Simplified engineering method of suspension two-span pedestrian steel bridges with flexible and rigid cables under action of asymmetrical loads/ Asimetriskai apkrauto kabamojo vienajuoscio dvieju tarpatramiu plieninio pesciuju tilto su lanksciaisiais ir standziaisiais lynais inzinerine skaiciavimo metodika/Vienkarsota aprekinu metode nesimetriskai slodzei paklautam divlaidumu lentveida sistemas gajeju terauda tiltam ar stingu un....
Sandovic, Giedre ; Juozapaitis, Algirdas ; Kliukas, Romualdas 等
1. Introduction
Suspension bridges, due to their technical performance and
excellent architectural appearance are widely used both for large and
small spans to overlap (Gimsing 1997; Ryall et al. 2000; Troyano 2003).
One of the oldest and successfully used to this day pedestrian
suspension bridges is a steel stress-rIbbon bridge (Schlaich et al.
2005; Schlaich, Bergerman 1992; Troyano 2003). The main carrying
elements of such modern steel bridges are steel bands or high-strength
steel wire ropes (Juozapaitis et al. 2006; Michailov 2002; Strasky
2005). Construction depth of the above type buildings is the lowest one.
According to the operational requirements sag values of such bridge
suspension carrying elements are relatively mean (Schlaich et al. 2005).
It induces high tensile forces formed inside load-carrying structures,
requiring great steel rates and conditions the anchored foundation mass
of such bridges (Katchurin 1969; Kulbach 2007). One of the most serious
drawbacks of the suspension bridges is their excessive deformations
caused by the impact of asymmetric loads (Juozapaitis, Norkus 2004;
Katchurin 1969; Kulbach 1999; Moskalev 1981; Wollmann 2001). There are
methods known to be applied in order to reduce the deformability of such
sus pension structures. Commonly heavy reinforced concrete decks or
prestressed concrete structures are used for such steel stress-ribbon
(Caetano, Cunha 2004; Schlaich et al. 2005; Strasky 2005).
Recently the multi-span steel stress-ribbon pedestrian bridges are
applied (Schlaich, Bergerman 1992; Troyano 2003). Various design
solutions are known for structural suspension systems (Strasky 2005).
Due to horizontal displacements of standing (medium) pier and wider
range of situations to be applied for temporary load calculations, the
behaviour of such structures becomes more complex. Without any doubts
this fact complicates calculation of the structures facing geometrical
non-linearities (Katchurin 1969; Tarvydaite, Juozapaitis 2010).
It shall be noted that absolute flexibility of a suspension cable
is a theoretical concept, since the above elements of real structures
have a particular depth cross section and bending stiffness of a finite
size (not equal to zero) (Furst et al. 2001; Gimsing 1997; Katchurin
1969; Moskalev 1981; Wyatt 2004).
It is known that in order to reduce the displacements of suspension
bridges induced by asymmetric and local loads the so-called
"rigid" suspension elements shall be applied (Grigorjeva et
al. 2010; Juozapaitis et al. 2006, 2010). These retaining elements,
combining tension and bending elements abilities, not just stabilize the
initial form of the bridge effectively, but allow "to prevent"
the application of expensive pretension or bulk reinforced concrete deck
(Juozapaitis et al. 2008). These "rigid" structural elements
are made of hot-rolled or welded steel cross sections. Due to the
potential stress concentration it is recommended to rest such suspension
bridge structures upon flexible, i.e. to design elements as split
(Juozapaitis et al. 2006; Kala 2008; Prato, Ceballos 2003).
It shall be noted that behaviour of multi-span steel stress-ribbon
bridges is not completely considered, particularly in view of supporting
element bending stiffness.
The article describes pedestrian two-span suspension split-type
steel stress-ribbon bridges with a bending stiffness, analyzes the
behaviour of such structures under the asymmetric load. It deals with
kinematic displacements of bearing suspension cable of such bridges, and
provides them in the form of displacement calculation analytic
expressions. The efficiency of steel stress-ribbon bridge displacement
stabilization through the bending stiffness is being discussed. Methods
of an engineering design of tension and displacement of the
asymmetrically loaded suspension bridge steel structures, evaluating the
impact of bending stiffness. Numerical experiments show the basis of the
accuracy of the developed simplified analytical method.
