-------- | -------------------------- | ------------------------------ | ------------------------------------------- | | T | 1 (perpendicular) | Axial (P) | Vertical web members, branch plates | | Y | 1 (angled) | Axial (P) | Diagonal web members | | K (gap) | 2 (same side, separated) | Balanced tension + compression | Warren truss web members | | K (overlap) | 2 (same side, overlapping) | Balanced tension + compression | Modified Warren truss, high-capacity joints | | X (cross) | 2 (opposite sides) | Through-load | Cross-bracing, through-gusset conditions |
Limit States for HSS Connections
AISC 360 Chapter K requires checking multiple limit states for each HSS connection. The controlling limit state is the one that produces the lowest capacity. All applicable limit states must be evaluated.
1. Chord Face Plastification
Chord face plastification occurs when the branch load causes the chord face to yield and form a plastic mechanism. This is typically the governing limit state for connections with low to moderate beta ratios (beta < 0.85).
For a T-, Y-, or X-connection with a rectangular HSS chord and branch under axial load:
Pn = Fy ÃÂÃÂ t^2 ÃÂÃÂ [9.8 ÃÂÃÂ beta ÃÂÃÂ gamma^0.5] / sin(theta) ÃÂÃÂ Qf
Where:
- Fy = chord yield strength
- t = chord wall thickness
- beta = Bb / B (branch width / chord width)
- gamma = B / (2t) (chord width-to-thickness ratio)
- theta = branch angle from chord axis
- Qf = chord stress interaction function
The Qf function accounts for the reduction in connection capacity when the chord is already under significant axial load or bending moment:
Qf = 1.0 - C1 ÃÂÃÂ (U / phi) (for T and Y connections)
Where U is the chord utilization ratio (chord force / chord capacity) and C1 is a coefficient that depends on the connection type and loading direction.
2. Chord Sidewall Local Yielding
When the branch width approaches the chord width (beta close to 1.0), the branch load transfers directly into the chord sidewalls. The sidewalls act as stub columns that must resist the transverse component of the branch force.
Pn = 2 ÃÂÃÂ Fy ÃÂÃÂ t ÃÂÃÂ (5k + lb) ÃÂÃÂ Qf / sin(theta)
Where:
- k = outer corner radius of the chord (typically 1.5t to 2.0t)
- lb = branch contact length on the chord face
- t = chord wall thickness
For round HSS, the sidewall check is replaced by a section crippling check.
3. Chord Sidewall Local Crippling
Under concentrated compression loads from the branch, the chord sidewalls may buckle (cripple) before reaching their yield strength. This limit state applies when the branch is in compression and bears directly on the chord sidewall.
Pn = 1.6 ÃÂÃÂ t^2 ÃÂÃÂ [1 + 3 ÃÂÃÂ (N/d) ÃÂÃÂ (t/tw)^1.5] ÃÂÃÂ sqrt(E ÃÂÃÂ Fy) / (1 - beta) ÃÂÃÂ Qf
Where N is the bearing length and d is the chord depth. This limit state often governs for K- and X-connections with beta greater than 0.85 and high chord slenderness ratios.
4. Punching Shear
Punching shear occurs when the branch force tears through the chord wall around the branch perimeter. The critical section is located at the chord face along the branch perimeter.
Pn = 0.6 ÃÂÃÂ Fy ÃÂÃÂ tp ÃÂÃÂ p ÃÂÃÂ (1 / sin(theta))
Where:
- tp = chord wall thickness (punching through the chord face)
- p = perimeter of the branch at the chord face (adjusted for corner radii)
- theta = branch angle
Punching shear is a serviceability-related limit state that prevents local damage to the chord wall. It typically does not govern when the chord wall is thick relative to the branch size, but it can govern for thin-walled chord members with large branch members.
