外文翻译译文

丹江口重力坝后期加高的安全稳定性研究

周伟，常晓林，卢淑媛

（武汉大学水资源与水利水电学院，中国，湖北，武汉430072）

摘要：通过采用以传统库仑摩擦模型和相应的接触元件为基础的一种大位移改善接触摩擦模型，本文对丹江口重力坝后期加高的新老混凝土结合面进行接触分析，以便研究开裂后大坝的稳定和变形状态。计算结果表明，接触面的摩擦应力和正应力的分布及大小符合一般规律，沿坝基面的抗剪断安全系数也满足重力坝设计规范要求。

关键词：安全稳定性研究;接触模型;增广拉格朗日乘子法。

1.概述

为了满足南水北调中线工程，丹江口（DJK）重力坝从162米加高至177.6米。通过温度场的模拟，丹江口重力坝新老混凝土结合面将会产生厚度变化的裂缝。由于温度变化裂缝必然会产生在结合面。开裂后，研究大坝稳定性和变形状态，本文假定大坝建设后有缝隙，然后采用以增广拉格朗日乘子法为基础的大位移改善接触摩擦模型进行新老混凝土的接触分析。只有在研究了不同条件组合下大坝的整个安全能力和变形状态，设计师才能明白如何将丹江口重力坝新老混凝土接触面坚固地结合，以及是否有足够的安全能力，以抵抗增加的水的反作用力。

2接触模型

2.1接触问题综述

接触问题是高度非线性的，需要有效的计算机资源来解决。我们理解这个问题的物理性和花时间去建立尽可能高效运行的实际模型是非常重要的。接触问题呈现了两个显著的困难。首先，我们通常不知道接触的区域，直到我们已经遇到问题。根据载荷，材料，边界条件和其他因素的影响，结合面可以彼此渗入和分

离以不可预知的和突然的方式。然后，大多数接触问题需要考虑摩擦。有几种摩擦法和模型可供选择，并且都是非线性的。摩擦反应是无规律的，使得求解收敛困难。为了模拟一个接触问题，我们首先需要确定其可能的相互作用的部分进行分析。如果该相互作用中的一部分是在一个点上，模型对应的部件是一个节点。

如果该相互作用中的一部分是在一个表面上，模型中的对应的部件是一个元素：一个梁，壳，或固态元件。本文只讨论面面接触问题和模型，其中有限元模型通过特定接触元素的存在来识别可能的接触对。这些接触元件被叠加在正被用于分析相互作用模型的各个部分。

图一 简化摩擦模型 2.2库仑摩擦模型

在基本的库仑摩擦模型中，其结合面彼此开始滑动前，两个接触面可以承受剪应力在一定程度的升高。该状态被称为粘滞。库仑摩擦模型（见图 1 ）被定义为：

lim

lim p ττμτ≤?= (1)

其中lim τ是极限剪切应力，μ为摩擦系数，τ是等效的剪应力， p 为接触常压。 一旦等效剪应力超过lim τ ，结合面和目标表面就会相对于彼此滑动。这 状态被称为滑动。粘滞/滑动计算决定从粘滞到滑动的过渡点，反之亦然。

2.3改进库仑摩擦应力模型

摩擦是一个复杂的物理现象，它涉及到接触面的特性如表面粗糙度，相对速 度，正应力，甚至有时还与温度有关。摩擦的实际物理性仍然是一个研究课题。因此，摩擦的数值模拟已经被简化为两种理想化模型。

最流行的摩擦模型是粘着摩擦或库仑摩擦应力模型。本文采用了一种在传统

的库仑摩擦模型基础上改进的接触摩擦模型[]1，该模型是： t arctan 2r n

???

