Procedure 4 Large Structures Construction Stage Analysis)
4.5.5 Use of the Load Types FX0, LX0 for Cable Stayed Bridges
The „Stressing Process“ for a cable corresponds physically to a „Shortening“ of the ca- ble. This shortening is simulated by applying an initial force. Several different Load Types are provided in RM2000 to simulate this process. They are specified in the func- tion "LOADS AND CONSTR. SCHEDULE #LOADS !LSET.
# Initial Stress/Strain % Initial normal force FX0
# Initial Stress/Strain % Stress free element length LX0
# Initial Stress/Strain % Uniform temperature load T
# Actions on Element End % Element end deformation VGA
# Stressing % Cable/external tendon stressing FCAB Only the the use of the recommended Load Types „FX0“ and „LX0“ is discussed in this chapter, the other types are typically used in other contexts.
4.5.5.1 Relation between FX0 and LX0
The system length Lsys of a cable element is generally defined as the straight distance between the start and end points (i.e. the start and end nodes if no eccentric connection is defined).
Stressing this cable with the force N0 yields an elongation ∆Lsys. Applying the Load Types FX0 and LX0 means placing this stretched element with the original length into the structural system, i.e. the elongation is cut away and the original (stress-free) length is assumed to be smaller than Lsys.
Lsys
N=0
L0≡LX0 ∆L0
FX0 = N0
∆Lsys
N0
Lsys
0 sys
0 EA N
L L EA
++++
==== ⋅⋅⋅⋅
) L L L ( 0 EA
FX sys 0
0
−−−−
⋅⋅⋅⋅
====
Relation: with: E ... E-Modulus of the cable A ... Cross-section area of the cable
L0 ... stress-free element length (=LX0)
Lsys ... un-deformed element length N0 ... initial normal force (=FX0)
The „initial normal force FX0“ (=N0) is defined as the force, which is required to stretch the cable with the original length L0 by the amount of ∆L0 in order to reach the system length Lsys, which is necessary to place the element in the un-deformed system.
Relation: with: E ... E-Modulus of the cabel A ... Cross-section area of the cable
L0 ... stress-free element length (=LX0)
Lsys ... un-deformed element length FX0 .. initial normal force
This shows that the 2 Load Types are totally equivalent.
The load type „initial normal force FX0“ is a further possibility of defin- ing a reduction of the original length with respect to the system length Lsys.
4.5.5.2 Relationship between FX0 and the final normal force N in the cable
One of most frequently asked questions is, why - after calculating a load case with FX0 - the resulting normal force is not the same than the initial force FX0.
The answer is, that this is due to the flexibility of the remaining structural system. In the special case, that the support on both ends of the cable is absolutely rigid, the applied normal force remains in the cable (i.e. FX0=N), otherwise the reaction of the structural system causes a relaxation in the cable and N will be smaller than FX0.
The following example simulates the flexibility of the structural system by one spring (with spring constant k) applied in one of the 2 nodes:
A length reduction is applied by using the Load Type FX0 to the element with the sys- tem length Lsys. This causes a force FX0 acting on the structural system represented by the spring. The system is deformed by this force (the spring is stressed), causing a re- duction of the actual length of the cable (Lakt < Lsys).
This means, that the effective elongation (∆L0,akt = Lakt - L0) is smaller than the applied initial value ∆L0. The remaining normal force in the cable is the force, which is neces- sary to strain the cable by the amount of ∆L0,akt instead of ∆L0.
Equilibrium exists when
( 0 0,akt)
0 akt ,
0 L L
L EA L
N = ⋅ ∆ −∆
∆
⋅∆
= k
We see, that due to the deformation of the primary structural system the initial normal forces FX0 does not anymore act in the cable, but a reduced force corresponding to the equilibrium condition.
If, as it is often the case, a predefined normal force should act in the cable after the stressing process (stressing process against the structural system), then it is necessary to use an iterative process to define the stressing load by LX0 or FX0. TDV provides in RM2000 the function "LOADS AND CONSTR.SCHEDULE #ADDCON (see chap.
6.11 Additional Constraints).
4.5.5.3 Load Type FCAB
Cable jacking simulation in RM2000 can also be performed by using the Load Type Lsys
N=0
L0≡LX0 ∆L0
FX0
N≠FX0
L0≡LX0 ∆L0,akt
FX0
right results without iteration, and the applied normal force will fully be in the cable after the load case has been calculated. However, this load type is not applicable in non- linear analyses, and – due to the fact, that stressing a cable influences the force in all previously stressed elements - for determining the required jacking forces with
"LOADS AND CONSTR.SCHEDULE #ADDCON to have a certain normal force in the cable after all cables have been jacked.
The following procedure is performed internally in the program: The stressed element is extracted from the system (deactivated) and replaced by the specified force acting at the begin and the end of the cable. This calculation does not give a result for the considered element itself, because it is not active. Then the cable element is – without a new redis- tribution analysis - with the prescribed normal force as result value again integrated in the non-deformed system.
A possibly known cable force can therefore be directly introduced in the structural sys- tem. Other than in function FX0, the specified force will be in the cable after the calcu- lation of the Load Case.
This Load Type must not be used for complex, non-linear calculations (espe- cially not for non-linear cable elements). Cable sagging, large displacements, 2nd order theory etc. are not compatible with FCAB. Due to the cable element being inactive, the consideration of the non-linear behaviour will be incomplete.
4.5.5.4 TDV Recommendation – conclusion
In the case the cable force is known (checking a certain situation), then FCAB is appli- cable if the non-linearity is not considered. In the case that the erection sequence respec- tively stressing strategy should be evaluated, then FX0 should be used together with ADDCON. FCAB is not applicable in this case, because every cable stressing Load Case is calculated on a different system and the interaction is not taken into account.
Also in the case of a non-linear behaviour FCAB should not be used, because the used cable end forces are related to the direction of the original cable and a rotation of the normal force is not considered.