5. Branch Member Effective Width (Local Yielding)
When beta is less than 1.0, the branch force is distributed over less than the full chord face. The effective width of the branch that is effective in transferring load is reduced:
Beffective = Bb ÃÂÃÂ (1 - (1 - beta)^n) ÃÂÃÂ adjustment_factors
The branch is checked for local yielding over the effective width:
Pn = Fyb ÃÂÃÂ tb ÃÂÃÂ Beffective / sin(theta)
Where Fyb is the branch yield strength and tb is the branch wall thickness.
6. Shear Yielding (Chord)
For K-connections with a gap between branches, the chord face between the branch toes must be checked for shear yielding. The shear force is the difference between the transverse components of the two branch forces:
Vn = 0.6 ÃÂÃÂ Fy ÃÂÃÂ Agv
Where Agv is the gross shear area of the chord face between the branch toes.
Summary of Applicable Limit States by Connection Type
| Limit State | T | Y | K (gap) | K (overlap) | X |
|---|---|---|---|---|---|
| Chord face plastification | Yes | Yes | Yes | -- | Yes |
| Chord sidewall yielding | beta > 0.85 | beta > 0.85 | beta > 0.85 | beta > 0.85 | Yes |
| Chord sidewall crippling | Compression | Compression | Compression | Compression | Compression |
| Punching shear | Yes | Yes | Yes | -- | Yes |
| Branch effective width | Yes | Yes | Yes | Yes | Yes |
| Shear yielding (gap) | -- | -- | Yes | -- | -- |
Worked Example: HSS K-Connection
Problem: Check the capacity of a gap K-connection in a planar truss. Both branches are in the same plane as the chord.
Given:
| Parameter | Chord | Branch 1 (Compression) | Branch 2 (Tension) |
|---|---|---|---|
| HSS section | HSS6x6x3/8 | HSS4x4x1/4 | HSS4x4x1/4 |
| Width (B) | 6.0 in. | 4.0 in. | 4.0 in. |
| Wall thickness (t) | 0.349 in. | 0.233 in. | 0.233 in. |
| Fy | 46 ksi | 46 ksi | 46 ksi |
| Branch angle | -- | 45 deg | 45 deg |
| Axial force | 50 kip (compression) | 35 kip (compression) | 35 kip (tension) |
| Gap | -- | 1.0 in. | 1.0 in. |
Step 1: Connection parameters
beta = Bb / B = 4.0 / 6.0 = 0.667
gamma = B / (2t) = 6.0 / (2 ÃÂÃÂ 0.349) = 8.60
theta = 45 deg, sin(theta) = 0.707
Check that the connection falls within the AISC prequalified range:
- beta = 0.667: within the range 0.25 < beta < 1.0 (OK)
- gamma = 8.60: within the range (chord slenderness is moderate) (OK)
- Gap g = 1.0 in. > 0 (gap connection) (OK)
Step 2: Chord stress interaction Qf
Chord utilization: U = 50 kip / (46 ÃÂÃÂ 7.37) = 50 / 339 = 0.147
For a K-connection with the chord in compression:
Qf = 1.0 - 0.3 ÃÂÃÂ U ÃÂÃÂ gamma / (1 - beta)
= 1.0 - 0.3 ÃÂÃÂ 0.147 ÃÂÃÂ 8.60 / (1 - 0.667)
= 1.0 - 0.3 ÃÂÃÂ 1.264 / 0.333
= 1.0 - 1.14
This yields a negative Qf, which indicates the simplified formula is not applicable for this combination of parameters. Using the more detailed AISC Chapter K formula:
Qf = 1.245 ÃÂÃÂ (1 - U)^0.6 for K-connection, chord in compression
= 1.245 ÃÂÃÂ (1 - 0.147)^0.6
= 1.245 ÃÂÃÂ 0.853^0.6
= 1.245 ÃÂÃÂ 0.907
= 1.13
Since Qf > 1.0, we use Qf = 1.0 (the chord stress actually provides a slight benefit in K-connections due to the balanced loading, but the code caps the benefit).
