In the triangular truss system shown above, elements (2) and (3) do not line up with the structure's global coordinate system.\u00a0 Deformations and reaction forces will be computed relative to the local member and later converted to the global coordinate system, as seen in the following section, ''Assembly''.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[Image:DSMTrussBreakdown.jpg]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The basic 2D global->local coordinate transformation scheme can be derived by four basic equations that relate local horizontal and vertical coordinates to the global system, where <math>\\begin{matrix}u\\end{matrix}</math> represents an element's global coordinate displacement vector, and <math>\\begin{matrix}\\bar u\\end{matrix}</math> represents the same element displacement vector in local coordinates:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{matrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xi} = u_{xi}cos(\\phi) + u_{yi}sin(\\phi)\\;\\;\\;\\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xj} = u_{xj}cos(\\theta) + u_{yj}sin(\\phi)\\;\\;\\;\\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yi} = -u_{xi}cos(\\phi) + u_{yi}cos(\\phi)\\;\\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yj} = -u_{xj}sin(\\phi) + u_{xj}cos(\\phi)</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{matrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">These local<-->global transformations are geometrically-derived relations that lay the basis for a general transformation matrix: <math>\\bar u = T * u</math> or <math>\\bar f = T </ins>* <ins class=\"diffchange diffchange-inline\">f</math> where T is the transformation matrix:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xj} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yi}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">cos(\\phi) & sin(\\phi) & 0 & 0 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">-sin(\\phi) & cos(\\phi) & 0 & 0 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">0 & 0 & cos(\\phi) & sin(\\phi) \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">0 & 0 & -sin(\\phi) & cos(\\phi)</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u_{xi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u_{yi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u_{xj} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u_{yj}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Once displacements in the individual element are calculated, the truss's reaction forces at its two nodes can be found with a linear displacement->force relationship.\u00a0 Forces on a truss can be related to the truss's resulting displacement using a ''stiffness matrix'':</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 f_{x1} \\\\ f_{y1} \\\\ f_{x2} \\\\ f_{y2}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 k_{11} & k_{12} & k_{13} & k_{14} \\\\ </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 k_{21} & k_{22} & k_{23} & k_{24} \\\\ </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 k_{31} & k_{32} & k_{33} & k_{34} \\\\ </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 k_{41} & k_{42} & k_{43} & k_{44} \\\\ </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 u_{x1} \\\\ u_{y1} \\\\ u_{x2} \\\\ u_{y2}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">In local coordinate analysis, force and elongation can be expressed in terms of joint forces and displacements</ins>:</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>F = \\bar f_{xj} = -\\bar f_{xi}, d = \\bar u_{xj} - \\bar u_{xi}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">This leads to the stiffness matrix necessary for relating an element's elongation to its joint forces:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar f_{xi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar f_{yi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar f_{xj} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar f_{yj}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"> = \\frac{E A} {L}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 1 & 0 & -1 & 0 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 0 & 0 & 0 & 0 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -1 & 0 & 1 & 0 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 0 & 0 & 0 & 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yi} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{xj} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\bar u_{yj}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">= \\bar K \\bar u</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><</ins>/<ins class=\"diffchange diffchange-inline\">math>,</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">where E is the element's elastic modulus, A is its cross-sectional area, and L is its length.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Assembly===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Assembly involves two sub-steps: ''globalization'' and ''merging''.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">''Globalization'' is through which the member stiffness equations are transformed back to the global coordinate system</ins>. <ins class=\"diffchange diffchange-inline\"> Relations connecting joint displacements and forces in the local and global coordinate systems can be established in terms of transformation matrices</ins>. <ins class=\"diffchange diffchange-inline\"> </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Now that the local<-->global coordinate transformation is understood, an element's stiffness matrix, K, can be re-written in terms of global coordinates:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>K^{(e)} = \\frac{E^{(e)}A^{(e)}}{L^{(e)}}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">cos^2(\\phi) & sin(\\phi)cos(\\phi) & -cos^2(\\phi) & -sin(\\phi)cos(\\phi) \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">sin(\\phi)cos(\\phi) & sin^2(\\phi) & -sin(\\phi)cos(\\phi) & -sin^2(\\phi) \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">-cos^2(\\phi) & -sin(\\phi)cos(\\phi) & cos^2(\\phi) & sin(\\phi)cos(\\phi) \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">-sin(\\phi)cos(\\phi) & -sin^2(\\phi) & sin(\\phi)cos(\\phi) & sin^2(\\phi) </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Once the member stiffness equations are written in terms of global coordinates, they can be ''merged'' into a stiffness matrix of the complete structure.