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Effect of characteristic parameters of pitting on strength and stress concentration factor of cable steel wire
Construction and Building Materials 240 (2020) 117915
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Effect of characteristic parameters of pitting on strength and stress concentration factor of cable steel wire Rou Li a,b,⇑, Changqing Miao a,b,⇑, Jie Yu b a b
Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing 210096, China School of Civil Engineering, Southeast University, Nanjing 210096, China
h i g h l i g h t s The effect of pits on the stress concentration factor of steel wires is investigated. Parametric analyses were performed on steel wire with different pits parameters. The formula of stress concentration factor of steel wire with pits was established.
a r t i c l e
i n f o
Article history: Received 20 June 2019 Received in revised form 29 November 2019 Accepted 19 December 2019
Keywords: Corrosion pit Steel wire Strength Stress concentration factor Calculation model
a b s t r a c t In order to investigate the influence of characteristic parameters of pits on the strength and stress concentration factor of cable steel wire, 198 cable steel wires with pits were manually prepared. The variation law of stress distribution of steel wire was analyzed, and its relationships with strength and stress concentration factor were studied through tensile test and finite element analysis. On this basis, the calculation model of yield strength and stress concentration factor of steel wire with pits was established. The influences of secondary pits on stress distribution and stress concentration factor of steel wire were also analyzed. The results showed that the strength of steel wire decreased gradually and the stress concentration factor increased with the increase of pit depth and the decrease of pit width. The change of pit clearance on the same side had no obvious effect on the strength and stress concentration factor of steel wire, but it had significant effect when located on the opposite side. The strength and stress concentration factor of steel wire with adjacent pits usually depended on the depth of the larger pits. The pits with depth to width ratio of 1 to 2 had the most significant effect on the stress concentration factor. Moreover, secondary pit would change the stress distribution, and the position of maximum stress changed from near the mouth to the bottom of the pit. The stress concentration factor of secondary pit was obviously higher than that of steel wire with only the primary pit. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Modern cable-stayed bridges, known as greater capacity for spanning, easy construction and beautiful radial shape, have been widely applied to bridge construction [1–3]. The stay cables are the main structural components that transfer the weight and load of the main girder and bridge deck to the pylon. Whether they can be used normally and safely directly determines the safety and reliability of the whole cable-stayed bridge [4,5]. However, the polyethylene jackets will inevitably crack due to the bending stress originated from the vibrations of cable or relative movements ⇑ Corresponding authors at: School of Civil Engineering, Southeast University, Nanjing 210096, China. E-mail addresses: [email protected] (R. Li), [email protected] (C. Miao). https://doi.org/10.1016/j.conbuildmat.2019.117915 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.
between the main cable and the girder under long-term corrosion environment [6,7], then the high strength steel wire inside the cable are usually difficult to avoid corrosion through protective measures, structural measures and maintenance systems, resulting in a serious corrosion problem [8–10]. Stafford and Watson [11] found that nearly 200 cable-stayed bridges around the world were facing danger due to cable corrosion based on the external investigation of cable-stayed bridges. Kohlbrand Estuary Bridge in Germany had to replace all the cables at a cost of up to $6 million, due to serious corrosion at the anchorage end of the lower part caused by water immersion into the cables [11]. In addition, the cables of Haiyin Bridge in Guangzhou adopted more advanced protection system, electrochemical corrosion still took place, and eventually a sudden fall occurred [12]. On the whole, the above collapse accidents found that corrosion
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Nomenclature ABC D E fy fu fy’ fu’ H H1 H2 Hs Kt L Py PyFEA Pu
all functions containing variable H/D clearance of pits on the same side elastic modulus yield strength ultimate strength average yield strength average ultimate strength depth of pit depth of primary pit depth of secondary pit depth of adjacent pits on the same side stress concentration factor distance of pits centerlines on the different side yield load yield load by FEA ultimate load
caused degradation of the performance of high-strength steel wire, which would further reduce the carrying capacity of cables, thus bringing a severe test to the safe service and normal use of cable-stayed bridges [13–15]. Therefore, it is necessary to investigate the degradation law of mechanical properties of corroded cable steel wires and establish the relationship between mechanical properties and corrosion characteristic parameters, which can be used to accurately evaluate the degradation degree of mechanical properties of corroded cable steel wire by corrosion characteristic parameters. Previous researches were mainly focus on the mechanical properties of corroded steel plate and rebar, hence limited data is available for the mechanical properties of cables steel wire with corrosion damage. Garbatov et al. [16] found that with corrosion damage increasing, significant degradations in strength and deformation of corroded steel plates can be observed. Xu et al. [17] and Xia et al. [18] investigated the effect of corrosion pits on the strength and ductility of the corroded steel, and the strength and ductility decreased significantly with the increase of pits depth. The damage parameters of corrosion steel and their effects on the mechanical properties of the steel were studied from the corrosion surface, and the constitutive model of corrosion steel was established and the variation of parameters was also analyzed [19]. However, corrosion may cause different influences on different kinds of steel and thus it is important to investigate the mechanical properties of steel wire with corrosion damage. Moreover, although a little research had been done on the degradation of mechanical properties of corroded steel wires in cables, there was no generally accepted definition. R. Betti et al. [20] used acetic acid to regulate Nacl and Na2SO4 solution as corrosive medium to carry out corrosion test of high strength galvanized steel wire. It was found that corrosion resulted in decrease of tensile strength and yield strength of steel wire. Nakamura et al. [21] obtained four grades of steel wire by accelerated corrosion, and determined the corresponding relationship between corrosion degree and mass loss ratio. The research showed that the actual tensile strength of steel wire after corrosion had not decreased. Ma et al. [22] tested the mechanical properties of steel wires with different corrosion degrees, and found that the tensile strength and elongation decreased sharply when the corrosion degree was serious. Mayrbaurl et al. [23] systematically studied the crack propagation and fracture of steel wire after water entry into cable structure, which could be used as the basis for the strength study of cable structure. However, the overview of the state of the art showed that most studies on the corrosion degree of steel wires usually took mass
PuFEA T W W1 W2 Z
r rnominal rpeak ry r0 y e c
4u 4y
ultimate load by FEA diameter of steel wire width of pit width of primary pit width of secondary pit difference between ultimate strength and yield strength stress nominal stress peak stress yield strength of non-corroded steel wire yield strength of corroded steel wire strain poisson’s ratio error of ultimate load error of yield load
loss ratio (weight loss ratio) as a characterization method, but its essence was the average mass loss ratio of steel, difficult to reflect the impact of local corrosion. Furthermore, those researches were mainly based on the statistics results of tensile test, and seldom investigating the effect of local pit distribution on mechanical properties and stress distribution of steel wire. The relationship between mechanical properties and pit distribution parameters of steel wire was still not established. The size of the pits directly determines the stress concentration factor, which directly leads to the reduction of the fatigue strength. Cerit [24,25] studied the stress distribution of semi ellipsoid pits by finite element analysis. It was found that the depth to width ratio of pits was the main factor affecting the stress concentration coefficient, and the existence of secondary pits had a significant effect on the stress concentration coefficient. Sankaran et al. [26] pointed out that the effect of pitting on the fatigue life of 7075-T6 aluminum alloy members was related to the equivalent stress concentration coefficient. Xiang et al. [27] proposed that the stress concentration factor of the hemispherical pits was about 3, and pointed out that the stress intensity factor of the crack at the root of the structural groove can be calculated by using SCF as a function of variable. Makhlou et al. [28] studied the corrosion crack growth of duplex stainless steel in air and artificial seawater environment. The results showed that the stress intensity factor was a function of the load applied to the structure, the width and thickness of the specimen, and the ratio of the length and width of the surface crack of the specimen. Nakamura et al. [29] found that the stress concentration was larger for sharper pit shapes, indicating that this was the major cause for the decrease of fatigue strength by conducted fatigue tests of wire specimens with artificial pits. To investigate the influence of pit characteristic parameters on the strength and stress concentration factor of cable steel wire, 198 cable steel wires with different pit characteristic parameters were obtained by manually prefabricating pits. The influence of pit depth, width, pitch and location on the strength of cable steel wire was discussed through monotonic tensile test. The stress distribution law of cable steel wire with different pit characteristic parameters was compared by finite element numerical analysis, and the relationship between pit characteristic parameters and stress concentration factor was also investigated. On this basis, the calculation model of yield strength and stress concentration factor of steel wire with corrosion pits were established. Finally, the effect of secondary pits on stress distribution and stress concentration factor of cable steel wire were also investigated.
R. Li et al. / Construction and Building Materials 240 (2020) 117915
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Fig.1. The size of tensile specimen.
