# STI Limit State Table Derivations

The derivations shown below are intended to demonstrate the equivalence of limit state equations between the current AISC 360-22 Specification and 16th Ed. Manual Part 9 with the prescriptive tables that were previously shown in AISC 360-10 Specification Chapter K.

The following limit state derivations are included:

- 1)
**Limit State = Local Yielding of HSS Chord Sidewalls**- STI Limit State Table = Moment Table
- Connection = Rectangular HSS-to-Rectangular HSS under In-Plane Moment

- 2)
**Limit State = Local Yielding of Branches Due to Uneven Load Distribution**- STI Limit State Table = Moment Table
- Connection = Rectangular HSS-to-Rectangular HSS under In-Plane Moment

- 3)
**Limit State = Local Yielding of HSS Chord Sidewalls**- STI Limit State Table = Moment Table
- Connection = Rectangular HSS-to-Rectangular HSS under Out-of-Plane Moment

- 4)
**Limit State = Local Yielding of Branches Due to Uneven Load Distribution**- STI Limit State Table = Moment Table
- Connection = Rectangular HSS-to-Rectangular HSS under Out-of-Plane Moment

## 1) Limit State: Local Yielding of HSS Chord Sidewalls | In-Plane Moment for Rectangular HSS (Refer to Moment Limit State Table M2)

Refer to AISC 360-10 Eqn K3-7:

M_{n }= 0.5F*_{y }t (H_{b} + 5t)^{2} [AISC 360-10 Eqn K3-7]

Refer to AISC 360-22 Specification Section J10.2 to derive available strength to match AISC 360-10 Eqn K3-7, based on local yielding of rectangular HSS Sidewalls under in-plane moment.

R_{n }= F_{yw}t_{w}(5k + l_{b})^{ }For l_{end} > H [AISC 360-22 Section J10.2]

l_{b} = H_{b}

t_{w} = t_{des} of HSS chord = t

k = t Consider a conservative 1:1 slope for dispersion in lieu of k = 1.5t.

F*_{y} = F_{y} for HSS T-Connections

= 0.8F_{y }for HSS Cross-Connections

2 Sidewalls

Φ = 1.00, Ω = 1.50

Per Cidect Design Guide 3, pp. 62, moment derived from 2 stress blocks can be represented as:

## 2) Limit State: Local Yielding of Branches Due to Uneven Load Distribution | In-Plane Moment Conn for Rectangular HSS (Refer to Moment Limit State Table M2)

Refer to AISC 360-22 Specification Section F7 to derive available strength to match AISC 360-10 Eqn K3-8, based on local yielding of rectangular HSS branches due to uneven load distribution under in-plane moment.

M_{n} = F_{y}Z_{eff } [AISC 360-22 Eqn F7-1]

(The approximation ignores the HSS thickness in the last term only which results in a conservative value that underestimates Z_{eff})

Since the branch member is primarily an axial member rather than a flexural member, the following resistance/safety factors are applied for this limit state:

Φ = 0.95, Ω = 1.58

## 3) Limit State: Local Yielding of HSS Chord Sidewalls | Out-of-Plane Moment for Rectangular HSS (Refer to Moment Limit State Table M2)

Refer to AISC 360-10 Eqn K3-10:

M_{n }= F*_{y}(B – t) (H_{b} + 5t) [AISC 360-10 Eqn K3-10]

Refer to AISC 360-22 Specification Section J10.2 to derive available strength to match AISC 360-10 Eqn K3-10, based on local yielding of rectangular HSS Sidewalls under out-of-plane moment.

R_{n }= F_{yw}t_{w }(5k + l_{b}) For l_{end} > H^{ }[AISC 360-22 Section J10.2]

t_{w} = t_{des} of HSS chord = t

k = t Consider a conservative 1:1 slope for dispersion in lieu of k = 1.5t.

l_{b = }H_{b }/ sinθ For θ = 90, l_{b }= H_{b}

F*_{y} = F_{y }for HSS T-connections

= 0.8F_{y }for HSS Cross-Connections

Substituting to determine available strength:

R_{n }= F*_{y} t [5t + H_{b}] = F*_{y }t (H_{b} + 5t) for l_{end} > H

R_{n }= F*_{y} t [5t + H_{b}] = F*_{y }t (H_{b} + 2. 5t) for l_{end} < H

Φ = 1.00, Ω = 1.50

## 4) Limit State: Local Yielding of Branches Due to Uneven Load Distribution | Out-of-Plane Moment Conn for Rectangular HSS (Refer to Moment Limit State Table M2)

Refer to AISC 360-22 Specification Section F7 to derive available strength to match AISC 360-10 Eqn K3-11, based on local yielding of rectangular HSS branches due to uneven load distribution under out-of-plane moment.

M_{n} = F_{y}Z_{eff } [AISC 360-22 Eqn F7-1]

The steps to determine Z_{eff} are as follows:

- Find effective width of the two HSS branch walls transverse to the HSS chord
- Subtract plastic modulus of non-effective wall area from total HSS plastic modulus (Z
_{b})

Since the branch member is primarily an axial member rather than a flexural member, the following resistance/safety factors are applied for this limit state:

Φ = .95, Ω = 1.58

## STI Limit State Tables Per AISC 360

STI's free Limit State Tables have been updated to the AISC 360-22 Specification and 16th Edition Manual. These tables serve as a comprehensive guide to the HSS limit state checks required for a wide range of HSS connection conditions and loading scenarios, including shear, axial, moment and truss connections. This invaluable resource also serves as an excellent companion to STI's Spreadsheet Design Tools.