2. Simplified engineering method for kinematic displacements of
pedestrian two-span suspension split-type steel stress-ribbon bridge is
The main bearing element of a pedestrian two-span suspension
split-type steel stress-ribbon bridge is a flexible cable which is
calculated as a structure facing geometrical non-linearity. Supporting
steel structure of suspension two-span bridge is being estimated.
Suspension elements of this structure are flexibly and rigidly supported
in the area of end piers. Standing pier of the structure is horizontally
shiftable (Fig. 1).
When calculating the one band suspension bridge, it is assumed,
that bearing suspension cable is absolutely flexible cable, i.e. it is
free of bending stiffness EI. Under the own weight flexible cable takes
shape close to the square parable. Cable stressed by concentrated or
asymmetric loads will change its initial form. Such deformation is
determined by the kinematic origin shifts. It shall be noted that the
increase rate of cable curvature induced by kinematic displacements
exceeds the increase conditioned by the elastic deformation. This means
that kinematic displacements can be more dangerous for cable
deformability than vertical displacements caused by elastic
deformations. Simplified engineering method (estimated) has been
developed for the analysis of kinematic displacements of suspension
split-type steel stress-ribbon bridge (Tarvydaite, Juozapaitis 2010).
[FIGURE 1 OMITTED]
It is assumed, that axial stiffness of the cable is equal to EA =
[infinity] and the kinematic displacements caused by asymmetric load
will be considered
accordingly to the assumption. According to the ratio of the right
and left suspension elements length [s.sub.l] = [s.sub.r], the
horizontal kinematic displacement of the standing pier can be calculated
as follows (Tarvydaite, Juozapaitis 2010):
[DELTA]h = 4/3L [[([f.sub.0] + [DELTA][f.sub.l]).sup.2] -
[([f.sub.0] + [DELTA][f.sub.r]).sup.2]], (1)
where L--span length of suspension cable, m; [f.sub.0]--suspension
cable initial sag, m; [DELTA][f.sub.l]--vertical kinematic displacement
in the middle of the left span, m; [DELTA][f.sub.r]--vertical kinematic
displacement in the middle of the right span, m.
Then the thrust force equilibrium condition ([H.sub.l] = [H.sub.r])
is applied for calculation of left span vertical kinematic displacement:
[DELTA][f.sub.l] = -[f.sub.0] +
(p+g)([f.sub.0]+[DELTA][f.sub.r])/g, (2) where g--constant load, kN/m;
p--temporary load, kN/m.
With the help of geometric equations vertical kinematic
displacement of the right span can be determined:
[DELTA][f.sub.r] = -[f.sub.0] + g [square root of [f.sup.2.sub.0] +
3/8 L [DELTA]h] / (p+g). (3)
It shall be noted that the left kinematic displacement is downward,
and the right--upward (Fig. 1).
3. Simplified engineering method for general displacements of
pedestrian two-span suspension split-type steel stress-ribbon bridge is
It is known that general (total) displacements of the cable include
of Kinematic and Elastic displacements. Elastic displacements are caused
by cable elongation under the thrust (tension) force H. While
calculating the above displacements it is considered that bearing cables
of steel stress-ribbon bridge under the asymmetric loads experience, in
particular, kinematic, And only then elastic displacements. In order to
determine general (kinematic and elastic) displacements the well known
equilibrium Eq of a deformed state shall be applied:
[DELTA][s.sub.g] - [DELTA][s.sub.el = 0, (4)
where:
where [s.sub.1] and [s.sub.k1]--accordingly the cable length after
the elastic deformation and before it, m; H--thrust force, kN;
E--elastic modulus, MN/[m.sup.2]; A--cross-sectional area, [m.sup.2].
Length of the cable prior the elastic deformation
[s.sub.k1]=L+8[f.sup.2.sub.0]/3L, (7)
and after the elastic deformation [s.sub.1]:
[s.sub.1]=L-[DELTA]h+8[([f.sub.0]+[[DELTA][f.sub.el]).sup.2]/3(L-[DELTA]h), (8)
where [DELTA]h--the horizontal displacement of a standing pier.
