10.2 Design of Threaded Fasteners, Welded & Riveted Joints

Design of Threaded Fasteners, Welded & Riveted Joints

Mechanical connections join separate structural and machine elements together. They are broadly categorized into separable/demountable fasteners (bolts, screws, studs) and permanent fasteners (welded and riveted joints). In mining machinery, crushers, and pressure vessels, improper connection design can lead to catastrophic joint separation, bolt fatigue rupture, or brittle weld fracturing.

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1. Threaded Fasteners & Bolted Joint Mechanics

1.1 Thread Geometry & Design Parameters

Standard ISO metric screw threads are designated as M d × p (e.g., M20 × 2.5 represents nominal major diameter $d = 20\text{ mm}$, pitch $p = 2.5\text{ mm}$ with $60^{\circ}$ included thread angle).

• Nominal / Major Diameter $(d)$: Largest diameter of the screw thread.
• Minor / Core Diameter $(d_1 / d_c)$: Smallest diameter at the root of the thread. For ISO metric threads: $d_1 = d - 1.2268p$.
• Pitch Diameter $(d_2)$: Theoretical diameter where thread width equals space width: $d_2 = d - 0.6495p$.
• Tensile Stress Area $(A_t)$: The effective cross-sectional area resisting axial tensile loading: $A_t = \frac{\pi}{4}\left(\frac{d_2 + d_1}{2}\right)^2 \approx \frac{\pi}{4}(d - 0.9382p)^2$

1.2 Bolt Preload, Tightening Torque & Proof Strength

Bolts must be properly tightened during assembly to clamp joint members together.

• Proof Load $(F_p)$: $F_p = A_t \cdot S_p$, where $S_p$ is proof strength (typically $S_p \approx 0.85 S_{yt}$).
• Initial Tightening Preload $(F_i)$:
  • Reusable joints (service disassembly required): $F_i = 0.75 F_p = 0.75 A_t S_p$
  • Permanent / static joints: $F_i = 0.90 F_p = 0.90 A_t S_p$
• Tightening Torque Equation: $T = K \cdot F_i \cdot d$ where $K$ is the non-dimensional torque coefficient ($K \approx 0.20$ for standard as-received black-finish steel bolts; $K \approx 0.15$ for lubricated threads; $K \approx 0.28$ for zinc-plated unlubricated threads).

1.3 Bolted Joint Elastic Interaction & Stiffness Ratio

When an external tensile force $P$ is applied to a preloaded bolted joint, the bolt stretches while the clamped members expand elastically from their compressed state.

   +-------------------------------------------------------------+
   |                  Bolted Joint Elastic Model                 |
   |                                                             |
   |   Bolt (Tension Spring, kb)    Clamped Members (Spring, km) |
   |   ======[ /\/\/\/\/\ ]======    ======[ /\/\/\/\/\ ]======   |
   |                                                             |
   |   Resultant Bolt Load:          Resultant Member Force:     |
   |   Pb = Fi + C * P               Pm = -Fi + (1 - C) * P      |
   +-------------------------------------------------------------+

• Bolt Stiffness $(k_b)$: $k_b = \frac{A_b E_b}{L_b}$, where $A_b$ is nominal shank area and $L_b$ is grip length.
• Member Stiffness $(k_m)$: Determined using the equivalent conical frustum method ($30^{\circ}$ half-apex angle) across clamped plates: $k_m = \frac{\pi E_m d \tan 30^{\circ}}{\ln\left[ \frac{(2t \tan 30^{\circ} + D - d)(D + d)}{(2t \tan 30^{\circ} + D + d)(D - d)} \right]}$
• Joint Stiffness Factor $(C)$: $C = \frac{k_b}{k_b + k_m}$
• Resultant Bolt Tension $(P_b)$: $P_b = F_i + C \cdot P$
• Resultant Clamping Force on Members $(P_m)$: $P_m = -F_i + (1 - C) \cdot P$
• Joint Separation Condition: When clamped members lose all compression ($P_m = 0$): $P_{\text{sep}} = \frac{F_i}{1 - C}$
• Fatigue Implications (Gasketed vs Non-Gasketed Joints):
  • In rigid metal-to-metal joints, member stiffness is very high relative to the bolt ($k_m \gg k_b$), giving a low stiffness ratio $C \approx 0.15 - 0.25$. When an external dynamic cyclic force $P$ fluctuates, the bolt experiences only $15-25%$ of the load amplitude, shielding it from fatigue failure.
  • In soft gasketed joints (e.g., asbestos/elastomer seals in low-pressure pipe flanges), gasket stiffness is extremely low, causing $k_m \ll k_b$ and $C \to 1.0$. Almost $100%$ of the external fluctuating load is transferred directly into the bolt shank, severely exacerbating fatigue susceptibility.

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2. Design of Welded Connections

Welded joints are permanent fusions classified into butt welds (groove welds) and fillet welds (lap joints, tee joints, corner joints).

2.1 Fillet Weld Geometry & Critical Throat Section

A fillet weld cross-section forms an approximate right-angled isosceles triangle where the perpendicular sides are the leg size $(s)$.

• Throat Thickness $(t)$: The minimum distance from the root of the weld to the hypotenuse face: $t = s \cdot \cos 45^{\circ} = \frac{s}{\sqrt{2}} \approx 0.707 s$
• Throat Area ($A_{\text{throat}}$): For a weld of length $L$: $A = t \cdot L = 0.707 s \cdot L$ (Note: For design calculations, effective length $L_{\text{eff}} = L_{\text{total}} - 2s$ to account for starting and stopping craters).

