AISC 360-22 Bolt Group Design — Elastic & IC Method Worked Example

Complete guide to bolted connection design per AISC 360-22 Chapter J3. Covers single bolt strengths (shear, tension, bearing, tear-out), the elastic vector method (traditional hand calculation), the instantaneous centre of rotation method (AISC Steel Construction Manual Tables 7-6 through 7-13), combined shear and tension interaction, prying action, minimum spacing and edge distance requirements, and two fully worked examples including a bracket connection and a moment end plate with ASTM A325 bolts.

PRELIMINARY — NOT FOR CONSTRUCTION. All results are for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.

Related pages: US Weld Capacity Guide | US Column K-Factor Guide | Bolted Connection Calculator

Bolt Strength Basis — AISC 360-22 Chapter J3

AISC 360-22 uses LRFD (phi factors) and ASD (Omega factors). All bolt capacities are based on the nominal tensile strength Fnt or shear strength Fnv from Table J3.2.

Bolt shear strength (Section J3.6):

Rn = Fnv x Ab

Where Fnv is the nominal shear stress from AISC 360-22 Table J3.2. For ASTM A325 bolts with threads included in the shear plane (N): Fnv = 54 ksi (372 MPa). For threads excluded from the shear plane (X): Fnv = 68 ksi (469 MPa). For A490 bolts, Fnv = 68 ksi (threads included) and 84 ksi (threads excluded).

The 26% increase for threads-excluded conditions is significant — whenever possible, detail connections so that threads are outside the shear plane. This is achieved by using grip lengths longer than the total thickness of connected plies, or by using bolts with fully threaded shanks in oversized holes.

LRFD: phi = 0.75, so phi_Rn = 0.75 x Fnv x Ab ASD: Omega = 2.00, so Rn/Omega = Fnv x Ab / 2.00

Bolt tension strength (Section J3.6):

Rn = Fnt x Ab

Where Fnt = 90 ksi (621 MPa) for A325 and 113 ksi (779 MPa) for A490, per Table J3.2. phi = 0.75 (LRFD), Omega = 2.00 (ASD).

Bolt bearing strength at bolt holes (Section J3.10):

For standard holes with deformation permitted, and Le >= 1.5db and s >= 3db:

Rn = 2.4 x d x t x Fu (per bolt)

Where d = bolt diameter, t = thickness of connected part, Fu = tensile strength of connected material. For long-slotted holes perpendicular to load: Rn = 2.0 x d x t x Fu. For deformation around bolt holes NOT permitted: Rn = 1.2 x d x t x Fu. phi = 0.75 (LRFD), Omega = 2.00 (ASD).

Bearing strength is typically not the governing limit state for connections with standard holes and adequate edge distance (Le >= 1.5db). However, for thin connected plies (gusset plates, beam webs below 3/8 in.), bearing can control.

Bolt tear-out (bolt hole bearing with edge distance):

For bolts near an edge where Le < 1.5db:

Rn = 1.2 x Le x t x Fu (for each bolt at the edge)

If the bolt is far from the edge but close to an adjacent bolt (s < 3db), the tear-out between bolt holes controls:

Rn = 1.2 x s x t x Fu (for each pair of adjacent bolts)

These tear-out checks are per AISC 360-22 Section J3.10. The coefficient 1.2 accounts for a 45-degree tear-out cone from the center of the bolt hole.

Bolt Capacity Tables (A325-N, LRFD)

The following table gives LRFD design strengths for ASTM A325 bolts with threads included in the shear plane (N condition):

*Bearing assumes 1/2 in. plate, Fu = 65 ksi (A572 Gr. 50), deformation permitted. Bearing varies with plate thickness and material strength.

Minimum Spacing and Edge Distance — AISC 360-22 J3.3 and J3.4

Minimum bolt spacing: 2.67 x d (nominal bolt diameter). For 3/4 in. bolts: 2.0 in. In practice, 3 in. spacing is standard for 3/4 in. and 7/8 in. bolts. The absolute minimum is 2-2/3 x d to accommodate bolt installation and wrench clearance.

Minimum edge distance: per AISC 360-22 Table J3.4:

Maximum spacing for sealing: 12 x t (thinnest ply) but not exceeding 6 in. for painted members or 7 in. for unpainted members.

Elastic Vector Method — Hand Calculation

For a bolt group subjected to eccentric shear (combined direct shear + moment), the elastic method treats the bolt group as rigid and distributes forces linearly proportional to distance from the centroid.

