Mechanical properties of Precontraint 1202S coated fabric under biaxial tensile test with different load ratios

Construction and Building Materials 80 (2015) 210–224

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Mechanical properties of Precontraint 1202S coated fabric under biaxial tensile test with different load ratios Andrzej Ambroziak ´ sk, Poland Department of Structural Mechanics, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdan

h i g h l i g h t s  The author specified initial material parameters for dense net model and orthotropic model.  Variation of initial mechanical properties of coated fabric under cyclic tests is investigated.  Laboratory tests necessary for identification of non-linear elastic initial properties are described.  The material parameters are specified on the basis of the uniaxial and biaxial tensile tests.

a r t i c l e

i n f o

Article history: Received 12 May 2014 Received in revised form 3 November 2014 Accepted 28 January 2015

Keywords: Coated fabrics Fabric/textiles Mechanical properties Mechanical test Tensile test

a b s t r a c t The paper describes a method of laboratory tests necessary for identifying the mechanical properties of polyester coated fabrics named Precontraint 1202S with PVDF surface treatment. Two sets of initial material parameters for dense net model and orthotropic model are specified. Material parameters for Precontraint 1202S coated fabric are specified on the basis of the biaxial tensile tests for different load ratios. In order to observe a change of initial mechanical properties, the biaxial cyclic tests are carried out. Additionally, a short survey of literature concerning the coated woven fabrics description and numerical simulations is presented. This paper is also proposed as an introduction to the comprehensive investigation of the mechanical properties of coated woven fabrics. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Coated fabrics (named architectural fabrics or technical woven fabrics) are commonly used for both temporary and permanent types of constructions. They are frequently used in engineering structures for wide-span surfaces, membrane structures and pneumatic constructions. The coated fabrics consist of the thread net which is coated on both sides with materials like PTFE (polytetrafluoroethylene), PVC (polyvinyl chloride) or PVDF (polyvinyl difluoride). Thread nets are made of cotton fibre, polyamide, polyester, fibreglass or aramid fibre and usually have two families, named warp and weft (fill), see e.g. [1]. For the description of the technical woven fabric behaviour a number of theoretical models have been developed [2]. The proper determination of the mechanical properties strongly depends on the increase of the analysis precision, economical design based on the advantages of the materials used, see e.g. [3,4]. The mathematical description of material behaviour and its constitutive characteristics enables the full

E-mail address: [email protected] http://dx.doi.org/10.1016/j.conbuildmat.2015.01.074 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.

utilisation of its properties in the design of fabric or membrane structures. First of all, a short literature survey of the papers concerning the description of coated fabrics is presented. The study of technical woven fabric starts with the very early work of Hass [5] and is followed by the paper of Pierce [6]. In these papers the crimp theory is presented, describing geometrical and mechanical force models of a membrane structure made of woven fabric. This theory is focused primarily on the cross-section shape of threads. Reinhardt [7] presented biaxial and uniaxial tensile tests results for polyester fabrics coated with PVC and concluded that the results of tensile strength in both cases were equal. Ansell et al. [8] investigated structure and performance of several commercial PTFE-coated fabric systems using scanning electron microscopy, mechanical testing, and weathering in artificial and natural environments. Mott et al. [9] performed an experimental study to develop a test method and software for the purpose of characterising the material behaviour of coated glass fabrics for application in large roof structures. Day [10] investigated biaxial tensile tests for PTFE fabric for different stress ratios. Testa and Yu [11] proposed a method for

