Finite element method in structural mechanics

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class="vector-toc-link" href="#Theoretical_overview_of_FEM-Displacement_Formulation:_From_elements,_to_system,_to_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Theoretical overview of FEM-Displacement Formulation: From elements, to system, to solution</span> </div> </a> <ul id="toc-Theoretical_overview_of_FEM-Displacement_Formulation:_From_elements,_to_system,_to_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpolation_or_shape_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpolation_or_shape_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Interpolation or shape functions</span> </div> </a> <ul id="toc-Interpolation_or_shape_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_virtual_work_in_a_typical_element" class="vector-toc-list-item vector-toc-level-1 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id="toc-Element_virtual_work_in_terms_of_system_nodal_displacements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Element_virtual_work_in_terms_of_system_nodal_displacements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Element virtual work in terms of system nodal displacements</span> </div> </a> <ul id="toc-Element_virtual_work_in_terms_of_system_nodal_displacements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-System_virtual_work" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#System_virtual_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>System virtual work</span> </div> </a> <ul id="toc-System_virtual_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assembly_of_system_matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Assembly_of_system_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Assembly of system matrices</span> </div> </a> <ul id="toc-Assembly_of_system_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main 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</div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b><a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a></b> (FEM) is a powerful technique originally developed for numerical solution of complex problems in <a href="/wiki/Structural_mechanics" title="Structural mechanics">structural mechanics</a>, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate <b>finite elements</b> interconnected at discrete points called nodes. Elements may have physical properties such as thickness, <a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">coefficient of thermal expansion</a>, <a href="/wiki/Density" title="Density">density</a>, <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> and <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The origin of finite method can be traced to the matrix analysis of structures <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> where the concept of a displacement or stiffness matrix approach was introduced. Finite element concepts were developed based on engineering methods in 1950s. The finite element method obtained its real impetus in the 1960s and 1970s by <a href="/wiki/John_Argyris" title="John Argyris">John Argyris</a>, and co-workers; at the <a href="/wiki/University_of_Stuttgart" title="University of Stuttgart">University of Stuttgart</a>, by <a href="/wiki/Ray_W._Clough" class="mw-redirect" title="Ray W. Clough">Ray W. Clough</a>; at the <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a>, by <a href="/wiki/Olgierd_Zienkiewicz" title="Olgierd Zienkiewicz">Olgierd Zienkiewicz</a>, and co-workers <a href="/wiki/Ernest_Hinton" title="Ernest Hinton">Ernest Hinton</a>, <a href="/wiki/Bruce_Irons_(engineer)" title="Bruce Irons (engineer)">Bruce Irons</a>;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> at the <a href="/wiki/Swansea_University" title="Swansea University">University of Swansea</a>, by <a href="/wiki/Philippe_G._Ciarlet" title="Philippe G. Ciarlet">Philippe G. Ciarlet</a>; at the <a href="/wiki/Pierre-and-Marie-Curie_University" class="mw-redirect" title="Pierre-and-Marie-Curie University">University of Paris</a>; at <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>, by Richard Gallagher and co-workers. The original works such as those by Argyris <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and Clough <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> became the foundation for today’s finite element structural analysis methods. </p><p>Straight or curved one-dimensional elements with physical properties such as axial, bending, and torsional stiffnesses. This type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames. Straight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes. The elements are positioned at the <a href="/wiki/Centroid" title="Centroid">centroidal</a> axis of the actual members. </p> <ul><li>Two-dimensional elements that resist only in-plane forces by membrane action (plane <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>, plane <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a>), and plates that resist transverse loads by transverse shear and bending action (plates and <a href="/wiki/Thin-shell_structure" class="mw-redirect" title="Thin-shell structure">shells</a>). They may have a variety of shapes such as flat or curved <a href="/wiki/Triangle" title="Triangle">triangles</a> and <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilaterals</a>. Nodes are usually placed at the element corners, and if needed for higher accuracy, additional nodes can be placed along the element edges or even within the element. The elements are positioned at the mid-surface of the actual layer thickness.