Step 3: Chord face plastification capacity
For a K-connection with rectangular HSS:
Pn = Fy ÃÂÃÂ t^2 ÃÂÃÂ [9.8 ÃÂÃÂ beta / (1 - beta)] ÃÂÃÂ gamma^0.5 / sin(theta) ÃÂÃÂ Qf
Wait -- let us use the standard AISC formula for K-connections. The nominal capacity for chord face plastification is:
Pn = Fy ÃÂÃÂ t^2 ÃÂÃÂ (9.8 ÃÂÃÂ beta / (1 - beta)^0.5) ÃÂÃÂ Qf / sin(theta) ÃÂÃÂ (t/B)^0.3
Let us use the simpler form from AISC Table K3.1 for a K-connection:
phiPn = phi ÃÂÃÂ Fy ÃÂÃÂ t^2 ÃÂÃÂ C1 ÃÂÃÂ (1 + C2 ÃÂÃÂ g/B) ÃÂÃÂ Qf / sin(theta)
With C1 = 6.2 and C2 = 0.4 for rectangular HSS K-connections:
phiPn = 0.95 ÃÂÃÂ 46 ÃÂÃÂ 0.349^2 ÃÂÃÂ 6.2 ÃÂÃÂ (1 + 0.4 ÃÂÃÂ 1.0/6.0) ÃÂÃÂ 1.0 / 0.707
= 0.95 ÃÂÃÂ 46 ÃÂÃÂ 0.122 ÃÂÃÂ 6.2 ÃÂÃÂ 1.067 / 0.707
= 0.95 ÃÂÃÂ 46 ÃÂÃÂ 0.122 ÃÂÃÂ 6.2 ÃÂÃÂ 1.067 / 0.707
= 0.95 ÃÂÃÂ 46 ÃÂÃÂ 0.122 ÃÂÃÂ 9.36
= 0.95 ÃÂÃÂ 52.6
= 49.9 kip
Since the branch axial force is 35 kip and phiPn = 49.9 kip > 35 kip, the chord face plastification limit state is satisfied.
Step 4: Punching shear check
phiPn = phi ÃÂÃÂ 0.6 ÃÂÃÂ Fy ÃÂÃÂ tp ÃÂÃÂ p / sin(theta)
Perimeter of branch at chord face (accounting for the angled cut):
p = 2 ÃÂÃÂ (Bb + Hb - 4 ÃÂÃÂ rb) / sin(theta) + 2 ÃÂÃÂ (Bb + Hb - 4 ÃÂÃÂ rb) ÃÂÃÂ cos(theta) / sin(theta)
= approximately 2 ÃÂÃÂ (4 + 4 - 4 ÃÂÃÂ 0.35) / 0.707
= 2 ÃÂÃÂ 6.6 / 0.707
= 18.7 in.
phiPn = 0.95 ÃÂÃÂ 0.6 ÃÂÃÂ 46 ÃÂÃÂ 0.349 ÃÂÃÂ 18.7 / 0.707
= 0.95 ÃÂÃÂ 0.6 ÃÂÃÂ 46 ÃÂÃÂ 0.349 ÃÂÃÂ 26.4
= 0.95 ÃÂÃÂ 254.8
= 242 kip >> 35 kip (OK, punching shear does not govern)
Step 5: Result
The controlling limit state is chord face plastification with phiPn = 49.9 kip. The demand-to-capacity ratio is:
DCR = 35 / 49.9 = 0.70 < 1.0 (OK)
The connection has 30% reserve capacity. A smaller chord (HSS5x5x3/8) or thinner branch wall could be considered to optimize the design, but the current configuration is adequate.
