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1 FKM-Guideline ANALYTICAL STRENGTH ASSESSMENT OF COMPONENTS IN MECHANICAL ENGINEERING 5 th , revised edition, 2003, English Version Translation by E. Haibach Title of the original German Version: RECHNERISCHER FESTIGKEITSNACHWEIS FÜR MASCHINENBAUTEILE 5., überarbeitete Ausgabe, 2003 Editor: Forschungskuratorium Maschinenbau (FKM) Postfach 71 08 64, D-60498 Frankfurt / Main Phone ρ49 - 69 - 6603 - 1345

Transcript of FKM - commentary

Page 1: FKM - commentary

1 FKM-Guideline ANALYTICAL STRENGTH ASSESSMENT OF COMPONENTS IN MECHANICAL ENGINEERING 5th, revised edition, 2003, English Version Translation by E. Haibach Title of the original German Version: RECHNERISCHER FESTIGKEITSNACHWEIS FÜR MASCHINENBAUTEILE 5., überarbeitete Ausgabe, 2003 Editor: Forschungskuratorium Maschinenbau (FKM) Postfach 71 08 64, D-60498 Frankfurt / Main Phone ρ49 - 69 - 6603 - 1345

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2 Preface of the English Version of the 5th Edition. For engineers concerned with construction and calculation in mechanical engineering or in related fields of industry the FKM-Guideline for analytical strength assessment is available since 1994. This guideline was elaborated by an expert group "Strength of components" of the “Forschungskuratorium Maschinenbau (FKM), Frankfurt/Main,” with financial support by the Bundesministerium für Wirtschaft (BMWi), by the “Arbeitsgemeinschaft industrieller Forschungsvereinigungen ´Otto von Guericke´" and by the “Forschungskuratorium Maschinenbau.

Based on former TGL standards and on the former guideline VDI 2226, and referring to more recent sources it was developed to the current state of knowledge.

The FKM-Guideline

- is applicable in mechanical engineering and in related fields of industry,

- allows the analytical strength assessment for rod-shaped (1D), for shell-shaped (2D) and for block-shaped (3D) components under consideration of all relevant influences,

- describes the assessment of the static strength and of the fatigue strength, the latter according to an assessment of the fatigue limit, of the constant amplitude fatigue strength, or of the variable amplitude fatigue strength according to the service stress conditions,

- is valid for components from steel, cast steel, or cast iron materials at temperatures from -40 ºC to 500 ºC, as well as for components from aluminum alloys and cast aluminum alloys at temperatures from -40 ºC to 200 ºC,

- is applicable for components produced with or without machining, or by welding,

- allows an assessment in considering nominal stresses as well as local elastic stresses derived from finite element or boundary element analyses, from theoretical mechanics solutions, or from measurements.

A uniformly structured calculation procedure applies to all of these cases of application. The calculation procedure is almost completely predetermined. The user has to make some decisions only.

The FKM-Guideline is a commented algorithm, consisting of statements, formulas, and tables. Most of the included figures have an explanatory function only.

Textual declarations are given where appropriate to ensure a reliable application.

Its content complies with the state of knowledge to an extend that may be presented in a guideline and it enables quite comprehensive possibilities of calculation. The employed symbols are adapted to the extended requirements of notation. The presented calculation procedure is complemented by explanatory examples.

Practically the described procedure of strength assessment should be realized by means of a suitable computer program. Presently available are the PC computer programs "RIFESTPLUS" (applicable for a calculation using elastically determined local stresses, in particular with shell-shaped (2D) or block-shaped (3D) components) and "WELLE" (applicable for a calculation using nominal stresses as it is appropriate in the frequently arising case of axles or shafts with gears etc).

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The preceding editions of the FKM-Guideline observed a remarkably great interest from which the need of an up to date guideline for analytical strength analyses becomes apparent. Moreover the interest of users was confirmed by the well attended VDI conferences on "Computational Strength Analysis of Metallic Components", that were organized for presentation of the FKM-Guideline at Fulda in 1995, 1998 and 2002.

The contents-related changes introduced with the third edition from 1998 were mainly concerned with the consideration of stainless steel and of forging steel, with the technological size factor, with the section factor for assessing the static strength, with the fatigue limit of grey cast iron and of malleable cast iron, with additional fatigue classes of welded structural details and with the local stress analysis for welded components, with the specification of an estimated damage sum smaller than one for the assessment of the variable amplitude fatigue strength, with the assessment of multiaxial stresses, and with the experimental determination of component strength values.

An essential formal change in the third edition was a new textual structure providing four main chapters, that describe the assessment of the static strength or of the fatigue strength with either nominal stresses or local stresses, respectively. For ease of application each of these chapters gives a complete description of the particular calculation procedure, although this results in repetitions of the same or almost the same parts of text in the corresponding sections.

