Bolted Joint Calculator

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The bolted joint is among the most common joining methods -- other common joining methods include riveting, welding, press fits, pins, keys, and adhesives. The primary components of a bolted joint include the threaded fastener as well as the parts to be joined together (the clamped parts). The bolted joint works by inducing an initial clamping force ("preload") on the joint by threading the fastener into either a nut or into threads that have been tapped into one of the parts. This preload ensures that the clamped parts remain in contact and in compression throughout the life of the joint.

Washers are typically used in the joint and serve many purposes. They minimize embedment of the bolt head and nut into the clamped parts, and they aid in tightening. Since bolt holes could have sharp edges or burrs, washers are used to protect the fillet under the bolt head from scratching since this is a critical area that is prone to failure. Washers also serve to distribute the preload and applied force over a larger area, both on the bolt head and on the faces of the clamped parts. This reduces bearing stresses, helps to prevent pull-through, and helps to prevent damage to the surface of the clamped parts.

Contents

Thread Dimensions

When analyzing a joint, it is necessary to know the characteristic dimensions of both the external thread and internal thread. A thread size is specified based on a nominal (major) diameter and either the number of threads per inch (for unified inch threads) or the pitch (for metric threads). The thread sizes for coarse thread and fine thread can be found in tables located in any standard machine design handbook, as well as in the thread size tables in the Appendix. The pitch, \(P\) is the distance between the threads. When the pitch is in units of inches, it is related to the threads per inch, \(TPI\), by:

$$ TPI = { 1 \over P } $$

Thread Dimensions (Internal and External Thread):

Symbol US Units SI Units
Nominal (Major) Diameter \( d_{nom} \) in mm
Threads Per Inch \( TPI \) in-1 ---
Pitch \( P \) in mm

The tables below provide equations for some of the thread profile dimensions of interest for both unified inch threads and ISO metric threads. In the case of metric threads, the thread profile is based on a parameter \(H\), the height of the fundamental triangle. The value of \(H\) is related to the thread pitch, \(P\) by:

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

External Thread (Bolt) Dimensions:

Equation, US Units [in] Equation, Metric Units [mm]
Minor Diameter
$$ d_{m.ext} = d_{nom} - {1.299038 \over TPI} $$

(Machinery's Handbook)

$$ d_{m.ext} = d_{nom} - 1.226869 P $$

(Shigley)

Pitch Diameter
$$ d_{p.ext} = d_{nom} - {0.64951905 \over TPI} $$

(ASME B1.1, Section 10.1p)

$$ d_{p.ext} = d_{nom} - 0.75 H = d_{nom} - 0.64951905 P $$

(Machinery's Handbook)


Internal Thread Dimensions:

Equation, US Units [in] Equation, Metric Units [mm]
Minor Diameter
$$ d_{m.int} = d_{nom} - {1.08253175 \over TPI} $$

(ASME B1.1, Section 10.1s)

$$ d_{m.int} = d_{nom} - 1.25 H = d_{nom} - 1.08253175 P $$

(Machinery's Handbook)

Pitch Diameter
$$ d_{p.int} = d_{nom} - {0.64951905 \over TPI} $$

(ASME B1.1, Section 8.3)

$$ d_{p.ext} = d_{nom} - 0.75 H = d_{nom} - 0.64951905 P $$

(Machinery's Handbook)


External Thread (Bolt) Areas:

Equation, US Units [in] Equation, Metric Units [mm]
Nominal Area $$ A_{nom} = {\pi \over 4} d_{nom}^2 $$ $$ A_{nom} = {\pi \over 4} d_{nom}^2 $$
Tensile Stress Area
$$ A_{t} = {\pi \over 4} \left( d_{nom} - {0.9743 \over TPI} \right)^2 $$

(ASME B1.1, Appendix B)

$$ A_{t} = {\pi \over 4} \left( d_{nom} - 0.9382 P \right)^2 $$

(ASME B1.13M, Appendix B)

Minor Area
(Shear Area)
$$ A_{m} = {\pi \over 4} d_{m.ext}^2 $$ $$ A_{m} = {\pi \over 4} d_{m.ext}^2 $$

In the tables above, US units are in inches and metric units are in millimeters.

Preload

Bolts are installed with a preload that ensures that the joint members remain clamped and in compression throughout the life of the joint. Preload is also important for joints with a cyclically applied load. The preload will increase the mean stress, but it will reduce the alternating stress.

Preload Values

The preload is commonly specified as a percentage of the bolt material's tensile yield strength, \( S_{ty} \). To calculate preload force as a percentage of yield strength, use:

$$ F_{PL} = \%_{yld} \cdot S_{ty} A_t $$

where \(\%_{yld}\) is the preload percent of yield, \(S_{ty}\) is the yield strength, and \(A_t\) it the tensile stress area.

In general, the preload force should be no less than the maximum tensile force that will be applied to the joint. This will ensure that the clamped parts always remain in contact and in compression. Because some of the tensile force applied to the joint will act to relieve compression in the clamped parts, the joint will separate at a value of applied force that is somewhat higher than the preload. This will be discussed in a later section.

Because the tensile force that will be applied to the joint dictates the required preload, then the maximum utility is obtained from a bolt by preloading it to the highest possible value. The ductility of the bolt material dictates how close to the yield strength the bolt can be preloaded. Shigley and Lindeburg both recommend the following (conservative) values of preload:

$$ F_{PL} = \left| \begin{array}{ll} 0.75 F_{proof} & \text{for nonpermanent joint (reused fasteners)} \\ 0.90 F_{proof} & \text{for permanent joints} \end{array} \right. $$

where \( F_{proof} \) is the proof load of the fastener. The relationship between the proof load and the proof strength is:

$$ F_{proof} = S_{proof} A_t $$

Per Shigley, the proof strength is approximately equal to 85% of the tensile yield strength, \( S_{ty} \). Based on \( S_{proof} = 0.85 S_{ty} \), the recommended preload force as a function of yield strength is:

$$ F_{PL} = \left| \begin{array}{ll} 0.64 S_{ty} A_t & \text{for nonpermanent joint (reused fasteners)} \\ 0.77 S_{ty} A_t & \text{for permanent joints} \end{array} \right. $$

Considering that the above values are conservative, a general rule of thumb is to preload the fastener to 2/3 of the yield strength (i.e. \(\%_{yld}\) = 66.7%).

Preload Relaxation

There are many factors which can result in a "relaxation" or a loss of the preload applied during installation. Temperature fluctuations will result in expansion and contraction of the joint members and can cause either an increase or a decrease in the preload force depending on the relative lengthening and shortening between the fastener and the clamped parts. If the preload is achieved by applying torque to the bolt or nut, then this will result in torsion on the fastener which will act to increase the bolt tension and thus the preload force. Over time this torsion will dissipate and causing relaxation of the preload force. Other factors which contribute to preload relaxation include embedment and creep.

