Shear behavior of steel plate girders considering variations in geometrical properties

Journal of Constructional Steel Research 153 (2019) 567–577

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Journal of Constructional Steel Research

Shear behavior of steel plate girders considering variations in geometrical properties A. Ghadami, V. Broujerdian ⁎ School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 6 July 2018 Received in revised form 9 October 2018 Accepted 11 November 2018 Available online xxxx Keywords: Steel plate girder Imperfection Shear strength Flange-to-web stiffness ratio Slenderness Finite element analysis

a b s t r a c t This paper presents equations to determine the ultimate shear strength reduction factor of web plate in steel plate girders due to initial imperfection. The proposed equations are based on a parametric study of shear behavior of steel plate girders considering variation in geometrical properties of the section. 112 models with various initial imperfections, flange-to-web stiffness ratios, and web slenderness were analyzed using finite element software ABAQUS. Both geometric and material nonlinearity have been considered. According to the results, the ultimate shear strength proposed by AISC code is conservatively less than the obtained results by FEM, especially for slender plates; the difference between these two values could reach up to 60% in some cases. That is while, for compact and non-compact web plates, maximum difference is about 10%. Furthermore, it is observed that AISC elastic shear buckling strength is conservatively less than the results derived from FEM. This difference in slender and compact web plates reaches to almost 40% and 20%, respectively. This study reveals that the initial geometrical imperfection is an effective parameter on the shear strength of web plates. According to the results, the sensitivity of girders to initial imperfection is greater in non-compact-web girders than the slender or compact ones, and the maximum shear strength reduction is observed about 23% for maximum allowable imperfection as recommended in part1–5 of EC3. Based on the obtained results, a maximum permissible construction tolerance is proposed as well. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction

imperfection sensitivity studies. A brief review of each of them is presented as follows.

1.1. Background 1.2. Literature review Steel plates are widely used in engineering structures. Plate girders of bridges and buildings, gas and liquid containment structures, shelters, offshore structures, vessel fuselages, and steel plate shear walls are among the engineering applications of such elements. Generally, a large portion of the shear force and bending moment in a plate girder is resisted by web panel and flange plates, respectively. It is common to design plate girders with deep slender webs so as to obtain a high strength-to-weight ratio. This results in slender web plates which are prone to shear buckling. Various parameters affect the shear buckling behavior of the plates. Some of these parameters include material properties, web panel boundary conditions, aspect ratio, curvature, initial imperfections, and slenderness of web and flange plates. Accordingly, many researchers in the past century have dealt with the evaluation of these parameters. In view of the discussion above, and considering the scope of the present paper, studies on the web panel behavior of steel plate girders may be categorized into shear behavior studies and ⁎ Corresponding author. E-mail address: [email protected] (V. Broujerdian).

https://doi.org/10.1016/j.jcsr.2018.11.009 0143-974X/© 2018 Elsevier Ltd. All rights reserved.

1.2.1. Studies on the shear behavior of steel plate girders Steel plate girders are generally composed of compact or noncompact flanges and compact, non-compact, and slender webs. Any flat rectangular plate subjected to in-plane shear may buckle before, after or simultaneously with its yielding [1]. A slender plate under shear loading buckles elastically at early stages of loading, and may experience geometric and material nonlinearities during its post-buckling behavior [2]. The load–displacement history for such a plate is divided into four stages of elastic buckling, formation of diagonal yield zone, ultimate capacity, and softening [1]. In contrast, a stocky or compact plate yields prior to buckling, and no post-buckling capacity is expected in this respect [3]. The applied loading history of compact plates is divided into three stages, namely, material reaching the limit of proportionality, plastic buckling and post-plastic buckling softening [1]. In regard to non-compact plates, material yielding and geometric instability are nearly simultaneous [3], and the applied loading history of these plates is divided into two stages, namely, material reaching the limit of proportionality, followed by inelastic post-buckling softening stage [1].

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The earliest study on the post-buckling behavior and strength of web panels of plate girders is credited to Wilson [4,5]. After the pioneering work of Wagner [6] who proposed a diagonal tension theory developed for very thin web panels made from aluminum with applications in aeronautical engineering, a number of studies have been conducted on the tension field action of plate girders. A summary of these can be found in SSRC [7]. Basler et al. [8] studied the behavior of girders experimentally. Their tests were designed to investigate the combined bending and shear actions. They concluded that the post-buckling strength of girders cannot be simply expressed on the basis of the critical stress as is often expected. This implies that the strength of plate girders is not merely a function of the web slenderness ratio. The first plastic design method for predicting the ultimate shear strength of plate girders was proposed by Basler [9]. He assumed the flanges to be flexible and unable to withstand the lateral loading imposed by the inclined tension field. Noteworthy is that the Basler's model [8,9] is now the base of AISC 360–16 [10] to evaluate the shear strength of steel plate girders. Hoagland [11] investigated the rotated stress field method with some modifications, and summarized the results of 366 tests on steel plate girders as well as 93 tests on aluminum alloy plate girders in shear. In that research, the rotated stress field method was also adopted in Eurocode 3 [12]. In 2006, Yoo et al. [5] examined a fundamental assumption used in most classical failure theories for post-buckled web plates under shear. Based on this assumption, the compressive stresses developed in the direction perpendicular to the tension diagonal are not increased after elastic buckling. According to their results, the diagonal compression continuously increases in close proximity to the edges after buckling, thereby producing a self-equilibrating force system in the web panel. This force system does not depend on the flanges and stiffeners. Lee et al. [13] investigated the post-buckling behavior of web panels with high aspect ratios through nonlinear finite element method (FEM) analysis and developed a new approach of predicting the ultimate shear strength. According to the results, Basler's equation [8,9] is not applicable to web panels with a/D ≥ 3 (a: the distance between transverse stiffeners, D: web plate depth). In such panels, Basler's equation underestimates both the post-buckling strength and elastic buckling strength. Alinia et al. [14] evaluated the nonlinearity in the post-buckling behavior of shear plates. According to that research, although material yielding starts on one face prior to the other one, both faces have full yield bands at the ultimate load. In addition, a large portion of the post-buckling behavior of slender shear panels is governed by geometric nonlinearity, while material nonlinearity is activated only in the final stages. Gheitasi and Alinia [1] used the nonlinear large deformation FEM analysis to describe the shear buckling behavior of plates, and qualitatively and quantitatively categorized commonly utilized unstiffened metal shear plates into slender, moderate and stocky plates. Alinia et al. [15] investigated the state of stresses in steel girder web plates. It was found that the principal tensile and compressive stresses on the two sides throughout the web plate increase during the post-buckling stage, Furthermore, all available theories are conservative in predicting the elastic buckling strength of web plates. However, except EC3 [16], they overestimate the ultimate strength of plate girders. Daley et al. [17] conducted a research on the shear behavior of unstiffened Igirders and compared the results of experimental tests with predictions from models recommended by Basler [8,9], Höglund [18,19], and Lee et al. [20]. According to the results, Basler's model is shown to be accurate for webs with slenderness between 60 and 100. However, it is very conservative for girders with typical web slenderness ratios. Höglund's methods are found to be accurate to conservative. The method proposed by Lee et al., which aims to predict the post-buckling shear strength of webs with widely spaced stiffeners, is moderately accurate. 1.2.1.1. Elastic shear buckling strength 1.2.1.1.1. Basler's method. Basler [8,9] was the first to develop a method for considering the reserve strength of plate girders constructed

