Discrete analysis of elastic cables/Lanksciuju lynu diskrecioji analize/Elastigu kabelu diskreta analize/Elastsete trosside diskreetne analuus.
Kiisa, Martti ; Idnurm, Juhan ; Idnurm, Siim 等
1. Introduction
A method is presented here to calculate an elastic cable using the
numerical discrete analysis and the basis of the analytical discrete
analysis is presented also. The geometrical non-linearity is taken into
account. The supporting nodes of the cable can locate on different
levels and the cable is loaded by static concentrated forces. The
following assumptions were made: the stress-strain dependence of the
material is linear, the cross-sectional area of the cable remains
unchanged and the flexural rigidity of the cable is not taken into
account. Focus in other studies may be on the utilization of cables with
flexural rigidity and developing the corresponding calculation methods
(Furst et al. 2001; Grigorjeva et al. 2004, 2010a, 2010b; Juozapaitis et
al. 2010).
Despite the fact that the calculations in the discrete method
require more computational power than in the continual model, it makes
it possible to apply all kinds of loads, such as distributed or
concentrated ones. A geometrically non-linear continual model is
especially useful for simpler loading types (e.g. a uniformly
distributed load). Both of the methods give quite similar results (Aare,
Kulbach 1984; Idnurm 2004; Kulbach 1999, 2007; Kulbach et al. 2002;
Leonard 1988). The biggest problem of the discrete analysis is the huge
amount of cubic and quartic equations that should be calculated.
Extensive simplifications have been made in previous studies to solve
this problem. This paper presents a new algorithm to increase the
accuracy of the calculation results.
Under the action of concentrated forces the cable takes the form of
a string polygon. Discrete analysis is based on the equilibrium of the
balanced condition composed for every nodal point of the cable.
Elongation of the cable is determined using the equation of deformation
compatibility for every straight section of the cable. These conditions
generate a nonlinear equation system, the solution of which gives all
node displacements and internal forces in the cable. The final solution
(displacements and internal forces) is found by describing the initial
and the final balance of the cable (before and after the loading).
2. Initial balance of the cable
The initial balance (state) of the cable is the situation before
deflection. The initial balance is marked with subscript "0".
The cable is loaded with concentrated forces, i.e. the cable takes the
configuration of a string polygon and the cable segments between the
nodal points are as straight lines.
Let us define a cable whose neighbouring nodes are denoted by
indices i - 1, i and i + 1 (Fig. 1). Let us observe the nodal point i of
the cable. The nodal point is in equilibrium under the action of the
internal forces of two consecutive cable segments and the external
concentrated load. Then the condition of equilibrium for the initial
state may be presented as (Kulbach, Oiger 1986)
[F.sub.0,i] + [[H.sub.0][[z.sub.0,i+1] - [z.sub.0,i]]/[a.sub.0,i]]
+ [[H.sub.0][[z.sub.0,i-1] - [z.sub.0,i]/[a.sub.0,i- 1]]] = 0, (1)
where [F.sub.0,i]--initial nodal load; [H.sub.0]--initial
horizontal component of the cable's internal force; [z.sub.0,i-1],
[z.sub.0,i], [z.sub.0,i+1]--initial ordinates of the cable nodes;
[a.sub.0,i-1], [a.sub.0,i]--initial horizontal distance between the
nodes.
For a cable that has supporting nodes on different levels,
[H.sub.0] is calculated as (Gimsing 1997)
[H.sub.0] = [a.sub.0,0][n.summation over (i=1)]
[F.sub.0,i]([L.sub.0] - [x.sub.0,i])/[[L.sub.0]([z.sub.0,1] -
[z.sub.0,0]) + [a.sub.0,0]([z.sub.0,0] - [z.sub.0,n+1])] (2)
where [z.sub.0,0] and [z.sub.0,n+l]--initial ordinates of the
cable's start-and endpoint; [z.sub.0,1]--initial ordinate of node
1; [L.sub.0]--span of the cable; [x.sub.0,1]--initial horizontal
distance between node i and the starting point of the cable;
[a.sub.0,0]--initial horizontal distance between node 1 and the starting
point of the cable.
