A CALCULATIONOF PISTON RODSTRENGTH

M. C. BennettDepartmentEngineer, AdvancedDesign

J I CASEConstruction EquipmentDivision

625Third AvenueRock Island, Illinois 61201

Summary

Formulasare derived for maximumdeflection andstress conditions existing in the piston rod of a typ-ical mobile type of hydraulic cylinder underaxial com-pressiveloads. Theformulascanbeusedona rela-tively small sized computer. Verification test datais presentedandit showsgoodagreementwith calcu-lated values.

Background

Our requirementsfor a calculationof thestrengthof the cy1i nder rod of a hydrau1ic cyli nder are thefoll owing:

a. A basic design tool to provide a rod with ade-quatestrength, yet preventexcessiveover-design.

b. A tool to help evaluatefailures andwarrantycosts, particularly to help identify thoseapplications whichmayhavepermittedforcesor pressuresin excessof thedesignparam-eters.

c. Besimpleenoughin executionthat draftsmencanuseit.

d. Besmallenoughthat it can be run on a pro-grammablecalculator, i.e., not haveto use avery largecomputer.

It is realized that since these formulaswereoriginally developed,other peoplehavepublishedre-sults of similar effort, andto the extent that theyoverlap, hopefully they will reinforce the confidencelevel in both. If they shoulddisagree, then at leasta basis of comparisonwill exist to helpresolvedif-ferences. In addition to the test data presented,thereare manyapparentlysuccessfulcylinderapplica-tions in use whichweresized by these calculations.

Scope

- Thecalculations presentedherearederivedfor,andlimited to, thesimpletypeof cylinderhavingasingle rod, piston, andpivoted mountingsfor both thecap androd endsof the cylinder. Perhapssomeexten-sion to coverothertypesof constructioncouldbe

madebymOdifyingthe formulas,however,theyarenotcoveredhere. Only the stress anddeflection of therod portion of the cylinder is calculated. This isbecausethe size rangethat I aminvolved with gener-ally has a momentof inertia of the cylinder tube fargreater than the rod, andalthoughanydeflection ofthe cylinder tubemustincrease the stress in the rod,its magnitudeseemsgenerally very small for our typeof productand its effect is lumpedinto a safety fac-tor along with potential material imperfections. Fieldexperienceindicatesthat this is anacceptableap-proachfor the cylinders weproduce. Thesize rangefor whichthese calculations weredevelopedencompasses2-1/8"borex 8" stroke to 7"borex 54"stroke. Theseare all the typical mobilehydraulics type. That is,they are of weldedsteel design, not the tie bolt type,andare generally applied to systemswith averageworkingpressuresaround2500PSI.

I havetried to useexistingNFPAterminologyinregard to the detail elementsof cylinder construction,althoughseveral differences exist betweenthat andourpast "in-house"practices. Figure1 illustrates thebasic cylinder design to be discussed.

Calculations

A surveyof thestandardtextbookbeamandcolumnformulasshowedthatnoneof themwere developedwiththe specific set of operatingparametersthat fit ourcylinderapplications. NeitherweretheEulerequa-tions satisfactory as presentedin the texts researched,however,the general pattern of their derivation wasusedwith different boundryconditions. Thesignifi-cant forces assumedto be acting on a cylinder areshownin figure 2.

Theyare generally axial force (F) acting along aline betweenthemountingpointsandrepresentingtheprimaryoutput of the cylinder, a moment(Mf) causedby friction at the rod mountingpin, andtransverseforces (F ) causinga coupleeffect betweentheheadof the cy~inderandthe rod mountingpin representingreaction componentsdueto the weightof the cylinder.

Figure1. CylinderTerminology

157

Figure2.

