HSS Truss Connection Worked Example — K-Connection Chord Plastification per AISC 360 Chapter K

Complete step-by-step design of a welded HSS gap K-connection for a Warren truss. HSS 8x8x1/2 chord with HSS 5x5x3/8 braces per AISC 360-22 Chapter K. Chord plastification, punching shear, sidewall yielding, and local brace yielding — all checked with actual numbers.

Problem Statement

PRELIMINARY — NOT FOR CONSTRUCTION. All results are for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.

A Warren truss with 6 ft panel points uses an HSS 8x8x1/2 (ASTM A500 Gr C) top chord and HSS 5x5x3/8 (ASTM A500 Gr C) web members. The connection at a typical interior panel point is a K-connection (gap joint) with one brace in tension and one in compression. All connections are welded.

Chord: HSS 8x8x1/2, ASTM A500 Gr C (Fy = 50 ksi, Fu = 62 ksi) Braces: HSS 5x5x3/8, ASTM A500 Gr C (Fy = 50 ksi, Fu = 62 ksi) Connection type: Welded gap K-connection

Section Properties:

Property Chord (8x8x1/2) Brace (5x5x3/8) Units
B (width) 8.00 5.00 in.
H (height) 8.00 5.00 in.
t (design wall) 0.465 0.349 in.
A 13.2 6.18 in.²
B/t 17.2 14.3 —
H/t 17.2 14.3 —

Design Forces (factored, LRFD):

Step 1: Geometry Classification and Validity Check

AISC 360-22 K2 — Limits of Applicability:

(1) Chord wall slenderness: B/t_c = 8.0/0.465 = 17.2 ≤ 35. OK. (2) Brace wall slenderness: B_b/t_b = 5.0/0.349 = 14.3 ≤ 35. OK. (3) Width ratio: β = B_b/B = 5.0/8.0 = 0.625 ≥ 0.25 and ≤ 1.0. OK. (4) Aspect ratio of brace: H_b/B_b = 5.0/5.0 = 1.0. 0.5 ≤ 1.0 ≤ 2.0. OK.

Gap check:

For a K-connection, the minimum gap is g ≥ t_1 + t_2 = 0.349 + 0.349 = 0.698 in.

Select gap g = 1.5 in. (provides adequate access for welding and satisfies code minimum).

Eccentricity check: For K-connections with gap, the joint eccentricity e must satisfy:

e = [(B/2)/tan(theta_1) × (sin(theta_1) × sin(theta_2)/sin(theta_1 + theta_2))] × [1/(tan(theta_1) + tan(theta_2))]
  - g/2 × [sin(theta_1) × sin(theta_2)/sin(theta_1 + theta_2)]

For equal brace sizes and angles (theta_1 = theta_2 = 45°):

e = (4.0/tan(45)) × (sin(45)×sin(45)/sin(90)) / (2 × tan(45)) - g/2 × (sin(45)×sin(45)/sin(90))
  = 4.0 × (0.5/1.0) / 2 - 0.75 × (0.5/1.0)
  = 2.0/2 - 0.75 × 0.5
  = 1.0 - 0.375
  = 0.625 in.

Eccentricity limit per AISC K1.3: -0.55 ≤ e/H ≤ 0.25.

e/H = 0.625/8.0 = 0.078 < 0.25. **Eccentricity OK.**

Step 2: Chord Plastification — Compression Brace (AISC K2-4)

The governing limit state for K-connections is chord plastification. Available axial strength per AISC Eq. K2-4:

P_n × sin(theta) = Fy × t_c² × [9.8 × β^0.5 × gamma^0.5] × Q_f

Where:

Calculate Q_f:

For chord in compression, compute U = P_r/(Fy × A) + M_r/(Fy × S):

P_r = |P_u_chord| = 180 kips
phi P_n = 0.90 × Fy × A = 0.90 × 50 × 13.2 = 0.90 × 660 = 594 kips

U = (P_r/phi_c) / (phi P_n) simplified: U = |P_r| / (Fy × A) = 180 / 660 = 0.273

Per AISC K2-3b for chord in compression:

Q_f = 1.3 - 0.4 × U/β = 1.3 - 0.4 × 0.273/0.625
    = 1.3 - 0.4 × 0.437
    = 1.3 - 0.175
    = 1.125

But Q_f ≤ 1.0 per the specification for compression chord. Use Q_f = 1.0.

Chord plastification capacity:

P_n × sin(theta) = 50 × 0.465² × [9.8 × 0.625^0.5 × 8.60^0.5] × 1.0

Calculate intermediate terms:

beta^0.5 = sqrt(0.625) = 0.791
gamma^0.5 = sqrt(8.60) = 2.933

P_n × sin(45°) = 50 × 0.2162 × [9.8 × 0.791 × 2.933] × 1.0
               = 10.81 × [9.8 × 2.320]
               = 10.81 × 22.74
               = 245.8 kips

Since sin(45°) = 0.707:

P_n = 245.8 / 0.707 = 347.7 kips (nominal)

Design strength (LRFD, phi = 0.90 for Chapter K connections):

phi P_n = 0.90 × 347.7 = 312.9 kips

Check compression brace: P_u_compression = 78 kips < 312.9 kips. Chord plastification OK. D/C = 0.249.