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Knowing that <math>\\begin{matrix}f^{(e)} = K^{(e)}u{(e)}\\end{matrix}<</ins>/<ins class=\"diffchange diffchange-inline\">math>, we can merge multiple elements into a larger stiffness matrix, a force vector, and a displacement vector that represent all of the nodes of the structure, all in global coordinates</ins>:</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>\\begin{matrix}f = f^{(1)} + f^{(2)} + ... + f^{(n)} = (K^{(1)} + K^{(2)} + ... + K^{(n)})u\\end{matrix}</math>,</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">where n represents the total number of elements in the structure.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Solution===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Once the master stiffness equation has been formed, a method can be chosen for solving it.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Solving for the overall structure's unknown reaction forces and resulting nodal displacements is a simple matter of solving for the unknowns in the global structure's force-displacement relation, <math>\\begin{matrix}f = Ku\\end{matrix}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">See the following section for an example of this algorithm, carried out step-by-step with MATLAB.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Example Problem===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">''This is a sample problem taken from Exercise 3.7 of Felippa's Finite Element Methods book. ''</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Consider the two-member arch-truss structure shown below.\u00a0 Take span S = 8, height H = 3, elastic modulus E = 1000, cross section areas A(1) = 2 and A(2) = 4, and horizontal crown force P = 12, as shown here.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[Image</ins>:<ins class=\"diffchange diffchange-inline\">DSMExampleTruss.JPG</ins>]<ins class=\"diffchange diffchange-inline\">]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">'''Problem'''</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Compute nodal displacements, nodal reaction forces, and internal truss forces for the structure above.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">'''Solution'''</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The two-truss structure is held together at three nodes: two on the ground ''(nodes 1 and 3)'' and one node that connects member (1) to member (2) ''(node 2)''.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Nodes 1 and 3 are stationary, and can have no displacements in the x or y direction.\u00a0 However, node 2 is not affixed to ground and is free to move as a result of any imposed forces.\u00a0 The hinged nodes 1 and 3 cannot move and instead transfer any forces imposed on the structure into the ground as reactions, expressed in x-y vector components - the same way displacements are express in x-y vector components.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">MATLAB Direct Stiffness analysis begins by creating a ''global force vector'', f, which represents the degrees of freedom of each of the nodes that compose our structure.\u00a0 Since our structure has three nodes and each node has two 2-dimensional degrees of freedom (x and y), the master force and displacement vectors, f and u, respectively, will contain 6 elements: </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>f = </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 f_{x1} \\\\ f_{y1} \\\\ f_{x2} \\\\ f_{y2} \\\\ f_{x3} \\\\ f_{y3}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 - \\\\ - \\\\ 12 \\\\ 0 \\\\ - \\\\ -</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">and </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>u = </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 u_{x1} \\\\ u_{y1} \\\\ u_{x2} \\\\ u_{y2} \\\\ u_{x3} \\\\ u_{y3}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 0 \\\\ 0 \\\\ - \\\\ - \\\\ 0 \\\\ 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">where the reaction forces at nodes 1 and 3 and the displacement at node 2 are initially unknown.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">In addition to the initial loading and displacement vectors, a master stiffness matrix, K, is constructed.\u00a0 A master stiffness matrix is generated by combining the two elements' globalized matrices.\u00a0 The following is the MATLAB driver function for generating the two elements' global-coordinate stiffness matrices and merging them into a master stiffness matrix:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><matlab>%***********************************************************************</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">%\u00a0 function AssembleMasterStiffOfExTruss</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">%</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">%\u00a0 Forms the 6x6 global stiffness matrix of the example problem</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">%******************************************************************</ins>*<ins class=\"diffchange diffchange-inline\">****</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">function K = AssembleMasterStiffOfExTruss</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% global matrix</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">K = zeros(6);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% element 1:\u00a0 E = 1000 A = 2</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Ke = ElemStiff2DTwoNodeBar(-4, 0, 0, 3, 1000, 2);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">K = MergeElemIntoMasterStiff(Ke, [1,2,3,4], K);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% element 2:\u00a0 E = 1000 A = 4</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Ke = ElemStiff2DTwoNodeBar(0, 3, 4, 0, 1000, 4);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">K = MergeElemIntoMasterStiff(Ke, [3,4,5,6], K);</matlab></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">After assembling the master stiffness matrix, displacement boundary conditions are placed on nodes (1) and (3), which restrict movement in these ground-hinged nodes.\u00a0 Then the force vector, f, is modified to become fmod.\u00a0 fmod accounts for pre-described displacements in any of the nodal elements.\u00a0 In our case, neither of the ground nodes will have any kind of pre-described displacements.