IMPORTANCE OF INDUCTION
HARDENED CASE DEPTH IN
TORSIONAL APPLICATIONS
Induction hardened case
depth plays a very
important part in
determining the static and
fatigue properties of shafts.
Both effective and total case
must be considered to
optimize shaft performance.
Gregory A. Fett*
Dana Corp.
Maumee, Ohio
*Member of ASM International and member,
ASM Heat Treating Society
T
his article updates work originally published in February
1985 Metal Progress, which examines the relationship between induction hardened case depth
and torsional strength and fatigue life.
This relationship is especially important when designing shafts to transmit
torque, such as automotive and truck
axle shafts. The recent work also examines the effect of both different steel
grades and prior microstructures on
the relationship.
Induction hardened shafts lend
themselves very well to most torsional
applications because induction hardening increases the hardness near the
surface where it is most needed and it
leaves the surface in compression,
which improves fatigue life. When a
shaft is loaded torsionally, the shear
stress is highest at the surface and zero
at the center. In the absence of a stress
concentration factor, stress increases
linearly from the center to the surface.
Thus, only the surface needs to be
hardened to a depth to adequately exceed the applied stress. When the surface layer is hardened, martensitic
1,200
Case depth A
Case depth B
Torsional strength and stress, MPa
1,000
800
HRC
52
50
40
30
20
10
0
HB
514
481
371
286
226
187
150
ULT
1793
1690
1276
966
759
655
517
Tensile
Yield
1614
1524
1152
941
614
503
366
Torsional
Yield
966
917
690
503
366
303
221
From SAE J413 (units are MPa)
600
Case depth B
Case depth A
400
200
Applied stress
0
0
10
20
30
40
Percent of bar diameter
Fig. 1 — Case depth versus torsional strength and stress.
HEAT TREATING PROGRESS • OCTOBER 2009
50
60
transformation causes it to expand
leaving the surface in compression, as
opposed to through hardening where
the core also expends leaving the surface in tension.
The depth to which a shaft must be
hardened can be determined theoretically. Figure 1 shows two different induction case depths (A and B) with a
surface hardness of 52 HRC and a core
of 12 HRC. The applied stress is shown
as a straight line from zero at the center
to a maximum at 52 HRC at the surface with no stress concentration factor.
Also shown are the corresponding torsional yield strength values which
were derived by converting hardness
to tensile and yield and then using a
factor of 0.6 to convert yield to torsional yield.
Clearly, case depth A will fail first at
the case-core interface. The applied
stress curve exceeds the strength curve
at the case-core interface. However,
case B is able to take full advantage of
the 52 HRC surface hardness. The applied stress curve just touches the
strength curve at the surface and at the
case core interface. Thus, it may fail at
the surface or at the case-core interface.
Hardening deeper than case B in this
situation will do no good because it
will fail from the surface even if the
strength curve is shifted farther to the
right. This is the optimum case depth.
In addition, by hardening too deep, the
residual surface compressive stress
may be reduced.
Case depth A has an effective depth
measured to 40 HRC equal to 15% of
the bar diameter and a total case depth
to 20 HRC equal to 25%. Case depth B
has an effective depth of 23% and a
total depth of 31%. Although case
depth B (the optimum case depth)
takes full advantage of the situation,
case depth A may be sufficient in many
applications where the stress is not excessively high. Case depth A could also
be improved if necessary by increasing
the core hardness possibly through a
quench and temper operation.
To examine the correlation of actual
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shaft torsional performance to case
depth, test shafts splined at both ends
were induction hardened to varying
case depths. The shafts had diameters
of 28.58 and 38.86 mm (1.125 and 1.530
in.) in the center and a slightly larger
spline on both ends, which caused the
16
failure to occur in the middle. Different
hardenability steels were used to look
at the relationship of effective and total
case depth. The effect of core hardness
also was studied by using shafts made
of hot-rolled (HR), cold-drawn (CD),
and quench and tempered (Q&T) steels.
