SME
Advanced Solid Mechanics (Solid Mechanics II)
Energy methods
Use this for “A 2 m steel cantilever with I = 8 × 10⁶ mm⁴ carries 5 kN at its tip. How much strain energy does it store, and how far does the tip drop?”
Diagram
ΣF = 0: RA = 5,000 N, the total load.
The wall at A holds MA = 1 × 10⁷ N·mm, counterclockwise.
The bent shape is drawn bigger than life; it meets every support.
I is taken about the dashed bending axis.
EI is how stiff the beam is in bending: the larger it is, the less the beam bends.
This is an example. Type your own values to start a new problem.
U (J)
=
(
P (kN)
)
2
×
(
L (m)
)
3
6
×
E (GPa)
×
I (mm⁴)
δ (mm)
=
P (kN)
×
(
L (m)
)
3
3
×
E (GPa)
×
I (mm⁴)
Formulas
U
=
P
2
×
L
3
6,000 ×
E
×
I
20.8333
=
5,
000
2
×
2,
000
3
6,000 × 200,000 × 8,000,000
δ
=
P
×
L
3
3 ×
E
×
I
8.3333
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
Formulas are worked in N, mm, MPa. The step-by-step shows the conversions.
Assumptions
•
Linear elastic, small deflections; bending energy only, shear energy neglected.
•
The beam is fixed at the wall and the load stands at the free tip.
•
One material and one section along the length: E and I stay the same.
Step-by-step
WE KNOW
P = 5 kN, L = 2 m, E = 200 GPa, I = 8,000,000 mm⁴
FIND
strain energy (U), tip deflection (δ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to N, mm, MPa).
P = 5 kN = 5,000 N (1 kN = 1000 N)
L = 2 m = 2,000 mm (1 m = 1000 mm)
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find strain energy (U)
Use U = P² × L³ ÷ (6,000 × E × I)
U
=
∫
M
2
dx
2
EI
with
M
=
Px
from
the
tip
gives
P
2
L
3
6
EI
in
N·mm;
divide
by
1000
for
joules.
U
=
5,
000
2
×
2,
000
3
6,000 × 200,000 × 8,000,000
U
=
5,
000
2
×
2,
000
3
1,200,000,000 × 8,000,000
U
=
5,
000
2
×
2,
000
3
9.6 ×
10
15
U
=
25,000,000
×
8,000,000,000
9.6 ×
10
15
U
=
2 ×
10
17
9.6 ×
10
15
U
=
20.8333
J
Step 3 · Find tip deflection (δ)
Use δ = P × L³ ÷ (3 × E × I)
Castigliano:
the
tip
deflection
is
∂
U
∂
P
;
differentiate
P
2
L
3
6
EI
with
respect
to
P
to
get
PL
³
3
EI
.
δ
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
δ
=
5,000
×
2,
000
3
600,000 × 8,000,000
δ
=
5,000
×
2,
000
3
4,800,000,000,000
δ
=
5,000
×
8,000,000,000
4,800,000,000,000
δ
=
40,000,000,000,000
4,800,000,000,000
δ
=
8.3333
mm
Check
Put the numbers back in. Both sides should match.
20.8333
=
5,
000
2
×
2,
000
3
6,000 × 200,000 × 8,000,000
✓
8.3333
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
✓
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Plane strain: the out-of-plane stress
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Truss joint deflection by the unit-load method
Topic check
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