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Article(id=1228046474269881339, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.02.015, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1656864000000, receivedDateStr=2022-07-04, revisedDate=1661270400000, revisedDateStr=2022-08-24, acceptedDate=null, acceptedDateStr=null, onlineDate=1770718695466, onlineDateStr=2026-02-10, pubDate=1709049600000, pubDateStr=2024-02-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770718695466, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770718695466, creator=13701087609, updateTime=1770718695466, updator=13701087609, issue=Issue{id=1228046469559681568, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='2', pageStart='191', pageEnd='364', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770718694343, creator=13701087609, updateTime=1770795432451, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368332575928712, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368332575928713, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=326, endPage=335, ext={EN=ArticleExt(id=1228046474584453122, articleId=1228046474269881339, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Experimental research on shearing properties of small lead-core rubber bearings, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Through shear tests on small lead-core rubber bearings，the effects of bearing shear strain，compressive stress and loading frequency on the bearing's equivalent horizontal stiffness，post-yield stiffness，yield force and equivalent damping ratio are studied. The results show that： With the increase of shear strain，the equivalent horizontal stiffness and the post-yield stiffness of the bearing decrease，the yield force increases，and the equivalent damping ratio decreases linearly； with the increase of the vertical compressive stress，the equal the effective horizontal stiffness decreases，the stiffness decreases linearly after yielding，and the yield force and the equivalent damping ratio increase linearly； the loading frequency has basically no effect on the shear performance of the bearing and can be ignored； within 100% of the shear strain，apply the shear strain and compressive stress did not cause damage to the isolation bearing； large deformation and large compressive stress above 100% shear strain will cause the bearing to enter into plastic，causing partial permanent damage to the bearing. Suggestions that the working conditions need to be designed according to the actual needs of the shaking table test are put forward. The correction coefficient of the Japanese code equivalent horizontal stiffness calculation formula is proposed，which is compared with the measured value of the shaking table test to improve the accuracy.

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通过对小型铅芯橡胶支座（LRB）进行剪切试验，研究了支座剪应变、压应力和加载频率对支座等效水平刚度、屈服后刚度、屈服力和等效阻尼比的影响。结果表明：随着剪应变的增大，支座的等效水平刚度和屈服后刚度减小，屈服力增大，等效阻尼比呈线性减小；随着竖向压应力的增大，支座的等效水平刚度减小，屈服后刚度呈线性减小，屈服力和等效阻尼比呈线性增大；加载频率对支座的剪切性能基本没有影响，可以忽略；100%剪应变以内，施加的剪应变和压应力没有对隔震支座造成损伤；剪应变超过100%的大变形和大压应力会使支座进入塑性，造成支座部分永久性损伤。建议根据振动台试验的工况顺序及支座在振动台试验的最大实际变形设计拟静力试验的工况。提出日本规范等效水平刚度计算公式的修正系数，并与振动台试验实测值进行对照，证明了修正后的计算公式计算精度有所提高。

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杜红凯（1981—），男，博士，高级实验师。E-mail：。
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韩淼（1969—），男，博士，教授。E-mail：。

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韩淼（1969—），男，博士，教授。E-mail：。

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韩淼（1969—），男，博士，教授。E-mail：。

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caption=不同剪应变和压应力下的试验值和修正值, figureFileSmall=B/K5wzpXssNWS7cBAMZVIw==, figureFileBig=VwdrWAnR71z68w00W0TZ+Q==, tableContent=null), ArticleFig(id=1228046509854356321, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Fig.15, caption=Shaking table experimental model （Unit： mm）, figureFileSmall=GY5URAK+tpNQUVJPXhOpJg==, figureFileBig=FP5FQhbAZnHcNyVWOYmnSA==, tableContent=null), ArticleFig(id=1228046509921465188, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=图15, caption=振动台试验模型 （单位：mm）, figureFileSmall=GY5URAK+tpNQUVJPXhOpJg==, figureFileBig=FP5FQhbAZnHcNyVWOYmnSA==, tableContent=null), ArticleFig(id=1228046510009545574, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.1, caption=

Design specifications of bearing

, figureFileSmall=null, figureFileBig=null, tableContent=
支座总高叠层钢板外径铅芯直径内部橡胶厚度内部钢板厚度上下封钢板厚度上下连接钢板厚度剪切模量
102.5150152.375122111221420.392
), ArticleFig(id=1228046510089237353, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表1, caption=

支座设计规格

, figureFileSmall=null, figureFileBig=null, tableContent=
支座总高叠层钢板外径铅芯直径内部橡胶厚度内部钢板厚度上下封钢板厚度上下连接钢板厚度剪切模量
102.5150152.375122111221420.392
), ArticleFig(id=1228046510164734825, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.2, caption=

