Experimental investigation on the flexural-torsional buckling behavior of pultruded GFRP angle columns

Thin-Walled Structures 125 (2018) 269–280

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Full length article

Experimental investigation on the flexural-torsional buckling behavior of pultruded GFRP angle columns

T

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Daniel C.T. Cardoso , Barbara S. Togashi Department of Civil and Environmental Engineering, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marques de Sao Vicente, 225, Rio de Janeiro 22451900, RJ, Brazil

A R T I C L E I N F O

A B S T R A C T

Keywords: Angle columns Flexural-torsional buckling Generalized Beam Theory Post-buckling Pultruded glass-fiber reinforced polymer

In this paper, the findings from an experimental investigation on the flexural-torsional buckling behavior of pultruded glass-fiber reinforced polymer (GFRP) angle columns are reported and discussed. The program included the study of two sizes of equal-leg angles made with different resins (polyester and vinylester). Prior to testing, a detailed material characterization was carried out and signature curves (critical load x length) were obtained using a generalized beam theory (GBT) software for predicting critical loads. Lengths were selected in order to ensure ‘pure’ flexural-torsional buckling, in a range of slenderness not studied in previous works. Twenty-two members with fixed ends and clamped walls were tested in concentric compression and had their motions measured during loading. Load-deflection curves are presented and the influences of post-buckling reserve of strength, damage and differential rotation of legs are discussed. Experimental critical loads obtained using Koiter's method are reported and shown to be in good agreement with GBT predictions. Finally, a design recommendation through the use of a Winter-type equation accounting for the plate-like behavior is made.

1. Introduction The advantages of pultruded glass-fiber reinforced polymers (GFRP) have made the material attractive in applications where low self-weight and corrosion resistances are required. Due to the combination of its relatively high strength-to-weight ratio and low modulus of elasticity, engineers have preferred to use GFRP in trussed structural systems, where members are mainly subject to axial forces. In these systems, angle sections are sometimes preferred over other cross-sections because of its ease of connection. However, despite the simple geometry, angles may exhibit quite complex behavior when subject to concentric compression forces. Due to their reduced torsional stiffness, buckling modes mainly characterized by torsion may occur and couple with flexure during buckling, affecting significantly the member capacity. The high imperfection sensitivity, the strong influence of end-conditions and the plate-like behavior with significant elastic post-buckling have also been studied recently in works dedicated to steel members (e.g. [1,2]). In GFRP angles, other issues make the problem even more complicated, such as the orthotropic nature of material, inhomogeneity of fibers distribution throughout cross section, influence of in- and out-ofplane shear deformations due to low shear properties and influence of damage. However, whereas studies on metal angles have been carried

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at least over a century, only a few works intended to understand the behavior of GFRP angles can be found in literature in the last two decades. This difference is reflected in the provisions of codes and standards under development: while the recently released NorthAmerican Specification for Cold-Formed Steel Members [3] provides Direct Strength Method (DSM) equations including all phenomena highlighted in the previous paragraph for steel, as well as residual stresses and plasticity [4], the codes dedicated to GFRP released or under development [5,6,7] either provide an old-fashioned approach in which critical loads are assumed as limit loads or recommend the strength to be determined through numerical or experimental analyses. In fact, most of the advances in codes dedicated to GFRP have been based on findings for box- and I-sections, briefly discussed throughout this paragraph. Many authors have investigated failure modes and capacity of pultruded FRP columns with different slenderness, intending to obtain/validate design strength equations (e.g. [8,9,10,11]). The influence of the interaction between material damage, local buckling and global buckling in the behavior of compression members has been highlighted in several works and it is clear that capacity is highly influenced by material strength and critical loads. Whereas there exists a consensus in literature on the most suitable approach for global buckling, some gaps remain on the understanding of local buckling behavior of such members, which is strongly affected by boundary conditions of

Corresponding author. E-mail address: [email protected] (D.C.T. Cardoso).

https://doi.org/10.1016/j.tws.2018.01.031 Received 17 November 2017; Received in revised form 17 January 2018; Accepted 24 January 2018 Available online 22 February 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.

