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J. W . Kanne l
Battelle Columbus Laboratory,
Columbus, Ohio 43201
T. A. Dow
Precision Engineering Laboratory,
North Carolina State University,
Raleigh, N.C. 27695-79 10
Analysis of Traction Forces in a
Precision Traction Drive
A theory for the shear stress between a rough elastic cylinder and a cylinder w ith a
soft laye r has been developed. The theory is based on a Fourier tran sform approach
for the elasticity equations coupled with surface deflection equations for transient
contacts. For thick layers (h > .001 in.) the shear stress on the surface approaches
the shear of the layer alone. The elastic shear deflection (-100 \xin.) as a result of
the tangential load is significant and increases if a surface layer such as a thin
coating is added to one or both cylinders. The predicted interfacial shear stresses are
considerably altered by surface roughn ess on uncoated surfaces and these effects
are ameliorated by the addition of a thin soft surface coating.
Intr oduc t ion
Two crit ical aspects of a precision machine are l inear
location of one part relative to another and smooth motion
between lim its. One m eth od of achieving this result [1, 2] is to
use a traction drive on the slideway as i l lustrated in Fig. 1.
Positioning accuracy below the microinch level is typically
required. W hen accu racies of this level are involved, virtually
all factors which affect motio n must be considered in order to
minimize errors in the system. One such factor is the shear
deformation of the drive system, especially the elasticity of
the traction interface. The elasticity is affected by many
factors, including the Young's modulus of the traction
components, surface layers on the rollers (such as solid fi lm
layers) and the roughn ess of the rollers and slideway.
The most extensive work reported on the analysis of the
traction interface is by Kalker [ 3 , 4 ] . Kalker traces the traction
interface between two extremes: the Cattaneo [5] problem and
the Carter [6] problem. The Cattaneo problem occurs when a
cylinder is rotated slightly, while in contact with a stationary
surface. The Carter problem occurs when both the cylinder
and the mating surface are moving but at sl ightly different
speeds. Kalker 's study traces the traction forces through the
transients between the two extremes.
Bentall and Johnson [7] analyzed the slip between two
dissimilar cylinders in rolling contact. This research allowed
for tangential deflections due to microslip. Barber [8] con
ducted research similar to Kalker 's, only he analyzed three-
dimensional contacts of rollers under misalignment. Poritsky
[9] derived basic equations for cylinders in contact and
discussed the problem of rough surfaces. Krause and Senuma
[10] did experimental studies with rollers which developed
surface corrugations. The surface corrugations notably af
fected the tractio n beh avior of the cylinders.
The work presented here is an extension of the Kalker and
Poritsky work with allowances for surface layers. The surface
layer algorithm is developed from the work of Sneddon [11],
and Gupta and Walowit [12].
Contributed by the Tribology Division of THE
AM ERICAN SOCIETY
OF
MECHANICAL ENGINEERS and presented at the A SM E/A SL E Joint Tr ibology
Conference, A tlanta, Ga ., October 8-10, 1985. M anuscript received by the
Tribology Division, A pril 19, 1985. Paper N o. 85-Trib-45.
Leaf spring
Hydrostatic bearings
Drive motor
Traction bar
Idler roller
Capstan
Hydrostatic
bearings
The resolution of the slide drive will be
0.2pin. (0.005yumj. This is the equivalent to
0.05 arc.
Fig. 1 Illustration of traction control system for precision engineering
A p p r o a c h
The same general approach used for the normal stress
analysis [13] can be used for the shear stress computa tions. A s
will be shown subsequently, the matrix equation is almost
identical to that developed by Gupta-Walowit for the normal
stress computations. That is the deformation equations can be
put in the matrix form:
CF=e
(1)
where C is the matrix equation with elements c,y that relate the
tangential de flection, e, to the tangential forces. A s with the
normal stress equations, a relationship between point-loads
and po int stresses can be developed.
Matrix Coeff ic ients
For a solid body, Po ritsky derived a similar relationship for
load-d eflec tion coefficients as for nor ma l stress coefficients,
that is
4(1
v
2
)
7rE
In I/-1
(2)
Journal of Tr ibology JULY 1986, Vol . 108/4 03Copyright 1986 by ASME
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1
-s
2
1
- z
Z( v
x
' -
_
(1 + " '
where
z-
? =
1)
)
3 - z -
(1 +
1-
=lsl
=
exp(z)
0
I
\-z
\'(\-z)
)-2z/s
2
1
s
1
7
2
zt
1
( l - v , ' ) z ?
