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BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x MAXIMUM DEFLECTION 1. Cantilever Beam – Concentrated load P at the free end 2 2 Pl EI θ= ( ) 2 3 6 Px y l x EI = 3 max 3 Pl EI δ = 2. Cantilever Beam – Concentrated load P at any point 2 2 Pa EI θ= ( ) 2 3 for 0 6 Px y a x x a EI = < < ( ) 2 3 for 6 Pa y x a a x l EI = < < ( ) 2 max 3 6 Pa l a EI δ = 3. Cantilever Beam – Uniformly distributed load ω (N/m) 3 6 l EI ω θ= ( ) 2 2 2 6 4 24 x y x l lx EI ω = + 4 max 8 l EI ω δ = 4. Cantilever Beam – Uniformly varying load: Maximum intensity ω o (N/m) 3 o 24 l EI ω θ= ( ) 2 3 2 2 3 o 10 10 5 120 x y l lx lx x lEI ω = + 4 o max 30 l EI ω δ = 5. Cantilever Beam – Couple moment M at the free end Ml EI θ= 2 2 Mx y EI = 2 max 2 Ml EI δ =

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Page 1: Beam formulas

BEAM DEFLECTION FORMULAE

BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x MAXIMUM DEFLECTION1. Cantilever Beam – Concentrated load P at the free end

2

2PlEI

θ = ( )2

36Pxy l xEI

= − 3

max 3PlEI

δ =

2. Cantilever Beam – Concentrated load P at any point

2

2Pa

EIθ =

( )2

3 for 06Pxy a x x aEI

= − < <

( )2

3 for6Pay x a a x l

EI= − < <

( )2

max 36Pa l a

EIδ = −

3. Cantilever Beam – Uniformly distributed load ω (N/m)

3

6lEIω

θ = ( )2

2 26 424

xy x l lxEI

ω= + −

4

max 8lEIω

δ =

4. Cantilever Beam – Uniformly varying load: Maximum intensity ωo (N/m)

3o

24lEI

ωθ = ( )

23 2 2 3o 10 10 5

120xy l l x lx xlEI

ω= − + −

4o

max 30lEI

ωδ =

5. Cantilever Beam – Couple moment M at the free end

MlEI

θ = 2

2Mxy

EI=

2

max 2Ml

EIδ =

Page 2: Beam formulas

BEAM DEFLECTION FORMULAS

BEAM TYPE SLOPE AT ENDS DEFLECTION AT ANY SECTION IN TERMS OF x MAXIMUM AND CENTER DEFLECTION

6. Beam Simply Supported at Ends – Concentrated load P at the center

2

1 2 16Pl

EIθ = θ =

223 for 0

12 4 2Px l ly x xEI

⎛ ⎞= − < <⎜ ⎟

⎝ ⎠

3

max 48Pl

EIδ =

7. Beam Simply Supported at Ends – Concentrated load P at any point

2 2

1( )6

Pb l blEI−

θ =

2(2 )

6Pab l b

lEI−

θ =

( )2 2 2 for 06Pbxy l x b x alEI

= − − < <

( ) ( )3 2 2 3

6for

Pb ly x a l b x xlEI b

a x l

⎡ ⎤= − + − −⎢ ⎥⎣ ⎦< <

( )3 22 2

max 9 3

Pb l b

lEI

−δ = at ( )2 2 3x l b= −

( )2 2 at the center, if 3 448

Pb l bEI

δ = − a b>

8. Beam Simply Supported at Ends – Uniformly distributed load ω (N/m)

3

1 2 24lEI

ωθ = θ = ( )3 2 32

24xy l lx xEI

ω= − +

4

max5

384lEI

ωδ =

9. Beam Simply Supported at Ends – Couple moment M at the right end

1 6MlEI

θ =

2 3MlEI

θ =

2

216Mlx xy

EI l⎛ ⎞

= −⎜ ⎟⎝ ⎠

2

max 9 3Ml

EIδ = at

3lx =

2

16Ml

EIδ = at the center

10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ωo (N/m)

3o

17360

lEI

ωθ =

3o

2 45lEI

ωθ =

( )4 2 2 4o 7 10 3360

xy l l x xlEI

ω= − +

4o

max 0.00652 lEIω

δ = at 0.519x l=

4o0.00651 l

EIω

δ = at the center

Page 3: Beam formulas

file:///G|/BACKUP/Courses_and_seminars/0MAE4770S12/url%20for%20beam%20formulas.txt[1/23/2012 12:15:35 PM]

http://www.advancepipeliner.com/Resources/Others/Beams/Beam_Deflection_Formulae.pdf

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