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Effect of geometric discontinuities on failure of pultruded GFRP columns in axial compression
Composite Structures 136 (2016) 171–181
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Effect of geometric discontinuities on failure of pultruded GFRP columns in axial compression Yail J. Kim ⇑, Kenny Zongxi Qian Department of Civil Engineering, University of Colorado Denver, Denver, CO, United States
a r t i c l e
i n f o
Article history: Available online 23 October 2015 Keywords: Axial compression Column Composite Glass fiber reinforced polymer Pultruded shape
a b s t r a c t This paper presents the failure characteristics of pultruded glass fiber reinforced polymer (GFRP) columns loaded in concentric compression. Of interest is the effect of geometric discontinuities on the behavior of the columns, including a bolted splice or utility holes. The local- and global-responses of the columns are examined through a hybrid approach consisting of laboratory experiment and finite element modeling: the test encompasses a total of 50 columns and the model has varying slenderness ratios up to 150. Bolted connections at column supports result in stress concentrations accompanied by crack initiation and propagation, thereby reducing the load-bearing capacity. Tear-up or edge-splitting dominates failure modes at the bolted connections. Spliced members experience fiber misalignment and out-of-plane behavior because of displacement incompatibility between the inner and outer GFRP members bolted together. A weak link provided by side holes, whether symmetric or asymmetric configurations, causes a reduction in axial stiffness and diagonal shearing. The results of modeling indicate that buckling capacity of the column significantly decreases beyond a slenderness ratio of 75 and the presence of side holes is not a contributing factor to the stability of the columns. The design expressions proposed are simpler than existing ones, while provide sufficiently accurate capacity prediction for practice. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Glass fiber reinforced polymer (GFRP) composites are broadly used in engineering disciplines with many benefits [1]. Contrary to conventional load-bearing structural materials such as concrete and steel, pultruded GFRP shapes demonstrate linear behavior until failure occurs. Two types of failure modes are observed for GFRP members loaded in axial compression: material crushing and stability, depending upon the extent of slenderness. Material-level failure happens when the applied stress exceeds the compressive strength of the GFRP and accompanies local fiber buckling or resin fracture. Hashem and Yuan [2] studied the failure mechanism of GFRP stubs (L = 305 mm) in axial load. Microbuckling controlled the crushing of the composite, followed by surface delamination. Wong et al. [3] reported the failure of short GFRP channels (L = 30 mm) subjected to compression at elevated temperatures, including fiber kinking or micro-buckling. The buckling of pultruded GFRP columns has been examined by a number of researchers. Zureick and Scott [4] investigated the stability of wide-flange and box sections made of GFRP having a slenderness ⇑ Corresponding author. E-mail addresses: [email protected] (Y.J. Kim), zongxi.qian@ucdenver. edu (K.Z. Qian). http://dx.doi.org/10.1016/j.compstruct.2015.10.016 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
ratio of 36–103. The composite material encompassed a fiber volume fraction of 30%. Geometric imperfection was measured to evaluate the degree of out-of-straightness for each test specimen, while their deviations were within an acceptable range as per an ASTM standard. Failure loads were recorded and compared to critical load equations proposed by others. Design guidelines were discussed as well. Tommaso and Russo [5] tested I- and H-shape pultruded GFRP members monotonically loaded in compression. These columns had slenderness ratios ranging from 9 to 195 and 22 to 63 for the I- and H-shapes, respectively. Test results were compared with a classical Euler buckling load and finite element prediction. The H-shape profiles were more sensitive to the interaction between local and global buckling compared to the I-shape ones. Mottram [6] developed analytical closed-form equations for concentrically loaded GFRP columns, including a review of existing equations, and recommended that more test data and universal column design procedures be required. Nguyen et al. [7] developed a finite element model to predict the global buckling of GFRP columns. An emphasis was given to the influence of boundary conditions and imperfection when associated with a slenderness ratio between 10 and 50. The level of end-warping fixity was reduced with an increase in specimen length. A maximum out-of-straightness of span/200 was recommended for practice.
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Utility holes may be cut in structural members to accommodate electrical conduits or analogous purposes. It is generally suggested that holes be located where a minimum bending moment takes place when a member is loaded. Concerns, however, may arise if the member is subjected to compression because the presence of the holes can affect the load-bearing or buckling response of the member. Although some research programs discussed bolted connections for pultruded GFRP members [8,9], limited effort was made when various geometric configurations (e.g., discontinuity in structural members) are imposed to GFRP shapes loaded in compression. As such, current practice does not appropriately address technical consequences along with this important issue that can alter the failure characteristics of the shapes. As far as buckling is concerned, existing equations are complex and rigorous endeavors may be required for the practitioners to use [10]. Straightforward design expressions are, therefore, required to facilitate the design of pultruded GFRP members in axial compression, including sufficient accuracy in terms of predicting a critical load that controls the load-bearing capacity. This paper discusses the failure characteristics of concentrically loaded GFRP columns having geometric discontinuities, which are defined as physical separation of the continuum columns by the presence of utility holes or splices. A hybrid approach is implemented with laboratory testing and finite element modeling to elucidate the behavior of the columns, which can complement to one another (i.e., a wide variety of failure modes associated with the discontinuities and slenderness effects linked with global buckling). Design equations are proposed based on probability-based simulation and further compared with existing expressions. 2. Research significance The state-of-the-art of pultruded GFRP design encompasses stress prediction using classical laminate theory, global and local buckling, interaction between compression and bending, and serviceability. The effect of a geometric discontinuity on the failure characteristics of GFRP columns is, however, restrictedly known and thus such a research gap needs to be filled. Transition between material crushing and stability failure dependent upon the slenderness of a GFRP column is an important consideration in design. A dearth of adequate understanding of physical behavior (e.g., failure modes and buckling capacity) often leads to unreliable design and may cause unsatisfactory performance on site. The research is dedicated to exploring the behavior of GFRP columns having various geometric discontinuities and to proposing practical design information.
Table 1 Mechanical properties reported by manufacturer.
a
Property
Value
ASTMa
Tensile stress (longitudinal) Tensile stress (transverse) Tensile modulus (longitudinal) Tensile modulus (transverse) Compressive stress (longitudinal) Compressive stress (transverse) Compressive modulus (longitudinal) Compressive modulus (transverse) Shear modulus Poisson’s ratio Density
207 MPa 48 MPa 17 GPa 6 GPa 207 MPa 17 GPa 17 GPa 7 GPa 3 GPa 0.33 1.8 kg/mm3
D638 D638 D638 D638 D695 D695 D695 D695 – D3039 D792
Test methods by American Society for Testing and Materials.
coupon test results agreed with the manufacturer-reported data at room temperature [11]). 3.2. Specimen details and testing scheme Five geometric configurations were designed to examine the effect of a splice and utility holes, as shown in Fig. 1: plain columns, columns with two- and four-bolted splices, and columns having one- and two-side holes. For the spliced columns, a piece of GFRP tube was inserted and bolted (63.5 mm 63.5 mm 6.4 mm and 76.2 mm 76.2 mm 6.4 mm splice tubes were employed for the 76.2 mm and 88.9 mm HSS specimens, respectively). The utility holes had a diameter of 40 mm at a center-to-center spacing of 60 mm. The identification of these specimens is as follows (Table 2): T75 and T90 (76.2 mm and 88.9 mm HSS specimens, respectively), PC (plain columns), 2BS and 4BS (two- and four-bolted splices, respectively), and 1SH and 2SH (one- and two-side utility holes, respectively). Each column category had five specimens that were repeatedly tested to ensure that their average failure load would represent the load-bearing capacity of the individual category. The GFRP columns were positioned in a universal testing machine (900 kN capacity) using custom-made steel fixtures (Fig. 1) and high strength bolts with nuts (13 mm in diameter). The length of all specimens was 1830 mm which was the longest dimension that the test machine can accommodate. The columns were monotonically loaded in compression at a rate of 2 mm/min until failure occurred. A data acquisition system was used to measure the applied load (builtin load cell) and axial displacement (stroke of the machine head: it is worth noting that the effect of machine compliance at an early loading stage on the displacement was calibrated using a linear potentiometer and compensated).
3. Experimental program 4. Test results The test program conducted was comprised of 10 categories, depending upon geometric configurations such as the presence of a splice or utility holes, and a total of 50 columns were loaded to failure. A concise description is provided below to explain the material used and specimen details.
In subsequent sections, the results of the experimental investigation are discussed: the capacity of the GFRP columns as a loadbearing element and their failure characteristics along with test parameters. Stability-oriented failure (buckling) was not observed in all tested specimens from a structural perspective.
