AISC 360-22 Gusset Plate Design — Full Worked Example

Complete worked example for a gusset plate in a braced frame connection per AISC 360-22 and the AISC Steel Construction Manual, 15th Edition. Covers Whitmore section analysis, Thornton uniform force method, block shear rupture, bolt shear and bearing, and gusset buckling checks. All calculations in US customary units with metric equivalents.

Related pages: AISC Connection Design Guide | Block Shear Guide | Bolted Connection Calculator | Brace Connection Guide

Problem Statement

Design a gusset plate connection for a diagonal brace in a concentrically braced frame (CBF). The brace is an HSS 6x6x1/2 (ASTM A500 Grade C, Fy = 50 ksi) carrying a factored axial tension of Tu = 180 kips and factored axial compression of Cu = 160 kips. The connection is at a beam-to-column moment frame joint in a 4-storey office building in Seismic Design Category D.

Design data:

Step 1 — Whitmore Section Analysis (AISC 360 Section J4.2)

The Whitmore effective width defines the gusset plate section that resists axial force from the brace. Construction: from the first bolt in the connection (closest to the brace), project lines at 30 degrees on each side of the brace axis until they intersect a line perpendicular to the brace axis through the last bolt.

Bolt group geometry:

Whitmore width calculation:

The 30-degree projection from the first bolt extends to the last bolt row:

Whitmore width, Ww = 2 x Lb x tan(30 deg) + gauge = 2 x 12 x 0.5774 + 4.0 = 13.86 + 4.0 = 17.86 in.

Use Ww = 17.9 in.

Whitmore area: Ag_whit = Ww x tg = 17.9 x 0.50 = 8.95 in^2

Tension yielding on Whitmore section (AISC Eq. J4-1):

Rn = Fy x Ag_whit = 50 x 8.95 = 447.5 kips phi x Rn = 0.90 x 447.5 = 402.8 kips

Tu = 180 kips << 402.8 kips — OK, utilisation = 180 / 402.8 = 0.447 (44.7%)

Tension rupture on Whitmore section (AISC Eq. J4-2):

The net area through the last bolt row subtracts bolt holes. Standard hole for 3/4 in. dia. bolt = 13/16 in. dia.

Net width per row: Wn = Ww - 2 x dh = 17.9 - 2 x 0.8125 = 17.9 - 1.625 = 16.28 in. Ae = Wn x tg = 16.28 x 0.50 = 8.14 in^2 (assume U = 1.0 for Whitmore section with uniform stress)

Rn = Fu x Ae = 65 x 8.14 = 529.1 kips phi x Rn = 0.75 x 529.1 = 396.8 kips

Tu = 180 kips << 396.8 kips — OK, utilisation = 180 / 396.8 = 0.454 (45.4%)

Step 2 — Thornton Uniform Force Method (AISC Manual Part 9)

The Uniform Force Method (UFM) distributes the brace force to the beam and column interfaces through the gusset plate, ensuring no moment at the connection interfaces. For a brace at angle theta from the beam:

Geometry:

Brace force resolution:

Horizontal component: Hu = Tu x cos(theta) = 180 x 0.7071 = 127.3 kips Vertical component: Vu = Tu x sin(theta) = 180 x 0.7071 = 127.3 kips

UFM force distribution (from equilibrium):

Force at gusset-to-beam interface:

Force at gusset-to-column interface:

These forces are transferred through welds at each interface. For a 5/16 in. fillet weld, E70XX electrode:

Weld capacity per inch: phi x Rn = 1.392 x D = 1.392 x 5 = 6.96 kips/in. (AISC Eq. J2-4, theta = 0 deg)

Gusset-to-beam weld length required:

Resultant at beam interface: R_b = sqrt(63.6^2 + 63.6^2) = 89.9 kips

Load angle on weld: atan(63.6 / 63.6) = 45 deg. Weld strength increases for non-parallel loading per AISC Eq. J2-5:

phi x Rn = 1.392 x D x (1.0 + 0.50 x sin^1.5(theta)) = 6.96 x (1.0 + 0.50 x sin^1.5(45 deg)) = 6.96 x (1.0 + 0.50 x 0.595) = 6.96 x 1.298 = 9.03 kips/in.

