# Beams - Fixed at One End and Supported at the Other - 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 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

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

### • Mechanics

The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Statics

Forces acting on bodies at rest under equilibrium conditions - loads, forces and torque, beams and columns.

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