Brace Connection — Engineering Reference

Whitmore section, block shear, net section fracture, UFM interface forces, and AISC 341 SCBF gusset requirements with an interactive check.

Overview

Brace connections transfer axial force from a diagonal bracing member into the beam-column joint through a gusset plate. The design must address the gusset plate itself (Whitmore section tension, Thornton compression buckling, block shear), the gusset-to-beam and gusset-to-column interfaces (bolts or welds transferring the resolved brace force components), and the beam and column at the joint (web yielding, web crippling, panel zone). For seismic applications (SCBF per AISC 341), additional requirements for overstrength, ductile behavior, and gusset plate hinging must also be satisfied.

The two primary analysis methods for distributing gusset interface forces are the Uniform Force Method (UFM) from AISC Manual Part 13 and the parallel force method. The UFM produces a set of forces at the gusset-to-beam and gusset-to-column interfaces that are statically consistent with the brace force without introducing additional moments on the interfaces. This simplifies the interface connection design.

Whitmore section — gusset tension capacity

The Whitmore effective width defines how much of the gusset plate participates in resisting the brace tension force:

W_w = 2 x L_w x tan(30) + s_g

where L_w is the length from the first bolt row to the last bolt row along the brace axis, and s_g is the bolt gage perpendicular to the brace axis. The tension capacity is:

phi x P_n = 0.90 x F_y x W_w x t_g (yielding) or 0.75 x F_u x W_n x t_g (net rupture)

where W_n = W_w minus hole deductions and t_g is the gusset plate thickness.

Thornton method — gusset compression buckling

When the brace is in compression, the gusset plate can buckle. The Thornton method models the gusset as an equivalent column with an effective length determined by the geometry:

1. Identify the three distances from the Whitmore section corners (two corners and the midpoint) perpendicular to the nearest gusset edge (beam flange, column flange, or free edge).
2. The effective length L_eff is the average of these three distances.
3. The radius of gyration r = t_g / sqrt(12).
4. The slenderness ratio KL/r = K x L_eff / r, where K = 0.65 for a gusset restrained by framing on two edges.
5. Calculate F_cr from AISC E3 and phi x P_n = 0.90 x F_cr x W_w x t_g.

Uniform Force Method (UFM)

The UFM distributes the brace force P into horizontal (H) and vertical (V) components at each interface without eccentricity moments:

• alpha = e_b x tan(theta) - e_c + (distance to beam interface centroid)
• beta = e_c x cot(theta) - e_b + (distance to column interface centroid)

where e_b and e_c are the half-depths of the beam and column at the work point, and theta is the brace angle. When alpha and beta satisfy the UFM equilibrium equations, the gusset-to-beam interface carries H_b and V_b, and the gusset-to-column interface carries H_c and V_c, with no additional moment.

Worked example — HSS 6x6x3/8 brace to corner gusset

Given: HSS 6x6x3/8 brace (A500 Gr C, A = 7.58 in^2, F_y = 46 ksi, F_u = 62 ksi), brace angle theta = 45 degrees, P_u = 200 kip (tension), gusset plate 1/2 in. A36, four 3/4 in. A325-N bolts in a single line at 3 in. spacing on the brace.

1. Whitmore width: L_w = 3 x 3 = 9.0 in. (4 bolts). W_w = 2 x 9.0 x tan(30) + 0 = 2 x 9.0 x 0.577 = 10.4 in. (single line, gage = 0).
2. Gusset tension yielding: phi x P_n = 0.90 x 36 x 10.4 x 0.50 = 168.5 kip < 200. NG.** Increase to 5/8 in. plate: phi x P_n = 0.90 x 36 x 10.4 x 0.625 = **210.6 kip > 200. OK.
3. Gusset net rupture (5/8 in. plate): W_n = 10.4 - 1 x (13/16 + 1/16) = 9.525 in. phi x P_n = 0.75 x 58 x 9.525 x 0.625 = 258.9 kip. OK.
4. Bolt shear: 4 bolts x 17.9 kip = 71.6 kip < 200. NG. Need bolts in double shear or larger bolts. With 7/8 in. A325-N: phi x r_n = 0.75 x 54 x 0.6013 = 24.4 kip. 8 bolts (double line of 4): 8 x 24.4 = 195 kip. Still marginal — use 7/8 in. A490-N: phi x r_n = 0.75 x 68 x 0.6013 = 30.7 kip. 8 bolts = 245 kip. OK.

