Prestressed bolt connection.

The calculation is designed for a geometrical design and strength check of a prestressed bolt connection, loaded by static or cyclic loading resp., acting both in the axis of the bolt and in the plane of the connected parts. The application solves the following tasks:

1. Automatic design of a connection bolt of standard design.

2. Calculation and check of connections fitted with special shanks.

3. Design and calculation of necessary mounting prestressing of the connection and fastening torque.

4. Calculation of force conditions of a loaded connection.

5. Static and dynamic strength check.

6. The application includes a table of commonly used materials of bolts according to ISO, SAE and ASTM, and a selection of materials of the connected parts according to AISI/SAE, DIN, BS, AF and others.

7. Support of 2D CAD systems.

The calculations use data, procedures, algorithms and data from specialized literature and standards ANSI, ISO, DIN.
List of standards: ANSI B1.1, ANSI 273, ANSI B18.2.1, ANSI B18.2.2, ANSI B18.3, ANSI B18.6.2, ANSI B18.6.3, ANSI B18.22.1, ISO 273, ISO 1207, ISO 4016, ISO 4032, ISO 4035, ISO 4762, ISO 8738, VDI 2230

Control, structure and syntax of calculations.

Information on the syntax and control of the calculation can be found in the document "Control, structure and syntax of calculations".

Information on the project.

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document  "Information on the project".

Process of calculation.

Prestressed bolt connections form the majority of bolt and threaded connections used in practice. These connections are loaded with great internal axis force (mounting prestressing) during assembly. This prestressing provides the necessary force bond of contact surfaces of the connected materials.

In principle, the designed structural junction should behave as a compact formation. A correctly prestressed connection forms during operation of a cohesive unit with guaranteed force closing in the contact surfaces and unchanged mutual position of the connected parts. The mounting prestressing of a bolt connection thus includes two basic functions. In case of connections loaded in the plane of the connected parts, the prestressing meets the requirement of shear loading capacity with utilization of friction forces; in case of connections loaded in the axis of the bolt, it meets the requirement of compactness or tightness of the connection resp.

From the above-mentioned, the individual steps which are necessary for successful design of a prestressed bolt connection result:

• Selection of a suitable connection bolt
• Calculation of the necessary prestressing
• Strength check of the designed connection

It is advisable to use the following procedure during design and check:

1. In paragraph [1] select the type of connection, mode and size of loads.
2. Set the necessary operational and mounting parameters of the connection. [2.1 - 2.13]
3. Set a factor of implementation of the working force for the connection which is loaded in the axis of the bolt. [2.15]
4. For a connection with dynamic loading, select a thread design, desired service life, reliability and safety of the connection. [2.19 - 2.22]
5. Set the method of design, material and dimensions of the connected parts in the paragraph. [3]
6. Select the material of the bolt [4.3] and the type of thread [4.10]. As a hint, the table of recommended min. diameters of threads in the paragraph may be used. [4.1].
7. Pressing the button in the row [4.11] activates a design of dimensions of threads.
8. If you wish to use a non-prismatic alternatively fitted screw instead of a common bolt, specify its geometry in the paragraph. [4.16]
9. Similarly, you can modify the geometry of the connection in chapter [4.23] if you wish to use a bolt with a head other than with a hexagonal or cylinder head or a ball seating surface, or with a different diameter of hole.
10. In chapter [5] check force conditions in the connection and/or modify the designed amount of mounting prestressing. [5.12]
11. Check results of the strength checks in chapter [6]. In case any of the mentioned checks are not sufficient, enlarge dimensions of the thread [4.11] or modify parameters in chapters [4.16, 4.23].
12. Save the workbook with the designed solution under a new name.

Note: This calculation is meant for the design of a connection with one bolt. If you need to calculate bolt connections with more bolts, first it is necessary, according to general principles (see the chapter "Calculation of bolt fields") to specify the maximum load falling on one (identically or the most loaded) bolt. This will be then be solved as an independent bolt connection according to the above-mentioned procedure.

Loading of the connection, basic parameters of the calculation. [1]

In this paragraph it is first necessary to enter basic input parameters, characterizing the manner, mode and amount of loading and the type of connection.

1.1 Calculation units.

In the selective list, select the desired system of calculation units. All values will be recalculated immediately after switching to other units.

1.3 Design of the bolt connection.

In view of the construction, there are two basic methods of bolt connection design:

1. Bolt connection with a bolt stud
2. Bolt connection with a through bolt

In the selective list, select the connection design.

1.4 Loading of the bolt connection.

Depending on the type of loading, a prestressed bolt connection must meet different requirements; this results in a different method of calculation of the mounting prestressing. There are three different ways of loading for purposes of calculations of bolt connections:

1. Loading in the bolt axis.
   The bolt connection is loaded with the axial force F_a. Here the mounting prestressing meets the requirement of compactness, or more precisely, tightness of the connection during operation. The prestressing of the connection must, therefore, be high enough to provide sufficient residual prestressing of the connected part after loading of the connection with the operational force and thus the necessary force bond of the contact surfaces.
2. Loading perpendicular to the bolt axis.
   The bolt connection is loaded with a radial force F_r , acting in the plane of the connected parts. The mounting prestressing meets the requirement of shear loading capacity of the connection using friction forces. The shear acting on the connection must therefore be carried between the connected parts using friction, which is created by the tightening prestressing of the bolts.
3. Combined loading.
   With connections exposed to combined loading, the requirement of compactness and a requirement of shear loading capacity must be met as well.

