# 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})*