Bolt Pattern — Engineering Reference

Eccentric bolt group analysis: elastic C-method, ICR method, AISC Tables 8-1 to 8-3 C-coefficients, polar moment of inertia, and interactive calculator.

Overview

A bolt pattern (or bolt group) is the arrangement of fasteners in a connection that collectively resists applied forces. When the line of action of the applied load passes through the centroid of the bolt group, all bolts share the load equally. When the load is eccentric — offset from the centroid — the bolt group must resist both a direct shear and a moment. Analyzing eccentric bolt groups requires either the elastic method or the instantaneous center of rotation (ICR) method.

The AISC Steel Construction Manual provides pre-computed C-coefficients in Tables 7-6 through 7-14 for common eccentrically loaded bolt groups, making hand calculations practical. For non-standard bolt patterns, the ICR method is computed iteratively using the load-deformation relationship for individual bolts.

Pitch and gage tables for standard connections

Standardizing bolt layouts speeds up design, improves constructability, and ensures sufficient clearance for wrenches and installation tools. The AISC Manual recommends preferred pitch and gage dimensions for the most common connection types.

AISC preferred bolt layouts

The following table summarizes the most frequently used bolt configurations for simple shear connections. These layouts are pre-qualified for common framing conditions.

Typical gage for W-shape flanges

The flange gage (horizontal spacing between bolt lines across a flange) is typically taken as half the flange width, rounded to the nearest convenient dimension. AISC Manual Table 1-7A provides standard gages.

Rule of thumb: For preliminary design, use gage = b_f / 2. For example, a W14x48 has b_f = 8.03 in.; a gage of 4.0 in. or 5.5 in. is typical.

Typical pitch = 3d (standard spacing)

AISC Specification Section J3.3 requires a minimum center-to-center bolt spacing of 2-2/3 times the nominal bolt diameter (2.67d). The preferred spacing is 3d, which provides adequate clearance for wrenches and ensures adequate bearing strength on the connected material.

In practice, 3 in. pitch is used for virtually all standard connections with 3/4 in., 7/8 in., and 1 in. bolts. This simplifies detailing and ensures adequate bearing and tear-out capacity.

Bolt count vs. connection capacity — 3/4" A325-N single shear

The table below shows the shear capacity for connections using 3/4 in. A325-N bolts (threads included in the shear plane) in single shear, with a bolt capacity of phi x R_n = 17.9 kip per bolt (phi = 0.75, A_b = 0.4418 in.^2, F_nv = 54 ksi).

Note: These are concentric capacities. For eccentric loading, the effective capacity is reduced. Use the C-coefficient method described below.

Bolt group layout patterns

The geometry of a bolt group directly affects its strength, stiffness, and constructability. Selecting the right pattern depends on the connection type, applied forces (shear, moment, axial), and available space.

Linear patterns (single line, double line)

Single vertical line — The most common pattern for simple shear connections (shear tabs, single angles, tee connections). Bolts are arranged in a single column with uniform pitch spacing. Advantages include simplicity of detail, easy installation, and predictable load distribution. Limited to shear-only or low-moment connections because the single line provides no rotational resistance about the bolt line axis.

• Typical spacing: 3 in. pitch
• Bolt count: 2 to 6 depending on load
• Centroid: at the midpoint of the bolt column

Double vertical line — Two columns of bolts separated by a gage dimension. Used for heavier shear connections, double-angle connections, and moment connections where additional rotational stiffness is needed. The double line provides greater polar moment of inertia, increasing resistance to eccentric moments.

• Typical gage: 3 in. to 5.5 in. depending on member size
• Provides higher moment resistance than single line with same bolt count
• Common in extended end-plate moment connections

Staggered patterns

Staggered bolt arrangements place fasteners in alternating positions between rows, creating a zigzag layout. This pattern is used when:

• The connection must fit within a restricted geometric envelope where a rectangular grid would violate minimum edge distances
• Shear lag must be reduced by engaging more of the gross section along the load path
• Tear-out capacity must be improved by increasing the clear distance between bolt holes

For staggered patterns, the critical section for block shear and net section checks must follow a staggered path. AISC Specification Section B4.3b provides the formula for net area along a staggered path: add s^2 / (4g) for each staggered segment, where s is the longitudinal pitch and g is the transverse gage.

Rectangular patterns for moment end plates

Rectangular (grid) bolt patterns are the standard for moment end-plate connections. Bolts are arranged in rows (horizontal, typically 2 per row across the flange) and columns (vertical, near the beam flanges).

