How does the modified slenderness ratio account for shear flexibility in built-up columns?

A built-up column is weaker than a solid column of the same overall slenderness because the lacing or batten connection system introduces shear flexibility — the individual components can slide relative to each other between connector points. AISC 360-22 Section E6 accounts for this through the modified slenderness ratio: (KL/r)m = √[(KL/r)o² + (a/ri)²], where (KL/r)o = overall slenderness ratio of the built-up member about the built-up axis, a = distance between connectors (lacing/batten spacing) along the member length, and ri = minimum radius of gyration of an individual component between connectors. The term (a/ri)² represents the local slenderness penalty — wider connector spacing increases (KL/r)m. AISC requires a/ri ≤ 0.75 × (KL/r)o to prevent local buckling of individual components between connectors from governing before global buckling of the built-up member. This limit ensures the component buckling mode does not precede the global mode. Example: double C12×20.7 column with 13.5 in lacing spacing, ri = 0.799 in, a/ri = 16.9, (KL/r)o = 42.7, giving (KL/r)m = √(42.7² + 16.9²) = 45.9 — only a 7.5% increase over the unmodified slenderness. If lacing were spaced at 24 in instead: a/ri = 30.0, (KL/r)m = √(42.7² + 30²) = 52.1 — a 22% increase, reducing column capacity by approximately 18%. The penalty increases quadratically with connector spacing because the a/ri term is squared — doubling the spacing quadruples the penalty term.

What is the 2% lacing rule and how is lacing bar force calculated?

The '2% rule' per AISC 360 Section E6.2 requires lacing bars to resist a shear force equal to at least 2% of the axial compression in the column. This is a simplified representation of the shear demand from: (1) initial geometric imperfections (out-of-straightness of the column), (2) accidental eccentricity of load application, and (3) shear from moment gradient along the column. V_lacing = 0.02 × Pu (or 0.02 × φPn for design). For a column carrying Pu = 300 kips: V_lacing = 6 kips total across the column cross-section. For single lacing at 45°: force in each diagonal = V_lacing / (2 × cos θ) where two planes of lacing (one each side) share the shear. At θ = 45°: F_diagonal = 6 / (2 × 0.707) = 4.24 kips. The lacing bar must be checked as a compression member with KL/r ≤ 140 per AISC E6.2. Flat bar example: 2 in × 3/8 in bar, L_diagonal = h/sin(45°) where h is the separation between component centroids. For h = 13.4 in: L = 18.95 in. r_bar = t/√12 = 0.375/3.464 = 0.108 in. KL/r = 18.95/0.108 = 175 > 140 → FAILS. Increase to 2.5 in × 1/2 in: r = 0.5/3.464 = 0.144 in, KL/r = 131.6 < 140 → OK. Double lacing (X-pattern on each face) shares the shear between two diagonals in each plane, halving the force per bar but increasing fabrication cost. Lacing bar connections: minimum 2 bolts or equivalent weld per end per AISC E6.2 — single-bolt connections are not permitted because they don't provide sufficient fixity for buckling resistance.

How do batten plates compare to lacing bars for built-up columns?

Batten plates (flat plates welded or bolted perpendicular to the column axis) and lacing bars are alternative connector systems for built-up columns, each with distinct characteristics: Lacing bars — diagonal bars connecting main components. Advantages: higher shear stiffness (truss action), lighter connectors, more efficient for long columns. Disadvantages: more complex fabrication (angle cuts, multiple pieces), more connections to make, appearance may be industrial. Each bar acts in tension or compression. Batten plates — flat plates perpendicular to column axis. Advantages: simpler fabrication (straight cuts only), cleaner appearance preferred for architectural columns, fewer pieces, better access between components for maintenance. Disadvantages: less efficient shear resistance (Vierendeel/moment-frame action instead of truss action), heavier connectors required, introduce local bending moments in main components between battens. AISC E6.2 requirements for batten plates: (1) batten plate length ≥ 2/3 of distance between lines of fasteners, (2) batten spacing ≤ 1.5 × distance between lines of fasteners (closer than lacing), (3) end battens required at each end of the member. Batten plate design forces: V_batten = V_lacing × a/h (shear) and M_batten = V_lacing × a/2 (moment), where a = batten spacing and h = centroid-to-centroid distance between main components. Batten plates are effectively moment-frame beams between the main component 'columns.' For the same column, battens are typically 30-50% heavier than lacing but reduce fabrication labor. Selection guidance: lacing for heavy industrial columns where efficiency matters, battens for architectural columns or where simpler fabrication is preferred.

