Mechanical properties of high strength steel strand at low temperatures: Tests and analysis

Construction and Building Materials 189 (2018) 1076–1092

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Mechanical properties of high strength steel strand at low temperatures: Tests and analysis Jian Xie, Xueqi Zhao, Jia-Bao Yan ⇑ School of Civil Engineering, Tianjin University, Tianjin 300350, China Key Laboratory of Coast Civil Structure Safety of Ministry of Education, Tianjin University, Ministry of Education, Tianjin 300350, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Low temperature of 20 to 160 °C

increases strength but reduces ductility of steel wires.
Proposed design equations predicts well strengths of wires at low temperatures.
Theoretical models predict well nonlinear strength behaviours of steel strands.
FE models simulate well stress-strain behaviours of strands at low temperatures.
Proposed empirical equations offer means to determine strengths of strands at low temp.

a r t i c l e

i n f o

Article history: Received 8 March 2018 Received in revised form 6 September 2018 Accepted 10 September 2018

Keywords: Low temperatures Steel strands Prestress concrete structures Concrete structures Steel wires Mechanical properties Elastic-plastic behaviours Finite element method

a b s t r a c t This manuscript studied mechanical properties of steel strands for prestressed concrete (PC) structures at different low temperatures ranging from 20 °C to 160 °C. 21 tensile tests were performed to obtain the stress-strain curves of steel wires in strand at different low temperatures. Empirical prediction formulae were developed to incorporate the influences of low temperatures on mechanical properties of steel wires. Based on the stress-strain curves of single wire, theoretical and numerical models were developed to predict the mechanical properties of multi-layer steel strands. The accuracies of these theoretical and numerical models were validated by the test results. Based on the test and analysis results, empirical models were developed to predict the mechanical properties of multi-layer steel strands at different low temperatures including the elastic modulus, yield and ultimate strengths. This offers useful means to calculate the mechanical properties of steel strands at low temperatures since their properties varied with their geometry and layout of steel wires. Finally, recommended prediction procedures are given to determine the mechanical properties of steel strands at different low temperatures. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction High strength steel strands are widely used in civil engineering constructions, e.g., prestressed concrete (PC) structures, long-span ⇑ Corresponding author. E-mail address: [email protected] (J.-B. Yan). https://doi.org/10.1016/j.conbuildmat.2018.09.053 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

structures and bridges [1] as shown in Fig. 1. Due to their excellent mechanical properties, steel strands were also used in engineering constructions in harsh environments with low temperatures, e.g., infrastructures in cold regions, the Arctic onshore and offshore platforms, liquefied natural gas (LNG) containers. In northern China and Tibet, the recorded lowest temperature was 53.4 °C [2]. The lowest temperature in the Arctic could drop to about

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

Nomenclature A0 Ai E, E1 Es Fi F Hi IE If u If y G0i N N0i Pi Ri T T0 Ti Xi

cross sectional area of core wire cross sectional area of helical wires of layer i elastic and plastic modulus of steel wire elastic modulus of steel strand total axial force of helical wires of layer i total axial force of steel strand twisting moment in tangential direction of helical wires of layer i factor for elastic modulus factor for ultimate strength factor for yield strength bending moment in binormal direction of helical wires of layer i number of steel wires in a strand force in binormal direction of helical wires of layer i pitch length helical wires of layer i helix radius of steel strand different temperature level ambient temperature orce in tangential direction of helical wires of layer i line load per unit length in normal direction of helical wires of layer i

70 °C [3,4]. In the scenario of leakage of LNG, the external concrete structure of LNG containers may suffer low temperature of about 165 °C [5–6]. Since these infrastructures suffer low temperatures produced by these harsh environments, the mechanical properties of steel strands used in these structures at low temperatures need to be carefully considered for the evaluation on their structural performances. There is extensive reported research on mechanical properties of steel materials at low temperatures. Elices et al. [7] reported ten-

fy, fu f ys , f us f ya ,f ua L mi r0 ri si

ai e0 ei ey , eu , eF w

ui j0i

ji ci vi m

yield and ultimate strength of steel wire yield and ultimate strength of steel stand yield and ultimate strength of steel wire at ambient temperature length of core wire number of helical wires of layer i radius of core wire radius of helical wires of layer i length of helical wires length on the centreline of layer i lay angle of helical wires of layer i axial strain of core wire axial strain of helical wires of layer i yielding, ultimate and fracture strain of steel wire cross-sectional area of steel wire polar angle of helical wires of layer i curvature in the binormal direction of helical wires of layer i curvature in the normal direction of helical wires of layer i torsional strain of helical wires of layer i twists of helical wires of layer i Poisson’s ratio of steel wire

sile tests on hot rolled steel reinforcements at 20 °C, 80 °C and 180 °C. It showed that as the temperature decreased, both yield and ultimate strengths of steel reinforcements increased but their ductility was slightly affected. Lahlou et al. [8] studied the mechanical properties of mild steel at ambient temperature and low temperature 195 °C. The test results showed that strengths and elastic modulus increased but ductility significantly decreased at low temperatures. Yan et al. [9] carried out tensile tests on mild and high strength steel plates within temperature ranges of

Prestressed concrete beam

Prestressed concrete bridge

Strand

Helical wire Helical wire Strand Core wire Fig. 1. The application of a multi-layer strand.

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+30 °C to 80 °C. They found that fracture strain, yield and ultimate strengths were all increased at low temperatures. Filiatrault et al. [10] reported the mechanical properties of steel reinforcements at temperatures varying from +20 °C to 40 °C. They observed that strengths of steel reinforcements increased at low temperatures. Planas et al. [11] reported the tensile tests on steel strands at 20 °C and 195 °C and test data showed that strengths and elongations of steel strands were both increased at 195 °C. These above experimental results show that most of previous studies focused on mild and high strength steel plates. There is still limited information on the high strength steel strands at low temperatures. Thus, as the key components in the prestressed concrete structures, it is necessary to investigate the mechanical properties of steel strands at low temperatures that will provide useful information on the analysis and design of infrastructures especially the prestressed concrete structures built in cold regions. Multi-layer steel strands typically consist of a straight core wire surrounded by several layers of helical wires in a symmetrical way as shown in Fig. 1. Different from the reinforcement or steel plate, mechanical properties of steel strands are more complex due to their geometric patterns and deforming mechanisms. Several geometric and analytical models have been developed to study the mechanical properties of strands based on Love’s curved beam theory [12]. Machida and Durelli [13] reported the stiffness matrix of steel strands that included the influences of bending and torsion stiffness of wires. Knapp [14] proposed prediction equations and investigated the deformation of core wire. Costello et al. [15,16] presented linearized equilibrium equations that considered the influence of Poisson’s ratio, curvature and lay angle variations. Kumar et al. [17] extended Costello’s theory and offered a closedform solution for axial-torsional stiffness matrix of strands. These developed models were based on the assumption of linearelastic-constitutive law of the material, and the plastic deformation in wires were neglected. Several researchers used finite element methods to study the global response of strands that considered plastic deformation of wires. Jiang et al. [18,19] developed a finite element model to predict the global behaviours of strands under axial tension. Yu et al. [20] proposed a finite element model to study behaviours of seven-wire strands under axial tension that incorporate the effect of friction on the stiffness of strands. Abdullah et al. [21] proposed a full-scale finite element model to observe the breakage response of seven-wire strands that considered contact and frictions among the steel wires. Judge et al. [22] presented a 3D elastic-plastic finite element model for multilayer spiral strands to simulate their global responses and local deformations. Recently, an analytical model was developed by Foti et al. [23] to predict the elastic–plastic behaviour of seven-wire strands under axial torsion. It can be concluded from these previous studies that most of previous analytical models focused on the elastic behaviours of the strands, and there are still few proposed analytical models that could predict their plastic behaviours. Due to the limited computational efficiency, some of the FE models with complex contacts were not suitable for large-scale structural analyses. Moreover, few FE models considered the stress redistribution and contact among different wires in steel strands. Thus, it is necessary to develop accurate theoretical and numerical models to predict global responses of multi-layer strands that considered both geometric and material nonlinearities as well as contact algorithms among different steel wires in the strands. Multi-wire strands are made of several twisted wires. Stressstrain curves of strands are quite different from steel wires due to complex geometric patterns. Under tension, the nominal stress-strain behaviour of the multi-wire strand behaves differently from that of single wire due to stress redistribution and interactions among the wires in strand. And the tensile stress-strain behaviours of steel strands vary with the layout of the wires in

