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enStrain Energy Method (Castigliano’s Theorem) | Beam Deflection
https://mathalino.com/reviewer/strength-materials/beam-deflection-strain-energy-method-castigliano-s-theorem
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><div class="tex2jax"><div class="picture"><img src="/sites/default/files/reviewer-strength/06-beam-deflections/alberto_castigliano.jpeg" width="200" height="234" alt="Alberto Castigliano (Catigliano's Theorem)" /><br />Engr. Alberto Castigliano</div>
<p>Italian engineer Alberto Castigliano (1847 – 1884) developed a method of determining deflection of structures by strain energy method. His <em>Theorem of the Derivatives of Internal Work of Deformation</em> extended its application to the calculation of relative rotations and displacements between points in the structure and to the study of beams in flexure.<br />
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<!--break--><p><em>Energy</em> of structure is its capacity of doing work and <em>strain energy</em> is the internal energy in the structure because of its deformation. By the principle of conservation of energy,<br />
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<div align="center">$U = W_i$</div>
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<p>where $U$ denotes the strain energy and $W_i$ represents the work done by internal forces. The expression of strain energy depends therefore on the internal forces that can develop in the member due to applied external forces.<br />
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<p><strong>Castigliano’s Theorem for Beam Deflection</strong><br />
For linearly elastic structures, the partial derivative of the strain energy with respect to an applied force (or couple) is equal to the displacement (or rotation) of the force (or couple) along its line of action.<br />
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<div class="equation" align="center">$\delta = \dfrac{\partial U}{\partial P}$ or $\theta = \dfrac{\partial U}{\partial \bar{M}}$</div>
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<p>Where $\delta$ is the deflection at the point of application of force $P$ in the direction of $P$, $\theta$ is the rotation at the point of application of the couple $\bar{M}$ in the direction of $\bar{M}$, and $U$ is the strain energy.<br />
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<p>The strain energy of a beam was known to be $U = \displaystyle \int_0^L \dfrac{M^2}{2EI}dx$ . Finding the partial derivative of this expression will give us the equations of Castigliano’s deflection and rotation of beams. The equations are written below for convenience.<br />
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<div class="equation" align="center">$\displaystyle \delta = \int_0^L \left( \dfrac{\partial M}{\partial P} \right) \dfrac{M}{EI} \, dx$ and $\displaystyle \theta = \int_0^L \left( \dfrac{\partial M}{\partial \bar{M}} \right) \dfrac{M}{EI} \, dx$</div>
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</div></div></div><div class="field field-name-taxonomy-vocabulary-3 field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tag/reviewer/energy-method" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">energy method</a></div><div class="field-item odd"><a href="/tag/reviewer/strain-energy" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">strain energy</a></div><div class="field-item even"><a href="/tag/reviewer/work" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Work</a></div></div></div>Fri, 30 Nov 2012 00:06:34 +0000Jhun Vert1703 at https://mathalino.comhttps://mathalino.com/reviewer/strength-materials/beam-deflection-strain-energy-method-castigliano-s-theorem#comments