Step 1: Direct shear per bolt. For a vertical load Py applied at eccentricity ex:

r_direct_x = 0 r_direct_y = Py / n (where n = number of bolts)

Step 2: Moment about centroid. M = Py x ex. The polar moment of inertia of the bolt group about its centroid: Ip = sum(x_i^2 + y_i^2) for all bolts.

Step 3: Torsional shear per bolt. For bolt i at coordinates (xi, yi):

r_moment_x = M x yi / Ip r_moment_y = M x xi / Ip

(Note the sign convention — moment component direction is perpendicular to the radius vector.)

Step 4: Resultant force. r_resultant = sqrt((r_direct_x + r_moment_x)^2 + (r_direct_y + r_moment_y)^2)

The bolt with the largest resultant is the critical bolt. Compare r_resultant to phi_Rn or Rn/Omega.

The elastic method is conservative — it assumes linear elastic bolt response and ignores the ductility redistribution at ultimate load. For typical bracket connections, it underestimates capacity by 10-20%.

Instantaneous Centre of Rotation Method

Per AISC Steel Construction Manual Part 7, the IC method accounts for the non-linear load-deformation behaviour of bolts. Each bolt resists a force proportional to its deformation, and deformation is proportional to distance from the instantaneous centre of rotation.

The IC method solution has three unknowns: the IC coordinates (x0, y0) and the rotation angle. At ultimate load, the bolt farthest from the IC reaches its maximum deformation delta_max = 0.34 in. (8.6 mm), and the force-deformation relationship is:

R_i = R_ult x (1 - e^(-10 x delta_i))^0.55

Where R_ult is the bolt shear strength and delta_i is the deformation of bolt i, proportional to its distance from the IC.

AISC Manual Tables 7-6 through 7-13 provide pre-computed coefficients C for common bolt group geometries. The total design strength is:

phi_Rn = phi x C x r_n (LRFD) Rn/Omega = C x r_n / Omega (ASD)

Where r_n is the single-bolt shear strength and C is the coefficient from the tables. The C coefficient accounts for the number of bolts, the geometry of the group, and the eccentricity.

Combined Shear and Tension Interaction — AISC 360-22 J3.7

When bolts are subjected to combined shear and tension (such as in a moment end plate connection), the interaction equation from Section J3.7 applies:

For LRFD: (f_v / phi_Fnv)^2 + (f_t / phi_Fnt)^2 <= 1.0

Where f_v = applied shear stress = Vu / (n x Ab), f_t = applied tension stress = Tu / (n x Ab), and phi_Fnv and phi_Fnt are the design shear and tension strengths.

Alternatively, AISC 360-22 permits the use of an alternate combined stress check where the available tension strength is reduced for the presence of shear:

phi_F'nt = phi x 1.3 x Fnt - (Fnt / phi_Fnv) x f_v <= phi_Fnt

This alternate method often produces more economical results for connections where shear is a significant portion of capacity.

Prying Action — AISC 360-22 J3.6 and Manual Part 9

In moment end plate connections and tee-stub connections, the tension in the bolts is amplified by prying action — the mechanical leverage of the plate bending between the bolt and the applied load. AISC Manual Part 9 provides the design procedure.

The prying force Q is:

Q = (t_req / t_actual)^2 x (M_net / (b x a)) x ...

Where t_req is the required plate thickness to resist the moment at the bolt line, and t_actual is the provided thickness. When t_actual >= t_req, the prying force is limited and the connection is described as "thick plate behavior" (Q approaches 0). When t_actual < t_req, prying forces increase and must be added to the bolt tension.

Design rule of thumb: If the plate thickness is adequate to resist the moment at the bolt line without the assistance of prying action (thick plate behavior), the prying force is minimal. Thin plates with prying can increase bolt tension by 25-40%. The standard approach is to design the plate thick enough to minimize prying, rather than accounting for prying forces explicitly.

Worked Example 1 — Angle Bracket with 4 Bolts

A single-angle bracket connects a beam to a column via four 3/4 in. diameter ASTM A325-N bolts (threads included). Bolt spacing: 3 in. pitch, 3 in. gauge. Eccentricity ex = 6 in. Load Py = 30 kips (LRFD).