Select the desired type of loading in the selective list. After selection, the calculation will be modified into the configuration which corresponds with the selected type of loading - the parameters, which have no meaning for the selected type, will be hidden.

1.5 The course of loading.

This list allows users to define the type (course) of loading which acts on the connection. The bolt connection can be designed for the following types of loading:

1. Quiet (static)
2. Pulsating
3. Repeated
4. Reversed asymmetric
5. Reversed symmetric

With connections exposed to cyclic loading (loading B to E) it is also necessary to carry out, in addition to common strength checks, checks of the connection bolt in view of fatigue strength.

Select the desired course of loading in the selective list. After selection, the calculation will be modified into the configuration which corresponds with the selected type of loading - the parameters, which have no meaning for the selected type, will be hidden.

Note: With bolt connections exposed to radial forces in the plane of the connected parts (see [1.4]), the option of the course of loading has no effects on the proper calculation of the connection. The connection is dimensioned for the maximum amount of radial force. For a cyclic course of loading it is advisable to select higher values of safety against side shift in the row [2.2].

1.6 Loading of the connection.

In this row, enter operational forces acting on the bolt connection. In the row [1.7] enter the amount of static axial (axis) force or the upper value of amplitude of the force with cyclic loading. In the row [1.8] enter the lower value of amplitude of the force with cyclic loading. The row [1.9] can be used to enter the radial force, but always enter the maximum amount of this force.

Warning: Input fields for entering forces are accessible according to settings of the loading mode in rows [1.4, 1.5].

Operational and mounting parameters of the connection. [2]

This paragraph is designed for setting various operational and mounting parameters which are necessary for the design and calculation of a prestressed bolt connection.

2.1 Desired coefficient of tightness (prestressing) of the connection.

A correctly prestressed bolt connection forms during operation a cohesive unit with guaranteed force bond on the contact surfaces and unchanged mutual position of the connected parts. This requirement of compactness, which is important particularly with connections exposed to variable loading, is extended in some cases by the requirement for connection tightness. The pressure on the contact surfaces caused by the prestressing must guarantee a hermetic bond of the connection during operation.

The requirement of compactness or tightness of the connection is applied with the design of the connection through this coefficient, which gives the ratio between the residual prestressing of the clamped parts of the connection and the maximum operational force. The option of this coefficient thus affects the amount of mounting prestressing of the designed connection. The coefficient is usually selected in the limits according to the following recommendations:

Requirement of compactness of the connection

Requirement of tightness of the connection (higher values used with a variable force or with sealing of a dangerous medium)

Note: This coefficient has no meaning for connections loaded with a radial force only.

2.2 Desired safety against side shift.

With a properly designed connection loaded in the plane of the connected parts, the entire radial force must be transferred using friction between the connected parts, which arises from the mounting prestressing. This safety coefficient gives the ratio between the actual residual prestressing in the connection and the minimum (calculated theoretically) clamping force necessary for entire transfer of the radial force. The requirement of the shear loading capacity of the connection should meet a safety level higher than 1, however, in fact with regards to technological properties of the operation and possible inaccuracy of theoretical determination of coefficients of friction between the connected surfaces, it is advisable to specify the safety against side shifts in a range from 1.5 to 3. The upper values are selected with connections exposed to a variable loading. With combined loading (see [1.4]) or loading with shocks, it is possible to use a higher safety level.

Note: This coefficient has no meaning for connections loaded with axis forces only.

2.3 Desired safety of the bolt at the yield point.

The minimum permissible ratio of the yield point of the selected material of the bolt and the maximum reduced stress in the bolt core. The lower limit of safety at the yield point with connection bolts is usually selected with regards to the type of loading, importance of the connection, quality of production, operational conditions and accuracy of calculation, in a range from 1.5 to 3. The lower values are selected for connections exposed to a static loading, the upper values are selected for connections exposed to a variable loading. With important connections, connections exposed to shocks, connections working in a corrosive environment or at high operational temperatures, there are usually selected even higher values of safety (3 ... 6). General procedures of setting the safety coefficient can be found in the document "Coefficients of safety".

Warning: In case the calculation does not respect any effects of additional bending stresses [2.7] or effects of operational temperature [2.9] as the case may be, despite the fact that these effects appear in reality, it is advisable to design the bolt connection with an adequately increased level of safety.

2.4 Friction coefficient in threads.

The size of the coefficient of friction in threads depends on the material, roughness, surface treatment and angle of slope of the thread sides. Orientation values of the coefficient of friction for a sharp thread are given in the table. The coefficient of friction is usually lower with flat threads.

Non-lubricated thread (without any special lubrication, however, not degreased)

Lubricated thread

2.5 Friction coefficient in seating face of the head (nut) of the bolt.

The size of the coefficient of friction under the head (nut) of the bolt depends on the material of the nut and the clamped parts, roughness, surface treatment and lubrication. Orientation values of the coefficient of friction for a steel head of the bolt (nut) are given in the table.

2.6 Friction coefficient between he connected surfaces.

The size of the coefficient of friction between the connected surfaces depends on the material of the connected parts, roughness, surface treatment and degreasing of the connected surfaces. Orientation values of the coefficient of friction are given in the table.