2
(1 + I V ) ?
2
v'=(2-v)/(\-v)
v
=v/(\-
V
)
Table 1
0
s
-,2
( 1 + z ) ?
2
[2+ ( l + z ) ( l - V ) J '
2
[(l + v
x
) + 2z/s
2
]l
2
0
0
- ?
z?
( 1 - K
2
' ) Z ? ( y
2
'
- ( 1 - K
2
" ) 7 ? . 2z7? /s'
3)7?
s
2
c,
Di
A
2
B
J
--
-i
=
=
1
0
0
0
^oj
/3= ( 1 - ^
2
) E , / ( 1 - . ,
2
) E
3
7 = ( l -
J
' 2
2
) E i / ( l - " i
2
) E
2
wherer is the distance between a point tangential load and the
corresp onding tange ntial deflections. A s with the norm al
stress equations, a relationship between point loads and
stresses over a small region is:
F:
rdx
(3)
where Tis the surface shear stress acting over a distance dx (or
A x in finite term inology). Then
-4(1
-v
2
)
7rE
j -
\n\x:x\dx (4)
Inlet
F ig .2 S l ippage in the contac t zone
which allows a solution to be found whenx
;
= x
h
A n expression has been developed for the influence-
coefficient matrix for shear stresses in a layered solid (see
A ppendix A ) . The so lu t ion (assuming no normal pressure on
the surface) can be written as:
7rE
[ J o
2 5
=i
where e,(0) is the assumed initial deflection relative to a fixed
poin t (x
0
= - 106). For subsequent t imes equation (7) is used
with AV At being a constant that is added to each time step to
produce a given traction.
In the computations the coefficients c,y were set and the
shear stress computed by a matrix solution of equation (12).
A t some poin ts
Tj >f'Pj
wh ere / is a coefficient of friction an dP j is the local pressure
computed using the technique given in reference [12 and 13].
For these points the matrix was adjusted as follows:
n
Y, l--8jj
)(c
u
-c
0J
)TjAx=e
i
/ =
n
-H h
f
fPj^iJ-
c
oj)-r
Jf
=fPj
f
(13)
where
I
1
J=Jj
j
are the po in ts where r
y
>
fpj.
The essential size of the
matrix will be reduced by one row for each value ofjf. The
computations involved a simple i teration starting with a full
matrix and subsequently reducing the matirx for each value of
Tj > fPj until further i terations produced no changes.
D isc uss ion
Figure 3 i l lustrates the shear stress at the interface for th e
case of a stationary lower cylinder (Vt/b = 0) and for a series
of relative slip values
(Vt/b
> 0). The slip values are ex
pressed in terms of the half width, b. I f Vt/b = 0.8, a point
on the upper cylinder has moved a distance equivalent to 80
percent of the half-width of con tact between the cylinders.
For the case of a stationary lower disk (analogous to the
Cattaneo problem), the shear stresses are the lowest in the
center of contact and rise toward the edges. This rise is due to
the contribution of each element to support the shear
deformation outside the contact region. The rise in shear
stress is limited at
x/b =
0.6 by slip between the cylinder
surfaces; tha t is, the shear stress becomes equal to the friction
coefficient t imes the normal stress. Thus, the shear stress
curve has the same shape as the normal stress distribution
where slip is present. For this example the horizontal load was
105 N /m m (600 lb / in . ) , and to produ ce th is force, the upper
cylinder was rotated 2 X
10
~
4
rad (the upper surface moved 2
/mi).
Cont inued ro tat ion of the upper cy l inder would produce
rotation of the lower disk because the loading was assumed to
remain constant (analogous to Carter problem). Different
values of the upper cylinder motion are also i l lustrated in Fig.
3 . For the largest rotation of the upper cylinder shown in Fig.
