PlateSection

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The plate section model implements a complex cross section built from multiple PlaneStress material class instances

\[\varepsilon_{kl}(z) = \varepsilon^0_{kl} - z \,\phi_{kl}\]

\[n^{ij} = \int_{-h/2}^{h/2} \sigma^{ij}(z)\, dz \qquad \qquad m^{ij} = -\int_{-h/2}^{h/2} z\sigma^{ij}(z)\, dz\]

Stress \(\sigma^{ij}(z)\) is obtained from a PlaneStress material class material layer at distance z. Through-the-thickness integration is performed numerically as a sum over layers.

The PlateSection material will further compute the tangent stiffness tensors defined as:

\[dn^{ij} =: \mathbb{D}^{ijkl} \, d\varepsilon^{0}_{kl} +\mathbb{C}^{ijkl} d\phi_{kl}\]

\[dm^{ij} =: \mathbb{C}^{ijkl} \, d\varepsilon^{0}_{kl} + \mathbb{B}^{ijkl} \, d\phi_{kl}\]

where

\(\varepsilon^0_{ij}\)	membrane strain (input)
\(\phi_{ij}\)	curvature change (input)
\(n^{ij}\)	component of resulting membrane force (output)
\(m^{ij}\)	component of resulting moment (output)
\(\mathcal{C}^{ijkl}(z)\)	material tangent stiffness of a plane-stress material layer at distance z
\(\mathbb{D}^{ijkl}\)	axial stiffness
\(\mathbb{B}^{ijkl}\)	flexural stiffness
\(\mathbb{C}^{ijkl}\)	coupling stiffness