Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

7.8 Second Moments of Areas 185

EXAMPLE 7.3 Estimate the inside volume of a soda can. We have used a ruler to measure the height and the

diameter of the can, as shown in Figure 7.19.

We may approximate the inside volume of the soda can using the vol-

ume of a cylinder of equal dimensions:

Compared to the 355-mL value shown on a typical soda can, the approx-

imated value seems reasonable. The difference between the approximated

value and the indicated value may be explained in a number of ways. First,

the soda container does not represent a perfect cylinder. If you were to

look closely at the can you would note that the diameter of the can reduces

at the top. This could explain our overestimation of volume. Second, we

measured the outside diameter of the can, not the inside dimensions. However, this approach

will introduce smaller inaccuracies because of the small thickness of the can.

We could have measured the inside volume of the can by filling the can with water and then

pouring the water into a graduated cylinder or beaker to obtain a direct reading of the volume.

Finally, it is worth noting here that numerical solid modeling is an engineering topic that

deals with computer generation of the surface areas and volumes of an actual object. Solid-

modeling software programs are becoming quite common in engineering practice. Computer-

generated solid models provide not only great visual images but also such information as

magnitude of the area and the volume of the model. To generate numerical solid models of

simple shapes, area and volume primitives are used. Other means of generating surfaces include

dragging a line along a path or rotating a line about an axis, and, as with areas, you can also gen-

erate volumes by dragging or sweeping an area along a path or by rotating an area about a line.

We will discuss computer solid modeling ideas in more detail in Chapter 16.

7 7..8 8 SSeeccoonndd MMoommeennttss ooff AArreeaass

In this section, we will consider a property of an area known as thesecond moment of area. The

second moment of area, also known as thearea moment of inertia, is an important property of

an area that provides information on how hard it is to bend something. Next time you walk by

a construction site, take a closer look at the cross-sectional area of the support beams, and notice

how the beams are laid out. Pay close attention to the orientation of the cross-sectional area of

an I-beam with respect to the directions of expected loads. Are the beams laid out in the ori-

entation shown in Figure 7.20(a) or in Figure 7.20(b)?

Steel I-beams, which are commonly used as structural members to support various loads,

offer good resistance to bending, and yet they use much less material than beams with rectangu-

lar cross sections. You will find I-beams supporting guard rails and I-beams used as bridge cross

members and also as roof and flooring members. The answer to the question about the orienta-

tion of I-beams is that they are oriented with respect to the loads in the configuration shown in

Figure 7.20(a). The reason for having I-beams support loads in that configuration is that about

VpR

2

h 1 3.14152a

6.3 cm

2

b

2

1 12.0 cm 2 374 cm

3

374 mL

diameter = 6.3 cm

12 cm

■Figure 7.19

Soda can in Example 7.3.

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