Determination of critical loads for global buckling of axially loaded pultruded fiber-reinforced polymer members with doubly symmetric cross sections

Abstract

This article proposes a new closed-form equation to determine the reduction factor for global buckling of concentrically loaded pultruded fiber-reinforced polymer struts based on the Ayrton–Perry formula and observed initial out-of-straightness of pultruded fiber-reinforced polymer members measured by other researchers, which makes the original solution recommended by Eurocode 3 easy to be used to predict the global buckling loads of doubly symmetric pultruded fiber-reinforced polymer members subjected to axial compression. The influence of the geometric imperfections of pultruded fiber-reinforced polymer profiles is considered in this new closed-form equation. Validation of the solution including the parameter of the reduction factor for global buckling of pultruded fiber-reinforced polymer columns is performed by comparison with published experimental evidence. In addition, compared with the five closed-form solutions available in the literature, this solution exhibits higher accuracy in predicting the global buckling capacity of concentrically loaded pultruded fiber-reinforced polymer struts with doubly symmetric cross sections. The solution implemented into the new reduction factor equation for global buckling of pultruded fiber-reinforced polymer members can be conveniently used by structural engineers at the preliminary engineering design stage for accurately assessing the reliability and safety of composite structures under concentric compressive loading.

Keywords

column initial out-of-straightness column strength reduction factor doubly symmetric pultruded fiber-reinforced polymer members global buckling loads

Introduction

Pultruded fiber-reinforced polymer (PFRP) profiles are being increasingly used as primary structural members in civil engineering applications such as pedestrian bridges, electricity transmission towers, and offshore structures, due to their favorable properties of lightweight, relatively high tensile strength, excellent corrosion resistance, good fatigue properties, freeze–thaw resistance, low conductivity, ease of handling, transportation and erection, low life-cycle costs, and—increasingly important—positive environmental aspects, such as lower energy consumption and carbon dioxide emission than comparable material systems (Bakis et al., 2002; Castro and De Keller, 2010; Zhan et al., 2014). PFRP shapes consist of thin plates pultruded to form open or closed cross-sectional shapes typically mimicking counterparts found in the structural steelwork (Bank, 2006). From a macro-mechanical viewpoint, PFRP thin-walled profiles can be considered to be linear elastic, homogeneous and transversely isotropic, with the plane of isotropy being normal to the longitudinal axis (i.e. the axis of pultrusion). Generally, due to the thin-walled sectional geometry and relatively low stiffness-to-strength ratio, PFRP shapes are particularly susceptible to buckling before reaching their material strength limit states, that is, the full capacity of the material typically may not be realized. Therefore, buckling is the dominant failure mode for PFRP members. Accurate prediction of buckling strength is essential for the reliable, efficient, and safe design of thin-walled PFRP elements. Determination of a compression member’s concentric buckling load is therefore a fundamental first step in design.

Previous studies of axial behavior of PFRP have identified the following general conclusions as summarized by Cardoso et al. (2014): (a) short columns, those having plate relative slenderness ratios, λ_p = (F_c/F_crl)^0.5 ≤≈0.7, where F_c is the material compressive strength, and F_crl is the local buckling critical stress, are dominated by local buckling of plate elements (e.g. Barbero and Tomblin, 1994); (b) long columns, those having λ_p ≥≈1.3, are dominated by global buckling (e.g. Barbero and Tomblin, 1993; Zureick and Scott, 1997); and (c) the so-called intermediate columns, having slenderness falling between short and long columns, exhibit a complex interaction between local and global buckling (e.g. Barbero and Tomblin, 1994; Cardoso et al., 2014; Lane and Mottram, 2002). The focus of the present study is long columns with doubly symmetric cross sections whose behavior is dominated by global buckling.

For PFRP struts of sufficient slenderness that fail by global buckling, Barbero’s group (Barbero, 2000; Barbero and DeVivo, 1999; Barbero and Raftoyiannis, 1993; Barbero and Tomblin, 1993; Barbero and Trovillion, 1998), Hashem and Yuan (2001), and Seangatith and Sriboonlue (1999) have demonstrated through experimental, numerical, and analytical work that the critical global buckling load can be predicted with reasonable accuracy using the classical Euler formula (Euler, 1933) given by equation (1)

P_{E} = {π^{2} E_{L C} I_{m i n} / (K L_{e f f})}^{2} = π^{2} E_{L C} A_{g} / λ^{2}

(1)

where E_LC is the longitudinal compressive modulus, I_min is the weak axis moment of inertia of section, A_g is the gross cross-sectional area, K is the end-restraint coefficient, L_eff is the effective length, λ = KL_eff/r is the slenderness ratio, and r is the weak axis radius of gyration of section. Currently, the classical Euler formula is adopted by the EUROCOMP Design Code Handbook (Clarke, 1996), Bedford Reinforced Plastics, Inc. (2010), and Creative Pultrusions, Inc. (2004). In general, PFRP members typically have a relatively high ratio of longitudinal elastic modulus to in-plane shear modulus (E_LC/G_LT). Lee and Hewson (1978) presented experimental data for PFRP members subjected to concentric loading and proposed that the critical buckling capacity of PFRP struts can be better estimated using the Engesser shear correction formula (Engesser, 1889) given by equation (2)

