One Three hundred and Eighty Fourth

John Read@johndavidread

𝑥

𝑥

2

𝑥

𝑥

+ =

+ =

=

2

=

+

= + 𝑥

𝑥

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

∫𝑞 𝑣Shear force

∫𝑣 𝑚Bending moment

∫𝑚 𝑠slope

∫ 𝑠 𝑤deflection

q

q

q

q

q

w

w

w

w

w

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

The coefficients are the reciprocals of the double factorials !! of n, for n = 0 to 4