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\rhead{Erku\c{s}, B.}
\rhead{}
\lhead{Shear Displacement}

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\begin_body

\begin_layout Title
A Note on Virtual Work Principle Regarding Shear Rotations
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\begin_layout Author
Barış Erkuş
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Assist.
 Prof.
 Dr., Civil Eng.
 Dept., Istanbul Technical University, Istanbul, Turkey; bariserkus@itu.edu.tr
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\begin_layout Date
10 November 2018
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\begin_layout Abstract
Virtual work principle can be applied to find the deflections of beams and
 frames.
 However, it is noted that rotations of frame elements estimated by conventional
 virtual work principle includes rotations due to bending but due to shear
 deformations since moment and shear does not do work through shear rotations.
 Shear rotation at a given point of a frame element is straightforward to
 estimate and can be added to the total rotation afterwards the application
 of the virtual work principle.
 Two simple examples are given, where it is shown the procedure does not
 include shear rotations in the estimated rotation of a given point on the
 frame.
\end_layout

\begin_layout Section
Introduction
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\begin_layout Standard
Virtual work principle is fundamental apprach that can be used to estimate
 deflection values of a point on a frame element.
 The most general form of the principle states that work done by the external
 forces, inluding reactions, of the virtual loading system through the correspon
idng deflections of the original system is equal to the work done by the
 internal forces, moment, shear and axial, of the virtual system through
 the corresponding internal deformations of the original system, rotations,
 shear and axial deformations.
 For typical frame structures, shear and axial deformations often ignored
 since deflection values (both rotations and displacements) due to bending
 are significantly larger than deflection values caused by shear and axial
 forces 
\begin_inset CommandInset citation
LatexCommand cite
key "Warburton1980"
literal "false"

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.
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\begin_layout Section
Review of Virtual Work Principle for Frames
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In this section, a review of the frame element internal forces and corresponding
 internal deformations are given.
 Then, virtual work principle is stated, and its formulation is presented.
 A proof of the formulation is not given as it is well-documented in the
 literature.
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A frame element and internal forces and deformations at a given point
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Consider a portion of a frame element and internal forces and corresponding
 deformations at a given section (Figure 
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).
 Internal stresses of a beam element are represented as internal force for
 structural analysis.
 These, bending moment 
\begin_inset Formula $M$
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, shear force 
\begin_inset Formula $V$
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, and axial force 
\begin_inset Formula $N$
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.
 The deformations associated with these internal forces are bending rotations
 
\begin_inset Formula $d\theta$
\end_inset

, shear displacements 
\begin_inset Formula $dv$
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 and axial displacements 
\begin_inset Formula $d\delta$
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.
 Therefore, corresponding strains can be represented by 
\begin_inset Formula 
\begin{align}
\epsilon_{\text{bending}} & =\dfrac{d\theta}{dx} & \epsilon_{\text{shear}} & =\dfrac{dv}{dx} & \epsilon_{\text{axial}} & =\dfrac{d\delta}{dx}
\end{align}

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\begin_layout Standard
\noindent
Note that shear strain corresponds to the shear rotation 
\begin_inset Formula $\gamma$
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, i.e.
 
\begin_inset Formula $\gamma=\epsilon_{\text{shear}}=dv/ds$
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.
 For a given frame element, internal forces are related to the internal
 deformation as follows:
\begin_inset Formula 
\begin{align}
d\theta & =\dfrac{M(x)}{EI}dx & dv & =\dfrac{V(x)}{GA_{\text{s}}}dx & d\delta & =\dfrac{N(x)}{AE}dx\label{eq:def-int-rel}
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A frame element and internal forces and deformations at a given point
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Consider that a frame structure is under the effect of a set of external
 forces, support settlements and temperature changes, which are collectively
 called System A.
 Here, 
\begin_inset Formula $\Delta t_{\text{rel}}^{\text{A}}$
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 is the relative temperature between the bottom and top fiber of sections,
 
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 is the uniform temperature change along the section of the frame elements
 and 
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 is a given support movement value.
 The resulting internal forces of the 
\begin_inset Formula $i^{\text{th}}$
\end_inset

 frame element are represented by 
\begin_inset Formula $M_{i}^{\text{A}}(x)$
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, 
\begin_inset Formula $V_{i}^{\text{A}}(x)$
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 and 
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.
 Corresponding internal deformations are shown by 
\begin_inset Formula $d\theta_{i}^{\text{A}}$
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, 
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 and 
\begin_inset Formula $d\delta_{i}^{\text{A}}$
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, where Equations 
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 can be used to relate them to internal forces.
 Likewise, consider a new set of external effects, which is called System
 B.
 For the sake of simplicity, it is assumed that System B does not include
 temperature and support movement effects.
 Replacing the deformation terms with their representation in terms of internal
 forces and temperature change, this equation becomes
\begin_inset Formula 
\begin{multline}
F_{\text{a}}^{\text{B}}\delta_{\text{a}}^{\text{A}}+M_{\text{d}}^{\text{B}}\theta_{\text{d}}^{\text{A}}+R^{\text{B}}\delta_{\text{sm}}^{\text{A}}+\int_{\text{b}}^{\text{c}}w^{\text{A}}(x)\delta_{\text{bc}}^{\text{A}}(x)dx=\\
\sum_{i}\Big(\int_{L}M_{i}^{\text{B}}(x)\dfrac{M^{\text{A}}(x)}{EI}dx+\int_{L}V_{i}^{\text{B}}(x)\dfrac{V^{\text{A}}(x)}{GA_{\text{s}}}dx+\int_{L}N_{i}^{\text{B}}(x)\dfrac{N^{\text{A}}(x)}{AE}dx\\
+\int_{L}M_{i}^{\text{B}}(x)\dfrac{\alpha_{i}\Delta t_{\text{rel}}^{\text{A}}}{d_{i}}dx+\int_{L}N_{i}^{\text{B}}(x)\alpha_{i}\Delta t_{\text{uni}}^{\text{A}}dx\Big)\label{eq:virtual-work}
\end{multline}

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\begin_layout Standard
Displacements estimated using virtual work principle includes shear deformations.
 On the other hand, rotations found from this equation does not include
 shear rotation 
\begin_inset Formula $\gamma(x)$
\end_inset

 since there is no work term associated with the shear rotation.
 Shear rotation can be included in the Equation 
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reference "eq:virtual-work"
plural "false"
caps "false"
noprefix "false"

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 by considering it as an existing structural deformation similar to fabrication
 errors or support settlements.
 In this case, external moments of System B does work through shear rotations.
 In Figure 
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reference "fig:frame-structure"
plural "false"
caps "false"
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, shear rotation at joint d will be the difference between the shear rotations
 at the connecting beam and column.
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