TO TANK

.F

Fw

Illustration of ForcesAssumed

Fw

Of course, it is possible that other forces mightexist dueto misalignmentconditions in the mountingof the cylinder, interference conditions along thelength of the cylinder, or possible forces fromthehoseor tube connectionsto the ports. However,theselatter forces are not supposedto be present in theapplication of our cylinders and if determinedto bepresent, wenormally require modification to preventthemrather than trying to designsufficient strengthinto the cylinder to accommodatethem. In somecaseswhereeither the mountingpin alignmentsare very dif-ficult to hold or somelateral movementmustoccur dueto the geometryof the connectinglinkage, sphericalball mountingsockets are usedto minimizetheir ef-fects. It can also be seenthat if the direction ofthe friction momentwere reversed, whichmayvery wellhappendependingon the geometryof the linkage con-nectedto, then the friction momentandthe coupledueto the weightof the cylinder maytend to cancel oneanother, thus decreasingthe maximumstress conditionin the rod. Althoughsomeof these effects can behandledeither by entering a negativevalue for theweight componentor modifyingother values in the pro-gram,usually they are not considered,but the follow-ing assumptionsare made:

a. Thereare no side loads exceptfor normalweight of the cylinder andreactions duetopin friction.

b. Thereis rotationof the rodeyepin.

c. Thedirectionof pin rotationis suchthat themomentdueto pin friction andtheweightofthecylinderare additiveinsofaras deflec-tion andstressof thecylinderrodis con-cerned.

Theforegoing assumptionssimplify the calcula-tions on a routine basis without requiring a largeamountof addedinformationregardingsystemgeometryandmodesof operation as well as standardizingthedesign so that onceit is in productionit can usuallybe reapplied to newsystemshavingdifferent geometrybut similar pressureandflow requirements.

L,L,

Fw

L,

L, CLOSEO C'NOTH L, STROKE

Figure3. Symbologyof TermsUsedIn Calculations

Figure3 illustrates themeaningof the termsusedin derivingandusingtheformulas. Inchandpoundunits areusedthroughout.

Summaryof Formulas

The formulasfor the bendingmomentsin the rod atany point, as designatedby the dimensionX, are asfollows:

Mf =- F~d3 (Dueto friction at the rod2 eyepin)

MA=-Ffh(L6+X\- U - Yl (Dueto axialL ~6 +L3) L3 J force F)

MW=-W(L3- X) (Dueto designatedweightcomponentFW)

MT =Mf + MA+ MW (Total combinedbendingmoment)

1.

2.

3.

4.

Usingthesebasicmomentequations,theformulasfor deflectiondueto bending,total offset of the rodeye,andlocationof thehigheststresspointalongtherodwerederivedas shownin AppendixB. Theyareas follows:

y =(hL6 + ~d3 + WL3)(

1 - cos (JQ X~+. \L6+L3 2 F )

(u - h +~)(sinLlQll )+

L3 L6 +L3 F ~

( h - C - W) XL6 + L3 [3 F 5.

h =~-f+~)(1 - cos ( \fQL~))l (L6 +L3) +~ L6cos(JQL3)+sin (fQL3~IQ

~~c + *) (

sin( ~L3))- WL3

J(L6 + L3)

3 JQ F 6.

L6 cos ( {qL3)+ sin ~~L3

.[Q

158

-1(x= ~ -

k2

)(For location 7.

~ of maximumstress)Thesymbolsusedaredefinedin AppendixA. Some

of the individualtermsin theseformulasarerathercomplexanddonot lendthemselvesto manualcomputa-tion. However,onceprogrammedin a computertheyarereadilyhandled.MYdepartmenthasa desktopcalcu-lator, MonroeModel1880-88.Theseformulasarepro-grammedon it andpresentlyuse3575bytesof RAMmem-ory in additionto its ROM.

