Equivalent Stress Concentration Factor Theory
Mechanical Monograph
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
The “Collegium” All World Academy of Sciences Publishers
Munich (Germany)
Third Edition
(2010)
Second Edition (2006)
First Edition (2005)
0. Introduction. Known Stress Concentration Factor
When comparing machine parts with and without essential changes in their geometric forms, these cause changing both cross-sections (which are usually decreased) and the character of stress distributions (which often becomes strongly inhomogeneous with maximum stresses possibly much greater than homogeneous nominal ones).
To determine the maximum stress σmax via the nominal stress σnom, the so called stress concentration factor (SCF) [1, 2]
(0.1)
C = σmax/σnom
can be used.
1. Maximum and Nominal Stresses
The maximum stresses are real and hence unambiguous. The nominal stresses are conditional, considered homogeneous (with ignoring stress concentration), and can be defined in different reasonable ways two of which are most commonly used.
In elasticity theory [3, 4], these are so called far-field stresses, i.e., stresses at places very (theoretically infinitely) far from stress concentrators with ignoring cross-sections changes caused by concentrators (considered relatively infinitely small in comparison with the whole machine parts). This approach consequently ignores the both above influences of stress concentrators and is further used in this work.
In design studies [5, 6], the nominal stresses are so called ligament stresses with taking into account real cross-sections changed by concentrators (considered finite). This approach inconsequentially ignores one of the both above influences of stress concentrators with taking into account the remaining one, can lead to seemingly decreasing the stress concentration factor, which is very dangerous, and needs considering the precise sense of the nominal stresses.
of the Stress Concentration Factor
Circular holes are common stress concentrators in machine parts.
The simplest design scheme is a plate infinite in the both directions, having a finite (i.e., infinitely small in comparison with the both plate dimensions besides the thickness) circular hole, and subjected to two-directional uniform loading with equal normal stresses σx = σy in the infinity (Fig. 1).
Fig. 1. The Generalized Lame-Kirsch Problem
The third normal stress σz vanishes. Both the geometry and loading are axially symmetric. This is a particular case of the classical Lame problem for a thick-walled cylinder under an external press -σx = -σy [7], namely the limiting case holding for a cylinder with a finite internal and the infinite external radius. By zero internal press throughout the present work, the maximum stresses 2σx are homogeneous on the hole surface. In this case
(2.1)
σmax = 2σx ,
σnom = σx ,
C = σmax / σnom = 2
where naturally σx is implicitly considered positive only.
A much more complicated design scheme is a plate having the above axially symmetric geometry but subjected to one-directional uniform loading with normal stress σx in the infinity (see Fig. 1). The third normal stress σz vanishes. Loading is not axially symmetric. By the classical Kirsch solution [8], the maximum stresses are 3σx and hold at the ends of the hole diameter normal to this direction whereas stresses -σx hold at the ends of the hole diameter parallel to this direction. In this case
(2.2)
σmax = 3σx ,
σnom = σx ,
C = σmax / σnom = 3
where naturally σx is implicitly considered positive only.
The well known generalizations [9-11] of the above classical solutions [7, 8] consider two-directional loading with generally unequal σx and σy in the infinity (see Fig. 1). The third normal stress σz vanishes. The geometry is axially symmetric but loading is not axially symmetric. All normal and shear stresses at each point of the plate are precisely determined due to the principle of superposition [1, 2]. Then
(2.3)
σmax = 3σx - σy ,
σnom = σx ,
C = σmax /σnom = (3σx - σy)/σx
if
σx ≥ σy
and
(2.4)
σmax = 3σy - σx ,
σnom = σy ,
C = σmax /σnom = (3σy - σx)/σy
if
σx ≤ σy
where the maximum stresses hold at the ends of the hole diameter normal to the direction with the maximum stress in the infinity (see Fig. 1).
Formulae (2.3) and (2.4) hold by positive σx and σy only, otherwise many additional cases with also considering their signs and relations between their algebraic values including zero σz have to be taken into account.
A typical and hence important particular case of the last example is pure shear in the infinity with
σx = -σy
separately analyzed [4]. Because of the different signs of σx and σy , formulae (2.3) and (2.4) may not be applied to this case without further consideration. But it can be additionally shown [11] that they hold in this specific situation. We have
(2.5)
σmax = 4σx ,
σnom = σx ,
C = σmax /σnom = 4σx /σx = 4
if
σx > 0
and
(2.6)
σmax = 4σy ,
σnom = σy ,
C = σmax /σnom = 4σy /σy = 4
if
σy > 0
where the maximum stresses hold at the ends of the hole diameter normal to the direction with the maximum stress in the infinity (see Fig. 1).
3. Analysis of the Known Definition
The known definition of the stress concentration factor commonly used practically always implicitly considers the positive values of individual stresses only.
