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Hertz Wheel-Rail Contact Stress Calculator engineering
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Hertz Wheel-Rail Contact Stress Calculator

Railway geomechanics & vehicle-track interaction: Calculate elliptical Hertzian contact patch semi-axes ($a, b$), peak normal pressure ($p_0$), subsurface shear stress ($\tau_{max}$), and rolling contact fatigue (RCF) shakedown index.

Wheel & Rail Geometry

Contact Patch & Stress Results

Max Contact Pressure $p_0$
--
MPa (peak normal stress)
Longitudinal Semi-Axis $a$
--
mm (contact patch half-length)
Transverse Semi-Axis $b$
--
mm (contact patch half-width)
Contact Patch Area $A_c$
--
$\text{mm}^2$ ($\pi a b$)
Max Subsurface Shear $\tau_{max}$
--
MPa (@ depth $z_{max}$)
Shakedown Ratio $p_0 / k_e$
--
RCF plasticity threshold

Contact Patch Ellipse & Subsurface Shear Depth

Contact Patch (Top View) a = -- mm, b = -- mm Shear Stress $\tau(z)$ Profile $\tau$ Depth $z$ z_max
Contact patch size is typically about the size of a coin (~150–250 mm²), concentrating axle loads up to 1,500 MPa. Peak shear stress is subterranean, triggering deep squat and head check microcracks.

Hertz Contact & Rolling Contact Fatigue Theory

The contact between a cylindrical railway wheel ($R_{w1} = R_w, R_{w2} = \infty$) and transversely curved rail head ($R_{r1} = \infty, R_{r2} = R_r$) forms an elliptical contact patch:

$$p_0 = \frac{3 Q}{2 \pi a b}, \quad E^* = \left( \frac{1 - \nu_w^2}{E_w} + \frac{1 - \nu_r^2}{E_r} \right)^{-1}$$ $$\tau_{\max} \approx 0.31 \, p_0 \quad \text{at depth } z \approx 0.48 \, b$$

Under Johnson's shakedown map, when the contact pressure ratio $p_0 / k_e > 4.0$ (where shear yield limit $k_e = \sigma_y / \sqrt{3}$), plastic ratchetting occurs on every wheel pass, leading to rolling contact fatigue (RCF) defects like squats and spalling.

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