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.
Given an area, A, of any shape, and division of that area into n number of very small, elemental areas (dAi). Let xi and yi be the distances (coordinates) to each elemental area measured from a given x-y axis. Now, the first moment of area in the x and y directions are respectively given by:
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:
- Qj,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
- q - the shear flow through a particular web section of the cross-section
- Vy - the shear force perpendicular to the neutral axis x through the entire cross-section
- Sx - the first moment of area about the neutral axis x for a particular web section of the cross-section
- Ix - 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:
- Shigley's Mechanical Engineering Design, 9th Ed. (Page 96)