SME
Steel Design · Problem type
Tension member: yielding or rupture
Use this for “An 8 in × ¾ in plate with Fᵧ = 50 ksi and Fᵤ = 65 ksi has two 1 in holes across it. Find φPₙ.”
Diagram
Ag = wt = 6 × 0.5 = 3 in².
Aₙ = (w − nₕdₕ)t
Aₙ = (6 − 2 × 1) × 0.5 = 2 in²
The lit line is the net section through the holes. φPₙ = min(0.90FᵧAg, 0.75FᵤAₑ) = min(97.2, 87) = 87 kips: rupture at the holes governs.
This is an example. Type your own values to start a new problem.
Ag (in²)
=
w (in)
×
t (in)
Aₙ (in²)
=
(
w (in)
−
nₕ
×
dₕ (in)
)
×
t (in)
φPₙ (kip)
=
min(0.90
×
Fᵧ (ksi)
×
Ag (in²)
,
0.75
×
Fᵤ (ksi)
×
U
×
Aₙ (in²)
)
Aₑ
Effective net area
calculated · in²
Formulas
Ag
=
w
×
t
3 = 6 × 0.5
Aₙ
=
(
w
−
nₕ
×
dₕ)
×
t
2 = (6 − 2 × 1) × 0.5
Aₑ
=
U
×
Aₙ
2 = 1 × 2
φPₙ = 0.75 × Fᵤ × Aₑ
87 = 0.75 × 58 × 2
Assumptions
•
AISC 360-22 (LRFD): φ = 0.90 for yielding of the gross area, 0.75 for rupture of the net area.
•
One straight row of holes across the plate; no staggered path.
•
Hole size is the bolt diameter plus ⅛ in: the standard hole plus 1/16 in for damage.
•
Shear lag factor (U): 1 when the plate is bolted across its whole width, less for an angle.
Step-by-step
WE KNOW
w = 6 in, t = 0.5 in, nₕ = 2, dₕ = 1 in, U = 1, Fᵧ = 36 ksi, Fᵤ = 58 ksi
FIND
gross area (Ag), net area (Aₙ), effective net area (Aₑ), design tensile strength (φPₙ)
Step 1 · Find gross area (Ag)
Use Ag = w × t
The gross area: the whole width times the thickness.
Ag = 6 × 0.5
Ag = 3 in²
Step 2 · Find net area (Aₙ)
Use Aₙ = (w − nₕ × dₕ) × t
The net area: the steel left on the line through the holes.
Aₙ = (6 − 2 × 1) × 0.5
Aₙ = 4 × 0.5
Aₙ = 2 in²
Step 3 · Find effective net area (Aₑ)
Use Aₑ = U × Aₙ
Shear
lag:
only
the
share
U
of
the
net
area
carries
load
fully.
Aₑ = 1 × 2
Aₑ = 2 in²
Step 4 · Find design tensile strength (φPₙ)
Use φPₙ = 0.75 × Fᵤ × Aₑ
Rupture through the holes gives the smaller strength, so it governs, with φ = 0.75.
0.75 × 58 × 2 < 0.90 × 36 × 3: rupture governs
φPₙ = 0.75 × 58 × 2
Yielding: 0.90 × 36 × 3 = 97.2
Rupture: 0.75 × 58 × 2 = 87
φPₙ = 87 kip
Check
Put the numbers back in. Both sides should match.
3 = 6 × 0.5 ✓
2 = (6 − 2 × 1) × 0.5 ✓
2 = 1 × 2 ✓
87 = 0.75 × 58 × 2 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.