3. Final balance of the cable
3.1. Exact analysis
After loading the cable with additional loads [DELTA][F.sub.i].,
the nodes have horizontal displacements [u.sub.i] and vertical
displacements [w.sub.i](Fig. 2). The condition of equilibrium in the
final balance of the nodal point i is
[F.sub.i] + [H[[z.sub.0,i+1] + [w.sub.i+1] - [z.sub.0,i] -
[w.sub.i]]/[[a.sub.0,i] + [u.sub.i+1] - [u.sub.i]]] + [H[[z.sub.0,i-1] +
[w.sub.i-1] - [z.sub.0,i] - [w.sub.i]]/[[a.sub.0,i-1] + [u.sub.i] -
[u.sub.i-1]]] = 0, (3)
where [F.sub.i]--final nodal load ([F.sub.i-] = [F.sub.0,i] +
[DELTA][F.sub.i]); H--final horizontal component of the cable's
internal force; [u.sub.i-1], [u.sub.i], [u.sub.i+1]--horizontal
displacements of the nodes; [w.sub.i-1], [w.sub.i],
[w.sub.i+1]--vertical displacements of the nodes.
There are three unknown parameters in Eq (3): u, w and H that need
extra equations to calculate them. It is done using the relative
deformation of the cable. The relative deformation of the cable's
segment has been found by using Hooke's law and the displacements
of the cable's nodal points. Equalizing them, the equation of
deformation compatibility obtains in the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [E.sub.c]--cable's modulus of elasticity;
[A.sub.c]--cable's cross-sectional area.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
3.2. Simplified analysis
An exact analysis in the final balance is complicated because there
is a need to calculate numerous cubic and quartic equations to find
[u.sub.i+1] - [u.sub.i] and there is no usable analytical solution. Eqs
(3) and (4) is simplified using the numerical analysis.
Provided that ([u.sub.i+1] - [u.sub.i]) << [a.sub.0,i], Eq
(3) takes the form of (Kulbach, Oiger 1986)
[F.sub.i] + [H[[[z.sub.0,i+1] + [w.sub.i+1] - [z.sub.0,i] -
[w.sub.i]]/[a.sub.0,i]]] + (5) [H[[[z.sub.0,i-1] + [w.sub.i-1] -
[z.sub.0,i] - [w.sub.i]]/[a.sub.0,i-1]]] = 0. (5)
Before Eq (4) is simplified, it is expressed as follows:
[t.sup.4][D.sub.4,j] - [t.sup.3][D.sub.5,j] + [t.sup.2][D.sub.6,j]
- t[D.sub.7,j] + [D.sub.8,j] = 0, (6)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[FIGURE 3 OMITTED]
To calculate Eq (6) it occurs that if ([u.sub.i+1] - [u.sub.i])
<< [a.sub.0,i], the following simplifications are suitable for use
(a satisfying result is attained if [[u.sub.i+1] -
[u.sub.i]]/[a.sub.0,i] <0.1; Fig. 3):
(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This simplification leads us to the following formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Taking into account that (Kulbach, Oiger 1986)
[n.summation over (i=1)] ([u.sub.i+1] - [u.sub.i]) = [u.sub.n+1] -
[u.sub.0] (8)
where [u.sub.n+1], [u.sub.0]--horizontal displacements of the
support nodes of the cable, Eq (7) may be written in the form of
[u.sub.n+1] - [u.sub.0] = [n+1.summation over (j=1)] [D.sub.13,j].
(9)
The final internal force of the cable and the displacements of the
nodes may be calculated using a one- or a two-level iterative process
(respectively steps 1...5 and 6...8 below). The solution algorithm is
presented bellow.
1. Use [F.sub.i] in Eqs (1) and (2) to calculate estimated H.
2. Use Eq (5) to calculate [w.sub.i].
3. Use Eq (9) to calculate [u.sub.n+1] - [u.sub.0].
4. Compare the calculated [u.sub.n+1] - [u.sub.0] to the exact
value (for example--if the supports are fixed, then [u.sub.n+1] -
[u.sub.0] = 0). If the difference between them is not small enough,
modify the value of H and repeat the calculation from step 2.
5. Use Eq (7) to calculate [u.sub.i]. The first level of the
iterative process is completed.
6. Here starts the second level of the iterative process. Take H
and [u.sub.i] from the first iterative process and use Eq (3) to
calculate [w.sub.i].
7. Use Eqs (7) and (9) to calculate corrected H and [u.sub.i].
8. Repeat steps 6 and 7 until H, [u.sub.i] and [w.sub.i] are
converged to the required precision.
If the deflections of the cable are relatively small (the
experiments and the calculations showed that the vertical deflection
should be less than L/200, then it is accurate enough to use the
one-level iterative process (further: the first simplification). It is
not recommended to calculate the horizontal displacements of the
cable's nodal points (except the cable's supports) using the
first simplification.
The full two-level iterative process (further: the second
simplification) requires a high computational efficiency, but if
([u.sub.i+1] - [u.sub.i])/[a.sub.0,i] < 0.1, this method is exact in
practice.