2. Specimen design The galvanized steel wire with strength grade of 1670 MPa was selected as the test material. Its nominal diameter of section was 5 mm and total length was 400 mm. The size of tensile specimen was shown in Fig. 1. The mechanical properties of steel wire were shown in Table 1. The size of the pits directly determines the stress concentration factor, which directly leads to the reduction of the fatigue strength [29]. Therefore, it can be considered that the bearing capacity of corroded steel wire was closely related to the pit size parameters of corroded steel wire. Cerit et al. [25] through experiments found that the shape of the pit in real bridge was broad and shallow hemispherical or narrow and deep hemispherical. The hemispheric pit was really a special case with the same length, width and depth of ellipsoidal pit. Elliptical pits with different lengths, widths and depths were more common in practical engineering [30–32]. According to the statistics of pit shape and study of pit formation process, the artificial prefabricated pits could be used in the tests
to represent the corrosion pits in real bridge and the pit shape was simplified to semi-elliptic sphere. The characteristic parameters of artificial prefabricated pits were shown in Fig. 2. The pit size for pit depth was that keeping the width of 5 mm and changing the pit depth, and for pit width was that keeping the pit depth of 1 mm, changing the width (See Fig. 2(a)). Fig. 2(b) was the net distance between the edges of the pits on the same side, that keeping the pit size (Width 5 mm and Depth 1 mm), and changing the net distance between the edges of the pits. Fig. 2(c) showed the distance between the centerlines of different pits, that keeping the pit size (Width 5 mm and Depth 1 mm), and changing the distance between the centerlines of different pits. Fig. 2(d) was the depth of adjacent pits on the same side that keeping one pit size (Width 5 mm and Depth 1 mm), and only changing the depth of the anther adjacent pits (Width 5 mm). There were 114 tensile specimens in total, of which 3 were parallel in each group. The specific values of the different characteristic parameters of pits were shown in Table 2. 3. Tensile test of corroded cable steel wire
Table 1 The mechanical properties of steel wire.
3.1. Test device
Material
E/GPa
fu/MPa
fy /MPa
c
Galvanized steel wire
205
1670
1540
0.3
The test was carried out on CMT5105 computer controlled electronic universal testing machine (See Fig. 3). The loading speed of
Fig. 2. The characteristic parameters of artificial prefabricated pits.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Table 2 The different characteristic parameters of pits. Characteristic parameters
Number
Value/mm
Invariant parameter
Pit depth H
H-0.0 H-0.2 H-0.4 H-0.6 H-0.8 H-1.0 H-1.5 H-2.0 W-2.0 W-3.0 W-4.0 W-5.0 W-6.0 W-10 Hs-0.0 Hs-0.2 Hs-0.4 Hs-0.6 Hs-0.8 Hs-1.0 Hs-1.5 Hs-2.0 D-0 D-1 D-2 D-3 D-4 D-5 D-10 D-15 L-0 L-1 L-2 L-4 L-5 L-10 L-15 L-20
0 0.2 0.4 0.6 0.8 1 1.5 2 2 3 4 5 6 10 0 0.2 0.4 0.6 0.8 1 1.5 2 0 1 2 3 4 5 10 15 0 1 2 4 5 10 15 20
W=5
Pit width W
Depth of adjacent pits Hs
Net distance of pits edge on the same side D
Distance of pits centerlines on the different side L
H=1 Fig. 4. Tensile specimens.
H1 = 1, W1 = 5 W2 = 5
H = 1, W = 5
H = 1, W = 5
Fig. 3. Tensile test device.
the test was 0.5 mm/min until the specimen was broken, which marked the end of the test. According to ‘‘Hot Galvanized Steel Wire for Bridge Cables” [33], the special extensometer with a standard distance of 250 mm was adopted. Tensile specimens were shown in Fig. 4. 3.2. Results and discussion The tensile strength of specimens with different geometric parameters, the breaking force divided by the original cross sec-
tional area, was shown in Table 3. It can be found that all the size, spacing, location and number of corrosion pits will lead to the reduction of ultimate load and yield load compared with the non-corrosion steel wire. When the pit depth reached 2 mm in particular, the ultimate load and yield load were about 23 kN and 12 kN, respectively, which were 30.3% and 61.3% lower than that of the non-corrosion steel wire. The relationship between nominal strength and geometric characteristic parameters of pits was shown in Fig. 5. The influence on the strength of steel wire varied from different pit characteristic parameters. Fig. 5(a) showed the effect of pit depth on nominal ultimate and yield strength of steel wire. It can be seen that the pit depth had a significant effect on the ultimate strength and yield strength of steel wire. The value of Z in the Fig. 5(a) was the difference between the ultimate strength and yield strength. With the increase of pit depth, the gap of Z to be further enlarged, which indicated that the effect of pits depth on yield strength was greater than ultimate strength. Therefore, the pit depth would affect the plastic development process of specimens. Fig. 5(b) showed that the effect of pit width on nominal yield and ultimate strength of steel wire, contrary to the effect of pit depth. With the increase of pit width, the load-carrying capacity of steel wire increased. The reason may be that the larger the pit width, the more gentle the stress development near the pit, resulting in less stress concentration effect. In addition, with the increase of pit width, the difference between yield strength and ultimate strength decreased, but not significant, which indicated that the pit width had little influence on the plastic development process of specimen. The effect of adjacent pit depth on nominal strength of steel wire was shown in Fig. 5(c). The horizontal dashed line represented the ultimate and yield strength of steel wire with single pit of W = 5 mm and H = 1 mm. It can be seen that the nominal strength of steel wire gradually decreased with the depth of adjacent pits increasing. When the value of Hs was less than 1 mm, the change of ultimate strength and yield strength was not obvious, equivalent to the strength of single pit steel wire. However, when the value of Hs exceeded 1 mm, the bearing capacity of steel wire decreased rapidly with the increase of pit depth H, far below the strength of single pit steel wire. So when two adjacent pits existed, the change of the depth of smaller pits had no obvious effect on the bearing capacity of steel wire, always mainly determined by the deeper pits. In addition, the pit with larger depth in adjacent pits had also an influence on the plastic development process of specimens. Fig. 5(d) showed the effect of net distance of pits edge at the same side on nominal strength of steel wire. The horizontal line represented the ultimate strength and yield strength of steel wire with single pit of W = 5 mm and H = 1 mm. With the increase of the distance between adjacent pits, the ultimate strength and yield strength remain unchanged, and basically located near the hori-
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R. Li et al. / Construction and Building Materials 240 (2020) 117915 Table 3 The tensile strength of the specimens with different geometric parameters. Number
Pu /kN
Py /kN
fu /MPa
fy /MPa
fu’/MPa
fy’/MPa
Number
Pu /kN
Py /kN
fu /MPa
fy /MPa
fu’/MPa
fy’ /MPa
H-0
35 33 34 31 33 30 31 30 28 29 30 31 28 28 29 26 27 25 22 23 24 17 15 14 24 25 23 24 25 24 25 25 26 26 27 25 27 27 26 29 28 29 26 27 25 27 24 25 27 25 25 28 25 25 27 23 25
29 31 31 27 25 27 21 23 23 20 18 19 15 20 13 15 12 14 7 6 5 4 6 5 12 9 12 14 12 13 13 15 15 14 16 16 17 16 17 17 20 19 15 12 14 15 13 13 15 13 13 13 15 13 14 13 14
1796 1662 1732 1588 1656 1528 1560 1544 1552 1464 1512 1560 1416 1432 1472 1320 1368 1272 1125 1165 1220 880 776 720 1206 1244 1169 1232 1277 1219 1268 1278 1306 1320 1368 1272 1365 1376 1328 1450 1429 1446 1320 1368 1272 1385 1233 1267 1384 1277 1245 1405 1279 1258 1365 1178 1276
1480 1590 1550 1379 1289 1349 1052 1187 1150 1002 912 944 766 1006 647 750 602 698 367 280 265 200 309 262 610 470 590 700 600 640 670 765 737 730 820 807 870 790 844 870 1000 963 750 604 695 750 657 645 740 648 670 650 751 666 730 645 704
1730
1540
Hs-1 Hs-1.5
463
1130
Hs-2
792
306
1512
953
D-0
1281
650
1440
806
D-1
1282
648
1320
683
D-2
1319
656
1170
304
D-3
1267
645
792
257
D-4
1343
667
1206
557
D-5
1285
645
1243
647
D-10
1297
652
1284
724
D-15
1273
647
1230
786
L-0
1052
826
1356
835
L-1
1008
787
1442
944
L-2
906
733
1320
683
L-4
895
683
1295
684
L-5
882
671
1302
686
L-10
903
671
1314
689
L-15
920
671
1273
693
L-20
720 672 702 501 410 478 350 260 308 701 602 647 700 659 585 600 695 673 700 590 645 710 610 681 690 592 653 706 610 640 701 592 648 900 770 808 830 750 781 700 770 729 730 640 679 711 643 659 703 630 680 634 670 709 640 600 776
1052
1552
1357 1234 1321 1055 999 1102 860 762 754 1222 1352 1269 1278 1302 1266 1344 1291 1323 1272 1216 1314 1355 1363 1312 1254 1272 1328 1297 1344 1251 1198 1316 1304 1168 1017 972 1076 913 1035 908 934 875 791 852 1043 887 795 963 926 798 986 865 964 931 952 869 861
698
1339
14 13 14 10 8 10 7 5 6 14 12 13 14 13 12 12 14 13 14 12 13 14 12 13 14 12 13 14 12 13 14 12 13 18 15 16 16 15 15 14 15 14 14 13 13 14 13 13 14 12 13 13 13 14 13 12 15
1304
1591
27 24 26 21 20 22 17 15 15 24 27 25 25 26 25 27 25 26 25 24 26 27 27 26 25 25 26 26 27 25 24 26 26 23 20 19 21 18 20 18 18 17 16 17 21 18 16 19 18 16 19 17 19 18 19 17 17
894
672
H-0.2
H-0.4
H-0.6
H-0.8
H-1
H-1.5
H-2
W-2
W-3
W-4
W-5
W-6
W-10
Hs-0
Hs-0.2
Hs-0.4
Hs-0.6
Hs-0.8
zontal straight line. The results showed that the load-carrying capacity of steel wires with two pits on the same side had nothing to do with the increase of pit number and pit distance, closely related to the pit size on the same side. The reason may be that the existence of double pits did not cause excessive stress concentration, so the load-carrying capacity of steel wire had not been reduced compared with that of steel wire with single pit. The influence of centerlines distance of pits at the different side on the yield strength and ultimate strength was shown in Fig. 5(e). When the pits centerlines distance at the different side was relative small, the strength of steel wire decreased obviously with the increase of distance. However, when the distance exceeded
5 mm, the average strength of steel wire remained basically unchanged. The reason may be that the distance between the centerlines of the two pits was close and approximately symmetrical, which weakened the stress concentration at the pit bottom and beneficial to the uniform stress of the steel wire. But with the distance increase, the enlarged stress concentration slightly weakened the tensile strength of the steel wire. In addition, compared with single pit (the tensile strength was 1320 MPa), the existence of double pits on the different side significantly reduced the tensile strength of steel wire. The gap between yield and ultimate strength increased and then tended to remain unchanged as the pit distance changed.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 5. The relationship between nominal strength and geometric characteristic parameters.
