框架结构内力位移计算算例

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1、结构力学课程大作业多层多跨框架结构内力及位移计算班级学号姓名华中科技大学土木工程与力学学院年月日结构力学课程大作业多层多跨框架结构内力与位移计算一、任务1、计算多层多跨框架结构在荷载作用下的弯矩和结点位移。2、计算方法要求：(1)用迭代法、D值法、反弯点法及求解器计算框架结构在水平荷载作用下的弯矩，并用迭代法的结果计算其结点位移。(2)用迭代法、分层法、二次力矩分配法及求解器计算框架结构在竖向荷载作用下的弯矩，并用迭代法的结果计算其结点位移。3、分析近似法产生误差的原因。、计算简图及基本数据本组计算的结构其计算简图如图1所示，基本数据如下。混凝土弹性模量：Eh3.0107kN/m2构件尺寸：柱

2、：底层：bh3040cm2其它层：bh3030cm2梁：边梁：bh2560cm2中间梁：bh2530cm2水平荷载：一一Fp=15kN,Fp=30kN(见图2)竖向均布包载：q顶=17kN/mq其它=21kN/m(见图8)各构件的线刚度:EI.,bh3一，其中IL12i1L6梁：I10.250.63124.5103m4EIi733.0104.51022500kNm中间梁:0.250.33445.62510m12底层柱:其它层柱:i2I4i4i3EI2L0.3EI4L0.3-743.0105.625106750kNm0.43121.63.01072.53410m1.61039600kNm0.33

3、126.75104m4EI33.0L1076.7510三、水平荷载作用下的计算（一）用迭代法计算1、计算各杆的转角分配系数角分配系数计ikiik27k(i)1”:22500122(6136.422500)6136.4152(6136.422500)22500213.3ik算公式0.3930.1072(67506136.422500)0.3186136.4kNmFP=15kNi1i2i11134i3i3i3i3FP=30kNi1i2i15678i3i3i3i3FP=30kNi1i2i19101112i3i3i3i3FP=30kNi1i2i113141516i3i3i3i3Fp=30kNi1i2i

4、117181920i4i4i4i4n6J5JJ图22367502(67506136.422500)0.095266136.42(67506136.422500)0.087由于该结构是对称结构，因此结点“3”的分配系数应该等于结点“2”的，结点“4”的分配系数应该与结点“1”的相等，所以本题只需计算1、2、5、6、9、10、13、14、17、18结点的分配系数。结点“5”：516136.42(6136.4222500)0.08856225002(6136.4222500)0.324590.088结点“6”225002(6136.42225006750)0.2716767502(6136.4222

5、5006750)0.081626136.42(6136.42225006750)0.0746100.074由于该框架第二、三、四层结构一样，故结点9、13与结点“5”相应杆的分配系数相同，结点10、14与结点“6”相应杆的分配系数相同。结点“17”：17136136.42(6136.4225009600)0.081718225002(6136.4225009600)0.294172196002(6136.4225009600)0.126结点“18” ：318172(6136.42250096006750)6136.418142(6136.42250096006750)960018222(613

6、6.42250096006750)675018192(6136.42250096006750)225000.0680.1070.0752、计算各柱的侧移分配系数ik0.25侧移分配系数计算公式:ik“2iik(r)此结构各层柱的线刚度均相等,因此所有柱的侧移分配系数都相同:26136.4ik0.3753(6136.44)3、计算各杆的周端弯矩、楼层剪力及楼层弯矩由于各杆没有节间荷载,因此各杆的固端弯矩均为零，故各结点的不平衡力矩Mig0o楼层剪力的计算公式:Fpr楼层弯矩的计算公式:MrhrrFpr具体计算如下:FQ515kNFq445kNM4FQ375kNM33Fq2105kNM2135kN

