Effect of inflation pressure on the constitutive response of coated woven fabrics used in airbeams

Composites: Part B 42 (2011) 526–537

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Effect of inflation pressure on the constitutive response of coated woven fabrics used in airbeams Jean Paul Kabche a,⇑, Michael L. Peterson b, William G. Davids c a

Advanced Engineered Wood Composites Center, University of Maine, Orono, ME 04469-5793, USA Civil Engineering, Boardman Hall, Orono, ME 04469-5711, USA c Mechanical Engineering, Boardman Hall, Orono, ME 04469-5711, USA b

a r t i c l e

i n f o

Article history: Received 18 May 2010 Accepted 11 November 2010 Available online 17 November 2010 Keywords: A. Fabrics/textiles B. Elasticity D. Non-destructive testing Airbeams

a b s t r a c t Textile fabrics have gained considerable attention as reinforcement for inflatable structures. A natural challenge arising from their implementation is the proper characterization of their constitutive behavior, which involves complex inter-tow contact interactions under different loading conditions, and the effect of inflation pressure on the fabric stiffness. This paper presents an experimental investigation aimed to independently quantify the effective constitutive properties of coated, woven textile fabrics used as reinforcement for pressurized fabric tubes. When used as structural members, these tubes are commonly known as airbeams. To investigate the influence of inflation pressure on the effective fabric properties, tension/torsion experiments of airbeams were performed at various inflation pressures and under various applied axial and torsional load conditions. Good test repeatability was achieved and the results showed that the effective fabric moduli increase with internal pressure. Effective constitutive properties obtained from these experiments were used as material inputs for beam finite element models to predict the load– deflection response of woven airbeams loaded in four-point bending. The FE model results correlated well with the experimental data over the full range of loading, thus demonstrating that the test method presented is well-suited to determine independent material-level fabric properties which account for the effect of inflation pressure. Additionally, tension/torsion results of braided fabric tubes tests are presented, where good repeatability of the material properties was obtained, highlighting that the test procedure can be extended to different fabric architectures. Ó 2011 Published by Elsevier Ltd.

1. Introduction In recent decades, inflatable structures have been typically used in space-based applications, sports facilities, recreational tents, and military shelters [1]. When compared to conventional composite structures, their main advantages are: light weight, low transportation costs, and ease of deployment and storage. Inflatable structures used as load-carrying members in the form of pressurized tubes and beams are generally known as airbeams, as shown in Fig. 1. Some lightly loaded airbeams used in space and other applications are made of a single layer of Mylar or plastic. The more common configuration is, however, a fabric tube with an internal compliant bladder enclosed by a separate continuous weave tube which forms the structure. Modern structures usually employ seamless, woven fabric tubes to alleviate the assembly difficulties encountered with seamed tubes and to increase allowable ⇑ Corresponding author. Address: ESSS North America Inc., 11200 Westheimer Road, Suite 760, Houston, Texas 77042, USA. Tel.: +1 832 339 0931. E-mail address: [email protected] (J.P. Kabche). 1359-8368/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.compositesb.2010.11.007

pressures and load-carrying capacity. Plain-weaves and braids are commonly used as reinforcing materials, ultimately increasing the airbeam’s structural stiffness. The internal pressure of the tube is balanced by the fabric’s inter-tow contact forces and by the resulting fabric pre-tension in the axial and hoop directions. Understanding the fabric behavior in pressurized tubes is fundamental for assessing the contribution of the fabric stiffness to the structural behavior of inflated beams and arches. A woven fabric consists of two groups of intertwined tows (yarns) arranged in an orthogonal fashion. Tows aligned with the axial direction are referred to as ‘‘warp tows’’ and those aligned with the hoop direction are known as ‘‘weft tows’’. A study by Cavallaro et al. [2] suggested that due to this orthogonal arrangement, the axial and hoop stiffnesses of uncoated woven fabrics are nearly uncoupled and shear deformations are only countered by rotational inter-tow friction due to inflation pressure. Textile composite structures are typically sub-divided into four hierarchical levels: yarn (tow), fabric (woven, braided, etc.), composite unit cell, and composite part. The meso-structural response of the fabric refers to kinematics at the tow level [3]. At this level, the fabric

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Parallel airbeam arches Guy lines Fig. 1. Schematic of a typical airbeam tent.

response is primarily driven by crimp interchange, locking, and resistance to inter-tow rotation [4]. Crimping refers to the undulation (or wave) of the tow. Crimp interchange refers to tow straightening in one direction of the fabric as crimping decreases, while crimping in the other fabric direction increases. Locking is the fabric’s resistance to deformation as the interwoven tows jam against each other. Mechanical testing for characterization of the fabric behavior used in structural components is a challenging task due to the complex inter-yarn contact interactions. Further, the level of complexity increases for pressurized structures such as airbeams, where the mechanical behavior of the fabric is largely influenced by inflation pressure [2]. Research by Wicker [5] and Davids et al. [6] suggests that the axial and shear moduli of the fabric are strongly dependent on the inflation pressure and pre-tensioning of the fabric. When subjected to in-plane shear loading, the dominant response mechanism of the fabric is its resistance via relative inter-tow rotation. When locking occurs, local shear and inplane forces increase and they are compensated by out-of-plane deformation, or buckling. A critical shearing angle, also known as the locking angle, is reached just prior to the onset of buckling [7]. During the course of the present study, experimental testing procedures for fabric material testing were reviewed [3,8–10] for possible implementation/adaptation. Methods addressed in this previous research are typically used for performance and qualification of clothing and upholstery textiles, which are non-structural materials and in turn do not require constitutive property characterization. Given that textile fabrics used in airbeams primarily carry membrane forces, these methods are not directly applicable for quantification of the fabric’s elastic moduli, due to the omission of inflation pressure effects. Similarly, standardized tests such as strength testing of woven materials [11] are not suited for textile airbeam fabric characterization, given that the critical failure mode of airbeams as structural members may be deflection of the beam, which would be categorized as global failure and not fabric failure. Hence, for assessing airbeam structural capacity, it is imperative to characterize the fabric’s elastic moduli rather than the strength. Lomov et al. [3] devised a testing method for determining the longitudinal and shear constitutive properties of fabric materials using the Kawabata Evaluation System (KES) picture frame test. This test is typically used to quantify fabric shear stiffness, by gripping two edges of a sample and pulling at opposing corners, while allowing rotation at the hinged corners. A peak shear angle of 75° can be achieved in pure shear. The main drawbacks of this method, as they relate to our investigation, are: (1) variation of shear effects between the center of the sample and the corners due to clamping conditions; (2) its setup is intended for woven textiles only; and (3) if used for braided fabric testing, the fabric would need to be adjusted to the braid angle and fiber orientation. Cavallaro et al. [8] employed a biaxial test fixture designed to determine the longitudinal and shear moduli of the fabric. The