Preload relaxation can be mitigated through the use of thread-locking mechanisms including locking adhesives, lock nuts, lock washers, lock wire, and locking pellets/patches. Barret provides a comprehensive treatment of thread locking mechanisms.

According to the Machinery's Handbook, preload relaxation occurs within hours after installation, and a preload loss allowance of approximately 10% is sufficient as a general rule.

Preload Uncertainty

The accuracy of the preload that is applied during installation is highly dependent on the tightening method employed. The following table is adapted from the Machinery's Handbook and from Barrett:

Tightening Method Accuracy
By feel ±35%
Torque wrench ±25%
Turn-of-the-nut ±15%
Load indicating washer ±10%
Bolt elongation ±3-5%
Strain gages ±1%
Ultrasonic sensing ±1%

Torque to Obtain Preload

Many of the common tightening methods achieve the preload force by applying a torque to the nut or to the bolt head. When tightening a fastener with a torque wrench, which is one of the easiest and most common methods, the fastener is considered to be properly tightened once the specified torque is achieved. In this case, it is necessary to determine the torque value necessary to achieve the desired preload force in the bolt. This torque is calculated using:

$$ T = K_T d_{nom} F_{PL} $$

where \(d_{nom}\) is the nominal bolt diameter and \(F_{PL}\) is the bolt preload force. \(K_T\) is the torque coefficient and is calculated by:

$$ K_T = \left( {r_t \over d_{nom}} \right) \left( { \tan\lambda + f_t \sec\alpha \over 1 - f_t \tan\lambda \sec\alpha } \right) + {f_c r_c \over d_{nom}} $$

where \(r_t\) is the mean thread radius (the effective location of at which the thread friction acts), \(r_c\) is the mean collar radius (the effective location at which the friction on the bearing face acts), \(f_t\) is the friction coefficient between the thread surfaces, \(f_c\) is the friction coefficient between the collar (bearing face) surfaces, \( \lambda \) is the lead angle, and \( \alpha \) is the thread half angle ( \( \alpha = 30^{\circ} \), per ASME B1.1, 10.1b).

The value for \(r_t\) is calculated as half of the mean bolt diameter, which is the average of the minor diameter and nominal diameter:

$$ r_t = { { \left( d_{nom} + d_{minor} \right) /~ 2 } \over 2 } = { d_{nom} + d_{minor} \over 4 } $$

The collar area is the area of the bearing face of the part being rotated during installation (either the nut or the bolt head). The width across flats of a nut is typically 1.5 times the nominal diameter. In this case, the mean collar radius is calculated as:

$$ r_c = { { \left( d_{nom} + 1.5 d_{nom} \right) /~ 2 } \over 2 } = 0.625 d_{nom} $$

The lead angle, \( \lambda \), is calculated by:

$$ \tan\lambda = { l \over 2 \pi r_t } = { 1 \over 2 \pi r_t \left( TPI \right) } $$

where \(l\) is the lead per revolution ( \( = 1 / TPI \) ).

Shigley provides a table of torque coefficients based on bolt condition, which has been adapted as shown below. When the bolt condition is unknown, a value of 0.2 is recommended for \(K_T\).

Bolt Condition \(K_T\)
Nonplated, black finish 0.30
Zinc-plated 0.20
Lubricated 0.18
With Anti-Seize 0.12

Because of the many variables that affect the value of the torque coefficient, any tightening method that measures a preload force indirectly via a torque value will be inherently inaccurate. It is for this reason that there is such a large uncertainty in preload accuracy when using a torque wrench.



Joint Stiffness

The joint can be considered as a set of springs. The parts within the grip act as a set of springs in series, and the grip and the bolt act as springs in parallel. The joint can be modeled as shown below. Note that in the joint shown below there are only 2 parts in the grip; however, the number of parts is not limited to 2, and each part in the joint would be represented by a spring.

Joint as Springs

Each spring from the figure above has a stiffness of:

$$ k = {AE \over L} $$

where A is the area, E is the elastic modulus of the material, and L is the length.

Bolt Stiffness

When a joint is assembled properly, the full shank of the bolt will be in the grip along with some length of threads. The stiffness of the shank is given by:

$$ k_{shank} = {A_{nom} E_{bolt} \over L_{shank} } $$

where \(A_{nom}\) is the bolt nominal area, \(E_{bolt}\) is the elastic modulus of the bolt material, and \(L_{shank}\) is the length of the bolt shank.

The stiffness of the threaded portion in the grip is given by:

$$ k_{thd} = {A_t E_{bolt} \over L_{thd.g} } $$

where \(A_t\) is the tensile stress area and \(L_{thd.g}\) is the length of the threaded portion within the grip.

The shank and the threaded portion of the bolt will act as springs in series, so that the effective stiffness of the portion of the bolt within the grip is given by:

$$ k_{bolt} = {1 \over 1/k_{shank} + 1/k_{thd}} = { k_{shank} k_{thd} \over k_{shank} + k_{thd} } $$

Per ASME B18.2.1, the nominal thread length of inch-series bolts is found by:

$$ L_{thd} = \left| \begin{array}{ll} 2 d_{nom} + {1 \over 4} \text{ in} & L \le 6 \text{in} \\ 2 d_{nom} + {1 \over 2} \text{ in} & L \gt 6 \text{in} \end{array} \right. $$

The bolt shank length can then be found by:

$$ L_{shank} = L - L_{thd} $$

The length of thread in the grip is found by:

$$ L_{thd.g} = L_g - L_{shank} $$

where \(L_g\) is the grip length.

Grip Stiffness

The stiffness of the grip is determined by calculating based on a simplified pressure-cone method as presented by Shigley. This method predicts the pressure distribution throughout the thickness of the grip. The pressure cone for a joint can be visualized in the diagram below.

Grip Pressure Cone

The portion of a part within the pressure cone is called a frustum. Every part in the grip will contain either 1 or 2 frustums. The stiffness of an individual frustum is given by:

$$ k_{fr} = { \pi E d \tan\alpha \over \ln \left[ { (2t \tan\alpha + D - d) (D + d) \over (2t \tan\alpha + D + d) (D - d) } \right] } $$

where \(d\) is the inner diameter of the frustum, \(D\) is the smallest value of the frustum outer diameter, \(t\) is the frustum thickness, \(E\) is the elastic modulus of the material, and \( \alpha \) is the angle of the pressure cone. Shigley recommends a value of 30° for \( \alpha \).

The height of the pressure cone depends on the grip length, \(L_g\), which is the combined thickness of the parts being clamped in the joint (see the figure above). In a joint with a nut, the pressure cone starts under the head of the bolt and ends under the nut. The frustum diameters in this case can be easily determined using the diameters of the bearing faces. In a tapped joint, the pressure cone starts under the head of the bolt and ends in the threaded portion of the final plate. Per Shigley, the effective grip thickness of the final plate is given as:

$$ L_{g.p2}^{'} = { \min (t, d_{nom}) \over 2} $$

where \(t\) is the plate thickness and \(d_{nom}\) is the bolt nominal diameter. The frustum diameter at the end of the pressure cone is assumed to be \(1.5 d_{nom}\).

