國立台灣海洋大學河海工程 研究所 BEM2004 第 8 次作業 博三 錢榮芳 D91520006...
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Transcript of 國立台灣海洋大學河海工程 研究所 BEM2004 第 8 次作業 博三 錢榮芳 D91520006...
國立台灣海洋大學河海工程研究所 BEM2004 第 8次作業
博三 錢榮芳 D91520006博一 周家慶 D93520007碩一 吳安傑 M93520008碩一 李文愷 M93520030
Filename: BEM08-2004-ppt by A. C. Wu
單雙層解法
Fixedend
0M
0F
L
Governing equation:
Lxdx
xud 0,0
)(4
4
Boundary conditions:
00 )(,)(
0)0(,0)0(
FLvMLm
u
Solved problem.
Solve the exact solution of cantilever case subject to the end moment and shear.
Solve the exact solution by using mathematic software (Mathematica).
DSolve[{u’’’’[x]==0,u[0]==0,u’[0]==0,u’’[L]==M0,u’’’[L]=F0},u[x],x]
→{{u[x]→1/6(-3Lx2F0+x3F0+3x2M0)}}
Solve the exact solution by using direct method.
, 0<x<1域內點的邊界積分方程式, x>1 or x<0域外點的邊界積分方程式
0 Us, xvs s, xms Ms, xs Vs, xuss 1s 0
0 Us, xvs s, xms Ms, xs Vs, xuss 1s 0
0 Ums, xvs ms, xms Mms, xs Vms, xuss 1s 0
0 Uvs, xvs vs, xms Mvs, xs Vvs, xuss 1s 0
域外點的積分方程式 x>1 or x<0
u0Us, 0vs s, 0ms Ms, 0s Vs, 0uss 1s 0
u1Us, 1vs s, 1ms Ms, 1s Vs, 1uss 1s 0
0Us, 0vs s, 0ms Ms, 0s Vs, 0uss 1s 0
1Us, 1vs s, 1ms Ms, 1s Vs, 1uss 1s 0
Us, xsx312, s x
sx312
, s xUs, x Us,x
xsx24
, s xsx24
, s x
s, x Us,xs
sx24, s x
sx24
, s xs, x s,x
x
Us,xs
sx2
, s xsx2
, s x
Ms, x s,xs
sx2
, s x
sx2
, s xMs, x Ms,x
x
s,xs
12, s x
12, s x
Vs x Ms,xs
12, s x
12, s x
Vs x Vs,xx
Ms,x
s0, s x
0, s x
Ums, x Us,xx
sx2
, s x
sx2
, s xUvs, x Ums,x
x 1
2, s x
12, s x
ms, x s,xx
Ums,x
s1
2, s x
12, s x
vs, x ms,xx
Uvs,x
s0, s x
0, s x
Mms, x Ms,xx
ms,x
s0, s x
0, s x Mvs, x Mms,xx
vs,x
s0, s x
0, s x
Vms x Vs,xx
Mms,x
s0, s x
0, s x Vvs x Vms,xx
Mvs,x
s0, s x
0, s x
u0 112
F0 14M0
121 1
2u1 0
u1 12u1 1
12v0 1
4m0 0
0 14F0
12M0
121 0
1 121 1
4v0 1
2m0 0
u1 13F0
12M0, 1
12F0 M0, m0 F0 M0, v0 F0
uxUs, xvs s, xms Ms, xs Vs, xuss 1s 0
U1, xv1 1, xm1 M1, x1 V1, xu1 U0, xv00, xm0 M0, x0 V0, xu0
1 x3
12F0 1 x2
4M0
1 x2
12F0 M0
12 1
3F0
12M0 x3
12F0
x2
4F0 M0
x22M0 x36
x2
2F0
Solve the problem by the indirect method.By choosing .
uxj1
2
Psj, xj j1
2
Qsj, xj
ux Us, xs s, xss 1
s 0
x Us,xx
s s,xx
ss 1s 0 Us, xs s, xss 1
s 0
mx Us,xx
s s,xx
ss 1s 0 Ums, xs ms, xss 1
s 0
vx Ums,xx
s ms,xx
ss 1s 0 Uvs, xs vs, xss 1
s 0
取源點s要特別小心 當場點x已固定在0 and 1, 源點s不能進入其域內
,U
Case 1 s=0,s=1 (by A. C. Wu)
0 2F0, 1 0, 0 2F0 2M0, 1 0
ux Us, xs s, xss 1s 0
U1, x1 1, x1 U0, x0 0, x0 x3122F0x242F0 2M0
x22M0 x36
x2
2F0
u0 112
1 141 0
0 141 1
21 0
m1 121 1
20 1
20 M0
v1 121 1
20 F0
0101
2F00
2F0 2M00
Case 2 s=-1,s=2 (by 錢榮芳 )
Case 3 s=2,s=4 (by 周家慶 )
24244F0 3M02F0 6M0
16M02F0 5M0
Case 4 s=-1,s=-3 (by 李文愷 )
Detect the rank of [A] matrix.
A1212
00F0M0
0 1
120 1
4
0 14
0 12
12
12
0 0
12
0 12
12
010100F0M0
RankA 4
Case 1 s=0,s=1 (by A. C. Wu)
Case 2 s=-1,s=2 (by 錢榮芳 )
RankA 4
Case 3 s=2,s=4 (by 周家慶 )
Case 4 s=-1,s=-3 (by 李文愷 )
RankA 4
RankA 4
Indirect method for 1-D beam problem
THE END