Beam Formulas

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Beam formulas may be used to determine the deflection, shear and bending moment in a beam based on the applied loading and boundary conditions.

Reference Documents:

  1. Formulas for Stress and Strain, Roark & Young, McGraw-Hill
  2. Formulas for Stress, Strain, and Structural Matrices, Pilkey, John Wiley & Sons.

Common Beam Formulas:

 
PINNED-PINNED BEAM WITH UNIFORM LOAD

 

 
 FIXED-FIXED BEAM WITH UNIFORM LOAD

 

 
PINNED-FIXED BEAM WITH UNIFORM LOAD

 


FREE-FIXED BEAM WITH UNIFORM LOAD 

 
 PINNED-PINNED BEAM WITH POINT LOAD

 
See definitions of < > step functions below.
 
 FIXED-FIXED BEAM WITH POINT LOAD

 
 See definitions of < > step functions below.
 
 PINNED-FIXED BEAM WITH POINT LOAD


See definitions of < > step functions below. 

 
FREE-FIXED BEAM WITH POINT LOAD
  
See definitions of < > step functions below.

Legend:

w = uniform load (force/length units)
V = shear
M = moment
d = deflection
E = modulus of elasticity
I = moment of inertia

Definitions of step functions:

If x < a then < x - a >0 = 0
If x < a then < x - a >n = 0
If x > a then < x - a >0 = 1
If x > a then < x - a >n = ( x - a )n
 

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