MOMENT OF INERTIA Type of moment of inertia

MOMENT OF INERTIA

Type of moment of inertia � Moment of inertia of Area � Moment of inertia of mass Also known as second moment Why need to calculate the moment of Inertia? To measures the effect of the cross sectional shape of a beam on the beam resistance to a bending moment Application � Determination of stresses in beams and columns Symbol � I – symbol of area of inertia Ix, Iy and Iz

Application : Design Steel ( Section properties)

Moment of Inertia of Area y y dx dy h C b dy x h C b Area of shaded element, Moment of inertia about x-axis Moment of inertia about y-axis Integration from h/2 to h/2 Integration from b/2 to b/2 x

Table 6. 2. Moment of inertia of simple shapes Shape J = polar moment of inertia y 1. Triangle h x b y 2. Semicircle r 3. Quarter circle x y r x y 4. Rectangle h 5. Circle y b x r x

PARALLEL - AXIS THEOREM There is relationship between the moment of inertia about two parallel axes which is not passes through the centroid of the area. y h x b From Table 6. 1; Ix = The centroid is, ( Moment of inertia about x-x axis, Ixx = Ix + Ady 2 � where dy is distance at centroid y Moment of inertia about y-y axis, Iyy = Iy + Asx 2 where , and Iy = ) = (b/2, h/2) sx is distance at centroid x

Example 6. 2 y y 140 mm 60 mm 1 3 2 x x 60 mm 160 mm Determine centroid of composite area PART AREA(mm 2) y(mm) x(mm) 1 60(200) = 12000 160(60)=9600 200/2 = 100 60/2 = 30 60(200) = 12000 200/2 = 100 2 3 Σ: 33 600 Ay (103)(mm 3) Ax (103) (mm 3) 60/2 = 30 1200 360 60 + [160/2] = 140 220 +60/2= 250 288 1344 1200 3000 Σ: 2688 x 103 Σ: 4704 x 103

Second moment inertia PART 1 AREA (A)(mm 2) 60(200) = 12000 Ix = bh 3/12 (106) (mm 4) dy = |y-y|(mm) 60(2003)/12 = 40 |100– 80| = 20 2 3 160(60)=9600 60(200) = 12000 Ady 2(106)(mm 4) 4. 8 160(603)/12 = 2. 88 |30 – 80| = 50 24 |100 – 80| =20 4. 8 60(2003)/12 = 40 Σ: 33 600 = = PART 1 2 3 [Ix + Ady 2]1 + [Ix + Ady 2]2 +[Ix + Ady 2]3 [44. 8 + 26. 88 + 44. 8] x 106 116. 48 x 106 mm 4 AREA(mm 2) 60(200) = 12000 160(60)=9600 60(200) = 12000 Iy = b 3 h/12 (106) (mm 4) 603(200)/12 =3. 6 1603(60)/12 =20. 48 603(200)/12 =3. 6 Σ: 33 600 [Iy + As 2]1 + [Iy + As 2]2 +[Iy + As 2]3 = [148. 8 + 20. 48 + 148. 8] x 106 = 318. 08 x 106 mm 4 Sx=|x-x| (mm) |30 -140|=110 Asx 2(106)(mm 4) 145. 2 |140 – 140 |= 0 0 |250 – 140|=110 145. 2