3 (Vt/b = 1.2, correspon ding to 3.2 x 10
2
rad (about 2 deg)
of rotation), a point on the upper cylinder has moved a total
of 2.5 ^m from a point on the lower cylinder which was
adjacent at no load. This means that the driving cylinder will
rota te 2.7 x 1 0
4
rad more than the driven cylinder for an
average rotation of 3.2 X 10 ~
2
ra d against a load of 6001b.
One purpose of the analytical traction model was to
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-0. 8 -0.6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8
Distance From Contact Center, x/ b
Fig .
4 Con tact zone shear stress distribution (0.01 in. layer)
b = .89 mm (.035 in.) Load = 543 N/mm (3099 lb/in.)
F, = 14.5 N/mm (83 lb/in.) R=9.44 m m (.37 in.)
E T =2 GPa
290 ksi)
p
h
=.44 GPa
64
ksi)
e
= 2.5 pm (100 ^in.)
evalaute the role of surface layers on traction drive per
forma nce. Figures 4 to 6 show the shear stress distribution for
the condition of a uniform tangential displacement of 2.5 xim
but with a soft surface layer (E = 2 GP a) on one of the
cylinders. Surface layer thicknesses of 250 /xm, 25 ttm and 2.5
/xm are illustrated.
A ll curves show peak shear stresses at the edges of contac t
and reasonably uniform shear stress in the center. For the
thicker layers this center shear stress should approach the
shearing of a soft layer of known thickness a known amount.
That is:
T
G
m
y G
m
e/h (14)
where
7 is the shear strain
G, is the shear modulus of the layer (G, = E /2 ( l + v) =
7 80 M Pa)
For the 250 /xm layer shown in Fig. 4, the predicted shear
stress using equation 14 is 7.8 M Pa (1.125 ksi). This predicted
stress is consistent with the stress near the center of contact of
Fig. 4. A t the edges of the contact (x = b) the stress rises
considerably above this level to compensate for the forces
required to tange ntially deflect the layer outside of the contac t
region. Very near the edges, the shear stress is l imited by the
coefficient of friction times the normal pressure.
In Fig. 5 the predicted shear stress using equation (14)
would be 78 M Pa (11.25 ksi). This stress is higher than the
stress at the center of contact given in Fig. 5. That is, for thin
films, the shear stresses tend to be high enough to deflect the
substrate as well as the surface. When the substrate is
deflected this deflection must be subtracted from
e
in com
puting the surface shear. For very thin fi lms (h = 2.5 /xm) as
shown in Fig. 6 the center shear stress is considerably lower
than predicted by shearing of the layer alone. For this case the
equatio n (14) shear stress would be 780 M Pa (112.5 ksi) or
about four t imes that given in Fig. 6. Clearly then the
evaluation of this surface films requires the use of com
prehensive theories and cannot be achieved by simple
analyses.
Figures 7 and 8 i l lustrate the compariso n between a layered
and a nonlayered body. For the conditions given here a
surface "wind-up" of 2.5 / t in produces a tangential load of
105 N /m m w hen the soft layer (E = 2 GP a) is in place. If
there were no layer, 105 N /m m could be obtained with a
- 0 .8 - 0 .6 - 0 .4 - 0 .2 0 0 .2 0 .4 0 .6 0 .8
Distance From Contact Center, x/ b
Fig . 5 Con tact zone shear stress distribution (1000 in. layer)
b
= .44 mm (.0173 in.) Load = 490 N/mm (2800 lb/in.)
F
(
= 45 N/mm (258 lb/in.)
R =
9.4(.37 in.)
E = 2 GPa 290 ksi) p
h
=.8 GPa 117ksi)
e= 2 . 5 / i m ( 1 0 0 x 1 0 % i n . )
45
40
35
p
s
" o
30
S
t
e
S 20
CO
15
10
5
-
~ / ^\
I I I
-
y \ -
\ -
i i i
- I .0 -0 .8 -0.6 -0. 4 -0. 2 0 0.2 0.4 0.6 0.8 I .O
Distance From Contact Center, x/ b
Fig . 6 Con tact zone shear stress distribution (100 p in. layer)
b = .29 mm (.0115 in.) Load = 544 N/mm (3110 lb/in.)
F
(
= 103 N/mm
591
lb/in.) R= 9.4 mm (.37 in.)