4. Finite element analysis 4.1. Finite element model Due to the limitation of test conditions and the number of specimens, the effect of pits on the strength of cable steel wire cannot be reflected more comprehensively. Therefore, based on the analysis of experimental result, this paper used finite element software ABAQUS to investigate the effect of the depth, width and spacing of pits on the stress development, and the relationship between characteristic parameter and stress concentration factor. The material properties of cable steel wire were shown in Table 1. The elastic modulus of cable steel wire was 205 GPa, and Poisson’s ratio was 0.3. The plastic section was determined according to the stress– strain relationship (See Fig. 6) and converted into real stress and strain. Flexible damage criterion was selected to define the damage, in which the fracture strain (equivalent fracture strain at damage
Fig. 6. The original stress–strain relationship of steel wire.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
seen, the maximum stress was located at the maximum depth of the pit, and finally steel wire broke at the pit with a larger depth. Therefore, the influence of adjacent pits on the stress development of steel wire always depended on the larger depth pits. The stress distribution of steel wire with different pit distances was shown in Fig. 8(c). The stress distribution and fracture location of steel wire varied from different pit distances. With the increase of distance, the fracture mode of steel wire changed alternately from flat to oblique mode, and the stress development path determined the fracture modes of steel wire.
initiation) was 0.113, the stress triaxiality was 0.33 and the strain ratio (equivalent plastic strain rate) was 0.12. The element type was 10 nodes C3D10, and the mesh was divided freely. Considering the irregularity of corrosion surface morphology, and in order to improve the accuracy and reduce the cost of calculation, the size of corrosion surface element was about 0.05 mm, and the other dimensions were 0.1 mm. The model was loaded by displacement, which moving the one end of the model by limiting the displacement of other end. The specific loading mode and meshing were shown in Fig. 7. Due to the large number of specimens, only a part of the steel wires were listed and compared with the test results, shown in Table 4. The results of tensile test and finite element analysis had a good coincidence, and the model can describe the monotonic tensile properties of steel with different characteristic parameter. However, the size deviation existed in specimen processing, which leaded to a certain error between the finite element analysis and the experimental result. But the error was within the allowable range of the engineering application.
4.2.2. Stress concentration factor Based on the finite element analysis, the existence of corrosion pits leaded to excessive local stress in the pit, which resulted in stress concentration, and seriously affected the full exertion of the strength of steel wire. However, the degree of stress concentration varied with characteristic parameters of different pits. The stress concentration factor Kt was usually selected to calculate the severity of stress concentration (See Eq. (1)). When calculating the stress concentration factor in the finite element analysis, the material properties only retained the elastic section, and the tensile load was set to 4 kN (200 MPa). rpeak was the maximum value of Mises stress and rnominal was the ratio of the applied load P (4 kN) to the section area S0 of the stainless steel wire.
4.2. Finite element results 4.2.1. Stress analysis of pits The stress distribution of steel wire with different pit parameters was shown in Fig. 8. The stress distribution at the pit of each specimen was similar. Taking the specimen H-2 as an example, according to the Fig. 8(a), the stress at the bottom of the pit was always higher than that of other parts. The stress expanded from the center of the pit bottom to the specimen bottom with the loading, and finally the whole section reached the same stress level, and broke at the pit. Fig. 8(b) showed the stress distribution of steel wires with different pit depths adjacent to each other. As can be
Kt ¼
rpeak rnominal
ð1Þ
where Kt was the stress concentration factor, rpeak was the peak stress and rnominal was the nominal stress without stress concentration. The relationship between stress concentration factor and characteristic parameters of different pits was shown in Fig. 9. As can
Fig. 7. The specific loading mode and meshing.
Table 4 Comparison of finite element results with experimental results. Number
Pu/kN
PuFEA/kN
4u/%
Py/kN
PyFEA/kN
4y/%
H-0 H-2 Hs-1 W-2 L-5 D-5
34 15.3 25.7 24.0 17.7 25.3
34 14.7 26.5 24 17.2 24.8
0 3.9 3.1 0 2.8 1.9
30.3 5 13.6 11 13.3 13
29.8 4.8 14.2 11.5 13.1 12.6
1.6 4 4.4 4.5 1.5 3.1
8
R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 8. Stress distribution of steel wire with different pit parameters.
be seen from Fig. 9(a) that for single pit, the stress concentration factor Kt increased significantly with the pit depth H, when the pit width remained unchanged (W = 5); For double pits on the same side, the pit size remained unchanged (W = 5, H = 1). With the increase of pit distance D, the stress concentration factor Kt did not change, and equal to that of single pit size (W = 5, H = 1). It showed that the pit distance on the same side did not affect the stress concentration factor, but only depended on the pit size; For adjacent pit depth Hs, when the value of Hs was lower than the adjacent pit depth (H = 1), the value of Kt remained basically unchanged and the same as that single pit. When the value of Hs exceeded the depth of adjacent pits, the stress concentration factor increased with the value of Hs (H > 1), and the same as the Kt of single pit. It showed that the stress concentration factor of the depth of adjacent pits on the same side depended on the maximum pit, and the same as that of the maximum size under the single pit. Therefore, the stress concentration factor mainly depended on the larger size of pits. As can be seen from Fig. 9(b), the stress concentration factor Kt decreased gradually with the increase of pit width while the pit depth remained unchanged (H = 1). Fig. 9(c) showed that the stress concentration factor increased rapidly with the increase of the distance between the centerlines of pits, but when the distance reached a certain distance (L = 6), the stress concentration factor basically remained unchanged and converged to about 2.3. The value was equal to that of the single pit with the same size
(W = 5, H = 1). Therefore, the stress concentration factor was closely related to the pit size. 4.3. Parameter analysis 4.3.1. The depth-width ratio and relative pit depth From the previous experiments and finite element analysis, it can be concluded that the most critical factor affecting the stress concentration factor of corroded steel wire was the pit size (width and depth). Therefore, in order to establish the quantitative relationship between different pit parameters and stress concentration, and to study the influence on stress concentration factor, 84 finite element models of steel wire with different characteristics parameters were established. Specific parameters were shown in Table 5. The relationship between H/W (depth-width ratio) and Kt under different H/T (the relative pit depth) was shown in Fig. 10. It can be found that under the same H/T of pit, the stress concentration factor increased gradually with the H/W increase. Then the stress concentration factor reached the maximum when the H/W was at 1 ~ 2, which most seriously affected the stress concentration factor. However, when greater than 2, the stress concentration factor tended to be stable and converged to about 2.7. In addition, under the same H/W of pit, the greater the H/T, the greater the stress concentration factor Kt is, that is, the more obvious the stress concentration. When the H/T was less than 0.12,
R. Li et al. / Construction and Building Materials 240 (2020) 117915
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Fig. 9. Relation between stress concentration factor and pits characteristic parameters.
Table 5 Corrosion pit parameters. Parameters
Value/mm
H T H/W H/T
0.2, 0.4, 0.6, 1, 1.5, 2 5 0 ~ 10 0.04, 0.08, 0.12, 0.2, 0.3, 0.4
Model
Note: Pit width was appropriately selected according to the H/W.
there was little difference between the trend of stress concentration factor Kt with the H/W of pit; while the depth-width ratio was higher than 0.12, the difference of trend became larger. The main reason for the difference was that the change of pit shape from wide-shallow to deep-narrow. Therefore, within a certain range of depth-width ratio, the effect of deep and narrow pits on stress concentration factor was the most sensitive. The stress distribution of four different types of pits was shown in Fig. 11. It can be seen that different pit types had different stress distributions and maximum stress locations. For shallow-wide and deep-wide pits (Fig. 11(a) and (b)), the maximum stress was near the bottom of the pit, while for narrow-shallow and deep-narrow pits (Fig. 11(c) and (d)), far away from the bottom of the pit and near the edge of the pit. At the same time, the red area (high stress) was mainly concentrated on the bottom section of the pit (the weakest section), so it can be predicted that the fracture was basically on the weak section. In conclusion, the stress concentration factor Kt was related to depth-width ratio H/W and relative pit depth H/T of pit. According to Fig. 10, the expressions of depth-width ratio H/W of pit and stress concentration factor Kt conformed to the same type of functional relationship. The specific expressions were as follows:
K t ¼ A þ BeðW Þ=C H
Fig. 10. The relationship between H/W and Kt under different H/T.
