Roymech engineering encyclopedia

Online Beam Calculator Index



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Simple Beam Distributed Load

  • Uniformly Distributed Load
  • Load Uniformly Increasing to One End
  • Load Uniformly Increasing to Centre
  • Partially Distributed Loads At Each End
  • Partially Distributed Uniform Load
  • Partially Distributed Uniform Load At One End
  • Uniformly Distributed Load & Variable End Moments

Simple Beam Concentrated Load

Fixed Beam One End, Supported at Other End

  • Uniformly Distributed Load
  • Concentrated Load At Centre
  • Concentrated Load At Any Point
  • Free Vertically But Not Rotate - Uniform Distributed Load
  • Free Vertically But Not Rotate - Concentrated Load


Fixed Beam Both Ends

  • Concentrated Load At The Centre
  • Concentrated Load At Any Point
  • Uniformly Distributed Load

Cantilever Beam

  • Concentrated Load At Free End
  • Concentrated Load At Any Point
  • Left Guided End & Right Fixed End with a Concentrated Load
  • Load Increasing Uniformly To Fixed End
  • Uniformly Distributed Load

Beam Overhanging Loads

  • Uniformly Distributed Load On Overhang
  • Uniformly Distributed Load
  • Concentrated Load At End Of Overhang
  • Uniformly Distributed Load Between Supports
  • Unequal Overhanging Both Supports - Uniformly Distributed Load



Explanation of Beam Theory

Euler-Bernoulli Beam Theory

The Euler-Bernoulli equation describes the relationship between the applied load and the resulting deflection of the beam and is shown mathematically as:

Euler-Bernoulli Beam Equation

Where w is the distributed loading or force per unit length acting in the same direction as y and the deflection of the beam Δ(x) at some position x. E is the modulus of elasticity of the material under consideration and I is the second moment of area calculated with respect to the axis which passes through the centroid of the cross-section and is perpendicular to the applied load. If EI or the flexural rigidity does not vary along the beam then the equation simplifies to:
Simplified Euler-Bernoulli Beam Equation

Once the deflection due to a given load has been determined the stresses in the beam can be calculated using the following expressions:
The bending moment in the beam:
Bending Moment in Beam Equation

The shear force in the beam:
Shear Force in Beam Equation

Assumptions and Limitations of Beam Calculations

  • The beam is initially straight and any deflection of the beam follows a circular arc with the radius of curvature considered to remain large compared to the dimension of the cross section.
  • Deflections are assumed to be very small compared to the overall length of the beam.
  • The centroidal plane or neutral surface is subjected to zero axial stress and does not undergo any change in length.
  • Self-weight of the beam has been ignored and should be taken into account in practice.
  • The cross-section of the beam is considered small compared to its length meaning that the beam is long and thin.
  • The cross-section remains planar and perpendicular to the longitudinal axis during bending.
  • The response to strain is one dimensional stress in the direction of bending.
  • The material of the beam is homogenous and isotropic and has a constant Young’s modulus in all directions in both compression and tension.
  • Loads act transverse to the longitudinal axis and pass through the shear centre eliminating any torsion or twist.

Accuracy of the Beam Theory Calculations

From assumptions, a general rule of thumb is that for most configurations, the equations for flexural stress and transverse shear stress are accurate to within about 3% for beams with a length-to-height ratio greater than 4. The conservative nature of structural design (load factors) in most instances compensate for these inaccuracies.