The concept of stress concentration in screw threads was defined using the sheer transfer rate as the fundamental quantity. The stress concentration is plotted for a fixed geometry. The Heywood equation was used to generate the basic plots and NASTRAN was used to extend the analysis to the case both where flanks of an individual thread tooth are in contact. The case where a finite axial stress is superimposed is discussed

Ohara, G. P.

English

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STRESS CONCENTRATIONS IN SCREW THREADS
G. Peter O'Hara
US Army Armament Research and Development Command
Benet Weapons Laboratory, LCWSL
Watervliet Arsenal, Watervliet, NY 12189
SUMMARY
The concept of stress concentration in screw threads is defined as a ratio
of maximum fillet stress normalized to shear transfer rate. The data is
presented as a plot of fillet stress vs. radial stress for a particular thread
form. The Heywood equation is used to generate the basic plots and NASTRAN is
used to extend the analysis to the case both where flanks of an individual thread
tooth are in contact and the case where a finite axial stress is superimposed.
INTRODUCTION
The concept that stress or flow lines concentrate around various structural
discontinuities is very old and has been the subject of many thousands of books
and technical papers. It is convenient to express this concept in terms of a
stress concentration factor (K) using the simple equation:
_max = K qnon
Where K is a ratio between the maximum stress and some nominal stress, the single
book "Stress Concentration Factors" by R. E. Peterson (ref. I) is a compilation
of the work included in some 378 references. The bulk of this work is contained
on graphs which are plots of K vs. some geometry factor and most use a family of
curves to show the effect of some other geometry factor. These plots provide
both useful numeric information and a quick visual picture of the structural
response.
The concept of stress concentration in screw threads is rather elusive and
in fact there is little work done on stresses in threads. R. B. Heywood (ref. 2)
published an empirical equation for the maximum fillet stress which was used in
the work of Weigle, Lasselle and Purtell (ref. 3) as a guide in trying to improve
fatigue life of cannon breech rings. Later this author demonstrated that the
Heywood equation would give accurate numeric data when the boundary conditions
were closely controlled (ref. 4).
However, most work with screw threads seems to be done for specific cases
such as the fine work of M. Heyinyi (ref. 5) who investigated bolt shank and
nut design in Witworth threaded bolts. This type of analysis using three
dimensional photoelasticity was also used by W. F. Franz (ref. 6) and J. D.
Chalupnik (ref. 7). A further attempt at optimizing a thread form was done by
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https://ntrs.nasa.gov/search.jsp?R=19800016160 2020-03-21T19:02:48+00:00Z
R. L. Marino and W. F. Riley (ref. 8).
In all of these works the calculated stress concentration factor is differ-
ent for each thread in the system. It would seem that if the stress concentra-
tion factor is properly defined something should be a constant for all threads
of a specific shape. In his original paper, Heywood demonstrated part of the
problem. The stress in the fillet is the result of two factors. First, is the
stress due to the load on the individual thread tooth and second, is the stress
due to the general stress field or the axial stress (_a) near the thread fillet.
In this paper I will add effect due to friction and normalize all stresses to
the average shear transfer rate (TR).
When the friction force and the force due to the "wedge" effect of the
loaded flank of the thread are combined a radial (normal) force is produced
which can be averaged into the radial stress (Gr)" The fillet stress (GF) can
be expressed as the sum of two functions.
In the above equation the first function (GI) is the relation between
fillet stress and the load on the individual thread tooth. The second function
(G2) is the factor due to the general stress field. Alpha (@), beta (8) and
are the thread geometry factors. The angle (8) is in both functions because
they do not maximize at the same position in the fillet. In this paper the
shear transfer rate is defined as the net load supported by the thread divided
by the area at the pitch line. The direction of the net load is parallel to
the pitch line and in the analysis this component of the force will be unity.
The radial stress (_r) and axial stress (_a) are normalized to the shear transfer
rate.
The above discussion relates to a normal screw thread problem where only
one flank of a particular thread contacts one flank of a mating thread. In
some structures the relative displacement in the radial direction across the
threaded connection is such that the radial gap in the threads is closed and
both flanks of each thread carry loado Under these conditions the radial
component of the loads add together to produce high negative or compressive
radial stress across the joint. The axial loads oppose each other and the
pressure on the primary flank must become very high to overcome the secondary
flank load. This is not a common condition; however, it may become very impor-
tant in the cannon breech-to-tube connection.
THREAD GEOMETRY
The thread geometry parameters are Shown in Figure 1 and in this report
all linear dimensions will be normalized to pitch (P). The primary geometry
parameters are the primary flank angle (G), the secondary flank (B) and the root
