The permanent state of stress in a cable-stayed bridge subject to its dead load is determined by the tension forces in the cable stays. They are introduced to reduce the bending moments in the main girder and to support the reactions in the bridge structure. The cable tension should be chosen in a way that bending moments in the girders and the pylons are eliminated or at least reduced as much as possible. Hence, the deck and pylon would be mainly under compression under the dead load. In case of a concert deck, this reduces creep-induced deflections and the corresponding uncertainties in concrete members. Furthermore, second-order effects decrease [14]. The designer first selects the final condition and then evaluates the initial cable forces. The initial tension forces, which must be applied to the cable-stays at the time of installation, are determined by a precise construction stage analysis.
Chapter 3: General description of a Construction Stage Analysis 30
To estimate the cable forces, the simplest method is to assume a bridge segment to be a simple beam supported by cables. The method of the simple supported beam can be used in the preliminary design stage to estimate the cable area. Another method is to assume that, under the dead load, the main girder behaves like a continuous beam and the inclined stay cables provide rigid supports for the girder. The vertical component of the forces in stay cables are equal to the support reactions calculated on this basis. Virlogeux [56] describes the first procedure by the pendulum method: The cable tensions are evaluated from the key section to the pylon by balancing the load to produce no bending moment, e.g. at the cable anchorage. Distributing the weight of each segment at the two corresponding anchorages, it follows that the cable force Si is (Figure 3-1):
and from the equilibrium of horizontal forces, the normal force Ni:
i
Figure 3-1: Illustration of the pendulum rule [13]
Herzog [9]and Gimsing [8] developed easy hand-formulae to predict the cable forces by taking the stiffness of the girder and pylon into consideration.
Since the cable stayed structure is a highly undetermined system, there is no unique solution for calculating the initial cable forces directly. Usually it is an iterative process to find an economical solution.
As mentioned before, the moment and displacement distribution along the girder and the pylon can reach the ideal state by adjusting the cable stresses. Using vector and matrix calculations, the moment or the displacement of an ideal state I can be written as:
[
i i in]
TI = 1 2 ... (3-6)
n is the total number of the targets that need to be satisfied and T stands for the transformation of a matrix or a vector. The approach to the ideal state is to make Equation 3-6 as close to a designated value as possible. The result cable stresses S can be written as
[
s s sm]
TS = 1 2 ... , (3-7)
in which m is the number of cables to be adjusted.
By analysing the response of the unit prestress applied to each tuning cable, the influence values of all the targets can be obtained. When m rounds of the analysis are done, the influence matrix T can be written as: relation can be set down as
I S
T* = (3-9)
If the number of cables that are to be tuned is the same as the number of targets, the setting I to the designated target values, the cable stresses S can be obtained accordingly by solving the linear Equation 3-9. In this case, engineering experience is required to select the proper target values. In this method, m cannot be greater than n. If, as in most cases, m is less than n, the
Chapter 3: General description of a Construction Stage Analysis 32
cable stresses can be optimized so that the error of the target value and the designated state is kept to a minimum. The method of square error minimization is an effective way to obtain the optimal I. A is the adjustment value which has the same form as I. E describes the error between A and I, and can be written as:
I A
E= − (3-10)
The optimization of the cable stress is to minimize Ω, the square of E. As a definition, Ω can be written as:
(
A−I) (
T A−I)
=
Ω * (3-11)
The principal condition to minimize Ω is:
=0
∂ Ω
∂
Si ,i=1,2,3...,m (3-12)
Using the matrix differential and considering Equation 3-9 and 3-11, the following equation can be obtained:
A T S T
TT * = T (3-13)
After calculating S from the linear equation group in Equation 3-13, the optimized target value can be calculated by Equation 3-9. An example is presented in a later chapter. A very similar method can be applied for adjustments of errors during the erection process and is explained in detail in Chapter 3.7.
Many analysis programmes use an optimization method for determining the cables forces.
Under permanent loads, a criterion (objective function) is chosen in a way that the internal forces, mainly the bending moments, are evenly distributed and small. The deflection of the structure can be limited to prescribed tolerances, too.
Bruer and Pircher [17] favour a numerical approach to reduce the calculation effort, the Unit Load Method. For the final stage structure including its total dead load, unit load cases as well as the ideal moment diagram must be defined. Commonly, the selected unit forces are:
• A unit shortening of the cable or a unit tensioning causing an axial cable shortening
• A unit translation of a rigid support or an element. A transverse or longitudinal movement changes the moments in the deck by changing the cable forces which act on the deck.
The same number of unit loading cases must be defined as the number of “Fixed Moment”
points, chosen on the structural model to represent the ideal moment diagram. Figure 3-2 illustrates this procedure.
Figure 3-2: Unit Load Case Method for determining the ideal state 0
The ideal dead load bending moment diagram is defined for the deck girder. As shown in the figure, the bending moments for nine points along the girder are described from position A to I.
Nine unit load cases are selected for setting up the simultaneous equations – eight unknown stay cable forces and one unit translation at the end support.
In this case, the linear equation system follows to be:
9 8
1 8 1
1
1 *x ... M *x M *x
M M
MA = P + T = + + T = + TJ (3-14)
:
9 8
1 8 1
1
1 *x ... M *x M *x
M M
MI = P + T = + + T = + TJ
where MA to MI is the final stage moment at positions A to I including tensioning and movement at the end supports, MP the permanent load moment at the current position without cable
Chapter 3: General description of a Construction Stage Analysis 34
tensioning and support movement, MT1=1 to MT 8 =1 the bending moments due to each unit tensioning at position A to I, and MJ the bending moment due to unit jacking of the end support.
The solution of the equations, the unknown xi, is the factor by which the unit loads must be multiplied to achieve the defined moment distribution. This basic solution defines the cable forces and the jacking force for the final stage, but does not include the effect of any chosen construction sequence, creep, the 2nd Order Theory and the non linearity of the cables due to the sagging effects.
However, in the analysis programme RM2000, developed by Pircher, a method which allows the consideration of non-linear problems in the optimisation process is introduced. This approach is called the AddCon Method [43] [45] (Additional Constrain Method) and is an extension of the Unit Load Method. It is possible to include time effects (creep and shrinkage) and non-linear structural behaviours (non-linearity of cable elements and P-delta effects) in the calculation. An iterative process is used to appropriately factorise the user-defined unit loading cases so that the defined constrains are achieved. User-defined constrains can be a set of forces/moments, stresses or displacements or a combination of both, which must be fulfilled under the applied loading.
The analysis programme MiDAS also provides the Unknown Load Factor function, which is based on an optimization technique. Similar to the described method above, this can be used to calculate the load factors that satisfy specific boundary conditions (constrains) defined for a system. An example of this procedure is given in Chapter 3.3.3. There, it is explained in detail how to apply this method in the analysis of a structural system.