7、M11313131311545751051353.33.33.33.316.5kNm49.5kNm82.5kNm115.5kNm5225kNm转角迭代公式：MikikMig(MkiMik)侧移迭代公式：MikikMr(MkiMik)r由于只有水平荷载，因此先迭代侧移，后迭代转角。循环路线：侧移：五层五层四层三层二层一层转角：1”“2”“3”“4”“5”“6”“7”8”“9”“10”“11”“12”13”“14”“15”“16”“17”“18”“19”“20”具体迭代过程见图3，杆端弯矩计算见图4，弯矩图见图弯矩图见图5。例：第五层第1轮的侧移迭代如下：M15M51M26M62M37M73M48

8、M840.37516.5(00)6.19r“1”结点第一轮的转角迭代如下：M120.3930(06.19)2.43M150.1070(06.19)0.66例：第五层第3轮的侧移迭代如下：M15M51M26M62M37M73M48M840.37516.5(0.652.290.461.640.481.770.652.21)9.99r“1”结点第3轮的转角迭代如下：M120.3930(1.672.299.99)2.37M150.1070(1.672.299.99)0.65“18”结点第4轮的转角迭代如下：M18170.250(40.335.199.06106.8366.6)29.71M18140.0

9、680(40.335.199.06106.8366.6)8.08M18190.0750(40.335.199.06106.8366.6)8.91M18220.1070(40.335.199.06106.8366.6)12.726555546666QaQQ399773333La012rd5201255555bu另047Q二0n97333677-7-735A3圈9099i74小I砥.65544Fo-ro0ZD/D-flu,ueg1g40211354444333841811423441ndMf*5l-l-.50,01,30岁,344占&-IB.Zib川-25.96?ll-?7.39-755-2756

10、148548工44#7BBB0-0.080亚I阜-a0883-07Qy1?22222tJ7587715555CM16OJ32409?-B5542nJ222-x-EO4go二i10989.6O3333117,也?5,而方48170-IL-fl-&R98吼419/b3-ej3J77aaa3a7503365777rd3mm30-B如,&3,a5,14,19,19工4iri2ySEQ4c77oo.IBJoo二807-,9529,1S.18Z4mu-$ltSlt5iri2987g4.7fzbrotHAaLltS8“388366!6riuitri3uinCD14-nr3442aaaaMiniI-3034目

11、711?11-45.16-45-94-45.975572131ggg3&61fi-Is!_75087,r8888/b/Dyuz&-/o，舞4氐7634141919345G1613町1c,d/gDo3789975fto77J555590虱al13h，g847424nunuCJTc二6,1T=,4D783.371627r7OJ-nnW?3344,03火为,89制9-tt-AQ/Qy2332CUs.华010-K=124307642977己a己22186747ruDs589q&aOJZD-11TJ6Jg9s&氏8.11-j8J900_o-Qwn37.5乃鼠al84593H就,74ML?-nc?oTL1

12、LIL1186I8DCQ3772工3g*2?NmJ22MltiI-43.31目-6527T|-G6J04-66j60-66j65MmiI-84.3S-102.76Tl-106,25-106,27-106.831IIo324332s535222EPu31436J043CULpjQQQR34444依88k向阜-0.098?s7672HOOS，受oLooCJ-UM.naUH7rfcjnuQu7r218r3,&.s6.&/&60o*o153863K3225z4z7.B229M-10.048,717?7鸵M-27.56/1.73-6,117,11.732”一豆中:7,0w工:TbiI心41478L/3t

13、2)4.9C3,77丽f班ITS:，需t.3334$1?3J4.77H7；篇标说叨j.-IB.416E7-3453-”6635D774Idn5r9o0/i?尼乃ut.87-353-45.97t,97-双F253044JI19,0HJ.M-66.65Y-4B.17b.1717,04-3213171C.96-43.82g-b3X402&/rl0-72.36Tio巧,787O018I?,748.9211.：.,。加2177T7T后弯W44/16第U3-.1GMi-53.3O17.8.39,8：70,4131阴友,M-83TrogeDo明或4哂70.070271724-布丘-a:M-106.88OJO