Whenoriginally programmed,we hadonlythe inte-gral printer availableas anoutputdevicewhichprintsnumbersprimarily, ona 3-7/16"widepapertape. Aformwasdevisedas shownin figure4 to keeptrackofthedatainputfor eachcalculation,illustrate themeaningof someof thesignificant inputdimensions,provideinstructionsontheuseof theprogram,andprovidean identificationof eachoutputnumber.Thecalculatortapewasfoldedandpositionedovertheoperatinginstructionsandstapledin placeas shownin figure 5. This really workedquitewell, however,wenowhavea MonroeModel350Input/OutputWriteralso, andhaveprogrammedit in conjunctionwiththesecalculations.

As seen in figure 6, all of the input data is pre-sentedon the left half of the sheetandthe calculatedresults on the right half of the sheet. This providesa somewhatmorelegible copyandmorespaceexists todefine the meaningof the numbers. The3575bytes ofRAMmemorypreviously mentionedinclude the amountrequired for the Input/OutputWriter.

E .c: '".. ~o.c +' '"u.. ..'"'' ...0"- 0...'''''' """.......,.+'0 =''''''''' '"'"'' ''''"'~c: c: u'O 0...~ .. .. o.+, ...'"+' ",, c:

.-3,1 00 .0000 OperatingPressure - P

4 .50 00 Cyl. BoreDia. - 012.2500 Piston RodDia. - 02

52 .6 200 ClosedLength - Ll2.00 00 RodPin Dia. - 034 .1200 RodEye4? to GI. Brg.- L4

4 0 .1:20 0 Stroke - L23'5000 GI. Brg. to Piston Brg- LS0.1600 Coefficient of Friction-/A

5 5.0000 Weight on RodEye' '- W0 .0000 Locationof StressCal.- 'X..

CylinderNo. Vate

Thevaluesto use in this programshouldbe obtainedfromthe cylinderassemblydrawingper the sketchbelow. DataReg-Svmbol Description Value istel",

1.4 9,2 7 6 .3750 Axia1 RodForcp

2.

01

02

Ll

03

L4

LY Max. 2

. LS

/-'

- F

0.049 1 Deflectionat 'X. - y3.

1.6 49 7 Deflectionat RodEye-4 .

56,926' 9543 Momentof Axial Force- MA5.

6,670' 1075 Momentof Friction at eye-Mf6 .67,966 .6758 Total BendingMoment- MT7.12,400. 0000 Stress fromAxial Force- SA

6 .60.904. 7906 StressfromBending- SB9 .73,304'7906 TotalStress - ST

1 0 . (90 000)1.2 27 7 Safety Factor. (T)II.4.4251 "X.@Max.BendingPoint

12.

-.2' 1657 ,TotalRodEyeOffset - H. . . , . . . . . . . .

Figure5.

Machine Name ):y 1. Type,

Calculator Input DataSheetfor Program.10A

W

"X.

('X-will not be used unless 'Flag" depressedprior to run)

Exampleof Initial FormWith Data

DATE- BY CYLINDER No.CYLINDER ROD STRENGTH PROGRAH - II OA

VEHICLE

INPUT DATA

Operating Pressure

Cylinder Bore Diameter

3, OOU. 0000

4.0000

Piston Rod Diameter 2.0000

47.5000Closed Length

Rod Pin Diameter 1.7500

4.3800Rod Eye Ctrln to Gland Brg Ctrln

Cylinder Stroke

Gland Brg to Piston Brg Ctrln

35.0000

3.2500

Coeficient of Friction 0.1800

50. 0000weight on Rod Bearing

Location of Stress Calculation -eX) 3.3574

PresentDataFormatFromInput/OutputWriter

Total Rod Eye Offset

Figure6.

