SME
Statics
Moments of inertia
Use this for “Find I of a 50 × 100 mm rectangle about an axis 150 mm from its centroid.”
Diagram
A = 5,000 mm²
Ī = 4,167,000 mm⁴ about the centroidal axis
About x′: I = Ī + Ad² = 1.167 × 10⁸ mm⁴
This is an example. Type your own values to start a new problem.
A (mm²)
=
b (mm)
×
h (mm)
Ī (mm⁴)
=
b (mm)
×
(
h (mm)
)
3
12
I (mm⁴)
=
Ī (mm⁴)
+
A (mm²)
×
(
d (mm)
)
2
Formulas
A
=
b
×
h
5,000 = 50 × 100
Ī
=
b
×
h
3
12
4.1667 ×
10
6
=
50
×
100
3
12
I
=
Ī
+
A
×
d
2
1.1667 ×
10
8
=
4.1667 ×
10
6
+
5,000
×
150
2
Assumptions
•
d is measured from the rectangle’s own centroidal axis to the parallel axis x′.
•
The parallel-axis theorem only adds to the centroidal Ī: I = Ī + Ad² is never less than Ī.
•
Sizes in mm give the area in mm² and I in mm⁴; sizes in inches give in² and in⁴.
Step-by-step
WE KNOW
b = 50 mm, h = 100 mm, d = 150 mm
FIND
area (A), moment of inertia about the centroidal axis (Ī), moment of inertia about the parallel axis (I)
Step 1 · Find area (A)
Use A = b × h
A rectangle’s area is its width times its height.
A
=
50
×
100
A
=
5,000
mm²
Step 2 · Find moment of inertia about the centroidal axis (Ī)
Use Ī = b × h³ ÷ 12
A
rectangle
about
its
own
centroidal
axis:
Ī
=
∫
y
2
dA
over
the
area,
which
comes
to
bh
³
12
.
Ī
=
50
×
100
3
12
Ī
=
50
×
1,000,000
12
Ī
=
50,000,000
12
Ī
=
4.1667 ×
10
6
mm⁴
Step 3 · Find moment of inertia about the parallel axis (I)
Use I = Ī + A × d²
The
parallel-axis
theorem:
add
the
area
times
the
offset
squared
to
Ī
.
I
=
4.1667 ×
10
6
+
5,000
×
150
2
I
=
(4.1667
×
10
6
)
+
5,000
×
22,500
I
=
(4.1667
×
10
6
)
+
(1.125
×
10
8
)
I
=
1.1667 ×
10
8
mm⁴
Check
Put the numbers back in. Both sides should match.
5,000 = 50 × 100 ✓
4.1667 ×
10
6
=
50
×
100
3
12
✓
1.1667 ×
10
8
=
4.1667 ×
10
6
+
5,000
×
150
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
I of a composite T-section
Use this for “Find I about the centroid of a T with a 100 × 20 mm flange on an 80 × 20 mm web.”
Polar moment of a shaft or tube
Use this for “Find J for a tube 60 mm outside and 40 mm inside.”
Radius of gyration
Use this for “Find the radius of gyration of a 20 × 100 mm bar about its strong axis.”