Bolt Preload and Torque: Method

Preloaded Bolted Joints: Method and Conventions

What the tool does

A tension joint is a bolt (or cap screw) clamping two or more members with a preload F_i, then loaded by a separating force P. The tool follows Chapter 8 of Shigley's Mechanical Engineering Design (10th ed.): thread areas and bolt strengths from the standard tables, the preload and the torque to reach it, the bolt and member stiffnesses, the division of the external load between bolt and members, the separation, proof-load and overload factors, the number of bolts for a required load factor, and the fatigue factors for a fluctuating load. Its default inputs are Shigley's Example 8-4, six 5/8 in-11 UNC grade 5 bolts holding a cast-iron pressure-vessel cover against 36 kip, which it reproduces (k_b 5.21 and k_m 8.95 Mlbf/in, C 0.368, F_i 14.4 kip, 5.52 bolts for a load factor of 2, and with six bolts n_L 2.18, n_p 1.16, n₀ 3.80).

Thread and strength tables

Unified inch threads (ASME B1.1, Shigley Table 8-2): tensile-stress area A_t = (π/4)(d − 0.9743/N)² and minor-diameter area A_r = (π/4)(d − 1.299038/N)² for N threads per inch. ISO metric threads (ISO 724, Table 8-1): pitch diameter d_p = d − 0.649519 p, minor diameter d_r = d − 1.226869 p, A_t = (π/4)[(d_p + d_r)/2]². The tool computes the areas from these formulas, which reproduce the printed tables to their last digit. Minimum proof, tensile and yield strengths come from SAE J429 (Table 8-9), ASTM A307, A325, A354, A449, A490 (Table 8-10) and ISO 898-1 (Table 8-11) by diameter range; a diameter outside a grade's range takes the nearest range and says so. The fully corrected endurance strength of rolled threads (Table 8-17: SAE 5 18.6, SAE 7 20.6, SAE 8 23.2 ksi; ISO 8.8 129, 9.8 140, 10.9 162, 12.9 190 MPa) is used for the fatigue check when the grade has one; other grades need an entered value. A custom thread or grade can be entered.

Grip, lengths and bolt stiffness

For a bolt and nut the grip l is the whole clamped stack: member 1 under the head, an optional gasket, member 2, and a hardened washer under the nut. For a cap screw in a tapped hole the clamped thickness above the thread is h and the effective grip is l = h + t₂/2 when the tapped member is thinner than the bolt diameter, else h + d/2 (Shigley Fig. 8-21). The bolt length may be entered; blank, the shortest standard length that leaves two threads beyond a regular hex nut (grip + 7/8 d + 2 p rounded up to 1/4 in or 5 mm, as Example 8-4 does with Table A-17) or h + 1.5 d for a cap screw is used. The thread length is 2d + 1/4 in for bolts up to 6 in long (2d + 6 mm up to 125 mm, eqs. 8-13, 8-14); the unthreaded length in the grip is l_d = L − L_T and the threaded length l_t = l − l_d. The bolt is two springs in series, k_b = A_d A_t E / (A_d l_t + A_t l_d) (eq. 8-17), with A_d the shank area and A_t the tensile-stress area.

Member stiffness

Shigley's pressure-cone model: the clamped material is a pair of conical frusta of half-apex angle 30°, each opening from its bearing face (diameter 1.5 d, the washer face) toward the mid-plane of the grip. A frustum of height t, small diameter D and modulus E has k = 0.5774 π E d / ln[(1.155 t + D − d)(D + d) / ((1.155 t + D + d)(D − d))] (eq. 8-20); the tool splits each cone at every change of material (plate, gasket, washer, tapped member), so stacks of different moduli are handled exactly as in Example 8-5 (steel washer and cover plate over a cast-iron base). For a one-material grip this reduces to k_m = 0.5774 π E d / [2 ln(5 (0.5774 l + 0.5 d)/(0.5774 l + 2.5 d))] (eq. 8-22). The Wileman, Choudhury and Green (1991) finite-element fit k_m = E d A exp(B d/l) with the constants of Table 8-8 (steel A 0.78715, B 0.62873; aluminum 0.79670, 0.63816; copper 0.79568, 0.63553; gray cast iron 0.77871, 0.61616; general 0.78952, 0.62914) is printed beside it and can be chosen instead; it is for identical members, and with a mixed stack the tool evaluates it with the series-equivalent modulus l / Σ(t_i/E_i). A gasket is treated as a confined layer; an unconfined soft gasket needs the gasket-stiffness method of Shigley Section 8-10, which the tool does not include.

