歪と図心変位の関係式

$ \gdef\d{\mathrm{d}}\varepsilon_{zz}=\frac{\d u_{cz}}{\d z}-y\frac{\d^2 u_{cy}}{{\d z}^2}

導出

$ \pmb{u}_c:z\rightarrow\pmb{u}(0,0,z)

と定義される

簡単のため、以下の記号を定義する

$ \varepsilon_{zz}:=[\pmb{\varepsilon}]^\mathcal{SS}_{zz}

$ u_{cy}:=[\pmb{u}_c]^\mathcal{S}_{y}

$ u_{cz}:=[\pmb{u}_c]^\mathcal{S}_{z}

$ [\pmb{u}(y,z)]^\mathcal{S}= [\pmb{u}_c(z)]^\mathcal{S}+\begin{pmatrix}0\\0\\y\mathrm{d}\theta(z)\end{pmatrix}

https://kakeru.app/e529892c15f9ea4f4b225e1627ea64be https://i.kakeru.app/e529892c15f9ea4f4b225e1627ea64be.svg

$ \varepsilon_{zz}\pmb{e}_z\otimes\pmb{e}_z=\frac{1}{2}\pmb{\nabla}(\pmb{u}_c(z)+y\mathrm{d}\theta(z)\pmb{e}_z)+\frac{1}{2}\pmb{\nabla}(\pmb{u}_c(z)+y\mathrm{d}\theta(z)\pmb{e}_z)^\top

$ \implies\begin{dcases}\varepsilon_{zz}=\frac{\partial}{\partial z}(u_{cz}(z)+y\mathrm{d}\theta(z))\\0=\frac{1}{2}\left(\frac{\partial u_{cy}(z)}{\partial z}+\frac{\partial}{\partial y}(u_{cz}(z)+y\mathrm{d}\theta(z))\right)\end{dcases}

$ \iff\begin{dcases}\varepsilon_{zz}=\frac{\partial u_{cz}(z)}{\partial z}+y\frac{\partial \mathrm{d}\theta(z)}{\partial z}\\0=\frac{\partial u_{cy}(z)}{\partial z}+\mathrm{d}\theta(z)\end{dcases}

$ \implies \varepsilon_{zz}=\frac{\partial u_{cz}(z)}{\partial z}-y\frac{\partial^2 \partial u_{cy}(z)}{{\partial z}^2}

$ \therefore \gdef\d{\mathrm{d}}\varepsilon_{zz}=\frac{\d u_{cz}}{\d z}-y\frac{\d^2 u_{cy}}{{\d z}^2}

引数は省略した

常微分なので何の函数かは分母から自明

Reference