圆柱形洞室在反平面冲击荷载作用下的动力响应研究
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摘要
在车辆隧道、地铁隧道等此类地下交通道路中,刹车、车辆碰撞以及意外爆炸时有发生,这些情况发生时外界对洞室的冲击力不但有径向的,沿洞室切向的冲击力也不容忽视,切向冲击力对结构和土体的影响也很大,很有必要对其安全性能进行评估。截止目前为止国内外学者对于地下洞室受径向冲击荷载作用下的动力响应的研究已趋于成熟,但是对于洞室在反平面冲击力下的动力响应的研究报道较少,因此,基于弹性动力学理论和波的传播理论对地下洞室在外界轴向冲击力作用下的动力响应问题的研究不但具有深远的理论意义,同时对实际工程也具有巨大的现实价值。
本文将土体视为弹性均质的各向同性材料,将外界冲击力简化为施加在圆柱形洞室表面沿轴线方向的均布荷载,基于波的传播理论,讨论了弹性土体内地下圆柱形洞室在轴线方向冲击荷载的作用下的动力响应问题,包括稳态响应和瞬态响应,主要研究内容和成果如下：
首先假设源函数是简谐的,用解析的方法求解了弹性全空间内地下洞室表面在轴向冲击荷载作用下土体的稳态动力响应,绘制了土体应力和位移的时程曲线和随波的传播距离的衰减曲线,然后在全空间解答的基础上运用镜像原理,计算得到了半空间内土体的动力响应,分析结果显示：土体内应力和位移都是轴对称的,全空间内应力和位移值仅仅取决于离开震源圆柱轴的距离,距离波源越远,应力和位移值越小；半空间内,波源的频率越高,应力和位移的幅值较大,沿圆周的空间变化越复杂,随着频率的增大,应力和位移的最大值出现的位置由靠近地表的位置向远离地表的位置移动；洞室埋深越深,应力和位移幅值也越小,应力和位移随角度的空间变化越小。
其次研究了无限弹性空间内地下洞室表面在轴向冲击力作用下土体的瞬态响应。利用拉普拉斯积分变换法在频域内求得了土体位移和应力的一般解析表达式,然后结合围道积分,运用拉普拉斯逆变换的方法,在时域内求得了土体应力和位移的一般解析解答式,并将这种解析方法得出的计算结果、地下洞室在静力作用下的土体应力和位移与采用Durbin的数值逆变换方法得出的结果做了比较,验证了本文结果的正确性,Durbin的数值逆变换方法对本文研究问题是适用的,最后分析了应力和位移随时间的变化规律,得出以下结论：波到达后,该点土体的应力和位移均瞬间增大,随后慢慢减小并逐渐趋于稳定值。
然后考虑到自由地面的存在,研究了弹性半空间内圆柱形洞室在轴向荷载作用下的瞬态响应。借助于镜像方法将半无限空间内的求解问题转化为无限体内的求解问题,先利用积分变换法,在频域内求得应力和位移的解答,然后运用Durbin的拉普拉斯逆变换法求得数值的计算结果,从而在半空间内获得了土体的位移应力分布,分析了半空间内弹性波的传播规律,土体应力和位移随时间、空间、传播距离以及洞室埋深的变化。
接着将洞室衬砌的作用考虑进去,利用积分变换法,在频域范围内分别求的了衬砌和土体位移和应力的表达式,采用Durbin拉普拉斯数值逆变换得到了衬砌和土体在时域内的动力响应,分别绘制了衬砌边界上和土介质内一点上应力和位移的时程曲线,讨论了衬砌相对剪切模量和相对厚度对响应的影响,结果表明：衬砌与土介质的相对剪切模量越大,衬砌外边界上和土介质的中应力和位移就越小,应力和位移衰减的越快；位移和应力随衬砌相对厚度增加而减小。
最后基于全空间内圆柱形衬砌洞室在轴向荷载作用下的瞬态动力响应的研究,借助于镜像方法,在频域内分别求得半空间内衬砌和土介质应力和位移的解答,然后运用数值逆变换在时域内得到了衬砌和土介质在冲击荷载作用下的瞬态动力响应,绘制了半空间内应力和位移的时程曲线和极坐标曲线,讨论了洞室埋深、衬砌相对剪切模量和相对厚度对动力响应的影响。
The brakes, the vehicle collision and accidental explosion occur frequently in underground traffic roads such as vehicles tunnels and subway tunnels. When these situations occur the impact load exerted on the cavity is not only radial, tangential impact load along the cavity can not be ignored. It is necessary to evaluate its safety performance because the influence of the tangential impact on the structure and the soil is also great. So far the theory of dynamic response for underground cavity under radial impact loads has developed sophisticatedly at home and abroad. However, the study on anti-plane dynamic response is reported less. Therefore the study on the dynamic response of a cylindrical cavity subjected to an axial impact load is not only of profound theoretical significance, but also of great practical value in the actual project based on elastodynamics and wave propagation theory.
