有横缝高拱坝的非线性地震响应分析
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摘要
随着我国经济的发展，小湾、溪洛渡等一批300米级世界超高拱坝即将在我国西部地区进行建设。这些大坝所在地区山高谷深，岸坡陡峻，河道狭窄，地处高地震烈度区，高坝的抗震性能关系到下游广大地区工农业生产和人民生命财产的安全，考虑地震因素成为大坝设计中的控制工况。拱坝是分段浇筑的、各坝段之间沿拱向设有均匀排列的伸缩横缝。由于地震作用时的动拉应力将抵消并超过静载作用下的压应力，导致横缝面沿法向张开，加上地震的交变作用，横缝可能会呈现反复的“渐开渐合”现象。横缝张合效应将削弱大坝的整体刚度、延长其振动周期、引起显著的应力重分配。因而横缝的非线性特性是高拱坝地震反应分析中至关重要的因素，在动力分析模型中合理计及这类非线性特性具有特别重要的学术意义和工程意义。
以高278m的溪落渡拱坝为工程背景，进行了考虑横缝张开、行波效应等复杂因素的非线性动力分析。基于流固耦合理论，推导了可压缩水体的耦合有限元方程。为适应有横缝高拱坝非线性分析的复杂性，采用不可压缩水体和无质量地基假定进行有限元处理，以水体附加质量和地基附加刚度近似代替水体和地基的影响，通过静凝聚提高计算效率。论述了三维Fenves缝单元模型及缝单元的本构关系，可用以模拟横缝的张开、闭合和滑移，并可考虑缝面的初始压力和抗拉强度。为有效控制误差的传递，提高计算精度，对Wilson-θ法进行改进并分析了改进Wilson-θ法的稳定性。鉴于横缝非线性数值分析需要在每一时间步内迭代多次，推导了Newmark常加速度增量平衡方程及其等效迭代格式。对行波理论进行了阐述，用单向行波模式分析了整体坝的动力反应。对比了横河向和顺河向行波输入时坝体位移、应力的反应和分
四川大学工学博士学位论文
布异同。研究了溪落渡分缝拱坝的地震反应。分析了整体坝和分缝坝地震反
应的异同，得出分缝坝较之整体坝在加速度、应力的大小与分布上的诸多不
同点：分析了水深对有缝拱坝地震响应的影响，得出分缝坝在不同水深下缝
的张开深度、宽度和频次的一系列结论。分析了分缝坝行波地震输入的地震
响应，比较了常规均匀输入和行波输入，行波不同输入方向对应力的大小和
分布，缝的张合深度和宽度、缝的张合时序等诸多方面的表现特性。通过以
上分析，得到了如下的一些主要结论：
1．计算结果分析表明：在高拱坝流固耦合动力分析中，水体不可压缩和无
质量地基假定可显著地消减方程复杂度和计算工作量，但带来的数值误
差较小。
2．改进了wilson习法，在国内外首次分析了改进了wilson习法的数值稳定性，
证明了改进ilsonE法与wilson习法具有相同的稳定性，用改进wilson习
法求解线性整体坝是可行的。提出了增量形式Newmark法的等效迭代格
式，该等效格式求解分缝拱坝非线性问题简便可行。
3．进行了顺河向常规输入情况下整体坝和分缝坝两种计算模型的抗震动力
特性对比研究。结果表明：分缝坝由于缝的开合，其横河向和顺河向的
加速度要大于整体坝，同时，分缝坝本身的横河向加速度有可能大于顺
河向加速度。在靠近基础的部位，两种计算模型的动拱应力改变不大，
但动梁应力变化明显。分缝坝较之整体坝，上游坝面明显表现出总拱应
力下降，总梁应力增加的趋势；而对下游坝面，其总压应力梁向增加，
拱向减小，而总拉应力的梁向和拱向则增有减。整体坝的控制拉应力出
现在下游顶拱的梁向，控制压应力出现在上游顶拱的拱向；分缝坝的控
制拉应力出现在上游坝底的梁向，控制压应力却出现在下游坝底的梁向。
4．研究了不同水位有缝坝在常规地震输入情况下横缝的开合规律和应力反
应特点。无论运行低水位还是正常蓄水位，横缝张开最深最宽的是上游
坝面中缝，张开宽度从中缝到边缝依次减小；正常蓄水位时缝的张开宽
度明显小于运行低水位时缝的张开宽度。从控制裂缝张开深度和宽度的
角度而言，低水位是控制工况；而从坝体应力的角度考虑，高水位是控
制工况，
5．深入研究了整体坝在常规输入和行波输入两种地震输人方式下的动力反
四川大学工学博士学位论文
应。行波横河向输入时顺河向位移反应在拱冠梁处有所增大，其它点有
所削弱：行波J’顷河向输入时非拱冠梁节点的横河向位移反应有所削弱。
常规输入时，顺河向输入较横河向输入的应力反应大，说明常规输入时
的控制方向是顺河向。但在行波输入时，行波横河向输入的应力反应比
行波顺河向输入时大，说明行波输入的控制方向是横河向。
6．在国内外首次进行了分缝坝行波输入的抗震分析，研编了计算程序，深
入研究了常规输入与行波输入两种地震输入方式对有缝高拱坝地震响应
的影响。行波顺河向输入时缝的张汗宽度普遍大于常规顺河向输入时的
张开?
With the development of economy, Many arch dams such as Xiao wan, Xi
Luodu and so on whose height are more than 300m are going to be constructed in
westem regions of China. The arch dams locate at such areas which have high
moUnain, deep valley, steep slope, narrow riverway and high seismic intensity.
The aseismic capabilities of high arch dams have relationship with lives and
properties of people and productions of industry and agriculture, so the eanhquake
becomes the main factor of arch darns design. Arch dams are built by parts, and