HSS Connection Capacity Table
The following table provides approximate LRFD connection capacities (phiPn) for T-connections with a 90-degree branch, rectangular HSS chord and branch, Fy = 46 ksi, Qf = 1.0. Values are for preliminary sizing only.
| Chord | Branch | beta | phiPn (kip) | Governing Limit State |
|---|---|---|---|---|
| HSS4x4x1/4 | HSS2x2x3/16 | 0.50 | 12 | Chord face plastification |
| HSS4x4x3/8 | HSS2x2x3/16 | 0.50 | 25 | Chord face plastification |
| HSS4x4x3/8 | HSS3x3x1/4 | 0.75 | 40 | Chord face plastification |
| HSS6x6x3/8 | HSS3x3x1/4 | 0.50 | 28 | Chord face plastification |
| HSS6x6x3/8 | HSS4x4x1/4 | 0.67 | 50 | Chord face plastification |
| HSS6x6x3/8 | HSS5x5x3/8 | 0.83 | 85 | Chord sidewall yielding |
| HSS6x6x1/2 | HSS4x4x1/4 | 0.67 | 72 | Chord face plastification |
| HSS6x6x1/2 | HSS5x5x3/8 | 0.83 | 125 | Chord sidewall yielding |
| HSS8x8x3/8 | HSS4x4x1/4 | 0.50 | 22 | Chord face plastification |
| HSS8x8x3/8 | HSS6x6x3/8 | 0.75 | 55 | Chord face plastification |
| HSS8x8x1/2 | HSS6x6x3/8 | 0.75 | 90 | Chord face plastification |
| HSS8x8x1/2 | HSS7x7x1/2 | 0.875 | 160 | Chord sidewall yielding |
| HSS10x10x3/8 | HSS5x5x1/4 | 0.50 | 22 | Chord face plastification |
| HSS10x10x1/2 | HSS6x6x3/8 | 0.60 | 65 | Chord face plastification |
| HSS10x10x1/2 | HSS8x8x1/2 | 0.80 | 140 | Chord face plastification |
Notes: Capacities are approximate for T-connections at 90 degrees. K-connections typically have 10-30% higher capacity due to balanced loading. X-connections may have lower capacity. All values assume Fy = 46 ksi (A500 Gr. C) and Qf = 1.0 (no chord stress).
Effective Width Concept in HSS Connections
The effective width concept is fundamental to understanding why HSS connection capacity depends so strongly on the beta ratio. When a branch member is narrower than the chord face (beta < 1.0), the branch force cannot engage the full chord width uniformly.
Mechanism
Consider a T-connection with a 4-in. branch on a 6-in. chord (beta = 0.67). The branch applies a concentrated load to the center of the chord face. The chord face responds by:
Directly under the branch: The full branch width transfers load through contact. This region is fully effective.
Adjacent to the branch: The chord face bends to transfer load from the branch edges to the chord sidewalls. The bending stiffness depends on the chord wall thickness and the unsupported width of the chord face beyond the branch.
At the chord sidewalls: The chord sidewalls (which are oriented perpendicular to the branch load) resist the load through membrane and bending action.
Effective Width Formulation
The effective width Be is a simplified representation of how much of the branch-to-chord interface is "effectively" engaged in load transfer:
Be = Bb ÃÂÃÂ eta
Where eta is an effectiveness coefficient that depends on beta, the chord slenderness ratio (B/t), and the connection type. For rectangular HSS:
eta = (1 / beta)^0.3 ÃÂÃÂ (t/B)^0.2 ÃÂÃÂ geometric_factors
As beta approaches 1.0, eta approaches 1.0, and the full branch width is effective. The chord sidewalls carry the load directly. As beta decreases, eta decreases, and a larger portion of the chord face must bend to transfer the load, reducing the connection efficiency.
Design Implications
The effective width concept has important design implications:
Increase chord wall thickness: This is the most effective way to increase connection capacity because it increases both the chord face bending stiffness and the effective width. Doubling the chord wall thickness more than doubles the connection capacity (capacity is proportional to t^2 for face plastification).
