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

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Where Ad,L is the cross-sectional sectional area, yd,L y is the cross-sectional ordinate and hd,L is the distance of the considered wire in layer L from the cable axis (see also Fig. 2.12). Equation Equation 2.27 2.27 may may also be be written in the following manner manner for for an an individual wire in the the layer: layer: By comparing (2.28) and (2.29) and by solving the first integral of (2.28) for the cross-section cross section of the wire with σb,L as in (2.20), we obtain (EJ) (EJ)min,d,L, , the wire stiffness affecting how an individual wire bends around its own transverse axis: The corresponding cable stiffness, denoted (EJ) (EJ)min, , is obtained by summing the individual wire stiffnesses acc. to (2.30): where EK is Young's modulus and δδK is the diameter of the core wire. This This would would be be the the only only stiffness stiffness of of the the cable cable if the individual wires wires were lying lying on top of each other without friction. The friction forces between the individual wires however, analogous to elongation and stress, Section 2.3, give rise to secondary stiffness (EJ) (EJ)zus,d,L as in (2.32). Since Since the the secondary stiffness stiffness arises from the secondary stress, which which is constant over the cross-section section of the wire, it is not actually nnecessary ecessary to integrate in order to determine the second part in (2.28). The secondary stiffness of an individual wire in layer L is therefore: 23
Depending Depending on on the the slip slip in in the considered wire, the secondary stresses stresses from from (2.17) or or (2.22) must be inserted in this equation, as well as the distance hhd,L of the centre of mass of the cross-section cross of the wire wire from from the the neutral neutral axis axis of of the the bent bent cable cable acc. to (2.19). (2.19). This secondary secondary stiffness stiffness is dependent on on the angle ϕ in both cases, i.e. it varies as ϕ varies aries along an individual wire or yields different results for the different wires of the same cable cross cross-section, depending on the ϕ “position” of each wire under consideration. The secondary stiffness acc. to (2.22) for an individual wire and when the wires have not slipped (called Region I) is therefore given by: and by (2.17) when the wires are fully displaced (called Region II): Summing Summing equation equation (2.33) (2.33) over over all all wires wires and layers yields the the secondary secondary stiffness stiffness of of the the cable cable with undisturbed wires: In In this this case, case, the the cable cable has has the the maximum maximum bending bending stiffness, stiffness, called called the the cable cable stiffness (EJ) (EJ)max, (EJ) calculated from from the the sum sum of of the the wire wire stiffness stiffness acc. acc. to to (2.31) (2.31) and and the the secondary secondary stiffness stiffness acc. to to (2.35): (2.35): This result is similar to the case where the ca cable ble is considered a homogeneous, rigid beam and its area moment moment of inertia is is calculated calculated using using Steiner’s theorem. theorem. This This is also a a plausibility plausibility check check on the correctness of the approach used here to obtain the secondary stiffness. Equation (2.34), however, for or the the first time time describes describes an additional additional contribution to to wire stiffness, not considered considered to to date date in in cases cases of of fully fully slipped slipped wires. This additional stiffness, approaching zero as the cable curvature increases, arises from the maximum secondary stress allowed by friction forces in the wires wires acc. to to (2.17), (2.17), and and opposes opposes the external moment of the the load, load, in in addition addition to to the the wire wire stiffness stiffness acc. to (2.31). 24
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 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: Fig. 2.12 Distribution of the cable
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
Page 69 and 70: To determine the stiffnesses, momen
Page 71 and 72: Assumption 1: Undisturbed isturbed
Page 73 and 74: Fig. 3.7d M-B B diagram diagram und
Page 75 and 76: Phase 1 The The relative position p
Page 77 and 78: Fig. 4.2c Bending of a cable under
Page 79 and 80: Phase 0 The strain energy W0D = W4D
Page 81 and 82: Fig. 4.3 M-B B diagram for the anal
Page 83 and 84:
The The balance balance of of momen
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76 In the next step, for an “arbi
Page 87 and 88:
Input module The cable data input,
Page 89 and 90:
( x )E*I = f ( k ) 80 Stiffness for
Page 91 and 92:
82 The same process is now repeated
Page 93 and 94:
84 Fig. 4.13 Flow chart of the SEIL
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(a) Fig. 5.1 Cable bending: (a) for
Page 97 and 98:
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
Page 113 and 114:
Sag [m] Fig. 6.1 Measured and calcu
Page 115 and 116:
Sag [m] Fig. 6.4 Measured and calcu
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108 Fig. 6.6 also clearly shows tha
Page 119 and 120:
Fig. 6.9 Hysteresis for Cardinal at
Page 121 and 122:
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
Page 131 and 132:
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
Page 141 and 142:
Annexure I The The tensile stress s
Page 143 and 144:
Considering Considering the the abo
Page 145 and 146:
It follows that: If the angle is do
Page 147 and 148:
The position of the catenary in the
Page 149 and 150:
The The following following matrice
Page 151 and 152:
For single component conductors: Fo
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144 CIGRE: Study Committee 22 - Wor
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146 Papailiou, Müller, Roll: Anwen
Page 157 and 158:
List of symbols Latin symbols 148 a
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ma mi mL 150 Friction summation fac
Page 161 and 162:
Z0 Z1 Za Z c,n Zd Z d,a Zd,i Zd,L Z
Page 163 and 164:
ρ Curvature radius of the cable ax
Page 165 and 166:
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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