Canadian Bolt Group Capacity — Eccentric Load Design per CSA S16

Complete reference for bolt group capacity with eccentric loading per CSA S16-19. Covers the elastic vector method for simple cases, the instantaneous centre (IC) method for rigorous analysis, C-values from the CISC Handbook, and a worked example for a 4-bolt bracket connection.

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CSA S16 Bolt Group Philosophy

CSA S16-19 does not prescribe a specific method for eccentrically loaded bolt groups. The CISC Handbook of Steel Construction recommends:

Elastic (vector) method: Conservative, suitable for preliminary design and simple patterns
Instantaneous centre (IC) method: More accurate, accounts for ductile load redistribution
C-value tables: Pre-computed IC capacities from CISC Handbook Part 5

Elastic Vector Method

Per the elastic method, eccentric shear is resolved into direct shear and torsional components:

Direct shear per bolt: V_dir = P / n

Torsional shear per bolt: V_tor = P × e × r_i / J_group

Where:

P = applied load
e = eccentricity (distance from load to bolt group centroid)
n = number of bolts
r_i = distance from bolt group centroid to bolt i
J_group = polar moment of inertia of bolt group = sum(r_i^2)

Resultant shear on critical bolt: V_resultant = vector sum of V_dir and V_tor

Elastic Method — Common Bolt Patterns

4-Bolt Pattern (Rectangular)

For 4 bolts forming a rectangle of a × b (spacing):

J = 4 × (a/2)^2 + 4 × (b/2)^2 = a^2 + b^2

Max V_to = P × e × sqrt((a/2)^2 + (b/2)^2) / (a^2 + b^2)

8-Bolt Pattern (2 rows × 4)

For bolts at 70 mm spacing with 2 rows at 80 mm gauge:

J = 4 × (35^2 + 105^2) + 4 × (40^2) × 2 = 4 × (1225 + 11025) + 4 × 1600 × 2 = 49,000 + 12,800 = 61,800 mm^2

Instantaneous Centre (IC) Method

The IC method (also called the ultimate strength method) accounts for the non-linear bolt load-deformation relationship:

Steps

Assume IC location: The centre of rotation is not at the bolt group centroid for eccentric loads
Calculate bolt deformations: Delta_i = r_i × theta where r_i = distance from IC to bolt i
Determine bolt forces: R_i from load-deformation curve: R_i = R_ult × (1 - e^(-10 × Delta_i))^0.55
Rotate forces: Each bolt force acts perpendicular to its radius from IC
Check equilibrium: Sum of bolt moments = P × e (external moment)
Iterate: Adjust IC location until equilibrium is achieved

C-Value Method (CISC Handbook)

The C-value is the ratio of the ultimate eccentric load to the bolt shear capacity:

P_ult = C × Vr_per_bolt

Where:

C = C-value from CISC Handbook Table 5-1 through 5-6
Vr_per_bolt = factored shear resistance per bolt

C-Values for Common Patterns

For A325M bolts, standard holes, threads excluded from shear plane:

4-Bolt Pattern (Vertical line)

e (mm)	C-value (vertical spacing = 75 mm)	C-value (vertical spacing = 100 mm)
100	2.11	2.13
150	1.68	1.73
200	1.38	1.44
250	1.17	1.23
300	1.01	1.07

For a 4-bolt bracket with e = 200 mm, vertical spacing = 75 mm, C = 1.38. If Vr per A325M M20 AA = 81.3 kN: P_ult = 1.38 × 81.3 = 112 kN per bracket web (one vertical line of 4 bolts).

Worked Example — 4-Bolt Bracket

Given: Bracket connection with 4-M20 A325M bolts in a vertical line at 75 mm centres. Bracket projects 200 mm horizontally. Eccentric load P (factored) = 100 kN at the bracket tip. Threads excluded (AX). All bolts in 350W steel.

