3. Principle of virtual work#
3.1. Deformation work#
In 3D the expression of deformation work is written:
\(\begin{array}{c}{W}_{\text{def}}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\tilde{\varepsilon }}_{\text{ij}}{\tilde{\sigma }}_{\text{ij}})\text{dV}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\tilde{\varepsilon }}_{\text{ij}}{\tilde{C}}_{\text{ijkl}}{\tilde{\varepsilon }}_{\text{kl}})\text{dV}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\varepsilon }_{\text{rs}}{P}_{i}^{r}{P}_{k}^{s}{\tilde{C}}_{\text{ijkl}}{P}_{k}^{p}{P}_{l}^{q}{\varepsilon }_{\text{pq}})\text{dV}\\ \mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\varepsilon }_{\text{rs}}{C}_{\text{rspq}}{\varepsilon }_{\text{pq}})\text{dV}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\varepsilon }_{\text{ij}}{\sigma }_{\text{ij}})\text{dV}\end{array}\)
We check that this expression is invariant with respect to the base in which the tensors are expressed. For the rest of this document, we choose to express everything in the local \(({T}_{k})\) base, knowing that we are going from the local behavior tensor to the global behavior tensor through the relationship \({C}_{\text{rspq}}\mathrm{=}{P}_{i}^{r}{P}_{k}^{s}{\tilde{C}}_{\text{ijkl}}{P}_{k}^{p}{P}_{l}^{q}\).
The general expression of the 3D deformation work for the shell element is:
\({W}_{\text{def}}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}(\tilde{\varepsilon }\tilde{\sigma })\text{dV}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}(\tilde{\varepsilon }\tilde{C}\tilde{\varepsilon })\text{dV}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\tilde{\varepsilon }}_{\text{mf}}\tilde{H}{\tilde{\varepsilon }}_{\text{mf}})\text{dV}+\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{h\mathrm{/}2}{\mathrm{\int }}}({\tilde{\varepsilon }}_{\gamma }{\tilde{H}}_{\gamma }{\tilde{\varepsilon }}_{\gamma })\text{dV}\)
where \(S\) is the average area and the position in the thickness of the shell varies between \(–h\mathrm{/}2\) and \(+h\mathrm{/}2\). In the expression of deformation work, there appears a deformation contribution in membrane‑flexure and a deformation contribution in transverse shear.
3.1.1. Elastic internal energy of shell#
It is expressed in the following way:
\({\Phi }_{\text{int}}\mathrm{=}\frac{1}{2}\underset{S}{\mathrm{\int }}\mathrm{[}\frac{E}{1\mathrm{-}{\nu }^{2}}({\tilde{\varepsilon }}_{\text{11}}^{2}+{\tilde{\varepsilon }}_{\text{22}}^{2}+2\nu {\tilde{\varepsilon }}_{\text{11}}{\tilde{\varepsilon }}_{\text{22}})+G({\tilde{\gamma }}_{\text{12}}^{2}+k({\tilde{\gamma }}_{1}^{2}+{\tilde{\gamma }}_{2}^{2}))\mathrm{]}\text{dV}\)
where k is the transverse shear correction factor defined in paragraph 2 and \(G\mathrm{=}\frac{E}{2(1+\nu )}\).
3.1.2. Expression of the resulting efforts#
We note:
\(N\mathrm{=}(\begin{array}{c}{N}_{\text{11}}\\ {N}_{\text{22}}\\ {N}_{\text{12}}\end{array})\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}(\begin{array}{c}{\tilde{\sigma }}_{\text{11}}\\ {\tilde{\sigma }}_{\text{22}}\\ {\tilde{\sigma }}_{\text{12}}\end{array})\text{dz}\); \(M\mathrm{=}(\begin{array}{c}{M}_{\text{11}}\\ {M}_{\text{22}}\\ {M}_{\text{12}}\end{array})\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}(\begin{array}{c}{\tilde{\sigma }}_{\text{11}}\\ {\tilde{\sigma }}_{\text{22}}\\ {\tilde{\sigma }}_{\text{12}}\end{array})z\text{dz}\); \(T\mathrm{=}(\begin{array}{c}{T}_{1}\\ {T}_{2}\end{array})\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}(\begin{array}{c}{\tilde{\sigma }}_{\text{13}}\\ {\tilde{\sigma }}_{\text{23}}\end{array})\text{dz}\).