SCBF seismic requirements (AISC 341 F2.6)

For Special Concentrically Braced Frames, gusset plate connections must satisfy additional requirements:

• Design force: The connection must resist the expected brace capacity in tension: R_y x F_y x A_g = 1.30 x 46 x 7.58 = 453 kip (for HSS A500 Gr C). This is much larger than the code-level force.
• Compression design: The connection must also resist 1.14 x F_cre x A_g (expected compression capacity) without buckling.
• Gusset plate hinging: The gusset must be detailed to allow the brace to buckle out-of-plane. The standard detail provides a 2t_g linear clearance (clear distance of twice the gusset thickness) between the end of the brace and the beam/column re-entrant corner. This clearance allows a yield line to form in the gusset plate during brace buckling.
• Net section reinforcement: If the brace connects with a slotted gusset, the reduced net section through the slot must develop the expected brace capacity. Reinforcing plates may be required at the slot.

Code comparison — brace connections

Common mistakes to avoid

• Designing the gusset for the code-level brace force instead of the expected capacity — in SCBF, the connection must resist R_y x F_y x A_g, which is typically 2-3 times the code-level seismic force. Under-designed gusset plates are the most common failure in braced frames during earthquakes.
• Ignoring the Thornton compression check — a gusset plate that is adequate in tension can buckle in compression if it is too thin relative to its unbraced length. The Thornton method must be checked for all braces that carry compression.
• Not providing the 2t clearance for SCBF — the standard SCBF gusset detail requires a straight line free distance of 2t_g from the brace end to the nearest restraint (beam flange/column flange re-entrant). This allows the gusset to form a ductile yield line during brace out-of-plane buckling. Without this clearance, the gusset may fracture instead of yielding.
• Omitting the beam-to-column interface check — the UFM distributes forces to both the gusset-to-beam and gusset-to-column interfaces. The beam web must be checked for the vertical and horizontal components at the gusset, and the column must be checked for any additional axial or shear demand from the gusset connection.
• Not checking the beam for the "frame action" forces — in a chevron braced frame, the beam at the brace intersection must resist an unbalanced vertical force when one brace yields and the other buckles. This force can be very large and may govern beam sizing.

AISC 341 brace connection requirements for seismic systems

AISC 341-22 (Seismic Provisions for Structural Steel Buildings) imposes significantly more stringent requirements on brace connections than AISC 360. The fundamental philosophy is capacity design: the connection must be stronger than the brace, forcing the brace to yield (tension) or buckle (compression) as the energy-dissipating fuse. The connection itself must remain essentially elastic.

Seismic force levels for brace connections

where R_y is the ratio of expected yield stress to specified minimum yield stress (AISC 341 Table A3.1), and F_cre is the expected critical buckling stress.

R_y values for common steel grades

For an HSS 6x6x3/8 (A500 Gr. C) brace: the expected tension capacity is Ry * Fy * A*g = 1.30 * 50 _ 7.58 = 493 kip. This is 2.5 times the nominal yield capacity (50 * 7.58 = 379 kip) and may be 3-4 times the code-level seismic design force. This is why SCBF connections are often much larger and more heavily welded than the engineer initially expects.

SCBF gusset plate hinging requirements

AISC 341 Section F2.6c requires that SCBF gusset plates be detailed to accommodate brace out-of-plane buckling. The standard detail provides a "2t linear clearance" between the end of the brace and the re-entrant corner of the beam-column intersection:

• 2t_g clearance: The distance from the end of the brace to the nearest restraint (beam flange or column flange) measured along a line perpendicular to the brace must be at least 2 * t_g, where t_g is the gusset plate thickness.
• Yield line formation: This clearance allows a plastic hinge (yield line) to form in the gusset plate when the brace buckles out-of-plane. The gusset yields in bending rather than fracturing, providing ductile energy dissipation.
• Free edge length: The free (unsupported) edge of the gusset plate between the brace connection and the beam or column interface must be long enough to allow the yield line to develop. AISC 341 Commentary recommends that this free length be at least 2 times the Whitmore section width.

Net section requirements for slotted HSS connections

When an HSS brace is slotted to fit over a gusset plate, the net section through the slot is reduced. AISC 341 F2.6b requires:

phi * R_n (net section) >= R_y * F_y * A_g (brace)

For an HSS 6x6x3/8 with a 1/2 in. gusset slot: the slot removes material from two walls of the HSS. The net area through the slot is Ag - 2 * t _ t_g = 7.58 - 2 _ 0.375 _ 0.50 = 7.58 - 0.375 = 7.205 in.^2. If this is insufficient to develop R_y _ Fy * A_g, reinforcing plates must be welded to the HSS walls on each side of the gusset.

Uniform Force Method — detailed procedure

The Uniform Force Method (UFM) is the AISC-recommended method for distributing brace forces at gusset plate interfaces. It was developed by Richard et al. and codified in the AISC Steel Construction Manual Part 13. The UFM produces interface forces that are statically consistent without introducing eccentricity moments, simplifying the design of the gusset-to-beam and gusset-to-column connections.

