The following example explains techniques for modelling a simple cable-stayed bridge using the structural analysis programme MiDAS/Civil. A short introduction to the programme is given first. Then, the calculation of the ideal cable forces using the Unknown Load Factor function is demonstrated. Furthermore, a backward and a forward analysis are performed in Chapter 3.3.4 and 3.3.5. respectively.
3.3.1 The analysis programme
MiDAS/Civil is an analysis programme for modelling and analysing structural systems. It offers special features for analysing civil structures, such as box girder or composite bridges.
The programme provides inter alia the Construction Stage Analysis feature, which allows the creation of construction stages by changing the structural system. This function offers the addition or the removal of elements. The boundary condition can be changed and loadings can be activated or deactivated in order to simulate the real condition for each construction stage. It is capable of carrying out analysis reflecting the erection and the dismantlement (backward analysis) of structures. In the construction stage input, time steps can be defined to represent the loading and unloading time for load changes within a given construction stage without changes in the structural system.
For modelling structural systems, the programme offers various element types. The main elements used for the analysis of cable stayed bridges are General Beam elements and Truss elements or Tension-Truss Cable elements (catenary cable element). However, numerous other structural elements exist, too, such as the plate and shell elements.
Initial pre-stressing forces can be calculated through optimizing the equilibrium state. The calculation of the ideal cable prestressing forces by the optimization is restricted to the linear analysis as the different loadings are superposed. The initial cable pre-stressing forces are obtained by the Unknown Load Factor function and the initial equilibrium state analysis of a completed cable-stayed bridge.
Chapter 3: General description of a Construction Stage Analysis 38
For static systems, linear and non-linear analyses can be performed by using truss and cable elements. Geometric non-linearities can be considered by including P-delta effects in the calculation or by performing a large displacement analysis.
The programme also offers to consider structural non-linear behaviour in the construction stage analysis. In the Construction Stage Analysis Data, it can be defined to perform a large displacement analysis. It was tried to use these functions and to perform a non-linear construction stage analysis. However, when contacting the MiDAS support-team, it turned out that these functions were not available at the time of working out this thesis. The main e-mail contact with Mr. Lee from MiDAS is listed in the appendix. These details are reported in more detail in later chapters.
Furthermore, material behaviors as creep and shrinkage can be modelled. These features are generally explained and also tested in the construction stage analysis for a simple cable-stayed system in Chapter 4.5.
More information on this programme is given in the following chapters and can also be found in detail in the analysis reference [60] or in the on-line manuals [61].
3.3.2 Structural Data
The following structural system is used for the calculation in the example to illustrate a simplified modelling of a construction stage analysis. Table 3-1 gives a list of the input data which are assumed in the model.
Figure 3-4: Cable-stayed example
1 2 3 4 5
1
2
3
7
Data Value Data Value Area A Deck 4.38 m2 Area A Cable 1 0.0115 m2 Stiffens I Deck 0.92 m4 Area A Cable 2 0.0062 m2 Poisson Ratio υ Deck 0.3 Area A Cable 3 0.0208 m2 Modulus of Elasticity ECable /EDeck 5.25
Table 3-1: Input data
3.3.3 Unknown Load Factor
The first step of analysing a cable-stayed bridge, in order to perform a construction stage analysis, is to evaluate the ideal cable forces for the final structure under its self-weight. The general procedure of calculating these ideal cable forces by the Unknown Load Factor function in MiDAS is outlined in Figure 3-5. The function allows the determination of superposition factors for previously calculated load cases to obtain a prescribed state in the structure by combining those load cases. As far as a solution for realizing the user defined conditions exists, the factors will be calculated.
To determine the unknown load factors for each cable stay to achieve an ideal state, a unit pretension load is applied for each cable. Performing a linear analysis, the programme computes the influence on the structure due to each unit tension load. In the Unknown Load Data the unit load cases are then defined as an Unknown Load. Furthermore, the structural restrictions for e.g.
moment or displacement values, which are to be realized through the load factors in the combined load case, must be defined.
