SME
Steel Design · Problem type
Live-load deflection
Use this for “A floor beam spans 20 ft with Iₓ = 518 in⁴ and carries a service live load of 1.2 kip/ft. Is its sag within L ÷ 360?”
Diagram
ΣF = 0: RA + RB = 12 + 12 = 24 kip, the total load.
The bent shape is drawn bigger than life; it meets every support.
This is an example. Type your own values to start a new problem.
Δ (in)
=
5
×
w (kip/ft)
×
(
L (ft)
)
4
÷
(384
×
E (ksi)
×
Iₓ (in⁴)
)
Δallow (in)
=
L (ft)
÷
360
Δ ≤ Δallow?
Within the limit?
worked out (not typed)
Formulas
Δ
=
5
×
w
×
L
4
384 ×
E
× Iₓ
0.2876
=
5
×
0.1
×
240
4
384 × 29,000 × 518
Δallow
=
L
360
0.6667
=
240
360
Δ
≤
Δallow?
=
within
L
360
when
Δ
≤
Δallow
0.2876
≤
0.6667,
so
within
L
360
Formulas are worked in kip/in, in. The step-by-step shows the conversions.
Assumptions
•
Service loads, not factored: the sag is checked under the live load alone.
•
Simply supported with w over the whole span; steel E = 29,000 ksi; steps in kip, in and ksi.
•
L ÷ 360 is the usual live-load limit for a floor beam under a plastered ceiling.
Step-by-step
WE KNOW
w = 1.2 kip/ft, L = 20 ft, E = 29,000 ksi, Iₓ = 518 in⁴
FIND
midspan deflection (Δ), deflection limit (Δallow), within the limit? (Δ ≤ Δallow?)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to kip/in, in).
w = 1.2 kip/ft = 0.1 kip/in (1 kip/in = 12 kip/ft)
L = 20 ft = 240 in (1 ft = 12 in)
Step 2 · Find midspan deflection (Δ)
Use Δ = 5 × w × L⁴ ÷ (384 × E × Iₓ)
Integrate
EIy″
=
M(x)
twice
with
y
=
0
at
both
supports;
the
midspan
sag
is
5
w
L
4
384EI
.
Δ
=
5
×
0.1
×
240
4
384 × 29,000 × 518
Δ
=
5
×
0.1
×
240
4
11,136,000 × 518
Δ
=
5
×
0.1
×
240
4
5,768,448,000
Δ
=
5
×
0.1
×
3,317,760,000
5,768,448,000
Δ
=
0.5
×
3,317,760,000
5,768,448,000
Δ
=
1,658,880,000
5,768,448,000
Δ
=
0.2876
in
Step 3 · Find deflection limit (Δallow)
Use Δallow = L ÷ 360
Under live load a floor beam may sag at most its span over 360.
Δallow
=
240
360
Δallow = 0.6667 in
Step 4 · Find within the limit? (Δ ≤ Δallow?)
Use Δ ≤ Δallow? = within L ÷ 360 when Δ ≤ Δallow
The sag is no more than the limit: the beam passes.
0.2876
≤
0.6667:
within
L
360
Δ
≤
Δallow?
=
within
L
360
Check
Put the numbers back in. Both sides should match.
0.2876
=
5
×
0.1
×
240
4
384 × 29,000 × 518
✓
0.6667
=
240
360
✓
0.2876
≤
0.6667,
so
within
L
360
✓
Topic check
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