 # Online Beam Calculator Index  • Load Uniformly Increasing to One End
• Load Uniformly Increasing to Centre
• Partially Distributed Loads At Each End
• Partially Distributed Uniform Load At One End
• Uniformly Distributed Load & Variable End Moments

##### Fixed Beam One End, Supported at Other End

• 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

##### 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 On Overhang
• 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: 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: 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: The shear force in the beam: #### 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.