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fixture consists of an in-plane, multi-axial and shear testing apparatus, with two independently rotating arms which can be oriented at any angle from 0 to 180° from the other arm. Each of the arms contains a fabric grip and a load cell to monitor the load at each axis. Once tension is applied to the fabric specimen, the arms rotate with respect to one another to produce shear deformation. Though a combined loading state and different stress ratios can be accommodated with this test, this method requires an additional loading axis to incorporate the effects of inflation pressure applied by the internal bladder. As described by Wicker [5] and Turner [12], beam testing can be used to back-calculate material-level properties of airbeam fabrics. Although this technique allows pressure-dependence of the properties to be accounted for, it does not allow the shear and elastic moduli to be determined independently. In other words, determining one of the moduli depends upon providing some information about the other. Another difficulty posed by this method is that it requires that a specimen of appropriate scale be made for each fabric configuration used in computational modeling efforts. For cases where the assumptions of Timoshenko beam theory (i.e. linear shear deformable behavior prior to wrinkling) may not be satisfied, caution must be used when obtaining the average effective elastic and shear moduli from this type of beam test. For instance, Malm et al. [13] used an experimental value for the shear modulus (G), determined from a tension/torsion experiment, to back-calculate the axial modulus of woven fabrics (E). An objective function was minimized to calculate an optimal value of E, noting that only load-deformation data prior to wrinkling was used. It is interesting to note that their back-calculations resulted in the same optimal value of 866 N/mm for both inflation pressures of 138 and 207 kPa. Such result deviates from previous observations by Wicker [5] and Turner et al. [14], which indicated that fabric moduli increase with increasing inflation pressure. Although each of the reviewed methods provides important fabric material data, none of them is able to provide a complete set of independent elastic properties or fully assess the influence of inflation pressure on the fabric response. To address this issue, the experimental investigation presented in this paper focuses on quantifying the effective axial and shear moduli of pressurized, coated fabric tubes. A tension/torsion testing methodology previously developed by Turner et al. [14], which can achieve combined tension and torsion loading states while regulating the airbeam inflation pressure, was adopted for this purpose. While their methodology provided a set of independent fabric properties which cannot be readily obtained using existing material tests, their procedure was reviewed and modified to mitigate some of the main difficulties encountered. In particular, the following issues were identified: (1) lack of a rigorous fabric pre-conditioning procedure; (2) the unloading behavior of the fabric was not addressed (i.e. data was recorded up to a peak load only); and (3) significant variability of the computed effective axial modulus was observed. The present methodology includes fabric pre-conditioning as a key modification in an attempt to achieve higher repeatability of the computed fabric moduli. While the paper focuses on test results for woven fabrics, braided fabric results are also included, noting that axial pre-tension was not considered in the torsion tests. It is also necessary to consider braided fabrics which are currently used for beams and arches [15,16]. The ability to quantify effective material properties of braided fabrics, in particular the shear stiffness, is of primary importance. Effective fabric properties obtained from these tests were used as inputs for a beam finite element analysis code developed by Davids and Zhang [17] to generate load-deformation curves of coated woven airbeams loaded in four-point bending at various inflation pressures. Finite element model predictions are then compared with the experimental load-deformation response of coated woven

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airbeams reported by Malm et al. [13]. For the work presented herein, it is assumed that the woven fabric behaves homogenously and thus can be treated as a continuum. The rationale behind this assumption is that, upon inflation, an internal urethane bladder expands and transfers the loads to the fabric. Furthermore, given that the fabric is composed of two families of continuously distributed and intertwined tows, it is assumed that these two families are constrained to occupy a common surface in three dimensional spaces, thus acting as a continuum. 2. Experimental testing setup Two types of tests were conducted for this experimental program, namely axial tension and torsion loading. Fig. 2 shows a typical coated, woven fabric tube specimen and its geometry. Each tube had a nominal diameter of 102 mm and a nominal length of 460 mm prior to pressurization and application of external loads. Specimens consisted of a fabric tube placed over an integral urethane bladder, which acts as a compliant and non-structural membrane when pressurized. Woven fabric specimens were procured from Federal Fabrics-Fibers, Inc., of Lowell, MA and manufactured from 3000 denier dimensionally stable polyester (DSP) fibers with a 0–90° continuous weave. The filament count for the woven fabric specimen was 750–800 per end, and each filament was approximately 10–15 lm in diameter. Braided fabric specimens were procured from Vertigo, Inc., and were made from Vectran tows with a braid angle of 60°. Each tow is comprised of five ends of 200 filaments each and each filament was approximately 23 lm in diameter. It is noted that only coated specimens were considered for this study. Coatings are typically used for protection against environmental degradation of the structural fibers. Additionally, they reduce relative tow translations and rotations, thereby increasing the in-plane stiffness of the fabric. For instance, Cavallaro et al. [2] stated that for a given deflection, a coated airbeam specimen can support twice the load as an uncoated specimen. The experimental setup used for both tension and torsion testing is shown in Fig. 3. Fabric tube specimens were gripped, pretensioned with internal pressure, and loaded in a tension/torsion, servo-hydraulic actuator testing machine (8874, Instron, Norwood MA, USA). Aluminum couplings were attached to the load head and the testing bed. Inflation pressure was provided by an existing 414 kPa air supply line which was controlled by a pressure regulator (PRG101, Omega Engineering, Stamford CT, USA), and monitored with a pressure transducer (PX209-100G5V, Omega

