Calculating Hertzian Stress: Equation and Unknown Terms Explained

[adrag10]

I'm trying to calculate the hertzian stress between a cylinder and rail. I've looked up the formula in Shigley and Mischke's "Mechanical Engineering Design 6th edition". In chapter 2 they give the equation to calculate the width of the area of contact(b) and the max load(Pmax). Once those are known, the equation for stress in the z direction is given as sigma=-Pmax/(1+(z^2/b^2))^.5

I got to that point. But I don't know what the z term is. It's not defined in the book. It's probably something silly I'm overlooking. Could anyone help?

Thanks

Al

 

[FredGarvin]

It looks like it is the radius measured in the z direction, but it's tough to tell without a picture. None of my references has the formulation you quote.

 

[adrag10]

I should have given the entire formula. Due to deformation, the area of contact is a narrow rectangle of width 2b and length l. The first equation calculates b.

b={(2F/pi*l)*[(1-v1^2)/E1 + (1-v2^2)/E2]/(d1^-1+d2^-1)}^.5

The diameter of the cylinders is taken into account here. For a cylinder rolling on a flat plane, d2 becomes infinity and so that term goes to zero. The second equation calculates Pmax.

Pmax= 2F/pi*b*l

At that point, I have the load and the area, so I thought it should just be a simple P/A calculation to get the stress. But instead, the formula given is:

stress=-Pmax/[1+(z^2/b^2)]

That is for stress in the z direction which is the maxima. This is for principal stress. There are two other fomulas given for sigma(y) and sigma(x). As for coordinate system, if you imagine two circles, one on top of the other as the side view for contact stress, the z coordinate is the vertical. The line of action for the forces pushing the circles together is the z direction. The y coordinate is the horizontal and is perpendicular to the z. The x-coordinate is coming out of the page toward you.

Al

 

[adrag10]

Or could some provide a simplified equation for Hertzian stress? Someone that has to calculate contact stresses must be using it at work.

Al

 

[FredGarvin]

In Advanced Mechanics of Materials by Cook and Young, they give a solution to a sphere on a flat plate (sphere with an infinite radius) as:

Contact width being 2a:

$$a = 1.109\left[\frac{PR}{E}\right]^{(1/3)}$$

The maximum pressure and thus the max stress, $$\sigma_z$$:
$$p_o = .388\left[\frac{PE^2}{R^2}\right]^{(1/3)}$$

Where:
P = Applied load
R = Radius of sphere
E = Young's Modulus

In Roark's Formulas for Stress and Strain a cylinder on a flat plate is given a max stress as:

$$\sigma_c = .798\sqrt{\frac{p}{K_D*C_E}}$$

The width of contact (b) is calculated using:

$$b = 1.60\sqrt{p*K_D*C_E}$$

If the two materials are identical the above reduces to:

$$\sigma_c = .591\sqrt{\frac{p*E}{K_D}}$$

$$b = 2.15\sqrt{\frac{p*K_D}{E}}$$Where:
p = Applied Load

$$K_D$$ = Diameter of the cylinder (in this case)

$$C_E$$ = $$\left[\frac{1-\nu_1^2}{E_1}\right] + \left[{\frac{1-\nu_2^2}{E_2}\right]$$

 

[vijidj69]

Good evening friends

my name is vijay from India i am doing my project on one way clutch so i am stuck in deriving hertz stress.

i started the deriving through TOE timoshenkao contact stress and now i am not able to understand in that equation po=.591sqroot ... and after using E Valvue of steel and
we get the value 2531 and later these value is converted to 1743.6 don't no how it was converted.

Kindly help me out through these.

Thanking you
vijay kumar
9964080943