In this paper the material of the soil is assumed to be homogeneous, isotropic and linearly elastic. The external impact load excited on cylindrical cavity is simplified to uniform sudden load along the axial direction. Based on wave propagation theory, the dynamic response of an elastic soil body containing a cylindrical cavity subjected to the anti-plane load is studied including steady-state response and transient response. The main research contents and conclusions are as follows:
First of all, assuming that the wave source function is harmonic, the steady dynamic response of an elastic full space containing a cylindrical cavity subjected to the axial impact load is solved using analytical methods. The soil stress-time curve and displacement-time curve and the decay curve with wave propagation distance are drawn. And then on the base of the theory of the full space, according to the image method, the dynamic response of the soil half-space is obtained, The calculation results show that, the stress and displacement of the soil are axially symmetric, the farther the wave is from the source, the smaller the stress and the displacement are, and the stress and the displacement are just functions of the radial distance from the axis in full space. In half space, the stress and displacement and the complexity of spatial variations along the circumference increase with the increasing of the frequency of the wave source, and the position of maximum displacement and maximum stress moves away from the ground surface. The bigger the depth is, the smaller the stress and displacement are, and the less the spatial variation with angle of stress and displacement is.
Secondly, the transient response to sudden anti-plane shock load of an infinite soil body containing a cylindrical cavity is studied. Using Laplace transform method, the frequency domain solutions are obtained. Using Laplace inverse transform combining with contour integral, the general analytical expressions of the soil displacement and stress are obtained in the time domain, respectively. And the calculation results are compared with those from numerical inversion proposed by Durbin and the static results, one observes good agreement between analytical and numerical inversion results, which lending the further support to the method presented. Durbin's method is applicable to this paper. Finally the stress and displacement variation with time is analyzed, the conclusions are gained:after the wave arrival the stress and the displacement at this point increase abruptly, then reduce and tend to the static value gradually at last.
Then, taking into account the ground surface, the transient response of an elastic soil body half-space containing a cylindrical cavity subjected to the axial impact load is investigated. Then infinite space solutions are translated into solutions for the elastic half-space using image method. Using integral transform method, the stress and displacement are obtained in the frequency domain. And the numerical solutions of the problem are presented by the numerical inversion proposed by Durbin. Thus the displacement and stress distribution of the soil in half-space are gained. The propagation of elastic waves, the variation of the stress and displacement with respect to the time, angle, the radial distance and depth of the cavity are presented.
After that, taking into account the lining, using the integral transform frequency solutions of the lining and soil of the displacement and stress are obtained. The dynamic response in time domain of the lining and the soil using the numerical inverse Laplace transform proposed by Durbin is attained. Displacement-time curves and stress-time curves are drawn on the outer boundary of the lining and a point in the soil. The influence of the relative shear modulus and relative thickness of the lining on the response is analyzed. The results show that, lining and soil media the relative shear modulus, the displacement and stress of the lining and the soil decrease with the increasing of the relative shear modulus and relative thickness.
