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

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This figure also shows the definition of the lay angle consideration, comprising of nL round round wires wires of the the same same diameter in the cables investigated here. Fig. 2.2 shows the components of an external force S tensioning a helically tw twisted isted cable, as they act on individual individual wires, wires, ignoring ignoring the transverse force, force, caused by by bending bending and torsion of the the wire (caused (caused by the external external tensile tensile force). force). This This transverse transverse force force is very small small because the lay lay angle and layer radii of the cable change very y little under tensile load [Feyrer, 1990]. Fig. 2.2 Distribution of the tensile force to individual wires The wire tensioning force Zd,L in every individual wire of an arbitrary layer L of the cable is therefore: The total longitudinal tensile force in the cable is given by the sum: wires,layers where the summation also includes the core wire ( (βK = 0). In the determination of the tensile forces Z Zd,L in the individual wires, it is assumed that the cable cross-sections sections remain remain the the same same as as a a tensile tensile force force S, S, lengthening lengthening an an arbitrary arbitrary length length of of cable cable ℓs by ∆ℓs all the wires are in the linear-elastic elastic region, i.e. Hooke’s law law is applicable. Finally, the effects of the tangential force Ud,L are are assumed very very small. The The changes in the the lay angle angle and the transverse transverse contraction of the wire cross-sections sections are ne negligible in this context since the resultant stresses amount to to only only 1% 1% [CEA, [CEA, 1986] 1986] or or 2% 2% [Feyrer, [Feyrer, 1990] 1990] of of the the total total tensile stress in the the wire. The corresponding formulae are summarised in Annexure I. 9 the definition of the lay angle βL and the coil diameter dL of the layer under and also lengthening the associated wire length ℓd,L by ∆ℓd,L, is applied. It is furthermore assu , is applied. It is furthermore assumed that
Figure 2.3 explains these relationships as wires and cable are stretched under the influence of a tensile force, with no change in the cable cross-section. cross Fig. 2.3 Wire and cable elongation The elongation of a wire εd,L is smaller than the associated cable elongation εs by a factor of cos Since the wire tension Zd,L is is the the same same for for all all wires wires in in the the same same layer, layer, Hooke’s law states states that: or where Ed,L is Young’s modulus for a wire in position L and with cross-sectional area Ad,L. . The combination of (2.1), (2.2), (2.3) and (2.4) gives the tensional force Zd,L on an arbitrary wire in layer L, , as a function of the total cable tension S: 10 by a factor of cos 2 βL
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 and 14: 4 This device measures the deflecti
Page 15 and 16: 6 From early years, overhead transm
Page 17: 2. Basic principles 2.1. The tensil
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
Page 65 and 66: By applying the Zi just just achiev
Page 67 and 68: 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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