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_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.

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:

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

• 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:

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

• ${\displaystyle \tau }$ - 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)