结构力学答案部分(8)

由约束条件v1??z1??z3?0,R2??k2v2??入?K?得

?18?20?l2?EI?0l???6?2??l012?6l?3ql??6??2?2l?v??2??2??6????ql??z2????? l??4????v3??6?ql??2??l????20EIl2，划去1、2、6行列，将k2代

7.7 题

a) 写出各杆元对总体坐标之单元刚度矩阵

?A??0???0E??K(1)???K(3)??????l??A??0???0??K22(1)??(1)?K12?K21??K33(3)???(1)(3)?K43K11???(1)012Il206Il4I0?6Il2I(3)?A00A000?12Il26Il012Il2?6Il012Il26IlK34?6Il?K22(2)?(2)?K320??6I?l??2I?? 0???6I?l??4I??K23??K(2)???K(2)? ?(2)??????K33??(2)?2l代l??以???(3)K44????cos?2??sin?t????2?0????sincos0?2?2??0?0??1?????01????0??1000??t?0 ∴?T?????01??0?? t?(1)?1?K(1)???K(3)???T??K??T? ?????????0?1??0???????100001010?A??0??????0?E???10?l??A?00??0?01????0?012Il206Il4I0?6Il2I?A00A000?12Il26Il0?12Il2?6Il012Il26Il?6Il0??6I?01?l????10?2I??00??0????6I???l???4I??00101?1000???? ?0?0??1??12I?l2??0??6I?E?ll??12I?2?l?0??6I??l0A00?A0?6Il04I6Il02I?12Il20?A00A006Il12Il206Il?6I?l??0??2I??K22(1)???6I??K12(1)?l?0??4I??K21??K33(3)??(3)(1)?K11??K43(1)? (3)?K44?K34(3)b）集成总刚度矩阵

(1)?K11?(1)K?K???21????K12(1)(1)(2)K22?K22K32(2)K23(2)(2)(3)K33?K33K43(3)????(3)K34??(3)K44??6Il02I?6I?A20012Il2?12I?l2?0??6I?l?12I??2l??0??6I?l???????????????0A00?A06Il04I?6Il02I?12Il06Il?0A26I4l220?A00?A?12Il2l3I2l0??A26I4l203I2lI??6Il3I2l?12Il06Il12Il06Il226I4l3I2??6IlA200?3I2l06I4l3I2l26I0?3I2lI??2l0?A3I2l0?06I0?A00A0?6Il0l12Il06Il2?6I6Il02I2I6Il04I?A0?????????????????????????????

c）写出节点位移及外载荷列阵

???T??u1固端力：

v1?z1?u2v2?z2?u3v3?z3?u4v4?z4????1?2T?3?4?

T?F?(1)T局???0??F局Q2(3)TQl12?0Q2Ql??? 12?T?F?(2)T局????0?

?F?(1)T总??T?F局?(1)??0?1??0???????100001010?100?0?Q??2??Ql????12??0?0??0??Q?21???Ql??12??Q????2???0???Ql?????F?212?????????????? ??Q??F????2??1???0???Ql?????12????P?总?P1????P2??????P3??P??4?Ty1??M?R1??Q???R1x??2???Ql????12?Q20?Ql12?000?R4xR4y?MR4??T约束处理

(1)(2)?K22?K22?(2)K32?K23K33(2)???2??P2????? (3)???K33???3??P3?(2)7.8 题

由对称性，计算图示两个单元即可。

但A12?A/2 P2取

???P/2 ?x,x????450

???K???K?(1)(1)?1?E?A/2??0???1l??0?10000?10100??(1)0?K11???(1)0??K21?0?(1)K12?(1)?K22?

?K(2)???T??K(2)??T??????1?1?1?2?0??0?1100001111?1?10??1??0EA0???1?2l??1??1??0?1?1110000?10100??1??01?1??0?2?0??0??01100001?10??0?1??1??1?EA?1?22l??1???1?1??(2)?1?K33???(2)1??K23?1?(2)K32?(2)?K22?(1)?K11?(1)?K???K21??K12(1)(1)(2)K12?K22K320000??1?121(2)?(2)?K11?(2)K33??00?????????????????1?0????1?EA??02l??????????10121212??1212??12121212??12121212

212结构节点位移列阵为

?????u1,v1,u2,v2,u3,v3?T其中u1(1)?Tx1??(1)?EA?Ty1???(1)?2l?Tx2??T(1)??y2??v1?u3?v3?0,v2?0

所以在总刚度矩阵中划去1，2，4，5，6组列，设平衡方程为：

?1?0???1??00000?10100??0???u2???1????????0?0?0?EAP/2??0??????????0??u2?2l?u2??1??1??1?????0???0??02???????0?

由于实际12杆受力为图示对称情况，

(1)所以Tx(21)??Tx1?P1???1??2???0.586P?1.172tf，

对32杆

?U2?1?11??U2?1?U2??1?U2????t????????? ???2??11??0?2??U2??V2??V2??Tx3????Ty3???Tx2???T??y2?(2)?1?EA?0?2l??1??00000?101000?????1??????00?P/2??0???????0??u2/2??1??1??1?????0???u2/2??2???0??

所以23杆内力为

P/21???1??2???0.586tf

7.9 题

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