The formula for the moment of inertia I=∫(y^2)da reveals that the resistance of any differential area da increases with its distance y from the neutral axis squared, forming a parabolic distribution. For a beam of rectangular cross-section, the resistance of top and bottom fibers with distance y = d/2 from the neutral axis is (d/2)^2. Thus, the moment of inertia, as geometric resistance, is the volume of all fibers under a parabolic surface, which is 1/3 the volume of a cube of equal dimensions, or I= bd ((d/2)^2)/3, or
the moment of inertia of a rectangular beam of homogeneous material. A formal calculus derivation of this formula is given in Appendix A. The section modulus gives only the maximum bending stress, but the moment of inertia gives the stress at any distance c from the neutral axis as f= Mc/I. This is useful, for example, for bending elements of asymmetrical cross-section, such as T- and L-shapes.
1 Bending stress distribution over beam cross-section
2 Moment of inertia visualized as volume under parabolic surface
3 T-bar with asymmetrical stress: max. stress at c2 from the neutral axis
4 Angle bar with asymmetrical stress distribution about x, y, and z-axes: maximum resistance about x-axis and minimum resistance about z-axis
1 Bending stress distribution over beam cross-section
2 Moment of inertia visualized as volume under parabolic surface
3 T-bar with asymmetrical stress: max. stress at c2 from the neutral axis
4 Angle bar with asymmetrical stress distribution about x, y, and z-axes: maximum resistance about x-axis and minimum resistance about z-axis
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