Therefore, this definition has a number of sources of defects:
1) individual stresses can be zero or negative, which naturally brings ambiguity to defining the maximum stresses (either by algebraic values or by moduli, i.e. absolute values);
2) there are different choices of individual stresses in a non-uniaxial stress state;
3) the roles of the chosen individual stresses in complex stress states at different places with maximal and conditionally nominal stresses can be very distinct, which can lead to great errors in determining the stress concentration factor;
4) no individual stress alone can evaluate the danger of a complex stress state.
The further defects are natural:
1) straightforwardly correcting the known definition leads to the necessity to consider many different cases caused by the number of the individual stresses, their signs, and the relations between the algebraic values of the individual stresses;
2) stress concentration factor analysis is very difficult;
3) the obtained results are not suitable for practical applications;
4) no general results can be obtained.
4. Definition of the Equivalent Stress Concentration Factor
To correct the common definition of the stress concentration factor via avoiding such shortcomings, it is now proposed to use the equivalent stresses [1, 2] instead of the individual ones by defining both the maximum and the nominal stresses.
To determine the maximum equivalent stress σemax via the nominal equivalent stress σenom, the so called equivalent stress concentration factor (ESCF)
(3.1)
Ce = lim σemax/σenom
as σenom approaches its real value is proposed to be used instead of (0.1).
Remark.
Introducing the above limit allows to give a definition that holds in the general case, i.e. also if
σenom = 0.
Otherwise, i.e. by nonzero σenom , we simply have
Ce = σemax/σenom .
of the Equivalent Stress Concentration Factor
For the above plate subjected to two-directional uniform loading with equal normal stresses σx = σy (Fig. 1)
(5.1)
σemax = 2|σx|,
σenom = |σx|,
Ce = lim σemax / σenom = 2
as |σx| approaches its real value which can also vanish.
For the above plate subjected to one-directional uniform loading with normal stress σx (see Fig. 1)
(5.2)
σemax = 3|σx|,
σenom = |σx|,
Ce = lim σemax / σenom = 3
as |σx| approaches its real value which can also vanish.
These results hold for any σx, namely positive, zero, and negative.
For the above plate subjected to two-directional loading with generally unequal σx and σy (see Fig. 1)
(5.3)
σemax = max{|3σx - σy|, |3σy - σx|},
σenom = max{|σx|, |σy|, |σy - σx|},
Ce = lim σmax /σnom = lim max{|3σx - σy|, |3σy - σx|}/max{|σx|, |σy|, |σy - σx|}
as max{|σx|, |σy|, |σy - σx|} approaches its real value which can also vanish by the Tresca criterion of the maximum shear stress [12] and
(5.4)
σemax = max{|3σx - σy|, |3σy - σx|},
σenom = {[σx2 + σy2 + (σy - σx)2]/2}1/2 = (σx2 - σxσy + σy2)1/2 ,
Ce = lim σmax /σnom = lim max{|3σx - σy|, |3σy - σx|}/(σx2 - σxσy + σy2)1/2
as (σx2 - σxσy + σy2)1/2 approaches its real value which can also vanish by the Huber-von Mises-Hencky criterion of the octahedral shear stress (or distortion energy) [13-15].
In the particular case of pure shear σx = -σy (see Fig. 1) by the Tresca criterion of the maximum shear stress [12]
(5.4)
σemax = 4|σx|,
σenom = 2|σx|,
Ce = lim σmax /σnom = 2
as |σx| approaches its real value which can also vanish and by the Huber-von Mises-Hencky criterion of the octahedral shear stress (or distortion energy) [13-15]
(5.4)
σemax = 4|σx|,
σenom = 31/2|σx|,
Ce = lim σmax /σnom = 4/31/2
as |σx| approaches its real value which can also vanish, both instead of the above stress concentration factor (2.6)
C = 4.
This great distinction naturally holds due to the very different roles of the individual stresses chosen to determine the maximum stresses and the nominal ones in order to obtain the stress concentration factor:
1. The individual stress 4σx chosen to determine the maximum stresses is the only nonzero stress at the places of the maximum stress concentration, and its modulus (absolute value) equals the equivalent stress at these places.
2. The individual stress σx chosen to determine the nominal stresses is only one of the two nonzero stresses with equal moduli (absolute values) at the infinity, and its modulus (absolute value) equals only a certain part (fraction) (determined by a specific strength criterion) of the equivalent stress at the infinity.
6. Advantages of the Equivalent Stress Concentration Factor
The introduced definition of the equivalent stress concentration factor has the following advantages:
1) the definition holds in the general case of a complex stress state with different individual components having any signs independently from the relations between the algebraic values of these components;
2) there are no different choices of individual stresses in a non-uniaxial stress state;
3) the roles of individual stresses at different places of a machine part are completely taken into account;
4) the equivalent stresses much more adequately evaluate the danger of a complex stress state than any individual stress alone;
5) the choice of a strength criterion to determine the equivalent stress brings no danger due to consequently using the only chosen criterion for the whole machine part and due to the nearness of the results given by different specific criteria;
6) the choice of a strength criterion even plays a very positive role due to the possibility to select the strength criterion most suitable for the specific material and loading and to further use improving the strength criteria;
7) it is not necessary to consider many different cases caused by the number of the individual stresses, their signs, and the relations between these stresses;
8) there is no ambiguity when defining the maximum stresses (either by algebraic values or by moduli, i.e. absolute values);
9) in determining the stress concentration factor, great errors caused by the very distinct roles of the chosen individual stresses in complex stress states at different places with maximal and conditionally nominal stresses are completely excluded;
10) the equivalent stress concentration factor analysis is very suitable for definition, determination, analysis, practical application, and generalization.