When should a built-up column be used instead of a single rolled section?

Built-up columns are used when: (1) Required radius of gyration exceeds available rolled sections — a single W14 cannot achieve rx = ry = 6+ in, but two C12 channels separated by 12 in can achieve r ≈ 6.75 in about the built-up axis. The built-up section's r scales with separation distance, not just section depth. (2) Architectural or functional requirements demand an open cross-section — crane runway columns need access openings between components for trolley rails or maintenance access. Built-up columns provide this naturally. (3) Available rolled sections are too small for the applied load — for very heavy columns (Pu > 2,000 kips), built-up box sections (four plates or four angles with batten plates) can achieve areas exceeding the largest W14×730 (A = 214 in²). (4) Transportation or erection constraints limit maximum section size — a built-up column can be shipped as components and assembled on site, whereas a single W14×730 is difficult to transport and erect. (5) Biaxial bending or torsional requirements — cruciform (cross-shaped) columns from four angles provide balanced properties in all directions for biaxial loading or seismic ductility. Limitations: built-up columns have higher fabrication cost (lacing/batten fabrication and connection labor), the modified slenderness penalty reduces efficiency, and they require more inspection. For typical building columns under 500 kips and 15 ft length, single W10-W14 shapes are more economical. Built-up columns are most common in: heavy industrial buildings, crane runway support, bridge piers, transmission towers, and architecturally expressed structures.

How do different design codes handle built-up compression members?

Four major design codes address built-up compression members with different approaches: AISC 360-22 (US) Section E6 — uses the sqrt method: (KL/r)m = √[(KL/r)o² + (a/ri)²]. Minimum shear force = 2% of axial load. Lacing bar KL/r ≤ 140. Connector spacing limit: a/ri ≤ 0.75×(KL/r)o. AS 4100:2020 (Australia) Clause 6.5 — similar sqrt-based modified slenderness approach but with: (1) minimum shear force = 2.5% of axial load (25% higher than AISC), (2) more restrictive connector spacing: a/ri ≤ 0.5×λn where λn is the modified slenderness, (3) explicit provisions for battened members in Clause 6.5.3 with effective stiffness reduction. EN 1993-1-1 (Eurocode) Section 6.4 — most refined approach: (1) lambda_eff calculated considering shear flexibility of the connector system through a shear stiffness parameter Sv, (2) minimum shear force = 2.5% + additional shear from initial bow imperfection eo = L/500, (3) lacing bars: KL/r ≤ 150 for tension members, ≤ 70 for compression members, (4) battened columns get an additional effective imperfection amplification. EN 1993 more accurately captures battened column behavior than the AISC single-formula approach. CSA S16-19 (Canada) Clause 13.3.4 — similar to AISC with minor differences: (1) modified slenderness uses same sqrt formula, (2) lacing bar KL/r ≤ 140 (same as AISC), (3) minimum stitch bolt requirement: at least 2 bolts per component at each connector location. For heavy columns in seismic zones, AISC 341 adds special requirements: lacing and batten connections must develop the expected yield strength of the components to prevent connector failure before column buckling.

Steel Lattice Column Design — Built-Up Members, Lacing, and Batten Plates

Q: How do batten plates compare to lacing bars for built-up columns?