The major change in the forth edition from 2002 is the possibility of considering structural components made from aluminum alloys or cast aluminum alloys by applying the same calculation procedure that was developed for components from steel, cast steel and cast iron materials so far.

The decisions necessary to include aluminum materials were derived from literature evaluations. It had to be recognized, however, that some of the relevant factors of influence were not yet examined with the desirable clearness or that available results could not be evaluated objectively due to large scatter. In these cases the decision was based on a careful consideration of substantial relations.

Concerning an analytical strength assessment of components from aluminum alloys or from cast aluminum alloys this guideline is delivered to the technical community by supposing that for the time being it will be applied with appropriate caution and with particular reference to existing experience so far. The involved research institutes and the “Forschungskuratorium Maschinenbau (FKM)” will appreciate any reports on practical experience as well as any proposals for improvement.

Further improvements may also be expected from ongoing research projects concerning the procedure of static strength assessment using local elastic stresses, Chapter 3, and the fatigue assessment of extremely sharp notches.

Last not least the fifth edition of the FKM-Guideline is a revision of the forth edition and it is presented in both a German version and an English version with the expectation that it might observe similar attention as the preceding editions on a broadened international basis of application.

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4 Notes of the translator The English translation is intended to keep as close as possible to the original German version, but by using a common vocabulary and simple sentences. If the given translation is different from a literal one, the technical meaning of the sentence and/or of the paragraph is maintained, however.

The translation observes an almost identical structure of the headlines, of the chapters, of the paragraphs and of the sentences, and even of the numbering of the pages.

Also the tables and the figures as well as their numbering and headlines are adapted as they are, while only the verbal terms have been translated.

In particular the original German notation of the mathematical symbols, indices and formulas, as well as their numbering, has not been modified in order to insure identity with the German original in this respect.

The applier of this guideline is kindly asked to accept the more or less unusual kind of notation which is due to the need of clearly distinguishing between a great number of variables. In particular the applier is pointed to the speciality, that a comma ( , ) is used with numerical values instead of a decimal point ( . ), hence 1,5 equals 1.5 for example.

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Contents

Page

0 General survey 0.1 Scope 9 0.2 Technical background 0.3 Structure and elements

1 Assessment of the static strength using nominal stresses 1.0 General 19 1.1 Characteristic stress values 1.2 Material properties 22 1.3 Design parameters 30 1.4 Component strength 33 1.5 Safety factors 34 1.6 Assessment 36

2 Assessment of the fatigue strength using nominal stresses 2.0 General 41 2.1 Parameters of the stress spectrum 2.2 Material properties 47 2.3 Design Parameters 50 2.4 Component strength 57 2.5 Safety factors 68 2.6 Assessment 70

3 Assessment of the static strength using local stresses 3.0 General 73 3.1 Characteristic stress values 3.2 Material properties 76 3.3 Design parameters 85 3.4 Component strength 89 3.5 Safety factors 90 3.6 Assessment 93

4 Assessment of the fatigue strength using local stresses 4.0 General 97 4.1 Parameters of the stress spectrum 4.2 Material properties 103 4.3 Design parameters 106 4.4 Component strength 113 4.5 Safety factors 125 4.6 Assessment 127

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6 Page 5 Appendices 5.1 Material tables 131 5.2 Stress concentration factors 178 5.3 Fatigue notch factors 187 5.4 Fatigue classes (FAT) for welded components of structural steel and of aluminum alloys 195 5.5 Comments about the fatigue strength of welded components of structural steel 209 5.6 Adjusting the stress ratio of a stress spectrum to agree to that of the S-N curve and deriving a stepped spectrum 216 5.7 Assessment using classes of utilization 218 5.8 Particular strength characteristics of surface hardened components 222 5.9 An improved method for computing the component fatigue limit in the case of synchronous multiaxial stresses 223 5.10 Approximate assessment of the fatigue strength in the case of non-proportional multiaxial stresses 226 5.11 Experimental determination of component strength values 227 5.12 Stress concentration factor for a substitute structure 230 6 Examples 6.1 Shaft with shoulder 231 6.2 Shaft with V-belt drive 236 6.3 Compressor flange made of grey cast iron 241 6.4 Welded notched component 245 6.5 Cantilever loaded by two loads 250 6.6 Component made of a wrought aluminum alloy 256

7 Symbols and basic formulas 7.1 Abbreviations 259 7.2 Indices 7.3 Lower case characters 7.4 Upper case characters 260 7.5 Greek alphabetic characters 261 7.6 Basic formulas 262

8 Subject index 263

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7 0 General survey 0.1 Scope This guideline is valid for components in mechanical engineering and in related fields of industry. Its application has to be agreed between contracting parties.

For components subjected to mechanical loadings it allows an analytical assessment of the static strength and of the fatigue strength, the latter as an assessment of the fatigue limit or of the variable amplitude fatigue strength according to the service stress conditions.