from hot-rolled plates in their post-buckling region. This reserve strength was caused by the action of tension field. According to the proposed assumptions, the pure shear continues to act on the web until shear buckling load reaches. After this stage, the principal compressive stress remains constant [5,9,17]. One may arrive at the renowned concept that states in plate girders reinforced with transverse stiffeners, the tension field action is required to be restrained with the aid of flanges and stiffeners so that the webs are able to develop their desired post-buckling strength [5]. However, in a series of analytical and experimental studies, Lee and Yoo [21,22] and Lee et al. [23,24], showed that the flanges and transverse stiffeners do not necessarily behave as anchors. In Basler's method, the elastic shear buckling strength of web plates, Vcr, subjected to pure shear loading is computed using the classical plate buckling equation, which is given by Timoshenko and Gere [25] as. Vcr ¼ K

π2 E tw 3 2 12 ð1−υ Þ D

ð1Þ

where, E is the Young's modulus, v is the Poisson's ratio, tw and D are the web plate thickness and depth, respectively, φ is the web panel aspect ratio, a is the transverse stiffeners distance relative to each other, and K is the elastic shear buckling coefficient which is dependent on the boundary conditions and aspect ratio of the web panel [21,26]. It must be noted that the elastic shear buckling coefficient, Kss, of a web with simply supported conditions at all four edges was originally used in Basler's method [17,21,26] as. Kss ¼ 4 þ

5:34 ϕ2

Kss ¼ 5:34 þ

ϕ ¼ a=Db1

for

4 ϕ2

for

ð2aÞ

ϕ ¼ a=D≥1

ð2bÞ

Basler assumed that web panels behave similar to simply supported plates, since flanges are not stiff enough to provide a desirable torsional rigidity. 1.2.1.1.2. Basler's modified method. According to Lee et al. [20] who investigated the shear behavior of steel I-girders, a new method named as Basler's modified method was proposed to calculate the elastic shear buckling coefficient which depends on the ratio of flange thickness to web thickness, tf/tw. The real boundary condition of web panel was found to be closer to the clamped case in the range of practical design parameters of plate girders. In addition, to calculate the elastic shear buckling load using Basler's method, they proposed the following relations: Ksf ¼

5:34 ϕ2

þ

Ksf ¼ 8:98 þ

2:31 þ 8:39 ϕ−3:44 ϕ 5:61 ϕ2

−

1:99 ϕ3

for

for

ϕ ¼ a=Db1

ϕ ¼ a=D ≥1


 4 2 t 2− f KLee ¼ Kss þ ðKsf −Kss Þ 1− 5 3 tw 4 KLee ¼ Kss þ ðKsf −Kss Þ 5

for

ð3aÞ

tf ≥2 tw

for

ð3bÞ

0:5 ≤

tf b2 tw

ð4aÞ

ð4bÞ

where Ksf is the elastic shear buckling coefficient of plates with two opposite simply supported edges and two other fixed edges. Eq. (3b), which signifies the usual case in steel plate girders, may imply that the web-flange junction is 80% clamped. 1.2.1.2. Ultimate shear strength. Höglund [11] recommended a modified version of his previous method [18,19], which is the basis of the

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569

In addition, KAISC is the elastic shear buckling coefficient according to AISC specifications written as. KAISC ¼ 5 þ

5 ϕ2

for ϕ ¼ a=Db3

ð5eÞ

KAISC ¼ 5 for ϕ ¼ a=D≥3

Fig. 1. Boundary condition and loading pattern of models in the FE analysis (1 in. = 25.4 mm).

Eurocode 3 [16] shear-strength provisions. An adaptation of Höglund's method [11] for non-rigid end posts was proposed by the Daley et al. [17], vetted by the AISC Specification Committee, and adopted into the AISC 360–16 [10] specification. According to AISC 360–16 (Chapter G) [10] which is inspired by Basler's theory, the design shear strength of web plates, Vu = ϕvVn, with relatively small initial geometric imperfection, D/120000 [21], shall be determined as. " V n ¼ 0:6σ yw h t w

1:51 K AISC E ðD=t w Þ2 σ yw

# for

λw N1:37

ðElastic BucklingÞ ð5aÞ

" V n ¼ 0:6σ yw h t w

1:1 ðD=t w Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi# K AISC E for 1:1bλw ≤1:37 ðInelastic BucklingÞ σ yw ð5bÞ