? ??=

RV υπ

μστ (2)

其中n σ是正应力，τ是切向（摩擦）力，μ是摩擦系数， RV 被理想化的滑动速度，t 是在相对速度的方向上的切向矢量 r

r

t V V =

(3) r V 是相对滑动速度。

物理上，RV 值是滑动发生时的相对速度值。在决定数学模型如何密切地表示阶梯函数时，RV 值很重要。 一个非常大的RV 值会导致有效摩擦的作用降低。一个非常小的RV 值则会导致收敛不佳。建议的RV 值应为典型相对滑动速度的1 ％或10 ％。正因为如此平滑的过程，接触节点总是有一些打滑。摩擦模型的反正切表示这个非线性摩擦行为的数学理想化，这使得解决方案容易收敛。

2.4接触元素

可以使用非线性弹簧单元[]2进行改善接触摩擦模型的实施。注意一个非线性弹簧的行为，如图2所示，由下式给出：

??

?

???=??????????

??--2121u u F F K K K K

(4) 其中K 为弹簧刚度,2112u u F F 和，，是点1和2的位移和力，在速度方面的等价式很容易理解

??

?

???=??????????

??--t 1t 121F F K K K K

νν (5)

图二 弹簧元件

由于K 是相对速度的一个非线性函数， 方程5被逐步得到解决，其中，每

个增量内的迭代次数可能是必要的。对于一个典型的迭代i ，则方程解决后的形式

??

?

?????=????????????--i t i t i i i i

i i

F F K K K K 2121δνδν (6) 2121ννδνδν和被用来替代和其中i

i

i i i

i i i 2

12

2

1111δν

ν

νδννν+=+=-- (7)

注意，1

211--i i νν和对应于迭代的开始。对于可变形的刚性接触，很容易看到： 02=i δν (8)

在静态分析中，速度必须根据位移和时间增量计算，将时间增量表示为t ?，我们可以这样写：

t

u i

i

?=

11

δδν （9）

其中i u 1δ表示位移增量i u 1?对于迭代i 的修正（另见公式7） ：

δ+?=?-1

11i i u u

（10)将式8和式9代入到方程6中得到

i i

i F u K t

111?=?δ (11)通过考虑先前增量结束时的速度，方程ll 可以得到改进，来得到第一次迭代的增量。然后方程9可以改写为： p 111

11

νδδν—t

u ?= （12）

因此方程11可以修改为： )(12111

11p p r K F u K t

ννδ-?=?—） （13）

对于后续的迭代将调整到方程11 。

在方程13中，p r ν表示在以前的增量结束时，点1和2之间的相对速度。必须注意的是，如果所述刚性体的速度与相对速度差别极大，那么方程13的附加

术语就特别重要。

3基于增广拉格朗日乘子法的接触算法

对于这种面面接触的问题，它可以用增广拉格朗日乘子法或者用罚方法。

增广拉格朗日乘子法是一系列罚参数迭代的更新，以找到准确的拉格朗日乘子（即接触力）。相较于罚方法，增广拉格朗日方法通常会导致更好的调节，并且对接触罚参数的大小不太敏感[]3。以下内容是以增广拉格朗日乘子法为基础而进行的讨论。

一般的接触可以被定义为点A 和B 的限制位移 ：

TOL n u u B A

其中，A 在一个主体，B 在另一个主体， n 是两个点之间一个矢量的方向余弦，TOL 是闭合距离（图3）。在拉格朗日乘子过程中，变分方法是通过约束增广的[]4。虚功原理产生了传统的系统方程

f u K =? （15） 约束条件可以表示为：

0=?u C （16） 通过对增广功能的最小化：

图三 可能接触体之间的正常间隙

u C f u u K u T T T

??+-?=λψ2

1 （17） 我们得到

???

???=????????????00f u C C K T λ

（18） 其中K 是刚度矩阵，C 是约束矩阵，f 是节点力。方程8可以同时解决位移（u ）

和拉格朗日乘子（ ）。

4丹江口重力坝后期加高的接触分析

4.1计算条件

图4给出了丹江口重力坝的几何尺寸和接触界面的位置。原来坝的下游坡度是1:0.75 ；在海拔150米上部的后期坝下游坡度是1:0.35，而在海拔150米下部的下游坡度则为1:0.6 。表1给出了坝体和基础材料的参数值。在研究中，考虑水压力（水位），沙压力，坝体自重等荷载。