4. Numerical results
To characterize the behaviour of the cable under the concentrated
loads, the numerical results were calculated. For that purpose a cable
with a span of 50 m was chosen. Supports of the cable are on different
levels (the vertical distance between them is 15 m). In the initial
balance all 4 nodes of the cable were loaded by a concentrated load of
50 kN and in the final balance 100 kN was added. The cross-sectional
area of the cable was [A.sub.c] = 2228 [mm.sup.2] and the modulus of
elasticity [E.sub.c] = 1.25 x [l0.sup.5] N/[mm.sup.2]. The design scheme
of the structure in the initial balance is presented in Fig. 4.
Results of the calculation (vertical and horizontal displacements
of the nodal points and internal force of the cable) using the first and
the second simplification of the discrete analysis are presented in
Table 1 (Fig. 5). The geometrically non-linear behaviour of the cable
under concentrated nodal loads between 50...150 kN is illustrated in
Fig. 6.
5. Experimental investigation
The numerical calculation method presented in this paper was
checked experimentally. The cable structure is the same as presented in
Fig. 4. The scale of the model is [alpha] = 1/25 = 0.04. Parameters of
the model are presented in Table 2 and some pictures of the model in
Figs 7-10. The vertical displacements of the nodal points and the
internal force of the cable were measured.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
Comparison between the test results proved the reliability of the
model. Results among each testing varied at a max of [+ or -]1% from the
average. The average difference between single tests and mean results
was 0.0%. That was the main reason why only five tests were done.
Comparison between the average experimental vertical displacement
of each nodal point and the calculations was in the range of
+0.4...-5.1% using the first simplification and +2.7...-3.1% using the
second simplification (Fig. 11). The experimental horizontal component
of the cable's internal force was 6.7% and 5.7% smaller than the
calculations based on two simplified algorithms showed. Measured
displacements of nodal points were on average 1.9% and 0.2% smaller
(accordingly, compared to the first and the second simplification) than
the calculations predicted, which proves that the numerical calculation
method worked out in this paper is usable.
6. Conclusion
This article provides an algorithm to calculate internal forces and
deflections of an elastic cable using the discrete analysis. Three
solutions are presented--an exact analysis (analytical) and two
simplified methods (numerical). An experimental investigation was also
carried out to verify the simplified calculation methods.
Using the exact analysis in the final balance is complicated
because it leads to non-linear equations that have no usable analytical
solutions. The idea of a simplification is that some parameters that
have inconsiderable influence on the final result are eliminated from
the equations. As a result, all cubic and quartic equations are
transformed to quadratic equations.
The consequences of the numerical calculation methods.
1. If the vertical deflection of the cable is relatively small (the
experiments and the calculations showed that it should be less than
L/200, it is accurate enough to use the first simplification (one-level
iterative process) to calculate the vertical deflections and the
internal forces of the cable. It is not recommended to calculate the
horizontal displacements of the cable's nodal points (except the
cable's supports) using this method.
2. The full two-level iterative process uses simplifications only
in the equations of the relative deformation of the cable's
segments. If ([u.sub.i+1] - [u.sub.i])/[a.sub.0,i] < 0.1, this method
is exact in practice. The disadvantage of this method is that it
requires high computational efficiency.
The consequences of load-testing.
1. The test results verified the reliability of the test model. The
results of each test varied at a max of [+ or -]1% from the average. The
average difference between single test results and mean results was
0.0%.
2. Measured vertical displacements of the nodal points were on
average 1.9% and 0.2% smaller (accordingly, compared to the first and
the second simplification) than the calculations predicted. The
experimental horizontal component of the cable's internal force was
6.7% and 5.7% smaller than the calculations based on two simplified
algorithms showed. This proves that the numerical calculation method
worked out in this paper is usable.
The geometrically nonlinear numerical discrete analysis presented
in this paper enables adequate determination of deflections and internal
forces of the elastic cable. The numerical example demonstrated a very
good agreement between the results of both the simplified discrete
methods and the experimental investigation. The biggest advantage of the
discrete analysis is that it is easy to describe different load types
and load combinations. The most important disadvantage is the necessity
to calculate complicated systems of equations and very often these
systems converge slowly. Because the accuracy of the calculations is
high and the development of the digital computers is fast, the discrete
analysis has a significant role in the calculations of the long-span
cable-supported structures.
DOI: 10.3846/bjrbe.2012.14
References
Aare, J.; Kulbach, V. 1984. Accurate and Approximate Analysis of
Statical Behaviour of Suspension Bridges, Journal of Structural
Mechanics 3(17): 1-12.
Furst, A.; Marti, P.; Ganz, H. R. 2001. Bending of Stay Cables,
Structural Engineering International 11(1): 42--16.
http://dx.doi.org/10.2749/101686601780324313
Gimsing, N. J. 1997. Cable Supported Bridges. Concept and Design.
