For axial tension or bending, Eq. For torsion of a round bar shear stress , by Eq. As explained in Section 1. The factor Kt f represents a calculated estimate of the actual fatigue notch factor K f. Naturally, if K f is available from tests, one uses this, but a designer is very seldom in such a fortunate position. The expression for Kt f and Kts f , Eqs.

Fatigue tests of unnotched specimens by Gough and Pollard and by Nisihara and Kawamoto are in excellent agreement with the elliptical relation. Fatigue tests of notched specimens Gough and Clenshaw are not in as good agreement with the elliptical relation as are the unnotched, but for design purposes the elliptical relation seems reasonable for ductile materials. A cyclic fluctuating stress Fig. This appears to be a reasonable procedure and is in conformity with test data Houdremont and Bennek such as that shown in Fig.

The limitations discussed in Section 1. By plotting minimum and maximum limiting stresses in Fig. However, one can also use a simpler representation such as that of Fig. If, in Fig. This is the equation for the lower full curve of Fig. Note that in Fig. That is, the factor of safety n from Eq.

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It is interesting to note that, for unnotched torsion specimens, static torsion up to a maximum stress equal to the yield strength in torsion does not lower the limiting alternating torsional range. It is apparent that further research is needed in the torsion region; however, since notch effects are involved in design almost without exception , the use of Eq. Even in the absence of stress concentration, Eq. Then Eq. For alternating stress only, Eq. For normal stress only, Eq. For torsion only, Eq. In tests by Ono , and by Lea and Budgen the alternating bending fatigue strength was found not to be affected by the addition of a static steady torque less than the yield torque.

Other tests reported in a discussion by Davies indicate a lowering of the bending fatigue strength by the addition of static torque. Hohenemser and Prager found that a static tension lowered the alternating torsional fatigue strength; Gough and Clenshaw found that steady bending lowered the torsional fatigue strength of plain specimens but that the effect was smaller for specimens involving stress concentration.

Further experimental work is needed in this area of special combined stress combinations, especially in the region involving the additional effect of stress concentration. In the meantime, while it appears that use of Eq. For combined shear and normal stresses, data are very limited.

For combined alternating bending and static torsion, Ono reported a decrease of the bending fatigue strength of cast iron as steady torsion was added. The following approximation which satisfies Eqs. These relations were based on an average of available test data and therefore apply to polished test specimens 0. If the member being designed is not too far from this size range, the formulas may be useful as a rough guide, but otherwise they are questionable, since the number of cycles required for a crack to propagate to rupture of a member depends on the size of the member.

Fatigue failure consists of three stages: crack initiation, crack propagation, and rupture. So much progress has been made in the understanding of crack propagation under cyclic stress, that it is believed that reasonable estimates can be made for a number of problems.

The stress intensity factor K represents the strength of the elastic stress fields surrounding the crack tip Pilkey It would appear that there might be a relationship between the stress concentration factor and the stress intensity factor. Creager and Paris analyzed the stress distribution around the tip of a crack of length 2a using the coordinates shown in Fig. Discarding all terms higher than second order, the approximation for mode I fracture Pilkey , Sec.

Substituting this condition into Eq. Due to the rapid development of fracture mechanics, a large number of crack configurations have been analyzed, and the corresponding results can be found in various handbooks. These results may be used to estimate the stress concentration factor for many cases. It is not difficult to apply Eq. Use known stress intensity factors. For this problem, choose the stress intensity factor for the case of radial cracks emanating from a circular hole in a rectangular panel as shown in Fig.

Further results are listed below. It would appear that this kind of approximation is reasonable. Boresi, A. Cox, H. London, Vol. Creager, M. Davies, V. Nimhanmimie and W. Huitt Battersea Polytechnic , Proc. Davis, E. ASTM, Vol. Draffin, J. Durelli, A. Eichinger, A. Findley, W. Gough, , Trans. ASME Appl. Fralich, R. Gough, H. Gunn, N. R, , Fatigue properties at low temperature on transverse and longitudinal notched specimens of DTDA aluminum alloy, Tech. Note Met. Hencky, H. Hohenemser, K. Houdremont, R. Howland, R. London Ser. A, Vol.

Ku, T. E, No. Kuhn, P. Lazan, B. Lea, F. Leven, M. SESA, Vol. Little, R. Ludwik, P. Marin, J.