V n ¼ 0:6σ yw h t w

for

λw ≤1:1

ðPlastic BucklingÞ

ð5cÞ

ð5fÞ

1.2.2. Imperfection sensitivity studies Initial imperfections such as deviations from nominal geometric parameters, load eccentricities, and variations in material properties are inevitable in metallic structures [27]. Owing to the manufacturing process, plate girders are also subjected to geometric imperfections. The amplitude and shape of the aforementioned imperfections have random nature and can reduce the load bearing capacity of structural elements in compression [28–30]. Various experimental studies have been carried out on shell buckling [31,32]. The obtained buckling loads were significantly lower than those obtained through analytical means that used simplified bifurcation approaches on perfect shells having ideal supports and nominal dimensions [33]. A major source of difference between experimental and analytical methods initiates from induced nonconformities with respect to a geometrically perfect shell, as noted by Kármán and Tsien [34], Donnell and Wan [35], and Koiter [36]. In addition, developing a general theory on sensitivity to imperfection has been addressed by Koiter [36,37], Budiansky and Hutchinson [38], Stein [39] and Arbocz [40] among others such as [30]. Koiter [41] developed a general theory for the buckling and post-buckling behavior of elastic systems incorporating imperfection sensitivity [42]. This theory presented an approximate sensitivity law to estimate the reduced buckling load due to small shape imperfections. Later, Koiter's law was implemented to find the maximum supported load up to the onset of buckling and to relate the buckling strength to the magnitude and shape of initial imperfections [43]. A nearly theoretical study on the effect of initial imperfections on the buckling load was then proposed by Featherston [41]. According to the Koiter's imperfection power law [36], the reduction of the critical load (~f ) can be approximated as a multiple of the powered imperfecc

where Vn is nominal shear strength, ϕv = 0.9 is the resistance factor for shear, σyw is the web plate yield stress, h is the girder overall depth, and λw is the non-dimensional web slenderness parameter determined as. λw ¼

pffiffiffiffiffiffiffiffiffi σ yw D pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tw KAISC E

ð5dÞ

tion amplitude (ϵ ρ) as fc ≈ Cϵρ , in which the power (ρ) is equal to 1 for a limit point and equal to 2/3 for a simple unstable symmetric bifurcation point, and C is a constant. Hutchinson [44] investigated the postbuckling behavior of structures undergoing plastic deformation, and pointed out the importance of imperfections in plastic buckling. For a slender web under patch loading, Bergfelt [45] recognized that an initial shape deformation which has the same shape as the future buckling mode may decrease the strength more than other shapes. However, an initial deformation counteracting the buckling mode might increase

Table. 1 Model characteristics (1 in. = 25.4 mm). Model

a (mm)

φ = a/D

D (mm)

tw (mm)

bf (mm)

tf (mm)

λwa

Web plate classificationb

Flange plate classificationb

PG1 PG2 PG3 PG4 PG5 PG6 PG7 PG8 PG9 PG10 PG11 PG12 PG13 PG14

750

1

750

300

1500

2

5 6 5 8 8 12 12 5 6 5 8 8 12 12

10 10 20 10 20 10 20 10 10 20 10 20 10 20

1.646 1.409 1.646 1.09 1.029 0.753 0.703 1.858 1.608 1.858 1.269 1.161 0.895 0.804

S S S NC NC C C S S S NC NC C C

NC NC C NC C NC C NC NC C NC C NC C

a b

Slenderness parameter of web plate. C = compact, NC = non-compact, S = slender.

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300

Shear Load (kN)

250 200 150

FEM Failure mode at ultimate load Current paper FEM

100

Test [22]

50 0 0

1

2

3

4

5

6

7

8

Displacement (mm)

9

10

Test Failure mode at ultimate load [22]

Fig. 2. Comparing the experimental and finite element analysis failure modes.

the strength. It was also found that four factors should be considered when analyzing initial shape imperfections, i.e. magnitude, shape, slenderness of the element and the flange-to-web stiffness ratio. Lee et al. [21], studied the effects of initial out-of-flatness on the shear strength of transversely stiffened plate girder web panels subjected to pure shear. According to obtained results, for web panels having low web slenderness ratios (D/tw), the ultimate shear strength was significantly reduced as the initial distortion increased. Here, D represented the depth of the web plate, and tw was the web plate thickness. On the one hand, examination of FEM data reveals that, when the elastic shear buckling strength is greater than the shear yield strength, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vp = 0.6 σywhtw (i.e. when D=t w ≤1:4 K E=σ yw with K representing the elastic shear buckling coefficient), the ultimate shear strength is considerably reduced (approximately by 20%) due to large initial deformations as much as D/120, as a reasonable upper-bound value of the

permissible out-of-flatness of web panels. On the other hand, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi strength reduction in web panels with D=t w ≥2:8 K E=σ yw as a result of larger initial deformations is found to be so inconsequential that it may be safely neglected. For the range of web slenderness ratios between the two above-mentioned ranges, the strength reduction varies almost linearly from 20 to 0%. Various tolerances for the amplitude of imperfections are suggested in the existing specification codes based on different experiences. EN 1993-1-5 Annex C [16] recommends using imperfections based on the plate's critical buckling mode and the maximum amplitude of 0.005 times the smallest dimension of the panel. Based on investigations carried out by Moffiin and Dwight [34], this value was assumed to be an upper bound for geometric imperfections in a welded aluminum plate. According to Ref. [16], the largest amplitude of this critical shape can be scaled to 80% of the fabrication tolerances. Currently, fabrication

Fig. 3. PG1, PG8, and PG13 buckling mode shapes (1 in. = 25.4 mm, 1 kip = 4.45 kN, 1 psi = 6.895 kPa).