UFM force resolution

Given a brace force P at angle theta from horizontal, the UFM resolves P into four interface force components:

Gusset-to-beam interface:
  H_b = alpha * H / e_b    (horizontal force on beam)
  V_b = V * (1 - alpha / e_b)  ... simplified

Gusset-to-column interface:
  H_c = H * (1 - beta / e_c)   ... simplified
  V_c = beta * V / e_c    (vertical force on column)

where:

• H = P * cos(theta) (horizontal component of brace force)
• V = P * sin(theta) (vertical component of brace force)
• e_b = d_b / 2 (half-depth of beam)
• e_c = d_c / 2 (half-depth of column)
• alpha = horizontal distance from column face to the centroid of the gusset-to-beam connection
• beta = vertical distance from beam top flange to the centroid of the gusset-to-column connection

The UFM equilibrium condition requires:

alpha * beta = e_b * e_c

When this condition is satisfied, the interface forces are purely axial (no moments), and each interface connection is designed for its resolved horizontal and vertical components.

UFM worked values for a typical corner connection

Given: Brace force P = 200 kip at theta = 40 degrees. W18x50 beam (d_b = 18.0 in., e_b = 9.0 in.). W14x68 column (d_c = 14.0 in., e_c = 7.0 in.).

H = 200 * cos(40) = 153.2 kip
V = 200 * sin(40) = 128.6 kip

alpha * beta = e_b * e_c = 9.0 * 7.0 = 63.0

Assume alpha = 9.0 in. (typical for beam web connection)
Then beta = 63.0 / 9.0 = 7.0 in.

Gusset-to-beam interface:
  H_b = alpha * H / (alpha + e_c) = 9.0 * 153.2 / (9.0 + 7.0) = 86.2 kip
  V_b = beta * V / (beta + e_b) = 7.0 * 128.6 / (7.0 + 9.0) = 56.3 kip

Gusset-to-column interface:
  H_c = H - H_b = 153.2 - 86.2 = 67.0 kip
  V_c = V - V_b = 128.6 - 56.3 = 72.3 kip

The gusset-to-beam connection must resist 86.2 kip horizontal and 56.3 kip vertical. The gusset-to-column connection must resist 67.0 kip horizontal and 72.3 kip vertical. These are purely axial forces — no moments — which is the key advantage of the UFM.

Gusset plate design — Whitmore section and block shear

Whitmore section evaluation

The Whitmore section defines the effective width of gusset plate resisting the brace force at the last row of bolts (or the end of the weld). It was developed by Whitmore (1952) and is universally accepted in steel connection design.

Whitmore width calculation:

W_w = L_spread + s_g

where Lspread = 2 * Lw * tan(30 degrees) = 1.155 * L_w, L_w is the distance from the first to last bolt along the brace axis, and s_g is the bolt gage perpendicular to the brace (zero for a single line of bolts).

Material limits on W_w:

• W_w cannot extend beyond the free edges of the gusset plate.
• W_w cannot extend beyond the intersection of the gusset plate with the beam or column flange.
• If the Whitmore section crosses a free edge, the effective width must be truncated.

Block shear at the brace-to-gusset connection

Block shear is a combined failure mode where a "block" of material tears out of the gusset plate through a combination of shear and tension planes. For a single line of bolts perpendicular to the brace axis:

phi * R_n = 0.75 * (0.6 * F_u * A_nv + U_bs * F_u * A_nt)

where A_nv is the net shear area (along the bolt lines parallel to the force), A_nt is the net tension area (across the bolt line perpendicular to the force), and U_bs = 1.0 for uniform tension stress distribution.

The block shear check must be performed on both the gusset plate and the brace (for bolted brace connections). For slotted HSS connections, the block shear may also include the slot dimensions.

Brace connection to beam and column interface

The interface connections between the gusset plate and the framing members (beam and column) are designed for the UFM-resolved forces. The connection type depends on the framing geometry and the force magnitude.

Common interface connection types

Interface force transfer checklist

1. Verify the gusset-to-beam weld capacity exceeds the UFM beam interface forces (H_b, V_b).
2. Verify the gusset-to-column weld capacity exceeds the UFM column interface forces (H_c, V_c).
3. Check the beam web for local yielding from the vertical interface force V_b (AISC J10.2).
4. Check the beam web for local crippling if V_b is applied near the column face (AISC J10.3).
5. Check the column web for local yielding from the horizontal interface force H_c (AISC J10.2).
6. Check the column web panel zone shear from the unbalanced brace forces (AISC J10.6).
7. Check the supporting member for the net axial force from the resolved brace components.