</li> <li><a href="/wiki/Torus" title="Torus">Torus</a>-shaped elements for axisymmetric problems such as membranes, thick plates, shells, and solids. The cross-section of the elements are similar to the previously described types: one-dimensional for thin plates and shells, and two-dimensional for solids, thick plates and shells.</li> <li>Three-dimensional elements for modeling 3-D solids such as <a href="/wiki/Machine" title="Machine">machine</a> components, <a href="/wiki/Dam" title="Dam">dams</a>, <a href="/wiki/Embankment_(transportation)" class="mw-redirect" title="Embankment (transportation)">embankments</a> or soil masses. Common element shapes include <a href="/wiki/Tetrahedral" class="mw-redirect" title="Tetrahedral">tetrahedrals</a> and <a href="/wiki/Hexahedral" class="mw-redirect" title="Hexahedral">hexahedrals</a>. Nodes are placed at the vertexes and possibly in the element faces or within the element.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Element_interconnection_and_displacement">Element interconnection and displacement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=2" title="Edit section: Element interconnection and displacement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible. Nodes will have nodal <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">(vector) displacements</a> or <a href="/wiki/Degrees_of_freedom_(engineering)" class="mw-redirect" title="Degrees of freedom (engineering)">degrees of freedom</a> which may include translations, rotations, and for special applications, higher order <a href="/wiki/Derivative" title="Derivative">derivatives</a> of displacements. When the nodes displace, they will <i>drag</i> the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be <a href="/wiki/Interpolation" title="Interpolation">interpolated</a> from the nodal displacements, and this is the main reason for the approximate nature of the solution. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_considerations">Practical considerations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=3" title="Edit section: Practical considerations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the application point of view, it is important to model the system such that: </p> <ul><li>Symmetry or anti-symmetry conditions are exploited in order to reduce the size of the model.</li> <li>Displacement compatibility, including any required discontinuity, is ensured at the nodes, and preferably, along the element edges as well, particularly when adjacent elements are of different types, material or thickness. Compatibility of displacements of many nodes can usually be imposed via constraint relations.</li> <li>Elements' behaviors must capture the dominant actions of the actual system, both locally and globally.</li> <li>The element mesh should be sufficiently fine in order to produce acceptable accuracy. To assess accuracy, the mesh is refined until the important results shows little change. For higher accuracy, the <a href="/wiki/Aspect_ratio_(image)" title="Aspect ratio (image)">aspect ratio</a> of the elements should be as close to unity as possible, and smaller elements are used over the parts of higher stress <a href="/wiki/Gradient" title="Gradient">gradient</a>.</li> <li>Proper support constraints are imposed with special attention paid to nodes on symmetry axes.</li></ul> <p>Large scale commercial software packages often provide facilities for generating the mesh, and the graphical display of input and output, which greatly facilitate the verification of both input data and interpretation of the results. </p> <div class="mw-heading mw-heading2"><h2 id="Theoretical_overview_of_FEM-Displacement_Formulation:_From_elements,_to_system,_to_solution"><span id="Theoretical_overview_of_FEM-Displacement_Formulation:_From_elements.2C_to_system.2C_to_solution"></span>Theoretical overview of FEM-Displacement Formulation: From elements, to system, to solution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=4" title="Edit section: Theoretical overview of FEM-Displacement Formulation: From elements, to system, to solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the theory of FEM can be presented in different perspectives or emphases, its development for <a href="/wiki/Structural_analysis" title="Structural analysis">structural analysis</a> follows the more traditional approach via the <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> principle or the <a href="/wiki/Minimum_total_potential_energy_principle" title="Minimum total potential energy principle">minimum total potential energy principle</a>. The <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> principle approach is more general as it is applicable to both linear and non-linear material behaviors. The virtual work method is an expression of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>: for conservative systems, the work added to the system by a set of applied forces is equal to the energy stored in the system in the form of strain energy of the structure's components. </p><p>The principle of <a href="/wiki/Virtual_work" title="Virtual work">virtual displacements</a> for the structural system expresses the mathematical identity of external and internal virtual work: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{External virtual work}}=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>External virtual work</mtext> </mstyle> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>δ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{External virtual work}}=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4fc60d6a9d452593d4494b67f649e525dc6342c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.514ex; height:5.676ex;" alt="{\displaystyle {\mbox{External virtual work}}=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>In other words, the summation of the work done on the system by the set of external forces is equal to the work stored as strain energy in the elements that make up the system. </p><p>The virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work done on the individual elements. The latter requires that force-displacement functions be used that describe the response for each individual element. Hence, the displacement of the structure is described by the response of individual (discrete) elements collectively. The equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum). The latter would result in an intractable problem, hence the utility of the finite element method. As shown in the subsequent sections, Eq.