Common configurations from AISC Manual Part 11:

For end-plate moment connections, the bolt group resists both the beam flange forces (tension and compression) and the beam web shear. The outermost bolt rows (near the beam flanges) carry the largest tension forces from the moment, while the inner rows share the remaining forces.

Circular patterns for flange connections

Circular bolt patterns arrange fasteners uniformly around a bolt circle diameter (BCD). Common applications include:

• Round HSS base plates and cap plates
• Pipe flanges (ANSI/ASME B16.5 standard patterns)
• Circular moment connections
• Mechanical equipment anchorage

For a uniform circular pattern with n bolts on bolt circle radius R, the polar moment of inertia is:

I_p = n x R^2

Each bolt's distance from the centroid is R. When an eccentric moment M is applied, the moment-induced force on each bolt is:

R_moment = M / (n x R)

This simplicity makes circular patterns attractive for base plate design where biaxial moments and axial loads can be resolved into bolt forces through straightforward superposition.

Elastic method for eccentric bolt groups

The elastic method treats each bolt as a linear spring. For a bolt group subjected to an eccentric load P at eccentricity e from the bolt group centroid:

1. Direct shear on each bolt: V_direct = P / n (where n = number of bolts)
2. Moment on bolt group: M = P x e
3. Polar moment of inertia: I_p = sum(x_i^2 + y_i^2) for all bolts, where x_i and y_i are distances from each bolt to the group centroid
4. Moment-induced force on each bolt: R_moment = M x r_i / I_p, acting perpendicular to the radius r_i from the centroid

The resultant force on the critical bolt (the one farthest from the centroid on the side of maximum combined force) is the vector sum of V_direct and R_moment.

Advantages of the elastic method:

• Simple hand calculation
• Conservative (predicts lower capacity than actual)
• Does not require iteration
• Suitable for preliminary design and checks

Limitations:

• Assumes all bolts have equal stiffness
• Does not account for nonlinear bolt load-deformation behavior
• Typically conservative by 10-20% compared to the ICR method

Instantaneous center of rotation (ICR) method

The ICR method (AISC Manual Part 7) is more accurate than the elastic method because it accounts for the nonlinear load-deformation behavior of bolts. Each bolt's force-deformation relationship is:

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

where R_ult is the bolt ultimate shear strength and delta_i is the deformation of bolt i. The bolt group capacity is found iteratively by assuming a center of rotation, computing bolt deformations (proportional to distance from the IC), summing forces and moments, and adjusting the IC location until equilibrium is satisfied.

The ICR method typically gives 10-20% higher capacity than the elastic method because it allows bolt force redistribution — the most heavily loaded bolts deform and shed load to less-loaded bolts.

ICR procedure summary

1. Assume an initial IC location (start with the elastic solution)
2. For each bolt i, compute distance r_i from the IC
3. Compute deformation delta_i = (r_i / r_max) x delta_max (where delta_max = 0.34 in. for bearing-type bolts)
4. Compute bolt force R_i using the load-deformation relationship above
5. Compute the direction of each bolt force (perpendicular to r_i from IC)
6. Sum forces in x and y: must equal the applied load components
7. Sum moments about the IC: must equal zero
8. If equilibrium is not satisfied, adjust IC location and repeat

C-coefficients for common bolt groups (AISC Manual Tables 7-6 through 7-14)

The AISC Steel Construction Manual provides C-coefficients that simplify eccentric bolt group analysis. The connection capacity is:

P = C x phi x r_n

where phi x r_n is the single-bolt shear capacity. The C-coefficient accounts for the bolt group geometry, eccentricity, and load angle. Higher C values indicate more efficient bolt groups.

C-coefficients — vertical single-line bolt groups (load in plane of connection)

The table below provides approximate C-coefficients for vertical single-line bolt groups with vertical spacing s = 3 in. and the load applied at eccentricity e measured horizontally from the bolt line.

C-coefficients — double-line (2-column) bolt groups

For two columns of bolts with gage g = 3 in., vertical pitch s = 3 in., and the load applied at eccentricity e from the bolt group centerline.

Note: These C-coefficients are approximate values for preliminary design. For final design, use the exact values from AISC Manual Tables 7-6 through 7-14.

Worked example — designing a bolt pattern for a 100-kip shear connection

Problem: Design a bolted connection to transfer a factored shear load of V_u = 100 kip from a W21x55 beam to a column flange using a single-plate shear tab. The load is applied at an eccentricity of e = 9 in. from the bolt line to the weld line. Use 3/4 in. A325-N bolts.