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section sizes studied. For all tests, end rotations about both minor and major axes were released whereas rotation of walls was prevented by steel bars bolted to the end plates in L-shaped configuration. Failure by flexural-torsional and flexural buckling were respectively reported for shorter and longer members; very short members failed by crushing. According to author, load-axial shortening curves are typically characterized by an initial linear response up to 80–95% of the critical load, followed by an increase in shortening with constant load, i.e. flat postbuckling response. Evernden and Smith [27] conducted a series of 17 tests in 76 × 6.4 and 102 × 9.6 mm pultruded GFRP equal-leg angles having lengths ranging from 940 to 1740 mm. The columns were subject to concentric and eccentric compression (load applied at one leg) and the authors reported failure modes by flexural-torsional buckling, flexural buckling or interaction between both. End conditions adopted were similar to those reported by Zureick and Steffen [22]. Coupled buckling described by the authors can be clearly observed from the following characteristic marks: i) load-lateral deflection curves with descending branch after peak load; and ii) Southwell plots exhibiting two distinct slopes. No post-buckling was observed within the range of slenderness studied. Comparison was made to classical equations from Theory of Stability and considering lower and upper acceptable values of material properties. It was shown that the ultimate loads for the smaller angles (76 × 6.4) mm fell quite below the prediction, which the authors attributed to variations and properties and geometry, as well as to unavoidable eccentricities that exert more influence over members with lower radius of gyration. Finally, the authors pointed out that members with greater slenderness were less influenced by eccentricity. Recently, works dedicated to the study of GFRP transmission line towers have also investigated the behavior of angle members. Selvaraj et al. [28] studied the use of FRP pultruded members in towers having equal-leg angles for bracing. The authors carried out compression tests in short and long angles, as well as a full-scale test in a braced frame. Results were compared to finite element method (FEM) models and differences up to 5.5% were reported. To assess the behavior of angles struts as bracing members of transmission towers, compression tests were carried out by Godat et al. [29] in six angle-sections 76 × 6.4 mm in a braced frame configuration. In the setup adopted, a typical bolted connection using a steel gusset plate attached to one of the legs was used, therefore corresponding to an eccentric loading condition. According to the authors, all columns failed by flexural buckling and, after comparing with theoretical expressions, it was concluded that failure loads were similar to those obtained using Euler critical load with buckling coefficient set to 0.75. Rao et al. [30] carried out full-scale tests on three 24 m tall triangular lattice communication towers made of GFRP angle members. The behavior of members with internal angles (between legs) of 60° and 90° subject to compression was studied. Tests on isolated 60° and 90° members were carried out and, based on reported load-deflection curves for those experiencing flexural-torsional buckling, moderate post-buckling reserve of strength can be observed. Finally, failure modes observed in isolated members were consistent to those observed in the full-scale tower tests. It is important to mention that significant experimental and numerical studies have been carried out in the last two decades for steel angles, intending to gather more information about the performance of such members and to propose reliable design approaches. To allow better interpretation of the results so far obtained with pultruded GFRP angles, some recent findings are reported in the following paragraphs. Dinis et al. [31] used generalized beam theory (GBT) to investigate local and global buckling behavior of thin-walled steel angle, cruciform and ‘tee’ sections, intending to provide fresh insight on the mechanics of such members. With respect to angle columns, the authors observed that buckling modes can be described by combination of local, flexural and torsional modes: “for very short columns, buckling takes place in mixed local-torsional modes. For very short to short lengths, columns buckle in pure torsional modes. In intermediate length columns, buckling occurs in mixed

constituent walls at loaded edges, elastic properties and cross-section geometry, as well as by damage accumulation and web-to-flange rotational stiffness (e.g. [12,13,14,15,16]). Some of new findings on the local buckling, such as the use of recently developed closed-form equations for critical loads of columns and beams [17,18], are likely to be incorporated in the next update of the Italian code [19]. Regarding the linear coupling of flexure and torsion in columns, it may occur for Isections with locally unsymmetric layup due to coupling stiffness [20] or to members with shear center not coinciding with the centroid [21,22]. As will be discussed in the next section, in most of tests previously conducted by other authors, angle members governed by flexural-torsional (FT) buckling exhibited some degree of interaction with flexure about minor axis, experiencing failure with nearly flat post-buckling response or a descending branch after peak load. However, if no coupling occurs with FT buckling, angles may exhibit plate-like behavior and, therefore, pronounced post-buckling reserve of strength may be observed, as reported in literature for GFRP plates and members experiencing local buckling [23,24]. In the present work, a ‘pure’ FT condition is ensured by selecting adequately lengths and end conditions, in a range of slenderness not studied previously. This allowed a detailed study on the particularities of such failure mode, including the significant post-buckling behavior, influence of damage and differential rotation of legs. In a first step, Generalized Beam Theory (GBT) is used to study the buckling modes and critical loads and a map of slenderness is used to predict failure modes and help selecting testing lengths. In a second stage, the results of an experimental program intended to gather information about the flexural-torsional behavior of angles subject to concentric compression are reported and discussed. The immense postbuckling reserve of strength not previously reported in experimental works addressing pultruded GFRP angle columns is highlighted and quantified and the influence of damage in the behavior is clearly identified. Finally, comments towards a consistent design approach for angle members are made and the use of a Winter-type equation is proposed. Throughout this paper, equal-leg angle sections are designated as b x t, where b and t refer to nominal leg width and wall thickness, respectively. 2. Previous studies on angles McClure and Mohammadi [25] conducted the first work dedicated to pultruded GFRP angles subject to compression. The authors carried out stub tests on 152 mm long and 51 × 6.4 mm equal-leg angles as part of a study on the creep buckling behavior of such members. Thick wood plates with L-shaped slots were used at both ends to ensure uniform loading and to restrain end rotations. Material characterization tests have shown that one of the legs was consistently stronger and stiffer than the other, probably resulting from manufacturing variability. Load-deflection curves were not reported, but failure modes of the stub columns were described as ‘buckling of the weakest leg’. Zureick and Steffen [22] tested 25 angle struts in concentric compression. Ranges of leg sizes, thicknesses and lengths were considered in the study, as well as two different types of resins: polyester and vinylester. For the end conditions, angles were simply seated on steel plates with rotations released about the weak axis and restrained about the strong axis. For the longer specimens, a monotonically increase in deflections with load was reported, with failure governed by flexural buckling (global). For the shorter specimens tested, the load-lateral deflection curve exhibited a descending branch after reaching a peak load. Although this kind of curve is characteristic of coupled buckling [1,2], failure mode was described as flexural-torsional buckling. No post-buckling behavior was reported by the authors. A large experimental program intended to investigate the buckling behavior of equal-leg angle members subject to compression was also conducted by Seangatith [26]. In all, 32 columns were tested, considering a range of slenderness ratios for each of the three different 270

