Explicit Finite Element Formulations for the Free Vibrations and Buckling of Partial-Interaction Composite Beams with Two Sub-components

Abstract

The variational principles and corresponding finite element formulations are developed in this work for the free vibration and buckling analysis of partial-interaction composite beams under complex engineering situations, such as axial loading conditions, non-uniform material distributions, multi-span boundaries. The composites structures consist of sub-components connected with shear connections of finite rigidities and thus suffer from the partial-interaction phenomenon, significantly affecting the structural performance. The macroscopic elements for dynamic and buckling behavior are explicitly derived by simultaneously considering the sub-elements’ deformations and interlayer slips, avoiding the stress concentrations in the contact areas existing in the commercial packages. The generated frequencies and critical loads of several orders for the single- or two-span composite structures with different boundary conditions are then validated against the analytical and numerical results in the literature with well-matched agreement. The interfacial shear connections and structural parameters are also studied of their influences on the structural dynamic and buckling behavior. The present theory is finally implemented in the theoretical analysis of the steel box-girder viaduct with non-uniformly-distributed shear braces and multi-span continuous composite beams of Jiubao Bridge. The explicit expressions of the elements are provided and can be readily implemented into the commercial software with high computational efficiency.

Appendices

Appendix 1

The elements of the symmetric mass matrix ${\mathbf{M}}^{e}$ for the dynamic analysis of partial-interaction composite beams are

$$m_{11} = \frac{1}{210}l_{e} (70 - 7m + m^{2} )\rho A_{0} + \frac{{6m^{2} \rho I_{0} }}{{5l_{e} }},$$

$$m_{12} = \frac{1}{840}l_{e}^{2} (35 - 7m + 2m^{2} )\rho A_{0} + \frac{1}{10}m(5 + 6m)\rho I_{0} ,$$

$$m_{13} = \frac{1}{420}l_{e} ( - 7 + 2m)p\rho A_{0} + \frac{{6mp\rho I_{0} }}{{5l_{e} }},$$

$$m_{14} = - \frac{1}{210}l_{e} ( - 35 - 7m + m^{2} )\rho A_{0} - \frac{{6m^{2} \rho I_{0} }}{{5l_{e} }},$$

$$m_{15} = \frac{1}{840}l_{e}^{2} ( - 35 - 7m + 2m^{2} )\rho A_{0} + \frac{1}{10}m(5 + 6m)\rho I_{0} ,$$

$$m_{16} = \frac{1}{420}l_{e} ( - 7 + 2m)p\rho A_{0} + \frac{{6mp\rho I_{0} }}{{5l_{e} }},$$

$$m_{22} = \frac{1}{840}l_{e}^{3} (7 + m^{2} )\rho A_{0} + \frac{1}{30}l_{e} (10 + 15m + 9m^{2} )\rho I_{0} ,$$

$$m_{23} = \frac{1}{420}l_{e}^{2} mp\rho A_{0} + \frac{1}{10}(5 + 6m)p\rho I_{0} ,$$

$$m_{24} = \frac{1}{840}l_{e}^{2} (35 + 7m - 2m^{2} )\rho A_{0} - \frac{1}{10}m(5 + 6m)\rho I_{0} ,$$

$$m_{25} = \frac{1}{840}l_{e}^{3} ( - 7 + m^{2} )\rho A_{0} + \frac{1}{30}l_{e} (5 + 15m + 9m^{2} )\rho I_{0} ,$$

$$m_{26} = \frac{1}{420}l_{e}^{2} mp\rho A_{0} + \frac{1}{10}(5 + 6m)p\rho I_{0} ,$$

$$m_{33} = \frac{1}{210}l_{e} p^{2} \rho A_{0} + \frac{{6p^{2} \rho I_{0} }}{{5l_{e} }},$$

$$m_{34} = \frac{1}{420}l_{e} (7 - 2m)p\rho A_{0} - \frac{{6mp\rho I_{0} }}{{5l_{e} }},$$

