Internal Contact Mechanics of 61-Wire Cable Strands

Abstract

Background

We provide an insight into the internal mechanics of parallel wire strands, as found in suspension bridges ranging from the historic Brooklyn to the state-of-the-art Akashi Kaikyō Bridges. The confinement stresses of wires in a bridge cable are critical in generating sufficient friction for load transfer in the case of wire fracture or creep/relaxation due to fire damage.

Objective

We aim to non-destructively probe the internal stress distribution of the constituent wires of a 61-wire bridge cable strand model. These internal contact forces will, in turn, inform numerical models with realistic boundary conditions.

Methods

A parallel 61-wire strand is measured in the VULCAN Engineering Materials Diffractometer under various clamping loads. During the experiment, each of the 5 mm diameter constituent wires is interrogated using a neutron beam to measure four independent strain components, including in-plane shear strain, ultimately providing a full in-plane stress tensor.

Results

The measured strains show that confinement stresses vary significantly among the wires of the cross section. Under standard clamping loads, only half of the wires in the cross section carry significant compression load while the remainder of the wires are largely unstressed with respect to clamping.

Conclusions

We show that this heterogeneity is caused by micron-scale dimensional variations of the constituent wires due to manufacturing tolerances during wire production, resulting in a stochastic stress field among the various wires. Finally, we present a genetic algorithm optimization code that is capable of back-solving the inter-wire stiffnesses of strands for various loading scenarios and boundary conditions.

Appendix

Alignment Strategy

The alignment strategy for this experiment closely follows the procedures outlined in [18]. For each clamping load case and orientation ($\varphi$), select wire rows are strafed with the neutron beam at 1 mm increments in $\overrightarrow{x}$ with integrated intensity counted on Bank I. For areas where a clear peak of each wire is not apparent on the first run, the sweep increment is reduced to 0.5 mm. Ultimately, 19 benchmark wires of the strand are aligned in $\overrightarrow{x}$ and $\overrightarrow{z}$ using Gaussian peak fitting and the 61 wire array fitted using a geometric least squares fit with four free parameters of $\overrightarrow{x}$, $\overrightarrow{y}$, $\varphi$, and $\rho$, the latter being a packing constant that adjusts the density of the packing matrix.

Considering that this sample was, at the time, the most complex multi-body system measured on the VULCAN diffractometer, this procedure required extreme care and precision to prevent any errors in measurement. The triply redundant measurement of ${\varepsilon }_{22}$ serves as an error check to ensure that the neutron beam is in fact reliably centered in each wire and that the beam is not partially buried in the sample. A partially buried beam volume results in spurious strain readings that are incredibly difficult to deconvolute after the fact [18]. Since nearly the same gauge volume (and therefore the same atomic lattice) is illuminated for ${\varepsilon }_{22}$ at different rotations $\varphi =0^\circ , 45^\circ , 90^\circ$, any outliers in the measured lattice parameters in $\overrightarrow{y}$ for any wire implies bad alignment or strong texture. With an average fitting error of 35 με, any measurement with a deviation in precision greater than 60 was flagged, the wire realigned, and measured again. In the case of the High Clamping Load Case, this resulted in seven out of 61 wires being realigned and measured again. In all cases, the realignment resulted in data with good statistics.

Rietveld Fitting Error Optimization

A counting optimization was performed at the onset of the experiment, considering wires #5.1 and #5.13 as representatives of the best and worst case scenarios regarding neutron beam sample penetration path length. Recall that an engineering diffractometer’s setup has two collimated banks at ± 90° to the incident beam (Fig. 2). At VULCAN, Bank I, which measures ${\varepsilon }_{22}$ in this experiment, is also called the transmission bank, in that the path length of the diffracted neutron beam through the sample remains invariant with respect to the $\overrightarrow{x}$ position of the sample stage. In contrast, the through-sample path length of a neutron that is captured by Bank II (measuring ${\varepsilon }_{11}, {\varepsilon }_{33}, {\varepsilon }_{13}$, depending on sample rotation, $\varphi$) depends heavily on $\overrightarrow{x}$ position, varying from near zero to twice that of the invariant path length of Bank I. Bank II is therefore subject to significant attenuation for far measurements, requiring longer counting times for accurate peak fitting for that strain component. The experiment’s counting test is shown in Fig. 11 for the near wire (#5.1) and the far wire (#5.13).

Fig. 11

Rietveld peak fitting optimization, showing fitting error in microstrain vs. counting time for both Bank I and Bank II. The error bars signify standard deviation of error (1σ) over the measured ensemble

The sample was subsequently broken into three counting zones, with counting set to 7, 10, and 15 min for Zone 1, 2, and 3, respectively. The VULCAN diffractometer incident beam monitor 2 (bm2) was used as the metric for measurement time (measurement time is defined in neutron counts to account for flux variation and temporary outages).

Large grain size and texture were considered as potential sources of error during peak fitting. Micrographs of etched samples of the Al wire confirmed that the grain size in both longitudinal and transverse directions was sufficiently small such that the diffracting volume would reliably generate a lattice strain reading representative of the bulk material strain state. Furthermore, multi-peak fitting Rietveld refinement was used to prevent any specific crystallographic orientation from skewing the derived macro stress state. Figure 12 shows the TOF diffraction spectra of an Al wire sample and its Rietveld multi-peak fit. In the transverse direction (Bank II), the wire shows no preferred orientation. In the longitudinal direction, (Bank I), [200] texture can be observed.

Fig. 12

TOF spectra of one of the Al wires showing FCC phase in longitudinal Bank I (left) and transverse Bank II (right). Neutron counts are shown as red ticks, the Rietveld fit in green, and residual (fit error) in purple. [200] texture is evident in the longitudinal direction. Bank II, however, shows a well-structured TOF spectrum with all major peaks

However, the strong solution-finding stability and adherence of the resultant fits to physical parameters (rather than arbitrary fits with no physical meaning) demonstrate that the global refinement achieves a robust solution that is representative of the bulk material’s elastic stress state.

Further, the statistics of the triply redundant ${\varepsilon }_{22}$ measurements provide a clear measure beyond simple fitting error, as provided by the Rietveld algorithm (Fig. 11). Measurement precisions (1σ of each ensemble of three ${\varepsilon }_{22}$ readings) are provided in Table 3.

Table 3 Average measurement precision of all 61 wires for each load case

Yield Stress Piecewise Linear Curve for Aluminum Wires

The following plastic strain-yield stress curve was used in the FEM model of the 61-wire Al strand model, Table 4.

Table 4 Yield stress piecewise linear curve for Al wires