To further improve estimating the danger of stress concentration, the very essentially corrected and generalized strength criteria [16-26] in fundamental mechanical and strength sciences of the author universally holding for any materials and loading conditions can be successfully used in order to determine the equivalent stresses for this factor.
7. Main Results and Conclusions
1. The common stress concentration factor has many evident lacks and can lead to very great errors in estimating the danger of stress concentration.
2. The equivalent stress concentration factor introduced now has many very important advantages and adequately estimates the danger of stress concentration.
3. To further improve determining the equivalent stresses for this factor, fundamental mechanical and strength sciences of the author can be successfully used.
[1] Timoshenko S. P. y Young D. H.: Elementos de resistencia de materiales. Traducción de Jesús Ibáñez Gar. 2ª ed. Montaner y Simón, Barcelona, DL 1979
[2] Pisarenko, G. S. y otros : Manual de Resistencia de Materiales. Editorial MIR, Moscú, 1989
[3] Love, A. E. H.: A Treatise on the Mathematical Theory of Elasticity. Vols. I, II. Cambridge University Press, Cambridge, 1892, 1893
[4] Timoshenko, S. P.: Theory of Elasticity, 3rd ed. McGraw-Hill, New York, 1970
[5] Military Handbook. Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5H, 1998
[6] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss- Struktur-Berechnungsunterlagen, Bremen, 1998
[7] Lamé, G.: Lecons sur la theorie mathematique de l’élasticite des corps solides. Gauthier-Villars, Paris, 1852
[8] Kirsch, G.: Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. VDI Z., 42 (1898), 797-807
[9] Muskhelishvili, N. I.: Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations, Plane Theory of Elasticity, Torsion, and Bending, 4th corr. and augmented. Ed. P. Noordhoff, Groningen, 1963
[10] Lur'e, A. I.: Three-Dimensional Problems of the Theory of Elasticity. Interscience Publishers, N. Y., L., Sydney, 1964
[11] Lev Gelimson: Generalized Analytic Methods of Solving Strength Problems. Dr. Sc. Dissertation. Institute for Strength Problems after G. S. Pisarenko, Academy of Sciences of the Ukraine, Kiev, 1994
[12] Tresca, H. E.: Memoire sur l'ecoulement des corps solides soumis a de fortes pressions. Comptes Rendus de l’Academie des Sciences, Paris, 59 (1864), 754-758
[13] Huber, M. T.: Die spezifische Formänderungsarbeit als Maß der Anstrengung eines Materials. Czasopismo Techniczne, Lemberg (Lwow), 20 (1904), 81-83
[14] von Mises, R.: Mechanik der festen Körper im plastisch-deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttingen, 4 (1913), 582-592
[15] Henky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Zeitschrift angewandter Mathematik und Mechanik, 4 (1924), 323-334
[16] Lev Gelimson: General Strength Theory. Monograph. Drukar Publishers, Sumy, 1993
[17] Lev Gelimson: The generalized structure for critical state criteria. Transactions of the Ukraine Glass Institute, 1 (1994), 204-209
[18] Lev Gelimson: Basic New Mathematics. Monograph. Drukar Publishers, Sumy, 1995
[19] Lev Gelimson: Yield and Fracture Laws of Nature (Universal Yield and Failure Criteria in the Relative Stresses). Strength Monograph. The ”Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2001
[20] Lev Gelimson: Theory of Measuring Stress Concentrations. Proceedings of the ”Collegium” All World Academy of Sciences, Munich (Germany), Mechanics, 1 (2001), 1
[21] Lev Gelimson: Theory of Measuring Inhomogeneous Distributions. Physical Journal of the ”Collegium” All World Academy of Sciences, Munich (Germany), 2 (2002), 1
[22] Lev Gelimson: General Strength Theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), Berlin
[23] Lev Gelimson: Elastic Mathematics. General Strength Theory. Mathematical, Mechanical, Strength, Physical, and Engineering Monograph. The ”Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2004
[24] Lev Gelimson: Providing Helicopter Fatigue Strength: Flight Conditions [Overmathematics and Other Fundamental Mathematical Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416
[25] Lev Gelimson: Providing Helicopter Fatigue strength: Unit Loads [Fundamental Mechanical and Strength Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 589-600
[26] Lev Gelimson: Equivalent Stress Concentration Factor. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 30-32