Step 1: Create a model of the structural system
↓
Step 2: Generate the load conditions for the dead load and a unit pretension load for each cable
↓
Step 3: Assign the dead loads and the unit loads to the elements
↓
Step 4: Define the load combination for the dead loads and the unit loads after the performance of a linear analysis
Step 5: Define boundary conditions which have to be fulfilled
↓
Step 6: Calculate Unknown Load Factors for each cable Figure 3-5: Flowchart for cable initial prestress calculation
In this example, constraints are specified to restrict the displacement of the main girder and the pylon. The displacements of Node 2, 3 and 7 are limited. In the same way, it is also possible to
Chapter 3: General description of a Construction Stage Analysis 40
define constraints, for example, for moment values or tension forces in the cables. Since the unknown load factors are calculated on the basis of the superposition of different load cases, truss elements must be defined for the cables; it is not possible to use cable elements.
Figure 3-6 shows the moment distribution under the structural self-weight and a unit pretension load in each of the cables. The figure clearly demonstrates that the structural performance can be improved by a higher pretension in the first cable.
Figure 3-6: Moment self-weight & unit pretension load [tonf]
In order to fulfil the defined restrictions, the load factors are calculated. Figure 3-7 illustrates the result table given by MiDAS.
Figure 3-7: Results of the Unknown Load Factor calculation
For this example, the factors for Cable 1, 2 and 3 are 5.829, 3.105 and 10.724 respectively. The values can be found in Figure 3-7 in the first column. Figure 3-8 shows the resulting moment distribution including the factors for the tension forces in the cable stays 1 to 3.
Figure 3-8: Moment self -weight & initial pretension load [tonf]
The distribution of the bending moments is changed into the direction of a continuous beam condition. Thus, the moment distribution is more equal and the maximum moment is reduced.
For the application of the Unknown Load function, the displacement has been restricted, whereas the main target in the structural design is an equal distribution of the bending moments.
However, the example shows the correlation of both parameters.
As the bridge turns into a more complex structure containing a higher number of cables, it becomes more difficult to define the essential restrictions, which are needed in order to calculate the most efficient ideal cable forces. Experience is helpful in this case. Care must be taken in the selection of sensible and unrelated restrictions. If a specified condition is in conflict with another defined requirement, a singularity will result and there will be no solution.
3.3.4 Backward Analysis
After the determination of the ideal cable forces, the principles of a backward analysis by MiDAS are explained.
In the previous calculation, only one element has been used per segment. In the backward and forward calculation the model is refined using five elements per girder.
The following table shows the steps carried out to perform a backward analysis for the described example.
Step 1: Define the initial cable forces (internal forces)
↓
Step 2: Define each construction stage and the names
↓
Step 3: Define the elements by group which are added or deleted
↓
Step 4: Define the boundary condition by group which are added or deleted Step 5: Define the loading conditions by group which are added or deleted
↓
Step 6: Define the elements, the boundary and the loading belonging to each stage Table 3-2 Flowchart for backward analysis
The construction stages which are defined for this simple example are shown in Figure 3-9. The numbering for the cables and the girder elements starts from the left to the right. In the backward analysis the complete structure is dismantled step by step. The initial cable forces,
Chapter 3: General description of a Construction Stage Analysis 42
which must be applied at the time of installing the cables in the forward analysis, can be obtained from the backward analysis before removing the corresponding cable.
Construction stage 0 Construction stage 1
-/- - remove support
Construction stage 2 Construction stage 3
- remove girder 1 - remove cable 1
Construction stage 4 Construction stage 5
- remove girder 2 - remove cable 2
Construction stage 6 Construction stage 7
- remove cable 3 - remove girder 3
Figure 3-9: Sequence for backward analysis
The calculated cable forces for each construction stage are used as external prestress loads for the forward analysis. The cable forces are shown in Table 3-3.
CS 0 [tonf]
CS 1 [tonf]
CS 2 [tonf]
CS 3 [tonf]
CS 4 [tonf]
CS 5 [tonf]
CS 6 [tonf]
CS 7 [tonf]
562.19 835.04 302.82 -/- -/- -/- -/- -/-
288.94 292.20 243.06 577.98 217.57 -/- -/- -/-
1000.06 1348.44 625.97 577.38 217.57 118.78 -/- -/-
Table 3-3: Cable forces [tomf]
A simple way to check whether or not the loads are correctly considered in the analysis is to control the reaction forces in the final state.
The applied loads are:
Ftot 11ton m ⋅80m
:= Ftot 880ton=
The reaction forces in the MiDAS programme are:
F1 86.43ton:= ⋅ F2 1414.28ton:= ⋅ F3:=−620.71ton
Control:
Ftot_Mi:=F1 F2+ + F3 Ftot_Mi 880ton=
The calculation is correct as far as the reaction forces are identical. The correct consideration of the influence of the construction sequence is not proved here, but will be shown later on.
3.3.5 Forward Analysis
In the forward analysis, the calculated cable forces from the backward analysis are applied as external initial tension forces. That means that the forces are treated as external loads to support the structure at the construction stage of installing and pretensioning the cables. In the case of internal forces, as used in the backward analysis, the tension values reduce, depending on the loads and the support of the structure based on its stiffness. The applied cable forces are shown in the next figure.
Chapter 3: General description of a Construction Stage Analysis 44
Figure 3-10: Applied cable forces [tonf]
The construction sequence in the forward analysis is the same as in the backward analysis. The next figure shows the moment distribution before adding the support on the left side of the bridge.
Figure 3-11: Moment distribution; forward analysis before adding the support [tonfm]
Even if the support is being added in the next step, there still is a vertical deformation in the system. It is possible to add a boundary condition to the deformed or the undeformed structure.
In both cases, the results are different from the target values as shown in Construction Stage 0 in the backward analysis. The problem can be solved by applying the reaction force F1 instead of the support. The resulting moment distribution for adding the support and using a force are illustrated in the figure below.
a) Construction stage 0 b) Construction stage 0
- adding support to the original structure - adding reaction force F1
Figure 3-12: Moment distribution in the last step of the forward analysis [tonfm]
In addition to the moment distribution, the reaction forces and deformations of both analytical methods (backward and forward analyses using the reaction force F1) are compared. The values are identical within a very limited range.
This simple example explains the purpose of performing a backward and a forward analysis to control the construction stages. It also demonstrates the importance of taking remaining deformations into consideration when structural groups are activated or deactivated and the boundary conditions are changed. Otherwise, it is not possible to perform a back- and forward analysis with satisfying results.
A more complex example is given in Chapter 4, illustrating the analysis of a cable-stayed bridge in detail.
3.3.6 Forward Method = Backward Method
As far as no creep or shrinkage is considered in the calculation, the obtained results from the forward and backward analyses are identical. However, in addition to the already mentioned problems and considerations which are required to perform a construction stage analysis, a few more remarks have to be given.
The backward method requires a stress-free situation that has to be achieved for the elements before their deactivation. In case of a support, it requires some movement or balance loads until the reaction becomes zero before its removal. A segment may be removed by applying its weight in an upward direction first to create a zero stress condition.
In MiDAS, the results of all prior construction stages are accumulated and applied to the current stage. Once activated, elements, boundary conditions and loads remain active until they are deactivated. When an element is being removed, the internal forces are internally imposed to the contiguous remaining elements in the opposite directions. Therefore, in MIDAS, it is not necessary to apply the loads in the opposite direction before removing the loads or the elements.
Figure 3-13: Discontinuity between two segments
Chapter 3: General description of a Construction Stage Analysis 46
Forward and backward analyses give different results in case both are applied straight forward. The forward method automatically applies some discontinuities at every joint between two segments as it can be seen in Figure 3-13.
The backward method assumes that the new segments are added to the previous stage in the tangential direction. Therefore, when comparing both results, there are differences in the deformed shape. The moment distribution stays the same as before, since the internal forces are not influenced by this circumstance. Using the Initial Tangent Displacement for Erected Structures option in MiDAS, the real displacement as well as the rotational angle is calculated for the elements installed in the following stage. This option allows to install new segments tangentially and to avoid any discontinuities.
In programmes which do not offer this function, it may be a solution to install all segments of the main girder in the fist stage and to apply only the self-weight in the corresponding construction stages. In this way, the segments are always considered tangentially to the already existing structure.