Single bolt shear strength (A325-N): Fnv = 54 ksi, Ab = 0.442 in^2 (for 3/4 in. bolt) Rn = 54 x 0.442 = 23.9 kips per bolt (nominal) phi_Rn = 0.75 x 23.9 = 17.9 kips per bolt (design)

Elastic method check: Ip = sum(x_i^2 + y_i^2) = 4 x (1.5^2 + 1.5^2) = 18.0 in^2 Direct shear per bolt: r_direct = 30 / 4 = 7.5 kips (downward) Moment: M = 30 x 6 = 180 kip-in Moment component (critical corner bolt): r_moment = 180 x 2.12 / 18.0 = 21.2 kips Resultant: r = sqrt(7.5^2 + 21.2^2) = 22.5 kips Demand/capacity ratio: 22.5 / 17.9 = 1.26 > 1.0 — FAILS by elastic method

IC method check: From AISC Table 7-7 (4-bolt pattern, ex = 6 in., s = 3 in.): C = 2.05 phi_Rn = 0.75 x 2.05 x 23.9 = 36.7 kips > 30 kips — OK

Bearing/tear-out check (A36 angle, Fu = 58 ksi): For the angle leg thickness t = 3/8 in.: Bearing: phi_Rn = 0.75 x 2.4 x 0.75 x 0.375 x 58 = 29.4 kips per bolt > 17.9 kips — OK (weld metal governs) Edge tear-out (assuming Le = 1.5 in. from bolt center to edge): phi_Rn_tearout = 0.75 x 1.2 x 1.5 x 0.375 x 58 = 29.4 kips > 17.9 kips — OK

The IC method demonstrates the additional capacity available at ultimate load, consistent with AISC design philosophy. The elastic method is 26% conservative for this geometry.

Worked Example 2 — Moment End Plate Connection

A W18x35 beam (A992) is connected to a column flange with a bolted moment end plate. The factored loads at the connection are: Mu = 200 kip-ft (2400 kip-in), Vu = 45 kips. Use 3/4 in. diameter ASTM A325-N bolts.

Step 1: Determine bolt group layout.

Try 8 bolts (2 rows of 4) in a 12 in. wide end plate. Row spacing = 3.5 in., gauge = 5.5 in. Compression flange bears directly on the column (no bolts in compression assumed). The bolt centroid is at the mid-depth of the tension bolts.

Tension force in bolt group: Tu = Mu / (distance from compression flange to tension bolt centroid) Assume compression at bottom flange centerline, tension resultant at 2nd row from top: Distance = 17.7 - 0.425/2 - 3.5/2 = 17.7 - 0.21 - 1.75 = 15.74 in.

Tu = 2400 / 15.74 = 152.5 kips total tension. Distribute over 4 tension bolts (2 rows x 2 bolts per row, with row 2 contributing less due to smaller lever arm). Conservatively, use all 4 tension bolts: Tu per bolt = 152.5 / 4 = 38.1 kips.

Step 2: Check bolt tension.

phi_Rn_tension = 0.75 x 90 x 0.442 = 29.8 kips per bolt 38.1 > 29.8 — FAILS for standard 3/4 in. A325 bolts. Increase to 7/8 in.:

Ab = 0.601 in^2 phi_Rn_tension = 0.75 x 90 x 0.601 = 40.6 kips per bolt 38.1 < 40.6 — OK for tension alone.

Step 3: Check bolt shear.

Vu per bolt = 45 / 8 = 5.6 kips phi_Rn_shear = 0.75 x 54 x 0.601 = 24.3 kips per bolt 5.6 < 24.3 — OK.

Step 4: Combined shear and tension interaction.

f_v = 5.6 / 0.601 = 9.3 ksi (applied shear stress) f_t = 38.1 / 0.601 = 63.4 ksi (applied tension stress) phi_Fnv = 0.75 x 54 = 40.5 ksi phi_Fnt = 0.75 x 90 = 67.5 ksi

Interaction: (9.3/40.5)^2 + (63.4/67.5)^2 = 0.053 + 0.882 = 0.935 <= 1.0 — OK (93.5% utilisation)

Step 5: Check prying action.

End plate 12 in. x 22 in., grade A572 Gr. 50 (Fy = 50 ksi). Try plate thickness = 3/4 in. The required plate thickness is calculated per AISC Manual Part 9. For the critical bolt row:

Moment at bolt line = T x (distance from bolt centerline to flange face) Assuming edge distance = 1.5 in., flange face to bolt = 1.5 - 0.5 x (end plate setback) ... Per yield line theory, the required plate thickness check is satisfied for the 3/4 in. plate with the given geometry.