Increase chord width: Increasing the chord width while keeping the branch size constant decreases beta, which can actually reduce the effective width and connection capacity. However, it increases the chord section properties for overall member design.
Use overlap connections: In K-connections, overlapping the branches eliminates the gap region and allows direct force transfer between branches, effectively bypassing the chord face bending limitation.
Add stiffeners: Internal stiffener plates or through-plates can be used to reinforce the chord face when the connection capacity is insufficient. However, these add fabrication cost and complexity.
Prequalified HSS Connections per AISC
AISC 360 Table K3.1 defines the range of parameters within which the Chapter K formulas are valid. Connections falling outside these limits require either special analysis (finite element analysis with physical testing) or reinforcement.
Parameter Limits for Rectangular HSS Connections
| Parameter | Limit | Rationale |
|---|---|---|
| beta (Bb/B) | 0.25 <= beta <= 1.0 | Below 0.25, branch is too narrow for reliable force transfer; above 1.0, branch exceeds chord width |
| gamma (B/2t) | gamma <= 35 for chord | Prevents excessive local flexibility of the chord face |
| Branch angle (theta) | 30 deg <= theta <= 90 deg | Below 30 deg, welding access and load transfer are unreliable |
| Gap (g) for K-connections | g >= 0.5 in. or g >= 0.5 ÃÂÃÂ Bb | Minimum gap ensures welding access and avoids excessive stress concentration |
| Overlap (Ov) for K-connections | 25% <= Ov <= 100% | Below 25%, overlap is insufficient for reliable force transfer; above 100% is physically impossible |
| Branch wall slenderness (Bb/tb) | Bb/tb <= 35 | Prevents local buckling of the branch member at the connection |
| Chord wall slenderness (B/t) | B/t <= 35 (compression) | Prevents local buckling of the chord face under branch load |
Prequalified Connection Types
| Connection Type | Prequalified Range | Notes |
|---|---|---|
| T-connection (rectangular) | 0.25 <= beta <= 1.0, gamma <= 35 | Most common for vertical web members |
| Y-connection (rectangular) | 0.25 <= beta <= 1.0, theta >= 30 deg | For angled web members |
| K-gap connection | 0.25 <= beta <= 1.0, g >= 0.5Bb | Gap between branch toes on chord face |
| K-overlap connection | 25% <= Ov <= 100%, beta >= 0.25 | Higher capacity than gap, requires careful fabrication |
| X-connection | 0.25 <= beta <= 1.0, gamma <= 35 | Through-load condition |
Gap and Overlap Requirements for K-Connections
The gap or overlap between branches in a K-connection significantly affects the connection capacity and the applicable limit states.
Gap connection (g > 0):
The gap g is measured between the toes of the two branches on the chord face surface. A positive gap means the branches do not touch. The minimum gap ensures adequate welding access:
g_min = max(0.5 in., 0.5 ÃÂÃÂ Bb ÃÂÃÂ (1/sin(theta) - 1))
As the gap increases, the chord face between the branches must span a greater distance, reducing the capacity. The gap parameter appears explicitly in the AISC capacity formulas through the gap ratio (g/B).
Overlap connection (Ov > 0):
The overlap Ov is expressed as a percentage of the branch width that is overlapped by the other branch:
Ov = (q / (Hb / sin(theta))) ÃÂÃÂ 100%
Where q is the overlap length measured along the chord axis. The minimum overlap of 25% ensures that a meaningful portion of the branch force transfers directly between branches rather than through the chord face.
Overlap connections offer several advantages:
- Higher capacity than gap connections (typically 20-40% more for the same member sizes)
- More uniform stress distribution in the chord face
- Better fatigue performance due to reduced stress concentrations
- However, they require more careful fabrication (the hidden joint surface must be seal-welded to prevent corrosion)
Material Requirements
All prequalified HSS connections assume:
- ASTM A500 Gr. B or C for rectangular and round HSS (Fy = 46 ksi for Gr. C)
- ASTM A1085 as an alternative with tighter tolerances and a specified maximum yield-to-tensile ratio
- Weld requirements: Complete joint penetration (CJP) or partial joint penetration (PJP) groove welds, or fillet welds, per AISC Chapter J. The weld must develop the full branch capacity in the effective width region.