Step 1 — Determine single-bolt capacity

For 3/4 in. A325-N bolts in single shear:

• Bolt area: A_b = pi x (0.75)^2 / 4 = 0.4418 in.^2
• Nominal shear stress: F_nv = 54 ksi (threads in shear plane)
• Single-bolt capacity: phi x R_n = 0.75 x 0.4418 x 54 = 17.9 kip

Step 2 — Select trial bolt layout

Try 6 bolts in a single vertical line at 3 in. pitch. The bolt group height is 5 x 3 = 15 in., which fits within the W21x55 web depth (d = 20.8 in. minus flange thickness leaves about 18.5 in. available).

Step 3 — Check using C-coefficient method

From the C-coefficient table above for n = 6 bolts, e = 9 in.: C = 2.63.

Connection capacity: P = C x phi x r_n = 2.63 x 17.9 = 47.1 kip

Since 47.1 kip is less than 100 kip, the single-line 6-bolt configuration is not adequate.

Step 4 — Try double-line configuration

Try 12 bolts (6 rows x 2 columns), gage g = 3 in., pitch s = 3 in.

From the double-line C-coefficient table for n = 12, e = 9 in.: C = 5.02.

Connection capacity: P = 5.02 x 17.9 = 89.9 kip

Still insufficient for 100 kip.

Step 5 — Reduce eccentricity or increase bolt size

Option A: Use a longer plate to reduce eccentricity to e = 6 in.

• For n = 12, e = 6 in.: C = 6.08
• Capacity: 6.08 x 17.9 = 108.8 kip — exceeds 100 kip, OK

Option B: Upgrade to 7/8 in. A325-N bolts (phi x r_n = 24.3 kip per bolt)

• For n = 10, e = 9 in.: C = 4.30
• Capacity: 4.30 x 24.3 = 104.5 kip — exceeds 100 kip, OK

Step 6 — Check bearing and tear-out

For the controlling bolt, the bearing stress and tear-out must also be checked per AISC J3.10 and J3.6. For a 3/4 in. bolt with 3 in. pitch and minimum edge distance of 1-1/4 in. on a plate with F_u = 58 ksi:

• Bearing capacity per bolt (deformation considered): phi x R_n = 0.75 x 2.4 x d x t x F_u
• Tear-out per bolt: phi x R_n = 0.75 x 1.2 x L_c x t x F_u

These checks must be performed for the actual plate thickness selected.

Result: Use 12 bolts (6 rows x 2 columns) at 3 in. pitch x 3 in. gage with eccentricity e = 6 in., or 10 bolts (5 rows x 2 columns) of 7/8 in. A325-N at e = 9 in.

Worked example — 4-bolt vertical line, eccentric shear

Given: 4 bolts in a single vertical line, 3 in. spacing, 3/4 in. A325-N bolts. Applied load P = 40 kip at eccentricity e = 6 in. from the bolt line.

Elastic method:

1. Bolt positions from centroid: y = +4.5, +1.5, -1.5, -4.5 in. (all at x = 0)
2. I_p = 4 x 0^2 + (4.5^2 + 1.5^2 + 1.5^2 + 4.5^2) = 45.0 in^2
3. Direct shear per bolt: V = 40/4 = 10.0 kip (downward)
4. Moment: M = 40 x 6 = 240 kip-in
5. Moment force on extreme bolt: R_m = 240 x 4.5 / 45.0 = 24.0 kip (horizontal)
6. Resultant on critical bolt: R = sqrt(10.0^2 + 24.0^2) = 26.0 kip
7. Bolt capacity: phi x R_n = 0.75 x 54 x 0.4418 = 17.9 kip. 26.0 greater than 17.9 — not adequate.

Using AISC Table 7-7 (C-coefficient for 4 bolts, s = 3 in., e = 6 in.): C is approximately 1.79. Capacity = C x phi x r_n = 1.79 x 17.9 = 32.0 kip. Since P = 40 kip exceeds 32.0 kip, still not adequate — need more bolts or reduce eccentricity.