Engineering, Stamford CT, USA). The tube’s internal urethane bladder was attached to the pressure regulator and the pressure transducer. The pressurized fabric tube specimen prior to testing is shown in Fig. 4. We emphasize that the same test setup and testing machine was used for both the tension and torsion tests, although the instrumentation differed as discussed below. For the axial tension tests, strain in the axial direction was measured using a 50-mm gage length extensometer (2630–112, Instron, Norwood MA, USA). Extensometer grips were attached at the mid-span of the tube specimen using two rubber bands. For the torsion loading tests, circumferential displacements (arc lengths) were measured at three distinct locations on the tube by using string potentiometers (SP-25 Celesco, Chatsworth, CA), with a 625-mm range. String pots were evenly spaced over a total gage length of 154 mm. String pot cables were then placed around the circumference of the tube and the end of the cables were attached to pins glued to the surface of the test specimen. Recorded arc lengths were used to compute rotations and their corresponding twist per unit length, and finally the shear strain over the specified gage length. String pot data yielded more accurate shear strain results than the rotation measurements obtained from the load head of the tension–torsion test machine, given that rotations were computed for a fixed gage length. Also, load head rotation values may not correspond to the rotation of the fabric because of backlash in the clamp and grip system. Therefore, by measuring the rotation using string potentiometers, the contribution of coupling backlash and fabric slipping to the rotation calculations was eliminated. Test data was recorded at a rate of ten samples per second (0.10 kHz). String potentiometer and pressure readings were collected by a 16 bit vertical resolution data acquisition chassis (PXI Platform, National Instruments, Austin TX, USA). Axial force, rotation, torque, and extensometer strain readings from the tension– torsion test machine were output to the chassis for data acquisition. 3. Fabric tube specimen pre-conditioning Results from the tension/torsion tests conducted by Turner et al. [14] showed good repeatability of the effective shear modulus computed from the torsion load testing, but significant variability of the effective axial modulus, which was apparent from the high coefficients of variation (COVs). It was hypothesized that such variability was likely caused by the lack of proper fabric pre-condi-

102

Coated woven fabric r

460

Urethane bladder

Fig. 2. Fabric tube specimen configuration and geometry. (a) Typical woven tube specimen. (b) Fabric tube geometry, in mm.

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Instron 8874 force and torque Aluminum coupling

Fabric tube specimen

Fire hose coupling

Celesco SP-25 string potentiometers

Instron 2630-113 extensometer

Omega PRG101 pressure regulator

Omega PX209 pressure transducer

Instron 8874 control tower

National Instruments data acquisition chassis

PC

414 kPa air supply line Testing bed Fig. 3. Experimental setup for tension/torsion testing.

Fig. 4. Pressurized fabric tube specimen prior to testing.

tioning. In an attempt to alleviate this issue, a fabric pre-conditioning procedure was implemented as a key aspect of the present test methodology, by subjecting the inflated fabric tube to peak loading conditions before actual test runs were conducted, thereby allowing self-arrangement and stabilization of the tows prior to loading used to evaluate the material.

The initial state of the tube specimen was taken to be the state at which the fabric was pre-tensioned due to inflation pressure before any external (tension or torsion) loads were applied. For coated woven fabric tubes, a peak inflation pressure of 276 kPa was selected, which is about four times the usual operating pressure for woven airbeams (83 kPa). Similarly, a peak pressure of 345 kPa was chosen for coated braided tubes, to support concurrent component level testing efforts of braided arches. Fabric pre-conditioning was performed by inflating the tube to the specified peak pressure, allowing the bladder and fabric to stabilize, and then applying axial or torsion loads in a cyclic fashion until the stress–strain response stabilized. It is noted that preconditioning runs were conducted in descending order. For example, for woven fabrics, the tube was pre-conditioned at an inflation pressure of 276 kPa, after which testing was conducted using the following sequence: three trials at 276 kPa, three trials at 207 kPa, and so on. The rationale behind this approach was that such sequence would allow self-arrangement of the tows at peak levels of inflation pressure and external loading. To pre-condition the fabric axially, tension was applied to the tube specimen cyclically until the slope of the axial stress–axial strain curves (effective axial modulus) stabilized. For woven fabric tubes, the specimen was loaded up to a peak magnitude of about 2234 N at the end of each cycle. For braided fabric tubes, axial pre-conditioning was conducted by applying a strain of approximately 2%. Preliminary trials showed that the axial stiffness of coated braided fabric was about 1/10th of that of the woven fabric for a given inflation pressure. Hence, it was necessary to monitor axial strain rather than axial force, given that large strains were achieved with relatively low axial loads. Fig. 5 shows a typical plot of axial stress versus axial strain pre-conditioning curves (15 cycles, total) for woven tubes inflated to 276 and braided tubes inflated to 345 kPa, respectively. In a general sense, these plots show that after the first 2–3 cycles of loading, the response of the tube begins to stabilize until each loop virtually lies on top of the previous one. A similar approach was employed to pre-condition the fabric in torsion, by applying a rotation to the tube in a cyclic fashion until the slope of the shear stress-shear strain curves (effective shear

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4. Tension/torsion testing procedure Tension/torsion test procedures of woven fabric tubes were conducted for inflation pressures of 69, 138, 207, and 276 kPa. For braided fabric tubes, a pressure range of 69–345 kPa was used, in increments of 69 kPa. The experimental plan was sub-divided into two groups: (1) axial loading tests, where the specimen was subjected to tensile loading using various initially applied rotations, to study the effect of shear pre-stress on the effective axial modulus of the fabric for a given inflation pressure and (2) torsional loading tests, where the specimen was subjected to pure torsion under fixed axial loading magnitudes, to study the effect of axial pre-stress on the effective shear modulus of the fabric for a given inflation pressure. Nine trials were performed for each pressure and axial load combination: three each on three nominally identical specimens. Specimen tube diameter was measured at five different locations equally spaced along the span of the tube, at the tube’s inflated state. The average of these measurements was then used for all membrane stress related computations.