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(major) flexural–torsional modes. Finally, the longer columns buckle in pure (minor) flexural modes.” According to the authors, torsion mode plays a key role in the behavior of very short to intermediate members. Such dependence makes critical loads of these members to be nearly the same, regardless the length, resulting in a ‘plateau’ in the signature curve (critical load × member length). Dinis et al. [1] presented a discussion on the post-buckling behavior and strength of short-to-intermediate angle columns having pinned and fixed end conditions. Numerical analyses carried out for steel equal-leg angles 70 × 1.2 mm with different lengths but similar critical loads showed that the elastic post-buckling response may present distinct characteristics. While shorter members exhibited clearly stable postbuckling path with significant reserve of strength, relatively longer members within the ‘plateau’ exhibited considerable shear center displacements at mid-span (flexural displacements) and stable post-buckling path characterized by negligible post-critical strength and occurrence of limit points. This difference in the behavior may be explained by closeness between flexural-torsional and flexural critical loads experienced by longer columns, indicating an interaction between the two buckling modes. This coupling effect was also investigated by Mesacasa et al. [2], who conducted numerical analyses on angle members with different end-conditions but having equivalent flexural-torsional and flexural critical loads. The authors concluded that failure modes were characterized by the presence of elastic peak loads, highly influenced by the shape and amplitude of imperfections. A significant post-buckling reserve of strength in locally slender plain angles with fixed ends and buckling governed by the flexuraltorsional mode was observed by Shifferaw and Schafer [32]. The authors highlighted the “unique similarity of local plate buckling and global torsional buckling”, creating “unusual post-buckling performance and extreme sensitivity to boundary conditions”. The conclusions drawn in previous works on steel angles have led to the development of design approaches based on Direct Strength Method (DSM) [33,34]. The core idea of the approach is to adopt an empirical Winter-type curve (originally developed for plates) to obtain the flexural-torsional strength and, then, use this result in a global buckling strength equation to account for the interaction between the buckling modes.

minor axis, Pcr,Fz, neglecting influence of shear deformation and transverse bending of walls are available elsewhere in literature (e.g. [22]) and will not be herein reproduced. In the present work, GBTul – a Generalized Beam Theory (GBT)based software developed at IST-Lisbon [35] – is used to study the changes in buckling modes and critical loads with length. GBT, originally developed by Schardt [36], is based on the decomposition of buckling modes into a linear combination of cross-section deformation modes. A system of equilibrium equations (one per deformation mode) can, then, be derived and the critical stresses are obtained from eigenvalue problem. More details on GBT equations and developments are presented in Camotim and Basaglia [37]. It is important to note that original GBT formulation for isotropic material has already been successfully modified to study buckling behavior of arbitrary sections made of orthotropic materials [38]. In Fig. 2, typical deformation modes used for GBT analysis for an angle section with their respective numbers are reproduced. Since angles exhibit a mix of plate and column-like behavior, boundary conditions for the constituent walls and for the overall column must be properly defined to obtain the signature curve. In this work, fixed ends and clamped walls (F+C) are considered, preventing overall rotation of member ends and rotation of legs at the loaded edges. Regarding the elastic properties to be used, it is important to note that flexural buckling is associated with overall member bending and, for thin-walled members, resulting stresses are approximately uniform through wall thickness (membrane stresses). On the other hand, buckling modes governed by torsion also involve bending of walls, producing stress gradients through thickness. Therefore, longitudinal modulus in tension or compression (EL,t or EL,c) must be used for flexural buckling whereas longitudinal plate bending modulus (EL,f) must be adopted for plate-like modes. It has been pointed out that these moduli may differ greatly depending on the fiber architecture [9,39,40]. In Fig. 3a, a signature curve (critical force versus member length) for the 51 × 4.8 mm angle section studied in the present work (material properties defined in the next section) with fixed ends and clamped walls at loaded edges is presented. Critical forces obtained using classical equations given in Zureick and Steffen [22] are also plotted in the graph and the ‘plateau’ described by Dinis et al. [31] is indicated. It can be noted that differences between classical equations less than 5% with respect to GBT for a wide range of column lengths. For the angle size and material properties studied, differences increased for members shorter than approximately 200 mm. This can be explained by means of the participation of deformation modes with length, shown in Fig. 3b, from which the strong influence of shear deformation (mode 12) can be observed for very short members. It can also be seen that, for short to intermediate lengths (200–2750 mm), flexural-torsional buckling governs the behavior and participations of torsion (mode 4) and flexure about major axis (mode 2) respectively reduces and increases with length. For members longer than 2750 mm, buckling is governed by flexure about minor axis (mode 3). Influence of transverse wall bending seems to be negligible within the range of lengths studied. Similar