each of parts has vertical contraction joints spaced regular intervals. The dynamic
tensile stresses of earthquake may balance or exceed to the precompression
stresses due to static watef, this may arouse opening of joints along the normal
direction. If the altemate actions of earthquake join, the joints may behave gradual
opening and closing. The effect of opening and closing of joints will weaken the
integral stiffness of arch dams, prolong their vibration periods and remarkably
readjusting of stresses. In general, the nonlinear character of joints is the
significant factor of the analysis of high arch dams seismic response. Reasonably
taking into account nonlinear character of dynarnic model has very important
significance of theory and engineering.
Nonlinear dynamic analysis of high arch dams is carried out considering the
opening of joints and the effect of traveling wave with engineering background of
Xi Luodu arch dam whose height is 278 meteL Based on the theory of interaction
of fluid-solid, the coupled FEM equation considering compressible water is
deduced. In order to reduce the equation complexity and adapt to complicaied
nonIinear ana1ysis of high arch dams with joints, finite element treatments are
used with incompressibIe water and massless foundation. The added mass of
water and added stiffness of foundation are respectively used to substitute the
influence of water and fOundations. The efficiency in the computations is
improved by means of static condensation. Fenves' 3-D joint element modeI and
its constitutive relations are demonstrated and used to simulate the opening,
closing and sliding of joints,and the initial pressure and tensile strength at the
interface. In order to control the propagation of errors and enhance the accuracy of
numerical solution, the wilson- 0 method is modified and the numerical stability
of modified wilson- e method is analyzed. Because of need of much iteration in
each time step while carrying on nonlinear numerical anaIysis, the incremefltal
equilibrium formulation and its equivalent iterative equation of the constant
average acceleration Newnark method are deduced. The theory of traveling wave
input is discussed, and the dynatnic response of linear whole dams is analyzed by
use of single-direction traveling wave model. The difference and commonness
between cross-direction and longitude direction traveling wave that exists in the
response and distribution of dam displacement and stress are obtained. The
