SME
Machine Design · Problem type
Life on the S–N line
Use this for “A steel part with Sᵤₜ = 600 MPa and Sₑ = 210 MPa sees a fully reversed 300 MPa. How many cycles will it last?”
Diagram
The line runs from fSᵤₜ = 540 MPa at 10³ cycles to Sₑ = 210 MPa at 10⁶: a = 1,389 MPa, b = −0.1367.
At
Sf
=
300
MPa:
N
=
(
Sf
a
)
1 ÷ b
=
73,600
cycles.
This is an example. Type your own values to start a new problem.
Sf (MPa)
=
a (MPa)
×
N
Sᵤₜ
Ultimate tensile strength
example
Sₑ
Endurance limit
example
f
Fatigue strength fraction
example
Formulas
a
=
(
f
× Sᵤₜ)
2
÷
Sₑ
1,388.5714
=
(0.9 × 600)
2
÷
210
b
=
−
log₁₀(
f
× Sᵤₜ ÷ Sₑ)
3
−0.1367
=
−
log₁₀(0.9 × 600 ÷ 210)
3
Sf
=
a
×
N
b
300
=
1,388.5714
×
73,
630
−0.1367
Assumptions
•
The same stress cycle repeats; stresses at the critical point stay under the yield strength.
•
Fully reversed stress (σₘ = 0); Sf = aNb runs from fSᵤₜ at 10³ cycles to Sₑ at 10⁶.
•
The fraction f is typed from its chart (0.9 for most steels here).
•
Basquin’s σₐ = σf′(2Nf)b is the same line counted in reversals: a = σf′2b.
Step-by-step
WE KNOW
Sᵤₜ = 600 MPa, Sₑ = 210 MPa, f = 0.9, Sf = 300 MPa
FIND
S–N coefficient (a), S–N exponent (b), cycles to failure (N)
Step 1 · Find S–N coefficient (a)
Use a = (f × Sᵤₜ)² ÷ Sₑ
Putting
both
ends
of
the
line
into
Sf
=
aN^
b
and
eliminating
b
leaves
a
=
(fSᵤₜ)
2
÷
Sₑ.
a
=
(0.9 × 600)
2
÷
210
a
=
540
2
210
a
=
291,600
210
a
=
1,388.571429
MPa
Step 2 · Find S–N exponent (b)
Use b = −log₁₀(f × Sᵤₜ ÷ Sₑ) ÷ 3
The slope on log–log axes: the fall from fSᵤₜ to Sₑ over three decades of cycles.
b
=
−
log₁₀(0.9 × 600 ÷ 210)
3
b
=
−
log₁₀(540 ÷ 210)
3
b
=
−
log₁₀(2.571)
3
b
=
−
0.4101
3
b
=
−0.1367248
Step 3 · Find cycles to failure (N)
Use Sf = a × Nᵇ
Divide
by
a
,
then
raise
to
the
power
1
b
to
undo
N
b
.
N
=
(
Sf
a
)
1 ÷
b
N
=
(
300
1,388.571429
)
1 ÷ (−0.1367248)
N
=
0.21605
−7.314
N
=
73,630
cycles
Check
Put the numbers back in. Both sides should match.
1,388.571429
=
(0.9 × 600)
2
÷
210
✓
−0.1367248
=
−
log₁₀(0.9 × 600 ÷ 210)
3
✓
300
=
1,388.571429
×
73,
630
−0.1367248
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Main lesson
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