Screw thread mechanics

There are always three major components in practical applications of the screw thread mechanism :

A nut can spin and move freely along a screw without contacting another component, ie. without the need for any thrust bearing, but a thrust bearing comes into existence immediately contact occurs and the mechanism is put to practical use.

Clearly screw jack there is relative motion in the thrust bearing, and also between the nut and the screw - and where there is relative motion there is   friction. We now examine the role of friction since it dominates the behaviour of the mechanism unless special ( read 'expensive' ) means are taken to minimise its effects. When considering friction it doesn't matter which component rotates and which is stationary - it's the   relative motion which is important. We shall therefore analyse the jack shown here to deduce the general effect of friction on screw thread behaviour.

The jack's screw is fixed; the nut is rotated by a spanner and translates vertically. The thrust collar's only motion is vertical translation as it is prevented from rotating by contact with the load, one corner only of which is pictured. Since there is relative rotation between contacting nut and collar, the contacting surface assumes the role of thrust bearing.

The torques on nut nut shown here in plan is in contact with three bodies :

We are interested in the tightening torque   T, and, if the nut is in equilibrium then
( i)         T   =   Tt + Tb
                                        from which we can evaluate   T once   Tt and   Tb are found individually. friction in the thrust bearing

Consider the thrust bearing first. We shall assume that the contact surface of area   A is in the form of a narrow annulus of mean radius   rb on which the uniform pressure is   W/A, where   W is the load supported by the mechanism. If the coefficient of friction in the bearing is   μb then the torque exerted by the frictional force on an area element δA is   δTb   =   μb δN rb   =   μb rb ( W/A ) δA. Integrating over all the contact area
( ii)       Tb   =   W μb rb

Consider now the thread which is square, of mean radius   rm and lead angle   λ. The nut engages the screw with friction coefficient   μ corresponding to a   friction angle φ = arctan μ. The static and kinetic coefficients of friction are taken to be essentially equal for this preliminary analysis.

We wish to find the torque   Tt which must be exerted on the nut to offset thread friction and maintain the load   W in equilibrium - that is either static or moving at constant speed. A torque which tends to raise the load is reckoned positive; a negative torque is one which tends to lower the load.

A small element of the nut is shown below sliding or about to slide on the inclined ramp of the thread, this motion being either up or down the thread. Both motion directions are sketched. δP is the small force on the element due to the torque, and the element supports a small part of the load   δW. The contact force components are the normal and friction forces,   δN and   δF - the latter opposing motion or motion tendency.
  The nut element free body
requires for equilibrium . . .
element going up   element going down
      normal to plane : δN = δP sinλ + δW cosλ
      parallel to plane :     δF =   δP cosλ - δW sinλ       δF = - δP cosλ + δW sinλ
  If the element is moving, or on the point of motion, then   δF = μ δN   and so
      eliminating   δF, δN :     δP = δW tan ( λ + φ )       δP = δW tan ( λ - φ )
  The total torque   Tt is the sum   Σ rmδP over all the identical nut elements, ie.
( iii)     . . . . if motion (tendency) is     Tt = W rm tan ( λ + φ )
      Tt = W rm tan ( λ - φ )

When raising a load the thread inclination   λ is thus effectively steepened by the thread friction angle   φ.

The upward tightening torque from ( iii) is plotted here against friction coeffficient for a lead angle,   λ = 0.25 ( 14.3o ). thread torque behaviour
When there is negligible thread friction ( φ = 0 ) then   Tt /Wrm = tanλ = 0.255, the point a.
The equilibrium line a-d corresponds to incipient motion up the thread; the equilibrium line a-b-e corresponds to incipient motion down the thread. The area between these lines represents stasis ( δF < μ δN ); the areas beyond the lines correspond to non-equilibrium, ie. to acceleration - though it should be realised that only thread friction is being considered at this point.

Consider what happens if the coefficient of friction is 0.5 for example. Before any torque is applied ( Tt = 0 ) the nut is at rest, point c. The torque is thereafter increased until at d the nut breaks away and rotates up the thread at constant velocity. If the torque were to increase further then the nut would accelerate.
If however at c the torque were to increase in a negative sense - to 'undo' the nut, assisted by the load - then when point e was reached the nut would break away and move down the thread at uniform speed. Any further increase of torque in the negative sense would lead to downward acceleration.

The point b is significant. It corresponds to the critical coefficient of friction for the lead concerned,   μc = tanλ,   at which the nut is on the point of moving down the thread of its own volition without any torque applied ( Tt = 0 ). Further, if . . .
  μ>μc then the thread is   self-locking in that the nut cannot undo by itself, it needs to be unscrewed by a definite negative torque; or
  μ<μc then the thread is   overhauling in that the nut will unscrew by itself under the action of the load unless prevented by a positive tightening torque.
Clearly self-locking behaviour is essential for threaded fasteners. Car lifting jacks would not be of much use if the load fell as soon as the operating handle was released - so they too demand self-locking characteristics.
archimedean drill
pump action screwdriver

Some applications of power screws require overhauling behaviour. The Archimedean drill (above) and pump action screwdrivers (below) incorporate very large lead angles which guarantee that the critical friction coefficients are larger recirculating ball screw than any actual coefficient likely to be encountered. Sensitive linear actuators may incorporate recirculating ball screws such as that illustrated on the left to reduce thread friction to levels which go hand- in- hand with overhauling.