them. Moreover, tensile tests on steel strands are more difficult to perform compared with that on steel wires especially at various low temperatures. Thus, this manuscript aimed to develop prediction methods on the mechanical properties of multi-layer strands at different low temperatures based on tensile tests of steel wires at ambient temperature. This manuscript reported a study on mechanical properties of steel strands at different low temperatures by experimental, theoretical and numerical methods. To achieve this objective, tensile tests were firstly carried out on steel wires at different temperatures ranging from +20 °C to 160 °C to obtain their mechanical properties at low temperatures. Based on the test results of steel wires, empirical prediction formulae were then developed to predict the yield and ultimate strengths of steel wires at low temperatures. In order to predict the mechanical properties of multi-layer steel strands, accurate theoretical and numerical models were proposed. Based on theoretical and numerical analysis results, regression analyses were finally carried out to develop empirical models to predict the tensile stress-strain behaviours of the multi-layer strands. Thus, with the given mechanical properties of steel wires at ambient temperatures, the tensile behaviours of the complex multi-layer steel strands at ambient and low temperatures can be determined by these developed models. 2. Tests on high strength steel wires at low temperatures 2.1. Specimens Twenty-one high strength steel wire coupons in total were prepared for tensile tests at seven different temperature levels of +20 °C, 40 °C, 70 °C, 100 °C, 120 °C, 140 °C, 160 °C. Three identical steel wires were prepared for each temperature level. All the high strength steel wires measured 5.2 mm in diameter and 250 mm in length, and they were cut from central core wires in seven-wire strands. The chemical compositions of these high strength steel wires are listed in Table 1. The two ends of each steel wire were enlarged to facilitate the anchorage of the tested specimens to the testing frame. Fig. 2 shows a representative steel wire coupon for tensile tests. 2.2. Test setup and loading procedures A 100-ton loading machine working with a cooling chamber was used for the tensile tests on steel wires at low temperatures as shown in Fig. 2. Designed target temperatures for high strength steel wires included +20 °C, 40 °C, 70 °C, 100 °C, 120 °C, 140 °C and 160 °C. To simulate the low temperature, a cooling chamber with an automatic control system was used. The temperature in the cooling chamber can be lower to 190 °C. Cooling rate, stabilization duration and target temperature can be pre-set in integrated system running on a controlling PC. During tensile tests, liquid nitrogen was sprayed into the cooling chamber through an electromagnetic valve to low down the temperatures of testing specimens to the target temperatures. Several thermocouples were placed at different locations in the chamber to monitor the environmental temperatures. Meanwhile, thermocouples worked together with electromagnetic valve to maintain the target temperature by controlling the injection velocity of liquid nitrogen. Insulation measures were taken at the conjunction between the testing machine and cooling chamber to reduce heat loses. Two ends of steel wires were installed on the testing machine through two clamps. The bottom clamp was fixed to the solid bottom frame whilst the top clamp moved upwards that transferred the tensile force to the steel wires. Three thermocouples were directly attached to the specimens at different locations to monitor

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092 Table 1 Chemical composition of the steel wire. C (%)

Si (%)

Mn (%)

P (%)

S (%)

Cr (%)

Ni (%)

Cu (%)

V (%)

0.801

0.227

0.75

0.009

0.006

0.248

0.014

0.012

0.003

100ton MTS Loading system

Cooling chamber

Frame

Tensile force

Liquid Nitrogen

Detail A 250mm

Extensometer

Steel wire Coupon

PT100 Thermolcouples

Data acquisition system

(a) Detail A

(b) Test Setup Fig. 2. Tensile test setup of steel wires at low temperatures.

the temperatures and achieve a balanced cooling effect. Once the target temperature in the cooling chamber was achieved, it should be maintained for 15 min and temperature variations should be controlled within 1 °C to ensure that all tests were conducted under steady state condition. After that, direct tensile tests were conducted. The loading rate was determined according to ASTM: A370-13 [24] for low temperature tensile tests. Two strain gauges were attached to the middle region of high strength steel wires to record their strains. An extensometer was also used to record the plastic strains after the strain gauges spoiled. The reaction forces, displacement, temperature, strain gauges and extensometer were recorded through a data acquisition system during the tests.

900

Stress (MPa)

fy 600

E

300

1 0 0

0.2

0.4

0.6

0.8

Strain (%)

2.3. Test results

Fig. 3. Determination of the elastic modulus and yield strength.

Test results presented herein included tensile stress-strain curves at different low temperature levels, yield strength, ultimate strength, elastic modulus, yielding strain, fracture strain, and reduction in cross-sectional area. Since the stress-strain curves of high strength steel wires have no transparent yielding plateaus, 0.2% offset method was used to determine the yield strength and elastic modulus according to ASTM: A370-13 [24] as shown in Fig. 3. The indexes of yielding strain ey , ultimate strain eu , fracture strain eF and reduction in the cross-sectional area w, were used to describe the plasticity of steel that are defined as the following:

ey

Ly  L ¼ L

ð1aÞ

Lu  L L

ð1bÞ

eu ¼

eF ¼

LF  L L

ð1cÞ

where L denotes the original length of the measured region by extensometer; Ly denotes the length of the measured region at the yield; Lu denotes the length of the measured region at the ultimate; LF denotes the length of the measured region at the fracture.

w¼

A  AF A

ð2Þ

where A denotes the area of the cross section before test; AF denotes the area of the cross section at the fracture point. 2.3.1. Stress-strain curves Fig. 4 shows the stress-strain curves of representative tests at seven low temperatures of +20 °C, 40 °C, 70 °C, 100 °C,

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

Fig. 5(a) also shows that the E exhibits low correlation ratio of 0.33 with the low temperatures that implies the effect of T on E is quite limited and can be neglected.

2500

Stress

MPa

2000

1500

1000

20

-40

-70

-100

-120

-140

500 -160 0 0

1

2

Strain %

3

4

5

Fig. 4. Stress-strain curves of steel wires at different low temperatures.

120 °C, 140 °C and 160 °C. For high strength steel wires, the stress increases linearly until it reaches yield strength. It can be observed that there are no yielding plateaus in the stress-strain curves. Beyond yielding point, the decreasing stiffness of specimens resulted in the decrease of the slope of stress-strain curves. After achieving ultimate strength, stress-strain curves show obvious decrease in stress accompanied with necking taking place in the middle region of the specimen. Finally, fractures occurred to all the tested specimens. Fig. 4 also shows that both yield and ultimate strengths are increased as the temperature decreases. 2.3.2. Elastic modulus The elastic modulus of high strength steel wires is determined from the initial elastic portion of the stress-strain curves. Table 2 lists the elastic modulus of high strength steel wires at different low temperatures. Fig. 5 (a) shows the influence of the low temperature T on the elastic modulus E. It shows that the elastic modulus of high strength steel wires increases slightly with the decreasing temperature. As the temperature decreases from +20 °C to 160 °C, the average E value is only increased by 5%. Moreover,

2.3.3. Yield strength and ultimate strength Yield strength fy and ultimate strength fu of the high strength steel wires can be determined from the corresponding stress– strain curves by the methods as specified in ASTM: A370-13 [24]. ‘‘0.2% strain offset” method is used to determine the yield strength as shown in Fig. 3. The determined values of yield and ultimate strengths at various low temperatures are listed in Table 2. Fig. 5 (b) shows the effects of low temperature T on yield strength fy and ultimate strength fu. They show that as the temperature decreases from +20 °C to 160 °C, both yield and ultimate strengths increase with the decreasing temperature, and the increasing rate of yield strength is smaller than that of ultimate strength. As the temperature decreases from +20 °C to 40 °C, 70 °C, 100 °C, 120 °C, 140 °C and 160 °C, the yield strength is averagely increased by 6%, 12%, 13%, 14%, 16% and 18%, respectively; and the ultimate strength is averagely increased by 6%, 12%, 14%, 16%, 18% and 20%, respectively. The correlation coefficients of yield and ultimate strengths with temperature are 0.97 and 0.96, respectively, which are much higher than that of the elastic modulus. This implies both fy and fu of steel wires are more significantly influenced by and they are more sensitive to low temperatures. Thus, the influences of the low temperatures on fy and fu should be carefully considered. 2.3.4. Yielding strain and ultimate strain The measured yielding strain ey and ultimate strain eu are listed in Table 2. Fig. 5 (c) plots the influences of the temperature T on yielding strain ey and ultimate strain eu . These table and figure show that both ey and eu increase as the temperature decreases from +20 °C to 100 °C. As T decreases from +20 °C to 40 °C, 70 °C, 100 °C, the values of ey (oreu ) are averagely increased by 6% (3%), 12% (7%), and 15% (15%), respectively. At the temperature