Batten plates (flat plates welded or bolted perpendicular to the column axis) and lacing bars are alternative connector systems for built-up columns, each with distinct characteristics: Lacing bars — diagonal bars connecting main components. Advantages: higher shear stiffness (truss action), lighter connectors, more efficient for long columns. Disadvantages: more complex fabrication (angle cuts, multiple pieces), more connections to make, appearance may be industrial. Each bar acts in tension or compression. Batten plates — flat plates perpendicular to column axis. Advantages: simpler fabrication (straight cuts only), cleaner appearance preferred for architectural columns, fewer pieces, better access between components for maintenance. Disadvantages: less efficient shear resistance (Vierendeel/moment-frame action instead of truss action), heavier connectors required, introduce local bending moments in main components between battens. AISC E6.2 requirements for batten plates: (1) batten plate length ≥ 2/3 of distance between lines of fasteners, (2) batten spacing ≤ 1.5 × distance between lines of fasteners (closer than lacing), (3) end battens required at each end of the member. Batten plate design forces: V_batten = V_lacing × a/h (shear) and M_batten = V_lacing × a/2 (moment), where a = batten spacing and h = centroid-to-centroid distance between main components. Batten plates are effectively moment-frame beams between the main component 'columns.' For the same column, battens are typically 30-50% heavier than lacing but reduce fabrication labor. Selection guidance: lacing for heavy industrial columns where efficiency matters, battens for architectural columns or where simpler fabrication is preferred.

Q: When should a built-up column be used instead of a single rolled section?

Built-up columns are used when: (1) Required radius of gyration exceeds available rolled sections — a single W14 cannot achieve rx = ry = 6+ in, but two C12 channels separated by 12 in can achieve r ≈ 6.75 in about the built-up axis. The built-up section's r scales with separation distance, not just section depth. (2) Architectural or functional requirements demand an open cross-section — crane runway columns need access openings between components for trolley rails or maintenance access. Built-up columns provide this naturally. (3) Available rolled sections are too small for the applied load — for very heavy columns (Pu > 2,000 kips), built-up box sections (four plates or four angles with batten plates) can achieve areas exceeding the largest W14×730 (A = 214 in²). (4) Transportation or erection constraints limit maximum section size — a built-up column can be shipped as components and assembled on site, whereas a single W14×730 is difficult to transport and erect. (5) Biaxial bending or torsional requirements — cruciform (cross-shaped) columns from four angles provide balanced properties in all directions for biaxial loading or seismic ductility. Limitations: built-up columns have higher fabrication cost (lacing/batten fabrication and connection labor), the modified slenderness penalty reduces efficiency, and they require more inspection. For typical building columns under 500 kips and 15 ft length, single W10-W14 shapes are more economical. Built-up columns are most common in: heavy industrial buildings, crane runway support, bridge piers, transmission towers, and architecturally expressed structures.

Built-up (lattice) columns consist of two or more main components (typically channels, angles, or W-shapes) connected by lacing bars, batten plates, or perforated cover plates. They are used when a single rolled section cannot provide the required radius of gyration, or when architectural/functional considerations demand an open cross-section (such as columns supporting crane runways). AISC 360-22 Section E6 provides the modified slenderness ratio approach for their design.

Modified slenderness ratio per AISC 360-22 Section E6

The key principle: a built-up column is weaker than a solid column of the same overall slenderness because the lacing or battens introduce shear flexibility. AISC accounts for this through a modified slenderness ratio:

(KL/r)_m = sqrt[(KL/r)_o^2 + (a/r_i)^2]

where:

(KL/r)_o = slenderness ratio of the overall built-up member about the built-up axis
a = distance between connectors (lacing/batten spacing) along the member length
r_i = minimum radius of gyration of an individual component between connectors

This formula penalizes wide connector spacing. The term (a/r_i) represents the local slenderness of each component between lacing points. AISC requires a/r_i <= 3/4 * (KL/r)_o to prevent local buckling of individual components from governing before global buckling of the built-up member.

Worked example — double-channel lattice column

Given: Built-up column made from two C12x20.7 channels, back-to-back with 12 in. separation (face-to-face). Lacing bars at 45 degrees, single-lacing system. Column length = 24 ft, pinned-pinned (K = 1.0). A36 steel (Fy = 36 ksi).

Step 1 — Individual channel properties: A = 6.09 in.^2 per channel, I_x = 129 in.^4, I_y = 3.88 in.^4, r_y = 0.799 in. x-bar (distance from web back to centroid) = 0.698 in.