2.2 Transverse vs. Parallel (Longitudinal) Fillet Welds

1. Parallel Fillet Weld: The external load $P$ acts parallel to the longitudinal axis of the weld. The throat plane fails exclusively in longitudinal shear: $\tau = \frac{P}{0.707 s \cdot L} \quad (\text{Single weld}) \implies \tau = \frac{P}{1.414 s \cdot L} \quad (\text{Double parallel fillet})$
2. Transverse Fillet Weld: The external load $P$ acts perpendicular to the weld axis. The internal stress state is a combination of normal tensile stress and shear stress. However, by international standard design codes (AWS, IS 816), fillet welds are always designed based on the maximum shear stress at the $45^{\circ}$ throat section: $P_{\text{allow}} = 1.414 s \cdot L \cdot \tau_{\text{allow}} \quad (\text{Double transverse fillet})$ (Experimental Fact: Under static loading, transverse fillet welds are approximately $17%$ to $20%$ stronger than parallel fillet welds due to multi-axial constraint, but have lower ductility).

2.3 Eccentric Welded Connections in the Plane of Welds (Torsion)

When a load $P$ acts at an eccentricity $e$ in the plane of the weld group, it generates a combined primary direct shear force and a secondary torsional twisting moment $T = P \cdot e$.

          +=============================+
          |                             |
          |          [ G (c.g.) ]-------|---------> P (Load at eccentricity e)
          |             /               |
          |           r/                |
          |           /                 |
          |         [Weld Corner]       |
          +=============================+

1. Determine the Center of Gravity $(G)$ of the weld throat area.
2. Primary Direct Shear Stress $(\tau_1)$: Distributed uniformly across the total throat area $A = 0.707 s \sum L$: $\tau_1 = \frac{P}{A} = \frac{P}{0.707 s \sum L} \quad (\text{Acts parallel to load } P)$
3. Secondary Torsional Shear Stress $(\tau_2)$: Acts perpendicular to the radius vector $r$ drawn from $G$ to the point of interest: $\tau_2 = \frac{T \cdot r}{J_w} = \frac{(P \cdot e) \cdot r}{0.707 s \cdot J_u}$ where $J_u$ is the unit polar moment of inertia of the weld contour treated as lines $(J_w = 0.707 s \cdot J_u)$.
4. Resultant Shear Stress $(\tau_R)$: Evaluated at the most critically loaded corner (maximum $r$ and minimum angle $\theta$ between vectors): $\tau_R = \sqrt{\tau_1^2 + \tau_2^2 + 2\tau_1 \tau_2 \cos \theta} \le \tau_{\text{allow}}$

2.4 Eccentric Welded Connections Out of Plane (Bending)

When load $P$ acts at eccentricity $e$ perpendicular to the weld plane, it causes primary shear and bending: $\tau = \frac{P}{A} = \frac{P}{0.707 s \sum L}, \quad \sigma_b = \frac{M \cdot y}{I_w} = \frac{(P \cdot e) y}{0.707 s \cdot I_u}$ $\tau_{\max} = \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau^2} \le \tau_{\text{allow}}$

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3. Design of Riveted Joints & Sealing Mechanics

Riveted joints are permanent mechanical connections widely employed in structural steel trusses, boiler shells, and bridge spans.

3.1 Terminology & Proportions

• Pitch $(p)$: Distance between centers of adjacent rivets in the same row.
• Margin $(m)$: Distance between the center of a rivet hole and the nearest plate edge ($m = 1.5 d$).
• Transverse Pitch / Back Pitch $(p_b)$: Distance between adjacent rows in multi-row joints.
• Diagonal Pitch $(p_d)$: Distance between centers of adjacent rivets in staggered rows: $p_d = \frac{2p + d}{3}$ or $p_d = \sqrt{p_b^2 + (p/2)^2}$.
• Unwin's Formula: Determines rivet hole diameter $d$ from plate thickness $t$ (for $t \ge 8\text{ mm}$): $d = 6 \sqrt{t} \quad (\text{where } d \text{ and } t \text{ are in mm})$ (For thin plates $t < 8\text{ mm}$, $d$ is determined by equating crushing strength to shearing strength).

3.2 Failure Modes & Strength Formulations (per Pitch Length $p$)

1. Tearing Failure of the Plate along Rivet Line: The plate tears across the weakened cross-section containing rivet holes. $P_t = (p - d) \cdot t \cdot \sigma_t$
2. Shearing Failure of the Rivet:
   • Single Shear (Lap joint or single-strap butt joint): $P_s = n \cdot \frac{\pi}{4} d^2 \cdot \tau$
   • Double Shear (Double-strap butt joint):
     • Theoretical: $P_s = n \cdot 2 \cdot \frac{\pi}{4} d^2 \tau$
     • Indian Boiler Regulations (IBR): $P_s = n \cdot 1.875 \cdot \frac{\pi}{4} d^2 \tau$
3. Crushing / Bearing Failure of Rivet or Plate: High compressive bearing pressure crushes the cylindrical contact area between rivet shank and hole. $P_c = n \cdot d \cdot t \cdot \sigma_c$
4. Margin Shear-Out / Splitting of Plate Edge: Shearing of the plate margin behind the rivet. Prevented by maintaining standard margin $m \ge 1.5 d$.

3.3 Joint Efficiency ($\eta$)

The efficiency of a riveted joint is the ratio of the lowest joint strength to the strength of an unpunched solid plate of width equal to the pitch $p$: $P_{\text{solid}} = p \cdot t \cdot \sigma_t$ $\eta = \frac{\min(P_t, P_s, P_c)}{P_{\text{solid}}} \times 100\% = \frac{\min(P_t, P_s, P_c)}{p \cdot t \cdot \sigma_t} \times 100\%$

3.4 Caulking and Fullering Operations

In pressure vessels and boiler shells, riveted joints must be leak-proof.