Stress curves for body 1 and body 2 maximum principal stress. The equations for two spheres in contact are also valid for. E youngs modulus f load r 1, r 2 radii v poissons ratio. Hertz 1881 considered the stresses and deformations in two perfectly smooth t ellipsoidal, contacting elastic solids much like those shown in figure 2. This kind of stress may not be significant most of the time, but may cause. These stresses were first studied by the german physicist heinrich hertz in 1881. Initiation of plastic flow starts at a lower pressure and occurs when the max shear stress reaches the shear yield stress k for the material. In addition to normal stress that was covered in the previous section, shear stress is an important form of stress that needs to be understood and calculated. Contact pressure and yield stress phoenix tribology. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of curvature such as a toroid. Hertzian contact stress on ball bearing the hertzian theory of elastic. The stress field under the surface is termed hertzian contact stress reference hertz 1881 has done the hard work to derive the equations we need to calculate the stress when there is contact between two bodies that are conformal or nonconformal. Although this point is regarded as a proportional limit, the joint will have a slight set on the removal.
Here the aluminum plate being r2 and having no radius was set to infinity. Pa, the average shear stress is defined as the the shear load divided by the area. Tutorial on hertz contact stress university of arizona. In discusssing the bearing strength of wood under bolts it is convenient to use the term proportionallimit stress, which is taken as the average stress under the bolt when the slip in the joint ceases to be proportional to the load. Hertz contact stress of deep groove ball bearing sciencedirect. This tutorial addresses the basics of hertzian contact stress theory and it relates to certain aspects of optomechanical design. Railway, wheel, rail, contact stress, hertz theory 1.
It uses a color function to visualize the contact zone between a statically loaded pair of contact models, a sphere and a torus. Excessive contact pressure can lead to a typical bearing failure such as a plastic deformation similar to peening. Between the head of the bolt and the top plate a between the surfaces of the shanks and hole which they pass b b b a f. However, for 12th and 10thpower relations reflected by modern bearing steels, the zaretsky model based on the weibull equation is superior. Lesson 9 hertz contact problem patran 322 exercise workbook 93 model description. Introduction wheel set is the most important component among the different components of a car and this. Lewis formula 1 is generally employed for flexural stress on dedendum. Basic calculation principles are mentioned in this chapter, to give an overview of.
This problem is also referred to as bearing resistance. Character of fatigue failure of gear teeth and of ball and roller bearings. It is important to remember that va is only the average shear stress. Bearing calculation when selecting an axlebox bearing or unit, skf should be contacted for assistance with necessary calculations. Since the ultimate usefulness of any formula depends upon experimental verification, we have compiled results of pertinent experiments and various empirical formulae. What is the difference between normal stress and bearing. For 9th and 8thpower hertz stress life exponents for ball and roller bearings, respectively, the lundbergpalmgren model best predicts life. Itis an honor and quite gratifying to correspond with the many individuals who call attention to errors andor convey useful and practical suggestions to incorporate in future editions. Well, technically, a bearing stress is the stress between two surfaces in compression. This page focuses mainly on the normal direction, i. Whats the difference between bearing, shear, and tearout. Apr 12, 2018 bearing stress can also be said as crushing stress. This article aims to calculate the contact stresses in different parts on the wheel surface by using hertz theory.
Hertz contact stress mechanical engineering general. A theoretical background to the hertz contact theory can be found here. Oct are represented for at the line of contact under the surfaces. Plain bearing journal bearing, sleeve bearing, rifle bearing, composite bearing a plain bearing in railroading sometimes called a. Bearing stress compressive normal stress while the amount of the force compression load, the area depends on the mode of the contact examples. Using the hertzs surface pressures and the mohrs circles the substance. The contact area will change depending on the magnitude of the contact stress. Using hertz contact formula i calculated contact stress for cylinder in contact with cylinderical socket but i dont know to which stress to compare this contact stress. This demonstration presents the hertzian contact stress distribution and elastic deformations. Calculation of contact stress is the basis of analysis of rolling bearing. In this example problem, a steel cylinder with a radius of 5 is pressed against a 2 deep aluminum base. The calculation procedure consists of calculation of maximum contact pressure at different loads using analytical formulas and comparison of results with mesys tool to predict the contact pressure between inner race and ball of single row deep. In bearing engineering, the equalized terms are between balls and an inner or outer ring.
Missouri university of science and technology scholars mine. Therefore, it is very imporant to calculate the actual stress at the point of contact, the socalled contact stress. Pdf extended hertz theory of contact mechanics for case. Implementation of the hertz contact model and its volumetric modication on modelica ivan kosenko evgeniy alexandrov russian state university of tourism and service, department of engineering mechanics glavnaya str.
