Authored by:Moo-Zung Lee West Hills, Calif. Edited by Kenneth J. Korane ken.korane@penton.com Key points:• Engineers traditionally use a laborious, trial-and-error process to calculate bolt stress from torque. • The universal torque table and torque coefficients link preload torque, stress, and clamping force. Moo-Zung Lee has a BSME from National Taiwan Univ., MSME from Univ. of Houston, and a PhD. from NYSU at Stony Brook. He has nearly 40 years experience in power-plant construction and dynamic and stress analyses of nuclear power-plant piping and aerospace and defense systems. |

Engineers must frequently calculate the bolt preload torque needed to securely hold parts together without overstressing joints. They mainly rely on torque equations and torque tables, time-consuming tools that were developed more than a half-century ago. Here is a user-friendly alternative.

**Generalized torque coefficient**

Bolt-preloading torque, *T*, consists of three components that share a common factor.

*T = (C _{L} + C_{t} + C_{c})(DF_{i}), *

where torque coefficients for lifting, thread friction, and collar friction, respectively, are:

*C _{L} = L/2πD,*

*C _{c} = µ_{c}D_{c}/2D*

Denoting the overall torque coefficient as *C = C _{L} + C_{t} + C_{c}*, the equation reduces to

*T = C(DF*.

_{i})This equation is similar to the classical approximation *T* = 0.2 *DF _{i}*that applies to coefficient of friction (c.o.f.) μ

_{c}= μ

_{t}= 0.15 for small screws. Torque coefficient,

*C*, however, is generalized to embrace all screw sizes and c.o.f.s. Only two factors affect the equations — bolt size and friction — provided μc = μt and

*D*= 1.25

_{c}*D*. The generalized torque-coefficients table lists values for

*C*for selected screw sizes and c.o.f.s. The data directly link torque and clamping force, thus eliminating the need to toil over the torque equation.

For example, let’s determine the clamping forces produced from 70 to 80 lbf-in. torque on a 0.250-28UNF screw. Assume a c.o.f. of 0.10 to 0.15 for both threads and nut collar.

From the table, torque coefficients for a 0.250-in.-diameter screw are *C _{max}* = 0.195915 at maximum friction

*µ*= 0.15; and

_{max}*C*= 0.138035 at minimum friction

_{min}*µ*= 0.10.

_{min}*F _{imin} = -T_{min}/C_{max}D* = 1,429 lbf, and

*F*= 2,318 lbf.

_{imax}= T_{max}/C_{min}DNote that data in the table are based on the assumptions that, µ* _{c}*= µ = µt and

*D*=1.25

_{c}*D*. If either of these conditions is violated, adjust the torque coefficient using:

*C’ = C* – 0.625µ + 0.5*µ _{c}D_{c}/D*.

The accompanying “Torque-coefficients” graph puts values from the table in perspective. The classical equation *T* = 0.2 *DF _{i}*is reasonably accurate for c.o.f. = 0.15 and size 0.25 in. and smaller screws; that is,

*C*≈ 0.2. Bolt size has a moderate effect on C but the effect of c.o.f. is substantial. Average torque coefficients for all sizes in the generalized table are close to a straight line:

*C*= 0.02 + 1.17μ. Therefore, engineers can use linear interpolation to determine

*C*across c.o.f.s for a specific bolt size. Typical values of c.o.f. are about 0.1 to 0.2.

Universal torque table

Applied torque concurrently produces both tensile and shear stresses in a bolt. Shear stress originates from the bolt twisting torque *T _{b}* where:

*T _{b} = C_{b}T/C* and

*C*– 0.625µ.