Parameters of bearings under different shear strains

, figureFileSmall=null, figureFileBig=null, tableContent=
支座编号剪应变/%等效水平刚度/(kN·m-1)屈服后刚度/(kN·m-1)支座屈服力/kN等效阻尼比/%
LRB-110466.05364.090.2913.16
20413.01332.620.4611.57
50338.18279.550.8410.13
100283.35236.291.349.33
150246.43210.441.548.63
200215.08187.561.577.19
250208.21184.841.677.19
LRB-210523.02412.880.3112.61
20424.35336.020.5012.09
50345.40282.900.899.94
100291.43242.751.399.79
150251.60215.061.568.70
200228.32198.491.707.18
250210.78188.121.616.65
平均值10494.53388.490.3012.88
20418.68334.320.4811.83
50341.79281.220.8610.04
100287.39239.521.369.56
150249.02212.751.558.66
200221.70193.031.637.19
250209.50186.481.646.92
), ArticleFig(id=1228046510240232302, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表2, caption=

不同剪应变下支座的参数

, figureFileSmall=null, figureFileBig=null, tableContent=
支座编号剪应变/%等效水平刚度/(kN·m-1)屈服后刚度/(kN·m-1)支座屈服力/kN等效阻尼比/%
LRB-110466.05364.090.2913.16
20413.01332.620.4611.57
50338.18279.550.8410.13
100283.35236.291.349.33
150246.43210.441.548.63
200215.08187.561.577.19
250208.21184.841.677.19
LRB-210523.02412.880.3112.61
20424.35336.020.5012.09
50345.40282.900.899.94
100291.43242.751.399.79
150251.60215.061.568.70
200228.32198.491.707.18
250210.78188.121.616.65
平均值10494.53388.490.3012.88
20418.68334.320.4811.83
50341.79281.220.8610.04
100287.39239.521.369.56
150249.02212.751.558.66
200221.70193.031.637.19
250209.50186.481.646.92
), ArticleFig(id=1228046510345089905, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.3, caption=

Parameters of bearings under different compressive stresses

, figureFileSmall=null, figureFileBig=null, tableContent=
支座编号压应力等效水平刚度/(kN·m-1)屈服后刚度/(kN·m-1)支座屈服力/kN等效阻尼比/%
LRB-10.5287.56246.781.168.16
1.0283.35236.291.349.33
1.5283.35229.991.5210.92
2.0277.50214.511.8013.06
2.5274.83200.772.1115.54
LRB-20.5297.83254.011.258.23
1.0291.43242.751.399.79
1.5290.84235.321.5811.45
2.0287.30223.031.8313.75
2.5281.25209.442.0515.48
平均值0.5292.69250.401.218.19
1.0287.39239.521.369.56
1.5287.09232.661.5511.18
2.0282.40218.771.8113.41
2.5278.04205.102.0815.51
), ArticleFig(id=1228046510521250679, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表3, caption=

不同压应力下支座的参数

, figureFileSmall=null, figureFileBig=null, tableContent=
支座编号压应力等效水平刚度/(kN·m-1)屈服后刚度/(kN·m-1)支座屈服力/kN等效阻尼比/%
LRB-10.5287.56246.781.168.16
1.0283.35236.291.349.33
1.5283.35229.991.5210.92
2.0277.50214.511.8013.06
2.5274.83200.772.1115.54
LRB-20.5297.83254.011.258.23
1.0291.43242.751.399.79
1.5290.84235.321.5811.45
2.0287.30223.031.8313.75
2.5281.25209.442.0515.48
平均值0.5292.69250.401.218.19
1.0287.39239.521.369.56
1.5287.09232.661.5511.18
2.0282.40218.771.8113.41
2.5278.04205.102.0815.51
), ArticleFig(id=1228046510642885500, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.4, caption=

Equivalent horizontal stiffness of bearings under different shear strains and different compressive stresses（Unit： kN/m）

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剪应变压应力
0.51.01.52.02.5
10%410.01449.33545.71645.45699.88
20%349.41379.19412.02473.35495.51
50%300.34312.52323.93328.93331.96
100%277.84271.44264.91256.20251.25
150%241.60236.38225.22212.52196.02
200%214.83207.15197.49183.65175.64
250%202.28196.35182.73167.58162.09
), ArticleFig(id=1228046510756131712, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表4, caption=

不同剪应变和压应力下支座的等效水平刚度（单位：kN/m）

, figureFileSmall=null, figureFileBig=null, tableContent=
剪应变压应力
0.51.01.52.02.5
10%410.01449.33545.71645.45699.88
20%349.41379.19412.02473.35495.51
50%300.34312.52323.93328.93331.96
100%277.84271.44264.91256.20251.25
150%241.60236.38225.22212.52196.02
200%214.83207.15197.49183.65175.64
250%202.28196.35182.73167.58162.09
), ArticleFig(id=1228046510844212099, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.5, caption=