Further reading useful solutions for standard problems mike ashby engineering department trumpington street, cambridge cb2 1pz, uk 8th edition, march 2010. In case of two cylinders in contact with radii, as shown in fig. The bearing stress is the stress in the material due to the force imparted by the fastener itself. Contact stresses and deformations curved surfaces in contact. Bending stress lewis equation in 1893, wilfred lewis provided a formula to estimate the bending stress in a tooth. The calculation procedure consists of calculation of maximum contact pressure at different loads using analytical formulas and comparison of results with mesys tool to predict the contact pressure between inner race and ball of single row deep groove ball bearing. Therefore, ifthe deformation around the hole is a critical design consideration, section e3. A hypothetical value of stress at failure for exactly n f cycles as determined from an sn diagram. When this happens, complex stresses are induced under the surface at the contact. Hertz theory assumes that surfaces are continuous and nonconforming, strains are small, each solid can be considered as an elastic halfspace, and surfaces are frictionless. Rolling bearing life prediction, theory, and application. Extended hertz theory of contact mechanics for casehardened steels with implications for bearing fatigue life article pdf available in journal of tribology 1402. Pdf contact stress distribution of deep groove ball bearing. To properly evaluate the contact stress in such systems, finite element analysis fea of this mechanical system is required.
Linear bearings that use balls or rollers to carry a load are subjected to hertz contact stresses a type of material stress that plays a significant. Pdf calculation of hertz contact stress between the deep groove ball bearing and inner race using analytical and mesys tool analysis. Contact mechanics is the study of the deformation of solids that touch each other at one or more points. Hertzian contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. Spheres in contact vertical stress distribution at center of contact area plot shows material with poissons ratio. Background 1 contact between two continuous, nonconforming solids is initially a point or line. On the compression of a cylinder in contact with a plane surface. Chapter 6, is expanded, presenting more coverage on electrical strain gages and providing tables of equations. Stress life curves, sn continued common terms used with the sndiagram are fatigue life, n f. In the early stage, approximate modelling establishes whether the concept will work at all, and identifies the combination of material properties that maximize performance.
For systems that need to survive extreme environments or have no room for failure would need to carefully examine risks and benefits of allowing these high contact stresses. Useful solutions for standard problems dartmouth college. A b characteristic area perpendicular to compressing load acting. R 1 and r 2 are the principal radii of curvature of the two bodies convex positive effective curvature.
Elastic contact stress formulae newcastle university. Calculation of hertz contact stress between the deep groove ball bearing and inner race using analytical and mesys tool analysis. Damage phenomenon like pitching and wear will occur on tooth surface, and hertz formula 2 is generally. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of. Even though the derivation of the theory is relatively difficult, the final solution is a set of simple analytical equations relating the properties of the system to the developed stress. The maximum shear stress in a hertz contact is buried at 0. However, for this basic solid mechanics course, it is assumed that the shear load is uniformly distributed across the shear surface. Pdf hertz contact stress of deep groove ball bearing.
The application performs all calculations automatically. On the compression of a cylinder in contact with a plane. While normal stress is either tension or compression and acts normal to a particular axis. It can be used to calculate mechanical point and line contact dimensions and material stresses. Hertzian contact equations for elliptical, spherical and. Stresslife curves, sn continued common terms used with the sndiagram are fatigue life, n f. The number of cycles of stress or strain that a specimen sustains before failure occurs. Unusually high stresses may occur when a load is applied over a small area of contact. Calculating maximum contact pressure on a heavily loaded journal bearing p average load diameterxwidth e eef p hertz load x e eef. The problem is linear except the contact condition at the base which is modeled using the contact pair approach.
The proposed equations retrieve the hertz contact type and offer some realistic. Under the action of a load the solids deform and a contact area is formed as shown in figure 1. Take for instance this single lap joint where we have two plates connected by a rivet under load p. Example for bearing stress is a pin in hole under load and for normal stress is a beam in bending. What are hertz contact stresses and how do they affect linear. A device that supports, guides, and reduces the friction of motion between. Hertz contact stress calculations and graphs with hertzwin.
Hertzs analysis is applied only to surface stresses caused by a concentrated force applied. By theory of elasticity, developed by hertz, in 1881, when he was 23 years old. Mathematically bearing stress can be defined as the compressive load divided characteristic area perpendicular to it. Like bearing stress, it is also an average stress and the maximum shear will be higher.
In actual shear stress distribution, it is not uniform and the maximum shear stress will be higher than the average shear stress. The following equation will determine the contact stress between a sphere and a flat surface with known youngs modulus and applied load. Hertzian contact stress wolfram demonstrations project. Hertzwin is a free hertzian contact stress calculator. Calculating the contact stress resulting from lateral. Under these assumptions, only normal pressure acts between two bodies due to hertz contact producing normal displacements of the surfaces. Skf can provide a portfolio of different calculation methods to optimize bearing selection for any kind of application.
A bearing is a small machine element which constrain relative motion to only desire motion and reduces friction between moving parts. The depth of maximum shear stress was calculated only. Contact stress sphere in contact flat surface equation. The calculated stresses far exceed anything 6061 can withstand. Contact stresses two spheres in contact same material and differnet spherical radius. Ball bearing thrust bearing roller bearing contact stress causes deformation, plastic or elastic.
Useful solutions for standard problems preface modelling is a key part of design. In rolling bearing analysis, hertzian contact theory is used to compute local contact stresses. Most structures need to be designed for both normal and shear stress limits. Take turntable bearing 3645k for example, normal contact stresses between steel ball. For bolted joints without a preload shear, stress is calculated like bearing stress.
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