With the help of (4)--(8) expressions the equation for general
cable displacement in the middle of the span will be as follows:
[DELTA][f.sup.2.sub.el]+2[f.sub.0][[DELTA]f.sub.el]-0.375[DELTA]hL-
[HL.sup.2]3/EA8=0. (9)
Thrust forces of the right [H.sub.l] and left [H.sub.r] spans are
as follows:
[H.sub.l]=(g+p)[L.sup.2]/8([f.sub.0]+[DELTA][f.sub.l]), (10)
[H.sub.r]=[gL.sup.2]/8([[f.sub.0]+[DELTA][f.sub.r]), (11)
where [DELTA][f.sub.l] and [DELTA][f.sub.r]--accordingly general
displacements of the left and right spans.
Solving the (9), (10) and (11) Eqs, the 3rd degree expressions will
be delivered for the calculation of the left and right displacement
values:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Horizontal displacement of the standing pier induced by the
kinematic and elastic displacements can be calculated applying (14) and
(15) Eqs:
[s.sub.0]=[s.sub.l]-[DELTA][s.sub.el,l], (14)
[s.sub.0]=[s.sub.r]-[DELTA][s.sub.el,r]. (15)
Then:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
To solve the Eq of the 3rd degree with the help of modern
mathematical methods, or mathematical programming operators (Mathcad,
Maple, Matlab, etc.) is not difficult. However, the design of suspension
bridges shall take into the consideration the operational requirements
(for example, the threshold bridge displacements), and to determine
parameters of the cable cross sections. In this case, the 3rd de gree Eq
is not helpful and the calculation will become quite complicated.
In order to obtain simplified engineering (estimated) expressions
of suspension bridge displacements applying Eq (9) and taking into
account Eq (10), the simplified formula to be applied for the
calculation of the general displacement of the left cable will be as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
This formula allows, at a known horizontal displacement of a
standing pier to reduce the volume of the iterative calculation.
From the thrust force equilibrium condition ([H.sub.l]; =
[H.sub.r]), the right cable displacement can be calculated:
[DELTA][f.sub.r]=-[f.sub.0]+g([f.sub.0]+[DELTA][f.sub.l])
[(L+[DELTA]h).sup.2]/ (p+g)[(L-[DELTA]h).sup.2]. (18)
It shall be noted, that in this case the horizontal displacement of
the standing pier is calculated according to the Eq (16).
4. Numerical analysis of a general displacement of two-span
suspension structure
In order to determine the accuracy of the developed engineering
techniques the numerical experiment has been performed. For the
numerical analysis the two-span split-type structure flexibly and
rigidly supported in the area of end piers with spans equal to 40 m, and
the total length of the suspension bridge structure equal to 80 m, has
been selected. The initial sag values of the cable are
accordingly--[f.sub.0]= L/50=0.8, [f.sub.0]=L/40=1.0 and
[f.sub.0]=L/32=1.25. The analytical model of the retaining element is
presented in Fig. 2.
Calculating with the program Cosmos/m, the each span retaining
element was composed of 80 straight finite elements. Uniformly
distributed loads have been replaced in points (nodes) by concentrated
forces.
[FIGURE 2 OMITTED]
During the numerical analysis of the corresponding bridge structure
all values of evenly distributed asymmetric load are set in the context
of temporary and permanent changes in the load ratio range. Table 1
provides the results of the corresponding structure.
Analysis of the results presented in Table 1, shows that general
vertical cable displacements (at the middle of the left and right spans)
and the horizontal displacement of the standing pier ([DELTA]t),
calculated according to the 3rd degree and engineering formulas
delivered by Cosmos/m program are almost coinciding. It shall be noted
that displaceMent of the left cable at span quarters (i.e. if [x.sub.l]
= L/4 and [x.sub.l] = 3L/4) and general displacements are as well the
same, and the greatest difference is less than 0.33%.