2.7 Additional bending loading of the bolt.

An additional bending stress appears in the bolt core in case of uneven seating of the head of the bolt or nut on the seating surfaces. Bending stress appears usually due to inaccuracy of production (seating surfaces below the head and nut are not parallel and perpendicular to the axis of the bolt) or due to a deformation of the clamped parts under loading. The bending stress may be several times higher than the tensile loading in the bolt core and often causes breaking of the bolt in the thread output. Additional bending is always very dangerous as far as strength of the bolt (particularly in case of a variable loading) and you must prevent this by careful machining surfaces and/or using levelling or ball pads. The size of bending loading can also be reduced by reducing the diameter of the connection bolt shaft or increasing its length.

The size of bending stress depends on the angle deviation of the seating surface of the bolt head from perpendicularity to the bolt axis [2.8]. In precision engineering, the maximum permissible deviation is approx. d=5' (=0.085°).

Note: In case you do not wish to include the possible effects of bending stresses in the calculation, it is advisable to increase safety at the yield point [2.3] by 20 to 50%.

2.9 Influence of the operational temperature on prestressing of the connection.

If the connected parts and the bolt are made of materials with the same thermal expansivity, heating of all parts of the connection to the same temperature has no noticeable effects on force conditions in the connection.

However, some bolt connections get to a situation where heat from the heated parts is transferred to the bolt with difficulties, so that its temperature is lower than the temperature of adjacent parts. This increases the operational prestressing of the connection and thus also loading of the connecting bolt. Operational prestressing is also changed if the bolt and the connected part have different thermal expansivities. In both these cases, the correct design must include effects of operational temperature in the calculation of the connection.

Note: If you decide to include possible effects of the operational temperature in the calculation, the necessary mounting prestressing of the connection [5.12] will be designed with regards to its possible change if the connection is heated to operational temperature. Otherwise, it is advisable to compensate the possible increase in operational prestressing of the connection (and thus higher loading of the bolt) using a higher level of safety [2.3] in the design of the connection.

Warning: The application does not solve effects of operational temperature on the strength of the material. The strength, however, decreases at higher temperatures in real operation; this may cause defects of the bolt. Effects of temperature are obvious particularly with cyclic loading of the connections. This fact must be taken into account when determining desired safety coefficients [2.3, 2.22].

2.12 Reduction of mounting prestressing using permanent deformation (settlement) of the connection.

Certain permanent (plastic) deformation of the connection occurs in prestressed bolt connections in operation. This "settlement" of the connection is caused e.g. by squeezing of the threads on the bolt and/or the nut, squeezing of contact surfaces of the connected parts and the sealing insert, permanent elongation of the bolt, etc. This deformation may cause a slow decrease in prestressing of the connection in operation and may also cause possible leakage or non-compactness of the connection.

Guide values in [mm] for permanent squeezing of the clamped parts (including threads) are given in the following tables (values in [in] are given in parenthesis):

Tensile/pressure loading of the connection

Shear loading of the connection

Note: If you decide not to include the possible effects of "settlement" of the connection in the calculation, it is advisable to design the bolt connection with a higher coefficient of prestressing [2.1], increased by 50 to 80 %.

2.14 Factor of implementation of operational force.

Places of inputs of external axial loading do not always have to be located in the bearing surfaces under heads and nuts of the bolts. On the contrary, the axial forces usually act in places inside the clamped parts. The factor of implementation of operational force gives a ratio between distances of actual points of actions of the operational force and the total height of the clamped parts, and with regards to possible limiting conditions it reaches values in the range [0...1]. This ratio may significantly affect distribution of loading from the operational force between the connection bolt and the clamped parts and thus the amount of necessary mounting prestressing of the connection.

Factor of implementation of operational force - limiting conditions

Whilst the position of the point of action of the operational force is quite obvious with some connections, such positions are more intuitive with other connections and its exact determination may be quite difficult. The existence of two limiting conditions, between which reality can be found, is a certain guideline. If determination of the coefficient of implementation of the operational force is not obvious from the geometry of the connection, it is presupposed as n=0.5 in connections with bolts through and n=0.75...0.25 according to the design of the connection in connections with stud bolts.

Determination of coefficient of implementation of operational force in flange connections

More precise values of the factor of implementation of operational force can be obtained e.g. from the following table according to VDI 2230:

where the meaning of individual dimensions and the type of design are defined on the following illustration:

Note: Parameters of implementation of operational force can be described in the calculation in two ways. The first way is to enter a coefficient of implementation of operational force [2.15]; the second is to enter the exact distance of the point of action of the operational force from the head/nut of the bolts [2.16, 2.17]. The way is determined by setting the switch at the respective row. Both ways are equivalent as regards calculations, however, the second way gives more exact results in case of connection with more connected parts made of different materials.

2.19 Thread design.

The way of calculation of production of the thread significantly affects the fatigue strength of the connection bolt. As far as fatigue loading, the cut thread gives the worst results.

Recommendations: If you do not know the method of production of the thread, select a cut thread.

2.20 Desired service life of the connection in thousands of cycles.

With bolt connections loaded repeatedly we can differentiate between two fields of fatigue loading of connection bolts. In the first field with limited service life of the connection (the desired service life is lower than approx. 10⁶ operational cycles) the fatigue strength of the bolt decreases with an increasing number of operational cycles. In the field of unlimited service life (the desired service life of the bolt is higher than 10⁶ operational cycles) the fatigue limit of the material and thus the strength of the connection bolt remains approximately constant.

