| 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:
- In-plane eccentricity: force in plane of bolt pattern → additional rotational shear component
- 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):
- Assume location of IC (try offset on e-line from centroid)
- Calculate distance r_i from IC to each bolt i
- 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]
- 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)
- Check moment equilibrium about IC: Σ (R_i × r_i) = P × e_IC
- Check force equilibrium: Σ R_i,x = P_x; Σ R_i,y = P_y
- 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
- Calculate J_m = Σ (x_i² + y_i²)
- M = P × e
- Critical bolt: V_R = √[(P/N + M×y_max/J_m)² + (M×x_max/J_m)²]
- Check V_R ≤ φ × F_nv × A_b → select bolt size/grade
- 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).
