Modeling and Simulation for Material Selection and Mechanical Design Part 15 pot

To obtain maximum assemblytorque for overelastic tightening method, take maximum friction coefficientsand highest screw strength.. Mechanical Loading If a threaded fastening system is ti

Wpfor full plastified cross-section is used (Wp¼ pdo =12) Step 7 offers thelead angle of the thread profile j Step 8 formulates the theory of maximumdistortion energy for producing a material failure (this is also called ‘‘vMisestheory of failure’’) This is the background-formula for step 1 to combineequivalent stress and axial stress General information about theories of fail-ure can be found in Ref [3].

In general, the friction coefficient m is defined as the ratio of normalforce acting over produced tangential frictional force in a sliding motion

of two bodies (Fig 17) The frictional force is always directed against thedirection of motion For a screw, the normal force is the preload Fp Thetangential force can be formulated as mtFpin the thread contact zone and

as mhFp in the head support area These tangential forces cause frictionaltorques, because of the radii of thread and head contact zones due to screwaxis (diameters Deb resp d2, see also Fig 16) Therefore, the frictional

Figure 17 Definition of friction coefficients mhand mt

coefficients define the part of preload, which acts tangentially in the contactareas of a screw.

Table 4 proposes classes of frictional coefficients valid for boltedjoints, based on the VDI 2230 guideline [70] and experience [17] If no exactvalue is available, one can select a value from this table which is valid forlow surface roughness But one must always remember that the friction co-efficient depends on complex influences like materials surfaces, lubricationincl homogeneity, hardness ratio of the two surfaces in contact, local stresspeaks or stress distribution in contact zone, tolerances for contact geometry

as well as tightening level and number of (re) tightenings A selection tablecan only provide rough approximations The supplier of screws can provideinformation related to friction behavior

In practice, all parameters for calculations ofFig 16have deviations.Main influences are based on minimum and maximum strength of screwmaterial (e.g heat treatment process) as well as minimum and maximumfriction coefficients (e.g roughness and lubricant) Geometry is usually veryprecise, so tolerances from diameters are not significant for screw tightening.This situation is shown schematically inFig 18with two correspond-ing diagrams for highest material stressing in the screw shank The uppercase A refers to conditions with minimum friction mmin (both, mtmin and

mhmin) and maximum screw strength Rmsmax On the abscissa axis, the

Table 4 Values for Guidance of frictional Coefficients mtand mhin

Classes A–E [17,70]

A 0.04–0.10 Hard polished surfaces, thick lubrication with wax or grease,

high pressure lubricants, anti-friction coatings, e.g polishedmagnesium and screw with PTFE-low friction coating andMoS2, no peak pressure at edges of support area

B 0.08–0.16 Commonly used conditions with defined friction by optimized

lubricants, such as oil, wax, grease for fasteners; suitable forferritic steel metallic blank, phosphate, zinc and microlayersurfaces as well as nonferrous metals with relevant lubricant

C 0.14–0.24 Usual conditions with only thin or inhomogeneous lubricant,

austenitic steel screws with suitable lubricant; zinc, zinc alloy,and nonelectrolytical applied surfaces without lubricant

D 0.20–0.35 Austenitic steel with oil, rough surfaces and Zn=Ni coating

without lubricant

E 0.30–0.45 Austenitic steel, aluminum, and nickel alloys blank without

lubricant

plastification of the screw shank (transition of the strong gradient of thetightening curve in Fig 18 to the low gradient in the range of plastifiedscrew).

Another possibility to reach high tightening levels is using the angularcontrolled tightening method (also called ‘‘turn-of-the-nut-method’’): Afterapplying a snug-torque Tsan additional, fixed defined tightening angle Du isadded, so the screw is plastified to a certain grade in any case (comp mark-ings in Fig 18)

For yield point controlled tightening and angular controlled tighteningthe ratio of Fpymax=Fpyminresp Fpanmax=Fpanminis about 1.1–1.3 The devia-tion in practice is reduced drastically For this reason, the greatest advan-tage of overelastic tightening methods is a significant increase of theminimum preload and a slight increase of the maximum preload But onemust always note the resulting torque value can vary extremely for overelas-tic tightening methods, because torque is no controlled parameter

