Beams  Fixed at Both Ends  Continuous and Point Loads
Stress, deflections and supporting 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 } )
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