SME
Mechanics of Materials (Solid Mechanics I) · Problem type
Deflection anywhere along a cantilever
Use this for “A 2 m steel cantilever with I = 8 × 10⁶ mm⁴ carries 5 kN at its tip. How far has it dropped 1 m from the wall?”
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.
y = 2.604 mm at x = 1 m: the bent shape’s drop there.
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.
y (mm)
=
P (kN)
×
(
x (m)
)
2
6
×
E (GPa)
×
I (mm⁴)
×
(3
×
L (m)
−
x (m)
)
Formulas
y
=
P
×
x
2
×
3 ×
L
−
x
6 ×
E
×
I
2.6042
=
5,000
×
1,
000
2
×
3 × 2,000 − 1,000
6 × 200,000 × 8,000,000
Formulas are worked in N, 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 (x = 0) with the load P at the free tip (x = L); y is measured downward.
•
At the tip, x = L, the curve gives PL³ ÷ (3EI), the tip deflection.
Step-by-step
WE KNOW
P = 5 kN, L = 2 m, E = 200 GPa, I = 8,000,000 mm⁴, x = 1 m
FIND
deflection at x (y)
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)
x = 1 m = 1,000 mm (1 m = 1000 mm)
Step 2 · Find deflection at x (y)
Use y = P × x² × (3 × L − x) ÷ (6 × E × I)
Integrate
EIy″
=
P
(
L
−
x
)
twice;
y
=
0
and
y
′
=
0
at
the
wall
leave
y
=
Px²
3
L
−
x
6
EI
.
y
=
5,000
×
1,
000
2
×
3 × 2,000 − 1,000
6 × 200,000 × 8,000,000
y
=
5,000
×
1,
000
2
×
6,000 − 1,000
1,200,000 × 8,000,000
y
=
5,000
×
1,
000
2
×
5,000
9,600,000,000,000
y
=
5,000
×
1,000,000
×
5,000
9,600,000,000,000
y
=
5,000,000,000
×
5,000
9,600,000,000,000
y
=
25,000,000,000,000
9,600,000,000,000
y
=
2.6042
mm
Check
Put the numbers back in. Both sides should match.
2.6042
=
5,000
×
1,
000
2
×
3 × 2,000 − 1,000
6 × 200,000 × 8,000,000
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Beam deflection
Main lesson
Sag of a simply supported beam under a uniform load
Use this for “A 6 m steel beam on two supports carries 10 kN/m, with I = 80 × 10⁶ mm⁴. Does its sag stay within L ÷ 360?”
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?”