3. Buckling behavior and procedure for length selection As mentioned previously, the behavior of relatively short angles is mainly characterized by linear coupling of flexure about major axis and torsion about shear center, the so-called flexural-torsional mode (FT), which is likely to occur in thin-walled open cross-sections with low torsional rigidity and shear center S not coinciding with the centroid O, e.g. angles and channels. As length increases, flexural buckling about the weakest axis governs the behavior (Fz). For very short members, transverse bending of walls may also be observed. The aforementioned buckling modes are illustrated in Fig. 1, along with nomenclature and reference axes adopted in this work. Expressions to compute critical loads for flexural-torsional buckling, Pcr,FT, and flexural buckling about

Fig. 1. Geometry parameters and reference axes; and buckling modes observed in angle members subject to compression.

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Fig. 2. Main GBTul cross-section deformation modes (degrees of freedom) linearly combined to obtain buckling modes for angles.

4. Experimental program

behavior can be observed for the 102 × 6.4 mm angles. An interesting conclusion can be observed if the same study is carried out with steel angles having similar geometries. Due to the isotropic nature of steel and its much greater shear modulus, flexural buckling about minor axis occurs for shorter lengths than the ones reported for GFRP, i.e. a shorter plateau can be observed. In real angle struts, i.e. those containing geometric imperfections and unavoidable loading eccentricities, interaction between crushing (material strength), flexural-torsional and flexural buckling may occur, similarly to what has been observed in literature for GFRP I-section and square tube columns (e.g. [8,9]). The interaction is stronger when individual failure loads (material strength and critical stresses) are closer to each other and, to quantify the ‘distance’ between them, flexuraltorsional and flexural buckling relative slendernesses, λFT and λFz, are introduced as follows:

λFT =

PL, c Pcr , FT

(1)

λFz =

min(PL, c , Pcr , FT ) Pcr , Fz

(2)

The experimental program intended to investigate the flexural-torsional buckling behavior of pultruded GFRP angle members was divided into two stages: material characterization and member compression tests. Two groups of equal-leg angle sizes with nominal dimensions 51 × 4.8 mm (2 × 3/16″) and 102 × 6.4 mm (4 × 1/4″), made respectively with polyester and vinylester resins reinforced with E-glass fibers, were studied. Throughout the text, designation L2 and L4 will be used in reference to L 51 × 4.8 mm and L 102 × 6.4 mm, respectively. 4.1. Material characterization Material characterization tests were carried out to obtain the relevant properties necessary for a good correlation between theory and experiments. Before testing, all coupons had their dimensions measured with digital caliper. A brief description of the performed tests is made in the following paragraphs and a summary of the results obtained is presented in Table 2. Although materials are made of different resins, little differences in mechanical properties obtained seem to be more dependent on the fiber content and architecture.

in which PL,c = A FL,c, where A is the cross-section area and FL,c is the material compressive strength. Adapting the proposal made by Cardoso et al. [9], actual failure modes can be predicted according to the values of λFT and λFz, as shown in Table 1. In the same table, a suggestion for column classification is proposed, but limits are only indicative as interaction between individual failure modes occurs continuously. Therefore, to study flexural-buckling behavior with lesser influence of crushing and flexure about minor axis, lengths for each cross-section must be selected for in order to make λFT > 1.3 and λFz < 0.7. In this work, despite the limits presented, lengths are selected to obtain make λFz ~ 0.1–0.2.

4.1.1. Fiber burnout Three 20 × 20 mm samples extracted from each angle size were exposed to a temperature of 600 °C in a muffle furnace for three hours, according to the recommendations of EN ISO 1172:1998. From the burnout test, it could be seen that L2 and L4 are constituted by two and three roving layers, respectively, intercalated by CSM layers. 4.1.2. Longitudinal Tensile Tests Five dog-bone shaped samples with 250 mm in length with long

Fig. 3. Signature curve (a) and participation of deformation modes (b) for angle 51 × 4.8 mm studied considering fixed ends and clamped walls at loaded edges.

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Table 1 Expected failure modes based on combination of flexural-torsional and flexural buckling relative slendernesses.

control rate of 2 mm/min up to a total deflection of 10 mm. A significant coefficient of variation was observed for L2, which may be attributed to deviations on the position of roving layers due to manufacturing process and the stronger influence this imperfection has in thin coupons [9,40].

