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 .
First moment of area is commonly used to determine the centroid of an area.
Definition
Given an area A of any shape and a division of that area into n very small elemental areas, let xᵢ and yᵢ be the distances to each elemental area measured from a given x–y axis. 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. In the American Engineering and Gravitational systems the unit is a cubic foot or more commonly inch3.
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:
whereQj,x – the first moment of area "j" about the neutral x axis of the entire body ;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
- 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
- – 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