SME
Mechanics of Materials (Solid Mechanics I) · Problem type
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?”
Diagram
ΣF = 0: RA + RB = 30,000 + 30,000 = 60,000 N, the total load.
The bent shape is drawn bigger than life; it meets every support.
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.
δₘₐₓ (mm)
=
5
×
w (kN/m)
×
(
L (m)
)
4
384
×
E (GPa)
×
I (mm⁴)
δallow (mm)
=
L (m)
360
δallow is the sag allowed, L ÷ 360.
Formulas
δₘₐₓ
=
5
×
w
×
L
4
384 ×
E
×
I
10.5469
=
5
×
10
×
6,
000
4
384 × 200,000 × 80,000,000
δallow
=
L
360
16.6667
=
6,000
360
Formulas are worked in N/mm, mm, MPa. The step-by-step shows the conversions.
Assumptions
•
Linear elastic, small deflections; the steps work in N, mm and MPa.
•
A pin at one end, a roller at the other, and w over the whole span.
•
The largest sag is at midspan; L ÷ 360 is the usual limit for a floor beam.
Step-by-step
WE KNOW
w = 10 kN/m, L = 6 m, E = 200 GPa, I = 80,000,000 mm⁴
FIND
midspan deflection (δₘₐₓ), deflection limit (δallow)
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, mm, MPa).
w = 10 kN/m = 10 N/mm (1 kN/m = 1 N/mm)
L = 6 m = 6,000 mm (1 m = 1000 mm)
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find midspan deflection (δₘₐₓ)
Use δₘₐₓ = 5 × w × L⁴ ÷ (384 × E × I)
Integrate
EIy″
=
M(x)
twice
with
y
=
0
at
both
supports;
at
midspan
the
sag
is
5
w
L
4
384
EI
.
δₘₐₓ
=
5
×
10
×
6,
000
4
384 × 200,000 × 80,000,000
δₘₐₓ
=
5
×
10
×
6,
000
4
76,800,000 × 80,000,000
δₘₐₓ
=
5
×
10
×
6,
000
4
6,144,000,000,000,000
δₘₐₓ
=
5
×
10
×
1,296,000,000,000,000
6,144,000,000,000,000
δₘₐₓ
=
50
×
1,296,000,000,000,000
6,144,000,000,000,000
δₘₐₓ
=
6.48 ×
10
16
6,144,000,000,000,000
δₘₐₓ = 10.5469 mm
Step 3 · Find deflection limit (δallow)
Use δallow = L ÷ 360
A floor beam may sag at most its span over 360.
δallow
=
6,000
360
δallow = 16.6667 mm
Check
Put the numbers back in. Both sides should match.
10.5469
=
5
×
10
×
6,
000
4
384 × 200,000 × 80,000,000
✓
16.6667
=
6,000
360
✓
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
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?”
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?”