First moment of area

The first moment of area, sometimes misnamed as the first moment of inertia, is based in the mathematical construct moments in metric spaces, stating that the moment of area equals the summation of area times distance to an axis [Σ(a × d)]. It is a measure of the distribution of the area of a shape in relation to an axis.

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_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:

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

$q={\frac {V_{y}S_{x}}{I_{x}}}$

• 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:

${\tau }={\frac {q}{t}}$

• ${\tau }$ - 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]

See also

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

References

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