Table 2 Details and tensile test results of the steel wires. Item

T (°C)

E (GPa)

E (GPa)

fy (MPa)

fy (MPa)

fu (MPa)

fu (MPa)

ey (%)

ey (%)

eu (%)

eu (%)

eF (%)

eF (%)

w (%)

w (%)

A-1 A-2 A-3 B-1 B-2 B-3 C-1 C-2 C-3 D-1 D-2 D-3 E-1 E-2 E-3 F-1 F-2 F-3 G-1 G-2 G-3

20 20 20 40 40 40 70 70 70 100 100 100 120 120 120 140 140 140 160 160 160

211.7 209.0 198.8 212.3 202.4 204.1 208.6 199.8 206.5 205.7 207.9 210.6 213.5 212.7 204.1 213.7 217.5 221.2 215.9 209.7 221.1

206.5

1799.4 1817.9 1808.2 1931.3 1903.3 1919.2 2035.0 2003.4 2012.9 2043.8 2039.0 2027.7 2079.2 2064.3 2048.3 2093.4 2089.9 2104.6 2127.7 2139.9 2131.7

1808.5

1937.1 1961.8 1942.6 2100.3 2080.5 2028.8 2200.1 2168.0 2190.2 2218.0 2223.0 2222.9 2280.2 2267.7 2226.7 2284.3 2297.0 2317.2 2349.4 2315.9 2316.8

1947.2

1.03 1.03 1.04 1.08 1.13 1.08 1.15 1.14 1.17 1.21 1.18 1.17 1.17 1.17 1.19 1.20 1.18 1.16 1.18 1.19 1.15

1.03

3.41 3.64 3.58 3.57 3.80 3.53 3.73 3.75 3.85 4.03 4.00 4.19 3.87 3.93 3.91 4.17 3.82 4.07 3.98 3.66 3.83

3.54

3.78 3.76 3.94 3.79 4.02 3.79 3.92 3.92 4.08 4.28 4.25 4.56 4.16 4.46 4.20 4.41 4.18 4.38 4.30 4.22 4.06

3.83

0.20 0.19 0.18 0.20 0.21 0.18 0.17 0.19 0.15 0.21 0.19 0.20 0.17 0.19 0.11 0.19 0.17 0.15 0.15 0.16 0.09

0.19

206.3

205.0

208.1

210.1

217.4

215.6

1917.9

2017.1

2036.9

2063.9

2096.0

2133.1

2069.9

2186.1

2221.3

2258.2

2299.5

2327.4

1.10

1.15

1.19

1.18

1.18

1.17

3.63

3.78

4.07

3.90

4.02

3.83

3.87

3.97

4.36

4.27

4.33

4.19

0.20

0.17

0.20

0.16

0.17

0.13

Number in three identical specimens

A-1

E, E, fy, fy, fu, fu denote elastic modulus, average elastic modulus, yield strength, average yield strength, ultimate strength,

Temperature level: 1~7 denote 20, -40, -70, -100, -120, -140, -160ćˈrespectively. average ultimate strength of steel wires, respectively. ey, ey, eu, eu, eF, eF denote yield strain, average yield strain, ultimate strain, average ultimate strain, facture strain, average facture strain of steel wires, respectively. w and w denote reductions in cross-sectional area and average reductions in cross-sectional area, respectively.

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fy

250

fu

2500 y = -0.061x + 204.54 R² = 0.33

y = -2.150x + 1999.7 R² = 0.96

2300

Strength (MPa)

E (GPa)

300

2100

200

1900

150 -200 -160 -120 -80

T(

-40

0

1700

40

-200 -160 -120 -80

)

T(

(a) Effect on E

u

y = -1.775x + 1855.8 R² = 0.97 -40

0

40

0

40

)

(b) Effect on fy and fu

F

y

0.21

4.00

0.18

1.20

(%)

Strain (%)

3.25

1.10

2.50 1.75

0.15

1.00 -200-160-120 -80 -40 0

40

0.12

1.00 -200 -160 -120 -80

T( (c) Effect on y,

-40

0

40

-200 -160 -120 -80

T(

) u and

-40

)

(d) Effect on

F

Fig. 5. Effect of low temperature on different mechanical properties of steel wires.

of 100 °C, both ey and eu of high strength steel wires are 1.15 times of those values at ambient temperature. However, once the temperature is below 100 °C, i.e., as T decreases from 100 °C to 160 °C, both ey and eu slightly decreased. As T decreases from 100 °C to 120 °C, 140 °C, 160 °C, ey (oreu ) are averagely reduced by 0.95% (4.14%), 0.57% (1.21%), and 1.30% (6.05%), respectively. Thus, it can be concluded that as the temperature is below 100 °C, the influences of the low temperature on ey and eu are marginal and even can be neglected. 2.3.5. Fracture strain and reductions in cross-sectional area The fracture strain eF and reductions in cross-sectional area w are used to describe the plasticity of steel at low temperatures. Fig. 5 (c) and (d) depicts the influences of low temperature T on eF and w, respectively. Table 2 lists the test results of eF and w at various low temperatures. As T decreases from 20 °C to 100 °C, eF increases with the decrease of temperature. As T decreases from +20 °C to 40 °C, 70 °C, 100 °C, the average eF value is increased by 1%, 4%, and 14%, respectively. However, as T is below 100 °C,eF exhibits slight changes. As T decreases from 100 °C to 120 °C, 140 °C, 160 °C, the eF is averagely reduced by 2.03%, 0.81%, and 3.90%, respectively. Meanwhile, w ratio almost remains at the same level as the temperature decreases from +20 °C to 100 °C, which implies the effect of temperature T on w are quite limited within this temperature interval. As T decreases from 100 °C to 120 °C, 140 °C, 160 °C, w is averagely reduced by 22%, 16%, and 33%, respectively.

Fig. 6 shows the typical failure modes of the tested specimen and fracture surfaces of the cross section. It can be observed that obvious necking occurred to the specimens after the tests at temperatures of +20 °C to 100 °C. However, as temperature T is below 100 °C, failure mode of the tested specimens changes from ductile mode to brittle mode. 2.4. Proposal stress-strain curves for the single wire at low temperatures The effects of low temperatures on yield strength fy and ultimate strength fu of the single wire are shown in Fig. 5 and Table 2. Based on the reported test data in Table 2, the mathematical relationships between the mechanical properties of steel wires and the low temperature T are developed, along with regression analyses. Thus, mathematical relationships between the yield (or ultimate) strength and temperatures can be given as follows:

f y ¼ f ya eað1=T1=T 0 Þ

ð3Þ

f u ¼ f ua ebð1=T1=T 0 Þ

ð4Þ

where f ya and f ua express the yield strength and ultimate strength of steel wires at ambient temperature, respectively; f y and f u express the yield strength and ultimate strength of steel wires at temperature T, respectively, in K; T0 is the ambient temperature; a and b are

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70 °C, 100 °C, 120 °C, 140 °C and 160 °C were used for the determinations on A, B, a, b and k at different low temperatures as listed in Table 3. Hence, the equation of regression analysis for predicting the yield and ultimate strengths are recommended with the given yield and ultimate strengths at ambient temperature as follows:

f y ¼ f ya e50:17ð1=T1=T 0 Þ

ð10Þ

f u ¼ f ua e54:31ð1=T1=T 0 Þ

ð11Þ

where T0 and T denote the ambient temperature and low temperature, in K, and 114 K  T  293 K. According to Eqs. (10) and (11), the simplified equations can be obtained to predict the yield and ultimate strengths of steel wires as follows:

(a) Fracture modes of the steel wires Ductile fracture 20

-40

A-2

B-2

-70

C-3

-100

D-1

-120

E-2

-140

F-1

-160

G-3

Fig. 6. Fracture of the steel wires at different low temperatures.