Preload and torque

Proof load F_p = A_t S_p; recommended preload 0.75 F_p for a connection that will be reused, 0.90 F_p for a permanent one (eq. 8-31). Torque by the nut factor T = K F_i d (eq. 8-27), with K from Table 8-15 (0.30 nonplated black finish, 0.20 zinc-plated, 0.18 lubricated, 0.16 cadmium-plated, 0.12 anti-seize, 0.09 Bowman-Grip nuts) or entered. Torque by the friction equation (eq. 8-26): T = (F_i d_m/2)(tan λ + f sec α)/(1 − f tan λ sec α) + F_i f_c d_c/2 with d_m the pitch diameter, λ the lead angle, α = 30° the thread half-angle, f and f_c the thread and collar friction coefficients and d_c = 1.25 d the collar mean diameter; f = f_c = 0.15 gives K ≈ 0.20 (Example 8-3: 3.55 kip·in against 3.75 kip·in by K = 0.20). Motosh (1976) wrote the same torque as three terms, pitch F_i p/(2π), thread friction F_i f d_m/(2 cos α) and underhead friction F_i f_c d_c/2, which shows that about 90 % of the torque is spent on friction; the tool prints that split. Because friction is uncertain, a torque wrench holds the preload only within about ±25 % (turn-of-nut ±15 %, bolt stretch or ultrasonic ±5 %, Bickford); the tool takes the entered scatter, checks the bolt against the proof strength at the high preload and the separation factor at the low one, and reports VDI 2230's tightening factor α_A = F_max/F_min.

External load and factors

With the joint constant C = k_b/(k_b + k_m) (eq. 8-24) the bolt carries F_b = C P + F_i and the members keep F_m = F_i − (1 − C) P of clamp force (eq. 8-25); they separate at P₀ = F_i/(1 − C). Shigley's factors: yielding factor n_p = S_p A_t/(C P + F_i) (eq. 8-28), load factor n_L = (S_p A_t − F_i)/(C P) (eq. 8-29), the multiple of P that brings the bolt to its proof load, and separation factor n₀ = F_i/[P(1 − C)] (eq. 8-30). For a total load on N bolts the number needed for a required load factor is N = C n_L P_total/(S_p A_t − F_i) (Example 8-4). Margins of safety are written as n − 1 on each load path. The joint diagram (Figure 1) draws the bolt line of slope k_b from the origin and the member line of slope −k_m from the preload point; an external load moves both to the same added deflection P/(k_b + k_m) and the vertical gap between them is P.

Fatigue

For a load cycling between P_min and P_max per bolt the bolt stress has amplitude σ_a = C(P_max − P_min)/(2 A_t) and mean σ_m = C(P_max + P_min)/(2 A_t) + σ_i, σ_i = F_i/A_t (eqs. 8-35, 8-36; Shigley's repeated load has P_min = 0). The load line starts at the preload point (σ_i, 0), and each fatigue factor is the ratio of the alternating stress where that line meets the criterion to σ_a: Goodman σ_a/S_e + σ_m/S_ut = 1, which for P_min = 0 gives n_f = S_e(S_ut − σ_i)/[σ_a(S_ut + S_e)] (eq. 8-45); Gerber σ_a/S_e + (σ_m/S_ut)² = 1 (eq. 8-46); ASME-elliptic (σ_a/S_e)² + (σ_m/S_p)² = 1 (eq. 8-47). The tool solves the intersection for any P_min, so the three closed forms are the special case it checks against. The Langer first-cycle yield factor S_p/(σ_a + σ_m) (eq. 8-51) is printed with them. Goodman is the design value; Example 8-5 (5/8 in-11 grade 5 cap screws, 0 to 5 kip) gives 2.32 Goodman, 3.48 Gerber, 2.91 ASME-elliptic. The high preload is what makes a bolt fatigue-tolerant: the bolt sees only C times the load swing, and a larger k_m or smaller k_b (a longer, slimmer bolt) lowers C.

Conventions and limits

Forces are per bolt; a total load is divided equally among the bolts (eccentric loads and prying are not considered). Tension on the joint is positive. The strengths are the standards' minimum values; the endurance strengths of Table 8-17 already include the thread stress concentration and the Marin factors for rolled threads. Not covered: eccentric loads and prying action, unconfined gaskets (Section 8-10), shear joints (Section 8-12), thermal preload changes, embedding relaxation, thread stripping (Section 8-2 length-of-engagement rule), bolt bending under a non-parallel face, and the plastic-range (yield-controlled) tightening of VDI 2230.