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Appendix B: Mechanics
Fig. B.1 Plane surface with
centroid S
B.2 Second Moment of Area
The second moment of area
2 or the second area moment is a geometrical property
of a surface which reflects how its area elements are distributed with regard to an
arbitrary axis. The second moments of area for an arbitrary surface with respect to
an arbitrary Cartesian coordinate system (see Fig. B.1) are generally defined as:
I Y =
A
Z
2 dA ,
(B.5)
I Z =
A
Y
2 dA .
(B.6)
These quantities are normally used in the context of plane bending of symmetrical
cross sections. For unsymmetrical bending, the product moment of area is additionally required:
I Y Z = −
A
Y ZdA .
(B.7)
B.3 Parallel-Axis Theorem
The parallel-axis theorem gives the relationship between the second moment of area
with respect to a centroidal axis (Z 1 , Y 1 ) and the second moment of area with respect
to any parallel axis
3
(Z , Y ). For the rectangular shown in Fig. B.2, the relations can
be expressed as:
2 The second moment of area is also called in the literature the second moment of inertia. However,
the expression moment of inertia is in context of properties of surfaces misleading since no mass
or movement is involved.
3 This arbitrary axis can be for example the axis trough the common centroid S of a composed
surface.