Procedure 4 Large Structures Construction Stage Analysis)
5.4.2 Basics of the Geometry Calculation
The ensuing description is made for internal tendons, where the geometry is specified via constraint points of the type normal. The procedure used for external tendons or tendon segments is partially different. The deviations are described in detail in chapter 5.5 “External Tendons”.
The drawing below shows an example for user defined constraint conditions (P1, P3, P2, P4, P5, F1, F3). In this example the tangent vectors at points P2, P4 and P5 are “free”, i.e.
not prescribed.
Note: The following figures are drawn in 2D. The presented polygons and curves are however general curves and polygons in space.
i No. of the constraint point
Li Length between two constraint points
Ei Fictitious E-Modulus of the “tendon members”
Pi Constraint points
Fi Prescribed tangent vectors with fixed angles at a specified constraint point Step 1:
The basic reference geometry is the polygon formed by the straight connections be- tween the constraint points. The tendon part between 2 constraint points is called “ten- don segment”. The straight connection forms a beam element (“spare beam”).
The first approximation of the tendon geometry is now calculated assuming a prescribed P1
P2 P4
P3
F1
F3
L1, E1 L2, E2 L3, E3 L4, E4
F2=Free F4=Free
P5
end rotation, and the resultant bending lines of the spare beams form the basic tendon geometry. The initially prescribed tangent directions are at the start and end points the directions of the connection lines to the next or former constraint point respectively.
The initial tangent direction in intermediate points is the median line between the 2 di- rections from the former and to the next point.
This initial geometry fits the requirement of continuous direction changes and the curve has at the beginning and at the end tangents in the direction of the spare beams and in the intermediate points in the direction of the median. The compensating bending line of the total spare beam sequence is the superimposed in the 2nd step to this initial geome- try.
The vectors and angles are presented in the figure below.
Step 2:
A compensation calculation is in the 2nd step made for the total sequence of the spare beams (bending line of a continuous beam). The tangent directions are in the calculation adapted to match the prescribed direction conditions and the minimum energy condi- tion. The minimum energy condition automatically implies, that the friction losses be- come a minimum, because both, the deformation energy and the friction losses are de- termined by the curvature integral over the tendon length.
The bending lines are calculated separately for the X-Y-plane and the X-Z-plane. The resulting curves are superimposed to the initial curve. The resulting space curve of the tendon geometry is then a cubic spline curve.
Bending line in the X-Y-plane (y is normal to the spare beam in the X-Y-plane):
: y
x o 3 3
2 2 1
0
0) ( )
( S S y x a a x a x a x
y − = = + ⋅ + ⋅ + ⋅
Bending line in the X-Z-plane (z is normal to the spare beam in the X-Z-plane):
: z
x o 3 3
2 2 1
0
0) ( )
( S S z x b b x b x b x
z − = = + ⋅ + ⋅ + ⋅
P5
P1
P2 P4
P3
V1, D1B
α2/2 α2/2
α3/2 α3/2
α4/2 α4/2
1
2 3 4
α5
α1
V2
V3
V4
V5, D4E
F3
D1E
D2B
D2E
D3B
D3E
D4B
In order to calculate the position of the tendon in intermediate cross-sections between the constraint points, the intersection between the analytically given space curve and the cross-section plane is calculated. The position s* of the corresponding point (S*) on the spare beam is the calculated by using the orthogonality condition.
The below presented figure shows schematically the geometry.
S* Any point along the “tendon members”.
P(S*) Calculated point of the tendon based on S* (in 3D) (see figure above).
s* Length from P1 to S* along the “tendon elements”
( ) *
....
* s1 s 1 P S
s = + + i− + i
Pi
Pi+1
s* S*
P(S*) (x/y)
x z y
P1
P2 P4
P3
F1
F3
s1=L1 s2=L2 s3=L3 s4=L4
P5
nodes
s
Longitudinal axis
Element X/L s
1 1 2 2
0 1 0 0.5
0.00 5.52 5.52 6.43
2 1 7.34
EL 1 EL 2 EL 3
Summary of the procedure:
• Determination of the stiffness matrices [kTMj] of the different segments and cal- culation of the fixed end forces of the different spare beams {pj}= [kTMj] * {dj}
• Specification of the additions stiffness terms [kZi] for the points with prescribed tangent direction and calculation of the equivalent forces: {pz,i}= [kZi] * {∆vi}
• Assembling the element stiffness matrices to the total matrix [K], considering the additional stiffness at the points, where the tangent direction is prescribed.
• Solving the equation system: { }p +[ ]K ⋅( )∆α =0
Notations in the above formulae:
i No. of the constraint point
j No. of the segment between 2 constraint points pj, pz,i Equivalent load vectors (fixed end values)
kTM.j Stiffness matrix of the segment j (straight spare beam)
kZ,i Rotation stiffness of the constraint point (with prescribed tangent direction) dj Deformation vector (element end rotations) of the considered spare beam
∆vi Prescribed deviation of the vector Fi from Vi.
∆α Resultant deviations of the final tangent directions from Vi. Fi Prescribed direction vector in the constraint point
Vi Vector in the constraint point; orientation in the median line direction of 2 subsequent spare beam directions
Straight Parts:
The above procedure is slightly changed in order to consider straight parts of a cable:
1. The vectors Vi at the beginning and at the end of the straight part are not pre- scribed in the median direction, but both in the direction of the connection line (spare beam direction). Therefore the initial geometry has in this part no devia- tion from the straight connection line.
2. The fictitious stiffness of the spare beam in the straight part is considerably in- creased compared to the other segments. This guarantees that the segment re- mains straight in the compensatory bending line calculation.
Attention has to be drawn to the fact that 2 straight segments must not be arranged im- mediately one after the other, and that the tangent direction in the straight part cannot be prescribed because it is unconditionally determined by the direction of the connection line.