--------- | ------------------------------------ | -------------------------------------------- | | T-connection | One branch perpendicular to chord | Vertical bracing, truss web at right angle | | Y-connection | One branch at angle to chord | Diagonal bracing, truss web | | Cross (X) | Two branches opposite sides of chord | Through members, X-bracing nodes | | K-connection | Two branches same side, gap between | Warren truss joints with gap | | N-connection | Two branches same side, overlap | Pratt truss joints with overlapping branches |

The chord is the continuous member (typically the larger section). The branch is the member that terminates at the chord face (typically the smaller section). All limit state equations in Chapter K use the ratio of branch width to chord width (B_b/B or H_b/B for rectangular HSS) to capture the effect of load concentration.

The Four Primary Limit States (K1-K4)

When a branch member loads an HSS chord, the chord wall must resist the concentrated force. The failure mode depends on the relative stiffness of the chord wall, the branch angle, and whether the load is tension or compression. AISC 360 Section K2 defines the four limit states.

K1 — Chord Wall Plastification

Chord wall plastification occurs when the branch member punches through the chord face. This is the most common limit state for T-, Y-, and cross-connections where the branch width is significantly smaller than the chord width (beta < 0.85). The chord face yields in flexure as the branch pushes or pulls on it.

For a rectangular HSS chord under branch axial load P (perpendicular to chord):

P_n = Fy x t^2 / (1 - beta) x ((2 x H_b / B) / sin(theta) + 4 x sqrt(1 - beta)) x Q_f

where:
  beta = B_b / B  (branch width ratio, <= 0.85 for K1 to govern)
  theta = branch angle from chord axis (degrees)
  Q_f = chord stress interaction factor, accounting for chord axial stress
  t = chord wall design thickness (0.93 x t_nom for HSS)
  Fy = chord specified minimum yield stress (ksi)
  B = chord width (in.)
  B_b = branch width (in.)
  H_b = branch depth (in.)

The phi factor for K1 is 0.90 (LRFD). The Q_f term reduces capacity when the chord carries significant axial stress — see AISC 360 Table K2.1 for the Q_f expression based on the chord utilization ratio U = |P_r/P_c| + |M_r/M_c|.

K1 is the baseline check. If K1 capacity is sufficient, the connection passes. If not, you must reinforce the chord face with a stiffening plate or increase the chord wall thickness.

K2 — Chord Sidewall Yielding (or Crippling)

Sidewall yielding occurs when the branch member bears directly on the chord sidewall. This limit state governs when the branch width B_b approaches the full chord width B (beta > 0.85), so the load bypasses the chord face and transfers to the sidewalls.

For branch axial compression:

P_n = Fy x t x (5k + L_b)    (local yielding, similar to web yielding in W-shapes)

where:
  k = outside corner radius of HSS (from AISC Manual Table 1-12)
  L_b = effective bearing length of branch on chord sidewall

For branch tension:

P_n = Fy x t x (5k + L_b)    (same equation but check sidewall in tension)

The phi factor is 1.00 per AISC 360 Section K2.2. When beta > 0.85, K2 capacity usually exceeds K1 capacity because the load transfers more efficiently through the sidewalls than through the chord face in bending.

K2 also applies to the branch connection to the chord — the branch wall itself must be checked for local yielding where it meets the chord. This local check uses the same form as K1 but with the branch wall properties.

K3 — Chord Shear Yielding (Punching Shear)

Shear yielding (punching shear) occurs when the branch member shears through the chord wall around the branch perimeter. K3 is the brittle failure mode — it must be checked even when K1 or K2 appears to govern.

For a rectangular HSS branch:

P_n = 0.6 x Fy x t x L_p   (punching shear around branch perimeter)

where:
  L_p = 2 x H_b / sin(theta) + 2 x B_e   (effective punching perimeter)
  B_e = effective width = (10 / (B/t)) x Fy x t / (Fy_b x t_b) x B_b
       but B_e <= B_b

The phi factor is 0.95. K3 becomes critical when the chord wall is thin relative to the branch dimensions (high B/t ratio). A typical warning sign: B/t > 30 for the chord. If K3 controls, options include increasing the chord wall thickness locally via a reinforcing plate or reducing the branch width at the connection through a tapered transition.

Key design note: K3 is independent of beta. Even when K1 is satisfied by a small beta, K3 can fail because the thin chord wall cannot sustain the shear demand around the branch footprint. Always check K3 for thin-walled HSS chords (B/t > 25).

K4 — Local Yielding of Branch(es) Due to Uneven Load Distribution

K4 addresses local yielding in the branch member itself caused by non-uniform stress distribution at the connection interface. When a branch loads the chord at an angle or with combined axial and moment, the stress concentrates at the heel and toe of the branch wall. K4 checks that the branch can transmit the force without local buckling or yielding.