(<b><a href="#math_1">1</a></b>) leads to the following governing equilibrium equation for the system: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\mathbf {Kr} +\mathbf {R} ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\mathbf {Kr} +\mathbf {R} ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c7549c27fa986dd634b3c217679f98192f2907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.171ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} =\mathbf {Kr} +\mathbf {R} ^{o}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></span> = vector of nodal forces, representing external forces applied to the system's nodes.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368b3827262016c64b340a761d9b95e0b031d6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.094ex; height:2.176ex;" alt="{\displaystyle \mathbf {K} }"></span> = system stiffness matrix, which is the collective effect of the individual <i>elements' stiffness matrices</i> :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66dca73b1d04ed138886e159f51fd97fd672c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.409ex; height:2.343ex;" alt="{\displaystyle \mathbf {k} ^{e}}"></span>.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> = vector of the system's nodal displacements.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b780ac201428945b71472557909470520ceed868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.033ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{o}}"></span> = vector of equivalent nodal forces, representing all external effects other than the nodal forces which are already included in the preceding nodal force vector <b>R</b>. These external effects may include distributed or concentrated surface forces, body forces, thermal effects, initial stresses and strains.</dd></dl> <p>Once the supports' constraints are accounted for, the nodal displacements are found by solving the <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a> (<b><a href="#math_2">2</a></b>), symbolically: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {K} ^{-1}(\mathbf {R} -\mathbf {R} ^{o})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {K} ^{-1}(\mathbf {R} -\mathbf {R} ^{o})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75bd307904e495fb1a54e7ebd0336d9625b1a2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.313ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} =\mathbf {K} ^{-1}(\mathbf {R} -\mathbf {R} ^{o})}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>Subsequently, the strains and stresses in individual elements may be found as follows: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\epsilon } =\mathbf {Bq} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\epsilon } =\mathbf {Bq} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fada64c336814969af31c6714a5abfa1e9543b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.509ex;" alt="{\displaystyle \mathbf {\epsilon } =\mathbf {Bq} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\sigma } =\mathbf {E} (\mathbf {\epsilon } -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}=\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\sigma } =\mathbf {E} (\mathbf {\epsilon } -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}=\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253fff6c97ceaa47704c66ce1ca7a3ae019da2ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.949ex; height:2.843ex;" alt="{\displaystyle \mathbf {\sigma } =\mathbf {E} (\mathbf {\epsilon } -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}=\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }"></span> = vector of a nodal displacements--a subset of the system displacement vector <b>r</b> that pertains to the elements under consideration.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.901ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} }"></span> = strain-displacement matrix that transforms nodal displacements <b>q</b> to strains at any point in the element.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> = elasticity matrix that transforms effective strains to stresses at any point in the element.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\epsilon } ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\epsilon } ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc16803fa648b2e1314ce82cb60d6b73426270c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.974ex; height:2.343ex;" alt="{\displaystyle \mathbf {\epsilon } ^{o}}"></span> = vector of initial strains in the elements.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\sigma } ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\sigma } ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f530f0abfeea1df64a94b09c9daad06aa9308bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.36ex; height:2.343ex;" alt="{\displaystyle \mathbf {\sigma } ^{o}}"></span> = vector of initial stresses in the elements.