Other analytical assessments, for example of safety against brittle fracture, of stability, or of deformation under load, as well as an experimental assessment of strength *1, are not subject of this guideline.

It is presupposed, that the components are professionally produced with regard to construction, material and workmanship, and that they are faultless in a technical sense.

The guideline is valid for components produced with or without machining or by welding of steel, iron or aluminum materials that are intended for use under normal or elevated temperature conditions, and in detail

- for components with geometrical notches, - for components with welded joints, - for static loading, - for fatigue loading with more than about 104 constant or variable amplitude cycles, - for milled or forged steel, also stainless steel, cast iron materials as well as aluminum alloys or cast aluminum alloys, - for component temperatures from − 40°C to 500°C for steel, from − 25°C to 500°C for cast iron materials and from − 25°C to 200°C for aluminum materials, - for a non-corrosive environment.

If an application of the guideline is intended outside the mentioned field of application additional specifications are to be agreed upon.

The guideline is not valid if an assessment of strength is required according to other standards, rules or guidelines, or if more specific design codes are applicable, as for example for bolted joints.

0.2 Technical Background Basis of the guideline are the references listed on page 7, in particular the former TGL-Standards, the former VDI-Guideline 2226, as well as the regulations of DIN 18 800, the IIW-Recommendations and Eurocode 3. Moreover the guideline was developed to the current state of knowledge by taking into account the results of more recent investigations.

1 Subject of Chapter 5.11 "Experimental determination of component strength values" is not the realization of an experimental assessment of strength, but the question how specific and sufficiently reliable component strength values suitable for the general procedure of strength assessment may be derived experimentally.

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0.3 Structure and elements Contents Page

0.3.0 General 9

0.3.1 Procedure of calculation 10

0.3.2 Stress parameters

0.3.3 Methods of strength assessment 11 0.3.3.0 General 0.3.3.1 Assessment of the static strength

using nominal stresses, Chapter 1 0.3.3.2 Assessment of the fatigue strength 12 using nominal stresses, Chapter 2 0.3.3.3 Assessment of the static strength using local stresses, Chapter 3 0.3.3.4 Assessment of the fatigue strength 13 using local stresses, Chapter 4

0.3.4 Kinds of components 13 0.3.4.0 General 0.3.4.1 Rod-shaped (1D) components 0.3.4.2 Shell-shaped (2D) components 14 0.3.4.3 Block-shaped (3D) components 15

0.3.5 Uniaxial and multiaxial stresses 16

0.3.0 General An assessment of the static strength is required prior to an assessment of the fatigue strength.

Before applying the guideline it has to be decided

- what cross-sections or structural detail of the component shall be assessed *2 and - what service loadings are to be considered.

The service loadings are to be determined on the safe side, that is, with a sufficient probability they should be higher than most of the normally occurring loadings *3.

The strength values are supposed to correspond to an anticipated probability of 97,5 % (average probability of survival PÜ = 97,5 %).

2 In particular, what critical points of the considered cross-sections or component. 3 Usually this probability can hardly be quantified, however.

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0.3.1 Procedure of calculation The procedure of calculation for an assessment of the static strength is presented in Figure 0.0.1, the almost identical procedure for an assessment of the fatigue strength in Figure 0.0.2 *4.

Figure 0.0.1 Procedure of calculation for an assessment of the static strength.

Figure 0.0.2 Procedure of calculation for an assessment of the fatigue strength.

4 A more detailed survey on the procedures of assessment referring to the equations of the guideline may be found in the Appendix, Chapter 7.6.

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At the assessment stage (box at bottom of either Figure) the characteristic values of service stress occurring in the component (box at top on the left) and the component strength values derived from the mechanical material properties and the design parameters (middle column) are compared by including the required safety factors (box at bottom on the right). In specifying component fatigue strength values the mean stress and the variable amplitude effects are regarded as essential factors of influence. The assessment of strength is successful if the degree of utilization is less or equal 1,00 , where the degree of utilization is defined by the ratio of the characteristic service stress to the component strength value that has been reduced by the safety factor.

In Figure 0.0.1 and Figure 0.0.2 the arrangements of the individual boxes from top to bottom illustrate the sequential procedure of calculation.

0.3.2 Characteristic service stresses For an application of the guideline the stresses resulting from the service loadings have to be determined for the so-called reference point of the component, that is the potential point of fatigue crack initiation at the cross-section or at the component under consideration. In case of doubt several reference points are to be considered, for example in the case of welded joints the toe and the root of the weld.

There is a need to distinguish the names and subscripts of the different components or types of stress, that may act in rod-shaped (1D), in shell-shaped (2D) or in block-shaped (3D) components, respectively.

The stresses are to be determined according to known principles and techniques: analytically according to elementary or advanced methods of theoretical mechanics, numerically after the finite element or the boundary element method, or experimentally by measurement.

All stresses, except the stress amplitudes, are combined with a sign, in particular compressive stresses are negative.

