天然橡胶支座大变形压剪性能的双非线性超弹性理论和实验研究

杨静; 潘文; 苏何先; 何颖成; 管庆松; 张岩岩

doi:10.6052/j.issn.1000-4750.2021.04.0325

天然橡胶支座大变形压剪性能的双非线性超弹性理论和实验研究

doi: 10.6052/j.issn.1000-4750.2021.04.0325

杨静^{1, 2, ,},
潘文^{1, 2,},
苏何先^{1, 2,},
何颖成^1,,
管庆松^3,,
张岩岩^4,

1.
昆明理工大学建筑工程学院，云南，昆明 650500
2.
云南省抗震工程技术研究中心，云南，昆明 650500
3.
震安科技股份有限公司，云南，昆明 650000
4.
云南省设计院第五建筑设计有限公司，云南，昆明 650000

详细信息

作者简介:
潘　文(1968−)，男，江苏人，教授，博士，博导，主要从事结构抗震和防灾减灾研究(E-mail: 18987884341@189.com)

苏何先(1982−)，男，四川人，高级实验师，博士，主要从事结构抗震及实验研究(E-mail: 47762702@qq.com)

何颖成(1963−)，男，云南人，硕士，主要从事结构抗震研究(E-mail: 970158864@qq.com)

管庆松(1982−)，男，河南人，高工，硕士，主要从事建筑减隔震技术研究(E-mail: qsguandream@126.com)

张岩岩(1986−)，女，山东人，高工，硕士，主要从事结构减隔震设计研究(E-mail: 503901986@qq.com)

通讯作者:
杨　静(1982−)，女，河南人，高工，博士生，一级注册结构工程师，主要从事结构隔震研究(E-mail: 314679765@qq.com)

中图分类号: TU352.1
计量
- 文章访问数: 282
- HTML全文浏览量: 164
- PDF下载量: 58
- 被引次数: 0
出版历程
- 收稿日期: 2021-04-29
- 修回日期: 2021-07-18
- 网络出版日期: 2021-07-28
- 刊出日期: 2022-08-01

DOUBLE NONLINEAR HYPERELASTIC THEORY AND EXPERIMENTAL RESEARCH ON THE LARGE DEFORMATION OF NATURAL RUBBER BEARING IN COMPRESSION AND SHEAR

YANG Jing^{1, 2
, ,},
PAN Wen^{1, 2
,},
SU He-xian^{1, 2
,},
HE Ying-cheng^1
,,
GUAN Qing-song^3
,,
ZHANG Yan-yan^4
,

1.
School of Civil Engineering, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
2.
Earthquake Engineering Researching Center of Yunnan, Kunming, Yunnan 650500, China
3.
Zhenan Technology Co., Ltd., Kunming, Yunnan 650000, China
4.
The Fifth Architectural Design Co., Ltd. of Yunnan Design Institute, Kunming, Yunnan 650500, China

