法律资料6渐近法

结构力学

Structuralmechanics

渐近法successiveapproximationmethod§6-1引言foreword渐近法Chapter11successiveapproximationmethod逐次逼近approachgradually→收敛convergence→精确解accuratesolution特点：characteristic①不组成求解方程

notformequation生动形象，重复步骤，易于手算

vivid,repeatedstep,easytomanualcalculation渐近法Successiveapproximationmethod力矩分配法momentdistributionmethod无剪力分配法no-sheardistributionmethod迭代法iterationmethod§11-2力矩分配法的根本原理

basicprincipleofmomentdistributionmethod一、根本概念basicconcept1、劲度系数(转动刚度)S:stiffnesscoefficient(rotationstiffness)——近端〔转动端〕仅发生单位转角时，引起的近端杆端弯矩。Bar-endmomentofnearendwhenonlyunitangleofrotationappearsinnearend(rotatingend)AAAiABiBAiBA2、传递系数C：carry-overcoefficient——当近端有转角时，远端弯距与近端弯距的比值。theratiooffar-endmomentandnear-endmomentwhennearendhasangleofrotation

AAAiABiBAiBA远

端

约

束far-endrestrictionSC固

定fix0.5铰

支hinge0滑

动slip-1劲度系数S与传递系数C：stiffnesscoefficientandcarry-outcoefficient3、节点不平衡力矩：unbalancemomentofjoint4、分配系数：distributioncoefficient

5、分配弯矩：distributionmoment

6、传递弯矩：carry-outmoment

(CoupleMactedatjointAisdistributedtonearendofeachmemberbydistributioncoefficient,andfar-endmomentisequaltothatnear-endmomentmultipliescarry-overcoefficient)

i14—节点转动

momentofZ1anglewhenjointrotates角产生的弯矩Z1M12m于是得(so)

近端弯矩可写为：near-endmomentwriteas

——分配系数distributioncoefficient分配弯矩distributionmoment固端弯矩fix-endmoment—FM12®将不平衡力矩变号后按劲度系数大小的比例分配给各近端reversetheunbalancemomentanddistributetoeverynear-endbytheproportionofstiffnesscoefficient—1FjMåFMM1212+=mFFjMMM1211212)(+-=åm远端弯矩：far-endmoment

同理(assameas)：同理(assameas)：

distributeeverynear-endmomenttoeveryfar-endbytheratioofcarry-outcoefficient归纳:conclusion

fixjoint,everybarhasfix-endbendingmomentandunbalancemomentofjointrelaxjoint,distributebendingmoment,carry-outbendingmoment二、计算步骤：calculationsteps例(e.g.)计算步骤结点BAD

杆端BAABACADDA

1求各杆分配系数/1/21/61/3/

2求固端弯矩0400-75-253力矩分配与传递035/235/635/3-35/34计算最后杆端弯矩057.55.8-63.3-36.7结点A不平衡力矩为：

总结：conclusion

应用条件：applicationcondition只有角位移的刚架和连续梁rigidframeandbeamonlywithangulardisplacement

三概念：threeconcepts刚度系数、分配系数、传递系数stiffness、distribution、carry-outcoefficient

符号规定：signstipulation单结点力矩分配法根本原理：basicprincipleofmomentdistributionmethod加刚臂——去刚臂——叠加addrigidarm——removerigidarm——superposition步骤:step1、计算固端弯距，不平衡力矩calculatefix-endbendingmoment,unbalancemoment2、计算分配系数、传递系数calculatedistributioncoefficient、carry-overcoefficient3、分配传递distributeandcarryover4、绘内力图drawinternalforcediagram与位移法一致identicaltodisplacementmethod§11-3力矩分配法计算连续梁和无侧移刚架

calculatebeamandrigidframewithoutsideswaywithmomentdistributionmethod一、计算步骤的形象化介绍visualizedintroductionofcalculationstep锁住结点1、2：lockjoint1、2

放松结点1：relaxjoint1P121212PP1211'<重新锁住1结点，放松2结点：lockjoint1again,relaxjoint2然后再依次放松1、2结点，进行第二、三…轮计算。relaxjoint1、2onebyone,calculatebythesecond、thirdturn…最后：finally

P1212'2'<二、举例说明example例1

计算分配系数：calculatedistributioncoefficient0.50.54/7

3/7分配和传递Distributeandtransfer-8080-200200

30←60

60→30

-66←-131-9917←

3333

→17

-5←

-10-7

2←

3

2

-31176-176106-106先放松结点不平衡力矩的绝对值大的结点——加快收敛。firstlyrelaxthejointwhoseabsolutevalueofunbalancemomentisbiggest.