HEAT TREATING PROGRESS • OCTOBER 2009
1,800
Ultimate
1040 CD ultimate
1040 CD yield
1541 Q&T ultimate
1541 Q&T yield
1541 HR ultimate
1541 HR yield
1050M HR ultimate
1050M HR yield
4140 HR ultimate
4140 HR yield
1038 HR ultimate
1038 HR yield
1040 HR ultimate
1040 HR yield
1,600
Torsional strength, MPa
1,400
1,200
Yield
1,000
800
600
4140 steel
400
All others
200
0
0
5
10
15
20
Effective case depth, % of bar diameter
25
30
Fig. 2 — Effective case versus torsional strength.
1,800
Ultimate
1,600
1040 CD ultimate
1040 CD yield
1541 Q&T ultimate
1541 Q&T yield
1541 HR ultimate
1541 HR yield
1050M HR ultimate
1050M HR yield
4140 HR ultimate
4140 HR yield
1038 HR ultimate
1038 HR yield
1040 HR ultimate
1040 HR yield
Torsional strength, MPa
1,400
1,200
Yield
1,000
800
600
400
1541, 1541
Q&T, 4140
200
All others
0
0
10
20
30
40
Total case depth, % of bar diameter
50
60
Fig. 3 — Total case depth versus torsional strength.
1,400
1040 steel
1541 steel
1050M steel
4140 steel
1,200
Torsional strength and stress, MPa
Static Torsional Test Results
Table 1 lists results of static torsional
tests, which also includes data from
production axle shafts made of SAE
1038 and 1040 steel. It should be noted
that the yield strength was determined
by the JEL (Johnson elastic limit)
method, which is defined by a 50%
change in slope. Also, total case was
defined as 20 HRC, or the total visual
case if the core was 20 HRC or greater.
Effective case versus torsional strength
(Fig. 2). The torsional yield and torsional ultimate strengths increase with
case depth up to a certain point, and
level off at the optimum case depth.
The bottom line of each curve represents the minimum strength versus
case depth values for the steels listed.
The minimum yield for the optimum
effective case depth of 23% of the bar
diameter is approximately 795 MPa
(115,000 psi). The minimum ultimate
strength for the same 23% effective
case depth is about 1,379 MPa (200,000
psi). SAE 4140 provides a lower torsional strength for any given case
depth compared with the rest of the
steels, except at the far right portion of
the curve. This is because 4140 has
higher hardenability, and, hence, a
lower total case depth for the same effective case, indicating that total case
depth is also a factor in determining
torsional strength. At the far right of
the curve, all of the steels are about
equal, indicating that only effective
case is important in this area of the
curve.
Total case depth versus torsional
strength (Fig. 3). This curve is similar to
that in Fig. 2, except case depth values
for any given strength are greater (as
expected) and there appears to be
more variation in the data. The optimum case depth where the yield
strength levels off is at 31% of the bar
diameter. Again, there is a difference
in the minimum strength depending
on the grade of steel. In these tests, SAE
1541 and 4140 provided a higher torsional strength for any given case
depth compared with the other steels,
especially in the right hand portion of
the curve. This is because the higher
hardenability steels have a deeper effective case depth for any given total
case depth compared with the rest of
the steels. Also, SAE 1541 has a quench
and tempered core with a greater hardness. The higher core hardness is similar to a deeper total case depth. Both
effective case and total case are important in determining torsional strength,
but effective case seems to be a more
1,000 1050M
4140
800
1040 steel
HRC
52
50
40
30
20
10
0
HB
514
481
371
286
226
187
150
ULT
1793
1690
1276
966
759
655
517
Tensile
Yield
1614
1524
1152
941
614
503
366
Torsional
Yield
966
917
690
503
366
303
221
From SAE J413 (units are MPa)
600
400
1541 Q&T
200
Applied stress
0
0
10
20
30
40
Percent of bar diameter
50
60
Fig. 4 — Case depth providing 621 MPa minimum torsional yield strength.
accurate predictor.
