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Bolted joints

VDI 2230 is a guideline for the calculation of bolted joints that has been internationally recognized and applied in practice for many years. It provides designers and calculation engineers a systematic procedure that uses calculation steps to enable functionally and operationally stable designs with extensive utilization of the bolt load capacity.

Strictly speaking, the specifications of this guideline only apply for steel bolts (screw thread with 60° flank angle) in highly-stressed and high-strength bolted joints (i.e.; for strength classes between 8.8 and 12.9 or 70 and 80) and force transmission of the operating load which is always introduced via the tensioned components. This generally consists of a static or dynamic axial force (i.e.; direction of action parallel to the bolt axis). Bending moments and shear forces can also occur. The underlying material properties only apply at room temperature. Furthermore, corrosion or impact and stochastic loads are not considered. The further away the calculated bolted joint is from the validity range of the guideline, the more uncertain the results and the greater the deviations.

The guideline generally does not eliminate the need for experimental or numerical FEM investigations to verify the results of the calculations. This is especially recommended for critical joints.

Calculation steps

All of the necessary factors are systematically analyzed in the calculation steps in order to avoid failure of the bolted joint. Knowledge of the operating force FB acting on the bolted joint is a basic prerequisite of the calculation. This is divided into an axial operating force FA, a shear force FQ, a torque MT, and possibly a bending moment MB, which are assumed to be known in the following.

R0 - rough calculation d

Table A7 can be used to determine recommended nominal bolt diameters d for strength classes 12.9, 10.8, and 8.8 based on the load (axial force, shear force), the load type (static, dynamic), the type of tensioning (centric, eccentric), and the accuracy of the tightening method used.

R1 - tightening factor αA

The tightening factor αA takes into account the scatter of the achievable assembly preload between FMmin and FMmax. It can be determined with Table A8.

Equation 221.
αA=FMmaxFMmin\alpha_A=\frac{F_{M\max}}{F_{M\min}}


R2 - minimum clamp load FKerf

The required minimum clamp load FKerf is determined while taking into account the required friction grip to transmit a transverse load FKQ, the sealing against a medium FKP, and the prevention of opening FKA of the joint.

Equation 222.
KKerf≥max(FKQ;FKP+FKA)K_{Kerf}\ge\max(F_{KQ};F_{KP}+F_{KA})


R3 - Dividing of the working load into FSA, FPA, Φ

Calculation of the elastic resiliences of the connecting elements, bolt δS, and the clamped parts δP make up the core of the bolt calculation. An estimation of the load introduction factor n is also required.

This can be used to calculate the load factor φ, which is generally calculated from:

Equation 223.
ϕen*=n⋅δP**+δPZuδS+δP*\phi_{en}^*=n\cdot\frac{\delta_P^{**}+\delta_{PZu}}{\delta_S+\delta_P^*}


The load factor φ can be used to calculate the additional bolt force FSA and the additional load of the clamped parts FPA:

Equation 224.
FSA=ϕ⋅FAF_{SA}=\phi\cdot F_A


Equation 225.
FPA=(1-ϕ)⋅FAF_{PA}=(1-\phi)\cdot F_A


R4 - preload changes FZ, ΔF'Vth

Embedding of all joining surfaces occurs after tightening and brief operation. The following applies for the losses of preload FZ of a bolted joint as a result of embedding:

Equation 226.
FZ=fZ(δS+δP)F_Z=\frac{f_Z}{(\delta_S+\delta_P)}


Guide values for the amounts of embedding fZ together with the occurring roughness height Rz can be determined from Table 5. In the case of thermally stressed bolted joints, changes in preload ΔF'Vth can occur as a result of different coefficients of thermal expansion and heating of bolt and clamped components. The temperature influence is neglected in this first implementation of the bolt calculation.

R5 - minimum assembly preload FMmin

The required minimum assembly preload FMmin is obtained while taking into account preload changes and assuming the greatest possible relief of the joint.

Equation 227.
FMmin=FKerf+(1-ϕen*)⋅FAmax+FZ+ΔFVth′F_{M\min}=F_{Kerf}+(1-\phi_{en}^*)\cdot F_{A\max}+F_Z+\Delta F_{Vth}'


R6 - maximum assembly preload FMmax

Depending on the tightening method, the possible maximum assembly preload is calculated as:

Equation 228.
FMmax=αA⋅FMminF_{M\max}=\alpha_A\cdot F_{M\min}


R7 - assembly stress σredM, FMzul

The aim is to utilize the bolt strength to the greatest possible extent. In the event that only a proportion of the minimum yield point Rp 0.2 min of the bolt (normally 90 %) is allowed to be utilized for the comparative stress in the assembly state, the following applies with the utilization factor ν:

Equation 229.
σred,Mzul=ν⋅Rp0.2min\sigma_{red,Mzul}=\nu\cdot R_{p0.2\min}


The assembly preload permitted for the bolt selected is calculated as:

