引水隧道在地震波入射时的动力响应解析解
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摘要
跨流域调水工程的建设能够有效缓解我国淡水资源空问分布极不均衡的矛盾,然而,长距离的调水工程不可避免地需要穿越地震多发区域,研究调水工程构筑物的地震响应问题对调水工程的成功建设和安全运营具有重要的意义。本文正是针对引水隧道这一调水工程中的常见结构,进行了一系列特殊场地中引水隧道动力响应的解析解研究。
首先,本文提出在两斜交柱坐标系间谐波函数空间坐标变换的辅助平面方法,运用该方法,得到谐波函数在斜交柱坐标系间的坐标变换公式,作为研究隧道交叉等特殊场地地震响应问题的基础。通过建立与空间点对应的一系列辅助平面,将地震波三维散射研究中的谐波函数表达式从柱坐标系变换至辅助平面内的极坐标系,然后在辅助平面内运用Graf加法公式,将柱坐标系下的波函数表达式变换至与该柱坐标系斜交的另一柱坐标系下,从而将三维问题转换成二维问题进行处理,最终得到谐波函数在斜交柱坐标系下内域问题和外域问题的坐标变换公式。
其次,本文建立了完全充水引水隧道场地模型,并采用波函数展开法给出了P波和SV波入射时引水隧道平面地震响应和三维地震响应的解析解。通过算例分析,重点讨论了入射波性质、隧道尺寸、衬砌材料参数、地层泊松比等因素对引水隧道动力响应的影响。算例分析结果表明,完全充水引水隧道衬砌的动力响应与入射波振幅成正比；在入射波无量纲频率不变时,衬砌的动应力峰值不受隧道内径变化的影响,而衬砌的动位移峰值则与隧道内径成正比；完全充水引水隧道衬砌的动应力峰值均随着衬砌厚度的增大而减小,动位移峰值则随着衬砌厚度的增大而增大；随着隧道衬砌弹性模量的增大,衬砌受到的动水压力峰值和衬砌的动位移峰值均逐渐减小,而切向动应力峰值和轴向动应力峰值均增大；土层泊松比的改变使得引水隧道的动力响应发生明显改变。此外,当P波和SV波倾斜入射时,入射波传播方向与引水隧道横断面之间的夹角对引水隧道的动力响应有着显著的影响。
再次,本文建立了水下完全充水引水隧道场地模型,给出了P波和SV波入射时水下引水隧道平面地震响应的解析解。通过算例分析,重点讨论了入射波性质、地表水体深度、隧道埋深等因素对水下引水隧道动力响应的影响。研究表明,P波入射时水下引水隧道的动力响应随着地表水深的增大而增大,且低频P波入射时地表水深变化对衬砌动力响应的影响程度要大于高频P波入射情况;而在SV波入射时,地表水深改变对水下引水隧道衬砌的动应力峰值影响不大；随着隧道埋深的增大,低频P波入射引起的水下引水隧道衬砌动应力逐渐减小,动位移则逐渐增大,而低频SV波入射时,水下引水隧道衬砌的动应力和动位移随着隧道埋深的增大而减小。
最后,本文建立了引水隧道下穿地铁区间隧道局部复杂场地模型,并采用波函数展开法给出了P波和SV波入射下该复杂场地地震响应的解析解。通过算例分析,讨论了上层地铁隧道和下层引水隧道之间的相互影响。研究表明,下层引水隧道对上层地铁隧道产生屏障作用,使得P波入射和SV波入射时上层地铁隧道定区域内的动应力峰值和动位移峰值均有所减小,而上层地铁隧道的存在则使得引水隧道在一定区域内的动应力峰值和位移峰值有所增大。两座隧道在交叉点两侧大致4-6倍的内半径范围内存在明显的相互影响。
:Long distance water diversions projects are constructed to alleviate the unbalanced distribution of water resources in China. However, it is inevitable that the long distance water diversions projects need through the seismic regions. Therefore, the study on the seismic response of water diversions projects has a vital significance to the projects construction and operation. This paper investigates the analytical solutions about the dynamic responses of convey tunnel, a commonly used structure in long distance water diversions projects, in a series of special fields.
Firstly, this paper presents an auxiliary plane method for harmonic function transformation between two skew cylindrical coordinate systems. By creating a series of auxiliary plane corresponded with spatial point, the3-D harmonic wave function is transformed from cylindrical coordinate system to polar coordinate. Then the wave function under a cylindrical coordinate system is transformed into another which is skewed with the former by using the Graf's addition formula on the auxiliary plane method. Thus the problem is transformed from3-D to2-D. At last, the coordinate transformation formula between two skew cylindrical coordinate systems is given.
