# Difference between second moment of area and section modulus

1. Aug 25, 2016

### Peter10

Hi everyone!!

Please help, I have spent some considerable time to understand the two concepts and still this is nagging at me.... I am relating to Structural Engineering, just to let you know guys. My question is ..
Moment of inertia is about distribution of mass, the further away from the axis the higher the resistance to applied moment ( I believe I know the concepts of angular acceleration and etc.). Likewise in case of Elastic Section Modulus if the distribution of area of section is larger away from the axis the resistance to elastic deformation is greater... So in terms of Structural Engineering what is the difference between the two? I know that moment of inertia is related to mass (althou it is in cm^4 which is brings another question, why is it not in kg?), but as in practical sense why do we need one or the other to determine the strength of the beam, lets say up to elastic limit???

2. Aug 25, 2016

### SteamKing

Staff Emeritus
You are confusing two different concepts, namely the second moment of area and the mass moment of inertia, which are unfortunately and inaccurately known by the grab-all term "moment of inertia".

Let's take the "mass moment of inertia" of a body first.

We know from Newton's laws of motion that it takes a certain applied effort to change the motion of a body. For bodies in rectilinear motion, the equation F = ma describes how the acceleration of a body of mass m will change when a certain external force F is applied.

For bodies in rotational motion, the corresponding equation is T = J α, where J is the mass moment of inertia of the body (and which has units of ML2) and α is the angular acceleration (units of radians per second2) produced by the applied torque T (units of force times distance, or M × L × T-2 × L = ML2T-2). The mass moment of inertia J is an intrinsic property of the body, the value of which is influenced by the distribution of mass about the center of gravity.

The mass moment of inertia is used to calculate the dynamics of bodies undergoing motion.

The other moment of inertia, the second moment of area, is the one which is most commonly encountered in structural engineering. The second moment of area, usually denoted as I, arises from studying the bending of beams. The bending stress σ in a beam is given by the equation σ = M y / I, where M is the applied bending moment, y is the distance from the neutral axis to where the bending stress is sought, and I is the second moment of area of the cross section of the beam. This inertia is roughly analogous to J in that it describes the distribution of section area about the centroid of the section. The units of I are A × L2 (compare the units of J = M × L2), which is why tables of structural properties have units like cm4.

In beam analysis, the maximum bending stress is encountered at the outer fiber of the beam, which is located at the greatest distance from the neutral axis.

Thus for a given structural section, I can be calculated and y is known from the geometry, so the bending stress formula can be re-written as σ = M / SM, where SM is the section modulus of the beam cross section and SM = I / y (units of L3. By knowing the section modulus of a beam at a given location, it is thus a simple calculation to find out how much bending moment will produce a given level of bending stress. If you take a beam with a limiting elastic stress σ, then the maximum bending moment is M = σ × SM.

3. Aug 26, 2016

Thank you!!