Murakami, Y. Nadai, A. Neuber, H. Department of Commerce, Washington, DC, p. Newmark, N. Newton, R. Nichols, R. Nishida, M. Nisihara, T. Ono, A. Kyushu Imp. Peterson, R. Stress Anal. Pilkey, W. Prager, W. Roark, R. Sachs, G. VDI, Vol. Shin, C. Fatigue, Vol. Sih, G. Smith, J. Soderberg, C. ASME, Vol. APM 52—2. Steele, M. Templin, R. Timoshenko, S.

Van den Broek, J. Von Mises, R. Goettingen Jahresber. Von Phillipp, H. Department of Commerce, Washington, DC. Savin, G. Young, W. Web Site for This Book www. A thin flat element with in-plane loading. This is a sheet that is sometimes referred to as a membrane. It occurs in machine elements such as in turbine rotors between blade rows and at seals. Other examples are found in a variety of shafts Fig. The round-bottomed V-shaped notch or circumferential groove, and to a lesser extent the U-shaped notch, are conventional contour shapes for stress concentration test pieces in the areas of fatigue, creep-rupture, and brittle fracture.

A threaded part may be considered an example of a multigrooved member. As mentioned in Chapter 1, two basic Kt factors may be defined: Ktg , based on the larger gross section of width H and Ktn , based on the smaller net section of width d Fig. Unless otherwise specified, Kt refers to Ktn in this chapter. Neuber found the theoretical stress concentration factors for the deep hyperbolic notch Fig. These results will be given in this chapter. Consider, for example, a flat bar with opposite notches of arbitrary shape, with notch depth t, notch radius r , and a minimum bar width d.

If Kte is the stress concentration factor of a shallow elliptical notch of Chart 2. Recent investigations have provided more accurate values for the parameter ranges covered by the investigations, as will be presented in the following sections. If the actual member being designed has a notch or groove that is either very deep or shallow, the Neuber approximation will be close. More accurate values have been obtained over the most used ranges of parameters. These form the basis of some of the charts presented here.

Some charts covering the extreme ranges are also included in this book. Another use of the charts of Neuber factors is in designing a test piece for maximum Kt , as detailed in Section 2. Some guidance may be obtained by referring to the introductory remarks at the beginning of Chapter 4.

## Peterson's Stress Concentration Factors, 2nd Edition

The Kt factor for a notch can be lowered by use of a reinforcing bead Suzuki Chart 2. For practical purposes, the solid curve of Chart 2. Ling has provided the following summary of Kt factors: Maunsell Isibasi Weinel Ling Yeung Mitchell 3. All such notches, U-shaped, keyhole circular hole connected to edge by saw cut , and so on have very nearly the same Kt as the equivalent elliptical notch.

It is not applicable for shear. Stress concentration factors have been approximated by splitting a thin element with a central hole axially through the middle of the hole Fig. From Eq. The factors for the U-shaped slot Isida are practically the same. A comparison of the curves for notches and holes in Chart 2.

The difference between Ktg and Ktn is illustrated in Chart 2. Consider a bar of constant width H and a constant force P. An important check is provided by including in Chart 2. Barrata has compared empirical formulas for Ktn with experimentally determined values and concludes that the following two formulas are satisfactory for predictive use. Referring to Chart 2. Equation 2. For the dashed curves not the dot-dash curve for semicircular notches , Eq.

In the absence of better basic data, the dashed curves, representing higher values, should be used for design. In Chart 2. The effect is small and is not shown on Chart 2. For shallow or deep notches the error becomes progressively smaller. Some specific photoelastic tests Liebowitz et al. Plots for opposite narrow edge notches are given in Peterson In comparing these results with Chart 2. Corresponding Neuber Ktn factors obtained by use of Chart 2. Considered from the standpoint of flow analogy, a smoother flow is obtained in Fig. For the infinite row of semicircular edge notches, factors have been obtained mathematically Atsumi as a function of notch spacing and the relative width of a bar, with results summarized in Charts 2.

For infinite notch spacing, the Ktn factors are in agreement with the single-notch factors of Isida and Ling , Chart 2. For a finite number of multiple notches Fig. The maximum stress concentration occurs at the end notches Charts 2. Sometimes a member can be designed as in Fig. The results Charts 2. Fatigue tests Moore of threaded specimens and specimens having a single groove of the same dimensions showed considerably higher strength for the threaded specimens. In this paper, the shape of the notch is replaced by an equivalent ellipse or hyperbola. The closed-form relations are based on finite element analyses.

Noda et al. For equal biaxial stress Fig. Dimpling is often used to remove a small surface defect. These values also apply for equal biaxial stressing. The calculated values of Chart 2.