E=2G Pa 290ksi) p
h
= 1.28 GPa 187ksi)
c
= 2 .5 / im(100/i in.)
"wind-up" of 2/xm, as indicated in Fig. 8. The presence of the
soft layer also increases the amount of "wind-up" required to
move the driven cylinder against the load. For example, for
the bare cylinder (Fig. 8) a " w in d- up " of 2.5 /xm occurs in
traversing through 1.2 half widths 0.3 mm). For the layered
cylinder (Fig. 3) a "wind-up" of 3.5 tun occurs in traversing
the same distance.
For the above examples the surface of the drive cylinder
would move 2.5 t tm farther than the driven cylinder over 0.3
mm traverse. If one of the cylinders contained a soft layer the
differential tra verse w ould be increased by ab out 1 /xm. In a
precision control system, all errors must be minimized to
reduce the level of error compensation required of the control
system. Based on error minimization alone then, i t would
seem that bare cylinders would be superior to coated cylin
ders. However, when surface roughness factors are included
40 6/V ol . 108, JULY 1986 Transact ions of the ASME
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-1.0 -0 8 -0, 6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8 10
Distance From Contact Center, x/b
Fig.
7 Conta ct zone shear stress distribution (100p in. layer)
b = .29 mm (.0115 in.) Load = 544 N/mm (3110 lb/in.)
F
t
=105 N/mm (600 lb/in.) R=9.4 mm (.37 in.)
D
=
2
G Pa
290
ksi)
p
h
=
1.28 GPa
187
ksi)
-1.0 -0.B -0.6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8 I.0
Distance From Contact Center, x/ b
Fig.
9 Shear stress distribution with 1 p in. CLA surface roughness (no
layer)
b =.25mm (.01 in.)
F, = 105 N/mm (600 lb/in.)
E = 200GPa(29Mp s i )
= 1.9 pm (75 pin.)
Load:
R-
Pma x
=
:639 N/mm (3649 lb/in.)
9.4 mm (.37 in.)
2 GPa (295 ksi)
-I.O -0.8 -0.6 -0. 4 -0.2 0 0.2 0.4 0.6
Distonce From Contact Center, x/b
Fig .8 Conta ct zone shear stress distribution (no layer)
b
= .25 mm (.01 in.) Load = 607 N/mm (3470 lb/in.)
F, = 105 N/mm (600 lb/in.) R=9.4 mm (.37 in.)
=
200
G Pa
29M
psi)
p
h
= 1.5 GPa
219
ksi)
in the shear stress examinations the value of a surface layer
becomes clear. Figure 9 indicates the shape of the shear stress
distribution for a stationary lower cylinder without a layer but
with a 0.025
pm
center line average (cla) surface roughness.
The increased deformations due to the surface roughness
produces peaks in the shear stress distribution in the slip
regions
(x/b >
0.7). The addition of a soft, thin layer (2.5
nm )cushions the surface asperities and produces a smoother
shear stress curve as indicated in Fig. 10. Based on the results
of Figs. 9 and 10, it would be difficult to compensate for the
erratic stresses for a nonlayered cylinder. However, it is quite
reasonable to attempt to predict the stresses where a layer is
present.
- 10 - OB - 0 6 - 0 .4 - 0 .2 0 0 .2 0 .4 0 6
OB
10
Distance From Contact Center, x/ b
Fig . 10 Shear stress distribution with 1 p in. CLA surface roughness
(100 pi n. layer)
b = .
29 mm (.0115 in.)
F
(
= 100 N/mm 601lb/in.)
=
2
G Pa
29
ksi)
e= 2.5pm 100pm)
Conclusions
Load = 545 N/mm (3114 lb/in.)
R = 9.4mm(.37 in.)
Pmax =
1
-
3 3 G P a
Traction drive systems represent reasonable devices for
traversing slideways in precision machining. However, one
inherent problem with traction is that the traction interface
must incur sizable elastic "wind-up" before the driven
cylinder will move against a given load. For one specific case
analyzed the "wind-up" was on the order of 2 ^m for a 105
N /mm traction load. The amount of "w ind-u p" increased as
the driven cylinder moved.
