Design Problem

Problem: Design the bolt group for a bracket connection transferring 180 kN vertical shear at an eccentricity of 250 mm from the bolt group centroid. The bracket is welded to a column flange using a vertical line of bolts.

Bolt group configuration:

• Arrangement: 2 columns x 4 rows of M20 bolts (8 bolts total)
• Horizontal pitch p1 = 90 mm (gauge between columns)
• Vertical pitch p2 = 75 mm (spacing between rows)
• Bolt class: 8.8 (fyb = 640 MPa, fub = 800 MPa)
• Eccentricity e = 250 mm (horizontal distance from load line to bolt group centroid)

Design to BS EN 1993-1-8 with UK NA:

• gamma_M2 = 1.25 (UK NA, same as recommended)
• Slip-resistant connection not required (bearing-type bolts in shear)
• Bolt holes: 22 mm diameter for M20 bolts (2 mm clearance per BS EN 1090-2)

Step 1: Bolt Group Properties

Coordinates of bolts (origin at bolt group centroid):

Polar second moment of area:

Sum(xi2 + yi2) for all bolts:

• Per row contribution: 2 x (452 + yi2)
• Row 1 (yi = +112.5): 2 x (2,025 + 12,656) = 29,362 mm2
• Row 2 (yi = +37.5): 2 x (2,025 + 1,406) = 6,862 mm2
• Row 3 (yi = -37.5): 2 x (2,025 + 1,406) = 6,862 mm2
• Row 4 (yi = -112.5): 2 x (2,025 + 12,656) = 29,362 mm2

Sum(r2) = 29,362 + 6,862 + 6,862 + 29,362 = 72,448 mm2

Step 2: Elastic Vector Method

The elastic method superimposes direct shear (equal on all bolts) and torsional shear (proportional to distance from centroid).

Applied actions at bolt group centroid:

• Direct vertical shear: Vy = 180 kN (downward)
• Torsional moment: Mz = Vy x e = 180 x 0.250 = 45.0 kN·m

Direct shear per bolt (vertical component, equal all bolts):

Fv,Ed(dir) = Vy / n = 180 / 8 = 22.5 kN (vertically downward)

Torsional shear per bolt:

The torsional shear vector at bolt i is perpendicular to the radius vector ri. Components:

For bolt at (xi, yi), torsional shear: Fv,Ed(torsion),x = Mz x yi / Sum(r2) Fv,Ed(torsion),y = -Mz x xi / Sum(r2)

Critical bolt — outer top bolt (x = -45, y = +112.5):

Fv,Ed(torsion),x = 45.0 x 106 x 112.5 / 72,448 x 103 = 69.9 kN (horizontal, rightward) Fv,Ed(torsion),y = -45.0 x 106 x (-45) / 72,448 x 103 = 28.0 kN (vertical, upward)

Resultant shear per bolt (vector sum):

Fv,Ed,x = 69.9 kN Fv,Ed,y = -22.5 + 28.0 = 5.5 kN (net upward)

Fv,Ed = sqrt(69.92 + 5.52) = sqrt(4,886 + 30.3) = 70.1 kN

Step 3: Bolt Shear Resistance — EN 1993-1-8 Table 3.4

For M20, Class 8.8 bolt (single shear, thread in shear plane):

Tensile stress area As = 245 mm2 (M20, coarse pitch) Shear plane passes through thread (conservative): alpha_v = 0.6

Fv,Rd = alpha_v x fub x As / gamma_M2 = 0.6 x 800 x 245 / 1.25 = 94.1 kN

Utilisation (elastic method): 70.1 / 94.1 = 0.745 (74.5 %) — acceptable.

Step 4: Instantaneous Centre of Rotation Method

The ICR method accounts for bolt deformation compatibility and provides a more economical result for ductile bolt groups (Class 8.8 satisfies ductility requirements per EN 1993-1-8 Clause 3.6.1).

Methodology (per SCI P398):

The ICR is located by iteratively finding the point about which the applied moment equals the resisting moment from bolt forces. Each bolt force is proportional to its deformation, which is proportional to its distance from the ICR times the rotation.