Table 2 Summary of properties for angle sections studied (average ± standard deviation [coefficient of variation - %]). Property Fiber volume content of roving layers (%) Fiber volume content of CSM layers (%) Number of roving layers Longitudinal tensile modulus (GPa) Longitudinal tensile strength (MPa) Longitudinal compressive modulus (GPa) Longitudinal compressive strength (MPa) Longitudinal bending modulus (GPa) Transverse bending modulus (GPa) Shear modulus (GPa) Leg size (mm) Thickness (mm)

L2

L4

Vf,rov

28.9 ± 0.3 [1.0]

33.9 ± 2.1 [6.2]

Vf,csm

0.05–0.10

0.05–0.10

N EL,t

2 24.6 ± 0.9 [3.7]

3 27.8 ± 2.5 [9.0]

FL,t

312 ± 19 [6.2]

360 ± 26 [7.2]

EL,c

28.7 ± 3.9 [13.6]

30.4 ± 4.7 [15.5]

FL,c

290 ± 54 [18.6]

286 ± 84 [29.4]

EL,f

13.2 ± 2.0 [15.2]

20.1 ± 1.3 [6.5]

ET,f

6.88 ± 0.71 [10.3]

8.62 ± 0.80 [9.3]

GLT b t

1.73 ± 0.10 [5.8] 50.7 ± 0.8 [1.6] 4.68 ± 0.07 [1.5]

2.47 ± 0.40 [16.2] 101.6 ± 0.39 [0.4] 6.37 ± 0.05 [0.8]

4.1.5. Transverse bending tests To determine the transverse plate bending modulus, ET,f, a nonstandard test was performed on three L-shaped coupons 20 mm wide cut out from each angle group. Each coupon was fixed at one end and transversally loaded at the other end, therefore producing bending in the transverse direction. For each load applied, curvature at the constant moment region was measured using back-to-back strain gages. Specimens were loaded to produce strains in the range of 0.0005–0.0025 mm/mm. The modulus was obtained after correlation between moment and curvature. It is important to note that this configuration also produces axial force in the instrumented section, which was found to result in negligible strains if compared to those produced by bending. 4.1.6. Torsion tests Three longitudinally-oriented specimens 25 mm wide × 250 mm long were extracted from each group and tested in torsion using a nonstandard procedure. At one end, specimen was fixed with restrained twist; at the other end, an adapter was used to connect the specimen to a torsion head (free to twist). Balanced loads were applied at the torsion head in order to produce pure torsion in the coupon. Specimens were loaded up to a torque of 6000 N mm. 45°-oriented strain gages were positioned at the center of each specimen and 50 mm distant from the specimen's end near the torsion head. In one specimen of each group, a rectangular strain gage rosette (0–45–90°) was added to confirm principal directions at approximately ± 45°. Assuming linear elastic behavior, shear modulus, GLT, for each group was obtained using the expressions suggested by Turvey [42].

edges parallel to pultrusion direction and dimensions as recommended by ASTM D638 were extracted from each angle size group. A clip gage used at mid-length to measure strains and all tests were conducted at a rate of 0.5 mm/min until failure in a universal testing machine, as shown in Fig. 4a. Reinforcing tabs were not adopted and failure occurred close to ends in some cases. Since tensile strength is not a relevant property for the present study, tests were not repeated. 4.1.3. Longitudinal compression tests Five prismatic samples 155 mm long and 12 mm wide with long edges parallel to pultrusion direction were extracted from each angle size and tested in compression according to the combined loading compression (CLC) method described by ASTM D6641, as presented in Fig. 4b. To measure the strains, one strain gage was applied at one of the faces of each coupon within the adopted 15 mm gage length. One specimen of each group was instrumented with back-to-back strain gages. The negligible difference in strains observed between the gages confirmed that tests were not affected by buckling or possible loading eccentricity. The larger coefficients of variation (COV) obtained may be explained by the adoption of narrow specimens in combination with the inhomogeneous distribution of fibers throughout the cross section [41], as well as by the presence of voids.

4.2. Compression tests Twenty-two angle columns with lengths ranging from 150 to 500 mm and 190–1000 mm were respectively extracted from L2 and L4 profiles supplied by the same manufacturer. Critical loads were obtained using GBTul considering fixed ends and clamped walls, with material properties given in Table 2. Lengths were determined in order to make λFT > 1.3 and λFz ~ 0.1–0.2, as described previously. It is important to mention that effective lengths (total length minus 38 mm) were used for the theoretical predictions in order to account for test fixtures described in the next paragraph. To identify specimens, nomenclature adopted includes angle group (L2 or L4), resin used (PE = polyester or VE = vinylester), sample length in mm and a number to differentiate specimens with the same characteristics (1 or 2), as presented in Table 3. In Fig. 5, a plot of λFT versus λFz including the specimens tested in current study as well as by other authors is

4.1.4. Longitudinal bending tests Three-point bending tests were carried out to obtain the longitudinal plate bending modulus, EL,f, following recommendations of ASTM D790. 25 mm wide × 250 mm long specimens were cut out from each angle group and tested over a 200 mm span under a displacement 273

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Fig. 4. Material characterization tests: a) longitudinal tension; b) longitudinal compression; c) longitudinal bending; d) transverse bending; e) torsion.

