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.