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 1）清华大学土木工程系，北京 100084；2）北京工业大学城市与工程安全减灾教育部重点实验室，北京 100124；3）华北科技学院，北京 101601
[收稿日期]： 2018-04-28

1 椭圆形柱体动水压力求解

 图 1 椭圆柱体与水体相互作用分析模型 Fig. 1 Analytical model of the interactiin of elliptic cylinder and water
1.1 控制方程和边界条件

 $x=\mu \cos h\xi \cos \eta$ (1)
 $y=\mu \text{sin}h\xi \text{sin}\eta$ (2)
 图 2 椭圆坐标系 Fig. 2 Elliptical cylindrical coordinate

 ${{\xi }_{0}}=\text{ta}{{\text{n}}^{-1}}(b/a)$ (3)

 $\frac{2}{{{\mu }^{2}}(\text{cos}h2\xi-\text{cos}2\eta)}\left(\frac{{{\partial }^{2}}p}{\partial {{\xi }^{2}}}+\frac{{{\partial }^{2}}p}{\partial {{\eta }^{2}}} \right)+\frac{{{\partial }^{2}}p}{\partial {{z}^{2}}}=0$ (4)

 $\frac{\partial p}{\partial z}\left| _{z=0} \right.=0$ (5)
 $p\left| _{z=h} \right.=0$ (6)
 $p\left| _{\xi \to \infty } \right.=0$ (7)

 $\frac{\partial p}{\partial \eta }\left| _{\eta =0} \right.=0$ (8a)
 $p\left| _{\eta =0.5\pi } \right.=0$ (8b)

 $\frac{\partial p}{\partial \eta }\left| _{\eta =0.5\pi } \right.=0$ (9a)
 $p\left| _{\eta =0} \right.=0$ (9b)

 $\frac{\partial p}{\partial \xi }\left| _{\xi ={{\xi }_{0}}} \right.=-\rho \ddot{u}b\text{cos}\eta$ (10)
 $\frac{\partial p}{\partial \xi }\left| _{\xi ={{\xi }_{0}}} \right.=-\rho \ddot{u}a\text{sin}\eta$ (11)

1.2 分离变量求解

 $p=R(\xi)G(\eta)Z(z)$ (12)

 ${Z}''+\lambda _{j}^{2}Z=0$ (13)
 ${G}''+({{a}_{0}}+2q\text{cos}2\eta)G=0$ (14)
 ${R}''-({{a}_{0}}+2q\text{cos}h2\xi)R=0$ (15)

 $c{{e}_{2n}}(\eta, -q)={{(-1)}^{n}}\sum\limits_{k=0}^{\infty }{{{(-1)}^{k}}A_{2k}^{(2n)}}\text{cos}2k\eta$ (17a)
 $c{{e}_{2n+1}}(\eta, -q)={{(-1)}^{n}}\sum\limits_{k=0}^{\infty }{{{(-1)}^{k}}B_{2k+1}^{(2n+1)}}\text{cos}(2k+1)\eta$ (17b)
 $s{{e}_{2n+2}}(\eta, -q)={{(-1)}^{n}}\sum\limits_{k=0}^{\infty }{{{(-1)}^{k}}B_{2k+2}^{(2n+2)}}\text{sin}(2k+2)\eta$ (17c)
 $s{{e}_{2n+1}}(\eta, -q)={{(-1)}^{n}}\sum\limits_{k=0}^{\infty }{{{(-1)}^{k}}A_{2k+1}^{(2n+1)}}\text{sin}(2k+1)\eta$ (17d)

1.3 柱体的动水压力

 $p=-\rho b\sum\limits_{j=1}^{\infty }{{{u}_{j}}\frac{B_{1}^{(1)}K{{e}_{1}}(\xi, -q)}{K{{e}_{1}}^{\prime }({{\xi }_{0}}, -q)}c{{e}_{1}}(\eta, -q)Z}$ (21)

 图 5 地面位移时程和傅里叶谱 Fig. 5 The time history of displacement of the ground motion and its Fourier spectrum
 图 6 参考解和集中附加质量模型位移时程的对比 Fig. 6 The comparison of time history of displacement of the cylinder obtained by reference solution and lumped added mass method

 $({\bf{M}}+\alpha {{{\bf{M}}}_{\text{g}}}){{{\bf{\ddot{u}}}}_{\text{s}}}+{\bf{C}}{{{\bf{\dot{u}}}}_{\text{s}}}+{\bf{K}}{{{\bf{u}}}_{\text{s}}}=-({\bf{M}}+{{{\bf{M}}}_{\text{g}}}){{{\bf{\ddot{u}}}}_{\text{g}}}$ (35)