表1 坝体和基础材料参数

参数弹性模量泊松比内聚力摩擦角（度）原来部分混凝土25 0.167 2.5 45

加高部分混凝土20 0.167 2.5 45 基岩21 0.22 0.9 41

4.2计算案例

由于温度变化的影响，开裂必然发生在接触界面，为了研究开裂后大坝的稳定性和变形状态，本文假定大坝建成有缝隙后，缝隙的宽度变化从0.0mm至1.1mm。本文的详细假设情况介绍如下：

（1）工况一：假设大坝建成后高程为177.6米且无裂缝，上游水位为172.2米；（2）工况二：假设大坝已经开裂，垂直方向开裂宽度为1.1mm ，斜向开裂宽度0.9mm，上游水位高程为172.2米;

（3）工况三：假设大坝已经开裂，开裂的宽度是所有0.25米在洗澡的垂直缝，斜缝，上游水位为172.2米。

4.3有限元法计算模型

该模型采用块体平面中4个节点元素和上述接触元素。图5给出了大坝有限元法（FEM）的网格。坝节点的总数为2650；元素总数为2732。

图四 DJK重力坝模型图五 DJK重力坝有限元网格

4.4计算结果分析

σ和水平应通过有限元计算，表2和表3分别给出了基础界面的垂向应力

y

σ，表4给出了沿坝基面的抗剪断安全系数，而图6和图7分别给出了结合力

x

面的摩擦应力和正应力分布。此外，为了与有限元计算对比，本文还利用刚体极限平衡法（RBLEM）计算了沿坝基面的抗剪断安全系数。

计算结果表明，在接触面上的摩擦应力和正应力的分布与大小符合一般规律，从表4可以看出，利用有限元法得到的沿坝基面的抗剪断安全系数也大于3.0 ，这满足“重力坝设计规范”的要求，就刚体极限平衡法的结果而言，它表明了有限元的结果是可信的，因为两种方法都在理论上相等的。

σ /MPa

表2 基础界面的垂向应力

y

情形坝踵前期坝趾后期坝趾

情形一-0.364 -0.989 -1.410

情形二-0.211 -1.130 -2.040

情形三-0.200 -0.070 -1.150

注意：“+”表示拉应力。

/MPa

表3 基础界面的水平应力

x

情形坝踵前期坝趾后期坝趾

情形一0.052 -0.628 -1.060

情形二-0.025 -0.556 -0.860

情形三0.139 -0.473 -0.911

注意：“+”表示拉应力。

表4 沿坝基面的抗剪断安全系数

情形整个坝基面前期坝基面后期坝基面

情形一 4.24 ————

情形二 4.64 5.79 2.89

情形三 4.81 6.20 2.97

图六结合面摩擦应力/Pa 图七结合面正应力/Pa

5结语

本文采用以传统库仑摩擦模型和相应的接触元件为基础的一种大位移的改善接触摩擦模型，同时还给出了以增广拉格朗日乘子法为基础的接触算法。通过

运用该模型对丹江口重力坝新老混凝土结合面进行接触分析，归纳得出以下结论：

（1）与传统的接触模型相比，改进的接触摩擦模型可以处理大位移的接触问题。（2）接触界面上的摩擦应力和正应力的分布及大小满足一般规律。

（3）在初始大坝顶部的下游部位有一小部分塑性屈服区域，这非常紧密地符合长江科学研究院进行现场试验的结果。

（4）利用有限元法得到的沿坝基面的抗剪断安全系数都大于3.0 ，这满足重力坝设计规范的要求。

（ 5 ）刚体极限平衡法和有限元法的结果是非常相似的，这表明有限元分析结果是可信的。

参考文献

1Peric D和Owen D R J，基于罚方法的3D接触摩擦问题的计算模型，工程数值方法国际期刊，1992，35:1289-1309

2 Cescotto,S和Charilier R，基于混合变分原则的摩擦接触有限元，工程数值方法国际期刊,1992,36:1681-1701

3 Simo J C和laursen T A，接触摩擦问题的一种增广拉格朗日处理方法，电脑与结构，1992,42（1）:97-116

4 Laursen TA和Simo J C ，使用增广拉格朗日的库仑摩擦问题均衡算法，计算机方法在力学与工程系的应用，1993,108（1＆2）:133-146

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