dynamic seismic response of Xi Luodu dam is studied. The difference and
commonLness between dams with joints and without are analyzed. The magnitude
and distribution of acceleration and stress of dams with joint and without possess
many differences. The influence of different water depth on seismic response of
dams with joint is studied, the conclusion in term of opening width, opening depth
and opening frequency is attained. The characteristics about the magnitude and
distribution of stress, the width and depth of joint and time sequence of joint
opening are compared on condition of different traveling wave directions. The
difference between general unifOrm wave input and traveling wave inPut is also
compared. Though the above analysis, some conclusion are drew as followings f
l. The computational results show that the assumptions of incompressible
water and massless fOundation can considerably simplify the complexity of
equation
以高278m的溪落渡拱坝为工程背景，进行了考虑横缝张开、行波效应等复杂因素的非线性动力分析。基于流固耦合理论，推导了可压缩水体的耦合有限元方程。为适应有横缝高拱坝非线性分析的复杂性，采用不可压缩水体和无质量地基假定进行有限元处理，以水体附加质量和地基附加刚度近似代替水体和地基的影响，通过静凝聚提高计算效率。论述了三维Fenves缝单元模型及缝单元的本构关系，可用以模拟横缝的张开、闭合和滑移，并可考虑缝面的初始压力和抗拉强度。为有效控制误差的传递，提高计算精度，对Wilson-θ法进行改进并分析了改进Wilson-θ法的稳定性。鉴于横缝非线性数值分析需要在每一时间步内迭代多次，推导了Newmark常加速度增量平衡方程及其等效迭代格式。对行波理论进行了阐述，用单向行波模式分析了整体坝的动力反应。对比了横河向和顺河向行波输入时坝体位移、应力的反应和分
四川大学工学博士学位论文
布异同。研究了溪落渡分缝拱坝的地震反应。分析了整体坝和分缝坝地震反
应的异同，得出分缝坝较之整体坝在加速度、应力的大小与分布上的诸多不
同点：分析了水深对有缝拱坝地震响应的影响，得出分缝坝在不同水深下缝
的张开深度、宽度和频次的一系列结论。分析了分缝坝行波地震输入的地震
响应，比较了常规均匀输入和行波输入，行波不同输入方向对应力的大小和
分布，缝的张合深度和宽度、缝的张合时序等诸多方面的表现特性。通过以
上分析，得到了如下的一些主要结论：
1．计算结果分析表明：在高拱坝流固耦合动力分析中，水体不可压缩和无
质量地基假定可显著地消减方程复杂度和计算工作量，但带来的数值误
差较小。
2．改进了wilson习法，在国内外首次分析了改进了wilson习法的数值稳定性，
证明了改进ilsonE法与wilson习法具有相同的稳定性，用改进wilson习
法求解线性整体坝是可行的。提出了增量形式Newmark法的等效迭代格
式，该等效格式求解分缝拱坝非线性问题简便可行。
3．进行了顺河向常规输入情况下整体坝和分缝坝两种计算模型的抗震动力
特性对比研究。结果表明：分缝坝由于缝的开合，其横河向和顺河向的
加速度要大于整体坝，同时，分缝坝本身的横河向加速度有可能大于顺
河向加速度。在靠近基础的部位，两种计算模型的动拱应力改变不大，
但动梁应力变化明显。分缝坝较之整体坝，上游坝面明显表现出总拱应
力下降，总梁应力增加的趋势；而对下游坝面，其总压应力梁向增加，
拱向减小，而总拉应力的梁向和拱向则增有减。整体坝的控制拉应力出
现在下游顶拱的梁向，控制压应力出现在上游顶拱的拱向；分缝坝的控
制拉应力出现在上游坝底的梁向，控制压应力却出现在下游坝底的梁向。
4．研究了不同水位有缝坝在常规地震输入情况下横缝的开合规律和应力反
应特点。无论运行低水位还是正常蓄水位，横缝张开最深最宽的是上游
坝面中缝，张开宽度从中缝到边缝依次减小；正常蓄水位时缝的张开宽
度明显小于运行低水位时缝的张开宽度。从控制裂缝张开深度和宽度的
角度而言，低水位是控制工况；而从坝体应力的角度考虑，高水位是控
制工况，
5．深入研究了整体坝在常规输入和行波输入两种地震输人方式下的动力反
四川大学工学博士学位论文
应。行波横河向输入时顺河向位移反应在拱冠梁处有所增大，其它点有
所削弱：行波J’顷河向输入时非拱冠梁节点的横河向位移反应有所削弱。
常规输入时，顺河向输入较横河向输入的应力反应大，说明常规输入时
的控制方向是顺河向。但在行波输入时，行波横河向输入的应力反应比
行波顺河向输入时大，说明行波输入的控制方向是横河向。
6．在国内外首次进行了分缝坝行波输入的抗震分析，研编了计算程序，深
入研究了常规输入与行波输入两种地震输入方式对有缝高拱坝地震响应
的影响。行波顺河向输入时缝的张汗宽度普遍大于常规顺河向输入时的
张开?
