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Why can a cable not be modelled

A wet sweater (5kg ≅ 0,05kN) is hanging on a steel string which spans 5m. What would happen if we’d consider the the string as a beam and as a cable?

If we would consider the string as a pinned beam, the bending moment would be

    \[\frac{F\cdot l}{4} = \frac{0,05kN \cdot 5m}{4} = 0,0625 kNm\]

That seems small. But the string is only 5 mm in diameter (section modulus W = 12,27 mm³). This means that in order to carry the weight of the sweater, the string should withstand the stress \sigma of

    \[\frac{M}{W} = \frac{0,0625 kNm}{12,27 mm^3} = 5093 N/mm^2\]

Only 20 times more than S235 steel… We wonder were that gigantic strength is hidden. And yet it works…

The above example is a typical linear approach to a problem. Perhaps you noticed that we avoided the topic ‘deflection’. In the beam-approach the deflection \delta would be

    \[\frac{F \cdot l^{3}}{48 \cdot E \cdot I_y } = \frac{0,05kN \cdot (5m)^{3}}{48 \cdot 210 000 N/mm^2 \cdot 30,68mm^4} = 20,2 m\]

Think how much the string would have to elongate to deform that much (5m string deflecting 20m?!?). This question brings us closer to understanding nonlinear geometry… where large deformations theory is being used.

The string cannot elongate 20m as linear approach suggests, but it will elongate some for sure. This elongation will cause a tensile force in the string. In turn, this tensile forse will stabilize the structure, reducing deflections. Using literature, we can approach the tensile force and resulting stresses, deformations:

    \[\theta =\left ( \frac{F}{E \cdot A} \right )^{1/3}=1,316°\]

    \[N = \frac{F}{2 \cdot tan(\theta)} = 1,088kN\]

    \[\delta = sin(\theta) \cdot \frac{l}{2} = 57mm\]

    \[\sigma=\frac{P \cdot cos(\theta)}{A} =55,4 N/mm^2\]

It is more to believe the string will deform 57mm under a tensile stress of 55,4 N/mm², instead 20m under a stress of 5093 N/mm².

Conclusion

  • A cable doens not behave like a beam.
  • Cables cannot be modelled in Diamonds. Not with a beam, nor with a tie rod.
    There is no element in Diamonds supporting the tensile forces needed to describe the cable-behaviour. Diamonds is not suitable for large deformation theory.
  • Even though tensile forces arise in tie rods, you cannot load tie rods perpendicular to their axis in Diamonds. Diamonds will simply refuse to calculate the model.

(Source: Roark’s Formulas for Stress and Strain, ISBN 0-07-072542-X)

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