P_{E s h 1} = P_{E} / [1 + β P_{E} / (G_{L T} A_{g})]

(2)

where P_E is given by equation (1); β is the cross section shape-dependent shear coefficient. Thereafter, Zureick’s group (Zureick, 1998; Zureick and Scott, 1997), Roberts (2002), Mottram et al. (2003), Bank’s group (Bank, 2006; Vanevenhoven et al., 2010), and Boscato et al. (2014) all advocated using the Engesser shear correction formula to calculate the buckling capacity of PFRP members, since it has more accurate predictions. Barbero and DeVivo (1999) claimed that the effect of shear deformation is usually small on weak axis buckling and can be neglected accordingly. Kardomateas and Dancila (1997) derived a three-dimensional elasticity solution for the problem of an orthotropic hollow cylinder under axial compression and concluded that the classic Euler formula may overestimate the critical capacity of PFRP profiles, whereas the Haringx shear correction formula (Haringx, 1948) may always underestimate the buckling capacity of PFRP sections. The Haringx shear correction formula can be given as

P_{E s h 2} = [\sqrt{1 + 4 β P_{E} / (G_{L T} A_{g})} - 1] [G_{L T} A_{g} / 2 β]

(3)

where P_E is given by equation (1). On the other hand, the critical buckling load equation recommended by Fiberline Composites (2003) for global buckling is

P_{E F} = \frac{N_{C}}{(1 + N_{C} / P_{E})}

(4)

in which

N_{C} = F_{L C} A_{g}

(5)

where P_E is given by equation (1), and F_LC is the longitudinal compressive strength related to the material strength. Based on in-house experimental tests, Strongwell Corporation (2013) also developed empirical equations according to different cross sections for slender doubly symmetric PFRP members, as follows:

For I-sections and wide-flange sections (W-sections)

P_{E S} = 4.9 E_{L C} A_{g} / (K L_{e f f} / r)^{1.7}

(6a)

For square tubes

P_{E S} = 1.3 E_{L C} A_{g} / (K L_{e f f} / r)^{1.3}

(6b)

Although substantial studies have been performed addressing the global buckling behavior of PFRP shapes under concentric compression and five corresponding closed-form solutions (i.e. classical Euler formula (Euler, 1933), Engesser (1889), and Haringx (1948) shear correction formulae and design equations provided by Fiberline Composites (2003) and Strongwell Corporation (2013)) have been proposed, there is little consensus among researchers on the best calculation method for such applications. Moreover, no study has been reported up to the present to evaluate the existing five solutions.

In this article, a new closed-form equation to determine the reduction factor for global buckling of axially loaded PFRP struts was proposed based on the Ayrton–Perry formula (Ayrton and Perry, 1866) and the initial crookedness of PFRP columns measured by other researchers. In this case, the original solution recommended by Eurocode 3 (EC3) (EN 1993-1-1, 2005) can be used to calculate the global buckling loads of doubly symmetric PFRP members subjected to axial compression. This solution considering the effect of geometric imperfections of PFRP profiles was then validated in light of published experimental data. In addition, the performance of this solution was compared with predictions based on the existing five solutions.

New closed-form equations

To develop a closed-form equation of reduction factor for global buckling of PFRP members, the following assumptions are made: (a) the PFRP profiles are considered to be linear elastic, homogeneous, and transversely isotropic, with the plane of isotropy being normal to the longitudinal axis (i.e. the axis of pultrusion); (b) shear strains across the thickness of the PFRP sections are neglected; (c) compared to the stress parallel to the longitudinal axis, the other two normal stresses are considered very small, and thus neglected; and (d) all deflections are small. According to Mottram et al. (2003), there is no evidence to suggest that the residual stresses in PFRP shapes affect column behavior. Therefore, it is assumed that the imperfection of PFRP shapes only includes the initial out-of-straightness (i.e. geometric imperfection).

The well-known Ayrton–Perry formula (Ayrton and Perry, 1866) can be written as

\begin{matrix} σ_{c r} = [F_{L C} + (1 + ε_{0}) σ_{E}] / 2 \\ - \sqrt{{[F_{L C} + (1 + ε_{0}) σ_{E}] / 2}^{2} - F_{L C} σ_{E}} \end{matrix}

(7)

By rearranging equation (7), the ratio of critical compression stress to material compression capacity (i.e. the reduction factor for global buckling, χ) may be expressed as

\begin{matrix} χ = σ_{c r} / F_{L C} = [(1 + ε_{0} + λ_{n}^{2}) - \sqrt{{(1 + ε_{0} + λ_{n}^{2})}^{2} - 4 λ_{n}^{2}}] / 2 λ_{n}^{2} \end{matrix}

(8)

where

ε_{0} = v_{0} A_{g} / S

(9)

λ_{n} = (\frac{λ}{π}) \sqrt{\frac{F_{L C}}{E_{L C}}}

(10)

where σ_cr is the critical buckling stress, σ_E is the Euler buckling stress, ε₀ is the relative initial bending (imperfection), λ_n is the universal slenderness ratio, v₀ is the maximum initial deflection at the midspan of the PFRP member, and S is the section modulus.

The global buckling capacity of PFRP members is therefore

P_{χ} = χ F_{L C} A_{g}

(11)

Equation (5) is originally recommended by the EC3 (EN 1993-1-1, 2005) to calculate the buckling capacity of steel members. Note that the partial factor for equation (11) is not applied. For PFRP profiles, however, an explicit expression for the relative initial bending (ε₀) for global buckling is required.

According to GB 50018-2002 (2002), the relative initial bending can be assumed to take the following form

ε_{0} = A + B λ_{n}^{2}

(12)

To derive the values of A and B in equation (12), experiments by Zureick and Scott (1997), Lane and Mottram (2002), and Mottram et al. (2003), which presented the measured values of initial crookedness of PFRP struts with doubly symmetric cross sections (i.e. wide-flange I-sections and square tubes) that failed with global buckling, were used in this investigation.

Figure 1 plots the variation of relative initial bending (derived from initial deflection) with the universal slenderness ratio of the doubly symmetric PFRP shapes presented in the three papers (Lane and Mottram, 2002; Mottram et al., 2003; Zureick and Scott, 1997). Coefficients A = 0.146 and B =−0.003 were determined from regression analysis, in which the least square method was utilized, although the test data exhibited considerable scatter (Figure 1).

Figure 1.

Variation of relative initial bending with universal slenderness ratio.

Substituting equation (12) into equation (8), the reduction factor for global buckling of PFRP members with doubly symmetric cross sections can be expressed as

χ = σ_{c r} / F_{L C} = [(1.146 + 0.997 λ_{n}^{2}) - \sqrt{{(1.146 + 0.997 λ_{n}^{2})}^{2} - 4 λ_{n}^{2}}] / 2 λ_{n}^{2}

(13)

Considering the degree of scatter inherent in test data, equation (13) can be reasonably simplified as

χ = σ_{c r} / F_{L C} = [(1.15 + λ_{n}^{2}) - \sqrt{{(1.15 + λ_{n}^{2})}^{2} - 4 λ_{n}^{2}}] / 2 λ_{n}^{2}

(14)

Due to this simplification, the prediction accuracy of equation (11) implemented into equation (14) has been changed compared with that of equation (11) implemented into equation (13). To quantify this difference, the comparison of the predictions of equation (11) implemented into equations (13) and (14), respectively, is performed in terms of the related test database (the test database will be introduced in subsequent section), as shown in Table 1. It is clear that the values of average absolute error (AAE) and mean of equation (11) implemented into equation (14) are approximately 2.2% and 0.5% lower, respectively, than those of equation (11) implemented into equation (13). While the standard deviation (SD) values of equation (11) implemented into equations (13) and (14), respectively, are the same. That is, the discrepancy of prediction accuracies between equation (11) implemented into equations (13) and (14), respectively, is less than 2.5%.

Table 1.

Comparison of the predictions of equation (11) implemented into equations (13) and (14), respectively.

Equation	AAE (%)	Mean (%)	SD (%)
Equation (11) (implemented into equation (13))	9.3	103.9	12
Equation (11) (implemented into equation (14))	9.1	103.4	12
Error (%)	−2.2	−0.5	0

AAE: average absolute error; SD: standard deviation.

Error = (equation (11) (implemented into equation (14)) – equation (11) (implemented into equation (13))/equation (11) (implemented into equation (13)).

As a consequence, equation (11) provided by the EC3 (EN 1993-1-1, 2005) can be used to predict the critical global buckling capacity of PFRP struts with doubly symmetric cross sections using equation (12) to account for initial geometrical imperfections.

Experimental validation

To validate equation (11), which takes into account initial geometrical imperfection described by equation (12) and is implemented into equation (14), a comparison with the global buckling capacity of specimens obtained from experiments tested by other researchers was performed. Three series of specimens with doubly symmetric cross sections (W- and I-sections and square tubes) totalizing 18 specimens were selected, as listed in Table 2. The second column in Table 2 describes the details of test specimens. The first character in the specimen name indicates the cross-sectional geometry. The second character in the specimen name designates the ordinal number of specimen. For example, the specimen I3 is the third test PFRP member with I-section. The expression of relative initial bending (ε₀) that is of paramount importance for equation (11) was taken as that defined by equation (12), in which A = 0.146 and B =−0.003 as described previously. Consequently, the test data from which A and B were derived is not used to validate equation (11). The experimental results of the 18 specimens listed in Table 2 were employed to validate equation (11), as shown in Figure 2. It is clear from Figure 2 that the experimental results were in good agreement with equation (11). Meanwhile, Table 2 presents the comparison between the individual test results and predictions from equation (11). The column strength predicted by equation (11) varied from approximately 12.7% higher to 10.8% lower than the corresponding test results. The values of AAE and SD were 7.5% and 7.9%, respectively, while the mean indicated an over-prediction of 3.1%. It can be concluded that the predictions of equation (11) were in reasonable agreement with the corresponding experimental results, indicating the validity of equation (11) as modified by equations (12) and (14).

Table 2.

Typical doubly symmetric PFRP members and comparison with equation (11) capacity.

Source	Specimen’s ID	Specimen details		Specimen properties			Experimental and predicted results
		Cross section (mm)^a	L_eff (mm)	E_LC (MPa)	G_LT (MPa)	F_LC (MPa)	P_Exp (kN)	P_χ (kN)	Error_χ (%)
Cardoso et al. (2014)	S1	S 25.4 × 3.2	416	22,100	2100	224	24.8	27.0	8.9
	S2	S 76.2 × 6.35	1311	31,100	2700	333	210.2	228.0	8.5
	S3	S 76.2 × 6.35	1412	31,100	2700	333	179.8	200.5	11.5
	S4	S 76.2 × 6.35	1677	31,100	2700	333	130.8	146.6	12.1
	S5	S 88.9 × 6.35	1775	29,900	2600	320	186.6	204.3	9.5
	S6	S 88.9 × 6.35	1969	29,900	2600	320	158.7	168.9	6.4
Hashem and Yuan (2001)	S7	S 132 × 9.53	2630	26,700	3738	293.7	401	384.2	−4.2
	S8	S 132 × 9.53	3850	26,700	3738	293.7	191	186.3	−2.5
Seangatith and Sriboonlue (1999)	S9	S 76.2 × 6.35	1527	22,500	3150	218.8	143.9	128.3	−10.8
	S10	S 101.6 × 6.35	1520	22,500	3150	218.8	299.1	286.6	−4.2
Nunes et al. (2013)	I1	I 60 × 120 × 6	1540	29,200	3890	360	24.8	26.5	6.9
Seangatith and Sriboonlue (1999)	I2	I 50.8 × 101.6 × 6.35	1512	22,500	3150	218.8	12.9	13.2	2.5
	I3	I 76.2 × 152.4 × 9.53	2717	22,500	3150	218.8	18.4	20.7	12.7
	I4	I 101.6 × 203.2 × 9.53	906	22,500	3150	218.8	427.4	382.6	−10.5
	W1	W 101.6 × 6.35	2184	22,500	3150	218.8	48.3	50.0	3.5
	W2	W 152.4 × 9.53	2125	22,500	3150	218.8	230.8	255.6	10.7
	W3	W 152.4 × 9.53	2723	22,500	3150	218.8	156.9	159.8	1.9
	W4	W 203.2 × 9.53	2751	22,500	3150	218.8	395.6	364.4	−7.9
					AAE				7.5
					Mean (%)				103.1
					SD (%)				7.9

PFRP: pultruded fiber-reinforced polymer; AAE: average absolute error; SD: standard deviation.

L_eff is the effective length about the minor axis; P_Exp is the global buckling load obtained from experiment; P_χ is the prediction obtained from equation (5); Error_χ = (P_χ – P_Exp)/P_Exp.

Breadth × thickness (b × t) for square tubes (S) or wide-flange I-sections (W); breadth × depth × thickness (b × h × t) for I-sections (I).

Figure 2.

Comparison between equation (11) and the experimental results of typical PFRP specimens.

Comparison of equation (11) and existing closed-form solutions

A database collected from the literature was created to evaluate equation (11) and the existing closed-form solutions, as shown in Table 3. It contains experimental results of 121 PFRP columns with doubly symmetric cross sections, including 61 square tubes, 9 I-section columns, and 62 W-section columns. Note that specimens that only provided the buckling capacity derived from Southwell’s method (Southwell, 1932) were excluded from the database due to anticipated errors (Nunes et al., 2013; Spencer and Walker, 1975).

Table 3.

Test database for global buckling of doubly symmetric PFRP struts.

