One Three hundred and Eighty Fourth
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Transcript of One Three hundred and Eighty Fourth
𝑑2
𝑑𝑥2 (𝐸𝐼 𝑑2𝑤𝑑𝑥2 )=𝑞
Daniel Bernoulli1700 - 1782
Leonard Euler 1707 - 1783
The Euler Bernoulli Beam Theory 1750
Eiffel Tower under construction 1887-1889
q
q
wx
For q(x) = a constant, setting q = 1 , and setting both E = 1 and I = 1 and for a unit length beam l = 1
= 1
For q(x) = a constant, setting q = 1 , and setting both E = 1 and I = 1 and for a unit length beam l = 1
= 1
w=1
24𝑥4+
16𝑐1 𝑥
3+12𝑐2 𝑥
2+𝑐3 x+𝑐4
w=14 !𝑥4+
13 !𝑐1𝑥
3+12!𝑐2𝑥
2+11 !𝑐3𝑥
1+10 !𝑐4 𝑥
0
For q(x) = a constant, setting q = 1 , and setting both E = 1 and I = 1 and for a unit length beam l = 1
= 1
w=1
24𝑥4+
16𝑐1 𝑥
3+12𝑐2 𝑥
2+𝑐3 x+𝑐4
w= 14 ! ( 1
2 )4
+ 13 !𝑐1( 1
2 )3
+ 12!𝑐2( 1
2 )2
+ 11!𝑐3( 1
2 )1
+ 10 !𝑐4( 1
2 )0
At the centre of the beam where x = ½
w= 14 ! ( 1
2 )4
+ 13 !𝑐1( 1
2 )3
+ 12!𝑐2( 1
2 )2
+ 11!𝑐3( 1
2 )1
+ 10 !𝑐4( 1
2 )0
At the centre of the beam where x = ½
w=1
24 4 !+
1
23 3 !𝑐1+
1
22 2 !𝑐2+
1
211 !𝑐3+
1
20 0 !𝑐4
w= 14 ! ( 1
2 )4
+ 13 !𝑐1( 1
2 )3
+ 12!𝑐2( 1
2 )2
+ 11!𝑐3( 1
2 )1
+ 10 !𝑐4( 1
2 )0
At the centre of the beam where x = ½
w=1
24 4 !+
1
23 3 !𝑐1+
1
22 2 !𝑐2+
1
211 !𝑐3+
1
20 0 !𝑐4
w=1
384+
148𝑐1+
18𝑐2+
12𝑐3+
11𝑐4