Secondly, it is given that the plane and3-D analytical solutions of the dynamic responses of water-filled convey tunnel induced by plane P waves and SV waves by using the wave function expansion method in this paper. The example analysis shows the effects of the incident wave parameters, the tunnel dimension, the material of tunnel lining, and the Poisson's ratio of half space on the seismic response of the water-filled convey tunnel. It is proved that the dynamic responses of water-filled diversion tunnel are proportional to the amplitude of incident wave. And the maximums of the dynamic displacements of tunnel lining are proportional to the tunnel radius, but the maximums of the dynamic stresses are irrelevant to the radius. With the increasing of the tunnel lining thickness, the maximums of the dynamic stresses of tunnel are decrease, but the maximums of the dynamic displacements are increase. Moreover, the elastic modulus of the lining and the Poisson's ratio of the soil have a significant effects on the dynamic responses of water-filled convey tunnel.
Thirdly, this paper establishes a model of underwater water-filled convey tunnel, and gives the plane analytical solution of the dynamic responses of the model caused by the incident P waves and SV waves. The example analysis shows the effects of the incident wave parameters, the depth of surface water, and the buried depth of underwater tunnel on the seismic response of the tunnel lining. It can be found from the results that the dynamic responses of underwater convey tunnel is increase with the increasing of the surface water depth when the P waves incidences. Moreover, with the increasing of the buried depth of underwater convey tunnel, the maximums of the dynamic stresses of tunnel lining caused by incident P waves with a low frequency are decrease, but the maximums of the dynamic displacements are increase. While, when the incident wave is SV waves, the dynamic responses of underwater convey tunnel is decrease with the increasing of tunnel buried depth.
Finally, this paper gives the analytical solution of the dynamic responses of a complicated partial field in which a convey tunnel is beneath and pass through a subway tunnel caused by incident P waves and SV waves. Through the example analysis, the interaction between the convey tunnel and the subway tunnel is investigated. The results show that the lower convey tunnel acts as a barrier to the subway tunnel, and it makes the dynamic responses of the upper subway tunnel reduced around the intersection. Inversely, the upper subway tunnel makes the dynamic responses of the lower convey tunnel magnified around the intersection.
首先,本文提出在两斜交柱坐标系间谐波函数空间坐标变换的辅助平面方法,运用该方法,得到谐波函数在斜交柱坐标系间的坐标变换公式,作为研究隧道交叉等特殊场地地震响应问题的基础。通过建立与空间点对应的一系列辅助平面,将地震波三维散射研究中的谐波函数表达式从柱坐标系变换至辅助平面内的极坐标系,然后在辅助平面内运用Graf加法公式,将柱坐标系下的波函数表达式变换至与该柱坐标系斜交的另一柱坐标系下,从而将三维问题转换成二维问题进行处理,最终得到谐波函数在斜交柱坐标系下内域问题和外域问题的坐标变换公式。
其次,本文建立了完全充水引水隧道场地模型,并采用波函数展开法给出了P波和SV波入射时引水隧道平面地震响应和三维地震响应的解析解。通过算例分析,重点讨论了入射波性质、隧道尺寸、衬砌材料参数、地层泊松比等因素对引水隧道动力响应的影响。算例分析结果表明,完全充水引水隧道衬砌的动力响应与入射波振幅成正比；在入射波无量纲频率不变时,衬砌的动应力峰值不受隧道内径变化的影响,而衬砌的动位移峰值则与隧道内径成正比；完全充水引水隧道衬砌的动应力峰值均随着衬砌厚度的增大而减小,动位移峰值则随着衬砌厚度的增大而增大；随着隧道衬砌弹性模量的增大,衬砌受到的动水压力峰值和衬砌的动位移峰值均逐渐减小,而切向动应力峰值和轴向动应力峰值均增大；土层泊松比的改变使得引水隧道的动力响应发生明显改变。此外,当P波和SV波倾斜入射时,入射波传播方向与引水隧道横断面之间的夹角对引水隧道的动力响应有着显著的影响。
再次,本文建立了水下完全充水引水隧道场地模型,给出了P波和SV波入射时水下引水隧道平面地震响应的解析解。通过算例分析,重点讨论了入射波性质、地表水体深度、隧道埋深等因素对水下引水隧道动力响应的影响。研究表明,P波入射时水下引水隧道的动力响应随着地表水深的增大而增大,且低频P波入射时地表水深变化对衬砌动力响应的影响程度要大于高频P波入射情况;而在SV波入射时,地表水深改变对水下引水隧道衬砌的动应力峰值影响不大；随着隧道埋深的增大,低频P波入射引起的水下引水隧道衬砌动应力逐渐减小,动位移则逐渐增大,而低频SV波入射时,水下引水隧道衬砌的动应力和动位移随着隧道埋深的增大而减小。
最后,本文建立了引水隧道下穿地铁区间隧道局部复杂场地模型,并采用波函数展开法给出了P波和SV波入射下该复杂场地地震响应的解析解。通过算例分析,讨论了上层地铁隧道和下层引水隧道之间的相互影响。研究表明,下层引水隧道对上层地铁隧道产生屏障作用,使得P波入射和SV波入射时上层地铁隧道定区域内的动应力峰值和动位移峰值均有所减小,而上层地铁隧道的存在则使得引水隧道在一定区域内的动应力峰值和位移峰值有所增大。两座隧道在交叉点两侧大致4-6倍的内半径范围内存在明显的相互影响。