Source	Specimen details				Specimen properties			P_Exp (kN)
Source	Composite type	Cross section (mm)	L_eff (mm)	M	E_LC (MPa)	G_LT (MPa)	F_LC (MPa)	P_Exp (kN)
Cardoso et al. (2014)	G/P	S 25.4 × 3.2	304		22,100	2100	224	51.6
	G/P	S 25.4 × 3.2	325		22,100	2100	224	38.6
	G/P	S 25.4 × 3.2	374		22,100	2100	224	27.3
	G/P	S 25.4 × 3.2	416		22,100	2100	224	24.8
	G/P	S 25.4 × 3.2	445		22,100	2100	224	20.8
	G/P	S 25.4 × 3.2	464		22,100	2100	224	19.3
	G/P	S 25.4 × 3.2	495		22,100	2100	224	17.5
	G/P	S 25.4 × 3.2	563		22,100	2100	224	13.6
	G/P	S 25.4 × 3.2	653		22,100	2100	224	12.4
	G/P	S 50.8 × 3.2	935		23,700	2200	240	43
	G/P	S 50.8 × 3.2	1182		23,700	2200	240	32.2
	G/P	S 50.8 × 3.2	1716		23,700	2200	240	14.1
	G/P	S 76.2 × 6.35	897		31,100	2700	333	415.5
	G/P	S 76.2 × 6.35	989		31,100	2700	333	280.4
	G/P	S 76.2 × 6.35	1234		31,100	2700	333	196.5
	G/P	S 76.2 × 6.35	1255		31,100	2700	333	242.6
	G/P	S 76.2 × 6.35	1311		31,100	2700	333	210.2
	G/P	S 76.2 × 6.35	1412		31,100	2700	333	179.8
	G/P	S 76.2 × 6.35	1677		31,100	2700	333	130.8
	G/P	S 88.9 × 6.35	1310		29,900	2600	320	320.3
	G/P	S 88.9 × 6.35	1441		29,900	2600	320	287.4
	G/P	S 88.9 × 6.35	1775		29,900	2600	320	186.6
	G/P	S 88.9 × 6.35	1969		29,900	2600	320	158.7
	G/P	S 88.9 × 6.35	2077		29,900	2600	320	143.8
	G/P	S 101.6 × 6.35	1699		28,200	2400	302	246.8
	G/P	S 101.6 × 6.35	1892		28,200	2400	302	208.5
	G/P	S 101.6 × 6.35	1936		28,200	2400	302	198.7
	G/P	S 101.6 × 6.35	2141		28,200	2400	302	170
Hashem and Yuan (2001)	G/P	S 132 × 9.53	2630		26,700	3738	293.7	401
	G/P	S 132 × 9.53	2630		26,700	3738	293.7	397
	G/P	S 132 × 9.53	3850		26,700	3738	293.7	191
	G/P	S 132 × 9.53	3850		26,700	3738	293.7	185
Zureick and Scott (1997)	G/V	S 76.2 × 6.35	2524		27,700	4750	342	55
	G/V	S 76.2 × 6.35	2543		27,900	4700	347	56
	G/V	S 76.2 × 6.35	2232		27,300	5720	344	76
	G/V	S 76.2 × 6.35	1931		28,200	5400	349	97
	G/V	S 76.2 × 6.35	1620		28,300	5120	320	135
	G/V	S 76.2 × 6.35	1310		26,700	5070	357	201
	G/V	S 101.6 × 6.35	2546		26,800	4530	365	140
	G/V	S 101.6 × 6.35	2543		26,600	4610	396	139
	G/V	S 101.6 × 6.35	2229		25,200	4600	359	174
	G/V	S 101.6 × 6.35	1929		26,400	4170	366	229
	G/V	S 101.6 × 6.35	1613		26,700	3940	375	316
Seangatith and Sriboonlue (1999)	G/P	S 44.45 × 6.35	610	C	22,500	3150	218.8	153.9
	G/P	S 44.45 × 6.35	1521	C	22,500	3150	218.8	23.2
	G/P	S 50.8 × 6.35	609	C	22,500	3150	218.8	237.2
	G/P	S 76.2 × 6.35	1527	C	22,500	3150	218.8	143.9
	G/P	S 101.6 × 6.35	1520	C	22,500	3150	218.8	299.1
	G/P	S 101.6 × 6.35	1520	C	22,500	3150	218.8	356.6
	G/P	S 101.6 × 6.35	4862	C	22,500	3150	218.8	42.9
Nunes et al. (2013)	G/P	I 60 × 120 × 6	1540	F	29,200	3890	360	24.8
Seangatith and Sriboonlue (1999)	G/P	I 50.8 × 101.6 × 6.35	1512	C	22,500	3150	218.8	12.9
	G/P	I 76.2 × 152.4 × 9.53	1512	C	22,500	3150	218.8	54.4
	G/P	I 76.2 × 152.4 × 9.53	2112	C	22,500	3150	218.8	33.9
	G/P	I 76.2 × 152.4 × 9.53	2717	C	22,500	3150	218.8	18.4
	G/P	I 101.6 × 203.2 × 9.53	906	C	22,500	3150	218.8	390.3
	G/P	I 101.6 × 203.2 × 9.53	906	C	22,500	3150	218.8	427.4
	G/P	I 101.6 × 203.2 × 9.53	1208	C	22,500	3150	218.8	237.0
	G/P	I 101.6 × 203.2 × 9.53	2112	C	22,500	3150	218.8	81.9