2.21 Desired reliability of the connection.

The coefficient of reliability is, in principle, a percentage of the service life and expresses the probability of trouble-free operation of the connection. In mechanical engineering, reliability is usually considered to be between 80 and 99.9%. A value of desired reliability higher than 99.9% is used only with very important equipment, whose failure could pose a threat to human lives or high material losses. In case of common bolt connections with a variable loading, the value of reliability between 95 and 99.5% is usually chosen.

2.22 Desired dynamic (fatigue) safety.

It is necessary to check bolt connections loaded with variable loading regarding fatigue strength. Resistance against possible fatigue breakage of the connection bolt is considered in the resulting coefficient of dynamic safety. This level of safety is a value composed of two partial safety levels. One component is represented by own "dynamic" safety considering the position of the bolt in view of variable tensile stress; the other component is the level of safety with regards to permanent shear stress arising in the bolt from mounting prestressing of the connection.

With regards to the accuracy and credibitility of input data, constructional design of the connection, character of loads and quality of production and operational conditions, there is usually selected a value of dynamic safety in the range from 1.5 ... 2.5. The following safety values are recommended for connections used in a non-corrosive environment at operating temperatures up to 100 °C:

Constructional principles for design of connections exposed to variable loading are given in the chapter "Causes of failures of bolt connections, increasing loading capacity of bolts".

Note 1: For connections where additional bending stresses appear (see [2.7]), higher safety values are required (2 ... 3).

Note 2: Substantially higher safety values are used for connections exposed to shocks, used in a corrosive environment or at high temperatures.

Design, dimensions and material of connected parts. [3]

The calculation allows users to design a prestressed bolt connection of up to five parts using different materials. This paragraph can be used for a description of the geometry and selection of materials of the connected parts.

3.1 Design of connected parts.

It is necessary to know the stiffness of the connected parts to determine force conditions in a prestressed bolt connection. In view of calculating stiffness, the design of connections can be divided into two basic model situations:

1. Connection of plates.
   The clamped parts are plates with cross dimensions so large that the area exposed to the stress (having approximately the shape of a truncated cone) has no marginal limitation. The amount of stiffness then depends only on the material and total height of the clamped parts.
2. Connection of thick-walled cylinders.
   The connected parts do not have such large cross dimensions as to create a "pressure cone" without any marginal limitation. The area exposed to stress is partly or fully bordered by the outer walls of the clamped parts. All such cases are, with certain simplification, replaced by a virtual thick-walled cylinder with an adequate outer diameter. The amount of stiffness of the clamped parts depends, in addition to others, also on this diameter.

Select the design of the connection part in the selective list. In case of connection of cylinder parts you must also enter their diameters [3.2].

3.2 Number of clamped parts.

Select the number of clamped parts in the selective list. Define their dimensions in the table [3.5].

3.4 Total height of the clamped parts.

The total height of the clamped parts is considered as the distance between the head of the bolt and the nut. If the bolt connection is provided with washers, it is necessary to include the thickness of the washers in the total clamping height.

3.5 Material and height of the connected parts.

Enter the height and material of the clamped parts in the table. The connected parts are arranged in the table successively going down from the head of the bolt.

The meaning of parameters in the table:

The first five rows of the list is reserved for materials defined by the user. Information and settings of proper materials can be found in the document "Workbook (calculation) modifications". Other rows of the list include a selection of materials for the actually specified standard in the sheet "Material".

Note 1: If the connection is provided with washers, add the thickness of the washers to the height of the marginal clamped parts.

Note 2: If a washer is used with a material with modulus of elasticity different from the material of clamped parts and its thickness is more than approx. 10% of the total clamped height, it is advisable to define the washer in the calculation as an independent part.

Design of connecting bolt. [4]

This paragraph can be used for selection of material and design of suitable dimensions of bolts for the above-mentioned specified design and loading of a prestressed bolt connection. The connection bolt can be designed manually, or use the automatic design function using the pushbutton in the row [4.11].

Note: The function of the automatic design is functional only for standardized types of threads (see [4.9]).

4.1 Preliminary design of minimum thread diameters.

Values of minimum diameters of the thread are calculated additionally in the table depending on various combinations of materials of bolts and types of threads. Each column of the table concerns the specified material of the bolt and each row concerns the specified thread. Individual strength classes of bolts are selected according to the standard and specified in a list of standards in the sheet "Material". The used specification of the type of thread has the following meaning:

Note: The values of minimum diameters of threads specified in this table are orientation values only, designed in view of minimum safety of the connection, and do not respect the required coefficient of safety defined in rows [2.3, 2.22].

4.2 Material of the bolt.

Material of the connection bolt can be selected in the pop-up list in the row [4.3]. The table "recommended diameters of the thread" [4.1] can be used as a guideline for selection of a suitable material. The first five rows of the list are reserved for materials defined by the user. Information on setting proper materials can be found in the document "Workbook (calculation) modifications". Other rows of the list include the selection of materials for the actually specified standard in the sheet "Material".

4.9 Parameters of the thread.

This part can be used for selection of the type and dimensions of the thread of the connection bolt. Select the type of thread in the list [4.10]. The first five items of the list are reserved for standardized threads. Select the desired nominal size of the thread for the selected type of thread in the list [4.11]. Other necessary dimensions of the thread [4.12 to 4.15] are calculated automatically according to the standard. For a design of suitable size of the thread you can use the pushbutton "Design" [4.11] for automatic design of the connection bolt.