In order to achieve precision control in a traction drive,
some type of compensation algorithm must be employed to
eliminate "wind-up" errors. It would be expected that a
compensation algorithm of the type presented herein could be
employed provided good reproducibility of the traction in-
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te r face could b e achieved. O n ep r o b l e m in r e p r oduc i b i l it y is
surface roughness . Even smal l levels o f roughness (0 .025 t im
cla)
c a n
cause wi ld var ia t ion s
in
in ter fac ia l shear s t resses .
A
th in coat ing such as a 2 .5fim mo lyb den um disul f ide coat ing
c a n a bs o r b t h e r oughne s s a n d crea te a muc h s moo t he r ( a n d
hence more reproducible) shear s t ress d is t r ibut ion . T h e
pr e s e nc e o f t h e soft layer wou ld cau se a slight (0. 6 i tm)
increase
i n
" w i n d - u p "
b u t
p r e s uma b l y
t h e
" w i n d - u p " w o u l d
be predic table .
.Body 3
,v
. 4 $ 2 b *
T ( X )
) h Body 1
t
w
7
y
r
E..
'
Body 2
References
1 Bryan, J . B., "De sign and Construction of an Ultraprecision 84 Inch Dia
mond Turn ing M achine,"
Precision Engineering,
Vol. 1, N o. 1, 1979, pp.
13-17.
2 Barkman, W. E. , "M achine and Tool Drive Sys tem,"
Precision Engineer
ing,
Vol. 2, N o. 3, 1980, pp. 141-146.
3 Kalker, J . J . , "Trans ien t Ro l l ing Contact Phenomen a,"
Trans. ASLE,
Vol. 14, 1971, pp. 177-184.
4 K alker, J . J ., "R oll ing With Slip and Spin in the Presence of Dry Fric
t i o n , "
Wear,
9, 1966, pp. 20-38.
5 Cattan eo, C , "Sul Contatto di du Corpi Elast ici: Destribuzione Locale
Degli Sfoizi ," fiend. Acad.
Lincei,
Series 6, Vol. 27, 1938, pp. 342-348,
434-436, 474-478.
6 C ar ter , F . W., "O n the A ct ion of a Locomot ive Driving Whe el ,"
Proc.
Royal Soc, a 112, 1926, pp . 151-157.
7 B entall , R . H., and John son, K. L., "Slip in the Roll ing Contact of Two
Dissimilar Elast ic R ollers ,"
J. Mech. Eng. Sci.,
Vol. 9, 1967, pp. 389-404.
8 Barber, J . R., " The R oll ing Contact of M isaligned Elast ic Cylin ders, " / .
Mech.
Eng. Sci.,
(I . M ech. E.), Vol. 22, N o. 3, 1980, pp.. 125-128.
9 Poritsky, H., "Stress and Deflections of Cylindrical Bodies in Contact
Wi th Appl icat ion to Contact Gears and L ocomot ive Wheels ," A SM E
Journal
of Applied Mechanics, 1950, pp . 191-201.
10 Krause, H., and Senuma, T., " Investigation of the Influence of Dynamic
Forces on the Tribological Behavior of Bodies in R oll ing/Sliding Contac t W ith
Par t icu lar Regard to Surface Corrugat ions ," A SM E
JOURNAL
OF
LUBRICATION
TECHNOLOGY Vol. 103, 1981.
11 Sneddon, I . N . , Fourier Transforms, M cGraw-Hill , 1951.
12 Gupta, P . K., and Walowit , J . A ., "C ontac t Stresses Between a Cylinder
and a L ayered Elast ic Sol id ," A SM E
JOURNAL
OF
LUBRI CATION TECHNOLOGY,
A pr. 1974, pp. 250-257.
13 Kannel, J . W., and D ow, T. A ., "E valuatio n of Contact Stresses Between
a R ough Elastic and a L ayered Cylinder," to be presented at the Leeds-L yon
Conference, Sept. 1985.
