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MODULE 1. BASIC CONCEPTS

MODULE 2. SYSTEM OF FORCES

MODULE 3.

MODULE 4. FRICTION AND FRICTIONAL FORCES

MODULE 5.

MODULE 6.

MODULE 7.

MODULE 8.

## LESSON 10. Triangular Area

**Triangular Area**

From Eq. (11.7), moment of inertia of a triangle about its base *x*-*x* as shown in Fig.12 is

* I _{x}* = \[{{b{d^3}} \over {12}}\]

Fig.12 Moment of inertia of a triangle about its centroidal axis

The moment of inertia about centroidal axis is

*I _{Gx}* =

*I*

_{x}_{ }–

*Ah*

^{2}= Ah

^{2}\[{{b{d^3}} \over {12}} - {{bd} \over 2}{\left( {{d \over 3}} \right)^3}\]

* I _{Gx}* = \[{{b{d^3}} \over {36}}\] (11.13)

**Semicircular Area**

Equation (11.9) gives the moment of inertia of semicircular area about *x* and *y* axes as given in Fig.13. That is

*I _{x}* =

*I*= \[{{{R^4}} \over 8}\]

_{y}Fig.13 Moment of inertia of a semicircular about its centroidal axis

However, *y*-axis passes through the centroid *G*, hence, moment of inertia about centroidal *y*-axis is same as

*I _{Gy }*=

*I*= \[{{{R^4}} \over 8}\]

_{y}Moment of inertia about centroidal

*x*-axis is

*I _{Gx}* =

*I*–

_{x}*Ah*

^{2}= \[{{{R^4}} \over 8} - {{{R^2}} \over 2}{\left( {{{4R} \over {3}}} \right)^2}\]

*I*

_{Gx}_{ }= 0.11

*R*

^{4}(11.14)

**Quarter Circular Area**

Moment of inertia of quarter circular area about *x* and *y* axes [Eq.(11.10)] as shown in Fig.14 is

*I _{x}*

_{ }=

*I*= \[{{{R^4}} \over 16}\]

_{y}

Fig.14 Moment of inertia of a quarter circle about its centroidal axis

Therefore, moment of inertia about centroidal axes (*G _{x} *and

*G*) is determined as

_{y}*I _{Gx}* =

*I*

_{x}_{ }–

*Ah*

^{2}= \[{{R^4}} \over {16}}\] - \[{{{R^2}} \over 4}{\left( {{{4R} \over {3}}} \right)^2}\]

*I*= 0.055

_{Gx}*R*

^{4 }(11.15a)

Similarly, *I _{Gy}*

_{ }will be obtained and it is equal to

*I*

_{Gx}_{ }itself.

* I _{Gy}* = 0.055

*R*

^{4 }(11.15b)

Moment of Inertia of hollow rectangular and circular sections about their centroidal axes are given in Fig.15.

Fig.15 (a) Hollow rectangular section and (b) Hollow circular section

Hollow rectangular section:

* I _{Gx}* = \[{1 \over {12}}\left( {B{D^3} - b{d^3}} \right)\] (11.16a)

* I _{Gy}* = \[{1 \over {12}}\left( {D{B^3} - d{b^3}} \right)\] (11.16b)

Hollow circular section:

* I _{Gx}* =

*I*= \[{ \over 4}\left( {{R^4} - {r^4}} \right)\] (11.17)

_{Gy}