</dd></dl> <p>By applying the <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> equation (<b><a href="#math_1">1</a></b>) to the system, we can establish the element matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.901ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66dca73b1d04ed138886e159f51fd97fd672c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.409ex; height:2.343ex;" alt="{\displaystyle \mathbf {k} ^{e}}"></span> as well as the technique of assembling the system matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b780ac201428945b71472557909470520ceed868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.033ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{o}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368b3827262016c64b340a761d9b95e0b031d6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.094ex; height:2.176ex;" alt="{\displaystyle \mathbf {K} }"></span>. Other matrices such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\epsilon } ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\epsilon } ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc16803fa648b2e1314ce82cb60d6b73426270c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.974ex; height:2.343ex;" alt="{\displaystyle \mathbf {\epsilon } ^{o}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\sigma } ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\sigma } ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f530f0abfeea1df64a94b09c9daad06aa9308bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.36ex; height:2.343ex;" alt="{\displaystyle \mathbf {\sigma } ^{o}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> are known values and can be directly set up from data input. </p> <div class="mw-heading mw-heading2"><h2 id="Interpolation_or_shape_functions">Interpolation or shape functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=5" title="Edit section: Interpolation or shape functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }"></span> be the vector of nodal displacements of a typical element. The displacements at any other point of the element may be found by the use of <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> functions as, symbolically: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =\mathbf {N} \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =\mathbf {N} \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d955ed182aaa813478b90e83012f2d46aaebc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.091ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} =\mathbf {N} \mathbf {q} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span>)</b></td></tr></tbody></table> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span> = vector of displacements at any point {x,y,z} of the element.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }"></span> = matrix of <i><a href="/w/index.php?title=Shape_functions&action=edit&redlink=1" class="new" title="Shape functions (page does not exist)">shape functions</a></i> serving as <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> functions.</dd></dl> <p>Equation (<b><a href="#math_6">6</a></b>) gives rise to other quantities of great interest: </p> <ul> <li>Virtual displacements that are a function of virtual nodal displacements: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {u} =\mathbf {N} \delta \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {u} =\mathbf {N} \delta \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805aa7abb93de5b3aa88635cf28660c9c1b6319a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.188ex; height:2.676ex;" alt="{\displaystyle \delta \mathbf {u} =\mathbf {N} \delta \mathbf {q} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6b" class="reference nourlexpansion" style="font-weight:bold;">6b</span>)</b></td></tr></tbody></table> </li> <li>Strains in the elements that result from displacements of the element's nodes: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\epsilon } =\mathbf {Du} =\mathbf {DNq} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">N</mi> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\epsilon } =\mathbf {Du} =\mathbf {DNq} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13f99857db1badde6c1af8cd8e8acde1f576217" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.233ex; height:2.509ex;" alt="{\displaystyle \mathbf {\epsilon } =\mathbf {Du} =\mathbf {DNq} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2345293072878db24e119c580def49ad582e3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.05ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} }"></span> = matrix of <a href="/w/index.php?title=Strain-displacement_relations&action=edit&redlink=1" class="new" title="Strain-displacement relations (page does not exist)">differential operators</a> that convert displacements to strains using <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear elasticity</a> theory. Eq.