\({N}_{\text{11}},{N}_{\text{22}},{N}_{\text{12}}\) are the widespread membrane efforts (in \(N\mathrm{/}m\));
\({M}_{\text{11}},{M}_{\text{22}},{M}_{\text{12}}\) are the generalized flexural forces or moments (in \(N\));
\({T}_{1},{T}_{2}\), are generalized shear forces or shear forces (in \(N\mathrm{/}m\));
The expression of the resulting forces that is given here is an approximate expression that does not take into account the curvature of the shell (cf. p.316 of [bib3]). The error made on these efforts is then in \({h}^{2}\mathrm{/}R\) where \(1\mathrm{/}R\) is the mean curvature. When the shell becomes flat, the expressions given above are accurate and the meaning of the resulting efforts can be found in [R3.07.03]. We will not further develop this otherwise well-documented aspect in [bib3] because the shell theory used here is not based on a generalized deformations/resultant forces formulation but on a three-dimensional deformations/stresses formulation.
3.2. Work of forces and external couples#
The work of the forces exerted on the solid shell is expressed in the following way:
\({W}_{\text{ext}}\mathrm{=}\underset{S}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}{F}_{v}\text{.}U\text{dV}+\underset{S}{\mathrm{\int }}{F}_{s}\text{.}U\text{dS}+\underset{C}{\mathrm{\int }}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}{F}_{c}\text{.}U\text{dz}\text{ds}\)
where \({F}_{v},{F}_{s},{F}_{c}\) are the volume, surface, and contour forces exerted on the shell, respectively. \(C\) is the part of the shell contour to which the \({F}_{c}\) contour forces are applied.
a) |
Loads given in the global coordinate system: |
With the [§2.2.1] kinematics, we determine as follows:
\(\begin{array}{c}{W}_{\text{ext}}\mathrm{=}\underset{S}{\mathrm{\int }}({f}_{i}{u}_{i}+{c}_{i}{\beta }_{i})\text{dS}+\underset{C}{\mathrm{\int }}({\phi }_{i}{u}_{i}+{\chi }_{i}{\beta }_{i})\text{ds}\mathrm{=}\underset{S}{\mathrm{\int }}({f}_{i}{u}_{i}+{c}_{i}({\tilde{\theta }}_{2}{t}_{\mathrm{1i}}\mathrm{-}{\tilde{\theta }}_{1}{t}_{\mathrm{2i}}))\text{dS}\\ +\underset{C}{\mathrm{\int }}({\phi }_{i}{u}_{i}+{\chi }_{i}({\tilde{\theta }}_{2}{t}_{\mathrm{1i}}\mathrm{-}{\tilde{\theta }}_{1}{t}_{\mathrm{2i}}))\text{ds}\mathrm{=}\underset{S}{\mathrm{\int }}({f}_{i}{u}_{i}+{c}_{i}({\tilde{\theta }}_{2}{t}_{\mathrm{1i}}\mathrm{-}{\tilde{\theta }}_{1}{t}_{\mathrm{2i}}))\text{dS}+\underset{C}{\mathrm{\int }}(\phi u+\chi \beta )\text{ds}\end{array}\)
where are present on the shell:
\({f}_{1},{f}_{2},{f}_{3}\): |
surface forces acting along the axes of the global Cartesian coordinate system |
\({f}_{i}\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}{F}_{v}\text{.}{e}_{i}\text{dz}+{F}_{s}\text{.}{e}_{i}\) |
where \({e}_{i}\) are the vectors of the global cartesian base. |
\({c}_{1},{c}_{2},{c}_{3}\): |
surface pairs acting around the axes of the global coordinate system. |
\({c}_{i}=\underset{-h/2}{\overset{+h/2}{\int }}{\mathrm{zF}}_{v}\text{.}{e}_{i}\text{dz}\pm \frac{h}{2}{F}_{s}\text{.}{e}_{i}\) |
where \({e}_{i}\) are the vectors of the global cartesian base. |
and where are present on the outline of the shell:
\({\phi }_{\mathrm{1,}}{\phi }_{\mathrm{2,}}{\phi }_{3}\): |
linear forces acting along the axes of the global Cartesian coordinate system. |
\({\phi }_{i}\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}{F}_{c}\text{.}{e}_{i}\text{dz}\) |
where \({e}_{i}\) are the vectors of the global cartesian base. |
\({\chi }_{\mathrm{1,}}{\chi }_{\mathrm{2,}}{\chi }_{3}\): |
linear couples acting around the axes of the global coordinate system. |
\({\chi }_{i}\mathrm{=}\underset{\mathrm{-}h\mathrm{/}2}{\overset{+h\mathrm{/}2}{\mathrm{\int }}}{\mathit{zF}}_{c}\text{.}{e}_{i}\text{dz}\) |
where \({e}_{i}\) are the vectors of the global cartesian base. |
Note:
We also note \(\phi\) and \(\chi\) the linear force and moment distributions applied to the outline of the finite element.
