Flange local buckling of pultruded GFRP box beams

Abstract

An experimental program investigating the flange local buckling (FLB) behavior of pGFRP box-sections is reported. The commonly accepted design equation based on plate theory was validated although importance of accurate assessment of the rotational stiffness of the web-flange junctions was identified. It is concluded that the lower bound solution, assuming the flange is a simply-supported plate subject to uniform compressive stress, results in uniformly conservative predictions of the critical FLB moments. The theoretical solution accounting for flange plate edge support stiffness based only on web stiffness, material and geometric properties of the cross section over predicts the support stiffness resulting in unconservative predictions of FLB behavior. The rotational stiffness of flange-web junction of the pGFRP box-section is also investigated experimentally. It is found that the actual rotational stiffness of flange-web junction is relatively low, closer to the simply-supported boundary condition. The role of fiber architecture at the web-flange junction is identified as affecting this behavior. The conclusions of this study support the use of the lower bound solution for design of pGFRP box-sections.

Introduction

Pultruded glass fiber reinforced polymer (pGFRP) structural profiles are seeing wider adoption in the field of civil infrastructure applications, including pedestrian bridges, cooling towers and low-rise modular structures. As a result, pGFRP section are being applied in a wide variety of axial and flexural load-carrying applications. In the direction of pultrusion (longitudinal axis of member), pGFRP composite materials have tensile and compressive strength comparable to that of mild steel, although the modulus of elasticity is only about one-tenth that of steel. In the transverse direction, strength and modulus vary, but are typically three to five times lower than in the longitudinal direction. The low modulus and high anisotropy, as well as the commonly used thin-walled profiles, result in pGFRP structural members that tend to exhibit large deflections and buckling instabilities before the material strength limit state is achieved. pGFRP flexural members, such as pGFRP I- and box-sections, may experience flange and/or web local buckling (FLB and WLB), global lateral torsional buckling (LTB) or, at intermediate lengths, an instability characterized by the interaction between local and global buckling. The present work concentrates on FLB dominated buckling behavior; in order to mitigate interaction with LTB, the case of flexure about the weak axis of a rectangular box-section is considered.

In the past decades, flange local buckling behavior of pGFRP sections has been investigated by many researchers. Nonetheless, only eight tests reporting FLB of box sections (all 100 × 100 × 4.3 mm) are known [10]. Since few flexural tests on pGFRP box-sections are known in the literature, it is informative to consider studies of the axial load behavior of pGFRP box-sections exhibiting local buckling and flexural tests of I-sections in order to understand some of the factors affecting FLB behavior of pGFRP.

Short axial load carrying pGFRP box-sections (stub columns) exhibit either crushing (not relevant to present study) or local buckling (e.g., [30], [26], [18]). As would be expected for column members, the cross sections studied previously typically have compact flange plates and none of the cited works establish a relationship between wall slenderness and failure mode. Cardoso et al. [14] reported 74 concentric axial load tests of square pGFRP box-sections. Of these, 15 short columns (all having KL/r < 30) having flange width-to-thickness ratios, b/t, equal to 13.9 or 15.8 exhibited FLB-dominated behavior. Cardoso et al. also present a method and examples of the calculation of axial capacity of pGFRP box sections accounting for both local plate (i.e., wall) and global member slenderness and addressing both local and global imperfections. Cardoso et al. [15] presents related closed form equations for local buckling of pGFRP box shape and other sections.

Analysis of previous experimental investigations of FLB behavior of I-sections subject to flexure identify flange slenderness (e.g., [24], [16]) and the relatively poor rotational restraint provided by the web [9] as affecting FLB behavior. Both effects suggest that box-sections may be a more efficient shape for pGFRP flexural members. A limitation of most available previous studies on I-sections is that only two flange slenderness ratios, b/2t ≈ 8 and 12 have been reported. More recently, Vieira et al. [29] reported 62 four-point flexure tests on pGFRP I-sections having flange slenderness ratios ranging from b/2t ≈ 6 to 12. In order for FLB to be observed, b/2t > 8 and the unbraced length, L_b/r_y < 50. Vieira et al. also demonstrated that extant code/standard-based equations for calculating FLB capacity significantly underestimate that seen in experiments on I-sections.

To predict the critical flange local buckling load of pGFRP box-sections, plate theory and the energy method are typically used. Barbero et al. [10] adopted the classic expression of composite plate bending (Eq. (1)) and used an approximate shape function (Eq. (2)) to calculate critical FLB stress.

$$D_{11}\frac{{\partial }^{4}w}{\partial x^2}+2(D_{12}+2D_{66})\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+D_{22}\frac{{\partial }^{4}w}{\partial y^4}+f_x\frac{{\partial }^{2}w}{\partial x^2}=0$$

Where D_ij are the flexural stiffness parameters for a homogenous orthotropic plate (given in list of notations) and w is the out of plane shape given by Eq. (2) in which x is the longitudinal direction of the member and y is the transverse dimension of the compression flange.

$$w(x\text{,}y)={\displaystyle \sum _{m-1}^{\infty }}{f}_m(y)\sin \frac{m\pi x}{a}$$

Prior to Barbero et al., the flange plates of pGFRP sections were commonly assumed to be either simply-supported or fully restrained at the flange edges, which under- or over-estimated the true critical FLB stress, respectively. To overcome this limitation, Barbero et al. proposed to use the transverse plate bending stiffness of the web, $D_{22}^w$, to simulate the elastic rotational restraint of the compression flange at its edges. This significantly improved the accuracy of critical FLB stress predictions. Using Barbero’s approach, however, a system of transcendental equations must be solved, making the approach cumbersome.

