ABAQUS培训讲义教案Lecture 3 nonlinearity

ABAQUS培训讲义教案Lecture 3 nonlinearity CopyrightxxABAQUS,Inc.ABAQUS/S tandard中的非线性分析第三讲Introduction to ABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.概述?非线性结构力学?求解平衡方程?非线性输入文件?非线性悬臂梁分析的输出CopyrightxxABAQUS,Inc.非线性结构力学Introduction to ABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.非线性结构力学一些材料非线性的例子?非线性的?材料非线性?非线性弹性?塑性?材料损伤?失效机制?等等。 ?注意如果预定义了温度或场变量，则材料与温度或场变量的相关性不引入非线性。 Introduction to ABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.非线性结构力学自接触的例子例子手册1.1.16，压缩减震器?边界条件非线性?接触问题?在分析过程中边界条件变化。 ?严重不连续形式的非线性。 Video ClipIntroduction to ABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.非线性结构力学?几何非线性?大绕度和大变形?大旋转?结构不稳定（屈曲）?预载荷效应几何非线性例子聚合物键盘罩Video ClipCopyrightxxABAQUS,Inc.求解平衡方程Introduction to ABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程?典型的非线性问题具有所有三种形式的非线性。 ?在方程中必须包括非线性项。 ?一般的，每个自由度的非线性方程是耦合的。 ?静态平衡的基本表达式为由单元应力引起的加在节点上的内力，I，与外力，P，必须平衡，即0)()(=?u I u P(Eq.3.1)Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程?为了求解非线性平衡问题，在ABAQUS/Standard中使用基于牛顿-拉普森技术的增量迭代法。 ?假定前一步载荷增量的解，u0，为已知。 ?假定在第i次迭代之后，得到近似解u i。 设c i+1为离散平衡方程的精确解和当前解之差（Equation3.1），所以有?将方程3.2的左手边在近似解u i附近，以泰勒级数方式展开，可以得到(Eq.3.2)(Eq.3.3).0)()(11=+?+i i i ic u IcuP1()()()().0i ii i iPu I uP uIucu u+?+?+=?.Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.?略去高阶项，方程可以写为其中为切线刚度。 ?解的下一次近似为?注意如果载荷与位移相关（比如，旋转的表面压力），刚度矩阵中包含载荷刚度的贡献。 求解平衡方程,)()(1iii iuIuP c K?=+uu PuuIKi ii?=)()(.11+=iiic uuIntroduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程?网格中的静力平衡1.施加规定的“载荷增量”。 2.迭代，直到每个节点的所有节点力的和非常小。 3.满足平衡后，更新模型状态。 4.回到第1步，施加下一个载荷增量。 Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.?单自由度例子?非线性弹簧?求解u(P)或P(u)典型的，在分析步中，载荷从0递增到P FINAL。 ?时间经常从0变到1。 求解平衡方程Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程?牛顿-拉普森求解技术?代第一次迭代(i=1)?假定前面收敛增量步的解u0，P0，为已知的。 ?在当前增量步中，将一个小的增量P，载荷施加到结构上。 ?ABAQUS基于u0处的切线刚度K0确定位移修正c1；前一增量步结束时，总载荷P TOTAL和内力间的关系为.:0100I Pc KITOTAL?=Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.?ABAQUS更新模型的状态为u1，形成K1并计算I1。 ?总载荷P TOTAL与内力I1的差称为残差，R1:R1=P TOTAL?I1.?如果R1在模型的每个自由度上都非常小(在容差范围之内)，结构就是平衡的。 ?默认的容差R1必须小于结构对时间平均力的0.5%。 ?ABAQUS自动计算时间平均力。 ?如果迭代不能得到收敛的解，ABAQUS执行另外的迭代，以找到收敛的解。 求解平衡方程Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.?代第二次迭代(i=2)?基于更新的刚度K1，计算新的位移纠正c2，并且?把新的残差R2与容差进行比较，察看在u2处是否得到收敛解，。 求解平衡方程.1211:I PcKITOTAL?=Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.求解平衡方程?该过程将一直重复，直到力的残差在允许的容差之内。 每次迭代i需要1.形成切线刚度K i。 2.求解系统方程组，得到位移修正c i+1。 ?修正位移的估计值u i+1=u i+c i+1。 3.基于u i+1计算内力向量I i+1。 4.进行平衡收敛判断?是否R i+1在容差之内?