The stiffness of the grip is calculated by considering the frustums to act as springs in series:

$$ k_{grip} = {1 \over 1/k_{fr.1} + 1/k_{fr.2} + ... + 1/k_{fr.N}} $$

Grip Stiffness Approximation

Shigley provides an equation which calculates the correct grip stiffness in the case that every part in the grip has the same elastic modulus:

$$ k_{grip} = {\pi E d \tan\alpha \over 2 \ln \left[ { (L_g \tan\alpha + d_{bh} - d) (d_{bh} + d) \over (L_g \tan\alpha + d_{bh} + d) (d_{bh} - d) } \right] } $$

In the equation above, \(E\) is the elastic modulus of the material, \(L_g\) is the grip length, \( \alpha \) is the frustum angle (30°), \(d_{bh}\) is the diameter of the bearing face under the bolt head, and \(d\) is the inner frustum diameter.

The value for \(d\) can either be the nominal bolt diameter (for a more approximate result), or it can be the hole diameter of the most central part in the joint (for a more exact result).

If the parts in the grip all have the same elastic modulus, then the grip stiffness calculated using the equation above will be the same as the grip stiffness calculated using the full procedure. However, if the parts have different elastic moduli, a close approximation can still be achieved by calculating an effective elastic modulus for the grip:

$$ E_{eff} = { \sum E_p L_p \over \sum L_p } $$

where \(E_p\) is the elastic modulus of a part in the grip and \(L_p\) is the thickness of a part in the grip.

An even more simplified equation for the grip stiffness can be found if it is assumed that the bolt head diameter is 50 percent larger than the bolt nominal diameter (i.e. \(d_{bh} = 1.5 d_{nom} \) ):

$$ k_{grip} = {\pi E d_{nom} \tan\alpha \over 2 \ln \left[ 5 { (L_g \tan\alpha + 0.5 d_{nom}) \over (L_g \tan\alpha + 2.5 d_{nom}) } \right] } $$

Bolt Load vs Applied Load

The preload elongates the bolt and compresses the clamped parts. When a tensile load is applied to the joint, some portion of the applied load acts to relieve the compression in the clamped parts and the other portion further elongates the bolt. The portion of the applied load that is carried by the bolt is dependent on the relative stiffness of the bolt and the clamped parts. This relative stiffness is known as the joint constant, \(C\):

$$ C = { k_{bolt} \over k_{bolt} + k_{grip} } $$

The following is a representative diagram of bolt load as a function of the applied joint load:

Bolt Load Diagram

In the figure above, the x-axis is the tensile load applied to the joint, and the y-axis is the tensile load that exists in the bolt. When the applied load is zero, the tensile load on the bolt is equal to the preload force. As load is applied to the joint, the some of the applied load acts to relieve compression in the clamped parts and some of the applied load acts to increase the tension in the bolt. The bolt load line has a constant slope equal to the joint constant, \(C\).

The nominal bolt load curve is shown as a thick solid blue line. The two lighter blue lines running parallel to the nominal curve are the minimum and maximum bolt load curves. The minimum curve accounts for preload uncertainty due to installation method as well as preload relaxation. The maximum curve accounts for preload uncertainty. In the curves above, a preload uncertainty of ±25% was used (corresponding to installation with a torque wrench) along with a preload relaxation of 10%.

Another curve to note in the figure above is the bolt tensile yield load line, shown as a horizontal red dashed line. This line indicates the value of tensile load on the bolt which will result in yielding. This line accounts for shear and bending, so any shear force or bending moments acting on the bolt will lower this line.

Joint Separation

The knee in the curve in the bolt load diagram above shows the point where the joint separates. At this point, the applied load is sufficient to separate the parts in the joint (all of the compression in the clamped parts has been relieved), and after this point any load applied to the joint is taken entirely by the bolt. The force that will result in separation of the joint is found by:

$$ F_{sep} = { F_{PL} \over 1 - C } $$

Note that the separation force will always be somewhat higher than the preload force.

Separation of the joint is a failure criteria, and a joint should be designed such that it will not separate during service. The factor of safety on separation is found by:

$$ FS_{sep} = { F_{sep} \over F_{t.app} } $$

Forces on Bolt

The total tensile force on the bolt is due to 2 components: the preload force and the applied tensile load. The values of these components for each portion of the bolt load curve are found by:

Joint Not Separated Joint Separated
Bolt tension due to preload, \(F_{b.PL}\): \(F_{PL}\) \(0\)
Bolt tension due to applied load, \(F_{b.t.app}\): \(C \cdot F_{t.app}\) \(F_{t.app}\)

The total tensile force on the bolt is the sum of the tension due to preload and the tension due to the applied load, as determined from the table above:

$$ F_{b.t} = F_{b.PL} + F_{b.t.app} $$

Another way to express the total tensile force on the bolt is:

$$ F_{b.t} = \left| \begin{array}{ll} F_{PL} + C \cdot F_{t.app} & \text{joint not separated} \\ F_{t.app} & \text{joint separated} \end{array} \right. $$

The shear force on the bolt is equal to the shear load applied to the joint:

$$ F_{b.s} = F_{s.app} $$

A bending moment could exist in the bolt if there is a gap between the plates (i.e. due to a gasket) or if there are long spacers used in the joint:

$$ M_b = { F_{b.s} a \over 2} $$

where \(a\) is the moment arm. More discussion on the bending moment on the bolt is given in the Appendix.

Bolt Stresses

The stresses in the bolt are calculated per the equations shown in the table below:

Preload Stress Tensile Stress Shear Stress Bending Stress
$$ \sigma_{PL} = {F_{b.PL} \over A_t} $$ $$ \sigma_{t} = {F_{b.t.app} \over A_t} $$ $$ \tau_{sh} = {F_{b.s} \over A_s} $$ $$ \sigma_{bnd} = {32 M_b \over \pi d^3} $$

where \(A_t\) is the tensile stress area, \(A_s\) is the shear area (either the nominal area if the shear plane is in the shank or the minor area if the shear plane is in the threads), and \(d\) is the either the nominal diameter if the maximum moment is in the shank or the minor diameter if the maximum moment is in the threads. Since the maximum moment will occur under the head and at the start of the internal threads, the maximum moment will typically occur in the bolt threads and so the minor diameter should be used to calculate bending stress.