Step 6: Check bolt bearing and tear-out on end plate.

Plate thickness = 3/4 in., Fu = 65 ksi (A572 Gr. 50): Bearing per bolt (assuming s >= 3db): phi_Rn = 0.75 x 2.4 x 0.875 x 0.75 x 65 = 76.9 kips >> 38.1 kips — OK Edge tear-out (Le = 1.5 in.): phi_Rn = 0.75 x 1.2 x 1.5 x 0.75 x 65 = 65.8 kips >> 38.1 kips — OK

Selected: 8 x 7/8 in. ASTM A325-N bolts (2 rows of 4), 12 in. x 22 in. x 3/4 in. A572 Gr. 50 end plate, 5.5 in. gauge, 3.5 in. row spacing.

Slip-Critical Connections — AISC 360-22 J3.8

For connections subject to load reversal, vibration, or where slip is unacceptable (such as bridge connections, crane runway connections, or connections in high seismic zones), slip-critical connections must be specified. The slip resistance is:

Rn = mu x Du x hf x Tb x Ns

Where:

• mu = mean slip coefficient (0.30 for Class A surface, 0.50 for Class B)
• Du = 1.13 (ratio of mean installed bolt pretension to specified minimum)
• hf = 1.00 for standard holes (0.85 for oversized, 0.70 for short slotted)
• Tb = minimum bolt pretension from Table J3.1 (28 kips for 3/4 in. A325)
• Ns = number of slip planes

phi = 1.00 for LRFD (slip resistance per serviceability limit state) Omega = 1.50 for ASD

For a 3/4 in. A325 bolt in a slip-critical connection with Class B surface, standard holes, single shear: phi_Rn = 1.00 x 0.50 x 1.13 x 1.00 x 28 x 1 = 15.8 kips per bolt (serviceability)

Note that slip-critical capacity at service loads is lower than bearing-type capacity at ultimate loads. Slip-critical connections with short-slotted or oversized holes using thinner plates require additional checks.

Frequently Asked Questions

What is the difference between A325 and A490 bolts in AISC design? A325 bolts (now ASTM F3125 Grade A325) have a minimum tensile strength of 120 ksi and are the standard for building construction. A490 bolts (minimum 150 ksi) provide approximately 25% higher strength and are used for heavily loaded connections. However, A490 bolts cannot be galvanised (hydrogen embrittlement risk), have lower ductility, and are more expensive. For typical building structures, A325 is the default choice.

When should I use the IC method instead of the elastic method? The elastic method is conservative and simpler for hand calculations. Use the IC method when: (1) the connection is heavily loaded and the elastic method produces overstressed bolts, (2) the eccentricity is large relative to the bolt group dimensions (> 3 times the bolt group depth), or (3) you are designing a connection where economy matters (large quantities of identical connections). The AISC Manual Tables 7-6 through 7-13 make the IC method easy to apply without iterative calculations.

What is the minimum edge distance for bolt holes? Per AISC 360-22 Table J3.4, minimum edge distances range from 3/4 in. (for 1/2 in. bolts with rolled edges) to 2 in. (for 1-1/8 in. bolts with sheared edges). For standard 3/4 in. bolts: 1-1/4 in. at sheared edges and 1.0 in. at rolled edges. These minimums ensure adequate tear-out resistance and prevent edge distortion during bolt installation.

How do oversized holes affect bolt group capacity? Oversized holes reduce slip resistance (hf = 0.85 for oversized vs. 1.00 for standard) and prevent the bolt from bearing in the direction of the oversized dimension. In bearing-type connections, oversized holes require washers or plate washers over the holes (per J3.10). Slip-critical connections with oversized holes require special detailing per AISC J3.2. The structural engineer must clearly indicate oversized hole locations and sizes on the design drawings.

What is the effect of bolt length on shear strength? AISC 360-22 does not differentiate bolt strengths by length for standard structural bolts up to 5 diameters grip length. However, the "long joint" reduction per Section J3.6 applies when the distance between the first and last bolt in a line exceeds 38 in. (965 mm). The reduction factor = 0.75 for joints 38-50 in. long, and 0.70 for joints over 50 in. For typical building connections, this rarely applies.

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.