8

Axial stress (N/mm)

7 6 5 4

4.1. Axial tension testing procedure

3 2 1 0 0.000

0.005

0.010 0.015 Axial strain (mm/mm)

0.020

Fig. 5. Pre-conditioning axial stress–strain curves of 102-mm tubes. (a) Woven fabric tube at 276 kPa. (b) Braided fabric tube at 345 kPa.

modulus) stabilized. For woven fabric tubes, a peak rotation of about 20° (17.75 N-m of torque) was applied. For braided tubes, a peak rotation of about 3° (76.11 N-m of torque) was applied. Compared to woven tubes, this relatively small rotation was applied in light of the braid’s large torsional stiffness and to avoid out-of-plane fabric buckling. For example, when inflated to 138 kPa, 1/10th of the rotation applied to a woven fabric tube generated approximately 4.3 times the applied torque in a braided fabric tube. For both axial tension and torsion cases, preliminary runs showed that a minimum of 6–7 loading cycles was required to attain stability of the load versus deformation response of the fabric at peak pressures, and a minimum of 3 min was required to allow the fabric to stabilize at each inflation pressure. A total of ten loading cycles was selected as appropriate for fabric pre-conditioning for both woven and braided fabric tubes. Although not formally presented in this paper, it is important to address whether the effect of fabric pre-conditioning is permanent after the test specimen has been unloaded completely. This may likely be an important issue when real-world construction is considered, given that fabric pre-conditioning may not always be possible prior to assembly of the structure. In that light, preliminary testing conducted during our experimental program suggested that pre-conditioning was necessary for specimens that had been depressurized after a tension or torsion test run. That said, it was possible to obtain repeatable results after a considerable amount of time (between 60 and 90 min) had gone by, on condition that the tube specimen remained inflated. However, a complete data set was not generated to assess this particular issue and a definite conclusion cannot be drawn on the basis of these preliminary studies.

The objective of the axial tension test was to quantify the fabric’s effective axial modulus (E) and to investigate the effect of inflation pressure on the modulus’ variation. Fabrics are typically assumed to behave as tension-only materials, providing stiffness in their taut state, but little to no resistance in compression – their slack state. In the case of inflated tubes, fabric compression begins to take place when the fabric pre-tension is countered by the applied axial compression load, and the fabric begins to wrinkle. Axial compression trials at low pressures (69, 138 kPa) were successfully performed. However, a comprehensive study of the tubes loaded in compression was not conducted due to three main factors: (1) severe out-of-plane bending of the tube specimen during compression (internal bladder bending); (2) safety concerns with applying high compressive loads at inflation pressures higher than 138 kPa and corresponding out-of-plane bending, which occurred more suddenly and severely at higher inflation pressures; and (3) relative motion between the clamp used on the fabric and the testing machine grip. The relative motion of the clamp and grip resulted in a ‘‘gap’’ between the tensile and compressive regions of the stress–strain response, making it difficult to interpret the data in this region. Therefore, compressive loading results have not been included in this study. Prior to testing, the tube specimens were pressurized and preconditioned at peak pressure values. Pressurization of the tube, assuming full compliance of the internal bladder, results in axial and circumferential pre-tensioning of the fabric. To investigate the response of the fabric beyond its pre-tensioned state, woven tube specimens were loaded in tension from their pressurized state up to a load value of 2234 N. A challenging aspect of testing braided tubes in tension is the braid’s low resistance to axially applied forces, given that the braids are angled to primarily resist torsional loads. For braided fabric tubes, large strains occur at relatively low axial force; hence, it was necessary to load the specimen by monitoring axial strain rather than axial force. A strain of about 2% was deemed reasonable to apply to braided tubes for this study. A small axial load of about 25–30 N was applied to the specimens at the beginning of loading to minimize slip between the fabric clamp and the grip on the machine. Axial tension was applied in position-control mode using a linear ramp generator rate of 5.08 mm/min. Preliminary studies showed that this loading rate was appropriate to prevent creep of the fabric (i.e. significant drops in load due to relaxation of the tows).

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4.2. Torsion testing procedure The objective of the torsion test was to quantify the fabric’s effective shear modulus (G) and to investigate the effect of inflation pressure on the modulus’ variation. Prior to testing, the tube specimens were pressurized and pre-conditioned at peak pressure values. To investigate the torsional response of the fabric beyond pretension, the tube specimen was loaded from its pressurized state by applying a rotation of about 20° at a rate of 10 deg/min for woven tubes, and a rotation of about 3° at a rate of 2 deg/min for braided tubes. Preliminary loading rate studies showed that these rates were appropriate for loading the specimens while precluding creep in the fabric. An issue encountered during preliminary tests was the influence of axial load restraint on the effective shear stiffness. If axial load was allowed to develop during the torsion test, tension developed while the specimen was being twisted and the effective shear modulus increased, particularly at lower pressures. This was attributed to the coupling between shear and axial stiffnesses of the fabric. To ensure that axial force would not develop during torsion, the actuator was set to axial load control and rotation displacement control. Thus, by holding the axial force constant at 0 N, a state of pure shear was developed in the fabric. The high shear stiffness in the braided specimen gave rise to a secondary challenge when measuring the arc lengths resulting from applied rotations of 2–3°. Even when using a lower range string potentiometer (121 mm range versus a 635 mm range) it was not possible to accurately capture the fabric response. The shear strains were therefore obtained as a simpler gross value using the load head rotations. The effect of coupling backlash was monitored for every test, and data recorded during the start of each test, which reflected the coupling backlash effect, was not included when calculating the moduli reported later in this paper. For woven fabric specimens, two trials were conducted at each inflation pressure: 69, 138, 207, and 276 kPa, with initial axial loads of 0 N and 1780 N, for a total of 48 trials. For braided tube specimens, three trials were conducted at each inflation pressure: 69, 139, 207, 276, and 345 kPa, with no initial axial load, for a total of 45 trials. The influence of axial pre-stress on the shear modulus of braided fabrics was not considered. However, it is expected that applying axial pre-stress would have a moderate influence on the shear response of the braid up to its critical shear angle. When this angle is reached, locking occurs and the resistance to loading may be compensated by out-of-plane buckling, which is expected to be insensitive to inflation pressure.