The von Mises stress is calculated by:

$$ \sigma_{VM} = \sqrt{ [ \sigma_{PL} + n (\sigma_t + \sigma_{bnd}) ]^2 + 3 (n \tau_{sh})^2 } $$

In the equation above, \(n\) is a load factor which is applied to the tensile, bending and shear stress but is not applied to the preload stress. 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 factor of safety can be solved for iteratively by finding the value of the load factor at which the von Mises stress equals the allowable stress:

$$ FS = [\text{value of } n \text{ that results in } \sigma_{VM} = \sigma_{allow}] $$

Thread Shear

Thread shear is an important failure mode for a bolted joint, and occurs when the threads shear off of either the bolt (external thread shear) or off of the nut or tapped part (internal thread shear). There should be enough engagement between the bolt threads and internal threads that the bolt fails in tension before the threads shear. This will ensure that the full strength of the bolt is developed (and therefore there is no "wasted" bolt strength), and it will avoid the task of drilling and re-tapping the internal thread. Thread shear should be considered for both the external (bolt) thread and the internal thread.

Length of Thread Engagement

The length of thread engagement is a dominant factor that determines whether the threads will experience shear failure. A general rule of thumb is that a length of engagement equal to the bolt diameter is sufficient to protect against thread shear. However, shear calculations should always be performed as per the following sections to ensure safety.

In a bolted joint with a nut, as long as the bolt protrudes beyond the end of the nut then the length of thread engagement can be estimated by the nut height, \(h_{nut}\). In reality, there will be some loss of engagement due to the chamfering around the threaded hole in the nut.

In a tapped joint, the depth of the threads in the final part should be equal to the minimum of the tapped part thickness, \(t_p\), or the bolt nominal diameter, \(d_{nom}\), so the length of thread engagement can be estimated as the minimum of those values. Note that these estimates do not account for chamfering at the end of the bolt or around the threaded hole in the part.

The estimates for length of thread engagement in a bolted joint are summarized by the following equation. This value of \(L_E\) should then be used to calculate factors of safety on external and internal thread shear.

$$ L_E = \left| \begin{array}{ll} h_{nut} & \text{estimate for bolted joint with nut} \\ \min(t_p, d_{nom}) & \text{estimate for tapped joint} \end{array} \right. $$

External Thread Shear

The thread shear area for the external thread is determined from a cylindrical area with a height equal to the length of thread engagement, \(L_E\), and with a diameter equal to the pitch diameter, \(d_{p.ext}\). According to the Federal Standard, the thread shear area for an external thread is calculated by:

$$ A_{ts.ext} = {5 \over 8} \pi d_{p.ext} L_E $$

The shear stress in the external threads is calculated by:

$$ \tau_{ts.ext} = {F_{b.t} \over A_{ts.ext} } $$

where \(F_{b.t}\) is the total tensile force on the bolt, accounting for preload and the portion of the applied tensile load carried by the bolt.

The factor of safety on external thread shear with respect to the shear yield strength, \(S_{sy}\) of the thread material is calculated by:

$$ FS_{ts.ext} = {S_{sy} \over \tau_{ts.ext} } $$

The shear yield strength can typically by estimated as \(0.577 S_{ty}\).

Internal Thread Shear

Internal thread shear is calculated in a similar manner as external thread shear. According to the Federal Standard, the thread shear area for an internal thread is calculated by:

$$ A_{ts.int} = {3 \over 4} \pi d_{p.int} L_E $$

where \( d_{p.int} \) is the pitch diameter of the internal thread and \(L_E\) is the length of thread engagement, which is calculated in the same manner as for the external thread shear.

The shear stress in the internal threads is calculated by:

$$ \tau_{ts.int} = {F_{b.t} \over A_{ts.int} } $$

The factor of safety on internal thread shear with respect to the shear yield strength of the thread material is calculated by:

$$ FS_{ts.int} = {S_{sy} \over \tau_{ts.int} } $$

Clamped Part Stresses

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 in the following sections.

Pull Through

The tensile force applied to the joint will act to pull the parts above the location of the applied force through one another. In the figure below, the bolt head will act to pull through Washer 1, Washer 1 will act to pull through Plate 1, the Washer 2 will act to pull through Plate 2, and the nut will act to pull through Washer 2.

Surface Under Pull Through

The relevant equations 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 shear yield strength can typically by estimated as \( 0.577 S_{ty} \).

Bearing

The preload force will act to cause each part to bear on the adjacent parts. For example, in the previous figure the bolt head and Washer 1 will bear against one another, as will Washer 1 and Plate 1.

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 through-hole 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. 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. In the p revious figure, Surface 3 is inside of the applied tensile force locations, and so that surface does not experience an increase due to the applied force. However, Surfaces 1, 2, 4, and 5 are all outside of the applied force locations and do experience an increase. 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 and the maximum bearing force experienced by that surface is equal to the tensile force in the bolt.

$$ F_{bear} = \left| \begin{array}{ll} F_{PL} & \text{surface inside of applied force} \\ F_{PL} + F_{b.t.app} & \text{surface outside of applied force} \end{array} \right. $$

Pin Bearing

If the joint is loaded in shear, then the bolt may be pressed against the inner walls of the through-holes in the clamped parts.

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 bearing yield strength can typically by estimated as \( 1.5 S_{ty} \).



Appendix


Hardware Sizes: Unified Inch Screw Thread

This section contains tables of sizes for inch thread hardware.


Bolt Thread Sizes

The following table of thread sizes for coarse and fine thread was adapted from ASME B1.1:

Size Nominal (Major)
Diameter [in]
Coarse Thread (UNC) Fine Thread (UNF)
Threads
Per Inch
Tensile Stress
Area [in2]
Minor
Area [in2]
Threads
Per Inch
Tensile Stress
Area [in2]
Minor
Area [in2]
#0 0.0600 --- --- --- 80 0.00180 0.00151
#2 0.0860 56 0.00370 0.00310 64 0.00394 0.00339
#4 0.1120 40 0.00604 0.00496 48 0.00661 0.00566
#5 0.1250 40 0.00796 0.00672 44 0.00830 0.00716
#6 0.1380 32 0.00909 0.00745 40 0.01015 0.00874
#8 0.1640 32 0.0140 0.01196 36 0.01474 0.01285
#10 0.1900 24 0.0175 0.01450 32 0.0200 0.0175
1/4" 0.2500 20 0.0318 0.0269 28 0.0364 0.0326
5/16" 0.3125 18 0.0524 0.0454 24 0.0580 0.0524
3/8" 0.3750 16 0.0775 0.0678 24 0.0878 0.0809
7/16" 0.4375 14 0.1063 0.0933 20 0.1187 0.1090
1/2" 0.5000 13 0.1419 0.1257 20 0.1599 0.1486
9/16" 0.5625 12 0.182 0.162 18 0.203 0.189
5/8" 0.6250 11 0.226 0.202 18 0.256 0.240
3/4" 0.7500 10 0.334 0.302 16 0.373 0.351
7/8" 0.8750 9 0.462 0.419 14 0.509 0.480
1" 1.0000 8 0.606 0.551 12 0.663 0.625
1-1/8" 1.1250 7 0.763 0.693 12 0.856 0.812
1-1/4" 1.2500 7 0.969 0.890 12 1.073 1.024
1-3/8" 1.3750 6 1.155 1.054 12 1.315 1.260
1-1/2" 1.5000 6 1.405 1.294 12 1.581 1.521
1-3/4" 1.7500 5 1.90 1.74 --- --- ---
2" 2.0000 4.5 2.50 2.30 --- --- ---