To perform an assessment it is necessary to decide about the kind of stress determination for the reference point considered: The stresses can be determined - as nominal stresses *5 (notation S and T), - as elastically determined local stresses, effective notch stresses or structural (hot

spot) stresses *6 (notation σ and τ).

Correspondingly the component strength values are to be determined - as nominal strength values or - as local strength values of the elastic local stress, of the effective notch stress or of the structural stress.

With the procedures of calculation structured uniformly for both types of stress determination it is intended that more or less identical results will be obtained from comparable strength assessments based on either nominal stresses or local stresses.

5 Nominal stresses can be computed for a well defined cross-section only. 6 The elastic notch root stress exceeds the nominal stress by a stress concentration factor. In the case of welded joints effective notch stresses are applied to the assessment of the fatigue strength only. Structural stresses, also termed geometrical or hot spot stresses, are normally in use with welded joints only. For further information see Chapter 5.5.

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0.3.3 Methods of strength assessment 0.3.3.0 General

In order to present the guideline clearly arranged and user-friendly, it is organized in four chapters, Figure 0.0.3: - Assessment of the static strength using nominal stresses, Chapter 1, - Assessment of the fatigue strength using nominal stresses, Chapter 2, - Assessment of the static strength using local stresses, Chapter 3, - Assessment of the fatigue strength using local stresses, Chapter 4.

Figure 0.0.3 Organization of the guideline.

In particular the procedure of calculation is completely presented in everyone of the four chapters, even if this results in repetitions of the same or almost the same parts of text in Chapter 1 and Chapter 3 or in Chapter 2 and Chapter 4, respectively.

The procedure of calculation using nominal stresses is to be preferred for simple rod-shaped (1D) and for shell-shaped (2D) components. The procedure of calculation using local stresses has to be applied to block-shaped (3D) components, and moreover in general, if the stresses are determined by a finite-element or a boundary-element calculation, if there are no well-defined cross-sections or no simple cross-section shapes, if stress concentration factors or fatigue notch factors are not known, or (concerning the assessment of the static strength) in the case of brittle materials.

0.3.3.1 Assessment of the static strength using nominal stresses, Chapter 1 Relevant nominal characteristic service stresses are the extreme maximum and extreme minimum values of the individual types of stress or stress components, e.g. nominal values of the axial (or tension-compression) stress, Szd, of the bending stress, Sb, and so forth *7 *8, Chapter 1.1.

Relevant material properties are the tensile strength and the yield strength (yield stress or 0.2 proof stress) as well as the strength values for shear derived from these. A technological size effect is taken into account if appropriate. The influence of an

7 According to rod-, shell- or block-shaped components, Chapter 0.3.4. 8 The extreme maximum or minimum stresses for the assessment of the static strength may be different from the maximum and minimum stresses for the assessment of the fatigue strength, that are determined from the largest amplitude and the related mean value of a stress spectrum.

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elevated temperature on the material properties - strength at elevated temperature and creep strength, yield strength at elevated temperature and 1% creep limit - is allowed for by means of temperature factors, Chapter 1.2.

Design parameters are the section factors, by which an experienced partial plasticity of the component is allowed according to yield strength, type of loading, shape of cross-section, and stress concentration factor. From the section factor and from further parameters an overall design factor is derived, Chapter 1.3.

The nominal values of the static component strength are derived from the tensile strength, divided by the respective overall design factor, Chapter 1.4.

As common in practice the safety factor against the tensile strength is 2,0. For materials with a yield strength less than 0,75 times the tensile strength, the safety factor is 1,5 against the yield strength, however. Under favorable conditions these safety factors may be reduced, Chapter 1.5.

The assessment is carried out by proving that the degree of utilization is less or equal 1,00 . The degree of utilization for an individual stress component or type of stress is the ratio of its nominal characteristic service stress value, divided by the allowable nominal static component strength value, which follows from the nominal static component strength divided by the safety factor.

If there are several stress components or types of stress their individual degrees of utilization are combined to obtain an entire degree of utilization. The interaction formula to be applied to that combination allows for the ductility of the material in question, Chapter 1.6.

For welded components the assessment of the static strength has to be carried out for the toe section as for non-welded components, and for the throat section with an equivalent nominal stress, that is computed from the components of nominal stress acting in the weld seam *9.

0.3.3.2 Assessment of the fatigue strength using nominal stresses, Chapter 2

Relevant nominal characteristic service stresses are the largest stress amplitudes in connection with the respective stress spectra and the related mean stress values. They are determined for the individual stress components or types of stress, e.g. amplitudes and mean values of the nominal axial (tension-compression) stresses, Sa,zd and Sm,zd, and so forth *7 *8, Chapter 2.1.

Relevant material properties are the fatigue limit for completely reversed axial stress and the fatigue limit for completely reversed shear stress of the material in question. A technological size effect is taken into account where appropriate. The influence of an elevated temperature is allowed for by means of temperature factors, Chapter 2.2.

