# 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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