Seismic waves for shaking table experiments

, figureFileSmall=null, figureFileBig=null, tableContent=
地震波编号地震名称地震时间台站名称PGA/g地震持时/s
165ImperialValley-061979Chihuahua0.2751.56
6ImperialValley-021940El Centro0.28153.69
185ImperialValley-061979Holtville Post Office0.25837.85
3475Chi-Chi_Taiwan-061999TCU0800.471107.985
), ArticleFig(id=1228046510932292487, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表5, caption=

振动台试验用地震波

, figureFileSmall=null, figureFileBig=null, tableContent=
地震波编号地震名称地震时间台站名称PGA/g地震持时/s
165ImperialValley-061979Chihuahua0.2751.56
6ImperialValley-021940El Centro0.28153.69
185ImperialValley-061979Holtville Post Office0.25837.85
3475Chi-Chi_Taiwan-061999TCU0800.471107.985
), ArticleFig(id=1228046511053927306, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=EN, label=Tab.6, caption=

Correction values compared with equivalent horizontal stiffness obtained from shaking table experiments（ Unit： kN/m）

, figureFileSmall=null, figureFileBig=null, tableContent=
地震波编号γeδheq/%试验值规范值[14]规范误差/%修正值修正误差/%
1650.380.940.778.68349.34424.9621.65327.46-6.26
60.480.940.835.68330.62378.5814.50312.8-5.39
1850.610.940.884.03313.82337.097.42297.07-5.34
34750.540.940.856.48314.91356.9213.34305.09-3.12
), ArticleFig(id=1228046511125230478, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046474269881339, language=CN, label=表6, caption=

修正值与振动台试验等效水平刚度对照（单位： kN/m）

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地震波编号γeδheq/%试验值规范值[14]规范误差/%修正值修正误差/%
1650.380.940.778.68349.34424.9621.65327.46-6.26
60.480.940.835.68330.62378.5814.50312.8-5.39
1850.610.940.884.03313.82337.097.42297.07-5.34
34750.540.940.856.48314.91356.9213.34305.09-3.12
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小型铅芯橡胶支座剪切性能试验研究
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韩淼 , 刘祥辉 , 杜红凯 , 蒋金卫 , 杨杰川
振动工程学报 | 2024,37(2): 326-335
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振动工程学报 | 2024, 37(2): 326-335
小型铅芯橡胶支座剪切性能试验研究
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韩淼 , 刘祥辉, 杜红凯 , 蒋金卫, 杨杰川
作者信息
  • 北京建筑大学土木与交通工程学院，北京　100044

    • 韩淼（1969—），男，博士，教授。E-mail：。

通讯作者:

杜红凯（1981—），男，博士，高级实验师。E-mail：。
Experimental research on shearing properties of small lead-core rubber bearings
Miao HAN , Xiang-hui LIU, Hong-kai DU , Jin-wei JIANG, Jie-chuan YANG
Affiliations
  • School of Civil and Transportation Engineering，Beijing University of Civil Engineering and Architecture， Beijing 100044，China
出版时间: 2024-02-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.02.015
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摘要
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通过对小型铅芯橡胶支座（LRB）进行剪切试验，研究了支座剪应变、压应力和加载频率对支座等效水平刚度、屈服后刚度、屈服力和等效阻尼比的影响。结果表明：随着剪应变的增大，支座的等效水平刚度和屈服后刚度减小，屈服力增大，等效阻尼比呈线性减小；随着竖向压应力的增大，支座的等效水平刚度减小，屈服后刚度呈线性减小，屈服力和等效阻尼比呈线性增大；加载频率对支座的剪切性能基本没有影响，可以忽略；100%剪应变以内，施加的剪应变和压应力没有对隔震支座造成损伤；剪应变超过100%的大变形和大压应力会使支座进入塑性，造成支座部分永久性损伤。建议根据振动台试验的工况顺序及支座在振动台试验的最大实际变形设计拟静力试验的工况。提出日本规范等效水平刚度计算公式的修正系数，并与振动台试验实测值进行对照，证明了修正后的计算公式计算精度有所提高。