14、O2223L.i.士-W2447777T7T算公式：MikFik2MikkiMik或：MikFikMik(MikMkiMik)5.845,6118-41-1绕L55与4邑二。69.62”跳上J/-86S9.&29412/J8464/94J4.M(kNm)5二ai-i1j.,A-)61?I?112,63.3256823,694.8118,411L.J3.9635.18*74心及6pg.on63-3181/0司5,2日y9.Au17.34j，am?6.818L36图55、计算各结点的转角和侧移引用公式:Miki为ikM ikhik由于结构是对称的，在水平荷载作用下，其位移应该是反对称的第五层：12

15、.365,45.2710rd2225000.52-5,233.85105rd2675010.043.34152637489.0010m66136.4第四层：58.44481.8810rd2225001.90,/“4,67-1.41104rd2675027.563.33596107118122.4710m66136.4第三层：916.814123.7410rd2225003.73八)A4.10112.76104rd2675045.973.33913101411151216cc,4.1210m66136.4第层：1325.34.165.6210rd162225005.68414154.21104r

16、d2675066.653.331317141815191620八c,5.9710m66136.4第一层：1740.264,208.9510rd2225008.93418196.61104rd1926750106.88531721182219232024$613649.2810m（二）用D值法计算1、求各柱的剪力值本结构由于对称，只需计算一半即可表1柱的剪力计算（kN）弟五层1-52-6D-222500k3667-22250026750,k476726136.4IXT.1J126136.4D-676136.4上23.6673.312=3970.739(2)3.3c4.767巾，12D6136.4

17、一224.7673.312=4322.775(2)3.31216587.028(-2)3.323970.739153.591Q16587.028FQ4322.775153.909Q16587.028第四层5-96-10D222500k3667丁22250026750,ek4767INJ.t26136.4.，、(/，26136.4c3.66712nR1QRAc4.767，12nR1AI/UIOU.*Iq23.6673.3212=3970.739(2)3.32I/UIOUJIq24.7673.3212=4322.775(-2)3.321216587.028(2)3.323970.739FQ4510

18、.77316587.028FQ754511.72716587.028第层9-131410D222500222500267501,amk3.66726136.4k4.76726136.4c3.66712c4.767，121216587.028(2)3.3D6136.423.6673.3212=3970.739(-2)3.3D6136.424.7673.3212=4322.775(一)3.33970.739FQ7517.955Q16587.028l4322.775.FQ7519.545Q16587.028第层13-1714-18D-222500k3.66726136.4-22250026750,k

19、4.76726136.4c3.66712c4.767，121216587.028(-2)3.32D6136.4223.6673.312=3970.739(2)3.3D6136.4224.7673.312=4322.775(.)3.33970.739FQ10525.137Q16587.0284322.77510527.363Q16587.02817-2118-22D第-22500k2.3449600225006750k3.0479600层c2.334“cc12D9600222.3345212=5180.110(/)c3.047“cc12D9600223.0475212=5795.760()521

20、221951.740()5180.110Fq13531.85721951.7405795.760Fq13535.64321951.7402、计算各柱反弯点高度比表2柱的反弯点高度比计算弟五层1-52-6k3.667k4.767y00.45y00.45,22500._I1y1022500,22500._I1y10225003H1y303.33H1y303.3y0.45y0.45第四层5-96-10k3.667k4.767y00.48y00.522500I1y102250022500I1y10225002双1y203.3221y203.33331y3033.33困1y3033.3y0.48y0.5

21、第9-1310-14层k3.667k4.767y00.5y00.522500八I1yi02250022500八I1y10225002Hiy203.32341y203.33Hiy303.33次1y303.3y0.5y0.5第层13-1714-18k3.667k4.767y00.5y00.522500I1y102250022500I1y102250023.31y203.323.31y203.331.515y03.331.515y03.3y0.5y0.5第层17-2118-22k2.344k3.047y00.55y00.5520.66y20520.66y205y0.55y0.553、求各柱杆端弯矩第

22、五层：M510.453.33.5915.33kNmM150.553.33.5916.52kNmM620.453.33.9095.80kNm第四层:第三层:第二层:第一层:M26M95M59M106M1390.553.33.9097.09kNm10.483.310.77317.06kNm20.523.310.77318.49kNmM1410M1713M1814M2117M1721M2218M1822M6100.53.311.72719.35kNmM913M1014M1317M14180.550.450.550.454、求各横梁的杆端弯矩第五层:M12M156.52kN0.53.317.9550.