A. Ghadami, V. Broujerdian / Journal of Constructional Steel Research 153 (2019) 567–577

tolerances limit this out-of-flatness to the lowest value between two magnitudes. The highest value of possible plate imperfection amplitude as a function of thickness and width was studied by Carlsen and Czujko [46] and Antoniou [47]. Carlsen and Czujko [46] took advantage of statistical and numerical methods regarding the geometrical specifications of distortions and compressive strength of plates in order to predict the tolerances of distortions in the post-buckling range in an easy fashion. Antoniou [47] analyzed a considerable number of measurements carried out on the panel central deflection of ships over time. The affecting parameters on this deflection were the plate aspect and slenderness ratios as well as the thickness of stiffeners and weld throat. Using a regression analysis, a new set of equations for maximum deflection was obtained for different cases. Featherston [30] investigated the effects of various shape and amplitude imperfections on the buckling and post-buckling behavior of flat panels subjected to combined compressive and shear loads. It was shown that initial imperfection dramatically affects the buckling load of curved plates under compression. Maiorana et al. [48] examined the stability of imperfect web plates subjected to patch loads and concluded that their first eigenmode can be assumed as the initial configuration in the post-buckling behavior studies. For small values of curvature, the buckling behavior of curved panels under shear is fairly similar to flat plates. Consequently, it is also sufficient to use initial imperfections based on only the first eigenmode shape for small-curvature panels. In order to estimate the ultimate load of specific structures considering geometric imperfections, some models have been developed [28]. Hassanein [49] investigated the shear behavior and strength of laterally unsupported I-shaped austenitic stainless steel plate girders of nonrigid end stiffeners. Two different initial imperfection magnitudes of D/100000 and D/100 are included in the model, where D denotes the web-depth of the plate girders. An arbitrary small initial imperfection value of D/100000 is incorporated in an attempt to determine a bifurcation-type buckling load. D/100 is the maximum value allowed by the Bridge Welding Code [50].The results showed that the design equations specified by the European specifications [12,51] are conservative for the design of austenitic stainless steel plate girders with the maximum imperfection amplitude, D/100. In 2011, Graciano et al. [28], conducted a parametric study by considering the position of the stiffener as well as the amplitude and shape of imperfection on the post-buckling behavior of longitudinally stiffened plate girder webs subjected to patch loading. They concluded that the amplitude of imperfections results in a reduction in patch loading resistance in most cases. Also, initial imperfection for patch loaded girder webs can be modeled using a shape resembling either the first eigenmode or a sinus-wave, while the effect of this imperfection diminishes if its maximum amplitude is maintained within limited values (w b D/200).

571

Table. 2 Results of the finite element analysis (FEA) for elastic shear buckling loads. Model tf/tw

PG1 PG2 PG3 PG4 PG5 PG6 PG7 PG8 PG9 PG10 PG11 PG12 PG13 PG14

2 1.667 4 1.25 2.5 0.833 1.667 2 1.667 4 1.25 2.5 0.833 1.667

λw-Lee Kss

1.646 1.409 1.646 1.09 1.029 0.753 0.703 1.858 1.608 1.858 1.269 1.161 0.895 0.804

Ksf

9.34 9.34 9.34 9.34 9.34 9.34 9.34 6.34 6.34 6.34 6.34 6.34 6.34 6.34

12.6 12.6 12.6 12.6 12.6 12.6 12.6 10.13 10.13 10.13 10.13 10.13 10.13 10.13

KAISC KLee

10 10 10 10 10 10 10 6.25 6.25 6.25 6.25 6.25 6.25 6.25

Elastic shear buckling load (kN)

11.948 11.368 11.948 10.644 11.948 9.92 11.368 9.375 8.701 9.375 7.858 9.375 7.014 8.701

VcrFEM

VcrAISC

VcrLee

434.54 742.11 438.34 1701.4 1751.04 5219.32 5598.9 293.89 493.58 307.37 1091.79 1224.58 3185.1 3881.5

301.27 520.6 301.27 1234.01 1234.01 4164.78 4164.78 188.29 325.37 188.29 771.25 771.25 2602.98 2602.98

359.96 591.84 359.96 1313.48 1474.39 4131.27 4734.7 282.44 452.95 282.44 969.62 1156.88 2921.36 3623.59

Note: 1 in. = 25.4 mm, 1 kip = 4.45 kN, 1 psi = 6.895 kPa

FEM models for a thorough parametric study are developed here. Using these models, the effect of the above-mentioned parameters on the elastic shear buckling strength and shear strength of girders are studied. • According to the literature, one observes that there is a lack of research focusing on shear strength of steel plate girders under pure shear in case of imperfections. Moreover, it was also found that four main factors should be considered when analyzing initial imperfections, including size, shape, and slenderness of the element and the flange-to-web stiffness ratio [28]. In this article, to evaluate the shear strength of steel plate girders in view of geometric imperfections, three parameters including the non-dimensional web plate slenderness parameter, flange-to-web stiffness ratio, and imperfection magnitude are selected. In addition, based on the analyses on a large number of web panels with different initial distortions, the equations for determining the shear strength reduction are formulated. Also, in order to achieve a reasonable and optimum strength, the maximum permissible construction tolerance is proposed.

2. Materials and methods 2.1. Details and geometric properties of models In order to investigate the research objectives, 112 models including 14 girders with slender, non-compact and compact webs, along with non-compact and compact flanges were constructed using the FEM

1.3. Aims and scopes Increasing use of thin-walled structures has augmented the need for more in-depth research on the shear buckling behavior of these elements. Various parameters affect the buckling behavior of plates including material properties, loading and boundary conditions, aspect ratio, curvature, initial imperfections, and slenderness of plates [52]. Considering these parameters, the key novelties of the present study are summarized as follows: • According to the literature, most researches on the shear behavior of plates have discussed their results based on web plate slenderness ratio parameter, D/tw, which is unable to take the effect of web plate boundary conditions on the shear behavior of web panel into account [13,17]. Accordingly, conducting a comprehensive study which considers the effects of the flange-to-web stiffness ratio, aspect ratio, and non-dimensional web plate slenderness parameter (λ) on the elastic and plastic shear buckling behavior is necessary. So several

AISC

Basler's Modified method

1.2 1 0.8 0.6 0.4 0.2 0 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

Slenderness Parameter (λw-Lee) Fig. 4. The ratio of VcrAISC and VcrLee to the FEM results vs. λw-Lee.

