# 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

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).