SME
Mechanics of Materials (Solid Mechanics I) · Problem type
Tip deflection of a cantilever by superposition
Use this for “A 2 m steel cantilever with I = 8 × 10⁶ mm⁴ carries 5 kN at its tip and 2 kN/m along it. How far does the tip drop?”
Diagram
ΣF = 0: RA = 9,000 N, the total load.
The wall at A holds MA = 1.4 × 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.
δ (mm)
=
P (kN)
×
(
L (m)
)
3
3
×
E (GPa)
×
I (mm⁴)
+
w (kN/m)
×
(
L (m)
)
4
8
×
E (GPa)
×
I (mm⁴)
δP
Drop from the tip load
calculated · mm
δw
Drop from the uniform load
calculated · mm
Formulas
δP
=
P
×
L
3
3 ×
E
×
I
8.3333
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
δw
=
w
×
L
4
8 ×
E
×
I
2.5
=
2
×
2,
000
4
8 × 200,000 × 8,000,000
δ
=
δP
+
δw
10.8333 = 8.3333 + 2.5
Formulas are worked in N, N/mm, mm, MPa. The step-by-step shows the conversions.
Assumptions
•
Linear elastic, small deflections; the steps work in N, mm and MPa.
•
Fixed at the wall; P stands at the free tip and w covers the whole length.
•
Superposition: each load bends the beam as if alone, and the drops add.
Step-by-step
WE KNOW
P = 5 kN, w = 2 kN/m, L = 2 m, E = 200 GPa, I = 8,000,000 mm⁴
FIND
drop from the tip load (δP), drop from the uniform load (δw), 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, N/mm, mm, MPa).
P = 5 kN = 5,000 N (1 kN = 1000 N)
w = 2 kN/m = 2 N/mm (1 kN/m = 1 N/mm)
L = 2 m = 2,000 mm (1 m = 1000 mm)
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find drop from the tip load (δP)
Use δP = P × L³ ÷ (3 × E × I)
The
tip-load
case
alone:
integrating
EIy″
=
M(x)
twice
from
the
wall
gives
PL
³
3
EI
at
the
tip.
δP
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
δP
=
5,000
×
2,
000
3
600,000 × 8,000,000
δP
=
5,000
×
2,
000
3
4,800,000,000,000
δP
=
5,000
×
8,000,000,000
4,800,000,000,000
δP
=
40,000,000,000,000
4,800,000,000,000
δP = 8.3333 mm
Step 3 · Find drop from the uniform load (δw)
Use δw = w × L⁴ ÷ (8 × E × I)
The
uniform-load
case
alone:
integrating
EIy″
=
M(x)
twice
from
the
wall
gives
wL
⁴
8
EI
at
the
tip.
δw
=
2
×
2,
000
4
8 × 200,000 × 8,000,000
δw
=
2
×
2,
000
4
1,600,000 × 8,000,000
δw
=
2
×
2,
000
4
12,800,000,000,000
δw
=
2
×
16,000,000,000,000
12,800,000,000,000
δw
=
32,000,000,000,000
12,800,000,000,000
δw = 2.5 mm
Step 4 · Find tip deflection (δ)
Use δ = δP + δw
Superposition: the beam is linear, so the two drops add.
δ
=
8.3333
+
2.5
δ
=
10.8333
mm
Check
Put the numbers back in. Both sides should match.
8.3333
=
5,000
×
2,
000
3
3 × 200,000 × 8,000,000
✓
2.5
=
2
×
2,
000
4
8 × 200,000 × 8,000,000
✓
10.8333 = 8.3333 + 2.5 ✓
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Sag of a simply supported beam under a uniform load
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Deflection anywhere along a cantilever
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Beam deflection
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