3.1. Material 4.1. Load-carrying capacity Two sizes of pultruded GFRP hollow structural sections (HSS) were used: 76.2 mm 76.2 mm 6.35 mm and 88.9 mm 88.9 mm 6.35 mm. The non-conductive filament-wound GFRP is made of E-glass fibers embedded in an isophthalic polyester resin at a fiber volume fraction ratio of 55%, and is surface-veiled with polyester fabric to enhance resistance to ultraviolet rays. Table 1 lists typical engineering properties of the GFRP used in the experimental program (previous research confirmed that
Table 2 summarizes the average load-carrying capacity of the tested specimens. The anticipated compression failure loads of the T75 and T90 plain columns due to material crushing (ffu = 207 MPa) were 367 kN and 434 kN, respectively; however, their actual failure loads were measured to be 31% and 32% lower because of stress concentrations in the vicinity of the bolted connection (further discussions will follow). This observation implies
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Plain
Holed
Plate (25 mm x 115 mm x 115 mm) 25 mm diameter rod
Bolted splice (4 bolts)
61 0 m m
1220 mm Utility holes
Bolted splice (2 bolts)
610 mm
Bolted support
1585 mm
GFRP column
245 mm
1830 mm
Bolted support
1220 mm
15 mm x 25 mm slotted hole
Spliced
Splice and fixture
Fig. 1. Geometric configurations of test specimens.
Table 2 Test specimens and load-carrying capacity. Specimen
T75-PC T75-2BS T75-4BS T75-1SH T75-2SH T90-PC T90-2BS T90-4BS T90-1SH T90-2SH a
Aca (mm2)
1774 1774 1774 1492 1210 2097 2097 2097 1815 1532
Ultimate load (kN)
Ultimate stress (MPa)
Ave
Stdev
COV (%)
Ave
Stdev
COV (%)
252.6 229.2 233.8 223.9 201.0 297.0 284.1 265.2 247.6 258.7
19.8 6.1 18.2 23.4 10.1 18.8 25.6 19.8 15.3 10.5
7.8 2.7 7.8 10.5 5.0 6.3 9.0 7.5 6.2 4.1
142.4 129.2 131.8 150.1 166.1 141.6 135.5 126.5 136.4 168.9
11.2 3.4 10.3 15.7 8.3 9.0 12.2 9.4 8.4 6.9
7.8 2.7 7.8 10.5 5.0 6.3 9.0 7.5 6.2 4.1
Cross-sectional area at a critical section.
that the load-carrying capacity of GFRP compression members in a structural system cannot be estimated by a simple engineering calculation based on ultimate strength and cross-sectional area when bolted connections are used. The coefficients of variation (COV) for these two test categories were similar to each other: 7.8% and 6.3% for the T75-PC and T90-PC, respectively. These COV values were relatively lower than that of conventional compression members (e.g., typical COVs for reinforced concrete, aluminum, and coldformed steel columns are 11.5%, 11.0%, and 17.5%, respectively [12,13]), which may be attributed to the fact that the manufacturing process of pultruded GFRP members is well controlled. The presence of the bolted splice decreased the load-carrying capacity of the columns. For example, the T75-2BS and T90-2BS categories showed 9% and 4% lower capacities relative to the plain columns, respectively. The effect of bolt numbers connecting the splice tube to the column member appears to be a function of the crosssection of the columns, given that T75-4BS exhibited a 2% higher failure load than T75-2BS, while T90-4BS demonstrated a 7% lower failure load than T90-2BS. The capacity of the T75-1SH and T90-1SH columns having utility holes on one side was 11% and 17% lower than that of the plain columns, respectively. A trend similar to the splice-bolt effect was noticed for the columns having two-side holes (T75-2SH and T90-2SH). 4.2. Load–displacement behavior The load–displacement behavior of representative columns is provided in Fig. 2(a) and (b) for the T75 and T90 series, respectively. The axial responses of the spliced columns in the T75
category (T75-2BS and T75-4BS) were analogous to that of the T75-PC specimen, as shown in Fig. 2(a), whereas those of the holed specimens were deviated from the plain column since the presence of the utility holes created a weak link in the load-carrying system. Unlike the case of the T75 specimens, the response of the T90 columns was virtually indistinguishable irrespective of the geometric configurations (Fig. 2(b)). These observations imply that the axial behavior of pultruded GFRP load-bearing members is sizedependent because the reduction of resistant area controls the behavior. Fig. 2(c) compares the elastic stiffness of the respective column category. Since the failure of the specimens was brittle without gradual load-softening (to be discussed), the recorded displacement by the stroke did not influence their axial stiffness. Although the difference between the 2- and 4-bolted splice cases was insignificant, the former revealed a 2% lower stiffness value, on average, in comparison to the latter. This appears to be the contribution of the enhanced shear resistance of the internal splice tube when connected with the four bolts. An average axial stiffness reduction of 11% and 2% was observed for the T75 and T90 holed columns relative to the plain columns, respectively. Fig. 2(d) illustrates the effect of column size in terms of column continuity (i.e., with and without the utility holes). The stiffness ratio (T75/T90 comparing their axial responses) of the plain columns was 0.89, while those of the spliced and holed columns were 0.86 and 0.81, on average, respectively. 4.3. Failure modes Various failure modes were observed depending upon the geometric configurations of the columns, as shown in Figs. 3–5. These failure modes were classified as three groups and were not affected by the slenderness of the columns: (i) failure adjacent bolted support, (ii) failure at splice, and (iii) failure near utility holes. The first failure group (Fig. 3) encompassed three sub-failure modes. The majority of the column specimens failed by tear-up (Fig. 3(a)) because of stress concentrations at the bolted region. Hair-line cracks developed around the connection bolts and coalesced with an increase in compression load. Local fiber crushing was followed and finally the column collapsed. Such a localized failure mode was the primary reason why the tested columns did not reach their material crushing strength (ffu = 207 MPa), as discussed in Section 4.1. In some cases, the tear-up failure was extended to local buckling (Fig. 3(b)): the fractured GFRP section was subjected to eccentric loading to a certain extent (possibly due to the geometric imperfection of the GFRP termination at the end of the column)
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Fig. 2. Load–displacement behavior: (a) 75 mm square columns; (b) 90 mm square columns; (c) stiffness comparison with respect to column size; (d) stiffness ratio.
(a)
(b)
(c)
Fig. 3. Failure modes adjacent to bolted support: (a) tear-up (T75-2BS); (b) tear-up-induced local buckling (T75-PC); (c) splitting at corners (T75-4BS).
and the longitudinal bursting cracks propagated as the load increased. The other sub-failure took place by edge-splitting of the columns (Fig. 3(c)), which was attributed to corner stresses. Fig. 4 exhibits the failure of the bolted splice. The edge-splitting failure similar to the one shown in Fig. 3(c) was observed
regardless of the number of connection bolts. This fact indicates that the applied axial load was transferred to the inner GFRP tube at the spliced location, while the geometric discontinuity between the two GFRP columns enclosing the tube entailed fiber misalignment and caused out-of-plane displacement. It may be
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(a)
(b)
Fig. 4. Failure modes at bolted splice: (a) two-bolt splice (T90-2BS); (b) four-bolt splice (T75-4BS).
Diagonal shear
Diagonal shear
(a)
(b)
(c)
(d)
Fig. 5. Failure modes near holed section: (a) diagonal shear of one-holed section (T75-1SH); (b) diagonal shear of two-holed section (T75-2SH); (c) delamination of one-holed section (T90-1SH); (d) delamination of one-holed section (T90-2SH).
recommended that adhesive-bonding be used between the inner and outer GFRP sections in addition to bolted connections so that the applied axial stress can be distributed to the primary column members, which then enhance the overall behavior of the column system. The failure of the columns having utility holes is summarized in Fig. 5. Typical shear failure was noticed at an approximate angle of 45° along the diagonal direction of the holed walls (Fig. 5 (a) and (b)) where the planes of maximum shearing stress progressed. Another failure mode observed was the delamination of the column wall perpendicular to the holed side (Fig. 5 (c) and (d)): the reduced cross section of the column resulted in local compression at the holes and, consequently, a lateral dilatation effect occurred. 5. Finite element modeling Three-dimensional finite element modeling was conducted using the general-purpose program ANSYS to expand the experimental investigations, particularly for stability issues that were not examined in the test program because of the limited specimen length, with a focus on plain and holed columns having various parameters such as wall thickness, GFRP modulus, and slenderness
ratio. Details on model development, validation, and predictive responses are provided in this section. 5.1. Model development 5.1.1. Element and geometric configurations Four-node shell elements (SHELL181) were employed to represent the thin-walled GFRP columns. Each node of the element has three-translational and three-rotational degrees of freedom. A wide variety of slenderness ratios from Le/r = 25 to 150 were modeled, as shown in Fig. 6, where Le is the effective column length and r is the radius of gyration of the column. The material properties of the GFRP described in Section 3 were used as input. A fixed-pin boundary condition was imposed to the model (the translational and rotational degrees of freedom were constrained at the fixed nodes, while the translational degrees of freedom were constrained at the pined nodes) because most tested columns revealed one-end rotation due to local tear-up (Fig. 3(a)). It is worth noting that the observed failure modes such as corner-splitting or delamination were not an interest of the current simulation, provided that the focus of the modeling was on global stability related to various slenderness ratios which was not available in the experimental
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the corresponding buckled shape. The eigenvalue of the column model can be expressed [15]:
½KfUi g ¼ ki ½SfUi g
One side
(a)
Two sides
(b)
Fig. 6. Column details for finite element modeling: (a) various slenderness ratios varying from 25 to 150; (b) presence of side holes.