Determined that general displacements values of the particular
structure calculated using the simplified engineering Eqs (12), (16) and
(18), practically coincide with displacement values obtained with the
help of Cosmos/m program, inaccuracies do not exceed 2.41%.
5. Engineering method for displacements of structure with bending
stiffness
Bending stiffness structure is the structure which takes over loads
both by stretching and bending. These retaining elements, combining
tension and bending elements abilities, stabilize the initial geometric
form (Juozapaitis et al. 2006).
Bending stiffness of the structure parameter is estimated by the
pliantness kL. The greater pliantness parameter kL is, the greater
flexibility of the structure, and vice versa, the less pliantness
parameter value kL, the more rigid the structure is. It can be assumed
that if the kL [approximately equal to] 1, the structure is very rigid,
and when the kL [approximately equal to] 10--the structure can be
considered absolutely flexible.
Pliantness coefficient k is calculated as follows:
K = [square root of H/EI]. (19)
For the analysis of general displacements of the two-span
suspension split-type steel stress-ribbon bridge the engineering formula
(estimated) has been developed. The analytical model is presented in
Fig. 1.
Thrust forces, acting on the left and right spans, are equal to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)
where I--cross-sectional moment of inertia, [m.sup.4].
Similar to the flexible cable, when substituting expression (9)
with (20) and (21) Eqs, the 3rd degree (cubic) equation needed for the
calculation of the left and right cable displacements will be obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)
Horizontal displacement of the standing pier, according to (14) and
(15) Eqs will be equal to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)
In order to simplify the calculation it is proposed to calculate
the general displacement of the right cable (in the middle of the second
span) through a thrust force equality condition ([H.sub.l]=[H.sub.r]),
as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)
Eqs (22)-(26) show that displacement of suspension bridge cables
depends not only on their cross-axial stiffness, but also on the bending
stiffness.
According to the engineering calculation method (using the beam
analogy) the bending moments of the left and right cables in the middle
of the span are:
[M.sub.l] [approximately equal to] 48/5
EI[DELTA][f.sub.l]/[L.sup.2], (27)
[M.sub.r] [approximately equal to] 48/5
EI[DELTA][f.sub.r]/[L.sup.2]. (28)
Inaccuracy of (27) and (28) expressions is high and does not exceed
8.86% (Fig. 3).
6. Numerical analysis of the suspension structure bending stiffness
general displacements
For the numerical analysis the same type of the suspension bridge,
as a two-span flexible cable structure has been selected. Pliantness
values of the selected parameter vary from 2 to 10. The analytical model
of the retaining element is presented in Fig. 2.
During the numerical analysis of the current structure the evenly
distributed asymmetric load values are adopted in the context of
temporary and permanent changes in the load ratio range y.
The results of calculation are given in Table 2.
Figs 4-6 show stabilisation of dispacements with the help of
bending stiffness depending on the initial sag ([f.sub.0]), when the
temporary and permanent in the load ratio of 1.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
The results in Table 2 show that general displacements of the left
and right cables and the horizontal displacements of the standing pier
([DELTA]h) calculated in accordance with the engineering calculation
formulas and delivered by the program Cosmos/m practically coincide. The
greatest inaccuracy of the general vertical displacement of the right
unloaded by the asymmetric load mid-span in case of the directed
downward elastic displacement is practically equal to the directed
upwards kinematic disPlacement.
It shall be noted that general displacements at the quarter of the
left loaded with asymmetric load span (when
[x.sub.l] = L/4 and [x.sub.l] = 3L/4 as well are practically the
same.
The greatest difference between the results obtained by program
Cosmos/M and after the engineering calculation is equal to 1.93%.