2nd edition. Chichester: John Wiley and Sons Ltd. 461 p.
Grigorjeva, T.; Juozapaitis, A.; Kamaitis, Z. 2004. Structural
Analysis of Suspension Bridges with Varying Rigidity of Main Cables, in
Proc. of the 8th International Conference "Modern Building
Materials, Structures and Techniques". May 19-21, 2004, Vilnius,
Lithuania. Vilnius: Technika, 469-472.
Grigorjeva, T.; Juozapaitis, A.; Kamaitis, Z. 2010. Influence of
Construction Method on the Behaviour of Suspension Bridges with Main
Rigid Cables, in Proc. of the 10th International Conference "Modern
Building Materials, Structures and Techniques". May 19-21, 2010,
Vilnius, Lithuania. Vilnius: Technika, 628-634.
Grigorjeva, T.; Juozapaitis, A.; Kamaitis, Z. 2010. Static Analysis
and Simplified Design of Suspension Bridges Having Various Rigidity of
Cables, Journal of Civil Engineering and Management 16(3): 363-371.
http://dx.doi.org/10.3846/jcem.2010.41doi:10.3846/jcem.2010.41
Idnurm, J. 2004. Discrete Analysis of Cable-Supported Bridges. PhD
thesis. Tallinn: TUT Press. 88 p.
Juozapaitis, A.; Idnurm, S.; Kaklauskas, G.; Idnurm, J.; Gribniak,
V. 2010. Non-Linear Analysis of Suspension Bridges with Flexible and
Rigid Cables, Journal of Civil Engineering and Management 16(1):
149-154. http://dx.doi.org/10.3846/jcem.2010.14
Kulbach, V. 1999. Half Span Loading of Cable Structures, Journal of
Constructional Steel Research 49(2): 167-180.
http://dx.doi.org/10.1016/S0143-974X(98)00215-6
Kulbach, V. 2007. Cable Structures. Design and Static Analysis.
Tallinn: Tallinn Book Printers Ltd. 224 p.
Kulbach, V.; Idnurm, J.; Idnurm, S. 2002. Discrete and Continuous
Modeling of Suspension Bridge, in Proc. of the Estonian Academy of
Sciences. Engineering 2: 121-133.
Kulbach, V; Oiger, K. 1986. Staticheskii raschet visiachikh sistem.
Tallinn: Tallinskii politekhnicheskii institut. 114 p.
Leonard, J. W. 1988. Tension Structures: Behavior and Analysis.
McGraw-Hill. 400 p. ISBN 0070372268.
Received 17 November 2010; accepted 27 April 2011
Martti Kiisa (1) ([mail]), Juhan Idnurm (2), Siim Idnurm (3)
Dept of Road Engineering, Tallinn University of Technology,
Ehitajate tee 5, 19086 Tallinn, Estonia
E-mails: (1) martti.kiisa@mnt.ee; (2) juhan.idnurm@ttu.ee; (3)
siim.idnurm@ttu.ee
Table 1. Numerical results of the example
Parameter Unit First simpl. Second simpl. [[S.sub.1]/
[S.sub.1] [S.sub.2] [S.sub.2]]
100%
H, N 1297733 1284067 1.1
[w.sub.1] mm 311.7 322.8 -3.4
[w.sub.2] mm 467.6 470.6 -0.6
[w.sub.3] mm 467.6 452.8 3.3
[w.sub.4] mm 311.7 288.1 8.2
[u.sub.1] mm 55.2 55.4 -0.4
[u.sub.2] mm 115.2 113.5 1.5
[u.sub.3] mm 147.8 140.2 5.4
[u.sub.4] mm 120.4 108.5 11
Table 2. Parameters of the structure and the model
Parameter Cable Coeff. Model
Over. Span 50 m [alpha] 2000 mm
dim. Height 15 m [alpha] 600 mm
Sect. area 2228 [[alpha].sup.2] 3.565
[mm.sup.2] [mm.sup.2]
Cable Mod. of 1.25 x 1 1.25 x
elasticity [10.sup.5] MPa [10.sup.5] MPa
Loads Init. bal. 50 kN [[alpha].sup.2] 80 N
Final bal. 150 kN [[alpha].sup.2] 160 N
Fig. 11. Comparison of max vertical displacements of
nodal points between experimental results and numerical
calculations (model scale 1:25)
Number of experimental theoritical-- theoritical--
nodal point simplif. 1 simplif. 2
1 12.5 12.5 12.9
2 18.6 18.7 18.8
3 18.3 18.7 18.1
4 11.8 12.5 11.5
Note: Table made from bar graph.