:Long distance water diversions projects are constructed to alleviate the unbalanced distribution of water resources in China. However, it is inevitable that the long distance water diversions projects need through the seismic regions. Therefore, the study on the seismic response of water diversions projects has a vital significance to the projects construction and operation. This paper investigates the analytical solutions about the dynamic responses of convey tunnel, a commonly used structure in long distance water diversions projects, in a series of special fields.
Firstly, this paper presents an auxiliary plane method for harmonic function transformation between two skew cylindrical coordinate systems. By creating a series of auxiliary plane corresponded with spatial point, the3-D harmonic wave function is transformed from cylindrical coordinate system to polar coordinate. Then the wave function under a cylindrical coordinate system is transformed into another which is skewed with the former by using the Graf's addition formula on the auxiliary plane method. Thus the problem is transformed from3-D to2-D. At last, the coordinate transformation formula between two skew cylindrical coordinate systems is given.
Secondly, it is given that the plane and3-D analytical solutions of the dynamic responses of water-filled convey tunnel induced by plane P waves and SV waves by using the wave function expansion method in this paper. The example analysis shows the effects of the incident wave parameters, the tunnel dimension, the material of tunnel lining, and the Poisson's ratio of half space on the seismic response of the water-filled convey tunnel. It is proved that the dynamic responses of water-filled diversion tunnel are proportional to the amplitude of incident wave. And the maximums of the dynamic displacements of tunnel lining are proportional to the tunnel radius, but the maximums of the dynamic stresses are irrelevant to the radius. With the increasing of the tunnel lining thickness, the maximums of the dynamic stresses of tunnel are decrease, but the maximums of the dynamic displacements are increase. Moreover, the elastic modulus of the lining and the Poisson's ratio of the soil have a significant effects on the dynamic responses of water-filled convey tunnel.
Thirdly, this paper establishes a model of underwater water-filled convey tunnel, and gives the plane analytical solution of the dynamic responses of the model caused by the incident P waves and SV waves. The example analysis shows the effects of the incident wave parameters, the depth of surface water, and the buried depth of underwater tunnel on the seismic response of the tunnel lining. It can be found from the results that the dynamic responses of underwater convey tunnel is increase with the increasing of the surface water depth when the P waves incidences. Moreover, with the increasing of the buried depth of underwater convey tunnel, the maximums of the dynamic stresses of tunnel lining caused by incident P waves with a low frequency are decrease, but the maximums of the dynamic displacements are increase. While, when the incident wave is SV waves, the dynamic responses of underwater convey tunnel is decrease with the increasing of tunnel buried depth.
Finally, this paper gives the analytical solution of the dynamic responses of a complicated partial field in which a convey tunnel is beneath and pass through a subway tunnel caused by incident P waves and SV waves. Through the example analysis, the interaction between the convey tunnel and the subway tunnel is investigated. The results show that the lower convey tunnel acts as a barrier to the subway tunnel, and it makes the dynamic responses of the upper subway tunnel reduced around the intersection. Inversely, the upper subway tunnel makes the dynamic responses of the lower convey tunnel magnified around the intersection.
引文
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