Step 2 — Built-up section properties about the lacing axis (y-y, the built-up axis):

Separation center-to-center = 12 + 2 * 0.698 = 13.40 in. (distance between channel centroids in the y-direction, assuming back-to-back with 12 in. gap)

Wait — for back-to-back channels with a 12 in. face-to-face gap:

Channel web thickness = 0.282 in.
Center-to-center of channels = 12 + 0.282 = 12.28 in. (approximate, measured centroid to centroid = 12 + 2 * (t_w/2) is not correct)
Actually: distance from back of web to centroid = x-bar = 0.698 in. The back-to-back gap is 12 in. So centroid-to-centroid = 12 - 2 * 0.698 = 10.60 in. No — if the channels face outward (toes out), the backs face each other across the 12 in. gap:

Centroid distance = 12 + 2 _ x-bar = 12 + 2 _ 0.698 = 13.40 in. (from centroid of left channel to centroid of right channel, measured perpendicular to webs).

I*y,built-up = 2 * [I_y + A _ (13.40/2)^2] = 2 _ [3.88 + 6.09 _ 6.70^2] = 2 * [3.88 + 273.4] = 554.6 in.^4

A_total = 2 * 6.09 = 12.18 in.^2 r_y,built-up = sqrt(554.6 / 12.18) = sqrt(45.5) = 6.75 in.

Step 3 — Overall slenderness about built-up axis: (KL/r)_o = 1.0 _ 24 _ 12 / 6.75 = 288 / 6.75 = 42.7

Step 4 — Lacing spacing and local slenderness: For single lacing at 45 degrees: lacing bar spacing along the column = separation _ tan(45) = 13.40 _ 1.0 = 13.40 in. Use a = 13.5 in. (practical rounding).

r_i = r_y of individual channel = 0.799 in. a/r_i = 13.5 / 0.799 = 16.9

Check: a/r*i <= 3/4 * (KL/r)_o = 0.75 _ 42.7 = 32.0. Since 16.9 < 32.0, OK.

Step 5 — Modified slenderness ratio: (KL/r)_m = sqrt(42.7^2 + 16.9^2) = sqrt(1,823 + 286) = sqrt(2,109) = 45.9

Step 6 — Column capacity using AISC Chapter E: F_e = pi^2 * 29,000 / 45.9^2 = 286,200 / 2,107 = 135.8 ksi

Since Fy/Fe = 36/135.8 = 0.265 < 2.25: F*cr = 0.658^(Fy/Fe) * Fy = 0.658^0.265 _ 36 = 0.896 * 36 = 32.3 ksi

phi _ P_n = 0.90 _ 32.3 * 12.18 = 354 kips

Without the modification: Fcr at KL/r = 42.7 would give phi * Pn = 0.90 * 33.0 * 12.18 = 362 kips. The modification reduces capacity by about 2% in this case because the lacing spacing is well-controlled.

Lacing bar design (the "2% rule")

Lacing bars must resist a shear force equal to at least 2% of the axial compression in the column (AISC 360 Section E6.2). For the example above:

V_lacing = 0.02 * P_u

If P_u = 300 kips: V_lacing = 0.02 * 300 = 6 kips

For single lacing at 45 degrees, the force in each lacing bar = V*lacing / (2 * cos(45)) = 6 / (2 _ 0.707) = 4.24 kips (two lacing planes, one on each side).

The lacing bar must be checked as a compression member with KL/r_lacing <= 140 (AISC Section E6.2). For a flat bar 2 in. x 3/8 in., L_lacing = 13.4 / sin(45) = 18.95 in.:

r_bar = t / sqrt(12) = 0.375 / 3.464 = 0.108 in. KL/r = 18.95 / 0.108 = 175 >> 140 (NOT OK)

Need a larger bar. Try 2.5 in. x 1/2 in.: r = 0.5/3.464 = 0.144 in., KL/r = 18.95/0.144 = 131.6 < 140 (OK).