In case you wish to use a bolt with another (special) type of thread in the connection, select the last item in the list [4.10] and in input fields [4.12 .. 4.15] enter all necessary dimensions of the thread.

4.11 Automatic design of the connection bolt.

Automatic design can be initiated using the pushbutton "Design" [4.11]. In case of automatic design, a bolt of the minimum dimensions for the selected material of the bolt [4.3] and the type of the thread [4.11] is selected so that the desired functionality of the connection is ensured and the bolt meets specifications of the desired level of safety [2.3] in view of strength resp. [2.22]. The application provides a design only for prismatic bolts of constant diameters (see [4.16]) with inter-circular contact surface [4.24]. Furthermore, the design also automatically includes dimensions of seating surfaces of the head and the diameter of the hole for the connection bolt (see [4.23]). If the application does not find a suitable bolt, this fact is indicated by a warning message.

The most frequent causes of insufficient designs and their possible solutions:

• Too high loading of the connection:
  Solution - The amount of input loading of the connection [1.6] can be reduced in a design of the connection with more bolts (see the chapter "Calculation of bolt fields").
• Low loading capacity of the connection bolt:
  Solution - use bolts of a material with a higher loading capacity [4.2].
• Too high additional bending loading of the bolt:
  Solution - A suitable constructional modification of the connection (see [2.7]).
• Increase of the operational prestressing of the connection due to heating of the connection to operating temperature:
  Solution - A suitable constructional modification of the connection, e.g..:
  - ensure uniform heating of the bolt and the connected parts
  - increase elastic pliability of the connection by reducing the diameter of the bolt shank or inserting a spacing pipe between the nut and the connected parts
  - use bolts with high strength

Note: Automatic design functions only for standardized types of threads (see [4.9]).

Warning: In case of use of automatic design of the bolt, the data are preset in the paragraphs [4.16, 4.23] according to the above-mentioned rules.

4.16 Design and geometry of the bolt.

Sometimes it may be suitable, from a technological or constructional viewpoint, to use a special bolt in the connection instead of a common prismatic bolt with several different cross- sections. For example in cases of request for accurate connection of the parts using fitted bolts or when using the connecting bolt with shortened stem to reduce the influence of the additional bending tensions. Pliable bolts with special treatment are also frequently used with connections exposed to a variable loading.

This paragraph can be used to define these special bolts. The number of sections of a bolt with different cross-sections is entered in the row [4.18], the length and diameter of the particular section is defined in rows [4.21, 4.22]. Individual sections of the bolt are numbered in increasing order from the nut of the bolt.

4.23 Geometry of the connection.

This part can be used to set the shape and dimensions of the seating surface of the head (nut) of the bolt and to determine the diameter of the hole for the connection bolt. When the check mark in the row [4.24] is enabled, all necessary dimensions are set automatically according to the following rules:

• Diameter of the hole for the connection bolt [4.25] - The minimum diameter permitted by the standard is selected for the open hole for the thread selected in paragraph [4.9] in case of common prismatic bolts. The diameter of the hole coincident with the maximum diameter of the bolt shank preset in the row [4.22] is selected in case of fitted bolts.
• Outer diameter of seating face (contact surface) [4.26] - The diameter is preset for a common standardized bolt with a hexagonal head according to the size of the thread selected in [4.9].
• Inner diameter of seating face (contact surface) [4.27] - The diameters coincident with the diameter of the hole is preset in accordance with the most frequently used design.

If you wish to enter your own values for some of the above-mentioned dimension, first it is necessary to disable the check mark box in the row [4.24].

4.24 Design of seating faces below heads (nuts) of bolts.

In case of bolt connections, three basic types of designs of seating surfaces below heads (nuts) of bolts are used.

1. Annulus seating face.
   The most common and natural case of contacts in bolt connections with normal nuts if the seating surfaces are perpendicular to the axis of the bolt.
2. Conical seating face.
   Special use with bolt connections where precise centring of the connected part against the axis of the bolt is required. This requires a special conical nut and conical seat in the hole for the bolt, hence higher requirement for precision in production.
3. Spherical seating face.
   Special use in connections where perpendicularity of seating surfaces to the axis of the bolt and hence its additional bending loading can be expected. This requires a special ball nut and a ball seat in the hole for the bolt. This is very demanding for production technology.

Note: Individual designs differ, above all, as regards the technology and areas of utilization. In view of the design of a prestressed bolt connection, various designs of contact surfaces have no substantial effects on the calculation itself and differ only in the amount of friction moment below the head (nut) of the bolt and hence in the amount of necessary tightening torque.

Prestressing, force conditions and operational diagram of the connection. [5]

In this paragraph you can find force conditions acting in the designed prestressed bolt connection. Constants of stiffness of the connection are first calculated additionally in the first part [5.1]. Once the constants have been calculated, the necessary mounting prestressing of the connection and the corresponding tightening moment are designed in the second part [5.6]. The force conditions in a fully loaded bolt connection for the given mounting prestressing are calculated additionally in the last part [5.14]. The force conditions are shown in the illustration in the lower part of this paragraph.

5.1 Stiffness constants of the connections.

Constants of stiffness express the linear dependence between the axis force acting in the connection and the deformations of individual parts of the bolt connection caused by this force. The constants are considered as guide data for determination of force conditions of the prestressed bolt connection. Distribution of actions of external axial force between the connection bolt and the clamped parts of the connection is determined depending on the ratio of the resulting stiffnesses [5.4, 5.5]. The resulting stiffnesses are determined using the stiffnesses [5.1, 5.2] based on the selected factor of implementation of the operational force [2.14].

