Beam Deflection Formulae PDF

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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 Pl 2 Px 2 Pl 3 θ= y= ( 3l − x ) δ max = 2 EI 6 EI 3EI 2. Cantilever Beam – Concentrated load P at any point Px 2 y= ( 3a − x ) for ...

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Beam Deflection Formulae Abhijit Deshmukh

Related papers Int roduct ion t o Finit e Element s dhirendra pandey St ruct ural Analysisin T heory Pract ice-Alan Williams Alex Ovcharenko Aircraft St ruct ures for engineering st udent s Rafael Ferreira

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BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x 1. Cantilever Beam – Concentrated load P at the free end

θ=

y=

Pl 2 2 EI

2. Cantilever Beam – Concentrated load P at any point

Px 2 ( 3l − x ) 6 EI

Px 2 ( 3a − x ) for 0 < x < a 6 EI Pa 2 y= ( 3x − a ) for a < x < l 6 EI

y=

Pa 2 θ= 2 EI

MAXIMUM DEFLECTION

δ max =

δ max

Pl 3 3EI

Pa 2 = ( 3l − a ) 6 EI

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

ωl 3 6 EI

θ=

y=

ωx 2 x 2 + 6l 2 − 4lx ) ( 24 EI

δ max =

ωl 4 8 EI

δ max =

ωo l 4 30 EI

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

θ=

ωol 3 24 EI

y=

ωo x 2 (10l 3 − 10l 2 x + 5lx2 − x3 ) 120lEI

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

θ=

Ml EI

y=

Mx 2 2 EI

δ max =

Ml 2 2 EI

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

Pl 2 θ1 = θ2 = 16 EI

⎞ Px ⎛ 3l 2 l − x 2 ⎟ for 0 < x < y= ⎜ 12 EI ⎝ 4 2 ⎠ Pbx 2 l − x 2 − b 2 ) for 0 < x < a ( 6lEI Pb ⎡ l 3 y= ( x − a ) + ( l 2 − b2 ) x − x3 ⎤⎥ ⎢ 6lEI ⎣ b ⎦ for a < x < l

δ max =

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

Pb(l 2 − b 2 ) θ1 = 6lEI Pab(2l − b) θ2 = 6lEI

y=

δ max = δ=

Pb ( l 2 − b 2 )

Pl 3 48 EI

32

at x =

(l

2

− b2 ) 3

Pb ( 3l 2 − 4b2 ) at the center, if a > b 48 EI

9 3 lEI

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

θ1 = θ2 =

ωl 3 24 EI

y=

ωx 3 l − 2lx 2 + x3 ) ( 24 EI

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

Ml θ1 = 6 EI Ml θ2 = 3EI

y=

Mlx ⎛ x 2 ⎞ ⎜1 − ⎟ 6 EI ⎝ l 2 ⎠

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

7ωol 3 360 EI ω l3 θ2 = o 45 EI

θ1 =

y=

ωo x ( 7l 4 − 10l 2 x 2 + 3x4 ) 360lEI

δmax =

δmax = δ=

5ωl 4 384 EI

Ml 2 l at x = 9 3 EI 3

Ml 2 at the center 16 EI

δ max = 0.00652 δ = 0.00651

ωo l 4 at x = 0.519 l EI

ωol 4 at the center EI...