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1
2021 (Vol. 22)
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21
Metal Forming CIS Iron and Steel Review — Vol. 22 (2021), pp. 20–25
According to the technique of M. F. Glushko [3],
According to the technique of I. P. Getman and Yu. A.
Ustinov [8],
where Е and G – modules of elasticity and shift of wire
material; F
i – square of cross section of layer wires; I i and
I
ρi – axial and polar cross section inertial moments; α i —
laying layer angle; r
i – average layer radius; k 1 – relation of
summarized square of wires cross sections ΣF
i and square of
rope cross section as a round cylinder F
К; a – rope radius as
a round cylinder; ν – Poisson coefficient for wires material;
α – laying angle for external layer wires.
Solutions of the equations (1) relating to deformations ε
and θ are presented as [3]:
where Δ = A ⋅B − C
2.
The existing calculation methods can’t be used uncon-
ditionally for SSS determination of steel ropes, e. g. because the techniques [3, 8] don’t take into account friction forces
and contact interaction between the rope elements.
Computer-aided simulation of ropes SSS based on the
finite element modeling (FEM) using different software
products can be considered as an efficient tool for in-
crease of workability during operation of multi-layer ropes
[5, 6, 10, 13–15]. This tool does not give up the experi-
mental and analytical methods. The number of researches
on computer-aided simulation of closed ropes SSS is re-
stricted [5, 16].
Comparison of SSS calculation results for closed rope dur-
ing extension and twisting via conventional algorithms and
computer-aided simulation methods is the aim of this work.
Materials and methods of investigations
Analysis of a closed rope SSS during axial extension
and twisting was conducted via FEM method using li-
censed software package SIMULIA/Abaqus, allowing to
obtain the results with required accuracy and productivity
[6, 17, 18]. The cross section of simulating closed rope is
shown on the Fig. 1. The core 1 of 1 + 7 + 7/7 + 14 con-
struction is presented by the rope with ordinary laying
from round wires with axes having shape of screw lines.
These wires are laid with three layers with the same pace
around the central straight wire.
Laying direction of the external layer 6 from Z-shaped
wires and of the layer 4 from round wires (positions 4 and 2
on the Fig. 1) was considered as right (positive), while laying
direction of the alternating round and H-shaped wires and
the core (positions 3 and 1 on the Fig. 1) was considered as
left (negative). If laying directions are different, the layers
from round wires contact in a points, while the contact of
shaped section wires can be considered as linear one [3].
Parameters and dimensions of 20.5 mm diameter (d) rope
elements that were used during simulation and the same as
those presented in the researches [5, 18]. The length simu-
lating rope sample l
о = 130 mm, elasticity module of wire
material Е = 2⋅10 5 MPa, friction coefficient μ = 0.1.
The edges of wires layers at the rear end of simulating
rope were firmly connected with the edge surface of a mov-
able auxiliary hard disk, and at the front end – with coaxi-
ally located fixed hard internal disk (connected with the core
1 + 7 + 7/7 + 14 — position 1 on the Fig. 1) and rings (con-
nected with the layers 4–6 — positions 2–4 on the Fig. 1).
The scheme of analytical SSS calculation of the spiral
rope includes the following steps [3]: determination of the
generalized rope stiffness coefficients (A, B and С) accord-
ing to the equations (2–4) и (5–7); calculation of rope de-
formation ε and θ with preset external load as axial force P
and torque M, according to the equations (8); evaluation of
distribution of force and torque load along the rope layers
and internal forces in transversal cross sections of ropes; de-
termination of stresses occurring in transversal cross sections
of rope wires.
Definition of SSS parameters for the closed rope ele-
ments during simulation was conducted for the following
loading variants: (2)
(3)
(4)
n
i
i
iii
n
i
i
i
ii i
i
n
i
i
i
ii i i i
n
i
i
i
ii i i
i
n
i
i
ii i AA
EF EI
r
GI
r
CC
EF r GI
r
EI
r
BB
EF r() =
=
ρ
=
ρ
= ==
⎡⎤
α
α+ ×
⎢⎥
⎢⎥
=
⎢⎥
α
⎢⎥
×α+ α
⎢⎥
⎣⎦
==
⎡⎤
α
αα+ ×
⎢⎥
⎢⎥
=
⎢⎥
α
⎢⎥
×α− + α α
⎢⎥
⎣⎦
==
α
=∑
∑
∑
∑
∑ i1
4
3
2
1
6
32
2
1
4
2
2
=1
323
1
2
sin
cos
sin
cos cos
cos
cos sin
cos
sin 1 cos sin
cos sin
n
iii
i
iiii GI
EI ρ
= ⎡⎤
α+ α+
⎢⎥
⎢⎥
+ααα
⎣⎦
∑
27
1
222 cos
(1 + cos ) sin cos
(8) BC
PM; ε= − ⋅
ΔΔ CA
PM
θ=− ⋅ +
ΔΔ
(5)
(6)
(7) AkaE
CkaEtg
BkaEtg⎡⎤
ν
⎛⎞
=π − + α
⎜⎟
⎢⎥
⎝⎠
⎣⎦
⎡⎤
⎛⎞
−+ν α
⎜⎟
⎢⎥
⎝⎠
⎣⎦
=π ⋅ α
⎡⎤
ν
⎛⎞
−+ α
⎜⎟
⎢⎥
⎝⎠
⎣⎦
=π ⋅ α 22
1
2
3
1
2
42
1 11 sin
2
4
1sin
3
2
33
1sin
24
3

CIS Iron and Steel Review — Vol. 22 (2021), pp. 41–47 Metal Science and Metallography
44
Table 3. Composition of clusters of actual inclusions found
in defects of welded joints
Clusters1234
O-Al-Mg-Fe Mg-O-Fe Fe-O-Al-MgO-Ca-Al-
Fe-Mg
Fraction in
defects, P.38 12.98 11.45 25.19
Elemental composition of NMIs clusters, %
Ca 4.86 1.48 4.17 20.43
O 41.74 36.62 25.42 36.90
S 0.00 0.12 0.23 3.41
Mg 11.17 51.02 7.02 5.58
Al 35.21 1.40 9.69 10.53
Si 0.60 0.03 1.78 4.54
Fe 5.34 9.14 47.82 10.02
K+Na 0.54 0.16 0.65 1.51
cluster to the centers of other clusters (distortion 2). As
the number of clusters increases, distortion 1 decreases and
distortion 2 increases (Fig. 3).