(<b><a href="#math_7">7</a></b>) shows that matrix <b>B</b> in (<b><a href="#math_4">4</a></b>) is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mathbf {DN} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {DN} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126bfc8207db3f1bf6e3652ee902867afed7c79f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.141ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} =\mathbf {DN} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span>)</b></td></tr></tbody></table> </li> <li>Virtual strains consistent with element's virtual nodal displacements: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\boldsymbol {\epsilon }}=\mathbf {B} \delta \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\boldsymbol {\epsilon }}=\mathbf {B} \delta \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befd67105ec88524974ccd45358d4fcc7f41cc5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.635ex; height:2.676ex;" alt="{\displaystyle \delta {\boldsymbol {\epsilon }}=\mathbf {B} \delta \mathbf {q} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span>)</b></td></tr></tbody></table> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Internal_virtual_work_in_a_typical_element">Internal virtual work in a typical element</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=6" title="Edit section: Internal virtual work in a typical element"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a typical element of volume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c412a6ac22071e6689a5c7484277e156387eb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.915ex; height:2.343ex;" alt="{\displaystyle V^{e}}"></span>, the internal virtual work due to virtual displacements is obtained by substitution of (<b><a href="#math_5">5</a></b>) and (<b><a href="#math_9">9</a></b>) into (<b><a href="#math_1">1</a></b>): </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{Internal virtual work}}=\int _{V^{e}}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV^{e}=\delta \ \mathbf {q} ^{T}\int _{V^{e}}\mathbf {B} ^{T}{\big \{}\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}{\big \}}\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Internal virtual work</mtext> </mstyle> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <mi>δ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{Internal virtual work}}=\int _{V^{e}}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV^{e}=\delta \ \mathbf {q} ^{T}\int _{V^{e}}\mathbf {B} ^{T}{\big \{}\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}{\big \}}\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6446013519aee4e0e470ab33d1da6dbd0bc3a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:77.233ex; height:5.676ex;" alt="{\displaystyle {\mbox{Internal virtual work}}=\int _{V^{e}}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}\,dV^{e}=\delta \ \mathbf {q} ^{T}\int _{V^{e}}\mathbf {B} ^{T}{\big \{}\mathbf {E} (\mathbf {Bq} -\mathbf {\epsilon } ^{o})+\mathbf {\sigma } ^{o}{\big \}}\,dV^{e}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Element_matrices">Element matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=7" title="Edit section: Element matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Primarily for the convenience of reference, the following matrices pertaining to a typical elements may now be defined: </p> <dl><dd>Element stiffness matrix</dd></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 3.2em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} ^{e}=\int _{V^{e}}\mathbf {B} ^{T}\mathbf {E} \mathbf {B} \,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} ^{e}=\int _{V^{e}}\mathbf {B} ^{T}\mathbf {E} \mathbf {B} \,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a2938e0b3ed72ecc56f84ac037a9163c32eddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.711ex; height:5.676ex;" alt="{\displaystyle \mathbf {K} ^{e}=\int _{V^{e}}\mathbf {B} ^{T}\mathbf {E} \mathbf {B} \,dV^{e}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span>)</b></td></tr></tbody></table> <dl><dd>Equivalent element load vector</dd></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 3.2em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{oe}=\int _{V^{e}}-\mathbf {B} ^{T}{\big (}\mathbf {E} \mathbf {\epsilon } ^{o}-\mathbf {\sigma } ^{o}{\big )}\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{oe}=\int _{V^{e}}-\mathbf {B} ^{T}{\big (}\mathbf {E} \mathbf {\epsilon } ^{o}-\mathbf {\sigma } ^{o}{\big )}\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f6b73ed5c03bc2660aff8d66dc2a0530e1ee34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.634ex; height:5.676ex;" alt="{\displaystyle \mathbf {Q} ^{oe}=\int _{V^{e}}-\mathbf {B} ^{T}{\big (}\mathbf {E} \mathbf {\epsilon } ^{o}-\mathbf {\sigma } ^{o}{\big )}\,dV^{e}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_12" class="reference nourlexpansion" style="font-weight:bold;">12</span>)</b></td></tr></tbody></table> <p>These matrices are usually evaluated numerically using <a href="/wiki/Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a> for <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a>. Their use simplifies (<b><a href="#math_10">10</a></b>) to the following: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {q} ^{T}{\big (}\mathbf {K} ^{e}\mathbf {q} +\mathbf {Q} ^{oe}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Internal virtual work</mtext> </mstyle> </mrow> <mo>=</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {q} ^{T}{\big (}\mathbf {K} ^{e}\mathbf {q} +\mathbf {Q} ^{oe}{\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808880a3b950ea43f2995be3599f3e7825cb2b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.832ex; height:3.343ex;" alt="{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {q} ^{T}{\big (}\mathbf {K} ^{e}\mathbf {q} +\mathbf {Q} ^{oe}{\big )}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_13" class="reference nourlexpansion" style="font-weight:bold;">13</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Element_virtual_work_in_terms_of_system_nodal_displacements">Element virtual work in terms of system nodal displacements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=8" title="Edit section: Element virtual work in terms of system nodal displacements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the nodal displacement vector <b>q</b> is a subset of the system nodal displacements <b>r</b> (for compatibility with adjacent elements), we can replace <b>q</b> with <b>r</b> by expanding