5. Axial tension test results A representative plot of the axial stress versus axial strain response for a coated woven fabric specimen inflated to 276 kPa with no initially applied rotation is shown in Fig. 6. The initial shallow slope shown in the figure corresponds to tow de-crimping and relative frictional sliding at the beginning of loading. The strain magnitude at which a change in slope was observed to occur – 0.36% for this case – varied with inflation pressure and with the number of loading cycles that the fabric had been subjected to. After this

10 slope 9 Initial corresponds to tow decrimping 8 at the beginning 7 of loading 6 5 4 3 2 1 0 0.000

Region used to compute effective axial modulus

Loading path

Unloading path

0.005 0.010 Axial strain (mm/mm)

0.015

Fig. 6. Representative axial stress versus axial strain response of a coated woven fabric tube specimen at 276 kPa.

initial region, the stress–strain response of the fabric is very linear during both loading and unloading. The resultant axial force applied by the actuator, F, is defined as the product of the axial stress times the cross-sectional area of the tube, 2prt. Fabric thickness, t, was very small and not well defined and was therefore not explicitly considered in the computation of the effective moduli. Instead, the effective moduli, E and G, were taken as membrane resultants, defined as the product of the true modulus times the small but unknown tube thickness. The effective axial stress, r, is computed according to:

F 2pr

r ¼

ð1Þ

where r is the measured radius of the tube. The effective axial modulus, E, was computed using a linear regression of the stress–strain curves during loading and unloading, according to:

E ¼

r e

ð2Þ

5.1. Axial tension results for woven fabric tubes Fig. 7 shows the axial stress versus axial strain response of a coated fabric tube for a range of pressures from 69 kPa to 276 kPa and no initial rotation applied. An increase in axial stiffness with increasing internal pressure is clearly observed. For instance, at an applied axial stress of 7.43 N/mm, the axial strain recorded was 0.0059 and 0.0112 for the 69 kPa and 276 kPa cases,

10 207 kPa

9 8 Axial stress (N/mm)

For woven tube specimens, three trials were conducted at each inflation pressure: 69, 138, 207, and 276 kPa with initially applied rotations of 0°, 10°, and 20°, for a total of 108 trials. For braided tube specimens, three trials were conducted at each inflation pressure: 69, 139, 207, 276, and 345 kPa, with no initially applied rotations, for a total of 45 trials.

Stress (N/mm)

J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537

7

138 kPa

276 kPa

6 5 4

69 kPa

3 2 1 0 0.000

0.002

0.004 0.006 0.008 Axial strain (mm/mm)

0.010

0.012

Fig. 7. Axial stress–strain curves of coated woven tube at various inflation pressures with no initial rotation.

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9

1300

10 deg

6

1008.15

1100

20 deg

5 4 3

1048.67

1200

7

0 deg

E (N/mm)

Axial stress (N/mm)

8

1000 876.40

900 800 Spec01

2 1 0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Axial strain (mm/mm)

700

Spec02

600

Spec03 Average

500 50

100

Fig. 8. Axial stress–strain curves of a coated woven tube at 69 kPa and with various initially applied rotations.

150 200 Pressure (kPa)

250

300

1300

5.2. Axial tension results for braided fabric tubes Fig. 10 shows the average axial stress-axial strain response of a braided specimen. The axial stiffness of the braid increases with inflation pressure, as crimp interchange and friction between the tows become more pronounced with increasing pressure. For example, at a strain of 0.01, the peak membrane stress values at 69 kPa and 345 kPa are 0.72 N/mm and 1.17 N/mm, respectively – a 38.46% stiffness increase. To provide a quantitative comparison between braided and woven fabrics, it is noted that at this strain level and at an inflation pressure of 69 kPa, the average membrane

1200

1024.09

E (N/mm)

1121.96

967.82

1100 1000 786.60

900 800

Spec01

700

Spec02

600

Spec03 Average

500 50

100

150 200 Pressure (kPa)

250

300

1200 1020.21

1100

975.09

1000

E (N/mm)

respectively. This corresponds to effective moduli of 726.03 N/mm at 69 kPa and 997.50 N/mm at 276 kPa, which results in a stiffness increase of about 27.22%. A slightly more linear response is also observed as the pressure increases due to the larger pre-tension in the fabric which alleviates its non-linear kinematic response at the beginning of loading. Fig. 8 shows the axial stress/axial strain response of a woven tube inflated to 69 kPa with initially applied rotations of 0°, 10°, and 20°. It is observed that due to coupling between the axial and shear stiffnesses, initial rotations induce additional axial stress which in turn increases the stiffness in that direction. This response is not representative of the response at all inflation pressures; depending on the pre-stress in the specimen, the effective axial modulus can also decrease with an increasing initial rotation. This is more clearly shown in Fig. 9, where the effective axial modulus for all specimens at all inflation pressures and rotations of 0°, 10°, and 20°, are presented. Averaged values of the three trials for the three specimens for each loading condition are enclosed by boxes. The general trend observed in these three plots is that, for the range of inflation pressures considered, the average axial modulus increases with an initially applied rotation of 10°. For a 20degree rotation, however, the average modulus at 207 kPa decreased by 0.1%, and by 6.5% at 276 kPa, when compared to a rotation of 10°. One likely explanation for this decrease is that the local strain is highly variable and thus needs to be averaged over a larger area. Such local strain variability may also explain the discrepancy in the axial modulus values computed for the three specimens at 138 kPa. Nonetheless, upon computing the average elastic modulus value for a fixed pressure value, the general trend of increasing elastic moduli with increasing rotation, persisted. The complete set of effective axial moduli with initial rotations for woven tubes is summarized in Table 1, where good result repeatability is observed, particularly at higher inflation pressure.

900

1049.09

793.17

800 Spec01

700

Spec02

600

Spec03 Average

500 50

100

150 200 Pressure (kPa)

250

300

Fig. 9. Effective axial modulus of coated woven tubes. (a) No initially applied rotation. (b) Applied rotation of 10°. (c) Applied rotation of 20°.

stress computed for woven tubes is 6.93 N/mm. Further, the averaged effective modulus computed for woven and braided tubes at this pressure level is 669.06 N/mm and 51.17 N/mm, respectively. The complete set of effective axial moduli for braided tubes is summarized in Table 2. At a pressure of 276 kPa, specimen 2 shows an axial modulus decrease from its previous value at 207 kPa. Nonetheless, average axial tension results for braided tubes follow the general trend of an increasing modulus with increasing inflation pressure. Also, very good repeatability is observed, with COVs under 2%. Although a fixed diameter of 102 mm was used for the fabric tube specimens presented in this work, it is important to address

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J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537 Table 1 Effective axial modulus of coated woven tubes at various inflation pressures and various initially applied rotations. Coated woven tubes

69 kPa Specimen

138 kPa Specimen

Trial

1

2

3

No rotation 1 2 3 Average StDev CoV (%)

