Beam Formulas PDF

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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 2

2

Pl 2EI

θ=

MAXIMUM DEFLECTION

y=

Px (3l − x ) 6 EI

3

δmax =

Pl 3EI

2. Cantilever Beam – Concentrated load P at any point

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

y=

2

Pa 2EI

θ=

2

δ max =

Pa ( 3l − a ) 6 EI

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

ωl3 6EI

θ=

ωx 2 ( x2 + 6l 2 − 4lx ) 24 EI

δmax =

ωl 4 8EI

ωo x 2 (10l3 −10l2 x + 5lx2 − x3 ) 120lEI

δ max =

ωo l 4 30EI

Mx 2 2 EI

δmax =

Ml 2 2EI

y=

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

θ=

ωol 3 24EI

y=

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

θ=

Ml EI

y=

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

θ1 = θ 2 =

Pl 2 16 EI

y=

Px ⎛ 3l 2 l ⎞ − x 2 ⎟ for 0 < x < ⎜ 12 EI ⎝ 4 2 ⎠

δ max =

Pl 3 48EI

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

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

Pbx 2 ( l − x2 − b2 ) for 0 < x < a 6lEI 3 Pb ⎡ l y= ( x − a ) + (l 2 − b 2 ) x − x 3 ⎤⎥ 6lEI ⎢⎣ b ⎦ for a < x < l y=

δmax = δ=

Pb ( l 2 − b2 )

32

9 3 lEI

at x =

(l

2

− b2) 3

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

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

θ1 = θ 2 =

ωl 3 24EI

y=

ωx 3 ( l − 2lx 2 + x 3 ) 24 EI

δmax =

5ωl 4 384 EI

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

Ml θ1 = 6EI Ml θ2 = 3 EI

y=

2 Mlx ⎛ x ⎞ ⎜ 1− 2 ⎟ 6EI ⎝ l ⎠

δmax = δ=

Ml 2 l at x = 3 9 3 EI

Ml 2 at the center 16EI

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

7ωol 3 360 EI ω l3 θ2 = o 45E I

θ1 =

y=

ωo x (7l 4 − 10l 2x 2 + 3x 4 ) 360lEI

δmax = 0.00652

ωo l 4 at x = 0.519 l EI

ω l4 δ = 0.00651 o at the center EI

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