小型铅芯橡胶支座  /  剪应变  /  压应力  /  加载频率  /  振动台试验

Through shear tests on small lead-core rubber bearings，the effects of bearing shear strain，compressive stress and loading frequency on the bearing's equivalent horizontal stiffness，post-yield stiffness，yield force and equivalent damping ratio are studied. The results show that： With the increase of shear strain，the equivalent horizontal stiffness and the post-yield stiffness of the bearing decrease，the yield force increases，and the equivalent damping ratio decreases linearly； with the increase of the vertical compressive stress，the equal the effective horizontal stiffness decreases，the stiffness decreases linearly after yielding，and the yield force and the equivalent damping ratio increase linearly； the loading frequency has basically no effect on the shear performance of the bearing and can be ignored； within 100% of the shear strain，apply the shear strain and compressive stress did not cause damage to the isolation bearing； large deformation and large compressive stress above 100% shear strain will cause the bearing to enter into plastic，causing partial permanent damage to the bearing. Suggestions that the working conditions need to be designed according to the actual needs of the shaking table test are put forward. The correction coefficient of the Japanese code equivalent horizontal stiffness calculation formula is proposed，which is compared with the measured value of the shaking table test to improve the accuracy.

small lead-core rubber bearing  /  shear strain  /  compressive stress  /  loading frequency  /  shake table experiment
韩淼, 刘祥辉, 杜红凯, 蒋金卫, 杨杰川. 小型铅芯橡胶支座剪切性能试验研究. 振动工程学报, 2024 , 37 (2) : 326 -335 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.02.015
Miao HAN, Xiang-hui LIU, Hong-kai DU, Jin-wei JIANG, Jie-chuan YANG. Experimental research on shearing properties of small lead-core rubber bearings[J]. Journal of Vibration Engineering, 2024 , 37 (2) : 326 -335 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.02.015
正文
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引言
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铅芯橡胶支座是在天然橡胶支座的中心压入铅芯而构成的。铅芯橡胶支座通过钢板和橡胶将铅芯紧紧地束缚住，使铅芯和橡胶协同变形，这种支座是由橡胶支座的复原装置和铅的能量吸收装置共同构成的阻尼机构一体型的隔震装置［1］。铅芯橡胶支座的剪切特性主要包括支座的等效水平刚度、屈服后刚度、屈服力和等效阻尼比等。
