# 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} ) *

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