是否#11iteri jj+=P05.000E-03SOFT CONTACTCONSTRAINT PATIBILITYTOLERANCEFORP=0.00.100DISPLACEMENT PATIBILITYTOLERANCEFORDCOUP ELEMENTS1.000E-05ROTATION PATIBILITYTOLERANCEFORDCOUP ELEMENTS1.000E-05非线性悬臂梁分析的输出Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.TIME INCREMENTATIONCONTROL PARAMETERS:FIRST EQUILIBRIUM ITERATION FORCONSECUTIVE DIVERGENCECHECK4EQUILIBRIUM ITERATIONAT WHICHLOG.CONVERGENCE RATECHECK BEGINS8EQUILIBRIUM ITERATIONAFTER WHICHALTERNATE RESIDUALIS USED9MAXIMUM EQUILIBRIUM ITERATIONS ALLOWED16EQUILIBRIUM ITERATIONCOUNT FORCUT-BACK INNEXT INCREMENT10MAXIMUM EQUILIB.ITERS INTWO INCREMENTSFOR TIME INCREMENT INCREASE4MAXIMUM ITERATIONSFOR SEVEREDISCONTINUITIES12MAXIMUM CUT-BACKS ALLOWEDIN ANINCREMENT5MAXIMUM DISCON.ITERS INTWO INCREMENTSFOR TIME INCREMENT INCREASE6CUT-BACK FACTORAFTER DIVERGENCE0.2500CUT-BACK FACTORFOR TOOSLOW CONVERGENCE0.5000CUT-BACK FACTORAFTER TOOMANY EQUILIBRIUM ITERATIONS0.7500CUT-BACK FACTORAFTER TOOMANY SEVEREDISCONTINUITY ITERATIONS0.2500CUT-BACK FACTORAFTER PROBLEMSIN ELEMENTASSEMBLY0.2500INCREASE FACTORAFTER TWOINCREMENTS THATCONVERGE QUICKLY1.500MAX.TIME INCREMENTINCREASE FACTORALLOWED1.500MAX.TIME INCREMENTINCREASE FACTORALLOWED(DYNAMICS)1.250MAX.TIME INCREMENTINCREASE FACTORALLOWED(DIFFUSION)2.000MINIMUMTIME INCREMENT RATIOFOR EXTRAPOLATIONTO OCCUR0.1000MAX.RATIO OFTIME INCREMENTTO STABILITY LIMIT1.000FRACTION OFSTABILITYLIMITFOR NEWTIME INCREMENT0.9500非线性悬臂梁分析的输出Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.PRINT OFINCREMENT NUMBER,TIME,ETC.,EVERY1INCREMENTSRESTART FILEWILL BEWRITTEN EVERY1INCREMENTSTHE MAXIMUMNUMBER OFINCREMENTS INTHIS STEPIS25LARGE DISPLACEMENTTHEORY WILL BE USEDEXTRAPOLATIONWILLBEUSEDCHARACTERISTIC ELEMENTLENGTH40.0PRINT OFINCREMENT NUMBER,TIME,ETC.,TO THEMESSAGE FILEEVERY1INCREMENTS非线性悬臂梁分析的输出Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.非线性悬臂梁分析的输出INCREMENT1STARTS.ATTEMPT NUMBER1,TIME INCREMENT0.100EQUILIBRIUM ITERATION1AVERAGE FORCE3.457E+03TIME AVG.FORCE3.457E+03LARGEST RESIDUALFORCE-1.288E+04AT NODE11DOF1LARGEST INCREMENTOF DISP.-3.07AT NODE11DOF2LARGEST CORRECTIONTO DISP.-3.07AT NODE11DOF2FORCE EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGE MOMENT1.200E+04TIME AVG.MOMENT1.200E+04LARGEST RESIDUALMOMENT133.AT NODE9DOF6LARGEST INCREMENTOF ROTATION-2.304E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION-2.304E-02AT NODE11DOF6MOMENT EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.EQUILIBRIUM ITERATION2AVERAGE FORCE63.2TIME AVG.FORCE63.2LARGEST RESIDUALFORCE-1.66AT NODE11DOF1LARGEST INCREMENTOF DISP.-3.07AT NODE11DOF2LARGEST CORRECTIONTO DISP.-2.797E-02AT NODE11DOF2FORCE EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGE MOMENT1.200E+04TIME AVG.MOMENT1.200E+04LARGEST RESIDUALMOMENT-1.59AT NODE5DOF6LARGEST INCREMENTOF ROTATION-2.304E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION2.998E-06AT NODE11DOF6THE MOMENT EQUILIBRIUM EQUATIONSHAVE CONVERGEDX0.00517.3X0.0050.3X0.00560X0.00560Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.EQUILIBRIUM ITERATION3AVERAGE FORCE62.9TIME AVG.FORCE62.9LARGEST RESIDUALFORCE-4.223E-05AT NODE11DOF1LARGEST INCREMENTOF DISP.