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coated fabrics to compute the path-dependent inelastic strain and the elastic strain for any loading once material parameters are evaluated from uniaxial tests in the principal directions of elasticity. Argyris et al. [12] described numerical modelling with rheological relations in time domain for PVC-coated fabrics. Mewes [13] presented the status of PVC coated fabrics and related the textile developments for architectural structures. Chen et al. [14] performed tensile tests on coated fabrics before coating and after coating. Szostkiewicz-Chatain and Hamelin [15] presented experimental stiffness characterisation methods for polyester fabrics and proposed inverse and experimental stiffness identification methods. Bassett et al. [16] gave a review of various experimental approaches towards the assessment of the mechanical properties of fabrics. Bridgens and Gosling [17] described the direct stress– strain representation for coated woven fabrics. Kłosowski et al. [18] presented viscoplastic properties of coated fabrics under uniaxial tensile tests. Kumazawa et al. [19] conducted biaxial tensile experiments of cruciform specimens with an open hole to evaluate the strength of coated plain-weave fabrics composed of specific high-tensile-strength fibres. Chen et al. [20] analysed the anisotropic behaviours of coated fabrics on elastic constants with the application of off-axial constitutive response of orthotropic and elastic materials. Pargana et al. [21] proposed a constitutive model to describe coated fabrics for the finite element analysis. Chen et al. [22] investigated tensile behaviour of PVC-coated woven fabrics under uniaxial and biaxial loads. Galliot and Luchsinger ([23,24]) described a non-linear material model to describe the yarn-parallel behaviour of PVC-coated polyester fabric under biaxial tension and the behaviour of coated fabrics using the shear ramp test method. Kłosowski et al. [25] presented a study of nonlinear viscoelastic description of the PVC-coated fabric under uniaxial tests. Zouari et al. [26] performed tensile tests on fabrics in seven directions of 15° increment with respect to the warp direction. Wang and Zhang [27] studied the tensile behaviour of four neat woven fabrics and two coated woven fabrics. The authors compared the tensile strength of the uncoated and coated woven fabrics. Zhang et al. ([28,29]) presented the mechanical properties of PTFE and PVC coated fabrics and proposed the temperature reduction factor for analysis and design. Wang et al. [30] reported the stab failure behaviour of coated and uncoated woven fabric in experimental and finite element analysis approaches. Bridgens et al. [31] described a comparison of two biaxial tests simulated by finite element analysis of the cruciform specimens and, then, discussed the difficulties in comparing the results. Ambroziak and Kłosowski [32] presented the collection of technical fabric testing methods and the results of these tests for different fabrics. Glaser and Caccese ([33,34]) investigated an experimental procedure to obtain the mechanical stiffness, strength and shear properties of orthotropic polyurethane-coated nylon fabric. Ambroziak and Kłosowski ([35,36]) described, firstly, a method of laboratory tests necessary for the identification of the mechanical properties of coated fabric and, secondly, a method of laboratory tests necessary to assess the influence of temperature on the mechanical properties of polyvinylidene fluoride-coated polyester fabric. Gosling et al. [37] presented the round robin analysis exercise. The results showed very high levels of variability in terms of stresses, displacements, reactions and material design strengths, and highlighted the need for future work to harmonise analysis methods and to provide val_ idation and benchmarking for membrane analysis software. Zerdzicki et al. [38] studied inelastic properties of the technical fabric Valmex that was used for 20 years as the roof structure of the Forest Opera in Poland. Chrós´cielewski et al. [39] described the assessment of tensile forces in the roofing membrane of the newly built Forest Opera in Poland, by the measurements in situ and the iterative numerical strategy for an inverse problem. Ambroziak and

Kłosowski [40] compared uniaxial and biaxial tensile test results and applied these results to a preliminary design of tensile structures made of PVC coated fabric. Chen et al. [41] presented an experimental study to determine the tensile properties of the envelope fabric Uretek321A under uniaxial and biaxial tests. Zhang et al. [42] described the tensile behaviour of PTFE-coated fabrics subjected to monotonous and cyclic loading. Lewis [43] discussed all the design stages requiring advanced computational modelling based on a thorough understanding of the relationship between form and stress. Chen et al. [44] evaluated the Poisson’s ratio variation with regards to three typical cases of tension ratio 1:1, 1:2 and 1:2 for the Uretek3216 coated fabric. Zhang et al. [45] presented the mechanical properties of membrane connections in tensioned membrane structure. Colman et al. [46] described a novel picture frame shear test design and associated a test protocol that aims to provide a practicable solution for the accurate determination of the in situ shear stiffness of architectural fabrics. Attention should be also paid to guidelines for designers of elaborate tension structures e.g. by TensiNet (see e.g. Forster and Mollaert [47]) and the American Society of Civil Engineers (see e.g. ASCE/SEI55-10 [48], Huntington [49]). It can be seen that the coated woven fabrics are still being tested and developed. New types of coated fabrics are manufactured. For a better recognition of coated fabric behaviour, the laboratory tests were performed. New tests method and constitutive models were developed. On the other hand, there is no universal model to fully describe the coated fabrics. The following investigation provides a step forward to better recognition of the coated fabrics. The following research and laboratory tests are focused on the coated woven fabrics made out of polyester fibres covered by PVC with the PVDF surface treatment, named Precontraint 1202S. According to technical data specified by a manufacturer [50], the basic properties were collected in Table 1. The lack of material parameters was commonly observed in the sets of technical data. These parameters are necessary for the calculations of real structures. If material parameters are required, laboratory tests should be carried out. In this paper, the mechanical properties under biaxial tensile tests are specified.

