Engineering task and calculation objective
The KRVE module calculates the elastic resiliences of a preloaded bolted joint: the resilience of the whole bolt from its individual sections (plain shank, reduced-shank element, free thread) and the resilience of the clamped parts via the equivalent cross-section of the clamped sleeve. From the preload of the bolt, which is at the same time the clamping force of the components, follow the elastic elongations of bolt and clamped parts, i.e. the joint diagram of the connection according to the systematic approach of the German guideline VDI 2230.
These quantities are the key to every highly stressed bolted joint in pressure equipment and mechanical engineering: the ratio of the resiliences determines the load factor, that is, which portion of an axial working load additionally stresses the bolt and how much clamping force is lost at the interface. Compliant, reduced-shank bolts combined with stiff clamped parts take up only a small portion of the working load and are therefore more favorable in fatigue; in addition, long, resilient bolts lose proportionally less preload through embedding of the contact surfaces.
The inputs are the geometry of the bolt (outside thread diameter, head bearing surface, lengths and cross-sections of shank, reduced shank and unengaged thread), the dimensions of the clamped sleeve with through hole and clamping length, as well as the materials and the design temperature, from which the moduli of elasticity of bolt and components follow.
Calculation workflow
- Break the bolt geometry down into sections: The bolt is modeled as a series connection of elastic sections: length and cross-sectional area of the plain shank, of the reduced-shank element and of the unengaged thread with its root cross-section; added to this are the proportional deformations of the bolt head and the engaged thread.
- Determine material properties at design temperature: From the bolt material and the material of the clamped parts, the moduli of elasticity at the design temperature are determined; they enter all resiliences directly.
- Calculate the resilience of the bolt: The elastic resilience of the whole bolt is obtained as the sum of the resiliences of all sections, each formed from section length, the corresponding cross-sectional area and the modulus of elasticity of the bolt material.
- Determine the equivalent cross-section and the resilience of the clamped parts: For the clamped components, the equivalent cross-section of the load-bearing compression cone or equivalent sleeve is determined from the outside diameter of the head bearing surface, the outside diameter of the clamped sleeve, the through hole and the clamping length; from this follows the elastic resilience of the clamped parts.
- Evaluate the preloaded state: With the preload, the elastic elongation of the bolt (extension) and of the clamped parts (compression) are calculated. These values form the joint diagram triangle and provide the basis for the load factor, the preload loss due to embedding and the assessment of axial working loads.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Prestress force of the bolt | FV | N |
| = Clamping force of the components | FKl | N |
| Outside thread diameter | d | mm |
| Limiting condition is valid | = + | mm |
| Length of the smooth shank | l1 | mm |
| Length of the expansion shank element | l2 | mm |
| Length of thread which has not been screwed in | l3 | mm |
| Nominal cross section of the bolt shank | AN | mm² |
| Cross-sectional area of the smooth shank | A1 | mm² |
| Cross-sectional area of the expansion shank | A2 | mm² |
| Cross section of the bottom of the thread | A3 | mm² |
| Outside diameter of the braced sleeve | DA | mm |
| Diameter of the pass hole | dh | mm |
| Limiting condition is valid | = + | mm |
| Modulus of elasticity of the bolt material | ES | N/mm² |
| Modulus of elasticity of the braced components | ET | N/mm² |
| Limiting condition is valid | = + | mm |
| Design temperature | T | °C |
| Bolt material | Schraube | - |
| Material of the braced components | Teile | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Equivalent cross section | Aers | mm² |
| Compliance of the whole bolt | δS | mm/N |
| Compliance of the braced components | δT | mm/N |
| Elastic longitudinal deformation of the bolt | fS | mm |
| Elastic longitudianl deformation of the braced | fT | mm |
Frequently asked questions
Why is a resilient bolt favorable for fatigue strength?
In an axially loaded, preloaded joint, the working load is shared in proportion to the stiffnesses: the more compliant the bolt is relative to the clamped parts, the smaller the load portion that reaches the bolt as an alternating stress amplitude. Reduced-shank bolts with a necked-down shank thus lower the dynamic stress on the thread considerably, even though their statically load-bearing cross-section is smaller.
What does the equivalent cross-section of the clamped parts describe?
The compressive stress under the bolt head spreads into the components in the shape of a cone; it is therefore not the entire component cross-section that carries the load, but only a limited region around the through hole. The equivalent cross-section is the area of an imaginary sleeve that has the same stiffness as this compression cone. It depends on the head bearing diameter, the clamping length, the hole diameter and the lateral extent of the component; with narrow flange lugs, the outside diameter of the clamped sleeve limits the cone.
What role do the resiliences play in the embedding behavior of the joint?
After assembly, roughness peaks in the thread, the head bearing surface and the interfaces flatten out; this embedding amount shortens the clamped length and reduces the preload. The loss of force is the embedding amount divided by the sum of the resiliences of bolt and clamped parts. Long, compliant joints therefore lose proportionally little preload, while short, stiff joints with few interfaces are particularly sensitive to embedding.
Is the calculation also valid at elevated temperature?
The resiliences themselves are determined with the moduli of elasticity at the design temperature, which the module covers via the material selection. In addition, at temperature it must be checked whether differing thermal expansions of bolt and clamped parts change the preload, and whether at high temperatures relaxation (creep) limits the retention of the preload; these effects must be assessed separately.