Analysis of Partial Beam Section:
Until now, we have seen forces & moments acting over the whole cross-section of the beam. Now, let us assume a part of the beam section and attempt to establish a relation for the normal force working on the partial beam section & the moment of the normal force regarding the neutral axis.
Normal Force
Consider a beam section as illustrated in Figure.
Consider σmax be maximum bending stress at the topmost layer and ymax be the distance of the top most layer from the neutral axis.
Let us discover the normal force on a partial area A, illustrated shaded in Figure. In that shaded area, consider da be an elemental area & y be its distance from the neutral axis.
Beam Section Bending Stress Distribution
Figure
The stress, σ on elemental area is proportional to the distance y.
σ = σmax ×(y/ ymax)
Normal force on the elemental area = σ × da
= σmax (y/ ymax)
Total normal force on the shaded area = ∑ σmax (y/ ymax)
= (y /ymax) ∑ y da
Let be the distance of the centroid of the shaded area from the neutral axis.
∴ Normal force on a partial beam section =