elastic curves of beams. basic equations for beam deflection
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Elastic Curves of Beams
Basic Equations for Beam Deflection
xx EI
My
yx
EI
M
1
2/32
2
2
1
1
dxdv
dxvd
2
2
dx
vd
EI
Mdx
dv
Elastic Curve by Direct Integration
EI
xM
dx
vd )(2
2
)(xdx
dv
EI
xw
dx
vd )(4
4
EI
xV
dx
vd )(3
3
wdx
vdEI
4
4
13
3
CwdxVdx
vdEI
212
2
CxCwdxdxMdx
vdEI
322
12
1CxCxCwdxdxdxEI
dx
dvEI
432
23
1 2
1
6
1CxCxCxCwdxdxdxdxEIv
Alternative Coordinates
Boundary Conditions of Beams
Example I-Cantilever Beam under an End Load
x
PxMdx
vdEI
2
2
12
2
1CPx
dx
dvEI 2
1 2
10)( PLCLx
223
2
1
6
1CxPLPxEIv 3
2 3
10)( PLCLxv
323 236
LxLxEI
Pv
Example II-Simple Beam under Uniform Load
x
4334
12
1
24
1CxCwLxwxEIv
212
212
2
2
1CxCwxCxCwdxdxM
dx
vdEI
0C0)0x(M 2 wL2
1C0)Lx(M 1
00)0( 4 Cxv 33 24
10)( wLCLxv
xLLxxEI
wv 334 224
Method of Superposition
Only suitable for linear problems!!!
Easy Mistake in Superposition
Cannot remove the supporting boundary
conditions!!! X
Statically Indeterminate Beams
212
2
2
2
1CxCwx
dx
vdEI
221 2
10)( wLCLCLxM
32213
26
1CxCx
Cwx
dx
dvEI
00)0( 3 Cx
422314
2624
1Cx
Cx
CwxEIv
00)0( 4 Cxv
221 4
130)( wLCLCLxv
Alternative Solution by Superposition (I)
232 3
6Lxx
EI
Rv B
x
22341 64
24xLLxx
EI
wv
wLRvv BLxRBLxwB B 8
3
2234 35248
xLLxxEI
wv
Alternative Solution by Superposition (II)
xLLxxEI
wv 3341 2
24 xLLxx
EIL
Mv A 2232 236
2234 35248
xLLxxEI
wv
2
00 8
1wLM AxMAxwA A
Table of Beam Elastic Curves
Table of Beam Elastic Curves