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