In this chapter, the influence of the element type for the cable stays, as well as the analysis method shall be investigated. During the construction, the cables may be installed with low initial tension values or, as the construction proceeds, this situation can occur in some other stages. A low tension value produces a higher deflection of the cable, which directly influences its stiffness. The effect of various sags to span ratio, according to the calculation method, is examined.
For the following calculations a different structural system than in the examples given before is used. The new system is a harp type system as it can be seen below. The bridge is analysed under the distributed service load of 40 kN/m and various initial tension forces.
Figure 4-42: Structural system of harp type cable stayed bridge (dimensions in [m])
Table 4-23 gives the properties of the structural elements used for the model.
Element Area
Table 4-23: Property table for harp system
The cables shall be installed with an initial sag to span ration of 1/60, 1/80, 1/100 and 1/120 respectively. This can be achieved by applying a pretension force on the cable elements. The required pre-stressing values are given in Table 4-24. The calculation for the top cable with a sag to span ratio of 1/120 is given in the following. The value is controlled by a single cable model.
Calculation for the top cable, sad to span ratio R of 1/120:
From the cable area ACtop and the cable density γ the weight per metre cable length L is calculated.
ACtop:=677cm2 γ 78kN
m3 :=
wtop:=ACtop⋅γ wtop 5.281kN
= m
The projected length ltop of the cable is:
ltop:=3 45.7⋅ m ltop 137.1m=
With the angle cosα t.m.b. between cable and girder, the weight qtop is determined (load per metre based on the projected length).
cosαt.m.b.
3 45.7⋅ 612+ (3 45.7⋅ )2
:= cosαt.m.b.=0.914
qtop wtop cosαt.m.b.
:= qtop 5.78kN
= m
For a sag to span ratio R120, the maximal sag is given as R120 1
:= 120 ftop120 ltop R120:= ⋅ ftop120 1.143m=
The required initial pre-stressing value Ttop120 is calculated from the horizontal component Htop120 of the cable force S(x).
R=1/120:
Htop120 qtop ltop⋅ 2 8 ftop120⋅
:= Htop120 11885.9kN=
Ttop120 Htop120 cosαt.m.b.
:= Ttop120 13009.3kN=
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 144
Control of a single cable
In order to control this calculation and the MiDAS programme, the top cable is modelled separately. Fixed boundaries are applied to the upper node and a nodal load of Htop120 of 11885.9 kN is applied in horizontal direction at the lower node. In the Element Table for each element, a pretension load Ttop120 of 13009.3 kN is entered. Figure 4-43 shows the generated model and the calculated maximal displacement perpendicular to the undeformed cable.
Figure 4-43: Non-linear analysis of a single cable (cable 6 in the model Figure 4-42) [m] and [kN]
The maximal sag, which is calculated as follows, shows only a small divergence to the intended value ftop120.
fMiDAS:=1.0467m
fmax fMiDAS cosαt.m.b.
:= fmax 1.146m=
Furthermore the cable force at the lower end shall be compared. With the vertical distance of the two anchorage points htop, the cable force is calculated as
htop:=61m
S0 Htop120 1 htop ltop
wtop ltop⋅ 2 Htop120⋅
−
2 +
⋅
:= S0 12866.45kN=
The value obtained from MiDAS is S0_MiDAS 12854.44kN:= ⋅
The formulas used for the calculation assume a parable shape of the deformed cable and a maximal sag at the mid span, whereas the computer calculates on the basis of a more exact catenary model. The small discrepancies in the results obtained from the formulas and the computation may be influenced by these differences.
After the calculation is controlled with satisfying results, the pretension values given in Table 4-24 are applied for the different cases of sag to span ratio.