5.12 Mounting prestressing of the connection.

Determination of "correct" mounting prestressing is one of the main tasks in the design of a prestressed bolt connection. A sufficient amount of mounting prestressing is decisive for correct functioning of the connection. At the same time, this also affects the resulting force acting in the connection bolt, hence the level of safety against a possible breakage of the bolt. The mounting prestressing must be designed to ensure the requirement of compactness or tightness of the connection in case of connections loaded in the axis of the bolt, and the requirement of shear loading capacity of the connection in case of connections loaded in the plane of the connected parts.

Mounting prestressing can be designed manually or an automatic design can be used. The automatic design can be initiated if the check mark box to the right of the input field is enabled. The programme then provides a design for the minimum mounting prestressing to meet the above-mentioned requirement of compactness or shear loading capacity of the connection. The condition of compactness of the connection is considered as fulfilled if the coefficient of prestressing of the connection [5.20] is higher or equal to the desired value [2.1]. For fulfilling of the condition of shear loading capacity of the connection, the safety against side shift [5.21] must be higher or equal to the desired safety [2.2].

Mounting diagram of the connection

In the course of tightening (prestressing) of the connection, the bolt in the connection elongates and at the same time, the clamped parts are squeezed. The ratio between deformation of the bolt and deformation of the clamped parts is given by the ratio of their particular stiffnesses. After implementation of the axis operational force into the connection, the loading of the clamped parts is reduced and the loading of the connection bolt is increased. For the purpose of a strength check, it is therefore necessary to determine the maximum internal axis force acting on the bolt. The mounting diagram of the connection is used for this purpose.

The mounting diagram of the connection is compiled for known prestressing and stiffness values of individual elements of the connection. Distribution of the action of external axial force between the connection bolt and the clamped parts of the connection is determined using this diagram.

where:

F₀ - mounting prestressing of the connection

DL₁ - deformation (elongation) of the bolt due to mounting prestressing

DL₂ - deformation (squeezing) of the clamped parts due to mounting prestressing

c₁ = tg y₁ - constant of stiffness of the bolt

c₂ = tg y₂ - constant of stiffness of the clamped parts

F_a - maximum operational axial force loading the connection

DF₁ - the part of the axis component of the operational force additionally loading the bolt

DF₂ - the part of the axis component of the operational force relieving the clamped parts

F₁ - maximum internal axial force in the bolt

F₂ - residual prestressing of clamped parts of the connection

The given diagram is compiled with the presupposition that the points of inputs of the external axial loading are situated at the ends of the clamping length, in seating surfaces below the head and nut of the bolt. However, in reality the axial forces act usually on points situated inside the clamped parts (see The factor of implementation of the operational force [2.14]). This causes changes in the ratio of stiffness between the loaded and relieved parts of the connection, hence changes in the angles y₁ a y₂.

5.20 Coefficient of tightness (prestressing) of the connection.

This coefficient gives the ratio between the residual prestressing of the clamped parts of the connection [5.18] and the maximum axis operational force [5.7]. Detailed information can be found in [2.1].

Note: The coefficient of tightness of the connection has no meaning for connections loaded with radial forces only.

5.21 Safety against side shift.

This coefficient of safety gives the ratio between the actual residual prestressing in the connection [5.18] and the minimum (theoretically calculated) clamping force [5.9] necessary for full transfer of the radial force. Detailed information can be found in [2.2].

Note: This safety has no meaning for connections loaded with axial forces only.

Strength checks of the connection. [6]

This paragraph gives results of basic strength checks of the bolt connection.

6.1 Strength checks of statically loaded bolt connections.

The strength check is executed by comparing the resulting reduced stress in the bolt core [6.5] with the yield point of the material of the bolt [6.6]. The resulting reduced (comparative) stress is calculated in the thinnest part of the bolt (for prismatic bolts for small diameter threads, in the weakened shaft with pliable bolts).

The reduced stress is calculated according to the formula:

where:

s - tensile stress in the bolt core for the maximum axis force

s_b - additional bending stress

t - torsional stress in the bolt core for the tightening torsional moment

If the designed bolt has to meet all requirements of the strength check, the resulting level of safety [6.7] must be higher or equal to the desired level of safety [2.3].

6.8 Check of pressure in seating face of the bolt head.

If the designed bolt connection has to meet specifications of the check, the pressure in the seating surface [6.9] must be lower than the permitted pressure in the marginal connected part [6.10]. If the designed connection does not meet the specifications, modify the design so that the head (nut) seating surface is enlarged.

6.11 Strength check of dynamically loaded bolt connection.

A bolt connection exposed to variable loading must be checked in view of fatigue strength. Fatigue breaks usually appear on bolts at points of stress concentrations (in places of constructional notches), most frequently in the section of the first load carrying thread.

Resistance against a possible fatigue break of the connection bolt is considered on the basis of the resulting coefficient of dynamic safety [6.22]. This level of safety is composed of two partial safeties. The first one represents own "dynamic safety" in tension [6.20], the other represents the level of safety with regard to the constant shear stress [6.21] which appears in the bolt as the mounting prestressing of the connection. Partial safety fatigue [6.20] evaluates the position of the bolt in view of variable tensile stress and is defined as the ratio between amplitude components of the stress of the limit cycle s_A and the operational cycle s_a. If the designed bolt has to meet all requirements of the strength check, the resulting safety level [6.22] must be higher or equal to the desired safety level [2.22].

