APPROPRIATE CORRELATIONS FOR ASSESSING EXPANDED

TUBE-TO-TUBESHEET JOINT STRENGTH

Stanley Yokell
MGT Inc.
Boulder, Colorado 80303-3607
Telephone 303-494-9608
Fax 303 499-1949
e-mail syokell@mgt-Inc.com

1. ABSTRACT

This paper advocates correlating rolling torque or hydraulic expanding pressure with shear load strength
instead of percent wall reduction as the criterion of expanding adequacy for thin-walled tubes. To establish
a basis for such advocacy, it discusses tube expanding theory, methods and annular grooves. The paper
compares the magnitudes of measurements required to estimate percent wall reduction with cumulative
tolerances for hole drilling and tube manufacture. A graphic illustrates how data scatter increases with ratio
of tube diameter to wall thickness. The paper comments on tube rolling procedure specifications and
concludes with recommendations for two-stage and hybrid expanding.

2. INTRODUCTION

Whether to correlate percent wall reduction or rolling torque with joint pullout of thin-walled, high-
strength, low-elastic modulus tubes expanded into tubesheets has proponents on both sides of the issue.
The author believes that for such tubes correlating torque with pullout (shear-load) strength is the superior
method. The following discussions of tube expanding theory and tube rolling practice underpins this
opinion.

3. TUBE EXPANDING THEORY

The vast majority of tubes expanded into tubular heat transfer equipment are roller expanded. However,
the complexity of analyzing the large numbers of variables that apply to tube rolling is insuperable.
Therefore, the basis of most published investigations of expanding tubes into tubesheets is a model of
applying pressure uniformly in the end of the tube.

3.1. Expanded Tube-to-Tubesheet Joint Model

A simplistic model of the tube and surrounding metal is as follows. The tube is a cylinder composed of an
infinite number of infinitesimally thin shells in contact with each other. Initially, there is clearance
between the outside of the cylinder and the tube hole. With application of pressure in the tube, the
assembly of shells deflects. The inside tube radius increases until the outside of the tube contacts the hole.
At this point, the deflection of the shells may be full elastic, fully plastic or the deformation of inner shells
may be plastic and outer ones elastic with an elastic plastic interface. The state of the deflection depends
upon the tube dimensions and its yield stress, σyt.

Theoretically, the tube will remain fully elastic up to a limiting applied pressure equal to σyt/√3 or 0.577σyt,
the so-called elastic limit. As the tube end undergoes elastic deformation its wall becomes thinner and its
length becomes shorter in response to the Poisson effect. As pressure exceeds this limit, plastic
deformation begins. Increasing the internal pressure moves the interface between elastically and plastically
deformed shells outward. After the tube contacts the tubesheet, a similar process takes place in the metal of
the tubesheet. The goal is to create fully plastic tube deformation, after which, upon relaxation of the tube
and surrounding tubesheet, residual stress remains at the location of the tube-tubesheet interface. The
analysis treats this residual stress as interfacial pressure between the tube and hole.

1
One analysis uses a plain-stress model based on applying pressure to a hole in a large plate to investigate
tube expanding. 1 The holes that surround a hole under investigation in actual tubesheets limit the utility of
this model. However, finite element analyses have shown the model of a hole under pressure in a large
plate offers a practical way to examine tube expanding. In this model, when the radius of the plastic zone
reaches 1.75 times the tube inside radius, it is not possible to attain further enlargement because the
innermost shells begin to extrude. The corresponding expansion pressure at this point is 2σyt/√3
(~1.155σyt.). Attempts to apply greater pressure spread the extrusion into successive shells.

As expanding pressure increases, the pressure that the tube exterior exerts on the hole-wall initiates
deflection in the surrounding tubesheet metal, which deflects in similar fashion to the tube. The boundary
between the elastic-plastic interface moves outward into the tubesheet. Upon pressure release, relaxation is
approximately elastic. In order to create a tube-to-tubesheet joint, stress at the tube-hole interface in the
residual stress continuum must be greater than zero after relaxation. We refer to the residual stress at the
interface as interfacial pressure. The various analyses for relating and optimizing interfacial pressure to
expanding pressure consider tube dimensions, tube and tubesheet yield stresses and other properties of the
tube and tubesheet.