ð2Þ
The coefficients A, B and C in the formula were all functions containing H/T. The relationship was shown in Fig. 12. Therefore, within the range of relative pit depth (H/T) of 0.4, the quantitative relationship between pit characteristic parameters and stress concentration factor was as follows:
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 11. Stress distribution of steel wire with different pit types.
Fig. 12. Relationships between coefficients A, B, C and H/T.
h i H H K t ¼ A þ BeðW Þ=C ¼ 2:70984 þ 8:83 104 eð T Þ=0:06237 2 H H H þ 1:7546 þ 0:482e76ð T 0:22078Þ eðW Þ=½lnð1:926832:28643T Þ ð3Þ The stress concentration factor of some test specimens was calculated by finite element result and Eq. (3), respectively. The results were shown in Fig. 13. It can be found that the finite element calculation was very close to the calculation result of Eq. (3), and the error was about 5%, which proved the correctness of the theoretical formula. Based on the above discussion, the main reason that affected the strength degradation of steel wire with corrosion pits was that the existence of corrosion pits leaded to excessive local stress and stress concentration. The pit size was the most important factor that affecting the stress concentration. Therefore, based on the relationship between stress concentration factor and characteristic parameters of pits, the formula for predicting residual nominal yield strength of steel wire with pits was proposed.
Fig. 13. Stress concentration factor between finite element and theoretical result.
R. Li et al. / Construction and Building Materials 240 (2020) 117915
r0 y ¼ ry =K t
¼ h
ry
i 2 H H 2:70984 þ 8:83 10 eð T Þ=0:06237 þ 1:7546 þ 0:482e76ð T 0:22078Þ 4
eðW Þ=½lnð1:926832:28643 T Þ H
H
ð4Þ
where ry is the yield strength of non-corroded steel wire and ry’ is the nominal yield strength of steel wire with corroded pits. The comparison between experimental results and theoretical results was shown in Fig. 14. It was found that the experimental result was basically in agreement with the theoretical result, which proved that the theoretical formula can basically predict the nominal yield strength of steel wire with corrosion pits.
Fig. 14. Comparison of yield strength between experimental and theoretical result.
Table 6 Secondary pit parameters. Parameters
Value/mm
H1 W1 H2 W2
0.4, 2 0.6, 1, 2, 6 0.1 0.1
Model with secondary pit
11
4.3.2. Secondary corrosion pit Material corrosion originated from the steel surface, accompanied by primary pits, secondary pits also occurred from the surface of primary pits. Cerit [24,25] studied the stress distribution of semi ellipsoid pits by finite element analysis. It was found that the depth to width ratio of pits was the main factor affecting the stress concentration coefficient, and the existence of secondary pits had a significant effect on the stress concentration coefficient. Therefore, in order to better carry out numerical simulation, this paper assumed that the secondary pit located at the bottom of the primary pit, as shown in Table 6. Taking the size H = 0.4 mm, W = 1 mm; H = 2 mm, W = 1 mm of primary pit as an example, the stress distribution was shown in Fig. 15. I was the top view of primary pits; II was the top view of secondary pits; III was the profile of primary pits; IV was the profile of secondary pits. It can be seen that for steel wires with only primary pits, the area of larger local stress was perpendicular to the direction of tensile load and distributed in strips along the weakest section (See I). The maximum stress value appeared near the pit mouth (See III). When secondary pits appeared at the bottom of primary pits, the area of larger local stress was perpendicular to the direction of tensile load and distributed in blocks at the junction of primary pits and secondary pits (See II). The maximum stress occurred at 1/2 of the depth of secondary pit (See IV). As can be seen from Fig. 15(b), for the primary pit steel wire, the area of larger local stress was perpendicular to the direction of tensile load, and distributed in block shape along the weakest section at the pit mouth (See I). The maximum stress appeared at 1/2 of the pit depth (See III). When secondary pits occurred at the bottom, the area of larger local stress was perpendicular to the direction of tensile load, and distributed in blocks along the weakest section, mainly concentrated at the mouth and the middle part of primary pit, and also the junction of primary pit and secondary pit (See II). The maximum stress occurred at 1/2 of distance from the secondary pit mouth to the pit bottom (See IV). All in all, secondary pits changed the stress distribution of pits, and the maximum stress position shifted from the mouth of pits to the bottom of pits. The stress concentration factor of steel wire with the same secondary pit size under different primary pits was shown in Fig. 16. As can be seen from Fig. 16, the stress concentration factor of steel wire with secondary pits was obviously higher than that of steel wire with only primary pits. When the depth of primary pit was 2 mm (H1 = 2.0 mm), the stress concentration factor of steel wire with secondary pit decreased significantly, compared with the
Fig. 15. Stress distribution of primary and secondary pits.
12
R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 16. Stress concentration factor of steel wire under different primary pit sizes.
concentration factor tended to be stable and converged to about 2.7. In addition, with the same H/W of pit, the greater the H/D, the greater the stress concentration factor Kt is, that is, the more obvious the stress concentration. Within a certain range of H/W, the effect of deep-narrow pits on stress concentration factor was the most sensitive. At the same time, the area of high stress was mainly concentrated on the bottom section of the pit (the weakest section), so it can be predicted that the fracture was basically on the weak section. Finally, the models of strength and stress concentration factor of steel wire with different depth-width ratio and relative depth of pit were established. (4) Secondary pits changed the stress distribution of pits, and the maximum stress position shifted from the mouth of pits to the bottom of pits. The stress concentration factor of steel wire with secondary pits was obviously higher than that of steel wire with primary pits only. When the depth of primary pit was 2 mm (H1 = 2.0 mm), the stress concentration factor of steel wire with secondary pit decreased significantly, compared with the steel wire with primary pit (H1 = 0.4 mm). Therefore, attention should be paid to the development of secondary pits in shallow primary pits.
steel wire with primary pit (H1 = 0.4 mm). Therefore, attention should be paid to the development of secondary pits in shallow primary pits.
CRediT authorship contribution statement
5. Conclusion
Rou Li: Data curation, Methodology, Software, Writing - review & editing. Changqing Miao: Conceptualization, Methodology, Supervision. Jie Yu: Visualization, Investigation, Software.
198 cable steel wires with different pit characteristic parameters were obtained by manual prefabrication. The influence of pit depth, width, spacing and location on the strength of steel wires was discussed through monotonic tensile test. The stress distribution of cable steel wires with different pit characteristic parameters was compared by finite element analysis, and the relationship between stress concentration factor and characteristic parameters was studied. The main conclusions were as follows: (1) The influence on the strength of steel wire varied from different characteristic parameters of pit. The strength of steel wire decreased gradually with the increase of pit depth and the decrease of pit width. When two adjacent pits existed, the change of the depth of smaller pits had no obvious effect on the strength of steel wire, only determined by the pits with larger depth. The change of pits clearance on the same side had no obvious effect on the strength of steel wire, but it had significant effect when located on the opposite side. However, when the clearance of pits centerlines exceeded 5 mm, the average strength of steel wire remained basically unchanged. In addition, the pit with larger depth in adjacent pits had an influence on the plastic development process of specimen. (2) The stress at the bottom of the pit was always higher than that of other parts. The stress expanded from the center of the pit bottom to the specimen bottom with the loading, and finally the whole section reached the same stress level, and broke at the pit. The influence of adjacent pits on the stress development of steel wire usually depended on the pit with a larger depth. The stress distribution and fracture location of steel wire varied from different distances of pit. With the increase of distance, the fracture mode of steel wire changed alternately from flat to oblique mode, and the stress development path determined the fracture modes of steel wire. (3) The stress concentration factor reached the maximum at 1 ~ 2 (H/W), which most seriously affected the stress concentration factor. While the H/W was greater than 2, the stress
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The financial support provided by National Natural Science Foundation of China under Grant Nos. NSFC-51078080 is highly acknowledged. The authors also would like to express gratitude to the reviewers for their comments. References [1] P.B. David, N. Aly, History and aesthetics of cable-stayed bridge, J. Struct. Eng. 117 (19) (1990) 3103–3134. [2] A.M. Ruizteran, A.C. Aparicio, Two new types of bridges: under-deck cablestayed bridges and combined, Can. J. Civil Eng. 34 (8) (2007) 1003–1015. [3] S.I. Nakamura, H. Tanaka, K. Kato, Static analysis of cable-stayed bridge with CFT arch ribs, J. Constr. Steel Res. 65 (4) (2009) 776–783. [4] S.C. Barton, G.W. Vermaas, P.F. Duby, Accelerated corrosion and embrittlement of high-strength bridge wire, J. Mater. Civil Eng. 12 (1) (2000) 33–38. [5] K. Furuya, M. Kitagawa, S. Nakamura, Corrosion mechanism and protection methods for suspension bridge cables, Struct. Eng. Int. 10 (3) (2000) 189–193. [6] Z.X. Liu, T. Guo, et al., Corrosion fatigue analysis and reliability assessment of short suspenders in suspension and arch bridges, J. Perform. Constr. Fac. 32 (5) (2018) 04018060. [7] Z.X. Liu, T. Guo, et al., Measurement and comparative study on movements of suspenders in long-span suspension bridges, J. Bridge Eng. 24 (5) (2019) 04019026. [8] L.L. Wang, W.J. Yi, Cases analysis on cable corrosion of cable-stayed bridges, Central South Highway Eng. 32 (1) (2007) 93–98 (in Chinese). [9] J.B. Ji, J. Tong, Corrosion rate and mechanical properties of 316l stainless steel wires in different corrosive conditions, Appl. Mech. Mater. 441 (2013) 48–52. [10] S. Wu, H. Chen, H.L. Ramandi, Effects of environmental factors on stress corrosion cracking of cold-drawn high-carbon steel wires, Corros. Sci. (2017) 11459. [11] I.H.R. Hamilton, J.E. Breen, K.H. Frank, Bridge stay cable corrosion protection. II: accelerated corrosion tests, J. Bridge Eng. 3 (2) (1998) 72–81. [12] D.G. Su, D.J. Han, Z.D. Han, Study on corrosion failure of cable wire of cablestayed bridge, J. South China Univ. Technol. (Nat. Sci.) 24 (8) (1996) 108–112 (in Chinese).