the sensitive coefficients of steel wires for yield strength and ultimate strength, respectively, in 1/K. The relationships between coefficient a and b can be obtained as follows:

k¼

lgðf 0 =f u Þ lgðf 0 =f y Þ

lgðf 0 =f u Þ lgðf 0 =f y Þ

ð5Þ

ð6Þ

where k denotes the constant value related to the mechanical properties of steel wires at different low temperatures. The fracture strength of the steel wires f0 is independent with the temperature that can be obtained by tensile tests. In Eqs. (3) and (4), the temperature is in Kelvin degree. In order to facilitate to engineering applications, the unit of temperature can be converted from Kelvin degrees to Celsius degrees. The simplified equations can be modified as the following:

f y ¼ f ya eAðT 0 TÞ

ð7Þ

f u ¼ f ua eBðT 0 TÞ

ð8Þ

where A and B are the sensitive coefficients of steel wires for yield strength and ultimate strength, respectively, in 1/.°C Similar to a and b, the relationships between coefficient A and B can be obtained according to Eqs. (5) and (6) and it is written as follows:

B ¼ kA

ð12Þ

f u ¼ f ua e0:0011ðT 0 TÞ

ð13Þ

Brittle fracture

(b) Fracture surfaces of the steel wires

b¼a

f y ¼ f ya e0:0010ðT 0 TÞ

ð9Þ

where the value of k in Eq. (9) can be calculated by Eq. (6). With the test data in Table 2, different coefficients, A, B, a, b and k are calculated by Eqs. (3)–(9). Due to the regression analysis and experimental studies focused on the mechanical properties of steel wires at low temperatures, the experimental results of 40 °C,

where T0 and T denote, respectively the ambient temperature and low temperature, in °C, and 160 °C  T  20 °C. Table 4 lists the predicted yield and ultimate strengths at different low temperatures by Eqs. (10)–(13). These predicted values are compared with those test results as shown in Fig. 7 (a) and (b). It can be observed that the predictions by Eqs. (10)–(13) show high correlations with the experimental values and the correlation coefficients R2 are all larger than 0.9. It can be also found that the correlation coefficients R2 for predictions by Eqs. (12) and (13) are both 0.98 that are larger than those by Eqs. (10) and (11). This implies Eqs. (12) and (13) offers more accurate predictions for yield and ultimate strengths of steel wires at different low temperatures. Fig. 8 (a) and (b) compares the predictions by Eqs. (10)–(13) with the test data of 21 specimens. For Fig. 8 (a), most of the predictions fall within the scope 10% error of the test results and the average test-to-prediction are both 0.98. The coefficient of variation (COV) listed in Table 6 are 0.04 and 0.05, respectively. Fig. 8 (b) shows that the predictions fall within the scope 5% error of test results. Furthermore, the average test-to-prediction for yield and ultimate strengths are about 1.0 and coefficients of variation (COV) are only 0.01. According to these figures and data in Table 4, it can be concluded that all of the empirical formulae Eqs. (10)–(13) can offer satisfactory predictions on the yield and ultimate strengths of steel wires at different low temperatures. In particular, Eqs. (12) and (13) can offer more accurate predictions. Thus, once the yield and ultimate strengths of steel wires at ambient temperature are given, Eqs. (10)–(13) can be used to determine the yield and ultimate strengths of steel wires at different low temperatures within 20 °C to 160 °C. In addition, according to the test results, the effects of low temperatures on elastic modulus and plastic modulus is quite limited and can be neglected. Thus, stress-strain curves for steel wires at low temperatures can be determined by the given modulus and strengths obtained by Eqs. (10)–(13).

Table 3 Different coefficients obtained from the regression analysis on test results. T (°C)

a (1/K)

b (1/K)

A (1/K)

B (1/K)

k

-40 -70 -100 -120 -140 -160 Mean

67.44 72.83 50.76 42.79 36.38 30.79 50.17

70.13 77.22 56.22 47.99 41.01 33.27 54.31

0.0010 0.0012 0.0010 0.0009 0.0009 0.0009 0.0010

0.0010 0.0013 0.0011 0.0011 0.0010 0.0010 0.0011

1.040 1.060 1.107 1.122 1.127 1.080 1.090

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092 Table 4 Predicted yield and ultimate strengths at different low temperatures by the developed design equations. T (°C)

fy (MPa)

fu (MPa)

fy1 (MPa)

fu1 (MPa)

fy/fy1

fu/fu1

fy2 (MPa)

fu2 (MPa)

fy/fy2

fu/fu2

20 20 20 40 40 40 70 70 70 100 100 100 120 120 120 140 140 140 160 160 160

1799.4 1817.9 1808.2 1931.3 1903.3 1919.2 2035.0 2003.4 2012.9 2043.8 2039.0 2027.7 2079.2 2064.3 2048.3 2093.4 2089.9 2104.6 2127.7 2139.9 2131.7

1937.1 1961.8 1942.6 2100.3 2080.5 2028.8 2200.1 2168.0 2190.2 2218.0 2223.0 2222.9 2280.2 2267.7 2226.7 2284.3 2297.0 2317.2 2349.4 2315.9 2316.8

1808.5 1808.5 1808.5 1889.3 1889.3 1889.3 1949.7 1949.7 1949.7 2034.0 2034.0 2034.0 2111.5 2111.5 2111.5 2216.5 2216.5 2216.5 2366.5 2366.5 2366.5

1947.2 1947.2 1947.2 2041.5 2041.5 2041.5 2112.3 2112.3 2112.3 2211.4 2211.4 2211.4 2302.7 2302.7 2302.7 2426.8 2426.8 2426.8 2605.2 2605.2 2605.2

0.99 1.01 1.00 1.02 1.01 1.02 1.04 1.03 1.03 1.00 1.00 1.00 0.98 0.98 0.97 0.94 0.94 0.95 0.90 0.90 0.90

0.99 1.01 1.00 1.03 1.02 0.99 1.04 1.03 1.04 1.00 1.01 1.01 0.99 0.98 0.97 0.94 0.95 0.95 0.90 0.89 0.89

1808.5 1808.5 1808.5 1920.3 1920.3 1920.3 1978.8 1978.8 1978.8 2039.1 2039.1 2039.1 2080.3 2080.3 2080.3 2122.3 2122.3 2122.3 2165.2 2165.2 2165.2

1947.2 1947.2 1947.2 2080.1 2080.1 2080.1 2149.8 2149.8 2149.8 2222.0 2222.0 2222.0 2271.4 2271.4 2271.4 2321.9 2321.9 2321.9 2373.6 2373.6 2373.6

0.99 1.01 1.00 1.01 0.99 1.00 1.03 1.01 1.02 1.00 1.00 0.99 1.00 0.99 0.98 0.99 0.98 0.99 0.98 0.99 0.98

0.99 1.01 1.00 1.01 1.00 0.98 1.02 1.01 1.02 1.00 1.00 1.00 1.00 1.00 0.98 0.98 0.99 1.00 0.99 0.98 0.98

0.98 0.04

0.98 0.05

1.00 0.01

1.00 0.01

Mean Cov

fy1, fy2 denote the yield strengths predicted by Eqs. (10) and (12), respectively. fu1, fu2 denote the ultimate strengths predicted by Eqs. (11) and (13), respectively.

Fig. 7. Comparisons of the scatters of predictions by the design equations with the test results for different steel wires.

Fig. 8. Comparisons of predictions by the design equations with test results for 21 steel wires.

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

3. Theoretical models on predicting stress-strain behaviours of steel strands Multi-layer strands were usually used in the prestressed concrete structures. Different from single wire, multi-layer strands were made through twisting several steel wires together as an integrity. These geometric and processing differences would produce the differences of nominal stress-strain curves of the steel strands from those of single straight wire. Compared with tensile tests on steel strands, tensile tests on wires are much easier to perform. This section makes efforts to develop the theoretical models to predict the stress-strain curves of multi-layer steel strands based on the tensile stress-strain curves of steel wires.

