# The bolt connects 2 flanges (torque load)

Serious bolt failures often occur in the operation of machines and machinery. In some cases, this phenomenon occurs even when the designer is convinced of sufficient safety of the bolted connection (given the external load conditions). The failure is most often caused by the underestimation of the limit state of the bolt connection or by the lack of reliable data for a thorough analysis of the strength conditions of the connection. These are especially experimentally obtained data, which are necessary both for the theoretical calculation and for the determination of the optimal design of the bolt connection. Fig. 1 the bolt connects 2 flanges (torque load)

Axial stress the bolt:

$\sigma =\frac{{F}_{Bmax}}{\frac{\pi }{4}{\left(\frac{{d}_{b2}+{d}_{b3}}{2}\right)}^{2}}$

σ - axial stress the bolt - [MPa]

FBmax - maximum axial force in the bolt - [N]

db2 - medium diameter - [mm]

db3 - smaller external thread diameter - [mm]

Shear stress the bolt:

$\tau =\frac{M}{\frac{\pi }{16}{\left(\frac{{d}_{b2}+{d}_{b3}}{2}\right)}^{3}}$

τ - shear stress the bolt - [MPa]

M - the bolt tightening torque - [Nm]

db2 - medium diameter - [mm]

db3 - smaller external thread diameter - [mm]

Bending stress the bolt:

${\sigma }_{B}=\frac{{M}_{B}}{\frac{\pi }{32}{\left(\frac{{d}_{b2}+{d}_{b3}}{2}\right)}^{3}}$

σB - bending stress the bolt - [MPa]

MB - bending moment the bolt - [Nm]

db2 - medium diameter - [mm]

db3 - smaller external thread diameter - [mm]

Bending moment the bolt:

${M}_{B}=\frac{{d}_{B3}^{2}*{F}_{Bmax}*arc\psi *\sqrt{E*\pi }}{8\sqrt{{F}_{Bmax}}*tanh\left(\frac{8s}{{d}_{B3}^{2}}*\sqrt{\frac{{F}_{Bmax}}{E*\pi }}\right)}$

MB - bending moment the bolt - [Nm]

db3 - smaller external thread diameter - [mm]

FBmax - maximum axial force in the bolt - [N]

ψ - angular displacement of perpendicular contact surface of the bolt head - [rad]

E - Young’s modules - [MPa]

s - flange thickness - [mm]

Maximal shear stress (Tresca) the bolt:

${\sigma }_{tresca}=\sqrt{{\sigma }^{2}+{\sigma }_{B}^{2}+4{\tau }^{2}}\le {\sigma }_{Call}$

σtresca - maximal shear stress (Tresca) the bolt - [MPa]

σ - axial stress the bolt - [MPa]

σB - bending stress the bolt - [MPa]

τ - shear stress the bolt - [MPa]

σCall - allowable combined stress - [MPa]

Allowable combined stress:

${\sigma }_{Call}=\frac{{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σCall - allowable combined stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Bearing stress the bolt:

${p}_{t}=\frac{4{F}_{Bmax}}{\frac{L}{P}*\pi *\left({d}^{2}-{d}_{b1}^{2}\right)}\le {\sigma }_{all\left(t\right)}$

pt - bearing stress the bolt - [MPa]

FBmax - maximum axial force in the bolt - [N]

P - thread pitch - [mm]

db1 - minor diameter - [mm]

σall(t) - allowable bearing stress the thread - [MPa]

${\sigma }_{all\left(t\right)}=\frac{0,9{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σall(t) - allowable bearing stress the thread - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Bearing stress the washer:

$p=\frac{4{F}_{Bmax}}{\pi *\left({d}_{w}^{2}-{d}_{1}^{2}\right)}\le {\sigma }_{all}$

p - bearing stress the washer - [MPa]

FBmax - maximum axial force in the bolt - [N]

dw - washer diameter - [mm]

d1 - the bolt nominal diameter - [mm]

σall - allowable bearing stress - [MPa]

Allowable bearing stress:

${\sigma }_{all}=\frac{0,9{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σall - allowable bearing stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

${\tau }_{t}=\frac{3{F}_{Bmax}}{2\pi {*d}_{b1}*\frac{L}{P}*\left(P-\frac{H}{2}\mathrm{tan}30\right)}\le {\tau }_{all\left(t\right)}$

τt - shear stress the thread - [MPa]

FBmax - maximum axial force in the bolt - [N]

db1 - minor diameter - [mm]

P - thread pitch - [mm]

H - height of basic triangle - [mm]

τall(t) - allowable shear stress the thread - [MPa]

${\tau }_{all\left(t\right)}=\frac{0,4{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

τall(t) - allowable shear stress the thread - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Height of basic triangle:

$H=\frac{\sqrt{3}}{2}P$

H - height of basic triangle - [mm]

P - thread pitch - [mm]

${\sigma }_{t}=\frac{3{F}_{Bmax}*\left({d-d}_{b1}\right)}{2\pi {*d}_{b1}*\frac{L}{P}*{\left(P-\frac{H}{2}\mathrm{tan}30\right)}^{2}}\le {\sigma }_{Ball\left(t\right)}$

σt - bending stress the thread - [MPa]

FBmax - maximum axial force in the bolt - [N]

db1 - minor diameter - [mm]

P - thread pitch - [mm]

H - height of basic triangle - [mm]

σBall(t) - allowable bending stress the thread - [MPa]

${\sigma }_{Ball\left(t\right)}=\frac{0,6{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σBall(t) - allowable bending stress the thread - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Literature:

F. Pospíšil: Závitové a šroubová spojení. 1968

MET-Calc: Allowable stress

F. Boháček a kol.: Části a mechanismy strojů I. 1984   