Summary tables of results are shown below. These tables give the factors of safety for the joint and for the clamped parts. Any factors of safety of at least 1 are shown in green, and below 1 are shown in red. It is up to the discretion of the engineer to determine the appropriate factor of safety to use in design.
Joint Summary
The table below summarizes the factors of safety for the joint corresponding to the nominal, maximum, and minimum bolt load (the bolt load varies based on preload uncertainty and relaxation).

FS_{separation} 
FS_{bolt yield} 
FS_{thread shear bolt} 
FS_{thd shear internal} 
Nominal: 




Max Bolt Load: 


 
 
Min Bolt Load: 


 
 
Clamped Parts Summary
The table below summarizes the factors of safety for each clamped part in the grip.

FS_{pull thru} 
FS_{pin bearing} 
FS_{bearing, top surface} 
FS_{bearing, bot surface} 
See full result details on the other tabs.
This section details the properties of the bolted joint components.
Inputs
Joint Type 
Tensile Force 
Shear Force 
Bolt Material 
Thread Size 
Preload % 
Torque Coefficient 
Preload Uncertainty 
Preload Relaxation 










Elastic Modulus 
Tensile Yield Strength 
Shear Yield Strength 
Bearing Yield Strength 
d_{nom} 
= 

nominal diameter 
TPI 
= 

threads per inch 
P 
= 

pitch 

d_{m} 
= 

minor diameter 
d_{p} 
= 

pitch diameter 

A_{t} 
= 

tensile stress area 
A_{s} 
= 

minor (shear) area 

L 
= 

length 
L_{t} 
= 

thread length 

d_{head} 
= 

head diameter 
Nut Properties
h_{nut} 
= 

nut height 
w_{nut} 
= 

width across flats 
Clamped Parts List

Grip Thk 
Hole Dia 
Width 
Material 
Grip: 

Grip length 
Threaded Member: 

Additional length to protrude through threaded member. 
2 threads: 

2 threads at the end to ensure full engagement 
Total: 

Length of the bolt before rounding up to the preferred size 
The bolt length was rounded up to the preferred size of .
The individual stiffnesses of the shank and the threads in the grip are calculated as:
Stiffness of shank: 
$$ k_{shank} = { A_{nom} E_{bolt} \over L_{shank} } = $$ 

Stiffness of threads in grip: 
$$ k_{thd} = { A_t E_{bolt} \over L_{thd.g} } = $$ 

Reference Values
A_{nom} 
= 

A_{t} 
= 

E 
= 

L_{shank} 
= 

L_{thd.g} 
= 

The bolt stiffness is calculated considering that the shank and threads in the grip act as springs in series:
$$ k_{bolt} = { 1 \over 1/k_{shank} + 1/k_{thd} } = $$ 

The stiffness of each part in the grip is calculated based on a simplified pressurecone method, as described here. The table below shows the stiffness for each part:

Grip Thk 
Hole Dia 
Elastic Modulus 
Stiffness 
The grip stiffness is calculated considering that the parts in the grip act as springs in series:
$$ k_{grip} = { 1 \over \sum (1/k_{part}) } = $$ 

The joint constant is the stiffness of the bolt relative to the total joint stiffness:
$$ C = { k_{bolt} \over k_{bolt} + k_{grip} } = $$ 

This section details the parameters relevant to installation of the joint. A summary of the installation parameters is given in the table below:
Preload (% Yield) 
Preload Force 
Torque Coefficient, K_{T} 
Install Torque 




The nominal preload force is calculated as:
$$ F_{PL} = \%_{yld} \cdot S_{ty} A_t = $$ 

Reference Values
%_{yld} 
= 

S_{ty} 
= 

A_{t} 
= 

Reference Values
K_{T} 
= 

d_{nom} 
= 

F_{PL} 
= 

This section details the force and stress analysis on the bolt itself.
Preload Force
The nominal preload force is calculated as:
$$ F_{PL.nom} = \%_{yld} \cdot S_{ty} A_t = $$ 

