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Beams - Fixed at One End and Supported at the Other - 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 L2)                               (1a)

where

M A = moment at the fixed end (Nm, lbf ft)

F = load (N, lbf )

M F = R b b                               (1b)

where

M F = moment at point of load F (Nm, lbf ft)

R b = support load at support B (N, lbf )

Deflection

δ F = F a3 b2(3 L + b) / (12 L3 E I)                                  (1c)

where

δ F = deflection (m, ft)

E = Modulus of Elasticity (Pa (N/m2), N/mm2, psi)

I = Area Moment of Inertia (m4, mm4, in4 )

Support Reactions

R A = F b (3 L2- b2) / (2 L3 )                                 (1d)

where

R A = support force in A (N, lbf )

R B = F a2(b + 2 L ) / (2 L3 )                                 (1f)

where

R B = support force in B  (N, lbf )

.

Beam Fixed at One End and Supported at the Other - Continuous Load

Bending Moment

M A = - q L2/ 8                               (2a)

where

M A = moment at the fixed end (Nm, lbf ft)

q = continuous load (N/m, lbf /ft)

M1 = 9 q L2/ 128                              (2b)

where

M1 = maximum moment at x = 0.625 L  (Nm, lbf ft)

Deflection

δ max = q L4 / (185 E I)                                  (2c)

where

δ max = max deflection at x = 0.579 L (m, ft)

δ 1/2 = q L4 / (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 L2/ 15                               (3a)

where

M A = moment at the fixed end (Nm, lbf ft)

q = continuous declining load (N/m, lbf /ft)

M1 = q L2/ 33.6                              (3b)

where

M1 = maximum moment at x = 0.553 L (Nm, lbf ft)

Deflection

δ max = q L4 / (419 E I)                                  (3c)

where

δ max = max deflection at x = 0.553 L   (m, ft)

δ 1/2 = q L4 / (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, lbf ft)

Deflection

δ max = M B L2/ (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

• Beams and Columns

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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