Bending of helically twisted cables under variable ... - Pfisterer

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5 For all these reasons, the first and most important step towards the solution of the described problems should be the study of conductor bending, including the effect of variable bending stiffness caused by the construction of the conductor and concomitant internal friction. This also defines the problem to be solved with this project. 1.2. Developments to date Virtually since the day cables were introduced as construction elements, engineers were preoccupied with their bending behaviour, since they soon recognised that this stress mode is of particular importance to the reliability in service of the cables. For this reason, many papers have been written on this topic, contributing to better comprehension of this problem. A full overview of all such work would exceed the scope of this project. Instead, only those projects are briefly covered below that specifically addressed the bending stiffness of cables and conductors, as does this project. This is because this parameter is decisive to the understanding and solution of cable bending problems and it was addressed in the technical literature at an early stage already. The literature overview below distinguishes between work on cables for conveyance, which deals mainly with steel cables, and work on overhead lines, which naturally focuses on conductors. The first significant work on cables was that carried out by Reuleaux [1861] and Isaachsen [1907], which also included the first meaningful attempts at calculating the bending stress and catenary of cables. The work by Ernst [1933] is particularly significant, since he was probably the first to quantify the effect of inner friction on cable bending behaviour. The work of Leider [1973, 1975] was a further milestone towards understanding the internal cable conditions, especially in view of the secondary stress, which he defined in detail. More recent work like that of Schiffner [1986] and Wang [1990], who refined and developed earlier theories, also bears mentioning. Also to be mentioned in this connection is the work by Hruska [1951], Czitary [1962], Zweifel [1969], Wiek [1973], Costello [1983] and Raoof and Hobbs [1988], who dealt with this problem in depth, producing valuable methods and solutions.
6 From early years, overhead transmission engineers have also worked on directly establishing the bending stiffness of conductors. In these studies, the effective mean stiffness was often calculated from measurements of the maximum sag of a test conductor clamped at both ends and under constant tension, assuming that the conductor behaves like a homogeneous beam with constant bending stiffness [Pape, 1930, Monroe and Templin, 1932, Sturm, 1936, Morisson, 1962, McConell and Zemke, 1980]. In parallel, [Bückner, 1960, Helms, 1964, Scanlan and Swart, 1968, Möcks and Swart, 1969, Brand, 1972, CEA, 1986, Ramey, 1987] carried out some work in which they, by means of measurements with strain gauges on the outer layer of wires of a conductor, attempted to calculate the internal stress conditions in the conductor or of individual wires. The ongoing interest of engineers in the bending stiffness of cables is clearly documented by new papers by haulage cable users [Raoof and Huang, 1992, Wiek, 1993] and others in the field of conductors [AIF, 1991, Zeitler, 1994]. But researchers in the experimental mechanics field [Lanteigne, 1985, Goudeau and Cardou, 1993] are also endeavouring to come to grips with this interesting problem. 1.3. Project objectives Based on the above, the objective of this project can be defined as follows: A cable model is developed and existing analytical approaches are augmented to allow the bending process in a tensioned helically twisted cable to be mathematically defined and predicted. In this process, a variable bending stiffness is introduced and analytically defined, caused by internal friction in the cable, and dependent on the external stress on the cable. A measurement method is designed and implemented to verify the theory, allowing the cable sag and curvature and ultimately the variable bending stiffness along the cable to be determined with good accuracy. This enables significant insights to be gained into the inner cable state, especially the relative displacements of the individual wires in the cable and the stresses prevailing in these. Despite the complexity of the problem, the theory is kept simple and is explained in a straightforward manner, with a view to extension in ongoing research and useful practical applications. With this in mind, the programs developed within the framework of this project will be written user friendly enough to be used as tools in the design of cables or conductors.
Page 1 and 2: Bending of helically twisted cables
Page 3 and 4: For my father
Page 5 and 6: Zusammenfassung Die Seilbiegung ste
Page 7 and 8: And another special word of thanks
Page 9 and 10: 5. Measurements....................
