Objective
To perform and obtain the dynamic behavior of a stay cable and compare RM Bridge results with the formulation presented in SETRA – Recommendations of French inter-ministerial commission on Prestressing.
Structural Model
A stay cable fixed at two anchorages with a horizontal distance of 200 m and a vertical distance of 100 m apart was considered.
A parallel-strand cable stay with a resisting section S consists of 15 mm strands of fclass = 1770 MPa.
Formulation
By SETRA – Recommendations of French inter-ministerial commission on Prestressing, in chapter 8 the following formulation was verified.
A cable-stay can be considered to be a straight element tensioned to a tension F, with a lineic mass m and with negligible stiffness.
Transverse vibration of cable stays is studied here with co-ordinate system (Axy). Where x is the direction of chord [AB] and y is a lateral direction.
It is assumed that vibrations are of low amplitude and that tension F is constant. If the cable-stay damping is overlooked, lateral displacement y(x,t) is given by the fundamental equation of dynamics applied to an infinitesimal element dx:
or,
defining c, the celerity of transverse waves as:
The equation is solved by obtaining y(x,t) in the form e(x) modal deformation and f(t), which makes it possible to separate variables x and t:
, where w2 is an integration constant.
Assuming , the spatial equation can be written:
Boundary conditions y(0) = y(l) = 0 are used to deduce the form of the nth order mode:
and the wave number:
The temporal equation is written
Its general solution is
This yields the taut-string formula for eigen period Tn and nth – order eigen frequency Nn:
The free vibrations of the cable-stay are finally a superposition of sinusoidal modes of the form:
Schedule Action
Only one schedule action is run to calculate the Cable eigen frequencies and eigenvectors (natural modes). The reference load case containing the definition of effective masses must be specified.
After the calculation, a number of n load cases is presented and can be accessed from the load case pool, as ‘Outputfilename#n’, being n the eigenmode number. The load cases contain normalized eigenvectors as displacements that were used in graphic presentation.
Results
In the verification example studied, the Cable model was discretized in 10 (ten) sub-cable elements and in the Recalculate path “Non-linear Stay Cables” was selected.
The calculated and obtained results are the ones presented in the following table:
The correspondent deformations for the first four eigenmodes are presented below: