Assignment #2 Solution 1 / 6 Spring 2012 Influence Lines...

BridgeDesign Assignment#2Solution 1/6Spring2012 InfluenceLinesforBridgeDeck2.Maximum+’veand–‘vemomentsduetomovingloads.(a)Two16kloads,spaced6ftapart.Setuptwo16kloadsonthe“MovingLoads”menu.Makesurebothloadsareonthebridgeforalltimeincrements.

• Addanode(N1A)forthestartingpositionoftherightwheelload;andaddanode(N5A)fortheendingpositionoftheleftwheel.

• Specifystartinglocations(1stJoint)andendinglocations(2ndJoint)sothattheloadsappearattheleftendsimultaneously,anddisappearattherightendsimultaneously.

StartingPosition:

...

IntermediatePosition:

...

EndingPosition:

BridgeDesign Assignment#2Solution 2/6Spring2012 InfluenceLinesforBridgeDeck

• Combinebothmovingloadsintooneloadcombination.SpecifyPDelta=“Y”sothat“DesignResults”canbedisplayedforfindinglocationsofmax.moments.“Solve”theloadcombinationanddisplaytheenvelopeofbendingmomentsfromthemovingloads.The“DesignResults”(CodeChecks)menushowsthatthelocationofthemaximumpositivemomentinSpan1is2.905ftfromtheleftsupport.

Therefore,max.momentsduetotwo16kwheelloads: Max.+’veM:23.7k‐ftat2.91ftfromtheexteriorsupport Max.–‘veM:23.2k‐ftat1stinteriorsupport(b)twopairsof16kwheelloads.Varyspacingbetween2ndand3rdwheelloadfrom4ft(minimum)to14ft.Movingloadmenufor4ftspacingbetween2ndand3rdwheelload:

BridgeDesign Assignment#2Solution 3/6Spring2012 InfluenceLinesforBridgeDeckStartinglocationsofwheelloads:

Results:max+’veM=21.8k‐ftat3.8995ftfromexteriorsupportmax–‘veM=‐24.5k‐ftatfirstinteriorsupport.

Spacingbetween2ndwheel&3rdwheel,ft

max.+’veM,k‐ft

Locationmax+’veM

max.‐’veM,k‐ft

Locationmax‐’veM

4 21.8 3.9ftfromext.sup. ‐24.5 1stint.support6 23.5 3.0ftfromext.support ‐21.2 1stint.support8 24.5 2.9ftfromext.support ‐20.9 1stint.support10 24.5 2.9ftfromext.support ‐21.9 1stint.support12 24.2 2.8ftfromext.support ‐23.1 1stint.support14 23.8 2.8ftfromext.support ‐23.8 1stint.support

3.InfluenceLineformaximum–‘vemomentatfirstinteriorsupport(nodeN10).3.1CalculatedeflectionsusingRISAmodel

• TheRISAmodelismodifiedsothattherearetwonodeswiththesamecoordinatesatthelocationforwhichtheinfluencediagramisdesired:N10LandN10R(seeJointCoordinatesscreenbelow).

• N10LisconnectedtothenodetotheleftviamemberM10,andN10RisconnectedtothenodetotherightviamemberM11.ThereisnomemberconnectingN10LtoN10R.(SeeMemberPrimaryDatascreenbelow).

• TheXandYdegreesoffreedom(DOF)ofnodeN10RareslavedtothecorrespondingDOFofnodeN10L.TherotationabouttheZaxisisnotslavedsothatthetwonodescanrotateindependentlyofoneanother.(SeeJointBoundaryConditionsscreenbelow).

BridgeDesign Assignment#2Solution 4/6Spring2012 InfluenceLinesforBridgeDeck

• ApairofequalandoppositemomentsareappliedtojointsN10LandN10R(seeJointLoadsscreenbelow).Themomentsareappliedtocauseacounter‐clockwiserelativerotation(slopeatN10R–slopeatN10R).TheresultingYdeflectionswillbeusedtoformtheinfluencediagramforbendingmomentatthefirstinteriorsupport.TheZrotationsofN10LandN10Rwillalsobeused.

3.2CalculatetheinfluencelineusingthedeflectionsfromRISA.FromBetti’slaw:

Mi

Px

Px

=!x

!i

!i

,!!!!or !!Mi

Px = Px

!x

!i

!i

Mi

Px = Px!!I,!!!!!where!!I =

!x

!i

!i

TheinfluencevaluesforthemomentatnodeN10isequaltothedeflectionsintheYdirection(Δ)dividedbytherelativerotationofthetwoendsofthebeam(θ).Forexample,Iat5ftfromtheexternalsupport:

I5 ft =!5 ft

!=

.2418"!1 ft

12in

(.009308rad

" (".01075rad))=1.0046 ft

BridgeDesign Assignment#2Solution 5/6Spring2012 InfluenceLinesforBridgeDeckTest:applya10kload5ftfromtheexternalsupportandcalculateMatthefirstinternalsupport(N10).

MN10 = (!10k)(1.0046 ft) = !10.046

k! ft (checks)3.3Adjusttheinfluencevaluesforfive‐spanbeamswithspanlengthsdifferentthan10ft.Sincemomentdueapointloadisdirectlyproportionaltothespanlength,theinfluencevaluesforsayabeamwithfiveequal8‐ftspansis:

I8 ft spans = I10 ft spans

8 ft

10 ft

Test:applya10kloadinthemiddleofthefirstspanoffive‐spanbeamwith8ftspans.

I8 ft spans = I10 ft spans

8 ft

10 ft

I8 ft spans =1.0046 ft

8

10= 0.8037

I4 ft = (!10k)(0.8037) = !8.037

k! ft

BridgeDesign Assignment#2Solution 6/6Spring2012 InfluenceLinesforBridgeDeck4.Testmax.MsusingInfluenceLinewithmax.MsusingmovingloadsinRISA.

Influencelineforfirstinteriorsupport,fromspreadsheet:

Mat1stinteriorsupportduetoonetruck(two1kloadsseparatedby6ft),fromspreadsheet:

ComparisonofRISAMovingLoadAnalysisvs.InfluenceLineMovingLoadAnalysis:

RISAMovingLoadAnalysis

‐‐‐‐‐‐‐‐InfluenceLineMovingLoadAnalysis‐‐‐‐‐‐‐‐

‐‘veM,duetoonetruck ‐23.2k‐ft ‐23.2k‐ft Leftwheel@4.82’fromextsupport ‐‘veM,duetotwotrucks ‐24.5k‐ft,

4’betweentrucks ‐24.5k‐ft

4’betweentrucksLeftwheel@4.20’fromext.support

+‘veM,duetoonetruck 23.7k‐ft 23.7k‐ft* Leftwheel@2.79’fromext.support +‘veM,duetotwotrucks 24.5k‐ft

8’to10’betweentrucks

24.6k‐ft*9’betweentrucks

Leftwheel@2.80’fromext.support

*Influencelinefor+Mconstructedat0.35L=2.8ftfromext.support