Static analysis

Because of the theoretical character of this dissertation, the static analysis is made with some simplification. Displacements and internal forces are obtained for the maximum combination

An Example of Deterministic Analysis of Cable-Stayed Bridges • 55

of the previously described loads: dead weight of the main parts of the system, moving loads of road bridge object, crowd and static wind pressures. In the case of real structure designing it would be necessary to include additionally: weight of the equipment elements, thermal im-pacts, installation load at different stages of construction, internal forces created as a result of changing the static scheme caused by damage of the cable, difference in supports subsidence, dynamic wind and rain influences (specific loads for the suspended bridge).

It should be noticed, that various combinations of the above cases need to be considered in real designing of this type of bridge. However the goal of this example is to show in details the dy-namic and sensitivity results for suspended bridge. Therefore, the statics is used to confirm the validity of the adopted scheme and cross-section areas of main elements. Because of symmetry of the bridge with respect to x axis, the results of displacements and internal forces obtained for the corresponding nodes should be equal.

Statics is the analysis which encounter many model problems associated with too large values of received displacements in the span. Using in the bridge only reinforced concrete plate sus-pended on the cables, makes the maximum vertical movement reach about 9 meters. To stiffen the main span, seven longitudinal rows of steel ribs were designed and spacing of transverse ribs ranges from 4 to 5 meters. Unfortunately, the treatment improved the situation but did not give fully expected results. By the council of experts instead of the reinforced concrete for the plate the modern composite material is used, with the properties assumed as follows: Young modulus — E = 100GPa, Poisson’s ratio — ν = 0, 25. The composite is generally the specific material strengthened by glass, graphite or carbon fibers. It is relatively expensive and therefore rarely used in civil engineering, but has much better mechanical and strength features and at the same time has a low specific gravity. However, in spite of the structural material change, deflection of the span was still to large.

Only assistance from the members of the Building Mechanic Unit in Szczecin West Pomeranian University of Technology allowed to find a solution of this issue. The large dimensions of the bridge cause the length of main cables reach even up to 150m. It turns out that the cables do not fulfill their function and work like springs. It proved necessary to apply initial tension of those structures to minimize the span’s displacements. To accomplish this, vertical forces imitating the initial tension were placed at the pylon and plate, by using the experimental method. Some methods about the modeling of bridges initial tension can be found in [46]. This treatment not only reduced the deflection of suspended part of the span, but also resulted in getting the greater part of the load by a plate’s fragment rested on supports. The summation of the final displacements in selected nodes obtained after assuming the composite plate strengthened by longitudinal and transverse ribs, is presented in table 2.7. The results include the implied initial tension to the suspended part of the bridge.

If we look at many model problems that were encountered in static analysis, we come to the conclusion that the initial up-deflection of the span, should be considered in this type of struc-ture. Properly matched bent arrow may reduce the values of vertical displacements. The results received for the nodes from plate that are given in table 2.7, confirm correctness of the adopted system — symmetrical points have the same values of movements. It can be also noticed that the closer to the fixed end of the pylon and the arches the lover values of x- and z-displacements are. On this basis we may conclude that the accepted model works properly. In tables 2.8 and 2.9 the internal forces obtained for the selected points in beam and plate elements are presented.

Table 2.7 Displacements of the representative nodal points in different part of the structure

Nodal X Y Z XX YY ZZ

point Translation Translation Translation Rotation Rotation Rotation

[cm] [cm] [cm] [rad] [rad] [rad]

a) pylon

635 9.411E-01 -9.150E-13 -4.451E-01 1.846E-16 3.228E-05 4.780E-16 610 7.061E-01 -4.390E-13 -3.888E-01 1.555E-16 1.005E-04 3.292E-16 580 2.969E-01 -9.370E-14 -2.321E-01 7.470E-17 1.229E-04 1.504E-16 b) plate — max. static displacements

