Please wait a minute...
Journal of ZheJiang University (Engineering Science)  2019, Vol. 53 Issue (3): 503-511    DOI: 10.3785/j.issn.1008-973X.2019.03.011
Civil Engineering     
Free vibration analysis of concrete beams with corrugated steel webs based on Zig-zag theory
Lin-yuan HU1(),Wei-qiu CHEN1,Zhi-cheng ZHANG2,Rong-qiao XU2,*()
1. School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
2. College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
Download: HTML     PDF(751KB) HTML
Export: BibTeX | EndNote (RIS)      

Abstract  

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.



Key wordscorrugated steel web      Zig-zag theory      variational principle of mixed energy      free vibration      frequency      mode shape     
Received: 07 January 2018      Published: 04 March 2019
CLC:  O 342  
Corresponding Authors: Rong-qiao XU     E-mail: hulinyuan@zju.edu.cn;xurongqiao@zju.edu.cn
Cite this article:

Lin-yuan HU,Wei-qiu CHEN,Zhi-cheng ZHANG,Rong-qiao XU. Free vibration analysis of concrete beams with corrugated steel webs based on Zig-zag theory. Journal of ZheJiang University (Engineering Science), 2019, 53(3): 503-511.

URL:

http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2019.03.011     OR     http://www.zjujournals.com/eng/Y2019/V53/I3/503


基于Zig-zag理论的波形钢腹板梁自由振动分析

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.