7. Conclusions
Presented simplified engineering method of suspension two-span
pedestrian steel bridge under action of asymmetrical loads allows
performing the relatively simple calculation of bearing suspension cable
thrust forces, vertical and horizontal displacements and bending
moments. Numerical analysis shows that proposed simplified engineering
method of steel stress-ribbon bridge is sufficiently precise. The
greatest inaccuracy of vertical displacement and thrust forces
calculation does not exceed 1.37%, and the determination of bending
moments--8.86%. It shall be noted that this method is suitable for
preliminary design of such bridges.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
doi: 10.3846/bjrbe.2011.34
Received 8 November 2009; accepted 4 August 2011
References
Caetano, E.; Cunha, A. 2004. Experimental and Numerical Assessment
of the Dynamic Behaviour of a Stress-Ribbon Footbridge, Structural
Concrete 5(1): 29-38. doi:10.1680/stco.2004.5.1.29
Furst, A.; Marti, P.; Ganz, H. R. 2001. Bending of Stay Cables,
Structural Engineering International 11(1): 42-46.
doi:10.2749/101686601780324313
Gimsing, N. J. 1997. Cable Supported Bridges: Concept and Design.
2nd edition. John Wiley & Sons, Chichester. 480 p. ISBN 0471969397
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.
doi:10.3846/jcem.2010.41
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. doi:10.3846/jcem.2010.14.
Juozapaitis, A.; Norkus, A.; Vainiunas, P. 2008. Shape
Stabilization of Steel Suspension Bridge, The Baltic Journal of Road and
Bridge Engineering 3(3): 137-144. doi:10.3846/1822-427X.2008.3.137-144
Juozapaitis, A.; Vainiunas, P.; Kaklauskas, G. 2006. A New Steel
Structural System of a Suspension Pedestrian Bridge, Journal of
Constructional Steel Research 62(12): 1257-1263.
doi:10.1016/j.jcsr.2006.04.023
Juozapaitis, A.; Norkus, A. 2004. Displacement Analysis of
Asymmetrically Loaded Cable, Journal of Civil Engineering and Management
10(4): 277-284. doi:10.1080/13923730.2004.9636320
Kala, Z. 2008. Fuzzy Probability Analysis of the Fatigue Resistance
of Steel Structural Members under Bending, Journal of Civil Engineering
and Management 14(1): 67-72. doi:10.3846/1392-3730.2008.14.67-72
Katchurin, V. K. 1969. Staticheskij raschet vantovykh sistem [[TEXT
NOT REPRODUCIBLE IN ASCII]]. Leningrad: Stroyizdat. 141 p.
Kulbach, V. 1999. Half-Span Loading of Cable Structures, Journal of
Constructional Steel Research 49(2): 167-180.
doi:10.1016/S0143-974X(98)00215-6
Kulbach, V. 2007. Cable Structures. Design and Analysis. Tallin,
Estonian Academy Publisher. 224 p.
Moskalev, N. S. 1981. Konstrukcii visyachikh pokrytij [[TEXT NOT
REPRODUCIBLE IN ASCII]]. Moskva: Stroyizdat. 335 p.
Michailov, V. V. 2002. Predvaritelno napryazhennyje
kombinirovannyje i vantovye konstrukciji [[TEXT NOT REPRODUCIBLE IN
ASCII]]. Moskva: ACB. 255 p.
Prato, C. A.; Ceballos, M. A. 2003. Dynamic Bending Stresses Near
the Ends of Parallel Bundle Stay Cables, Structural Engineering
International 13(1): 64-68. doi:10.2749/101686603777965008
Ryall, M.; Parke, G. A. R.; Harding, J. E. 2000. Manual of Bridges
Engineering. London: Tomas Telford Ltd. 1007 p. ISBN 0727727745
Schlaich, J.; Bergerman, R. 1992. Fu/igangerbrucken [Footbridges].
Zurich (ETH): Schwabische Druckerei GmbH.
Schlaich, M.; Brownlie, K.; Conzett, J.; Sobrino, J.; Strasky, J.;
Takenouchi, Mrs. Kyo. 2005. fib Bulletin 32. Guidelines for the design
of footbridges. Sprint-Digital-Druck, Stuttgart. p. 154. ISSN 1562-360,
ISBN 2-88394-072X
Strasky, J. 2005. Stress-Ribbon and Supported Cable Pedestrian
Bridges. London: Thomas Telford Ltd. 240 p. ISBN 9780727732828.
doi:10.1680/sracspb.32828
Tarvydaite, G.; Juozapaitis, A. 2010. The Kinematic Displacements
of the Two-Spans Single Lane Suspension Steel Footbridge and Their
Stabilization, Engineering Structures and Technologies 2(4): 155-162.