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Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Effect of characteristic parameters of pitting on strength and stress concentration factor of cable steel wire Rou Li a,b,⇑, Changqing Miao a,b,⇑, Jie Yu b a b
Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing 210096, China School of Civil Engineering, Southeast University, Nanjing 210096, China
h i g h l i g h t s The effect of pits on the stress concentration factor of steel wires is investigated. Parametric analyses were performed on steel wire with different pits parameters. The formula of stress concentration factor of steel wire with pits was established.
a r t i c l e
i n f o
Article history: Received 20 June 2019 Received in revised form 29 November 2019 Accepted 19 December 2019
Keywords: Corrosion pit Steel wire Strength Stress concentration factor Calculation model
a b s t r a c t In order to investigate the influence of characteristic parameters of pits on the strength and stress concentration factor of cable steel wire, 198 cable steel wires with pits were manually prepared. The variation law of stress distribution of steel wire was analyzed, and its relationships with strength and stress concentration factor were studied through tensile test and finite element analysis. On this basis, the calculation model of yield strength and stress concentration factor of steel wire with pits was established. The influences of secondary pits on stress distribution and stress concentration factor of steel wire were also analyzed. The results showed that the strength of steel wire decreased gradually and the stress concentration factor increased with the increase of pit depth and the decrease of pit width. The change of pit clearance on the same side had no obvious effect on the strength and stress concentration factor of steel wire, but it had significant effect when located on the opposite side. The strength and stress concentration factor of steel wire with adjacent pits usually depended on the depth of the larger pits. The pits with depth to width ratio of 1 to 2 had the most significant effect on the stress concentration factor. Moreover, secondary pit would change the stress distribution, and the position of maximum stress changed from near the mouth to the bottom of the pit. The stress concentration factor of secondary pit was obviously higher than that of steel wire with only the primary pit. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Modern cable-stayed bridges, known as greater capacity for spanning, easy construction and beautiful radial shape, have been widely applied to bridge construction [1–3]. The stay cables are the main structural components that transfer the weight and load of the main girder and bridge deck to the pylon. Whether they can be used normally and safely directly determines the safety and reliability of the whole cable-stayed bridge [4,5]. However, the polyethylene jackets will inevitably crack due to the bending stress originated from the vibrations of cable or relative movements ⇑ Corresponding authors at: School of Civil Engineering, Southeast University, Nanjing 210096, China. E-mail addresses: [email protected] (R. Li), [email protected] (C. Miao). https://doi.org/10.1016/j.conbuildmat.2019.117915 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.
between the main cable and the girder under long-term corrosion environment [6,7], then the high strength steel wire inside the cable are usually difficult to avoid corrosion through protective measures, structural measures and maintenance systems, resulting in a serious corrosion problem [8–10]. Stafford and Watson [11] found that nearly 200 cable-stayed bridges around the world were facing danger due to cable corrosion based on the external investigation of cable-stayed bridges. Kohlbrand Estuary Bridge in Germany had to replace all the cables at a cost of up to $6 million, due to serious corrosion at the anchorage end of the lower part caused by water immersion into the cables [11]. In addition, the cables of Haiyin Bridge in Guangzhou adopted more advanced protection system, electrochemical corrosion still took place, and eventually a sudden fall occurred [12]. On the whole, the above collapse accidents found that corrosion
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Nomenclature ABC D E fy fu fy’ fu’ H H1 H2 Hs Kt L Py PyFEA Pu
all functions containing variable H/D clearance of pits on the same side elastic modulus yield strength ultimate strength average yield strength average ultimate strength depth of pit depth of primary pit depth of secondary pit depth of adjacent pits on the same side stress concentration factor distance of pits centerlines on the different side yield load yield load by FEA ultimate load
caused degradation of the performance of high-strength steel wire, which would further reduce the carrying capacity of cables, thus bringing a severe test to the safe service and normal use of cable-stayed bridges [13–15]. Therefore, it is necessary to investigate the degradation law of mechanical properties of corroded cable steel wires and establish the relationship between mechanical properties and corrosion characteristic parameters, which can be used to accurately evaluate the degradation degree of mechanical properties of corroded cable steel wire by corrosion characteristic parameters. Previous researches were mainly focus on the mechanical properties of corroded steel plate and rebar, hence limited data is available for the mechanical properties of cables steel wire with corrosion damage. Garbatov et al. [16] found that with corrosion damage increasing, significant degradations in strength and deformation of corroded steel plates can be observed. Xu et al. [17] and Xia et al. [18] investigated the effect of corrosion pits on the strength and ductility of the corroded steel, and the strength and ductility decreased significantly with the increase of pits depth. The damage parameters of corrosion steel and their effects on the mechanical properties of the steel were studied from the corrosion surface, and the constitutive model of corrosion steel was established and the variation of parameters was also analyzed [19]. However, corrosion may cause different influences on different kinds of steel and thus it is important to investigate the mechanical properties of steel wire with corrosion damage. Moreover, although a little research had been done on the degradation of mechanical properties of corroded steel wires in cables, there was no generally accepted definition. R. Betti et al. [20] used acetic acid to regulate Nacl and Na2SO4 solution as corrosive medium to carry out corrosion test of high strength galvanized steel wire. It was found that corrosion resulted in decrease of tensile strength and yield strength of steel wire. Nakamura et al. [21] obtained four grades of steel wire by accelerated corrosion, and determined the corresponding relationship between corrosion degree and mass loss ratio. The research showed that the actual tensile strength of steel wire after corrosion had not decreased. Ma et al. [22] tested the mechanical properties of steel wires with different corrosion degrees, and found that the tensile strength and elongation decreased sharply when the corrosion degree was serious. Mayrbaurl et al. [23] systematically studied the crack propagation and fracture of steel wire after water entry into cable structure, which could be used as the basis for the strength study of cable structure. However, the overview of the state of the art showed that most studies on the corrosion degree of steel wires usually took mass
PuFEA T W W1 W2 Z
r rnominal rpeak ry r0 y e c
4u 4y
ultimate load by FEA diameter of steel wire width of pit width of primary pit width of secondary pit difference between ultimate strength and yield strength stress nominal stress peak stress yield strength of non-corroded steel wire yield strength of corroded steel wire strain poisson’s ratio error of ultimate load error of yield load
loss ratio (weight loss ratio) as a characterization method, but its essence was the average mass loss ratio of steel, difficult to reflect the impact of local corrosion. Furthermore, those researches were mainly based on the statistics results of tensile test, and seldom investigating the effect of local pit distribution on mechanical properties and stress distribution of steel wire. The relationship between mechanical properties and pit distribution parameters of steel wire was still not established. The size of the pits directly determines the stress concentration factor, which directly leads to the reduction of the fatigue strength. Cerit [24,25] studied the stress distribution of semi ellipsoid pits by finite element analysis. It was found that the depth to width ratio of pits was the main factor affecting the stress concentration coefficient, and the existence of secondary pits had a significant effect on the stress concentration coefficient. Sankaran et al. [26] pointed out that the effect of pitting on the fatigue life of 7075-T6 aluminum alloy members was related to the equivalent stress concentration coefficient. Xiang et al. [27] proposed that the stress concentration factor of the hemispherical pits was about 3, and pointed out that the stress intensity factor of the crack at the root of the structural groove can be calculated by using SCF as a function of variable. Makhlou et al. [28] studied the corrosion crack growth of duplex stainless steel in air and artificial seawater environment. The results showed that the stress intensity factor was a function of the load applied to the structure, the width and thickness of the specimen, and the ratio of the length and width of the surface crack of the specimen. Nakamura et al. [29] found that the stress concentration was larger for sharper pit shapes, indicating that this was the major cause for the decrease of fatigue strength by conducted fatigue tests of wire specimens with artificial pits. To investigate the influence of pit characteristic parameters on the strength and stress concentration factor of cable steel wire, 198 cable steel wires with different pit characteristic parameters were obtained by manually prefabricating pits. The influence of pit depth, width, pitch and location on the strength of cable steel wire was discussed through monotonic tensile test. The stress distribution law of cable steel wire with different pit characteristic parameters was compared by finite element numerical analysis, and the relationship between pit characteristic parameters and stress concentration factor was also investigated. On this basis, the calculation model of yield strength and stress concentration factor of steel wire with corrosion pits were established. Finally, the effect of secondary pits on stress distribution and stress concentration factor of cable steel wire were also investigated.