Some hints for selection of parameters considering deviations in tice are: for calculating the highest preload (related to the highest screwstressing) always take minimum friction coefficients and maximum screwstrength This is relevant for maximum contact pressure under head) Ifthe lowest preload has to be determined, maximum friction coefficientsand lowest screw strength are relevant To obtain maximum assemblytorque for overelastic tightening method, take maximum friction coefficientsand highest screw strength This is relevant for maximum screw driveloading

prac-If new tightening devices have to be designed for a production linewith screw assembly, these devices should be able to apply a high torquevalue for angular controlled tightening In practice, more than the doubletorque limit should be designed compared to torque controlledtightening

E Loading During Operation

1 Mechanical Loading

If a threaded fastening system is tightened, then screw, clamped part, andnut thread component are loaded mechanically by the flow of preload with-out external force(Fig 19).The preload leads to head contact pressure pchbetween screw head support and clamped part surface as well as to threadcontact pressure pct at engaged thread flanks Between clamped part andnut thread component, the component contact pressure pcc is generated(important for sealing) Following considerations due to force—elonga-tion-behavior which are based on Ref [70], details are discussed in Refs.[7,67,72]

This idealized model reduces all elastic contributions within the system torigid bodies and two springs with defined resilience: The screw shank ismodeled as one tensile spring with ds, the clamped part is represented by

a compressive spring with elastic resilience dp

Before tightening, all ‘‘springs’’ are unloaded (left side of Fig 20).After tightening, usually the tensile spring of the screw is elongated muchmore than the compression spring of the clamped part (right of Fig 20)

If an external axial force Fax is induced within the clamping length lc, theinducing factor n determines which part of the clamped part is additionallyloaded (towards the screw head) and which part is unloaded by Fax(towardsthe nut thread component) These parts of additional loading and unloading

by an external axial force Faxinfluence the relevant elastic resiliences of dsand dp, if the fastening system is loaded Therefore the resiliences varybetween tightening and operating, if n< 1

Fig 20 leads to the following force–elongation diagram shown inFig 21 The diagram shows on the x-axis the elongation of screw(left of ‘‘0’’) and clamped part with clamping length lc (right of ‘‘0’’) Onthe y-axis, the corresponding preload Fp in the screw shank is drawn For

Figure 21 Force–elongation-characteristics of screw and clamped part

the stable tightening level Fp0, a (positive) screw elongation of Fp0dsand atclamped part an (negative) elongation of Fp0dpis generated.

The representative curves of screw and clamped part are linear up tothe yield point of each material Here, the stable tightening level Fp0is com-pletely within the linear range If screw or clamped part show plastification,each nonlinear behavior has to be considered for force–elongation diagram(degressive dashed lines inFig.21)

If a tensile external axial force Fax is applied to the fastening system,

on the one hand, the screw is loaded additionally by nfFaxand on the otherhand the clamped part is unloaded by (1 nf)Fax, because the two springsare a parallel arrangement The consequence is that Faxreduces the residualclamping load and increases the tension in the screw shank, but always only

a part of Faxacts in any ‘‘spring’’

The additional operating force of screw (nfFax) besides the load factor

f is dependent on the inducing factor n For this reason,Fig 22gives someexamples for the value of n, which are approximations Some referencespropose a calculation of n [70], but an analytical solution is usually a lot

of work, and a simple approximation often gives the same range in practice

Figure 22 Examples for approximation of inducing factor n (From Ref 70.)

Numeric calculations like FEM are very suitable to determine

nf ¼ Faxscrew shank=Fax externaldirectly for a given geometry by selecting thenodes of the screw shank cross-section for Faxscrew shankand all nodes, whichare loaded externally for Fax external With the result of nf, the analyticalcalculation can be continued; therefore, FEM can be used to consider allinfluences from geometry and inhomogeneous stress distribution (e.g forclamped part)

The determination of the inducing factor n is an example, to show thatvery detailed design modifications lead to significant changing in screwloading In general, it is valid that a small inducing factor n decreases theadditional operating force of screw (interesting for increasing the fatigueloading capacity of the fastening system), and reduces also the residualclamping force under axial loading with an operating force (compare alsoFig 21)

If no numeric calculation is done, the load factor f can be mated with the analytical model ofFig 23,see also Ref [70,72] This loadfactor can be calculated from f¼ dp=(dsþ dp), if the axis of screw, clampedpart centerline and external axial force Fax is the same If these axes havedifferent positions, additional bending of the screw and clamped partoccurs, so that the elastic resiliences and in consequence the load factor fare changed

approxi-Figure 23 Linear model for determination of load factor F (From Ref 17.)