2. A description of laboratory test In the laboratory tests (Fig. 1a and b) the strength-testing computer-operated machine of Zwick type was applied. The machine consisted of four cross-shaped arms equipped with grips (Fig. 1a). Each arms was computer-operated by TextExpert program (Fig. 1c) and could be separately moved with e.g. constant force, displacement. The values of displacement and force on grips could be recorded. For measurement of displacement of specimen the optical extensometer (videoXtens extensometer) which was suspended under specimen (at the bottom of machine, Fig. 1a) was used. The videoXtens extensometer provided non-contact, high-resolution measurement of deformations. The gauge marks were automatically recognised and the displacement of the marks from image to image was converted to extension and transmitted to the measurement and control electronics. Table 1 Technical properties of Precontraint 1202S coated fabric. Properties

Precontraint 1202S

Weight [g/m2] Total thickness [mm] Yarn Tensile strength warp/weft [kN/m]

1050 0.78 1100/1670 Dtex PES HT 112/112

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Fig. 1. Biaxial tensile test stand: (a) a general view of the stand, (b) the specimen after the test, (c) screenshots taken during the test.

In the biaxial tensile tests the cross-section of specimen, the width was 100 mm, the grip separation 300 mm, and the gauge length of optical extensometer in both directions about 50 mm (see Fig. 2). Two types of tests were performed. Firstly, the specimens were subjected to tension in the warp and weft direction until rupture. Secondly, the biaxial cyclic tests were performed to reach the specified stress level. All tests were conducted with seven different constant load (stress) ratios 1:2, 1:4, 1:8, 1:1, 2;1, 4:1 and 8:1 (r1:r2, where r1 and r2 are stresses in the warp and weft directions respectively). The load rates for stress ratios were specified in Table 2. For each stress ratios two samples were tested. 2.1. Biaxial tensile test results Stress–strain curves for biaxial tensile tests for different load ratios are presented in Figs. 3–9. It should be explained that an average stress reduction factor (see e.g. Bridgens et al. [51]) nearby

Fig. 2. Specimen shape for biaxial tensile test.

Table 2 Load rate for different load ratio. Force ratio r1:r2

Warp direction [kN/m/s]

Weft direction [kN/m/s]

1:2 1:4 1:8 1:1 2:1 4:1 8:1

0.50 0.25 0.25 1.00 1.00 1.00 2.00

1.00 1.00 2.00 1.00 0.50 0.25 0.25

1.0 for the measurement points (central region) of the specimen is assumed. The comparison of biaxial tensile tests for different stress ratios were given in Figs. 10 and 11. The values of stress at failure of specimens were firstly determined and collected in Table 3. The results of laboratory tests were presented in the form of a sum  x  sx , where  x was the mean value and sx is the standard deviation (due to limited number of specimens determined values of standard deviation are not fully appropriate). It can be seen that the highest value of stress at failure of specimen was obtained for

Fig. 3. Biaxial tensile tests – 1:1 stress–strain curves.

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213

Fig. 4. Biaxial tensile tests – 1:2 stress–strain curves.

Fig. 7. Biaxial tensile tests – 2:1 stress–strain curves.

Fig. 5. Biaxial tensile tests – 1:4 stress–strain curves.

Fig. 8. Biaxial tensile tests – 4:1 stress–strain curves.

Fig. 6. Biaxial tensile tests – 1:8 stress–strain curves.

Fig. 9. Biaxial tensile tests – 8:1 stress–strain curves.

8:1 stress ratio. This value reached about 80% of uniaxial tensile strength given in Table 1 (the stress range for design of fabric structures is limited up to around 25% of the UTS). The smallest value was for 1:1 stress ratio. These results yielded from the failure mode of the specimen, see Fig. 12a. Generally, the damage of specimens began in the corner zone, next the cross arm is torn (Fig. 12a). The early failure of the sample under biaxial tests was strictly related with the cruciform shape of the sample (see e.g. [7,31]).The proper determination of the UTS value under biaxial tensile tests was not investigated in this paper. Nevertheless, the shape of the cross-shaped specimen with and without strippers

for the others coated fabric was tested by the author (see e.g. [35]). On the other hand, another type of specimen damage was observed during the tests. In this case, the damage of specimen arm was observed (Fig. 12b). In order to complement the stress–strain curves of the biaxial tests results the strain ratios ee12 ðr2 Þ and ee21 ðr1 Þ (where e1 and e2 are strains in warp and weft directions) were determined. The ratio curves with approximation functions in stress domain were presented in Figs. 13 and 14. The approximation functions were collected in Table 4. The calculated coefficients of determination R2 range from 0.907 to 0.969. It can be seen that the strain ratios were

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Fig. 10. Biaxial tensile tests for different stress ratios (1:1, 1:8, 1:4, 1:2). Fig. 13. Strain proportions for 1:1, 1:8, 1:4, 1:2 stress ratios.

Fig. 11. Biaxial tensile tests for different stress ratios (1:1, 8:1, 4:1, 2:1). Fig. 14. Strain proportions for 1:1, 2:1, 4:1, 8:1 stress ratios.

