工程科学与技术   2018, Vol. 50 Issue (3): 134-141

1. 广州大学 工程抗震研究中心，广东 广州 510405;
2. 广东省地震工程与应用技术重点实验室，广东 广州 510405;
3. 武汉大学 土木建筑工程学院，湖北 武汉 430072

A Simplified Method for Computing Mechanical Properties of High-performance Low-cost Seismic Isolators
TAN Ping1,2, LIU Han2,3, ZHANG Yafei1,2, ZHOU Fulin1,2,3
1. Earthquake Eng. Research & Test Center, Guangzhou Univ., Guangzhou 510405, China;
2. Key Lab. of Earthquake Eng. and Applied Technique of Guangdong Province, Guangzhou 510405, China;
3. School of Civil Eng., Wuhan Univ., Wuhan 430072, China
Abstract: An innovative high-performance low-cost isolator was presented for seismic isolation in economic underdeveloped regions. When this proposed isolator is applied to replace steel sheets with engineering plastic plates within a traditional isolator, the methods of flexible-reinforcement isolators must be used to calculate its mechanical properties . The analytical solutions for this type of isolators are complicated and not easy for engineering application, therefore a simplified method was put forward for computing the simplified solutions for analytical strength, stiffness and stress of the isolator under compression and bending, in which the weighted residual method was employed based on optimal selection of trial functions and the undetermined coefficients of the trial functions were determined by accuracy requirement. Therefore, the simplified formulas of stress distribution, compression stiffness, bending stiffness and strength were formulated and obtained herein. And a characteristic parameter was introduced to analyze the variation of these formulas. Parameter analysis showed that the error of the proposed formulas increases with the increase of the aspect ratio and the characteristic parameter. Subsequently, the valid application scope of these simplified formulas is given. Finally, systematic experimental tests of four different groups of the presented isolators were carried out. Test results showed that, for typical high-performance low-cost isolators, the deviation of the proposed formulas is close to that of the analytical formulas, which verifies the effectiveness of the simplified formulas and the rationality of the proposed method. The proposed method can be extended to isolators in any other shape and of different reinforcing materials, which is propitious for the popularization and application of high-performance low-cost isolation technology in less developed areas.
Key words: seismic isolation    isolator    mechanical property    simplified method    weighted residual method

1 高性能低造价隔震支座解析解 1.1 高性能低造价隔震支座

 图1 高性能低造价隔震支座构造 Fig. 1 Construction of high-performance low-cost isolators

1.2 隔震支座力学性能解析解

 图2 基本坐标系和变形示意图 Fig. 2 Coordinates and deformation

 ${\nabla ^2}p - (\frac{{{\alpha ^2} + {\beta ^2}}}{{{a^2}}})p = - \frac{{12G}}{{{t^2}}}\frac{\varDelta }{t}$ (1)
 ${E_{\rm c}} = 8K\sum\limits_{n = 1, 3, 5, \cdot\cdot}^\infty {\frac{{{\beta ^2}}}{{{\alpha ^2} + {\beta ^2} + {n^2}{{\text{π}} ^2}}}} \frac{1}{{{n^2}{{\text{π}} ^2}}}\left( {1 - \frac{{\tan\,{\rm h} (\lambda )}}{\lambda }} \right)$ (2)

 ${\nabla ^2}p - \left(\frac{{2{\alpha ^2} + {\beta ^2}}}{{{a^2}}}\right)p = \frac{{12G}}{{{t^2}}}\frac{{\theta x}}{t}$ (3)
 ${E_{\rm b}} = \sum\limits_{n, m = 1}^\infty {\frac{{576G{{\sin }^2}\left( {\displaystyle\frac{{m{\text{π}} }}{2}} \right){{(n{\text{π}} \cos (n{\text{π}} )- \sin (n{\text{π}} ))}^2}}}{{{m^2}{n^4}{{\text{π}} ^6}{t^2}[{{\left(\displaystyle\frac{{2n{\text{π}} }}{a}\right)}^2} + {{\left(\displaystyle\frac{{m{\text{π}} }}{{2b}}\right)}^2} + \displaystyle\frac{{2{\alpha ^2} + {\beta ^2}}}{{{a^2}}}]}}}$ (4)
2 加权残值法