关键词: 波形钢腹板,  Zig-zag理论,  混合能变分原理,  自由振动,  频率,  振型 
Fig.1 Geometric configuration of corrugated steel web (CSW)
Fig.2 Original and equivalent cross sections of CSW beam
n f/Hz δ/% f/Hz δ/%
有限元 本文方法 经典梁理论
1 17.191 17.218 0.16 19.950 16.05
2 51.768 51.904 0.26 79.800 54.15
3 88.945 89.264 0.36 179.550 101.90
4 126.248 126.572 0.26 319.200 152.80
5 162.577 163.840 0.78 498.749 206.80
6 199.196 201.377 1.10 718.199 260.50
7 235.990 239.461 1.47 977.549 314.20
8 273.130 278.316 1.90 1 276.800 367.50
Tab.1 Natural frequency of simply support (SS) beam with simply support (SS)
n f/Hz δ/% f/Hz δ/%
有限元 本文方法 经典梁理论
1 27.665 27.708 0.16 45.224 63.47
2 58.502 58.737 0.40 124.663 113.10
3 93.965 94.475 0.54 244.388 160.10
4 130.414 131.375 0.74 403.989 209.80
5 167.346 168.981 0.98 603.490 260.60
6 204.474 207.107 1.29 842.891 312.20
7 241.953 245.946 1.65 1 122.180 363.80
8 279.834 286.625 2.07 1 441.400 415.10
Tab.2 Frequencies of clamped-clamped (CC) beam with CSWs
n f/Hz δ/% f/Hz δ/%
有限元 本文方法 经典梁理论
1 22.435 22.473 0.17 31.166 38.91
2 55.357 55.537 0.33 100.997 82.45
3 91.464 91.880 0.46 210.723 130.40
4 128.171 128.980 0.63 360.348 181.10
5 164.940 166.390 0.88 549.874 233.40
6 201.835 204.220 1.18 779.299 286.10
7 238.937 242.671 1.56 1 048.61 338.90
8 276.454 281.931 1.98 1 357.84 391.20
Tab.3 Frequencies of clamped-simple (CS) beam with CSWs
n f/Hz δ/% f/Hz δ/%
有限元 本文方法 经典梁理论
1 6.593 6.599 0.10 7.107 7.80
2 30.762 30.913 0.49 44.539 44.79
3 67.243 67.647 0.60 124.713 85.47
4 104.546 105.350 0.77 244.384 133.70
5 142.189 143.544 0.95 403.989 184.10
6 179.391 181.642 1.26 603.490 236.40
7 216.701 220.171 1.60 842.890 289.00
8 254.029 259.211 2.04 1 122.180 341.80
Tab.4 Frequencies of clamped-free (CF) beam with CSWs
Fig.3 First three mode shapes of beam with CSWs
[1]   陈宝春, 黄卿维 波形钢腹板PC箱梁桥应用综述[J]. 公路, 2005, 7 (7): 45- 53
CHEN Bao-chun, HUANG Qing-wei Summary of application of prestressed concrete box-girder bridges with corrugated steel webs[J]. Highway, 2005, 7 (7): 45- 53
doi: 10.3969/j.issn.1002-0268.2005.07.011
[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
doi: 10.1016/j.compstruct.2015.07.099
[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
doi: 10.1016/j.tws.2012.04.005
[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
doi: 10.1016/j.tws.2017.02.011
[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
doi: 10.1142/S0219455417500237
[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
doi: 10.1007/BF02479523
[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]. Applied Mechanics Reviews, 2003, 56 (3): 287- 308
doi: 10.1115/1.1557614
[10]   AMBARTSUMIAN S A On a general theory of anisotropic shells[J]. Journal of Applied Mathematics and Mechanics, 1958, 22 (2): 305- 319
doi: 10.1016/0021-8928(58)90108-4
[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
doi: 10.1002/(ISSN)1097-0207
[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
doi: 10.1115/1.3171828
[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
doi: 10.1016/j.ijsolstr.2006.04.027
[15]   JOHNSON R P, CAFOLLA J, BERNARD C Corrugated webs in plate girders for bridges[J]. Proceedings of the Institution of Civil Engineers-Structures and Buildings, 1997, 122 (2): 157- 164
doi: 10.1680/istbu.1997.29305
[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.
[1] Li WANG,Shi-zhong LIU,Wei LU,Si-sheng NIU,Xin-lei SHI. Temperature effect of new-type composite box girder with corrugated steel webs[J]. Journal of ZheJiang University (Engineering Science), 2021, 55(4): 675-683.
[2] Bang-huang CAI,Hui-min SONG,Shan-guang GUO,Hai-deng ZHANG,Jia-ming SHENG. Control effect of radio frequency discharge plasma excitation on shock wave/boundary layer interference[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(9): 1839-1848.
[3] Tong-you CHEN,Jia-qiang YANG,Shi-da ZHENG,De-zhi MENG,Min KANG. IF starting smooth switching method of sensorless permanent magnet fan based on cosine function[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(8): 1572-1577.
[4] Yue WANG,Zhen-bang ZHOU,Yun PENG. Collaborative selective harmonic elimination pulse width modulation for dual-module parallel-connected three-level converters[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(7): 1425-1432.
[5] Xiao-shun WU,Chen-hui HUANG,Xin-tao WANG,Hua DENG. Fast measurement of static displacement field in cable-strut tensile structure[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(6): 1086-1094.
[6] Han-yun ZHOU,Shan-he HUANG. Application of broadband parametric array in sub-bottom profile measurement[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(5): 972-977.
[7] Jia-lei ZHAO,Ding ZHOU,Jian-dong ZHANG,Chao-bin HU. Free vibration characteristics of multi-cracked beam based on Chebyshev-Ritz method[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(4): 778-786.
[8] Cong-cong FU,Xing-fei LI,Shao-bo YANG,Hong-yu LI,Wen-bin TAN. Frequency splitting characteristics and maximum power point analysis of ICPT system with ocean buoys[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(3): 475-482.
[9] Xin YU,Xiao-jun JIN,Shi-ming MO,Wei ZHANG,Zhao-bin XU,Zhong-he JIN. Performance assessment of BeiDou B3-based real-time orbit determination for LEO satellites[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(3): 589-596.
[10] Hui XU,Chang-sheng ZHU. Active control with multi-frequency transmission force of multi-span rotor system[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(3): 597-605.
[11] Guang OUYANG,Tian-jun LI,Jiang-tao ZHANG,Jing-feng WANG,Rong-qiao XU. Tension analysis of hangers with stepped cross-section based on state space method[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(2): 257-263.
[12] Chang WANG,Yong-sheng ZHANG,Xu WANG,Ying YU. Remote sensing image change detection method based on deep neural networks[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(11): 2138-2148.
[13] Ping WANG,Le LIU,Chen-yang HU,Zheng GONG,Jing-mang XU,Zhi-xin WANG. Identification of switch rail brakeage in high speed railway turnout based on elastic wave propagation[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(10): 2038-2046.
[14] Ya-qin ZHAO,Yu-ting NIE,Long-wen WU,Yu-peng ZHANG,Sheng-yang HE. Multi-component signal separation using variational nonlinear chirp mode decomposition based on ridge tracking[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(10): 1874-1882.
[15] Hong-guang LI,Ying GUO,Ping SUI,Zi-sen QI. Frequency hopping modulation recognition of convolutional neural network based on time-frequency characteristics[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(10): 1945-1954.