Beams - Fixed at Both Ends - Continuous and Point Loads

• Beams - Supported at Both Ends - Continuous and Point Loads
• Beams - Fixed at One End and Supported at the Other - Continuous and Point Loads
• Beams - Fixed at Both Ends - Continuous and Point Loads

Beam Fixed at Both Ends - Single Point Load

Bending Moment

M _A = - F a b ² / L ² (1a)

where

M _A = moment at the fixed end A (Nm, lb _f ft)

F = load (N, lb _f )

M _B = - F a ² b / L ² (1b)

where

M _B = moment at the fixed end B (Nm, lb _f ft)

M _F = 2 F a ² b ² / L ³ (1c)

where

M _F = moment at the point load (Nm, lb _f ft)

Deflection

δ _F = F a ³ b ³ / (3 L ³ E I)                                  (1d)

where

δ _F = deflection at point load (m, ft)

E = Modulus of Elasticity (Pa (N/m ² ), N/mm ² , psi)

I = Area Moment of Inertia (m ⁴ , mm ⁴ , in ⁴ )

Support Reactions

R _A = F (3 a + b) b ² / L ³ (1f)

where

R _A = support force at fixed end A (N, lb _f )

R _B = F (a + 3 b) a ² / L ³ (1g)

where

R _B = support force at fixed end B  (N, lb _f )

Beam Fixed at Both Ends - Uniform Continuous Distributed Load

Bending Moment

M _A = M _B

= - q L ² / 12                                   (2a)

where

M = moments at the fixed ends  (Nm, lb _f ft)

q = uniform load (N/m, lb _f /ft)

M ₁ = q L ² / 24                         (2b)

where

M ₁ = moment at the center (Nm, lb _f ft)

Deflection

δ _max = q L ⁴ / (384 E I)                                  (2c)

where

δ _max = max deflection at center (m, ft)

E = Modulus of Elasticity (Pa (N/m ² ), N/mm ² , psi)

I = Area Moment of Inertia (m ⁴ , mm ⁴ , in ⁴ )

Support Reactions

R _A = R _B

=  q L / 2                               (2d)

where

R = support forces at the fixed ends  (N, lb _f )

Beam Fixed at Both Ends - Uniform Declining Distributed Load

Bending Moment

M _A = - q L ² / 20                                  (3a)

where

M _A = moments at the fixed end A  (Nm, lb _f ft)

q = uniform declining load (N/m, lb _f /ft)

M _B = - q L ² / 30                                 (3b)

where

M _B = moments at the fixed end B  (Nm, lb _f ft)

M ₁ = q L ² / 46.6                         (3c)

where

M ₁ = moment at x = 0.475 L (Nm, lb _f ft)

Deflection

δ _max = q L ⁴ / (764 E I)                                  (3d)

where

δ _max = max deflection at x = 0.475 L (m, ft)

E = Modulus of Elasticity (Pa (N/m ² ), N/mm ² , psi)

I = Area Moment of Inertia (m ⁴ , mm ⁴ , in ⁴ )

δ _1/2 = q L ⁴ / (768 E I)                                  (3e)

where

δ _1/2 = deflection at x = 0.5 L (m, ft)

Support Reactions

R _A = 7 q L / 20                               (3f)

where

R _A = support force at the fixed end A  (N, lb _f )

R _B =  3 q L / 20                               (3g)

where

R _B = support force at the fixed end B  (N, lb _f )

Beam Fixed at Both Ends - Partly Uniform Continuous Distributed Load

Bending Moment

M _A = - (q a ² / 6) (3 - 4 a / l + 1.5 (a / L) ² )                                  (4a)

where

M _A = moment at the fixed end A  (Nm, lb _f ft)

q = partly uniform load (N/m, lb _f /ft)

M _B = - (q a ² / 3) (a / L - 0.75 (a / L) ² )                                 (4b)

where

M _B = moment at the fixed end B  (Nm, lb _f ft)

Support Reactions

R _A = q a (L - 0.5 a) / L - (M _A - M _B ) / L                              (4c)

where

R _A = support force at the fixed end A  (N, lb _f )

R _B =  q a ² / (2 L) + (M _A - M _B ) / L                              (4d)

where

R _B = support force at the fixed end B  (N, lb _f )

Insert beams to your Sketchup model with the Engineering ToolBox Sketchup Extension

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Length m
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Area m²
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Volume m³
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us gal

Weight kg_f
N
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Velocity m/s
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Pressure Pa
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Flow m³/s
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