Moments of inertia for areasMoments of inertia for areas

Previously determined centroid for an area by Previously determined centroid for an area by considering first moment of area about an axis, considering first moment of area about an axis,

Moment of inertia Moment of inertia →→ second moment of an area second moment of an area (“moment of inertia” is a misnomer)(“moment of inertia” is a misnomer)

Moments of inertia of a differential planar area dA Moments of inertia of a differential planar area dA about the x and y axes are dIabout the x and y axes are dIxx = y = y22dA and dIdA and dIyy = x = x22dA, dA, respectively respectively SHOWSHOW

For the entire area, For the entire area, Second moment of differential area dA about the pole Second moment of differential area dA about the pole

O or z axis is called the polar moment of inertia, dJO or z axis is called the polar moment of inertia, dJoo = = rr22dAdA

For the entire area, For the entire area,

dAxA 2

dAxIdAyIA

yA

x 22 ,

yA

xo

Ao

IIdAyxJ

yxr

dArJ

)( 22

222

2

A

xdA

Parallel-Axis TheoremParallel-Axis Theorem

SHOWSHOW

However, x’ passes through the centroid of the areaHowever, x’ passes through the centroid of the area

Similarly, Similarly,

AAyy

AAyx

yx

dAddAyddAydAdyI

dAdydI222

2

'2')'(

)'(

A

dAy 0'

2

' yxxAdII

2

2

'

AdJJ

AdII

co

xyy

Radius of Gyration of an AreaRadius of Gyration of an Area

A quantity often used in the design of columnsA quantity often used in the design of columns

(units are length)(units are length)A

Jk

A

Ik

A

Ik O

Oy

yx

x ,,

Moments of inertia for an area by Moments of inertia for an area by integration (Case 1)integration (Case 1)

Specify the differential element dA (usually a Specify the differential element dA (usually a rectangle with finite length and differential width)rectangle with finite length and differential width)

Differential element located so it intersects the Differential element located so it intersects the boundary of the area at an arbitrary point (x,y)boundary of the area at an arbitrary point (x,y)

Case 1:Case 1:– Length of element is oriented parallel to axis, Length of element is oriented parallel to axis, SHOWSHOW

– IIyy determined by a direct application of since determined by a direct application of since element has an infinitesimal thickness dx (all parts of the element has an infinitesimal thickness dx (all parts of the element lie at the same moment-arm distance x from the element lie at the same moment-arm distance x from the y-axis)y-axis)

dAxIA

y 2

Moments of inertia for an area by Moments of inertia for an area by integration (Case 2)integration (Case 2)

Case 2:Case 2:– Length of element is oriented perpendicular to axis, Length of element is oriented perpendicular to axis,

SHOWSHOW

– does not apply, all parts of the element will not does not apply, all parts of the element will not lie at the same moment-arm distance from axislie at the same moment-arm distance from axis

– First calculate moment of inertia of the element about a First calculate moment of inertia of the element about a horizontal axis passing through the element’s centroidhorizontal axis passing through the element’s centroid

– Determine moment of inertia of the element about the x-Determine moment of inertia of the element about the x-axis by using parallel-axis theoremaxis by using parallel-axis theorem

– Integrate to yield IIntegrate to yield Ixx

EXAMPLES (pg 523)EXAMPLES (pg 523)

dAyIA

x 2

Moments of inertia for composite Moments of inertia for composite areasareas

Divide the area into its “simpler” composite partsDivide the area into its “simpler” composite parts Determine the perpendicular distance from the centroid Determine the perpendicular distance from the centroid

of each part to the reference axis (often centroid of entire of each part to the reference axis (often centroid of entire area)area)

Determine the moment of inertia of each part about its Determine the moment of inertia of each part about its centroidal axis (parallel to the reference axis)centroidal axis (parallel to the reference axis)

If the centroidal axis does not coincide with the reference If the centroidal axis does not coincide with the reference axis, apply the parallel-axis theorem to determine the axis, apply the parallel-axis theorem to determine the moment of inertia of the part about the reference axismoment of inertia of the part about the reference axis

The moment of inertia of the entire area about the The moment of inertia of the entire area about the reference axis is obtained by summing the results of its reference axis is obtained by summing the results of its parts (“holes” are subtracted)parts (“holes” are subtracted)

EXAMPLES (pg 530)EXAMPLES (pg 530)