2.5

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600

1200

500

1000

1600 1400

300 200 100

Shear Load (kN)

FEM-Vcr PG1- Imp=t/1000 PG1- Imp=t/500 PG1- Imp=t/200 PG1- Imp=t/100 PG1- Imp=t/50 PG1- Imp=t/10 PG1- Imp=D/200 PG1- Imp=0.8D/100

Shear Load (kN)

Shear Load (kN)

1200

400

800 600 PG4- Imp=t/1000 PG4- Imp=t/500 PG4- Imp=t/200 PG4- Imp=t/100 PG4- Imp=t/50 PG4- Imp=t/10 PG4- Imp=D/200 PG4- Imp=0.8D/100

400 200

0 5

10

15

800

PG6- Imp=t/1000 PG6- Imp=t/500 PG6- Imp=t/200 PG6- Imp=t/100 PG6- Imp=t/50 PG6- Imp=t/10 PG6- Imp=D/200 PG6- Imp=0.8D/100

600 400 200 0

0

0

1000

0

Out of Plane Displacement (mm)

5

10

0

15

5

10

15

Out of Plane Displacement (mm)

Out of Plane Displacement (mm)

Fig. 5. Shear Load vs out-of-plane displacement curves for PG1, PG4, and PG6 models (1 in. = 25.4 mm, 1 kip = 4.45 kN, 1 psi = 6.895 kPa).

software, ABAQUS. As shown in Fig. 1, two girders with span lengths of a = 750 mm and a = 1500 mm, equal web depth of D = 750 mm and equal flange width of bf = 300 mm are selected. Considering these two girders, 14 models with various values of web and flange thickness are generated. The details and names of the models are presented in Table 1. Selecting a wide range of thickness and slenderness parameters provides the possibility for studying the elastic/plastic buckling behavior of slender and stocky plates. To investigate the effect of geometric imperfection on the ultimate shear strength of steel plate girders, eight different initial imperfection magnitudes of tw/1000, tw/500, tw/200, tw/100, tw/50, tw/10, D/200, and 0.8D/100 are included in the models. It must be noted that tw/ 1000 is selected as a minimum for the bifurcation-type buckling and 0.8D/100 is selected as 80% of the maximum allowable imperfection, as recommended in part 1–2 of EC3 [53]. The plate girders are denoted by PGi − Imp = j, in which i is the plate girder number, and j is the geometric imperfection magnitude applied to the finite element model. As an example, PG2-Imp = tw/200 represents the PG2 girder with the details as shown in Table 1 and with an initial geometric imperfection magnitude of tw/200. As shown in Fig. 1, boundary conditions and loading pattern of the plate girders are chosen to be as close to pure shear condition as possible. In the current study, the aforementioned boundary conditions and loading patterns are inspired from the boundary conditions of flat plate as mentioned in references [54–56]. To eliminate the lateral torsional buckling phenomenon, the entire mid-nodes of flanges are constrained in the z direction. Furthermore, to avoid distortions in the vicinity of shear force and to evaluate the realistic behavior of web middle panels in girders, transverse stiffeners with the thickness of 12 mm are used.

2.2. Material properties The material behavior of steel is assumed to be elastic-perfectly plastic. The flange material has a Young's modulus of 204 GPa, normal yield stress of 274.5 MPa, and Poisson's ratio of 0.3. In addition, the web plate material has a Young's modulus of 200 GPa, normal yield stress of 287.8 MPa, and Poisson's ratio of 0.3. It must be noted that these material properties are selected from TG3 test model in Ref. [57]. The VonMises yield criterion and the associated flow rule are used so as to define the plasticity of the steel.

2.3. Numerical modeling and validation 2.3.1. Numerical modeling The numerical modeling and analysis are performed using ABAQUS [58]. The S4R four-node quadrilateral doubly curved general-purpose shell element with reduced integration and a large strain formulation is used for modeling the plates. Both geometric and material nonlinearity are considered in the analysis. The modified Riks method, which is capable of identifying the exact equilibrium path, is implemented in the analysis. In this method, the exact post-buckling problem cannot be directly analyzed due to the discontinuous response at the point of buckling. Thereby, by applying initial imperfections, the problem must be turned into one with continuous response rather than with bifurcation. In other words, if the model is defined as ideal without initial imperfection, the buckling phenomenon may not be observed even in slender elements and the system yields under a specific load [52]. The most commonly used technique for modeling imperfection distributions is either to adopt a sinusoidal wave or to use one of the eigenmodes obtained from elastic buckling analysis [27,59]. Generally, the

Table. 3 Results of the finite element analysis (FEA) for ultimate shear loads. Ultimate shear load (kN) Geometric imperfection magnitude Model

t/1000

t/500

t/200

t/100

t/50

t/10

D/200

0.8D/100

PG1 PG2 PG3 PG4 PG5 PG6 PG7 PG8 PG9 PG10 PG11 PG12 PG13 PG14

499.87 727.74 505.28 1001.52 1005.84 1500.49 1511.52 416.28 572.09 429.41 992.19 996.11 997.81 1505.1

499.78 722.54 505.17 998.98 1000.56 1504.06 1511.52 416.25 571.57 429.37 990 994.6 996.34 1503

499.51 712.69 504.91 999.55 1001.92 1499.87 1511.5 416.15 571.69 429.28 984.61 991.67 994.28 1500.72

499.06 702.74 504.47 998.78 1001.26 1502.26 1511.44 416 571.22 429.08 976.44 986.2 990.73 1500.63

498.18 690.58 503.59 997.77 999.5 1503.79 1511.17 415.67 570.28 428.75 962.43 975.83 983.57 1500.31

491.78 663.29 497.16 984.49 987.97 1495.85 1502.88 413.24 563.5 426.03 914.65 928.11 940.37 1492.11

450.77 597.37 455.8 902.52 906.6 1475.39 1478.92 392.61 528.4 402.91 844.33 841.31 881.19 1443.86

428.32 565.64 434.33 857.17 861.67 1440.01 1445.27 378.35 506.74 387.07 810.79 798.65 846.72 1389.17