The procedure of determining dynamic safety of the connection and the meaning of the rows [6.12 .. 6.19] is obvious in the following illustrations:

Working diagram of the connection exposed to repeated or alternate operational forces resp.

where:

F - maximum axial force loading the connection

F₀ - prestressing of the connection

F₁ - maximum internal axial force in the bolt

F₂ - minimum residual prestressing of the clamped parts of the connection

F_m - medium axis force of the cycle

F_a - amplitude of the axis force of the cycle

reduced Smith's fatigue diagram

where:

S_y - yield point of the material of the bolt

s_f - fatigue limit

s₀ - stress in the thread core from prestressing of the connection

s_A - amplitude component of limit fatigue strength of the bolt for the given course of loading

s_m - medium stress of the operational cycle in the thread core

s_a - amplitude component of the stress of the operational cycle in the thread core

Also in case of a dynamically loaded connection, the designed bolt must meet requirements of the "static" check at the yield point for loads from the maximum axis force [6.1].

Warning: This strength check does not take into account any effects of additional bending stress (see [2.7]), which may very negatively affect the fatigue strength of the bolt. Therefore, it is necessary, particularly with connections loaded with variable loading, to eliminate as much as possible the creation of bending stress by using suitable constructional modifications, or by taking its presence into account when considering the dynamic safety of the connection.

6.16 Basic fatigue limit in tension of the given bolt material.

Theoretically calculated fatigue limit in tension of a smooth rod with a circular diameter of the selected material of the bolt, loaded with an alternate axial loading.

6.17 Corrected fatigue limit in tension of the given bolt.

Fatigue limit in tension of the designed bolt. Corrected value of the basic fatigue limit [6.16] with regards to the selected design of thread [2.19], type and dimensions of thread [4.9] and desired reliability of the connection [2.21].

6.18 Fatigue limit in tension with a limited service life.

Fatigue limit in tension of the designed bolt for the desired service life of the connection [2.20]. In case of the requirement for unlimited service life is the fatigue limit [6.17].

Graphic output, CAD systems.

Information on options of 2D and 3D graphic outputs and information on cooperation with 2D and 3D CAD systems can be found in the document "Graphic output, CAD systems".

Supplements - This calculation:

For drawing the bolt connection it is necessary to set in this paragraph some details of the connection which were not determined with calculations of the connection.

7.2 Design of the bolt head.

Select the respective design of the bolt head in the selective list. The application provides 4 basic designs of bolt heads. However, with regards to commonly produced sizes of bolts, it is not possible to use all these types of heads for some diameters of threads. The dimensions of the selected head are determined using the selected type and diameter of thread [4.10, 4.11] according to the following standards: ANSI B18.2.1, ANSI B18.3, ANSI B18.6.2, ANSI B18.6.3, ISO 1207, ISO 4016, ISO 4762.

Note: This setting has no meaning if a type of bolt without a head is set in the row [4.17].

7.3 Design of the nut.

Select the respective design of the bolt nut in the selective list. The application provides 2 designs of hexagonal nuts. The dimensions of the selected nut are determined using the selected type and diameter of thread [4.10, 4.11] according to the standards: ANSI B18.2.2, ISO 4032, ISO 4035.

Note: This setting has no meaning in case of stud bolts with heads (see setting of rows [1.3, 4.17]).

7.4 Number of washers below the bolt head.

In the selective list, specify the number of washers below the bolt head. If the drawing has to include a connection without washer, select "0". The dimensions of the washer are determined using the selected type and diameter of thread [4.10, 4.11], according to the standards: ANSI B18.22.1, ISO 8738.

7.5 Number of washers below the nut.

In the selective list, specify the number of washers below the nut. If the drawing has to include a connection without washer, select "0". The dimensions of the washer are determined using the selected type and diameter of thread [4.10, 4.11], according to the standards: ANSI B18.22.1, ISO 8738.

Note: This setting has no meaning for stud bolts (see row [1.3]).

Causes of failures of bolt connections, increase in loading capacity of the bolt.

The most frequent causes of failures of bolt connections and their remedies

• Incorrect knowledge of actual occurrence and action of external forces.
  Solution - Increase the level of safety.
• Additional bending stress in the bolt due to shape and position deviations of the bolts and nuts.
  Solution - Bending loading can be eliminated using ball washers, longer bolts with shafts with reduced diameters and parallel machining of the connected surfaces.
• Loss of prestressing caused by thermal elongation or plastic deformation of the bolt and connected parts.
  Solution - Use high-strength pliable bolts. Do not use soft washers, but rather hard tempered washers to eliminate seating as much as possible.
• Spontaneous loosening due to shaking.
  Solution - Secure it using shaped washers.
• Chemical or electrochemical attack, corrosion.
  Solution - Suitable selection of material and surface treatment.
• Breaks of connection bolts exposed to variable loading.

  Solution - Suitable technological and constructional modifications to increase the loading capacity of the bolt (described in detail in the following chapter).

Technological and constructional modifications to increase the loading capacity of the bolt

Fatigue breaks occur in parts exposed to variable loading, usually at points of stress concentration (in places of constructional notches), though the value of the nominal stress is well below the ultimate strength. Statistics shows that of the total number of evaluated standardized bolts, breaks occur in 65% of bolts at the point of the first load carrying thread, in 20% at the run-out of the thread and in 15% at the place of the transition of the shaft into the head of the bolt. The above-mentioned classification of frequencies of fatigue breaks shows some evident critical points which must be taken into account with designs of dynamically loaded connections.

Constructional modifications of the connection:

• Use of pliable bolts instead of common standardized bolts.
  In view of reduction of the amplitude component of the stress, it is advisable to design a connection with as high a ratio of stiffness c₂/c₁as possible. Hence the bolt has to be elastic and the stiffness of the connected parts as high as possible. Pliable bolts are modified using a reduction in their diameters, or drilling or elongation of their shafts.

• Use of a tractive nut or a nut with a relieving notch.
  Distribution of the stress between the thread will be improved and the first load carrying thread will be relieved.

• Carry out the first load carrying thread of the bolt and the very first thread, with a transition into a reduced shaft and an overlap of the nut thread.
  The first load carrying thread will be relieved and the notch coefficient in the run-out of the thread will be decreased.

• Decreasing the notch coefficient in the run-out of the thread with a rounded transition into a shaft with a reduced diameter.

• Increasing the transition radius from the head to the shaft.

• If possible, do not use threads with fine spacing.

• If possible, do not use bolts of large diameters.
  It is advisable to use two smaller bolts instead of one big bolt.

• Suitable constructional design of connections with stud bolts.
  It is advisable to design a connection with overlapping threads of the bolts in open holes. In case of blind holes, it is better to use a stud bolt with a nut bearing against the hole bottom instead of a common bolt with a head. The most advantageous case is to use a stud bolt with a ball end or a small ball inserted.

Technological modifications of the connection:

• Use of a cylindrical thread instead of a cut thread.

• Rolling of the thread bottom.

• Hardening of the material using cold forming.

• Chemical and thermal treatment of the thread, nitriding, phosphating.

Calculation of bolt fields.

Bolt connections often do not occur individually, but in groups, where they provide combined transfer of external forces. Due to technological reasons the fields mostly include bolts of the same diameters, arranged in rectangular or circular shapes. The solution of these group connections is based on determination of the maximum load falling on one (concurrently or the most loaded) bolt, which is then designed as an independent bolt connection according to the above-mentioned procedures.

There are also given general procedures of solutions for several basic types of bolt fields (the symbol "n" used in the formulas indicates the number of bolts in the connection):

Loading of the connection by a force perpendicular to the contact surface running through the centre of gravity of the connection.

The external loading is distributed uniformly to all bolts of the connection.

Axial loading falling on one bolt:

The group connection is then designed as an independent bolt connection exposed to the axial force F_ai.

Loading of the connection by a force inclined to the contact surface running through the centre of gravity of the connection.

The external force is decomposed into component F_a, perpendicular to the contact surface, and component F_r, parallel to it. The connection bolts will be loaded uniformly, thus:

Axial loading falling on one bolt:

Radial loading falling on one bolt:

The group connection is then designed as an independent bolt connection exposed to a combined loading from the axis force F_ai and the cross force F_ri.

Loading of the connection by a force inclined to the contact surface not running through the connection gravity centre.

The external force is decomposed into component F_a perpendicular to the contact surface and component F_r parallel to it, acting on the centre of gravity of the connection:

At the same time, it also creates a moment on the centre of gravity:

The condition of non-deviation of the left margin of the basic plate causes the minimum needed stress:

where e.g. for a rectangular contact surface the contact surface "A" of the connection and the sectional module "W" can be determined using the relations:

where:

a - length of the connection

b - width of the connection

When designing a connection, it is also necessary to determine maximum values of partial forces acting on one bolt.

Axis forces in bolts from force F_a:

Axis forces in bolts from moment M:

Maximum axis force in the bolt from moment M:

Total maximum axis operational force:

Radial forces in bolts from component F_r:

The group connection is then designed as an independent bolt connection exposed to combined loading from axis force F_amax and cross force F_ri. It is also necessary not to forget to check the designed prestressing of the connection [5.17] with regards to minimum necessary prestressing F_0min, determined using the condition of non-deviation of the margin of the basic plate.

Loading of a connection by a force lying in the plane of the connection running through the centre of gravity of the connection.

The external loading is distributed uniformly to all bolts of the connection.

Radial loading falling on one bolt:

The group connection is then designed as an independent connection exposed to cross force F_ri.

Loading of the connection by a moment in the plane of the connection.

If circular flanges are exposed to a torsional moment only, individual bolts will be exposed uniformly to the radial force:

The group connection is then designed as an independent bolt connection exposed to cross force F_ri.

Loading of the connection by a force running through the centre of gravity and a moment in the plane of the connection.

The cross force acting in the centre of gravity of the connection is distributed uniformly to all bolts:

The torsional moment causes loading of each individual bolt by a radial force:

The resulting force acting on one bolt is a vector sum of partial forces F_ri, F_Mi. The group connection is further designed as an independent bolt connection exposed to maximum cross force F_rmax.

Setting calculations, change the language.

Information on setting of calculation parameters and setting of the language can be found in the document "Setting calculations, change the language".

Workbook modifications (calculation).

General information on how to modify and extend calculation workbooks is mentioned in the document "Workbook (calculation) modifications".