摘要

摘要: 基于非线性固体力学已经形成与材料微观结构紧密结合的发展局面，该文从微观上橡胶的超弹性本构方程出发，推导出单层橡胶内任一点的竖向应力满足调和函数，进一步对竖向应力积分得到单层橡胶的单轴等效弹性模量${E_{\rm c}}$与纯弯等效弯曲刚度${E_{\rm c}}I_{\rm s}$。将隔震橡胶支座等效为符合${E_{\rm c}}$与${E_{\rm c}}I_{\rm s}$的均质体，建立橡胶支座在两种外部荷载同时作用下，能宏观反映剪切变形与弯曲程度的偏微分平衡方程，并得到通用解答，解决了橡胶支座竖向与水平两种荷载耦合作用时大剪切变形的几何非线性问题。在此基础上，开展了足尺隔震橡胶支座压剪实验，依据实验剪切模量$G$和水平剪应变$\gamma$曲线，得到的支座水平推力${{F_{\rm H}}}$和$\gamma $的实验曲线与理论曲线几乎完全重叠，即通过将材料非线性引入以上微分平衡方程，实现了超弹性橡胶支座大剪切变形的双非线性问题。通过以上解答，得到了隔震橡胶支座内力分布规律，对判断支座薄弱部位有明确的指导意义。随后，对隔震橡胶支座比较重要的两个特性(剪应变相关性，轴压力相关性)进行了对比分析，结果表明：随着轴力增大，橡胶支座的P-Δ效应并不明显，而橡胶内应力的变化不可忽视，同时，剪应变越大，压力相关性越强。可以采用该文的理论方法，通过传感器监测到支座内部应力反演支座水平推力与位移，从而实现地震作用时隔震建筑的健康监测。
- 非线性连续介质力学 /
- 几何非线性 /
- 偏微分方程 /
- 材料非线性 /
- 超弹性本构方程 /
- 拉普拉斯算子
Abstract: Based on the fact that the development of nonlinear solid mechanics has been closely integrated with the material microstructure, it derived the vertical stress satisfying the harmonic function at any point from the super-elastic constitutive equation of the rubber; further, the uniaxial equivalent elastic modulus ${E_{\rm c}}$ and the pure bending equivalent bending stiffness ${E_{\rm c}}I_{\rm s}$ of the single-layer rubber were obtained from the vertical stress integration. The seismic isolation rubber bearing was equivalent to a homogeneous body conforming to ${E_{\rm c}}$ and ${E_{\rm c}}I_{\rm s}$, and the partial differential equilibrium equation macroscopically reflecting the degree of shear and bending deformation of the rubber bearing was established in the simultaneous action of two external load, and its general solution was obtained. The difficulties of the geometrical non-uniformity of large shear deformation of rubber support were solved in the vertical and horizontal loads. On this basis, a full-scale seismic isolation test of rubber bearing for compression and shear experiment was carried out. According to the experimental shear modulus $G$ and horizontal shear strain $\gamma$ curves, the experimental curves of the horizontal thrust ${{F_{\rm H}}}$ and $\gamma $of the bearing were obtained. It was almost completely overlapped with the theoretical curve. Therefore by introducing the material nonlinearity into the above differential equilibrium equation, the double nonlinear problem of large shear deformation of the superelastic rubber bearing was solved. Through the above answers, the internal force distribution law of the rubber bearing was obtained, which has a clear guiding significance for judging the weak part of the bearing. Subsequently, the comparative analysis of two more important characteristics (shear strain correlation and axial pressure correlation) of the rubber bearing was carried out, the results showed that as the axial force increased, the P-Δ effect of the rubber bearing was not obvious, but the change of internal stress in the rubber couldn't be ignored, Concurrently, the greater the shear strain was, the stronger the pressure dependence was. The horizontal thrust and displacement of the support can be obtained through the sensor monitoring the internal stress of the support, achieving the health monitoring of the isolated building during the earthquake.
- nonlinear continuum mechanics /
- geometric nonlinearity /
- partial differential equation /
- material nonlinearity /
- hyperelastic constitutive equation /
- Laplace operator

HTML全文

图 1 单层橡胶纯压变形图

Figure 1. Single-layer rubber pure pressure deformation diagram

下载: 全尺寸图片幻灯片

图 2 微元体受力图

Figure 2. Micro-body force diagram

下载: 全尺寸图片幻灯片

图 3 单层橡胶纯弯变形图

Figure 3. Single-layer rubber pure bending deformation diagram

下载: 全尺寸图片幻灯片

图 4 均质体支座数学物理简图

Figure 4. Mathematics and physics diagram of homogeneous body support

下载: 全尺寸图片幻灯片

图 5 有限大支座几何位移关系

Figure 5. Geometric displacement relations of finite bearings

下载: 全尺寸图片幻灯片

图 6 动态压剪试验机

Figure 6. Dynamic compression shear testing machine

下载: 全尺寸图片幻灯片

图 7 剪切模量G与剪应变$\gamma $计算程序框图

Figure 7. Block diagram for calculation program of shear modulus G and shear strain $\gamma $

下载: 全尺寸图片幻灯片

图 8 隔震橡胶支座压剪$G - \gamma $实验、理论曲线(P=12 MPa)