3. Description of fabric behaviour under biaxial tests

Table 3 The value of the stress at the specimens’ breaking point. Force ratio

Warp direction [kN/m]

Weft direction [kN/m]

1:2 1:4 1:8 1:1 2:1 4:1 8:1

39.5 ± 0.1 20.55 ± 0.05 10.1 ± 0.5 70 ± 1 83.6 ± 0.3 89 ± 4 91 ± 4

79.1 ± 0.3 82.2 ± 0.1 81 ± 4 70 ± 1 41.7 ± 0.1 22 ± 1 11.4 ± 0.5

nonlinear functions, dependent on the stress ratio. It can be observed that it is not possible to apply a constant value of the strain ratios for different load ratios.

Designer of the fabric structures needs simple and easily accessible guidelines to describe the coated fabric behaviour. The constitutive models have to be simple for numerical engineering calculations and straightforward to identify the material parameters. The author proposed two models (dense net model and orthotropic model) for a coated fabric description. It should be noted that the constitutive relation in the plane stress state, expressed by the thread forces, takes a form:

rxt ¼ rxtDt þ Dr ¼ rxtDt þ Dx  Dex

ð1Þ

where rxt are the stress components in actual time increment t, rxtDt are the stress components in the previous increment t  Dt,

Fig. 12. Specimens damaged after the biaxial tensile test.

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A. Ambroziak / Construction and Building Materials 80 (2015) 210–224 Table 4 Approximation functions for strain ratios. Force ratio

R2

Approximation functions

1:1 1:8 1:4 1:2

e1 e2 ðr2 Þ

0.92 + 1.9745  exp (8.4429  10  r2) + 5.1624  10  r2 0.3874 + 1.0383  exp (2.1234  104  r2) + 3.5048  106  r2 0.2709 + 0.8286  exp (3.0839  104  r2) + 3.7778  106  r2 0.1810 + 0.5968  exp (9.2725  105  r2) + 8.2003  106  r2

0.912 0.969 0.963 0.965

1:1 2:1 4:1 8:1

e2 e1 ðr1 Þ

0.0325 + 1.0971  (1  exp (6.6990  104  r1)) 1.2373 + 1.5835  exp (1.8337  105  r1) + 1.8148  105  r1 0.0295  0.4112  exp (8.3737  105  r1) + 1.5947  106  r1 0.0677  0.8918  exp (8.7336  105  r1) + 6.7756  107  r1

0.907 0.920 0.948 0.968

4

Dr and Dex are the increments of stress and strain components between these time increments, Dx is the elasticity matrix. 3.1. Preliminary description

Dr1 ¼ F 1 ðe1 Þ  De1 Dr2 ¼ F 2 ðe2 Þ  De2

ð2Þ

where F1(e1) and F2(e2) are the material functions of the threads and are called the longitudinal stiffnesses. They are usually specified on the basis of the uniaxial, or during more complex investigation, like e.g. the biaxial laboratory tests. Basing on the geometrical relationship, the threads force ri and strain ei can be expressed by the stress rx and strains ex components in the plane stress state as follows:

e en ¼ 1 e2 8 > <



 ¼ 9

1 cos2 a 2

rx1 > 1 = rx ¼ rx2 ¼ 6 40 > > : sx1 x2 ; 0

8 > <

9

ex2 > = ex2 ¼ Txn ex 2 > sin a sin a cos a > : cx1x2 ; 3 cos2 a   7 r1 2 ¼ ðTxn ÞT rn sin a 5 r2 sin a cos a 0

0

ð3Þ

where a is the actual inclination angle between the threads families during the deformation process. The angle between thread families a changes during deformation and is calculated in accordance with the current values of stress components rx2 and rx1x2 in the fabric from the relation:

a ¼ arctg

rx2 : sx1x2

where the elasticity matrix is expressed as:

2

In order to describe the behaviour of the coated fabrics for preliminary design the dense net model (see e.g. [2,52]) is proposed. Additionally, the range of load ratios in which the proposed model can be use is specified. The dense net model belongs to the group of continuum models, in which the woven fabric is treated as a continuum without explicit reference to its discrete microstructure. In this model, it is assumed that the fabric forces in the threads families depend on the uniaxial strain in the same family only (the friction between threads families and influence of the coating in this concept cannot be fully included). These assumptions do not provide a full description of the fabric, but they enable creation of the fabric numerical model, which has at least the most important properties of this material. Consequently, the threads forces increment of the warp Dr1 or weft direction Dr2 are calculated from the following equations:



7

ð4Þ

3 2 F 1 ðe1 Þ þ F 2 ðe2 Þ cos4 a F 2 ðe2 Þsin a cos2 a F 2 ðe2 Þ sin a cos3 a 6 7 2 4 3 2 Dx ¼ 6 F 2 ðe2 Þ sin a F 2 ðe2 Þ sin a cos a 7 4 F 2 ðe2 Þ sin a cos a 5: 3 2 F 2 ðe2 Þsin a cos3 a F 2 ðe2 Þ sin a cos a F 2 ðe2 Þsin a cos2 a ð6Þ Due to the observation (see Figs. 10 and 11) that a part of the stress–strain curves for biaxial tests under different load ratios are located nearby 1:1 loading carves the mean curve of the stress–strain curves for 1:1 load ratio is specified. Based on the concept described in [35] the results of the piece-wise linear approximation for 1:1 biaxial tensile tests are determined, presented in Table 5. The strain e specifies the range of applicability of the longitudinal stiffness values F. The mean values of non-linear elastic parameters describe the warp and weft 1:1 stress– strain loading curves with coefficients of determination R2 > 0.99. Such high value of the coefficient of determination proves correctness of fit tests data by means of the proposed model. What follows, all curves for different load ratios were compared to mean stress–strain curves of 1:1 load ratio. The calculated coefficients of determination R2 are given in Table 6. The following conclusions may be derived:  The shapes of the stress–strain curves for the warp direction for 1:2, 2:1, 4:1, 8:1 and weft direction for 2:1, 1:2, 1:4, 1:8 are similar for 1:1 mean stress–strain curve (see Table 6).  Only for the load ratio 1:4 and 1:8 for the warp and for load ratio 4:1 and 8:1 for the weft results significant differences to 1:1 stress ratio result can be observed. These load ratios exhibit negative strains for the warp (see Fig. 5 and 6) and for the weft (see Figs. 8 and 9). The highly loaded threads straighten while other ones become folded.  Generally, it can be concluded, that engineering calculations in their preliminary design stage can apply dens net model with nonlinear parameters given in Table 5 to describe biaxial behaviour of fabric structures in range 1:2–1:1–2:1 load ratios.

Table 5 Non-linear elastic parameters for 1:1 load ratio. Warp/Weft

Next, the relation between stress and strain in the plane stress state is written in the form:

rx ¼ ðTxn ÞT FTxn ex ¼ Dx ex

ð5Þ

F [kN/m] 1400 ± 10 310 ± 5 710 ± 30

e [–] 0  0.013 ± 0.001 0.013 ± 0.001  0.095 ± 0.001 >0.095 ± 0.001

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Table 6 Coefficient of determination R2. Force ratio

Warp

Weft

1:8 1:4 1:2 2:1 4:1 8:1

0.012 0.268 0.823 0.992 0.997 0.997

0.830 0.931 0.968 0.977 0.469 0.009

3.2. Orthotropic model In order to approximate the coated woven fabric behaviour an orthotropic model (see e.g. MSAJ/M-02-1995 [53]) is taken. In this case the elasticity matrix can be expressed as follows:

2 6 D¼6 4

F1 1m12 m21

F 1 m21 1m12 m21

F 2 m12 1m12 m21

F2 1m12 m21

0

0

0

3

7 07 5

ð7Þ

G

where F1 and F2 are the longitudinal stiffness established under uniaxial tensile test; m12, m21 are the Poisson’s ratios, G is the modulus of the volumetric elasticity. Symmetry of elasticity matrix yields additional condition:

F 2 m12 ¼ F 1 m21

ð8Þ

It should be noted, that this constraint is applied by MSAJ/M02-1995 [53] to the calculation of material constants. On the other hand, Bridgens et al. [31] suggested that this condition is not appropriate for coated fabric. In order to specify the longitudinal stiffness F1 and F2 uniaxial tensile tests are performed (see Fig. 15). The uniaxial tensile tests are carried out according to EN ISO 1421:2001 standard. The specimen width 50 mm, grip separation of 200 mm, and constant base

Fig. 15. Uniaxial tensile test stand.

of optical extensometer about 50 mm are taken. Specimens in the uniaxial tensile tests are subjected to tension with a displacement rate of a grip of 100 mm/min. Stress–strain curves for uniaxial tensile tests are shown in Fig. 16a. Additionally the comparison between the Precontraint 1202S fabric and two others polyester PVC coated is made on Fig 16b. It should be noted that behaviour of the fabric with Precontraint system is different than PVC coated polyester fabric made by other manufacturers. The results of the piece-wise linear approximation for tensile tests are presented in Table 7. The strain e specifies the range of applicability of F – individual longitudinal stiffness values. The detailed investigation of the Precontraint 1202S under uniaxial tensile tests is performed by Ambroziak [54], where comparison of the uniaxial tensile tests with 1:1 biaxial tests is presented. Stresses for warp and weft directions according to Eq. (7) can be calculated as follows:

r1 ¼ 1mF121 m21 ðe1 þ m21  e2 Þ r2 ¼ 1mF122 m21 ðe2 þ m12  e1 Þ

ð9Þ

Two types of conditions (r1 P r2 and r1 < r2) are distinguished to properly determine material parameters. It should be noted that the strain proportions (see Figs. 13 and 14) are variable in specific stress ranges. Therefore material parameters should be also variable in specific stress ranges (there are no possibilities to use determined constant values of material parameters for the description of fabric behaviour subjected for different load ratios). For r1 P r2 (8:1, 4:1, 2:1, 1:1 load proportions) a constant value is assumed of m12 = 0.11 and a variable value of m21 ¼ F 2  m12 =F 1 .