23、53.30.53.30.53.3531.85731.85735.64335.64329.63kNm19.54525.13727.36332.25kNm41.48kNm45.15kNm87.61kNm71.68kN98.02kN80.20kNM21M26i21i23225002250067507.095.45kNM23M26I2112367507.091.64kN第四层:M56M51M592250067505.3318.4923.82kNM65(M62M610)I6516722500225006750(5.8019.35)19.35kNmM67(M62M610)I6516767502250067

24、50(5.8019.35)5.80kNm第三层:M910M95M91317.6029.6346.69kNmM109：(M106M1014)i109i101122500225006750(19.3532.25)39.70kNmM1011i1011i109i1011(M106M1014)6750225006750(19.3532.25)11.90kN第二层:M1314M139M131729.6341.4871.11kNM1413；(M1410M1418)i143i141522500225006750(32.2545.15)59.54kNM1415.i1415(M1410M1418)i143i141

25、56750225006750(32.2545.15)17.86kN第一层:M1718M1713M172141.4871.68113.16kNM1817曳(M1814M1822)i1817i181922500225006750(45.1580.2)96.42kNM1819i1819(M1814M1822)6750i1817i1819225006750(45.1580.2)28.93kN5、绘弯矩图如图6所示图6(三)用反弯点法计算1、求各柱的剪力值由于本结构对称，取半结构计算即可。又由于各层柱侧向刚度均为12i/h,故同层各柱剪力值相等，为：FQr-Fpr4r表3柱剪力计算层数楼层剪力Fpr(k

26、N)r柱子剪力Fqr(kN)五153.75四4511.25三7518.75二10526.25一13533.752、各柱反弯点高度底层柱：反弯点高度位于距底部2/3层高处；其它层柱：反弯点位于层高的中点处。3、求各柱杆端弯矩第五层：M51M15M62M260.53.33.756.19kNm第四层:M95M59M106M6100.53.311.2518.56kNm第三层:M139M913M141coM10140.53.318.7530.94kNm第二层:M1713M1317M1814M14180.53.326.2543.31kNm2第一层：M2117M2218533.75112.5kNm31,M1

27、721M1822533.7556.25kNm34、求各横梁的杆端弯矩第五层:M12M156.19kNgmM21上一M26i21i23225002250067506.194.76kNM23i23i21；M26i2367506.191.43kN第四层:M56M51M592250067506.1918.5624.75kNM65(M62i65i67M610)22500225006750(6.1918.56)19.04kNmM67-J(M62i65i67M610)6750225006750(6.1918.65)5.71kNm第三层:M910M95M91318.5630.9449.5kNM109.i109

28、(M106M1014)i109i101122500(18.5630.94)22500675038.08kNmM1011i1011(M106M1014)6750第二层:M1314M1413M1415i109i1011225006750(18.5630.94)11.42kNM139M131730.9443.3174.25kN(M1410M1418)i143i141522500第一层:M1718M1817225006750(30.9443.31)57.12kN(M1410M1418)i143i14156750225006750(30.9443.31)17.13kNM1713M172143.3156.