Figure 8. Compression-shear $G - \gamma $ experiment and theoretical curve of vibration isolation rubber bearing (P=12 MPa)

下载: 全尺寸图片幻灯片

图 9 隔震橡胶支座${F_{\rm{H}}} - \gamma$理论、实验曲线(P=12 MPa)

Figure 9. Compression-shear ${F_{\rm{H}}} - \gamma$ experiment and theoretical curve of vibration isolation rubber bearing (P=12 MPa)

下载: 全尺寸图片幻灯片

图 10 沿截面高度内力$ M{\text{、}}V$及运动量$ \mu {\text{、}}\beta $的分布图

Figure 10. Distribution diagram of internal force $ M{\text{、}}V$ and movement amount $ \mu {\text{、}}\beta $ along the section height

下载: 全尺寸图片幻灯片

图 11 轴压不变、不同剪应变时橡胶支座竖向应力分布图

注：压为正，拉为负

Figure 11. The vertical stress distribution diagram of the rubber bearing when the axial pressure is constant and the shear strain is different

下载: 全尺寸图片幻灯片

图 12 不同轴压，支座顶端水平位移与内力变化图

Figure 12. The horizontal displacement of the top of the support and the internal force change diagram under different axial pressure

下载: 全尺寸图片幻灯片

图 13 剪应变100%、不同轴压支座顶部竖向应力变化图

注：压为正，拉为负

Figure 13. The vertical stress change diagram on the top of the support, with 100% shear strain and different axial compression

下载: 全尺寸图片幻灯片

图 14 剪应变400%、不同轴压支座顶部竖向应力变化图

Figure 14. The vertical stress change diagram on the top of the support, with 400% shear strain and different axial compression

下载: 全尺寸图片幻灯片

图 15 剪应变100%、400%，不同轴压下支座顶部竖应力对比图

注：压为正，拉为负

Figure 15. Comparison of the vertical stress on the top of the support, when the shear strain is 100%, 400%, and different axial compressions

下载: 全尺寸图片幻灯片

表 1 橡胶支座尺寸

Table 1. Size of rubber bearing

材料	单层厚度t/ mm	层数n/ 个	总厚度/ mm	支座半径R/ mm	支座总高h/ mm
橡胶	5.81	16	93	250	138
钢板	3.00	15	45	250	138

下载: 导出CSV

表 2 等截面不同材料属性对比

Table 2. Comparison of different material properties of equal section

材料参数(刚度)	等效均质体	HRB335钢	C40混凝土
G/(N/mm²)	0.89	8.32×10⁴	1.35×10⁴
E_c/(N/mm²)	2.47×10³	2.00×10⁵	3.25×10⁴
A(A_s)/mm²	2.91×10⁵	1.96×10⁵	1.96×10⁵
I(I_s)/mm⁴	4.55×10⁹	3.07×10⁹	3.07×10⁹
EI(E_cI_s)/(N·mm²)	1.13×10¹³	6.14×10¹⁴	9.97×10¹³
GA_s/h_r(GA/h) /(N/mm)	2.79×10³	1.18×10⁸	1.92×10⁷

下载: 导出CSV

表 3 不同轴力下，支座顶部截面内力与水平位移理论值

Table 3. The theoretical values of the internal force and horizontal displacement of the top section of the support, under different axial forces

竖向轴力P/MPa	顶部$\mu $/mm	支座M/(kN·m)	跨中V/kN
0	372.6	48.3	700.0
2	372.8	121.5	700.4
4	373.2	194.9	701.4
6	373.8	268.5	702.9
8	374.6	342.5	705.0
10	375.6	417.0	707.5
12	376.7	492.1	710.7
14	378.1	568.0	714.4
16	379.7	644.7	718.7
18	381.5	722.5	723.6
20	383.5	801.4	729.1
22	385.8	881.5	735.3
24	388.3	963.2	742.2

