Theories Of Structural Mechanics: First Moment Of Area And Moment Of Inertia

Introduction

The purpose of Structural Mechanics is to lay the foundations for a thorough understanding of the basic engineering principles that apply to all structural analysis and design. In this project, we were required to do the research on two theories. The First moment of area and Moment of inertia. Once we have read about these theories, we must summaries the theories and apply them to the beam section that was given.

First moment of area

The first moment of an area quantifies the distribution of an area about an axis. It is calculated as the summation of the product of each area and its distance from the axis. While it may seem mostly mathematical, one fantastic application of the first moment is to check if an area is "symmetrically distributed" about an axis. That is, all points are evenly spread out from that axis. This symmetry has deep implications in structural analysis.

The second application is in the calculation of the centroid of the area, which is that point about which the entire area may be "lumped"- if you hold a cut out of the area at that point only, it will be perfectly balanced. It is a sort of "average" measure of the spread of all points. It could be considered (in a hand wavy sense) as that point about which the area distributes itself equally in all directions. As with any average, you'd sum each area multiplied by its distance from an axis and divide by the total area.

Moment of inertia

The second moment of the area, which is also called the area moment of inertia, talks about the internal resistance or inertia of the area to rotation. As far as torques are considered, you'd need to calculate "MASS" moments of inertia, which can be sometimes derived from area moments of inertia. But they are completely different quantities. In a simple sense, the mass moment of inertia of a body about an axis, would be the amount of torque required to rotate that body with a unit angular acceleration about the axis. All this can get a bit overwhelming in the beginning, Area moments do not talk about mass of the areas or densities or how the material itself is distributed. They only talk about distribution of points of that area about an axis. Mass moments talk about how mass is distributed about an axis. So, they involve densities too. The mass moment is often more intuitive to understand- the further away a mass is from the axis of rotation, the tougher it is to rotate. It would require a greater torque to rotate.

Conclusion

The principles which were assigned to us serve as a very important tool in structures and in civil engineering as whole. The First moment of area helps us to find the centroid and the point of where the section will perfectly balance. The moment of inertia helps us to find the Ixx and Iyy of the cross section.