Optimal design of longitudinal stiffeners of unsymmetric plate girders subjected to pure bending

Highlights

• The optimum stiffener location of the plate with a single stiffener and two stiffeners is at around 0.42D_c and around 0.25D_c and 0.56D_c, respectively, regardless of the asymmetry of the girder section.

• γ_rq obtained according to the AASHTO LRFD requirements is too conservative in cases $\varphi$> 1.0 but not conservative in cases of $\varphi$$\le$1.0.

• The formulas for determining the minimum flexural rigidity requirement of a single and two stiffeners are suggested for practical design purposes.

Abstract

This paper aims at investigating the optimum positions and the required flexural rigidity (γ_rq) of a single and two longitudinal web stiffeners of unsymmetric plate girders subjected to pure bending by using gradient-based optimization algorithms. The optimization procedure is performed into two steps: The first step is to maximize the bend-buckling coefficient k_b generated from eigenvalue buckling analyses, and then the second step aims to minimize the flexural rigidity of the stiffeners (γ). This procedure is implemented using the Abaqus2Matlab toolbox which allows for the transfer of data between Matlab and Abaqus and vice versa. Based on this research, the optimum locations of the stiffeners are recommended for the unsymmetric plate girders under pure bending. Additionally, equations are suggested to determine γ_rq of the longitudinal web stiffeners of the unsymmetric plate girders. These proposed equations are more realistic in the practical designs than those recommended by AASHTO LRFD specifications and several works in the literature in which γ_rq of the longitudinal web stiffeners is determined based on the simply supported plates but not the unsymmetric plate girders under bending.

Introduction

In deep plate girders having slender webs, longitudinal stiffeners are widely used since the presence of the longitudinal stiffeners can enhance the web strength against bend-buckling due to bending stresses by controlling the lateral deflection of the girder webs. Additionally, longitudinally stiffened web can provide improved restraint to the rotation of the compression flanges and consequently result in greatly increased bending strength (Ziemian, 2010; Myung et al., 2016). Research related to longitudinal stiffeners of the plate girders, especially their optimum location and γ_rq under pure bending, has been extensively conducted in the literature. Plate girders with stiffeners are a common structural component in various structures, e.g. bridges and ship hulls. In the latter case, stiffened panels and box girders are usually of critical importance for the ultimate strength of conventional cargo ships under longitudinal bending, the design of SWATH (Small Waterplane Area Twin Hull) ships against transverse bending due to their special sectional configuration (Liu et al., 2018), and the twin-hull inland catamarans which require special design against longitudinal bending due to their shallow draught (Xu et al., 2019).

For the steel plate girder with a single longitudinal stiffener, many researches have proven that the optimum position of the stiffener is at 0.2D from the compression flange for the symmetric girder assuming longitudinal edges of girder webs are simply supported (Cooper, 1967; Azhari and Bradford, 1993; Frank and Helwig, 1995; Maiorana et al., 2011; Vu et al., 2019a). Among these researches, Frank and Helwig (1995) conducted a series of eigenvalue buckling analyses for the optimum stiffener position of the unsymmetric girder, in which the longitudinal edges of the web were presumed to be simply supported. Based on their research, they suggested that the optimum stiffener position is at 0.4D_c (D_c is the web depth in compression in the elastic range), regardless of the asymmetry of the girder section. In addition, they also separately proposed the formulations computing k_b of the stiffened webs depending on the stiffener location (d_s/D_c$\ge$0.4 and d_s/D_c$<$0.4), which have been adopted by AASHTO LRFD (AASHTO, 2012). The AASHTO LRFD (AASHTO, 2012) is the national standard for the design and construction of bridges in the United States published by the American Association of State Highway and Transportation Officials (AASHTO). The AASHTO LRFD specifications (AASHTO, 2012) for k_b of a longitudinally stiffened web plate girder are based on the research of Frank and Helwig (1995). The equations and the associated optimum stiffener position adopted by AASHTO LRFD (AASHTO, 2012) assume simply supported boundary conditions at the flanges.

Performing numerical investigation of the entire steel plate girder under pure bending, Cho and Shin (2011) reported that the optimum stiffener location is achieved when it is located at around 0.425D_c from the compression flange. They suggested new equations to calculate k_b for cases when d_s/D_c$\ge$0.425 and d_s/D_c$<$0.425. More recently, Elbanna et al. (2014) and Kim et al. (2018) recommended that the optimum position of the stiffener is at around 0.42D_c regardless of the asymmetry of the girder section. It is noteworthy that the optimum stiffener location suggested by Cho and Shin (2011), Elbanna et al. (2014), and Kim et al. (2018) is slightly different from AASHTO LRFD (AASHTO, 2012) since they considered the effect of rotational restraints along the web edges due to the presence of the compression flanges.

Regarding the steel plate girder with multiple longitudinal stiffeners, researches related to its optimum locations and γ_rq are very limited. Rockey and Cook, 1965a, 1965b conducted an investigation on the optimum locations of multiple longitudinal stiffeners for the plate girders with doubly-symmetric section subjected to pure bending. In their model the longitudinal edges of the girder web were assumed to be either simply supported or clamped while the vertical edges of the girder web were presumed to be simply supported. It was also assumed that the longitudinal stiffeners were symmetrically located about the mid-plane of the girder web and their flexural rigidity was finite while torsional rigidity was negligible. Based on their results, the optimum positions of multiple stiffeners (up to six) were suggested. Rockey and Cook (1965b) also proposed equations to calculate γ_rq of the stiffeners of the girder webs with aspect ratio lower than 1.6.

The AASHTO LRFD (AASHTO, 2012) bridge design specification only mentions about k_b, the optimum location and γ_rq of a single stiffener but not multiple stiffeners. It has been reported by Viet et al. (Vu et al., 2019a) that by using two stiffeners, k_b can increase by as much as 180%, while the web thickness necessary for design decreases at least by 61.76% compared to the case in which only a single stiffener is used. However, engineers still use the AASHTO LRFD provision to determine k_b of the web plates with multiple stiffeners. Therefore, in practical design, this provision leads to uneconomical design of plate girders with more than one stiffener.

As stated above, the optimum stiffener position has been completely investigated for the girders with a single stiffener but not for multiple stiffeners. However, the γ_rq of one stiffener for the steel plate girders has not been fully studied since the AASHTO LRFD (AASHTO, 2012) recommends γ_rq for the plate girders based on the results obtained from simply supported plates. Moreover, to the best of the authors’ knowledge, the issue of the optimum location and γ_rq of multiple stiffeners for the unsymmetric plate girders in which the influence of the flanges are taken into consideration has not yet been adequately addressed in the literature.

This research aims to examine the optimum locations and γ_rq of a single and two longitudinal stiffeners for the unsymmetric plate girders under pure bending using various optimization procedures. At first, an eigenvalue buckling analysis is performed using Abaqus (ABAQUS, 2014), after that the optimization algorithm is employed to firstly maximize k_b until the optimum stiffener positions are attained, and then the same algorithm is used to minimize γ until the optimum stiffener dimensions are achieved. The Abaqus2Matlab toolbox (Papazafeiropoulos et al., 2017a), which integrates between Abaqus (ABAQUS, 2014) and Matlab (MathWorks and Inc.R, 2017) in a loop, is utilized for the optimization procedure. Based on this study, the optimum locations of a single and two longitudinal stiffeners are suggested for the unsymmetric plate girders subjected to pure bending. In addition, formulas to determine γ_rq of the longitudinal stiffeners are recommended for practical design purposes. The results achieved from these formulas are compared with AASHTO LRFD requirement (AASHTO, 2012) and previous works to demonstrate their efficiency.