1 End Continuous Vs Both Ends Continuous Beams

• 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 b² / L² (1a)

where

M_A = moment at the fixed end A (Nm, lb_f ft)

F = load (N, lb_f)

M_B = - F a² b / L²                          (1b)

where

M_B = moment at the fixed end B (Nm, lb_f ft)

M_F = 2 F a² b² / L³                           (1c)

where

M_F = moment at the point load (Nm, lb_f ft)

Deflection

δ_F = F a³ b³ / (3 L³ E I)                                  (1d)

where

δ_F = deflection at point load (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 (3 a + b) b² / L³                                 (1f)

where

R_A = support force at fixed end A (N, lb_f)

R_B = F (a + 3 b) a² / L³                                 (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 L² / 12                                   (2a)

where

M = moments at the fixed ends  (Nm, lb_f ft)

q = uniform load (N/m, lb_f/ft)

M₁ = q L² / 24                         (2b)

where

M₁ = moment at the center (Nm, lb_f ft)

Deflection

δ_max = q L⁴ / (384 E I)                                  (2c)

where

δ_max = max deflection at center (m, ft)

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

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

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 L² / 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 L² / 30                                 (3b)

where

M_B = moments at the fixed end B  (Nm, lb_f ft)

M₁ = q L² / 46.6                         (3c)

where

M₁ = moment at x = 0.475 L (Nm, lb_f ft)

Deflection

δ_max = q L⁴ / (764 E I)                                  (3d)

where

δ_max = max deflection at x = 0.475 L (m, ft)

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

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

δ_1/2 = q L⁴ / (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 a² / 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 a² / 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 a² / (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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Source: https://www.engineeringtoolbox.com/beams-fixed-both-ends-support-loads-deflection-d_809.html

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