Equation 230.
FMzul=A0⋅ν⋅Rp0.2min1+3[32d2d0(Pπ⋅d2+1.155μGmin)]2F_{Mzul}=A_0\cdot\frac{\nu\cdot R_{p0.2\min}}{\sqrt{1+3[\frac{3}{2}\frac{d_2}{d_0}(\frac{P}{\pi\cdot d_2}+1.155\mu_{G\min})]^2}}


It must be verified that the bolt size roughly estimated in R0 can continue to be used:

Equation 231.
FMzul≥FMmaxF_{Mzul}\ge F_{M\max}


If this is not the case, either a larger bolt nominal diameter or a higher strength grade, a different tightening method, or a reduction in the friction or the external loading  must be selected.

R8 - working stress σredB, SF

If the yield point is exceeded, plastic tightening is used in the calculation in contrast to the typical elastic tightening, as shown in the following. For the working state, the total bolt load FSmax is calculated as:

Equation 232.
FSmax=FMzul+ϕen*⋅FAmax-ΔFVth′F_{S\max}=F_{Mzul}+\phi_{en}^*\cdot F_{A\max}-\Delta F_{Vth}'


The maximum tensile stress is obtained from:

Equation 233.
σzmax=FSmax/A0\sigma_{z\max}=F_{S\max}/A_0


and the maximum torsional stress is given by:

Equation 234.
τmax=MG/WP\tau_{\max}=M_G/W_P


with

Equation 235.
MG=FMzuld22(Pπ⋅d2+1.155μGmin)M_G=F_{Mzul}\frac{d_2}{2}(\frac{P}{\pi\cdot d_2}+1.155\mu_{G\min})


and

Equation 236.
WP=π16d03W_P=\frac{\pi}{16}d_0^3


For the reduced or comparative stress with a torsional stress reduced to kτ in service (recommendation: kτ = 0.5):

Equation 237.
σred,B=σzmax2+3(kτ⋅τmax)2\sigma_{red,B}=\sqrt{\sigma_{z\max}^2+3(k_{\tau}\cdot\tau_{\max})^2}


The following must apply:

Equation 238.
σred,B<Rp0.2min\sigma_{red,B}<R_{p0,2\min}


Equation 239.
SF=Rp0.2min/σred,B≥1.0S_F=R_{p0.2\min}/\sigma_{red,B}\ge1.0


R9 - alternating stress σa, σab, SD

For checking the alternating stress, the stress amplitude must be determined with the stress occurring with minimum and maximum axial force. General:

Equation 240.
σa=FSAo-FSAu2AS\sigma_a=\frac{F_{SAo}-F_{SAu}}{2A_S}


Eccentric:

Equation 241.
σab=σSAbo-σSAbu2\sigma_{ab}=\frac{\sigma_{SAbo}-\sigma_{SAbu}}{2}


The tolerable stress must also be calculated.

For bolts rolled before heat treatment (SV):

Equation 242.
σASV=0.85(150/d+45)\sigma_{ASV}=0.85(150/d+45)


For bolts rolled after heat treatment (SG):

Equation 243.
σASG=(2-FSm/F0.2min)⋅σASV\sigma_{ASG}=(2-F_{Sm}/F_{0.2\min})\cdot\sigma_{ASV}


The following must apply:

Equation 244.
σa/ab≤σAS\sigma_{a/ab}\le\sigma_{AS}


Equation 245.
SD=σAS/σa/ab≥1.0S_D=\sigma_{AS}/\sigma_{a/ab}\ge1.0


R10 - surface pressure pmax, SP

In the contact surface between the bolt head and nut on the one hand and the clamped part on the other hand, surface pressures which cause creep (time-dependent plastic flowing) in conjunction with a loss of preload should not become effective either as a result of assembly preload or the maximum load in service. The calculated surface pressure should therefore not exceed the limiting surface pressure of the clamped material.

Assembly state:

Equation 246.
pMmax=FMzul/Apmin≤pGp_{M\max}=F_{Mzul}/A_{p\min}\le p_G


Working state:

Equation 247.
pBmax=(FVmax+FSAmax-ΔFVth)/Apmin≤pGp_{B\max}=(F_{V\max}+F_{SA\max}-\Delta F_{Vth})/A_{p\min}\le p_G


The following must apply:

Equation 248.
SP=pG/pM/Bmax≥1.0S_P=p_G/p_{M/B\max}\ge1.0


R11 - minimum length of engagement meffmin

To prevent bolted joints failing due to stripping of mating threads, the bolt thread must be adequately covered by the nut thread or internal (female) thread. It follows from this that the maximum tensile force of the bolt must be less than the critical stripping force of the internal or bolt thread.