Design parameters to be considered in particular are the fatigue notch factors, allowing for the design of the component (shape, size and type of loading), as well as the roughness factor and the surface treatment factor, by which the respective surface properties are accounted for. By specific combination of all these factors a summary design factor is calculated, Chapter 2.3.

The nominal values of the component fatigue limit for completely reversed stresses follow from the derived fatigue limit values of the material, divided by the respective design factors, Chapter 2.4.1. From these fatigue limit values the amplitudes of the

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component fatigue limit according to the mean stress values (or the stress ratios) are to be derived, Chapter 2.4.2. The amplitudes that specify the variable amplitude fatigue strength of the component are obtained from the fatigue limit values multiplied by a factor depending on the parameters of the stress spectrum (total number of cycles and amplitude frequency distribution), Chapter 2.4.3.

The basic value of the safety factor is 1,5. Under favorable conditions this safety factor may be reduced, Chapter 2.5.

The assessment is carried out by proving that the degree of utilization is less or equal 1,00 . The degree of utilization for an individual stress component or type of stress is the ratio of its nominal characteristic service stress amplitude, divided by the allowable amplitude of the component fatigue limit or of the component variable amplitude fatigue strength. The allowable amplitude value follows from the nominal amplitude of the derived component fatigue strength divided by the safety factor.

If there are several stress components or types of stress their individual degrees of utilization are combined to obtain the total degree of utilization. The interaction formula to be applied to that combination allows for the ductility of the material in question, that is in the same way as for the assessment of the static strength, Chapter 2.6.

For the assessment of the fatigue strength of welded components using nominal stresses basic fatigue limit values for completely reversed stress are given. They are independent of the tensile strength of the base material (which is different to non-welded components). They are converted by design factors that follow from a classification scheme of structural weld details. The combined effect of mean stress and of residual stresses in welded components is considered by means of a mean stress factor together with a residual stress factor *9.

0.3.3.3 Assessment of the static strength using local stresses, Chapter 3 Relevant characteristic local service stresses are the extreme maximum and extreme minimum stresses of the individual types of stress or stress components, e.g. local values of the normal (axial and/or bending) stress, σ, and of the shear (shear and/or torsional) stress *7 *8, Chapter 3.1.

Relevant material properties are to be determined as for nominal stresses, Chapter 3.2.

Design parameters are the section factors, by which an experienced partial plasticity of the component is allowed according to yield strength, type of loading, and shape of the component. The section factors are calculated on the basis of Neuber's formula, but by observing an individual upper bound value that follows from the plastic limit load (plastic notch factor). From the section factors and from further parameters an overall design factor is derived, Chapter 3.3 *10.

The local values of the static component strength are derived from the tensile strength, divided by the respective overall design factor, Chapter 3.4.

9 The assessment of the fatigue strength for welded components makes reference to the IIW-Recommendations and Eurocode 3. As far as conditionally weldable steel, stainless steel, weldable cast iron materials or weldable aluminum alloys are concerned this kind of calculation is provisional and may be applied with caution only. 10 The assessment of the static strength using local stresses on the basis of Neuber's formula and the plastic limit load is an approximation which has to be regarded as provisional and is to be applied with caution only.

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The safety factors are to be determined as for nominal stresses, Chapter 3.5.

The assessment is carried out by means of the degree of utilization as for nominal stresses, but with the respective local values of the characteristic service stress and the local component strength values, Chapter 3.6.

For welded components the assessment of the static strength using local stresses is carried out using structural stresses (not with effective notch stresses), for the weld toe as for non-welded components, for the root of the weld using an equivalent structural stress, that is to be derived from the structural stress components acting in the weld seam *9.

0.3.3.4 Assessment of the fatigue strength using local stresses, Chapter 4 Relevant local characteristic service stresses are the largest stress amplitudes in connection with the respective stress spectra and the related mean stress values. They are determined for the individual stress components or types of stress, e.g. amplitudes and mean values of the local normal (axial and/or bending) stress, σa and σm , and so forth *7 *8, Chapter 4.1.

The relevant material properties are determined as for nominal stresses, Chapter 4.2.

Design parameters to be considered in particular are the Kt-Kf ratios, allowing for the design of the component (shape and size), as well as the roughness factor and the surface treatment factor, by which the respective surface properties are accounted for. By specific combination of all these factors a summary design factor is calculated, Chapter 4.3.

The local values of the component fatigue limit for completely reversed stresses follow from the derived fatigue limit values of the material, divided by the respective design factors, Chapter 4.4.1. The conversions to the amplitude of the component fatigue limit and to the amplitude of the component variable amplitude fatigue strength are as for nominal stresses, Chapter 4.4.2 to 4.4.3.

The safety factors are to be determined as for nominal stresses, Chapter 4.5.

