SUSPNDRS: a numerical simulation tool for the nonlinear transient analysis of cable support bridge structures, part 1: theoretical development Page: 103 of 138

                                
ized with an estimation of the final cable forces in order to render the initial global stiff-
ness positive definite.
Once the appropriate initial length, linear cable segments are defined, gravity can be
turned on with full bridge dead load, and the geometry of the main suspension cables can
be obtained as indicated in Figure 48. Once the dead-load geometry of the cables is deter-
mined, the required length of vertical suspenders can be calculated. The length of each
vertical suspender cable can be calculated based on the deformed location of the main sus-
pender cable and the required elevation of the bridge deck. The required final suspender
length to achieve the specified deck geometry, denoted L fin, in combination with the load
carried in the vertical suspender P,, can be used to determine the initial, unstretched verti-
cal cable length,
Lo = Lf/(1 +   -v                          (EQ 401)
EC
Where Lo is the desired initial length of the cable, E is the effective modulus and Ac is the
effective aggregate area of one group of vertical suspenders. In utilizing EQ. 401, it is
assumed that each suspender carries the same vertical load, which is determined by tribu-
tary area dead load of the deck and stiffening truss. This assumption is valid because of the
construction sequence in which the truss joints are not rigidly coupled until much of the
dead load is applied to the suspension system, and thus the stiffening truss does not appre-
ciably effect the distribution of forces in the vertical suspension cables.
With the initial, undeformed length of the main suspension cables and vertical suspenders
determined, a dead load analysis of the entire bridge can be computed. Step three of the
model generation sequence consists of the bridge dead load analysis (Figure 48). The ini-
tial geometry for the dead load analysis prescribes the simple linear geometry for the main
suspension cables and each vertical suspender is provided with the appropriate unde-
formed length as calculated from EQ. 401. For the deck stiffening truss, the vertical post
truss elements are inserted with a geometric stiffness to ensure positive definiteness of the
global stiffness. "Virtual" truss elements are defined for the deck truss chords and diago-
nals for the dead load analysis. The virtual elements allow definition of the element con-
nectivities for these elements, but the element stiffnesses and element forces are not
included in the global matrices. This allows the deck truss system to deform without intro-
ducing stresses in the diagonals and chords of the truss. The virtual elements are effec-
tively "along for the ride" in the dead load analysis and contribute nothing to the global
stiffness. This insures that only the vertical posts of the truss will be stressed under dead
load, and that the vertical suspenders and posts will be perfectly vertical after the applica-
tion of dead load.
For the dead load analysis, the main suspender cables are master-slaved to the tops of the
towers only in the vertical direction. This ensures that the cables can slip horizontally rela-
tive to the tower tops and thus the towers will be perfectly vertical, without any horizontal
shear forces, at the completion of the dead load process.
The indicated procedure for the bridge dead load initialization guarantees that;

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