R. Li et al. / Construction and Building Materials 240 (2020) 117915
3
Fig.1. The size of tensile specimen.
2. Specimen design The galvanized steel wire with strength grade of 1670 MPa was selected as the test material. Its nominal diameter of section was 5 mm and total length was 400 mm. The size of tensile specimen was shown in Fig. 1. The mechanical properties of steel wire were shown in Table 1. The size of the pits directly determines the stress concentration factor, which directly leads to the reduction of the fatigue strength [29]. Therefore, it can be considered that the bearing capacity of corroded steel wire was closely related to the pit size parameters of corroded steel wire. Cerit et al. [25] through experiments found that the shape of the pit in real bridge was broad and shallow hemispherical or narrow and deep hemispherical. The hemispheric pit was really a special case with the same length, width and depth of ellipsoidal pit. Elliptical pits with different lengths, widths and depths were more common in practical engineering [30–32]. According to the statistics of pit shape and study of pit formation process, the artificial prefabricated pits could be used in the tests
to represent the corrosion pits in real bridge and the pit shape was simplified to semi-elliptic sphere. The characteristic parameters of artificial prefabricated pits were shown in Fig. 2. The pit size for pit depth was that keeping the width of 5 mm and changing the pit depth, and for pit width was that keeping the pit depth of 1 mm, changing the width (See Fig. 2(a)). Fig. 2(b) was the net distance between the edges of the pits on the same side, that keeping the pit size (Width 5 mm and Depth 1 mm), and changing the net distance between the edges of the pits. Fig. 2(c) showed the distance between the centerlines of different pits, that keeping the pit size (Width 5 mm and Depth 1 mm), and changing the distance between the centerlines of different pits. Fig. 2(d) was the depth of adjacent pits on the same side that keeping one pit size (Width 5 mm and Depth 1 mm), and only changing the depth of the anther adjacent pits (Width 5 mm). There were 114 tensile specimens in total, of which 3 were parallel in each group. The specific values of the different characteristic parameters of pits were shown in Table 2. 3. Tensile test of corroded cable steel wire
Table 1 The mechanical properties of steel wire.
3.1. Test device
Material
E/GPa
fu/MPa
fy /MPa
c
Galvanized steel wire
205
1670
1540
0.3
The test was carried out on CMT5105 computer controlled electronic universal testing machine (See Fig. 3). The loading speed of
Fig. 2. The characteristic parameters of artificial prefabricated pits.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Table 2 The different characteristic parameters of pits. Characteristic parameters
Number
Value/mm
Invariant parameter
Pit depth H
H-0.0 H-0.2 H-0.4 H-0.6 H-0.8 H-1.0 H-1.5 H-2.0 W-2.0 W-3.0 W-4.0 W-5.0 W-6.0 W-10 Hs-0.0 Hs-0.2 Hs-0.4 Hs-0.6 Hs-0.8 Hs-1.0 Hs-1.5 Hs-2.0 D-0 D-1 D-2 D-3 D-4 D-5 D-10 D-15 L-0 L-1 L-2 L-4 L-5 L-10 L-15 L-20
0 0.2 0.4 0.6 0.8 1 1.5 2 2 3 4 5 6 10 0 0.2 0.4 0.6 0.8 1 1.5 2 0 1 2 3 4 5 10 15 0 1 2 4 5 10 15 20
W=5
Pit width W
Depth of adjacent pits Hs
Net distance of pits edge on the same side D
Distance of pits centerlines on the different side L
H=1 Fig. 4. Tensile specimens.
H1 = 1, W1 = 5 W2 = 5
H = 1, W = 5
H = 1, W = 5
Fig. 3. Tensile test device.
the test was 0.5 mm/min until the specimen was broken, which marked the end of the test. According to ‘‘Hot Galvanized Steel Wire for Bridge Cables” [33], the special extensometer with a standard distance of 250 mm was adopted. Tensile specimens were shown in Fig. 4. 3.2. Results and discussion The tensile strength of specimens with different geometric parameters, the breaking force divided by the original cross sec-
tional area, was shown in Table 3. It can be found that all the size, spacing, location and number of corrosion pits will lead to the reduction of ultimate load and yield load compared with the non-corrosion steel wire. When the pit depth reached 2 mm in particular, the ultimate load and yield load were about 23 kN and 12 kN, respectively, which were 30.3% and 61.3% lower than that of the non-corrosion steel wire. The relationship between nominal strength and geometric characteristic parameters of pits was shown in Fig. 5. The influence on the strength of steel wire varied from different pit characteristic parameters. Fig. 5(a) showed the effect of pit depth on nominal ultimate and yield strength of steel wire. It can be seen that the pit depth had a significant effect on the ultimate strength and yield strength of steel wire. The value of Z in the Fig. 5(a) was the difference between the ultimate strength and yield strength. With the increase of pit depth, the gap of Z to be further enlarged, which indicated that the effect of pits depth on yield strength was greater than ultimate strength. Therefore, the pit depth would affect the plastic development process of specimens. Fig. 5(b) showed that the effect of pit width on nominal yield and ultimate strength of steel wire, contrary to the effect of pit depth. With the increase of pit width, the load-carrying capacity of steel wire increased. The reason may be that the larger the pit width, the more gentle the stress development near the pit, resulting in less stress concentration effect. In addition, with the increase of pit width, the difference between yield strength and ultimate strength decreased, but not significant, which indicated that the pit width had little influence on the plastic development process of specimen. The effect of adjacent pit depth on nominal strength of steel wire was shown in Fig. 5(c). The horizontal dashed line represented the ultimate and yield strength of steel wire with single pit of W = 5 mm and H = 1 mm. It can be seen that the nominal strength of steel wire gradually decreased with the depth of adjacent pits increasing. When the value of Hs was less than 1 mm, the change of ultimate strength and yield strength was not obvious, equivalent to the strength of single pit steel wire. However, when the value of Hs exceeded 1 mm, the bearing capacity of steel wire decreased rapidly with the increase of pit depth H, far below the strength of single pit steel wire. So when two adjacent pits existed, the change of the depth of smaller pits had no obvious effect on the bearing capacity of steel wire, always mainly determined by the deeper pits. In addition, the pit with larger depth in adjacent pits had also an influence on the plastic development process of specimens. Fig. 5(d) showed the effect of net distance of pits edge at the same side on nominal strength of steel wire. The horizontal line represented the ultimate strength and yield strength of steel wire with single pit of W = 5 mm and H = 1 mm. With the increase of the distance between adjacent pits, the ultimate strength and yield strength remain unchanged, and basically located near the hori-
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R. Li et al. / Construction and Building Materials 240 (2020) 117915 Table 3 The tensile strength of the specimens with different geometric parameters. Number
Pu /kN
Py /kN
fu /MPa
fy /MPa
fu’/MPa
fy’/MPa
Number
Pu /kN
Py /kN
fu /MPa
fy /MPa
fu’/MPa
fy’ /MPa
H-0
35 33 34 31 33 30 31 30 28 29 30 31 28 28 29 26 27 25 22 23 24 17 15 14 24 25 23 24 25 24 25 25 26 26 27 25 27 27 26 29 28 29 26 27 25 27 24 25 27 25 25 28 25 25 27 23 25
29 31 31 27 25 27 21 23 23 20 18 19 15 20 13 15 12 14 7 6 5 4 6 5 12 9 12 14 12 13 13 15 15 14 16 16 17 16 17 17 20 19 15 12 14 15 13 13 15 13 13 13 15 13 14 13 14
1796 1662 1732 1588 1656 1528 1560 1544 1552 1464 1512 1560 1416 1432 1472 1320 1368 1272 1125 1165 1220 880 776 720 1206 1244 1169 1232 1277 1219 1268 1278 1306 1320 1368 1272 1365 1376 1328 1450 1429 1446 1320 1368 1272 1385 1233 1267 1384 1277 1245 1405 1279 1258 1365 1178 1276
1480 1590 1550 1379 1289 1349 1052 1187 1150 1002 912 944 766 1006 647 750 602 698 367 280 265 200 309 262 610 470 590 700 600 640 670 765 737 730 820 807 870 790 844 870 1000 963 750 604 695 750 657 645 740 648 670 650 751 666 730 645 704
1730
1540
Hs-1 Hs-1.5
463
1130
Hs-2
792
306
1512
953
D-0
1281
650
1440
806
D-1
1282
648
1320
683
D-2
1319
656
1170
304
D-3
1267
645
792
257
D-4
1343
667
1206
557
D-5
1285
645
1243
647
D-10
1297
652
1284
724
D-15
1273
647
1230
786
L-0
1052
826
1356
835
L-1
1008
787
1442
944
L-2
906
733
1320
683
L-4
895
683
1295
684
L-5
882
671
1302
686
L-10
903
671
1314
689
L-15
920
671
1273
693
L-20
720 672 702 501 410 478 350 260 308 701 602 647 700 659 585 600 695 673 700 590 645 710 610 681 690 592 653 706 610 640 701 592 648 900 770 808 830 750 781 700 770 729 730 640 679 711 643 659 703 630 680 634 670 709 640 600 776
1052
1552
1357 1234 1321 1055 999 1102 860 762 754 1222 1352 1269 1278 1302 1266 1344 1291 1323 1272 1216 1314 1355 1363 1312 1254 1272 1328 1297 1344 1251 1198 1316 1304 1168 1017 972 1076 913 1035 908 934 875 791 852 1043 887 795 963 926 798 986 865 964 931 952 869 861
698
1339
14 13 14 10 8 10 7 5 6 14 12 13 14 13 12 12 14 13 14 12 13 14 12 13 14 12 13 14 12 13 14 12 13 18 15 16 16 15 15 14 15 14 14 13 13 14 13 13 14 12 13 13 13 14 13 12 15
1304
1591
27 24 26 21 20 22 17 15 15 24 27 25 25 26 25 27 25 26 25 24 26 27 27 26 25 25 26 26 27 25 24 26 26 23 20 19 21 18 20 18 18 17 16 17 21 18 16 19 18 16 19 17 19 18 19 17 17
894
672
H-0.2
H-0.4
H-0.6
H-0.8
H-1
H-1.5
H-2
W-2
W-3
W-4
W-5
W-6
W-10
Hs-0
Hs-0.2
Hs-0.4
Hs-0.6
Hs-0.8
zontal straight line. The results showed that the load-carrying capacity of steel wires with two pits on the same side had nothing to do with the increase of pit number and pit distance, closely related to the pit size on the same side. The reason may be that the existence of double pits did not cause excessive stress concentration, so the load-carrying capacity of steel wire had not been reduced compared with that of steel wire with single pit. The influence of centerlines distance of pits at the different side on the yield strength and ultimate strength was shown in Fig. 5(e). When the pits centerlines distance at the different side was relative small, the strength of steel wire decreased obviously with the increase of distance. However, when the distance exceeded
5 mm, the average strength of steel wire remained basically unchanged. The reason may be that the distance between the centerlines of the two pits was close and approximately symmetrical, which weakened the stress concentration at the pit bottom and beneficial to the uniform stress of the steel wire. But with the distance increase, the enlarged stress concentration slightly weakened the tensile strength of the steel wire. In addition, compared with single pit (the tensile strength was 1320 MPa), the existence of double pits on the different side significantly reduced the tensile strength of steel wire. The gap between yield and ultimate strength increased and then tended to remain unchanged as the pit distance changed.