-3.07AT NODE11DOF2LARGEST CORRECTIONTO DISP.1.338E-04AT NODE11DOF2THE FORCE EQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERAGE MOMENT1.200E+04TIME AVG.MOMENT1.200E+04LARGEST RESIDUALMOMENT-1.950E-04AT NODE7DOF6LARGEST INCREMENTOF ROTATION-2.304E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION1.415E-06AT NODE11DOF6THE MOMENT EQUILIBRIUM EQUATIONSHAVE CONVERGEDITERATION SUMMARY FOR THE INCREMENT:3TOTAL ITERATIONS,OF WHICH0ARE SEVEREDISCONTINUITY ITERATIONSAND3ARE EQUILIBRIUM ITERATIONS.TIME INCREMENTPLETED0.100,FRACTION OFSTEP PLETED0.100STEP TIMEPLETED0.100,TOTAL TIMEPLETED0.100非线性悬臂梁分析的输出4次或更少次数的迭代（如果这样的情况再次出现，t可以增加）X0.0050.3X0.00560Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.INCREMENT2STARTS.ATTEMPT NUMBER1,TIME INCREMENT0.100EQUILIBRIUM ITERATION1AVERAGE FORCE127.TIME AVG.FORCE95.0LARGEST RESIDUALFORCE6.63AT NODE11DOF1LARGEST INCREMENTOF DISP.-3.06AT NODE11DOF2LARGEST CORRECTIONTO DISP.-5.585E-02AT NODE11DOF1FORCE EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGE MOMENT2.399E+04TIME AVG.MOMENT1.799E+04LARGEST RESIDUALMOMENT-6.30AT NODE5DOF6LARGEST INCREMENTOF ROTATION-2.302E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION1.768E-05AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGED非线性悬臂梁分析的输出没有增加Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.EQUILIBRIUM ITERATION2AVERAGE FORCE126.TIME AVG.FORCE94.3LARGEST RESIDUALFORCE-1.752E-03AT NODE11DOF1LARGEST INCREMENTOF DISP.-3.06AT NODE11DOF2LARGEST CORRECTIONTO DISP.8.349E-04AT NODE11DOF2THE FORCE EQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERAGE MOMENT2.398E+04TIME AVG.MOMENT1.799E+04LARGEST RESIDUALMOMENT1.554E-03AT NODE7DOF6LARGEST INCREMENTOF ROTATION-2.301E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION8.762E-06AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGEDTIME INCREMENT MAYNOW INCREASETO0.150ITERATION SUMMARY FOR THE INCREMENT:2TOTAL ITERATIONS,OF WHICH0ARE SEVEREDISCONTINUITY ITERATIONSAND2ARE EQUILIBRIUM ITERATIONS.TIME INCREMENTPLETED0.100,FRACTION OFSTEP PLETED0.200STEP TIMEPLETED0.200,TOTAL TIMEPLETED0.200非线性悬臂梁分析的输出4个连续增量步的迭代次数等于或小于4t=1.5t oldIntroduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.INCREMENT3STARTS.ATTEMPT NUMBER1,TIME INCREMENT0.150EQUILIBRIUM ITERATION1AVERAGE FORCE224.TIME AVG.FORCE138.LARGEST RESIDUALFORCE23.1AT NODE11DOF1LARGEST INCREMENTOF DISP.-4.58AT NODE11DOF2LARGEST CORRECTIONTO DISP.