Generally, the bending moment-curvature analysis is a means to accurately determine the load- deformation behaviour of a structure using nonlinear material stress-strain relationships. For a given axial load for a conductor there exists an extreme axial strain and a section curvature at which the nonlinear stress distribution is in equilibrium with the applied axial load. The extreme strain and section curvature can vary for a range of moment-curvature values. A conductor is considered as a composite structure with a peculiar internal sub-structure (the strands) that directly affects its overall mechanical response. This form of structural arrangement of strands makes the analysis of moment-curvature very difficult.
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The analysis of the moment-curvature relations, starting with the single layer conductor. Starting with a single layer conductor has the advantage of extending the concepts derived for the single layer conductors to the multilayer conductors. When a curvature is imposed on the conductor, the lay cylinder is deformed as shown in figure (4.6) and the curvature varies along the strand path over a pitch length. The axial load on a conductor generates a normal contact force at the contact region, see details in chapter 2. As the curvature is imposed, slip between core and helical strands are prevented by friction force, generated by the normal contact forces. According to Coulomb’s law, the no-slip condition implies that at each point on the contact lines, imposed shear force is less than the friction force. Hence, the conductor is assumed to behave like a solid continuous beam. This is the stick state and the classical Bernoulli-Euler beam theory can be applied. Using the Bernoulli-Euler hypothesis, one gets the bending moments as the strands moments are proportional to the distance from the conductor section neutral axis. A unique bending moment can be calculated at this section curvature for the stress distribution.
Also discussed in chapter 2, due to the imposed stress, the bending moment on a conductor strand has two components: the normal and secondary moments. The combined moments can be expressed as:
ondary
normal
M
M
M
sec ……….……. (4.15)Using the Bernoulli-Euler hypothesis the conductor bending stiffness takes its maximum value EImax. The corresponding bending moment is given by the usual expression and for the stick condition:
EI EIcomp EI
M max min ……….……. (4.16)
The total bending stiffness of the conductor before any slippage occurs must also include the strands own bending stiffness EImin. Thus the conductor maximum bending stiffness is:
EIcomp
EI
EImax min ……….……. (4.17)
Expressing the above equation in terms of the bending stiffness and curvature:
EImax EImin EIsec
M ……….……. (4.18)
When the strand is bent, strands tend to slip relatively over each other, as a consequence of the axial force gradient generated along their length. Relative displacements are contrasted by the friction forces, which are developed at the contact surfaces as a function of the internal geometry of the strand, the material properties of its components and the inter-layer contact pressures.
When the forces which tend to initiate the sliding are greater than the friction force, then strands undergoes relative displacements with respect to their neighbouring strands. If, at a certain location
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in given layer, i, strand (i) is in the slip regime, then its normal force Fi contributes a moment with respect to the neutral axis which is given by:
i, j F
e , sin 1
Rsin
i,j cosM T i j ……….……. (4.19)
e
i j RFM
i
j i T
Comp 2 cos 1sin ,
3
1
sin
,
……….……. (4.20)
For the slip condition
sec
min M
EI
EI ……….……. (4.21)
Figure 4.7: The graph of bending moment versus curvature [57]
The diagram for bending moments against curvature for the single conductor is shown in figure (4.7). The slope of a straight line corresponds to the residual bending stiffness EIi: EI corresponds to EImax, EI1I corresponds to the (n-1) sticking layers plus the layer, i, minimum bending stiffness when total slip has been reached.
As illustrated in the diagram, as long as k < kS, there is no slip, and the conductor is assumed to behave as a solid beam and indicated in diagram, region I illustrates the behaviour of the conductor in this state.