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In terms of the variables given in Chart 2. For comparison, for a groove having the same sectional contour as the dimple, Ktg is shown by the dashed line on Chart 2. Note that in Chart 2. The Ktg factors therefore essentially represent the loss of the section. This is an approximation. However, after comparison with the bending and torsion cases, the results seem reasonable. The maximum stress occurs at the bottom of the groove. The corresponding Ktn from Chart 2. Approximate Ktn factors, based on the Neuber formula, are given in Chart 2.

In Nisitani and Noda , a solution technique is proposed for finding the stress concentration factors for V-shaped grooves under tension, torsion, or bending. For several caes, the resulting factors are shown to be reasonably close to results available previously. A variety of stress concentration factors charts are included in this paper. Noda and Takase extend the formulas to cover grooves of any shape in bars under tension and bending. Troyani et al. These curves are obtained by increasing the photoelastic Ktn values Frocht , which as in tension are known to be low, to agree with the semicircular notch mathematical values of Chart 2.

Photoelastic Wilson and White and numerical results Kitagawa and Nakade are in good agreement. The Ktn value is for a U notch. The curves of Chart 2. The full curve represents a crack Wilson and the dashed curved a semicircular notch Leven and Frocht The correction factor for the crack is the ratio of the stress-intensity factors. In the small-radius, narrow-notch limit, the ratio is valid for stress concentration Irwin ; Paris and Sih The same effect, only of slight magnitude, was obtained by Isida for the bending case of a bar with opposite semicircular notches Chart 2.

In tension, since there is no nominal stress gradient, this effect is not obtained. As can be found from Chart 2. Although these results are for a specific case of a beam in bending, it is reasonable to expect that, in general, a considerable stress reduction can be obtained by use of the semielliptic notch bottom. Other uses of the elliptical contour are found in Chart 4. The factors were obtained for transverse bending. The bending of a semi-infinite plate with a V-shaped notch or a rectangular notch with rounded corners Shioya is covered in Chart 2.

For comparison, the corresponding curve for the tension case from Chart 2. This reveals that the tension Ktn factors are considerably higher. As the notch spacing increases, the Kt value for the single notch is approached asymptotically. The results are shown for the thin plate in Chart 2. No direct results are available for intermediate thicknesses. If we consider the tension case as representing maximum values for a thick plate in bending, we can use Chart 2.

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For intermediate thickness ratios, some guidance can be obtained from Chart 4. The net section on the groove plane is circular. Using finite element analyses, stress concentration factors of Chart 2. Example 2. First we determine Ktn. From Chart 2. Substituting in Eq.

These are not working stresses, since a factor of safety must be applied that depends on type of service, consequences of failure, and so on. Different factors of safety are used throughout industry depending on service and experience. The strength of a member, however, is not, in the same sense, a matter of opinion or judgment and should be estimated in accordance with the best methods available. Naturally, a test of the member is desirable whenever possible. In any event, an initial calculation is made, and this should be done carefully and include all known factors.

Shearing forces are applied to an infinite thin element with deep hyperbolic notches. These forces are parallel to the notch axis1 as shown in Chart 2. The case of a twisted sheet with hyperbolic notches has been analyzed by Lee The net section is circular in the groove plane. To avoid possible confusion with the combined shear and bending case, the countercouple is not shown in Chart 2. Mathematical results for semicircular grooves Hamada and Kitagawa ; Matthews and Hooke are in reasonably good agreement with Chart 2.

The Kts values of Chart 2. However, the photoelastic values are not in agreement with certain other published values Okubo , Find the maximum shear stress and the equivalent stress at the root of the notch. Finally, the equivalent stress Eq. The effect of the V angle may be compared with Charts 2. Another specimen design problem occurs when r and d are given. The smaller diameter d may, in some cases, be determined by the testing machine capacity. Atsumi, A. Barrata, F. Strain Anal. Barrata, F, I. Bowie, O. Brown, W. Cheng, Y.

Ching, A. Cole, A. Cowper, G. Denardo, B. Dixon, J. Eubanks, R. Flynn, P. Frocht, M. Gray, T. Grayley, M. Hamada, M. Heywood, R. Hooke, C. Inglis, C. Irwin, G. Isida, M. Kikukawa, M.

Kitagawa, H. Koiter, W. Lee, G. Leibowitz, H. M,, , Quantitative three-dimensional photoelasticity, Proc. Ling, C. Matthews, G. Moore, R. Edwards Co. Nishioka, K. Nisitani, H. Noda, N. Okubo, H. Paris, P. Reed, R. Rubenchik, V. Rushton, K. Schulz, K. Seika, M. Shioya, S. Slot, T.

