help · WIRE - wires and sag

Method and formulas

How the sag calculator calculates: the catenary, the state-change equation, the catenary's shared load, the heights.

The calculator (WIRE - wires and sag) uses the method of the SBB tool Dh_Hf v250526; every formula below is the one that tool uses. The full method statement, with each derivation, is the document SAG_THEORY that comes with the tool; vdp Software Tools sends it on request.

Symbols: c span, h how much lower B is than A, x distance from A, y sag below A, g weight per metre, H horizontal tension, E elastic modulus, A section, α thermal expansion, ϑ conductor temperature. Index 0 is the known state, x the state calculated.

One wire

A wire carrying its own weight hangs as a catenary; its lowest point lies at a/2, where a is the horizontal span of the symmetric curve that contains the real one:

y (x) = \frac{H}{g} [\cosh (\frac{g a}{2 H}) - \cosh (\frac{g (\frac{a}{2} - x)}{H})]

For the small sags of contact lines the parabola is enough - and its mid-span sag f is the familiar one:

y (x) = \frac{g x (c - x)}{2 H} + \frac{h x}{c} f = \frac{g c^{2}}{8 H}

The support forces follow from the catenary: the vertical force F_V loads the mast, the pull F_Z is the wire's resultant:

F_{V} = H \sinh (\frac{g a}{2 H}) F_{Z} = \sqrt{H^{2} + F_{V}^{2}}

How the tension follows the temperature

The wire's length from its sag must equal its unstressed length stretched by the tension and by the heat:

L_{x} - L_{0} = L_{0} [α (ϑ_{x} - ϑ_{0}) + \frac{H_{x} - H_{0}}{E A}] L \approx c (1 + \frac{g^{2} c^{2}}{24 H^{2}})

This gives a cubic in the new tension - it has exactly one positive root, the physical tension:

H_{x}^{3} + H_{x}^{2} (\frac{E A g_{0}^{2} c^{2}}{24 H_{0}^{2}} + E A α Δ ϑ - H_{0}) - \frac{E A g_{x}^{2} c^{2}}{24} = 0

The catenary: the load shared by messenger and contact wire

At 10 °C the contact wire is level and the messenger carries the whole weight g_K. In any other state the contact wire follows the change of the messenger's shape, and its tension H_F carries part of the load. Adding the equilibrium of both wires gives one equation - the messenger hangs like one wire with the tension messenger + contact wire under an effective load:

(H_{t x} + H_{F}) y_{T}'' = - (g_{K x} + g_{K} \frac{H_{F}}{H_{t 0}})

So the sags of messenger and contact wire are:

f_{T} = \frac{c^{2}}{8} \cdot \frac{g_{K x} + g_{K} \frac{H_{F}}{H_{t 0}}}{H_{t x} + H_{F}} f_{F} = \frac{c^{2}}{8} \cdot \frac{g_{K x} - g_{K} \frac{H_{t x}}{H_{t 0}}}{H_{t x} + H_{F}}

and the messenger's tension of an N-FL follows from the state-change equation of the VEM handbook (1975, p. 445), solved for H_tx:

{(\frac{g_{K x} + g_{K} \frac{H_{F}}{H_{t 0}}}{H_{F} + H_{t x}})}^{2} - {(\frac{g_{K}}{H_{t 0}})}^{2} = (\frac{24}{c^{2}} + {(\frac{g_{K}}{H_{t 0}})}^{2}) (α (ϑ_{x} - ϑ_{0}) + \frac{H_{t x} - H_{t 0}}{A E})

The N-FL table chains six states: the messenger alone, the contact wire hung (H_10, rounded to 10 N), the regulated catenary with the new and with the worn wire, and ice of 7 and 15 N/m at -5 °C.

The sag of a span

An inclined span is calculated from its virtual span L_T1, the span of the parabola whose vertex is the lowest point:

L_{T 1} = c + \frac{2 h (H_{t x} + Z_{f})}{c (g_{K x} + g_{K} \frac{Z_{f}}{H_{10}})} y_{\max} = \frac{L_{T 1}^{2}}{8} \cdot \frac{g_{K x} + g_{K} \frac{Z_{f}}{H_{10}}}{H_{t x} + Z_{f}}

For an R-FL both tensions are constant; only the load beyond the reference - ice ZL and the wear ZL_1, a negative load - is shared by both tensions:

κ = \frac{g_{K}}{F_{0}} + \frac{Z_{L 1} + Z L}{F_{0} + H_{F}} L_{T 1} = \frac{2 h}{c κ} + c y_{\max} = \frac{L_{T 1}^{2}}{8} κ

Design contact wire heights

After AB-EBV 2024 Art. 5.2.1 and 5.2.2, the heights are sums of allowances in mm:

h_{f, \min} = G_{f A} + k + b_{e} + f + H + f_{g} + t_{h u} + f_{u d} + f_{u v} + f_{F D m a x Z L} + f_{F D m a x 40}

h_{f, \max} = h_{f, \max, a b s} - (t_{h o} + f_{u d o} - f_{u v} + f_{h} - f_{F D m i n - 20}) f_{F D m a x Z L} = \frac{g_{z} c^{2}}{8 (H_{T S} + H_{F D})}

Numbers and rounding

The equations are solved exactly (bisection to 10⁻⁹ N); the N-FL table is rounded as the SBB tool does - the reference at 10 °C to 10 N, the table to 5 N, halves away from zero. The calculator was checked against twelve calculations of the SBB tool: every value is reproduced. Where the SBB tool is inconsistent, the calculator is not, and says so on the page.

References

Handbuch Energieversorgung elektrischer Bahnen, VEM Verlag Technik, Berlin 1975, p. 445 and formulas 7.14c, 7.20, 7.53, 7.66, 7.67, 7.75, 7.82.
Kiessling, Puschmann, Schmieder: Fahrleitungen elektrischer Bahnen, Siemens 1998.
AB-EBV 2024, Art. 5.2.1, 5.2.2, 5.9.2; SBB regulations 0161.1010.0011, 0161.1010.0012, 0161.1010.0201, 0161.1013.0004, 0161.1013.0005.1.