investigations. The local behavior of the splice was also not an interest in the numerical study because the elastic buckling simulation utilized would reveal the same behavior whether the splice was physically included or not from a finite element modeling standpoint, the nodes of the splicing-stub elements need to be merged to the nodes of the main column elements (nodesharing) so that force transfer between the two discrete objects can be achieved in the model (i.e., splice connection); unless the interest of a simulation is in the local behavior of a splice connection (this kind of simulation is computationally expensive and a lot of contact properties are required), refined modeling in the bolted splice is not necessary. A penalty method was used to engage the in-plane displacement of the shell element with the rotational degrees of freedom about its surface normal. The penalty method is a numerical tool to control the degrees of freedom by adjusting a stiffness matrix and further details are available in any finite element text such as Cook et al. [14]. The uniform reduced integration method was selected to save computational effort. Since the size of the formulated mesh was sufficiently dense as shown in Fig. 6, hourglassing issues were not a technical concern. Previous finite element modeling reports that a mesh-sensitivity analysis may not be necessary if an element size is 2–4 times less than the specimen width or depth [7]; the present model has a range between 3 and 3.6. The accuracy of the solved model was checked by monitoring the ratio between the total energy and the artificial energy of the constructed column models, which was maintained below the recommended limit of 5% [15]. The Bathe–Dvorkin method was adopted to mitigate the shear locking of the elements. This method deals with transverse shear and bending effects in an element stiffness matrix [16]. 5.1.2. Buckling simulation Eigen buckling analysis was carried out to predict the bifurcation of the axial load-bearing capacity (i.e., global buckling) and
ð1Þ
where [K] is the stiffness matrix of the GFRP column; [S] is the stress stiffness matrix; and fUi g is the eigenvector. The Block Lanczos algorithm was employed [17] to extract the first mode of the solved column models. A Sturm sequence check was performed so as not to miss related eigenfrequencies: the number of negative pivots is calculated between the maximum and minimum eigenvalues and compared with that of converged eigenvalues. The buckled shape of the GFRP columns was obtained by expanding the extracted mode whose significance level was greater than 0.001 (i.e., the ratio between the mode coefficient of the extracted mode and the maximum mode coefficient of all modes [15]). 5.1.3. Assessment Fig. 7 compares the buckling loads acquired from the finite element model (the T90 configuration) with theoretical counterparts (called ‘theory’ hereafter) consisting of the Euler critical loads and the ultimate capacity of the material in compression (Table 1). Also compared were the test results of the T90PC specimens having a slenderness ratio of 27. The theoretical buckling loads were consistently higher than those predicted by the models. Such a trend became more pronounced when the slenderness ratio of the column decreased, which is typical because the Euler critical load is inversely proportional to the square of a slenderness ratio and approaches infinity with a decrease in slenderness ratio, as evidenced by numerous test results [6]. The predicted buckling load at a slenderness ratio of 27 revealed a relative difference of 7% against the test data. 5.2. Parametric investigation A summary of parametric investigations related to several engineering properties is given in Fig. 8. For comparison, the sectional and engineering properties of the T90 plain column were consistently utilized for the parametric finite element models, unless otherwise stated. Fig. 9(a) shows the effect of wall thickness at a slenderness ratio of 27. The buckling load was essentially a linear function with respect to the wall thickness within the simulated range from 4 mm to 16 mm (i.e., 63–252% of the thickness of the experimental column, T90-PC). This can be explained by fundamental stability theory: the global buckling load of an axially loaded member linearly increases with its moment of inertia. The
Fig. 7. Comparison between theoretical and computational buckling loads of T90 plain columns.
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(a)
(b)
(c)
Fig. 8. Parametric study results: (a) effect of wall thickness; (b) effect of GFRP modulus; (d) effect of presence of holes.
Fig. 9. Buckled shape of parametric columns: (a) one-side holed; (b) two-side holed; (c) comparison at a slenderness ratio of 25.
Table 3 Statistical properties for simulation.
a b
Variable
COV (%)a
Distribution
Source
Longitudinal GFRP modulus Dimensions of GFRP Load-carrying capacity
10.2 1.3 7.0b
Normal Normal Normal
[21] [21] Experiment
COV = coefficient of variation. Average of T75-PC and T90-PC.
Fig. 10. Normality test for the critical stress of test columns (T75-PC and T90-PC).
effect of GFRP modulus on the critical load of the columns is depicted in Fig. 8(b), including typical elastic moduli available in commercial products. The buckling load of the columns was exponentially reduced with respect to the slenderness ratio,
irrespective of the modulus values. For the cases of 40 GPa and 60 GPa columns, crushing failure in compression controlled the critical load (note that the local failure issue due to bolted connection was not considered because the focus of the investigation lied on global behavior). The rate of load-drop was relatively significant up to a slenderness ratio of 75 beyond which a gradual decrease rate was observed; for instance, a load drop of 54% was predicted for the 60 GPa column when its slenderness ratio changed from 50 to 75, while a drop of 34% was obtained as the slenderness ratio varied from 100 to 125. It points out the needs for appropriate bracing members that can control the stability of GFRP columns loaded in axial compression, which is in particular crucial for such a low modulus structural material. Fig. 8(c) illustrates the influence of utility holes on the global stability of the columns. Unlike the cases of compression-controlled failure observed in the test program, the global buckling behavior of the columns was not affected by the presence of the side holes. It is because of the fact that the buckling-critical plane (e.g., midspan for a column subjected to a single curvature) where stability failure is controlled is away from the location of the holes; in other words, the lateral displacement of the column caused unstable behavior before the holed section near the support crushed in compression. This technical assertion is further supported by the profile of the lateral displacement of the columns, as shown in Fig. 9(a) and (b) for the one-side and two-side holed cases, respectively. For a comparison purpose, the vertical distance and lateral displacement of each column were normalized by the column height and the maximum lateral displacement of the column having a slenderness ratio of 150, respectively. The presence of the side holes insignificantly affected the lateral displacement when the buckling failure took place (Fig. 9 (c)) and it could reasonably be ignored in practice. Another thing to note is that the lateral displacement of the columns became insensitive to a slenderness ratio beyond Le/r = 75 (Fig. 9 (a) and (b)).
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Fig. 11. Calibration of design constants: (a) k; (b) C1; (b) C2.
(
6. Design recommendations
rcr
A simple yet practical design approach is proposed in this section for the failure of GFRP members subjected to axial compression. The proposal is validated against independent data obtained from literature [4,5,18,19] and compared with existing expressions. 6.1. Formulation The load-bearing capacity of a GFRP column in concentric compression without buckling (ru) may be obtained by:
ru ¼ krm
ð2Þ
where k is the connection factor near support and rm is the ultimate compression capacity of the GFRP. The elastic buckling stress (re) of the GFRP column may be expressed as:
re
Pe p2 El ¼ ¼ A ðLe =rÞ2
ð3Þ
where Pe is the elastic critical load of the column; A is the crosssectional area of the column; and El is the elastic modulus of the GFRP in the longitudinal direction. According to the AISC Specifications [20], the following relationships between the critical stress (rcr) and the elastic buckling stress (re) and ultimate compressive strength (ru) of the GFRP may be established:
¼ C a1 ru ¼ C 2 re
for inelastic buckling for elastic buckling
ð4Þ
where C1 and C2 are empirical constants and a is the ratio between the compressive capacity and the elastic buckling stress of the GFRP (i.e., a = ru/re). It is important to note that the format of the AISC Specifications was taken for Eq. (4) whose constants were calibrated based on the behavior of GFRP columns. The transition point between the inelastic and elastic buckling responses in Eq. (4) may be determined by their common critical stress (rcr) at that point:
ru ru log C 1 ¼ log C 2 log re re
ð5Þ
Combining Eqs. (3) and (5), the transition slenderness ratio of the GFRP column is acquired by:
Le ¼ C3 r
sffiffiffiffiffiffi El
ru
ð6Þ
The k factor becomes unity after a transition is made from inelastic to elastic buckling of the column (Eqs. (4) and (6)). 6.2. Calibration Monte-Carlo simulation (a technique that randomly samples potential responses from developed probability distributions
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program), while a range of slenderness ratios varying from 50 to 150 was employed for calibrating the C2 constant, along with the numerically predicted rcr and simulated re and ru components:
Table 4 Existing critical stress (rcr) equations for global buckling. Source
Equation
Zureick and Scott [4]
rcr ¼ PAe 1þðns p2 =ððL1e =rÞ2 ðE =GÞÞÞ l
ns = 2 for square hollow section G = shear modulus Tommaso and Russo [5] Roberts [19]
rcr ¼ PAe 1þKP1e =ðGAÞ
rcr ¼ pAL2 EI2 1 þ rGKcr 1 PAe 1 rGKcr K = shear coefficient (K = 1.0 for a box section, Boresi and Sidebottom [22])
numerous times) was conducted to calibrate the k, C1, and C2 design constants, which could overcome the limited deterministic finite element observations so that more generalized design guidelines can be proposed. Each category of the simulation (up to Le/r = 150) included 10,000 randomly generated data samples. Statistical properties were acquired from the current test results and literature, as listed in Table 3. Fig. 10 exhibits the normality test of the critical loads taken from the T75-PC and T90-PC columns. Fig. 11(a) reveals the simulated k factors with an average value of 0.69. It is recommended that k = 0.69 when considering bolted connection near supports where tear-up failure can happen, otherwise k = 1.0. The experimental results at a slenderness ratio of 27 were used to determine the C1 constant (such a value was typically selected because it was used in this experimental
log C 1 ¼ C2 ¼
log rcr log re ru =re
ð7Þ
rcr re
ð8Þ
As illustrated in Fig. 11(b) and (c), the design constants C1 and C2 were respectively determined to be 0.89 and 0.67. Upon calibration of these constants, the transition slenderness ratio associated with the C3 constant was calculated to be 2.69. 6.3. Proposal and assessment The calibrated constants were substituted into the aforementioned design equations and the following expressions are proposed:
sffiffiffiffiffiffi Le El If ; 6 2:69 r ru
rcr ¼ 0:89re ru
sffiffiffiffiffiffi Le El ; > 2:69 If r ru
rcr ¼ 0:67re
ru
ð9Þ
ð10Þ
Fig. 12. Assessment of the proposed design expression without considering bolted connection (inset unit in mm): (a) Z and S (Zureick and Scott [4]); (b) T and R (Tommaso and Russo [5]); (c) B and T (Barbero and Tomblin [18]); (d) Roberts (Roberts [19]).