the size of the element matrices with new columns and rows of zeros: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {r} ^{T}\left(\mathbf {K} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Internal virtual work</mtext> </mstyle> </mrow> <mo>=</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {r} ^{T}\left(\mathbf {K} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c078d1bb3edeb12dc232ee0604c5ae22e2202685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.27ex; height:3.176ex;" alt="{\displaystyle {\mbox{Internal virtual work}}=\delta \ \mathbf {r} ^{T}\left(\mathbf {K} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_14" class="reference nourlexpansion" style="font-weight:bold;">14</span>)</b></td></tr></tbody></table> <p>where, for simplicity, we use the same symbols for the element matrices, which now have expanded size as well as suitably rearranged rows and columns. </p> <div class="mw-heading mw-heading2"><h2 id="System_virtual_work">System virtual work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=9" title="Edit section: System virtual work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Summing the internal virtual work (<b><a href="#math_14">14</a></b>) for all elements gives the right-hand-side of (<b><a href="#math_1">1</a></b>): </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{System internal virtual work}}=\sum _{e}\delta \ \mathbf {r} ^{T}\left(\mathbf {k} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>System internal virtual work</mtext> </mstyle> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{System internal virtual work}}=\sum _{e}\delta \ \mathbf {r} ^{T}\left(\mathbf {k} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb0d7973d88a66375becf1e44b9e86319b1f47a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:85.848ex; height:7.509ex;" alt="{\displaystyle {\mbox{System internal virtual work}}=\sum _{e}\delta \ \mathbf {r} ^{T}\left(\mathbf {k} ^{e}\mathbf {r} +\mathbf {Q} ^{oe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_15" class="reference nourlexpansion" style="font-weight:bold;">15</span>)</b></td></tr></tbody></table> <p>Considering now the left-hand-side of (<b><a href="#math_1">1</a></b>), the system external virtual work consists of: </p> <ul> <li>The work done by the nodal forces <b>R</b>: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a260a5dbe3d10a661ed58416b948a062145223ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.124ex; height:2.676ex;" alt="{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} }"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_16" class="reference nourlexpansion" style="font-weight:bold;">16</span>)</b></td></tr></tbody></table> </li> <li>The work done by external forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e0008216ce0d10e0470cb975050c2442c40c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.858ex; height:2.343ex;" alt="{\displaystyle \mathbf {T} ^{e}}"></span> on the part <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb017a253ac8486bbda4884f1dd3377a8a673cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.484ex; height:2.343ex;" alt="{\displaystyle \mathbf {S} ^{e}}"></span> of the elements' edges or surfaces, and by the body forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dc9468869afed1ba53399bd517bec559337182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.123ex; height:2.343ex;" alt="{\displaystyle \mathbf {f} ^{e}}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{e}\int _{S^{e}}\delta \ \mathbf {u} ^{T}\mathbf {T} ^{e}\,dS^{e}+\sum _{e}\int _{V^{e}}\delta \ \mathbf {u} ^{T}\mathbf {f} ^{e}\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{e}\int _{S^{e}}\delta \ \mathbf {u} ^{T}\mathbf {T} ^{e}\,dS^{e}+\sum _{e}\int _{V^{e}}\delta \ \mathbf {u} ^{T}\mathbf {f} ^{e}\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1cb70ad22f092683dab67a2cdcc60136565479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.782ex; height:6.343ex;" alt="{\displaystyle \sum _{e}\int _{S^{e}}\delta \ \mathbf {u} ^{T}\mathbf {T} ^{e}\,dS^{e}+\sum _{e}\int _{V^{e}}\delta \ \mathbf {u} ^{T}\mathbf {f} ^{e}\,dV^{e}}"></span></dd></dl> Substitution of (<b><a href="#math_6b">6b</a></b>) gives: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \ \mathbf {q} ^{T}\sum _{e}\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}+\delta \ \mathbf {q} ^{T}\sum _{e}\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>+</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \ \mathbf {q} ^{T}\sum _{e}\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}+\delta \ \mathbf {q} ^{T}\sum _{e}\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da6e816a7cb20abed57a9c7135025c4b91e16b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.381ex; height:6.343ex;" alt="{\displaystyle \delta \ \mathbf {q} ^{T}\sum _{e}\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}+\delta \ \mathbf {q} ^{T}\sum _{e}\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}"></span></dd></dl> or <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\delta \ \mathbf {q} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\delta \ \mathbf {q} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a670ff155ac7f1b1ee29a0f6592c19c55ea24a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.499ex; height:6.009ex;" alt="{\displaystyle -\delta \ \mathbf {q} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_17a" class="reference nourlexpansion" style="font-weight:bold;">17a</span>)</b></td></tr></tbody></table> <p>where we have introduced additional element's matrices defined