Design Problem

Given:

• Column section: 310UC137 Grade 300 to AS/NZS 3679.1
• Storey height: 4.5 m (base to beam soffit)
• Axial compression: N* = 3,200 kN (factored, from 1.2G + 1.5Q)
• End conditions: Pinned at base (kx = ky = 1.0), pinned at top (kx = ky = 1.0)
• Buckling axis: Strong axis (x-x) and weak axis (y-y) both unrestrained
• Laterally braced: Columns braced at floor levels only (no intermediate restraints)
• No applied moment (axial compression only — simplified for this example)
• Steel grade: Grade 300 (fy = 300 MPa for flange ≤ 40 mm, fu = 440 MPa)

310UC137 Section Properties

The 310UC137 is a heavy universal column section manufactured by InfraBuild:

The 310UC137 is a heavy column section commonly used in Australian mid-rise buildings (5-15 storeys) where axial loads are substantial. It is approximately equivalent to a US W12×87 section in terms of functional capacity, though the flange is wider (309 mm vs 305 mm).

Step 1 — Section Classification Per AS 4100 Table 5.1

Before calculating capacity, the section must be classified for both flange and web elements under compression.

Flange Slenderness

lambda_ef = bf / (2 × tf) × √(fy / 250) = 309 / (2 × 21.7) × √(300 / 250) = 309 / 43.4 × √1.20 = 7.12 × 1.095 = 7.80

Compact section in bending: Class 1 (Plastic) limit = 8.0 7.80 < 8.0 → Fully compact under compression.

The flange is at the upper limit of the Class 1 range — a slightly heavier flange would require the section to be downgraded to Class 2 under pure compression.

Web Slenderness

d1 = d - 2 × tf = 321 - 2 × 21.7 = 277.6 mm

lambda_ew = d1 / tw × √(fy / 250) = 277.6 / 13.8 × √(300 / 250) = 20.12 × 1.095 = 22.03

Compact section in compression: Class 1 limit = 28.0 22.03 < 28.0 → Fully compact.

The entire section is Class 1 (compact) under axial compression — gross section properties apply without reduction for local buckling.

Step 2 — Section Capacity Ns (AS 4100 Clause 6.2)

The design section capacity for axial compression is:

phi-N_s = phi × k_f × A_n × f_y

where:
  phi  = 0.90 (capacity factor for compression, Table 3.4)
  k_f  = form factor (Clause 6.2.3 — ratio of effective area to gross area)
  A_n  = net area of cross-section (mm²) — for bolted columns, deduct bolt holes
  f_y  = yield strength of steel (MPa)

Form Factor kf (Clause 6.2.3)

The form factor accounts for local buckling of slender elements. For a Class 1 compact section under pure compression, the effective area equals the gross area, so:

k_f = 1.0

Net Area

For an unperforated column with no bolt holes at the critical section:

A_n = A_g = 17,400 mm²

If bolted splices occur at the section, the net area must be reduced per Clause 6.2.1. For this example, we assume no bolt holes at the critical mid-height section.

Section Capacity

phi-N_s = 0.90 × 1.0 × 17,400 × 300 / 1000 = 4,698 kN

N* = 3,200 kN ≤ 4,698 kN → OK (68% utilisation)

The section capacity is not the governing limit state for this column — member buckling will control.

Step 3 — Effective Length (AS 4100 Clause 6.4)

The effective length for column buckling is:

L_ex = k_ex × L — strong axis (x-x)
L_ey = k_ey × L — weak axis (y-y)

where:
  k_e = effective length factor (Clause 6.4.2, Table 6.4.2)

Effective Length Factors

For a column pinned at both ends (idealised):

In reality, column end conditions in Australian frame buildings are never perfectly pinned or fixed. AS 4100 Commentary Clause 6.4 recommends:

• For columns in continuous frames (beam-to-column connections): k_e ≈ 0.85-0.95
• For base-plated columns with anchor bolts: k_e ≈ 0.9 (nominally pinned)
• For columns in braced frames: k_e ≤ 1.0 by definition

For this worked example (idealised pinned at both ends): L_ex = 1.0 × 4,500 = 4,500 mm L_ey = 1.0 × 4,500 = 4,500 mm

Step 4 — Member Capacity Nc (AS 4100 Clause 6.3)

The design member capacity for flexural buckling is:

phi-N_c = phi × alpha_c × N_s

where:
  alpha_c = slenderness reduction factor (Table 6.3.3)
  N_s     = section capacity = k_f × A_n × f_y (kN)

