摘 要

由于如兵马俑等置地文物缺乏一定的抗震保护措施，所以它们在地震作用下容易发生倾覆破坏，造成难以挽回的损失。目前，国内尚缺失关于置地文物在地震作用下动力响应分析及隔震措施的研究。因此，亟需开展置地文物在随机地震作用下动力响应分析及其倾覆概率研究的工作。

本文将兵马俑简化成非对称刚性块体，建立了不同摆动方向的分段动力方程。对于不同的刚性块体，根据其高宽比的不同，对此控制方程进行适当简化(对于高宽比较大的刚性块体，可以将控制方程简化成两个线性的动力方程)，并采用龙格库塔数值方法对其进行了数值模拟。首先，对于高宽比较大的刚性块体，与解析解的对比验证了数值方法的正确及有效；然后，用数值算法对刚性块体进行确定性动力响应分析，研究了不同参数(块体尺寸、高宽比、外激励大小、恢复系数、竖向激励)对块体倾覆状态的影响。最后，发现了由于系统非线性导致的参数敏感性，即：参数发生微小改变，块体最终倾覆状态也完全不同。

进而，在统计意义上，对块体进行随机动力响应分析。假定随机地震动激励为调制过滤白噪声，采用蒙特卡罗模拟(Monte Carlo Simulation)方法研究了块体倾覆概率随着参数变化的规律，即：它随着块体尺寸的增大而减小，且随着块体高宽比、外激励幅值和恢复系数的增大而增大；竖向激励(本文中取竖向激励为水平激励的0.6倍)对倾覆概率并未产生较大影响，这一点与历史文献并不一致，值得继续深入考察。不仅如此，研究发现，块体在初始振荡阶段，位移集中在零附近，较不易发生倾覆；随着时间推移，其倾覆概率越来越大；在振荡的最后阶段，块体位移集中在零附近或块体发生倾覆。

基于以上理论分析，本文选取了3个具有代表性的兵马俑，在对其表面点几何数据进行重构后，对它们进行了确定性动力响应分析。随后，针对非对称特征最为明显的3号兵马俑进行了随机动力响应分析，得到其倾覆概率密度曲线图。研究发现了俑体倾向于朝重心一侧发生倾覆。

关键词：置地文物，刚性块体，确定性分析，随机振动分析，蒙特卡罗模拟，倾覆概率。

Abstract

The free-standing cultural relics such as the Terra Cotta Warriors tend to overturn under seismic action due to a lack of certain seismic protection measures. At present, the dynamic response analysis and corresponding isolation measures of the free-standing cultural relics have not been developed in China. Therefore, it’s quite essential to develop a research on the dynamic response analysis and the capsizing probability of the free-standing cultural relics under stochastic seismic excitation.

In this paper, the Terra Cotta Warrior could be simplified to an asymmetric rigid block. The respectively different piecewise dynamic equations are established for different oscillating direction. For the rigid blocks of different ratio, the dynamic equations could be simplified appropriately (for the slender blocks, the nonlinear equation could be simplified to linear one).This paper deals with the rocking response through Runge-Kutta algorithm. First of all, for slender rigid blocks, the validity of the numerical method is proved through making comparison among the analytical solution and numerical solution obtained by Runge-Kutta algorithm. Then, this paper analyses the dynamic response and the influences made on the capsizing state of the blocks with different parameters (the size and the ratio of the block, the ground motion intensity and the recovery coefficient, vertical excitation). Finally, the results presented show that the block is indeed sensitive to small changes in these parameters due to the nonlinearity of the system. That’s to say, the ultima capsizing state of the block is completely different with the small changes in these parameters

Then, from the statistical view, the analysis of random dynamic response is carried out on the block. Now suppose that the stochastic seismic excitation is the modulating filtered white noise. Systematic trend of the toppling probability varying with these parameters could be observed by Monte Carlo simulation. The toppling probability of the block increases with the increasing ratio, increasing intensity of excitation, increasing recovery coefficient and decreasing size of the block. Which is different from historic reference is that the vertical excitation (In this paper, the vertical excitation is 0.6 times of the horizontal motivation) does not make great influence. It’s worth of further study. And not only that, through the time-history analysis, it’s found that the block does not tend to overturn in the initial stage of oscillation, but with time going by, the toppling probability of the block is increasing. In the end, the displacement of the block concentrates near either zero or the critical angle.

Based on the analysis above, this paper chooses three representative Terra Cotta Warriors with their surface data reconstructed and investigates the dynamic response analysis under certain excitation. Random dynamic response analysis is carried out on one of the block with especially obvious asymmetric characteristics. The capsizing density probability plot is obtained. It’s found that the asymmetric block tends to overturn at the heavy side.

Key words：Free-standing cultural relics, Rocking rigid blocks, Deterministic analysis, Random vibration analysis, Monte Carlo simulation, Toppling probability.

目 录

摘 要 I

Abstract II

第1章 绪论 1

1.1. 引言 1

1.1.1. 目的及意义 1

1.1.2. 国内外研究现状 1

1.2. 浮放物的保护措施 3

1.2.1. 浮放物在地震作用下的破坏 3

1.2.2. 浮放文物的抗震保护措施 4

1.3. 本文研究内容与结构 8

1.3.1. 研究内容 8

1.3.2. 本文结构 9

第2章 刚性块体倾覆的确定性分析 11

2.1. 刚性块体运动方程的理论推导 11

2.1.1. 对称刚性块体未简化运动方程 11

2.1.2. 简化的运动方程 13

2.1.3. 恢复系数的推导 13

2.2. 刚性块体倾覆的近似分析 15

2.2.1. 刚性块体在单脉冲下的倾覆 15

2.2.2. 刚性块体在地震作用下的倾覆 19

2.3. 刚性块体动力响应的数值分析与验证 21

2.3.1. 数值分析方法 21

2.3.2. 龙格库塔算法 22

2.3.3. 刚性块体自由振动 26

2.3.4. 刚性块体谐波激励振动 28

2.3.5. 非对称结构计算方法 30

2.4. 刚性块体的动力响应分析 34

2.4.1. 地震波的选取 34

2.4.2. 不同激励下的动力响应分析 37

2.4.3. 块体自身参数的影响 38

2.4.4. 恢复系数的影响 42

2.4.5. 外激励大小的影响 44

2.4.6. 竖向地震作用的影响 47

2.5. 本章小结 48

第3章 随机动力响应分析 50

3.1. 蒙特卡罗方法 50

3.2. 非平稳过滤白噪声模型的产生 51

3.2.1. 调制函数 51

3.2.2. 过滤白噪声 51

3.3. 块体倾覆的累计概率密度 53

3.3.1. 块体自身参数的影响 53

3.3.2. 恢复系数的影响 55

3.3.3. 外激励大小的影响 57

3.3.4. 竖向地震作用的影响 58

3.4. 时程30s内块体倾覆的概率密度 59

3.4.1. 总体分析 59

3.4.2. 对比分析 59

3.5. 本章小结 61

第4章 兵马俑的倾覆分析 62

4.1. 兵马俑几何特性数据 62

4.2. 兵马俑的确定性分析 62

4.2.1. 1号兵马俑确定性分析 63

4.2.2. 2号兵马俑确定性分析 64

4.2.3. 3号兵马俑确定性分析 66

4.3. 兵马俑的随机振动分析 69

4.3.1. 3号兵马俑倾覆概率密度 69

4.3.2. 时程30s内块体倾覆的概率密度 70

4.4. 多点地震场对块体倾覆概率的影响 71

4.5. 本章小结 74

第5章 结论与展望 75