Required weld length: Lw_b = 89.9 / 9.03 = 10.0 in.

Provide 12 in. of 5/16 in. fillet weld each side (24 in. total), util = 10.0 / 12 = 0.833 (83.3%)

Gusset-to-column weld length required:

Similarly, R_c = 89.9 kips. Provide 12 in. of 5/16 in. fillet weld each side.

Step 3 — Block Shear Rupture (AISC 360 Section J4.3, Eq. J4-5)

Block shear checks the gusset plate at the bolt group connecting the brace to the gusset. The potential failure block is defined by the bolt layout.

Block shear geometry (brace-to-gusset bolt group):

Gross shear area: Agv = 2 x L_shear x tg = 2 x 12 x 0.50 = 12.0 in^2

Net shear area: Anv = 2 x (12 - 4.5 x 0.8125) x 0.50 = 2 x (12 - 3.656) x 0.50 = 2 x 8.344 x 0.50 = 8.34 in^2

Gross tension area: Agt = gauge x tg = 4.0 x 0.50 = 2.0 in^2

Net tension area: Ant = (4.0 - 1 x 0.8125) x 0.50 = 3.188 x 0.50 = 1.59 in^2

Block shear capacity (AISC Eq. J4-5):

Ubs = 1.0 (uniform tension stress for single-row bolts in tension)

Rn1 = 0.60 x Fu x Anv + Ubs x Fu x Ant = 0.60 x 65 x 8.34 + 1.0 x 65 x 1.59 = 325.3 + 103.4 = 428.7 kips

Rn2 = 0.60 x Fy x Agv + Ubs x Fu x Ant = 0.60 x 50 x 12.0 + 1.0 x 65 x 1.59 = 360.0 + 103.4 = 463.4 kips

Rn = min(428.7, 463.4) = 428.7 kips phi x Rn = 0.75 x 428.7 = 321.5 kips

Tu = 180 kips << 321.5 kips — OK, utilisation = 180 / 321.5 = 0.560 (56.0%)

Block shear is not critical for this gusset plate.

Step 4 — Bolt Group Capacity (AISC 360 Section J3)

The brace-to-gusset connection uses 10 bolts (5 rows of 2) of 3/4 in. dia. A325-N.

Single bolt shear capacity (AISC Table J3.2):

A325-N, 3/4 in. dia., single shear: phi x rn = 15.9 kips/bolt (from AISC Table 7-1)

Total shear capacity: phi x Rn = 10 x 15.9 = 159.0 kips

Tu = 180 kips > 159.0 kips — BUT, the bolt group is loaded eccentrically? No. For a concentric brace connection with UFM, the bolt group at the brace end is concentrically loaded. However, 180 kips exceeds 159 kips — the bolt group is undersized.

Revise to 12 bolts (6 rows of 2) or increase bolt diameter to 7/8 in.

Using 7/8 in. dia. A325-N: phi x rn = 24.3 kips/bolt (Table 7-1). Required bolts: n = Tu / (phi x rn) = 180 / 24.3 = 7.41 bolts. Provide 10 bolts — OK, util = 180 / (10 x 24.3) = 0.741 (74.1%)

Bolt bearing at gusset plate (AISC Eq. J3-6a):

For 7/8 in. dia. bolts, standard holes, edge distance = 2 in. (actual min = 1.25 in. per Table J3.4):

Lc = 2.0 - 0.9375/2 = 1.53 in. > 0 — bearing on edge bolts governed by tear-out.

rn = 1.2 x Lc x t x Fu = 1.2 x 1.53 x 0.50 x 65 = 59.7 kips/bolt (tear-out, Eq. J3-6a)

Also check: rn = 2.4 x d x t x Fu = 2.4 x 0.875 x 0.50 x 65 = 68.3 kips/bolt (bearing, Eq. J3-6a)

Control value: rn = min(59.7, 68.3) = 59.7 kips/bolt phi x rn = 0.75 x 59.7 = 44.8 kips/bolt

Total bearing capacity: phi x Rn = 10 x 44.8 = 448 kips >> 180 kips. Bearing is not critical.