振动台试验是用于研究结构地震破坏机理和破坏模式、评价结构整体抗震能力和衡量减震、隔震效果的重要方法［2］。根据模型试验的特点，上部结构的模型可按照一定缩尺比例进行制作，而对于隔震支座，由于加工工艺问题，支座直径很少能够按要求比例缩小，只能通过调节支座的钢板或橡胶层的厚度、层数及中孔直径等参数以满足试验所需刚度要求。因此只能单独设计小型支座来满足振动台试验的需求。由于隔震支座的水平刚度对振动台试验结果的影响较大，为保证后续振动台试验的顺利进行，对试验用支座的实际力学性能进行检测是必不可少的。
韩淼等［3］对小型叠层橡胶支座的剪切性能进行试验研究，表明剪应变和压应力的变化对支座剪切性能的影响较大，加载频率和加载次数对支座剪切性能的影响较小。何文福等［4］对LRB支座进行循环加载试验，证明因支座在往复运动中产生疲劳温度效应，屈服力下降29%~34%。Kalpakidis等［5］研究了铅芯橡胶支座在大位移多次往复荷载下的水平剪切性能，结果显示，由于铅芯耗散能量，铅芯温度升高，将导致铅芯橡胶支座强度退化。Robinson［6］对铅芯橡胶支座进行了系统的研究，给出了铅芯橡胶支座的水平剪切性能参数的计算公式。王建强等［7］采用低周反复循环加载试验方法对铅芯橡胶支座进行了剪切性能试验，研究了支座压应力和剪应变对支座屈服力、屈服后刚度、等效水平刚度和等效阻尼比的影响，根据试验结果给出了支座剪切性能的压力相关性经验公式。彭天波等［8］采用实时混合试验，研究了加载速率、竖向压应力和地震加速度峰值对叠层天然橡胶支座抗震性能的影响。朱腾宇等［9］基于铅芯橡胶支座基础隔震结构的模型振动台试验数据，对该隔震装置在不同地震加速度、不同地震强度、不同地震输入方向等工况下的滞回曲线进行对比分析，得到一些有益结论。
随着生产实践的发展和需要，对小型橡胶支座的应用日趋广泛，如将其用于振动台结构隔震试验、结构减震试验，或用于小型精密仪器的振动防护等［10］。目前学者对小型橡胶支座的研究较少。本文将就剪应变、压应力、加载频率、加载顺序对小型铅芯橡胶支座剪切性能的影响，以及规范公式对小型铅芯橡胶支座的适用性等问题进行研究。
1　支座剪切试验设计
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1.1　小型铅芯橡胶支座几何参数
选用河北衡水震泰公司生产的铅芯橡胶支座，有效直径为150 mm，具体参数如表1所示，剖面图如图1所示。小型铅芯橡胶支座设计压应力=2 MPa。选用2个铅芯橡胶支座进行试验，分别标记为LRB-1，LRB-2。
1.2　试验装置
试验加载设备包括：竖向加载采用1台150 kN的液压千斤顶，千斤顶与支座之间放置量程为150 kN的荷载传感器；水平加载采用1台250 kN的MTS作动器，行程为±125 mm。本试验对铅芯橡胶支座进行压剪试验，竖向加载采用荷载控制，水平加载采用位移控制，数据采集采用静态高速应变数据采集仪DH3820，最大采样频率为100 Hz。图2给出了试验装置示意图。图3为试验装置实物图。
1.3　加载制度
对小型铅芯橡胶支座进行不同剪应变、不同压应力、不同加载频率，以及不同因素加载顺序的4种加载试验，加载制度如下：
（1）不同剪应变试验。对支座施加恒定的设计压应力，剪应变依次取10%，20%，50%，100%，150%，200%和250%，每级剪应变循环加载4次，加载频率为0.2 Hz。
（2）不同压应力试验。对支座分别施加0.5，1.0，1.5，2.0和2.5的压应力，进行100%剪应变下的循环加载试验，每级压应力循环加载4次，加载频率为0.2 Hz。
（3）不同加载频率试验。支座在压应力及100%剪应变的条件下，进行加载频率分别为0.01，0.02，0.05，0.1和0.2 Hz的试验，每一加载频率循环加载4次。
（4）不同因素加载顺序试验。以压应力为，剪应变为100%，加载频率为0.2 Hz时的工况为标准工况，进行不同剪应变（10%~200%）、不同压应力（0.5~2.5）和不同加载频率（0.01~0.5 Hz）的试验，在剪应变或压应力改变前均进行标准工况试验。
2　支座力学性能
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2.1　剪应变相关性
铅芯橡胶支座的“荷载-位移”滞回曲线能够直观反映支座的耗能情况。按照《橡胶支座 第3部分：建筑隔震橡胶支座》（GB/T 20688.3—2006）［11］中的相关公式，利用滞回曲线可计算得到支座的等效水平刚度（）、屈服后刚度（）、支座屈服力（）和等效阻尼比（）等性能参数。性能参数示意图如图4所示，图中Q表示剪切，D表示位移。
性能参数计算公式如下式所示：
式中　 为最大剪力；为最小剪力；为最大位移；为最小位移；和分别为滞回曲线正向和负向与剪力轴的交叉点；和分别为正向和负向屈服后刚度；heq为等效阻尼比；为滞回曲线的包络面积；为弹性势能。
图5为LRB-1和LRB-2两个支座在压应力作用下，加载频率为0.2 Hz，剪应变分别为10%，20%，50%，100%，150%，200%和250%时，第3圈水平剪力⁃水平位移的滞回曲线［12］。由图5可以看出：支座的滞回曲线平滑且较为饱满，说明支座有着较为稳定的剪切性能和较好的耗能性能。随着剪应变的增大，滞回曲线的面积增大，支座的耗能能力增大。