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Fig. 13. Comparison with existing equations (BC = bolted connection near support; Z and S = Zureick and Scott [4]; Roberts = Roberts [19]; and T and R = Tommaso and Russo [5]): (a) average difference against test programs [4,5,18,19]; (b) comprehensive evaluation with slenderness ratios.
Eqs. (9) and (10) include the slenderness effect Le/r and thus these expressions can be used for any boundary conditions. Fig. 12 compares the critical stresses predicted by the proposed design expressions with experimental data reported by others [4,5,18,19], including the existing formulas listed in Table 4. Provided that the design proposal is established on the basis of sectional properties, its applicability is not limited to specific member shapes. Although some discrepancy was noticed between the predicted and the reported critical stresses (Fig. 12), their agreement was reasonable for design and practice with an average difference of 26.9%. Existing models tended to show good prediction for their own experimental programs; however, their overall average discrepancy against the four selected test programs was higher than that of the proposed model, as shown in Fig. 13(a). Further evaluation of the predictive models with slenderness ratios and the Euler buckling formula is available in Fig. 13(b). For illustration, the T90-PC column was used with the foregoing material properties. As opposed to the proposed equations, the existing expressions do not explicitly limit the application range of a slenderness ratio; nonetheless, these expressions were employed up to the compressive strength of the GFRP. The critical stresses predicted by Zureick and Scott [4], Tommaso and Russo [5], and Roberts [19] were close to those estimated by the classical Euler approach. Overall, Eqs. (9) and (10) provided sufficiently accurate critical stresses with much simpler design expressions. 7. Summary and conclusions This paper has discussed the failure of pultruded GFRP columns subjected to axial compression. Attention was paid to the influence of geometric discontinuities such as a bolted splice or utility holes on characterizing the column behavior with various slenderness ratios. A hybrid approach was adopted to examine the local- and globallevel failure modes, based on experimental and finite element investigations. Failure mechanisms in conjunction with the geometric discontinuities were elucidated. Simple, yet practical equations were formulated to predict the failure of such column members and were calibrated using probability-based simulation. The design expression showed an average difference of about 25% relative to the 61 test data independently reported by others. The following conclusions summarize technical findings from this research: Stress concentrations near the bolted connection at the column supports hindered achieving the full compression capacity of the GFRP or entailed tear-up and edge-splitting failure modes.
The load-bearing capacity of the column was also affected by the presence of the mechanical splice whose failure was predominantly attributable to the displacement incompatibility between the inner and outer GFRP members bolted together. The symmetric or asymmetric holed walls provided a weak link to the column members and hence a reduction in axial stiffness was noticed. The bolted connection at the supports resulted in hair-line cracks that propagated with local fiber kinking; in some cases, longitudinal bursting cracks were followed because of instantaneous regional load eccentricity. Fiber misalignment and outof-plane displacement were observed as the spliced columns were loaded to failure. The holed columns failed by diagonal shearing with a lateral dilatation effect. The validated finite element models indicated that the buckling capacity of the GFRP columns noticeably decreased beyond a slenderness ratio of 75, while the capacity was linearly improved with an increase in wall thickness. The presence of side holes, either one-side or two-side, did not contribute to the global buckling of the columns since the stability failure took place prior to the crushing of the holed section. The critical stress of the GFRP columns was categorized when inelastic and elastic buckling failure modes occurred and corresponding design expressions were proposed, along with a transition slenderness ratio. The proposed design equations were assessed against existing test data. The equations were much simpler than the ones suggested by others, whereas provided sufficiently accurate failure capacity for pultruded GFRP columns.
Acknowledgments The writers are grateful to acknowledge financial supports provided by International Cooling Tower and the University of Colorado Denver. References [1] Sousa JM, Correira JR, Cabral-Fronseca S, Diogo AC. Effects of thermal cycles on the mechanical response of pultruded GFRP profiles used in civil engineering applications. Compos Struct 2014;116:720–31. [2] Hashem ZA, Yuan RL. Experimental and analytical investigations on short GFRP composite compression members. Compos B 2000;31:611–8. [3] Wong PMH, Davies JM, Wang YC. An experimental and numerical study of the behavior of glass fibre reinforced plastics (GRP) short columns at elevated temperatures. Compos Struct 2004;63:33–43.
Y.J. Kim, K.Z. Qian / Composite Structures 136 (2016) 171–181 [4] Zureick A, Scott D. Short-term behavior and design of fiber-reinforced polymeric slender members under axial compression. J Compos Construct 1997;1(4):140–9. [5] Tommaso A, Russo S. Shape influence in buckling of GFRP pultruded columns. Mech Compos Mater 2003;39(4):329–40. [6] Mottram JT. Determination of critical load for flange buckling in concentrically loaded pultruded columns. Compos B 2004;35:35–47. [7] Nguyen TT, Chan TM, Mottram JT. Influence of boundary conditions and geometric imperfections on lateral-torsional buckling resistance of a pultruded FRP I-beam by FEA. Compos Struct 2013;100:233–42. [8] Boscato G, Casalegno C, Russo S. Performance of built-up columns made by pultruded FRP material. Compos Struct 2015;121:46–63. [9] Turvey GJ, Cerutti X. Flexural behavior of pultruded glass fibre reinforced polymer composite beams with bolted splice joints. Compos Struct 2015;119: 543–50. [10] Bank LC. Composites for construction. Hoboken (NJ): John Wiley & Sons, Inc.; 2006. [11] Khan F, Kim YJ. Time-dependent and residual behavior of pultruded GFRP beams subjected to sustained intensities and cold temperature. Cold Reg Sci Technol 2012;74–75:43–51. [12] Ellingwood B, Galambos TV, MacGregor JG, Cornell CA. Development of a probability based load criterion for American National Standard A58, national Bureau of Standards. Washington (D.C.): NBS Special Publication 577; 1980.