For the model shown inFig 23,the force Fax, the distances of axes sand a, the through-hole diameter dhas well as the elastic resiliences dsand dpfromFigs 12and15 and the substituted area Asubmust be known Fromthese, the substituted diameter Dsub can be calculated This constantdiameter corresponds to Asubfor the same resilience dp The model is assum-ing a linear stress distribution s(x) within Dsub.

For the use ofFig 23,it is necessary that the real stress distribution issimilar to the linear distribution in the model The size of the clampedpart may not be much larger than Dsub, so the moment of inertia Ifullkeepsvalid

Then, the moment of inertia Ifullcan be obtained and as a next step fcan be calculated Ifull does include the cross-section area of the screw,because the screw gives also a bending resistance during loading with Fax.After tightening, any threaded fastening system shows relaxationeffects This short time relaxation often is called ‘seating’: it leads to a pre-load reduction as demonstrated in Fig 24.Important influence for this isthe roughness and strain hardening of all surfaces in contact zones betweenscrew, clamped part(s) and nut thread component as well as the direction ofmechanical loading due to a normal vector on the contact area Under con-tact pressure, the high surface spots are deformed axially which leads to aseating distance fzof the fastening system and in consequence to a reduction

of preload down to a stable preload level Fp0

Significant short time relaxation always occurs if the fastening system

is partially overloaded, such as when thread engagement is too small

Figure 24 Preload reduction by seating (short time relaxation)

(seeFig 33)or if contact pressure under the head is too large (seeFig 39),material mismatch (e.g material strength of clamped part is too low) or geo-metric mismatch (e.g nonperpendicular nut thread or screw head, oversizedunderhead fillets) The approximational equation for fzgiven inFig 24can

be used if there is no partial overloading

An eccentric loading of a threaded fastening system can lead to ponent separating.Figure 25demonstrates this for an external force Faxact-ing with a distance a from the axis of symmetry 0–0 of clamped part Theconfiguration ofFig 25is the same as inFig 23

com-Figure 25 Mechanics of component separating as a result of eccentric loading by

Fax(From Ref 17.)

There exists a point of tilting on one side of clamped part; on the site side, of the first component separating occurs With the given values

oppo-Fp0, Fax, s, a, dh, Dpand f after calculating the area Apof clamped part inthe contact zone between components and the moment of inertia Ip, the pre-load for first separating Fpscan be estimated for a given axial force Fax If thepreload Fp0is larger than Fps, then component separation does not occur forloading with Fax

On the other hand, if a stable preload after tightening Fp0is given, F critdetermines the beginning of component separating, if Fax> Faxcrit Thisleads to two cases indicated inFig 25.Case 1 is determined by elastic screwloading regarding the force–elongation diagram of a threaded fastening sys-tem The additional operating load of screw Fsais equal to nFFax Case 2refers to the situation of a beam lever system, built by Fpand Fax and thelength values a, s, Dp

ax-Component seperation must be avoided (case 2) because it leads toextensive additional loading of the screw Fsaand to early failure either bystatic overloading or by fatigue fracture But in some cases, for optimizedcomponents with high resilience dp and with exactly defined tightening byloading, a partial component separation can be allowed without problems(e.g bolted joints at lightweight piston rods) For more details regardingcomponent separation under eccentric mechanical loading, see Refs [67,70]

Figure 26 Preload behavior for overelastic tightening

Figure 26 explains the preload behavior for overelastic tightening ofscrew The corresponding force–elongation diagram illustrates the screwplastification with a degressive curve for exceeded elastic limit under the ten-sile and torsional stressing during tightening.

The first preload level after tightening Fp1is reduced to the stable load Fp0by the reason of seating effects Besides this, a general aspect is thatafter tightening a screw the torsional stress is reduced significantly—to app.30–50% of the torsional stress under applied torque This leads to anincreased elastic limit of screw and leading to a higher preload limit duringoperating compared to tightening A screw, which was tightened overelastic,can be loaded by a large operating force Fax In practice, there is almost nodifference between the tightening methods due to the loading capacity dur-ing operation (for dynamic loading, see alsoFig 52)

pre-Up to now, no time dependence of mechanical load is considered.Fig 27displays the effects for an alternating axial force Fax For positive axial force

Fax (tensile loading), the preload in the screw shank will be increased andthe clamping force will be reduced, producing the same effect as for static load-ing If a threaded fastening system is loaded axially, the preload in the screwshank is not the same as the clamping force between components

For a negative axial forceFax, just the opposite aspects are true: thepreload in the screw shank will be reduced and the clamping force between

Figure 27 Preload behavior for mechanical dynamic axial loading