Table 2 and, in general, internal angles were found to be between 89° and 90° and observed out-of-flatness amplitude was less than 1.0 mm. Specimens had their ends machined flat and parallel and were carefully positioned in a universal testing machine with their centroid coinciding with the actuator axis. Base twisting was prevented by using a rigid steel beam connecting the base plates to the machine's frame. To reproduce the condition of clamped walls, 19 mm wide steel square bars were attached to the base plates forming an L-shaped slot. To eliminate

presented, showing that the experimental program covers a combination of slenderness not investigated previously. To obtain slendernesses for previous works, classical equations – without simplifications – reproduced by Zureick and Steffen [22] were used along with reported properties. Prior to testing, leg sizes and thicknesses, internal angles and out-offlatness were measured using a digital caliper, a protractor and a gap gage. Average dimensions and standard deviations are reported in

Table 3 Dimensions and summary of the test results for the stub columns. Specimen

L2. L2. L2. L2. L2. L2. L2. L2. L4. L4. L4. L4. L4. L4. L4. L4. L4. L4. L4. L4. L4. L4.

PE.150.1 PE.150.2 PE.300.1 PE.300.2 PE.400.1 PE.400.2 PE.500.1 PE.500.2 VE.190.1 VE.200.1 VE.300.1 VE.300.2 VE.400.1 VE.400.2 VE.500.1 VE.500.2 VE.600.1 VE.600.2 VE.800.1 VE.800.2 VE.1000.1 VE.1000.2

Theory

Experimental

Pcr,FT (kN)

λFT

Pcr,exp (kN)

Pu (kN)

Pcr,exp / Pcr,FT

Pu / Pcr,exp

χFT

38.3 38.3 14.4 14.4 11.3 11.3 9.91 9.91 127.6 117.5 59 59 38.3 38.3 28.8 28.8 23.8 23.8 18.8 18.8 16.6 16.6

1.87 1.87 3.05 3.05 3.44 3.44 3.67 3.67 1.67 1.74 2.46 2.46 3.05 3.05 3.52 3.52 3.88 3.88 4.36 4.36 4.64 4.64

33.2 30.1 13.6 16.0 11.0 11.6 11.3 11.6 70.5 36.2 59.2 58.5 32.0 34.9 22.9 30.2 9.5 22.9 18.1 19.0 19.6 17.0

37.6 35.4 24.0 21.7 18.0 19.3 20.5 19.6 109.2 102.0 89.4 79.2 83.9 84.4 72.0 79.2 69.6 76.3 65.5 57.8 58.9 59.9

0.867 0.787 0.945 1.114 0.969 1.029 1.137 1.170 0.553 0.308 1.003 0.992 0.837 0.912 0.795 1.048 0.400 0.963 0.965 1.013 1.182 1.023

1.133 1.176 1.765 1.351 1.639 1.656 1.820 1.689 1.547 2.822 1.512 1.353 2.618 2.417 3.144 2.623 7.319 3.329 3.611 3.038 3.003 3.527

0.281 0.265 0.180 0.162 0.134 0.144 0.154 0.147 0.306 0.285 0.250 0.222 0.235 0.236 0.201 0.222 0.195 0.213 0.183 0.162 0.165 0.168

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specimens exhibited significant torsion motions during loading with little flexure about major axis, followed by longitudinal tearing failure at the leg junction. As expected, longer members (Fig. 7c and f) exhibited larger deflections than the shorter ones (Fig. 7a and d) and audible cracking sounds could be heard during post-buckling range, indicating damage growth or formation. Intermediate-length members (Fig. 7b and e) constituted intermediate cases. Additionally, negligible rotation could be observed at the loaded walls, showing that the setup adopted intending to reproduce clamped end condition worked well. In Fig. 8a, a representative graph for L2. PE.500–2 shows the lateral deflection growth, δ, with load, P, for all transducers. It can be clearly seen that transducers 1 and 2, positioned at tips of the legs, captured large displacements, whereas transducers 3 and 4, positioned orthogonally at the corner, recorded very little displacements. This indicates, therefore, the predominance of torsion about the shear center in the buckling behavior. The curves also show that behavior can be divided into three stages: I) pre-buckling, in which negligible deformation can be observed with increasing load; II) post-buckling, characterized by significant deflections accompanying increases in load; and III) severe damage growth stage, characterized by progressive reduction in the P-δ curve slope after stage II. A notable post-buckling reserve of strength can be observed, a trademark of the plate-like behavior. The plot of the axial shortening, δv, with load for the same specimen also provides insightful information about the member behavior. The graph shows that the initial member stiffness (=P/δv) remains approximately constant during stage I and starts degradating after the critical load is reached (stage II). Finally, degradation process accentuates with damage (stage III). Stages I and II are equivalent to those observed in steel plates. However, stage III in steel would correspond to the condition in which yielding progresses and the ability of redistributing stress is reduced along with a progressive degradation of stiffness. For the shortest members, significant differential rotation of legs could be clearly observed for both L2 and L4 specimens, as can be seen in Fig. 9. This phenomenon, previously observed by McClure and Mohammadi [25], may be associated to the variability in the material properties and differences in geometric imperfections for each leg, as well as to the semi-rigid nature of the junction between adjacent walls in GFRP members [43]. Moreover, stress concentration near the

