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

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Average transition curvature Referring to Fig. 2.13, we may make the approximating assumption that, ign ignoring oring the transition region which lies between the initial curvature κa and the final curvature κe, , as shown in Section 2.3, the stiffness changes instantaneously at the average transition curvature κm. . This approximation significantly improves the clarity of of the the following explanations explanations and is therefore introduced at at this point. The index L is is furthermore furthermore dispensed dispensed with in the following, because the cable considered considered in this paragraph has only one layer. Table 2 below shows the effective bending stiffnesses and bending moments of the cable under these assumptions, in the two Regions I and II, Fig. 2.13. Region I (no wire slippage) Region II (complete wire slippage) (EJ) min + (EJ) zusl = (EJ) max (EJ)min + (EJ) (EJ)zusII = (EJ)min +MR/κ (EJ)minκ + MR Table 2.1 Bending stiffnesses and bending moments in Regions I and II The moment MR was provisionally introduced as an auxiliary parameter. This moment will gain physical meaning further down only. We have: Considering (2.34), we obtain: If, in (2.43), the secondary force Zd d (EJ)Total where Z(ϕ) is the total and Zd the “pure” wire tensile force, independent of ϕ, we find: 27 (EJ)maxκ MTotal , which is a function of ϕ, , is replaced by (Z(ϕ) (Z( - Zd),
We may derive the mean transition curvature κm from the following consideration: In the simplified presentation presentation (Fig. (Fig. 2.13), 2.13), the the cable moments in Regions Regions I I and and II should, per definition, be the same for this curvature, i.e. using Table 2.1: This yields the average transition curvature: Figure 2.14 .14 below below shows shows the the curve curve of of cable cable stiffness across the curvature. This figure shows the "exact" curve curve for for bending bending stiffness, stiffness, i.e. i.e. taking taking into account account a a finite finite transition transition region region between between curvatures κa and κe as in (2.24) and (2.23). For comparison, the figu figure re also shows the approximate stiffness curve discussed discussed above, above, which which features a a sudden transition from the undisturbed condition to to fully fully slipped slipped cable wires at the average transition curvature κm as in (2.46). This figure is of particular rticular practical signi significance since it quantifies the variable bending stiffness of a cable cable as as caused by by internal friction friction for for the first time. time. This This dependence of the bending stiffness on the curvature of the cable will be used later again, in the calculation of the cable catenary, characterising the results of this project. The associated transition moment MMm is calculated from (2.45) as: The intersection MR of the M-B B line line with with the the moment moment axis axis is is found from from Fig. 2.13 as follows: 28
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 and 32: Fig. 2.12 Distribution of the cable
Page 33 and 34: Depending Depending on on the the s
Page 35: this results in: In this equation,
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
Page 85 and 86: 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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