哈尔滨工程大学船舶结构力学课程习题集答案(13)

变形能 V?EI2?l0(v)dx?''2EI?2n?x??2n??acos??n??dx ??02l???n?1?l??l?422l2?2n?? ? a?n??2n?1?l?2EI?力函数 ??pv?c??pv?l?c??2pv?c?（对称）

? ?2p?an?1?cosn?1??2n?c??l?

EIl22n?l2n?c??4)?2p?1?cos?

l??由

??V????an?0 ，所以

?V?an????an 。即

an(所以， an?pl43pl344EI??2n?c??1?cos??l???4n

v?x??4?EI?n?11?2n?c??2n?x?1?cos1?cos??? 4?n?l??l? 2）6.40如图所示

?设v?x??a0x??ansinn?1n?xl

2???n?x?EI?n???v?l???V??asindx???n?????20?ll2A2????n?1?22EIll?a0l?2?n??a?n?l?2?2A ??n?1?42? ??pU?c??p?ansinn?1n?cl?a0pc

由

??V????a0?0

得 a0l2/A?pc ， 所以，a0?Apc/l2 由

??V????an?0， 得

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2plEIl?n??n?c 所以，a?a?psinn??n2?l?lEI?n?? v?x??43?4sinn?cl

Apcl2x?2plEI?34??n?11n4sinn?clsinn?xl

3）6.50如图所示 令v?x??ax2?l?x? 所以， V??EI2?l0vdx''2

2EI22?0?2al?6ax?3ldx

5192?2aEIl ??由

?

?l/20qU?x?dx??l0/2qax52?l?x?dxql4?qal4

??V????a?0 得 4aEIl?3192 所以，a?5ql768EI v?x??5ql768EIx2?l?x?

4）6.60所示如图， 设v?x??a1x2?a2x3，v?x??2?a1?3a2x?

V?EI2''?l0vdx?''2EI2?l04?a1?3a2x?dx222

2 ?2EI?l1a?31a2a?l3?a

2l3???ll/2qv?x?dx?q?ll/2?ax12?a2x?dx

ql?7a115a2l?? ???

8?38?3由

??V????a1??V????a2?0 得 2EIl?2a1?3a2l??7ql3/24

由

?0 得 6EIl?a1l?2a2l2??15ql4/64

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2?67qla???1384EI解上述两式得 ?

?a??13ql??2192EI? v?x??0.1745ql2EIx?0.06772qlEIx

36.4题

如图所示

设 v?x??a1sin?xl

?EIV?2??2?l/40vdx?''2E?2I?2?l/2l/4?vdx?

?''2 ?EI?

l/40l/2?????x??x?????2a???sindx?2EIasin1?????dx ?l/4l?l??l???l??2144242???l?31??EIa?????l4???2??21

?xldx?2qla1/?4???l0qv?x?dx?q?a1sin0l

由

??V????a1???l?31?2ql?0 得 EIa1?? ??????l?2?2??4ql?4所以， a1?5???3?21?EI????0.00718ql4EI

ql U?x??0.00718EI4?xsi nl6.5题

如图所示

?设 v?x???n?1ansin?2n?1??x2l

2?EI2l''?v?l????? V?vxdx?????20?2A2 48

24??????2n?1???2?EI??2n?1????3????? ?an???ansin?2l?n?1?2?2????n?1???其中，A??Vl3EI

4??2n?1???EI?2n?1?EI???2n?1???3???an?3??ansin?????sin?? ?anl?2l22?????n?1??????2l0qv?x?dx?q?2l0??n?1ansin?2n?1??x2ldx

?2l?q?an???2n?1??n?1??4ql1?cos2n?1??????????????n?1an2n?1

所以，

???an?4ql?2n?1??

44EI???EI?VEI?3??EI取前两项得 ?3??a1?3?a1?a2?，?3?a??a1?a2? ?23?a1l?2?l?a2l?2?l?V 由

??V????a1??V????a2?0 得

??EI?3l??????4??EI4ql??1a?a??????1232l?????????

由

?0 得

??EI?3?l???3??4??EI4ql??1a?a?????21?32l3?????????

4?4ql?7.088a1?a2???EI即： ? 4?a?494.133a??4ql12?3?EI?4?ql?a1?0.1798?EI解得 ? 4?a?0.001ql182??EI?x3?x?ql??0.0012sin ?v?x???0.180sin?2l2l?EI?ql?l? ?中点挠度v???0.17862EI??44 49

6.6题 取v1(x)?V?EI2EI2EI2EIl4??2ansinn?xl,v2(x)?'2?bnsinn?xl

??l0v1dx?1GAs2?l0v2dx2 ?l02?GAsn?x??n???asindx????n??ll?2??????n?x??n??bcos?n?l??dx?0?l????l2 ???lGAs2?n??an???l22??GAsl?n??2a???n4?l?44??l2?n??bn???l?222

??V?an?n??2??bn?l?GAsln?2EIln?4?V()an,?()bn 2l?bn2ll???0qv1dx?l0?l0qv2dxn?xl?1 ?q??ansindx?q?l0?bnsinn?xldx?1

(1?cosn?) ?q??n??an???l?(1?cosn?)?q??n??bn???l?∴???an?q??(V??)?an?(V??)?an?l??l?(1?cosn?),???b?qn???(1?cosn?) n?n?????2ql(1?cosn?)(n?)EI2ql(1?cosn?)(n?)GAs3254n为奇数由

?0得an??n为奇数4ql54(n?)EI4ql32

由

?0得bn??(n?)GAs

∴U?x??U1?x??U2(x)

?4ql54?EI?nn15sinn?xl?4ql32?GAS?nn13sinn?xl

(N?1,3,5,? ? ? ? ? )6.7题

1）图6.9 对于等断面轴向力沿梁长不变时，复杂弯曲方程为：

EIVIV?TV?q?0''

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