下载: 导出CSV

表 4 纯剪切内力与位移理论值

Table 4. Theoretical values of internal force and displacement in pure shear

竖向轴力 P/kN	水平力 F_H/kN	顶部水平位移最大值 ${\mu _{\max }}$/mm		支座弯矩 M/(kN·m)	跨中剪力 V/kN
竖向轴力 P/kN	水平力 F_H/kN	$弯曲位移{\mu _{\rm E}}$	$剪力位移{\mu _{\rm G}}$	支座弯矩 M/(kN·m)	跨中剪力 V/kN
0	700	2.553×10⁻³	372.5	48.3	700.0

下载: 导出CSV

参考文献(24)

[1]	Anil K Chopra. 结构动力学理论及其在地震工程中的应用[M]. 第四版. 谢礼立, 吕大刚, 等译. 北京: 高等教育出版社, 2016. Anil K Chopra. Structural dynamics theory and its application in earthquake engineering [M]. 4th ed. Translated by Xie Lili, Lü Dagang, et al. Beijing: Higher Education Press, 2016. (in Chinese)
[2]	Haringx J A. On highly compressible helical springs and rubber rods and their application for vibration-free mountings [J]. Philips Research Reports 4, 1949: 49 − 80, 206 − 220.
[3]	Kelly J M. Earthquake-resistant design with rubber [M]. Oxford: The Alden Press, 1993.
[4]	Chang C H. Modeling of laminated rubber bearings using an analytical stiffness matrix [J]. International Journal Of Solids and Structures, 2002, 39(24): 6055 − 6078. doi: 10.1016/S0020-7683(02)00471-7
[5]	Ding L, Zhu H P, Wu L. Analysis of mechanical properties of laminated rubber bearings based on transfer matrix method [J]. Composite Structures, 2017, 159: 390 − 396. doi: 10.1016/j.compstruct.2016.09.074
[6]	周福霖. 工程结构减震控制[M]. 北京: 地震出版社, 1997. Zhou Fulin. Seismic mitigation control of engineering structures [M]. Beijing: Seismological Press, 1997. (in Chinese)
[7]	日本建筑学会, 著. 隔震结构设计[M]. 刘文光, 译. 北京: 地震出版社, 2005. The Architectural Society of Japan. Seismic isolation structure design [M]. Translated by Liu Wenguang. Beijing: Earthquake Press, 2005. (in Chinese)
[8]	Takaoka E. Nonlinear mechanical model for laminated rubber bearings subjected to monotonic loading based on Haringx's theory [J]. Journal of Structural and Construction Engineering, 2014, 79(701): 913 − 921. doi: 10.3130/aijs.79.913
[9]	郑哲敏. 非线性连续介质力学[J]. 中国科学院院刊, 1993(4): 283 − 289. Zheng Zhemin. Nonlinear continuum mechanics [J]. Bulletin of the Chinese Academy of Sciences, 1993(4): 283 − 289. (in Chinese)
[10]	黄可智. 非线性连续介质力学[M]. 北京: 清华大学出版社, 1989. Huang Kezhi. Nonlinear continuum mechanics [M]. Beijing: Tsinghua University Press, 1989. (in Chinese)
[11]	Lindley Peter Brian. Effect of poisson's ratio on compression modulus [J]. Journal of Strain Analysis, 1968, 3(2): 142 − 145. doi: 10.1243/03093247V032142
[12]	Lindley Peter Brian. Engineering design with natural rubber [M]. Great Britain: The Malaysian Rubber Producers Research Association, 1978.
[13]	Gent AN. Elastic stability of rubber compression springs [J]. Mechanical Engineering Science, 1964, 6(318): 415 − 430.
[14]	徐芝纶. 弹性力学简明教程[M]. 第三版. 北京: 高等教育出版社, 2002. Xu Zhilun. A concise course of elasticity [M]. 3rd ed. Beijing: Higher Education Press, 2002. (in Chinese)