Fig. 5. Slenderness ‘map’ showing combinations of slenderness studied by other authors and in present work.

eventual gaps between the walls and the bars as well as to allow for uniform loading distribution throughout cross-sectional area, a small amount of plastic adhesive paste was used into the L-slots immediately before column positioning. Finally, after adhesive curing (20–30 min), the columns were tested in compression under displacement control at a rate of 0.6 mm/min up to failure. To capture the motions at midlength during testing, draw wire transducers (DWT) were attached perpendicular to each leg tip whereas two potentiometric displacement transducers were orthogonally positioned close to the angle corner. Aluminum tabs were glued to specimens to allow attachment of measurement devices. Fig. 6 illustrates the apparatus used for the compression tests. 4.3. Discussion of results In Fig. 7, pictures taken during test for columns L2 and L4 having different lengths show buckled shapes and typical failure modes. All

Fig. 6. Test setup adopted: a) overview of test; b) detail of displacement transducers; c) detail of end fixture.

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Fig. 7. Typical flexural-torsional buckling and failure modes observed for different lengths: a) L2.PE.150–1; b) L2.PE.300–1; c) L2.PE.500–2; d) L4.VE.200–1; e) L4.VE.300–2; f) L4.VE.1000–2.

25 1

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III

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25

20

30

0

40

0.0

1.0

2.0

3.0

4.0

δv (mm)

δ (mm)

(a)

(b)

Fig. 8. Typical behavior observed during loading: a) applied load versus lateral deflections at mid-length for each transducer; b) column axial shortening with load.

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Fig. 9. Nonsymmetrical lateral deflection growth with load for the shortest members of each group (transducers 1 and 2).

Fig. 10. Comparison of typical lateral deflection growth with load for different column lengths (in mm) (transducers 1 and 2).

nonlinear least squares regression was adopted, considering all data within the interval of δ = 0.1 mm and end of stage II, therefore avoiding low-load nonlinearities and influence of damage. Experimental critical and ultimate loads as well as theoretical predictions for each specimen are reported in Table 3. To allow a better comparison between critical loads obtained experimentally with those predicted using GBTul, experimental results are plotted along with the signature curve for each group studied in Fig. 11. It can be seen that, except for specimens L2. PE.150–2, L4. VE.190–1, L4. VE.200–1 and L4. VE.600–1, good agreement was achieved with an average difference of 0.998 ± 0.105 (removing the aforementioned specimens). The differences for the shortest members are related to differential buckling motions experienced by each leg. As for the specimen L4. VE-600-1, it corresponds to a preliminary test carried out without filling the gaps between the steel fixture and the angle member with adhesive paste, therefore experiencing wall rotations at the loaded edges and lower critical loads. The result of this specimen was reported for comparison purposes and to show the importance of adopting wellcontrolled end-conditions. Comparing experimental ultimate and critical loads (Table 3), it can be seen that ultimate loads were always greater, with differences ranging from 13.3% to 361%. It can also be observed that difference increases with length, i.e. post-buckling reserve of strength is more

junction contributes to premature tearing of legs, affecting severely the behavior. Comparing the load-deflection curves for columns with different lengths, shown in Fig. 10, it can be seen that shorter columns exhibited greater critical stresses, with a post-buckling path characterized by increased curvature and smaller deflections prior to failure. On the other hand, for the longer members tested, very large deflections and relatively flatter post-buckling paths can be observed. It can also be noted that the increase in strength provided by post-buckling reserve is relatively more significant for longer columns. Many different techniques have been proposed in literature to determine the experimental critical load, Pcr,exp, in members exhibiting load-deflection curves with pronounced post-buckling path, where the usual Southwell method [44] does not apply [45,12]. In the present work, the critical stresses are determined using Koiter's approach [46], in which the load-deflection behavior is described as follows:

P = Pcr,exp

1 + c1 δ + c2 δ 2 1 + δ0 / δ

(3)

in which c1, c2 are constants to be determined and δ0 is the initial outof-flatness. To determine the four unknown parameters (c1, c2, δ0 and Pcr,exp), Debski et al. [47] recommended selecting four points from the load-deflection curve, e.g. the one at the pre-buckling range, one near the bifurcation point and two at the post-buckling path. In this work, 277

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Fig. 11. Signature curves obtained with GBTul and experimental critical loads for both groups of angles studied.