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[13] Nowak AS, Rakoczy AM, Szeliga E. Revised statistical resistance models for R/C structural components, ACI-SP-284-6. Farmington Hills (MI): American Concrete Institute; 2012. [14] Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and applications of finite element analysis. New York (NY): John Wiley & Sons; 2002. [15] ANSYS. Theory reference for the mechanical APDL and mechanical applications. Canonsburg (PA): ANSYS; 2014. [16] Bathe KJ, Dvorkin EN. A four node plate bending element based on Mindlin– Reissner plate theory and a mixed interpolation. Int J Numer Meth Eng 1985;21:367–83. [17] Rajakumar C, Rogers CR. The Lanczos algorithm applied to unsymmetric generalized eigenvalue problems. Int J Numer Meth Eng 1991;32(5):1009–26. [18] Barbero E, Tomblin J. A phenomenological design equation for FRP columns with interaction between local and global buckling. Thin-Walled Struct 1994;18:117–31. [19] Roberts TM. Influence of shear deformation on buckling of pultruded fiber reinforced plastic profiles. J Compos Construct 2002;6(4):241–8. [20] AISC. Steel construction manual. 14th ed. Chicago (IL): American Institute of Steel Construction; 2011. [21] Vanevenhoven LM. LRFD factors for FRP pultruded wide-flange columns (MS thesis). Madison (WI): University of Wisconsin-Madison; 2007. [22] Boresi AP, Sidebottom OM. Advanced mechanics of materials. Hoboken (NJ): John Wiley and Sons Inc.; 1985.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Effect of geometric discontinuities on failure of pultruded GFRP columns in axial compression Yail J. Kim ⇑, Kenny Zongxi Qian Department of Civil Engineering, University of Colorado Denver, Denver, CO, United States
a r t i c l e
i n f o
Article history: Available online 23 October 2015 Keywords: Axial compression Column Composite Glass fiber reinforced polymer Pultruded shape
a b s t r a c t This paper presents the failure characteristics of pultruded glass fiber reinforced polymer (GFRP) columns loaded in concentric compression. Of interest is the effect of geometric discontinuities on the behavior of the columns, including a bolted splice or utility holes. The local- and global-responses of the columns are examined through a hybrid approach consisting of laboratory experiment and finite element modeling: the test encompasses a total of 50 columns and the model has varying slenderness ratios up to 150. Bolted connections at column supports result in stress concentrations accompanied by crack initiation and propagation, thereby reducing the load-bearing capacity. Tear-up or edge-splitting dominates failure modes at the bolted connections. Spliced members experience fiber misalignment and out-of-plane behavior because of displacement incompatibility between the inner and outer GFRP members bolted together. A weak link provided by side holes, whether symmetric or asymmetric configurations, causes a reduction in axial stiffness and diagonal shearing. The results of modeling indicate that buckling capacity of the column significantly decreases beyond a slenderness ratio of 75 and the presence of side holes is not a contributing factor to the stability of the columns. The design expressions proposed are simpler than existing ones, while provide sufficiently accurate capacity prediction for practice. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Glass fiber reinforced polymer (GFRP) composites are broadly used in engineering disciplines with many benefits [1]. Contrary to conventional load-bearing structural materials such as concrete and steel, pultruded GFRP shapes demonstrate linear behavior until failure occurs. Two types of failure modes are observed for GFRP members loaded in axial compression: material crushing and stability, depending upon the extent of slenderness. Material-level failure happens when the applied stress exceeds the compressive strength of the GFRP and accompanies local fiber buckling or resin fracture. Hashem and Yuan [2] studied the failure mechanism of GFRP stubs (L = 305 mm) in axial load. Microbuckling controlled the crushing of the composite, followed by surface delamination. Wong et al. [3] reported the failure of short GFRP channels (L = 30 mm) subjected to compression at elevated temperatures, including fiber kinking or micro-buckling. The buckling of pultruded GFRP columns has been examined by a number of researchers. Zureick and Scott [4] investigated the stability of wide-flange and box sections made of GFRP having a slenderness ⇑ Corresponding author. E-mail addresses: [email protected] (Y.J. Kim), zongxi.qian@ucdenver. edu (K.Z. Qian). http://dx.doi.org/10.1016/j.compstruct.2015.10.016 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
ratio of 36–103. The composite material encompassed a fiber volume fraction of 30%. Geometric imperfection was measured to evaluate the degree of out-of-straightness for each test specimen, while their deviations were within an acceptable range as per an ASTM standard. Failure loads were recorded and compared to critical load equations proposed by others. Design guidelines were discussed as well. Tommaso and Russo [5] tested I- and H-shape pultruded GFRP members monotonically loaded in compression. These columns had slenderness ratios ranging from 9 to 195 and 22 to 63 for the I- and H-shapes, respectively. Test results were compared with a classical Euler buckling load and finite element prediction. The H-shape profiles were more sensitive to the interaction between local and global buckling compared to the I-shape ones. Mottram [6] developed analytical closed-form equations for concentrically loaded GFRP columns, including a review of existing equations, and recommended that more test data and universal column design procedures be required. Nguyen et al. [7] developed a finite element model to predict the global buckling of GFRP columns. An emphasis was given to the influence of boundary conditions and imperfection when associated with a slenderness ratio between 10 and 50. The level of end-warping fixity was reduced with an increase in specimen length. A maximum out-of-straightness of span/200 was recommended for practice.
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Utility holes may be cut in structural members to accommodate electrical conduits or analogous purposes. It is generally suggested that holes be located where a minimum bending moment takes place when a member is loaded. Concerns, however, may arise if the member is subjected to compression because the presence of the holes can affect the load-bearing or buckling response of the member. Although some research programs discussed bolted connections for pultruded GFRP members [8,9], limited effort was made when various geometric configurations (e.g., discontinuity in structural members) are imposed to GFRP shapes loaded in compression. As such, current practice does not appropriately address technical consequences along with this important issue that can alter the failure characteristics of the shapes. As far as buckling is concerned, existing equations are complex and rigorous endeavors may be required for the practitioners to use [10]. Straightforward design expressions are, therefore, required to facilitate the design of pultruded GFRP members in axial compression, including sufficient accuracy in terms of predicting a critical load that controls the load-bearing capacity. This paper discusses the failure characteristics of concentrically loaded GFRP columns having geometric discontinuities, which are defined as physical separation of the continuum columns by the presence of utility holes or splices. A hybrid approach is implemented with laboratory testing and finite element modeling to elucidate the behavior of the columns, which can complement to one another (i.e., a wide variety of failure modes associated with the discontinuities and slenderness effects linked with global buckling). Design equations are proposed based on probability-based simulation and further compared with existing expressions. 2. Research significance The state-of-the-art of pultruded GFRP design encompasses stress prediction using classical laminate theory, global and local buckling, interaction between compression and bending, and serviceability. The effect of a geometric discontinuity on the failure characteristics of GFRP columns is, however, restrictedly known and thus such a research gap needs to be filled. Transition between material crushing and stability failure dependent upon the slenderness of a GFRP column is an important consideration in design. A dearth of adequate understanding of physical behavior (e.g., failure modes and buckling capacity) often leads to unreliable design and may cause unsatisfactory performance on site. The research is dedicated to exploring the behavior of GFRP columns having various geometric discontinuities and to proposing practical design information.
Table 1 Mechanical properties reported by manufacturer.
a
Property
Value
ASTMa
Tensile stress (longitudinal) Tensile stress (transverse) Tensile modulus (longitudinal) Tensile modulus (transverse) Compressive stress (longitudinal) Compressive stress (transverse) Compressive modulus (longitudinal) Compressive modulus (transverse) Shear modulus Poisson’s ratio Density
207 MPa 48 MPa 17 GPa 6 GPa 207 MPa 17 GPa 17 GPa 7 GPa 3 GPa 0.33 1.8 kg/mm3
D638 D638 D638 D638 D695 D695 D695 D695 – D3039 D792
Test methods by American Society for Testing and Materials.
coupon test results agreed with the manufacturer-reported data at room temperature [11]). 3.2. Specimen details and testing scheme Five geometric configurations were designed to examine the effect of a splice and utility holes, as shown in Fig. 1: plain columns, columns with two- and four-bolted splices, and columns having one- and two-side holes. For the spliced columns, a piece of GFRP tube was inserted and bolted (63.5 mm 63.5 mm 6.4 mm and 76.2 mm 76.2 mm 6.4 mm splice tubes were employed for the 76.2 mm and 88.9 mm HSS specimens, respectively). The utility holes had a diameter of 40 mm at a center-to-center spacing of 60 mm. The identification of these specimens is as follows (Table 2): T75 and T90 (76.2 mm and 88.9 mm HSS specimens, respectively), PC (plain columns), 2BS and 4BS (two- and four-bolted splices, respectively), and 1SH and 2SH (one- and two-side utility holes, respectively). Each column category had five specimens that were repeatedly tested to ensure that their average failure load would represent the load-bearing capacity of the individual category. The GFRP columns were positioned in a universal testing machine (900 kN capacity) using custom-made steel fixtures (Fig. 1) and high strength bolts with nuts (13 mm in diameter). The length of all specimens was 1830 mm which was the longest dimension that the test machine can accommodate. The columns were monotonically loaded in compression at a rate of 2 mm/min until failure occurred. A data acquisition system was used to measure the applied load (builtin load cell) and axial displacement (stroke of the machine head: it is worth noting that the effect of machine compliance at an early loading stage on the displacement was calibrated using a linear potentiometer and compensated).
3. Experimental program 4. Test results The test program conducted was comprised of 10 categories, depending upon geometric configurations such as the presence of a splice or utility holes, and a total of 50 columns were loaded to failure. A concise description is provided below to explain the material used and specimen details.
In subsequent sections, the results of the experimental investigation are discussed: the capacity of the GFRP columns as a loadbearing element and their failure characteristics along with test parameters. Stability-oriented failure (buckling) was not observed in all tested specimens from a structural perspective.