For branch axial compression or tension:

P_n = Fy_b x t_b x (2 x H_b / sin(theta) + 2 x B_ei - 4 x t_b)

where:
  B_ei = effective width for branch = (10 / (B_b/t_b)) x (Fy x t) / (Fy_b x t_b) x B_b
         but B_ei <= B_b

The phi factor is 0.95. K4 limits the effective width of the branch that can participate in load transfer. The effective width B_ei accounts for shear lag — portions of the branch cross-section remote from the chord face cannot fully engage. For wide, thin branch sections (B_b/t_b > 25), the effective width may govern over the nominal yield strength.

Connection Geometry Validity Ranges

AISC 360 Table K2.1A establishes validity ranges for which the K1-K4 equations are experimentally validated. Connections outside these ranges require physical testing or advanced FEA:

Most domestic HSS sections (ASTM A500 Grade B, Fy = 46 ksi, Grade C, Fy = 50 ksi) fall within these limits. High-strength HSS (ASTM A1085, Fy = 50 ksi) satisfies all ranges and provides superior ductility for seismic applications.

Q_f — Chord Stress Interaction Factor

The chord stress interaction factor Q_f reduces the K1 capacity when the chord carries significant axial load or bending moment. This is the most frequently missed factor in HSS connection design.

Per AISC 360 Table K2.1, for rectangular HSS chords with axial load only:

Q_f = 1.0 - U^2    (for chord in tension)
Q_f = 1.0 - 0.3 x U x (1 + U)    (for chord in compression, connecting face)

where U = P_r / P_c + M_r / M_c (the chord utilization ratio at the connection location).

Example Q_f values:

The table shows that chord compression is less penalizing than tension at high utilization because compression stabilizes the chord face against inward punching. However, at very high compression (U > 0.9), the Q_f expression should be used with caution — the AISC provisions assume the chord is not near its buckling limit.

Design insight: Locate HSS connections away from points of maximum chord moment or axial load whenever possible. Shifting a brace connection 2 ft along a truss chord can reduce U from 0.8 to 0.4, doubling the K1 connection capacity.

Worked Example — HSS T-Connection (K1 Governs)

Given: HSS8x8x1/2 chord (ASTM A500 Gr. C, Fy = 50 ksi), HSS6x6x3/8 branch (ASTM A500 Gr. B, Fy = 46 ksi). T-connection (theta = 90 degrees). Branch axial compression P_u = 85 kip (LRFD). Chord axial load at connection: P_r = 120 kip compression (U = 0.40). Assume chord is adequately braced for global buckling.

Chord properties (HSS8x8x1/2):

• B = 8.00 in., H = 8.00 in.
• t_nom = 0.465 in. (design thickness per AISC Manual)
• t = 0.93 x 0.465 = 0.432 in. (design wall thickness per AISC 360 B3.12)
• A = 13.5 in.^2
• P_c = phi x Fy x A = 0.90 x 50 x 13.5 = 608 kip
• U = 120 / 608 = 0.197

Branch properties (HSS6x6x3/8):

• B_b = 6.00 in., H_b = 6.00 in.
• t_b_nom = 0.349 in.
• t_b = 0.93 x 0.349 = 0.325 in.

Step 1 — Geometry check: beta = B_b / B = 6.00 / 8.00 = 0.75 (within 0.25-1.0 range, OK) B/t = 8.00 / 0.432 = 18.5 <= 35 (OK) B_b/t_b = 6.00 / 0.325 = 18.5 <= 35 (OK) theta = 90 degrees >= 30 (OK) Connection is within AISC validity range.

Step 2 — K1 Chord Wall Plastification: Q_f for compression on connecting face with U = 0.197: Q_f = 1.0 - 0.3 x U x (1 + U) = 1.0 - 0.3 x 0.197 x 1.197 = 1.0 - 0.071 = 0.929

P_n = Fy x t^2 / (1 - beta) x (2 x H_b / B + 4 x sqrt(1 - beta)) x Q_f
P_n = 50 x 0.432^2 / (1 - 0.75) x (2 x 6.00 / 8.00 + 4 x sqrt(1 - 0.75)) x 0.929
P_n = 50 x 0.1866 / 0.25 x (1.50 + 4 x 0.50) x 0.929
P_n = 9.33 / 0.25 x 3.50 x 0.929
P_n = 37.32 x 3.50 x 0.929
P_n = 121.3 kip

phi x P_n = 0.90 x 121.3 = 109.2 kip

Step 3 — Check P_u against phi P_n: P_u = 85 kip < phi P_n = 109.2 kip. OK. K1 utilization = 85 / 109.2 = 0.78.

Step 4 — K2 Sidewall Yielding (beta > 0.85 only): beta = 0.75 < 0.85. K2 does not govern for this connection. (If beta were 0.90, K2 would be checked with the bearing length over the sidewall.)

Step 5 — K3 Punching Shear:

B_e = (10 / (B/t)) x (Fy x t) / (Fy_b x t_b) x B_b
B_e = (10 / 18.5) x (50 x 0.432) / (46 x 0.325) x 6.00
B_e = 0.541 x 21.6 / 14.95 x 6.00 = 0.541 x 1.445 x 6.00 = 4.69 in.
B_e <= B_b = 6.00. B_e = 4.69 in.