 $l=\frac{D}{h}$ (36)
 $\delta =\frac{a}{b}$ (37)
 图 7 集中附加质量修正系数 Fig. 7 Correction factors for the lumped added mass matrix

 图 8 参考解和修正集中附加质量模型的位移时程比较 Fig. 8 The comparison of time history of displacement of the cylinder obtained by reference solution and modified-lumped added mass method

3 椭圆形柱体均布附加质量简化公式

 ${{C}_{M}}=\sum\limits_{j=1}^{\infty }{\frac{8{{S}_{j}}}{{{(2j-1)}^{2}}{{\text{ }\pi\text{ }}^{2}}}}$ (38)

 ${{C}_{M1}}=0.6{{\text{e}}^{-0.93l}}+0.403{{\text{e}}^{-0.156l}}$ (39)

 ${{C}_{Mx}}/{{C}_{M1}}={{p}_{11}}{{\delta }^{2}}+{{p}_{12}}\delta +{{p}_{13}}$ (40)
 ${{p}_{11}}=0.00367{{l}^{1.554}}+0.0221$ (41a)
 ${{p}_{12}}=-0.185{{l}^{0.507}}-0.041$ (41b)
 ${{p}_{13}}=0.157{{l}^{0.505}}+1.037$ (41c)

 ${{C}_{My}}/{{C}_{M1}}={{p}_{21}}{{\delta }^{{{p}_{22}}}}+{{p}_{23}}$ (42)
 ${{p}_{21}}=-0.277{{\text{e}}^{-0.0186l}}+0.293{{\text{e}}^{-1.102l}}$ (43a)
 ${{p}_{22}}=-0.008{{l}^{2}}+0.186l-1.056$ (43b)
 ${{p}_{23}}=1.295{{{\rm{e}}}^{-0.0106l}}-0.31{{{\rm{e}}}^{-1.052l}}$ (43c)

 图 9 附加质量系数解析解和简化公式的对比 Fig. 9 The comparison of coefficient of the added mass obtained by analytical solution and simplified formula
 图 10 附加质量系数简化公式的误差 Fig. 10 The error of the simplified formula for the added mass coefficient
4 结语

(1) 基于椭圆坐标系，采用分离变量法将拉普拉斯方程转换为马蒂厄方程。通过求解马蒂厄方程，提出了椭圆柱体结构地震动水压力的解析解。

(2) 建立了地震作用下椭圆柱体结构与水体相互作用的动力有限元方程，结构的动水力通过附加质量矩阵施加，该矩阵是满阵的。

(3) 为便于椭圆柱体结构附加质量矩阵在商业有限元中实现，提出了集中附加质量矩阵的方法，该方法中柔性运动引起的附加质量为集中附加质量矩阵和修正系数的乘积。

(4) 基于刚性柱体结构动水力的解析解，通过曲线拟合的方法建立了椭圆柱体结构动水力的均布附加质量简化公式，公式中的系数仅与无量纲参数宽深比和长短轴比相关。

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The Simplified Method for the Earthquake Induced Hydrodynamic Pressure on Elliptical Cylinder
Wang Piguang*1,2), Huang Yiming2), Zhao Mi*2), Du Xiuli2), Zhang Lihua3)
 1) Department of Civil Engineering, Tsinghua University, Beijing 100084, China；2）Key Lab of Urban rban Security and Disaster Engineering of the Ministry of Education, Beijing University of Technology, Beijing 100124, China；3）North China Institute of Science and Technology, Beijing 101601, China
Abstract

Based on the radiation theory, the analytical solutions for the earthquake-induced hydrodynamic force on an elliptical cylinder caused by surrounded water are accurately derived in elliptical coordinate system. The dynamic equation for the dynamic interaction of water with elliptical cylinder is developed by finite element method, where the water-cylinder interaction is replaced by a full added mass matrix. However, the fully added mass matrix is difficult to be implemented in commercial software. Therefore, a lumped added mass matrix is presented to replace the fully added mass matrix with a correction factor, which is relevant to dimensionless parameters including width-depth ratio and ratio of long to short axis of the ellipse. In addition, the simplified formulas for the uniform added mass of the rigid elliptical cylinder are proposed by curve fitting method and these simplified formulas are the functions about the dimensionless parameters including width-depth ratio and ratio of long to short axis of the ellipse.

《震灾防御技术》, DOI：10.11899/zzfy20190103