7.2 – Matrizenformulierung

 

Auszug aus dem Skript der Vorlesung Finite Elemente bei Dr.-Ing Philipp Höfer an der UniBw München.

Spannungen:

{\left\{ \sigma \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{\sigma _{xx}}}&{{\sigma _{yy}}}&{{\sigma _{xy}}} \end{array}} \right\}

Verzerrungen:

{\left\{ \varepsilon \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{\varepsilon _{xx}}}&{{\varepsilon _{yy}}}&{{\varepsilon _{xy}}} \end{array}} \right\}

{\left\{{\delta \varepsilon } \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\delta {\varepsilon _{xx}}}&{\delta {\varepsilon _{yy}}}&{\delta {\varepsilon _{xy}}} \end{array}} \right\}

Verschiebungen:

{\left\{ u \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{u_x}}&{{u_y}} \end{array}} \right\}

{\left\{{\delta u} \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\delta {u_x}}&{\delta {u_y}} \end{array}} \right\}

Volumen- und Randkräfte:

{\left\{ k \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{k_x}}&{{k_y}} \end{array}} \right\},\quad \quad {\left\{ s \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{s_x}}&{{s_y}} \end{array}} \right\}

Einsetzen in die Anteile an den virtuellen Verschiebungen:

\delta {W_\sigma } = \int\limits_A {{{\left\{{\delta \varepsilon } \right\}}^T}\left\{ \sigma \right\}hdA}

\delta {W_a} = \int\limits_A {\rho {{\left\{{\delta u} \right\}}^T}\left\{ k \right\}hdA} +\oint\limits_C {{{\left\{{\delta u} \right\}}^T}\left\{ s \right\}hds}

\delta {W_T} = -\int\limits_A {\rho {{\left\{{\delta u} \right\}}^T}\left\{{\ddot u} \right\}hdA}

Annahme: \mathbb{E}{\vec e_3} = 0\quad \Rightarrow \quad {\varepsilon _{3j}} = {\varepsilon _{j3}} = 0\quad \Rightarrow \quad {\varepsilon _{zz}} = {\varepsilon _{xz}} = {\varepsilon _{yz}} = 0

Virtuelle Arbeiten

Arbeit der Spannungen an den virtuellen Verzerrungen:

\delta {W_\sigma } = \int\limits_A {\left( {{\sigma _{xx}}\delta {\varepsilon _{xx}}+{\sigma _{yy}}\delta {\varepsilon _{yy}}+{\sigma _{xy}}\delta {\varepsilon _{xy}}} \right)hdA}

Virtuelle Arbeit der äußeren Kräfte:

\delta {W_a} = \int\limits_A {\rho \left( {{k_x}\delta {u_x}+{k_y}\delta {u_y}} \right)hdA} +\oint\limits_C {\left( {{s_x}\delta {u_x}+{s_y}\delta {u_y}} \right)hds}

Virtuelle Arbeit der Trägheitskräfte:

\delta {W_T} = -\int\limits_A {\rho \left( {{{\ddot u}_x}\delta {u_x}+{{\ddot u}_y}\delta {u_y}} \right)hdA}