pronounced reserve experienced for L4 members, as a result of the larger b/t ratio discussed in the previous paragraph. As a design strategy, an empirical Winter-type strength equation – usually adopted in cold-formed steel design guides – may be used to account for both phenomena. In Fig. 12, the dashed line represents a strength curve fitting experimental results using an Winter-type equation described as follows:

significant for longer members. This may be explained by the fact that shorter members usually have critical stresses closer to the material compressive strength, therefore experiencing lower deflections and lesser stress redistribution prior to failure. In other words, failure load in short members is reached with less pronounced migration of compressive stress towards the junction and, thus, relatively lower contribution of post-buckling reserve of strength. It can also be seen that, for similar slenderness of L2 and L4, angles L4 exhibited a significantly greater post-buckling reserve, which may be attributed to the larger b/t relationship, increasing the capacity of stress redistribution throughout the leg. From a design perspective, it is important to understand how the relative strength χFT = Pu/Pcr,FT decreases with slenderness. A plot of χFT x λFT is presented in Fig. 12, comparing experimental results with those expected for a perfect column with no post-buckling (solid line). For the case of a perfect column, failure by crushing is expected for λFT ≤ 1.0 and by flexural-torsional buckling for λFT > 1.0. For the range of slenderness studied, it can be noted that values of χFT for members with λFT < 2.0 fell below the perfect member curve, as a consequence of previously described differential buckling between leg and interaction between crushing and flexural-torsional buckling. On the other hand, members with λFT > 2.0 exhibited strengths significantly greater than those expected for perfect members, as a result of the post-buckling reserve of strength associate to the plate-like behavior, with a more

χFT =

A B + 2 λFT λFT

(4)

in which A and B are constants. In the curve plotted in Fig. 12, A = 0.578 and B = −0.072 were obtained from nonlinear least squares regression. To obtain reliable parameters to be used in design, more tests are necessary for calibration, considering the appropriate confidence limits. In the same figure, the dotted line represents the usual strength curve recommended in codes for steel structures (e.g. [3]) with A = 1 and B = −0.22, indicating a much more pronounced postbuckling reserve of strength. 5. Conclusions This work presented the results obtained from an experimental program on angle members subject to concentric compression governed flexural-torsional buckling mode. The following findings can be highlighted: 1. GBTul was successfully used for predicting critical loads and it is shown that the classical equations also provide good estimates for all but very short lengths, for which influence of shear deformation is not negligible. The slenderness map has also proven to be a useful tool for predicting the actual failure mode and selecting appropriate testing lengths. 2. In this study, an attempt to clamp the walls was made using a steel fixture in order to control adequately the end conditions. For the sole case where the adhesive paste was not used to fill in the gaps between the fixture and the profile (L4. VE.600–1), rotation of walls at the loaded edges were observed, affecting significantly the results. 3. With respect to typical buckling behavior, it was mainly characterized by torsion about the shear center with little corner motions and three distinct stages could be clearly identified: pre-buckling (I); post-buckling (II); and severe damage growth (III). Longer members experienced larger deflections with flatter post-buckling path whereas increased post-buckling path curvature and smaller

Fig. 12. Relative strength versus flexural-torsional slenderness: experimental results and curves for real and perfect members.

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4.

5.

6.

7.

deflections were observed for shorter members. However, postbuckling reserve provided a greater contribution to the final strength for the longer members. This important post-buckling performance confirms the similarity with the plate-like behavior observed recently in works addressing cold-formed thin-walled steel angles (e.g. [32]). It is also important to mention that such significant contribution of the post-buckling reserve of strength to the final strength (up to 3.6 times the buckling load) has not been previously reported in experimental works addressing behavior of pultruded GFRP columns, in accordance with the contribution reported in literature for GFRP plates and thin-walled non-pultruded members (e.g. [23,24]). For the very short specimens, differential rotation of legs was observed, leading to reduction in critical load, which can be explained by variability in properties and geometry of legs, as well as to the semi-rigid nature of the junction between legs. Moreover, stress concentration near the junction contributes to a premature tearing of legs, affecting severely the behavior. This phenomenon must be investigated in future works, as it affects significantly the behavior of members with lower slenderness. Although different resins have been used for L2 and L4, similar mechanical properties were obtained and differences observed seem to be associated to fiber content and architecture. Since all comparisons between theory and experiments were made in terms of elastic properties, a relationship between type of resin used and critical loads could not be established. However, each matrix may behave differently regarding damage accumulation and this effect may be investigated in future studies. The Koiter method adopted for determining the experimental critical load allowed considering the stiffening provided by the stress redistribution associated to plate-like behavior and was proven to be easy-to-use and more adequate than the Southwell technique, which is applicable for the sole case of hyperbolic load-deflection curves. The adoption of a Winter-type curve consists in an interesting strategy for design approach, which accounts for interaction between crushing and flexural-torsional buckling, damage, postbuckling reserve of strength and differential buckling of legs. Comparing to the original Winter equation for steel plates, it is clear that post-buckling is much more pronounced in steel. To obtain reliable design equations, more tests are necessary, considering different b/t relationships and end conditions, as they may affect the post-buckling reserve. The relative flexural-torsional strength obtained from the Winter equation, χFT, may, then, be used as input in a global buckling design equation, similarly to the two-step procedure used in the direct strength method (DSM) adopted for cold formed steel members. However, to accomplish this task, more experimental studies are necessary to investigate buckling interaction.

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