Simple mathematics of cables

Transversely and analogously loaded cable

For a cable spanning amid two supports the simplifying acceptance can be fabricated that it forms a annular arc (of ambit R).

Catenary cable diagram.svg

By equilibrium:

The accumbent and vertical reactions :

H = \frac{{wS^2}}{{8d}}

V = \frac{{wS}}{{2}}

By geometry:

The breadth of the cable:

L = 2R\arcsin\frac{{S}}{{2R}}

The astriction in the cable:

T = \sqrt{H^2+V^2}

By substitution:

T = \sqrt{\left(\frac{wS^2}{8d}\right)^2 + \left(\frac{wS}{2}\right)^2}

The astriction is aswell according to:

T = wR

The addendum of the cable aloft getting loaded is (from Hooke's Law, breadth the axial stiffness, k, is according to k = \frac{{EA}}{{L}}):

e = \frac{{TL}}{{EA}}

where E is the Young's modulus of the cable and A is its cross-sectional area.

If an antecedent pretension, T0 is added to the cable, the addendum becomes:

e = L - L_0 = \frac{{L_0(T-T_0)}}{{EA}}

Combining the aloft equations gives:

{\frac{{L_0(T-T_0)}}{{EA}}}+L_0 = \frac{{2T\arcsin(\frac{{wS}}{{2T}})}}{{w}}

By acute the larboard duke ancillary of this blueprint adjoin T, and acute the appropriate duke ancillary on the aforementioned axes, aswell adjoin T, the circle will accord the absolute calm astriction in the cable for a accustomed loading w and a accustomed affectation T0.

edit Cable with axial point load

Point-loaded cable.svg

A agnate band-aid to that aloft can be acquired where:

By equilibrium:

W = \frac{{4Td}}{{L}}

d = \frac{{WL}}{{4T}}

By geometry:

L = \sqrt{S^2 + 4d^2} = \sqrt{S^2 + 4\left(\frac{{WL}}{{2T}}\right)^2}

This gives the afterward relationship:

L_0 + \frac{{L_0(T-T_0)}}{{EA}} = \sqrt{S^2 + 4\left(\frac{{W(L_0+\frac{{L_0(T-T_0)}}{{EA}})}}{{4T}}\right)^2}

As before, acute the larboard duke ancillary and appropriate duke ancillary of the blueprint adjoin the tension, T, will accord the calm astriction for a accustomed pretension, T0 and load, W.