Dynamic Analysis

The road suspended bridge is a special example of structure, therefore in practice it may be exposed to various dynamic loads, for example:

— influence of wind and rain,

— forces from a sudden movement of vehicle,

— vehicles colliding into bridge’s structure,

— vibrations induced by a crowd of pedestrians,

— earthquakes, etc.

These problems are very complex and their analysis requires comprehensive knowledge, expe-riences and especially an engineering instinct. However not only the above-mentioned types of loads should be included. In the age of uncertain times, terroristic threat is equally likely as the earthquake, therefore dynamic analysis in this kind of objects seems to be necessary. In research we decided to consider a case of a sudden hit of a constant force during 40seconds. The impulse with the value of 10000kN is applied at the top of the pylon, parallel to the longitudinal axis of the bridge (cf. Fig. 2.35). We justify this choice by the fact, that the loads given this way may cause the largest damage of the structure, in our opinion.

t

s

0 10 20 30 40

f(t)[kN]

10⁴

Figure 2.35 Dynamic force

The eigenproblem is solved for the first 12 eigenpairs and convergence is reached at the iteration step no. 11. The number of most dominated eigenvalues is 12. Maximum number of iteration required during the data processing is adopted as 20. The tolerance convergence is equal 1e-05.

The results of the first seven circular frequencies and their corresponding periods are presented in Table 2.10, compare [64].

Dynamic analysis is made by the mode superposition method with 10000 time steps 1t with 0, 004s, each. The obtained results are presented for two approaches, with and without in-cluding the modal damping coefficient λ = 0.01. Dynamic longitudinal displacements are the largest for the top of the pylon (node 635), where the impulse of excitation is put. This seems to be natural for two reasons, the time-dependent force acts on this node and it is the highest point in the pylon, that is fixed in the ground. What was predictable, under the dynamic force the most significant vertical vibrations occur in the middle of the suspended part of the plate, while in other points are slight. The static analysis shows that the largest z-axis displacement takes place in the middle point of the plate — node 315, however the most significant dynamic movement turns out to be in node 358, therefore this point is chosen to present further compu-tation results.

Table 2.10 Circular frequencies and periods of the undamped system Mode Circular frequency Period

number [rad/s] [s]

1 4.68 1.34

2 4.97 1.26

3 7.14 0.88

4 7.77 0.81

5 8.06 0.78

6 10.60 0.59

7 11.52 0.55

Fig. 2.35 shows the time-dependent vertical displacement for the chosen node of the plate and longitudinal displacement for the selected point of the pylon. Looking at these graphs, the spe-cific course of vibrations can be noticed. Namely, the amplitude is changing periodically in time. This may indicate the presence of the beat effect in our model. This phenomenon is often observed in structures with regular, repetitive segments of geometry and for this type of objects it is a consequence of overlapping vibrations with almost equal frequencies. Considering the results from Table 2.10, it can be seen that the two adjacent frequencies have very similar values.

From the point of view of the material fatigue the beat phenomenon may be treated as an un-desirable effect, and therefore in some cases it must be eliminated. It is commonly known, that if we consider the work of a real structure it is impossible to observe the time-dependent displacements without damping influence, because of many factors that are involved in. During the numerical computations, the elements that inhibit the vibrations are taken into account by the modal damping coefficient λ = 0.01.

Looking at the Fig. 2.35, one can mistakenly draw a conclusion about damping being sufficient to eliminate the beat effect. However if we see at the graphs that present time-dependent in-ternal forces (compare figs. 2.31-2.32), it is noticeable that the use of coefficient λ results in gradual decay of vibration but the periodical changes of the amplitude are still present. Further attempts to avoid the beat phenomenon are taken by analysis of the symmetrical bar dome and are described in details in section 3.6.

Fig. 2.37 shows the internal forces for the selected beam elements in the pylon. The component no. 10 is in one third of the height measured from the fixed end in the ground, while the 64’th is on the top.

An Example of Deterministic Analysis of Cable-Stayed Bridges • 59

a)

b)

!

Figure 2.36 a) Vertical displacements at the mid point of the plate; b) Longitudinal displacements at the top of the pylon

a)

b)

Figure 2.37 Dynamic internal forces in selected pylon’s beam elements a) shear forces; b) YY-bending moments

An Example of Deterministic Analysis of Cable-Stayed Bridges • 61

a)

b)

Figure 2.38 Dynamic bending moments in selected plate’s elements a) XX; b) YY

Figure 2.39 XY-torsion moments in selected plate’s elements

As it is seen in the graph, the shear forces are larger for the members which are located near the basis, however the difference between the values is not so significant. The same regularity can be observed in the case of the second type of internal forces, except that the bending moment for the beam component lying at a lower height is several times greater than at the top. Obtained result seems to be obvious due to higher distance between the 10’th element and the point of application of the dynamic load.

In Figs. 2.38 and 2.39 are presented the comparisons of the internal moments received for two plate member, from the middle of the suspended part of the span. 315’th element is adjacent to the longitudinal axis of the bridge while the 311’th is on the edge of the plate. Looking at fig. 2.38 we can conclude that the XX- and YY-bending moments in cross-section of the plate are larger for the components that are lying closer to the center of the span. In contrast to the previously described internal forces, XY-torsional moment, exposed in fig. 2.39, achieves the maximum values for elements placed on the edge of the plate, while the minimum for the mid-dle part of the span.

As it is shown in the mentioned graphs, the XY-torsional moment’s values are twice smaller than bending one’s. The described internal forces for the span behave the same as for normal plate in a complex stress state, namely, the larger displacements in nodes the higher values of bending moments, the smaller distance of the considered element from the symmetry axis of the span the lower torsional moments.

An Example of Deterministic Analysis of Cable-Stayed Bridges • 63