 图3 加权残值法流程图 Fig. 3 Flow chart of weighed residual method

 ${\textit{z}}= \sum\limits_{i = 1}^m {{C_i}{N_i}(x, y)}$ (5)

 ${N_i}\left( \pm \frac{a}{2}, y\right) = 0; \;\;{N_i}(x, \pm b) = 0$ (6)

 $Lp - f = 0$ (7)

 ${R_v} = L{\textit{z}} - f \ne 0$ (8)

 $\int {{W_i}} {R_V}{\rm d}V = 0$ (9)

 {W_i} = \delta (x-{x_i})\delta (y-{y_i}) = \left\{ \begin{aligned}& \infty, \;x = {x_i}{\text{且}}y = {y_i}{\text{；}}\\& 0,\;\;x \ne {x_i}{\text{或}}y \ne {y_i}\;\end{aligned} \right. (10)

 ${R_{\rm V}}\left( {{x_i}, {y_i}} \right) = 0$ (11)

3 轴压分析的简化计算 3.1 应力分布简化公式

 ${{\textit{z}}_1} = {C_1}\left( {{{\left( {x - \frac{a}{2}} \right)}^2} + {C_2}} \right)\left( {{{\left( {x - \frac{a}{2}} \right)}^2} - \frac{{{a^2}}}{4}} \right)\left( {{y^2} + {C_3}} \right)\left( {{y^2} - {b^2}} \right)$ (12)

 $\iint_\Omega {{{\textit{z}}_1}}{\rm d}x{\rm d}y = P$ (13)

 \begin{aligned}[b] & \frac{1}{{450}}{b^5}{a^5}{C_1} + \frac{2}{{45}}{b^5}{a^3}{C_1}{C_2} + \frac{1}{{90}}{b^3}{a^5}{C_1}{C_3} + \\ & \quad\quad \frac{2}{9}{b^3}{a^3}{C_1}{C_2}{C_3} = P \end{aligned} (14)

 ${C_1} \!=\! \frac{{450\left( {5{\gamma ^2} \!-\! 2\displaystyle\frac{{{a^2}}}{{{b^2}}} \!+\! 48} \right)\left( {5{\gamma ^2} \!+\! 12\displaystyle\frac{{{a^2}}}{{{b^2}}} \!-\! 8} \right)P}}{{{a^5}{b^5}\left( {5{\gamma ^2} \!+\! 348\displaystyle\frac{{{a^2}}}{{{b^2}}} \!+\! 48} \right)\left( {5{\gamma ^2} \!+\! 12\displaystyle\frac{{{a^2}}}{{{b^2}}} \!+\! 1\;392} \right)}}\!\!\!\!$ (15)
 ${C_2} = \frac{{70{a^2}}}{{5{\gamma ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} - 8}}$ (16)
 ${C_3} = \frac{{70{a^2}}}{{5{\gamma ^2} - 2\displaystyle\frac{{{a^2}}}{{{b^2}}} + 48}}$ (17)
 $Q = \frac{{ab\left( {5{\gamma ^2} + 348\displaystyle\frac{{{a^2}}}{{{b^2}}} + 48} \right)\left( {5{\gamma ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} + 1\;392} \right)}}{{78\;750\left( {5{\gamma ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} + 48} \right)}}\!\!\!\!$ (18)

3.2 轴压强度简化公式

 图4 正方形支座沿轴线应力分布 Fig. 4 Stress distribution on a $x$ is

 ${u_{1, x}} = {v_{1, y}} = \frac{p}{{({E_{\rm f}}{t_{\rm f}}/t)}}$ (19)

 ${\sigma _{1x}} = {\sigma _{1y}} = p\frac{t}{{{t_{\rm f}}}}$ (20)

 ${p_{\rm m}} = \frac{{{t_{\rm f}}}}{t}{\sigma _{{\rm fm}}}$ (21)

 ${\sigma\!_{{\rm vm}}} = \frac{{{P_{\max }}}}{A}$ (22)

