The formulas for determining stresses in simple structural members and machine elements are based on the assumption that the distribution of stress on any section of a member can be expressed by a mathematical law or equation of relatively simple form. For example, in a tension member subjected to an axial load the stress is assumed to be distributed uniformly over each cross section; in an elastic beam the stress on each cross section is assumed to increase directly with the distance from the neutral axis; etc.
The assumption that the distribution of stress on a section of a simple member may be expressed by relatively simple laws may be in error in many cases. The conditions that may cause the stress at a point in a member, such as a bar or beam, to be radically different from the value calculated from simple formulas include effects such as:
- abrupt changes in section such as occur at the roots of the threads of a bolt, at the bottom of a tooth on a gear, at a section of a plate or beam containing a hole, and at the comer of a keyway in a shaft
- contact stress at the points of application of the external forces, as, for example, at bearing blocks near the ends of a beam, at the points of contact of the wheels of a locomotive and the rail, at points of contact between gear teeth, or between ball bearings and their races
- discontinuities in the material itself, such as nonmetallic inclusions in steel, voids in concrete, pitch pockets and knots in timber, or variations in the strength and stiffness of the component elements of which the member is made, such as crystalline grains in steel, fibers in wood, aggregate in concrete
- initial stresses in a member that result, for example, from overstraining and cold working of metals during fabrication or erection, from heat treatment of metals, from shrinkage in castings and in concrete, or from residual stress resulting from welding operations
- cracks that exist in the member, which may be the result of fabrication, such as welding, cold working, grinding, or other causes
The conditions that cause the stresses to be greater than those given by the ordinary stress equations of mechanics of materials are called discontinuities or stress raisers. These discontinuities cause sudden increases in the stress (stress peaks) at points near the
stress raisers. The term stress gradient is used to indicate the rate of increase of stress as a stress raiser is approached. The stress gradient may have an influence on the damaging effect of the peak value of the stress.
Often, large stresses resulting from discontinuities are developed in only a small portion of a member. Hence, these stresses are called localized stresses or simply stress concentrations. In many cases, particularly in which the stress is highly localized, a mathematical analysis is difficult or impracticable. Then, experimental, numerical, or mechanical methods of stress analysis are used.
Whether the significant stress (stress associated with structural damage) in a metal member under a given type of loading is the localized stress at a point, or a somewhat smaller value representing the average stress over a small area including the point, depends on the internal state of the metal such as grain type and size, state of stress, stress gradient, temperature, and rate of straining; all these factors may influence the ability of the material to make local adjustments in reducing somewhat the damaging effect of the stress concentration at the point.
The solution for the values of stress concentrations by the theory of elasticity applied to members with known discontinuities or stress raisers generally involves differential equations that are difficult to solve. However, the elasticity method has been used with success to evaluate stress concentrations in members containing changes of section, such as that caused by a circular hole in a wide plate. In addition, the use of numerical methods, such as finite elements, has lead to approximate solutions to a wide range of stress concentration problems. Experimental methods of determining stress
concentrations may also prove of value in cases for which the elasticity method becomes excessively difficult to apply.
Some experimental methods are primarily mechanical methods of solving for the significant stress; see for example, the first three of the list of methods given in the next paragraph. These three methods tend to give values comparable with the elasticity method. Likewise, when a very short gage length is used over which the strain is measured with high precision, the elastic strain (strain-gage) method gives values of stress concentration closely approximating the elasticity value. In the other methods mentioned, the properties of the materials used in the models usually influence the stress concentration obtained, causing values somewhat less than the elasticity values.
Each experimental method, however, has limitations, but at least one method usually yields useful results in a given situation. Some experimental methods that have been used to evaluate stress concentrations are 1. photoelastic (polarized light), 2. elastic membrane (soap film), 3. electrical analogy, 4. elastic strain (strain gage), 5. brittle coating, 6. MoirC methods, and 7. repeated stress; see HetCnyi (1950), Doyle and Phillips (1989), Kobayashi (1993), and Pilkey and Peterson (1997).
In many practical engineering situations, the failure of a structural member or system is due to the propagation of a crack or cracks that occur in the presence of large stress gradients. The state of stress in the neighborhood of such geometrical irregularities is usually three dimensional in form, thus increasing the difficulty of obtaining complete analytical solutions. Generally, powerful mathematical methods are required to describe the stress concentrations.
The results for computation of stress gradients play a fundamental role in the analysis of fracture and the establishment of fracture criteria. In particular, stress concentrations coupled with repeated loading (fatigue loading) cause a large number of the failures in structures. The reason for this is fairly clear, since stress concentrations lead to local stresses that exceed the nominal or average stress by large amounts.
The concept of a stress concentration factor is often employed by designers to account for the localized increase in stress at a point, with the nominal stress being multiplied by a stress concentration factor to obtain an estimate of the local stress at the point.
Critical to the success of any component, especially in an impact situation, is the elimination of all stress risers or stress concentrations. These are points in the component where the cross sectional area changes abruptly thereby increasing the strength or intensity of applied stress. If a micro-void would happen to be located at this point of highest stress, it would be the initiation site for a crack. It would propagate rapidly under continued stress and lead to premature failure. Ductile materials are not as susceptible to stress risers since they can yield plastically or deform at the points of localized stress and not exhibit immediate failure.
A minor modification in the shape of the part can reduce the stress concentration considerably. For example, utilizing the largest possible radius when transitioning from one diameter to another will minimize the stress concentration in roundtool parts. Never allow internal sharp corners to exist. On external surfaces, add the note on technical drawings to "Break all sharp edges".
It is quite difficult to design a machine without permitting some changes in the cross sections of the members. Rotating shafts must have shoulders so that the bearings can be properly seated and so that they will take thrust loads; and the shafts must have key slots machined into them for securing pulleys and gears. A bolt has a head on one end and screw threads in the other end, both of which account for abrupt changes in the cross section. Other parts require holes, oil grooves, and notches of various kinds. Any discontinuity in a machine part alters the stress distribution in the neighborhood of the discontinuity so that the elementary stress equations no longer describe the state of stress in the part at these locations. Such discontinuities are called stress raisers, and the regions in which they occur are called areas of stress concentration.
The distribution of elastic stress across a section of a member may be uniform as in a bar in tension, linear as a beam in bending, or even rapid and curvaceous as in a sharply curved beam. Stress concentrations can arise from some irregularity not inherent in the member, such as tool marks, holes, notches, grooves, or threads. The nominal stress is said to exist if the member is free of the stress raiser. This definition is not always honored, so check the definition on the stress-concentration chart or table being used.
Related
▪ L - Stress Concentration Factor