R=1/60 1/80 1/100 1/120 Table 4-24: Initial pretension according to the sag to span ratio [kN]
For a non-linear analysis, the effect of the initial cable sag, which is a function of the initial pretension of the cable, is shown in Figure 4-44. The differences between the deflection values for R=1/60 and R=1/80 are higher than those for R=1/100 and R=1/120. On the other hand, the differences between the initial cable forces are the same. For example, for the top cable, the increase in the initial tension is constantly 2168 kN. This behaviour shows the non-linearity of cables.
Figure 4-44: Deflected shape of the girder due to non-linear analysis and different initial tension [m]
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 146
Figure 4-45 shows the results from a linear analysis (truss elements), a linear cable analysis (Ernst truss) and those from a non-linear cable analysis (elastic catenary cable analysis). In the case of a linear analysis, especially the displacement values obtained for high and low sag to span ratio show a large discrepancy to the values determined by the elastic catenary cable analysis.
Figure 4-45: Comparison of deflected shapes,
Elastic centenary cable (point + joint line), Ernst truss (point), elastic truss (line) [m]
In practice, an initial sag ratio R less than 1/100 is usually used [13]. For these ratios, the results illustrated for the equivalent cable stiffness proposed by Ernst show close results to the catenary cable elements. For this reason, the method has been adopted by many investigators.
Nevertheless, judging from the assumptions used in the theory, the method holds in limited cases for cables with low sag to span ratio. Therefore, particularly in order to perform more reliable and reasonable analyses for each stage of the construction, a more exact method should be used instead of using the Ernst truss elements.
However, at the time of working out this document and performing a construction stage analysis for the Second Jindo Bridge, MiDAS did not offer the option of using catenary cable elements, which would imply the performance of non-linear analyses included in the construction stage analysis. A non-linear analysis is not possible in combination with a construction stage analysis.
The MiDAS support stated that they are trying to solve the related problems and they may be able to present a version providing these functions in a couple of month (see appendix).
For that reason, for a final structural analysis, the decision is whether to choose a linear analysis with truss elements or cable elements in a linear analysis which considers an equivalent stiffness.
From the results given above, it seems to be reasonable to use the Ernst truss formulation.
However, it must be mentioned that, in order to obtain the above given reasonable results, all loads have to be stored in one load case. In the construction stage analysis various loads are
-0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4
Deflection [m]
R=1/60 R=1/80 R=1/100 R=1/120
Tower CL
removed and activated during the different construction stages. It must be proved wheter the loads, which are activated and deactivated in one construction step, are treated separately or if they are automatically stored in one load case. The last condition is a requirement in order to use the Ernst truss elements in a construction stage analysis; otherwise, the results will not be reliable.
In order to prove the approach used by MiDAS to consider loads in a construction stage analysis, the cable system of the harp type is investigated in more detail. A simplified construction stage analysis is modelled, consisting of 11 steps. The cable stays are modelled by Tens–Truss Cable elements (Ernst truss). In the first stage, the pylon, the elements of the left side of the bridge and the first segment of the main span are activated. In the following stages, the remaining segments and cables are added separately step by step. For each construction stage, the self-weight of the structural member plus a construction load of 40 kN/m is applied. The following three different calculations are performed:
1. The self-weight and the construction load are stored in one load case
2. The self-weight is stored in one load case. The construction load is splitted in two extra load cases, all loads are activated the same day
3. The different loads are modelled as given in 2. Additional time steps are generated in the construction stages and for each stage, the loads are activated on different days for the same structural system.
The examined results show the same values for the first two cases. It signifies that the load activation for each construction stage is treated as one load case. Consequently, the equivalent stiffness is calculated on the basis of the load changes in the system, which means that Ernst truss elements can also be used in a construction stage analysis. However, the results produced by the analysis of the third case are highly different. Therefore, in order to use the Ernst formulation, the loads must be activated on the same day. Load activations on different days are treated as different load cases and the results are superposed, which causes wrong results.
