SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Impact of a dropped weight on a bar
Use this for “A 100 N weight drops 50 mm onto the collar of a 1 m steel bar of 100 mm². What are the peak stretch and stress?”
Diagram
The
rod
stretches
by
δ
=
PL
AE
.
The stretch is drawn bigger than life.
This is an example. Type your own values to start a new problem.
δₛₜ (mm)
=
W (N)
×
L (mm)
A (mm²)
×
E (MPa)
δₘₐₓ (mm)
=
δₛₜ (mm)
×
(1
+
1
+
2
×
h (mm)
δₛₜ (mm)
)
σ
Peak stress
calculated
Formulas
δₛₜ
=
W
×
L
A
×
E
0.005
=
100
×
1,000
100 × 200,000
δₘₐₓ
=
δₛₜ
×
(1
+
√(1
+
2
×
h
δₛₜ
))
0.7121
=
0.005
×
(1
+
√(1
+
2
×
50
0.005
))
σ
=
E
×
δₘₐₓ
L
142.4249
=
200,000
×
0.7121
1,000
Assumptions
•
The bar stays linear elastic; its own mass and any energy lost in the impact are neglected.
•
The weight falls from rest through h, then moves with the bar until it stops.
•
N, mm and MPa throughout.
Step-by-step
WE KNOW
W = 100 N, h = 50 mm, L = 1,000 mm, A = 100 mm², E = 200,000 MPa
FIND
static stretch (δₛₜ), peak stretch (δₘₐₓ), peak stress (σ)
Step 1 · Find static stretch (δₛₜ)
Use δₛₜ = W × L ÷ (A × E)
The
stretch
if
the
weight
were
set
down
gently:
the
bar’s
stiffness
is
k
=
AE
L
.
δₛₜ
=
100
×
1,000
100 × 200,000
δₛₜ
=
100
×
1,000
20,000,000
δₛₜ
=
100,000
20,000,000
δₛₜ = 0.005 mm
Step 2 · Find peak stretch (δₘₐₓ)
Use δₘₐₓ = δₛₜ × (1 + √(1 + 2 × h ÷ δₛₜ))
The
work
done
by
the
falling
weight,
W
(
h
+
δₘₐₓ),
is
stored
in
the
bar
as
1
2
kδₘₐₓ²;
take
the
positive
root.
δₘₐₓ
=
0.005
×
(1
+
√(1
+
2
×
50
0.005
))
δₘₐₓ
=
0.005
×
(1
+
√(1
+
100
0.005
))
δₘₐₓ
=
0.005
×
(1
+
√
1 + 20,000
)
δₘₐₓ
=
0.005
×
(1
+
√
20
,001)
δₘₐₓ = 0.005 × (1 + 141.4)
δₘₐₓ = 0.005 × 142.4
δₘₐₓ = 0.7121245 mm
Step 3 · Find peak stress (σ)
Use σ = E × δₘₐₓ ÷ L
The
peak
force
is
kδₘₐₓ
with
k
=
AE
L
,
so
σ
=
kδₘₐₓ
A
=
Eδₘₐₓ
L
.
σ
=
200,000
×
0.7121245
1,000
σ
=
142,424.9
1,000
σ
=
142.4249
MPa
Check
Put the numbers back in. Both sides should match.
0.005
=
100
×
1,000
100 × 200,000
✓
0.7121245
=
0.005
×
(1
+
√(1
+
2
×
50
0.005
))
✓
142.4249
=
200,000
×
0.7121245
1,000
✓
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