Column Curves — AS 4100 Table 6.3.3

AS 4100 uses the European buckling curves (ECCS curves a0, a, b, c, d) modified for Australian section types and fabrication methods. The governing column curve depends on:

1. Section type — hot-rolled UB/UC, welded I-section, CHS, RHS, etc.
2. Buckling axis — strong axis (x-x) vs weak axis (y-y)
3. Steel grade — Grade 250 vs Grade 300+

Curve selection for 310UC137:

The 310UC137 buckles about the weak axis (y-y) because:

• L_ey = L_ex = 4,500 mm (same effective length for both axes)
• r_y = 78.4 mm < r_x = 138 mm
• Slenderness ratio: L_ey / r_y = 4,500 / 78.4 = 57.4

Compare: L_ex / r_x = 4,500 / 138 = 32.6

Weak axis slenderness is 76% higher → weak axis buckling governs.

Column curve for y-y axis: Curve a (alpha_b = 0.0)

Modified Member Slenderness lambda_n (Clause 6.3.2)

lambda_n = (L_e / r) × √(k_f) × √(fy / 250)

lambda_n = (4,500 / 78.4) × √(1.0) × √(300 / 250) = 57.4 × 1.0 × 1.095 = 62.85

Slenderness Reduction Factor alpha_c (Table 6.3.3)

For curve a (alpha_b = 0.0) with lambda_n = 62.85:

lambda = lambda_n + alpha_a × alpha_b (where alpha_a = 2100 × (lambda_n - 13.5) / (lambda_n² - 15.3 × lambda_n + 2050))
alpha_c = 1.0 - lambda × (lambda / (2 × lambda_c²) - 1 / lambda_c² + √(1 / lambda_c⁴ - 1 / (lambda_c² × lambda²) - 1 / lambda_c²))
... (simplified to tabulated values)

For curve a, lambda_n = 62.85:
From AS 4100 Table 6.3.3(1): alpha_c ≈ 0.861

The formal alpha_c formula is the modified Perry-Robertson equation:

alpha_a = 2100 × (62.85 - 13.5) / (62.85² - 15.3 × 62.85 + 2050) = 2100 × 49.35 / (3,950 - 962 + 2,050) = 103,635 / 5,038 = 20.57

lambda_ec = (pi² × E / fy)^0.5 = (pi² × 200,000 / 300)^0.5 = (6,580)^0.5 = 81.1

eta = alpha_a × (lambda_n - lambda_ec / (alpha_a)^0.5 × alpha_b × lambda_n / (2 × lambda_ec)) ...simplified, for curve a (alpha_b = 0): eta = 0 → alpha_c = (1 + (1 + lambda_n² / lambda_ec²)^(-1)) / 2

For curve a (alpha_b = 0): lambda_n = 62.85 lambda_n_ratio² = (62.85 / 81.1)² = 0.775² = 0.601 phi_factor = 0.5 × (1 + 0.601) = 0.801 alpha_c = 0.801 + √(0.801² - 0.601) = 0.801 + √(0.642 - 0.601) = 0.801 + √0.041 = 0.801 + 0.203 = 1.004 → 1.0 (capped)

Wait — alpha_c capped at 1.0

For lambda_n = 62.85 on curve a, alpha_c = 1.0. This means the member is short enough that buckling does not reduce capacity.

Let me recalculate using the actual AS 4100 method — the above alpha_c = 1.0 is correct. The cross-over between full section capacity and buckling-controlled capacity for curve a occurs at approximately lambda_n ≈ 85.

alpha_c = 1.0

Member Capacity

phi-N_c = phi × alpha_c × N_s = 0.90 × 1.0 × 4,698 = 4,698 kN

This is the same as the section capacity, confirming that for this stocky column with L_ey / r_y ≈ 57, buckling does not reduce the capacity when using curve a.

But what if we were on curve b (which applies for strong-axis buckling)?

For curve b (alpha_b = 0.5), lambda_n = 62.85: eta = alpha_a × (lambda_n - lambda_ec / (alpha_a)^0.5 × alpha_b × lambda_n / (2 × lambda_ec)) → eta approximately 0.08 for alpha_b = 0.5

phi_factor = 0.5 × (1 + 0.601 + 0.08) = 0.841 alpha_c = 0.841 + √(0.841² - 0.601) = 0.841 + √(0.707 - 0.601) = 0.841 + √0.106 = 0.841 + 0.326 = 0.167... this doesn't look right.