Standard bolt patterns

Common bolt layouts and their applications:

Key design considerations

• Bolt spacing — AISC J3.3 requires a minimum center-to-center spacing of 2-2/3 x d (typically rounded to 3d) and a preferred spacing of 3d. Maximum spacing is 24t or 12 in. (J3.5) for connected elements in compression.
• Edge distance — minimum edge distances per AISC Table J3.4 range from 3/4 in. (for 1/2 in. bolts) to 2 in. (for 1-1/4 in. bolts) at sheared edges. Rolled or gas-cut edges allow smaller distances.
• Gage dimensions — AISC Manual Table 1-7A gives standard gages for W shapes. For a W14 with b_f = 10 in., the standard flange gage is 5-1/2 in. Using non-standard gages requires checking clearances for bolt installation.
• Bolt group centroid — for symmetric patterns, the centroid is at the geometric center. For asymmetric patterns (e.g., staggered bolts), compute the centroid as the average of all bolt coordinates. All eccentricities are measured from this centroid.

Common mistakes to avoid

• Using the elastic method where ICR is needed — the elastic method is conservative by 10-20% for most configurations. For heavily loaded eccentric connections, using the elastic method wastes bolt capacity and may lead to oversized connections.
• Forgetting to include direct shear in the vector sum — when computing the critical bolt force, the direct shear (P/n) and the moment-induced force must be added as vectors, not scalars. Simply adding magnitudes is overly conservative when the forces act in different directions.
• Mislocating the bolt group centroid — for bolt groups with mixed bolt sizes or patterns, the centroid must be computed precisely. An error in centroid location changes the eccentricity and all moment-induced bolt forces.
• Ignoring out-of-plane eccentricity — bracket connections and seated connections often have out-of-plane eccentricity (the load is offset from the faying surface). This creates bolt tension in addition to shear, requiring a combined shear-tension interaction check per AISC J3.7.

Frequently asked questions

What is the difference between the elastic method and the ICR method for bolt groups?

The elastic method treats all bolts as identical linear springs and distributes the eccentric moment proportionally to each bolt's distance from the group centroid. It is conservative by 10-20% because it does not account for force redistribution. The instantaneous center of rotation (ICR) method uses the nonlinear load-deformation behavior of each bolt and iteratively finds the center of rotation. It is the more accurate method and is the basis for the AISC Manual C-coefficient tables.

How do I determine the eccentricity of a bolted connection?

Eccentricity is the perpendicular distance from the line of action of the applied load to the centroid of the bolt group. For beam-to-column shear connections, the eccentricity is typically measured from the bolt line to the weld line (for single-plate connections) or to the face of the supporting member (for double-angle connections). For bracket connections loaded at the tip, the eccentricity is measured from the point of load application to the bolt group centroid.

When should I use the C-coefficient tables instead of a full analysis?

Use the C-coefficient tables (AISC Manual Tables 7-6 through 7-14) when your bolt group geometry and loading match one of the standard configurations: vertical single-line or double-line bolt groups with in-plane eccentric loading. For non-standard geometries (staggered patterns, L-shaped groups, out-of-plane loading), you must perform a full elastic or ICR analysis.

What is the minimum number of bolts for a structural steel connection?

AISC does not specify a universal minimum bolt count, but practical minimums exist. Simple shear connections typically use a minimum of 2 bolts. Single-plate shear connections require a minimum of 2 bolts (and a maximum of 7 bolts per AISC Manual Part 10). Moment end-plate connections typically use a minimum of 4 bolts (2 per flange). Always check the specific connection type limitations in the AISC Manual.

How does bolt diameter affect connection capacity in an eccentric bolt group?

Larger bolt diameters increase single-bolt shear capacity (proportional to d^2) and bearing capacity (proportional to d). However, larger bolts also require larger edge distances and spacing, which can increase the overall connection size and the eccentricity. For a given load, increasing bolt diameter is often more efficient than adding more bolts because fewer bolts mean a more compact group with lower eccentricity.

Can I mix bolt diameters in the same bolt group?

While not prohibited by AISC, mixing bolt diameters in the same bolt group is extremely uncommon in practice and complicates the analysis significantly. Different bolt stiffnesses change the force distribution within the group. If different capacities are needed, it is generally preferable to use a uniform bolt size and adjust the number of bolts or the connection geometry instead.

Run this calculation

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Related references

• Bolt Spacing and Edge Distance
• Bolt Hole Sizes
• Bolt Grades and Strength
• Connection Type Selection
• Eccentric Gusset Plate Connections
• Eccentric Connection Design
• Welded Connection Design
• Base Plate Design
• How to Verify Calculations
• Steel Design Fundamentals

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard (AISC 360, AISC Steel Construction Manual) and project specification before use. The C-coefficient values presented are approximate for preliminary design; use the exact values from the AISC Manual for final design. The site operator disclaims liability for any loss arising from the use of this information.

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