647.41 616.24 666.32 643.33 25.29 3.93

630.00 643.52 639.92 637.82 7.00 1.09

690.04 718.21 769.33 726.03 40.47 5.57

812.65 873.98 843.89 843.51 30.66 3.63

889.45 907.59 898.87 898.64 9.07 1.01

858.39 879.65 923.08 887.04 32.97 3.71

10-deg rotation 1 2 3 Average StDev CoV (%)

787.64 792.96 811.01 797.20 12.25 1.54

705.41 764.03 728.71 732.72 29.52 4.02

874.71 830.90 783.99 829.87 45.37 5.46

891.61 935.24 959.30 928.72 35.31 3.69

1057.82 1046.61 1052.29 1052.24 5.60 0.52

20-deg rotation 1 2 3 Average StDev CoV (%)

721.58 723.82 777.52 740.97 31.67 4.27

849.35 862.41 854.00 855.25 6.62 0.77

853.16 762.14 734.52 783.27 62.08 7.93

1053.07 1032.22. 1059.08 1048.12 14.10 1.35

1003.93 986.48 978.45 989.62 13.02 1.32

1

207 kPa Specimen 2

3

1.8

Stress (N/mm)

2

3

941.46 967.39 984.49 964.45 21.66 2.24

1018.57 1108.39 1093.04 1073.34 48.04 4.47

989.68 1053.54 916.79 986.67 68.42 6.93

943.99 962.63 860.92 922.51 54.15 5.87

1004.79 1020.27 1052.48 1025.85 24.33 2.37

1046.19 1138.20 1119.42 1101.27 48.61 4.41

846.47 890.89 925.26 887.54 39.50 4.45

950.91 989.10 1008.57 982.86 29.33 2.98

1117.17 1126.51 1062.02 1101.90 34.85 3.16

1

2

3

956.35 997.82 989.87 981.35 22.01 2.24

1189.19 1157.22 1155.06 1167.16 19.11 1.64

968.67 1026.25 997.59 997.50 28.79 2.88

891.74 966.18 977.54 945.15 46.60 4.93

1071.36 1031.62 1048.40 1050.46 19.95 1.90

1240.97 1249.88 1250.05 1246.97 5.19 0.42

1118.09 1067.47 1019.77 1068.45 49.17 4.60

959.72 1045.43 922.43 975.86 63.07 6.46

976.65 1000.90 1014.63 997.39 19.23 1.93

1136.77 1142.93 1154.95 1144.88 9.24 0.81

935.66 1031.36 1047.99 1005.00 60.63 6.04

dence on the internal pressure is generally attributed to tow decrimping and inter-tow friction. For instance, an increase in the shear modulus with increasing internal pressure is likely due to crimp interchange between overlapping tows, as the tows stretch in both axial and hoop directions, generating a greater frictional force at their interfaces.

2.0 P = 345 kPa

1.6

1

276 kPa Specimen

1.4 1.2 1.0

6. Torsion tests results

0.8 P = 69 kPa

0.6

A representative plot of the shear stress versus shear strain response for a coated woven tube inflated to 138 kPa with no axially applied load is shown in Fig. 11. A small initial torque of about 0.85–1.00 N-m was applied to the specimen to minimize coupling backlash at the beginning of testing. After the slightly steeper slope observed for a strain range of 0–0.005, the shear response of the fabric is very linear during loading and unloading. Coupling backlash was carefully monitored at the start of the test runs in an attempt to generate more reliable strain data. To this effect, the strain portion of the recorded data which corresponds to the start of the test run, and which may reflect coupling backlash, was not used in the effective shear moduli calculations. Effective shear moduli values were computed as the average of the stress–strain slopes during loading and unloading, with the thickness variable, t, removed, according to Eqs. (3)–(6). The polar moment of inertia, J, was computed as follows:

0.4 0.2 0.0 0

0.005

0.01 0.015 Strain (mm/mm)

0.02

0.025

Fig. 10. Average axial stress–strain curves of an unstrapped braided tube at various inflation pressures.

the effect of internal pressure and beam diameter on the stiffness of the airbeams. As described by Davids [17], sizing an airbeam involves a trade-off between its operating pressure and diameter; that is, as internal pressure increases, the diameter can be decreased, and vice versa. Similarly, at the fabric level, an increase in inflation pressure results in larger longitudinal stresses which in turn increase the effective fabric moduli. The moduli’s depen-

J ¼ 2  p  r3  t

ð3Þ

Table 2 Effective axial modulus of coated braided tubes at various inflation pressures. Coated woven tubes

69 kPa Specimen

138 kPa Specimen

207 kPa Specimen

276 kPa Specimen

345 kPa Specimen

Trial

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1 2 3 Average StDev CoV (%)

49.48 50.11 50.16 49.92 0.38 0.76

52.10 52.49 52.44 52.34 0.21 0.40

50.98 51.33 51.40 51.24 0.23 0.44

58.68 59.74 60.26 59.56 0.80 1.35

60.28 61.99 62.49 61.58 1.16 1.88

60.29 60.81 61.16 60.75 0.44 0.73

69.14 70.64 70.86 70.21 0.94 1.33

73.52 73.37 71.72 72.87 1.00 1.37

71.01 71.27 71.34 71.21 0.18 0.25

78.07 77.80 78.30 78.06 0.25 0.32

70.48 70.30 70.24 70.34 0.12 0.18

82.51 82.10 81.79 82.13 0.36 0.43

82.02 83.22 83.60 82.94 0.83 1.00

79.38 80.49 80.37 80.08 0.61 0.77

94.01 92.51 91.42 92.65 1.30 1.41

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J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537