-0.104AT NODE11DOF1FORCE EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGE MOMENT4.192E+04TIME AVG.MOMENT2.597E+04LARGEST RESIDUALMOMENT-20.1AT NODE5DOF6LARGEST INCREMENTOF ROTATION-3.444E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION7.751E-05AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGED非线性悬臂梁分析的输出t=1.5t oldIntroduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.EQUILIBRIUM ITERATION2AVERAGE FORCE220.TIME AVG.FORCE136.LARGEST RESIDUALFORCE-1.994E-02AT NODE11DOF1LARGEST INCREMENTOF DISP.-4.57AT NODE11DOF2LARGEST CORRECTIONTO DISP.2.805E-03AT NODE11DOF2THE FORCE EQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERAGE MOMENT4.190E+04TIME AVG.MOMENT2.596E+04LARGEST RESIDUALMOMENT9.413E-03AT NODE7DOF6LARGEST INCREMENTOF ROTATION-3.441E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION2.937E-05AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGEDTIME INCREMENT MAYNOW INCREASETO0.225ITERATION SUMMARYFOR THEINCREMENT:2TOTAL ITERATIONS,OF WHICH0ARE SEVEREDISCONTINUITY ITERATIONSAND2ARE EQUILIBRIUMITERATIONS.TIME INCREMENTPLETED0.150,FRACTION OFSTEP PLETED0.350STEP TIMEPLETED0.350,TOTAL TIMEPLETED0.350RESTART INFORMATIONWRITTEN INSTEP1AFTER INCREMENT3非线性悬臂梁分析的输出4or fewerIntroduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.INCREMENT4STARTS.ATTEMPT NUMBER1,TIMEINCREMENT0.225EQUILIBRIUMITERATION1AVERAGE FORCE382.TIME AVG.FORCE198.LARGEST RESIDUALFORCE114.AT NODE11DOF1LARGEST INCREMENTOF DISP.-6.80AT NODE11DOF2LARGEST CORRECTIONTO DISP.-0.229AT NODE11DOF1FORCE EQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGE MOMENT6.863E+04TIME AVG.MOMENT3.663E+04LARGEST RESIDUALMOMENT-69.7AT NODE5DOF6LARGEST INCREMENTOF ROTATION-5.132E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION2.932E-04AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGED非线性悬臂梁分析的输出t=1.5t oldIntroduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.EQUILIBRIUMITERATION2AVERAGE FORCE361.TIME AVG.FORCE193.LARGEST RESIDUALFORCE-0.324AT NODE11DOF1LARGEST INCREMENTOF DISP.-6.79AT NODE11DOF2LARGEST CORRECTIONTO DISP.1.145E-02AT NODE11DOF2THE FORCEEQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERAGEMOMENT6.857E+04TIME AVG.MOMENT3.661E+04LARGEST RESIDUALMOMENT7.458E-02AT NODE7DOF6LARGEST INCREMENTOF ROTATION-5.120E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION1.183E-04AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGEDTIMEINCREMENT MAYNOW INCREASETO0.338ITERATION SUMMARYFOR THEINCREMENT:2TOTAL ITERATIONS,OF WHICH0ARE SEVEREDISCONTINUITY ITERATIONSAND2ARE EQUILIBRIUMITERATIONS.TIMEINCREMENTPLETED0.225,FRACTION OFSTEP PLETED0.575STEP TIMEPLETED0.575,TOTAL TIMEPLETED0.575非线性悬臂梁分析的输出Introduction toABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.INCREMENT5STARTS.ATTEMPT NUMBER1,TIMEINCREMENT0.338EQUILIBRIUMITERATION1AVERAGE FORCE663.TIME AVG.FORCE287.LARGEST RESIDUALFORCE544.AT NODE11DOF1LARGEST INCREMENTOF DISP.