3.1.1. Equal tube and tubesheet yield stresses

In the circumstance in which the tube and tubesheet yield stresses are equal, the plastic zone is essentially
continuous. In this circumstance, the optimum expanding pressure is the maximum that will not cause the
tube to begin to extrude. That pressure is equal to 1.155σyt. Analysis to determine the location of the
plastic-elastic interface for this circumstance has been adapted to determine the thickest tube capable of
being expanded into a tubesheet.

3.1.2. Tubesheet yield stress higher than tube yield stress

If tubesheet yield stress, σyp, is higher than that of the tube, a new elastic-plastic boundary is established in
the tubesheet metal after full tube plasticity has been achieved. As the ratio of tubesheet to tube yield
stress, σyt/ σyp increases, radius of the elastic-plastic boundary in the tubesheet moves closer to the tube. At
a ratio of two, the location of the elastic-plastic interface is at the tube-tubesheet contact surface and the
tubesheet metal remains fully elastic.

3.1.3. Tubesheet yield stress lower than tube yield stress

Steam surface condensers tubed with high-strength, thin-walled, low-elastic-modulus tubes expanded into
clad steel or non-ferrous tubesheets exemplify this situation. Here, at ratios of σyt/ σyp greater than one, the
relative thickness of the tube governs the maximum permissible expanding pressure. The simplest way to
understand the effect of pressure on the tube-hole structure is to imagine a tube of zero thickness in
intimate contact with the hole. In this imaginary circumstance, the pressure at the tube OD would be the
same as the pressure at the tube ID. Consequently, the limiting pressure in the tube would be that of the
tubesheet limiting pressure equal to 2σyp/√3. The plate yield stress would therefore govern the maximum
expanding pressure.

In a tube with some thickness, which is in the fully plastic condition, there is a calculable pressure drop
across the tube. The pressure that the tube exerts on the hole is the tube expanding pressure, less the
pressure drop across the fully plastic tube. In order to prevent tube hole extrusion, the external tube
pressure must not exceed the tube plate plastic limit of 2σyp/√3. Therefore, the maximum expanding
pressure to apply in the tube cannot exceed the sum of the plate plastic limit and the pressure drop across
the fully plastic tube. This provides a means of determining the maximum permissible pressures for tube-
wall, tube-diameter, and tube-plate constructions.

3.2. Effect of Tube Rolling on the Tube End

1
See S. Yokell, “Expanded, and Welded-and-Expanded Tube-to-Tubesheet Joints” for this analysis.

2
When the taper mandrel of the tube rolling tool forces the hardened pins into contact with tube surface, the
contact area is infinitesimally narrow and approximates a line. Theoretically, lines have no width. The
contact area surface is zero and the pressure that the pins apply is infinitely large. However, after contact,
as the process drives the mandrel forward, the pins first deform the tube elastically in the form of
depressions with finite contact area. The positions of the pins rotate counter to the direction of mandrel
rotation. At this point the tube end length becomes shorter and the wall becomes thinner because of the
Poisson effect. Very rapidly, the process begins to deform the tube plastically. The unit pressure that the
pins apply to the tube surface surpasses the tube plastic limit. Therefore, in the model described above, the
shells that make up the tube progressively extrude as the pressure on each shell exceeds the plastic limit.
The extrusion takes place at the expense of wall thickness, and the plastic thinning and thinning due to tube
metal extrusion is the total wall reduction.

In a physical tube-tubesheet structure, one cannot readily measure the actual percent wall reduction.
Consequently, the practice is to deduce it from measurements of the hole and tube dimensions before
expanding and measurement of the tube ID after expanding. The increase in the tube ID measurement
includes some unaccounted-for movement of the surrounding ligaments; hence, the practice is to call the
deduced wall reduction, “apparent wall reduction”.

3.3. Correlation of Tube End Deformation with Tube Pullout Strength and Tightness.

Early attempts to correlate tube-end extrusion with joint pullout strength were abandoned because of
measurement difficulties and widely scattered results. Greater success has been achieved in relatively thick
tubes by correlating joint pullout strength with measurements of percent wall reduction. The percent wall
reduction criterion has become the most common method used to establish tube rolling adequacy for
process and power heat exchangers. However, there is no theoretical basis for a general correlation of
pullout strength with percent wall reduction. Consequently, the universal practice is to perform shear load
tests (pullout tests) in models to establish the percent wall reduction to use in production equipment.

