SME
Mechanics of Materials (Solid Mechanics I) · Problem type
Thermal stress in a restrained bar
Use this for “A 1 m steel bar of 1,000 mm² held between two walls warms by 40 °C. What stress builds up, and what force does it put on the walls?”
Diagram
Free, the bar would grow δT = αΔTL; the walls push it back to its length, so σ = −EαΔT.
The growth is drawn bigger than life.
This is an example. Type your own values to start a new problem.
σ (MPa)
=
−
E (GPa)
×
α (/°C)
×
ΔT (°C)
F (kN)
=
σ (MPa)
×
A (mm²)
L
Length
example
δT
Free growth
calculated · mm
Formulas
δT
=
α
×
ΔT
×
L
0.48
=
1.2 ×
10
-5
×
40
×
1,000
σ
=
−
E
×
α
×
ΔT
−96
=
−200,000
×
(1.2
×
10
-5
)
×
40
F
=
σ
×
A
−96,000 = −96 × 1,000
Formulas are worked in MPa, N. The step-by-step shows the conversions.
Assumptions
•
The walls don’t move, so they push the bar back by all of its free growth.
•
ΔT is in °C and α is per °C (a change of 1 °C is a change of 1 K).
•
Compression is negative; the bar stays elastic and doesn’t buckle.
•
Steel by default: E = 200 GPa, α = 12 × 10⁻⁶ per °C; worked in N, mm and MPa.
Step-by-step
WE KNOW
α = 1.2 × 10⁻⁵ /°C, ΔT = 40 °C, L = 1,000 mm, E = 200 GPa, A = 1,000 mm²
FIND
free growth (δT), thermal stress (σ), force on the walls (F)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to MPa, N).
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find free growth (δT)
Use δT = α × ΔT × L
Free,
the
bar
would
grow
by
α
for
each
degree,
for
each
millimeter
of
its
length.
δT
=
1.2 ×
10
-5
×
40
×
1,000
δT = 0.00048 × 1,000
δT = 0.48 mm
Step 3 · Find thermal stress (σ)
Use σ = −E × α × ΔT
The
walls
squeeze
out
the
strain
αΔT
the
bar
would
take,
so
the
stress
is
−
E
times
it.
σ
=
−200,000
×
(1.2
×
10
-5
)
×
40
σ
=
−2.4
×
40
σ
=
−96
MPa
Step 4 · Find force on the walls (F)
Use F = σ × A
The
stress
acts
over
the
whole
section,
so
the
walls
feel
σ
times
the
area
(−
for
a
squeeze).
F
=
−96
×
1,000
F
=
−96,000
N
Step 5 · Convert the answers
Change each answer back to the unit you chose.
F = −96,000 N = −96 kN (1 kN = 1000 N)
Check
Put the numbers back in. Both sides should match.
0.48
=
1.2 ×
10
-5
×
40
×
1,000
✓
−96
=
−200,000
×
(1.2
×
10
-5
)
×
40
✓
−96,000 = −96 × 1,000 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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