Moment of area

Cross sectional area

Moments of area are cross-sectional parameters : they describe how the shape of the cross-sectional area influences the properties of elongated components .

The moments of area up to the 2nd degree ( area moment of inertia ) are used in particular for the strength calculation of components and structures. Moments of area from the 1st degree are related to a certain orientation of the cross-section, so they have different values ​​in different directions (i.e. depending on which axis they are related to).

Moments of area of ​​the nth degree (general definition)

In the following brief descriptions, the bar is parallel to the x-axis of the coordinate system and has the cross-sectional area in the yz-plane. The bar lies symmetrically in the coordinate system (more precisely: the main axes of inertia are parallel to the y and z axes).

The nth degree moment of area has the following general form:

${\displaystyle A_{i,j}=\int _{A}y^{i}\;z^{j}\;\mathrm {d} A}$         With       ${\displaystyle n=i+j}$

Moment of area of ​​0th degree

The moment of area of ​​the 0th degree corresponds to the cross-sectional area and has the unit  m ² :

${\displaystyle A=\int _{A}\mathrm {d} A}$

Moment of area 1st degree

The area moments of the 1st degree are also referred to as the static moment and have the unit  m ³ .

In relation to the y-axis:

${\displaystyle S_{y}=\int _{A}z\;\mathrm {d} A}$

In relation to the z-axis:

${\displaystyle S_{z}=\int _{A}y\;\mathrm {d} A}$

The two coordinates of the center of gravity can be calculated from these moments (if the area is known ) : ${\displaystyle A}$

${\displaystyle z_{s}={\frac {S_{y}}{A}};\quad y_{s}={\frac {S_{z}}{A}}}$

Second moment of area

→ Main article : Area moment of inertia

The area moments of the 2nd degree are also known under the term area moment of inertia and have the unit [m ⁴ ].

They indicate the influence of the shape of the cross-sectional area of ​​a beam on its stiffness . The moment of resistance , which describes the influence of the shape of the cross-sectional area on the strength , is also derived from the area moment of inertia .

Second moment of area related to the y-axis:

${\displaystyle I_{y}=\int _{A}z^{2}\;\mathrm {d} A}$

Second moment of area related to the z-axis:

${\displaystyle I_{z}=\int _{A}y^{2}\;\mathrm {d} A}$

biaxial area moment of inertia:

${\displaystyle I_{yz}=I_{zy}=-\int _{A}yz\ \mathrm {d} A}$

polar area moment of inertia:

${\displaystyle I_{P}=I_{y}+I_{z}=\int _{A}r^{2}\ \mathrm {d} A}$

Individual evidence

1. ↑ Jens Göttsche, Maritta Petersen: Strength theory - in a nutshell . Hanser Verlag, 2006, ISBN 978-3-446-40415-1 ( limited preview in Google book search).