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 �''ad'' First moment of area is commonly used to determine the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

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:

S_x = A \bar y = \sum_^n  = \int_A y \, dA

and

S_y= A \bar x = \sum_^n  = \int_A x \, dA.

The SI unit for first moment of area is a cubic

metre The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...

(m³). In the American Engineering and Gravitational systems the unit is a cubic

foot The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg mad ...

(ft³) or more commonly

inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth ...

³. 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 Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...

. By definition:

Q_ = \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 The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate: * a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960) ...

in a particular web section of the cross-section of a

semi-monocoque The term semi-monocoque or semimonocoque refers to a stressed shell structure that is similar to a true monocoque, but which derives at least some of its strength from conventional reinforcement. Semi-monocoque construction is used for, among ot ...

structure is:

q = \frac

*''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 The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...

about the neutral axis ''x'' for the entire cross-section

Shear stress Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...

may now be calculated using the following equation:

\tau = \frac

\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 measuredShigley's Mechanical Engineering Design, 9th Ed. (Page 96)

References

{{Reflist Solid mechanics Moment (physics)

Definition

Shear stress in a semi-monocoque structure

See also

References