For the last rod, however,anLVDTtransducerwasmountedin placeof thedial indicatorso simultaneousdeflection,strain andpressurerecordingsweremade.Torqueat therodpinwasinducedandmonitoredbyman-ually rotatingthepin underloadwitha torquewrenchdrivingthrua 4;1multiplier. Thestraingagesweremonitoredbothbya light beam,oscillographicrecorderandanx-y plotter wherestrain wastheordinateaxisandpressuretheabscissa. Sideloadingeffectswere

OUTPUT DATA

Axial Rod Force

Deflection at X

Deflection at Rod Eye

Homent of Axial Force

Homent of Friction at Rod Eye

Total Bending Homent

Stress Due to Axial Force

Stress Due to Bending

Total Stress

Safety Factor = :9050,0:X @ Hax. Bending' Point T

FUNCTION

observedbystrain gagesmountedoneachsideof therodandtheir averagevalueusedto verify the axialforce in therod.

ThecylinderconfigurationtestedhadaA" bore,2" rod, closedlengthof 47.5"anda 35.12"stroke.

Thetest procedurewasto set a pressurevalue inthe cap endof the test cylinder, record deflection

\160

Designatedmaximumoperatingpressure 3100 19

Cy1inderborediameter 4.5 11Pistonroddiameter Z.Z5 12

Closed1ength Si!..'-Z 03

Mountingpin diameter--rodend 2.0 13

Rodeye centerlineto glandbearing -I,/Z 06

CylinderStroke -10./2 04

Glandbearingto piston bearing

~ff'r

\~~",~~~:; .

, ' , \ \\ ' ' ' \ \ \\ " ,,'.'" ' \

~~

Figure7. CalculatorandInput/OutputWriterInstallation

Figure8. Test Fixture

andstrain values,thenwhilemaintaininga constantpressure,veryslowlyrotatetherodpin whilesimul-taneouslymonitoringdeflection,strains andtorquetorotatethe pin. Thepressurewasthenincrementedandthe abovesequencerepeated.

As is frequentin testingbynewmethods,manypro-blemsdevelopedbut after severaltries, it wasfeltthat the desiredtest hadbeenimplemented.Dataana-lysis, however,showedthe stress levels muchhigherthanthe calculationspredictedtheyshouldbe. Itwasthenobservedthat the test rodhada permanentbendin it, so it wasreplacedwith a secondone. Seefigure 9. I didn't knowwhatcausedit to yield, butthoughtwemayhaveinadvertentlyexceededthepres-cribedtest pressuresduringour start upproblems,sowerepeatedthefirst test, this timeinsuringthat

specified pressureswerenot exceeded,andput a per-manentbendin the secondrod also. However,this timeweobservedthat the yielding occurredsomeplacebe-tween3500PSI and4000PSI while the rod pin wasbeingrotated. Therefore, while the third rod wasbeingtested, wedid not exceed3300PSI while rotating thepin anddid not yield that rod.

Twobeneficial results developedfromthe failuresof the first twdrods. The first being that the na-ture andlocation of the bent region could be observedandmeasured. It wasa very gradualbendwith the max-imumcurvaturelocation approximately10-3/4" from itsattachmentpoint to the piston. That is about8" fromthe bearingin theheadastested. Secondly, since thereasonfor failure wasnot immediatelyknown,it wasdecidedto havethe first rod examinedmetallurgically

161

to verify thedrawing,requirE!ll1entsWeremet,particu-1ar1ythesteel chemistry~ndthe inductionhardenedcase. Bothwereacceptable..however,thehardenedcasewasonlymarginallyadequateandsomeferrite extendedto thesurface. This, of course,wasan ideal situa-tion fromthetestingstandpoint,but did not imme-diatelyexplainthefailures either. Latera crosssectionalplot of thebest~st!mateavailablefromourmetallurgists,of theyielctstrengthof the materialwasmadeandcomparedto tlj~strE!sslevelsproducedduringthis test. As ca

..n...

.

.

.. ...

b.

..

..

.

.

.

.

.

.

.

.

e.

~..~~n.

.

.

..

.

.

.

..

..

. .....

.

..

..in

.

fi gu.

r...

~..

1...0.,some

of mypreviouscalcu1ati91l~~~~9m~da surfaceyieldstrength of 21.0,.0.0.0PSImit- .

