CHAPTER 9: Moments of Inertia

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1 HPTER 9: Moments of nerti! Moment of nerti of res! Second Moment, or Moment of nerti, of n re! Prllel-is Theorem! Rdius of Grtion of n re! Determintion of the Moment of nerti of n re ntegrtion! Moments of nerti of omposite res! Polr Moment of nerti

2 9. Moment of nerti: Definition d(d(d ( ( d d O

3 9. Prllel-is Theorem of n re d entroidl is G d entroidl is d ( ' + d d [( ' + ( '( d + ( d ] d ( ' d + ( '( d d +, ( d d O + d ' d + d d + + d + + d JO J + d

4 9. Rdius of Grtion of n re k The rdius of grtion of n re with respect to the is is defined s the distnce k, where k. With similr definitions for the rdii of grtion of with respect to the is nd with respect to O, we hve O k k ko JO

5 9. Determintion of the Moment of nerti of n re ntegrtion The rectngulr moments of inerti nd of n re re defined s d d d These computtions re reduced to single integrtions choosing d to e thin strip prllel to one of the coordinte es. The result is d d d d 5

6 Moment of nerti of Rectngulr re. d d d (/d h d h/ d / d ' d h ( d h ( d ( h ( h/ h h 6

7 7 / h/ h d hd ( h ( d ' h d h ( / ( h h h d d d hd d (h/d

8 h/ h/ h h + d h + h h + h ( h( h 8

9 h d d Moment of nerti of Tringulr re. / d l d Using similr tringles, we hve l / h- d ntegrting d from to h, we otin h d h h d h [ h ] h h + h d d h h ( ( h h ( h h 6 d l h h l h h h d h d 9

10 Emple 9. Determine the moment of inerti of the shded re shown with respect to ech of the coordinte es. k

11 Moment of nerti. d k ( d d (-d k k k or d Sustituting nd / / ( / d / / / 7 ( 7 ( 7 / d 7 / 5/ 7 / d

12 Moment of nerti. k d d d d d ( d d 5 ( ( 5 ( ( 5 5 5

13 Emple 9. Determine the moment of inerti of the shded re shown with respect to ech of the coordinte es. (,

14 Moment of nerti. (, d ( d d d ( - d / ( 5/ ( d d ( d 7 7 / 7 7 /

15 Moment of nerti. d (, d ( d d ( - d ( d ( d ( d

16 9.5 Moment of nerti of omposite res d c similr theorem cn e used with the polr moment of inerti. The polr moment of inerti J O of n re out O nd the polr moment of inerti J of the re out its o centroid re relted to the distnce d etween points nd O the reltionship J O J + d The prllel-is theorem is used ver effectivel to compute the moment of inerti of composite re with respect to given is. 6

17 Emple 9. ompute the moment of inerti of the composite re shown. mm 5 mm 75 mm 75 mm 7

18 SOLUTON mm 5 mm 75 mm 75 mm (d ir h ( Rect ( + d ir [ ((5 ] Re [ π (5 + ( π 5 (75 ct ] ir 6 mm 8

19 Emple 9. Determine the moments of inerti of the em s cross-sectionl re shown out the nd centroidl es. 6 Dimension in mm 9

20 SOLUTON d B d D D 6 Dimension in mm ( + d + ( + d + ( + d B [ (( + ( ( ] + [ (6( + [ (( + ( ( ] + ].9 9 mm

21 Dimension in mm 6 B d d d ( ( ( B ] (5 ( (( [ ] ((6 [ ] (5 ( (( [ mm d d D D

22 Emple 9.5 (Prolem 9., Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es. mm mm 8 mm O 6 mm mm mm 6 mm mm mm

23 8 mm SOLUTON O mm mm B d mm mm [ (6( 6. mm d ] 6 mm mm mm 6 mm k + [ (8(8 ] B + [ (6(8 ] ( + d + ( + d + ( + d 9 mm 9 [( 6 + (8 8 + (8 6]. 9 ( + d + ( + d + ( + d B 6. k mm [( 6 + (8 8 + (8 6] [ ((6 + ( 6(7 ] + [ (8(8 + ] B + [ (8(6 + (8 6(7 B ] mm

24 Emple 9.6 (Prolem 9., Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es..5 m m m.5 m m O m m m m.5 m.5 m

25 .5 m m m.5 m ( + d 5 6 ( + d B ( + d B O d B d m m m [ (5(6 + ] [ (( [ (( + ( (.5 ] + ( ( ] B.5 m m m.5 m 6 m 6 k. 599 [(5 6 ( ( ] m ( + d ( + d ( + d B [ (6(5 ] [ (( ] B [ (( ] 6.5 m 6.5 k. 67 [(5 6 ( ( ] m 5

26 Emple 9.7 Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es nd t the centroidl es. cm cm 5 cm cm 5 cm 6

27 cm cm Moments of inerti out centroid d 5 (5(.5 5 cm cm Y.5 5 cm G.5 Y OR 5.5 cm [( ((5 + (5 ( ] + [( (5( + (5 ( ] 5.5 cm Y [(.5(5 ] + (.5( 5 (5.5 cm Moments of inerti out is [( (5( 5.5 cm + (5 ( ] + ((5 [( ((5 5 cm + (5 (.5 ] + (5( 5.5 k k. 88 cm 5 7

28 Emple 9.8 The strength of W6 57 rolled-steel em is incresed ttching 9 mm 9 mm plte to its upper flnge s shown. Determine the moment of inerti nd the rdius of grtion of the composite section with respect to n is which is prllel to the plte nd psses through the centroid of the section. 9 mm 9 mm 58 mm 7 mm 8

29 9 mm SOLUTON 9 mm Moment of nerti ' + ( ' plte ( ' wide flnge d Y 58 mm O 88.5 mm ( + d + ( + Y ' plte (9(9 + ' 6 [ 6. + (7(7.8 ] mm wide flnge + (5( mm entroid The wide-flnge shpe of W6 57 found referring to Fig mm 6. mm plte (9(9 5 mm YΣ Σ ' Rdius of Grtion k ' 6 57 mm ' k ' 9 mm (5+ 7 Y ( 5+ 7 (88.5(5 + ((7 Y 7. 8 mm 9

30 9.6 Polr Moment of nerti d The polr moment of inerti of n re with respect to the pole O is defined s r O J O r d The distnce from O to the element of re d is r. Oserving tht r +, we estlished the reltion J + O

31 Emple 9.9 ( Determine the centroidl polr moment of inerti of circulr re direct integrtion. ( Using the result of prt, determine the moment of inerti of circulr re with respect to dimeter. r O

32 SOLUTON. Polr Moment of nerti. dj O u d d πu du r O u du J O r djo u πu du ( π r u du J O π r. Moment of nerti with Respect to Dimeter. J + O π r dimeter π r

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