The literature on tube expanding is not replete with correlations of apparent percent-wall-reduction with
tube joint tightness. HydroPro, Inc. of San Jose, Calif. has done some shear-load strength and tightness
testing to examine expanded joint tightness of hybrid expanded tubes and validate the expanding
procedures. (Hybrid expanding is hydraulically expanding tubes at a pressure that creates a joint with
modest interfacial pressure, followed by tube rolling to increase the interfacial pressure required for the
desired joint strength.) For this testing, HydroPro correlated hydraulic expanding pressure and rolling
torque with joint strength and tightness. The pullout and tightness tests met the specified criteria for the
validation.

3.4. Typical Percent Wall Reductions

Tool manufacturers’ suggested increases in tube diameters after the tube contacts the wall of the hole, vary
with tube material, diameter and gage. A typical recommendation for a stainless steel tube 19-mm OD x
0.5-mm thick wall (18 mm ID) would be an increase of 0.1016 mm. Assuming a clearance of 0.254 mm,
and neglecting tube wall thinning to the point of tube-hole contact, the tube OD at point of contact would
be 19.254 mm. The ID at contact would be 18 + 0.254 = 18.254 mm. After expanding to achieve an
increase in ID of 0.1016 mm, the tube ID would be 18.254 + 0.1016 -mm = 18.3556 mm. The calculated
new wall thickness would then be (19.254 - 18.3556)/2 = 0.4492 mm. The percent wall reduction would be
100(0.5 - 0.4492)/0.5 = 5.08 percent.

3.5. Limits on the Applicability of Correlations of Percent Wall Reduction with Pullout
Strength

The utility of correlating percent wall reduction with pullout strength rests on several assumptions:

(1) The ODs and thicknesses of the tubes in each heat are uniform.
(2) The tube hole diameters are uniform and do not vary throughout the thickness of the tubesheet.

3
(3) Pullout strength varies as some definable function of tube-wall-thickness-reduction, and is
constant for the specified percent wall reduction in tubes from the same heat.
(4) The instruments used for measuring the unexpanded tube OD and ID, the hole ID, and the
expanded tube ID are adequate to the task of making the measurements precisely enough to
minimize data scatter.

The facts do not support these assumptions because of the following:

(1) It is customary to require application of the drilling tolerances in the TEMA Standards for special
close fit holes to feedwater heaters and surface condensers. For 19.1 mm OD tubes these are
19.25 mm ± 0.05 for 96% of the tubes, with 4% of the tubes permitted to be 0.25 mm oversized.
(2) Diameters of tube holes vary within the permissible tolerances through the thickness of the
tubesheet.
(3) Mill tolerances apply to tube manufacture; for example, Titanium tubes produced to ASME
specification SB-338 in sizes under 25.4 mm may vary ± 0.102 mm in OD from the specified
diameter and ± 10% from the specified wall thickness.
(4) Other than one paper that presented measurements of percent wall reduction VS joint strength for
boiler tubes, the author has found no theoretical or experimental data to validate that joint strength
varies in some uniform way with percent wall reduction. There was considerable scatter in the
results.
(5) Different individuals experienced in using three ball snap micrometers to measure tube and hole
IDs will read substantially different values, notwithstanding the typical instrument calibration to
0.00254 mm.

In practical terms the third assumption is moot because pullout loads in the models used for qualifying the
process are held constant after deciding the percent wall reduction to be used for a given assembly.

Although the percent-wall-reduction criterion is widely used, as Figure 1 shows, it becomes less reliable as
the tube wall thickness declines relative to diameter. Most process heat exchangers have wall thickness in
the range of 1.25 to 2.8 mm thick. Depending upon feedwater pressure and material selection, feedwater
heater tubes may vary from 0.712 to 2.8 mm thick. For very thin-walled tubes, the range of data scatter is
very wide between the sum of plus and minus tolerances as a fraction of ID to be measured to determine
apparent percent wall reduction. As the wall thickness increases, and D/t decreases, the range of data scatter
narrows. This shows graphically that correlating percent apparent wall reduction with pullout loads is a
reasonably good practice for most process exchangers where typical D/t values range from ~13 to ~7.
However, it is not a suitable way to determine the adequacy of tube expansion for thin-walled tubes, such
as those used in low-pressure feedwater heaters and in steam surface condensers.

