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

Ix = ${{b{d^3}} \over {12}}$

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

The moment of inertia about centroidal axis is

IGx = Ix Ah2 = Ah2 ${{b{d^3}} \over {12}} - {{bd} \over 2}{\left( {{d \over 3}} \right)^3}$

IGx = ${{b{d^3}} \over {36}}$           (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

Ix = Iy = ${{{R^4}} \over 8}$

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

IGy = Iy = ${{{R^4}} \over 8}$

Moment of inertia about centroidal x-axis is

IGx = IxAh2 = ${{{R^4}} \over 8} - {{{R^2}} \over 2}{\left( {{{4R} \over {3}}} \right)^2}$

IGx = 0.11R4                                                            (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

Ix = Iy${{{R^4}} \over 16}$

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

Therefore, moment of inertia about centroidal axes (Gx and Gy) is determined as

IGx = Ix Ah2 = ${{R^4}} \over {16}}$ - ${{{R^2}} \over 4}{\left( {{{4R} \over {3}}} \right)^2}$

IGx = 0.055R4                                                           (11.15a)

Similarly, IGy will be obtained and it is equal to IGx itself.

IGy = 0.055R4                                                     (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:

IGx = ${1 \over {12}}\left( {B{D^3} - b{d^3}} \right)$                                                     (11.16a)

IGy =  ${1 \over {12}}\left( {D{B^3} - d{b^3}} \right)$                                                     (11.16b)

Hollow circular section:

IGx = IGy =  ${ \over 4}\left( {{R^4} - {r^4}} \right)$                                                     (11.17)