表2给出了两个小型铅芯橡胶支座在标准压应力下，10%~250%剪应变时的等效水平刚度、屈服后刚度、支座屈服力和等效阻尼比，以及相应的平均值。
在不同剪应变下两个小型铅芯橡胶支座的等效水平刚度、屈服后刚度、支座屈服力和等效阻尼比的平均值如图6所示。
由图6（a）可知，在标准压应力下，支座的等效水平刚度随着剪应变的增大而减小。支座的等效水平刚度在100%剪应变以内剧烈下降，从10%剪应变时的494.53 kN/m减小到100%剪应变时的287.39 kN/m，减小幅度为100%剪应变时等效水平刚度的72.08%；100%剪应变以上等效水平刚度下降渐缓，从100%剪应变时的287.39 kN/m减小到250%剪应变时的209.50 kN/m，减小幅度为100%剪应变时等效水平刚度的27.60%。同样可以看出，在标准压应力下，支座的屈服后刚度随着剪应变的增大而减小，屈服后刚度的减小趋势与等效水平刚度的减小趋势相同，先剧烈后缓慢，100%剪应变时的屈服后刚度相比10%剪应变时的屈服后刚度，减小幅度为100%剪应变时屈服后刚度的72.08%；250%剪应变时的屈服后刚度相比100%剪应变时的屈服后刚度，减小幅度为100%剪应变时屈服刚度的22.14%。剪应变在100%到250%之间时的等效水平刚度和屈服后刚度的变化幅度远小于剪应变在10%到100%之间时的刚度变化幅度。
橡胶支座的屈服力在双线性模型中定义为滞回曲线与剪力轴的交点，不同的支座滞回曲线对应不同的支座屈服力［13］。由图6（b）可知，在标准压应力下，支座的屈服力随着剪应变的增大而增大。支座的屈服力在100%剪应变以内上升幅度较大，从10%剪应变时的0.30 kN增大到100%剪应变时的1.36 kN，增大幅度为100%剪应变时屈服力的77.94%，说明100%剪应变以内支座铅芯未全部发生屈服；100%剪应变以上支座的屈服力上升幅度较小，从100%剪应变时的1.36 kN增大到250%剪应变时的1.64 kN，增大幅度为100%剪应变时屈服力的20.59%，特别是从150%剪应变时的1.55 kN增大到250%剪应变时的1.64 kN，增幅仅为100%剪应变时屈服力的6.62%，说明100%剪应变以上支座的屈服力基本保持稳定，铅芯几乎完全屈服。同时，在标准压应力下，支座的等效阻尼比随着剪应变的增大整体上呈线性减小。从10%剪应变时的12.88%减小到250%剪应变时的6.92%，减小幅度为100%剪应变时等效阻尼比的62.34%，剪应变对阻尼比的影响较大。
随着剪应变的增大，滞回曲线的面积增大，消耗的能量增大，但同时滞回曲线较为细长，最大变形处的势能的增大速度比滞回曲线面积的增大速度快，故由式（4）可知等效阻尼比减小。
2.2　压应力相关性
在地震作用下支座承受的竖向压应力会产生较大变化，故研究支座的压应力变化对支座性能参数的影响是有必要的。本小节研究了剪应变为100%，加载频率为0.2 Hz，压应力分别为0.5，1.0，1.5，2.0和2.5时，小型铅芯橡胶支座的参数变化情况。
图7为100%剪应变，不同压应力下，LRB-1和LRB-2两个支座第3圈水平剪力-水平位移的滞回曲线。从图7中可以看出，支座的滞回曲线平滑且较为饱满，说明支座有着较为稳定的剪切性能和较好的耗能性能。随着支座承受的竖向压应力的增大，支座的滞回环面积增大，说明随着压应力的增大，支座的耗能能力增大。
表3给出了两个小型铅芯橡胶支座在100%剪应变下，0.5~2.5时的等效水平刚度、屈服后刚度、支座屈服力和等效阻尼比，以及相应的平均值。
在不同压应力下两个小型铅芯橡胶支座的等效水平刚度、屈服后刚度、支座屈服力和等效阻尼比的平均值如图8所示。
由图8（a）可知，在100%剪应变下，支座的等效水平刚度随着竖向压应力的增大而减小，但是总体来说减小幅度较小，从0.5时的292.69 kN/m减小到2.5时的278.04 kN/m，仅减小了5.01%，特别是竖向压应力在1.0和1.5之间时，支座的等效水平刚度基本上不变，故在不同压应力下支座的等效水平刚度相对较为稳定。同时，在100%剪应变下，支座的屈服后刚度随着竖向压应力的增大整体上呈线性减小。从0.5时的250.40 kN/m减小到2.5时的205.10 kN/m，减小幅度达到18.09%，远远大于等效水平刚度的减小幅度。
由图8（b）可知，在100%剪应变下，支座的屈服力随着压应力的增大整体上呈线性增大，从0.5时的1.21 kN增大到2.5时的2.08 kN，增大幅度为71.90%，当压应力有变化时，橡胶支座的模型化要考虑不同压应力的影响。支座的屈服力主要由内部铅芯决定，竖向压应力的增大使内部铅芯受到了更大的约束作用，提高了铅芯的屈服应力，从而使支座的屈服力增大。如图9所示，铅芯橡胶支座的滞回曲线为橡胶的滞回曲线加上铅芯的滞回曲线，橡胶的滞回曲线近似为一直线且刚度较小，铅芯的滞回曲线为上下水平的平行四边形，即铅芯在屈服前刚度无穷大，屈服后刚度为零，因此铅芯橡胶支座的屈服后刚度是由橡胶决定的。当压应力增大时，铅芯的屈服力增大，从而铅芯的等效水平刚度增大，但铅芯的屈服后刚度依然近似为零，而随着压应力的增大，橡胶部分的等效水平刚度和屈服后刚度减小，但因为铅芯部分的等效水平刚度增大，所以铅芯橡胶支座整体的屈服后刚度的减小幅度远大于整体的等效水平刚度的减小幅度。同样可以看到，在100%剪应变下，支座的等效阻尼比随着压应力的增大整体上呈线性增大，从0.5时的8.19%增大到2.5时的15.51%，增大幅度为89.38%。由式（4）可知，支座的等效阻尼比主要由支座的耗能和等效水平刚度决定，随着压应力的增大，橡胶分子之间的摩擦力增大，进而引起耗能的增大，同时铅芯的耗能增大，使得支座的总耗能增大，而支座的等效水平刚度则随着压应力的增大而减小，因此支座的等效阻尼比随着压应力的增大而明显增大。