Figure10.

comparedto a linear~tress distribution as inducedbyoperating loads, permitteda significant area of therod underneaththe hardenedcase to haveits yieldpoint exceededwhenthe design usedonly surfacestrength in calculations. Becauseof this, the usablesurface stress for calculations hasbeendroppedfrom21.0,.0.0.0PSI to 9.0,.0.0.0PSI. Thefailures also empha-sized the significance of rod pin rotation on stresslevels producedin the rod, since in excessof 45.0.0PSIhadbeenapplied to the test cylinder with no rod pinrotation andfar less deflection andstrain valueswereobservedthanwith the 35.0.0PSI with rotation.

Duringactual testing, care wasexercised to keepthe piston fromcontacting the inner surface of the

CrossSectionalStrengthRelationsIn A PistonRod

Whilethis apparentlydid exist in thefailed rod,thesharpdropin yield stressbelowthesurfaceas

162

Also, the initial deflection dueto the weightofthe cylinder andto whateverclearancesandeccentrici-ties existed at the headandpiston areas of the cylin-der to not be indicated. Therefore, the mostvalidcomparisonof the calculated values to the test dataavailable wouldbe to shift either curve vertically sothat both test andcalculated curves intersect the or-dinate axis at the samepoint. If this weredone,youwouldobservethat quite goodcorrelation exists be-tweenthe test andcalculated values, both in magnitudeof values andshapeof the curves.

Since a few of our cylinder applications do nothaveany rotation of the mountingpins, sometest runsweremadewithout intentionally rotating the pins.However,dueto t he geometryof our test fi xture andthe compressibility of the oil in the load cylinder,andpossibly someslight deflection of the fixtureparts, a slight rotation of the pin apparentlyexisted.This wasevidencedby observinga negativedeflectionindication at the higher pressurelevels whenpres-surizing the test cylinder directly with the pump. Thefriction effort wasgreat enoughto offset the weight

I

I

I I ' i'I Iii ..,.. I ..', iI . ' I I

N=PA=13,500PSI,!x12.56IN2=43.960tBS. 'IT : ,. .F=_=L940 LB FT x 12IN/FT.-: 'i6.44:6-LBS - - .--2 r I 2x .875IN I'!"!

(FACTOR 2 USED BECAUSE PI. N. WAS R.OTAT.ED)IN B9TH RODiEYE & ADJACE NTI BEARING

'I I: 1 i, I

-'fL=~i=4369~~6;=.146: 1-'---; 1--:--- i ,'o~ .. I' I I I '

I

'0'" ~}>-". \.~.=7.268.6,=165 "

I

' " ~ +~" ~43960 " "\. ' ~ \ \. 8. I ',.. ~ ~ t.- 8,091.4, ' .. t ",;

wherethe location of the highest stress along the rodis calculated andthe stress values at this point ap-pear in the output data. A safety factor, which isalwayscalculated, has the mostsignificance of coursewhenbasedon the highest stress in the rod.

Also, our mostcommonuse of the calculations isto assumea worsecase condition insofar as workingpressure, extendedlength, coefficient of friction,design tolerances regardingheadandpiston clearancesandeccentricities, andthat the weightof the cylinderanddirection of rotation of the rod pin are additivein bendingeffects.

90 i.I i .

i..-;

!

I.i

" -.. ._n. . -. . . . .....

80 ALL CALCULATEDVALUESUSED i .

~~L~U~T~~sAT ~N'I~PU~N~VALUE!. EQUAL.TOLOCA.TIOJjOE.STRAIN..;

GAGEflOfIITORED .

, ..-'-'-'''''''''''' ...n.

