SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Thick-walled cylinder (Lamé)
Use this for “A cylinder 50 mm inside and 100 mm outside in radius holds 50 MPa. Find the hoop stress at the bore and both stresses 75 mm from the axis.”
Diagram
Lamé: the hoop stress is largest at the bore and falls across the wall.
This is an example. Type your own values to start a new problem.
σθi (MPa)
=
p (MPa)
×
(
rᵢ (mm)
)
2
+
(
rₒ (mm)
)
2
(
rₒ (mm)
)
2
−
(
rᵢ (mm)
)
2
σₜₕᵢₙ
Thin-wall estimate
calculated
r
Radius of the point
example
σθ
Hoop stress at r
calculated
σᵣ
Radial stress at r
calculated
Formulas
σθi
=
p
×
rᵢ
2
+
rₒ
2
rₒ
2
−
rᵢ
2
83.3333
=
50
×
50
2
+
100
2
100
2
−
50
2
σₜₕᵢₙ
=
p
×
rᵢ + rₒ
2 × (rₒ − rᵢ)
75
=
50
×
50 + 100
2 × (100 − 50)
σθ
=
p
×
rᵢ
2
×
r
2
+
rₒ
2
r
2
× (
rₒ
2
−
rᵢ
2
)
46.2963
=
50
×
50
2
×
75
2
+
100
2
75
2
× (
100
2
−
50
2
)
σᵣ
=
−
p
×
rᵢ
2
×
rₒ
2
−
r
2
r
2
× (
rₒ
2
−
rᵢ
2
)
−12.963
=
−50
×
50
2
×
100
2
−
75
2
75
2
× (
100
2
−
50
2
)
Assumptions
•
Lamé: a long cylinder with open ends, inside pressure p only, the material elastic.
•
Tension is +, compression −: σᵣ runs from −p at the bore to 0 at the outside.
•
The thin-wall estimate is p × rₘ ÷ t, with mean radius rₘ and wall t = rₒ − rᵢ.
•
Stresses in MPa and radii in mm; the radii enter only as ratios.
Step-by-step
WE KNOW
p = 50 MPa, rᵢ = 50 mm, rₒ = 100 mm, r = 75 mm
FIND
hoop stress at the bore (σθi), thin-wall estimate (σₜₕᵢₙ), hoop stress at r (σθ), radial stress at r (σᵣ)
Step 1 · Find hoop stress at the bore (σθi)
Use σθi = p × (rᵢ² + rₒ²) ÷ (rₒ² − rᵢ²)
Lamé’s
hoop
stress
at
r
=
rᵢ,
where
it
is
largest.
σθi
=
50
×
50
2
+
100
2
100
2
−
50
2
σθi
=
50
×
2,500 + 10,000
10,000 − 2,500
σθi
=
50
×
12,500
7,500
σθi
=
625,000
7,500
σθi = 83.3333 MPa
Step 2 · Find thin-wall estimate (σₜₕᵢₙ)
Use σₜₕᵢₙ = p × (rᵢ + rₒ) ÷ (2 × (rₒ − rᵢ))
The
thin-wall
rule
pr
t
with
the
mean
radius
rᵢ + rₒ
2
and
the
wall
rₒ
−
rᵢ.
σₜₕᵢₙ
=
50
×
50 + 100
2 × (100 − 50)
σₜₕᵢₙ
=
50
×
50 + 100
2 × 50
σₜₕᵢₙ
=
50
×
150
100
σₜₕᵢₙ
=
7,500
100
σₜₕᵢₙ = 75 MPa
Step 3 · Find hoop stress at r (σθ)
Use σθ = p × rᵢ² × (r² + rₒ²) ÷ (r² × (rₒ² − rᵢ²))
Lamé:
σθ
=
A
+
B
r
2
with
A
=
p
rᵢ²
rₒ² − rᵢ²
and
B
=
A
rₒ²,
from
σᵣ
=
−
p
at
the
bore
and
0
outside.
σθ
=
50
×
50
2
×
75
2
+
100
2
75
2
× (
100
2
−
50
2
)
σθ
=
50
×
50
2
×
75
2
+
100
2
75
2
× (10,000 − 2,500)
σθ
=
50
×
50
2
×
75
2
+
100
2
75
2
× 7,500
σθ
=
50
×
50
2
×
5,625 + 10,000
5,625 × 7,500
σθ
=
50
×
50
2
×
15,625
42,187,500
σθ
=
50
×
2,500
×
15,625
42,187,500
σθ
=
125,000
×
15,625
42,187,500
σθ
=
1,953,125,000
42,187,500
σθ = 46.2963 MPa
Step 4 · Find radial stress at r (σᵣ)
Use σᵣ = −p × rᵢ² × (rₒ² − r²) ÷ (r² × (rₒ² − rᵢ²))
Lamé:
σᵣ
=
A
−
B
r
2
,
the
same
A
and
B
as
the
hoop
stress;
it
is
−
p
at
the
bore
and
0
outside.
σᵣ
=
−50
×
50
2
×
100
2
−
75
2
75
2
× (
100
2
−
50
2
)
σᵣ
=
−50
×
50
2
×
100
2
−
75
2
75
2
× (10,000 − 2,500)
σᵣ
=
−50
×
50
2
×
100
2
−
75
2
75
2
× 7,500
σᵣ
=
−50
×
50
2
×
10,000 − 5,625
5,625 × 7,500
σᵣ
=
−50
×
50
2
×
4,375
42,187,500
σᵣ
=
−50
×
2,500
×
4,375
42,187,500
σᵣ
=
−125,000
×
4,375
42,187,500
σᵣ
=
−
546,875,000
42,187,500
σᵣ = −12.963 MPa
Check
Put the numbers back in. Both sides should match.
83.3333
=
50
×
50
2
+
100
2
100
2
−
50
2
✓
75
=
50
×
50 + 100
2 × (100 − 50)
✓
46.2963
=
50
×
50
2
×
75
2
+
100
2
75
2
× (
100
2
−
50
2
)
✓
−12.963
=
−50
×
50
2
×
100
2
−
75
2
75
2
× (
100
2
−
50
2
)
✓
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