_{b}= C – C_{c}= CTensile stress, *σ _{t} = F_{i}/A_{t} = T/CDA_{t}.*

Shear stress τ = *T _{b}*(

*d*/2)/(π

_{r}*dr*

^{4}/32) = 16

*Tb*/π

*dr*

^{3}= (16/π

*dr*

^{3})

*(Cb/C)T.*

Engineers cannot judge a design’s safety by comparing this two-dimensional stress with material tensile strength. Conventional methods combine tension and shear stresses into an equivalent stress and compare it with material strength. A general notion is that if this equivalent stress reaches the material strength, the part fails. The maximum distortion energy criterion of failure stipulates that inelastic failure occurs when von Mises stress approaches the material yield stress.

For most bolt materials, von Mises stress criteria applies. Von Mises stress during preloading is:

*σ _{v}* = (

*σ*

_{t}^{2}+ 3τ

^{2})

^{0.5.}

It is straightforward, though laborious and error-prone, to calculate *σ _{v}* from

*T*using the

*σ*, τ, and σv equations, but there is no explicit method to calculate

_{t}*T*for a given

*σ*. Engineers traditionally use a trial-and-error method: Initially guess a

_{v}*T*to calculate

*σ*; compare the result with the target value; and if unsatisfactory, try again.

_{v}This is one of the most taxing tasks in bolt-stress analysis. And it happens to be important because engineers are often challenged to determine a torque that uses every bit of allowable stress based on material strength and the safety factor.

The following embedding technique circumvents the hurdle. First, embed τ into *σ _{t}*.

The proportional constant *ξv* is a multiplier to convert *σ _{t}* to

*σ*, while τ is embedded into

_{v}*σ*. Second, substitute

_{t}*T = C(DF _{i})* and

*Fi = A*

_{t}σ_{t}.The proportional constant, Kv, depends on bolt size and c.o.f., and could be tabulated in the same way as torque coefficients. Alternatively, universal torque values Tu can be tabulated for a fixed value of von Mises stress σu based on the relation Tu = Kvσu. The universal torque table does just that for σu = 100 ksi. This method replaces the traditionally laborious trial-and-error preload calculations with bidirectional data scaling:

*σ _{v}* =

*σ*, or

_{u}(T/T_{u})*T*=

*T*.

_{u}(σ_{v}/σ_{u})Here’s how it works. Say a 0.250-28UNF screw is preloaded to a maximum stress of 90 ksi. Assume c.o.f. ranges from 0.10 to 0.15 and torque tolerance is 0 to –15%. Determine the maximum and minimum torques and clamping forces. Von Mises stress in the universal torque table is σu = 100 ksi; for a 0.250-28UNF screw and c.o.f. = 0.10, *T _{u}* = 102 lbf-in.; and maximum allowable von Mises stress at preloading is σ

*= 90 ksi.*

_{max}Calculate maximum and minimum preload torques from:

*T _{max} = T_{u}(σ_{max}/σ_{u})* = 102(90/100) = 91.8 lbf-in. and

*T*= (1 – 0.15)

_{min}*T*= 0.85 × 91.8 = 78 lbf-in.

_{max}For a 0.250-28UNF screw, from the generalized torque coefficients table, *C _{max}* = 0.195915 for c.o.f. = 0.15 and

*C*= 0.138035 for c.o.f. = 0.10.

_{min}Maximum and minimum clamping forces are:

*F _{imax} = T_{max}/C_{min}D* = 91.8/(0.138035 × 0.25) = 2,654 lbf,

*F*= 78/(0.195915 × 0.25) = 1,593 lbf.

_{imin}= T_{min}/C_{max}DThe universal torque and torque coefficient link preload torque, stress, and clamping force. Given any one, determine the other two through simple math.

**Additional concerns**

**Clamping capability. **Determine the theoretical upper bound of clamping capability by pushing von Mises stress at assembly to the yield stress of the material.