The assessment by means of the degree of utilization is as for nominal stresses, but with the respective local values of the characteristic stress amplitude and the value of the component fatigue limit or of the component variable amplitude fatigue strength, Chapter 4.6.

For the assessment of the fatigue strength of welded components using structural stresses or effective notch stresses the same basic fatigue limit values for completely reversed stresses apply as for nominal stresses. They hold for effective notch stresses without conversion, but for structural stresses they have to be converted by factors given for some typical weld details. The combined effect of mean stress and of residual stresses in welded components is to be considered as for nominal stresses by means of a mean stress factor together with a residual stress factor *10.

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0.3.4 Kinds of components

0.3.4.0 General Rod-shaped (1D), shell-shaped (2D) and block-shaped (3D) components are to be distinguished, as in each case other stress components or types of stresses, identified by differing symbols and subscripts, are of concern. The distinction is only a formal one, however, and the procedure of calculation is the same in all cases. Specific particulars apply to welded components.

0.3.4.1 Rod-shaped (1D) components For rod-shaped (1D) components – rod, bar, shaft, or beam for example - the following system of co-ordinates is introduced: x-axis is the longitudinal center line of the component, y- and z-axes are the main axes of the cross-section that are to be specified so, that for the moments of inertia Iy≥ Iz is valid, Figure 0.0.4.

Figure 0.0.4 Rod-shaped (1D) component (round specimen with groove) in bending. Nominal stress Sb and maximum local stress σmax at the reference point W.

Calculation using nominal stresses If the assessment of rod-shaped (1D) components is carried out by using nominal stresses, Chapter 1 and 2, the nominal stresses to be computed at the reference point are Szd from an axial load, Sb from a bending moment, Ts from a shear load, and/or Tt from a torsional moment acting at the respective section.

For the equations given in Chapter 1 and 2 it is provided, that both the bending stress Sb and the shear stress Ts act in the x-z-plane. Otherwise stress components Sb,y and Sb,z , Ts,y and Ts,z are to be considered *11.

In case of rotationally symmetrical cross-sections with circumferential notches a resultant bending stress and a resultant shear stress can be calculated from these stress components, Sb = S Sb y b z, ,

2 2+ , (0.3.1)

Ts = T Ts y s z, ,2 2+ .

The equations given in Chapter 1 and 2 may be applied to Sb and Ts.

11 The indices y and z describe the direction of the related vectors of the bending moments My, Mz and of the lateral loads Fy, Fz .

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In the general case of not rotationally symmetrical cross-sections a calculation using local stresses is normally to be preferred.

Additional stresses at notches (as for example the circumferential stress associated with an axial stress of a shaft with groove) may be included in the stress concentration factor, otherwise they will be neglected.

Calculation using local stresses If the calculation of rod-shaped (1D) components is carried out using local stresses *12, Chapter 3 and 4, the local normal stresses at the reference point from axial and from bending loading (in x-direction), σzd = σ as well as the local shear stresses τs = τ from shear and from torsion (normal to the x-direction) are considered.

If the local stresses are calculated from the nominal stresses by multiplication with the respective stress concentration factors, the equations given in Chapter 3 and 4 are applicable.

However, if the calculation yields the complete local state of stress at the reference point (as for example a finite-element calculation does), the principle stresses σ1, σ2, σ3 are computed *13 and treated as described for block-shaped (3D) components.

Rod-shaped (1D) welded components For rod-shaped (1D) welded components *14 the notations σ and τ apply to structural stresses and the notation σK and τK apply to effective notch stresses *15.

0.3.4.2 Shell-shaped (2D) components For shell-shaped (2D) components - disk, plate, or shell for example - the following system of coordinates is introduced: The x- and y-axis are placed in the surface at the reference point, the z-axis is normal to the surface in thickness direction. The normal stress and the shear stress in thickness direction are supposed to be negligible, Figure 0.0.5.

Calculation using nominal stresses If the assessment of shell-shaped (2D) components is carried out using nominal stresses, Chapter 1 and 2, the nominal stresses at the reference point to be computed are the normal stresses Szdx = Sx and Szdy = Sy from loadings in the x- and y-directions and Ts = T from a shear loading. 12 The assessment of rod-shaped (1D) components should preferably be carried out using nominal stresses whenever possible. 13 Principle stresses are independent of the chosen coordinate system. In the special case of a proportional loading the directions of the principle stresses remain fixed to the coordinates of the component. In the more general case of non-proportional loading the directions and the amounts of the three principle stresses will change with time, see Chapter 0.3.5. 14 Rod-shaped (1D) welded components are rolled sections with circular, tube, I-, box or other cross-sections connected or joined with butt welds and/or fillet welds. 15 Structural stresses can be applied to the assessment of the static strength and to the assessment of the fatigue strength. Effective notch stresses can be applied to the assessment of the fatigue strength, but not to the assessment of the static strength.

