Neara uses the following calculations to calculate the strength capacity of a rectangular beam.
The maximum bending moment of an object is calculated using the general formula:
As the section modulus, and consequentially the moment capacity, is different for different directions of applied force, a diamond plot formula is used that takes the angle of applied force (or the angle of the neutral axis, which is assumed perpendicular to the angle of applied force), and section moduli around principal axes as the inputs. This formula determines the resultant section modulus and max moment about the neutral axis.
Section moduli and bending moment capacity about principal axes
The section modulus for a rectangular beam with an arbitrary cross section assuming the bending moment is result of a tip load force perpendicular to the neutral axis and parallel to one of the principal axes of a cross section is calculated by:
For example consider a wooden beam that is 100mm x 200mm that has a force applied on the 100mm face and a maximum bending stress (modulus of rupture) of 100MPa.
The bending moment capacity for a force applied to the 100mm face (about vertical principal axis) is calculated as:
And the maximum bending moment for a force applied to the 200mm face is calculated as:
To calculate the maximum bending moment for forces acting along any orientation on the beam, a polar plot is formed where the magnitude/length of the vector is the bending moment capacity in that direction.
To illustrate, consider the case above where the force is acting on the 100mm face and the magnitude of the bending capacity is 66.7kNm. This can be seen in the diagram below as the length of the vector in the polar plot.
Now consider the case where the force is applied to the beam at a 45° angle. The bending moment capacity is the length of the vector from the neutral axis's origin to the intersection point of the diamond plot and a line at a 45° angle.
Using trigonometry, the magnitude of this vector is calculated to be 31.41 kNm. From this example it is clear that the maximum strength of a rectangular beam against bending stresses will be obtained when the force is perpendicular to one of its faces (perpendicular to the major principal axis).