Step 5 — Gusset Plate Buckling (AISC 360 Section J4.4)

Under compression (Cu = 160 kips), the gusset plate must be checked for flexural buckling of the unsupported length between the end of the brace and the beam/column interfaces.

Unsupported length: The gusset plate free edge between the brace end and the beam or column constitutes the buckling length. Typically, the Whitmore width at the first bolt row is the critical section.

Lc = distance from last bolt row to the beam-column intersection along the buckling plane. For a 45-degree brace, this is the longest unsupported edge, approximately Lc = 9 in. (by geometry, measured along the free edge).

Effective length factor: K = 1.2 (AISC Manual Part 9 recommends K = 1.2 for gusset plates with one free edge and two edges welded).

Slenderness check (approximate):

The plate buckles as a column strip of width = 1.0 in. (unit width approach):

r = tg / sqrt(12) = 0.50 / 3.464 = 0.1443 in. (radius of gyration for rectangular section)

KLc / r = 1.2 x 9.0 / 0.1443 = 10.8 / 0.1443 = 74.8

Elastic buckling stress (AISC Eq. E3-4):

Fe = pi^2 x E / (KLc/r)^2 = 9.8696 x 29,000 / 74.8^2 = 286,218 / 5,594 = 51.2 ksi

Critical stress (AISC Eq. E3-2):

Fy/Fe = 50 / 51.2 = 0.977 <= 2.25 — inelastic buckling governs.

Fcr = (0.658^(Fy/Fe)) x Fy = (0.658^0.977) x 50 = 0.666 x 50 = 33.3 ksi

Buckling capacity per unit width:

phi x Pn = 0.90 x Fcr x Ag (per inch width) = 0.90 x 33.3 x (1.0 x 0.50) = 14.98 kips/in.

Whitmore width capacity in compression:

phi x Pn_whit = 14.98 kips/in. x Ww = 14.98 x 17.9 = 268.2 kips

Cu = 160 kips << 268.2 kips — OK, utilisation = 160 / 268.2 = 0.597 (59.7%)

Step 6 — Gusset-to-Beam and Gusset-to-Column Interface Checks

Gusset plate yielding at beam interface:

The gusset plate is welded to the beam flange along its length. The plate must transfer H_b = 63.6 kips and V_b = 63.6 kips.

Plate gross area on beam interface: Ag_beam = Lw_beam x tg = 12 x 0.50 = 6.0 in^2

Shear yielding: phi x Vn = 0.60 x Fy x Ag_beam x 1.0 = 0.60 x 50 x 6.0 = 180 kips

Combined shear and axial interaction is negligible since the plate is thick relative to the demand — utilisation < 0.5 for each component.

Summary of Checks

Limit State Reference phi x Rn (kips) Demand (kips) Utilisation
Whitmore tension yielding AISC J4-1 402.8 180 44.7%
Whitmore tension rupture AISC J4-2 396.8 180 45.4%
Block shear rupture AISC J4-5 321.5 180 56.0%
Bolt shear (10 x 7/8 in. A325-N) AISC Table J3.2 243.0 180 74.1%
Bolt bearing (gusset) AISC J3-6a 448.0 180 40.2%
Gusset buckling AISC J4.4 / E3 268.2 160 59.7%
Weld (beam interface) AISC J2-4 108.4 89.9 83.3%
Weld (column interface) AISC J2-4 108.4 89.9 83.3%

Conclusion: The 1/2 in. thick ASTM A572 Grade 50 gusset plate with 10 x 7/8 in. dia. A325-N bolts and 5/16 in. fillet welds (12 in. each interface) satisfies all AISC 360-22 limit states. The critical check is weld capacity at the beam and column interfaces (83.3% utilisation), followed by bolt shear (74.1%). The Whitmore section is well within capacity, confirming that a thinner gusset could be considered.

Practical Design Notes