Lane and Mottram (2002)	G/P	W 203.2 × 9.53	3800	C	24,300	3402	267.3	218.5
	G/P	W 203.2 × 9.53	3800	C	24,400	3416	268.4	215
	G/P	W 203.2 × 9.53	3800	C	24,600	3444	270.6	217
	G/P	W 203.2 × 9.53	3500	C	24,300	3402	267.3	254.5
	G/P	W 203.2 × 9.53	3500	C	24,400	3416	268.4	261
	G/P	W 203.2 × 9.53	3500	C	24,600	3444	270.6	253.5
	G/P	W 203.2 × 9.53	3200	C	24,300	3402	267.3	297.5
	G/P	W 203.2 × 9.53	3200	C	24,400	3416	268.4	322
	G/P	W 203.2 × 9.53	3200	C	24,600	3444	270.6	308.5
	G/P	W 203.2 × 9.53	3100	C	24,300	3402	267.3	307.5
	G/P	W 203.2 × 9.53	3100	C	24,600	3444	270.6	319
Mottram et al. (2003)	G/P	W 203.2 × 9.53	6300	C	23,200	3500	291	76.1
	G/P	W 203.2 × 12.7	6300	C	23,200	3500	291	94.4
	G/P	W 203.2 × 12.7	6300	C	23,200	3500	291	100
	G/P	W 152.4 × 9.53	6300	C	23,200	3500	291	28.2
	G/P	W 152.4 × 9.53	6300	C	23,200	3500	291	30.2
	G/P	W 203.2 × 9.53	4800	C	23,200	3500	291	131
	G/P	W 203.2 × 12.7	4800	C	23,200	3500	291	173
	G/P	W 203.2 × 12.7	4800	C	23,200	3500	291	175
	G/P	W 152.4 × 9.53	4800	C	23,200	3500	291	57
	G/P	W 152.4 × 9.53	4800	C	23,200	3500	291	57.6
	G/P	W 203.2 × 12.7	3300	C	23,200	3500	291	354
	G/P	W 203.2 × 12.7	3300	C	23,200	3500	291	349
	G/P	W 203.2 × 12.7	3300	C	23,200	3500	291	348
	G/P	W 203.2 × 12.7	3300	C	23,200	3500	291	351
	G/P	W 152.4 × 9.53	3300	C	23,200	3500	291	112
	G/P	W 152.4 × 9.53	3300	C	23,200	3500	291	113
Zureick and Scott (1997)	G/V	W 101.6 × 6.35	2532		20,100	4330	331	31
	G/V	W 101.6 × 6.35	2515		21,000	4240	320	34
	G/V	W 101.6 × 6.35	2243		21,200	4550	314	42
	G/V	W 101.6 × 6.35	1930		21,600	4270	322	54
	G/V	W 101.6 × 6.35	1634		20,800	4420	331	76
	G/V	W 152.4 × 9.53	2542		17,700	4190	317	138
	G/V	W 152.4 × 9.53	2544		17,800	4470	323	136
	G/V	W 152.4 × 9.53	2209		17,400	4070	304	173
	G/V	W 152.4 × 9.53	1933		18,000	4790	319	221
	G/V	W 152.4 × 9.53	1635		18,400	4440	311	308
	G/V	W 152.4 × 9.53	1313		17,700	4250	315	389
Seangatith and Sriboonlue (1999)	G/P	W 101.6 × 6.35	1214	C	22,500	3150	218.8	170.3
	G/P	W 101.6 × 6.35	1520	C	22,500	3150	218.8	103.1
	G/P	W 101.6 × 6.35	1520	C	22,500	3150	218.8	108.6
	G/P	W 101.6 × 6.35	2184	C	22,500	3150	218.8	48.3
	G/P	W 101.6 × 6.35	2763	C	22,500	3150	218.8	33.9
	G/P	W 152.4 × 9.53	1519	C	22,500	3150	218.8	442.5
	G/P	W 152.4 × 9.53	1519	C	22,500	3150	218.8	449.3
	G/V	W 152.4 × 9.53	1519	C	22,500	3150	218.8	486.2
	G/V	W 152.4 × 9.53	1519	C	22,500	3150	218.8	512.0
	G/V	W 152.4 × 9.53	2125	C	22,500	3150	218.8	230.8
	G/V	W 152.4 × 9.53	2723	C	22,500	3150	218.8	156.9
	G/V	W 152.4 × 9.53	2723	C	22,500	3150	218.8	174.8
	G/V	W 152.4 × 9.53	3632	C	22,500	3150	218.8	96.4
	G/V	W 152.4 × 9.53	4841	C	22,500	3150	218.8	56.0
	G/P	W 203.2 × 9.53	2751	C	22,500	3150	218.8	364.0
	G/P	W 203.2 × 9.53	2751	C	22,500	3150	218.8	395.6
	G/V	W 203.2 × 9.53	2751	C	22,500	3150	218.8	368.5
	G/V	W 203.2 × 9.53	2751	C	22,500	3150	218.8	377.5
	G/V	W 203.2 × 9.53	3653	C	22,500	3150	218.8	222.6
	G/P	W 254 × 12.7	2733	C	22,500	3150	218.8	710.5
	G/P	W 254 × 12.7	2733	C	22,500	3150	218.8	733.0
	G/P	W 254 × 12.7	3642	C	22,500	3150	218.8	397.8
	G/P	W 254 × 12.7	3642	C	22,500	3150	218.8	420.3
	G/P	W 254 × 12.7	4858	C	22,500	3150	218.8	307.7