Bolt Thread Dimensions

The following equations can be used to calculate dimensions for Unified Inch threads:

Equation, US Units [in] Source
Minor Diameter $$ d_{m.ext} = d_{nom} - {1.299038 \over TPI} $$ Machinery's Handbook
Pitch Diameter $$ d_{p.ext} = d_{nom} - {0.64951905 \over TPI} $$ ASME B1.1, Section 10.1p
Nominal Area $$ A_{nom} = {\pi \over 4} d_{nom}^2 $$
Tensile Stress Area $$ A_{t} = {\pi \over 4} \left( d_{nom} - {0.9743 \over TPI} \right)^2 $$ ASME B1.1, Appendix B
Minor Area
(Shear Area)
$$ A_{m} = {\pi \over 4} d_{m.ext}^2 $$

In the table above, \(d_{nom}\) is nominal diameter in inches and \(TPI\) is threads per inch.


Minimum Clearance Hole Diameters

The following table of clearance holes was adapted from ASME B18.2.8. The minimum hole diameters are given.

Bolt Size Bolt Dia. [in] Normal Fit [in] Close Fit [in] Loose Fit [in]
#0 0.0600 0.076 0.067 0.094
#2 0.0860 0.102 0.094 0.116
#4 0.1120 0.128 0.120 0.144
#5 0.1250 0.156 0.141 0.172
#6 0.1380 0.170 0.154 0.185
#8 0.1640 0.196 0.180 0.213
#10 0.1900 0.221 0.206 0.238
1/4" 0.2500 0.281 0.266 0.297
5/16" 0.3125 0.344 0.328 0.359
3/8" 0.3750 0.406 0.391 0.422
7/16" 0.4375 0.469 0.453 0.484
1/2" 0.5000 0.562 0.531 0.609
5/8" 0.6250 0.688 0.656 0.734
3/4" 0.7500 0.812 0.781 0.906
7/8" 0.8750 0.938 0.906 1.031
1" 1.0000 1.094 1.031 1.156
1-1/8" 1.1250 1.219 1.156 1.312
1-1/4" 1.2500 1.344 1.281 1.438
1-3/8" 1.3750 1.500 1.438 1.609
1-1/2" 1.5000 1.625 1.562 1.734


Bolt Thread Length

Per ASME B18.2.1, the nominal thread length of inch-series bolts can be found by:

$$ L_{thd} = \left| \begin{array}{ll} 2 d_{nom} + {1 \over 4} \text{ in} & L \le 6 \text{ in} \\ 2 d_{nom} + {1 \over 2} \text{ in} & L \gt 6 \text{ in} \end{array} \right. $$

where \(L\) is the total bolt length and \(d_{nom}\) is the nominal bolt diameter.


Hex Bolt Head Dimensions

The following table of hex bolt head dimensions was adapted from ASME B18.6.3, Table 29, "Dimensions of Plain (Unslotted) and Slotted Regular and Large Hex Head Screws." This table is used for smaller size hardware.

Size Nominal (Major)
Diameter [in]
Width Across Flats Head Height
Minimum [in] Maximum [in] Minimum [in] Maximum [in]
#2 0.0860 0.120 0.125 0.040 0.050
#4 0.1120 0.181 0.188 0.049 0.060
#6 0.1380 0.244 0.250 0.080 0.093
#8 0.1640 0.244 0.250 0.096 0.110
#10 0.1900 0.305 0.312 0.105 0.120
1/4" 0.2500 0.367 0.375 0.172 0.190
5/16" 0.3125 0.489 0.500 0.208 0.230
3/8" 0.3750 0.551 0.562 0.270 0.295


The following table of hex bolt head dimensions was adapted from ASME B18.2.1, Table 2, "Dimensions of Hex Bolts."

Size Nominal (Major)
Diameter [in]
Width Across Flats Head Height
Nominal [in] Minimum [in] Nominal [in] Minimum [in]
1/4" 0.2500 7/16"(0.438) 0.425 11/64" 0.150
5/16" 0.3125 1/2"(0.500) 0.484 7/32" 0.195
3/8" 0.3750 9/16"(0.562) 0.544 1/4" 0.226
7/16" 0.4375 5/8"(0.625) 0.603 19/64" 0.272
1/2" 0.5000 3/4"(0.750) 0.725 11/32" 0.302
5/8" 0.6250 15/16"(0.938) 0.906 27/64" 0.378
3/4" 0.7500 1-1/8"(1.125) 1.088 1/2" 0.455
7/8" 0.8750 1-5/16"(1.312) 1.269 37/64" 0.531
1" 1.0000 1-1/2"(1.500) 1.450 43/64" 0.591
1-1/8" 1.1250 1-11/16"(1.688) 1.631 3/4" 0.658
1-1/4" 1.2500 1-7/8"(1.875) 1.812 27/32" 0.749
1-3/8" 1.3750 2-1/16"(2.062) 1.994 29/32" 0.810
1-1/2" 1.5000 2-1/4"(2.250) 2.175 1" 0.902
1-5/8" 1.6250 2-7/16"(2.438) 2.356 1-3/32" 0.978
1-3/4" 1.7500 2-5/8"(2.625) 2.538 1-5/32" 1.054
1-7/8" 1.8750 2-13/16"(2.812) 2.719 1-1/4" 1.130
2" 2.0000 3"(3.000) 2.900 1-11/32" 1.175


Hex Nut Dimensions

The following table of hex nut dimensions was adapted from ASME B18.2.2, Table 1-1, "Dimensions of Square and Hex Machine Screw Nuts." This table is used for smaller size hardware.

Size Nominal (Major)
Diameter [in]
Width Across Flats Thickness
Nominal [in] Minimum [in] Minimum [in] Maximum [in]
#0 0.060 5/32"(0.156) 0.150 0.043 0.050
#2 0.086 3/16"(0.188) 0.180 0.057 0.066
#4 0.112 1/4"(0.250) 0.241 0.087 0.098
#6 0.138 5/16"(0.312) 0.302 0.102 0.114
#8 0.164 11/32"(0.344) 0.332 0.117 0.130
#10 0.190 3/8"(0.375) 0.362 0.117 0.130
1/4" 0.250 7/16"(0.438) 0.423 0.178 0.193
5/16" 0.312 9/16"(0.562) 0.545 0.208 0.225
3/8" 0.375 5/8"(0.625) 0.607 0.239 0.257


The following table of hex nut dimensions was adapted from ASME B18.2.2, Table 4, "Dimensions of Hex Nuts and Hex Jam Nuts."

