Winkler Railway Track Modulus Calculator
Track superstructure & geotechnical design: Model rail bending as a Beam on Elastic Foundation (BOEF) per the Winkler-Talbot-Zimmermann equations.
Rail Section & Track Parameters
Track Structural Response
Max Rail Deflection $y_0$
--
mm (under wheel load)
Max Bending Moment $M_{max}$
--
kN·m (sagging moment)
Max Base Bending Stress $\sigma$
--
MPa (tension at base)
Max Tie Reaction $q_{tie}$
--
kN (rail seat load)
Characteristic Length $L_c$
--
meters ($(4EI/u)^{0.25}$)
Tie Load Distribution Ratio
--
% of wheel load carried by tie
Rail Deflection & Bending Moment Basin
The deflection basin decays exponentially ($e^{-x/L_c}(cos x/L_c + sin x/L_c)$), reversing into slight upward wave lifting adjacent sleepers before dampening.
Talbot & Zimmermann BOEF Classical Theory
The Talbot method models the rail as an infinitely long Euler-Bernoulli beam supported by a continuous elastic Winkler foundation of modulus $u$:
$$E I \frac{d^4 y}{d x^4} + u y = 0, \quad L_c = \sqrt[4]{\frac{4 E I}{u}}$$
$$y(x) = \frac{Q}{2 u L_c} e^{-x/L_c} \left( \cos \frac{x}{L_c} + \sin \frac{x}{L_c} \right)$$
$$M_{\max} = \frac{Q L_c}{4}, \quad q_{\text{tie}} = y_0 \cdot u \cdot s$$
Excessive deflection ($y_0 > 5 \text{ mm}$) causes rapid ballast pulverization, sleeper churning, and accelerated track geometry degradation.
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