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Finally, solution vectors can be obtained.\u00a0 Vector u corresponds to the final nodal displacements, vector f corresponds to reactional forces at the nodes, and vector p corresponds to the trusses internal reaction forces.\u00a0 All values are relative to the global coordinate system specified in the above figure.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The following is the main MATLAB driver function for solving this truss structure problem:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><matlab>function driver</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Function driver</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Driver function to find the nodal displacements, reaction forces, and</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% internal truss forces of Felippa Exercise 3.7.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Set up a global force vector, corresponding to the x and y values of</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% each of the three nodes to be analyzed, and save it to variable f.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">f = [0; 0; 12; 0; 0; 0];</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Assemble the master stiffness matrix, K, and save it to variable K</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">K = AssembleMasterStiffOfExTruss;</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Modify the global K matrix to impose Displacement Boundary Conditions -</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% [1, 2, 5, 6] restricts x- and y- movement for nodes 1 and 3.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Kmod = ModifiedMasterStiffForDBC( [1, 2, 5, 6], K);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Modify the global load vector to impose homogeneous displacement BCs - in</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% our case, there will be no pre-described nodal displacements.\u00a0 So the two</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% ground hinge nodes, 1 and 3 - which correspond to elements 1, 2, 5, and 6</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% in the force and displacement vectors - will all be set to 0.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">fmod = ModifiedMasterForceForDBC( </ins>[<ins class=\"diffchange diffchange-inline\">1, 2, 5, 6], [0, 0, 0, 0], f, K);</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">fprintf('\\nComputed nodal displacements:')</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Since Kmod is symmetric and positive definite, operator \\ calls</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% chol(Kmod) and do a forward and back substitution, solving </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Kmod * u = fmod for u, the global displacement vector.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u = Kmod\\fmod</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Multiply the resulting global node displacements by the master stiffness</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% matrix, K, to compute the resulting node reaction forces.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">fprintf('\\nExternal node forces including reactions:')</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">f = K * u</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% Calculate the internal forces of the two trusses, based on global node</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">% displacements, u.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">fprintf('\\nInternal member forces</ins>:<ins class=\"diffchange diffchange-inline\">')</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">p = IntForcesOfExampleTruss( u )<</ins>/<ins class=\"diffchange diffchange-inline\">matlab></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">MATLAB's output from running driver.m :</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><pre>>> driver</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Computed nodal displacements:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">u =</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 \u00a0 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 \u00a0 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 0.0176</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 0.0078</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 \u00a0 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 \u00a0 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">External node forces including reactions:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">f =</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -6.0000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -4.5000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 12.0000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 \u00a0 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -6.0000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 4.5000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Internal member forces:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">p =</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 7.5000</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -7.5000<</ins>/<ins class=\"diffchange diffchange-inline\">pre></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">As seen from MATLAB's output above, elements 3 and 4 of the 6-element displacement vector u have non-zero values, which correspond to Node 2 horizontal displacement and Node 2 vertical displacement, respectively:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>u = </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 u_{x1} \\\\ u_{y1} \\\\ u_{x2} \\\\ u_{y2} \\\\ u_{x3} \\\\ u_{y3}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 0 \\\\ 0 \\\\ 0</ins>.<ins class=\"diffchange diffchange-inline\">0176 \\\\ 0</ins>.<ins class=\"diffchange diffchange-inline\">0078 \\\\ 0 \\\\ 0</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Node 2 is displaced 0.0176 m in the positive x-direction and 0.0078 m in the positive y-direction.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Nodal reaction forces are also listed in MATLAB's output for vector f:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>u = </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 f_{x1} \\\\ f_{y1} \\\\ f_{x2} \\\\ f_{y2} \\\\ f_{x3} \\\\ f_{y3}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 -6 \\\\ -4.5 \\\\ 12 \\\\ 0 \\\\ -6 \\\\ 4.5</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The first two elements correspond to reactions at Node 1: 6 to the left and 4.5 down.