$$m_{35} = \frac{1}{420}l_{e}^{2} mp\rho A_{0} + \frac{1}{10}(5 + 6m)p\rho I_{0} ,$$

$$m_{36} = \frac{1}{210}l_{e} p^{2} \rho A_{0} + \frac{{6p^{2} \rho I_{0} }}{{5l_{e} }},$$

$$m_{44} = \frac{1}{210}l_{e} (70 - 7m + m^{2} )\rho A_{0} + \frac{{6m^{2} \rho I_{0} }}{{5l_{e} }},$$

$$m_{45} = - \frac{1}{840}l_{e}^{2} (35 - 7m + 2m^{2} )\rho A_{0} - \frac{1}{10}m(5 + 6m)\rho I_{0} ,$$

$$m_{46} = \frac{1}{420}l_{e} (7 - 2m)p\rho A_{0} - \frac{{6mp\rho I_{0} }}{{5l_{e} }},$$

$$m_{55} = \frac{1}{840}l_{e}^{3} (7 + m^{2} )\rho A_{0} + \frac{1}{30}l_{e} (10 + 15m + 9m^{2} )\rho I_{0} ,$$

$$m_{56} = \frac{1}{420}l_{e}^{2} mp\rho A_{0} + \frac{1}{10}(5 + 6m)p\rho I_{0} ,$$

$$m_{66} = \frac{1}{210}l_{e} p^{2} \rho A_{0} + \frac{{6p^{2} \rho I_{0} }}{{5l_{e} }}.$$

Appendix 2

The additional stiffness matrix ${\mathbf{K}}_{{P_{A} }}^{e}$ for dynamic analysis of partial-interaction composite beams considering the influence of axial forces is

$${\mathbf{K}}_{{P_{A} }}^{e} = \frac{P}{{60l_{e} }}\left[ {\begin{array}{*{20}c} {60 + 12m^{2} } & {6l_{e} m^{2} } & {12mp} & { - 60 - 12m^{2} } & {6l_{e} m^{2} } & {12mp} \\ {6l_{e} m^{2} } & {5l_{e}^{2} + 3l_{e} m^{2} } & {6l_{e} mp} & { - 6l_{e} m^{2} } & { - 5l_{e}^{2} + 3l_{e} m^{2} } & {6l_{e} mp} \\ {12mp} & {6l_{e} mp} & {12p^{2} } & { - 12mp} & {6l_{e} mp} & {12p^{2} } \\ { - 60 - 12m^{2} } & { - 6l_{e} m^{2} } & { - 12mp} & {60 + 12m^{2} } & { - 6l_{e} m^{2} } & { - 12mp} \\ {6l_{e} m^{2} } & { - 5l_{e}^{2} + 3l_{e} m^{2} } & {6l_{e} mp} & { - 6l_{e} m^{2} } & {5l_{e}^{2} + 3l_{e} m^{2} } & {6l_{e} mp} \\ {12mp} & {6l_{e} mp} & {12p^{2} } & { - 12mp} & {6l_{e} mp} & {12p^{2} } \\ \end{array} } \right]$$

in which $m = \frac{1}{1 - \mu }\frac{{\left( {1 + 10\gamma_{2} } \right)l_{e}^{2} }}{{120\gamma_{2} \gamma_{3} h^{2} }}$, $p = \frac{1}{1 - \mu }\frac{{l_{e} }}{{24\gamma_{2} h}}$ and $\mu = \frac{{\left( {1 + 12\gamma_{1} } \right)\left( {1 + 10\gamma_{2} } \right)l_{e}^{2} }}{{120\gamma_{2} \gamma_{3} h^{2} }}$,$\gamma_{1} = \frac{{\overline{D}}}{{Cl_{e}^{2} }}$, $\gamma_{2} = \frac{{\overline{EA} }}{{k_{s} l_{e}^{2} }}$, $\gamma_{3} = \frac{{\overline{EA} }}{C}$, $\overline{D} = EI_{0} + h^{2} \overline{EA}$.