Fig. 16. Uniaxial tensile tests results (a) Precontraint 1202S, (b) comparison with polyester coated fabrics.

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A. Ambroziak / Construction and Building Materials 80 (2015) 210–224 Table 7 Non-linear elastic parameters for uniaxial tensile tests. Warp

Weft

F1 [kN/m] 1170 ± 20 300 ± 4 897 ± 12

e1 [–]

F2 [kN/m]

e2 [–]

0  0.0164 ± 0.0002 0.0164 ± 0.0002  0.1002 ± 0.0004 >0.1002 ± 0.0004

650 ± 26 288 ± 3 841 ± 8

0  0.030 ± 0.001 0.030 ± 0.001  0.1144 ± 0.0007 >0.1144 ± 0.0007

Fig. 17. 8:1 biaxial tensile tests results comparison.

Fig. 20. 1:1 biaxial tensile tests results comparison.

lations (according to Eq. (9)) and tests results for 8:1, 4:1, 2:1 and 1:1 load rates are given in Figs. 17–20. A good compatibility between simulations and test results is noticed. For r1 < r2 (1:8, 1:4, 1:2 load proportions) a constant value is assumed of m21 = 0.40 and a variable value of m12 ¼ F 1  m21 =F 2 . The longitudinal stiffness F1 and F2 are taken according to Table 7. In this case additional conditions should be imposed that F1 = 300 kN/m for e1 2 h0.0; 0.1002i. The value of m12 is constant in specific strain ranges, its averages are: 0.1846, 0.4167 and 0.1427. The comparison of numerical simulations and tests results for 1:8, 1:4 and 1:2 load rates are given in Figs. 21–23. A good compatibility between simulations and test results is noticed. 4. Application to finite element analysis Fig. 18. 4:1 biaxial tensile tests results comparison.

Fig. 19. Biaxial tensile tests results comparison.

In order to show the possibilities of finite element application and to verify the results of the identification, the numerical simulation of the biaxial tension tests was performed. In numerical simulations the commercial program MSC.Marc was used with a possibility to implement the user-defined HOOKLW subroutine. In the analysis 4-node isoparametric membrane elements were applied (see e.g. [55]). The following geometrical parameters were taken: a = 300 mm (cross length), b = 100 mm (width) and t = 1 mm (thickness). Analysis in MSC.Marc system with the nonlinear elastic properties for the warp and weft according to orthotropic model was performed. The results of Huber–von Mises–Hencky (HMH) equivalent stress and strain are shown in Figs. 24 and 25, respectively. The maximum values of the equivalent stress are located near the corners, where the damage began during experiments. This simple test confirms that FEM may be applied using determined material parameters. This procedure was also tested by the author for real structures calculations, see e.g. [56]. 5. Biaxial cyclic tests

The longitudinal stiffness F1 and F2 are taken according to Table 7. The value of m21 is constant in specific strain ranges and its average are: 0.0611, 0.2383 and 0.797. The comparison of numerical simu-

In order to observe a change of initial mechanical properties of Precontraint 1202S fabric biaxial cyclic tests are carried out.

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Fig. 21. 1:8 biaxial tensile tests results comparison. Fig. 24. Map of equivalent stress [1e + 6 = 1 kN/m].

Fig. 22. 1:4 biaxial tensile tests results comparison.

Fig. 25. Map of equivalent total strain [–].

Table 8 Stress range cyclic tests.

Fig. 23. 1:2 biaxial tensile tests results comparison.

5.1. Tests for 1:1 stress ratio The specimens were subjected to cyclic tension in warp and weft directions with a load (stress) ratio 1:1 (r1:r2). These tensile cyclic tests were carried out with a load rate of 200 N/s in warp and weft directions. The shapes of specimens were the same in the tests described in the previous chapter. In each test 20 cycles of loading up to one of the load levels specified in Table 8 and nearly full unloading were carried out and, finally, after this sequence, the specimens were loaded up to rupture.