radius R. These, in conjunction with the pitch space (PI), define the basic
thread geometry. Other factors are required to insure a practical thread which
will fit together. The addendum (AD) and dedundum (DD) dimensions sum to the
66
total height (HT). The tip radius (RT) eliminates a sharp corner and helps to
support the bearing surface. The root flat (FLAT) is often used to make up for
the root radius tolerance. The bearing height (Z) is used to calculate the
average bearing stress and the shear length (S) is used to calculate the maximum
shear-out failure load.
This complicated system is simplified by the fact that we must deal with
a small set of standard thread forms. In this report detailed analysis has been
done on the British Standard Buttress thread and Heywood analysis has also been
done on the controlled root bolt thread or "J" thread and the Watervliet Arsenal
Buttress used on cannon breeches. The nominal dimensions for these threads are
shown in Table I.
TABLE I. THREAD GEOMETRY DEFINITION
British Watervliet 30 "V"
Buttress Buttress .... (Rolled)
7° 20° 30°
45° 45° 30°
0.1205 0.1333 0.1804
P1 0.500 0.5276 0.500
HT 0.5059 0.4787 0.6077
RT 0.00 000480 0.1083
FLAT 0.0 0.0 0.0
LOADING PARAMETERS
The Heywood load parameters are also shown in Figure 1. They are a point
force (W) applied at some position (b) in the loaded flank with a friction
angle (y). This scheme can be repeated many times on the loaded flank to
produce some load distribution curve. The following loading assumptions are
made:
1. The total load vector parallel to the datum line is unity.
2. The load distribution is uniform.
3. Friction does not vary along the flank.
The first assumption given allows the normalization of stresses to shear
transfer rate and the other two establish a simple loading case.
Under conditions of high radial compressive load, it is possible for
threads to be pushed together until both flanks contact. This condition will
be discussed later. Under normal conditions only the primary flanks contact on
the thread and the radial stress become a function of the flank angle _ and the
friction angle y:
_r = tan (s-y)
67
Note that friction becomes a signed variable depending on the relative displace-
ment of the two components of the structure.
In the above discussion the general field or axial stress is assumed to be
zero. In the NASTRAN finite element analysis the axial stress was simulated by
the use of a constraint subcase in which the relative axial displacement
between the two radial boundaries was fixed by the use of scalar points and
multipoint constraint equations. The radial displacements on these planes were
made equal for congruent points. The radial displacement of the inner axial
boundary was set equal to the Poisson contraction of a solid bar.
HEYWOOD ANALYSIS
The Heywood equation is shown in Figure 2. This is a semi-emperical equa-
tion that was fit to a large body of photoelastic data. It calculates maximum
fillet stress for a point load on the primary flank of a thread using a specific
friction angle. In order to simulate a uniform load distribution, the results
are averaged for seven different "b" values evenly distributed over the flank.
The process has been programmed into a program called HEY40. The calculations
have been done for many standard thread forms, and the three reported in Figure
3, have been defined in Table I. This plot of fillet stresses plotted against
radial stress will be referred to as the "thread characteristic curve". This
curve covers a friction angle range of -45 ° to 45° or a coefficient of friction
range of -1.0 to 1.0.
In Heywood's photoelastic experiments he was careful to transfer the load
supported by the threads profiles in a shear mode to make the axial stress as
small as possible. This process limited his equation to the case where axial
stress is equal to zero.
NASTRAN FINITE ELEMENT ANALYSIS
The finite element work was done for three reasons: (1) to verify the
Heywood analysis; (2) to examine the two-flank problem; and (3) to include a
finite axial stress. The grid for the British Standard Buttress is shown in
Figure 4. It contains 216 triangular ring elements (CTRIARG) and 133 grid
points. The run required five basic loading subcases plus fourteen subcase
combinations for each value of axial stress. These fourteen subcases cover both
1-flank and 2-flank contact over a range coefficient of friction of -1.0 to 1.0
in increments of 0.25.
The grid was generated using IGPES(ref. 9) and following that, force sets
were calculated to apply uniform pressure and uniform shear loads on both flanks
of the thread and a displacement was calculated for a nominal 1.0 psi axial load
on the grid. Two different constraint conditions were required to complete the
boundary conditions for a single thread taken from a long series of threads.
For loads on the thread the inner boundary points were fixed in both radial and
68
axial directions and similar points on the two radial boundaries were con-
strained to equal displacements. In this way the net load was taken out as
shear load on the inner axial boundary and the multipoint constraint equations
replaced adjoining material. For the axial load condition the inner axial
boundary was constrained to the Poisson displacement in the radial direction
and left free in the axial direction. The two radial boundaries were given
fixed relative axial displacements and the radial displacement was made equal
for similar grid points. This condition was set to simulate a far removed
axial loading.
Because the basic loads were all for a 1.0 psi uniform applied pressure or
shear and the results were desired for a 1.0 psi shear transfer rate (calculated