b) |
Loads given in the local coordinate system: |
We then have:
\(\begin{array}{cc}{W}_{\text{ext}}\mathrm{=}& \underset{S}{\mathrm{\int }}(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\tilde{f}}_{\alpha }{t}_{\alpha i}{u}_{i}+{\tilde{c}}_{1}{\tilde{\beta }}_{1}+{c}_{2}{\tilde{\beta }}_{2})\text{dS}+\underset{C}{\mathrm{\int }}(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\tilde{\phi }}_{\alpha }{t}_{\alpha i}{u}_{i}+{\chi }_{1}{\tilde{\beta }}_{1}+{\chi }_{2}{\tilde{\beta }}_{2})\text{ds}\mathrm{=}\\ & \underset{S}{\mathrm{\int }}(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\tilde{f}}_{\alpha }{t}_{\alpha i}{u}_{i}+{\tilde{c}}_{1}{\tilde{\theta }}_{2}\mathrm{-}{\tilde{c}}_{2}{\tilde{\theta }}_{1})\text{dS}+\underset{C}{\mathrm{\int }}(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\tilde{\phi }}_{\alpha }{t}_{\alpha i}{u}_{i}+{\chi }_{1}{\tilde{\theta }}_{2}\mathrm{-}{\chi }_{2}{\tilde{\theta }}_{1})\text{ds}\end{array}\)
The expressions for \({\tilde{f}}_{1},{\tilde{f}}_{2},{\tilde{f}}_{3}\) and \({\tilde{c}}_{1},{\tilde{c}}_{2},{\tilde{c}}_{3}\) are the analogs of the expressions obtained for \({f}_{1},{f}_{2},{f}_{3}\) and \({c}_{1},{c}_{2},{c}_{3}\) by replacing the \({e}_{i}\) with the \({t}_{i}\).
Note:
For the couple \(c\) , the contribution \({\tilde{c}}_{3}\) associated with \(n\) is zero in shell theory.
3.3. Work of inertial forces#
The work due to acceleration quantities is written as:
\({W}^{\text{ac}}\mathrm{=}{\mathrm{\int }}_{\Omega }\rho \ddot{\mathit{OQ}}\text{'}\mathrm{\cdot }\mathit{OQ}\text{'}\mathit{dv}\)
where \(\rho\) is density.
We assume that \(\stackrel{\text{.}\text{.}}{\text{OQ}}\text{'}\), the acceleration vector of the point \(Q\text{'}\) is of the following form:
\(\stackrel{\text{.}\text{.}}{\text{OQ}}\text{'}\mathrm{=}{\ddot{U}}_{k}{e}_{k}+W\mathrm{\wedge }\left[W\mathrm{\wedge }{x}_{k}^{0}{e}_{k}\right]\)
where we neglected the Coriolis forces and the metric correction in thickness.
We note \({\ddot{U}}_{k}\mathrm{=}\frac{{d}^{2}{U}_{k}}{{\text{dt}}^{2}}\), and \(\Omega\) is the uniform rotation vector of the global coordinate system \((O,{e}_{k})\) (with respect to a Galilean coordinate system which has the same origin \(O\) as the global coordinate system).
We express \(\Omega\) in the global \(({e}_{k})\) base:
\(\Omega \mathrm{=}{\Omega }_{k}{e}_{k}\)
For virtual travel \(\text{OQ}\text{'}\), we have:
\(\text{OQ}\text{'}\mathrm{=}{U}_{k}{e}_{k}\)
The work due to the acceleration quantities then becomes:
\({W}^{\text{ac}}\mathrm{=}\underset{\Omega }{\mathrm{\int }}\rho {U}_{k}{e}_{k}\left[{\ddot{U}}_{k}{e}_{k}+\Omega \mathrm{\wedge }(\Omega \mathrm{\wedge }{x}_{k}^{0}{e}_{k})\right]\text{dv}\mathrm{=}{W}_{\text{mass}}^{\text{ac}}+{W}_{\text{cent}}^{\text{ac}}\)
with:
\({W}_{\text{masse}}^{\text{ac}}\mathrm{=}{\mathrm{\int }}_{\Omega }\rho {U}_{k}{\ddot{U}}_{k}\text{dv}\)
and:
\({W}_{\text{cent}}^{\text{ac}}\mathrm{=}\underset{\Omega }{\mathrm{\int }}{\mathit{\rho U}}_{k}{e}_{k}\left[\Omega \mathrm{\wedge }(\Omega \mathrm{\wedge }{x}_{k}^{0}{e}_{k})\right]\text{dv}\)
3.4. Principle of virtual work#
For a static load, it is written as follows: \(\delta {W}_{\text{ext}}\mathrm{=}\delta {W}_{\text{def}}\) where \({W}_{\text{ext}}\) is the sum of the various elementary jobs, corresponding to the various loads.
In harmonic dynamics (natural mode calculations), the principle of virtual works gives: \(\delta {W}_{\text{ext}}+\delta {W}_{\text{mass}}^{\text{ac}}\mathrm{=}0\)