Torsional strength versus applied stress
(Fig. 4). Case depth for each of the four
steel grades tested provided a minimum torsional yield of 621 MPa (90,000
HEAT TREATING PROGRESS • OCTOBER 2009
psi). The lower hardenability steels did
so with a shallower effective case and
a deeper total case compared with the
higher hardenability steels. The 1541
with its quench and tempered core did
17
so with a shallower effective and shallower total case. This demonstrates that
both effective and total case depths are
important in determining torsional
strength, and there are various ways to
achieve the same strength. In theory,
Fig. 4 indicates that the only two things
that should matter in keeping the
strength curve above the stress curve
are total case depth and surface hardness. However, in reality, both effective
and total case make a difference, and
the higher hardness of the 1050M material did not provide a higher strength
even at greater case depths.
1,200
SAE 1040 CD
1541 Q&T
4140 HR
Cycles at ±407 MPa, ×103
1,000
800
600
400
200
0
1
100
200
300
400
500
600
700
Average torsional yield strength, MPa
800
900
1,000
Fig. 5 — Fatigue life versus torsional yield strength.
Fatigue Characteristics
Figure 5 shows the fatigue characteristics of the SAE 1040, 1541, and
4140 test shafts. All shafts were run in
fully reversed torsional fatigue at a
stress of 407 MPa (59,000 psi). The data
show the correlation between fatigue
life and torsional yield strength. Considerable scatter, or variation, is
present, which is normal in fatigue
testing. It appears that the plain
carbon-steel grade 1040 reached suspension at 1,000,000 cycles before the
other two grades. A plausible explanation for this is shown in Fig. 4, where
all the case depths have an equal static
strength of 621 MPa (90,000 psi) minimum. However, if we look at the applied stress shown at various levels,
the 1040 has a deeper total case depth,
and the applied stress is somewhat
higher where it intersects the case-core
interface. This would seem to indicate
that total case depth may be more critical for fatigue life.
Fatigue life versus total case depth
is shown in Fig. 6. The data demonstrate that fatigue life increases with
increasing total case depth. SAE 1541
steel appears to provide higher fatigue
life for the same total case depth compared with the other two materials.
The reason for this is the higher hardness of the quench and tempered core,
which essentially acts the same as a
deeper total case depth. SAE 1040 and
4140 have the same fatigue life even
Table 2 — Required case depth versus torsional strength
Minimum torsional strength
Ultimate, MPa
966
1,138
1,379
Yield, MPa
483
621
793
1,172
1,310
1,517
621
758
862
Typical torsional strength
Ultimate, MPa
Yield, MPa
Diameter, mm
Case depth required (effective/total), mm
19.05
2.11/3.81
2.87/4.78
4.39/5.92
22.23
2.44/4.45
3.33/5.56
5.11/6.88
25.40
2.79/5.08
3.81/6.35
5.84/7.87
28.58
3.15/5.72
4.29/7.14
6.58/8.86
31.75
3.51/6.35
4.78/7.95
7.32/9.86
34.93
3.84/6.99
5.23/8.74
8.03/10.82
38.10
4.19/7.62
5.72/9.53
8.76/11.81
41.28
4.55/8.26
6.20/10.31
9.50/12.80
44.45
4.90/8.89
6.68/11.13
10.24/13.79
47.63
5.23/9.53
7.14/11.91
10.95/14.76
50.80
Any diameter
5.59/10.16
7.62/12.70
11.68/15.75
0.11(diam)/0.20(diam)
0.15(diam)/0.25(diam)
0.23(diam)/0.31(diam)
Note: torsional yield strength is valid for longer shafts with length/critical diameter ratio of 6 and greater. The critical diameter is the smallest diameter of the shaft where most of the deflection and
failure occurs. As shaft length decreases, the ratio of yield/ultimate increases. Torsional ultimate strength does not depend on length, and is valid for any length shaft.
