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hear Force and Bending Moment Diagrams

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Normallyabeamisanalysedtoobtainthemaximumstressandthisiscomparedtothematerialstrengthtodeterminethedesignsafetymargin.Itisalsonormallyrequiredtocalculatethedeflectiononthebeamunderthemaximumexpectedload.Thedeterminationofthemaximumstressresultsfromproducingtheshearandbendingmomentdiagrams.Tofacilitatethisworkthefirststageisnormallytodeterminealloftheexternalloads.

e=strain

=stress(N/m2)

E=Young'sModulus=/e(N/m2)

y=distanceofsurfacefromneutralsurface(m).R=Radiusofneutralaxis(m).

I=MomentofInertia(m4-morenormallycm4)

Z=sectionmodulus=I/ymax

(m3-morenormallycm3)

M=Moment(Nm)w=Distrubutedloadonbeam(kg/m)or(N/masforceunits)W=totalloadonbeam(kg)or(Nasforceunits)F=Concentratedforceonbeam(N)S=ShearForceonSection(N)L=lengthofbeam(m)x=distancealongbeam(m)

Toallowdeterminationofalloftheexternalloadsafree-bodydiagramisconstructionwithalloftheloadsandsupportsreplacedbytheirequivalentforces.Atypicalfree-bodydiagramisshownbelow.

Theunknownforces(generallythesupportreactions)arethendeterminedusingtheequationsforplanestaticequilibrium.

ForexampleconsideringthesimplebeamabovethereactionR2isdeterminedbySummingthemomentsaboutR

1tozero

R2.L-W.a=0ThereforeR

2=W.a/L

R1isdeterminedbysummingtheverticalforcesto0

W-R1-R

2=0ThereforeR

1=W-R

2

Theshearforcediagramindicatestheshearforcewithstoodbythebeamsectionalongthelengthofthebeam.Thebendingmomentdiagramindicatesthebendingmomentwithstoodbythebeamsectionalongthelengthofthebeam.ItisnormalpracticetoproduceafreebodydiagramwiththesheardiagramandthebendingmomentdiagrampositionbelowForsimplysupportedbeamsthereactionsaregenerallysimpleforces.Whenthebeamisbuilt-inthefreebodydiagramwillshowtherelevantsupportpointasareactionforceandareactionmoment....

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Thesignconventionusedforshearforcediagramsandbendingmomentsisonlyimportantinthatitshouldbeusedconsistentlythroughoutaproject.Thesignconventionusedonthispageisasbelow.

Ashearforcediagramissimplyconstructedbymovingasectionalongthebeamfrom(say)theleftoriginandsummingtheforcestotheleftofthesection.TheequilibriumconditionstatesthattheforcesoneithersideofasectionbalanceandthereforetheresistingshearforceofthesectionisobtainedbythissimpleoperationThebendingmomentdiagramisobtainedinthesamewayexceptthatthemomentisthesumoftheproductofeachforceanditsdistance(x)fromthesection.Distributedloadsarecalculatedbuysummingtheproductofthetotalforce(totheleftofthesection)andthedistance(x)ofthecentroidofthedistributedload.Thesketchesbelowshowsimplysupportedbeamswithonconcentratedforce.

ThesketchesbelowshowCantileverbeamswiththreedifferentloadcombinations.

Note:Theforceshownifbasedonloads(weights)wouldneedtobeconvertedtoforceunitsi.e.50kg=50x9,81(g)=490N.

Considerashortlengthofabeamunderadistributedloadseparatedbyadistancex.

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astUpdated15/02/2008

ThebendingmomentatsectionADisMandtheshearforceisS.ThebendingmomentatBC=M+MandtheshearforceisS+S.

Theequationsforequilibriumin2dimensionsresultsintheequations..Forces.

S-w.x=S+SThereforemakingxinfinitelysmallthen..dS/dx=-w

Moments..TakingmomentsaboutC

M+Sx-M-M-w(x)2/2=0

Thereforemakingxinfinitelysmallthen..dM/dx=S

Thereforeputtingtherelationshipsintointegralform.

Theintegral(Area)ofthesheardiagrambetweenanylimitsresultsinthechangeoftheshearingforcebetweentheselimitsandtheintegraloftheShearForcediagrambetweenlimitsresultsinthechangeinbendingmoment...

Sites Providing Relevant Information

1. Topic4Beams..Informativenotes(imperial)2. StaticsandStrengthofMaterials...UofWisconsinNotesincludingsectiononbeams3. Stressesinbeams...MITPaper

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