L_p = 2 x H_b / sin(90) + 2 x B_e = 2 x 6.00 / 1.0 + 2 x 4.69 = 12.00 + 9.38 = 21.38 in.

P_n = 0.6 x Fy x t x L_p = 0.6 x 50 x 0.432 x 21.38 = 277.1 kip
phi x P_n = 0.95 x 277.1 = 263.2 kip >> P_u. K3 does not govern.

K3 capacity far exceeds K1 because the chord wall is relatively thick (B/t = 18.5). For a thinner chord (B/t = 30), K3 could control.

Step 6 — K4 Branch Local Yielding:

B_ei = (10 / (B_b/t_b)) x (Fy x t) / (Fy_b x t_b) x B_b
B_ei = (10 / 18.5) x (50 x 0.432) / (46 x 0.325) x 6.00
B_ei = 0.541 x 1.445 x 6.00 = 4.69 in. (same numerical value here because B/t = B_b/t_b)

P_n = Fy_b x t_b x (2 x H_b / sin(90) + 2 x B_ei - 4 x t_b)
P_n = 46 x 0.325 x (12.00 + 9.38 - 1.30) = 14.95 x 20.08 = 300.2 kip
phi x P_n = 0.95 x 300.2 = 285.2 kip >> P_u. K4 does not govern.

Connection passes. K1 controls with 78% utilization.

Reinforcing HSS Connections

When K1 capacity is insufficient, several reinforcement strategies are available:

A stiffening plate is the most common retrofit. The plate thickness should be at least the branch wall thickness. The plate width should extend at least 2.5 x t beyond the branch footprint on all sides. The plate is fillet welded to the chord face around its perimeter.

Branch-to-Chord Weld Design

The weld connecting the branch to the chord must develop the branch force. For rectangular HSS, the effective weld length around the branch perimeter is:

L_weld = 2 x H_b / sin(theta) + 2 x B_b    (all-around weld)

For the worked example: L_weld = 2 x 6.00 / 1.0 + 2 x 6.00 = 24.0 in.

The required fillet weld size per unit length: D_req = P_u / (phi x 0.6 x F_exx x 0.707 x L_weld). For E70XX electrodes (F_exx = 70 ksi), phi = 0.75:

D_req = 85 / (0.75 x 0.6 x 70 x 0.707 x 24.0)
D_req = 85 / (0.75 x 42 x 0.707 x 24.0) = 85 / 534.5 = 0.159 (sixteenths of an inch)

A 3/16 in. fillet weld (D = 3) provides capacity of approximately 3 x 85 / 0.159 = 160 kip based on this weld length. Use 3/16 in. fillet weld all around.

Regional Standards Comparison

The international HSS connection provisions are largely harmonized — all derive from the same IIW (International Institute of Welding) CIDECT design guides. The primary differences are in the Q_f / k_n functions and the phi factors.

Frequently Asked Questions

What is the difference between K1 and K3 failure modes? K1 (chord wall plastification) is a flexural yield-line mechanism in the chord face — the face bends and forms plastic hinges. K3 (punching shear) is a shear failure through the chord wall thickness around the branch perimeter — the branch literally punches through the chord wall. K1 is ductile and provides warning through visible deformation. K3 is brittle and should be avoided as the controlling limit state. For thin-walled chords (B/t > 28), K3 often controls and should be carefully checked.

Why does Q_f matter so much in HSS connection design? Q_f accounts for the reduction in chord face capacity when the chord itself is highly stressed. A chord carrying 80% of its axial capacity has only about 36% of its face plastification capacity remaining for branch loads (Q_f = 0.36 in tension). This is the single most-commonly missed check in HSS truss design. A connection that passes at mid-span (low chord force) may fail near the support (high chord force). Always check connections at BOTH ends of the chord.

When should I use a through-plate instead of a direct welded connection? Use a through-plate when beta < 0.4 and branch loads are high. With very narrow branches on wide chords, the chord face is highly flexible and K1 capacity is low. A through-plate transfers the branch load to both chord sidewalls directly, bypassing the face entirely. This approximately triples the connection capacity. The plate passes through slots cut in the chord walls and is welded to both the branch and both chord sidewalls.

Can HSS connections resist moment in addition to axial load? Yes, but the interaction is complex. AISC 360 Section K3 provides interaction equations for combined axial and moment in the branch. The approach uses a generalized beta factor that accounts for the stress distribution from moment. For T-connections with in-plane moment M_ip: the moment capacity M_n-ip = P_n x H_b x (1 - beta) / 2, approximately. The interaction is linear: P_r/P_c + M_r/M_c <= 1.0 when K1 governs. For K-connections, AISC restricts moment transfer — K-connections should be designed as pinned unless tested.

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Related pages

• HSS Section Properties — AISC Manual Tables
• HSS Connection Types — General Overview
• Steel Fy & Fu Reference — Yield and Tensile Strength by Grade
• Brace Connection Design — AISC DG29
• Fillet Weld Size Chart — AISC J2.4
• Bolted Connections Calculator
• AISC Steel Construction Tables
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