HSS Connections Calculator

HSS truss and moment connection design per AISC Design Guide 24. Chord sidewall, punching shear, and local yielding limit state checks. Educational use only.

This page documents the scope, inputs, outputs, and computational approach of the HSS Connections Calculator on steelcalculator.app. The interactive calculator runs in your browser; this documentation ensures the page is useful even without JavaScript.

What this tool is for

• Checking limit states for welded HSS-to-HSS connections: chord face plastification, chord sidewall failure, punching shear, and branch failure.
• Screening connection adequacy for T, Y, X, and K-type HSS truss joints.
• Understanding the sensitivity of HSS connection capacity to beta ratio (branch/chord width ratio) and chord stress ratio.

What this tool is not for

• It does not design the welds between branch and chord members.
• It does not handle multiplanar connections, moment connections, or through-plate details.
• It does not check fatigue for cyclically loaded HSS joints.

Key concepts this page covers

• chord face plastification
• chord sidewall local yielding and crippling
• punching shear of the chord wall
• branch member effective width
• beta ratio and its effect on connection capacity

Inputs and outputs

Typical inputs: chord and branch HSS sizes (width, depth, wall thickness), connection type (T, Y, X, K), branch angle, chord axial stress, and branch axial force.

Typical outputs: capacity for each limit state, controlling limit state, demand-to-capacity ratio, and whether the connection is adequate without reinforcement.

Computation approach

The calculator evaluates the applicable limit states from AISC 360 Chapter K and AISC Design Guide 24 for the specified connection geometry. Each limit state produces a nominal capacity that is factored by phi. The controlling limit state is the minimum capacity. The chord stress interaction function Qf reduces the connection capacity when the chord is under significant axial load or bending.

HSS Connection Types per AISC Chapter K

AISC 360 Chapter K (also addressed in AISC Design Guide 24) classifies HSS-to-HSS connections by their geometry and load transfer mechanism. The connection type determines which limit states are checked and which capacity formulas apply.

T-Connection

A T-connection (also called a 90-degree branch plate connection) has a single branch member welded to the chord at a 90-degree angle. The branch transmits axial load (tension or compression) perpendicular to the chord axis. This is the simplest HSS connection geometry.

Branch angle: 90 degrees
Load: axial (tension or compression) perpendicular to chord
Limit states: chord face plastification, punching shear, branch effective width

Y-Connection

A Y-connection has a single branch member welded to the chord at an acute angle (less than 90 degrees). The branch load has components perpendicular and parallel to the chord, but only the perpendicular component is typically considered for the local connection check.

Branch angle: typically 30 to 75 degrees
Load: axial at angle to chord axis
Limit states: same as T-connection, with cos(theta) projection factor

K-Connection (Gap and Overlap)

A K-connection has two branch members on the same side of the chord, with one branch in tension and the other in compression. The balanced forces create an internal load path through the chord face. There are two subtypes:

• Gap K-connection: A gap exists between the toes of the two branch members on the chord face. The chord face between the branches must resist the net transverse force.
• Overlap K-connection: One branch partially overlaps the other on the chord face. The overlap transfers force directly between branches, reducing the demand on the chord face. Overlap connections generally have higher capacity than gap connections for the same member sizes.

Branch angle: 30 to 60 degrees typically
Load: one branch in tension, one in compression (balanced)
Key parameter: gap g (positive for gap, negative for overlap)

X-Connection (Cross)

An X-connection (cross-connection) has branch members on opposite sides of the chord, both loaded in the same direction (both compression or both tension). The load must pass completely through the chord, engaging both the chord face and sidewalls.

Branch angle: typically 90 degrees (cross)
Load: axial through chord (both sides)
Limit states: chord face plastification, chord sidewall crippling, punching shear

Connection Type Summary

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 × t^2 × [9.8 × beta × gamma^0.5] / sin(theta) × 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 × (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 × Fy × t × (5k + lb) × 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 × t^2 × [1 + 3 × (N/d) × (t/tw)^1.5] × sqrt(E × Fy) / (1 - beta) × 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 × Fy × tp × p × (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 × (1 - (1 - beta)^n) × adjustment_factors

The branch is checked for local yielding over the effective width:

Pn = Fyb × tb × 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 × Fy × Agv

Where Agv is the gross shear area of the chord face between the branch toes.

Summary of Applicable Limit States by Connection Type

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:

Step 1: Connection parameters

beta = Bb / B = 4.0 / 6.0 = 0.667
gamma = B / (2t) = 6.0 / (2 × 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 × 7.37) = 50 / 339 = 0.147

For a K-connection with the chord in compression:

Qf = 1.0 - 0.3 × U × gamma / (1 - beta)
   = 1.0 - 0.3 × 0.147 × 8.60 / (1 - 0.667)
   = 1.0 - 0.3 × 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 × (1 - U)^0.6  for K-connection, chord in compression
   = 1.245 × (1 - 0.147)^0.6
   = 1.245 × 0.853^0.6
   = 1.245 × 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 × t^2 × [9.8 × beta / (1 - beta)] × gamma^0.5 / sin(theta) × Qf

Wait -- let us use the standard AISC formula for K-connections. The nominal capacity for chord face plastification is:

Pn = Fy × t^2 × (9.8 × beta / (1 - beta)^0.5) × Qf / sin(theta) × (t/B)^0.3

Let us use the simpler form from AISC Table K3.1 for a K-connection:

phiPn = phi × Fy × t^2 × C1 × (1 + C2 × g/B) × Qf / sin(theta)

With C1 = 6.2 and C2 = 0.4 for rectangular HSS K-connections:

phiPn = 0.95 × 46 × 0.349^2 × 6.2 × (1 + 0.4 × 1.0/6.0) × 1.0 / 0.707
      = 0.95 × 46 × 0.122 × 6.2 × 1.067 / 0.707
      = 0.95 × 46 × 0.122 × 6.2 × 1.067 / 0.707
      = 0.95 × 46 × 0.122 × 9.36
      = 0.95 × 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 × 0.6 × Fy × tp × p / sin(theta)

Perimeter of branch at chord face (accounting for the angled cut):

p = 2 × (Bb + Hb - 4 × rb) / sin(theta) + 2 × (Bb + Hb - 4 × rb) × cos(theta) / sin(theta)
  = approximately 2 × (4 + 4 - 4 × 0.35) / 0.707
  = 2 × 6.6 / 0.707
  = 18.7 in.

phiPn = 0.95 × 0.6 × 46 × 0.349 × 18.7 / 0.707
      = 0.95 × 0.6 × 46 × 0.349 × 26.4
      = 0.95 × 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.

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:

1. Directly under the branch: The full branch width transfers load through contact. This region is fully effective.

2. 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.

3. 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 × 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 × (t/B)^0.2 × 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

Prequalified Connection Types

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 × Bb × (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))) × 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

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

Disclaimer (educational use only)

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