The multi-layer strand usually consists of a straight core wire with radius r0, surrounded by several layers of helical wires in a symmetrical way. The helix radius Ri and pitch length Pi in unloading stage can be calculated as follows: i1 X

2rj þ r i

ð14Þ

j¼1

Pi ¼

2pRi tanai

ð15Þ

where r0 is the radius of core wire, and ri is the radius of helical wires of layer i. ai denotes lay angle of helical wires of layer i as shown in Fig. 9. In unloading stage, the centerline of the helical wires are given by the following form:

rðui Þ ¼ Ri cosui eX þ Ri sinui eY þ Ri

ui eZ tanai

sinui sinai

1

0

cosui

1

0

sinui cosai

1

B C B C B C t i ¼ @ cosui sinai A; ni ¼ @ sinui A; bi ¼ @ cosui cosai A 0 cosai sinai ð17Þ where ti, ni and bi are tangent, normal and binormal unit vectors, respectively. The relationship between the local coordinate system and the global coordinate system is expressed as follows:

0

ni

1

0

cosui

B C B @ bi A ¼ @ sinui cosai sinui cosai ti

sinui cosui cosai cosui cosai

0

10

eX

1

CB C sinai A@ eY A cosai eZ

ð18Þ

The curvature in the normal direction, binormal direction and twists of helical line in Frenet-Serret frame before loading can be defined as follows:

3.1. Geometric model and elastic-plastic constitutive law of steel strands

Ri ¼ r 0 þ

0

ð16Þ

where a right-handed Cartesian frame {eX , eY , eZ } is used to describe the geometric structure of the steel strand, and eZ coincides with the strand centerline as shown in Fig. 9. ui denotes the polar angle of helix. This angle can be measured around the eZ relative to eX . A local coordinate system attached to the centerline of the helical wire is used to describe the local deformation as shown in Fig. 10 (a). Eq. (17) can be obtained as follows:

2

ji ¼ 0; j0i ¼

sin ai ; Ri

vi ¼

sinai cosai Ri

ð19Þ

The length of helical wires and core wire the centerline are defined as si and l. After axial deformation, the small change of helical wires, core wire, lay angle, polar angle and helix radius are defined as dsi , dl, dai , dui and dRi . The axial strain of core wire and helical wires can be expressed as e0 ¼ dl=l and ei ¼ dsi =si , respectively. The polar angle is expressed as follows: ui ¼ sinai si =Ri . According to [15], the helical wires strain ei and torsional strain ci of helical wires can be obtained as follows:

ei ¼ e0  tanai dai ci ¼ Ri

P r 0 e0 þ i1 dui ei j¼1 r j ej þ r i ei  dai þ m ¼ s tanai Ri tanai

ð20Þ ð21Þ

where m is the Poisson’s ratio of steel wire. The change of helix radius can be influenced by Poisson’s ratio and the deformation between helical wires and core wire. Due to the effect of interwire deformation on small-diameter steel strands is quite limited, for simplifying the calculation, interwire deformation is neglected, and thus, only the influence of Poisson’s ratio is considered in this work. The change of helix radius can be calculated as follows:

Fig. 9. A multi-layer strand with cross section.

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

t s

b

s s

s

n

(a)

(b)

(c )

Fig. 10. (a) Centerline of a helical wire. (b) Deformation through a wire acting on a helical wire [15]. (c) Force and moment resultants cross section [24].

dRi ¼ ðmr 0 e0 þ

i1 X

2mrj ej þ mr i ei Þ

ð22Þ

j¼1

The change of curvature in the normal direction, binormal direction and twists of helical line after axial deformation can be defined as follows:

dji ¼ 0

ð23aÞ 2

2

dj0i ¼

sin ai sin ai  Ri þ dRi Ri

ð23bÞ

dvi ¼

sinai cosai sinai cosai  Ri þ dRi Ri

ð23cÞ



Fig. 11. Bi-linear curve of steel wire.

where ai denotes the lay angle after deformation and can be 

expressed as ai ¼ ai þ dai . An elastic-plastic constitutive law is introduced in this theoretical models. Suppose that any point on the cross section of helical wires is Sðr s ; hs Þ, and its axial strain es and tangential strain cs as shown in Fig. 10 (b) can be determined as follows:

es ¼ ei  dj0i rs coshs

ð24Þ

cs ¼ rs dvi

ð25Þ

Accordingly, the axial strain es and torsional strain cs can be decomposed into purely elastic strainee , ce and plastic strainep , cp , i.e., es ¼ ee þ ep , cs ¼ ce þ cp . In the elastic region, the normal stresses r is obtained by adopting the generalized Hooke’s law as follows:

r ¼ ee E; s ¼

E c ¼ Gce ; 2ð1 þ mÞ e

ð26Þ

where E denotes the elastic modulus of high strength steel wires. In the plastic region, the normal stress r can be calculated as follows:

r ¼ rs þ Ep ep Ep ¼

EE1 E  E1

ð27Þ

f ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   3 sij  C epij sij  C epij  rs  0 2

where constant C is obtained by a simple tensile test and equals to 2/3Ep in Eq. (29). The symbols sij denotes the stress deviator, sij ¼ rij  rm dij . The symbols epij denotes the plastic strain and defined as follows:

0

eps

epij ¼ B @

0 1 2

p s sinhs

c

0 meps  12

c

p s coshs

 12 cps sinhs 1 2

1

cps coshs C A meps

ð30Þ

In the plastic region, the incremental plastic strain can be written as:

depij ¼ deij  deeij

ð31Þ

The incremental strain resulting from a stress increment that considered the classic Prandtl-Reuss associated flow rules is introduced as follows:

depij ¼ dk

ð28Þ

where E1 and rs denotes the plastic modulus and the first-yielding stress of the high strength steel wires are shown in Fig. 11. Subsequent yield criterion f is defined by the von-Mises yield criterion, along with a kinematic hardening law as follows:

ð29Þ

dk ¼ 2r2 s

9G

@f @ rij

@f @ rij

deij

ð3G þ Ep Þ

ð32Þ

ð33Þ

where the symbol k is a proportional coefficient that satisfies flow rules and consistency conditions.

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

Newton-Raphson method was used for calculations of incremental steps of elastic-plastic constitutive law [23,25]. Based on Love’s curved beam equations, the effect of Poisson’s ratio and lay angle variations are considered, and the effect of interwire friction and shear deformability is ignored. As shown in Fig. 10 (c), the following equilibrium equations are established:

N0i ðvi þ dvi Þ þ T i ðj0i þ dj0i Þ þ X i ¼ 0

ð34aÞ

G0i ðvi þ dvi Þ þ Hi ðj0i þ dj0i Þ  N 0i ¼ 0

ð34bÞ

where T i and Hi respectively express the force and the twisting moment in tangential direction of helical wires, N0i and G0i respectively express the force and the bending moment in binormal direction of helical wires and X i is line load per unit length in normal direction. The twisting momentHi and bending moment G0i in helical wires are derived through integration over the cross sectional areaAi :

Z

Hi ¼ G0i ¼ 

ð35aÞ

r s sdAi

Ai

Z Ai

ð35bÞ

rs coshs rs dAi

The force T i in tangential direction of helical wires can be obtained as follows:

Z

Ti ¼ Ai

ð35cÞ

rs dAi

The total axial force of each layer of helical wires can be calculated as follows:

F i ¼ mi T i cosðai þ dai Þ þ mi N0i sinðai þ dai Þ

ð36Þ

where mi represents the number of helical wires in each layer. The resultant axial force acting on the steel strand is the combination of the force on each wire of steel strand as follows:

Z F¼ A0

r0 dA0 þ

i X

ð37Þ

Fi

j¼1

where A0 is the cross sectional area of core wire. In summary, the value of axial strain of steel strand equal to axial strain of core wire and the axial stress of steel strand is obtained as follows:

rstrand ¼

A0 þ

F Pi

ð38aÞ

j¼1 mi Ai

estrand ¼ e0

ð38bÞ

The above equations define the elastic-plastic strain-stress behaviours of steel strands. The geometric model of steel strands is defined by Eqs. (14)–(19). In the solution process, the axial strain of steel strand is assigned, which equals to the axial strain of core wire e0 . The axial strain ei and torsional strain ci of helical wires defined by Eqs. (20)–(21) can be derived from the geometric relationship between core wire and helical wires. Within a local coordinate system, the normal stresses of core wire, the normal

stresses and tangential stresses of helical wires can be calculated through the elastic-plastic constitutive law defined by Eqs. (24)– (33) that is proposed by using the von-Mises yield criterion, along with a kinematic hardening law. Based on Love’s curved beam equations and Costello’s model [16], the equilibrium equations Eqs. (34) are established, and the resultant axial force can be calculated by Eqs. (35)–(37). Thus, the stress-strain behaviours of steel strands can be obtained by Eqs. (14)–(38). 3.2. Model validations To verify the accuracy of the elastic-plastic mechanical model, the theoretical results were compared with experimental data reported by Utting and Jones [26]. The geometric and material details of the tested steel strands are listed in Table 5 [18,26]. The proposed theoretical model in this manuscript is also compared with Costello’s theory that is based on linearly elastic constitutive law. The presented stress-strain curves of seven-wire and nineteen-wire steel strands by the developed theoretical models in this manuscript resemble well with curves proposed by Costello [15] in the linear portion, and these models also can give accurate predictions in the plastic portion, as depicted in Fig. 12. For sevenwire strand, the curve by the proposed model agrees well with the experimental curve [26] in both linear and plastic portions. The differences of elastic modulus between the theoretical results and test results are only about 1%. And in plastic range, the axial force is slightly larger than test results with a small difference of less than 3%. For nineteen-wire strand, it can be noted that the theoretical results are slightly higher than the test results of Utting and Jones [26]. That is due to the fact that contact deformation is ignored in the proposed model.

