euler bernouli beam
DESCRIPTION
Euler Bernouli BeamTRANSCRIPT
18/04/2023
CE502: Finite Element Analysis
By
Dr. A. ChakrabortyDepartment of Civil Engineering
Indian Institute of Technology Guwahati, India
1
EULER BERNOULI BEAM
18/04/2023
2
Euler-Bernouli Beam
22
200 0
22
20 0 0
2 2
2 20 0 0
2 2
2 20 0 0
2
2
= . . .
=
L L
x
L L
x
L L
x
L L
x
EI d u dudx q udx M
dx dx
EI d u dudx qudx M
dx dx
d u d u duEI dx qudx Mdx dx dx
d u d u duEI dx qudx Mdx dx dx
q
ML , EI
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3
2 2
2 20 0 0 0
2 3 4
2 3 400 0 0 0
2 3
2 3 00
. .
L L L
x
L L LL
x
LL
d u d u d d u d u duEI EI dx qudx Mdx dx dx dx dx dx
d u du d u d u duEI EI u EI udx q udx Mdx dx dx dx dx
d u du d uM EI EI u
dx dx dx
0
4
40 0
2 2
2 2
0
: , , ,
Secondary : EA , ,
L
x
d u duEI q dx u Mdx dx
u vPrimary Variables u v
x y
du d d u d uVariables EI EI
dx dx dx dx
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4
Strong Form
4
4
2
2
0,
4
40
3 3
3 30 00
0
0
0
0
L
L
L L L
d uEI qdx
d uEIdx
d uw EI q dx
dx
d u dw d uEIw EI dx wqdx
dx dx dx
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5
3 2 2 2
3 2 2 20 00 0
2 2 3 2
2 2 3 20 00 0
2 2
2 20 0
0
0
, = (w)
L L L L
L LL L
L L
d u dw d u d w d uEIw EI EI dx wqdx
dx dx dx dx dx
d w d u d u dw d uEI dx EIw EI wqdx
dx dx dx dx dx
d w d uEI dx wqdx
dx dx
B w u l
Strong Form
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6
20 1 2
0 1 20 0 0 ,
u a a x a x
u u L a a a L
2
1 2
2,
u a x a x
w x x
2 30 1 2 3
00 0 ,
u a a x a x a x
u u L a
21 2 3
2 3
0
, ,
a a L a L
w x x x
Example
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7
2 2
2 20 0
22 3
0 0
22 3
0 0
32
2 3
0
32
2 3
, = (w)
2 2 6
2 2 6
2 2 33
2 2 33
L L
L L
L L
L
B w u l
d w d uEI dx wqdx
dx dx
EI a a x dx x qdx
EI a a x dx x qdx
xEI a x a x q
qLEI a L a L
2w x
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2 2
2 20 0
32 3
0 0
2 32 3
0 0
42 3
2 3
21 3
22
2 343
, = (w)
6 2 6
6 2 6
6 24
01
0 4 63
0 6 12
4
L L
L L
L L
B w u l
d w d uEI dx wqdx
dx dx
EI x a a x dx x qdx
EI a x a x dx x qdx
L qEI a L a L
aL LL
EI L L a q
L L aL
3w x
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9
1
2
3
0.1250
0.2083
0.0833
a
a
a
EI=1
L=1
q=1
2 3
1
0.125 0.2083 0.0833
0
0.5
0.1042
x
u x x x
u
At L
u
450.0130
384th
qLu
EI
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10
Thank You!!!