A P P E N D I X
Development of Shear-Deflection Equations
Fourier Transform Equation. Th e objective of this
analysis is to develop a relationship between surface shear
stresses and tangential deflections in absence of applied
normal stresses at the boundary. The analyses are based on
elasticity theory using the Fourier transform approach given
by Sneddon and Gupta and Walowit . These equat ions are
given in the following form (see Fig. A l) .
d
2
^ I f
0 0
= \ c o
2
Gexp(- f'cox)da>
2ir J -oo
x
2
d
2
V
dy
2
d
2
V
J
d
2
G
dy
2
exp(- /cox)a ' co
dxdy 2ir J -
:
d G
t
w
(co
ex p( - /cox) aco
dy
( A l )
l-v
2
(" fd
3
G (2-v)\
2
dG-\ .
v = \ n (
)
u exp ( - /cox)
2TTE J-ooLcfy
3
\ l-v / dy 1
doi
u =
2TTE
p [d
2
G / v \ , -1. , . du
whe r e
V i s the
A i ry s t ress funct ion which sa t i s f ies
t h e
b i ha r mon i c e qua t i on a n d G i s the Fou r i e r t r a n s f o r m of Sf,
t ha t is
V
4
^
= 0
(A2)
Fig.
A1 Coo rdinate system for shear stress analysis
oo
$ exp(, wx)dx (A3)
Eliminating ^ from the above two_ equations and solving the
resulting differential equation in G, we get a solution of the
form
G= A+By)exp - \u\y) + (C+D y)exp(\o)\y)
(A4)
whe r e
A , B , C ,
a n d
D
a r e
c ons t a n t s
o f
i n t e g r a t i on
to be
e va l ua t e d a tt he bo unda r y .
Boundary Condit ions
The bounda r y c ond i t i ons fo r the t rac t ion analys isa r e :
1) bou nda ry s t ress i s t h eapp lied shear stress
2) stress a n d def lec t ions a r e c on t i nuous a c r o s s t h e layer
in ter face
3) stress goes
t o
zero a t oo;
at t h esur face ,
d
2
*
dxdy
dG
p
= -I 1 /'co exp(iu>x)du
2 i r J - = dy
(A6)
,dG
If is an odd function, Sneddon shows that:
dy
1 f
IT
JO
d G
J
co cos wxaco
dy
Based on Fourier Transform theory ;
dG
dy
o
co s
OJ Xdx
(A7)
(A8)
a t t he s ur fa c e v = 0 r = - r
0
. Let t ing T
0
be defined over the
interval A x and lett ing T
0
= 1/Ax then lim we have
dG
c o =1 (A9)
dy
L et t ing
s = hu, G
=
G/h
2
, r\ - y/h,
the first boun dary
condition becomes
dG
, f n
57- =
1
fo r n= 0
d{
(A10)
( A l l )
and
A lso for the case of no norma l stress on the surface
G = 0
Equat ions (A 4) , (A 10), and (A l l ) com bine to y ield
^ , + C , = 0
- ^ 4
1
5
2
+ 5
1
s - t - C
1
i '
2
+ i ?
1
i ' =l (A12)
where the co nstant in this equation are defined in body 1 of
Fig. 1.
4 0 8
/V ol . 108, JULY 1986
Transactions of the ASME
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7/25/2019 Analysis of Traction Forces in a... Kannel 1986
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The remaining boundary conditions can be met using the
same approach as used for the normal stress conditions [12].
These conditions yield the matrix given in Table 1.
Green's Function for Shear
The tangential deflection on the surface from equ ation ( A l)
can be expressed as:
( l - i >
2
) I"
30
d
2
G cos if
7rE
I;
d
v
2
'-ds
where
d
2
G
~ ~ h f
-2(5, -D
{
s
(A 13)
(A 14)
The Green's function can be written with reference to an
arbitrary displacement
V
x
at f = 1
1 _ 2
u
u,
=
r f H L
2 ( 5
'
-D,)s
co ssi;- co ss
efe-2j31nf]
(A 15)
It can be shown, using the matrix in Table 1 that for larger
values of s, (B, - > , ) - \/s. Equ ation (A 15) can be ex
pressed as two integrals (0 < s < s
0
) an d (s
0
< s), as given in
the text.
If we let
then this equation correspo nds to eq uation (5) in the text.
Because the shear deformation is calculated as a relative
displacement, a reference point must be selected. The
displacements c alculated from equation (A 15) becomes sm all
for large f; therefore a reasonable assumption for the
reference point is 5 contact widths (x = -1 0 6 ) . Th en t h e
relative ta nge ntial deflection (e,-) is:
e