2.3　加载频率相关性
图10为100%剪应变，压应力下，LRB-1和LRB-2两个支座在不同加载频率下的第3圈水平剪力-水平位移的滞回曲线。由图10可知，各个加载频率下支座的滞回曲线基本重合，故在100%剪应变，压应力下，加载频率对支座的剪切性能基本没有影响，可以忽略。
2.4　不同因素加载顺序对支座自身的影响
为了探究不同因素加载顺序对小型铅芯橡胶支座自身性能的影响，以压应力，100%剪应变时加载频率为0.2 Hz的工况为标准工况，在不同剪应变和不同压应力加载后均进行标准工况试验，对标准工况试验的等效水平刚度进行分析。
试验加载频率取值范围为0.01~0.5 Hz，压应力取值范围为0.5~2.5，剪应变取值范围为10%~200%。将5个压应力取值与6个剪应变取值组合，共30组。每一组均进行8个加载频率工况试验（有些剪应变和压应力组合下增加了其他加载频率工况，对本试验无影响，只增加工况数量），之后进行标准工况试验。每种工况均进行4次循环加载。
图11为剪切变形为小变形（剪应变为10%~100%）时，不同组合之间标准工况的等效水平刚度变化。图12为剪切变形为大变形（剪应变为150%~200%）时，不同组合之间标准工况的等效水平刚度变化。两图的横坐标为该工况之前工况的总数，如268为第268个工况。
由图11可知，第1到第33个工况之间，标准工况下的等效水平刚度不断下降，而第33到第268个工况之间，标准工况的等效水平刚度基本维持不变。开始阶段标准工况下的等效水平刚度下降是因为刚开始试验，支座处于磨合阶段，铅芯与橡胶未能达到协同变形，支座的性能未完全体现，随着循环次数的增加，支座的性能逐渐体现，在第33个工况时，支座已经进行了132次循环加载。后期等效刚度维持稳定说明在100%剪应变以内，剪应变、压应力和加载频率的变化均对铅芯橡胶支座无影响。说明小变形时支座的剪切性能试验较为准确。
由图12可知，在小变形工况结束后的大变形工况中，随着剪应变和压应力的增大，标准工况下的等效水平刚度逐渐下降，说明大变形使支座进入塑性，会对支座造成一定的损伤，在大变形的第1到第52个工况中，在150%剪应变时，随着压应力的增大，标准工况的等效水平刚度逐渐下降，特别是2.0和2.5压应力后的刚度下降幅度大于0.5~1.5压应力后的刚度下降幅度，说明大变形下大压应力对支座刚度有较大的影响。特别需要注意的是，做完大变形工况试验后第二天继续进行标准工况试验，支座的刚度相对前一天最后的标准工况的刚度有较大提升，但没有恢复到小变形之前的刚度，这可能是因为支座铅芯和橡胶的温度在大变形工况下对支座的影响较大，且大变形工况对支座部分结构造成了永久性的损伤。
综合以上分析，建议拟静力试验根据振动台试验的工况顺序及支座在振动台试验的最大实际变形设计工况。如果振动台试验工况顺序设定为前期支座的最大变形超过100%剪应变，后期仍有小变形，则拟静力工况先进行大变形工况试验，再进行小变形工况试验，充分考虑支座的损伤；如果振动台试验工况顺序设定为前期全是小变形，后期全是大变形，则拟静力工况先进行小变形工况试验，再进行大变形工况试验，以便拟静力试验结果更接近振动台试验。同时拟静力试验前先让支座进行100次100%剪应变的往复试验，以便支座充分磨合。
3　日本规范公式修正
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关于铅芯橡胶支座等效线性化模型的建立，中国规范《橡胶支座 第3部分：建筑隔震橡胶支座》（GB/T 20688.3—2006）［11］、日本桥梁免震设计条例［14］和美国桥梁隔震设计准则［15］都有相关研究。其中日本规范［14］中建议的等效水平刚度计算公式为：
式中　 为橡胶承压面积（钢板面积减去铅芯面积）；为铅芯截面积；G为橡胶剪切模量；为橡胶中的有效剪应变：；为有效设计位移；为铅芯中产生的剪应力，表示为：
另外再取两个与LRB-1和LRB-2相同型号的小型铅芯橡胶支座，进行不同剪应变（10%~250%）和压应力（0.5~2.5）的试验，取4个支座的试验平均值，将其列于表4。将试验用支座按规范公式计算的等效水平刚度值与0.2 Hz加载频率，1.0压应力下试验的平均值绘于图13。
由图13可知，规范公式计算值与试验值在100%和150%剪应变下相差不大，误差分别为-2.99%和-7.27%；而在其他剪应变下误差相对较大，最大误差发生在10%剪应变时，误差为-48.00%。
针对规范公式在一些剪应变下的等效水平刚度计算值与试验值相差较大，且规范公式尚未考虑不同支座竖向压应力情况下的等效水平刚度变化情况，本文对规范公式进行修正，引入竖向压应力相关参数，使其可以适用于该种类型的支座，并同时考虑水平剪应变和竖向压应力的影响，使计算值与试验值相一致，并更好地应用到有限元模拟中。