When the conductor experiences slip, it goes through a transition from region I to region II. At this stage moment-curvature (M vs k) relationship is indicated by region II. Beyond complete slip curvature, the total moment on the section can be obtained as:
sec
min M
EI
MII
……….……. (4.22)101
In figure (4.4), it will be imperative to determine the curvature for two limit cases of bending. The first correspond to a full stick state, in which friction forces are high enough to prevent any relative sliding among strands all along the strand. In this case, the cross sections of the strand remain plane and their bending stiffness takes the upper bound value, EImax, close to that of a solid circular beam with the same diameter as that of a similar conductor. The second limit case, instead, is attained when the friction forces are no longer able to contrast any relative displacement between stands, which as a consequence behave independently. In this case the plane section hypothesis is no longer valid and the cross section bending stiffness takes the lower bound value, EImin. The expressions to evaluate the bending stiffness limit values [52]:
i i i N
i i i
i N
i
i n R EA
EI n
EI
EI
2 3 1
1 0
max cos
cos 2 ……….……. (4.23)
i i N
i
i EI
n EI
EI
cos
1 0
min ……….……. (4.24)
The next step is to determine limit curvatures which have to be imposed on the conductor strands in order to attain these bending moments or bending stiffnesses. Based on work as documented in [57], the curvature imposed on a strand as the starting and end curvatures from start and completion of the strand displacement were determined for a conductor cross-section, based on the balance of forces on the individual affected strand elements. The imposed dynamic loading resulting in a change of curvature, and when a critical curvature is exceeded (k > kS), slip occurs over a bounded domain of each contact line. Because of the imposed curvature, normal force on a conductor cross section is a function of cross sectional angle 2 2 from the neutral axis.
The starting and end curvature can be evaluated as [58]:
i i
i
i i
S E R
2
cos
sin ……….……. (4.25)
i i
i i
E E R
e i
2 sin 2
cos 1
……….……. (4.26)
To extend the bending moment-curvature relation developed for single layer conductors to multi- layer conductors, will pose some challenges. This is because, how to determine the curvature necessary to initiates slip in different layers. This poses the question, how will slip be initiated in various layer in the multiple layer conductor. The problem encountered in this regard is how the slipping pattern from one layer to the other is. There are two hypotheses in this regard. The first is the notions that slip in a given layer start and stop, before being imposed on the next layer [58].
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This assumes that there is no overlap of the slip phases, meaning total slip is achieved in one layer before the next slip starts in that next layer. The second hypothesis is that slip in a given layer will come to an end after another slip has been initiated in the next layer as documented in the paper by K. Hong [13]. This slipping process will make it difficult to define the boundary conditions between stick and slip zones as a complete slip is never reached in any layer.
The first hypothesis will be used in this study which assumes slip occurs sequentially and indicate that slip will only starts in layer, i, only after it has been completed in layer, i -1,. The reason for this sipping process is that the slip always starts on the conductor layer where strand stress at those two locations keeps its initial at zero curvature. Because the inter-layer pressure is higher between inner layers, and also because the maximum tensile force on the strands (due to bending) is smaller due to their closeness to the neutral axis, slip phases start sequentially, from the outer layer to the inner ones.
As explained in [57] moment the curvature relationship for multiple layered conductors Mi
EI
M max
N
N1
1 ……….……. (4.27) Where the residual moment of friction is
1 ,
N i
i rf
i M
M
The expression for the total slip state for a given layer, i, when reached, the friction residual moments between layers, i, and i +1, is given as
2
1 sin ,
, 2 cos 1sin
i
i i
n
i
i i
i A i i
rf RF e
M
……….……. (4.28)This condition is the same as the one obtained for the single layer conductor. This yields the same equations for the limit curvatures, but the parameters are those pertaining to each layer, i.
Thus, for layer, i, the incipient slip curvature is given by the modified equations (4.25) and (4.26) respectively:
i i
i
i i i i
S E R
2
cos
sin ……….……. (4.29)
1
cos
sin 2
2
e i i
R
Ei i i
i i
E
……….……. (4.30)
The above two equations was used to determine the inception and completion of the slip in various layer as the slip start from the outer layer and propagate through various layer to the inner layer,
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thus the curvatures takes the form
N
N1...
2
1. The plot of the bending moment against the curvature is shown in figure (4.8).Figure 4.8: The plot of bending moment versus curvature for multi-layer conductor