Size Nominal (Major)
Diameter [in]
Width Across Flats Thickness
Minimum [in] Maximum [in] Minimum [in] Maximum [in]
1/4" 0.2500 0.428 0.438 0.212 0.226
5/16" 0.3125 0.489 0.500 0.258 0.273
3/8" 0.3750 0.551 0.563 0.320 0.337
7/16" 0.4375 0.675 0.688 0.365 0.385
1/2" 0.5000 0.736 0.750 0.427 0.448
9/16" 0.5625 0.861 0.875 0.473 0.496
5/8" 0.6250 0.922 0.938 0.535 0.559
3/4" 0.7500 1.088 1.125 0.617 0.665
7/8" 0.8750 1.269 1.312 0.724 0.776
1" 1.0000 1.450 1.500 0.831 0.887
1-1/8" 1.1250 1.631 1.688 0.939 0.999
1-1/4" 1.2500 1.812 1.875 1.030 1.094
1-3/8" 1.3750 1.994 2.062 1.138 1.206
1-1/2" 1.5000 2.175 2.250 1.245 1.317
1-5/8" 1.6250 2.350 2.430 1.364 1.416
1-3/4" 1.7500 2.538 2.625 1.460 1.540
1-7/8" 1.8750 2.722 2.813 1.567 1.651
2" 2.0000 2.900 3.000 1.675 1.763


Internal Thread Dimensions

The following equations can be used to calculate internal thread dimensions for Unified Inch threads:

Equation, US Units [in] Source
Minor Diameter $$ d_{m.int} = d_{nom} - {1.08253175 \over TPI} $$ ASME B1.1, Section 10.1s
Pitch Diameter $$ d_{p.int} = d_{nom} - {0.64951905 \over TPI} $$ ASME B1.1, Section 8.3

In the table above, \(d_{nom}\) is nominal diameter in inches and \(TPI\) is threads per inch.


Flat Washer Dimensions

The following table of flat washer dimensions was adapted from ASME B18.21.1, Table 11 for Type A Plain Washers. Type A washers come in 2 series: Narrow and Wide.

Size Basic Dia. [in] Series Inner Dia,
Basic [in]
Outer Dia,
Basic [in]
Thickness,
Basic [in]
#0 0.0600 --- 0.078 0.188 0.020
#2 0.0860 --- 0.094 0.250 0.020
#4 0.1120 --- 0.125 0.312 0.032
#6 0.1380 --- 0.156 0.375 0.049
#8 0.1640 --- 0.188 0.438 0.049
#10 0.1900 --- 0.219 0.500 0.049
1/4" 0.2500 Narrow 0.281 0.625 0.065
1/4" 0.2500 Wide 0.312 0.734 0.065
5/16" 0.3125 Narrow 0.344 0.688 0.065
5/16" 0.3125 Wide 0.375 0.875 0.083
3/8" 0.3750 Narrow 0.406 0.812 0.065
3/8" 0.3750 Wide 0.438 1.000 0.083
7/16" 0.4375 Narrow 0.469 0.922 0.065
7/16" 0.4375 Wide 0.500 1.250 0.083
1/2" 0.5000 Narrow 0.531 1.062 0.095
1/2" 0.5000 Wide 0.562 1.375 0.109
9/16" 0.5625 Narrow 0.594 1.156 0.095
9/16" 0.5625 Wide 0.625 1.469 0.109
5/8" 0.6250 Narrow 0.656 1.312 0.095
5/8" 0.6250 Wide 0.688 1.750 0.134
3/4" 0.7500 Narrow 0.812 1.469 0.134
3/4" 0.7500 Wide 0.812 2.000 0.148
7/8" 0.8750 Narrow 0.938 1.750 0.134
7/8" 0.8750 Wide 0.938 2.250 0.165
1" 1.0000 Narrow 1.062 2.000 0.134
1" 1.0000 Wide 1.062 2.500 0.165
1-1/8" 1.1250 Narrow 1.250 2.250 0.134
1-1/8" 1.1250 Wide 1.250 2.750 0.165
1-1/4" 1.2500 Narrow 1.375 2.500 0.165
1-1/4" 1.2500 Wide 1.375 3.000 0.165
1-3/8" 1.3750 Narrow 1.500 2.750 0.165
1-3/8" 1.3750 Wide 1.500 3.250 0.180
1-1/2" 1.5000 Narrow 1.625 3.000 0.165
1-1/2" 1.5000 Wide 1.625 3.500 0.180
1-5/8" 1.6250 --- 1.750 3.750 0.180
1-3/4" 1.7500 --- 1.875 4.000 0.180
1-7/8" 1.8750 --- 2.000 4.250 0.180
2" 2.0000 --- 2.125 4.500 0.180

Hardware Sizes: Metric Screw Thread

This section contains tables of sizes for metric thread hardware.


Bolt Thread Sizes

The following table of thread sizes for coarse and fine pitch thread was created using the standard sizes from ASME B1.13M. Coarse pitch threads are preferred and should be used whenever possible, as stated in ASME B1.13M. The thread equations given previously for tensile stress area and for minor area were used in constructing the table.

The thread size designation for metric thread is given as "M[dia] x [pitch]". For example, a thread with a nominal diameter of 6 mm and a pitch of 1 mm is designated as "M6 x 1."

Nominal (Major)
Diameter [mm]
Coarse Pitch Fine Pitch
Pitch
[mm]
Tensile Stress
Area [mm2]
Minor
Area [mm2]
Pitch
[mm]
Tensile Stress
Area [mm2]
Minor
Area [mm2]
1.6 0.35 1.270 1.076 --- --- ---
2 0.4 2.073 1.789 --- --- ---
2.5 0.45 3.391 2.980 --- --- ---
3 0.5 5.031 4.473 --- --- ---
3.5 0.6 6.775 6.000 --- --- ---
4 0.7 8.779 7.750 --- --- ---
5 0.8 14.18 12.68 --- --- ---
6 1 20.12 17.89 --- --- ---
8 1.25 36.61 32.84 1 39.17 36.03
10 1.5 57.99 52.29 1.25 61.20 56.30
12 1.75 84.27 76.25 1.25 92.07 86.04
14 2 115.4 104.7 1.5 124.5 116.1
16 2 156.7 144.1 1.5 167.2 157.5
20 2.5 244.8 225.2 1.5 271.5 259.0
24 3 352.5 324.3 2 384.4 364.6
30 3.5 560.6 519.0 2 621.2 596.0
36 4 816.7 759.3 2 914.5 883.8
42 4.5 1121 1045 2 1264 1228
48 5 1473 1377 2 1671 1629
56 5.5 2030 1905 2 2301 2252
64 6 2676 2520 2 3031 2975
72 6 3460 3282 2 3862 3799
80 6 4344 4144 1.5 4851 4798
90 6 5591 5364 2 6099 6020
100 6 6995 6740 2 7562 7473
110 --- --- --- 2 9182 9084


Bolt Thread Dimensions

The following equations can be used to calculate dimensions for ISO metric threads. The thread profile is based on a parameter \(H\), the height of the fundamental triangle. The value of \(H\) is related to the thread pitch, \(P\) by:

$$ H = { \sqrt{3} \over 2 } P $$
Equation, Metric Units [mm] Source
Minor Diameter $$ d_{m.ext} = d_{nom} - 1.226869 P $$ Shigley
Pitch Diameter $$ d_{p.ext} = d_{nom} - 0.75 H = d_{nom} - 0.64951905 P $$ Machinery's Handbook
Nominal Area $$ A_{nom} = {\pi \over 4} d_{nom}^2 $$
Tensile Stress Area $$ A_{t} = {\pi \over 4} \left( d_{nom} - 0.9382 P \right)^2 $$ ASME B1.13M, Appendix B
Minor Area
(Shear Area)
$$ A_{m} = {\pi \over 4} d_{m.ext}^2 $$

In the table above, \(d_{nom}\) is nominal diameter in millimeters and \(P\) is the thread pitch in millimeters.