Note: 1 in. = 25.4 mm, 1 kip = 4.45 kN, 1 psi = 6.895 kPa

A. Ghadami, V. Broujerdian / Journal of Constructional Steel Research 153 (2019) 567–577

573

t/1000

t/500

t/200

t/100

t/1000

t/500

t/200

t/100

t/50

t/10

D/200

0.8D/100

t/50

t/10

D/200

0.8D/100

1.2

1.2

1

1

K=KAISC

0.8

K=KLee

0.8

0.6

0.2

0.4

Elastic buckling

Inelastic buckling

0.4

Plastic buckling

Plastic buckling

Inelastic buckling

0.6

0.2

Elastic buckling

0

0 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

Slenderness Parameter (λw-AISC)

a)

0.5

2.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

2.5

Slenderness Parameter (λw-Lee)

b)

Fig. 6. The ratio of AISC to FEA shear strength vs. slenderness parameter.

probable constructional imperfection of steel plates is considered as a multiple of the first mode shape of buckling obtained by an elastic buckling analysis [60]. In this article, by applying the shear load using the traction load command of ABAQUS, the first eigenmode and the elastic shear buckling strength, Vcr, of each girder is obtained by eigenvalue analysis. Afterwards, this eigenmode is amplified with a certain magnitude (Imp) and then introduced as an initial girder shape for nonlinear static analysis (Riks method) [28]. According to this nonlinear analysis, the ultimate shear strength of each model is obtained. Concerning the convergence study on mesh size, a number of mesh sensitivity analyses are carried out based on the difference between the elastic shear buckling load obtained via ABAQUS and the results of Basler's method using shear buckling coefficient proposed by Lee et al. (see Eq. (1)). Moreover, the element size of 20 mm was taken as the minimum and optimum requirement in the analysis.

geometrical imperfection in the finite element model [61]. According to the test report, the total shear strength of G4 girder is equal to 268.8 kN, which is very close to Vu = 265.08 kN as the ultimate shear strength predicted by the FEM simulation, with a maximum difference of −1.38%. As seen in Fig. 2, there is a good agreement between the FEM and that of experimental test both in terms of failure mode and overall behavior.

2.3.2. Verification of the numerical modeling process Lee and Yoo [22] conducted an experimental study of steel plate girders with non-rigid end posts. In that research, 10 scaled plate girder models were tested so as to investigate the shear behavior of web panels up to failure. The girders were simply supported with a concentrated load applied at the mid-span. All the geometric and material properties as well as the support and loading conditions of the aforementioned experiment are available in Refs. [22, 61]. In this study, in order to evaluate the proposed simulation technique, the test model G4 is selected and the results obtained from numerical analysis are compared with those of experiments. The first eigenmode of G4 is amplified with a certain magnitude (tw/10) and then introduced as an initial

3.1. Effect of flange-to-web stiffness ratio on the elastic shear buckling load

The research results are discussed in three parts. In the first part, the elastic shear bulking load of models is discussed. In the second part, the ultimate shear strength of models is investigated. Finally, the third part is devoted to the strength reduction factor of girders with initial geometric imperfection.

In this section, the elastic shear buckling strength of models is discussed. The eigenmodes of each girder is obtained by linear eigenvalue buckling analysis using FEM. As an example, Fig. 3 illustrates the first three buckling mode shapes corresponding to PG1, PG8, and PG13 models. According to this figure, PG13 model experiences local flange buckling along with web buckling and the first three eigenmodes are not compatible with the buckling mode of a plate with simply supported condition along all edges. Among all models, because of the small flange-to-web stiffness ratio in PG11 and PG13 models, the flanges do not possess sufficient rotational stiffness to provide necessary AISC- ( K = K.Lee) t/500 t/100 t/10 0.8D/100

t/1000 t/200 t/50 D/200

0.8

0.8

0.6

0.6

0.2

Inelastic buckling

0.4

0.4 0.2

Elastic buckling

Elastic buckling

0

0 0.5

a)

t/1000 t/200 t/50 D/200

Inelastic buckling

1

Plastic buckling

1

Plastic buckling

AISC- ( K= K.AISC ) t/500 t/100 t/10 0.8D/100

3. Results and discussion

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

Slenderness Parameter (λw-AISC)

2.3

0.5

2.5

b)

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

Slenderness Parameter (λw-Lee)

Fig. 7. The ratio of ultimate shear strength to plastic shear strength vs. slenderness parameter.

2.3

2.5

A. Ghadami, V. Broujerdian / Journal of Constructional Steel Research 153 (2019) 567–577

3.2. Effect of flange-to-web stiffness ratio on the ultimate shear strength In this section, the ultimate shear strength of the models is addressed. The obtained diagrams of shear load versus out-of-plane displacement of the steel plate girders are shown in Fig. 5 for PG1, PG4, and PG6 as representative samples. According to these results, it is observed that the buckling behavior of models changes from an elastic mode (PG1) to a plastic mode (PG4 and PG6) as the web plate thickness increases. One can notice that by increasing the initial imperfection magnitude in all models, in addition to the unobservability of bifurcation buckling phenomena, the ultimate shear strength is decreased. The ultimate shear loads of the steel plate girders are reported in Table 3. To evaluate the AISC relationships for shear strength of girder plates (Eqs. (5a) − (5d)), the ratio of AISC ultimate shear strength to the FEM results, VuAISC/VuFEM, versus slenderness parameter, λw, is plotted in Fig. 6. To investigate the effect of flange-to-web stiffness ratio on the shear strength, for the two elastic shear buckling coefficients of AISC (λw-AISC) [10] and Lee et al. (λw-Lee) [20], the results are displayed in different diagrams of Fig. 6a and b, respectively. Here, λw-AISC is employed to calculate the slenderness parameter using the elastic shear buckling coefficient value in Eq. (5d), while λw-Lee is exploited to calculate this parameter by considering the flange thickness as in Eq. (4). According to Fig. 6a, in the models where the imperfection magnitude is less than D/200, the AISC ultimate shear strength is less than the FEM results. Also, by increasing the slenderness parameter, especially for slender plates (λw N 1.37), AISC yields more conservative values for shear strength such that the difference between the results of FEM and AISC reaches 60%. However, for compact and non-compact web plates, maximum difference is about 10%. With regard to Fig. 6b, by applying the effect of flange-to-web stiffness ratio on the AISC shear strength using the elastic shear buckling coefficient proposed by Lee et al. [20], in the models where the imperfection magnitude is less than D/200, AISC yields conservative values for shear strength. The maximum difference between the results of FEM and AISC for slender, compact, and non-compact web plates is about 40%, 20%, and 10%, respectively. Comparing the results of Fig. 6a and b shows that by applying the effect of flange-to-web stiffness ratio, the slenderness range which

