CHAPTER 5 GEOMETRICAL PROPERTIES OF SYMMETRICAL SECTIONS

Transcription

1 CHAPTER 5 GEOMETRCAL PROPERTES OF SYMMETRCAL SECTONS EXERCSE 7, Pge 0. Determine the second moments of re bout the centroid of the squres shown below. () b/ b/ b. dy. y b/ b/ b y / b b b/ (b) b b b b b b From similr tringles

2 .b B 0.707b 0.707b y i.e. (0.707b y) B 0.707b b B. dy. y b b b y dy y by y dy b b.b 0.707b 0.707b 0.707b.by y 0.707b.b 0.707b 0.707b b b b b b b b b b b 8 8 b b b

3 . Determine the positions of the centroidl xes nd the second moments of re bout these xes, for the sections of figures () to (d). () Section y i o Σ y cm Σ + Σi o y i.e. 5. cm.50 m (b)

4 Section y i o Σ y cm Σ + Σi o y i.e cm 5.70 m 0 (c) i.e cm 5.00 m (d)

5 Section y u o Σ y cm Σ + Σi o y i.e. 85 cm m. Determine the second moment of re of the section shown bout n xis pssing through the centroid nd prllel to the xis. Wht would be the percentge reduction in second moment of re if the bottom flnge were identicl to the top flnge? First cse

6 Section y i o Σ y m Σ + Σi o y i.e. 85 cm m Second cse i.e m 0 Percentge reduction %.05 5.%. Determine the second moment of re for the cross-section shown below bout the neutrl xis,.

7 Second moment of re bout the neutrl xis, bd d.8 m 5. Determine the second moment of re for the cross-section shown below bout the neutrl xis,. Section y i o Σ y..5.0 m Σ + Σi y i.e..0 m. Determine the second moment of re for the cross-section shown below bout the neutrl xis,.

8 Section y i o Σ y m Σ + Σi o y i.e m 7. Determine the second moment of re for the cross-section shown below bout the neutrl xis,.

9 Section y i o Σ y m Σ + Σi o y i.e. 0.0 m 8. Determine the second moment of re for the cross-section shown in Figure 5. bout the neutrl xis,. Figure 5. Section y i o Σ y m

10 Σ + Σi o y i.e. 0. m 9. Determine the second moment of re for the cross-section shown in Figure 5.5 bout the neutrl xis,. Figure 5.5 Section y i o Σ y mm Σ + Σi o y i.e..80 mm.80 m

11 0. Determine the second moment of re for the cross-section shown in Figure 5. bout the neutrl xis,, together with its polr moments of re. Figure 5. Second moment of re bout the neutrl xis, d () m Polr second moment of re, polr d ().57 m. Determine the second moment of re for the cross-section shown bout the neutrl xis,, together with its polr moments of re.

12 Second moment of re bout the neutrl xis, Polr second moment of re, polr d d. d d m 0. m

13 EXERCSE 8, Pge 9. Determine the second moment of re bout of the section enclosed by y x + x nd the x- xis between x 0 m nd x m s shown below by () the Nvl rchitects method, nd (b) by Ross s method. Compre the results with the exct solution obtined by integrtion. The re is shown in Figure 5. where x vries from 0 m to m. At x 0, y 0 m At x m, y m At x m, y 8 m () By the Nvl rchitects method, From eqution (5.7), 9 where Figure 5., h m, y 0, y m nd y 8 m Hence, 0 () m 9 (b) By Ross s method, from eqution (5.8), y y y y y y y y y y y h y y y y y y y () ()(8) 0 (8) (0)(8) () (0)() () (8) (0)()(8) 5 5

14 (7) m By integrtion, y y y dx dx nd from Figure 5., x x 5 dx 8x x x x dx x x x x m ntegrtion gives n exct vlue. Hence, Ross method is exct nd the Nvl rchitects method is in error by % 8.5% Clculte the second moment of re, bout the bse (where y 0), of the re enclosed by y e x nd the x-xis between the limits of x 0 nd x m, by the exct integrtion method. Hence, or otherwise, obtin the pproximte numericl vlues of this eqution using () the Nvl rchitects method, nd (b) Ross s method. By integrtion, nd from Figure 5., y y y dx dx x e x x e x 0 0 dx e dx e e e.7 m 0

15 ntegrtion gives n exct vlue. () By the Nvl rchitects method, From eqution (5.7), 9 where Figure 5., h m, y 0, y Hence, e m nd y 0 ( e ) e 9 (b) By Ross s method, from eqution (5.8), e m m 9 y y y y y y y y y y y h y y y y y y y e e 0 ( e ) ( e )( e ) 0 ( e ) (0)( e ) () (0)( e ) ( e ) ( e ) (0)( e )( e ) (0.08) m