During the step by step construction process, the stresses in the already erected cable stay change due to the different loading conditions. Thus, it must be finally proved if the effective stiffness of the cables is adapted to the various tension forces throughout the analysis. To investigate this condition, the generated construction stage model of the harp type is used for further calculations.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 148
Cable 4 is erected with a pretension load of 800 kN before Cable 3 is activated in the next stage with an pretension force of 1500 kN (bottom cables). The tension forces in these tow cables are compared using truss and cable elements (Ernst truss) for the stay cables.
In the linear truss model, the maximal tension load in Cable 4 is 818.95 kN after installing the cable. Due to its self-weight, the load is higher than the initially applied 800 kN. However, the load increases to 1488.49 kN after the erection of Cable 3.
Erection of cable 4 Erection of cable 3
Figure 4-46: Cable installation in the linear truss model [kN]
In the Ernst truss model the same initial pretension loads are used, but the tension value in Cable 4 is only 1390.16 kN after the activation of Cable 3. Because of a low initial pretension, this great difference is caused by the reduction of the cable stiffness due to the sag effect.
Erection of cable 4 Erection of cable 3
Figure 4-47: Cable installation in the Ernst truss model [kN]
With the tension force obtained from the Ernst truss model, the effective stiffness Keff can be recalculated. In order to achieve the same stiffness in the elastic truss model as in the Ernst model, the modulus of elasticity can be changed in the input data.
In the first analysis (Truss E1), the new modulus of elasticity Enew, which is applied in the truss model, is recalculated form the initial cable force of 818.95 kN.
With the tension force Sb (b stands for bottom cable) and the cable area ACb, the modulus of elasticity Eb, the weight per metre cable length wb and the projected length of the cable lb,
Sb:=818.95 kN⋅
the effective elasticity can be calculated using Formula 3-34
Etanb 1
The value Enew is entered in the truss model. Because of the changed elasticity, the cable force is reduced from 1488.49 kN to 1184.36 kN. Compared with the tension forces in the Ernst model, the cable force is too low, which means that the stiffness is too low.
For the second analysis (Truss E2), a new equivalent E-modulus is calculated with the tension force of Cable 4 after the erection of Cable 3. The new calculated modulus of elasticity is given in the result table below. Because of the higher tension force, the value Enew increases as well.
Table 4-25 gives tension forces obtained from the different calculations and the applied Table 4-25: Tension forces in cable 3 & 4 due to adapted stiffness
For the truss and the Ernst model, the result of the second analysis shows very similar tension forces for both construction steps. Since the effective stiffness and the tension load in the cable affect each other, the erection of Cable 3 will influence both of it in the already erected Cable 4.
Application of the same effective stiffness for Cable 4 in the linear truss model, as calculated from the tension force in the Ernst model after the installation of the second cable, results in the same tension. Therefore, it can be concluded that the effective stiffness of the Ernst truss elements is recalculated for each new construction stage considering the load changes in the stay cable. As a result, it seems to be possible to use the Ernst truss formulation as implemented in MiDAS in construction stage analysis.
Chapter 5: Model of the Second Jindo Bridge 150
5 Model of the Second Jindo Bridge
The following chapter describes the modelling and the construction stage analysis of the Second Jindo Bridge. After the description of the considerations made in order to generate an appropriate model for the analysis, the boundary conditions and the assumed loading during the construction and for the final state are given in detail. The unknown load function, as described above, is applied for attaining the ideal cable forces. Using various structural restrictions, different solutions of possible cable forces are determined. A backward and a forward analysis is performed and the results are compared. The influence and problems of the double activation of elements and the boundary activation is discussed. Furthermore, for the final analysis, the effect of the non-linearity of the stay cables is considered. The obtained results are compared with other independent construction stage analyses. The camber data is provided to achieve a construction with no deformation under permanent loads after the completion of the erection.
Additionally, in order to ensure a safe erection process, the occurring maximum and minimum stresses are controlled. The calculation of the unstressed cable length is exemplified. Finally, a closing conclusion on modelling the erection process of the Second Jindo Bridge is drawn.