Bending Equation Derivation

In applied mechanics, bending is defined as the deformation of a structure in one of the longitudinal planes due to some force. This force is also assumed to be applied in the direction of one of the longitudinal planes associated with the structure. Here we will derive the equation of bending of a beam. A structure is said to be a beam if two of its dimensions are smaller by a factor of 1/10 or less compared to the third dimension.

A beam is deformed when a transverse load is applied. This load acts in the direction perpendicular to the longitudinal plane of the beam. The stress experienced by the beam and the amount of load applied is assumed to be unchanging with time.

A classic example is a beam supported at its ends by some clamps. If a force is applied in the direction perpendicular to its longitudinal plane, then the upper part of the beam is compressed and the lower part of the beam is stretched.

Here, the beam has two kinds of stress associated with it. First is the shear stress that acts parallel to the direction of the lateral loading. The second is compression stress in the upper region of the beam and the tensile stress in the lower region. The compression and tensile stress form a couple or a moment. This moment is responsible for providing resistance to the sagging and deformation of the beam.

The sagging or deformation can be accurately determined if certain simplifying assumptions are made.

Assumptions

The assumptions made to simplify the calculation of the bending equations are:

The beam is assumed to be straight. Examples can be wooden beams used in the building of houses etc.
The beam is assumed to have a constant cross-sectional area. This means that the cross-sectional area of a beam should not change with its length.
The beam should have symmetrical longitudinal planes. This means that all the dimensions of the beam should be perpendicular to each other.
The vector sum of all the forces associated with the load should be directed in the planes of symmetry. This means that the forces should not have a resultant sum in any other direction than the direction of the symmetrical planes.
The primary cause of failure is assumed to be buckling.
E remains the same for tension and compression. E is Young’s modulus of the beam.
The cross-sectional area of the beam is assumed to be the same after the bending.

Derivation

To derive the bending equation, we shall consider the below-given arrangement:

The first image shows the beam without any applied pressure. We note that the beam does not have any strain. It is also free of any deformation. A, B, C, D are four points marked on the beam. The length AB = CD. The perpendicular distance between the lines AB and CD is y. CD is the neutral axis of the beam. The neutral axis is the locus where the beam experiences zero strain.

Now, consider the same beam when it is under pressure. We notice that the shape of the beam has deformed due to stress. It has taken the shape of an arc with the central angle θ. R is the radius of the given arc. The cross-sectional area is assumed to be unchanged. The points A’, B’, C’, D’ mark the changed positions of the points A, B, C, D.

Now, strain is the ratio of change in length experienced by a certain locus on the beam to the original length. Therefore the strain in AB can be expressed as: