S②?0?0.75???0?2.25??0.7501.1250.3750.750.75?2.25?0.3750??*1.0e0110.37500.375??
?????????
?1.1250?0.750?0.75?0.37500.375?2.25?0.375?2.25?0.3750?0??0.75?0.3751.31250.75?0.56250.375ke②??2.4375?0.375?0.1875??0.375?2.250.75?0?0.375?0.5625?0.375?10.56250??0.375?0.187500.1875?0.3750?
由ke和ke扩充KZ（总刚度阵）
?1.31250.75?0.56250.375?0.75?0.37500???0.752.4375?0.375?0.1875?0.375?2.2500????0.5625?0.3751.3125?0?0.75000.375??0.375?0.187502.43750?2.250.3750?kz?1.01e?011*???0.75?0.375?0.7502.06250.75?0.5625?0.375???0?2.250.754.6875?0.375?0.1875???0.375?2.25?0000.375?0.5625?0.375?0.56250???000.3750?0.375?0.187500.1875??? ?①②?Re??Rx1?而Re?kz.qe，其中
Ry110?210?2Rx2??Ry2??，
qe??00x1?x1??1.3125?y??0?1????x2???0.75????y2??0y1x2y200??，化简得：
02.43750?2.25?0.7502.06250.75?0?0??1??0.113???????2.25???2????0.5968?0.75??0??0.1947??????4.6875??1???0.4243????2??
则，
?0.75?0.375??0.1113???0.1481??Rx1???0.56250.375?Ry??0?????0.1875?0.375?2.25??1??????0.5968???0.9517??Rx2???0.750.375?0.5625?0.375??0.1947???0.1742?????????0?0.375?0.1875???0.4243??0.0482??Ry2??0.375
53.15如图所示有限元网格，a?4cm，单元厚度t?1mm，弹性模量E?2.0?10MPa，泊松比
??0.3。回答下述问题：
（1）结点如何编号才能使结构刚度矩阵带宽最小？
（2）如何设置位移边界条件才能约束结构的刚体移动？ （3）形成单元刚度矩阵并集成结构刚度矩阵。 （4）如果施加一定载荷，拟定求解步骤。
(1) (2) （3） 解：1、节点编号如图(2)所示；
2、如图(3)设置位移边界条件才能约束结构的刚体移动；
3、如图(2)所示各节点的坐标为(以m为单位)：1(0,0)，2(0.08,0)，3(0,0.04)，4(0.08,0.04 )，5(0,0.08)，6(0.08,0.08)，7(0,0.12)，8(0.08,0.12)
解：单元号 1 2 3 4 5 6 相邻结点 1 3 4 5 5 7 2 2 5 4 6 6 3 4 3 6 7 8
对于单元号1：
b1?y2?y3??0.04；b2?y3?y1?0.04；b3?y1?y2?0；
c1?x3?x2??0.08；c2?x1?x3?0；c3?x2?x1?0.08；
对于单元号2：
b3?y2?y4??0.04；b2?y4?y3?0；b4?y3?y2?0.04；
c3?x4?x2?0；c2?x3?x4??0.08；c4?x2?x3?0.08；
对于单元号3：
b4?y5?y3?0.04；b5?y3?y4?0；b3?y4?y5??0.04；
c4?x3?x5?0；c5?x4?x3?0.08；c3?x5?x4??0.08；
对于单元号4：
b5?y4?y6??0.04；b4?y6?y5?0；b6?y5?y4?0.04；
c5?x6?x4?0；c4?x5?x6??0.08；c6?x4?x5?0.08；
对于单元号5：
b5?y6?y7??0.04；b6?y7?y5?0.04；b7?y5?y6?0；
c5?x7?x6??0.08；c6?x5?x7?0；c7?x6?x5?0.08；
对于单元号6：
b7?y6?y8??0.04；b6?y8?y7?0；b8?y7?y6?0.04；
c7?x8?x6?0；c6?x7?x8??0.08；c8?x6?x7?0.08；
12??11x1x2
y1y2?0.0032m2
平面三角形单元的面积均为
x3 y3
?1?0?E?2.0?1011?D??10?2??0.911????0(1??)/2? 0 ?弹性矩阵均为
应变矩阵
?1?0.3???00.30??01?0.35?? 0
B(1)?B(3)?B(5)?b10b201??0c10c2?2???c1 b1 c2 b2b30???12.5012.5000????250c3???0?25000???25?12.500?b3?c3 ?? 12.5 25 ?