0.80

1.00 Region used to compute effective shear modulus

0.90

0.60 0.50 0.40 0.30 0.20

Unloading path

0.10 0.00 0.00

0.02

0.03 0.04 Shear strain

0.05

0.06

0.60 0.50 0.40 0.30

0.00 0.000

ð4Þ

where T is the applied torque recorded from the load cell. The shear strain, c, was computed as follows:

r/ L

ð5Þ

where / is the twist per unit length computed from the string potentiometer arc lengths and L is the gage length of 154 mm. Finally, the effective shear modulus was computed by dividing the membrane shear stress by the shear strain:

s G ¼ c

ð6Þ

6.1. Torsion tests results for woven fabric tubes Fig. 12 shows the shear stress versus shear strain response of a coated woven tube averaged over three trials, for cases of 0 N and 1780 N of applied load prior to torsion. As expected, these plots show the effective shear stiffness of the fabric increasing with inflation pressure. It is also observed that applying an initial axial load increases the shear stiffness of the fabric, likely due to increased crimp interchange and inter-tow friction, followed by tow interlocking during torsion. Fig. 13 shows plots of the effective shear modulus at various inflation pressures. Averaged values of the three trials for the three specimens for each loading condition are enclosed by boxes. The general trend observed in these three plots is that, for the range of inflation pressures considered, the average shear modulus increases with initially applied axial loads – increased fabric pre-tension. For instance, at an inflation pressure of 69 kPa, the shear modulus increases by 41% on average when an initial axial load of 1780 N is applied. This stiffness increase is 34% at 138 kPa, 30% at 207 kPa, and 24% at 276 kPa. In other words, the effect of an applied axial load is more pronounced at lower inflation pressures. Effective shear modulus results presented in Table 3 show that good repeatability was achieved for all inflation pressures considered and for both axial pre-tension conditions, with COVs lower than 2% consistently obtained for these results. 6.2. Torsion tests results for braided fabric tubes Fig. 14 shows the shear stress versus shear strain response of a braided tube averaged over three trials with no axial pre-tension

0.010

0.020 0.030 Shear strain

0.040

0.050

1.40 P = 276 kPa

1.20 Shear stress (N/mm)

T 2  p  r2

P = 69 kPa

0.10

where r is the measured outer radius of the inflated tube. The membrane shear stress, s, is calculated as follows:

c¼

0.70

0.20 0.01

Fig. 11. Representative shear stress versus shear strain curve for a coated woven specimen inflated to 138 kPa.

s ¼

P = 276 kPa

0.80

Loading path

Shear stress (N/mm)

Shear stress (N/mm)

0.70

1.00 0.80 0.60 0.40 P = 69 kPa 0.20 0.00 0.000

0.010

0.020

0.030 0.040 Shear strain

0.050

0.060

Fig. 12. Shear stress versus shear strain response of a coated woven tube at various inflation pressures. (a) No initial axial load applied. (b) Initial axial load of 1780 N applied.

applied. As for the results from the woven tubes, this plot shows that the effective shear stiffness of the braid increases with inflation pressure. Axial pre-tension was not applied to the braided tubes. Fig. 15 shows the effective shear modulus for all three specimens tested at various inflation pressures. Averaged values of the three trials for the three specimens for each loading condition are enclosed by boxes. The effective shear modulus results presented in Table 4 shows that very good test repeatability was achieved for the range of inflation pressures considered, with a peak COV of 2.77%. 7. Correlation of finite element beam model results with woven beam bending tests The primary purpose of the experimental work presented previously is to establish a method for obtaining constitutive properties for use in modeling of airbeams. The utility of the resulting data is demonstrated by using the effective axial and shear stiffnesses (E, G) obtained from the tension/torsion tests as inputs for a specialized beam finite element code developed by Davids and Zhang [18]. This model includes pressure-stiffening effects and fabric material wrinkling. Since its presentation in Ref. [18], it has been extended to include large deformation geometric non-linearity. This extended model is used to predict the load-deformation response of 160-mm diameter, coated woven airbeams supplied by Federal Fabrics-Fibers of Lowell, MA that were inflated to 69,

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J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537

7

20 15.61

Shear stress (N/mm)

15 G (N/mm)

6

13.69 11.48 8.81

10 Spec01

5

Spec02 Spec03

50

100

150 200 Pressure (kPa)

250

4 3 2 P = 69 kPa

0 0.000

300

25

0.002

0.004 Shear strain

0.006

0.008

Fig. 14. Shear stress versus shear strain response of an unstrapped braided tube at various inflation pressures.

20.76 19.59 17.67

20 15.31

G (N/mm)

5

1

Average

0

P = 345 kPa

1000

15

874.95

900

819.12

10

936.32

G (N/mm)

Spec01 Spec02

5

Spec03 Average

0 50

100

150 200 Pressure (kPa)

250

800

724.05

700 Spec01

300

Spec02

600

Spec03

Fig. 13. Effective shear modulus of coated woven tubes at various inflation pressures. (a) No axial load applied. (b) Axial load of 1780 N applied.

585.75

50 138, and 276 kPa. The coated beam fabric was identical to the fabric used for the tension–torsion tests reported in this paper. The simply-supported beams spanned 2.68 m, and were loaded in four-point bending with 102 mm-wide fabric load straps centered at 902 mm from the centerline of each support. Displacements were measured with string potentiometers located at mid-span and at the load points. Full details of the bend test and experimental results can be found in Davids and Zhang [18]. Previously, Davids and Zhang [18] used a pressure-dependent shear modulus value obtained from tension/torsion tests and an independent axial modulus, which was back-calculated from beam

Average

500 150

250 Pressure (kPa)

350

Fig. 15. Effective shear modulus of unstrapped coated braided tubes at various inflation pressures.

load-deformation response prior to wrinkling, to simulate beam response. The present test methodology eliminates the need for using back-calculated material properties. Simulations were run using the averaged axial and shear moduli given in Table 5 for the woven fabric tubes for the same inflation pressures at which the beam was tested.

Table 3 Effective shear modulus of coated woven tubes at various inflation pressures. Coated woven tubes

69 kPa Specimen

Trial

1

0 N axial load 1 2 Average StDev CoV (%) +1780 N axial load 1 2 Average StDev CoV (%)

138 kPa Specimen 2

3

207 kPa Specimen

276 kPa Specimen

1

2

3

1

2

3

1

2

3

8.75 8.74 8.75 0.01 0.07

9.47 9.63 9.55 0.11 1.20

8.02 8.21 8.12 0.13 1.62

11.27 11.10 11.19 0.12 1.10

12.17 12.40 12.29 0.17 1.37

10.85 11.09 10.97 0.17 1.51

13.34 13.35 13.35 0.01 1.51

14.16 14.26 14.21 0.07 0.08

13.26 13.78 13.52 0.37 0.52

13.96 14.19 14.07 0.16 1.14

16.83 16.91 16.87 0.06 0.33

15.67 16.12 15.89 0.32 1.99

15.88 16.02 15.95 0.10 0.62

15.85 15.77 15.81 0.06 0.36

14.14 14.18 14.16 0.03 0.22

17.99 18.05 18.02 0.04 0.21

17.84 18.05 17.98 0.20 1.13

17.04 18.12 17.01 0.04 0.22

20.11 20.37 20.24 0.19 0.92

19.89 20.04 19.97 0.11 0.92

18.58 18.55 18.57 0.03 0.56

19.69 19.69 19.70 0.02 0.09

21.73 22.10 21.92 0.26 1.17

20.43 20.88 20.66 0.31 1.52

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J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537