70 Ii i

,:.- +.IiI

00'---0~

X60

.,

I

~

i , :I 'I I

t t- i--l--~--'~---l. I . . ..Ii : i; ;--'-"'1

I~EASUREDSTRESS, 'PRESSURETO I.TEST CYLINDER..WITH PIli IROTATION. :. . -1

i! i. . /

....'1,/) .a..

i I'I,/):.VL.. .- .W0::I- 40I,/)

-w---.:>: tn 30

1,/)'W

-o::n__. .C-

. ~ 20

.0ICJ

50-1 . .:" "CALCUlATEDSTRESS~ITH'J! IIL.ROTATWN..Le.~u...=-la..-":". .. ,

10

. .-------.-

'--' '--

CALCULATEDSTRESS. NOPIN ROTATION; i.e..u " 0 .

00 1000 2000 40003000

. n . '-- .~... L...PRESSURE-PSI

Figure 12. Comparisonof MeasuredandCalculatedStress Values

Theseassumptionsminimizetheamountof informa-tion requiredfor a newapplicationandtendto makethedesignconservativeor increasethesafetyfactors.However,if all detail informationis availableandamoreprecisecalculationis desired,mostof it canbeenteredrathereasily, or at least their effects in-cludedbymodifyingotherinputitems,includingtheeffect of pin rotationat thecapend. Misalignmenteffects cannotbequantitivelydetermined,butweas-sumetherewill benonethat cannotbeaccommodatedby thesafetyfactor, as previouslymentioned.

Theformulasas presenteddonot properlyaccountfor theeffect of theweightof thecylinderif appliedto a trunnionmountedcylinder. Theclosedlength,L6'wouldbeassumedto bethedimensionfromtherodeyeto the trunnionin theclosedstate. In theextremecase,the trunnioncouldbe locatedat thebearingin

the head, anda large part of the cylinder's weightwouldactually producea negativemomentin the rodwhencomparedto the assumptionsused in deriving theequations. Therefore, somesort of modification,either in the equationsor in the magnitudeandpossi-bly sign, of the weightcomponentassignedshouldbemade.

Conclusion

Formulasare presentedfor calculating the deflec-tion andstress level in the piston rod of a simplepinnedendmountedcylinder. Theyhavebeenverifiedto agreevery well with laboratory tests set up speci-fically to reproducethe loading conditions uponwhich

. u_'-' . __n__" ,.. --- ...."-..---

1.6

1.4

w.>-'w

1.2a

.- 0-0::

~1.0

ALL CALCULATEDVALUESUSED

E,",~ .50..lbs., ~ " ,0.. --. . .

u. . .--....-. .L --. '-'-' . . -- ...'

II. .,

~8J-=::'T~~~E~~':~='". WITHPHI ROTATIOfI.'1 u " 0~.. -- -.- - "' --.E>-.6

-z-_.. 0

....I- .4CJW

-':J-"u..~.2

' '-""--""'-- "

,--.I . . .

r~:;~--- i. 2000: _.-L....PRESSURE - PSI

Figure 13. Comparisonof MeasuredandCalculatedDeflections At the RodEyeCenterline

they werebased. Althoughthey are not recommendedformanualuse, they are nowbeing usedin a desk top sizecomputer.

Rotationof the rodpin hasa verylargeeffect onthe stress level in therodfor ou~typical mobiletypeof cylinder. This suggeststhat onepossiblewaytoincreasethe loadcarryingcapacitywouldbeto applymountingpin bearingshavinga muchlowercoefficientof friction thantheonestestedhere.

References1i

1. Rothbart,H. A., "MechanicalDesignandSystemsHandbook", McGraw-Hi 11, NewYork (1964)

2. Thomas,G. B., "CalculusandAnalyticGeometry",Addison-Wesley,Reading,Massachusetts(1972) .1

;j.}

164 "

j

APPENDIXA

The symbolsused have the following meanings:

c - The offset of the rod eye centerline from theextended centerline of the cylinder tube whenthe rod is deflected without bending to takeup the clearances that mayexist at the headbearing and piston bearing and also allows foreccentricities that mayexist at the head andpiston. This is practically the offset exist-ing whenno force exists except the weight ofthe cylinder on its mounting pins in the hori-zontal position.

dl- The internal diameter of the cylinder tube.

dZ- The outside diameter of the piston rod.

d3- The outside diameter of the pin at the rod eyemounting.

