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

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Table of Contents Summary………….…………….…………..……………………………………………. Preamble…………...…………………………………………………………………….. 1. Introduction ...................................................................................................................................... 1 1.1. Status and problem definition.................................................................................................. 1 1.2. Developments to date ............................................................................................................. 5 1.3. Project objectives .................................................................................................................... 6 1.4. Limitations ............................................................................................................................... 7 1.5. Project structure ...................................................................................................................... 7 2. Basic principles ............................................................................................................................... 8 2.1. The tensile stress .................................................................................................................... 8 2.2. The secondary stress ............................................................................................................ 11 2.3. The bending stress ................................................................................................................ 14 2.4. The bending stiffness ............................................................................................................ 20 2.5. The cable bending moment ................................................................................................... 25 2.6. The cable state diagram ........................................................................................................ 32 2.7. The single layer helically twisted cable as a calculation example ........................................ 37 3. Multi-layer cables .......................................................................................................................... 45 3.1. Wire resting points ................................................................................................................ 45 3.2. Tensile force in the outer layer of multi-layer cables............................................................. 46 3.3. Tensile force in the inner layer of a two-layer cable ............................................................. 49 3.4. Tensile force in the inner layers of helically twisted cables with more than two layers ........ 58 3.5. The state diagram for multi-layer cables ............................................................................... 59 4. Calculation method ....................................................................................................................... 65 4.1. Bending of a cable under tensile load ................................................................................... 65 4.2. Analytical calculation of the cable catenary .......................................................................... 71 4.3. Determining the catenary with the finite element method ..................................................... 75 4.4. The SEIL program ................................................................................................................. 77
5. Measurements............................................................................................................................... 85 5.1. The cable scanner ................................................................................................................. 86 5.2. The test set-up ...................................................................................................................... 92 5.3. The test cables ...................................................................................................................... 98 5.4. The material properties ......................................................................................................... 99 5.5. The friction coefficients ....................................................................................................... 100 6. Analysis ....................................................................................................................................... 103 6.1. Test results for the steel cable ............................................................................................ 103 6.2. Test results for the ACSR conductor .................................................................................. 108 6.3. Sensitivity analysis .............................................................................................................. 110 6.4. Influence of the bending stiffness........................................................................................ 111 6.5. Effect of the sample length .................................................................................................. 114 7. Applications ................................................................................................................................. 117 7.1. Wire stresses in vibrating conductors ................................................................................. 117 7.2. Practical approach .............................................................................................................. 122 7.3. Effect of the friction corrosion and the static pre-stress ...................................................... 127 8. Prospects .................................................................................................................................... 130 Annexure I ........................................................................................................................................... 132 Annexure II .......................................................................................................................................... 135 Annexure III ......................................................................................................................................... 137 Annexure IV ........................................................................................................................................ 139 Annexure V.......................................................................................................................................... 141 References .......................................................................................................................................... 143 List of symbols .................................................................................................................................... 148 List of figures ....................................................................................................................................... 155 Resumé ............................................................................................................................................... 159
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 another special word of thanks
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
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
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
Page 95 and 96:
(a) Fig. 5.1 Cable bending: (a) for
Page 97 and 98:
The The sensor sensor is is mounted
Page 99 and 100:
Fig. 5.4 Single cross-section secti
Page 101 and 102:
5.2. The test set-up 92 To better o
Page 103 and 104:
Stiffness EJ [N*m^2] Fig. 5.11 Bend
Page 105 and 106:
This machine can handle tensile for
Page 107 and 108:
5.3. The test cables A four-layer h
Page 109 and 110:
The input values for the SEIL progr
Page 111 and 112:
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
Page 117 and 118:
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
Page 123 and 124:
6.5. Effect of the sample length 11
Page 125 and 126:
116 Force--displacement curve Fig.
Page 127 and 128:
This This is is based based on on e
Page 129 and 130:
Secondary stress [N/mm^2] Secondary
Page 131 and 132:
After After the the wire stresses s
Page 133 and 134:
124 The above is summarised in Fig.
Page 135 and 136:
The The trend trend determined here
Page 137 and 138:
128 Even if the conductor model pos
Page 139 and 140:
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
Page 153 and 154:
144 CIGRE: Study Committee 22 - Wor
Page 155 and 156:
146 Papailiou, Müller, Roll: Anwen
Page 157 and 158:
List of symbols Latin symbols 148 a
Page 159 and 160:
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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