Sobey, A. Suzuki, S. Solids Struct. Troyani, N. Tsao, C. Westergaard, H. Wilson, I. Wilson, W. Ktn 3. Tension loading in line with middle of minimum section. Ktn 2. Ktg 2. Ktg 3. Values are approximate. Ktn 1. Ktn 33 4. Dashed lines in above sketch 4. The net cross section is circular. Kt values are approximate. Ktn d 1. Shafts, axles, spindles, rotors, and so forth, usually involve a number of diameters connected by shoulders with rounded fillets replacing the sharp corners that were often used in former years. In tension Fig. Many of the fillet factors for tension and bending are based on photoelastic values.

Others are found from finite element analyses. For torsion, the fillet factors are from a mathematical analysis. The stress at the fillet surface at the middle of the panel thickness increases, and the stress at the panel surface decreases. Some of the stress concentration cases of Chapter 5 on miscellaneous design elements are related to fillets. These curves are modifications of the Kt factors determined through photoelastic tests Frocht whose values have been found to be too low, owing probably to the small size of the models and to possible edge effects.

These data fit well with the above results from Eq. Other photoelastic tests Fessler et al. Frequently one encounters a case in design where this shoulder width L Fig. The Baud rule, which was proposed as a rough approximation, has been quite useful. Although the Kt factors for bending of flat elements with narrow shoulders Section 3. Referring to Charts 3. Kumagai and Shimada state that their results are consistent with previous data Spangenberg ; Scheutzel and Gross obtained for somewhat different geometries.

Empirical formulas were developed by Kumagai and Schimada to cover their results. Round bar values are not available. It is suggested that Eq. The geometrical configuration is shown in the sketch in Chart 3. A finite difference method was used to find the stress concentration factor Derecho and Munse The dashed curve corresponds to a protuberance height where the radius is exactly tangent to the angular side; that is, below the dashed curve there are no straight sides, only segments of a circle.

See sketch in Chart 3. A similar comparison by Peterson, using the increased Kt fillet values of Chart 3. It may be noted here that had a further refinement of the spacing been possible in the previously discussed finite-difference solution, slightly higher values of the stress concentration factor could have been obtained.

Referring, for example, to Chart 3. The circular fillet does not correspond to minimum stress concentration. The variable radius fillet is often found in old machinery using many cast-iron parts where the designer or builder apparently produced the result intuitively. Sometimes the variable radius fillet is approximated by two radii, resulting in the compound fillet illustrated in Fig. Baud proposed a fillet form with the same contour as that given mathematically for an ideal, frictionless liquid flowing by gravity from an opening in the bottom of a tank Fig.

By means of a photoelastic test in tension, Baud observed that no appreciable stress concentration occurred with a fillet of streamline form. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation.

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For more information about Wiley products, visit our web site at www. Pilkey, Deborah F. The definitive guide to stress concentration, newly revised and updated. Stress concentration factors are used in the design of mechanical or engineering structures and machinery and have a tremendous impact on design of virtually every product or structure subject to environmental or use-related stress—from nuclear equipment, deep-sea vehicles, and aircraft to space vehicles, underground tunnels, and turbines. But the sheer volume of stress concentration data is now so large that analysts and designers may have difficulty integrating this information efficiently into design applications.

Now in its third edition, Peterson's Stress Concentration Factors establishes and maintains a system of data classification for all of the applications of stress and strain analysis and expedites their synthesis into CAD applications. This new edition of the book has been substantially revised and completely updated and presents stress concentration factors both graphically and with formulas. Structures and shapes of interest can be easily accessed via an illustrated table of contents that permits identification based on the geometry and loading of the location of a factor.

Completely updated and revised coverage of the most recent stress concentration factors, including geometric discontinuities in tubes and countersunk holes in plates. The how-to's of optimizing a shape to circumvent stress concentration problems and to achieve a well-balanced design of structures and machines that will result in reduced costs, lighter products, and improved performance. New guidelines for the likely stress decay away from the point of the stress concentration for various holes and notches.

New methods for analyzing and calculating the stress concentration factors based on the length of elements and on the location of loading on components. More comprehensive and up-to-date than previous editions, Peterson's Stress Concentration Factors is an essential addition to the professional libraries of engineers and designers working in the automotive, aerospace, and nuclear industries; for civil and mechanical engineers; and for students and researchers in these fields. Convert currency. Add to Basket. Book Description John Wiley and 38; Sons, Condition: New.

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