-10.0AT NODE11DOF2LARGEST CORRECTIONTO DISP.-0.494AT NODE11DOF1FORCEEQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGEMOMENT1.081E+05TIME AVG.MOMENT5.090E+04LARGEST RESIDUALMOMENT-215.AT NODE5DOF6LARGEST INCREMENTOF ROTATION-7.582E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION9.779E-04AT NODE11DOF6ESTIMATE OF ROTATION CORRECTION-1.931E-05MOMENTEQUILIB.ACCEPTED BASEDON SMALLRESIDUAL ANDESTIMATED CORRECTIONEQUILIBRIUMITERATION2AVERAGE FORCE575.TIME AVG.FORCE269.LARGEST RESIDUALFORCE-5.83AT NODE11DOF1LARGEST INCREMENTOF DISP.-9.95AT NODE11DOF2LARGEST CORRECTIONTO DISP.4.994E-02AT NODE11DOF2FORCEEQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGEMOMENT1.078E+05TIME AVG.MOMENT5.085E+04LARGEST RESIDUALMOMENT0.523AT NODE7DOF6LARGEST INCREMENTOF ROTATION-7.532E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION5.044E-04AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGED非线性悬臂梁分析的输出t=1.5t old残余应力在容差之内，但是转动位移纠正太大。 估计下次迭代的转动位移纠正会比较小。 IntroductiontoABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.EQUILIBRIUMITERATION3AVERAGE FORCE574.TIME AVG.FORCE269.LARGEST RESIDUALFORCE-1.901E-02AT NODE11DOF1LARGEST INCREMENTOF DISP.-9.95AT NODE11DOF2LARGEST CORRECTIONTO DISP.-3.270E-03AT NODE11DOF2THE FORCEEQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERAGEMOMENT1.078E+05TIME AVG.MOMENT5.085E+04LARGEST RESIDUALMOMENT-6.545E-03AT NODE7DOF6LARGEST INCREMENTOF ROTATION-7.534E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION-2.841E-05AT NODE11DOF6THE MOMENTEQUILIBRIUM EQUATIONSHAVE CONVERGEDITERATIONSUMMARYFORTHEINCREMENT:3TOTAL ITERATIONS,OF WHICH0ARE SEVEREDISCONTINUITY ITERATIONSAND3ARE EQUILIBRIUMITERATIONS.TIMEINCREMENTPLETED0.338,FRACTION OFSTEP PLETED0.913STEP TIMEPLETED0.913,TOTAL TIMEPLETED0.913非线性悬臂梁分析的输出IntroductiontoABAQUS/Standard andABAQUS/ExplicitCopyrightxxABAQUS,Inc.INCREMENT6STARTS.ATTEMPT NUMBER1,TIMEINCREMENT8.750E-02EQUILIBRIUMITERATION1AVERAGE FORCE632.TIME AVG.FORCE329.LARGEST RESIDUALFORCE18.1AT NODE11DOF1LARGEST INCREMENTOF DISP.-2.52AT NODE11DOF2LARGEST CORRECTIONTO DISP.-8.954E-02AT NODE11DOF1FORCEEQUILIBRIUMNOT ACHIEVEDWITHIN TOLERANCE.AVERAGEMOMENT1.178E+05TIME AVG.MOMENT6.201E+04LARGEST RESIDUALMOMENT-52.2AT NODE5DOF6LARGEST INCREMENTOF ROTATION-1.923E-02AT NODE11DOF6LARGEST CORRECTIONTO ROTATION3.032E-04AT NODE11DOF6ESTIMATE OFROTATION CORRECTION-1.367E-05MOMENTEQUILIB.ACCEPTED BASEDON SMALLRESIDUAL ANDESTIMATED CORRECTIONEQUILIBRIUMITERATION2AVERAGE FORCE629.TIME AVG.FORCE329.LARGEST RESIDUALFORCE-0.107AT NODE11DOF1LARGEST INCREMENTOF DISP.-2.52AT NODE11DOF2LARGEST CORRECTIONTO DISP.6.306E-03AT NODE11DOF2THE FORCEEQUILIBRIUM EQUATIONSHAVE CONVERGEDAVERA