Finally, based on transient response in full space containing a cylindrical cavity with lining subjected to the axial impact load, by means of image method, frequency solutions of the lining and soil of the displacement and stress are obtained respectively. Then using numerical inverse transform, the transient response of the lining and the soil subjected to impact load in the time domain is obtained. Displacement-time curves, stress-time curves and polar plots are drawn in half space. The influence of the depth, the relative shear modulus and relative thickness of the lining on the response is discussed.
本文将土体视为弹性均质的各向同性材料,将外界冲击力简化为施加在圆柱形洞室表面沿轴线方向的均布荷载,基于波的传播理论,讨论了弹性土体内地下圆柱形洞室在轴线方向冲击荷载的作用下的动力响应问题,包括稳态响应和瞬态响应,主要研究内容和成果如下：
首先假设源函数是简谐的,用解析的方法求解了弹性全空间内地下洞室表面在轴向冲击荷载作用下土体的稳态动力响应,绘制了土体应力和位移的时程曲线和随波的传播距离的衰减曲线,然后在全空间解答的基础上运用镜像原理,计算得到了半空间内土体的动力响应,分析结果显示：土体内应力和位移都是轴对称的,全空间内应力和位移值仅仅取决于离开震源圆柱轴的距离,距离波源越远,应力和位移值越小；半空间内,波源的频率越高,应力和位移的幅值较大,沿圆周的空间变化越复杂,随着频率的增大,应力和位移的最大值出现的位置由靠近地表的位置向远离地表的位置移动；洞室埋深越深,应力和位移幅值也越小,应力和位移随角度的空间变化越小。
其次研究了无限弹性空间内地下洞室表面在轴向冲击力作用下土体的瞬态响应。利用拉普拉斯积分变换法在频域内求得了土体位移和应力的一般解析表达式,然后结合围道积分,运用拉普拉斯逆变换的方法,在时域内求得了土体应力和位移的一般解析解答式,并将这种解析方法得出的计算结果、地下洞室在静力作用下的土体应力和位移与采用Durbin的数值逆变换方法得出的结果做了比较,验证了本文结果的正确性,Durbin的数值逆变换方法对本文研究问题是适用的,最后分析了应力和位移随时间的变化规律,得出以下结论：波到达后,该点土体的应力和位移均瞬间增大,随后慢慢减小并逐渐趋于稳定值。
然后考虑到自由地面的存在,研究了弹性半空间内圆柱形洞室在轴向荷载作用下的瞬态响应。借助于镜像方法将半无限空间内的求解问题转化为无限体内的求解问题,先利用积分变换法,在频域内求得应力和位移的解答,然后运用Durbin的拉普拉斯逆变换法求得数值的计算结果,从而在半空间内获得了土体的位移应力分布,分析了半空间内弹性波的传播规律,土体应力和位移随时间、空间、传播距离以及洞室埋深的变化。
接着将洞室衬砌的作用考虑进去,利用积分变换法,在频域范围内分别求的了衬砌和土体位移和应力的表达式,采用Durbin拉普拉斯数值逆变换得到了衬砌和土体在时域内的动力响应,分别绘制了衬砌边界上和土介质内一点上应力和位移的时程曲线,讨论了衬砌相对剪切模量和相对厚度对响应的影响,结果表明：衬砌与土介质的相对剪切模量越大,衬砌外边界上和土介质的中应力和位移就越小,应力和位移衰减的越快；位移和应力随衬砌相对厚度增加而减小。
最后基于全空间内圆柱形衬砌洞室在轴向荷载作用下的瞬态动力响应的研究,借助于镜像方法,在频域内分别求得半空间内衬砌和土介质应力和位移的解答,然后运用数值逆变换在时域内得到了衬砌和土介质在冲击荷载作用下的瞬态动力响应,绘制了半空间内应力和位移的时程曲线和极坐标曲线,讨论了洞室埋深、衬砌相对剪切模量和相对厚度对动力响应的影响。
The brakes, the vehicle collision and accidental explosion occur frequently in underground traffic roads such as vehicles tunnels and subway tunnels. When these situations occur the impact load exerted on the cavity is not only radial, tangential impact load along the cavity can not be ignored. It is necessary to evaluate its safety performance because the influence of the tangential impact on the structure and the soil is also great. So far the theory of dynamic response for underground cavity under radial impact loads has developed sophisticatedly at home and abroad. However, the study on anti-plane dynamic response is reported less. Therefore the study on the dynamic response of a cylindrical cavity subjected to an axial impact load is not only of profound theoretical significance, but also of great practical value in the actual project based on elastodynamics and wave propagation theory.