3.1. Material 4.1. Load-carrying capacity Two sizes of pultruded GFRP hollow structural sections (HSS) were used: 76.2 mm 76.2 mm 6.35 mm and 88.9 mm 88.9 mm 6.35 mm. The non-conductive filament-wound GFRP is made of E-glass fibers embedded in an isophthalic polyester resin at a fiber volume fraction ratio of 55%, and is surface-veiled with polyester fabric to enhance resistance to ultraviolet rays. Table 1 lists typical engineering properties of the GFRP used in the experimental program (previous research confirmed that
Table 2 summarizes the average load-carrying capacity of the tested specimens. The anticipated compression failure loads of the T75 and T90 plain columns due to material crushing (ffu = 207 MPa) were 367 kN and 434 kN, respectively; however, their actual failure loads were measured to be 31% and 32% lower because of stress concentrations in the vicinity of the bolted connection (further discussions will follow). This observation implies
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Plain
Holed
Plate (25 mm x 115 mm x 115 mm) 25 mm diameter rod
Bolted splice (4 bolts)
61 0 m m
1220 mm Utility holes
Bolted splice (2 bolts)
610 mm
Bolted support
1585 mm
GFRP column
245 mm
1830 mm
Bolted support
1220 mm
15 mm x 25 mm slotted hole
Spliced
Splice and fixture
Fig. 1. Geometric configurations of test specimens.
Table 2 Test specimens and load-carrying capacity. Specimen
T75-PC T75-2BS T75-4BS T75-1SH T75-2SH T90-PC T90-2BS T90-4BS T90-1SH T90-2SH a
Aca (mm2)
1774 1774 1774 1492 1210 2097 2097 2097 1815 1532
Ultimate load (kN)
Ultimate stress (MPa)
Ave
Stdev
COV (%)
Ave
Stdev
COV (%)
252.6 229.2 233.8 223.9 201.0 297.0 284.1 265.2 247.6 258.7
19.8 6.1 18.2 23.4 10.1 18.8 25.6 19.8 15.3 10.5
7.8 2.7 7.8 10.5 5.0 6.3 9.0 7.5 6.2 4.1
142.4 129.2 131.8 150.1 166.1 141.6 135.5 126.5 136.4 168.9
11.2 3.4 10.3 15.7 8.3 9.0 12.2 9.4 8.4 6.9
7.8 2.7 7.8 10.5 5.0 6.3 9.0 7.5 6.2 4.1
Cross-sectional area at a critical section.
that the load-carrying capacity of GFRP compression members in a structural system cannot be estimated by a simple engineering calculation based on ultimate strength and cross-sectional area when bolted connections are used. The coefficients of variation (COV) for these two test categories were similar to each other: 7.8% and 6.3% for the T75-PC and T90-PC, respectively. These COV values were relatively lower than that of conventional compression members (e.g., typical COVs for reinforced concrete, aluminum, and coldformed steel columns are 11.5%, 11.0%, and 17.5%, respectively [12,13]), which may be attributed to the fact that the manufacturing process of pultruded GFRP members is well controlled. The presence of the bolted splice decreased the load-carrying capacity of the columns. For example, the T75-2BS and T90-2BS categories showed 9% and 4% lower capacities relative to the plain columns, respectively. The effect of bolt numbers connecting the splice tube to the column member appears to be a function of the crosssection of the columns, given that T75-4BS exhibited a 2% higher failure load than T75-2BS, while T90-4BS demonstrated a 7% lower failure load than T90-2BS. The capacity of the T75-1SH and T90-1SH columns having utility holes on one side was 11% and 17% lower than that of the plain columns, respectively. A trend similar to the splice-bolt effect was noticed for the columns having two-side holes (T75-2SH and T90-2SH). 4.2. Load–displacement behavior The load–displacement behavior of representative columns is provided in Fig. 2(a) and (b) for the T75 and T90 series, respectively. The axial responses of the spliced columns in the T75
category (T75-2BS and T75-4BS) were analogous to that of the T75-PC specimen, as shown in Fig. 2(a), whereas those of the holed specimens were deviated from the plain column since the presence of the utility holes created a weak link in the load-carrying system. Unlike the case of the T75 specimens, the response of the T90 columns was virtually indistinguishable irrespective of the geometric configurations (Fig. 2(b)). These observations imply that the axial behavior of pultruded GFRP load-bearing members is sizedependent because the reduction of resistant area controls the behavior. Fig. 2(c) compares the elastic stiffness of the respective column category. Since the failure of the specimens was brittle without gradual load-softening (to be discussed), the recorded displacement by the stroke did not influence their axial stiffness. Although the difference between the 2- and 4-bolted splice cases was insignificant, the former revealed a 2% lower stiffness value, on average, in comparison to the latter. This appears to be the contribution of the enhanced shear resistance of the internal splice tube when connected with the four bolts. An average axial stiffness reduction of 11% and 2% was observed for the T75 and T90 holed columns relative to the plain columns, respectively. Fig. 2(d) illustrates the effect of column size in terms of column continuity (i.e., with and without the utility holes). The stiffness ratio (T75/T90 comparing their axial responses) of the plain columns was 0.89, while those of the spliced and holed columns were 0.86 and 0.81, on average, respectively. 4.3. Failure modes Various failure modes were observed depending upon the geometric configurations of the columns, as shown in Figs. 3–5. These failure modes were classified as three groups and were not affected by the slenderness of the columns: (i) failure adjacent bolted support, (ii) failure at splice, and (iii) failure near utility holes. The first failure group (Fig. 3) encompassed three sub-failure modes. The majority of the column specimens failed by tear-up (Fig. 3(a)) because of stress concentrations at the bolted region. Hair-line cracks developed around the connection bolts and coalesced with an increase in compression load. Local fiber crushing was followed and finally the column collapsed. Such a localized failure mode was the primary reason why the tested columns did not reach their material crushing strength (ffu = 207 MPa), as discussed in Section 4.1. In some cases, the tear-up failure was extended to local buckling (Fig. 3(b)): the fractured GFRP section was subjected to eccentric loading to a certain extent (possibly due to the geometric imperfection of the GFRP termination at the end of the column)
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Fig. 2. Load–displacement behavior: (a) 75 mm square columns; (b) 90 mm square columns; (c) stiffness comparison with respect to column size; (d) stiffness ratio.
(a)
(b)
(c)
Fig. 3. Failure modes adjacent to bolted support: (a) tear-up (T75-2BS); (b) tear-up-induced local buckling (T75-PC); (c) splitting at corners (T75-4BS).
and the longitudinal bursting cracks propagated as the load increased. The other sub-failure took place by edge-splitting of the columns (Fig. 3(c)), which was attributed to corner stresses. Fig. 4 exhibits the failure of the bolted splice. The edge-splitting failure similar to the one shown in Fig. 3(c) was observed
regardless of the number of connection bolts. This fact indicates that the applied axial load was transferred to the inner GFRP tube at the spliced location, while the geometric discontinuity between the two GFRP columns enclosing the tube entailed fiber misalignment and caused out-of-plane displacement. It may be
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(a)
(b)
Fig. 4. Failure modes at bolted splice: (a) two-bolt splice (T90-2BS); (b) four-bolt splice (T75-4BS).
Diagonal shear
Diagonal shear
(a)
(b)
(c)
(d)
Fig. 5. Failure modes near holed section: (a) diagonal shear of one-holed section (T75-1SH); (b) diagonal shear of two-holed section (T75-2SH); (c) delamination of one-holed section (T90-1SH); (d) delamination of one-holed section (T90-2SH).
recommended that adhesive-bonding be used between the inner and outer GFRP sections in addition to bolted connections so that the applied axial stress can be distributed to the primary column members, which then enhance the overall behavior of the column system. The failure of the columns having utility holes is summarized in Fig. 5. Typical shear failure was noticed at an approximate angle of 45° along the diagonal direction of the holed walls (Fig. 5 (a) and (b)) where the planes of maximum shearing stress progressed. Another failure mode observed was the delamination of the column wall perpendicular to the holed side (Fig. 5 (c) and (d)): the reduced cross section of the column resulted in local compression at the holes and, consequently, a lateral dilatation effect occurred. 5. Finite element modeling Three-dimensional finite element modeling was conducted using the general-purpose program ANSYS to expand the experimental investigations, particularly for stability issues that were not examined in the test program because of the limited specimen length, with a focus on plain and holed columns having various parameters such as wall thickness, GFRP modulus, and slenderness
ratio. Details on model development, validation, and predictive responses are provided in this section. 5.1. Model development 5.1.1. Element and geometric configurations Four-node shell elements (SHELL181) were employed to represent the thin-walled GFRP columns. Each node of the element has three-translational and three-rotational degrees of freedom. A wide variety of slenderness ratios from Le/r = 25 to 150 were modeled, as shown in Fig. 6, where Le is the effective column length and r is the radius of gyration of the column. The material properties of the GFRP described in Section 3 were used as input. A fixed-pin boundary condition was imposed to the model (the translational and rotational degrees of freedom were constrained at the fixed nodes, while the translational degrees of freedom were constrained at the pined nodes) because most tested columns revealed one-end rotation due to local tear-up (Fig. 3(a)). It is worth noting that the observed failure modes such as corner-splitting or delamination were not an interest of the current simulation, provided that the focus of the modeling was on global stability related to various slenderness ratios which was not available in the experimental
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the corresponding buckled shape. The eigenvalue of the column model can be expressed [15]:
½KfUi g ¼ ki ½SfUi g
One side
(a)
Two sides
(b)
Fig. 6. Column details for finite element modeling: (a) various slenderness ratios varying from 25 to 150; (b) presence of side holes.