\u00a0 The third element of f, 12, corresponds to the pre-defined horizontal force at Node 2.\u00a0 Finally, the last two elements of vector f, -6 and 4.5, refer to the reactions at Node 3: 6 to the left and 4.5 up.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Finally, MATLAB outputs its result for the internal forces in the two trusses as vector p, tension-positive</ins>:</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\"><math>p = </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 p_1 \\\\ p_2</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">=</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\begin{bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\u00a0 7.5 \\\\ -7.5</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">\\end{bmatrix}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The first element of vector p, 7.5, refers to the magnitude of reaction tensile force in Truss (1).\u00a0 The second element, -7.5, refers to the magnitude of compressive force in Truss (2).\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Software analysis using F-Tool confirms our resulting displacements and reaction forces in the hinges, as shown below:</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[Image:DSMTrussWithReactions.JPG]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">This example is clearly a very light sample of how the Direct Stiffness Method can be used to analyze truss structures.\u00a0 But the method is identical no matter how complicated the truss system may be.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===A Note on the Indeterminate Nature of Structures===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">By Lucas McGIll.\u00a0 Many simple truss problems like the one above can be solved in a few minutes merely by hand. That is, simply summing the forces and moments to solve for the reactions. A problem arises when there are more constraints than equilibrium equations. If the truss is over constrained, then the system of equations is deemed indeterminate. The beauty of this method, however, is that indeterminate can still easily be solved. For this reason, the DSM is used in many computer packages to quickly solve for deflection, forces, stress, etc. by generating meshes on not only truss structures but also solid parts.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">===Relevant MATLAB Files===</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The following are links to the MATLAB files necessary for running analysis of this example truss.\u00a0 </ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:driver.m|driver.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:AssembleMasterStiffOfExTruss.m|AssembleMasterStiffOfExTruss.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:ElemStiff2DTwoNodeBar.m|ElemStiff2DTwoNodeBar.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:MergeElemIntoMasterStiff.m|MergeElemIntoMasterStiff.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:ModifiedMasterStiffForDBC.m|ModifiedMasterStiffForDBC.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:ModifiedMasterForceForDBC.m|ModifiedMasterForceForDBC.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media:IntForcesOfExampleTruss.m|IntForcesOfExampleTruss.m]]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[[media</ins>:<ins class=\"diffchange diffchange-inline\">IntForce2DTwoNodeBar.m|IntForce2DTwoNodeBar.m</ins>]<ins class=\"diffchange diffchange-inline\">]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">==Conclusion==</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">The Direct Stiffness Method is a fairly straightforward algorithm that can be used to compute reactional forces, internal forces, and displacements in individual members of a structure.\u00a0 More complicated applications of the DSM involve complex structures in 2D or 3D that can carry distributed loads across members, carry moments, and carry rotational torsion.\u00a0 DSM is probably the most efficient way to quickly design buildings and structures for various purposes.\u00a0 All thanks to derivations and modifications from Finite Element Methods.</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">==References==</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Felippa, Carlos A. ''Introduction to Finite Element Method.'' 2001. University of Colorado. <</ins>[<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch00.d/IFEM</ins>.<ins class=\"diffchange diffchange-inline\">Ch00</ins>.<ins class=\"diffchange diffchange-inline\">pdf Available online here]></ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">Martha, Luis Fernando.\u00a0 [http:/</ins>/<ins class=\"diffchange diffchange-inline\">www.tecgraf.puc-rio.br</ins>/<ins class=\"diffchange diffchange-inline\">ftool</ins>/<ins class=\"diffchange diffchange-inline\">ftooleng.html Ftool - Two</ins>-<ins class=\"diffchange diffchange-inline\">dimensional Frame Analysis Tool</ins>]</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>[<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">en</ins>.<ins class=\"diffchange diffchange-inline\">wikipedia</ins>.org/wiki/<ins class=\"diffchange diffchange-inline\">Direct_stiffness_method Wikipedia.org - Direct Stiffness Method]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div><ins class=\"diffchange diffchange-inline\">[http</ins>:/<ins class=\"diffchange diffchange-inline\">/en.wikipedia.org/wiki/Finite_element_method Wikipedia.org - Finite Element Method</ins>]</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>[<ins class=\"diffchange diffchange-inline\">http</ins>://<ins class=\"diffchange diffchange-inline\">en</ins>.<ins class=\"diffchange diffchange-inline\">wikipedia</ins>.org/wiki/<ins class=\"diffchange diffchange-inline\">Structural_analysis Wikipedia.org - Structural Analysis]</ins></div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\">\u00a0</td><td class='diff-marker'>+</td><td class='diff-addedline'><div>Special <ins class=\"diffchange diffchange-inline\">thanks to [http</ins>:/<ins class=\"diffchange diffchange-inline\">/cern49.ce.uiuc.edu/strweb/duarte.html Professor C. Armando Duarte] and </ins>to <ins class=\"diffchange diffchange-inline\">the U of I TAM and CEE departments!</ins></div></td></tr>\n"
}
}