Step 1 — Bolt Capacities: Vr AX per M20 A325M = 100.0 kN (threads excluded)

Step 2 — Elastic Vector Method: Bolt group centroid at mid-height of 4 bolts. n = 4, total height = 225 mm, centroid at 112.5 mm from top/bottom. Average r_i = sqrt(112.5^2) for top/bottom = 112.5 mm. Middle bolts: r = 37.5 mm. sum(r_i^2) = 2 × 112.5^2 + 2 × 37.5^2 = 25,312 + 2,812 = 28,125 mm^2

Direct shear per bolt: 100/4 = 25.0 kN Torsional shear on critical bolt (top or bottom): r_max = 112.5 mm V_tor = P × e × r_max / sum(r_i^2) = 100 × 200 × 112.5 / 28,125 = 80.0 kN

Resultant (vectors perpendicular because r is perpendicular to eccentricity direction): V_res = sqrt(25.0^2 + 80.0^2) = 83.8 kN

Step 3 — Check: V_res = 83.8 kN ≤ Vr AX = 100.0 kN. Ratio = 0.84. OK (elastic method).

Step 4 — IC Method (for verification): From CISC Table 5-1, for e = 200 mm, vertical spacing = 75 mm, 4 bolts: C = 1.38. P_ult = 1.38 × 100.0 = 138 kN (IC method) With P = 100 kN: ratio = 100/138 = 0.72.

Result: The IC method (0.72) gives a better ratio than the elastic method (0.84), confirming that the elastic method is conservative.

Large Bolt Groups

For bolt groups with more than 12 bolts, the CISC Handbook provides extended C-values:

For vertical lines of 4-12 bolts with various eccentricities and spacings
For multiple vertical rows (2 rows, 3 rows)
For staggered patterns with offset bolts

Efficiency of Multiple Rows

Number of Bolts	1 Row C-value	2 Rows (100 mm spacing) C-value	Efficiency Increase
4	1.38	2.60	88% (vs 2×)
6	1.62	3.10	91%
8	1.76	3.40	93%
10	1.84	3.58	95%

Adding a second vertical row of bolts increases the C-value close to the theoretical 2× provided the rows are sufficiently far apart. The IC method accounts for the increased torsional resistance of a wider bolt pattern.

Connection Ductility

Per CSA S16 Clause 27.5, connections in seismic force resisting systems must provide adequate ductility:

Bolt groups in seismic connections must be proportioned so that the connection deformation capacity exceeds the demand
The IC method implicitly accounts for ductility through the non-linear bolt load-deformation model
For seismic design, the bolt group should be designed for the overstrength of the connected members

Frequently Asked Questions

What is the difference between elastic vector and IC methods for bolt groups? The elastic vector method assumes the bolt group rotates about its centroid, with bolt forces proportional to distance from centroid. The IC method recognises that the rotation centre shifts (the instantaneous centre) due to non-linear bolt behaviour. The IC method gives 15-30% higher capacity than the elastic method for typical eccentricities. Both are valid under CSA S16, but the IC method is more economical.

How do I determine the C-value for a bolt group from the CISC Handbook? C-values are found in CISC Handbook Part 5, Tables 5-1 through 5-6. Enter the table with: (a) number of bolts, (b) vertical spacing, (c) eccentricity e, (d) number of vertical rows, and (e) horizontal spacing between rows. The C-value is multiplied by the single-bolt shear resistance to get the group eccentric load capacity.

What eccentricity should I use for a bracket connection? The eccentricity e is measured from the load point (typically the bracket tip or beam reaction point) to the centroid of the bolt group. For a bracket loaded at the tip, use the full horizontal projection distance. For a beam-to-column connection (shear tab), use the distance from the bolt line to the column face plus the distance from the column face to the bolt line in the beam web.

Can the elastic method be used for all bolt group designs? Yes, the elastic method is always conservative and can be used for any bolt group configuration. It is simple to calculate by hand. However, for large bolt groups or connections where every kN of capacity matters, the IC method provides more economical results. For seismic connections or where ductility demands are high, the IC method is preferred because it better represents actual connection behaviour.

This page is for educational reference. Bolt group analysis per CSA S16-19 Clause 13.12 and CISC Handbook. Verify C-values against current edition. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.

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