Let σ_{v} = σ_{y} , because σ_{v} = ξ_{v}σ_{t}. Therefore, σ_{t}/σ_{y} = *F _{i}/A_{t}.*σ

_{y}= 1/ξ

_{v}. The “Clamping force” graph shows the clamping-to-yield-force ratio,

*F*σ

_{i}/A_{t}._{y}, depends on both bolt size and c.o.f. For example, for a 0.25-28UNF screw with c.o.f. = 0.15, the upper bound of clamping force is about 72% of the bolt yield strength in the threaded section. This percentage changes for other bolt sizes and c.o.f.s. Therefore, it would be fruitless to set one clamp/yield force ratio or tensile/yield stress ratio and hope to apply it uniformly. In contrast, one von Mises/yield stress ratio, σ

_{v}/σ

_{y}, applies to all sizes, c.o.f.s, and materials. In actual designs, the ratios are even smaller because of the safety factor.

**Adjustment for collar friction.** Torque values in the universal torque table are based on assumptions that µ_{t} = µ_{c} and *D _{c}*= 1.25

*D*. If one of these conditions is violated, first calculate C’ as shown in the previous torque-coefficient section and scale the torque accordingly. Adjusted torque is

*T’ = T(C’/C)*.

**Adjustment for mild steel bolts.** For lower-strength materials, the maximum shear-stress failure criterion applies. This states that when the maximum shear stress reaches half the material yield stress, the part fails. That leads to equivalent stress during preloading:

σ_{m} = (σ_{t}^{2} +4τ^{2})^{0.5}

Torque needs to be scaled down to maintain the same margin of safety.

For the screw sizes in the universal torque table, the torque reduction factor is: 0.98 > (ξv / ξm) > 0.88 for 0.08 ≤ µ ≤ 0.30.

Torque tables. Traditional torque tables typically list bolt-preload torques for a specific bolt-material strength and surface condition. It’s not uncommon for a project to require more than a dozen torque tables. Those tables are suitable for assembly shops but ineffective for stress analysis because they convey neither the von Mises stress produced from the torque nor the c.o.f. Engineers require the former to determine the margin of safety at assembly. And the latter is a key parameter to calculate stress, clamping force, and so on. The information must have been used to develop the torque tables but is not passed along with the tables.

A universal torque table, used in conjunction with torque coefficients, makes all essential information and processes transparent. Further, one universal torque table applies to all materials and surface conditions because preload control relates torque and von Mises stress irrespective of the material strength; and the c.o.f. is the core parameter in preload control irrespective of surface conditions.

**Margins of safety**. It’s good practice to limit preload to provide an adequate margin of safety. The safety margin = σy/(SFσv) – 1 > 0. SF is the safety factor at assembly, typically 1.1.

Design for preload in three steps:

1. Choose preload stress *σ _{v} < σ_{y}/S_{F}*.

2. Scale torque using the universal torque table.

3. Calculate clamping force using the torque coefficient.

Usually, it is not necessary to hold a portion of allowable stress in reserve to handle external loads because after preloading, there is no twisting shear and the bolt stress reduces from *σ _{v}* to

*σ*by the ratio shown in “Clamping force” chart. Also, the bolt shares only a fraction of the external load. Nevertheless, engineers must account for the effects of external load.

_{t}**Addressing external load**. Combine the initial force on the bolt from preload torque with the external force to determine the total load under service conditions. Bolt load in service is *F _{b}* =

*F*+ ϕFe; and clamp load is

_{i}*F*= -

_{c}*F*+ (1 – ϕ)

_{i}*F*.

_{e}Determine the bolt load-sharing ratio, ϕ, from stiffnesses of the bolt and clamped members. Typical values of ϕ are 0.2 to 0.5. Use the *F _{b}* equation to determine the margin of safety for the bolt. Use the

*F*equation to ensure that clamping force is sufficient to prevent joint leakage or separation. Generally, the clamping force has a margin of error resulting from tolerances in both c.o.f. and the wrench torque reading. Therefore, the

_{c}*F*equation uses the maximum

_{b}*F*, and the

_{i}*F*equation uses the minimum Fi in absolute terms. Preloading control is complete only after the margins of safety for both bolt load and clamping force are satisfied for all service conditions.

_{c}