eccentric-bolt-group

name	eccentric-bolt-group
description	Eccentric bolt group analysis — in-plane shear + moment, instantaneous center of rotation, elastic method, AISC plastic (ultimate strength) method, prying action, high-strength bolt groups, AISC design tables.
metadata	{"priority":7,"promptSignals":{"phrases":["eccentric bolt group","bolt group eccentricity","instantaneous center","eccentric shear","bolt group moment","AISC bolt group"],"minScore":3}}

Eccentric Bolt Group Analysis — Complete Skill

Problem Definition

Eccentric load: resultant force acts at eccentricity e from centroid of bolt group → creates shear + moment at bolt group Two types:

1. In-plane eccentricity: force in plane of bolt pattern → additional rotational shear component
2. Out-of-plane eccentricity: moment causing tension in some bolts, prying action

Method 1: Elastic Method (In-Plane)

Step 1: Centroid of bolt group: x̄ = Σ(n_i × x_i) / N_total; ȳ = Σ(n_i × y_i) / N_total

Step 2: Direct shear (direct component): V_x = P_x / N; V_y = P_y / N [P = total applied shear; N = number of bolts]

Step 3: Torsional shear (due to moment M = P × e): Polar moment of inertia: J_m = Σ (x_i² + y_i²) [summation over all bolts; dimensions in mm²] Torsional shear on bolt i: V_Tx_i = -M × y_i / J_m [x-component] V_Ty_i = +M × x_i / J_m [y-component] (x_i, y_i = distance from centroid to bolt i)

Step 4: Resultant shear on critical bolt: V_Ri = √((V_x + V_Tx_i)² + (V_y + V_Ty_i)²) Critical bolt: largest V_Ri (usually farthest from centroid in direction of maximum shear)

Design check: V_R_max ≤ φ × r_n [φ = 0.75 (AISC LRFD); r_n = nominal shear capacity per bolt] φ × r_n = φ × F_nv × A_b [F_nv = nominal shear stress [ksi]; A_b = bolt area [in²]]

Bolt shear capacity (AISC Table J3.2):

Grade	F_nv (threads in plane)	F_nv (shear plane outside threads)
A325 (ASTM F3125 Grade A325)	48 ksi (330 MPa)	60 ksi (414 MPa)
A490 (ASTM F3125 Grade A490)	60 ksi (414 MPa)	75 ksi (517 MPa)

Method 2: Instantaneous Center Method (IC Method — AISC, More Accurate)

Concept: the bolt group rotates about an instantaneous center (IC); each bolt's deformation proportional to distance from IC; force direction perpendicular to line from IC to bolt

Iterative procedure (AISC Commentary C-J3.6):

1. Assume location of IC (try offset on e-line from centroid)
2. Calculate distance r_i from IC to each bolt i
3. Calculate deformation of each bolt: Δ_i = Δ_max × (r_i / r_max) [Δ_max = 0.34 in for A325 in 3/4" dia typical]
4. Calculate force from load-deformation curve: R_i = R_ult × (1 - exp(-10 Δ_i))^0.55 × (r_i / r_max) Or: R_i = R_ult × Δ_i^0.3 / Δ_max^0.3 (simplified)
5. Check moment equilibrium about IC: Σ (R_i × r_i) = P × e_IC
6. Check force equilibrium: Σ R_i,x = P_x; Σ R_i,y = P_y
7. Iterate IC location until equilibrium satisfied

AISC Design Tables (Table 8-2 through 8-11): pre-computed C coefficients: φR_n = C × φr_n [C = tabulated coefficient; function of bolt pattern geometry and eccentricity e] C values range 1.0–12+ depending on pattern size and geometry

IC method gives 20–30% more capacity than elastic method (used in AISC design tables)

Out-of-Plane Eccentricity (Bolt Tension + Prying)

Applied moment M = P × e (out-of-plane): Bolt group: bolts in tension resisting moment; compression face = bearing against connected plate

Elastic neutral axis: For simple cases (symmetric bolt row): NA at bottom bearing surface Bolt tension: T_i = M × d_i / Σ (d_j²) [d_i = distance from NA to bolt i] Critical bolt: maximum T_i (farthest from NA)

Prying action (AISC): Additional tension from prying of connected plate: T_total = T_direct + Q [Q = prying force] Prying ratio: Q/T ≈ 0 (stiff plate) to 0.33 (flexible plate)

Prying moment (AISC manual approach): a = edge distance (from bolt to plate edge); b = bolt-to-web distance; t = plate thickness δ = 1 - d_bolt/p [p = tributary width of plate per bolt row] ρ = b/a; α = 1/(ρ(1+δ)) [if α ≤ 1: prying limited; if α > 1: full prying] Prying force Q = T × δ × α' [α' = min(α, 1)]

Bolt tension capacity: φ × T_n = φ × F_nt × A_b [F_nt = 90 ksi (A325); 113 ksi (A490); φ = 0.75]

Combined shear + tension (AISC J3-3a): F'_nt = 1.3 F_nt - F_nt/(φ F_nv) × f_v ≤ F_nt [F'_nt = reduced tensile capacity due to shear]

Slip-Critical vs. Bearing Type

Bearing type: bolts allowed to slip; shear at bolt shear plane; capacity = F_nv × A_b (as above)

Slip-critical: no slip allowed; shear by friction Design slip resistance: φ × R_n = φ × μ × D_u × h_f × T_b × N_s μ = 0.35 (Class A surface); 0.50 (Class B — roughened); D_u = 1.13 (mean/specified pretension ratio) T_b = bolt pretension (A325 ¾": T_b = 28 kips; A490 ¾": T_b = 35 kips per AISC Table J3.1) N_s = number of slip planes

Design Example Summary

Given: bolt group with N bolts at pattern; P kips at eccentricity e

1. Calculate J_m = Σ (x_i² + y_i²)
2. M = P × e
3. Critical bolt: V_R = √[(P/N + M×y_max/J_m)² + (M×x_max/J_m)²]
4. Check V_R ≤ φ × F_nv × A_b → select bolt size/grade
5. Verify with IC method or AISC tables for final design

Standards

Standard	Scope
AISC Steel Construction Manual	Bolt group design (Chapter J, Tables 8-2 to 8-11)
ASTM F3125	High-strength structural bolts (A325, A490)
ASTM F1554	Anchor bolts
AISC Design Guide 2	Steel and composite beams (connection examples)
AWS D1.1	Structural welding (companion to bolt connections)

Output

Provide: bolt group configuration (N bolts, coordinates x_i, y_i [in or mm]), eccentricity e [in or mm], applied load P [kips or kN] and direction, centroid location (x̄, ȳ), polar moment J_m [in⁴ or mm⁴], moment M [kip·in or kN·mm], critical bolt forces (V_x, V_y, V_Tx, V_Ty, V_R) [kips or kN] by elastic method, bolt shear capacity φ×r_n [kips or kN], AISC C coefficient (from Table 8-2), bolt grade (A325/A490), diameter, bearing vs. slip-critical designation, prying force Q [kips] (if out-of-plane), combined tension + shear check, and applicable standard (AISC Manual Chapter J, ASTM F3125).