investigations. The local behavior of the splice was also not an interest in the numerical study because the elastic buckling simulation utilized would reveal the same behavior whether the splice was physically included or not from a finite element modeling standpoint, the nodes of the splicing-stub elements need to be merged to the nodes of the main column elements (nodesharing) so that force transfer between the two discrete objects can be achieved in the model (i.e., splice connection); unless the interest of a simulation is in the local behavior of a splice connection (this kind of simulation is computationally expensive and a lot of contact properties are required), refined modeling in the bolted splice is not necessary. A penalty method was used to engage the in-plane displacement of the shell element with the rotational degrees of freedom about its surface normal. The penalty method is a numerical tool to control the degrees of freedom by adjusting a stiffness matrix and further details are available in any finite element text such as Cook et al. [14]. The uniform reduced integration method was selected to save computational effort. Since the size of the formulated mesh was sufficiently dense as shown in Fig. 6, hourglassing issues were not a technical concern. Previous finite element modeling reports that a mesh-sensitivity analysis may not be necessary if an element size is 2–4 times less than the specimen width or depth [7]; the present model has a range between 3 and 3.6. The accuracy of the solved model was checked by monitoring the ratio between the total energy and the artificial energy of the constructed column models, which was maintained below the recommended limit of 5% [15]. The Bathe–Dvorkin method was adopted to mitigate the shear locking of the elements. This method deals with transverse shear and bending effects in an element stiffness matrix [16]. 5.1.2. Buckling simulation Eigen buckling analysis was carried out to predict the bifurcation of the axial load-bearing capacity (i.e., global buckling) and
ð1Þ
where [K] is the stiffness matrix of the GFRP column; [S] is the stress stiffness matrix; and fUi g is the eigenvector. The Block Lanczos algorithm was employed [17] to extract the first mode of the solved column models. A Sturm sequence check was performed so as not to miss related eigenfrequencies: the number of negative pivots is calculated between the maximum and minimum eigenvalues and compared with that of converged eigenvalues. The buckled shape of the GFRP columns was obtained by expanding the extracted mode whose significance level was greater than 0.001 (i.e., the ratio between the mode coefficient of the extracted mode and the maximum mode coefficient of all modes [15]). 5.1.3. Assessment Fig. 7 compares the buckling loads acquired from the finite element model (the T90 configuration) with theoretical counterparts (called ‘theory’ hereafter) consisting of the Euler critical loads and the ultimate capacity of the material in compression (Table 1). Also compared were the test results of the T90PC specimens having a slenderness ratio of 27. The theoretical buckling loads were consistently higher than those predicted by the models. Such a trend became more pronounced when the slenderness ratio of the column decreased, which is typical because the Euler critical load is inversely proportional to the square of a slenderness ratio and approaches infinity with a decrease in slenderness ratio, as evidenced by numerous test results [6]. The predicted buckling load at a slenderness ratio of 27 revealed a relative difference of 7% against the test data. 5.2. Parametric investigation A summary of parametric investigations related to several engineering properties is given in Fig. 8. For comparison, the sectional and engineering properties of the T90 plain column were consistently utilized for the parametric finite element models, unless otherwise stated. Fig. 9(a) shows the effect of wall thickness at a slenderness ratio of 27. The buckling load was essentially a linear function with respect to the wall thickness within the simulated range from 4 mm to 16 mm (i.e., 63–252% of the thickness of the experimental column, T90-PC). This can be explained by fundamental stability theory: the global buckling load of an axially loaded member linearly increases with its moment of inertia. The
Fig. 7. Comparison between theoretical and computational buckling loads of T90 plain columns.
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(a)
(b)
(c)
Fig. 8. Parametric study results: (a) effect of wall thickness; (b) effect of GFRP modulus; (d) effect of presence of holes.
Fig. 9. Buckled shape of parametric columns: (a) one-side holed; (b) two-side holed; (c) comparison at a slenderness ratio of 25.
Table 3 Statistical properties for simulation.
a b
Variable
COV (%)a
Distribution
Source
Longitudinal GFRP modulus Dimensions of GFRP Load-carrying capacity
10.2 1.3 7.0b
Normal Normal Normal
[21] [21] Experiment
COV = coefficient of variation. Average of T75-PC and T90-PC.
Fig. 10. Normality test for the critical stress of test columns (T75-PC and T90-PC).
effect of GFRP modulus on the critical load of the columns is depicted in Fig. 8(b), including typical elastic moduli available in commercial products. The buckling load of the columns was exponentially reduced with respect to the slenderness ratio,
irrespective of the modulus values. For the cases of 40 GPa and 60 GPa columns, crushing failure in compression controlled the critical load (note that the local failure issue due to bolted connection was not considered because the focus of the investigation lied on global behavior). The rate of load-drop was relatively significant up to a slenderness ratio of 75 beyond which a gradual decrease rate was observed; for instance, a load drop of 54% was predicted for the 60 GPa column when its slenderness ratio changed from 50 to 75, while a drop of 34% was obtained as the slenderness ratio varied from 100 to 125. It points out the needs for appropriate bracing members that can control the stability of GFRP columns loaded in axial compression, which is in particular crucial for such a low modulus structural material. Fig. 8(c) illustrates the influence of utility holes on the global stability of the columns. Unlike the cases of compression-controlled failure observed in the test program, the global buckling behavior of the columns was not affected by the presence of the side holes. It is because of the fact that the buckling-critical plane (e.g., midspan for a column subjected to a single curvature) where stability failure is controlled is away from the location of the holes; in other words, the lateral displacement of the column caused unstable behavior before the holed section near the support crushed in compression. This technical assertion is further supported by the profile of the lateral displacement of the columns, as shown in Fig. 9(a) and (b) for the one-side and two-side holed cases, respectively. For a comparison purpose, the vertical distance and lateral displacement of each column were normalized by the column height and the maximum lateral displacement of the column having a slenderness ratio of 150, respectively. The presence of the side holes insignificantly affected the lateral displacement when the buckling failure took place (Fig. 9 (c)) and it could reasonably be ignored in practice. Another thing to note is that the lateral displacement of the columns became insensitive to a slenderness ratio beyond Le/r = 75 (Fig. 9 (a) and (b)).
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Fig. 11. Calibration of design constants: (a) k; (b) C1; (b) C2.
(
6. Design recommendations
rcr
A simple yet practical design approach is proposed in this section for the failure of GFRP members subjected to axial compression. The proposal is validated against independent data obtained from literature [4,5,18,19] and compared with existing expressions. 6.1. Formulation The load-bearing capacity of a GFRP column in concentric compression without buckling (ru) may be obtained by:
ru ¼ krm
ð2Þ
where k is the connection factor near support and rm is the ultimate compression capacity of the GFRP. The elastic buckling stress (re) of the GFRP column may be expressed as:
re
Pe p2 El ¼ ¼ A ðLe =rÞ2
ð3Þ
where Pe is the elastic critical load of the column; A is the crosssectional area of the column; and El is the elastic modulus of the GFRP in the longitudinal direction. According to the AISC Specifications [20], the following relationships between the critical stress (rcr) and the elastic buckling stress (re) and ultimate compressive strength (ru) of the GFRP may be established:
¼ C a1 ru ¼ C 2 re
for inelastic buckling for elastic buckling
ð4Þ
where C1 and C2 are empirical constants and a is the ratio between the compressive capacity and the elastic buckling stress of the GFRP (i.e., a = ru/re). It is important to note that the format of the AISC Specifications was taken for Eq. (4) whose constants were calibrated based on the behavior of GFRP columns. The transition point between the inelastic and elastic buckling responses in Eq. (4) may be determined by their common critical stress (rcr) at that point:
ru ru log C 1 ¼ log C 2 log re re
ð5Þ
Combining Eqs. (3) and (5), the transition slenderness ratio of the GFRP column is acquired by:
Le ¼ C3 r
sffiffiffiffiffiffi El
ru
ð6Þ
The k factor becomes unity after a transition is made from inelastic to elastic buckling of the column (Eqs. (4) and (6)). 6.2. Calibration Monte-Carlo simulation (a technique that randomly samples potential responses from developed probability distributions
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program), while a range of slenderness ratios varying from 50 to 150 was employed for calibrating the C2 constant, along with the numerically predicted rcr and simulated re and ru components:
Table 4 Existing critical stress (rcr) equations for global buckling. Source
Equation
Zureick and Scott [4]
rcr ¼ PAe 1þðns p2 =ððL1e =rÞ2 ðE =GÞÞÞ l
ns = 2 for square hollow section G = shear modulus Tommaso and Russo [5] Roberts [19]
rcr ¼ PAe 1þKP1e =ðGAÞ
rcr ¼ pAL2 EI2 1 þ rGKcr 1 PAe 1 rGKcr K = shear coefficient (K = 1.0 for a box section, Boresi and Sidebottom [22])
numerous times) was conducted to calibrate the k, C1, and C2 design constants, which could overcome the limited deterministic finite element observations so that more generalized design guidelines can be proposed. Each category of the simulation (up to Le/r = 150) included 10,000 randomly generated data samples. Statistical properties were acquired from the current test results and literature, as listed in Table 3. Fig. 10 exhibits the normality test of the critical loads taken from the T75-PC and T90-PC columns. Fig. 11(a) reveals the simulated k factors with an average value of 0.69. It is recommended that k = 0.69 when considering bolted connection near supports where tear-up failure can happen, otherwise k = 1.0. The experimental results at a slenderness ratio of 27 were used to determine the C1 constant (such a value was typically selected because it was used in this experimental
log C 1 ¼ C2 ¼
log rcr log re ru =re
ð7Þ
rcr re
ð8Þ
As illustrated in Fig. 11(b) and (c), the design constants C1 and C2 were respectively determined to be 0.89 and 0.67. Upon calibration of these constants, the transition slenderness ratio associated with the C3 constant was calculated to be 2.69. 6.3. Proposal and assessment The calibrated constants were substituted into the aforementioned design equations and the following expressions are proposed:
sffiffiffiffiffiffi Le El If ; 6 2:69 r ru
rcr ¼ 0:89re ru
sffiffiffiffiffiffi Le El ; > 2:69 If r ru
rcr ¼ 0:67re
ru
ð9Þ
ð10Þ
Fig. 12. Assessment of the proposed design expression without considering bolted connection (inset unit in mm): (a) Z and S (Zureick and Scott [4]); (b) T and R (Tommaso and Russo [5]); (c) B and T (Barbero and Tomblin [18]); (d) Roberts (Roberts [19]).
