# First moment of area

The **first moment of area** is based on the mathematical construct moments in metric spaces. It is a measure of the spatial distribution of a shape in relation to an axis.

The first moment of area of a shape, about a certain axis, equals the sum over all the infinitesimal parts of the shape of the area of that part times its distance from the axis [Σ(*a* × *d*)].

First moment of area is commonly used to determine the centroid of an area.

## Definition

Given an area, *A*, of any shape, and division of that area into *n* number of very small, elemental areas (*dA _{i}*). Let

*x*and

_{i}*y*be the distances (coordinates) to each elemental area measured from a given

_{i}*x-y*axis. Now, the first moment of area in the

*x*and

*y*directions are respectively given by:

and

- .

The SI unit for **first moment of area** is a cubic metre (m^{3}). In the American Engineering and Gravitational systems the unit is a cubic foot (ft^{3}) or more commonly inch^{3}.

The **static** or **statical moment of area**, usually denoted by the symbol *Q*, is a property of a shape that is used to predict its resistance to shear stress. By definition:

where

*Q*_{j,x}- the first moment of area "j" about the neutral*x*axis of the entire body (not the neutral axis of the area "j");*dA*- an elemental area of area "j";*y*- the perpendicular distance to the centroid of element*dA*from the neutral axis*x*.

### Shear stress in a semi-monocoque structure

The equation for shear flow in a particular web section of the cross-section of a semi-monocoque structure is:

*q*- the shear flow through a particular web section of the cross-section*V*_{y}- the shear force perpendicular to the neutral axis*x*through the entire cross-section*S*_{x}- the first moment of area about the neutral axis*x*for a particular web section of the cross-section*I*_{x}- the second moment of area about the neutral axis*x*for the entire cross-section

Shear stress may now be calculated using the following equation:

- - the shear stress through a particular web section of the cross-section
*q*- the shear flow through a particular web section of the cross-section*t*- the thickness of a particular web section of the cross-section at the point being measured[1]

## References

- Shigley's Mechanical Engineering Design, 9th Ed. (Page 96)