Physics: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 [Σad].
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 (dAi). Let xi and yi 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: [math]\displaystyle{ S_x = A \bar y = \sum_{i=1}^n {y_i \, dA_i} = \int_A y \, dA }[/math] and [math]\displaystyle{ S_y= A \bar x = \sum_{i=1}^n {x_i \, dA_i} = \int_A x \, dA. }[/math]
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: [math]\displaystyle{ Q_{j,x} = \int y_i \, dA, }[/math]
where
- Qj,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: [math]\displaystyle{ q = \frac{V_y S_x}{I_x} }[/math]
- 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
Shear stress may now be calculated using the following equation: [math]\displaystyle{ \tau = \frac{q}{t} }[/math]
- [math]\displaystyle{ \tau }[/math] – 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
References
- ↑ Shigley's Mechanical Engineering Design, 9th Ed. (Page 96)
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