Efficiency is important in rotary/ linear motion transformation such as provided by power screws. Whilst acting against a load, the efficiency of a screw thread alone ( ie. neglecting the thrust bearing ) is : thread efficiency
            η   =   Wfor a given Tt / Wfor the same Tt without friction
                  =   tan λ / tan( λ + φ )       from ( iii)

The effect of lead and friction on thread efficiency is plotted here - as expected, high efficiencies demand very low friction. Also shown is the line of critical friction.

Coefficients of friction around 0.1 to 0.2 may be expected for common materials under conditions of ordinary service and lubrication - steel on steel for fasteners, and steel on bronze or cast iron in the case of power screws. Special coatings and surface treatments can reduce friction significantly. Vibration also reduces friction and can cause fasteners to unloosen unless locked as mentioned above.
Lead angles around 3o are typical of single start power screws which are therefore somewhat inefficient - this is one reason for multiple start screws with their associated large leads.

From ( i), ( ii) and ( iii) the total torque necessary to raise a load   W whilst overcoming thread and thrust bearing friction is :

( 2)         T   =   W [ rm tan( λ + φ ) + μb rb ]

The foregoing analysis has been developed for square threads ( α = 0 ), but ( iii) and ( 2) may be applied with little error to Acme threads ( α = 29o ), and also to ISO Metric fasteners ( α = 60o ) provided the effect of wedging ( analogous to V-belts ) is included - thus the effective coefficient of thread friction to be used in ( 2) is   μ sec( arctan( cosλ tanα/2 )), which tends to   μ secα/2 for small lead angles.

A jack similar to the above is analysed in this example.

Static failure

    class no. 4.6 5.8 8.8 9.8 10.9 12.9
    Su MPa 400 500 800 900 1000 1200
    Sy MPa 240 400 640 720 900 1080
    Sp MPa 225 380 590 650 830 970
    elong'n % 22 20 12 10 9 8

We shall concentrate on steel fasteners, which, because of their strength and cheapness, constitute over 90% of all fasteners used. Steels for commercial fasteners are graded into property classes as in Table 2. Each class number consists of two figures Minimum material strengths are not less than nominal values.

It is difficult to determine the yield of a full size threaded connector ( as opposed to a cylindrical test piece ) because of the different strain rates of shank, thread and runout. For this reason the proof stress rather than the yield stress is used as a criterion for failure assessment - the   proof stress   Sp is the largest stress which does not lead to any permanent set. The proof load of a screw made from a particular material is the maximum load the screw can withstand without permanent deformation, and is given by ( 1) as the product of stress area ( Table 1 ) and proof stress ( Table 2 ). Selection of class 12.9 should not be undertaken lightly - its high strength begets other potential problems.
load & Stress variations
We have noted that the elastic behaviour of connected members has a significant effect upon the distribution of load between their connectors. In an elastic bolt and nut connected by a number of thread turns, the first 'thread' takes a disproportionate fraction of the total tensile load - Craddock   op cit reports the load distributed among the threads as sketched here. It is clear that there is little benefit from an engagement length exceeding half-a- dozen threads. Special nuts which alleviate non-uniform load distribution are available. The variation of normal stress   σ over a cross- section adjacent to the nut face ( as sketched ) is dominated by stress concentration induced by the thread root, so it is hardly surprising that bolt fractures usually occur in the exposed threads close to the nut face.
shear failure of nut

Another failure mechanism is stripping of the nut threads, which is essentially shear failure of the nut material on the cylindrical surface at the thread major diameter. Stripping of the bolt threads is a similar shear of the bolt material at the minor diameter - but this is rare. Other possible failure mechanisms, such as crushing of the nut bearing surface and dilation of a thin nut due to its riding up the thread flanks, are not critical in themselves, but contribute to other modes.

If any failure is to occur, then bolt fracture is the preferred mode. Bolt fracture is clearly discernable and often occurs whilst tightening, when torsional shear stresses ( which dissipate quickly after tightening ) are superimposed upon tensile stresses. So the operator replaces the bolt and learns to exercise care in subsequent tightening. Thread stripping on the other hand is insidious and progressive - the first thread fails putting more load onto the remaining threads, hence the second succumbs . . . and so on. When threads strip it is often difficult to separate components for fastener replacement, whereas a broken bolt requires no further separation. The consequences of a tapped hole in a car engine block being stripped by over- zealous spanner wielding is a case in point.

For a given load, bolt fracture tendency will clearly be reduced by a larger stress area ( ie. a larger bolt size ) and/or a higher class of material. Given these parameters ( load, size and bolt class ), nut stripping will be reduced by a longer nut and hence an increased cylindrical shear area, by a thicker walled nut to decrease dilation, and/or by a superior nut (property) class.

The first stage of fastener design is bolt selection - size, class and other geometric attributes. ISO Metric nuts have been dimensioned to bias any failure towards bolt fracture rather than nut stripping, on the tacit presumption that the nut class is equal to or greater than the bolt class - ie. the material of the chosen nut is at least as strong as the preselected bolt's material. It is for such reasons that regular nut lengths are about 80% of bolt size, and of the hexagonal dimensions shown in Table 1 - though there are various styles of nut which differ from these figures.

|   Notes contents   |   chapter index   |   previous   |   top of page   |   next   |

Valid HTML 4.0!     Copyright 1999-2005 Douglas Wright,
      last updated May 2005