Page 11 and 12: 2 Their acquisition value is estima
Page 13: 4 This device measures the deflecti
Page 17 and 18: 2. Basic principles 2.1. The tensil
Page 19 and 20: Figure 2.3 explains these relations
Page 21 and 22: Fig. 2.4 Determining the radial pre
Page 23 and 24: It It appears appears suitable suit
Page 25 and 26: Fig. 2.7 Geometry of the cable cros
Page 27 and 28: Fig. 2.9 Curves Curves for the seco
Page 29 and 30: 2.4. The bending stiffness As menti
Page 31 and 32: Fig. 2.12 Distribution of the cable
Page 33 and 34: Depending Depending on on the the s
Page 35 and 36: this results in: In this equation,
Page 37 and 38: We may derive the mean transition c
Page 39 and 40: The moment moment below the transit
Page 41 and 42: Fig. Fig. 2.15 2.15 finally shows s
Page 43 and 44: During During bending in in Region
Page 45 and 46: Fig. 2.18 Hysteresis in the M-B B d
Page 47 and 48: with: and: with: The tensile stress
Page 49 and 50: The The situation situation is is d
Page 51 and 52: As As the curvature curvature incre
Page 53 and 54: This shows that, based on the propo
Page 55 and 56: A wire of the outer layer therefore
Page 57 and 58: The Zn,a values values are are the
Page 59 and 60: According According to to the law o
Page 61 and 62: First First approximation: Continuo
Page 63 and 64: This means that the outer layer is
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By applying the Zi just just achiev
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3.4. Tensile Tensile force in in th
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To determine the stiffnesses, momen
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Assumption 1: Undisturbed isturbed
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Fig. 3.7d M-B B diagram diagram und
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Phase 1 The The relative position p
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Fig. 4.2c Bending of a cable under
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Phase 0 The strain energy W0D = W4D
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Fig. 4.3 M-B B diagram for the anal
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The The balance balance of of momen
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76 In the next step, for an “arbi
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Input module The cable data input,
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( x )E*I = f ( k ) 80 Stiffness for
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82 The same process is now repeated
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84 Fig. 4.13 Flow chart of the SEIL
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(a) Fig. 5.1 Cable bending: (a) for
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The The sensor sensor is is mounted
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Fig. 5.4 Single cross-section secti
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5.2. The test set-up 92 To better o
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Stiffness EJ [N*m^2] Fig. 5.11 Bend
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This machine can handle tensile for
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5.3. The test cables A four-layer h
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The input values for the SEIL progr
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Triggered by the dx signal, a 150 p
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Sag [m] Fig. 6.1 Measured and calcu
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Sag [m] Fig. 6.4 Measured and calcu
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108 Fig. 6.6 also clearly shows tha
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Fig. 6.9 Hysteresis for Cardinal at
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Figure Figure 6.10 6.10 shows shows
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6.5. Effect of the sample length 11
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116 Force--displacement curve Fig.
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This This is is based based on on e
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Secondary stress [N/mm^2] Secondary
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After After the the wire stresses s
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124 The above is summarised in Fig.
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The The trend trend determined here
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128 Even if the conductor model pos
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8. Prospects 130 In the last chapte
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Annexure I The The tensile stress s
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Considering Considering the the abo
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It follows that: If the angle is do
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The position of the catenary in the
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The The following following matrice
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For single component conductors: Fo
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144 CIGRE: Study Committee 22 - Wor
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146 Papailiou, Müller, Roll: Anwen
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List of symbols Latin symbols 148 a
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ma mi mL 150 Friction summation fac
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Z0 Z1 Za Z c,n Zd Z d,a Zd,i Zd,L Z
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ρ Curvature radius of the cable ax
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156 Fig. 3.3 Effect of the tensile
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158 assumptions; 1: (EJ)(κ); 2: (E
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Bending of helically twisted cables under variable ... - Pfisterer

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