0.8D/100

Reduction Factor ( VuImp/Vubf )

requirements for simply supported boundary conditions. According to AISC specifications (Section F13) [10], the ratio of the web area to the compression flange area should not exceed 10 for flexural members. In addition, according to the American Association of State Highway and Transportation Officials (AASHTO) specifications (Article 6.10.2.2) [62], flanges should be proportioned such that tf N 1.1 tw. Applying these limitations is ineffective in controlling the rotation of flanges. Therefore, there must be a cross-section proportion limit to insure that flanges behave as a simply support at the least. However, the authors did not observe this in the AISC specifications. This issue is out of the scope of this article and more in-depth research is needed. Accordingly, the results of PG11 and PG13 models are ignored in all sections of the current study. The finite element (VcrFEM), AISC (VcrAISC), and Lee et al. [20] (VcrLee) elastic shear buckling load of the steel plate girders are shown in Table 2. The ratio of predicted values based on AISC and Lee et al. [20] to the ones obtained by the proposed FEM model with respect to the slenderness parameter, λw-Lee, is plotted in Fig. 4. According to the results, it can be seen that the AISC yields more conservative values such that the difference between the results of FEM and AISC is about 40%. This difference reaches to 20% in lower slenderness values. However, when using elastic shear buckling coefficient of Lee et al. [20], the maximum and minimum difference are about 23% and 4%, respectively. Also, comparing the FEM results (VcrFEM) and the elastic shear buckling load of VcrLee indicates that the difference between VcrLee and VcrFEM is reduced that by increasing the web aspect ratio and reducing the effect of the rotational stiffness of transverse stiffeners on the web shear strength.

D/200

t/10

t/50

Proposed Equation

1.1 1.05 1 0.95 0.9 0.85 0.8 0.75

Plastic buckling

Elastic buckling

0.7 0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

Slenderness Parameter (λw-Lee) Fig. 8. Comparison of shear strength reduction factor of proposed equation and FEA results.

distinguishes the elastic, inelastic and plastic buckling is different from the proposed slenderness ranges of AISC. In other words, using the elastic shear buckling coefficients proposed by AISC, the slenderness limit of slender plates, λw = 1.37, turns out to be a conservative value. This may be due to neglecting the effect of flange-to-web stiffness ratio on shear strength in the proposed relationships of AISC, especially for slender web plates. 3.3. Effect of geometric imperfection on the plastic shear strength 3.3.1. Derivation of the proposed equation To investigate the effect of imperfection on the plastic shear strength of steel plate girder, the ratio of FEM ultimate shear strength to plastic shear strength, VuFEM/Vp, versus slenderness parameter, λw, is plotted in Fig. 7 and compared with the AISC predictions,VuAISC /Vp. According to this figure, increasing the imperfection results in a decrease in the ultimate shear strength. This reduction for plates with Imp b tw/50 is negligible such that the maximum reduction is close to 3.5%. Furthermore, it can be seen that the shear strength reduction in non-compact plates is more than that of slender and compact plates. In other words, by increasing or decreasing the web plate slenderness from a constant value of λw = 1.37, the effect of imperfection magnitude on the shear strength becomes smaller. It is also observed that up to an imperfection of Imp = tw/10, the maximum strength loss is about 10%. However, for geometric imperfections greater than tw/10, the shear strength of steel plate girder decreases nearly by 23%. Therefore, it is reasonable to reduce the maximum permissible construction tolerance up to tw/10. To investigate the effect of geometric imperfection on the shear strength of steel plate girders, a minimum value of tw/1000 for bifurcation-type buckling to 0.8D/100 as 80% of the maximum allowable imperfection is considered. The ultimate shear strength corresponding 30 25

Reduction (percent)

574

Lee et al. [21]

20

Proposed equation (6)

15 10 5 0 0.5

1

1.5

2

2.5

3

3.5

4

Slenderness Parameter (λw-Lee) Fig. 9. Comparing the FE analysis results of Ref. [21] and the results of Eq. (6) (Imp = D/120).

A. Ghadami, V. Broujerdian / Journal of Constructional Steel Research 153 (2019) 567–577 Table. 4 The elastic shear buckling load (kN) of FEM compared to the Ref [21] & Eq. (1). Web plate slenderness (λw-Lee)

Current paper FEM (1)

Lee et al. [21] (2)

Eq. (1) (3)

1.08

12,457.68

12,234.80 –

1.018

1.44 1.68 1.92 2.16 3.6

5288.04 3334.35 2236 1571.05 340.09

Not mentioned 5300.40 3337.95 2236.13 1570.41 339.09

5161.56 3250.43 2177.53 1529.35 330.34

1.025 1.026 1.027 1.027 1.030

Ratio Ratio (1)/(2) (1)/(3)