Batten plates as an alternative

Batten plates (flat plates welded or bolted perpendicular to the column axis) can replace lacing. They create a Vierendeel (moment frame) action between the main components. AISC Section E6.2 requires:

Batten plate length >= 2/3 of the distance between lines of fasteners
Batten spacing <= 1.5 * distance between lines of fasteners
End battens at each end of the member

Code comparison

| Aspect — AISC 360-22 — AS 4100:2020 — EN 1993-1-1 — CSA S16-19 | | ---------------------------- — --------------------- — -------------------- — -------------------------------- — ----------------- | | Modified slenderness formula — Sect. E6, sqrt method — Clause 6.5 (similar) — Sect. 6.4, lambda_eff — Clause 13.3.4 | | Minimum shear force — 2% of axial load — 2.5% of axial load — 2.5% + initial imperfection — 2% of axial load | | a/r_i limit — <= 0.75 * (KL/r)_o — <= 0.5 * lambda_n — <= 70 or 0.75 * lambda — <= 0.75 * (KL/r) | | Lacing bar KL/r limit — 140 — 140 — 150 (tension) / 70 (compression) — 140 | | Batten plate provision — Sect. E6.2 — Clause 6.5.3 — Sect. 6.4.3 — Clause 19 |

EN 1993-1-1 uses a more refined approach, modeling the built-up member with initial bow imperfections and calculating the shear force from the imperfection. The 2% rule in AISC is a simplification of this concept.

Key clause references

AISC 360-22 Section E6 — Built-up member design, modified slenderness ratio
AISC 360-22 Section E6.2 — Lacing and batten plate requirements
AISC 360-22 Section E6.1 — Connector spacing limits (a/r_i)
EN 1993-1-1 Section 6.4 — Uniform built-up compression members
AS 4100 Clause 6.5 — Laced and battened compression members

Topic-specific pitfalls

Ignoring the a/r_i limit — if the lacing or batten spacing is too wide relative to the component's radius of gyration, the individual components buckle between connectors before the overall column buckles. The a/r_i <= 0.75 * (KL/r)_o rule exists specifically to prevent this failure mode.
Designing lacing bars for tension only — in a single-lacing system, some bars are in compression. The KL/r <= 140 limit applies to these compression lacing bars. If only tension capacity is checked, the compression bars may buckle.
Neglecting end batten plates — without end battens, the load transfer from the main components to the gusset plate or base plate is eccentric, introducing local bending. AISC requires battens at each end.
Using the overall r for local checks instead of the individual component r_i — the individual channel's weak-axis r_y (0.799 in. in our example) controls the local buckling between lacing points, not the overall built-up section r_y (6.75 in.). Confusing these two is a common error.

Built-up column design per AISC E6 — detailed provisions

AISC 360-22 Section E6 covers the design of built-up compression members where two or more components are connected by lacing bars, batten plates, or perforated cover plates. The critical concept is that built-up members are less stiff in shear than solid members of the same overall cross-section, and this shear flexibility reduces the buckling load.

When to use built-up columns

Built-up columns are used when:

A single rolled section cannot provide the required radius of gyration about one or both axes
The architectural or functional requirements demand an open cross-section (e.g., crane runway columns with access openings)
The available rolled sections are too small for the applied load
Transportation or erection constraints limit the maximum section size

Common configurations include double angles (back-to-back), double channels (toe-to-toe or back-to-back), and built-up box sections (four plates or two channels with batten plates).

Laced column types and design

Single lacing: Diagonal bars in one direction on each face, forming a series of V or N patterns. Shear is resisted by tension in one diagonal and compression in the other.

Typical lacing angle: 30 to 60 degrees from the column axis (45 degrees is most common)
Bar spacing (along column): a = h × tan(theta), where h is the separation between components
AISC E6.2 requires the lacing bar slenderness KL/r <= 140 when loaded in compression

Double lacing: Diagonal bars in both directions on each face, forming an X pattern. Both diagonals can resist shear, and the effective stiffness is higher than single lacing.

More efficient in shear resistance
Higher fabrication cost (more pieces, more connections)
Preferred for heavy loads and larger column separations

Lacing bar connection: Each lacing bar must be connected to the main component by at least two bolts or a weld capable of developing the bar's full capacity. Single-bolt connections are not permitted because they do not provide sufficient fixity to develop the bar's buckling resistance.

Batten column types and design

Batten plates create a Vierendeel (moment-resisting) frame between the main components. They are flat plates welded or bolted perpendicular to the column axis at regular intervals.