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Fig. 13. Comparison with existing equations (BC = bolted connection near support; Z and S = Zureick and Scott [4]; Roberts = Roberts [19]; and T and R = Tommaso and Russo [5]): (a) average difference against test programs [4,5,18,19]; (b) comprehensive evaluation with slenderness ratios.
Eqs. (9) and (10) include the slenderness effect Le/r and thus these expressions can be used for any boundary conditions. Fig. 12 compares the critical stresses predicted by the proposed design expressions with experimental data reported by others [4,5,18,19], including the existing formulas listed in Table 4. Provided that the design proposal is established on the basis of sectional properties, its applicability is not limited to specific member shapes. Although some discrepancy was noticed between the predicted and the reported critical stresses (Fig. 12), their agreement was reasonable for design and practice with an average difference of 26.9%. Existing models tended to show good prediction for their own experimental programs; however, their overall average discrepancy against the four selected test programs was higher than that of the proposed model, as shown in Fig. 13(a). Further evaluation of the predictive models with slenderness ratios and the Euler buckling formula is available in Fig. 13(b). For illustration, the T90-PC column was used with the foregoing material properties. As opposed to the proposed equations, the existing expressions do not explicitly limit the application range of a slenderness ratio; nonetheless, these expressions were employed up to the compressive strength of the GFRP. The critical stresses predicted by Zureick and Scott [4], Tommaso and Russo [5], and Roberts [19] were close to those estimated by the classical Euler approach. Overall, Eqs. (9) and (10) provided sufficiently accurate critical stresses with much simpler design expressions. 7. Summary and conclusions This paper has discussed the failure of pultruded GFRP columns subjected to axial compression. Attention was paid to the influence of geometric discontinuities such as a bolted splice or utility holes on characterizing the column behavior with various slenderness ratios. A hybrid approach was adopted to examine the local- and globallevel failure modes, based on experimental and finite element investigations. Failure mechanisms in conjunction with the geometric discontinuities were elucidated. Simple, yet practical equations were formulated to predict the failure of such column members and were calibrated using probability-based simulation. The design expression showed an average difference of about 25% relative to the 61 test data independently reported by others. The following conclusions summarize technical findings from this research: Stress concentrations near the bolted connection at the column supports hindered achieving the full compression capacity of the GFRP or entailed tear-up and edge-splitting failure modes.
The load-bearing capacity of the column was also affected by the presence of the mechanical splice whose failure was predominantly attributable to the displacement incompatibility between the inner and outer GFRP members bolted together. The symmetric or asymmetric holed walls provided a weak link to the column members and hence a reduction in axial stiffness was noticed. The bolted connection at the supports resulted in hair-line cracks that propagated with local fiber kinking; in some cases, longitudinal bursting cracks were followed because of instantaneous regional load eccentricity. Fiber misalignment and outof-plane displacement were observed as the spliced columns were loaded to failure. The holed columns failed by diagonal shearing with a lateral dilatation effect. The validated finite element models indicated that the buckling capacity of the GFRP columns noticeably decreased beyond a slenderness ratio of 75, while the capacity was linearly improved with an increase in wall thickness. The presence of side holes, either one-side or two-side, did not contribute to the global buckling of the columns since the stability failure took place prior to the crushing of the holed section. The critical stress of the GFRP columns was categorized when inelastic and elastic buckling failure modes occurred and corresponding design expressions were proposed, along with a transition slenderness ratio. The proposed design equations were assessed against existing test data. The equations were much simpler than the ones suggested by others, whereas provided sufficiently accurate failure capacity for pultruded GFRP columns.
Acknowledgments The writers are grateful to acknowledge financial supports provided by International Cooling Tower and the University of Colorado Denver. References [1] Sousa JM, Correira JR, Cabral-Fronseca S, Diogo AC. Effects of thermal cycles on the mechanical response of pultruded GFRP profiles used in civil engineering applications. Compos Struct 2014;116:720–31. [2] Hashem ZA, Yuan RL. Experimental and analytical investigations on short GFRP composite compression members. Compos B 2000;31:611–8. [3] Wong PMH, Davies JM, Wang YC. An experimental and numerical study of the behavior of glass fibre reinforced plastics (GRP) short columns at elevated temperatures. Compos Struct 2004;63:33–43.
Y.J. Kim, K.Z. Qian / Composite Structures 136 (2016) 171–181 [4] Zureick A, Scott D. Short-term behavior and design of fiber-reinforced polymeric slender members under axial compression. J Compos Construct 1997;1(4):140–9. [5] Tommaso A, Russo S. Shape influence in buckling of GFRP pultruded columns. Mech Compos Mater 2003;39(4):329–40. [6] Mottram JT. Determination of critical load for flange buckling in concentrically loaded pultruded columns. Compos B 2004;35:35–47. [7] Nguyen TT, Chan TM, Mottram JT. Influence of boundary conditions and geometric imperfections on lateral-torsional buckling resistance of a pultruded FRP I-beam by FEA. Compos Struct 2013;100:233–42. [8] Boscato G, Casalegno C, Russo S. Performance of built-up columns made by pultruded FRP material. Compos Struct 2015;121:46–63. [9] Turvey GJ, Cerutti X. Flexural behavior of pultruded glass fibre reinforced polymer composite beams with bolted splice joints. Compos Struct 2015;119: 543–50. [10] Bank LC. Composites for construction. Hoboken (NJ): John Wiley & Sons, Inc.; 2006. [11] Khan F, Kim YJ. Time-dependent and residual behavior of pultruded GFRP beams subjected to sustained intensities and cold temperature. Cold Reg Sci Technol 2012;74–75:43–51. [12] Ellingwood B, Galambos TV, MacGregor JG, Cornell CA. Development of a probability based load criterion for American National Standard A58, national Bureau of Standards. Washington (D.C.): NBS Special Publication 577; 1980.
181
[13] Nowak AS, Rakoczy AM, Szeliga E. Revised statistical resistance models for R/C structural components, ACI-SP-284-6. Farmington Hills (MI): American Concrete Institute; 2012. [14] Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and applications of finite element analysis. New York (NY): John Wiley & Sons; 2002. [15] ANSYS. Theory reference for the mechanical APDL and mechanical applications. Canonsburg (PA): ANSYS; 2014. [16] Bathe KJ, Dvorkin EN. A four node plate bending element based on Mindlin– Reissner plate theory and a mixed interpolation. Int J Numer Meth Eng 1985;21:367–83. [17] Rajakumar C, Rogers CR. The Lanczos algorithm applied to unsymmetric generalized eigenvalue problems. Int J Numer Meth Eng 1991;32(5):1009–26. [18] Barbero E, Tomblin J. A phenomenological design equation for FRP columns with interaction between local and global buckling. Thin-Walled Struct 1994;18:117–31. [19] Roberts TM. Influence of shear deformation on buckling of pultruded fiber reinforced plastic profiles. J Compos Construct 2002;6(4):241–8. [20] AISC. Steel construction manual. 14th ed. Chicago (IL): American Institute of Steel Construction; 2011. [21] Vanevenhoven LM. LRFD factors for FRP pultruded wide-flange columns (MS thesis). Madison (WI): University of Wisconsin-Madison; 2007. [22] Boresi AP, Sidebottom OM. Advanced mechanics of materials. Hoboken (NJ): John Wiley and Sons Inc.; 1985.