Common brace connection configurations

Diagonal brace to corner gusset

The most common brace connection type. The gusset plate sits in the beam-column re-entrant corner, connecting the diagonal brace to both the beam and the column. The gusset outline is typically a trapezoidal or pentagonal shape with clipped corners for clearance.

Key dimensions: The gusset extends from the beam-column intersection along both members. The length along the beam and column is determined by the bolt layout and the Whitmore section requirements. The gusset is typically cut back from the re-entrant corner by 2t_g (for SCBF) or to the work point (for OCBF and non-seismic).

Chevron (V-brace) connection

In chevron-braced frames, two braces meet the beam at a single point from below (inverted V) or from above (V). The gusset plate is typically a single plate welded to the beam bottom flange with the two braces bolted to it. A critical design consideration is the unbalanced vertical force when one brace yields in tension and the other buckles in compression.

Unbalanced force: For a chevron brace with equal brace areas, the maximum unbalanced vertical force is approximately 0.3 _ R_y _ F_y * A_g (the difference between the full tension yield capacity and the post-buckling compression capacity of approximately 30% of P_n). This force can be very large and may require a heavier beam section at the brace intersection point.

X-brace connection (cross-bracing)

In X-braced frames, two diagonal braces cross at midspan. The connection at the crossing point must accommodate the force transfer between the two braces. The crossing connection is typically a simple bolted splice with a spacer plate between the overlapping brace members.

Key design consideration: In an X-brace system, only the tension brace is assumed to resist lateral force (the compression brace is assumed to have buckled). However, the connection at the crossing point must also be checked for the force transfer when the brace force reverses.

Worked example: gusset plate thickness for diagonal brace

Given: HSS 5x5x3/8 brace (A500 Gr. B, F_y = 46 ksi, F_u = 58 ksi, A_g = 6.52 in.^2). Brace angle theta = 38 degrees. Factored brace force P_u = 180 kip (tension). Gusset plate A36 (F_y = 36 ksi, F_u = 58 ksi). Four 7/8 in. A325-N bolts in a single line at 3 in. spacing on the brace.

Step 1 -- Required Whitmore width:

L_w = 3 * 3 = 9.0 in. (distance from first to fourth bolt along brace axis)
W_w = 2 * 9.0 * tan(30) + 0 = 10.39 in. (single bolt line, s_g = 0)

Step 2 -- Whitmore tension yielding (required plate thickness):

phi * P_n = 0.90 * F_y * W_w * t_g >= P_u
0.90 * 36 * 10.39 * t_g >= 180
336.6 * t_g >= 180
t_g >= 180 / 336.6 = 0.535 in.

Use 9/16 in. plate (tg = 0.5625 in.). phi * Pn = 0.90 * 36 _ 10.39 _ 0.5625 = 189.3 kip. DCR = 180 / 189.3 = 0.95. OK.

Step 3 -- Whitmore net rupture:

W_n = W_w - 1 * (15/16 + 1/16) = 10.39 - 1.0 = 9.39 in. (one hole deducted for single line)
phi * P_n = 0.75 * 58 * 9.39 * 0.5625 = 229.5 kip. DCR = 180 / 229.5 = 0.78. OK.

Step 4 -- Block shear at brace-to-gusset connection:

Gross shear area: A_nv = 2 * (1.25 + 2 * 3.0) * 0.5625 = 2 * 9.25 * 0.5625 = 10.41 in.^2
Net shear area: A_nv_net = 10.41 - 3.5 * (15/16 + 1/16) * 0.5625 = 10.41 - 3.5 * 1.0 * 0.5625 = 10.41 - 1.97 = 8.44 in.^2
Gross tension area: A_nt = 2.0 * 0.5625 = 1.125 in.^2
Net tension area: A_nt_net = 1.125 - 0.5 * 1.0 * 0.5625 = 1.125 - 0.281 = 0.844 in.^2

phi * R_n = 0.75 * (0.6 * 58 * 8.44 + 1.0 * 58 * 0.844)
         = 0.75 * (293.7 + 49.0)
         = 0.75 * 342.7 = 257.0 kip. DCR = 180 / 257.0 = 0.70. OK.

Step 5 -- Summary:

Governing limit state: Whitmore tension yielding at DCR = 0.95. The 9/16 in. gusset plate is marginally adequate. For additional reserve, use 5/8 in. plate (phi _ P_n = 0.90 _ 36 _ 10.39 _ 0.625 = 210.3 kip, DCR = 0.86).

Run this calculation

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Related references

• Steel Connection Design
• Bolt Capacity Table
• How to Verify Calculations
• base plate design
• SCBF and OCBF systems
• gusset plate design reference
• steel connection capacity calculator
• weld capacity for connection design
• Bolt Pattern
• Girder-to-column connections

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