Page 42 - Handbook of Civil Engineering Calculations, Second Edition

STATICS, STRESS AND STRAIN, AND FLEXURAL ANALYSIS 1.25

PRODUCT OF INERTIA OF AN AREA

Calculate the product of inertia of the isosceles trapezoid in Fig. 14 with respect to the
rectangular axes u and v.

Calculation Procedure:
1. Locate the centroid of
the trapezoid
Using the AISC Manual or another
suitable reference, we find h cen-
troid distance from the axis (Fig. 14)
(9/3)[(2 5 10)/(5 10)] 4
in. (101.6 mm).
2. Compute the area and
product of inertia P xy
The area of the trapezoid is A
1 /2(9)(5 10) 67.5 sq.in. (435.5 FIGURE 14
2
cm ). Since the area is symmetrically
disposed with respect to the y axis,
the product of inertia with respect to
the x and y axes is P xy 0.
3. Compute the product of inertia by applying the transfer
equation
The transfer equation for the product of inertia is P uv P xy Ax m y m , where x m and y m are
the coordinates of O
with respect to the centroidal x and y axes, respectively. Thus P uv
4
4
0 67.5( 5)( 4) 1350 in (5.6 dm ).

PROPERTIES OF AN AREA WITH RESPECT
TO ROTATED AXES

In Fig. 15, x and y are rectangular axes through the centroid of the isosceles triangle; x
and y
are axes parallel to x and y, respectively; x and y are axes making an angle of 30°
with x
and y
, respectively. Compute the moments of inertia and the product of inertia of
the triangle with respect to the x and y axes.

Calculation Procedure:
1. Compute the area of the figure
2
The area of this triangle 0.5(base)(altitude) 0.5(8)(9) 36 sq.in. (232.3 cm ).
2. Compute the properties of the area with respect to the
x and y axes
3
Using conventional moment-of-inertia relations, we find I x bd /36 8(9)3/36 16 2
4
4
4
4
3
in (0.67 dm ); I y b d/48 (8)3(9)/48 96 in (0.39 dm ). By symmetry, the product
of inertia with respect to the x and y axes is P xy 0.