29、2599.56kNm尬(M1814M1822)i1817i181922500225006750(43.3156.25)76.58kNii8i96750M1819(M1814M侬2)(43.3156.25)22.98kNmi1817i18192250067505、绘出弯矩图如图7所示图7四、竖向均布荷载作用下的计算计算简图如图8所示。（一）用迭代法计算1、各杆的转角分配系数及各柱的侧移分配系数由于结构对称，结点只有转角没有侧移,需计算转角分配系数（与水平荷载作用的相同）。2、计算周端弯矩Mi2ql21217621251kNm只MF1M：M43M%ql21251kNmql212_2172.5128

30、.85kNmm56M65M78M8f7ql22162121263kNm67m76ql212212.521210.94kNmM910M1g09M1112M1211ql21221621263kNmM1011M1314FM1415M1718M1819ql2212.5210.94kNm1212M1516M16152_2ql2216263kNm1212ql2212.5210.94kNm1212M1920M2019ql22162121263kNmql2_2212.510.94kNm1212FM1514M1110M1413M1817M19183、用迭代法计算如前所说由于结构对称，荷载对称，变形必然对称，因此

31、没有侧移，只需进行转角迭代。引用公式：MikikMig(Mki)循环路线：“1”“2”“3”“4”5”“6”“7”“8”“9”10”“11”“12”“13”“14”“15”“16”具体迭代过程见图“17”“18”9，杆端弯矩计算见图“19”10，弯矩图见图“20”11。例：“1”结点的第一轮转角迭代：M120.39351.0(0.00.0)20.04M150.10751.0(0.00.0)5.46“10”结点的第3轮转角迭代：M1090.27152.06(22.615.805.055.15)19.04M10110.08152.06(22.615.805.055.15)5.69M106M1014

32、0.07452.06(22.615.805.055.15)5.203-51.0920004222253566D5444fe-SZDfeaaaaa03333a,33315.茨忍出空Big嗜髻2也856-05-0.088-0,08873451100-CD一一一-S.03-MO-W)0-&0Q-627-6131-6J7-t.15-&14-6,27-&117-6.134-6.15-0.088阜回，。,。884口344E9DQ力416.&6一-78N.OS,10,10222E己222a263.0-2110旭72-祝湖的3610-33.07-23,22-22,71$63*-0JO8通-0.12624610

33、-20,57-ae.oi-22.18-9.61-25.20-3-22.20-9.50T51-9.5173打975354544a&R&101公49祖列加4gosc5%11IXIX部而那,03,004nr9X91411-11il11ao891114JG6.&.7677757.ii/oAb/o_rD1038797766*&一一5101-51.0u37214111111IX566s-_二.78副54373419.及魏茨馥-43378骋器G22222-a079316%35552M407SST386669RO0OX工U-5LfS5.制IQ9410.6-m-UJF_Jro880c0(0(4J4J1/535-

34、如.1+尿羽加5555617444J65554triiHuHuri9134423555工8urst一I.Tw909s3m35555Hru?恤,?.95.3224疝.19a555%Ds%QOTnJ55668o.oo4455ui.一-989必3J46息z7乱aS氏111449-78112oc23ssl_-Q08,10筐肚J5疆566669TJi一一%M%0055缢瑞踮308783s666455ds55398836666工5L5L覆区,?07-O.TT-5.-5.-S.-5.冷6-4w。6JDAzcd阴99J11-tl1-fl-95,32?420,19351/5853J0697915378.6E*0

35、cucu_oc花？一22241OB8囹窣-ftOB0,1O,22,H,15,1556666列a14T44Q9B36766*2区n-I.一.6O95D6D8叫34555sfctro94a8423Q0/DQz07_Q7Qz-QJ31工工,75800夕45E633&522700B4566nSSITJ81521式1943555一i尚皿ts,37北J05J07猊Tt-5-5-53J03-6口6555让dJc-JZDzb-0-ZD邮15即2O1S41HLHtriiH17830B80Jm_70D-o/_9TJnnull.76和以质发82222北2222e8加口7466A,87999&zz=一二22/7/5g