The following must apply:

Equation 249.
FmS≤(FmGM,FmGS)F_{mS}\le(F_{mGM},F_{mGS})


Equation 250.
Smeff=meff,vorh/meff,minS_{meff}=m_{eff,vorh}/m_{eff,\min}


R12 - slipping, shearing SG, τQmax

Transverse loads occurring in a bolted joint are to be transmitted by friction grip. Failure of the joint due to shearing or exceeding the permissible bolt bearing stress must be ruled out.

Occurring minimum residual clamp load:

Equation 251.
FKRmin=FMzulαA-(1-ϕen*)FAmax-fZ-ΔFVthF_{KR\min}=\frac{F_{Mzul}}{\alpha_A}-(1-\phi_{en}^*)F_{A\max}-f_Z-\Delta F_{Vth}


Required clamp load:

Equation 252.
FKQerf=FQmaxqF⋅μTmin+MYmaxpM⋅ra⋅μTminF_{KQerf}=\frac{F_{Q\max}}{q_F\cdot\mu_{T\min}}+\frac{M_{Y\max}}{p_M\cdot r_a\cdot\mu_{T\min}}


The following must apply:

Equation 253.
FKRmin>FKQerfF_{KR\min}>F_{KQerf}


Equation 254.
SG=FKRminFKQerf>1.0S_G=\frac{F_{KR\min}}{F_{KQerf}}>1.0


Equation 255.
τQmax=FQmax/Aτ\tau_{Q\max}=F_{Q\max}/A_{\tau}


The following must apply:

Equation 256.
τQmax<τB\tau_{Q\max}<\tau_B


or

Equation 257.
FQmax<tauB⋅Aτ=Aτ⋅Rm⋅(τB/Rm)F_{Q\max}<tau_B\cdot A_{\tau}=A_{\tau}\cdot R_m\cdot(\tau_B/R_m)


Equation 258.
SA=τBτQmax=τB⋅AτFQmax≥1.1S_A=\frac{\tau_B}{\tau_{Q\max}}=\frac{\tau_B\cdot A_{\tau}}{F_{Q\max}}\ge1.1


R13 - tightening torque MA

The tightening torque can be calculated as follows:

Equation 259.
MA=FMzul[0.16⋅P+0.58⋅d2⋅μGmin+DKm2μKmin]M_A=F_{Mzul}[0.16\cdot P+0.58\cdot d_2\cdot\mu_{G\min}+\frac{D_{Km}}{2}\mu_{K\min}]


This calculation is not applicable for torsion-free tightening.

Examples

The following examples are included in the FVA-Workbench under minimal models:

Notes and outlook

The current version is the initial implementation. It will be expanded as needed in the future. Possible next extensions include:

  • Different Young's moduli of clamped parts

  • Thermal preload changes

  • Washers

  • Dynamic strength within the fatigue strength range

Bolt and nut standards can be added by request. Optional inputs and outputs can be extended at any time.

Sources

Bolt calculation standards

  • VDI 2230:2015 Part 1, "Systematic calculation of highly stressed bolted joints - Joints with one cylindrical bolt," VDI Verlag Düsseldorf

  • VDI 2230:2014 Part 2, "Systematic calculation of highly stressed bolted joints - Multi bolted joints," VDI Verlag Düsseldorf

Hole standard

  • DIN EN 20273:1992, "Fasteners; clearance holes for bolts and screws“

Material properties

  • DIN EN ISO 898-1:2013, "Mechanical properties of fasteners made of carbon steel and alloy steel - Part 1: Bolts, screws and studs with specified property classes - Coarse thread and fine pitch thread“

Thread standards

  • DIN 13:1999 Parts 1 through 9, "ISO general purpose metric screw threads," DIN Deutsches Institut für Normung e. V., Berlin

Bolt standards

  • DIN EN ISO 4014:2011, "Fasteners - Hexagon head bolts - Product grades A and B“

  • DIN EN ISO 4017:2015, "Fasteners - Hexagon head screws - Product grades A and B"

  • DIN EN ISO 4762:2004, "Hexagon socket head cap screws"

  • DIN EN ISO 8676:2011, "Fasteners - Hexagon head screws, with fine pitch thread - Product grades A and B"

  • DIN EN ISO 8765:2011, "Fasteners - Hexagon head bolts, with fine pitch thread - Product grades A and B"

  • DIN EN ISO 12474:2011, "Hexagon socket head cap screws with metric fine pitch thread"

Nut standards

  • DIN EN ISO 4032:2022, "Fasteners - Hexagon regular nuts (style 1)"

  • DIN EN ISO 4033:2022, "Fasteners - Hexagon high nuts (style 2)"

  • DIN EN ISO 4035:2022, "Fasteners - Hexagon thin nuts (style 0)"

  • DIN EN ISO 8673:2022, "Fasteners - Hexagon regular nuts (style 1), with fine pitch thread"

  • DIN EN ISO 8674:2022, "Fasteners - Hexagon high nuts (style 2), with fine pitch thread"

  • DIN EN ISO 8675:2022, "Fasteners - Hexagon thin nuts (style 0), with fine pitch thread"

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