Troyano, L. F. 2003. Bridge Engineering: A Global Perspective.
London: Tomas Telford Ltd. 775 p. ISBN 0727732153
Wollmann, G. P. 2001. Preliminary Analysis of Suspension Bridges,
Journal of Bridge Engineering 6(4): 227-233.
doi:10.1061/(ASCE)1084-0702(2001)6:4(227)
Wyatt, T. A. 2004. Effect of Localised Loading on Suspension
Bridges, in Proc. of the Institution of Civil Engineers. Bridge
Engineering 157, Issue BE2. 55-63. doi:10.1680/bren.2004.157.2.55
Giedre Sandovic (1), Algirdas Juozapaitis (2), Romualdas Kliukas
(3)
(1, 2) Dept of Bridges and Special Structures, Vilnius Gediminas
Technical University, Sauletekio al. 11, 10223 Vilnius, Lithuania (3)
Dept of Strength of Materials, Vilnius Gediminas Technical University,
Sauletekio al. 11, 10223 Vilnius, Lithuania E-mails: (1)
giedre.tarvydaite@vgtu.lt; (1) algirdas.juozapaitis@vgtu.lt; (3)
romualdas.kliukas@vgtu.lt
Table 1. General displacement values delivered by Cosmos/M(C) program
and after the engineering calculation (A)
[f.sub.0] [gamma] [DELTA][f.sub.l] %
C A
0.8 0.5 -0.5035 -0.5017 0.36
1 -0.6100 -0.6077 0.38
2 -0.7130 -0.7102 0.39
3 -0.7604 -0.7576 0.37
4 -0.7869 -0.7840 0.37
5 -0.8036 -0.8007 0.36
1.0 0.5 -0.4539 -0.4515 0.53
1 -0.5703 -0.5674 0.51
2 -0.6805 -0.6773 0.47
3 -0.7300 -0.7266 0.47
4 -0.7570 -0.7535 0.46
5 -0.7738 -0.7702 0.47
1.25 0.5 -0.4219 -0.4192 0.64
1 -0.5530 -0.5498 0.58
2 -0.6746 -0.6711 0.52
3 -0.7275 -0.7240 0.48
4 -0.7557 -0.7521 0.48
5 -0.7728 -0.7692 0.47
[f.sub.0] [gamma] [DELTA]h %
C A
0.8 0.5 -0.03128 -0.03116 0.39
1 -0.04947 -0.04928 0.39
2 -0.06758 -0.06732 0.39
3 -0.07585 -0.07557 0.37
4 -0.08035 -0.08006 0.36
5 -0.08311 -0.08281 0.36
1.0 0.5 -0.03887 -0.03871 0.41
1 -0.06130 -0.06106 0.39
2 -0.08334 -0.08303 0.37
3 -0.09320 -0.09286 0.37
4 -0.09847 -0.09811 0.37
5 -0.10170 -0.10128 0.41
1.25 0.5 -0.05130 -0.05109 0.41
1 -0.08071 -0.08042 0.36
2 -0.1092 -0.10887 0.30
3 -0.1217 -0.12134 0.30
4 -0.1282 -0.12791 0.23
5 -0.1321 -0.13179 0.24
[f.sub.0] [gamma] [DELTA][f.sub.r] %
C A
0.8 0.5 -0.07148 -0.0705 1.39
1 -0.0924 -0.0927 -0.32
2 -0.2931 -0.2932 -0.03
3 -0.4077 -0.4076 0.02
4 -0.4807 -0.4807 0.00
5 -0.5310 -0.5310 0.00
1.0 0.5 -0.02791 -0.0286 -2.41
1 -0.2133 -0.2115 -0.09
2 -0.4363 -0.4362 0.02
3 -0.5645 -0.5643 0.04
4 -0.6460 -0.6458 0.03
5 -0.7021 -0.7019 0.03
