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Measurement of a shape about a certain axis
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.
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: S x = A y ¯ = ∑ i = 1 n y i d A i = ∫ A y d A {\displaystyle S_{x}=A{\bar {y}}=\sum _{i=1}^{n}{y_{i}\,dA_{i}}=\int _{A}y\,dA} and S y = A x ¯ = ∑ i = 1 n x i d A i = ∫ A x d A . {\displaystyle S_{y}=A{\bar {x}}=\sum _{i=1}^{n}{x_{i}\,dA_{i}}=\int _{A}x\,dA.}
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 = ∫ y i d A , {\displaystyle Q_{j,x}=\int y_{i}\,dA,}
where
The equation for shear flow in a particular web section of the cross-section of a semi-monocoque structure is: q = V y S x I x {\displaystyle q={\frac {V_{y}S_{x}}{I_{x}}}}
Shear stress may now be calculated using the following equation: τ = q t {\displaystyle \tau ={\frac {q}{t}}}
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