Minimum Clearance Hole Diameters

The following table of clearance holes was adapted from ASME B18.2.8. The minimum hole diameters are given. This table also matches the table of recommended clearance holes from ASME B18.2.3.1M.

Bolt Size Normal Fit [mm] Close Fit [mm] Loose Fit [mm]
M1.6 1.8 1.7 2
M2 2.4 2.2 2.6
M2.5 2.9 2.7 3.1
M3 3.4 3.2 3.6
M4 4.5 4.3 4.8
M5 5.5 5.3 5.8
M6 6.6 6.4 7
M8 9 8.4 10
M10 11 10.5 12
M12 13.5 13 14.5
M14 15.5 15 16.5
M16 17.5 17 18.5
M20 22 21 24
M24 26 25 28
M30 33 31 35
M36 39 37 42
M42 45 43 48
M48 52 50 56
M56 62 58 66
M64 70 66 74
M72 78 74 82
M80 86 82 91
M90 96 93 101
M100 107 104 112


Bolt Thread Length

Per ASME B18.2.3.1M, Table 7, "Thread Lengths," the nominal thread length of metric bolts can be found by:

$$ L_{thd} = \left| \begin{array}{ll} 2 d_{nom} + 6 \text{ mm} & L \le 125 \text{ mm}, & d_{nom} \le 30 \text{ mm} \\ 2 d_{nom} + 12 \text{ mm} & 125 \lt L \le 200 \text{ mm}, & d_{nom} \le 48 \text{ mm} \\ 2 d_{nom} + 25 \text{ mm} & L \gt 200 \text{ mm} \end{array} \right. $$

where \(L\) is the total bolt length and \(d_{nom}\) is the nominal bolt diameter.


Hex Bolt Head Dimensions

The following table of hex bolt head dimensions was adapted from ASME B18.6.7M, Table 14, "Dimensions of Hex Head Machine Screws." This table is used for smaller size hardware.

Nominal Diameter
and Thread Pitch
Width Across Flats Head Height
Minimum [mm] Maximum [mm] Minimum [mm] Maximum [mm]
M2 x 0.4 3.02 3.20 1.3 1.6
M2.5 x 0.45 3.82 4.00 1.8 2.1
M3 x 0.5 4.82 5.00 2.0 2.3
M3.5 x 0.6 5.32 5.50 2.3 2.6
M4 x 0.7 6.78 7.00 2.6 3.0
M5 x 0.8 7.78 8.00 3.3 3.8
M6 x 1 9.78 10.00 4.1 4.7
M8 x 1.25 12.73 13.00 5.2 6.0
M10 x 1.5 15.73 16.00 6.5 7.5
M12 x 1.75 17.73 18.00 7.8 9.0


The following table of hex bolt head dimensions was adapted from ASME B18.2.3.1M, Table 3, "Dimensions of Hex Cap Screws."

Nominal Diameter
and Thread Pitch
Width Across Flats Head Height
Minimum [mm] Maximum [mm] Minimum [mm] Maximum [mm]
M5 x 0.8 7.78 8.00 3.35 3.65
M6 x 1 9.78 10 3.85 4.15
M8 x 1.25 12.73 13.00 5.10 5.50
M10 x 1.5 15.73 16.00 6.17 6.63
M12 x 1.75 17.73 18.00 7.24 7.76
M14 x 2 20.67 21.00 8.51 9.09
M16 x 2 23.67 24.00 9.68 10.32
M20 x 2.5 29.16 30.00 12.12 12.88
M24 x 3 35.00 36.00 14.46 15.44
M30 x 3.5 45.00 46.00 17.92 19.48
M36 x 4 53.80 55.00 21.62 23.38
M42 x 4.5 62.90 65.00 25.03 26.97
M48 x 5 72.60 75.00 28.93 31.07
M56 x 5.5 82.20 85.00 33.80 36.20
M64 x 6 91.80 95.00 38.68 41.32
M72 x 6 101.40 105.00 43.55 46.45
M80 x 6 111.00 115.00 48.42 51.58
M90 x 6 125.50 130.00 54.26 57.75
M100 x 6 140.00 145.00 60.10 63.90


Hex Nut Dimensions

The following table of hex nut dimensions was adapted from ASME B18.2.4.1M, Table 1, "Dimensions of Hex Nuts, Style 1." For further reference, also see ASME B18.2.4.2M, Table 1, "Dimensions of Hex Nuts, Style 2.".

Nominal Diameter
and Thread Pitch
Width Across Flats Thickness
Minimum [mm] Maximum [mm] Minimum [mm] Maximum [mm]
M1.6 x 0.35 3.02 3.20 1.05 1.30
M2 x 0.4 3.82 4.00 1.35 1.60
M2.5 x 0.45 4.82 5.00 1.75 2.00
M3 x 0.5 5.32 5.50 2.15 2.40
M3.5 x 0.6 5.82 6.00 2.55 2.80
M4 x 0.7 6.78 7.00 2.90 3.20
M5 x 0.8 7.78 8.00 4.40 4.70
M6 x 1 9.78 10.00 4.90 5.20
M8 x 1.25 12.73 13.00 6.44 6.80
M10 x 1.5 15.73 16.00 8.04 8.40
M12 x 1.75 17.73 18.00 10.37 10.80
M14 x 2 20.67 21.00 12.10 12.80
M16 x 2 23.67 24.00 14.10 14.80
M20 x 2.5 29.16 30.00 16.90 18.00
M24 x 3 35.00 36.00 20.20 21.50
M30 x 3.5 45.00 46.00 24.30 25.60
M36 x 4 53.80 55.00 29.40 31.00


Internal Thread Dimensions

The following equations can be used to calculate internal thread dimensions for ISO metric threads:

Equation, Metric Units [mm] Source
Minor Diameter $$ d_{m.int} = d_{nom} - 1.25 H = d_{nom} - 1.08253175 P $$ Machinery's Handbook
Pitch Diameter $$ d_{p.ext} = d_{nom} - 0.75 H = d_{nom} - 0.64951905 P $$ Machinery's Handbook

In the table above, \(d_{nom}\) is nominal diameter in millimeters and \(P\) is the thread pitch in millimeters.