309 1.745E-02 4.472E-04 -8.985E-01 -3.453E-02 -8.047E-04 0.0 310 1.744E-02 3.585E-04 -9.561E+00 -3.407E-02 -8.493E-04 0.0 311 1.733E-02 2.388E-04 -2.503E+01 -2.776E-02 -6.448E-04 0.0 312 1.722E-02 1.419E-04 -3.693E+01 -1.871E-02 -3.805E-04 0.0 313 1.716E-02 4.747E-05 -4.289E+01 -5.256E-03 -3.120E-04 0.0 314 1.715E-02 -1.188E-15 -4.350E+01 -1.090E-16 -3.207E-04 0.0 315 1.716E-02 -4.747E-05 -4.289E+01 5.256E-03 -3.120E-04 0.0 316 1.722E-02 -1.419E-04 -3.693E+01 1.871E-02 -3.805E-04 0.0 317 1.733E-02 -2.388E-04 -2.503E+01 2.776E-02 -6.448E-04 0.0 318 1.744E-02 -3.585E-04 -9.561E+00 3.407E-02 -8.493E-04 0.0 319 1.745E-02 -4.472E-04 -8.985E-01 3.453E-02 -8.047E-04 0.0 c) arches

682 -2.376E-01 -2.573E-01 -6.056E-02 2.221E-04 3.834E-05 6.001E-05 683 -2.376E-01 2.573E-01 -6.056E-02 -2.221E-04 3.834E-05 -6.001E-05 702 1.726E+00 -1.200E+00 -9.793E-01 2.551E-04 4.810E-04 3.456E-04 703 1.726E+00 1.200E+00 -9.793E-01 -2.551E-04 4.810E-04 -3.456E-04 724 1.582E+00 -4.747E-01 -1.064E+00 2.106E-04 -2.043E-04 5.677E-04 725 1.582E+00 4.747E-01 -1.064E+00 -2.106E-04 -2.043E-04 -5.677E-04

Summing up this part of computations, despite many problems that were encountered in creat-ing the model, the issue of inputtcreat-ing the initial tension of the cables in suspended bridges and increasing stiffness of the plate, is very interesting and gives many prospects in future works. It is undoubtedly worth to be developed in further research.

Table 2.8 Internal forces in the representative beam elements

XX YY YY XX YY XY

Element Axial Shear Shear Bending Bending Torsion

number Force Force Force Moment Moment Moment

[kN] [kN] [kN] [kNcm] [kNcm] [kNcm]

10 2.677E+04 8.474E+01 1.446E-12 -7.966E-07 -3.648E-07 2.299E+05 -2.677E+04 -8.474E+01 -1.446E-12 7.966E-07 3.648E-07 -2.095E+05 64 0.000E+00 -1.310E-10 7.782E-12 -3.078E-10 -2.134E-09 8.475E-08

0.000E+00 1.310E-10 -7.782E-12 3.078E-10 -1.223E-09 1.546E-07 121 6.956E+01 5.870E+02 3.146E+01 5.285E+04 -2.266E+05 -2.544E+05

-6.956E+01 -5.870E+02 -3.146E+01 -5.285E+04 2.002E+05 7.473E+05

An Example of Deterministic Analysis of Cable-Stayed Bridges • 57

Table 2.9 Internal forces in the representative plate elements

XX YY YY XX YY XY

Element Membrane Membrane Membrane Bending Bending Torsion number Stress Stress Stress Moment Moment Moment

[kN/cm²] [kN/cm²] [kN/cm²] [kNcm] [kNcm] [kNcm]

85 1.784E-03 5.365E-03 -1.845E-03 -1.548E+02 4.970E+01 -2.259E+01 195 1.890E-02 1.989E-03 -2.307E-04 -1.926E+02 -3.908E+02 1.713E+01 315 1.214E-02 1.307E-03 -1.187E-04 -1.500E+02 -4.923E+02 -5.341E+00 475 7.661E-04 -7.193E-04 -1.208E-04 -3.522E+02 -9.312E+01 5.104E+00