Appendix B
Mechanics
B.1 Centroids
The coordinates (Z S , Y S ) of the centroid S of the plane surface shown in Fig. B.1
can be expressed as
Z S =
Z d A
d A
,
(B.1)
Y S =
Y d A
d A
,
(B.2)
where the integrals
Z d A and
Y d A are known as the first moments of area
1 . In
the case of surfaces composed of n simple shapes, the integrals can be replaced by
summations to obtain:
Z S =
n
i = 1
Z i A i
n
i = 1
A i
,
(B.3)
Y S =
n
i = 1
Y i A i
n
i = 1
A i
.
(B.4)
1 A better expression would be moment of surface since area means strictly speaking the measure
of the size of the surface which is different to the surface itself.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2
155
Mechanics
B.1 Centroids
The coordinates (Z S , Y S ) of the centroid S of the plane surface shown in Fig. B.1
can be expressed as
Z S =
Z d A
d A
,
(B.1)
Y S =
Y d A
d A
,
(B.2)
where the integrals
Z d A and
Y d A are known as the first moments of area
1 . In
the case of surfaces composed of n simple shapes, the integrals can be replaced by
summations to obtain:
Z S =
n
i = 1
Z i A i
n
i = 1
A i
,
(B.3)
Y S =
n
i = 1
Y i A i
n
i = 1
A i
.
(B.4)
1 A better expression would be moment of surface since area means strictly speaking the measure
of the size of the surface which is different to the surface itself.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2
155