Bolted joints under static and eccentric loading

Why it matters

Bolts hold motor mounts, gearbox covers, robot bases, linear-guide rails and machine frames together, and they are among the most common failure points in machines. A bolt that is too small, under-tightened or wrongly loaded will stretch, shear or work loose. Eccentric loads - a bracket carrying a load away from its bolt group - are the everyday case, and they load some bolts far more than others.

Key ideas

Thread basics. Metric threads are designated M d × pitch (e.g. M12 × 1.75, coarse). Tensile failure occurs at the threaded section, so the tensile stress area At (from the thread table; e.g. 84.3 mm² for M12 coarse), or approximately the core (minor) area π·dc²/4, is used - never the nominal shank area. Shear in a bolt is resisted by the shank area if the shank lies in the shear plane, otherwise by the core area.

Preload (initial tension). Tightening stretches the bolt and compresses the clamped parts. Good preload keeps the joint from separating, carries shear by friction, and greatly reduces the fluctuating load the bolt sees, improving fatigue life and resistance to loosening. Indian textbooks use the empirical estimate Pi = 2840·d (N, with d in mm) for a tight fluid joint; production design specifies torque or a target fraction of proof load from a standard.

Bolt and member stiffness. With bolt stiffness kb and member (clamped parts) stiffness km, an external separating load P is shared: the bolt sees only the fraction C = kb/(kb + km), the rest just reduces the clamp force. Because km is usually several times kb, C is small (typically 0.2-0.3). The joint separates when the clamp force reaches zero.

Eccentric load in the plane of the bolts (bracket bolted to a column face, bolts in shear). Move the load P to the centroid of the bolt group, adding a moment P·e:

• Primary (direct) shear: P/n on each bolt, parallel to P.
• Secondary (torsional) shear: proportional to the bolt's distance r from the centroid, perpendicular to that radius: P·e·r/Σr².
• Add the two vectorially; the most-loaded bolt is usually the one farthest from the centroid on the side where the two components act together. Assumptions: rigid bracket, identical bolts, elastic behaviour.

Eccentric load perpendicular to the bolt axis that tilts the bracket (bracket bolted to a wall, load parallel to the wall). The bracket tends to tip about an edge (the tilting edge). Bolt tension is proportional to its distance l from that edge: Fi = P·L·li/Σ(ni·li²) - where L is the distance of the load from the edge. The load also produces direct shear P/n in each bolt; combine tension and shear with a failure theory (e.g. max shear: τmax = √((σ/2)² + τ²)).

Other failure modes to check: stripping of nut threads, crushing (bearing) of plates, tearing of plates, and loosening under vibration (use preload, prevailing-torque nuts, thread-locking adhesive; plain washers do not prevent loosening).

Formulas

σt = P / At with At ≈ π·dc² / 4

• σt: tensile stress (Pa); P: bolt tension (N); At: tensile stress area from the thread table (m²); dc: core diameter (m).

τ = F / (π·d² / 4) (shank in shear)

• τ: shear stress (Pa); F: shear force on one bolt (N); d: shank diameter (m).

Pi = 2840·d (N, d in mm; empirical initial tension - check your data book)

C = kb / (kb + km) · Fb = Pi + C·P · Fm = Pi − (1 − C)·P · P_sep = Pi / (1 − C)

• kb, km: bolt and member stiffness (N/m); Fb: bolt load (N); Fm: clamp force on members (N); P_sep: external load that just separates the joint (N).

F′ = P / n · F″i = P·e·ri / Σr²

• F′: primary shear per bolt (N); F″i: secondary shear on bolt i (N); n: number of bolts; e: eccentricity of P from the group centroid (m); ri: distance of bolt i from the centroid (m).

F = √(F′² + F″² + 2·F′·F″·cos θ)

• θ: angle between the primary and secondary shear vectors on that bolt.

Fi = P·L·li / Σ(li²) (sum over all bolts)

• Fi: tension in bolt i (N); L: distance of P from the tilting edge (m); li: distance of bolt i from the tilting edge (m).

Worked examples

Example 1 (standard). An M12 bolt (At = 84.3 mm²) is preloaded to Pi = 10 kN. The external separating load is P = 8 kN and the joint constant C = 0.2. Find the bolt load, its stress, the remaining clamp force and the separating load.

1. Fb = Pi + C·P = 10 + 0.2 × 8 = 11.6 kN.
2. σt = Fb / At = 11,600 N / 84.3 mm² = 137.6 MPa.
3. Fm = Pi − (1 − C)·P = 10 − 0.8 × 8 = 3.6 kN (still clamped).
4. P_sep = Pi / (1 − C) = 10 / 0.8 = 12.5 kN.

Example 2 (GATE level). A bracket is fixed to a column by four identical bolts at the corners of a 100 mm × 100 mm square. A vertical load P = 10 kN acts in the plane of the joint at e = 200 mm from the group centroid. Find the largest bolt force and the bolt diameter for τ_allow = 80 MPa (shank in shear).

1. Primary shear: F′ = P/n = 10,000/4 = 2500 N (vertical, each bolt).
2. Radius of each bolt: r = √(50² + 50²) = 70.71 mm; Σr² = 4 × 70.71² = 20,000 mm².
3. Secondary shear: F″ = P·e·r/Σr² = 10,000 × 200 × 70.71 / 20,000 = 7071 N, perpendicular to each radius, i.e. at 45° to the vertical for every corner bolt.
4. The two bolts on the load side have F′ and F″ at θ = 45°: F = √(2500² + 7071² + 2 × 2500 × 7071 × cos 45°) = √(81.25 × 10⁶) = 9014 N.
5. Area: A = 9014 / 80 = 112.7 mm² → d = √(4 × 112.7/π) = 11.98 mm → choose a 12 mm shank (M12) or the next size up per your design table.

Common mistakes

• Using the nominal diameter area for tension instead of the stress area or core area.
• Adding primary and secondary shear as scalars without checking their directions.
• Treating an eccentrically loaded bolt group as "bending of one bolt" - the moment is resisted by the group through shear or tension distribution.
• Forgetting the factor that counts bolts in each row (Σ is over all bolts, so two bolts at the same li contribute 2·li²).
• Assuming the bolt carries the whole external load P when it is preloaded - it carries only C·P extra.
• Ignoring direct shear when the tilting-bracket bolts are designed for tension.

For GATE ME

Typical questions: the most-loaded bolt in an eccentric bolt or rivet group (primary plus secondary shear), bolt tension in a bracket tilting about an edge, and the bolt load or separation load of a preloaded joint given the stiffness ratio. Practise vector addition at the corner bolts and the Σr² bookkeeping quickly; these numericals are easy marks once the geometry is clear.

Quick check

1. In an in-plane eccentric bolt group, how does secondary shear vary with distance from the centroid?
2. Pi = 15 kN, C = 0.25. Find the external load that separates the joint.
3. Which area is used for the tensile stress in a bolt?
4. Do plain washers prevent loosening? Answers: 1. Directly proportional to r. 2. 15/0.75 = 20 kN. 3. Tensile stress area (≈ core area). 4. No - they spread the bearing load; loosening is prevented by preload and locking devices.