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HEAT TREATING PROGRESS • OCTOBER 2009
Applying the Knowledge
Table 2 illustrates how the results of
this study can be used, where an engineer is trying to design a series of
shafts to be good for three different torsional strength levels. The middle
column is intended to provide a minimum torsional yield of 621 MPa
(90,000 psi). From the data in Figs. 2
and 3, we found that for a plain-carbon
steel such as 1040, this would require
an effective case of 15% of the diameter and a total case of 25%. These percentages would also work for other
steels, although the strength may actually be greater. A higher hardenability steel, such as 4140, would require a 17% effective and 23% total
case. The 621 MPa (90,000 psi) static
yield strength could also be obtained
with a quench and tempered core of
approximately 20 HRC by using an effective case of 15% and a total case of
19%. However, the quench and temper
operation would also add a fair
amount of cost to the part.
A couple of other examples we
could look at would be minimum torsional yield strength of 485 MPa
(70,000 psi) minimum and the highest
strength shaft of 795 MPa (115,000 psi)
minimum. Figs. 2 and 3 show that the
485 MPa (70,000 psi) shaft would re-
1,200,000
1040 steel
1541 steel
4140 steel
1,000,000
Cycle life, ×103
800,000
600,000
1541 Q&T
1040 and 4140
400,000
200,000
0
0
10
20
30
40
Total case depth, % of bar diameter
50
60
25
30
Fig. 6 — Torsional fatigue life versus total case depth at ±59,000 psi (±407 MPa).
1,200,000
1040 steel
1541 steel
4140 steel
1,000,000
800,000
Cycle life, ×103
though these two steels are on the opposite ends of the hardenability spectrum. As long as the total case depth
is the same, the fatigue life is the same.
A situation encountered several years
ago serves to illustrate this. A production axle shaft made of SAE 1038 steel
was not providing the desired fatigue
life, so a more premium grade (SAE
4140) was substituted. The manufacturing plant induction hardened the
4140 to the same effective case depth
as the production parts and discovered
that the fatigue life actually decreased
rather than increased. The torsional fatigue life versus effective case depth
shown in Fig. 7 indicates the reason for
this. A 15% effective case with 1040
steel provides a fatigue life in excess of
200,000 cycles, while the same case
depth with 4140 provides a life of less
than half of that. The reason is that
4140 has a lower total case depth compared with 1040 due to the difference
in hardenability. To increase the fatigue
life of 4140, it was necessary to increase
the total case depth. This also means
increasing the effective case along with
it. In the end, 4140 did not really provide any benefit in fatigue over the current production parts.
600,000
1040 and
1541 Q&T
4140
400,000
200,000
0
0
5
10
15
20
Total case depth, % of bar diameter
Fig. 7 — Torsional fatigue life versus effective case depth at ±59,000 psi (±407 MPa).
quire 11% effective case and 20% total,
while the 795 MPa (115,000 psi) shaft
would need 23% effective and 31%
total. These numbers could be applied
to any diameter shaft, but as the size
increases, the hardenability of the steel
would also have to increase to be able
to obtain these case depths.
Conclusions
Induction case depth plays a very important part in determining the static
and fatigue properties of shafts. Torsional strength does increase with case
depth, but only to a point, then hardening deeper does no good. Both effective and total case must be considered
to optimize shaft performance. Effective case appears to be the best predictor
of torsional strength, while total case is
the best predictor of fatigue life. The relationship between case depth and torsional strength is certain, but there is a
HEAT TREATING PROGRESS • OCTOBER 2009
considerable amount of scatter or variation. It is easy to see that if the range
of case depths observed was not wide
enough, the relationship could be
missed. Core hardness must also be
considered because it has the same effect as changing the total case depth.
While fatigue life does correlate to shaft
strength to some degree, there is even
more scatter or variation then with the
case depth versus strength data. Below
200,000 cycles, the scatter from high-tolow life can be up to 10:1. Above that,
as we approach the fatigue limit, the
HTP
scatter can be in excess of 20:1.
Bibliography
Fett, G., Induction Case Depths for Torsional Applications, Metal Progress, p 4952, Feb., 1985.
For more information: Gregory Fett, Dana
Corp., Maumee, Ohio; tel: 419-887-3296; email: greg.fett@dana.com; www.dana.com.
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