In this paper the material of the soil is assumed to be homogeneous, isotropic and linearly elastic. The external impact load excited on cylindrical cavity is simplified to uniform sudden load along the axial direction. Based on wave propagation theory, the dynamic response of an elastic soil body containing a cylindrical cavity subjected to the anti-plane load is studied including steady-state response and transient response. The main research contents and conclusions are as follows:
First of all, assuming that the wave source function is harmonic, the steady dynamic response of an elastic full space containing a cylindrical cavity subjected to the axial impact load is solved using analytical methods. The soil stress-time curve and displacement-time curve and the decay curve with wave propagation distance are drawn. And then on the base of the theory of the full space, according to the image method, the dynamic response of the soil half-space is obtained, The calculation results show that, the stress and displacement of the soil are axially symmetric, the farther the wave is from the source, the smaller the stress and the displacement are, and the stress and the displacement are just functions of the radial distance from the axis in full space. In half space, the stress and displacement and the complexity of spatial variations along the circumference increase with the increasing of the frequency of the wave source, and the position of maximum displacement and maximum stress moves away from the ground surface. The bigger the depth is, the smaller the stress and displacement are, and the less the spatial variation with angle of stress and displacement is.
Secondly, the transient response to sudden anti-plane shock load of an infinite soil body containing a cylindrical cavity is studied. Using Laplace transform method, the frequency domain solutions are obtained. Using Laplace inverse transform combining with contour integral, the general analytical expressions of the soil displacement and stress are obtained in the time domain, respectively. And the calculation results are compared with those from numerical inversion proposed by Durbin and the static results, one observes good agreement between analytical and numerical inversion results, which lending the further support to the method presented. Durbin's method is applicable to this paper. Finally the stress and displacement variation with time is analyzed, the conclusions are gained:after the wave arrival the stress and the displacement at this point increase abruptly, then reduce and tend to the static value gradually at last.
Then, taking into account the ground surface, the transient response of an elastic soil body half-space containing a cylindrical cavity subjected to the axial impact load is investigated. Then infinite space solutions are translated into solutions for the elastic half-space using image method. Using integral transform method, the stress and displacement are obtained in the frequency domain. And the numerical solutions of the problem are presented by the numerical inversion proposed by Durbin. Thus the displacement and stress distribution of the soil in half-space are gained. The propagation of elastic waves, the variation of the stress and displacement with respect to the time, angle, the radial distance and depth of the cavity are presented.
After that, taking into account the lining, using the integral transform frequency solutions of the lining and soil of the displacement and stress are obtained. The dynamic response in time domain of the lining and the soil using the numerical inverse Laplace transform proposed by Durbin is attained. Displacement-time curves and stress-time curves are drawn on the outer boundary of the lining and a point in the soil. The influence of the relative shear modulus and relative thickness of the lining on the response is analyzed. The results show that, lining and soil media the relative shear modulus, the displacement and stress of the lining and the soil decrease with the increasing of the relative shear modulus and relative thickness.
Finally, based on transient response in full space containing a cylindrical cavity with lining subjected to the axial impact load, by means of image method, frequency solutions of the lining and soil of the displacement and stress are obtained respectively. Then using numerical inverse transform, the transient response of the lining and the soil subjected to impact load in the time domain is obtained. Displacement-time curves, stress-time curves and polar plots are drawn in half space. The influence of the depth, the relative shear modulus and relative thickness of the lining on the response is discussed.
引文
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