Flat Washer Dimensions

The following table of flat washer dimensions was adapted from ASME B18.22M, Table 1, "Dimensions of Metric Plain Washers (General Purpose)." Plain washers come in 3 series: Regular, Narrow, and Wide.

Nominal Size Series Inner Diameter Outer Diameter Thickness
Min [mm] Max [mm] Min [mm] Max [mm] Min [mm] Max [mm]
M1.6 Narrow 1.95 2.09 3.70 4.00 0.50 0.70
M1.6 Regular 1.95 2.09 4.70 5.00 0.50 0.70
M1.6 Wide 1.95 2.09 5.70 6.00 0.60 0.90
M2 Narrow 2.50 2.64 4.70 5.00 0.60 0.90
M2 Regular 2.50 2.64 5.70 6.00 0.60 0.90
M2 Wide 2.50 2.64 7.64 8.00 0.60 0.90
M2.5 Narrow 3.00 3.14 5.70 6.00 0.60 0.90
M2.5 Regular 3.00 3.14 7.64 8.00 0.60 0.90
M2.5 Wide 3.00 3.14 9.64 10.00 0.80 1.20
M3 Narrow 3.50 3.68 6.64 7.00 0.60 0.90
M3 Regular 3.50 3.68 9.64 10.00 0.80 1.20
M3 Wide 3.50 3.68 11.57 12.00 1.00 1.40
M3.5 Narrow 4.00 4.18 8.64 9.00 0.80 1.20
M3.5 Regular 4.00 4.18 9.64 10.00 1.00 1.40
M3.5 Wide 4.00 4.18 14.57 15.00 1.20 1.75
M4 Narrow 4.70 4.88 9.64 10.00 0.80 1.20
M4 Regular 4.70 4.88 11.57 12.00 1.00 1.40
M4 Wide 4.70 4.88 15.57 16.00 1.60 2.30
M5 Narrow 5.60 5.78 10.57 11.00 1.00 1.40
M5 Regular 5.60 5.78 14.57 15.00 1.20 1.75
M5 Wide 5.60 5.78 19.48 20.00 1.60 2.30
M6 Narrow 6.65 6.87 12.57 13.00 1.20 1.75
M6 Regular 6.65 6.87 18.28 18.80 1.20 1.75
M6 Wide 6.65 6.87 24.88 25.40 1.60 2.30
M8 Narrow 8.90 9.12 18.28 18.80 1.60 2.30
M8 Regular 8.90 9.12 24.88 25.40 1.60 2.30
M8 Wide 8.90 9.12 31.38 32.00 2.00 2.80
M10 Narrow 10.85 11.12 19.48 20.00 1.60 2.30
M10 Regular 10.85 11.12 27.48 28.00 2.00 2.80
M10 Wide 10.85 11.12 38.38 39.00 2.50 3.50
M12 Narrow 13.30 13.57 24.88 25.40 2.00 2.80
M12 Regular 13.30 13.57 33.38 34.00 2.50 3.50
M12 Wide 13.30 13.57 43.38 44.00 2.50 3.50
M14 Narrow 15.25 15.52 27.48 28.00 2.00 2.80
M14 Regular 15.25 15.52 38.38 39.00 2.50 3.50
M14 Wide 15.25 15.52 49.38 50.00 3.00 4.00
M16 Narrow 17.25 17.52 31.38 32.00 2.50 3.50
M16 Regular 17.25 17.52 43.38 44.00 3.00 4.00
M16 Wide 17.25 17.68 54.80 56.00 3.50 4.60
M20 Narrow 21.80 22.13 38.38 39.00 3.00 4.00
M20 Regular 21.80 22.32 49.00 50.00 3.50 4.60
M20 Wide 21.80 22.32 64.80 66.00 4.00 5.10
M24 Narrow 25.60 26.12 43.00 44.00 3.50 4.60
M24 Regular 25.60 26.12 54.80 56.00 4.00 5.10
M24 Wide 25.60 26.12 70.80 72.00 4.50 5.60
M30 Narrow 32.40 33.02 54.80 56.00 4.00 5.10
M30 Regular 32.40 33.02 70.80 72.00 4.50 5.60
M30 Wide 32.40 33.02 88.60 90.00 5.00 6.40
M36 Narrow 38.30 38.92 64.80 66.00 4.50 5.60
M36 Regular 38.30 38.92 88.60 90.00 5.00 6.40
M36 Wide 38.30 38.92 108.60 110.00 7.00 8.50


Maximum Bending Moment on Bolt

The maximum bending moment on a bolt is given by:

$$ M_b = {F_s a \over 2} $$

where \(F_s\) is the applied shear force and \(a\) is the moment arm.

A bolt can be modeled as a fixed-guided beam (i.e. a beam with a fixed boundary condition at one end and a guided boundary condition at the other end). In the model below, the left end of the bolt is fixed (all degrees of freedom fixed) and the right end is guided (rotation and x-translation are fixed, but free to translate in y). A shear force of 100 lbf is applied to the right end.

Bolt Model

The Free Body Diagram (FBD) and deformed mesh for this case are shown below. If the model above were a cantilever beam, the moment due to the applied force would be taken entirely at the single fixed end. However, because the model is fixed against rotation at both ends, the moment is shared between the two ends of the bolt.

Free Body Diagram
Deformed Mesh

The Shear-Moment Diagram for this case is shown below:

Shear-Moment Diagram


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References

General References:

  1. Barrett, Richard T., "Fastener Design Manual," NASA Reference Publication 1228, 1990.
  2. Budynas-Nisbett, "Shigley's Mechanical Engineering Design," 8th Ed.
  3. Machinery's Handbook, 27th Ed., Industrial Press Inc., 2004.
  4. Lindeburg, Michael R., "Mechanical Engineering Reference Manual for the PE Exam," 13th Ed.

Specifications and Standards:

  1. ASME B1.1, "Unified Inch Screw Threads (UN and UNR Thread Form)," The American Society of Mechanical Engineers, 2003.
  2. ASME B18.2.1, "Square, Hex, Heavy Hex, and Askew Head Bolts and Hex, Heavy Hex, Hex Flange, Lobed Head, and Lag Screws (Inch Series)," The American Society of Mechanical Engineers, 2012.
  3. ASME B18.2.2, "Nuts for General Applications: Machine Screw Nuts, Hex, Square, Hex Flange, and Coupling Nuts (Inch Series)," The American Society of Mechanical Engineers, 2010.
  4. ASME B18.2.8, "Clearance Holes for Bolts, Screws, and Studs," The American Society of Mechanical Engineers, 1999.
  5. ASME B18.21.1, "Washers: Helical Spring-Lock, Tooth Lock, and Plain Washers (Inch Series)," The American Society of Mechanical Engineers, 2009.
  6. FED-STD-H28/2B, "Screw-Thread Standards for Federal Services," Federal Standard, 1991.