Reference Values
%_{yld} 
= 

S_{ty} 
= 

A_{t} 
= 

Due to preload uncertainty, the actual preload applied to the bolt may be more or less than the nominal value. Due to preload relaxation, there will be some loss in the preload after the joint is installed.
The maximum value of preload accounts for the preload uncertainty, and is calculated as:
$$ F_{PL.max} = (1 + \%_{uncrt}) \cdot F_{PL.nom} = $$ 

The minimum value of preload accounts for the preload uncertainty as well as relaxation, and is calculated as:
$$ F_{PL.min} = (1  \%_{uncrt}  \%_{relax}) \cdot F_{PL.nom} = $$ 

Reference Values
F_{PL.nom} 
= 

%_{uncrt} 
= 

%_{relax} 
= 

The table below summarizes the preload values. The nominal value is the design target, and the min and max values account for preload uncertainty and relaxation.

Nominal 
Minimum 
Maximum 
Preload Force: 



Bolt Load Diagram
The Bolt Load Diagram below shows the tensile load on the bolt as a function of applied tensile load on the joint. The dark blue line gives the nominal bolt load, and the light blue lines account for preload uncertainty and relaxation. The knee in the curve shows the point at which the joint separates.
Before joint separation, only a portion of the applied load is carried by the bolt, and the other portion acts to relieve compression in the clamped parts. The bolt load line in this region has a constant slope equal to the joint constant. After separation, all applied load is taken by the bolt and so the bolt load line has a slope of 1.
Joint Constant
The value of the joint constant is determined by calculating the stiffness of the bolt, \(k_{bolt}\), and the stiffness of the clamped parts in the grip, \(k_{grip}\). The once the stiffnesses are known, the joint constant is calculated as:
$$ C = { k_{bolt} \over k_{bolt} + k_{grip} } $$
The joint constant for this joint was calculated in the Joint Properties tab as:
Joint Separation
The separation force is the force that will result in separation of the joint. A summary of the separation force values and their corresponding factors of safety is given in the table below. The nominal and minimum values in the table correspond to the nominal preload and the minimum preload, respectively.

Separation Force 
FS_{sep} 
Nominal: 


Minimum: 


Maximum: 


The minimum separation force dictates the adequacy of the joint and is calculated below:
$$ F_{sep.min} = { F_{PL.min} \over 1  C } = $$ 

The factor of safety on joint separation is calculated as the ratio of the separation force to the applied tensile load:
$$ FS_{sep.min} = { F_{sep.min} \over F_{t.app} } = $$ 

Reference Values
F_{sep.min} 
= 

F_{t.app} 
= 

Tensile Force on Bolt
The total (combined) tensile force on the bolt is the sum of the tension due to preload and the tension due to the applied load. The values of these tension components are dependent on whether the joint has separated. Refer to the Bolt Load Diagram above for a visual indication of joint separation. The combined tensile force on the bolt is calculated as:
$$
F_{b.t} =
\left \begin{array}{ll}
F_{PL} + C \cdot F_{t.app} & \text{joint not separated} \\
F_{t.app} & \text{joint separated}
\end{array} \right.
$$
The combined tensile forces on the bolt corresponding to the nominal, minimum, and maximum values of preload are given in the table below:

Separated? 
Combined Tensile Force 
Nominal: 

$$ F_{b.t.nom} = F_{PL.nom} + C \cdot F_{t.app} = $$
$$ F_{b.t.nom} = F_{t.app} = $$



Minimum: 

$$ F_{b.t.min} = F_{PL.min} + C \cdot F_{t.app} = $$
$$ F_{b.t.min} = F_{t.app} = $$



Maximum: 

$$ F_{b.t.max} = F_{PL.max} + C \cdot F_{t.app} = $$
$$ F_{b.t.max} = F_{t.app} = $$



Reference Values
F_{PL.nom} 
= 

F_{PL.min} 
= 

F_{PL.max} 
= 

C 
= 

F_{t.app} 
= 

The table below summarizes the tensile force components on the bolt. The nominal, minimum, and maximum values in the table correspond to the nominal, minimum, and maximum values of preload, respectively. Note that if the joint has not separated, the bolt will only take a portion of the applied tensile load.