4. Numerical model of stress-strain behaviours of steel strands 4.1. Finite element model 4.1.1. General A three-dimensional finite element model was developed using general commercial software ABAQUS to simulate the tensile behaviours of steel strands at different low temperatures. ABAQUS/ Standard type of implicit solver was used for the analysis on steel strands. 4.1.2. Elements and material properties All the components were modelled by three-dimensional eightnode continuum elements with reduced integration point and hourglass control (C3D8R) in ABAQUS element library. This element consists of three translation degrees of freedom at each node and one integration point. Convergence studies were also performed to find an appropriate mesh density that considered both computing accuracy and efficiency. Finally, the mesh sizes for the radical elements are 1/10 of the wire diameter and the longitudinal elements are 1/80 of helical pitch, respectively. Fig. 13 shows the FE model with finalized mesh size.

Table 5 The geometric and material data. Type

r0 (mm)

r1 (mm)

r2 (mm)

a1 (°)

a2 (°)

E (GPa)

E1 (GPa)

fy (MPa)

fu (MPa)

m

1*7 1*19

1.97 1.83

1.865 1.665

– 1.665

11.8 14.6

– 14.4

188 188

24.6 24.6

1540 1540

1800 1800

0.3 0.3

r0, r1, r2 denote the radius of core wire, first layer helical wires and second layer helical wires, respectively. a1 and a2 denote the lay angle of first layer helical wires and second layer helical wires, respectively. E, E1, m, fy and fu denote the elastic modulus, plastic modulus, Poisson’s ratio, the yield and ultimate strengths, respectively.

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092 Table 6 Predicted elastic modulus, yield and ultimate strengths at different low temperatures by theoretical and numerical models. N

T (°C)

D (mm)

a1(°)

a2(°)

fys,a(MPa)

fus,a (MPa)

Es,a(GPa)

fys,e (MPa)

fus,e(MPa)

Es,e(GPa)

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160 20 40 70 100 120 140 160

6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 11.4 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13

5 5 5 5 5 5 5 10 10 10 10 10 10 10 15 15 15 15 15 15 15 5 5 5 5 5 5 5 10 10 10 10 10 10 10 15 15 15 15 15 15 15 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15

– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 5 5 5 5 5 5 5 10 10 10 10 10 10 10 15 15 15 15 15 15 15 10 10 10 10 10 10 10

1801.7 1913.0 1971.3 2031.3 2072.4 2114.2 2156.9 1781.5 1891.6 1949.2 2008.6 2049.2 2090.6 2132.8 1749.2 1857.4 1913.9 1972.2 2012.1 2052.7 2094.2 1802.8 1914.3 1972.6 2032.7 2073.7 2115.6 2158.3 1785.8 1896.3 1954.0 2013.5 2054.2 2095.7 2138.0 1758.0 1866.8 1923.6 1982.2 2022.2 2063.1 2104.8 1796.8 1907.9 1966.0 2025.9 2066.8 2108.5 2151.1 1782.7 1892.9 1950.6 2009.9 2050.6 2092.0 2134.2 1761.4 1870.3 1927.2 1985.9 2026.1 2067.0 2108.7 1774.2 1883.9 1941.2 2000.3 2040.8 2082.0 2124.0

1939.4 2072.1 2141.3 2213.5 2263.4 2313.6 2365.1 1926.1 2059.4 2128.9 2200.3 2250.7 2300.6 2352.5 1887.3 2020.1 2088.6 2161.7 2210.6 2261.4 2313.3 1920.7 2051.0 2119.6 2190.5 2239.3 2289.0 2339.5 1897.0 2026.1 2093.7 2163.6 2211.7 2260.7 2310.4 1859.4 1985.5 2051.5 2119.8 2166.8 2214.7 2263.8 1933.5 2063.1 2130.7 2202.5 2251.6 2300.8 2352.2 1912.8 2043.5 2111.0 2181.4 2230.6 2279.3 2330.1 1884.7 2012.5 2078.8 2148.0 2195.6 2244.1 2293.9 1898.6 2025.8 2090.3 2162.5 2207.6 2259.5 2308.0

197.3 197.3 197.3 197.3 197.3 197.3 197.3 189.3 189.3 189.3 189.3 189.3 189.3 189.3 176.9 176.9 176.9 176.9 176.9 176.9 176.9 197.7 197.7 197.7 197.7 197.7 197.7 197.7 191.0 191.0 191.0 191.0 191.0 191.0 191.0 179.4 179.4 179.4 179.4 179.4 179.4 179.4 194.9 194.9 194.9 194.9 194.9 194.9 194.9 189.9 189.9 189.9 189.9 189.9 189.9 189.9 181.8 181.8 181.8 181.8 181.8 181.8 181.8 185.9 185.9 185.9 185.9 185.9 185.9 185.9

1766.5 1873.6 1928.2 1984.3 2032.3 2071.6 2111.6 1748.0 1858.2 1912.2 1967.8 2005.8 2044.5 2084.0 1700.0 1776.8 1866.4 1913.4 1942.5 2017.8 2056.4 1752.8 1834.5 1874.5 1931.0 1951.0 1982.7 2010.6 1723.5 1822.5 1859.3 1896.1 1951.4 1988.2 2025.0 1607.9 1709.7 1789.0 1827.0 1851.3 1874.1 1895.5 1737.5 1838.8 1888.6 1953.6 1989.5 2038.0 2077.4 1703.2 1822.3 1874.5 1928.1 1964.0 2000.0 2035.9 1681.7 1774.2 1821.2 1886.3 1919.5 1953.0 1986.3 1667.5 1748.2 1787.0 1824.4 1898.2 1926.4 1950.0

1890.7 2019.7 2087.4 2157.5 2205.4 2254.5 2304.6 1870.0 1997.6 2064.6 2133.9 2181.3 2229.8 2279.3 1838.4 1963.8 2029.6 2097.6 2144.1 2191.6 2240.0 1851.5 1971.0 2031.7 2096.7 2140.4 2184.4 2229.8 1809.1 1926.7 1987.9 2050.4 2093.9 2137.7 2182.3 1723.2 1854.2 1914.5 1977.1 2019.0 2060.1 2104.6 1812.8 1929.6 1990.9 2054.1 2096.7 2140.4 2182.0 1795.1 1911.7 1972.6 2035.5 2078.1 2121.7 2166.2 1763.5 1878.2 1938.5 2000.1 2042.1 2085.3 2129.1 1769.1 1884.0 1943.1 2001.5 2038.9 2071.0 2114.4

194.0 194.0 194.0 194.0 194.0 194.0 194.0 188.8 188.8 188.8 188.8 188.8 188.8 188.8 176.7 176.7 176.7 176.7 176.7 176.7 176.7 193.1 193.1 193.1 193.1 193.1 193.1 193.1 184.1 184.1 184.1 184.1 184.1 184.1 184.1 171.0 171.0 171.0 171.0 171.0 171.0 171.0 186.8 186.8 186.8 186.8 186.8 186.8 186.8 177.8 177.8 177.8 177.8 177.8 177.8 177.8 166.6 166.6 166.6 166.6 166.6 166.6 166.6 172.6 172.6 172.6 172.6 172.6 172.6 172.6

N denotes the number of wires. D is the diameter of steel wires. a1, a2 are the lay angle of the first layer steel wires and second layer steel wires. Es,a, fys,a, fus,a denote the elastic modulus, yield and ultimate strengths predicted by theoretical model. Es,e, fys,e, fus,e denote the elastic modulus, yield and ultimate strengths predicted by numerical model.