The optimal number of clusters was chosen based on the
analysis of these two distortions versus the number of clusters:
the separation of inclusions into groups was completed when
the change in these curves stabilized and did not lead to the
appearance of clusters with markedly different compositions.
Based on these considerations, for inclusions found in the
vicinity of defects and for indigenous NMIs found by ther-
modynamic simulation, the number of clusters was chosen
to four for each type of inclusions (Fig. 3).
The compositions of clusters — products of deoxidation and
modification in J55 API 5L (22GYu) steel are given in Table 2.
As follows from Table 2, three clusters out of four (No. 2, No. 3
and No. 4) contain compounds of the Ca-O-S system.
More over the cluster No. 4 consists almost only of
CaO-CaS with approximately equal concentrations of oxy-
gen and sulfur. Indeed, as follows from Fig. 1, at 1550 °C
and [Ca] ≥ 0.005 % almost only CaO-CaS inclusions are
formed.
Using the same clustering technique, all compositions
of inclusions found in defects of welded joints were pro-
cessed (Table 3).
Note that the first three clusters do not contain sulfur, or
its content does not exceed 0.23 %. Only the fourth cluster
contains 3.41 % of sulfur (Table 3), which is 2–6 times less
than the concentrations that should be in any of the clus-
ters of indigenous inclusions-products of deoxidation and
modification (Table 2). On the basis of this analysis, we can
already state that the contribution of indigenous inclusions
to the formation of defects in welded joints is insignificant.
For illustrative purposes, we use radar diagrams to com-
pare the compositions of the clusters of actual inclusions
found in defects of welded joints with the compositions of
indigenous NMIs clusters (Fig. 4a to Fig. 7a). The composi-
tions of all studied actual inclusions for each cluster and their
descriptive statistics are given in Fig. 4b — Fig. 7b.As follows from Table 3 and Fig. 4, more than 50 % of
all inclusions found in the vicinity of the studied defects
belong to cluster No. 1 and are pure magnesia spinel or
Al
2O3-MgO with a small content of calcium aluminates.
These NMIs are erosion products, which can be pure re-
fractory particles or the same particles but impregnated
with deoxidation and modification products. The presence
of alkaline earth metals indicates the interaction of these
NMIs with the slag in the tundish or in the crystallizer.
As a rule, these inclusions are located on the welded edge
near the fusion line in a discontinuity extended along the
lines of metal flow.
As follows from Fig. 5, cluster No. 2 is practically pure
magnesium oxide MgO, of which refractories (periclase or
magnesite) are made, less often it is impregnated with small
amounts of liquid calcium aluminates — products
of deoxi-
dation and modification of steel. Along with these inclusions,
NMIs from cluster No. 4 (calcium aluminates with magnesia
spinel) occur. A discontinuity with such inclusions is located
near the fusion line along the metal flow lines, usually with-
out reaching the external surface.
As follows from Fig. 6, cluster No. 3 is almost pure iron
oxide with admixtures of calcium aluminates and/or mag-
nesia spinel.
The defects are mainly located on the fusion line, related
to “cold welding” and are formed due to the low temperature
of the welded edges during pipe upsetting.
As follows from Fig. 7, exo-indigenous inclusions consist-
ing mainly of deoxidation and modification products (calcium
Table 2. The compositions of clusters — products of deoxidation
and modification in J55 API 5L (22GYu) steel
Clusters1234
O-Ca-Al-MgCa-O-Mg-
Al-S-SiCa-S-O-Mg Ca-O-S
Elements Elemental composition of NMIs clusters, %
Ca 24.42 36.75 49.09 63.61
O 39.57 31.97 18.79 17.82
S 0.07 6.65 20.53 17.27
Mg 8.78 10.96 6.11 0.67
Al 22.27 6.69 2.48 0.35
Si 2.33 6.07 2.66 0.24
Fig. 3. Distortions to choose the optimal number
of clusters of actual and indigenous NMIs
in J55 API 5L (22GYu) steel
Distortion
Distortion
Number of clusters
actual NMIs
indigenous NMIsdistortion 1
distortion 1 distortion 2
distortion 2