1.25 0.5 -0.1311 -0.1315 -0.30
1 -0.3430 -0.3428 0.06
2 -0.6032 -0.6026 0.10
3 -0.7511 -0.7505 0.08
4 -0.8450 -0.8444 0.07
5 -0.9095 -0.9090 0.06
Table 2. Cosmos/M(C) program and the displacement values received
after engineering calculation (A)
[f.sub.0] [kL] [DELTA][f.sub.l] %
C A
0.8 9.76 -0.5985 -0.5858 2.17
8.03 -0.5899 -0.5759 2.43
6.67 -0.5779 -0.5629 2.66
5.59 -0.5623 -0.5469 2.82
4.71 -0.5425 -0.5271 2.92
3.37 -0.4901 -0.4762 2.92
2.97 -0.4658 -0.4528 2.87
2.70 -0.4465 -0.4344 2.79
1.0 9.28 -0.5511 -0.5382 2.40
7.64 -0.5413 -0.5273 2.66
6.35 -0.5280 -0.5133 2.86
5.33 -0.5111 -0.4963 2.98
4.50 -0.4902 -0.4757 3.05
3.25 -0.4376 -0.4248 3.01
2.87 -0.4143 -0.4024 2.96
2.62 -0.3961 -0.3849 2.91
1.25 8.68 -0.5223 -0.5086 2.69
7.16 -0.5101 -0.4957 2.90
5.96 -0.4939 -0.4792 3.07
5.01 -0.4741 -0.4595 3.18
4.24 -0.4503 -0.4364 3.19
3.08 -0.3938 -0.3819 3.12
2.73 -0.3699 -0.3590 3.04
2.50 -0.3519 -0.3417 2.99
[f.sub.0] [kL] [DELTA]h
C A %
0.8 9.76 -0.04442 -0.04418 0.54
8.03 -0.04297 -0.04282 0.35
6.67 -0.04113 -0.04107 0.15
5.59 -0.03892 -0.03896 -0.10
4.71 -0.03630 -0.03643 0.36
3.37 -0.03003 -0.03033 -0.99
2.97 -0.02739 -0.02773 -1.23
2.70 -0.02539 -0.02577 -1.47
1.0 9.28 -0.05417 -0.05386 0.58
7.64 -0.05219 -0.05199 0.38
6.35 -0.04971 -0.04963 0.16
5.33 -0.04676 -0.04682 -0.13
4.50 -0.04333 -0.0435 -0.39
3.25 -0.03537 -0.03575 -1.06
2.87 -0.03211 -0.03254 -1.32
2.62 -0.02969 -0.03015 -1.53
1.25 8.68 -0.06986 -0.06943 0.62
7.16 -0.06692 -0.06666 0.39
5.96 -0.06327 -0.06319 0.13
5.01 -0.05903 -0.05911 -0.14
4.24 -0.05417 -0.05442 -0.46
3.08 -0.04336 -0.04385 -1.12
2.73 -0.03908 -0.03964 -1.41
2.50 -0.03596 -0.0655 -1.61
[f.sub.0] [kL] [DELTA][f.sub.r]
C A %
0.8 9.76 -0.03232 -0.0322 0.37
8.03 -0.02291 -0.0233 -1.67
6.67 -0.01123 -0.0124 -9.44
5.59 -0.00229 -0.00272 -15.70
4.71 -0.01753 -0.0147 19.25
3.37 -0.04982 -0.0455 9.49
2.97 -0.06148 -0.0567 8.43
2.70 -0.06937 -0.0 45 7.55
1.0 9.28 0.1396 0.1373 1.68
7.64 0.1260 0.1242 1.45
6.35 0.1091 0.1081 0.93
5.33 0.08952 0.0894 0.13
4.50 0.06748 0.0684 -1.35
3.25 0.0205 0.0234 -12.39
2.87 0.003103 0.00359 -13.57
2.62 -0.00889 0.000809 9.94
1.25 8.68 0.2539 0.2485 2.17
7.16 0.2345 0.2297 2.09
5.96 0.2107 0.2068 1.89
5.01 0.1834 0.1807 1.49
4.24 0.1531 0.1518 0.86
3.08 0.08965 0.0910 -1.48
2.73 0.06633 0.0685 -3.17
2.50 0.05003 0.0527 -5.07