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J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

105

240

70

Axial force (kN)

320

Axial force (kN)

140

160

Test [25] Analytical FEM Costello's model

35 0 0.000

0.005

Strain

0.010

0.015

Test [25] Analytical FEM Costello's model

80 0 0.000

(a) Seven-wire strand

0.005

0.010

Strain

0.015

0.020

(b) Nineteen-wire strand Fig. 12. Axial load–axial strain curves.

Loading end U(1)=Free U(2)=Fixed U(3)=Free

Fixed end U(1)=Free U(2)=Fixed U(3)=Fixed

Displacement directions: 1=radial 2=circumferential 3=longitudinal Fig. 13. FEM for strand.

A nonlinear isotropic model that adopted von-Mises yield criterion was used for steel strands. Typical bi-linear stress-strain curves with strain hardening was adopted in this model, as shown in Fig. 11. The values of the yield, ultimate strengths, elastic modulus and Poisson’s ratio were determined according to the test results. 4.1.3. Boundary conditions, and loading Boundary conditions of the strand used in the FE modelling are shown in Fig. 13. Two major steps that include 1) coupling the two ends of the steel strands to two reference points, and 2) defining the boundary condition to these two reference points at both ends. In order to simulate the boundary conditions, a local cylindrical coordinate system was created in radial, circumferential and longitudinal directions. Taking into account of the Poisson effect, the radial contraction of wires was allowed, and the translation along the radial direction was free at two ends. For circumferential motion, torsional rotation was fixed at both two ends of the strand to prevent unwinding of steel wires. Furthermore, the displacement along longitudinal direction was fixed at one end, but released at the loading end. The tensile force was applied by displacement-control on the loading end to simulate the loading condition of the tensile test.

A finite sliding algorithm allowing large rotations and deformations, was adopted to define the contact surface interactions. Surface of core wire was chosen as the master surface because of its axial stiffness was greater than the helical wires. In the contact interacting algorithm, the surface-to-surface contact can reduce the risk of the large, undetected penetrations of master surface into the slave surface and obtain a smoother contours. Furthermore, this approach of contact also can improve stability at the corners and edges. More accurate solution can be obtained via this approach, but required more computational time and space. Penalty contact algorithm was adopted to describe the interaction in the tangential direction to the contact surfaces. An isotropic Coulomb law was used to model the interwire frictional effect. However, preliminary studies have shown that the interwire frictional effect can be expressed by a constant friction coefficient varying from 0.1 to 0.2 [21]. Different finite models were established with different friction coefficients. The calculation results showed that the FE model with a constant friction coefficient 0.1 fitted test data very well than 0.2. 4.2. Model validations To verify the accuracy of the finite element (FE) model, the results from finite element analysis (FEA) are compared with the test results reported by Utting and Jones [26]. The geometric and material details of the steel strands are listed in Table 5, which are taken from [18,26]. The stress-strain curves from FEA are compared with the test results in Fig. 12. It can be observed that numerical results resemble well with those experimental curves. For seven-wire and nineteen-wire strands, the differences of elastic modulus between the numerical results and test results are both only about 1%. And for seven-wire strand, the numerical results in plastic range is slightly larger than test results with a small difference of less than 2%. Thus, it can be concluded that the developed FEM can give reasonable predictions on the stress-strain behaviours of multi-layer steel strands. 4.3. Discussions on localized analysis

4.1.4. Interactions A surface-to-surface type of contact was used to define the interactions among the surfaces of different wires in the steel strand. This contact type described the two interacting surfaces in both normal direction and tangential direction. Hard contact and penalty friction type contact algorithms were used to define the contacts in these two directions of the interacting surface, respectively.

In addition to global response, the exact finite model proposed in this manuscript can be used for localized analysis of strands with stressing. Taking seven-wire strands for example, from the contours of the equivalent plastic strain as shown in Fig. 14, it can be seen that the onset of plastic deformation is confirmed by the occurrence of equivalent plastic strain in core wire. Fig. 15(a) and (b) shows the von-Mises and normal stress distribution in

J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

Core wire

Helical wire

Fig. 14. Equivalent plastic strain at 0.76% elongation.

cross section at the onset of the plastic deformation in the strand. At that time, the axial force is 89kN and the axial strain of the strand is 0.0076. It can be observed that the plastic deformation initiates from the contacting surfaces between the helical wires and the core wire. The maximum stress also developed along the contact lines. As the axial strain of the strand equals to 0.0095, full cross section yield as shown in Fig. 15(c) and (d). From the normal stress contours as shown in Fig. 15 (b) and (d), it can be seen that the contacting stresses can be obtained as expected between the surfaces of the adjacent helical wires. 5. Development of stress-strain curves of steel strands at low temperatures Even though stress-strain curves of the multi-layer strands at low temperatures can be predicted by the developed theoretical and numerical models with the given curves of steel wires at dif-

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ferent low temperatures, it is still complex for the structural engineers. This section further simplified this procedure, and makes efforts to developed empirical formulae to predict the mechanical properties of the representative multi-layer strands used in the engineering constructions at different low temperatures with the only given test data of the steel wires at ambient temperature. Before carrying out this study, the representative steel strands, i.e., three-wire, seven-wire, and nineteen-wire strands were selected. Considering these steel strands were fabricated with different lay angles and pitch lengths, 70 cases were considered in this study. With the validated theoretical and numerical models, the yield strength, ultimate strength and elastic modulus of these different cases are determined and listed in Table 6. In addition, the material properties of these strands adopts the test values reported in this paper. The mathematical relationships between mechanical behaviours of the single wire at ambient temperature and the global response of steel strands at low temperatures were developed, along with multiple regression analyses. The factors are defined as follows:

IE ¼

Es Ea

ð39Þ

If y ¼

f ys f ya

ð40Þ

If u ¼

f us f ua

ð41Þ

where IE , If y and If u are the factors for elastic modulus, yield strength and ultimate strength, respectively. Ea , f ya and f ua are the elastic

Fig. 15. (a) von-Mises stress at 0.76% elongation. (b) Normal stress at 0.76% elongation. (c) von-Mises stress at 0.95% elongation. (d) Normal stress at 0.95% elongation.

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modulus, yield strength and ultimate strength of steel wires at ambient temperature, respectively. Es , f ys and f us are the elastic modulus, yield strength and ultimate strength of steel strand at different low temperatures, respectively. Considering the influences of temperature T, pitch length P and the number of wires N, the exponential models were developed in regression analysis. For nineteen-wire steel strands, the pitch length of helical two layers of steel wires are defined as P1 and P2. Therefore, the separate regression models for three-wire, seven-wire and nineteen-wire steel strands are established in this work. The assumed regression analysis equations are given as follows:

(

IE ¼

cN d P f egT

for 3 - wire and 7 - wire steel strands

f f cN d P11 P22 egT

for 19 - wire steel strands ð42Þ

( If y ¼

cN d P f egT f

for 3 - wire and 7 - wire steel strands

f

cN d P 11 P22 egT

for 19 - wire steel strands ð43Þ

( If u ¼

cN d P f egT

for 3 - wire and 7 - wire steel strands

f f cN d P11 P22 egT

for 19 - wire steel strands ð44Þ

where c, d, f1, f2 and g are constants that can be determined by regression analysis. T is the low temperature, in °C, and 160 °C  T  20 °C. In order to facilitate multiple linear regression analysis, Eqs. (42)–(44) needs a logarithmic transformation. Several regression analysis methods are available for this work, e.g., stepwise regression method, best subset method, forward selection method and backward elimination method. The best subset method was used in this study to evaluate the significant predictors for IE , If y and If u . The selection of proper subset of predictors for regression models must meet three evaluation criteria of Mallows Cp index, the correlation coefficient R2, and standard error of the regression S.