F - The axial force exertedby the cylinder.

Mf- Themomentcausedby rotation of the pin in therod eye.

Fw- The weight componenton the rod eye pin.

h - The offset of the centerline of the rod eyefrom the extended centerline of the cylindertube under any loading condition calculated.It is the sumof the offsets due to clearancesand eccentricities plus the actual bending de-formation of the rod.

Ll- The closed length of the cYlinder.

Lz- The stroke of the cylinder.

L3- A dimension from the bearing in the head to thecenterline of the rod eye pin at whateverstroke is used for calculation.

L4- A dimension from the head bearing to the cen-terline of the rod eye in the close positiononly.

L5- A dimension from the head bearing to the bear-ing on the piston at the state of calculation.

L6- A dimension from the centerline of the mount-ing pin in the cap to the bearing in the head.

P - The designated maximumoperating pressure.

w- Sameas Fw as used in formulas, i.e., trans-verse force on rod at the rod eye pin.

X - A dimension that is from the bearing in thehead to either a predesignated point along therod at which stress and deflection values aredesired or a calculated result which is thelocation of the highest stress along the rod.A switch setting on the calculator, set priorto a calculation, tells the calculator whichway to interpret it.

y - The offset of the centerline of the rod underany loading condition with respect to the cen-terline location when no bending deformationexists and at the location x along the rod.

y max- Theoffset of the rod pin centerline withrespect to the c dimensionanddueto bend-ing deformationof the rod only.

8 - Theanglebetweena line passingthru the cen-terline of both cylinder mountingpin~andthecenterline of the cylinder tube.

In addition to the above from figure 3, someothersymbols used in the derivation of the formulas are asfoll ows:

E - Modulus of elasticityI - Momentof inertia

k - Algebraic sumof clearances existing at thehead and piston, and of the eccentricities atthese samepoints

MA- Momentat location x along rod due to theaxial force F

MW-Momentat location x along rod due to theeffect of the FWforce at the rod pin

MT- Momentat location x along the rod due to thesumof Mf + MA+ MW

Q - FIT

SA- Resultant axial componentof stress at loca-tion x

SB- Resultant bending componentof stress at loca-tion x

ST- Resultant total combined stress at location x

Sy- Proportional limit value of stress for mate-ri al in the rod

SF- Safety factor

~ - Coefficient of friction

There are a few other symbols used in the derivationto simplify the expressions, but these are definedwhere first used in the derivations.

165

kl- hL6 + )Jd3+ WL3

L6+L3T

kZ- Zc- h + W

L3 L6+L3 F

APPENDIX B

Derivation of Formulas

The assumption is made that the sin a = tan a, since the angle

It can then be shown, using free body diagram principles, thatent at any point is given by the following expression for MB :

MB=- F[h + (X-L3) h ':. O~- Y + }ld31 - W(L3 - X)

L6+L3 L3 2J

a is less than 2 degrees.

the maximum bending mom-

B-1

Then after expanding and changing signs,

-M = F h

(

L6+X

)B --

L6+L3

or -MB = X(~L6 +L3

FOX F P.d3 WL3 W X - F Y- t + -L3 2

- F - ~ + F h L6L3 ) L6+L3

B-2

- F Y + F}l d3 + WL32

B-3

From Bernouli-Euler, y" = -=..1!E I

B-4

so, y" = X

(

F h - F - W

)

+ F h L6

E I F6+L3 L3 E I(L6+ L3)

!..z + F p. d3 + ~EI 2EI EI

B-5

y" +!..z = X

(

F h - F C - W\ + F(

h L6 + p.d3

)

