First moment of area

Given an area, A, of any shape, and division of that area into n number of very small, elemental areas (dA_i). Let x_i and y_i 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:

${\displaystyle S_{x}=A{\bar {y}}=\sum _{i=1}^{n}{y_{i}\,dA_{i}}=\int _{A}ydA}$

and

${\displaystyle S_{y}=A{\bar {x}}=\sum _{i=1}^{n}{x_{i}\,dA_{i}}=\int _{A}xdA}$.

The SI unit for first moment of area is a cubic metre (m³). In the American Engineering and Gravitational systems the unit is a cubic foot (ft³) 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:

${\displaystyle Q_{j,x}=\int y_{i}dA,}$

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.

• Second moment of area
• Polar moment of inertia
• Section modulus