For regression models established by p predictors from total n predictors (p < n), the best subset is that with the smallest value of Mallows Cp index. The correlation coefficient R2 is a value between 0 and 1, which describes the degree of correlation between the data and the regression results. Meanwhile, the accuracy of the regression models can be evaluated by the standard error of the regression S. Thus, the combinations and the number of predictors in the regression models can be determined. It is noteworthy that regression models should be developed with the lowest possible number of predictors that ‘‘adequately” describe the data. The regression analysis results are listed in Table 7. It can be observed that: (1) For three-wire and seven-wire steel strands, the regression models 4E, 3fy and 3fu are recommended considering the above evaluation criteria. In model 3fy and 3fu, the regression models are developed with the number of wires N, pitch length P and temperature T. However, model 4E is developed without considering temperature T. (2) For nineteen-wire steel strands, the regression models 6E, 6fy and 6fu are recommended. In model 6fy and 6fu, the regression models are developed with the pitch length P1, P2 and temperature T. However, model 6E is developed only with pitch length P1, P2. For nineteen-wire steel strands, the number of wires N shows lower correlations with data which is neglected in regression analyses. The recommended regression models for factors on elastic modulus, yield strength and ultimate strength are given as follows: (1) For three-wire and seven-wire steel strands

IE ¼

Es ¼ 0:64N0:055 P0:092 Ea

ð45Þ

If y ¼

f ys ¼ 0:87N0:058 P0:04 e0:001T f ya

ð46Þ

If u ¼

f us ¼ 0:86N0:047 P 0:038 e0:001T f us

ð47Þ

Table 7 The best subset regression analysis of lnIE, lnIfy, lnIfu on lnN, lnP and T. Response

Model

n

R2

Mallows Cp

S

lnN

IE

1E 2E 3E 4E 5E 6E 7E 8E

1 1 2 2 3 2 3 4

0.05 0.66 0.66 0.89 0.89 0.953 0.953 0.953

298.1 85.6 84.4 2.0 4.0 2.0 4.0 4.0

0.046 0.028 0.028 0.016 0.016 0.010 0.010 0.010

x*

1fy 2fy 3fy 4fy 5fy 6fy 7fy

1 2 3 1 2 3 4

0.76 0.80 0.95 0.869 0.121 0.990 0.990

129.0 102.5 4.0 292.4 2094.4 4.0 4.0

0.031 0.028 0.015 0.022 0.059 0.006 0.006

1fu 2fu 3fu 4fu 5fu 6fu 7fu

1 2 3 1 2 3 4

0.87 0.90 0.98 0.962 0.034 0.996 0.996

253.7 177.0 4.0 235.6 6586.0 4.0 4.0

0.025 0.022 0.009 0.012 0.063 0.004 0.004

Ify

Ifu

*

denotes considered predictors in the subset regression analysis. n is the number of considered predictors.

lnP1

lnP2

x

x x x x x x x

– – – – – x x x

x

x x

– – –

x

x x x

x x x

x

x x

– – –

x

x x x

x x x

x x

T

x x x x x x x x x x x x x x x x

J. Xie et al. / Construction and Building Materials 189 (2018) 1076–1092

Step 3. Based on the results in Step 2, the stress-strain curves of steel strands at low temperatures can be obtained by the theoretical models, i.e., Eqs. (14)–(38) and FE model. Step 4. For three-wire, seven-wire and nineteen-wire steel strands, with the obtained mechanical properties of steel wires at ambient temperature (Step 1), the mechanical properties of steel strands can be directly calculated by Eqs. (45)–(50).

(2) For nineteen-wire steel strands

IE ¼

Es ¼ 0:44P0:037 P 0:099 1 2 Ea

ð48Þ

If y ¼

f ys ¼ 0:56P0:082 P 0:032 e0:001T 1 2 f ya

ð49Þ

If u ¼

f us ¼ 0:72P0:032 P0:022 e0:001T 1 2 f ua

ð50Þ

7. Conclusions This manuscript firstly reported the test results on the mechanical properties of high strength steel wires at different low temperatures. Empirical formulae were also proposed to predict the lowtemp stress-strain behaviours of steel wires. Followed, theoretical and numerical models were developed to predict the stressstrain behaviours of multi-layer steel strands at low temperatures. Empirical prediction equations were finally developed to predict the tensile stress-strain behaviours of the multi-layer strand with the given mechanical properties of steel wires at ambient temperature. Based on these experimental, theoretical and numerical studies, the following conclusions can be both drawn:

In Eqs. (45)–(50), Ea denotes elastic modulus of steel wires at ambient temperature, in GPa, f ya and f ua denotes yield strength and ultimate strength of steel wires at ambient temperature, in MPa; T is the low temperature, in °C, and 160 °C T  20 °C. In Eqs. (45)–(47), the number of wires N is equal to 3 or 7; P denotes the pitch length of steel strands, in mm. In Eqs. (48)–(50), P1 and P2 denote the pitch lengths of two layers helical wires of steel strands, in mm. Otherwise, it is worth noting that the pitch length P can be defined with helix radius R and lay angle a through a geometric 2pR relationship P ¼ tan a.

(1) As the temperature decreased from +20 °C to 160 °C, yield and ultimate strengths of steel wires were increased by about 20%; However, the elastic modulus E of steel wires exhibited neglected marginal changes. As the temperature decreased from +20 °C to 100 °C, the values of yield strain ey , ultimate strain eu and failure strain eF were averagely increased by about 15%. As the temperature was below 100 °C, the influences of the low temperature on ey , eu and eF were marginal. The low temperature only affected w ratio as it was below 100 °C, and w ratio was averagely reduced by 33% as the temperature decreased from 100 °C to 160 °C. The fracture mode of steel strands changed from ductility to brittle mode at the low temperature of 100 °C.

6. Proposed prediction methods on stress-strain curves of steel strands at low temperatures In order to facilitate the engineering application of the steel strands at different low temperatures, the flowchart is shown in Fig. 16 and following procedures are recommended: Step 1. The tensile tests are carried out to obtain stress-strain curves of steel wires at ambient temperature. Step 2. Eqs. (10)–(13) can be used to predict the mechanical properties of steel wires at different low temperatures.

Start Determine

curves of steel wires at ambient temperature from tensile tests

Determine f y, f u of steel wires at different low temperatures by Eq.(10)-(13)

Determine mechanical properties of steel strand by theoretical method by Eq.(14)-(28)

Determine

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Determine mechanical properties of steel strand by FE method

Calculation of representative steel strand at low temperatures by Eq. (45)-(50)

curves of steel strands at different low temperatures End

Fig. 16. The flowchart of proposed prediction methods on stress-strain curves of steel strand at low temperatures.

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(2) Empirical prediction models, i.e. Eqs. (10)–(13), were developed to predict the mechanical properties of steel wires at low temperatures. Validations of the predictions against the test results proved their accuracies. (3) This manuscript developed theoretical models to predict the mechanical properties of multi-layer steel strands. These models considered the geometric and material nonlinearity of multi-layer strands. Their accuracies were checked by test results, and proved to be capable of predicting stress-strain behaviours of steel strands at low temperatures. (4) A three-dimensional finite element model was also developed to simulate the stress-strain behaviours of multilayer steel strands at low temperatures. It simulated the different geometric details and material nonlinearities. Through extensive validations of the FE predictions against the test results, the FE model was proved to be capable of simulating stress-strain curves of steel strands at low temperatures. (5) Based on the theoretical and numerical results, regression models were also developed to predict the mechanical properties of representative types of steel strands at low temperatures. Eqs. (45)–(50) can be used to predict the elastic modulus, yield and ultimate strengths of three-wire, seven-wire and nineteen-wire steel strands at low temperatures with the given mechanical properties of steel wires at ambient temperature. This may facilitate determination of mechanical properties of steel strands at low temperatures especially during the engineering design process. So far, the study was limited to ultimate tensile behaviours of steel strands and steel wires at low temperatures. More future works will be needed on their dynamic behaviours since these Arctic offshore structures are exposed to impacting ice loads. Conflicts of interest statement The authors declare that they have no conflicts of interests. Acknowledgment This work was financially funded by the National Natural Science Foundation of China (Grant No. 51608358). The authors gratefully express their gratitude for the financial supports. References [1] E. Stanova, G. Fedorko, S. Kmet, V. Molnar, M. Fabian, Finite element analysis of spiral strands with different shapes subjected to axial loads, Adv. Eng. Softw. 83 (2015) 45–58.

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