Step 3: Chord Plastification — Tension Brace Check

For a K-connection, the tension and compression braces interact through the chord face. Per AISC K2.2, the brace force components perpendicular to the chord must satisfy:

|P_1 × sin(theta_1) - P_2 × sin(theta_2)| ≤ phi × P_n × sin(theta)

Where P_1 and P_2 are the brace forces (positive for tension, negative for compression).

Tension brace perpendicular component: 85 × sin(45°) = 85 × 0.707 = 60.1 kips
Compression brace perpendicular component: -78 × sin(45°) = -78 × 0.707 = -55.1 kips

|60.1 - (-55.1)| = |60.1 + 55.1| = 115.2 kips

The available strength per the interaction equation:

phi P_n × sin(theta) = 0.90 × 245.8 = 221.2 kips (but this is per brace direction)

Actually, the K-connection interaction check per K2.2 states that the brace force component normal to the chord, times sin(theta), is checked against the capacity. For the combined action:

sum(P_i × sin(theta_i)) per side ≤ phi × P_n × sin(theta_interaction)

Since this is a balanced K-connection (tension ≈ compression), the net through-chord force is small. The larger individual brace governs. Using the same capacity from Step 2:

Tension brace check: 85 kips < 312.9 kips. Tension brace OK. D/C = 0.272.

Step 4: Punching Shear Check (AISC K2-5)

Punching shear occurs when the brace pulls through the chord wall. The available strength is:

P_n = 0.60 × Fy × t_c × pi × (B_b + t_c) × (1 + sin(theta))/ (2 × sin²(theta))

For square braces, the perimeter is calculated as the effective punching perimeter:

P_n = 0.60 × 50 × 0.465 × [4 × (B_b - 2 × t_b)] / (sqrt(3) × sin(theta))

Using AISC Eq. K2-5 for rectangular HSS:

P_n = 0.60 × Fy × t_c × (2 × B_b/sin(theta) + 2 × B_eop) where B_eop is effective width

B_eop = (10 × t_c) / (B/t_c) × (Fy × t_c) / (Fyb × t_b) × B_b ≤ B_b

But for equal strength material: B_eop = (10 × t_c) / (B/t_c) × (t_c/t_b) × B_b
        = (10 × 0.465) / 17.2 × (0.465/0.349) × 5.0
        = 4.65/17.2 × 1.332 × 5.0
        = 0.270 × 6.66
        = 1.80 in.

The effective perimeter for punching shear:

L_p = 2 × (B_b/sin(theta) + B_eop)
    = 2 × (5.0/sin(45°) + 1.80)
    = 2 × (5.0/0.707 + 1.80)
    = 2 × (7.072 + 1.80)
    = 2 × 8.872
    = 17.74 in.

P_n = 0.60 × 50 × 0.465 × 17.74
    = 0.60 × 50 × 8.250
    = 247.5 kips

Design strength (phi = 0.95 for punching shear per K2.3):

phi P_n = 0.95 × 247.5 = 235.1 kips

Check: 85 kips < 235.1 kips. Punching shear OK. D/C = 0.362.

Step 5: Sidewall Local Yielding (AISC K2-7)

For beta close to 1.0, sidewall yielding can govern. Here beta = 0.625 < 0.85, so sidewall yielding is not expected to govern, but check for completeness.

Sidewall yielding capacity per AISC Eq. K2-7:

P_n × sin(theta) = Fy × t_c × (5k + lb)

Where k = outside corner radius of HSS (k ≈ 1.5 × t_c = 1.5 × 0.465 = 0.698 in.), and lb = bearing length of brace on the sidewall.

For the inclined brace, the projection of brace contact on the sidewall:

lb = B_b / sin(theta) = 5.0 / sin(45°) = 5.0 / 0.707 = 7.07 in.

P_n × sin(45°) = 50 × 0.465 × (5 × 0.698 + 7.07)
               = 23.25 × (3.49 + 7.07)
               = 23.25 × 10.56
               = 245.5 kips

P_n = 245.5 / 0.707 = 347.2 kips

Design strength (phi = 1.00 for sidewall local yielding):

phi P_n = 1.00 × 347.2 = 347.2 kips > 85 kips. **Sidewall yielding OK.**

Step 6: Local Yielding of Brace (AISC K2-9)

The brace itself must resist local yielding due to non-uniform stress distribution at the connection. The effective width for the brace:

B_e = (10 × t_c) / (B/t_c) × (Fy × t_c) / (Fyb × t_b) × B_b ≤ B_b

For equal strength materials:
B_e = (10 × 0.465) / 17.2 × (50 × 0.465)/(50 × 0.349) × 5.0
    = 4.65/17.2 × 1.332 × 5.0
    = 0.270 × 6.66
    = 1.80 in. (same as B_eop from punching shear check)

Available brace local yielding:

P_n = Fyb × t_b × [2 × (B_e - 2 × t_b) + 2 × (B_b - 2 × t_b)]
    = 50 × 0.349 × [2 × (1.80 - 2 × 0.349) + 2 × (5.0 - 2 × 0.349)]
    = 17.45 × [2 × (1.80 - 0.698) + 2 × (5.0 - 0.698)]
    = 17.45 × [2 × 1.102 + 2 × 4.302]
    = 17.45 × [2.204 + 8.604]
    = 17.45 × 10.808
    = 188.6 kips

Design strength (phi = 0.95 for brace local yielding):

phi P_n = 0.95 × 188.6 = 179.2 kips > 85 kips. **Brace local yielding OK.**

Step 7: Chord Shear in Gap Region (AISC K2-10)

The chord must resist the vertical component of brace forces through shear in the gap region:

V_gap = |P_1 × sin(theta_1) + P_2 × sin(theta_2)|
      = |60.1 + (-55.1)| = |5.0| = 5.0 kips

The gap shear is very small because the K-connection is nearly balanced.

Chord shear capacity per AISC G4 for rectangular HSS:

A_w = 2 × H × t_c = 2 × 8.0 × 0.465 = 7.44 in.²
V_n = 0.60 × Fy × A_w × C_v

For HSS with h/t = (H - 3t)/t ≈ (8.0 - 1.395)/0.465 = 14.2 < 1.10 × sqrt(kv × E/Fy), C_v = 1.0:

V_n = 0.60 × 50 × 7.44 × 1.0 = 223.2 kips
phi V_n = 0.90 × 223.2 = 200.9 kips

Check gap shear: V_gap = 5.0 kips << 200.9 kips. Chord shear OK. D/C = 0.025.

Step 8: Weld Sizing

The brace-to-chord connection is a welded joint. The weld must develop the brace capacity or the connection capacity, whichever is less. For economy, size the weld for the actual brace force P_u with an appropriate safety factor.

Brace perimeter (for HSS 5x5): approximately 4 × 5.0 = 20 in.

Required weld strength per inch:

f_weld = P_u_brace / perimeter = 85 / 20 = 4.25 kip/in.

For fillet weld, use E70XX electrodes. Try 1/4 in. fillet (D = 4):

phi r_n = 1.392 × D = 1.392 × 4 = 5.57 kip/in. (theta = 50° for diagonal brace;
        directionality factor (1.0 + 0.5 × sin^1.5(50°)) = 1.0 + 0.5 × 0.694^1.5
        = 1.0 + 0.5 × 0.578 = 1.0 + 0.289 = 1.289

phi r_n_adjusted = 5.57 × 1.289 = 7.18 kip/in.)

Check: 7.18 kip/in. > 4.25 kip/in. 1/4 in. fillet weld OK.

Alternatively, use 3/16 in. fillet (D = 3) for economy:

phi r_n = 1.392 × 3 × 1.289 = 4.176 × 1.289 = 5.38 kip/in. > 4.25 kip/in. **OK.**

Use 3/16 in. fillet weld all around, E70XX.

Step 9: Summary — Pass/Fail

Limit State Reference Demand Capacity D/C Ratio Status
Chord wall slenderness (B/t_c) AISC K2.3a 17.2 35 — PASS
Brace wall slenderness (B/t_b) AISC K2.3a 14.3 35 — PASS
Width ratio (beta) AISC K2.3a 0.625 0.25–1.0 — PASS
Chord plastification (compression brace) AISC K2-4 78 k 312.9 k 0.249 PASS
Chord plastification (tension brace) AISC K2-4 85 k 312.9 k 0.272 PASS
Punching shear AISC K2-5 85 k 235.1 k 0.362 PASS
Sidewall local yielding AISC K2-7 85 k 347.2 k 0.245 PASS
Brace local yielding AISC K2-9 85 k 179.2 k 0.474 PASS
Chord shear (gap region) AISC G4 5.0 k 200.9 k 0.025 PASS
Joint eccentricity (e/H) AISC K1.3 0.078 0.25 — PASS
Weld capacity (3/16 in.) AISC J2.4 4.25 k/in. 5.38 k/in. 0.790 PASS

All checks pass. The governing limit state is the weld capacity at D/C = 0.790. Brace local yielding is the governing connection limit state at D/C = 0.474. The connection has significant reserve capacity.

Final Connection Details:

Step 10: Design Notes

The chord plastification capacity is high (312.9 kips vs 85 kips demand) because the HSS 8x8x1/2 chord has a relatively thick wall (t = 0.465 in.) relative to the 8 in. face width, giving a low gamma = 8.60. Chord plastification capacity scales with t_c², so even small increases in chord wall thickness yield large capacity increases.

For trusses where brace forces exceed chord plastification capacity, the designer has several options:

  1. Increase chord wall thickness (most effective — capacity ∝ t²)
  2. Increase chord width (reduces beta, but B/t must stay under 35)
  3. Use overlap K-connection instead of gap (force transfers between braces directly)
  4. Add stiffening plates to the chord face at the connection
  5. Reduce brace angle (reduces the perpendicular force component)

Related Calculators

Design HSS connections interactively with the HSS Connections Calculator. Check member capacities with the Beam Capacity Calculator and Column Capacity Calculator. For general HSS connection design guidance, see the HSS Connection Reference page.