[15]	GB 20688.3−2006, 建筑隔震橡胶支座[S]. 北京: 中国标准出版社, 2007. GB 20688.3−2006, Building vibration isolation rubber bearing [S]. Beijing: China Standard Press, 2007. (in Chinese)
[16]	GB/T 20688.1−2007, 隔震橡胶支座试验方法 [S]. 北京: 中国标准出版社, 2007. GB/T 20688.1−2007, Test method for vibration isolation rubber bearings [S]. Beijing: China Standard Press, 2007. (in Chinese)
[17]	朱宏平, 沈文爱, 雷鹰, 等. 结构减隔震控制系统性能监测、评估与提升[J]. 工程力学, 2020, 37(1): 1 − 16. doi: 10.6052/j.issn.1000-4750.2019.05.ST06 Zhu Hongping, Shen Wen'ai, Lei Ying, et al. Performance testing, evaluation and improvement of structural seismic isolation control system [J]. Engineering Mechanics, 2020, 37(1): 1 − 16. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.05.ST06
[18]	Makoto ohsaki, Tomoshi Miyamura, Masayuki Kohiyama, et al. Finite-element analysis of laminated rubber bearing of building frame under seismic excitation [J]. Earthquake Engineering and Structural Dynamics, 2015, 44: 1881 − 1898. doi: 10.1002/eqe.2570
[19]	朱宏平, 谭平, 叶昆. 极罕遇地震作用下铅芯橡胶隔震支座基础隔震结构抗震性能研究[J]. 建筑结构学报, 2019, 40(10): 122 − 131. Zhu Hongping, Tan Ping, Ye Kun. Investigation of seismic performance of LRB base-isolated structures subjected to extremely rare earthquakes [J]. Journal of Building Structures, 2019, 40(10): 122 − 131. (in Chinese)
[20]	袁涌, 魏威, 谭平. 一种基于改进超弹性 Zener 模型的高阻尼橡胶隔震支座速度相关性本构模型[J]. 土木工程学报, 2016, 49(3): 73 − 79. Yuan Yong, Wei Wei, Tan Ping. A rate-dependent constitutive model of high damping rubber bearing based on the improved hyperelastic Zener model [J]. China Civil Engineering Journal, 2016, 49(3): 73 − 79. (in Chinese)
[21]	李忠献, 高营, 李宁. 基于RSAPS平台的隔震单元模型[J]. 工程力学, 2016, 33(4): 144 − 149. doi: 10.6052/j.issn.1000-4750.2014.09.0768 Li Zhongxian, Gao Ying, Li Ning. RSAPS-based isolation element model [J]. Engineering Mechanics, 2016, 33(4): 144 − 149. (in Chinese) doi: 10.6052/j.issn.1000-4750.2014.09.0768
[22]	何文福, 许浩, 魏陆顺, 等. 多级性态隔震体系试验研究和结构动力响应分析[J]. 工程力学, 2018, 35(9): 107 − 116. doi: 10.6052/j.issn.1000-4750.2017.05.0370 He Wenfu, Xu Hao, Wei Lushun, el al. Experiment research and dynamic response analysis of high performance multi-level bearing [J]. Engineering Mechanics, 2018, 35(9): 107 − 116. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.05.0370
[23]	朱宏平, 周方圆, 袁涌. 建筑隔震结构研究进展与分析[J]. 工程力学, 2014, 31(3): 1 − 10. doi: 10.6052/j.issn.1000-4750.2013.05.ST05 Zhu Hongping, Zhou Fangyuan, Yuan Yong. Development and analysis of the research on base isolated structures [J]. Engineering Mechanics, 2014, 31(3): 1 − 10. (in Chinese) doi: 10.6052/j.issn.1000-4750.2013.05.ST05
[24]	彭天波, 李翊鸣, 吴意诚. 叠层天然橡胶支座抗震性能的实时混合试验研究[J]. 工程力学, 2018, 35(增刊): 300 − 306. doi: 10.6052/j.issn.1000-4750.2017.05.S058 Peng Tianbo, Li Yiming, Wu Yicheng. Real time hybrid test of seismic performance of laminated nature rubber bearings [J]. Engineering Mechanics, 2018, 35(Suppl): 300 − 306. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.05.S058