It is noteworthy that Table 5 of ASME Code Section II Part B, Specification SB-338, from which the
dimensional data used in constructing Figure 1 is abstracted, is titled, “Permissible Variations in Outside
Dimensions Based on Individual Measurements” (Italics added). Figure 1 illustrates the factor by which
the sum of maximum tube diameter and wall thickness, and hole drilling tolerances exceeds the increase in
tube ID that must be measured for five ratios of tube OD/t. It shows that the cumulative plus or minus
tolerances can be a substantial multiple of the increase in diameter required to create the specified apparent
percent wall reduction. The ratio (sum of tolerances)/(tube ID increase to be measured), decreases with
increasing percent average wall reduction. The combinations of plus and minus tolerances for individual
tubes and the difficulty of making accurate measurements of the tube and tube hole are what cause often-
observed wide data scatter.

3.6. Precautions for Correlating Pullout Load with Rolling Torque

There are some caveats for using correlations of rolling torque with pullout load as the determinant of
joining adequacy.

(1) The torque-sensing device in the rolling equipment cannot sense if resistance to rotation stems
from the effort to squeeze the tube wall, or if foreign matter in the tool or tool-to-hole region is

4
 Figure 1 Sum of + & - Tolerances/ID Increase to be Measured VS % Apparent Wall Reduction

2.2

1.7

1.2

D/t = 9.17 Sum of + Tolerances

0.7
D/t = 9.17 Sum of - Tolerances
D/t = 11.86 Sum of + Tolerances
0.2 D/t = 11.86 Sum of - Tolerances
D/t = 28.33 Sum of + Tolerances
-2 0 2 4 6 8 10 12 14 D/t = 28.33 Sum of - Tolerances
-0.3
D/t = 44 Sum of + Tolerances
D/t = 44 Sum of - Tolerances
-0.8 D/t = 62.5 Sum of + Tolerances
D/t = 62.5 Sum of - Tolerances

-1.3

-1.8

-2.3
% Apparent Wall Reduction

impeding rotation. Consequently, to achieve reliable correlations the surfaces and tool must be
extremely clean.

(2) Torque required to dry-roll tubes into tubesheets is considerably higher than that required for
rolling with lubrication. Therefore, qualification tests made with lubricated rolling do not apply to
dry rolling.

(3) Table 1 shows SB-338’s minimum and maximum allowable yield stresses for three grades of
annealed Titanium tubes. It includes the ratios of the maximum to the minimum allowable
stresses to highlight the magnitude of the range. Therefore, to use torque-pullout-strength
correlations, there must be a reasonably close match of the yield stresses reported for the tubes of
the surface condenser with the yield stresses of the tubes used to prepare the test models.
Depending upon the number of tubes to be installed several heats of tubing may be provided, and
a box may contain tubes from more than one heat. For this reason, it is prudent to require
manufacturers to provide heat maps of the installed tubes keyed to the tube heat numbers,

bearing in mind that one box of tubes may have several heats of tubing. Such heat maps are also valuable
for examining the potential causes of tube failures.

3.7. Comments on Rolling Procedure for Steam Surface Condensers Tubed with Thin
Titanium Tubes

Subject to the caveats stated above, the author’s opinion is that correlating pullout strength with rolling
torque of high-strength, thin-walled, low elastic modulus tubes is a superior method to correlating pullout
strength with percent wall reduction.