+ ~E 1 E 1(L6+L3) E I L3 E I) E I L6 +L3 2 E I

B-6

Now if we let Q =-E- ,E I

X

(

Q h - SLQ - W\ + Q(

h L6

L6 + L3 L3 E I) L6 + L3

then, B-7

y" + Q,.y

+ )' 2 d~) +

SE I

B-8

To solve for y , set y" + Q y =0 and using the auxiliary equation rl- + Q =0,cor D = ! If -Q = ! i {Q, then,

Yc = 1 eiVQ X + 2 e-iVQ X,

but from Euler, eie = cos 9' + i sin g'

B-9

B-I0

B-ll

"g'and e-1 = cos g' - i sin g' B-12

letting g , = (Q X and substituting, B-13

Yc 7 1 [cos(ifQ X) + i Sin(\jQ X)] + 2 [cos(v'QX)

= (1 + 02) cos(fQ X) + (i 1 - i 2) sin(.fQ X)now let A = 1 + 2 and B = i(Ol- 2)

y c = A cos(VQ X) + B Sin{fQ X)

- i Sin(fQ X)]B-14

B-15

B-16

then, B-17

166

Nowto solve for yp - observe that F(X) in equation B-8 is of the form COX+ Cl B-18

therefore, let yp =COX + Cl ' thenyp=~ =Cod X

Substituting into B-8, 0 + Q yp =0 + Q(COX + Cl) , and

Q(CoX + Cl) =X(

Q h - ~ - .JL)

+ Q

(h L6 + ~

)L6+ L3 L3 E I L6+ L3 2

and y" =0 B-19

B-20

+(~)B-21

Equating like coefficients of X, Q Co = Q h - ~ -.JL , divide B-21 by Q, then B-22L6+ L3 L3 E I

At X =0, andy =0, B-25 then gives A = -

(h L6 + ~

L6+ L3 2

+

(h L6 +~ + WL3\B-25

L6+L3 2 F )WL~ \

+ ~)' but since B-26

y =Yc + Yp = A cos(1Q X) + B sin(.JQ X) +(

h - 9-- !I)

X

L6+ L3 L3 F

y' =..z =- A sin({Q X){Q + B cos(J"QX)JQ + (h - Q - !I)d X \1'6+L3 L3 F

B-27

At X =0, y' =Q ,then Q = B {Q +---1!

L3 L3 L6+..L3

- Q -!I ,therefore,L3 F

B-28

B = 2 0

L3 JQ

h + 'vi- -L6+ L3 VQ F JQ

now substituting into equation B-25 gives, B-29

y =-

( h L6 +~ + 'viL3\ cos (fQ X) +f. 2~ - h r;::- +L6+ L3 2 F ) \L3 .J Q L6+ L3'V Q

(h - Q - !I

)X + h L6 + ~ + W L3

L6+L3 L3 F L6+L3 2 F

...L..\sin(~ X)F {Q)

+ B-30

or,

y = (h L6 + ~ + W L3)(1\L6+ L3 2 F cos(JQ X)\ +(LQ - h +!I\(Sin(JQ X)\ +J L3 L6+L3 F)\ \jQ J B-31

(h - C - 'vi

)X

L6+ L3 L3 F

Which is the formula used for deflection (y).

Nowto solve for h, observe that at X =L3' Y =h - 0 , and substituting these intoequationB-3l, weobtain,

167

Cl = h L6 + + , but Q =-L , so B-23L6+ L3 2 E I Q E I

C1 = h L6 + + , and00=(h

- C -),

therefore,B-24L6+ L3 2 F L6+L3 L3

h - C =~ - h L6 cos (VQ L3) - h sin(v'QL3 + h L3 +!pd3+WL~(1 - cos(.JQ L3~+

L6+L3 L6 + L3 (L6 + L3)IQ L6+L3 \ 2 F I )(~ +~ ) sin (VQL3) - (~ +~) L3 ' expanding and transposing terms,\L3 IQ F.JQ L3 F

h - h~L6 + L3) - L6 co,(;QL3) - 'in