Advantages of batten plates:

Simpler fabrication (straight cuts, no angle cuts required)
Cleaner appearance (often preferred for architectural columns)
Better access between components for maintenance or inspection

Disadvantages:

Less efficient shear resistance than lacing (requires heavier plates)
Introduce local bending moments in the main components between battens
AISC E6.2 requires battens at each end and at intermediate spacing not exceeding 1.5 times the distance between fastener lines

Batten plate design forces: Each batten plate and its connections must resist a shear force V_batten = V_lacing × a / h and a concurrent moment M_batten = V_lacing × a / 2, where a is the batten spacing and h is the perpendicular distance between component centroids.

Component slenderness limits

AISC 360-22 Section E6.1 requires that the local slenderness of each individual component between connectors satisfies:

a / ri <= 0.75 × (KL/r)o

where:

a = distance between connectors (lacing or batten spacing along the column)
ri = minimum radius of gyration of the individual component
(KL/r)o = overall slenderness of the built-up member about the built-up axis

This prevents the individual components from buckling between connector points before the overall built-up column buckles. If this limit is not satisfied, the local buckling capacity of the individual component controls, and the column capacity drops significantly.

For double-angle columns with welded or bolted stitch plates, the minimum connector spacing is typically controlled by this a/ri limit. For very slender individual components (small ri), this can require closely-spaced connectors that increase fabrication cost.

Effective length of built-up columns

The modified slenderness ratio in AISC E6 accounts for the shear flexibility of the connector system:

For laced columns (with tight-fitting connectors):

(KL/r)m = sqrt[(KL/r)o² + (a/ri)²]

For bolted or welded built-up members (with standard connectors): The same formula applies, but the interpretation of "a" differs — for bolted connections, a is measured between the centers of adjacent connectors. The shear flexibility of the bolt group is implicitly captured.

For battened columns: The modified slenderness may require additional adjustment because batten plates do not fully restrain rotation at the connection points. Some codes (EN 1993) apply an additional factor to the modified slenderness for battened columns. AISC treats laced and battened columns with the same formula but requires closer connector spacing for battens.

Connection requirements for lacing and batten plates

Requirement	Lacing Bars	Batten Plates
Minimum connections	2 bolts or weld per end	Weld or 2+ bolts per end
Bar/plate KL/r limit	<= 140 (compression bars)	Not directly limited
Minimum width	Per AISC Table 7-15 (bolted)	>= 2/3 of fastener line distance
Minimum thickness	Per KL/r = 140 limit	Per local bending check
End connections	Required at each end	Required at each end
Intermediate spacing (a)	Per a/ri <= 0.75×(KL/r)o	<= 1.5 × distance between lines
Weld requirement	Fillet weld both sides	CJP or fillet weld to main comp

Worked example: double-angle built-up column

Given: Two L6x4x1/2 angles (A36) with long legs back-to-back, separated by 3/8" spacer plates. Column length = 16 ft, fixed-pinned (K = 0.80). Axial load Pu = 180 kips.

Step 1 — Individual angle properties: L6x4x1/2: A = 4.75 in², Ix = 17.3 in⁴ (about long leg), Iy = 6.22 in⁴ (about short leg), ry = 1.14 in, x-bar = 0.981 in (from back of long leg).

Step 2 — Built-up section properties: Separation = 3/8 in between angle backs (long leg backs facing). Centroid-to-centroid = 3/8 in = 0.375 in (the long legs are parallel with 3/8" gap).

About the x-axis (parallel to long legs, same for both): Ix_built-up = 2 × Ix = 2 × 17.3 = 34.6 in⁴ (no parallel axis shift for this axis) A_total = 2 × 4.75 = 9.50 in² rx = sqrt(34.6 / 9.50) = 1.91 in

About the y-axis (perpendicular to long legs): Iy_built-up = 2 × [Iy + A × (0.375/2)²] = 2 × [6.22 + 4.75 × 0.0352] = 2 × 6.39 = 12.77 in⁴ ry_built-up = sqrt(12.77 / 9.50) = 1.16 in

Step 3 — Overall slenderness: (KL/r)x = 0.80 × 16 × 12 / 1.91 = 80.6 (KL/r)y = 0.80 × 16 × 12 / 1.16 = 132.4 (governs)

Step 4 — Connector spacing and modified slenderness: Spacer plates at a = 24 in spacing along the column. ri = ry of individual angle = 1.14 in (about its own weak axis perpendicular to built-up axis). a/ri = 24 / 1.14 = 21.1

Check: a/ri <= 0.75 × (KL/r)o = 0.75 × 132.4 = 99.3. OK.