Table 4 Effective axial modulus of coated braided tubes at various inflation pressures. Coated woven tubes

69 kPa Specimen

138 kPa Specimen

207 kPa Specimen

Trial

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1 2 3 Average StDev CoV (%)

537.04 556.31 539.61 547.96 11.81 2.15

564.81 583.91 587.06 585.48 2.23 0.38

613.04 624.61 623.02 623.81 1.13 0.18

697.06 698.64 701.44 700.04 1.98 0.28

715.17 732.18 733.88 733.03 1.20 0.16

723.40 738.50 739.69 739.10 0.84 0.11

779.40 803.62 804.83 804.22 0.85 0.11

784.61 815.33 821.26 818.29 4.19 0.51

818.70 836.06 833.63 834.85 1.72 0.21

859.65 852.71 820.00 836.35 23.13 2.77

873.93 879.73 882.93 881.33 2.26 0.26

870.91 908.23 906.12 907.17 1.50 0.17

892.15 923.23 935.99 929.61 9.02 0.97

931.68 930.49 918.61 924.55 8.40 0.91

943.64 953.98 955.61 954.79 1.15 0.12

Table 5 Effective fabric moduli used in finite element models. Po (kPa)

69 138 207 276 345

Axial modulus, E (N/mm)

Shear modules, G (N/mm)

Woven

Braided

Woven

Braided

669.06 876.39 1008.15 1048.67 –

51.17 60.63 71.43 76.84 85.22

8.81 11.48 13.69 15.61 –

585.75 724.06 819.12 874.95 936.32

900 Experimental Results

800

FE Model Theoretical Wrinkling Load

Applied Load (N)

700

p = 276 kPa

600 500

345 kPa Specimen

good, with the model successfully predicting both the increase in stiffness and capacity with inflation pressure as well as the loaddeformation response after the onset of non-linearity. The good correlation between the model and the experiments at both the load points and mid-span indicates that both the fabric E and G are accurate. For reference, the theoretical wrinkling load – the load at which beam theory first predicts that the fabric will lose tension – is shown for each pressure. The theoretical wrinkling loads correlate well with the onset of both model-predicted and measured nonlinear load-deformation response. Also of note is the fact that at the onset of wrinkling, shear deformations account for up to 32% of the total beam deflection. This highlights the importance of accurately determining the fabric shear modulus. Taken as a whole, the good correlation between the model and the simulations indicates that the pressure-dependent fabric moduli determined using the tension/torsion testing procedures detailed in this study are suitable for use in mechanics-based models of airbeams and airbeam-supported structures.

p = 138 kPa 400

8. Conclusions

300

p = 69 kPa

200 100 0

0

50

100 150 200 Midspan Displacement (mm)

250

900 800

p = 276 kPa

700

Applied Load (N)

276 kPa Specimen

600 500

p = 138 kPa 400 300

p = 69 kPa

200 100 0

0

50 100 150 200 Load Point Displacement (mm)

250

Fig. 16. Comparison of FE model and beam test results.

Fig. 16 compares the measured and predicted beam displacements at both mid-span and the load points. The results are quite

This paper presents an experimental tension/torsion procedure for testing inflated tubes reinforced with woven and braided fabrics. The primary objectives of these experiments were to independently quantify the effective constitutive properties of the fabric, and to assess the effect of internal pressure on the fabric stiffness. In contrast with previous studies, this method incorporates independent loading axes to achieve combined axial and shear loading states, while regulating the inflation pressure of the fabric tube. Thus, the effective properties obtained from these tests account for the effect of inflation pressure on the fabric constitutive response. Woven and braided specimens were tested for inflation pressures between 69 and 345 kPa by applying axial tension and torsion loads beyond the pre-tension induced by the inflation pressure. By incorporating pre-conditioning as a key aspect of the testing methodology, highly repeatable results were obtained for the range of pressures investigated. Further, by testing braided fabrics with good repeatability, it was demonstrated that the test procedure can be extended to other fabric architectures. Effective fabric moduli were computed as the average of the stress–strain slopes during loading and unloading. It was found that increasing the inflation pressure increased the axial and shear moduli of both woven and braided fabrics. A slightly more linear response was observed as the pressure increased, both axially and in shear. In comparison with braided fabric tubes, woven fabric tubes were very stiff axially but weaker in torsion, which is attributed to the orthogonal arrangement of the tows. For the axial loading case, applying an initial rotation resulted in an axial stiffness increase. Likewise, applying an initial axial load to the woven tube increased the torsion stiffness. This was explained by the coupling between axial and shear stiffness in fabrics, where applied torsion induced additional axial stress which in turn increased the stiffness

J.P. Kabche et al. / Composites: Part B 42 (2011) 526–537

in that direction, and vice versa. On the other hand, braided fabric tubes were very stiff in torsion but provided little axial resistance. Braided tubes were tested to quantify the axial and shear moduli of the braid. On average, the axial modulus of a braided tube was found to be about 1/10th of that of the woven tube for a given inflation pressure. Conversely, the shear modulus of the braid was estimated to be about 50 times larger than that of the woven fabric tubes. Effective woven fabric moduli were fed into beam finite element models to predict the load-deformation response of inflated beams loaded in four-point bending. A good comparison between the experimental and numerical load-deformation results was obtained over the full range of loading, including the onset of fabric wrinkling and subsequent softening of the load-deformation response. The good agreement between the model and the simulations demonstrates that pressure-dependent fabric moduli determined using the tension–torsion testing procedures detailed in this study can be confidently used in mechanics-based models of airbeams and airbeam-supported structures.

[2] [3] [4] [5] [6] [7] [8] [9]

[10]

[11]

[12]

Acknowledgement The research presented in this article was conducted under contract number W911QY-05-C-0043 with the US Army Natick Soldier Systems Center. The authors express their gratitude for this financial support.

[13]

[14] [15]

[16]

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