Four studies in the theory of stress concentration (Parts 1-4)

Abstract

PART I - The Effect of Holes on the Strength of Materials under Compiex Stress Systems. In a plane test piece pierced by a cylindrical hole the greatest stress is set up at some point in the periphery of the hole. This stress is a principal stress and both the other principal stresses, that normal to the contour of the hole and that parallel to the axis of the hole and normal to the free surface of the test piece, are zero. Therefore, failure of such a test piece under any system of applied loading depends almost entirely on the shape of the hole and on the properties of the material only in respect of a possible difference between its strengths in tension and in compression. On this basis criteria are developed for the failure of test pieces containing circular and elliptical cylindrical holes under systems of complex stress. The results are applicable to tests on pieces pierced by oil holes drilled either perpendicularly to the axis of the test piece or obliquely. The resulting criterion for circular holes perpendicular to the plane of stress is compared with some experimental results of tests under combined alternating bending and torsion. Criteria are also developed for elliptical holes oriented at random, and it is shown that these criteria do not in themselves accord with the results of tests on the majority of materials. It is concluded that internal flaws are unlikely to account for the mechanical properties of engineering materials. PART II - Stress Concentration due to Holes and Grooves other than Elliptical in Form. In order critically to compare the results of fatigue tests on pieces containing sharp V-notches and other abrupt changes of section with tile theoretical values of stress concentration factors, a need was apparent for detailed theoretical investigation of the effect of the form of the discontinuity of section. Following generally established methods of stress analysis the stress distributions round holes and grooves of a wide range of forms have been examined both under plane direct stress and under shear stress. These analyses have been applied to several particular cases and the results have been compared with approximate formulae based on the stress distribution round elliptical contours. From the results it appears that the approximate formulae based on elliptical holes afford a reasonably accurate estimate of the maximum stress at any hole or groove under plane direct stress, but that the stress concentration under shear is influenced to a much greater extent by the general form of the hole or groove. Under both types of stress system, certain cases of anomaly arising from application of the approximate formulae are examined, and it is shown that all these anomalies are resolved by the more accurate formulae here derived. Incidentally, in this examination it is demonstrated that abrupt changes of curvature of the contour of a hole or groove cause no concentration of stress. PART III - The Effect of Surface Irregularities on Fatigue Strength. It is perhaps not generally recognized that the approximate formulae 1 + ~/(a/e) and 1 + 2~/(a/e) for the stress concentrations under shear and under direct stress due to a groove of depth a and root radius 5 are applicable not only when the ratio ale is large but equally when it is small; indeed the accuracy of these approximate formulae improves as ale decreases. This is demonstrated by computation by exact theory of the stress concentration due to a continuous nearly sinusoidal undulation of the surface of a test piece, and it is shown incidentally that when a and e are both negative, so that the groove is inverted into a protrusion, the 'de-concentration' of stress is represented very closely by the approximate formulae 1 -- ~/(a/e) and 1 -- 2~/(a/Q). It is shown further that these factors applied as corrections to computed stress factors under torsion for an approximation to a square shaft with rounded corners suffice to reconcile these results to the established solution for a square shaft under torsion. PART IV - Stress Concentration in Twisted Shafts. straightforward method for computing the stress distribution in a twisted shaft of specified cross-section is developed, and the method is illustrated by application to a round shaft with a single flat on one side and to a six-splined shaft. In these applications use is made of the process of correction for local irregularities described in Part III, and some general comments are made on the means to represent complex boundaries by analytical forms, which supplement the techniques described in Part II. An approximate formula for the concentration of shear stress in the fillet at the root of the spline of a splined shaft under torsion is proposed and the accuracy of this formula is tested by three examples. One example of a hollow shaft with a lobed external contour and a wide variation of wall thickness is worked out; and it is shown that over the smooth inner boundary the shear stress is very nearly inversely proportional to the wall thickness, whereas at the lobed outer boundary marked concentration of stress occurs at the grooves between the lobes.