Tests type

Stress values [kN/m]

A B C D E F G

10 15 20 25 30 45 60

The stress–strain curves for biaxial cyclic tensile tests were shown in Figs. 26–32. Mean values of stresses at failure of specimens under cyclic tests were specified and an average value 74 ± 1 was determined. These values are comparable with the results given in Table 3. For a detailed comparison of a cyclic tests results residual , eN¼10 and eN¼20 (where N is a number of the cycle) strains eN¼1 I I I and longitudinal stiffness FN=1, FN=10 and FN=20 were determined. The results of all cyclic tests in the assumed range of loads in the form of stress–strain curves for initial state (N = 0) and cycles N = 1, N = 10 and N = 20 were shown in Figs. 33–39. The figures show stress–strain curves as linear function. The parameters F are determined form equation r = F  e. The results of this identification are collected in Tables 9 and 10. It should be explained that

A. Ambroziak / Construction and Building Materials 80 (2015) 210–224

Fig. 26. Biaxial cyclic tensile test results (1:1) – A test.

219

Fig. 29. Biaxial cyclic tensile test results (1:1) – D test.

Fig. 30. Biaxial cyclic tensile test results (1:1) – E test. Fig. 27. Biaxial cyclic tensile test results (1:1) – B test.

Fig. 28. Biaxial cyclic tensile test results (1:1) – C test.

the values of determination coefficient R2 for all parameters is great than 0.98. Visualisation of the determined results (N = 1 and N = 20) is shown in Fig. 40. Note that F is the value of initial longitudinal stiffness given in Table 5. The following remarks may be formulated:  An observation for biaxial cyclic tests and each stress level yields that the growth of cycle number makes the longitudinal stiffness values increase (FN=1 > FN=20).

Fig. 31. Biaxial cyclic tensile test results (1:1) – F test.

 The behaviour of the warp and weft directions under biaxial cyclic tests for both fabrics in the A and B stress range can be assumed elastic. The value eI is only about 0.003 []. To full confirmation of this behaviour the rheological test should be performed.  After the first cycle the loading curves (stress–strain curves) can be assumed linear in the assumed stress range. Load increment over the assumed stress range adjusts the stress–strain curves to the initial behaviour of the fabrics.

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Fig. 32. Biaxial cyclic tensile test results (1:1) – G test.

Fig. 33. A stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

Fig. 35. C stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

Fig. 36. D stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

Fig. 37. E stress range – loading curves for N = 0, N = 1, N = 10, N = 20. Fig. 34. B stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

 The values of the longitudinal stiffness for the warp direction are higher than for the weft direction (Fwarp > Fweft). The difference is 3–23% (see Table 9, Table 10). For A stress range the difference is highest (about 23%).  The values of FN=20 for the warp direction (see Fig. 40) decrease from about 1700 kN/m for A stress range to about 1150 kN/m for D stress range. For the next stress ranges the values FN=20 increase to about 1400 kN/m.  The values of FN=20 for the weft direction (see Fig. 40) decrease from 1400 kN/m for A stress range to about 1100 kN/m for D stress range, for the next stress ranges increase they to about 1400 kN/m.

 It can be observed that for weft and warp directions the longitudinal stiffness FN=20 tends towards about 1400 kN/m and corresponds to the first initial longitudinal stiffness determined in Table 5.  It should be noted that the initial mechanical properties of coated fabric vary significantly under cyclic loading.  It should be pointed that behaviour of polyester coated fabric is generally visco-elastic (see e.g. [18,25]) with large residual strains after testing. The viscous effect is not investigated in this paper. 5.2. Biaxial cyclic tests under different stress ratios Biaxial cyclic tensile tests for different stress ratio (1:2, 1:4, 2:1, 4:1, see Figs. 41–44) were carried out. The specimens were sub-

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Fig. 40. Change of warp and weft longitudinal stiffnesses – cyclic tensile test.

Fig. 38. F stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

Table 10 Weft parameters for 1:1 tensile cyclic tests. Weft

A B C D E F G

eN¼1 [-] I

eN¼10 [-] I

eN¼20 [-] I

FN=1 [kN/m]

FN=10 [kN/m]

FN20 [kN/m]

0.0019 0.0024 0.0041 0.0113 0.0263 0.0564 0.0718

0.0018 0.0029 0.0107 0.0230 0.0372 0.0704 0.0837

0.0022 0.0033 0.0119 0.0263 0.0405 0.0734 0.0865

1396 1302 1040 828 854 942 1082

1399 1303 1136 1017 1074 1184 1300

1412 1333 1211 1098 1123 1248 1364

Fig. 39. G stress range – loading curves for N = 0, N = 1, N = 10, N = 20.

jected to tension in the warp and weft directions with specified load ratios. The shapes of specimens are the same as in the tests described in the previous chapter. In each test 20 cycles of loading and nearly full unloading were carried out. Finally, after this sequence, the specimens were loaded up to failure. The results of the performed cyclic tests, in the form of stress– strain curves for initial state (N = 0) and cycles N = 1 and N = 20, are shown in Fig. 45–48. Comparison of the cyclic tests results gives residual strains eN¼1 , eN¼20 and the FN=1, FN=20 like in the previous I I chapter, presented in Table 11. The following conclusions may be formulated:  Similarly to the 1:1 cyclic tests the stress–strain curves may be assumed linear after cyclic tests. When a number of cyclic tests grows the stress–strain curves tend to a linear form.

