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

## Support loads, moments and deflections

- 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 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^{2} b^{2} (3 a + 4 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)*