36、qg67QOS66B60_Bnofiw-0O183回小065551111-iuJ1/5&3J017433934S89Dj45566-ftOBy26,3O,18,44,5O51z9997.21/：20.42-51J0T7443951D0JU4-6.11-11,3-5J94732典=1557l.P1315闿q.J5花j.JO羽乃1口l”977H17-：二器.即心377777-1623-10.9413F-54-IS,49：雨-ouoo辿F&3.001.,-3.563.00-2?.1O15637,li-Jl?0-LOO-1F.154S15F9个苦心判T咽4OJOO-5.68-.U._L10.305AB

37、ItZT”9屿卜-63.0063.00-6：5-】E5七：.3,63-519473包63.：34W3.537736-I.7-.519.：1；19二T，芸-163-V.631?-IS,44-10.364557TO例上8T.00-JEjJdc!.：.JO119_W-63.00519T踊15.4Gdtd-E丹-19.4363,00-在定-簸TtTTIt-：1.：T,明涉-1cm-7597-5：6-1027-1咽4DJ)O-16,弼107-53.0010.t7-47M-：D0M：63.00西?3OOD芍975S*/-6J0454TET5：0.02OjC：一问T5177777图103727_9J77TM

38、(kNm)图114、计算各结点的转角26.484,5.8810rd22250022.3344.96104rd22250022.0844.9110rd22250064.12104rd222500503104rd10114104.2210rd121322.6345.0310rd2225001415144.23104rd16171922.22225004.93104rd1822.3422250026.464.96104rd一一4,184.1510rd20（二）用分层法计算22250018.532225005.88104rd4.12104rd54.92104rd4.22104rd-495.0310rd

39、19.022225004.23104rd135.03104rd18.674.4.1510rd267504174.9310rd本结构可以分顶层、中间层和底层三个部分进行分别计算，如图12所示）,然后再迭加即可顶层中间层底层图121、顶层弯矩计算由于结构与荷载均为对称，因此可以取半结构进行计算，其计算简图如图13所小。1）计算分配系数柱的线刚度秉折减系数0.9 。结点上:12 4-22500 0.8034 (0.9 6136.4 22500)图13154 0.9 6136,0.1974 (0.9 6136.4 22500)结点“2”:214 225002 6750 4 0.9 6136.4 4 2

40、2500)0.717222 67502 6750 4 0.9 6136.4 4 225000.107262 67504 0.9 6136.44 0.96136.4 4 22500)0.1762）计算周端弯矩M 22M1F2ql21217 621251kN mql23171.25238.85kN mM2F2ql26_ 217 1.2564.43kN m3）弯矩分配结点51262，杆端5-11-51-22-12-62-26-22-20.1970.8030.7170.1760.107MF-5151-8.85-4.43顶层弯矩分配表4(kN.m)3.3510.0540.9520.48-22.45-44

41、.90-11.02-6.70-3.676.70分1.474.4218.039.02配-3.24-6.47-1.59-0.97-0.530.97与0.210.642.601.30传-0.47-0.93-0.23-0.14-0.080.14递0.030.090.380.19-0.140.03-0.02汇M5.0615.20-15.2029.55-12.87-16.68-4.283.382、中间层弯矩计算同理可取半结构进行计算，其计算简图如图14所示1）计算分配系数柱的线刚度秉折减系数0.9结点“5”:564225004(0.96136.4222500)0.67515940.96136.44(0.9

42、6136.4222500)0.165结点“6”:图14654225002675040.96136.42422500)0.60966267502675040.96136.424225000.0916240.96136.42675040.96136.42422500)0.152）计算周端弯矩M56M65ql22162121263kNm,22M66q-.一10.94kNm33.2一2fql211.25M66-5.47kNm663）弯矩分配表5中间层弯矩分配（kN.m）结点19566，102杆端1-59-55-15-95-66-56-106-26-66-610-62-60.1650.1650.670.6090.150.150.091MF-6363-10.94-5.473.473.4710.410.442.2121.1100-22.2-44.5-10.9-10.9-6.666.66-3.66-3.61.231.2386886分3.683.6814.937.47-0.680.68-0.37配0.130.13-2.28-4.55-1.12-1.12-0.3与0.380.381.530.77-0.070.07