5
 3.8. Comments on Tube Rolling Specifications

Specifications should include the following

(1) The specification should state, how the manufacturer proposes to clean the holes in the tube plate
before tubing. Common methods are to steam clean and blow-dry with filtered, dry air, or
alternatively, to wipe the holes with acetone-soaked felt plugs.
(2) The specification should require cleaning the tube ends before expanding and should require the
manufacturer to describe the proposed cleaning method.
(3) Wording like, “suitable protection”, is insufficiently specific. Specifications should require
workers not to open boxes of tubes until just before tubing the condenser. The specification
should require that tubes in boxes from which workers have loaded tubes into the condenser and
when loading is interrupted, be covered with plastic sheeting to prevent settling of foreign matter
on the tube surfaces.
(4) The specification should set a maximum time between removal of tubes from the box and their
installation. Workers should not open boxes of tubing until they are ready to install them in the
condenser. However, if the workload and schedule requires a period of more than one hour to
elapse after opening the box, the workers should cover the tubes with plastic sheeting.
(5) The specification should describe how the manufacturer proposes cleaning the tubes after
expanding in some detail.
(6) The specification should include a plan for the distribution throughout the tube field of bands or
clusters of tubes expanded into the tubesheet to stabilize the tubesheet before commencing
production rolling.
(7) The specification should have a description of how the manufacturer will calibrate the torque
controller, and the frequency of calibration.
(8) The specification should provide standards for maintaining the condition of the tube rolling
equipment.
(9) The specification should include a system for identifying tube positions.
(10) The specification should include a system for locating tube heats.

3.8.1. Rolling tools and equipment

The author is of the opinion that parallel 5-pin, hydraulically operated tube-rolling equipment is superior to
inclined-roll (self-feeding) 3-pin tube rolling equipment. Such equipment is available from Asian, North
American, and European manufacturers. The reason for this opinion is that, although the frequency with
which the rolls traverse the tube surface
is 67% greater, the amplitude of the
strains is much lower. Therefore, there is
less likelihood of fatiguing the tube metal
(See Figure 1). Because, it may be
impractical to manufacture 5-pin rollers
for use in tubing of inside diameter
smaller than 18 mm, for tube IDs 18 mm
and smaller, the tube rolling tool should
be 3-pin.

For most Titanium-tube surface

condensers, the tubes are harder than the
tubesheets. In this circumstance, self-
feeding (inclined) rolling tools create an
hour glass configuration of the metal
deformation. With such a configuration,
the interfacial pressure varies throughout Figure 2 Tube outside strain amplitude VS
the thickness of the tube end, effectively Frequency (Adapted from U. S. Patent No. 4,142,581,
producing a weaker, less tight joint than a Issued March 6, 1979 to Yuji Yoshitomi and others)
parallel roll tool would produce. Note

6
that parallel tools are not exactly parallel but are slightly inclined to assist the hydraulic ram in inserting
and retracting the tapered mandrel.

3.9. Suggested Alternative Method for Tube Expanding (Hybrid Expanding)

As the curve reproduced from U.S. Patent 4,142,581 shows, the greater number of pins is less likely to
cause tube ends to fatigue. If the number of pins were to be infinite, the tendency to fatigue would be
minimal. This (impossible) situation mimics what would happen if hydraulic pressure were to be applied in
the tube end. Applying hydraulic pressure directly in the tube end expands the tube without causing tube-
end extrusion. However, it has the disadvantage that it does not strain harden the inner shells of the tube
end sufficiently to restrain tube spring-back.

To the author’s knowledge, a major manufacturer of naval shipboard surface condensers has recently used
hybrid expanding to qualify models of tube-to-tubesheet joints. It consists of hydraulically expanding to
initial tightness, followed by tube rolling. Hybrid expanding has these advantages:

(1) The initial expanding increases the tube ligament efficiency thereby enhancing tubesheet stiffness.
Eq. 1 defines ligament efficiency.

p - d
h = 100 (1)
d
In this equation,

η = Ligament efficiency percent

p = Tube pitch, mm
d = hole ID, mm

The increase in ligament efficiency for 25-mm x 0.5-mm wall tubes is modest. For smaller diameter and
thicker-walled tubes, it is substantial. At either end of the wall-thickness range, such two-stage expanding
improves tubesheet stifness.

(2) Control of hydraulic expanding pressure currently available as standard equipment from the
manufacturer whose equipment the author is familiar is ± ~70 Bar. For a premium price, the
manufacturer can furnish a machine with a precision of approximately 35 Bar. In correlating tube
yields with hybrid expanding, the same requirements regarding tube yields apply as apply for
correlations of torque with pullout strength.
(3) The transition from expanded to unexpanded tube is more gradual than with tube rolling.
Therefore, the residual stress in the transition zone is lower than with tube rolling.