6
R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 5. The relationship between nominal strength and geometric characteristic parameters.
4. Finite element analysis 4.1. Finite element model Due to the limitation of test conditions and the number of specimens, the effect of pits on the strength of cable steel wire cannot be reflected more comprehensively. Therefore, based on the analysis of experimental result, this paper used finite element software ABAQUS to investigate the effect of the depth, width and spacing of pits on the stress development, and the relationship between characteristic parameter and stress concentration factor. The material properties of cable steel wire were shown in Table 1. The elastic modulus of cable steel wire was 205 GPa, and Poisson’s ratio was 0.3. The plastic section was determined according to the stress– strain relationship (See Fig. 6) and converted into real stress and strain. Flexible damage criterion was selected to define the damage, in which the fracture strain (equivalent fracture strain at damage
Fig. 6. The original stress–strain relationship of steel wire.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
seen, the maximum stress was located at the maximum depth of the pit, and finally steel wire broke at the pit with a larger depth. Therefore, the influence of adjacent pits on the stress development of steel wire always depended on the larger depth pits. The stress distribution of steel wire with different pit distances was shown in Fig. 8(c). The stress distribution and fracture location of steel wire varied from different pit distances. With the increase of distance, the fracture mode of steel wire changed alternately from flat to oblique mode, and the stress development path determined the fracture modes of steel wire.
initiation) was 0.113, the stress triaxiality was 0.33 and the strain ratio (equivalent plastic strain rate) was 0.12. The element type was 10 nodes C3D10, and the mesh was divided freely. Considering the irregularity of corrosion surface morphology, and in order to improve the accuracy and reduce the cost of calculation, the size of corrosion surface element was about 0.05 mm, and the other dimensions were 0.1 mm. The model was loaded by displacement, which moving the one end of the model by limiting the displacement of other end. The specific loading mode and meshing were shown in Fig. 7. Due to the large number of specimens, only a part of the steel wires were listed and compared with the test results, shown in Table 4. The results of tensile test and finite element analysis had a good coincidence, and the model can describe the monotonic tensile properties of steel with different characteristic parameter. However, the size deviation existed in specimen processing, which leaded to a certain error between the finite element analysis and the experimental result. But the error was within the allowable range of the engineering application.
4.2.2. Stress concentration factor Based on the finite element analysis, the existence of corrosion pits leaded to excessive local stress in the pit, which resulted in stress concentration, and seriously affected the full exertion of the strength of steel wire. However, the degree of stress concentration varied with characteristic parameters of different pits. The stress concentration factor Kt was usually selected to calculate the severity of stress concentration (See Eq. (1)). When calculating the stress concentration factor in the finite element analysis, the material properties only retained the elastic section, and the tensile load was set to 4 kN (200 MPa). rpeak was the maximum value of Mises stress and rnominal was the ratio of the applied load P (4 kN) to the section area S0 of the stainless steel wire.
4.2. Finite element results 4.2.1. Stress analysis of pits The stress distribution of steel wire with different pit parameters was shown in Fig. 8. The stress distribution at the pit of each specimen was similar. Taking the specimen H-2 as an example, according to the Fig. 8(a), the stress at the bottom of the pit was always higher than that of other parts. The stress expanded from the center of the pit bottom to the specimen bottom with the loading, and finally the whole section reached the same stress level, and broke at the pit. Fig. 8(b) showed the stress distribution of steel wires with different pit depths adjacent to each other. As can be
Kt ¼
rpeak rnominal
ð1Þ
where Kt was the stress concentration factor, rpeak was the peak stress and rnominal was the nominal stress without stress concentration. The relationship between stress concentration factor and characteristic parameters of different pits was shown in Fig. 9. As can
Fig. 7. The specific loading mode and meshing.
Table 4 Comparison of finite element results with experimental results. Number
Pu/kN
PuFEA/kN
4u/%
Py/kN
PyFEA/kN
4y/%
H-0 H-2 Hs-1 W-2 L-5 D-5
34 15.3 25.7 24.0 17.7 25.3
34 14.7 26.5 24 17.2 24.8
0 3.9 3.1 0 2.8 1.9
30.3 5 13.6 11 13.3 13
29.8 4.8 14.2 11.5 13.1 12.6
1.6 4 4.4 4.5 1.5 3.1
8
R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 8. Stress distribution of steel wire with different pit parameters.
be seen from Fig. 9(a) that for single pit, the stress concentration factor Kt increased significantly with the pit depth H, when the pit width remained unchanged (W = 5); For double pits on the same side, the pit size remained unchanged (W = 5, H = 1). With the increase of pit distance D, the stress concentration factor Kt did not change, and equal to that of single pit size (W = 5, H = 1). It showed that the pit distance on the same side did not affect the stress concentration factor, but only depended on the pit size; For adjacent pit depth Hs, when the value of Hs was lower than the adjacent pit depth (H = 1), the value of Kt remained basically unchanged and the same as that single pit. When the value of Hs exceeded the depth of adjacent pits, the stress concentration factor increased with the value of Hs (H > 1), and the same as the Kt of single pit. It showed that the stress concentration factor of the depth of adjacent pits on the same side depended on the maximum pit, and the same as that of the maximum size under the single pit. Therefore, the stress concentration factor mainly depended on the larger size of pits. As can be seen from Fig. 9(b), the stress concentration factor Kt decreased gradually with the increase of pit width while the pit depth remained unchanged (H = 1). Fig. 9(c) showed that the stress concentration factor increased rapidly with the increase of the distance between the centerlines of pits, but when the distance reached a certain distance (L = 6), the stress concentration factor basically remained unchanged and converged to about 2.3. The value was equal to that of the single pit with the same size
(W = 5, H = 1). Therefore, the stress concentration factor was closely related to the pit size. 4.3. Parameter analysis 4.3.1. The depth-width ratio and relative pit depth From the previous experiments and finite element analysis, it can be concluded that the most critical factor affecting the stress concentration factor of corroded steel wire was the pit size (width and depth). Therefore, in order to establish the quantitative relationship between different pit parameters and stress concentration, and to study the influence on stress concentration factor, 84 finite element models of steel wire with different characteristics parameters were established. Specific parameters were shown in Table 5. The relationship between H/W (depth-width ratio) and Kt under different H/T (the relative pit depth) was shown in Fig. 10. It can be found that under the same H/T of pit, the stress concentration factor increased gradually with the H/W increase. Then the stress concentration factor reached the maximum when the H/W was at 1 ~ 2, which most seriously affected the stress concentration factor. However, when greater than 2, the stress concentration factor tended to be stable and converged to about 2.7. In addition, under the same H/W of pit, the greater the H/T, the greater the stress concentration factor Kt is, that is, the more obvious the stress concentration. When the H/T was less than 0.12,
R. Li et al. / Construction and Building Materials 240 (2020) 117915
9
Fig. 9. Relation between stress concentration factor and pits characteristic parameters.