例2解：solve(3)计算分配系数、传递系数

calculatedistributioncoefficient、carry-outcoefficient(1)简化simplify(2)计算固端弯距calculatefix-endmoment查表checklist〔4〕分配、传递distribution、carry-over〔A-C-A-C-A次序〕〔5〕绘弯距图drawbendingmomentpicture例3〔EI=36000KN.m2〕,正、负最大弯矩绝对值相等，B、C支座应升降多少？solvethesedimentationofsupportB、C

设B、C支座下沉△B(ifthesettlementofsupportBisΔB)

18KN/mABCD

2/31/3

81-96-48

-3000△B0

10+2000△B5+1000△B

-5-1000△B令，ME=MB

得△B=0.019m例4作内力图drawinternalpicture

1、计算劲度系数：

calculatestiffnesscoefficient2、求分配系数：solvedistributioncoefficient结点(joint)B：

结点(joint)

C：

3、弯矩分配与传递：momentdistributionandcarry-over

4、作弯矩图M

drawbendingmomentpictureM例5:绘弯矩图。drawbendingpictureAi=2i=1i=180kn160kn30kn/mBD分配系数固端弯矩B点一次分、传C点一次分、传B点二次分、传C点二次分、传B点三次分、传C点第三次分配最后弯矩0.60.40.50.5+90.0-250.0+250.0-187.5+112.50.00.0+96.0+64.0+32.0-23.7-47.3-23.7-47.30.0+14.2+9.5+4.8-1.2-2.4-1.2-2.40.0+0.7+0.5+0.3-0.2-0.20.0+200.9-200.9+237.4-237.4+87.63m3m10m3m5mABCD120200.9375237.430087.6M图(kn.m)C本章小结conclusion1、力矩分配法momentdistributionmethod

是以杆件的杆端弯矩为计算对象，采用了“固定——放松——传递〞的物理概念，采取逐次逼近的方法，求得杆端弯矩的精确解。Objecttothebar-endbendingmoment,adoptthephysicsconceptof“fix——relax——carry-over〞,andthemethodofgraduallyapproachtosolvetheaccuratesolutionofbar-endbendingmoment.2、明确概念

clearconcept记住：remember4、力矩分配法与位移法之关系〔无结点线位移〕：therelationbetweenmomentdistributionmethodanddisplacementmethod(nojointlineardisplacement)在力矩分配法中：inmomentdistributionmethod

——远端（B）转动角引起的A端弯矩，即总的传递弯矩。totalcarry-overbendingmoment——固定结点时的杆端固端弯矩。fix-endbendingmomentwhenfixingjoint——近端（A）转动角引起的弯矩，即总的分配弯矩。totaldistributionbendingmoment所以分配弯矩、传递弯矩也是分次计算，叠加后的总值：sodistributionbendingmoment、carry-outbendingmomentalsocalculatethetotalvalueaftersuperposition分几次放松而得到的：relaxserioustimesthenobtainBar-endmoment=fixed-endmoment＋totaldistributionmoment＋totalcarry-overmoment1、力矩分配法是一种渐近法。momentdistributionmethodisaasymptoticmethod2、每次只放松一个结点。

relaxonejointeverytime

3、一般从不平衡弯矩绝对值较大的结点算起。

generallycalculatefromthejointwhoseabsolutevalueofnon-equilibriumbendingmomentisbigger

1、确定各结点处杆端力矩的分配系数、传递系数。confirmthebar-endmomentdistributioncoefficient、carry-outcoefficientofeveryjoint2、计算个杆端的固端弯矩。calculatefix-endbendingmoment3、逐次循环放松各结点，以使结点弯矩平衡，直至结点上的传递弯矩小到可以略去不计为止