616110310 8mohrscircleformomentsofinertia 120122232346 Phpapp02

10.8 Mohr’s Circle for Moments of Inertia


� It can be found that

� In a given problem, Iu and Iv are variables and Ix,

2

2

2

2

22I

III

III xy

yxuv

yxu +

−=+

+−

� In a given problem, Iu and Iv are variables and Ix, Iy and Ixy are known constants

� When this equation is plotted on a set of axes that represent the respective moment of inertia and the product of inertia, the resulting graph represents a circle

( ) 222 RIaI uvu =+−



� The circle constructed is known as a Mohr’s circle with radius

2

2

xyyx I

IIR +

+

=

and center at (a, 0) where

( ) 2/

2

yx

xy

IIa

IR

+=

+

=



Procedure for AnalysisDetermine Ix, Iy and Ixy

� Establish the x, y axes for the area, with the origin located at point P of interest and determine I , I and Idetermine Ix, Iy and Ixy

Construct the Circle� Construct a rectangular coordinate system such

that the abscissa represents the moment of inertia I and the ordinate represent the product of inertia Ixy



Procedure for AnalysisConstruct the Circle� Determine center of the circle O, which is located

at a distance (Ix + Iy)/2 from the origin, and plot the reference point a having coordinates (I , I )the reference point a having coordinates (Ix, Ixy)

� By definition, Ix is always positive, whereas Ixywill either be positive or negative

� Connect the reference point A with the center of the circle and determine distance OA (radius of the circle) by trigonometry

� Draw the circle



Procedure for AnalysisPrincipal of Moments of Inertia� Points where the circle intersects the abscissa

give the values of the principle moments of inertia I and Iinertia Imin and Imax

� Product of inertia will be zero at these points

Principle Axes� To find direction of major principal axis,

determine by trigonometry, angle 2θp1, measured from the radius OA to the positive I axis



Procedure for Analysis

Principle Axes

� This angle represent twice the angle from the x axis to the area in question to the axis of axis to the area in question to the axis of maximum moment of inertia Imax

� Both the angle on the circle, 2θp1, and the angle to the axis on the area, θp1must be measured in the same sense

� The axis for the minimum moment of inertia Imin

is perpendicular to the axis for Imax



Example 10.10

Using Mohr’s circle, determine the principle

moments of the beam’s cross-sectional area moments of the beam’s cross-sectional area

with respect to an axis

passing through the

centroid.



Solution

Determine Ix, Iy and Ixy

� Moments of inertia and the product of inertia have been determined in previous exampleshave been determined in previous examples

Construct the Circle

� Center of circle, O, lies from the origin, at a distance

( ) ( )( )

( ) 25.42/)60.590.2(2/

1000.3

1060.51090.2

49

4949

=+=+

−=

==

yx

xy

yx

II

mmI

mmImmI



Solution

� With reference point A (2.90, -3.00) connected to point O, radius OA is determined using Pythagorean theorem

( ) ( )

Principal Moments of Inertia

� Circle intersects I axis at

points (7.54, 0) and

(0.960, 0)

( ) ( )29.3

00.335.1 22

=

−+=OA



Solution

Principal Axes

� Angle 2θp1 is determined from

( ) ( )10960.01054.7 49min

49max == mmImmI

� Angle 2θp1 is determined from

the circle by measuring CCW

from OA to the direction of the

positive I axis

oo

o

2.11429.300.3

sin180

sin1802

1

11

=

−=

−=

−

−

OA

BApθ



Solution

� The principal axis for Imax = 7.54(109) mm4

is therefore orientated at an angle θp1 = 57.1°, 57.1°, measured CCW from the positive x axis

to the positive u axis

� v axis is perpendicular

to this axis

616110310 8mohrscircleformomentsofinertia 120122232346 Phpapp02

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