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# Beams - Fixed at One End and Supported at the Other - Continuous and Point Loads

### Beam Fixed at One End and Supported at the Other - Single Point Load

#### Bending Moment

M A = - F a b (L + b) / (2 L 2 )                               (1a)

where

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

F = load (N, lb f )

M F = R b b                               (1b)

where

M F = moment at point of load F (Nm, lb f ft)

R b = support load at support B (N, lb f )

#### Deflection

δ F = F a 3 b 2 (3 L + b) / (12 L 3 E I)                                  (1c)

where

δ F = deflection (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 b (3 L 2 - b 2 ) / (2 L 3 )                                 (1d)

where

R A = support force in A (N, lb f )

R B = F a 2 (b + 2 L ) / (2 L 3 )                                 (1f)

where

R B = support force in B  (N, lb f )

### Beam Fixed at One End and Supported at the Other - Continuous Load

#### Bending Moment

M A = - q L 2 / 8                               (2a)

where

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

q = continuous load (N/m, lb f /ft)

M 1 = 9 q L 2 / 128                              (2b)

where

M 1 = maximum moment at x = 0.625 L  (Nm, lb f ft)

#### Deflection

δ max = q L 4 / (185 E I)                                  (2c)

where

δ max = max deflection at x = 0.579 L (m, ft)

δ 1/2 = q L 4 / (192 E I)                                  (2d)

where

δ 1/2 = deflection at x = L / 2   (m, ft)

#### Support Reactions

R A = 5 q L / 8                            (2e)

R B = 3 q L / 8                            (2f)

### Beam Fixed at One End and Supported at the Other - Continuous Declining Load

#### Bending Moment

M A = - q L 2 / 15                               (3a)

where

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

q = continuous declining load (N/m, lb f /ft)

M 1 = q L 2 / 33.6                              (3b)

where

M 1 = maximum moment at x = 0.553 L (Nm, lb f ft)

#### Deflection

δ max = q L 4 / (419 E I)                                  (3c)

where

δ max = max deflection at x = 0.553 L   (m, ft)

δ 1/2 = q L 4 / (427 E I)                                  (3d)

where

δ 1/2 = deflection at x = L / 2   (m, ft)

#### Support Reactions

R A = 2 q L / 5                            (3e)

R B = q L / 10                            (3f)

### Beam Fixed at One End and Supported at the Other - Moment at Supported End

#### Bending Moment

M A = -M B / 2                               (4a)

where

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

#### Deflection

δ max = M B L 2 / (27 E I)                                  (4b)

where

δ max = max deflection at x = 2/3 L   (m, ft)

#### Support Reactions

R A = 3 M B / (2 L)                            (4c)

R B = - 3 M B / (2 L)                       (4d)

## Related Topics

### • Beams and Columns

Deflection and stress, moment of inertia, section modulus and technical information of beams and columns.

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Statics

Loads - forces and torque, beams and columns.

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