Fig. 41. Biaxial cyclic tensile test results (1:2).

Table 9 Warp parameters for 1:1 tensile cyclic tests. Warp

A B C D E F G

F N¼20 warp N=1

eN¼1 [-] I

eN¼10 [-] I

eN¼20 [-] I

F

0.0012 0.0016 0.0036 0.0096 0.0200 0.0509 0.0648

0.0014 0.0022 0.0063 0.0218 0.0362 0.0631 0.0770

0.0014 0.0026 0.0083 0.0235 0.0398 0.0659 0.0801

1736 1474 1165 885 870 1002 1103

[kN/m]

F

N=10

1718 1445 1186 1104 1174 1238 1332

[kN/m]

N=20

F

1739 1471 1269 1147 1217 1289 1423

[kN/m]

F N¼20 weft

1.23 1.10 1.05 1.04 1.08 1.03 1.04

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Fig. 42. Biaxial cyclic tensile test results (1:4).

Fig. 45. Biaxial cyclic tensile test results (1:2) – loading curves.

Fig. 43. Biaxial cyclic tensile test results (2:1).

Fig. 46. Biaxial cyclic tensile test results (1:4) – loading curves.

Fig. 44. Biaxial cyclic tensile test results (4:1).

Fig. 47. Biaxial cyclic tensile test results (2:1) – loading curves.

 Strains for weft and warp directions for cycle N = 20, for 1:2, 2:1, 4:1, are positive. The strain for warp is negative only for 1:4 ¼ 0:009½ and FN=20 = 2300 load ratio. The values of eN¼20 I [kN/m] are determined according to linear approximation.  For high proportion of loads (>1:4 and >4:1) the number of cyclic tests should be higher for stabilize the mechanical parameters.

 Graphical comparison with the results obtained for 1:1 stress ratio is given in Figs. 49 and 50. The values of longitudinal stiffnesses for high stress ranges are similar with the 1:1 results. Lower stress ranges result in the values of longitudinal stiffnesses higher but comparable with the presented results.

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A. Ambroziak / Construction and Building Materials 80 (2015) 210–224 Table 11 Identification for tensile cyclic tests. Warp

e 1:2 1:4 2:1 4:1

N¼1 [-] I

0.0006 – 0.026 0.070

Weft

e

N¼20 [-] I

0.001 0.009 0.044 0.085

N=1

F

[kN/m]

2220 – 760 1060

F

N20

[kN/m]

2045 2300 1060 1330

eN¼1 [-] I

eN¼20 [-] I

FN=1 [kN/m]

FN20 [kN/m]

0.028 0.075 0.0027 0.001

0.047 0.089 0.0045 0.0026

750 1040 1510 1690

1040 1250 1560 1750

6. Conclusions

Fig. 48. Biaxial cyclic tensile test results (4:1) – loading curves.

The author investigated mechanical properties of Precontraint 1202S coated fabric. The identification of material parameters was successfully performed on the basis of biaxial and uniaxial tension tests. In the future, a separate detailed investigation will be performed by the author to fully confirm the proposed approach to material parameters determination. Variation of initial mechanical properties of coated fabric under cyclic tests was investigated. The cyclic loads show variations of initial longitudinal stiffness and should be considered in the design. Nevertheless, determination of initial mechanical properties is necessary to properly predict the behaviour of fabric under initial stress state in time of a construction assembly. Additionally, it should be noted that after each of the cyclic tests range, behaviour of the fabric is similar to the stress–strain curves obtained for specimens without cyclic loading. The shape, dimension and loading of the fabric structures forced designers to select a proper type of coated fabrics. It can be proposed that to fully utilised of the Precontraint 1202S properties the fabric should applied for structures where stresses in the warp and weft directions are comparable (1:2–1:1–2:1). The stress and strains in warp and weft directions for these load ratios are similar. Material parameters, which have been established above, can be used in the direct way in the FEM to analysis of fabric structures, where geometric nonlinearity is supplemented by the physical elastic nonlinearity of a coated fabric material. The obtained results encourage the author to continue the outlined research, also on the basis of extended experimental and to apply another types of constitutive models. This study can provide engineers and designers with a theoretical basis for a wide use of the coated fabrics. Funding

Fig. 49. Change of warp FN=20 longitudinal stiffnesses under cyclic tensile test.

The authors would like to acknowledge the National Science Centre, Poland (Grant No. UMO-2011/03/B/ST8/06500) for the financial support of the research. References

Fig. 50. Change of weft FN=20 longitudinal stiffnesses under cyclic tensile test.

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