Separated? 
Preload, \(F_{b.PL}\) 
Applied Load, \(F_{b.t.app}\) 
Total (Combined), \(F_{b.t}\) 
Nominal: 




Minimum: 




Maximum: 




Shear Force on Bolt
The shear force on the bolt is equal to the shear load applied to the joint:
$$ F_{b.s} = F_{s.app} = $$ 

Bolt Stresses
The stresses in the bolt are calculated per the equations in the table below:
Preload Stress 
Tensile Stress 
Shear Stress 
Von Mises Stress 
$$ \sigma_{PL} = {F_{b.PL} \over A_t} $$ 
$$ \sigma_{t} = {F_{b.t.app} \over A_t} $$ 
$$ \tau_{sh} = {F_{b.s} \over A_s} $$ 
$$ \sigma_{VM} = \sqrt{ \left( \sigma_{PL.max} + n \cdot \sigma_{t.max} \right)^2 + 3 (n \cdot \tau_{sh})^2 } $$ 
where \(F_{b.PL}\) is the tensile force on the bolt due to preload, \(F_{b.t.app}\) is the tensile force on the bolt due to applied tensile load, \(F_{b.s}\) is the shear force on the bolt, \(A_t\) is the tensile stress area, and \(A_s\) is the minor (shear) area. The value \(n\) in the equation for von Mises stress is the load factor. The load factor is related to the factor of safety, with the difference being that it is a factor that is applied to the loads or stresses to ensure that the bolt stress remains below the allowable stress.
The stresses at the maximum bolt preload determine the adequacy of the joint. The preload stress at the maximum preload value is:
$$ \sigma_{PL.max} = { F_{b.PL.max} \over A_t } = $$ 

The tensile stress due to applied tensile load at the maximum preload value is:
$$ \sigma_{t.max} = { F_{b.t.app.max} \over A_t } = $$ 

Reference Values
F_{b.t.app.max} 
= 

A_{t} 
= 

The shear stress due to applied shear load is:
$$ \tau_{sh} = { F_{b.s} \over A_s } = $$ 

The von Mises stress at the maximum preload value, considering a load factor of 1, is:
$$ \sigma_{VM} = \sqrt{ \left( \sigma_{PL.max} + n \cdot \sigma_{t.max} \right)^2 + 3 (n \cdot \tau_{sh})^2 } = $$ 

Reference Values
σ_{PL.max} 
= 

σ_{t.max} 
= 

τ_{sh} 
= 

n 
= 
1 
The factor of safety when considering bolt stress is the value of the load factor, \(n\), at which the von Mises stress equals the allowable stress (the bolt yield stress in this case). At the maximum preload value, the factor of safety is:
The bolt stresses are summarized in the table below summarizes the tensile force components on the bolt. The nominal, minimum, and maximum values in the table correspond to the nominal, minimum, and maximum values of preload, respectively.

Preload Stress 
Tensile Stress (App) 
Shear Stress 
Von Mises Stress 
FS_{yld} 
Nominal: 





Minimum: 





Maximum: 





Thread Shear Results
Thread Shear is considered for both the bolt (external) threads and the internal threads.
Since this joint includes a bolt with a nut, the length of thread engagement is simply the nut height:
Since this is a tapped joint, the length of thread engagement is the minimum of the tapped part thickness, \(t_p\), or the bolt nominal diameter, \(d_{nom}\):
$$ L_E = \min(t_p, d_{nom}) = $$ 

External Thread Shear
The thread shear area for the external thread is calculated by:
$$ A_{ts.ext} = {5 \over 8} \pi d_{p.ext} L_E = $$ 

The shear stress in the external threads in the nominal case (where preload equals the nominal value) is:
$$ \tau_{ts.ext} = {F_{b.t} \over A_{ts.ext} } = $$ 

The factor of safety on external thread shear with respect to the shear yield strength, \(S_{sy}\), of the thread material is:
$$ FS_{ts.ext} = {S_{sy} \over \tau_{ts.ext} } = $$ 

Internal Thread Shear
The thread shear area for the internal thread is calculated by:
$$ A_{ts.int} = {3 \over 4} \pi d_{p.int} L_E = $$ 

The shear stress in the internal threads in the nominal case (where preload equals the nominal value) is:
$$ \tau_{ts.int} = {F_{b.t} \over A_{ts.int} } = $$ 