With the development of economy, Many arch dams such as Xiao wan, Xi
Luodu and so on whose height are more than 300m are going to be constructed in
westem regions of China. The arch dams locate at such areas which have high
moUnain, deep valley, steep slope, narrow riverway and high seismic intensity.
The aseismic capabilities of high arch dams have relationship with lives and
properties of people and productions of industry and agriculture, so the eanhquake
becomes the main factor of arch darns design. Arch dams are built by parts, and
each of parts has vertical contraction joints spaced regular intervals. The dynamic
tensile stresses of earthquake may balance or exceed to the precompression
stresses due to static watef, this may arouse opening of joints along the normal
direction. If the altemate actions of earthquake join, the joints may behave gradual
opening and closing. The effect of opening and closing of joints will weaken the
integral stiffness of arch dams, prolong their vibration periods and remarkably
readjusting of stresses. In general, the nonlinear character of joints is the
significant factor of the analysis of high arch dams seismic response. Reasonably
taking into account nonlinear character of dynarnic model has very important
significance of theory and engineering.
Nonlinear dynamic analysis of high arch dams is carried out considering the
opening of joints and the effect of traveling wave with engineering background of
Xi Luodu arch dam whose height is 278 meteL Based on the theory of interaction
of fluid-solid, the coupled FEM equation considering compressible water is
deduced. In order to reduce the equation complexity and adapt to complicaied
nonIinear ana1ysis of high arch dams with joints, finite element treatments are
used with incompressibIe water and massless foundation. The added mass of
water and added stiffness of foundation are respectively used to substitute the
influence of water and fOundations. The efficiency in the computations is
improved by means of static condensation. Fenves' 3-D joint element modeI and
its constitutive relations are demonstrated and used to simulate the opening,
closing and sliding of joints,and the initial pressure and tensile strength at the
interface. In order to control the propagation of errors and enhance the accuracy of
numerical solution, the wilson- 0 method is modified and the numerical stability
of modified wilson- e method is analyzed. Because of need of much iteration in
each time step while carrying on nonlinear numerical anaIysis, the incremefltal
equilibrium formulation and its equivalent iterative equation of the constant
average acceleration Newnark method are deduced. The theory of traveling wave
input is discussed, and the dynatnic response of linear whole dams is analyzed by
use of single-direction traveling wave model. The difference and commonness
between cross-direction and longitude direction traveling wave that exists in the
response and distribution of dam displacement and stress are obtained. The
dynamic seismic response of Xi Luodu dam is studied. The difference and
commonLness between dams with joints and without are analyzed. The magnitude
and distribution of acceleration and stress of dams with joint and without possess
many differences. The influence of different water depth on seismic response of
dams with joint is studied, the conclusion in term of opening width, opening depth
and opening frequency is attained. The characteristics about the magnitude and
distribution of stress, the width and depth of joint and time sequence of joint
opening are compared on condition of different traveling wave directions. The
difference between general unifOrm wave input and traveling wave inPut is also
compared. Though the above analysis, some conclusion are drew as followings f
l. The computational results show that the assumptions of incompressible
water and massless fOundation can considerably simplify the complexity of
equation
引文
1．高拱坝抗震结构工程措施研究报告(九五攻关)，中国水利水电科学研究院，1999．10
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28．李南生，周晶，林皋，有横缝高拱坝的动力接触模型及其数值解[J]，水利学报，1999 (12)：17-22
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