PFRP: pultruded fiber-reinforced polymer; G/P: glass fiber/polyester; G/V: glass fiber/vinyl ester; M: manufacturer; C: Creative Pultrusions, Inc.; F: fiberline composites.

To further investigate the performance of equation (11) and appraise the existing five solutions, equation (11) and the existing solutions are compared in Figure 3 by means of the experimental results from the database. Design equation recommended by Strongwell Corporation cannot be plotted due to its composition.

Figure 3.

Comparison between equation (11) and the existing equations.

As illustrated in Figure 3, the classical Euler formula overestimated the capacity of slender PFRP columns. This is because geometrical imperfections and second-order effects of members are not taken into account in this formula. While the unavoidable geometrical imperfections make the column resistance results lower than Euler curve. And the more slender the column, the greater influences of second-order effects on the prediction become. The other reason is that the PFRP profiles have a relatively higher ratio of longitudinal elastic modulus to in-plane shear modulus (E_LC/G_LT) than typical isotropic materials for which the Euler formula is based. The effect of E_LC/G_LT ratio on the performance of the Euler, Engesser, and Haringx shear correction formulae and equation (11) is shown in Figure 4 by presenting the relationship between the prediction errors of corresponding equations and the E_LC/G_LT ratio, based on the database (Table 3). And detailed results from Figure 4 are listed in Table 4. As expected, the prediction error of the Euler formula obviously increased with the increase of the E_LC/G_LT ratio. It is clear from Figure 4 and Table 4 that when the E_LC/G_LT ratio ranged from 7.0 to 10.0, the AAE of 8.1% was approximately 4.9% higher than the AAE of 8.1% as the E_LC/G_LT ratio was less than 7.0. When the E_LC/G_LT ratio exceeded 10.0, the AAE of 39.6% was nearly 388.9% higher than the AAE with the E_LC/G_LT ratio less than 7.0. This indicates that the effect of E_LC/G_LT ratio on the performance of the Euler formula becomes more pronounced with the increase of the E_LC/G_LT ratio, especially when the E_LC/G_LT ratio exceeds 10.0. Meanwhile, most of PFRP profiles’E_LC/G_LT ratio is greater than 7.0 (the E_LC/G_LT ratio of 94 PFRP columns (i.e. nearly 71% of the PFRP members) exceeds 7.0 in the database (Table 3)). The effect of transverse shear is therefore more remarkable in PFRP profiles, rendering the Euler formula non-conservative.

Figure 4.

Effect of E_LC/G_LT ratio on the performance of corresponding equations: (a) performance of the Euler and the Engesser shear correction formulae, (b) performance of the Euler and the Haringx shear correction formulae, and (c) performance of the Euler formula and equation (11).

Table 4.

Comparison of the prediction errors of corresponding equations according to E_LC/G_LT ratio.

E_LC/G_LT ratio	Euler	Engesser		Haringx		Equation (11)
	AAE (%)	AAE (%)	Decrease (%)^a	AAE (%)	Decrease (%)^a	AAE (%)	Decrease (%)^a
E_LC/G_LT < 7.0	8.1	6.3	22.2	6.3	22.2	5.8	28.4
7.0 < E_LC/G_LT < 10.0	8.5	7.3	14.1	7.3	14.1	7.3	14.1
E_LC/G_LT > 10.0	39.6	19.6	50.5	22.2	43.9	12.3	68.9

AAE: average absolute error; Euler: the classical Euler formula; Engesser: the Engesser shear correction formula; Haringx: the Haringx shear correction formula.

Decrease = (Euler – X)/Euler, in which X = Engesser, Haringx, or equation (11).

It is also evident from Figure 3 that the Engesser and Haringx shear correction formulae generally overlapped each other and predicted marginally lower capacity than the Euler formula. This is due to the fact that the Engesser and Haringx shear correction formulae are improved by accounting for shear. It can be observed from Figure 4 that the influences of E_LC/G_LT ratio on the performance of the Engesser and Haringx shear correction formulae decreased compared with that on the behavior of the Euler formula, especially when the E_LC/G_LT ratio exceeded 10.0. As illustrated in Table 4, the maximum AAE value of the Engesser shear correction formula was approximately 22.2% lower than that of the Euler formula with the E_LC/G_LT ratio of less than 10.0. When the E_LC/G_LT ratio exceeded 10.0, the AAE of the Engesser shear correction formula was nearly 50.5% lower than that of the Euler formula. The Haringx shear correction formula also displayed similar performance as that of the Engesser shear correction formula. This can explain the reason that the Engesser and Haringx shear correction formulae exhibit better performance than the Euler formula.

It appears that equation (11) could give the most accurate predictions, as shown in Figure 3. This could be attributed to the fact that the initial imperfections of PFRP members are taken into account indirectly. As observed in Figure 4(c) and Table 4, the influence of E_LC/G_LT ratio on the performance of equation (11) was not obvious. It could be inferred that the initial imperfection has more influence on the accuracy of prediction solution than that of the shear deformation for global buckling of doubly symmetric PFRP struts. Meanwhile, the equation recommended by Fiberline Composites was the most conservative compared to the other solutions. Nonetheless, the relationship among equation (11), the classical Euler formula, the Engesser and Haringx shear correction formulae, and the equation provided by Fiberline Composites appear to recommend its veracity. All equations are fundamentally functions of 1/λ_n² and are generally coincident for values of the universal slenderness ratio, λ_n > 4.0.