Pecce and Cosenza [24] proposed an empirical relationship for predicting the critical FLB stress of pGFRP I-sections, as:

$$f_{\mathit{cr}}=c_1{c}_2\left[\frac{{\pi }^2{D}_{11}^f}{{(b/2)}^2{t}_f}\right]$$

Where c₁ and c₂ are empirical coefficients accounting for rotational restraint at the flange edge support and orthotropy of the material, respectively. The term in brackets is the classic equation for the critical buckling stress of a simply-supported isotropic plate subject to uniaxial compression. Although Pecce and Cosenza successfully achieved a simple closed-formed equation, their approach is highly empirical and, due to the lack of available data on box-sections, is derived considering only pGFRP I-sections. Ascione et al. [3] and Cardoso and Vieira [12] also report closed form solutions for local buckling of pGFRP I-sections.

Qiao et al. [25] employed the same method used by Barbero et al. [10], although with a simpler shape function, to derive the equation for predicting the critical FLB stress of pGFRP box-sections. Qiao et al. proposed a modified expression for the elastic rotational restraint at the flange edge support: modifying $D_{22}^w$ by a factor that depends on the material and geometry of the web plates. Once again, this approach requires the solution of a transcendental equation.

Based on plate theory, Kollár [19], establishing a benchmark for the field, proposed a suite of explicit equations for predicting the critical FLB stress for pGFRP sections. In Kollár’s work, a more refined prediction of the elastic rotational restraint at the flange edge support of pGFRP box-sections was proposed, as:

$$k=\frac{4D_{22}^w}{b_w}\left[1-\frac{(t_f{f}_{\mathit{cr}\text{,}\mathit{ss}}^f)a_{11}^f}{(t_w{f}_{\mathit{cr}\text{,}\mathit{ss}}^w)a_{11}^w}\right]$$

Where $a_{11}^f$ and $a_{11}^w$ account for the thickness and material properties of the flange and web. For most available pultruded shapes, especially box-sections, $a_{11}^f=a_{11}^w$. $f_{\mathit{cr}\text{,}\mathit{ss}}^f$ and $f_{\mathit{cr}\text{,}\mathit{ss}}^w$ are the critical buckling stresses of the simply-supported orthotropic flange and web plates subject to uniform compression, respectively, given as [20]:

$$f_{\mathit{cr}\text{,}\mathit{ss}}^f=\frac{{\pi }^2}{b_f^2{t}_f}\left[2\sqrt{D_{11}^f{D}_{22}^f}+2(D_{12}^f+2D_{66}^f)\right]$$

$$f_{\mathit{cr}\text{,}\mathit{ss}}^w=\frac{{\pi }^2}{b_w^2{t}_w}\left[13.9\sqrt{D_{11}^w{D}_{22}^w}+11.1(D_{12}^w+2D_{66}^w)\right]$$

In Kollár’s expression of the so-called k-factor (Eq. (4)), not only the classic form, $k=D_{22}^w/b_w$, is adopted, but also the flexural behaviors of both flange and web plates are considered.

Existing consensus design guides address FLB of box-sections in variations of the same manner. Each adopts the analytic solution for an infinitely long plate, supported along its transverse edges subject to a uniform compression field.

$$f_{\mathit{cr}}=\frac{{\pi }^2}{t_f{b}_f^2}\left[\alpha \sqrt{D_{11}^f{D}_{22}^f}+\beta (D_{12}^f+2D_{66}^f)\right]$$

Each standard prescribes different values for α and β. EUR 27666 [2] provides lower and upper bounds of FLB critical stress values without additional guidance. The lower bound critical buckling stress corresponds to the simply-supported plate for which α = β = 2, while the upper limit corresponds to the case of a plate fixed against rotation along both transverse edges: α = 4.53 and β = 2.44. The 2010 ASCE Prestandard [1] adopted Kollar’s [19] equations to better define the critical stress between these limits by defining α and β to account for the rotational stiffness of the flange support. By rearranging Kollar’s equation, the following formulations for α and β are found:

$$\alpha =2\sqrt{1+\frac{4.139{\mathit{kb}}_f}{{\mathit{kb}}_f+10D_{22}^f}}$$

$$\beta =2+0.62{\left(\frac{{\mathit{kb}}_f}{{\mathit{kb}}_f+10D_{22}^f}\right)}^2$$

This approach was deemed unnecessarily complex for design equations and is unlikely to find its way into the anticipated ASCE Standard. In any case, the equations promulgated in design guides are based on plate theory solutions and – due to the lack of available data – have not been validated with experimental results.