结构力学答案部分

第2章 单跨梁的弯曲理论

２.1题

设坐标原点在左跨时与在跨中时的挠曲线分别为v(x)与v(x1)

１）图2.1? v(x)?M0x2EI2?N?x36EI2?l43p(x?l)4?6EIl23p(x?l)2?6EI3l4p(x?3l)4 6EI3原点在跨中：v1(x1)?v0?M0x12EI?N?x16EI3?l43?v(l)?0v'(l)?0p(x?l)12?124，? p'6EI?v1(0)?0N1(0)??2２）图2.2?v(x)??0x?Mx22EI?N?x36EIx0?l33p(x?l)3 6EI3p(x?l)2 6EI３）图2.3?v(xx)??0xx?N?x36EI??qxdx6EI3?l2２.２题

a) v1?vpppl?1131?pl?1131??vp?(3???)??(?2?)? ?6EI?41626EI?164444????pl333 =

512EIpl3

31?13?pl3?9pl???()??? ?192EI96EI1624??? V2??16EI?4? b) v(0)?'?Ml3EI?Ml6EI?2Pl296EI2(1?2)

32 =?0.1Pl6EI2?5Pl3?27EI2Pl2?73Pl1620EI

?(l)??Ml3EI?Ml6EI2?96EI2(1?1)

3 =?0.1Pl6EI?4Pl3?27EI22??107Pl21620EI

?l??2l?2p???l??3??3???3 vl33EIl6EI??1???11?1????m2?3?m1?3?

3?????? =

37pl22430EI

445ql c) vl ??3?2192EI768EI2304EI??'ql47ql v(0)??ql324EI??pl216EI???ql2166EI?l?ql11?ql3?1 ????96EI8EI?3612??3d)2.1图、2.2图和2.3图的弯矩图与剪力图如图2.1、图2.2和图2.3

图2.1

图2.2

图2.3

2.3题

1）

??右??M?Ml6EI13q1l?2ql324EI?Ml?l?q?q??0?21???45EI?2?3EI

2l2120 2）?0??Ml3EI3?q1l324EI?Ml?l? lq?1??180EI?2?6EI37l2ql?1q1l1713? =1?? ??????EI?18243606?120?80EI2.4 题

图2.5 ?v(x)?v0??0x??N0x36EI，

v0?A?p?N0?

?x3??v(x)?Ap??0x???A?N0

?6EI?如图2.4， 由v?l??v??l??0得

??l3?Ap??0l???A?N0?0???6EI??2l??0?N0?0?2EI?2?pl?Ap????0?l6EI??N?p3?0解出

33pl?3xx?3 ?v(x)??1?9EI?2l2l?? 图2.4 ? 图2.6

?v?x???1x?由v?l??0,M0x22EI?N0x6EI3v??l???2?N0l3得4EI2EI?M?????201? ?ll??N?6EI?????0122?l?2??0??2EI6EI?2M0lN0l?1????2??EI2EI??1l?M0l2解得?v?x???1x??2?1??2?x2l???1??2?x3l2.5题

图2.5：（剪力弯矩图如２．５）

??R1?pl?Mll3?p??2p32p3??pl322p39EI33v0?AR?6EIMlplpl5pl?l?vv???0????216EI18EI48EI144EI?2?v??0???0??v0l?Ml6EI??pl2

9EI?pl218EI??pl26EIM?pa?bb? ， 图2.5 A?1?l?KA?6l????将a?l,b?0A?l6,KA?16?13?12代入得：M?pl1???6312pl

2.7图：（剪力弯矩图如２．６）

v1?A1R1?v2?A2R2?0.05lEIl433???ql2ql24??ql440EIql450EI100EI5qlql?11??l?v????????2?384EI2EI?40100?ql?57?293ql?????EI?384400?9600EI44

图２．６

??0??ql3

24EIql3?v1?v2l?ql?111?2ql??????EI?2440100?75EI333??l????v1?v2l24EI?ql?111??17ql??????EI?2440100?300EI3

图2.8（剪力弯矩图如２．７）

21?b???M???12A??1???24KA?l?????Qa由Q?qa,a?l,b?0,??1,A?1824 KA?1?1?1?1,代入得 82432M?R1?qlql2224??2?12?1ql8?ql3ql84?24?1??ql28,v0?AR1?64EI 图２．７

5qlqlMl5ql?l??v??????384EI?2?384EI128EI16EIql34424ql?111??(0)????????24EIl6EIEI?246448?

3ql?192EIv0Ml3

?(l)???M??

l8EI?ql82??ql364EI2.6题

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