Draft

5.4Flexure 113

whereyis measuredfromtheaxisof rotation(neutralaxis). Thus strainsareproportionalto

thedistancefromtheneutralaxis.

64 (Greekletterrho) is theradiusof curvature. In sometextbook,thecurvature(Greek

letterkappa) is alsousedwhere

=

1



(5.14)

thus,

"x=y (5.15)

5.4.2 Stress-StrainRelations

65 Sofarwe consideredthekinematicof thebeam,yet lateronwe willneedto considerequi-

libriumin termsof thestresses. Hencewe needto relatestrainto stress.

66 For linearelasticmaterialHooke'slaw states

x=E"x (5.16)

whereEisYoung'sModulus.

67 CombiningEq.withequation5.15we obtain

x=Ey (5.17)

5.4.3 InternalEquilibrium;SectionProperties

68 Justas externalforcesactingona structuremustbe in equilibrium,theinternalforcesmust

alsosatisfytheequilibriumequations.

69 Theinternalforcesaredeterminedbyslicingthebeam. Theinternalforcesonthe\cut"

sectionmustbe in equilibriumwiththeexternalforces.

5.4.3.1 Fx= 0; NeutralAxis

70 The rstequationwe consideris thesummationof axialforces.

71 Sincetherearenoexternalaxialforces(unlike a columnor a beam-column),theinternal

axialforcesmustbe in equilibrium.

Fx= 0)

Z

A

xdA= 0 (5.18)

wherexwas givenby Eq.5.17, substitutingwe obtain

Z

A

xdA=

Z

A

EydA= 0 (5.19-a)