Bolted, riveted and welded joints

Why it matters

Machine frames, pressure vessels, brackets, cylinder heads and structural connections are held together by bolts, rivets or welds. A joint is often the weakest link of an assembly, so a designer must know how each type fails, how to calculate its strength and efficiency, and how eccentric loads share out among fasteners.

Key ideas

Choice of joint.

• Bolted (threaded) joints can be taken apart. Used for cylinder heads, flanges, machine bases and anything needing maintenance.
• Riveted joints are permanent, need holes that weaken the plates, and need caulking to be leak-tight. Still used where welding is unsuitable (aircraft aluminium alloys, some structures) and where a joint must not crack through as a weld can.
• Welded joints are permanent, need no holes, are lighter and leak-tight, and are cheaper to make in production; but they bring residual stresses, distortion, heat-affected zones and poor fatigue behaviour at the weld toe, and need inspection.

Riveted joints. Lap joints (plates overlap) or butt joints (plates meet, with one or two cover straps); single, double or more rows. Terms: pitch p (along a row), back pitch (between rows), margin (hole centre to plate edge, about 1.5d). A joint per pitch length can fail by:

• tearing of the plate between holes: P_t = (p − d)·t·σ_t,
• shearing of the rivets: P_s = n·(π/4)·d²·τ (×2 for double shear; data books often allow about 1.875 rather than 2),
• crushing (bearing) of rivets or plate: P_c = n·d·t·σ_c. The joint strength is the least of these; efficiency compares it with the solid plate.

Bolted joints under tension. The bolt is tightened with an initial preload P_i. When an external load P is applied, the bolt carries only a share of it, set by the stiffness ratio: P_b = P_i + C·P, with C = k_b/(k_b + k_c). Stiff members (small C) protect the bolt from fluctuating loads, which is why gaskets that soften the joint are avoided in fatigue. Stress is calculated on the tensile stress area or the core (minor) diameter, not the nominal diameter. Initial tightening torque T ≈ 0.2·P_i·d for dry steel threads (the 0.2 is an empirical torque coefficient; take it from your data book).

Eccentrically loaded joints (in the plane of the joint). A load P at eccentricity e from the centroid of a group of n identical fasteners is replaced by a direct load P at the centroid plus a moment P·e.

• Primary (direct) shear: P/n on each fastener, parallel to P.
• Secondary shear from the moment: proportional to the distance rᵢ from the centroid and perpendicular to it: Fᵢ = P·e·rᵢ/Σr².
• Add the two vectorially; the fastener with the largest resultant decides the size.

Welded joints. Butt welds carry tension or compression across the throat equal to the plate thickness. Fillet welds are designed on the throat, t = s·cos 45° = 0.707 s for an equal-leg weld of size s, and are usually assumed to fail in shear on the throat area. A parallel fillet weld is loaded in shear; a transverse fillet weld is somewhat stronger but is conventionally designed with the same shear formula. Eccentric welded connections are treated like eccentric bolt groups, using the polar moment of the weld lines.

Formulas

Riveted joint per pitch length: P_t = (p − d)·t·σ_t, P_s = n·(π/4)·d²·τ, P_c = n·d·t·σ_c; η = min(P_t, P_s, P_c)/(p·t·σ_t). Symbols: p pitch (mm), d hole/rivet diameter (mm), t plate thickness (mm), n rivets per pitch length (count each shear plane), σ_t, τ, σ_c permissible tensile, shear and crushing stresses (MPa); forces in N.

Bolt in tension: σ = P/A_s (A_s stress or core area); preloaded joint P_b = P_i + C·P, C = k_b/(k_b + k_c).

Eccentric bolt/rivet group: primary F₁ = P/n; secondary F₂ᵢ = P·e·rᵢ/Σr²; resultant R = √(F₁² + F₂² + 2F₁F₂ cos θ), θ the angle between them.

Fillet weld: throat t = 0.707·s; parallel fillet capacity P = 0.707·s·L·τ per weld line (L total weld length, mm).

Butt weld: P = σ_t·t·L.

Worked examples

Example 1 (standard). A single-riveted lap joint joins 10 mm plates with 20 mm rivets (hole diameter 20 mm) at 60 mm pitch. Permissible stresses: σ_t = 80 MPa, τ = 60 MPa, σ_c = 120 MPa. Find the joint strength per pitch and its efficiency.

1. Tearing: P_t = (60 − 20) × 10 × 80 = 32 000 N.
2. Shearing (one rivet, single shear): P_s = (π/4) × 20² × 60 = 18 850 N.
3. Crushing: P_c = 20 × 10 × 120 = 24 000 N.
4. Solid plate: p·t·σ_t = 60 × 10 × 80 = 48 000 N.
5. Strength = least = 18 850 N; η = 18 850/48 000 = 0.393.
6. Joint strength 18.85 kN per pitch, efficiency ≈ 39.3 % (rivet shear governs).

Example 2 (GATE level). A bracket is fixed to a column by four identical bolts at the corners of a 120 mm × 120 mm square. A vertical load of 10 kN acts 200 mm from the centroid of the bolt group. Permissible shear stress in the bolts is 80 MPa. Find the force on the most heavily loaded bolt and the minimum shear diameter.

1. Distance of each bolt from the centroid: r = √(60² + 60²) = 84.85 mm; Σr² = 4 × 84.85² = 28 800 mm².
2. Primary shear: F₁ = 10 000/4 = 2500 N each, vertical.
3. Secondary shear: F₂ = P·e·r/Σr² = 10 000 × 200 × 84.85/28 800 = 5893 N, perpendicular to the radius.
4. For the two bolts nearer the load, F₂ makes 45° with F₁ and adds to it: R = √(2500² + 5893² + 2 × 2500 × 5893 × cos 45°) = 7862 N.
5. d = √(4R/(π·τ)) = √(4 × 7862/(π × 80)) = 11.2 mm.
6. R ≈ 7.86 kN; use a bolt with at least 11.2 mm diameter in the shear plane (choose from the thread table, using the core diameter if threads lie in the shear plane).

Common mistakes

• Calculating bolt stress on the nominal diameter instead of the core or stress area.
• Forgetting that a butt joint with two straps puts each rivet in double shear.
• Adding primary and secondary shear as scalars without considering their angle.
• Using the fillet leg size s instead of the throat 0.707s.
• Assuming the bolt in a preloaded joint carries the whole external load.
• Using the efficiency of a joint as strength/(solid plate) with a different pitch length in each.

For GATE PI

Expect riveted-joint strength and efficiency, eccentric bolt groups (finding the worst bolt), fillet-weld capacity from size and length, and bolt load in a preloaded joint from a stiffness ratio. Practise drawing the primary and secondary force vectors at each bolt before calculating.

Quick check

1. What is the throat of a 10 mm fillet weld?
2. A 10 mm rivet in single shear with τ = 250 MPa. Shear capacity?
3. Two parallel 8 mm fillet welds, each 100 mm long, τ = 80 MPa. Capacity?
4. In a preloaded bolted joint with C = 0.25, how much of a 4 kN external load does the bolt take?

Answers: 1. 7.07 mm; 2. 19.6 kN; 3. 90.5 kN; 4. 1 kN.