stresses due to bending shear

7/24/2019 Stresses Due to Bending Shear

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Shear force and bendingShear force and bendingmoment diagrammoment diagram


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Stresses due to bendingStresses due to bending

Figure shows aFigure shows aslender memberslender member

(beam) subjected to(beam) subjected toshear force andshear force andbending moment.bending moment.We will beWe will beconsidering onlyconsidering onlysymmetrical beams.symmetrical beams.


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Concept of CurvatureConcept of Curvature

The curvature of aThe curvature of aplane curve is definedplane curve is definedas the rate of changeas the rate of change

of the slope angle ofof the slope angle ofthe curve with respectthe curve with respectto distance along theto distance along thecurve.curve.Curvature !Curvature ! "" ## $%$% ! ! $$ ss


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GeometryGeometry

of pureof purebendingbending


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Geometry of pure bendingGeometry of pure bending

From symmetry& we conclude thatFrom symmetry& we conclude thatin pure bending'in pure bending'

..lane sections remain plane.lane sections remain plane.

.. Initially parallel sections have aInitially parallel sections have acommon intersection. The beamcommon intersection. The beambends into the arc of a circle.bends into the arc of a circle.


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Figure shows aFigure shows asegment of asegment of abeam beforebeam before

deformation.deformation.The lines I* andThe lines I* and+, initially+, initiallye-ual in lengthe-ual in lengthdeform intodeform intoarcs of circles.arcs of circles.


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Stresses obtained from stress-Stresses obtained from stress-strain relationsstrain relations


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Equilibrium RequirementsEquilibrium Requirements

-uilibrium re-uires that the resultant of-uilibrium re-uires that the resultant of

the stress distribution over the crossthe stress distribution over the crosssection of the beam should e-ual thesection of the beam should e-ual thebending moment +bending moment + bb..


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Equilibrium Requirements on anEquilibrium Requirements on anelemental areaelemental area A A


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Stress and Deformation in symmetricalStress and Deformation in symmetricalelastic beams sub ected to pure bendingelastic beams sub ected to pure bending

The transverse stressesThe transverse stresses // yy&&// 00 andand 11y0y0 are assumed to be 0eroare assumed to be 0ero


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Equilibrium Requirements on anEquilibrium Requirements on anelemental areaelemental area A A


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2r&2r& 3344## // 44 ! # 5y! ! # 5y! ""Which gives'Which gives' // 44 # 5 y!# 5 y! ""

6ubstituting in the e-uilibrium6ubstituting in the e-uilibriume-uations'e-uations'

77 88// 44 d8# 5d8# 5 77 88 y!y! "" d8 # 9d8 # 9Which implies 5 !Which implies 5 ! "" 77 88 y d8 # 9y d8 # 9

Thus the neutral surface passes thruThus the neutral surface passes thruthe area centroid.the area centroid.


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6ubstituting6ubstituting // 44 # 5 y!# 5 y! "" in the secondin the seconde-uilibrium e-uation gives'e-uilibrium e-uation gives'

77 88// 44 0d8# 50d8# 5 77 88 y0!y0! "" d8 # 9d8 # 9Which implies 5 !Which implies 5 ! "" 77 88 y0 d8 # 9y0 d8 # 9

This is automatically satisfied since theThis is automatically satisfied since thesection is symmetrical about y a4is.section is symmetrical about y a4is.


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6ubstituting6ubstituting // 44 # 5 y!# 5 y! "" in the thirdin the thirde-uilibrium e-uation gives'e-uilibrium e-uation gives'

5577 88// 44 yd8#yd8# 77 88 yy )) ! ! "" d8 # +d8 # + bbWhich implies !Which implies ! "" 77 88 yy )) d8 # +d8 # + bb

:ut I:ut I 0000 ## 77 88 yy )) d8d82r& d2r& d %% !ds# ! !ds# ! "" # +# + bb ! I ! I 0000


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6ubstitution in the stress and strain6ubstitution in the stress and straine4pressions gives'e4pressions gives'33

44 # 5 +# 5 +

bby! Iy! I

0000

8nd8nd// 44 # 5+# 5+ bby! Iy! I 0000

Which e4press the strain and stressWhich e4press the strain and stressin terms of the applied bendingin terms of the applied bendingmoment.moment.


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Stress-Strain relationsStress-Strain relations


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6ubstituting e4pressions for stresses&6ubstituting e4pressions for stresses&we can obtain'we can obtain'33

yy## ;; ++

bby! Iy! I

0000

8nd8nd3300 ## ;; ++ bby! Iy! I 0000

8lso8lso


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ffect offfect oftransversetransversestrains on thestrains on the

deformeddeformedshape of ashape of arectangularrectangularbeambeam


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Stresses in symmetrical beams transmittingStresses in symmetrical beams transmittingbending moment and shear forcebending moment and shear force


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Stresses inStresses insymmetricalsymmetrical

beamsbeams

transmittingtransmitting

bendingbendingmoment andmoment and

shear forceshear force


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Stresses in symmetrical beams transmittingStresses in symmetrical beams transmittingbending moment and shear forcebending moment and shear force

The e-uilibrium e-uations for the segment is shown above.The e-uilibrium e-uations for the segment is shown above.6ubstituting stress5 moment relation in the above e-uation gives'6ubstituting stress5 moment relation in the above e-uation gives'


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Stresses inStresses insymmetricalsymmetrical

beamsbeams

transmittingtransmittingbending momentbending momentand shear forceand shear force

stresses due to bending shear

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