The factor of safety on internal thread shear with respect to the shear yield strength, \(S_{sy}\), of the thread material is:
$$ FS_{ts.int} = {S_{sy} \over \tau_{ts.int} } = $$ 

Thread Shear Summary
The table below summarizes the thread shear results:

Thread Shear Force 
Thread Shear Area 
Thread Shear Stress 
Thread Shear Allowable 
FS_{thd shear} 
External (Bolt) Thread 





Internal Thread 





Failure of the clamped parts must be investigated when analyzing a bolted joint. There are several principal failure mechanisms for the clamped parts which are described below.
Pull Through
The tensile force applied to the joint will act to pull the parts through one another. The relevant equations for analyzing pull through are:
Area 
Stress 
Factor of Safety 
$$ A_{pt} = \pi d_o t_p $$ 
$$ \tau_{pt} = { F_{t.app} \over A_{pt} } $$ 
$$ FS = { S_{sy} \over \tau_{pt} } $$ 
In the table above, \(d_o\) is the outer diameter of the part pulling through, \(t_p\) is the thickness of the part being considered, \( F_{t.app} \) is the applied tensile force, and \( S_{sy} \) is the shear yield strength of the material for the part being considered.
The table below summarizes the pull through results. The part listed in the "Pull Thru Part" column is acting to pull through the part listed in the first column. Stresses and factors of safety are calculated per the equations above. The allowable stress is the shear yield strength of the part in the first column.

Pull Thru Part 
Force 
Area 
Stress 
Allowable 
FS 
Pin Bearing
If the joint is loaded in shear, then the bolt may be pressed against the inner walls of the throughholes in the clamped parts. This is referred to as pin bearing, and the relevant equations are:
Area 
Stress 
Factor of Safety 
$$ A_{pb} = d_{nom} t_p $$ 
$$ \sigma_{pb} = { F_{s.app} \over A_{pb} } $$ 
$$ FS = { S_{by} \over \sigma_{pb} } $$ 
In the table above, \(d_{nom}\) is the bolt nominal diameter, \(t_p\) is the part thickness, \(F_{s.app}\) is the applied shear force, and \(S_{by}\) is the bearing yield strength of the material.
The table below summarizes the pin bearing results. Stresses and factors of safety are calculated per the equations above. The allowable stress is the bearing yield strength of the part.

Force 
Area 
Stress 
Allowable 
FS 
Bearing
The preload force will act to cause each part to bear on the adjacent parts. The relevant equations for analyzing bearing are:
Area 
Stress 
Factor of Safety 
$$ A_{bear} = {\pi \over 4}(d_{o.min}^2  d_h^2) $$ 
$$ \sigma_{bear} = { F_{bear} \over A_{bear} } $$ 
$$ FS = { S_{by} \over \sigma_{bear} } $$ 
In the table above, \(d_{o.min}\) is the minimum outer diameter of the two parts bearing against one another, \(d_h\) is the throughhole diameter of the part being considered, and \(S_{by}\) is the bearing yield strength of the part being considered.
\(F_{bear}\) is the bearing force, and the value for the bearing force on a surface depends on the location of that surface with respect to the location of the applied tensile force in the joint. If the bearing surface is inside of the applied force locations, then the maximum bearing force that surface experiences is simply the preload force. If the bearing surface is outside of the applied force locations, then the bearing force is increased by the applied force:
$$
F_{bear} =
\left \begin{array}{ll}
F_{PL} & \text{surface inside of applied force} \\
F_{PL} + F_{b.t.app} & \text{surface outside of applied force}
\end{array} \right.
$$
where \(F_{PL}\) is the preload force and \(F_{b.t.app}\) is the portion of the applied tensile force taken by the bolt.
Bearing, Top Surface
The table below summarizes the bearing results, considering the top surface of each part. The part listed in the "Bearing Part" column is bearing on the part listed in the first column. Stresses and factors of safety are calculated per the equations above. The allowable stress is the bearing yield strength of the part in the first column.

Bearing Part 
Force 
Area 
Stress 
Allowable 
FS 
Bearing, Bottom Surface
The table below summarizes the bearing results, considering the bottom surface of each part.

Bearing Part 
Force 
Area 
Stress 
Allowable 
FS 