基于Zig-zag理论的波形钢腹板梁自由振动分析
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摘要
根据波形钢腹板梁的特点,将波形钢腹板梁模拟为具有正交各向异性层的夹芯梁,并引入层间连续的Zig-zag位移假设和分层抛物线分布的横向切应力假设,基于混合能变分原理,建立波形钢腹板梁自由振动的Zig-zag理论.该理论的横向切应力满足上、下表面为0以及层间连续条件,因此无需引入剪切修正系数.导出波形钢腹板梁的频率方程,得到不同边界条件下的频率和振型.与其他方法的对比表明该理论能够精确地预测波形钢腹板梁的动力学特性.
Concrete beams with CSWs were idealized as sandwich beams with an orthotropic core, according to the characteristics of the corrugated steel web(CSW). The assumptions of a zig-zag displacement and a layer-wise parabolic distribution of the transverse shear stress were introduced to derive the governing equations for the free vibration of concrete beams with CSWs based on the variational principle of mixed energy. The transverse shear stress in the developed theory satisfied the traction-free condition at both top and bottom surfaces of the beam, as well as the continuity condition at the interfaces between adjacent layers. The shear correction factor is therefore not necessary. The frequency equation of the beam with CSWs was derived and solved analytically for the frequencies and corresponding mode shapes under various boundary conditions. Comparisons with other methods indicate that the present theory can accurately predict the dynamic behavior of such beams.
Concrete beams with CSWs were idealized as sandwich beams with an orthotropic core, according to the characteristics of the corrugated steel web(CSW). The assumptions of a zig-zag displacement and a layer-wise parabolic distribution of the transverse shear stress were introduced to derive the governing equations for the free vibration of concrete beams with CSWs based on the variational principle of mixed energy. The transverse shear stress in the developed theory satisfied the traction-free condition at both top and bottom surfaces of the beam, as well as the continuity condition at the interfaces between adjacent layers. The shear correction factor is therefore not necessary. The frequency equation of the beam with CSWs was derived and solved analytically for the frequencies and corresponding mode shapes under various boundary conditions. Comparisons with other methods indicate that the present theory can accurately predict the dynamic behavior of such beams.
引文
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[16]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981:423-424.
[17]XU R Q,WU Y F.Static,dynamic,and buckling analysis of partial interaction composite members using Timoshenko's beam theory[J].International Journal of Mechanical Sciences,2007,49(10):1139-1155.
[2]JIANG R J,KWONG AU F T,XIAO Y F.Prestressed concrete girder bridges with corrugated steel webs:review[J].Journal of Structural Engineering,2014,141(2):04014108.
[3]DAYYANI I,SHAW A D,FLORES E I S,et al.The mechanics of composite corrugated structures:a review with applications in morphing aircraft[J].Composite Structures,2015,133:358-380.
[4]OH J Y,LEE D H,KIM K S.Accordion effect of prestressed steel beams with corrugated webs[J].Thin-walled structures,2012,57:49-61.
[5]ZHOU W B,YAN W J.Refined nonlinear finite element modelling towards ultimate bending moment calculation for concrete composite beams under negative moment[J].Thin-Walled Structures,2017,116:201-211.
[6]CHEN X C,LI Z H,AU F T K,et al.Flexural vibration of prestressed concrete bridges with corrugated steel webs[J].International Journal of Structural Stability and Dynamics,2017,17(02):1750023.
[7]MO Y L,JENG C H,KRAWINKLER H.Experimental and analytical studies of innovative prestressed concrete box-girder bridges[J].Materials and Structures,2003,36(2):99-107.
[8]韦忠瑄,孙鹰,沈庆,等.波形钢腹PC组合箱梁的动力特性研究[J].固体力学学报,2011(s1):394-398.WEI Zhong-xuan,SUN Ying,SHEN Qing,et al.Study on dynamic properties of the prestressed concrete box-girder with corrugated steel webs[J].Chinese Journal of Solid Mechanics,2011(s1):394-398.
[9]CARRERA E.Historical review of Zig-zag theories for multilayered plates and shells[J].AppliedMechanics Reviews,2003,56(3):287-308.
[10]AMBARTSUMIAN S A.On a general theory of anisotropic shells[J].Journal of Applied Mathematics and Mechanics,1958,22(2):305-319.
[11]REISSNER E.On a certain mixed variational theorem and a proposed application[J].International Journal for Numerical Methods in Engineering,1984,20(7):1366-1368.
[12]REISSNER E.On a mixed variational theorem and on shear deformable plate theory[J].International Journal for Numerical Methods in Engineering,1986,23(2):193-198.
[13]MURAKAMI H.Laminated composite plate theory with improved in-plane responses[J].Journal of Applied Mechanics,1986,53(3):661-666.
[14]XU R Q,WU Y F.Two-dimensional analytical solutions of simply supported composite beams with interlayer slips[J].International Journal of Solids and Structures,2007,44(1):165-75.
[15]JOHNSON R P,CAFOLLA J,BERNARD C.Corrugated webs in plate girders for bridges[J].Proceedings of the Institution of Civil EngineersStructures and Buildings,1997,122(2):157-164.
[16]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981:423-424.
[17]XU R Q,WU Y F.Static,dynamic,and buckling analysis of partial interaction composite members using Timoshenko's beam theory[J].International Journal of Mechanical Sciences,2007,49(10):1139-1155.