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

620 Chapter 18 Mathematics in Engineering

an important role in design of structures. We explained that, for a small area elementAlocated

at a distancexfrom the axisy–y, as shown in Figure 18.17, the area moment of inertia is

defined by

(18.31)

We also included more small area elements, as shown in Figure 18.18. The area moment

of inertia for the system of discrete areas shown about they–yaxis is now

Iyyx

2

A

(18.32)

Similarly, we can obtain the second moment of area for a cross-sectional area, such as a rect-

angle or a circle, by summing the area moment of inertia of all the little area elements that

makes up the cross-section. However, for a continuous cross-sectional area, we use integrals

instead of summing thex

2

Aterms to evaluate the area moment of inertia. After all, the integral

sign, , is nothing but a big “S” sign, indicating summation.

(18.33)

We can obtain the area moment of inertia of any geometric shape by performing the

integration given by Equation (18.33). For example, let us derive a formula for a rectangular

cross-section about they–yaxes.

step 1 step 2 step 3 step 4

Step 1: The second moment of the rectangular cross-sectional area is equal to the sum (inte-

gral) of little rectangles.

Step 2: We substitute fordAhdx(see Figure 18.19).

Step 3: We simplify by taking outh(constant) outside the integral.

Step 4: The solution. We will discuss the integration rules later.

Iyy 

w /2

w /2

x

2

d A 

w /2

w /2

x

2

hdxh

w /2

w /2

x

2

dx

1

12

hw

3

u u u u

Iyyx

2

d A

Iyyx

2

1 A 1 x

2

2 A 2 x

2

3 A 3

A 1

y

y

x 1

x 2 A 2

■Figure 18.18 x 3 A 3

Second moment of area for three

small area elements.

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