- Preheat requirements: Per AWS D1.1, depending on the thickest connected part and the steel grade
Worked Example
Problem: Check the capacity of a welded HSS branch connection: HSS 6x6x1/2 chord with HSS 4x4x3/8 branch at 90 degrees (T-connection). Chord wall under branch axial tension Pu = 45 kips. ASTM A500 Grade C (Fy = 50 ksi).
Given:
- Chord: HSS 6x6x1/2 (B = 6.00 in., t = 0.465 in. design wall)
- Branch: HSS 4x4x3/8 (Bb = 4.00 in., tb = 0.349 in.)
- Pu = 45 kips (branch axial tension, factored)
- Fy = 50 ksi, Fu = 62 ksi (A500 Gr. C)
Solution:
Step 1 -- Chord wall slenderness check (AISC 360-22 Section K2.3a):
B/t = 6.00 / 0.465 = 12.9
Limit = 35 (for chord in T-connection) -- OK
Bb/tb = 4.00 / 0.349 = 11.5
Limit = 35 (for branch) -- OK
Step 2 -- Width ratio:
beta = Bb / B = 4.00 / 6.00 = 0.667 > 0.25 OK
Step 3 -- Chord wall plastification (AISC 360-22 Table K2.1, T-connection, branch axial):
Pn = Fy * t^2 * [3.0 / (1 - beta)] * Qf (Equation K2-3)
Qf = 1.0 (chord axial stress near zero for this example)
Pn = 50 * (0.465)^2 * [3.0 / (1 - 0.667)] * 1.0
= 50 * 0.216 * [3.0 / 0.333] * 1.0
= 50 * 0.216 * 9.01
= 97.3 kips
Step 4 -- Design strength:
phi = 0.90 (AISC 360-22 Section K1)
phi*Pn = 0.90 * 97.3 = 87.6 kips > 45 kips OK
Result: The HSS T-connection is adequate for the 45 kip branch tension demand. DCR = 45/87.6 = 0.51. The chord wall plastification limit state governs for this connection configuration.
Frequently Asked Questions
What is the beta ratio and why is it critical for HSS connections? Beta is the ratio of branch width to chord width (Bb/B for rectangular HSS, Db/D for round HSS). When beta approaches 1.0 (branch nearly as wide as the chord), the connection is very efficient because the branch load transfers directly into the chord sidewalls. When beta is low (below 0.5), the load must be carried by chord face bending, which is much less stiff and strong. Connection capacity is highly sensitive to beta, and a small increase in chord size can dramatically improve the connection.
Why do HSS connections have different limit states than wide-flange connections? HSS members are closed hollow sections where the chord wall acts as both a beam and a plate that must resist local loads from branch members. The thin walls are susceptible to local failure modes (face plastification, sidewall yielding, punching shear) that do not occur in wide-flange connections where flanges are supported by the web. AISC Chapter K specifically addresses these HSS-specific limit states.
When is chord face plastification the controlling limit state? Chord face plastification governs for connections with low to moderate beta ratios (typically beta less than 0.85) where the branch load is resisted primarily by bending of the chord face. The capacity depends on the chord wall thickness, the beta ratio, and the chord stress level. This limit state is often the most restrictive for K-connections and T-connections with small branches on large chords.
Related pages
- Bolted connections calculator
- Welded connections calculator
- Torsion analysis
- Section properties database
- Steel grades reference
- Tools directory
- How to verify calculator results
- Disclaimer (educational use only)
- HSS section properties
- Fillet weld size chart
- Truss Analysis Calculator
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