基于规范公式，采用线性回归的方法，引入修正系数，修正后的公式如下式所示：
式中　 为修正系数，取值如下式所示：
式中　 为支座承受的竖向压应力系数，，其中，为支座承受的竖向压力，，。
将10%~250%剪应变，0.5~2.5压应力下的试验值和修正公式计算值绘于图14。
由图14可知，修正公式计算值与试验值吻合较好，最大误差发生在150%剪应变、2.5压应力时，试验刚度为196.02 kN/m，公式计算刚度为207.74 kN/m，误差为5.98%，误差在可接受范围内。因此，当使用同种规格的铅芯橡胶支座时，等效水平刚度可以参照此修正公式计算确定。
4　振动台试验对照
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限于时间和损耗等条件，很难用振动台来进行大量的工况试验，更多的还是用振动台试验来校核分析模型及理论计算结果的正确性。
对于隔震结构，隔震层刚度需要设置得较为准确，才能使有限元模拟的结构响应与振动台试验较好吻合。故本节将上节提出的修正公式计算的等效水平刚度与振动台试验的实际刚度进行对比。
振动台试验模型为五层单跨钢框架基础隔震模型，平面尺寸为1.8 m1.8 m，层高为0.9 m，隔震支座采用与本文同种型号的小型铅芯橡胶支座，上部结构总质量为13.2378 t。试验模型照片如图15所示。
选取4条地震波，地震波的初始信息见表5。将地震波PGA调幅为0.15g，0.2g，0.3g和0.4g，作为振动台试验输入地震波。
采集4条地震波不同PGA输入工况下，振动台试验实测的小型铅芯橡胶支座的位移和剪力，绘出支座的滞回曲线，计算其等效水平刚度，与利用修正公式（7）计算的值对照，如表6所示。
由表6可知，考虑剪应变和压应力两个因素后的修正公式计算的等效水平刚度略小于振动台试验的支座实际刚度，修正公式计算值与规范计算值相比，能较好地体现支座的实际刚度特性。
5　结论
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通过试验分析了剪应变、压应力和加载频率对小型铅芯橡胶支座剪切性能的影响，考虑压应力参数的影响修正了日本规范对该种类型支座的水平刚度计算公式，并与振动台试验实测值进行了对照。
（1）在压应力下，随着剪应变的增大，支座的等效水平刚度减小，屈服后刚度减小，屈服力增大，等效阻尼比整体上呈线性减小。
（2）在100%剪应变下，随着竖向压应力的增大，支座的等效水平刚度减小，屈服后刚度呈线性减小，屈服力呈线性增大，支座的等效阻尼比呈线性增大。
（3）在100%剪应变和压应力下，加载频率对支座的剪切性能基本没有影响，可以忽略。
（4）在100%剪应变以内，剪应变和压应力的大小均不会对隔震支座造成损伤；在100%剪应变以上，大变形和大压应力会让支座进入塑性，对支座部分结构造成永久性的损伤。由此提出了根据振动台试验实际需要设计工况的建议。
（5）考虑剪应变和压应力的影响对日本规范等效水平刚度的计算公式进行修正，修正公式提高了该种类型小型铅芯橡胶支座的等效水平刚度计算精度。
基金
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  • 国家重点研发计划资助项目(2019YFC1509500)
  • 北京建筑大学高精尖创新中心项目(UDC2019032424)
文献
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2024年第37卷第2期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.02.015
  • 接收时间：2022-07-04
  • 首发时间：2026-02-10
  • 出版时间：2024-02-28
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  • 收稿日期：2022-07-04
  • 修回日期：2022-08-24
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国家重点研发计划资助项目(2019YFC1509500)
北京建筑大学高精尖创新中心项目(UDC2019032424)
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    北京建筑大学土木与交通工程学院，北京　100044

通讯作者:

杜红凯（1981—），男，博士，高级实验师。E-mail：。
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https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.02.015
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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