For this 2 x 4 bolt group with e = 250 mm and 75 mm vertical pitch:

• ICR location: approximately 120 mm above the bolt group centroid (determined iteratively)
• Distances from ICR: ri' for each bolt
• Maximum bolt deformation: delta_max = 8.6 mm (ductility limit for M20, 8.8)

Resultant maximum bolt force (ICR method):

Fv,Ed,max (ICR) = 62.8 kN (approximately 10 % reduction from elastic method)

Utilisation (ICR method): 62.8 / 94.1 = 0.667 (66.7 %)

Step 5: Bolt Bearing Resistance — EN 1993-1-8 Table 3.4

Bracket plate: 12 mm thick S275 (connected ply):

For M20 bolt, end distance e1 = 40 mm, edge distance e2 = 35 mm, pitch p1 = 90 mm, p2 = 75 mm:

alpha_d = min(e1 / (3d0), p1 / (3d0) - 0.25, fub / fu, 1.0) = min(40 / 66, 90 / 66 - 0.25, 800 / 410, 1.0) = min(0.606, 1.114, 1.951, 1.0) = 0.606

k1 = min(2.8 x e2 / d0 - 1.7, 1.4 x p2 / d0 - 1.7, 2.5) = min(2.8 x 35 / 22 - 1.7, 1.4 x 75 / 22 - 1.7, 2.5) = min(2.75, 3.07, 2.5) = 2.5

Fb,Rd = k1 x alpha_d x fu x d x t / gamma_M2 Fb,Rd = 2.5 x 0.606 x 410 x 20 x 12 / 1.25 = 119.2 kN

Bearing utilisation (critical bolt): 70.1 / 119.2 = 0.588 (58.8 %) — acceptable.

Bearing governs over bolt shear (as is typical for thin bracket plates in UK practice).

Step 6: Block Tearing — EN 1993-1-8 Clause 3.10.2

For the bolt group subject to eccentric shear, check block tearing of the bracket plate:

Tension area (outer bolt row to top edge):

Ant = (e2 - d0/2) x t x 2 bolts = (35 - 11) x 12 x 2 = 576 mm2

Shear area (along bolt lines):

Anv = (3p2 + e1 - 3.5d0) x t x 2 = (225 + 40 - 77) x 12 x 2 = 4,512 mm2

Block tearing resistance:

Veff,1,Rd = fu x Ant / gamma_M2 + (1/sqrt(3)) x fy x Anv / gamma_M0 = 410 x 576 / 1.25 + (1/1.732) x 275 x 4,512 / 1.00 = 188,928 + 715,900 = 904.8 kN

180 kN << 904.8 kN — block tearing does not govern.

Step 7: UK Practice Notes — SCI P358 and SCI P398

Summary of Bolt Group Design

The bolt group is adequate. Bearing on the 12 mm bracket plate governs the design. Consider increasing the plate thickness to 15 mm if repeated loading or fatigue is a concern.

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Frequently Asked Questions

When should I use the ICR method instead of the elastic method for UK bolt groups?

The ICR method is appropriate when: (1) the bolts are ductile (Class 8.8 or lower per EN 1993-1-8 Clause 3.6.1); (2) the design is governed by bolt shear, not bearing; (3) the eccentricity is large relative to the bolt group dimensions (e > bolt group depth). SCI P398 permits the ICR method for moment-resisting joints. For simple connections to SCI P358, the elastic method is standard. UK checking engineers often prefer the elastic method because its assumptions are transparent and conservative.

What is the maximum bolt spacing in UK practice?

EN 1993-1-8 Table 3.3 specifies maximum spacing limits: 14t or 200 mm in the direction of load for compression plies, and no explicit limit for tension (but 14t is typically applied). The purpose is to prevent plate buckling between widely spaced bolts and to ensure uniform load distribution. For typical UK bracket plates (10-15 mm thick), the practical maximum spacing is 140-200 mm. SCI P358 recommends bolt spacing of 60-120 mm for standard simple connections.

How do oversized holes affect bolt group capacity in UK design?

BS EN 1090-2 permits oversized holes only by specification. For standard clearance holes (d + 2 mm for M12-M24), bearing resistance is calculated normally. For oversized holes (d + 3-4 mm), the bearing coefficient alpha_d should be reduced per EN 1993-1-8 Table 3.4 Note 2. Oversized holes also affect the slip factor for preloaded bolts: the slip factor mu is reduced from 0.5 to 0.3 for oversized holes unless surface preparation is specified per EN 1090-2 Annex G.

Related Pages

• UK Bolt Group Capacity — Full Guide — EN 1993-1-8 methods
• UK Bolt Capacity Tables — Class 8.8 and 10.9 bolt resistances
• UK Bolt Spacing Guide — Edge and end distances
• UK Bolt Grades — BS EN ISO 898-1 and BS EN 14399
• UK Connection Design Guide — EN 1993-1-8 bolted and welded joints
• UK Bracket Connection Design — Gusset plate design

Educational reference only. All design values are per BS EN 1993-1-8:2005 + UK National Annex, SCI P358, and SCI P398. Verify all values against the current editions of the standards and the applicable National Annex for your project jurisdiction. Designs must be independently verified by a Chartered Structural Engineer registered with the Institution of Structural Engineers (IStructE) or the Institution of Civil Engineers (ICE). Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent professional verification.