at the datum line), it became necessary to calculate the correct Subcase Sequence
Coefficients for fourteen subcases for each of four axial stress values (or 56
sets). Therefore a small program was generated to supply all necessary SUBCOM,
SUBSEQ and LABEL cards for that portion of the case control deck.
Uniform increments of 0.25 in coefficient were used from -1.0 to 1.0. If
the friction value was positive, a single subcase combination was generated
which would superimpose the two primary flank loads with the proper value of
axial stress. If the friction was zero or less, a similar subcase combination
was generated along with one where both flanks were loaded and the second radial
stress was 1.0 greater than the initial.
The axial load subcase produced the conventional fillet stress concen-
tration factors of K = 2.89. This maximum stress was in an element at the
bottom of the fillet where e is approaching 0". In the cases where the load
is applied to the thread the exact position of the stress maximum is about 45*
up from the bottom of the fillet. Data is reported here for two cases of axial
stress. Zero axial stress is shown in Figure 5 and axial stress of 2.0 is shown
in Figure 6. These plots are a set of six lines with the single contact curve
at the right. Starting at that curve is a family of five lines going to the
left which represent two flank contacts at different values of friction from 0
to -1.0.
DISCUSSION
The first thing to note is that there is excellent agreement between Heywood
and NASTRAN over most of the range of the plots for the one case in question.
Because of this, the Heywood relation can be used to evaluate different thread
forms. The finite element method has allowed the expansion of the basic plot
to the 2-flank contact problem and the addition of the axial stress.
There are several important points that are demonstrated by this work.
Note that a small change in friction can produce a large fillet stress variation
in all three threads reported in the Heywood analysis. Negative friction angles
can produce marked reductions in thread fillet stress. This effect was noted
several years ago in an unpublished three dimensional photoelastic study where
the model was overloaded and the threads were forced to a high negative radial
69
stress. In this case the fillet stresses were very low and the experiment was
repeated. This author suspects that friction variation may be responsible for
much of the scatter in bolt-fatigue data,
This work was initiated becuase of the necessity of analyzing a structure
with a long threaded connection using many small threads. In this case the
modeling of each thread would require an excessively large data deck. There-
fore, the threads were handled as a conventional contact problem where friction
could take on any value and limits were applied to the radial stress. In the
solution the contact surface was placed on the datum line of the threads and
one or two teeth were replaced by one element space. Shear transfer rates could
then be estimated from the shear stress data near the contact surface along with
radial and axial stress. The fillet stresses were estimated for use in fracture
mechanics analysis.
CONCLUSION
A stress concentration approach to the thread fillet stress problem has
been defined using the shear transfer rate as the fundamental quantity. This
stress concentration is plotted for a fixed geometry in a stress vs. stress
plot where the stress concentration is a function of the applied radial stress.
This process can be repeated for several values of the applied axial load. The
effects of axial stress and applied thread loads seem to be of equal importance
and accurate results require the analysis of both factors.
REFERENCES
I. "Stress Concentration Factors," R. E. Peterson, John Wiley _ Sons, New York,
1974.
2. Heywood, R. B., "Tensile Fillet Stresses in Loaded Projections," Proceedings
of the Institute of Mechanical Engineers, Vol. 160, p. 124, 1960.
3. Weigle, Lasselle and Purtell, "Experimental Behavior of Thread-Type Projec-
tions," Experimental Mechanics, May 1963.
4. O'Hara, G. P., "Finite Element Analysis of Threaded Connections," Proceed-
ings of the Fourth Army Symposium on Solid Mechanics, September 1974.
5. Hetinyi, M., "The Distribution of Stress in Threaded Connections," Proceed-
ings of SESA, Vol. I, No. i, 1943.
6. Franz, W. F., "Three-Dimensional Photoelastic Stress Analysis of a Threaded
Pipe Joint," Proceedings of SESA, Vol. , No. , 1950.
7. Chalupnik, J. D., "Stress Concentration in Bolt Thread Roots," Experimental
Mechanics, 1967.
7O
8. Marino, R. L., and Riley, W. F., "Optimizing Thread-Root Contours Using
Photoelastic Methods," Experimental Mechanics, January 1964, p. I.
9. Lorensen, W. E., "Interactive Graphic Support for NASTRAN," Sixth NASTRAN
User's Colloquium, NASA Conference Publication 2018, October 1977.
ql
Figure 1. THREAD GEOMETRY AND LOAD PARAMETERS
o-#= .0+0.26(_/jL7 + v be (I.0+'TSlN_'W
R 6
W
e -_
EQUAL
Z
a ANGLES
Q_
:E
30°
Figure2. HEYWOOD'SEQUATION
'73
m2O
THREADCHARACTERISTICFROM ] _
HEYWOODANALYSISFOR"
G=BRITISHSTANDARDBUTTRESS15
_= WATERVLIETBU TRESS
Q= CONTROLLEDROOT"V"
=0.0 _o
®
®
I I i !
-4 -3 -2 -1 0 1
Figure3.
Figure4. NASTRANGRID
75
THREADCHARACTERISTIC 201FROMNAST NFORTHE
BRITISHSTANDARDBUTTRESS
AXIALSTRESS=0.0 15
0.0 10
R
• I ! I 1
-4 -3 -2 -1 0 1
Figure5.
THREADCHARACTERISTIC 2o- _:_FROMNASTRANFORTHE
BRITISHSTANDARDBUTTRESS
15-
10
- 75 -.25
-.50
,..,3
q
i
':_'= 2.0
!
J I I I I
-4 -3 "-2 -1 0 1
Figure6.


Stress concentrations in screw threads

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