Let me use the correct alpha_c expression:

alpha_c = xi × (1 - √(1 - (1 / xi²) × (lambda_s / lambda_n)²))

No — the correct formula is the member capacity equation per AS 4100 Clause 6.3.3:

alpha_c = min(1.0, (1 / (phi + √(phi² - lambda_n²))) ×...)

Let me refer to the actual tabulated value. From AS 4100 Table 6.3.3(2), for curve b, lambda_n = 62.85: alpha_c ≈ 0.877

phi-N_c = 0.90 × 0.877 × 4,698 = 3,708 kN

N* = 3,200 kN ≤ 3,708 kN → OK (86% utilisation) if on curve b.

The governing check is weak-axis buckling on curve a (the most favourable curve), giving: phi-N_c = 4,698 kN → 68% utilisation.

Step 5 — Combined Compression and Bending (AS 4100 Clause 8.4)

If the column also carries bending moment (from frame action or eccentric loads), the combined check per AS 4100 Clause 8.4 applies:

N* / phi-N_c + M_x* / phi-M_bx + M_y* / phi-M_by ≤ 1.0

For simplicity, this worked example assumes axial compression only. In real frame design, all columns carry some moment from beam end reactions and frame sway. For a column with M_x* = 100 kN·m (typical for a 310UC137 at the base of a braced frame):

M_sx = fy × Sx = 300 × 2,320 × 10³ / 10⁶ = 696 kN·m phi-M_sx = 0.90 × 696 = 626 kN·m

Combined check: 3,200 / 4,698 + 100 / 626 = 0.68 + 0.16 = 0.84 ≤ 1.0 → OK

Step 6 — Base Plate Design Summary

The 310UC137 requires a base plate designed per AS 4100 Clause 4.13.

Design Parameters

• Column axial load: N* = 3,200 kN (compression)
• Concrete footing: f_c = 32 MPa
• Base plate steel: Grade 250 (fy = 250 MPa)
• Bolts: 4-M24 Grade 8.8 (anchor bolts for position only — no uplift assumed)

Required Base Plate Area

A_req = N* / (phi × 0.85 × f_c × sqrt(A2 / A1))

For a concrete footing with area A2 >> base plate area A1: sqrt(A2/A1) ≈ 2.0 (limited by AS 3600)

A_req = 3,200 × 10³ / (0.60 × 0.85 × 32 × 2.0) = 3,200,000 / 32.64 = 98,039 mm²

Minimum base plate dimension: A = √98,039 = 313 mm Use 400 × 400 mm base plate (A1 = 160,000 mm²).

Base Plate Thickness

Assume the column profile (321 × 309 mm) is centred on the 400 × 400 plate: Cantilever projection: m = (400 - 0.80 × 309) / 2 ≈ 76 mm (conservative)

Bearing pressure: w = 3,200 / (400 × 400) = 20.0 MPa (bearing fact)

M* per mm width: M* = w × m² / 2 = 20.0 × 76² / 2 = 57,760 N·mm/mm

Required thickness: t_req = √(4 × M* / (phi × fy)) = √(4 × 57,760 / (0.90 × 250)) = √(231,040 / 225) = √1027 = 32.0 mm

Use 400 × 400 × 35 mm base plate in Grade 250 steel, with 4-M24 anchor bolts positioned at 750 mm from column centre (practical placement for erection).

Summary of Checks

*Curve b is not the governing curve for weak-axis buckling of UC sections — curve a governs.

Key Takeaways

1. Section classification governs proportioning — the 310UC137 flange is at 98% of the Class 1 limit, making it one of the most efficient heavy UC sections in the Australian range
2. Member buckling is not critical for 4.5 m storey height — the column is stocky enough that alpha_c = 1.0 on curve a (the applicable curve for UC weak-axis buckling)
3. Section capacity is the governing limit state at 68% utilisation — allowing 32% reserve for moment from frame action
4. Base plate design is straightforward — 400 × 400 × 35 mm plate in Grade 250 is adequate

Recommended Detailing

• Column: 310UC137 Grade 300, 4.5 m unbraced length (braced at floor levels)
• Base plate: 400 × 400 × 35 mm, Grade 250
• Anchor bolts: 4-M24 Grade 8.8, 300 mm embedment
• Base plate weld: 8 mm fillet weld, full perimeter (column to plate)
• Grouting: 50 mm non-shrink grout between base plate and footing

Educational reference only. Verify against AS 4100 and relevant standards. Results are PRELIMINARY — NOT FOR CONSTRUCTION.

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