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Figure 0. 0. 5 Shell-shaped (2D) component (shell with cutout detail). Local stresses σa,x at the reference point W (peak value) and σa,x,∆s at the neighbouring point B.

Calculation using local stresses If the assessment of shell-shaped (2D) components is carried out using local stresses, Chapter 3 and 4, the local stresses at the reference point σzdx = σx and σzdy = σy in the x- and y-directions and the local shear stress τs = τ are considered.

If the local stresses are computed from the nominal stresses by multiplication with the respective stress concentration factors, the equations given in Chapter 3 and 4 are applicable.

However, if the calculation yields the complete local state of stress at the reference point (as for example a finite-element calculation does), the principle stresses σ1, σ2, σ3 are computed *14 and treated as described for block-shaped (3D) components.

Figure 0.0.6 Shell-shaped (2D) welded component. Example: Strap with longitudinal stiffner. After Radaj /11/. Top: Joint, Centre: Stress distribution, Bottom: Profile. Relevant is the stress at the reference point W (at the toe line of the weld).

Calculation using nominal stresses: Stress Sx .

Calculation using structural stress: Maximum stress σx,max obtained from extrapolating the stress distribution towards the weld toe.

Calculation using effective notch stresses: Maximum stress σKx,max occurring at the weld toe, see Figure 0.0.7.

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18 Shell-shaped (2D) welded components For shell-shaped (2D) welded components the notations σx , σy and τ apply to structural stresses and the notations σKx , σKy and τK apply to effective notch stresses *16 .

Figure 0.0.7 Shell-shaped (2D) welded component. Example: Cruciform joint and butt weld. After Radaj /10/. Calculation using effective notch stresses: The maximum stress σKx,max occurring at the toe or at the root of the weld has to be computed by introducing a fictitious effective notch radius r = 1 mm, unless the real radius is r > 1 mm (the fictitious notch radius is intended for the assessment of the fatigue strength only).

The fictitious notch radius r = 1 mm applies to welded joints from structural steel. It is supposed, however, that it is applicable for other kinds of material as well, although this has to be considered as a preliminary specification for welded aluminum materials so far.

0.3.4.3 Block-shaped (3D) components In the general case block-shaped (3D) components are to be calculated using local stresses, Chapter 3 and 4 *17

For block-shaped (3D) components the coordinate system at the reference point may be of cartesian, cylindrical or spherical type.

The calculation is supposed to yield the complete state of local stress at the reference point (as for example a finite-element calculation does). From that the principle stresses σ1, σ2, σ3 are computed *14 , and for these the degrees of utilization are determined.

If the reference point W is located at a free surface of a block-shaped (3D) component, Figure 0.0.8, it is supposed that σ1 and σ2 are the principle stresses at the surface, while the principle stress σ3 is supposed to point normally to the surface inwards the component.

In general stress gradients exist for all three principle stresses, both normal to the surface and in either direction of the surface. However, only the stress gradients for σ 1 and σ2 normal to the surface can be considered in the procedure of calculation, while the stress gradients for σ1 and σ2 in any directions of the surface and the gradients of σ3 can not.

16 Structural stresses can be applied to the assessment of the static strength and to the assessment of the fatigue strength. Effective notch stresses can be applied to the assessment of the fatigue strength, but not to the assessment of the static strength.

17 For block-shaped components the determination of a nominal stress is not possible since there is no well defined cross-section.

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Block-shaped (3D) components can be calculated as shell-shaped (2D) components if the stresses σx , σy and τ at the load free surface are of concern only.

Figure 0.0.8 Block-shaped (3D) component (flange). Local longitudinal stress σ1 and circumferential stress σ2 at the reference point W (peak values), stresses σ1,�s and σ2,�s at neighboring point B.

Block-shaped (3D) welded components Welds at a load-free surface of block-shaped (3D) components having no inner defects can be assessed as shell-shaped (2D) welded components. Then the notations σx , σy and τ apply to structural stresses and the notations σKx , σKy and τK apply to the effective notch stresses at the surface, Figure 0.0.6.

0.3.5 Uniaxial and multiaxial stresses The stresses occurring in the cross-section or at the reference point of a component may be caused - by a single load or - by several loads acting simultaneously.

In both cases - an uniaxial stress or - multiaxial stresses may result at the reference point.

An uniaxial stress occurs under special circumstances only, as for example in a tension loaded prismatic bar, or at an unloaded edge of shell-shaped (2D) or block-shaped (3D) components, the latter even if several loads act on these components simultaneously, Figure 0.0.9. In addition an uniaxial stress may be assumed at the reference point if, by comparison, any further stresses are small.

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Figure 0.0.9 Uniaxial and multiaxial stresses.

Nominal stresses Sx, Sy and T.

Left: multiaxial stresses in a sheet section, Right: uniaxial stress in a sheet section at the edge of a cutout.

In general components are subject to multiaxial stresses, however. Then two or three normal stresses, or normal stresses and shear stresses occur at the reference point.