below: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{te}=-\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{te}=-\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe5604a69c4fbd42b6b0e262d4de118038e8ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.129ex; height:5.676ex;" alt="{\displaystyle \mathbf {Q} ^{te}=-\int _{S^{e}}\mathbf {N} ^{T}\mathbf {T} ^{e}\,dS^{e}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_18a" class="reference nourlexpansion" style="font-weight:bold;">18a</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{fe}=-\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{fe}=-\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c430a99e07b36599c7e25bd14d14600ad3e361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.38ex; height:5.676ex;" alt="{\displaystyle \mathbf {Q} ^{fe}=-\int _{V^{e}}\mathbf {N} ^{T}\mathbf {f} ^{e}\,dV^{e}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_18b" class="reference nourlexpansion" style="font-weight:bold;">18b</span>)</b></td></tr></tbody></table> <p>Again, <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a> is convenient for their evaluation. A similar replacement of <b>q</b> in (<b><a href="#math_17a">17a</a></b>) with <b>r</b> gives, after rearranging and expanding the vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{te},\mathbf {Q} ^{fe}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{te},\mathbf {Q} ^{fe}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca811701031a752b48726b9870febe816560be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} ^{te},\mathbf {Q} ^{fe}}"></span>: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545f27790608f16a1c0593a5dd39ac576ca0c55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.184ex; height:6.009ex;" alt="{\displaystyle -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_17b" class="reference nourlexpansion" style="font-weight:bold;">17b</span>)</b></td></tr></tbody></table> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Assembly_of_system_matrices">Assembly of system matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=10" title="Edit section: Assembly of system matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adding (<b><a href="#math_16">16</a></b>), (<b><a href="#math_17b">17b</a></b>) and equating the sum to (<b><a href="#math_15">15</a></b>) gives: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>−</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mi>δ</mi> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d844e9ea8f47b3030dcf54518c3f501daee2a257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.162ex; height:7.509ex;" alt="{\displaystyle \delta \ \mathbf {r} ^{T}\mathbf {R} -\delta \ \mathbf {r} ^{T}\sum _{e}\left(\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)=\delta \ \mathbf {r} ^{T}\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\delta \ \mathbf {r} ^{T}\sum _{e}\mathbf {Q} ^{oe}}"></span> </p><p>Since the virtual displacements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \ \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \ \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba3b1864275a8643c72d33d6779075e09223837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.731ex; height:2.343ex;" alt="{\displaystyle \delta \ \mathbf {r} }"></span> are arbitrary, the preceding equality reduces to: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d85b1df9f08c03b0c67ea63d55df8485976dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.776ex; height:7.509ex;" alt="{\displaystyle \mathbf {R} =\left(\sum _{e}\mathbf {k} ^{e}\right)\mathbf {r} +\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"></span> </p><p>Comparison with (<b><a href="#math_2">2</a></b>) shows that: </p> <ul><li>The system stiffness matrix is obtained by summing the elements' stiffness matrices: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} =\sum _{e}\mathbf {k} ^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} =\sum _{e}\mathbf {k} ^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2a0bcf9f0bf836b0f143f75bf8bfc41b0fb8f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.344ex; height:5.509ex;" alt="{\displaystyle \mathbf {K} =\sum _{e}\mathbf {k} ^{e}}"></span></dd></dl></li> <li>The vector of equivalent nodal forces is obtained by summing the elements' load vectors: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{o}=\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>e</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{o}=\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993613d8686b3bc4ad81d96a65a88bfd6e4ddf54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.644ex; height:6.009ex;" alt="{\displaystyle \mathbf {R} ^{o}=\sum _{e}\left(\mathbf {Q} ^{oe}+\mathbf {Q} ^{te}+\mathbf {Q} ^{fe}\right)}"></span></dd></dl></li></ul> <p>In practice, the element matrices are neither expanded nor rearranged. Instead, the system stiffness matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368b3827262016c64b340a761d9b95e0b031d6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.094ex; height:2.176ex;" alt="{\displaystyle \mathbf {K} }"></span> is assembled by adding individual coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k}_{ij}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {k}_{ij}^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae74f0993500b5211948eacac335c6a234578ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.688ex; height:3.176ex;" alt="{\displaystyle {k}_{ij}^{e}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}_{kl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {K}_{kl}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a543e9ee1f06ffcfa6df88e2cc75e2362fd99e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.509ex;" alt="{\displaystyle {K}_{kl}}"></span> where the subscripts ij, kl mean that the element's nodal displacements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {q}_{i}^{e},{q}_{j}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {q}_{i}^{e},{q}_{j}^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0010754e856d9728d8f1cf859e369c45ee39b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.19ex; height:3.176ex;" alt="{\displaystyle {q}_{i}^{e},{q}_{j}^{e}}"></span> match respectively with the system's nodal displacements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r}_{k},{r}_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {r}_{k},{r}_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745d4e91d5acd0a974689698ab8fcb4702ea37c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.942ex; height:2.009ex;" alt="{\displaystyle {r}_{k},{r}_{l}}"></span>. Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b780ac201428945b71472557909470520ceed868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.033ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{o}}"></span> is assembled by adding individual coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Q}_{i}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {Q}_{i}^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f01cf98a8329826e538d96f8f1b0464937e954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.837ex; height:2.843ex;" alt="{\displaystyle {Q}_{i}^{e}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {R}_{k}^{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {R}_{k}^{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bee0e1fc02a816b714d221844a722212a2c62cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.853ex; height:2.843ex;" alt="{\displaystyle {R}_{k}^{o}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {q}_{i}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {q}_{i}^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14729dbd0f924759a3c7d885f25f9d66151b3165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.078ex; height:2.843ex;" alt="{\displaystyle {q}_{i}^{e}}"></span> matches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {r}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14906324b1cba36c358be9071f7cde45bb144cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.137ex; height:2.009ex;" alt="{\displaystyle {r}_{k}}"></span>. This direct addition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k}_{ij}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {k}_{ij}^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae74f0993500b5211948eacac335c6a234578ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.688ex; height:3.176ex;" alt="{\displaystyle {k}_{ij}^{e}}"></span> into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}_{kl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {K}_{kl}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a543e9ee1f06ffcfa6df88e2cc75e2362fd99e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.509ex;" alt="{\displaystyle {K}_{kl}}"></span> gives the procedure the name <i><a href="/wiki/Direct_stiffness_method" title="Direct stiffness method">Direct Stiffness Method</a></i>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Flexibility_method" title="Flexibility method">Flexibility method</a></li> <li><a href="/wiki/Matrix_stiffness_method" class="mw-redirect" title="Matrix stiffness method">Matrix stiffness method</a></li> <li><a href="/wiki/Modal_analysis_using_FEM" title="Modal analysis using FEM">Modal analysis using FEM</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">List of finite element software packages</a></li> <li><a href="/wiki/Structural_analysis" title="Structural analysis">Structural analysis</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li> <li><a href="/wiki/Interval_finite_element" title="Interval finite element">Interval finite element</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_element_method_in_structural_mechanics&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Matrix Analysis Of Framed Structures, 3rd Edition by Jr. William Weaver, James M. Gere, Springer-Verlag New York, LLC, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-07861-3" title="Special:BookSources/978-0-412-07861-3">978-0-412-07861-3</a>, 1966</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jd6i0k4wvtQC&q=%22finite+element%22+OR+%22finite+method%22">Theory of Matrix Structural Analysis</a>, J. S. Przemieniecki, McGraw-Hill Book Company, New York, 1968</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintonIrons1968" class="citation journal cs1">Hinton, Ernest; Irons, Bruce (July 1968). "Least squares smoothing of experimental data using finite elements". <i>Strain</i>. <b>4</b> (3): 24–27. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1475-1305.1968.tb01368.x">10.1111/j.1475-1305.1968.tb01368.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Strain&rft.atitle=Least+squares+smoothing+of+experimental+data+using+finite+elements&rft.volume=4&rft.issue=3&rft.pages=24-27&rft.date=1968-07&rft_id=info%3Adoi%2F10.1111%2Fj.1475-1305.1968.tb01368.x&rft.aulast=Hinton&rft.aufirst=Ernest&rft.au=Irons%2C+Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+element+method+in+structural+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Argyris, J.H and Kelsey, S. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PCsDCAAAQBAJ&q=%22finite+element%22">Energy theorems and Structural Analysis</a> Butterworth Scientific publications, London, 1954</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Clough, R.W, “The Finite Element in Plane Stress Analysis.” Proceedings, 2nd ASCE Conference on Electronic Computations, Pittsburgh, Sep 1960</span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl 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