Table 5 Corrosion pit parameters. Parameters
Value/mm
H T H/W H/T
0.2, 0.4, 0.6, 1, 1.5, 2 5 0 ~ 10 0.04, 0.08, 0.12, 0.2, 0.3, 0.4
Model
Note: Pit width was appropriately selected according to the H/W.
there was little difference between the trend of stress concentration factor Kt with the H/W of pit; while the depth-width ratio was higher than 0.12, the difference of trend became larger. The main reason for the difference was that the change of pit shape from wide-shallow to deep-narrow. Therefore, within a certain range of depth-width ratio, the effect of deep and narrow pits on stress concentration factor was the most sensitive. The stress distribution of four different types of pits was shown in Fig. 11. It can be seen that different pit types had different stress distributions and maximum stress locations. For shallow-wide and deep-wide pits (Fig. 11(a) and (b)), the maximum stress was near the bottom of the pit, while for narrow-shallow and deep-narrow pits (Fig. 11(c) and (d)), far away from the bottom of the pit and near the edge of the pit. At the same time, the red area (high stress) was mainly concentrated on the bottom section of the pit (the weakest section), so it can be predicted that the fracture was basically on the weak section. In conclusion, the stress concentration factor Kt was related to depth-width ratio H/W and relative pit depth H/T of pit. According to Fig. 10, the expressions of depth-width ratio H/W of pit and stress concentration factor Kt conformed to the same type of functional relationship. The specific expressions were as follows:
K t ¼ A þ BeðW Þ=C H
Fig. 10. The relationship between H/W and Kt under different H/T.
ð2Þ
The coefficients A, B and C in the formula were all functions containing H/T. The relationship was shown in Fig. 12. Therefore, within the range of relative pit depth (H/T) of 0.4, the quantitative relationship between pit characteristic parameters and stress concentration factor was as follows:
10
R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 11. Stress distribution of steel wire with different pit types.
Fig. 12. Relationships between coefficients A, B, C and H/T.
h i H H K t ¼ A þ BeðW Þ=C ¼ 2:70984 þ 8:83 104 eð T Þ=0:06237 2 H H H þ 1:7546 þ 0:482e76ð T 0:22078Þ eðW Þ=½lnð1:926832:28643T Þ ð3Þ The stress concentration factor of some test specimens was calculated by finite element result and Eq. (3), respectively. The results were shown in Fig. 13. It can be found that the finite element calculation was very close to the calculation result of Eq. (3), and the error was about 5%, which proved the correctness of the theoretical formula. Based on the above discussion, the main reason that affected the strength degradation of steel wire with corrosion pits was that the existence of corrosion pits leaded to excessive local stress and stress concentration. The pit size was the most important factor that affecting the stress concentration. Therefore, based on the relationship between stress concentration factor and characteristic parameters of pits, the formula for predicting residual nominal yield strength of steel wire with pits was proposed.
Fig. 13. Stress concentration factor between finite element and theoretical result.
R. Li et al. / Construction and Building Materials 240 (2020) 117915
r0 y ¼ ry =K t
¼ h
ry
i 2 H H 2:70984 þ 8:83 10 eð T Þ=0:06237 þ 1:7546 þ 0:482e76ð T 0:22078Þ 4
eðW Þ=½lnð1:926832:28643 T Þ H
H
ð4Þ
where ry is the yield strength of non-corroded steel wire and ry’ is the nominal yield strength of steel wire with corroded pits. The comparison between experimental results and theoretical results was shown in Fig. 14. It was found that the experimental result was basically in agreement with the theoretical result, which proved that the theoretical formula can basically predict the nominal yield strength of steel wire with corrosion pits.
Fig. 14. Comparison of yield strength between experimental and theoretical result.
Table 6 Secondary pit parameters. Parameters
Value/mm
H1 W1 H2 W2
0.4, 2 0.6, 1, 2, 6 0.1 0.1
Model with secondary pit
11
4.3.2. Secondary corrosion pit Material corrosion originated from the steel surface, accompanied by primary pits, secondary pits also occurred from the surface of primary pits. Cerit [24,25] studied the stress distribution of semi ellipsoid pits by finite element analysis. It was found that the depth to width ratio of pits was the main factor affecting the stress concentration coefficient, and the existence of secondary pits had a significant effect on the stress concentration coefficient. Therefore, in order to better carry out numerical simulation, this paper assumed that the secondary pit located at the bottom of the primary pit, as shown in Table 6. Taking the size H = 0.4 mm, W = 1 mm; H = 2 mm, W = 1 mm of primary pit as an example, the stress distribution was shown in Fig. 15. I was the top view of primary pits; II was the top view of secondary pits; III was the profile of primary pits; IV was the profile of secondary pits. It can be seen that for steel wires with only primary pits, the area of larger local stress was perpendicular to the direction of tensile load and distributed in strips along the weakest section (See I). The maximum stress value appeared near the pit mouth (See III). When secondary pits appeared at the bottom of primary pits, the area of larger local stress was perpendicular to the direction of tensile load and distributed in blocks at the junction of primary pits and secondary pits (See II). The maximum stress occurred at 1/2 of the depth of secondary pit (See IV). As can be seen from Fig. 15(b), for the primary pit steel wire, the area of larger local stress was perpendicular to the direction of tensile load, and distributed in block shape along the weakest section at the pit mouth (See I). The maximum stress appeared at 1/2 of the pit depth (See III). When secondary pits occurred at the bottom, the area of larger local stress was perpendicular to the direction of tensile load, and distributed in blocks along the weakest section, mainly concentrated at the mouth and the middle part of primary pit, and also the junction of primary pit and secondary pit (See II). The maximum stress occurred at 1/2 of distance from the secondary pit mouth to the pit bottom (See IV). All in all, secondary pits changed the stress distribution of pits, and the maximum stress position shifted from the mouth of pits to the bottom of pits. The stress concentration factor of steel wire with the same secondary pit size under different primary pits was shown in Fig. 16. As can be seen from Fig. 16, the stress concentration factor of steel wire with secondary pits was obviously higher than that of steel wire with only primary pits. When the depth of primary pit was 2 mm (H1 = 2.0 mm), the stress concentration factor of steel wire with secondary pit decreased significantly, compared with the
Fig. 15. Stress distribution of primary and secondary pits.
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R. Li et al. / Construction and Building Materials 240 (2020) 117915
Fig. 16. Stress concentration factor of steel wire under different primary pit sizes.
concentration factor tended to be stable and converged to about 2.7. In addition, with the same H/W of pit, the greater the H/D, the greater the stress concentration factor Kt is, that is, the more obvious the stress concentration. Within a certain range of H/W, the effect of deep-narrow pits on stress concentration factor was the most sensitive. At the same time, the area of high stress was mainly concentrated on the bottom section of the pit (the weakest section), so it can be predicted that the fracture was basically on the weak section. Finally, the models of strength and stress concentration factor of steel wire with different depth-width ratio and relative depth of pit were established. (4) Secondary pits changed the stress distribution of pits, and the maximum stress position shifted from the mouth of pits to the bottom of pits. The stress concentration factor of steel wire with secondary pits was obviously higher than that of steel wire with primary pits only. When the depth of primary pit was 2 mm (H1 = 2.0 mm), the stress concentration factor of steel wire with secondary pit decreased significantly, compared with the steel wire with primary pit (H1 = 0.4 mm). Therefore, attention should be paid to the development of secondary pits in shallow primary pits.
steel wire with primary pit (H1 = 0.4 mm). Therefore, attention should be paid to the development of secondary pits in shallow primary pits.
CRediT authorship contribution statement
5. Conclusion
Rou Li: Data curation, Methodology, Software, Writing - review & editing. Changqing Miao: Conceptualization, Methodology, Supervision. Jie Yu: Visualization, Investigation, Software.
198 cable steel wires with different pit characteristic parameters were obtained by manual prefabrication. The influence of pit depth, width, spacing and location on the strength of steel wires was discussed through monotonic tensile test. The stress distribution of cable steel wires with different pit characteristic parameters was compared by finite element analysis, and the relationship between stress concentration factor and characteristic parameters was studied. The main conclusions were as follows: (1) The influence on the strength of steel wire varied from different characteristic parameters of pit. The strength of steel wire decreased gradually with the increase of pit depth and the decrease of pit width. When two adjacent pits existed, the change of the depth of smaller pits had no obvious effect on the strength of steel wire, only determined by the pits with larger depth. The change of pits clearance on the same side had no obvious effect on the strength of steel wire, but it had significant effect when located on the opposite side. However, when the clearance of pits centerlines exceeded 5 mm, the average strength of steel wire remained basically unchanged. In addition, the pit with larger depth in adjacent pits had an influence on the plastic development process of specimen. (2) The stress at the bottom of the pit was always higher than that of other parts. The stress expanded from the center of the pit bottom to the specimen bottom with the loading, and finally the whole section reached the same stress level, and broke at the pit. The influence of adjacent pits on the stress development of steel wire usually depended on the pit with a larger depth. The stress distribution and fracture location of steel wire varied from different distances of pit. With the increase of distance, the fracture mode of steel wire changed alternately from flat to oblique mode, and the stress development path determined the fracture modes of steel wire. (3) The stress concentration factor reached the maximum at 1 ~ 2 (H/W), which most seriously affected the stress concentration factor. While the H/W was greater than 2, the stress
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