Modified slenderness: (KL/r)m = sqrt(132.4² + 21.1²) = sqrt(17,530 + 445) = sqrt(17,975) = 134.1

Step 5 — Column capacity: Fe = pi² × 29,000 / 134.1² = 286,200 / 17,983 = 15.9 ksi Fy/Fe = 36/15.9 = 2.26 > 2.25, so use elastic formula: Fcr = 0.877 × Fe = 0.877 × 15.9 = 13.95 ksi

phi × Pn = 0.90 × 13.95 × 9.50 = 119.2 kips < 180 kips — FAILS.

Step 6 — Increase member sizes: Try double L6x4x3/4: A = 6.88 in² each, ry_individual = 1.08 in, Iy = 8.30 in². A_total = 13.76 in². Iy_built-up = 2 × [8.30 + 6.88 × 0.0352] = 16.88 in⁴ ry_built-up = sqrt(16.88 / 13.76) = 1.11 in (KL/r)y = 0.80 × 192 / 1.11 = 138.4

Modified: (KL/r)m = sqrt(138.4² + 21.1²) = sqrt(19,154 + 445) = 140.0 Fe = pi² × 29,000 / 140² = 14.6 ksi. Fcr = 0.877 × 14.6 = 12.8 ksi. phi × Pn = 0.90 × 12.8 × 13.76 = 158.5 kips — still short.

This shows that double-angle columns at 16 ft with K=0.8 are limited to relatively light loads. A W8 or W10 section would be more efficient for this load and length.

Buckling mode comparison

Mode	Description	Controls When
Global flexural	Overall column buckles about weak axis	Always checked (governs most cases)
Global torsional	Column twists about its longitudinal axis	Cruciform or thin-wall open sections
Local (component)	Individual component buckles between connectors	a/ri > 0.75 × (KL/r)o
Shear ratchet	Progressive connector failure under cyclic load	Seismic applications only
Connection failure	Lacing bar buckles or batten weld fractures	Undersized lacing or inadequate welds

For typical double-angle or double-channel columns, the global flexural mode about the built-up axis governs, modified by the shear flexibility penalty. The component local buckling mode is prevented by the a/ri limit.

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Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.

Column Buckling Theory

Euler Buckling

The Euler buckling load represents the theoretical critical load for an ideal elastic column:

Pcr = π²EI / (KL)²

Where:

E = modulus of elasticity (200 GPa for steel)
I = moment of inertia about the buckling axis
K = effective length factor
L = unbraced length

Real Column Behavior

Real columns deviate from Euler theory due to:

Initial out-of-straightness (typically L/1000)
Residual stresses from manufacturing (hot-rolling or welding)
Eccentricity of applied load
Inelastic material behavior

These effects are accounted for through column strength curves that reduce the theoretical Euler capacity based on slenderness ratio (KL/r) and section type.

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Frequently Asked Questions

What is the recommended design procedure for this structural element?

The standard design procedure follows: (1) establish design criteria including applicable code, material grade, and loading; (2) determine loads and applicable load combinations; (3) analyze the structure for internal forces; (4) check member strength for all applicable limit states; (5) verify serviceability requirements; and (6) detail connections. Computer analysis is recommended for complex structures, but hand calculations should be used for verification of critical elements.

How do different design codes compare for this calculation?

AISC 360 (US), EN 1993 (Eurocode), AS 4100 (Australia), and CSA S16 (Canada) follow similar limit states design philosophy but differ in specific resistance factors, slenderness limits, and partial safety factors. Generally, EN 1993 uses partial factors on both load and resistance sides (γM0 = 1.0, γM1 = 1.0, γM2 = 1.25), while AISC 360 uses a single resistance factor (φ). Engineers should verify which code is adopted in their jurisdiction.

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