B(2)?B(4)?B(6)应力矩阵
?b30b20b40???12.500012.50?1????0??0c25cc004?00?25032??2???12.5??0??? ?c3 b3 c2 b2c4 b4? ?12.5 ?25 0 25
S(1)?S(3)?S(5)?D?B(1)
16.4835??27.4725?16.483527.47250?0??1.0?1011??8.241854.9451?54.94518.241800????0??19.2308 ?9.6154 0? 9.6154 19.2308 S(2)?S(4)?S(6)?D?B(2)
??27.47250?0?16.483527.472516.4835??1.0?1011??8.241854.945100?54.94518.2418???9.6154??0? ?9.6154 ?19.2308 0 19.2308
单元刚度矩阵
K(1)?K(3)?K(5)?B(1)?S(1)?A?t
?1.31870.7143?0.5495?0.3846?0.7692?0.3297??0.7143??2.19782.3901?0.3297?0.1923?0.3846????0.5495?0.32970.54950.3297?00?1.0?108???0.38460?0.192300.19230.3846????0.7692?0.38460?00.38460.7692???0.32972.1978?? ?? ?2.1978 0.3297 0 0
TK(2)?K(4)?K(6)?B(2)?S(2)?A?t
T
?0.5495000.3297?0.5495?0.3297??0??0.19230.19230.38460?0.3846????0.3846?0.38460.76920?0.769280?1.0?10??0.3297?2.1978002.1978?0.3297????0.5495?0.3846?0.7692?0.32971.31870.7143??????0.3297 ?0.1923 ?0.3846 ?2.1978 0.7143 2.3901??
结构刚度矩阵为：
?1.31870.7143?0.5495?0.3846?0.7692?0.32970?0.71432.3901?0.3297?0.1923?0.3846?2.19780???0.5495?0.32971.3187000.7143?0.7692?2.39010.71430?0.3297??0.3846?0.19230??0.7692?0.384600.71432.08790?1.3187?04.5879?0.7143??0.3297?2.19780.71430?00?0.7692?0.3297?1.3187?0.71433.4066?0?0.3846?2.1978?0.7143?2.39011.42868?0K?1.0?10?000000.3297?0.5495??00000.384600?000000?0?0000000?000000?0?0000000??0000000?0? 0 0 0 0 0 0
00000000000000?0.3846000000?2.1978000000?0.714300.38460000?2.39010.3297000001.4286?0.54950?0.7692?0.3846006.97800?0.1923?0.3297?2.19780002.41770.7143?1.0990?0.7143?0.7692?0.3297?0.19230.71432.7747?0.7143?0.3846?0.3846?2.19780?0.5495?0.32971.3187000.71430?0.3846?0.192302.39010.714300?0.7692?0.384600.71431.318700?0.3297?2.19780.7143002.3901000?0.7692?0.3297?0.5495?0.38460 0 0 ?0.3846 ?2.1978 ?0.3297 ?0.19230?0?0??0??0?0??0?0??0?0???0.3846???2.1978??0.3297???0.1923?0.7143??2.3901??
若施加一定载荷，求解步骤为：
1、对单元编号，并列出各单元三个结点的结点号； 2、计算外载荷的等效结点力，列出结构结点载荷列阵； 3、计算单元刚度矩阵，组集结构整体刚度矩阵
4、引入边界条件，即根据约束情况修正结构有限元方程，特别是消除整体刚度矩阵的 奇异性，得到考虑约束条件的可解的有限元方程。
0000000000?0.7692?0.3297?0.5495?0.3846.31870.71431