3.10. Annular Tube-hole Grooves for Expanding Titanium Tubes

3.10.1. Grooves for rolling

TEMA standard grooves, 3.18 mm wide x 0.4 mm deep are not suitable for expanding thin-walled titanium
tubes and other materials with similar physical properties. This is because the tubes may crack at the
corners of the grooves as the tube metal deforms to form keys. Multiple shallow, Vee-shaped grooves
adequately enhance expanded joint strength and tightness for such materials.2 One firm that specializes in
retubing surface condensers, has found that refinishing the tube holes with serrations provides excellent
strength and leak tightness.3

2
Such grooves are pictured in Page 189 of S. Yokell, “A Working Guide to Shell-and-Tube Heat
Exchangers”, McGraw-Hill, New York, 1990.
3
Personal communication from Bob Hahn of Atlantic Group, Norfolk, Virginia.

7
 3.10.2. Grooves for hydraulic expanding

The current edition of the TEMA Standards requires grooves for hydraulic expanding to be 6.35 mm wide.
Although this width is suitable for most materials, use Eq. 2 to optimize the groove width.

W = 1 . 5 6 Rt (2)

In this equation, W is the optimal width, R is the mean tube radius, and t is the tube wall thickness.

4. CONCLUSIONS

The following conclusions are drawn from the foregoing discussions of expanding theory and the difficulty
of making methods adequate to predict deduction of percent wall reduction.

(1) For thin-walled Titanium tubes, correlating shear loads (pullout loads) with torque provides a
superior way to achieve consistent results.

(2) Designers should be aware that rolled joints with thin-walled Titanium tube may require annular
groove configurations different from TEMA standard grooves. They should be aware that grooves
for hydraulic expanding should be at least 6.35 mm wide.

(3) Two-stage expanding is desirable because of the stiffening effect of the first stage on the
tubesheet.

(4) Hybrid expanding takes advantage of fixing the tubes firmly in place and stiffening the tubesheet
in the hydraulic expanding stage, and strain hardening the inner shells of the tube end sufficiently
to overcome spring back. This results in tube-to-tubesheet joints less prone to fatigue failures and
with adequate strength for the service.

5. ACKNOWLEDGEMENTS

The author wishes to acknowledge the information that HydroPro, Inc. of San Jose, California provided
about their hydroexpanding equipment.

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 Tables

Table 1
Permissible variations in SB338 tubes

Nominal Diameter Variation Nominal Variation

OD Maximum Minimum Wall Maximum Minimum
mm mm mm mm mm mm
17.000 17.102 16.898 0.400 0.440 0.360
18.000 18.102 17.898 0.500 0.550 0.450
19.500 19.602 19.398 0.508 0.559 0.457
21.000 21.102 20.898 0.600 0.660 0.540
22.000 22.102 21.898 0.700 0.770 0.630
22.225 22.327 22.123 0.711 0.782 0.640
22.500 22.602 22.398
24.000 24.102 23.898 Range of Permissible Stress Variations
25.000 25.102 24.898 Grade Min Max Max/Min
25.400 25.527 25.273 Mpa Mpa
28.575 28.702 28.448 1 170 310 1.82
2 275 450 1.64
3 380 550 1.45

Table 2
Relative Thickness of Tubes
Tube OD Tube Relative
D mm Wall Thickness
t mm

17.000 0.400 42.50

17.000 0.500 34.00
17.000 0.600 28.33
18.000 0.400 45.00
18.000 0.400 45.00
18.000 0.500 36.00
18.000 0.600 30.00
19.000 0.400 47.50
19.000 0.500 38.00
19.000 0.600 31.67
20.000 0.400 50.00
20.000 0.500 40.00
20.000 0.600 33.33
20.000 0.700 28.57
21.000 0.400 52.50
21.000 0.500 42.00
21.000 0.600 35.00
21.000 0.700 30.00
22.000 0.400 55.00
22.000 0.500 44.00

9
 Table 2
Relative Thickness of Tubes
Tube OD Tube Relative
D mm Wall Thickness
t mm

22.000 0.600 36.67

22.000 0.700 31.43
23.000 0.400 57.50
23.000 0.500 46.00
23.000 0.600 38.33
23.000 0.700 32.86
24.000 0.400 60.00
24.000 0.500 48.00
25.000 0.400 62.50
25.000 0.500 50.00
25.000 0.600 41.67
25.000 0.700 35.71
25.400 0.500 50.80
25.400 0.600 42.33
25.400 0.700 36.29
28.575 0.500 57.15
28.575 0.600 47.63
28.575 0.700 40.82

10