0.998 0.999 1.000 1.000 1.003

Note: 1 in. = 25.4 mm, 1 kip = 4.45 kN, 1 psi = 6.895 kPa

to tw/1000 is shown as Vubf, and the other one is presented by VuImp. As already mentioned, the ultimate shear strength reduction magnitude for web plates with Imp b tw/50 is negligible. Therefore, in this section, the effect of geometric imperfection with Imp ≥ tw/50 is investigated. To derive a relation for predicting the shear strength reduction of a perfect web plate due to initial geometric imperfections, the ratio of ultimate shear strength of models with initial imperfections, VuImp, to the shear strength of the corresponding model with minimum initial geometric imperfection (Vubf), versus slenderness parameter, λw-Lee, is plotted in Fig. 8. According to this figure, increasing or decreasing the web plate slenderness from a constant value of λw = 1.37, as the distinguishing slenderness limit between elastic and inelastic buckling, shear strength reduction factor increases and converges to a constant value of 1 (i.e., the shear strength is increased). It can also be noted that the maximum strength reduction is about 23% for Imp = 0.8D/100. To derive simple equations for estimating the reduction factor as a function of imperfection and slenderness, a two-stage procedure of regression analysis and data fitting is performed here. In the first stage, a set of parallel V-shape two-segment lines are fitted to FEM curves of Fig. 8 so as to obtain conservative estimations with minimum error. In the second stage, an exponential function is fitted to relate the minimum reduction factors to imperfection values. The final equation is obtained as:
 VuImp tw 0:05 ¼ 1:03   ð1:15−0:3λLee Þ≤1; λLee b1:37 Vubf Imp ð6Þ
0:05 Vu Imp tw RF ¼ ¼ 1:03   ð0:34 þ 0:3λLee Þ≤1; λLee ≥1:37 Vubf Imp

RF ¼

It must be noted that, the correlation factors, R2, of the above equation for Imp = t/50, t/10, D/200, and 0.8D/100 are obtained as 0.925, 0.871, 0.938, and 0.945, respectively, with a mean correlation coefficient equal to 0.92. So, there is a good conformity between the reduction factors determined from Eq. (6) and the nonlinear FEM analysis for web panels with different initial out-of-flatness.

575

3.3.2. Verification of the proposed equation As previously mentioned in the introduction section, Lee et al. [21] studied the effects of initial imperfection on the shear strength of steel plate girder web panels subjected to pure shear. The reduction of ultimate shear strength due to large initial distortions was also reported for some girders. In the current study, in order to evaluate Eq. (6), six FEM models with a/D = 1, and D/tw = 90 (λw-Lee = 1.08), 120 (λw-Lee = 1.44), 140 (λw-Lee = 1.68), 160 (λw-Lee = 1.92), 180 (λw-Lee = 2.16), and 300 (λw-Lee = 3.6) are selected and the results obtained from Eq. (6) were compared with those reported in Ref. [21]. The dimensions and material properties were taken as D = 2032 mm (80 in), bf = 609.6 mm (24 in), ν = 0.3, E = 200 GPa (29,000 ksi), yield stress equal to 345 MPa (50 ksi), and tf /tw = 3. The results of FEM models and Eq. (6) are compared in Fig. 9. According to this figure, there is a good agreement between the results of Eq. (6) and that of FEM models for web plate slenderness greater than λw-Lee = 1.37. However, there is a mismatch between the results for one model with λw-Lee = 1.08. To evaluate this difference, according to numerical modeling procedure as described in previous sections and those one in Ref. [21], six FEM models are constructed. In order to evaluate the FEM models of the present article, the elastic shear buckling load (Vcr) of models are obtained and compared with those of Lee et al. [21] and Basler's modified method. According to Table 4, it can be seen that there is a good agreement between the results. Using this FEM model, the ultimate shear strength of model with λw-Lee = 1.08 by two initial imperfections of D/120000, and D/120 are obtained as 9130 and 7392 kN, respectively. According to these results, the obtained reduction is around 19%, which shows better conformity with the results of Eq. (6). The first eigenmode and the failure mode corresponding to the ultimate shear load (Vu) of the mentioned model with initial imperfections of D/120 are shown in Fig. 10. 4. Conclusions In the present study, in order to investigate the effects of web plate slenderness, flange-to-web stiffness ratio, and imperfection magnitude on the shear strength of steel plate girders, 112 models were simulated and analyzed. The obtained results are briefly discussed as follow: • By investigating the FEM outputs, the equations for determining the ultimate shear strength reduction factor of web plate with respect to initial imperfections (RF) are derived. This factor must be applied to the analytical or FEM ultimate shear strength of girders calculated with the assumption of small initial geometric imperfection. • According to the results, the AISC elastic shear buckling strength were conservatively less than the results derived from FEM. The differences were almost 40% and 20% in slender and compact web plates, respectively. However, when using elastic shear buckling coefficient of Lee et al. [20], the maximum and minimum differences were about 23% and 4%, respectively.

Fig. 10. a) First eigenmode, and b) failure mode of model with λw-Lee = 1.08.

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• With increasing web aspect ratio, the effect of the rotational stiffness of transverse stiffeners on the web elastic shear buckling strength decreases. • For steel plate girders whose geometric imperfection magnitude is b1/ 200 of web depth, the AISC proposed ultimate shear strength is conservatively less than the obtained results by FEM. Especially for slender plates, the difference between the results of FEM and AISC reached to almost 60%. However, for compact and non-compact web plates, the maximum difference is about 10%. • Applying the effect of flange-to-web stiffness ratio on the AISC shear strength based on Lee et al. [20], the elastic shear buckling coefficient, the maximum difference between the results of FEM and AISC for slender, compact, and non-compact web plates were obtained to be about 40%, 20%, and 10%, respectively. • According to the results, the AISC slenderness limit of λw = 1.37 which distinguishes the elastic and inelastic buckling is a conservative value. This may be due to neglecting the flange-to-web stiffness effect on the shear strength in AISC relationships. • Reduction of shear strength due to initial imperfection in noncompact web plates is more than the one in slender and compact web plates. • According to the results, it is observed that up to an imperfection of tw/ 10, the maximum shear strength loss is about 10%. However, for the geometric imperfections above tw/10, the shear strength decreases by nearly 23% for 80% of the maximum allowable imperfection, as recommended in parts 1–2 of EC3 [53]. • The maximum permissible construction tolerance is suggested to be tw/10, which corresponds to a maximum shear strength reduction of about 10%. • Further experimental study is recommended to more corroborate the proposed equations for determining the ultimate shear strength reduction factor (RF) of girder web panels with respect to initial imperfections.

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