SME
Aircraft Design · Problem type
Takeoff weight with an empty-weight fit
Use this for “Fixed load 4500 lbf, fuel fraction 0.2, and an empty-weight fit Wₑ/W₀ = 0.95W₀−0.05. Find the takeoff weight.”
Chart
Wₑ/W₀(W₀) = g(W₀)
The curves cross at W₀ ≈ 20,320.
This is an example. Type your own values to start a new problem.
Wₑ/W₀
=
A
×
W₀
W₀ (lbf)
=
Wₗ (lbf)
1
−
−
Wₑ
Empty weight
calculated · lbf
Formulas
Wₗ
=
W₀
×
(1
−
(Wf/
W₀
)
−
A
×
W₀
C
)
4,500
=
20,320
×
(1
−
0.2
−
0.95
×
20,
320
−0.05
)
Wₑ/
W₀
=
A
×
W₀
C
0.5785
=
0.95
×
20,
320
−0.05
W₀
=
Wₗ
1 − (Wf/
W₀
) − (Wₑ/
W₀
)
20,320
=
4,500
1 − 0.2 − 0.5785
Wₑ
=
(Wₑ/
W₀
)
×
W₀
11,760 = 0.5785 × 20,320
Assumptions
•
A and C are the course’s fit for this class of airplane, with W₀ in pounds.
•
The fixed load Wₗ is the crew and payload; W₀ = Wₗ + Wf + Wₑ.
•
Heavier airplanes have smaller empty fractions (C < 0), so one W₀ closes, or none.
Step-by-step
WE KNOW
Wₗ = 4,500 lbf, Wf/W₀ = 0.2, A = 0.95, C = −0.05
FIND
takeoff weight (W₀), empty fraction (Wₑ/W₀), empty weight (Wₑ)
Step 1 · Find takeoff weight (W₀)
Use Wₗ = W₀ × (1 − (Wf/W₀) − A × W₀C)
No
rearrangement
puts
W₀
on
its
own,
so
it
is
found
by
trial:
each
next
try
is
where
a
straight
line
through
the
last
two
meets
the
target.
Try
W₀
=
20,000:
20,000
×
(1
−
0.2
−
0.95
×
20,
000
−0.05
)
=
4,420.2
(want
4,500)
Try
W₀
=
30,000:
30,000
×
(1
−
0.2
−
0.95
×
30,
000
−0.05
)
=
6,978.9
(want
4,500)
Try
W₀
=
20,312:
20,312
×
(1
−
0.2
−
0.95
×
20,
312
−0.05
)
=
4,498.2
(want
4,500)
Try
W₀
=
20,319:
20,319
×
(1
−
0.2
−
0.95
×
20,
319
−0.05
)
=
4,500
(want
4,500)
W₀
=
20,320
lbf
Step 2 · Find empty fraction (Wₑ/W₀)
Use Wₑ/W₀ = A × W₀C
The fit: similar airplanes’ empty fraction at this takeoff weight.
Wₑ/
W₀
=
0.95
×
20,
320
−0.05
Wₑ/
W₀
=
0.95
×
0.609
Wₑ/
W₀
=
0.5785
Step 3 · Find empty weight (Wₑ)
Use Wₑ = (Wₑ/W₀) × W₀
The empty weight is its fraction of the takeoff weight.
Wₑ = 0.5785 × 20,319.0861
Wₑ = 11,760 lbf
Check
Put the numbers back in. Both sides should match.
4,500
=
20,320
×
(1
−
0.2
−
0.95
×
20,
320
−0.05
)
✓
0.5785
=
0.95
×
20,
320
−0.05
✓
20,320
=
4,500
1 − 0.2 − 0.5785
✓
11,760 = 0.5785 × 20,320 ✓
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Mission fuel fraction (Breguet cruise)
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Wing loading and thrust-to-weight (constraint diagram)
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