In this guideline a basic principle is defined both for an assessment of the static strength and of the fatigue strength in case of multiaxial stresses:

- the individual degrees of utilization for everyone of the computed types of stress or stress components have to be determined and assessed separately in a first step, and

- thereafter these individual degrees of utilization will be combined by means of an appropriate interaction formula to obtain the entire degree of utilization for assessment.

Assessment of the static strength For the assessment of the static strength the most unfavorable case to be considered is that the extreme values of all maximum and minimum stresses occur simultaneously. Accordingly the entire degree of utilization has to be computed. However, stresses of different sign that will decrease the entire degree of utilization are to be included only if they definitely occur together with the remaining stresses, Chapter 1.6 or 3.6.

Assessment of the fatigue strength For the assessment of the fatigue strength *18 multiaxial stresses varying with time have to be distinguished as follows: - proportional stresses, - synchronous stresses, or - non-proportional stresses.

18 Both for the assessment of the fatigue limit and for the assessment of the variable amplitude strength.

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21 Proportional stresses Normally proportional stresses result from a single loading acting on the component.

Examples of proportional stresses are the circumferential and the longitudinal stresses of a cylindrical vessel loaded by internal pressure, or the bending and torsional stresses of a round cantilever loaded eccentrically by a single load.

If this single acting loading is varying with time, all multiaxial stresses are varying proportionally to that loading and proportionally to each other, which also is true with regard to their amplitudes and their mean values. Further, as a consequence, the principle stresses observe non-changing directions relative to the component. The amounts of the stresses, also in the stress amplitude spectra, may be converted by constant factors. Hence all stress spectra are of similar shape, but may differ in intensity (amount of their characteristic maximum stress).

Proportional stresses my also result from several loadings that act on the component simultaneously and, for their part, change proportionally with time as well. Then several stresses of the same kind are to be overlaid additively.

For proportional multiaxial stresses, the interaction formulas given in Chapter 2.6 and 4.6 are exactly valid in the sense of material mechanics, if the related rules of signs are observed.

Synchronous stresses Synchronous stresses are a simple case of non-proportional stresses. They are proportional with regard to their amplitudes, however non-proportional with regard to their mean values.

Normally synchronous stresses result from a combined action of a constant loading with a second, different kind of loading, that is varying with time. Examples are a shaft with a non-changing torsional loading and a rotating bending loading. Or a long, lying cylindrical vessel under pulsating internal pressure, where the longitudinal stress is non-proportional to the circumferential stress because of the bending stress from the dead weight which is additively overlaid.

For synchronous multiaxial stresses, the interaction formulae given in Chapter 2.6 and 4.6 - if observing the related rules of sign - are valid as a useful approximation, because they are applied to the stress amplitudes, which are proportional to each other, and because the fatigue strength is determined by the stress amplitudes in the first place. Additional rules for considering the mean stresses are required, however.

An improved procedure for the assessment of the component fatigue limit in the case of synchronous multiaxial stresses is presented in Chapter 5.9.

Non-proportional stresses Non-proportional stresses result from the action of at least two loadings that vary non-proportionally with time in a different manner.

In this most general case of non-proportional loading different spectra apply to the individual types of stress that result from the combined loadings. In particular the amounts and the directions of the principle stresses are variable with time.

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The case of variable directions of the principle stresses can not be considered with the interaction formulas given in Chapter 2.6 and 4.6.

Appropriate methods of calculation proposed for the assessment of the fatigue strength in the case of non-proportional stresses, that have been developed from a material mechanics point of view, require much computing effort and are applicable with computer programs for short stress sequences only. Their plausibility is currently subject of investigations.

Therefore only an approximate way of calculation for the assessment of the fatigue strength in the case of non-proportional multi-axial stresses can be given, Chapter 5.10: As proportional stresses result from each of the acting loadings the degrees of utilization of these individual loadings can be correctly computed and assessed as described in Chapter 2.6 and 4.6. The so determined degrees of utilization for the individual loadings are then added linearly in order to estimate the entire degree of utilization. Compared to usual interaction formulas developed for proportional stresses the linear addition may be assumed to produce results on the safe side *19.

A necessary reservation for applying this approximate way of calculation is, that a thorough stress analysis is performed in every case and that careful evaluation of the result is performed finally.

In order to reach an optimum degree utilization of the component fatigue strength in the case of non-proportional multiaxial stresses, an experimental assessment of the fatigue strength has to be recommended according to the contemporary state of the art.

19 For non-proportional multiaxial loadings the reference point may be at different positions in the case of the combined loadings and in the case of each of the individual loadings, respectively. This is because the most damaging stresses from the combined loadings may occur at positions different from the positions of the maximum stresses from the individual loadings. By the above mentioned approximation, however, the full damaging effect of each loading may be assumed to be superimposed at the reference point in question.

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