Hertz Contact Stress

[harrai]

HI All:

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.Actally I comapred with material bearing stress & calculated stress was less that bearing stress(got margin of saftey 2), but I am not sure whether to comapre this with the baering stress so please some body tell me to which allowable stress to compare this contact stress.

Well to calculate the Hertz conatact stress i used the folowing formula.

stress=.591*SQRT(p*E/Kd)

thanks

 

[prost]

A couple of things to be careful:
1) the definition of "E" is often non standard. Meaning, it is possible that the "E" in this equation is not Young's modulus, but in fact an effective modulus E*
1/E*=(1-nu1^2)/E1+(1-nu2^2)/E2; nu1,E1 are Poisson's ratio and Young's modulus for the material of body 1, nu2,E2 are for the material of the other body.

I am very familiar with various formulae for contact, but I don't know this particular formula stress=.591*SQRT(p*E/Kd). What is "K"? Hertz contact stress is normally defined as "pressure=p0*sqrt(1-(x/a)^2)," where p0=(2*P)/(pi*a), '2a' is the contact width, P is load per length (that is, units are force per length).

2) The formulae for Hertz contact are valid only if the two bodies in contact are convex relative to each other. For instance, the formulae work for a cylinder in contact with a flat plate, or a cylinder in contact with another cylinder. However, they are not valid for a cylinder in contact with a groove, that is, when the two bodies are concentric--for instance, for a roller bearing in a race (is that the correct term, 'race'?).

 

[zekeman]

Quote". However, they are not valid for a cylinder in contact with a groove, that is, when the two bodies are concentric--for instance, for a roller bearing in a race (is that the correct term, 'race'?). "

The fact is that the formula quoted by the OP is almost correct and is valid for convex as well as concave interfaces. From the literature I get
Stress=0.564sqrt[(P(1/r1+1/r2)/{(1-mu1^2)/E1+(1-mu2^2)/E2}]
and for the case presented r2 is negative since it is convex and is the race radius and r2 is the cylindrical bearing radius
mu1, mu2 are the Poisson ratios
E1, E2 are the moduli of elasticity
To answer the OP question, that stress is the maximum stress developed and if it is less than the allowable bearing stress by a factor of 2 you should be OK.

 

[cbrn]

Hi,
AFAIK, Zekeman is correct.

Regards

 

[harrai]

Hi All

Thanks for ur reply.E is actually modulus of elasticity when both the material have same modulus of elasticity & poissson's ratio.I think kd is equivalent dia. These formuale are listed in Roarks formual for stress and strain book under chapter "Bodies under direct bearing and shear stress".

But if I want to compare this contact stress to shear stress then what would be the ralation.

contact stress=.591*SQRT(p*E/Kd)

thanks

 

[dimjim]

Check out some of the work by Arvid Palmgren.
You also need to check stresses at different
depths from the surface to check the orthogonal
stresses that will cause failure. Contact stress
is only one component that must be examined.
I think they can be in the magnitude of three
times the allowable tensile strength of the materials
based on the hardness of the materials.

 

[prost]

Looking in Johnson, Contact Mechanics (don't have the translation of the original German manuscript by Hertz, so I presume Johnson's text is accurate), it is apparent that what Johnson calls 'conforming contact' is the same as the 'concentric contact' here. Conforming contact is not Hertzian, as conforming contact violates a key assumption in the original derivation, namely, the size of contact region has to be 'small' relative to the characteristic dimension of the two contact bodies. Conforming contact clearly violates this condition for small initial gaps, as even for small loads, the contact region quickly spreads from the initial point of contact, thereby violating the "half space" approximations Hertz used.

Johnson, Section 5.3 "Conforming surfaces in contact frequently depart in another way from the conditions in which the Hertz theory applies. Under the application of load the size of contact area grows rapidly and may become comparable with the significant dimensions of the contacting bodies themselves...When the arc of contact occupies an appreciable fraction of the circumference of the hole neither the pin nor the hole can be regarded as an elastic half-space so that the Hertz treatment is invalid...."

Some of the Hertz formulae may work over a limited range of conditions, nevertheless in general these Hertzian formulae are not valid in general in conforming contact conditions.

I checked my old copy of Roarks, and the formula for "Max sigma-c" appears as 0.591*sqrt((p*E)/KD) in Table 33, where the Poissons ratio of each body is 0.3, and each has the same E. The Roarks table has a couple of drawings to the right, showing how "KD" constant is defined, and there is a drawing of concentric or conforming cylinders. It's curious to me that no references were given for this particular formula for KD. The text before Table 33 talks a bit about alternate sources for bearing problems like this, I would consult those.

 

[Bizcocho]

Hi,

I have seen that normally the maximum shear stress is about 0.3 of maximum contact pressure at a aprox depth of 0.786b (b is the width of the contact surface). Then, you need to apply a theory of failure. If you use the maximum shear stress theory for ductile materials, you need to compare your value with respect to the maximum tensile stress/2.