SME
Statics
Centroids
Use this for “A T-beam has a 100 × 20 mm flange on an 80 × 20 mm web. How high is its centroid?”
Diagram
ȳ
=
ΣAy
ΣA
=
244,000
3,600
=
67.78
mm
Flange area A₁ = 2,000 mm²
Web area A₂ = 1,600 mm²
This is an example. Type your own values to start a new problem.
A₁ (mm²)
=
bf (mm)
×
tf (mm)
A₂ (mm²)
=
tw (mm)
×
hw (mm)
ȳ (mm)
×
(
A₁ (mm²)
+
A₂ (mm²)
)
=
A₁ (mm²)
×
hw (mm)
+
tf (mm)
2
+
A₂ (mm²)
×
hw (mm)
2
A₁ = bf × tf is the flange’s area and A₂ = tw × hw the web’s; the flange’s center is hw + tf ÷ 2 above the bottom, and the web’s is hw ÷ 2.
Formulas
A₁
=
bf
×
tf
2,000 = 100 × 20
A₂
=
tw
×
hw
1,600 = 20 × 80
ȳ
=
(
A₁
×
(hw
+
tf
÷
2)
+
A₂
×
hw
÷
2)
÷
(
A₁
+
A₂
)
67.7778 = (2,000 × (80 + 20 ÷ 2) + 1,600 × 80 ÷ 2) ÷ (2,000 + 1,600)
Assumptions
•
The T is two rectangles: the flange sits on top of the web, both centered on one vertical line.
•
ȳ is measured up from the bottom of the web; by symmetry the centroid is on the center line.
•
Each rectangle’s area acts at its own center, so ȳ = ΣAy ÷ ΣA.
Step-by-step
WE KNOW
bf = 100 mm, tf = 20 mm, hw = 80 mm, tw = 20 mm
FIND
flange area (A₁), web area (A₂), centroid height (ȳ)
Step 1 · Find flange area (A₁)
Use A₁ = bf × tf
The flange is a rectangle bf wide and tf thick.
A₁
=
100
×
20
A₁
=
2,000
mm²
Step 2 · Find web area (A₂)
Use A₂ = tw × hw
The web is a rectangle tw wide and hw tall.
A₂
=
20
×
80
A₂
=
1,600
mm²
Step 3 · Find centroid height (ȳ)
Use ȳ = (A₁ × (hw + tf ÷ 2) + A₂ × hw ÷ 2) ÷ (A₁ + A₂)
Each
area
times
the
height
of
its
own
center
(the
flange’s
at
hw
+
tf
2,
the
web’s
at
hw
2
),
added,
then
divided
by
the
whole
area.
ȳ
=
(2,000
×
(80
+
20
÷
2)
+
1,600
×
80
÷
2)
÷
(2,000
+
1,600)
ȳ
=
(2,000
×
(80
+
10)
+
1,600
×
80
÷
2)
÷
(2,000
+
1,600)
ȳ
=
(2,000
×
90
+
1,600
×
80
÷
2)
÷
(2,000
+
1,600)
ȳ
=
180,000 + 128,000 ÷ 2
3,600
ȳ
=
180,000 + 64,000
3,600
ȳ
=
244,000
3,600
ȳ
=
67.7778
mm
Check
Put the numbers back in. Both sides should match.
2,000 = 100 × 20 ✓
1,600 = 20 × 80 ✓
67.7778 = (2,000 × (80 + 20 ÷ 2) + 1,600 × 80 ÷ 2) ÷ (2,000 + 1,600) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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