 ${\sigma\!_{{\rm vm}}} = \frac{{[5{\gamma ^2}{b^2} + 12{a^2} + 48{b^2}]}}{{84{{(a + 2b)}^2}}}\frac{{{E_c}}}{{G{S^2}}}\frac{{{t_{\rm f}}}}{t}{\sigma\!_{{\rm fm}}}$ (23)

 ${\sigma\!_{{\rm vms}}} = \frac{{{{({\gamma ^2} + 288)}^2}}}{{176\;400}}\frac{{{t_{\rm f}}}}{t}{\sigma\!_{{\rm fm}}}$ (24)

3.3 压缩模量简化公式

 $\frac{{{E_{\rm c}}}}{{G{S^2}}} = {\frac{{\left( {5{\gamma ^2} + 348\displaystyle\frac{{{a^2}}}{{{b^2}}} + 48} \right)\left( {5{\gamma ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} + 1\;392} \right)\left( {1 + \displaystyle\frac{{2b}}{a}} \right)}}{{13\;125\left( {5{\gamma ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} + 48} \right)}}^2}$ (25)

 $\frac{{{E_{\rm c}}}}{{G{S^2}}} = \frac{4}{{525}}\frac{{{{\left( {{\gamma ^2} + 288} \right)}^2}}}{{5{\gamma ^2} + 96}}$ (26)
 图5 压缩模量的参数分析 Fig. 5 Parameter analysis of compression modulus

4 弯曲模量的简化计算

 图6 压缩模量误差分析 Fig. 6 Error analysis of compression modulus

 ${\textit{z}} = {C_1}\left( {{x^3} -\frac{{{a^2}}}{4}x} \right)\left( {{y^2} + {C_2}} \right)\left( {{y^2} -{b^2}} \right)$ (27)

 $\frac{{{E_b}}}{{G{S^2}}} = \frac{{12\left( {2{\alpha ^2} + {\beta ^2} + 60\displaystyle\frac{{{a^2}}}{{{b^2}}} + 36} \right){{\left( {1 + \displaystyle\frac{{2b}}{a}} \right)}^2}}}{{25\left( {10{\alpha ^2} + 5{\beta ^2} + 12\displaystyle\frac{{{a^2}}}{{{b^2}}} + 180} \right)}}$ (28)

 ${\gamma _1}^2 = 2{\alpha ^2} + {\beta ^2}$ (29)

 $\frac{{{E_b}}}{{G{S^2}}} = \frac{{48\left( {{\gamma _1}^2 + 276} \right)}}{{25\left( {5{\gamma _1}^2 + 228} \right)}}$ (30)
 图7 弯曲模量误差分析 Fig. 7 Error analysis of bending modulus

5 试验验证

 图8 试验试件 B1和B3 Fig. 8 Specimen B1 and B3

 图9 竖向力与位移关系曲线 Fig. 9 Relativity between vertical force and displacement

6 结　论

1）提出了高性能、低造价隔震支座基本力学性能的简化计算方法。通过构造试函数，采用加权残值法求得该型支座在轴压和纯弯曲状态下橡胶层应力分布、支座极限面压、压缩模量和弯曲模量的一系列简化公式。

2）参数分析表明，本文给出的简化公式误差可控制在10%以内。简化公式随参数 $αb$ $βb$ 的变化规律与解析公式一致，随着长宽比和特征参数 $\gamma$ $\gamma_1$ 的增加，误差增大但可控。对于方形隔震支座：应力分布和压缩模量简化公式适用于 $\gamma$ 小于5.5，支座极限面压适用于 $\gamma$ 小于7，弯曲模量简化公式适用于 $\gamma_1$ 小于7.5，可覆盖大部分常用的高性能、低造价隔震支座，且精度的提高可通过优化应力试函数实现，如提高试函数多项式的次数或选取更合适的函数类型等。

3）通过4组典型支座的轴压试验，对压缩模量的简化公式进行了验证。试验结果表明简化公式误差小于6.3%，具有足够的精度。由于弯曲模量不能通过试验直接测得，本文中仅与解析公式对比。其简化公式的准确性有待进一步试验验证。

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