Chapter 7 Moment of Inertia Moment of Inertia

Chapter 7 Moment of Inertia

Moment of Inertia • A mathematical property of the area that describes the second moment of area about an axis • It describes an object’s resistance to bending to angular rotation

Moment of Inertia of Simple Areas • When an object is subjected to bending, a rotation occurs about some axis in the shape • The axis is referred to as the neutral axis • Note: the black and green bars represent bending forces developed by a beam

Moment of Inertia of Simple Areas (Cont’d) • Ix = y 2 A • Ix = Moment of inertia about the X axis • y = distance from the neutral axis or axis of bending • A = incremental area being considered

Parallel Axis Theorem • Ix = Ixc + Ad 2 or • Iy = Iyc + Ad 2 • Ix, Iy = moment of inertia desired about some parallel x or y axis, respectively

Parallel Axis Theorem (Cont’d) • Ixc, Iyc = moment of inertia about the shape’s centroidal x or y axis, respectively • A = area of shape • d = the perpendicular distance between the axis of interest and the centroidial axis

Moment of Inertia of Composite Areas X Shape # Ixc A d Ad 2 Ix total d Ad 2 Iy total Y Shape # Iyc A

Moment of Inertia of Composite Areas (Cont’d) X Shape # • • Ixc A = area of shape d = distance Ad 2 = Area * distance 2 Ix total = Ix + Ad 2 A d Ad 2 Ix total

Polar Moment of Inertia • The polar moment of inertia for a given shape represents that shape’s resistance to torsion or twisting forces • J = r 2 A • r = radial distance from the longitudinal axis to incremental area • A = incremental unit of area

Radius of Gyration • The relationship between a shape’s moment of inertia and its area