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

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Summary The bending of helically twisted cable is a severe loading case for these important structural components. During bending of such cables, slippage takes place between the individual wires of the cable, which is greatly influenced by the frictional forces acting at the inter-wire contacts. Because of this slippage, the cross-sections of the cable do not necessarily remain plane, which has to be considered in the formulation of the bending stiffness of the cable. The main subject of this work is to develop, based on generally accepted principles for bending of cables, an adequate model to quantify the bending behaviour of multilayered helically twisted cables. This model leads, under consideration of internal friction, to a variable effective bending stiffness which depends on the cable curvature and the tensile load acting on the cable. With this information it is now possible to calculate the deflection, curvature and the various components of stress in the wires of the cable analytically. With the newly developed “cable scanner”, the catenary and curvature of the cable axis can be determined by scanning the cable surface. The agreement between theory and measurement is quite good. The model is extended to cover also ACSR (Aluminium conductor, steel reinforced), a conductor which is widely used in overhead transmission lines, and is validated through corresponding measurements. An important application of this work is the calculation of dynamic bending stresses in the individual wires of such conductors, under wind induced (aeolian) vibrations. These calculations consider the ordinary alternating bending stresses plus the secondary tensile stresses (“zusatz” stresses) caused by interlayer friction between wires, which together constitute the longitudinal fatigue stress. The theory resolves a known discrepancy between measured stresses and stresses predicted by simpler conductor models in use today, which assume a constant bending stiffness. It also settles the long outstanding uncertainty on the proper choice of effective bending stiffness for vibrating conductors. This opens up a new way of determining the stress limits for such conductors under vibrating conditions.
Zusammenfassung Die Seilbiegung stellt einen wichtigen Betriebszustand von Spiralseilen dar. Während des Biegevorganges verschieben sich die Drähte des Seiles gegenseitig, wobei die Reibung zwischen den einzelnen Drähten eine entscheidende Rolle spielt. Durch die Biegeverformung bleiben auch die Querschnitte für das gesamte Seil nicht notwendigerweise eben. Dies muss in der Formulierung des Ansatzes für die Biegesteifigkeit des Seiles berücksichtigt werden. Unter Beachtung von allgemein anerkannten Grundsätzen für die Seilbiegung, wird im Rahmen dieser Arbeit ein Modell für mehrlagige Spiralseile vorgestellt, dass auf einer, durch die innere Reibung veränderlichen und von der Seilkrümmung und von der Seilzugkraft abhängigen, effektiven Biegesteifigkeit basiert. Damit werden die Seillinie, die Seilkrümmung und die verschiedenen Komponenten der in den einzelnen Drähten des Seiles herrschenden Spannungen berechnet. Mit dem neuentwickelten “Seiltomographen” werden durch Abtasten der Seiloberfläche, die Seillinie und die Seilkrümmung experimentell ermittelt. Dabei ist eine recht gute Übereinstimmung zwischen Rechnung und Messung festzustellen. Dieses Seilmodell wird auch auf Aluminium-Stahl-Verbundseile für die Stromübertragung erweitert und durch entsprechende Messungen bestätigt. Anschliessend wird damit, als wichtige Anwendung, die Ermittlung der Wechselbiegespannungen, die in den einzelnen Drähten von solchen Leiterseilen bei winderregten Schwingungen entstehen, durchgeführt und mit der heute üblichen Bemessungspraxis verglichen. Durch die Berücksichtigung der von der inneren Reibung herrührenden Zusatzspannung, welche zusammen mit der Biegespannung die schwellende Längsspannung in den Seildrähten ergibt, wird eine bekannte Diskrepanz zwischen den gemessenen und den mit einfachen Seilmodellen, die auf einer konstanten Biegesteifigkeit basieren, berechneten Wechselbiegespannungen geklärt, sowie die Unsicherheit über den für diesen Betriebszustand gültigen Ansatz für die effektive Biegesteifigkeit des Seiles behoben. Damit kann schliesslich ein neuer Weg zur Festlegung der zulässigen Schwingungsbeanspruchungen für Leiterseile prinzipiell aufgezeigt werden.
Page 1 and 2: Bending of helically twisted cables
Page 3: For my father
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 and 14: 4 This device measures the deflecti
Page 15 and 16: 6 From early years, overhead transm
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
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A wire of the outer layer therefore
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The Zn,a values values are are the
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According According to to the law o
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First First approximation: Continuo
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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
Page 167 and 168:
158 assumptions; 1: (EJ)(κ); 2: (E
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Bending of helically twisted cables under variable ... - Pfisterer

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