Table 5 and Figure 5 present comparisons of the prediction statistics obtained from equation (11) and the existing solutions according to the test results of all 121 specimens in the database. The performance of these solutions was quantified by the AAE, mean (M), and SD. Specifically, the AAE and SD values of the classical Euler formula were 18.1% and 23.2%, respectively, and on average, the classical Euler formula overestimated the experimentally obtained capacity by 16.7%. Because the effect of shear deformation is considered in the Engesser and Haringx shear correction formulae, the two solutions exhibited better predictive behavior, overestimating the capacity by only 8.4% and 9.5%, respectively. Empirical design equation recommended by Strongwell Corporation significantly overestimated the experimentally observed capacity (by 118.2%) and exhibited considerable scatter in terms of the veracity of predictions, with AAE = 118.2%. Such great error demonstrates that the prediction of the equation proposed by Strongwell Corporation is unreliable, and this generic equation cannot be used for the global buckling of PFRP profiles with doubly symmetric cross sections. However, the equation proposed by Fiberline Composites underestimated the experimentally measured capacity by 12.5%. As listed in Table 5, equation (11) exhibited the best results with an AAE value of approximately 9%. This solution predicted results very closely, overestimating by only 3.5% on average. Note that the predictions of the Euler formula and the Engesser and Haringx shear correction formulae are the linear buckling loads, while the predictions of the design equations recommended by Fiberline Composites and Strongwell Corporation and equation (11) are the failure loads of PFRP columns. Because of the linear and brittle nature of PFRP profiles, the linear buckling load is almost identical to the failure load. Therefore, the comparisons of the six solutions are reasonable for engineering application considering the main objective of this research.

Table 5.

Statistics of predictions of equation (11) and the existing closed-form solutions.

Closed-form solution	AAE (%)	Mean (%)	SD (%)
Classical Euler formula	18.1	116.7	23.2
Engesser shear correction formula	11.3	108.4	14.1
Haringx shear correction formula	12.0	109.5	15.4
Equations provided by Strongwell Corporation	118.2	218.2	42.2
Equation provided by Fiberline Composites	14.3	87.5	12.1
equation (11)	8.9	103.5	11.7

AAE: average absolute error; SD: standard deviation.

Figure 5.

Average absolute error in the predictions of equation (11) and the existing equations.

The predictive capacities of the solutions presented (except for the equation proposed by Strongwell Corporation) reflect the increased complexity of the parameters considered. The Engesser and Haringx shear correction formulae improve on the Euler formula by accounting for shear. Equation (11) implemented into equation (14) makes a further improvement by accounting for initial imperfections, although equation (11) requires a priori knowledge of the initial imperfections in the form of equation (12). This solution is suitable for structural design application due to its relatively simple and familiar form and satisfactory accuracy.

Summary and conclusion

In this article, the development of a new closed-form equation to determine the reduction factor for global buckling of concentrically loaded PFRP struts was presented. And the solution originally recommended by the EC3 (EN 1993-1-1, 2005) and modified by the above-described new equation was validated and compared with five existing closed-form solutions. Of the results of this investigation, the following should be emphasized:

The classical Euler formula overestimates the capacity of PFRP columns (except for the equation proposed by Strongwell Corporation) in predicting the global buckling loads of concentrically loaded PFRP struts with doubly symmetric cross sections. This is due to the fact that geometrical imperfections and second-order effects of members are not taken into account in this formula. While the unavoidable geometrical imperfections make the column resistance results lower than Euler curve. And the more slender the column, the greater influences of second-order effects on the prediction become.

The Engesser and Haringx shear correction formulae, which are improved by the additional consideration of shear, exhibit better performance than the classical Euler formula. The effect of shear can be more significant in highly orthotropic PFRP that has relatively large values of E_LC/G_LT compared to isotropic materials assumed by the Euler formulation.

The equation recommended by the manufacturer, Fiberline Composites, is observed to underestimate the actual buckling capacity and is the only conservative prediction of those considered. The manufacturer recommended equation proposed by Strongwell Corporation appears to be incapable of reasonably predicting the global buckling loads of doubly symmetric PFRP shapes subjected to axial compression.

Equation (11) that considers initial geometrical imperfection through the new closed-form equation (13) (simplified somewhat as equation (14)) exhibits the best performance compared with the existing closed-form equations and is suitable for structural design application due to its relatively simple and familiar form and satisfactory accuracy.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the National Natural Science Foundation of China (grant nos 51525801 and 51708259), the Australian Research Council (ARC) through an ARC Discovery Grant (grant no. DP160100739) and the Scientific Research Foundation of Jiangsu University of Science and Technology (grant no. 1122931604). Support from these sources is gratefully acknowledged.

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