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

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3 The economic importance lies in the fact that, especially with high voltage transmission lines, i.e. 400 kV lines in particular, the conductors are the most expensive single component and also the most important factor determining the overall cost of the line, since the choice of conductor material, cross sectional shape and area and the conductor tension are determining factors for the design of the rest of the transmission line components, i.e. the fittings, insulators, towers and foundations. There have been many unsuccessful attempts at solving this problem analytically because the overhead conductors, even though they appear to have a “simple” construction, are quite difficult to master mathematically, especially with respect to their mechanical behaviour [EPRI, 1979, Pg 53]. For this reason we have resorted to measuring the vibration stress on overhead conductors live in the field, using specially designed recorders [Papailiou, 1987], for over 30 years now. This trend has clearly intensified over the past years, assisted by the introduction (thanks to modern electronics) of high performance and compact recorders, Fig. 1.3, which greatly simplify storage and analysis of the measurements, to the point that such measurements are largely standardised today [CIGRE, 1994] and are being carried out worldwide. Suspension clamp Cable/conductor Fig. 1.3 Modern conductor vibration recorder in measurement position (dimensions in mm) 89
4 This device measures the deflection amplitude of the vibrating conductor at a short distance (3 1/2" = 89 mm) from the suspension point, which is assumed to be solidly clamped, Fig. 1.3, and converts this amplitude to a conductor bending stress. This conversion is based on the assumption of a simple quasi-static conductor model [Poffenberger and Swart, 1965]: the short conductor section (89 mm length) on which the deflection is measured is considered as a clamped cantilever influenced by the conductor tension and the displacement caused by the vibration amplitude. The bending stiffness assumed for this conductor cantilever is the sum of the bending stiffnesses of the individual wires (also wire stiffness) in the conductor. This assumption supposes that the individual wires of the conductor are each bent around their own longitudinal axis. In this way the strain (elongation) can ultimately be calculated for the outer layer of wires of the conductor. This model has the advantage of simple application, but its validity is often in doubt. It has, in particular, often been found (Hondalus, 1965, Claren and Diana, 1969, CEA, 1986, Ramey, 1987) that the wire stresses calculated in this way only represent a trend and do not always reflect the actual stresses in the individual wires of the conductor. There appears to be a large discrepancy between the stresses calculated as outlined above and the actual stress in the wires, especially in the inner layers, where wires are often found broken. In addition, the stresses calculated in this way must often be compared with the service strength characteristics of the conductor in order to estimate its service life. The latter are S-N curves found in the laboratory (CIGRE, 1985) or design limits derived therefrom (EPRI, 1979). A further problem arises from this comparison, demanding a better model for bending of conductors. This is because most of the S-N curves for conductors in the laboratory are measured on conductors excited to produce characteristic standing waves and then measuring the maximum bending amplitude of the conductor. Using the simple conductor cantilever model described above, these maximum bending amplitudes are then converted to bending stress at the conductor clamp. It is unfortunately not possible to directly compare these bending stresses with the bending stresses found from measurements in the field by converting the measured bending amplitude of the conductor near the suspension (clamp). We are even explicitly warned against this (EPRI, 1979, Pg 60) because it is known that the assumption of a constant bending stiffness of the conductor represented as a homogeneous cantilever does not take sufficient cognisance of the conditions as the conductor bends, which may lead to large uncertainties in the conversion of measured displacements (in the conductor as a whole) to stresses in the individual wires of the conductor.
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: 2 Their acquisition value is estima
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
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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
Page 147 and 148:
The position of the catenary in the
Page 149 and 150:
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
Page 157 and 158:
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
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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