结构力学答案部分(4)

32EI04EI0EI0??12q0l0??2??3??3?0??2??l0l02l01045EI0??则有?，得 ?23q0l016?8EI0??EI0??4EI0??2EI0???2q0l0???2223?l?33?1045EI2lll15000?00?4）

M12?M'12?M1234EI0??12q0l0?1257222??(?ql)??q0l0??0.246q0l0 ??00l0?1045EI0?51045M2138EI0??12q0l0?226222??ql?q0l0?0.0415q0l0 ??00l0?1045EI0?156273EI0??12q0l0?622???q0l0??0.0057q0l0 ??2l0?1045EI0?104534EI0??12q0l0?2EI0????l0?1045EI0?l032EI0??12q0l0?4EI0????l0?1045EI0?l03?16q0l0?11222??ql??0.0357ql??00003135?3?1045EI0?3?16q0l0?822??q0l0??0.0026q0l0 ??3135?3?1045EI0?M25M23

M32其余由对称性可知（各差一负号）：M65??M12，M56??M21，M52??M25，

M54??M23，M45??M32，M43??M34?M32；弯矩图如图5.1

5.4 题

（M14?M25?0）M12??pl8，M21?pl8，其余固端弯矩都为0

'? M412EIll?1，M14?''4EIll?1，M52?'2EIl?2，M25?'4EIl?2

M63?M12?M'23''2EI4EIl4EIl?3，M36??1?2EIl2EIl4EI?3

'?2， M21?2EIl2EIl?1??2?4EI??2??3， M32?'l4EIl?2 ?3

由1、2、3节点的平衡条件

?M14?M12?0?????M21?M25?M23?0 即?M??M32?M36?0???M14?M12???M14?M12?'''25?M'23'?M'21'???M23?M21?M25?

M32?M36???M32?M36?4EI2EI?4EI?????2?11?lll?4EI4EI?2EI?????2??12ll?l4EI4EI?2EI?????3?23?ll?lpl84EIl02EIlpl8?2??3??

解得：?1?27pl222?64EI，?2??5pl222?16EI，?3?5pl222?64EI

M14??M1224EI?27pl?27??pl?0.0767pl ??l?22?64EI?352M4122EI?27pl?27??pl?0.0383pl ??l?22?64EI?70424EI?5pl?5??pl?0.0142pl??M32 ??l?22?64EI?35222EI?5pl?5??pl?0.007pl ??l?22?64EI?70424EI?5pl?5????pl??0.0568pl ??l?22?16EI?88224EI?5pl?2EI?5pl?35?????pl??0.0497pl ????l?22?16EI?l?22?64EI?704M36M63M25M23M21??M25?M23?75704pl?0.1065pl

M5222EI?5pl?5????pl??0.0284pl ??l?22?16EI?176弯矩图如图5.2

5.5题

图5.2（单位：ql） 已知l12?l0?3m，l23?2.2l0?6.6m，l24?3l0?9m I0?0.3?104cm4，I12?2I0，I23?3I0，I24?8I0 Q0?12q2l12?12q0l0，q4?4q0，

12(3q0)3l0?6Q0?9Q03l0）? Q24?Q矩24?Q三角24?q（0

1）求固端弯矩

M21?Q0l010，M12??Q0l015，M32?M23?0

?(6Q0)(3l0)12??33Q0l010 M24??

M42(9Q0)(3l0)15

?(9Q0)(3l0)10?(6Q)(3l)0012?21Q0l05

2）转角弯矩

M'12?4E(2I0)l0?1?2E?2I0?l0?2，

M21?'2E(2I0)l0?1?4E?2I0?l0?2

' M23?4E(3I0)2?(2l0)?2?2E(3I0)2?(2l0)?3，

M32?'2E(3I0)2?(2l0)4E(8I0)(3l0)?2?4E(3I0)2?(2l0)?3

' M24??2，

M42?'2E(8I0)(3l0)?2

图5.3（单位：Q0l0） 3）对1、2、3节点列平衡方程

8EI04EI0????2?Q0l0151?l0l0??796EI030EI0?4EI0?16??0即：??1??2??3????Q0l0?33l011l0?5??l0?30EI060EI0???3?0?211l011l0??2M12?0?? ?M21?M24?M23?M32?0?

解得：?1??2234Q0l032880EI02??0.03397q0l02EI0，?2?209Q0l01370EI02?0.07628q0l02EI0，

?3??209Q0l02740EI0??0.03814q0l02EI0

4）求出节点弯矩 M21?????4?223432880?8?2091370?1??Q0l0?1.0487Q0l0 10?2096209??12M23??????Q0l0?0.6241Q0l01.213702.22740??33??32209M24?????Q0l0??1.6727Q0l0?3137010?21??14209M24?????Q0l0?5.0136Q0l0313705??

第6章 能量法

6.2题

1）方法一 虚位移法

考虑b),c)所示单位载荷平衡系统， 分别给予a)示的虚变形 ：

M(x)EIdx??d?

外力虚功为 ?W??虚应变能为

?V=?1??i?? ?1??j??M(x)MEI01l0(x)dx

?1?? =?EI?1??EI?0?Ril0ilx?Mi??0R?x1?idx

??Rx?M??Rx?dx0ii

?l???EI??=??l??EI???Mi3Mj3?Mj?l?1??M?M..........b)?j??i6?3EI?2?Mi?l?1?M?M...........c)??i??j6?3EI?2?

?由虚功原理：?W=?V 得：

?1??i?l? ?????3EIj??1????2?1?2??Mi???? Mj?1????2）方法二 虚力法（单位虚力法） ?　梁弯曲应力：???＝M?x?Iy

??＝?E＝M?x?EIy

Mx??Mi??Mi?Mlj?x

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