development of web slenderness limits for composite i ... - J-Stage

Doboku Gakkai Ronbunshuu A Vol.62 No.4, 854-864, 2006. 10

DEVELOPMENT OF WEB SLENDERNESS LIMITS FOR COMPOSITE I-GIRDERS ACCOUNTING FOR INITIAL BENDING MOMENT Vivek Kumar GUPTA1 , Yoshiaki OKUI2 and Masatsugu NAGAI3 1

Doctor Course Student, Dept. of Civil and Environmental Eng., Saitama University (255, Shimo-Okubo, Sakura, Saitama, 338-8570, Japan) E-mail: [email protected] 2 Member of JSCE, Assoc. Prof., Dept. of Civil and Environmental Eng., Saitama University (255, Shimo-Okubo, Sakura, Saitama, 338-8570, Japan) E-mail: [email protected] 3 Member of JSCE, Prof., Dept. of Civil and Environmental Eng., Nagaoka University of Technology (1603-1, Kami-Tomioka, Nagaoka, Niigata, 940-2188, Japan) E-mail: [email protected]

The positive bending moment capacity of composite steel girders is examined through parametric study employing elasto-plastic finite displacement analyses. The effects of initial bending moment on the bending moment capacity and on the web slenderness limit for section classification are investigated. Observations made during the numerical study indicate that the noncompact web slenderness limits in conventional design standards, which are based on tests of steel I-sections, are conservative for composite sections. Many sections, which are classified as slender by current specifications, demonstrate sufficient flexural capacity as noncompact. The conventional web slenderness limits for noncompact sections, independent of initial bending moment seems inappropriate for composite I-girders. The initial bending moment has considerable effect on the noncompact web slenderness limits. The web slenderness limits for compact and noncompact sections are proposed on the basis of the parametric study. Key Words : elasto-plastic finite displacement analysis, composite I-girder, web slenderness

1. INTRODUCTION

girders in flexure, first, sections are classified into several classes depending upon their ultimate bending moments and flexural deformation capacity. In AASHTO and Eurocode, the classification requirements for composite sections are originally derived from experimental5),6) and analytical7),8) studies for steel sections without concrete slabs. In fact, the same requirements as those for steel sections are used for composite sections in these specifications. For composite sections under positive bending, since concrete slabs inhibit buckling of the compression flange and the compressive part of the web plate, there is a fair possibility of relaxing the section classification requirements compared to those for steel sections only.

In Japan, construction of the first modern composite bridge1) in 1995, provided a new impetus for the development of an economical solution for medium span bridges. In this bridge, a continuous composite girder with two steel I-sections with a simple transverse stiffening system was employed to reduce the construction cost. The trend of using this structural system has been increasing since then. The economical efficiency can be achieved largely from cost savings in fabrication and erection of fewer members and elimination of temporary forms and shoring. In view of worldwide standardization for bridge design methods, the Limit State Design (LSD) method becomes the mainstream method as shown in current design codes such as Eurocode2) and AASHTO specifications3) , although the Japanese modern composite bridges are still designed in accordance with the Allowable Stress Design (ASD) method prescribed in Japanese Specifications for Highway Bridges4) (JSHB). In the LSD method for

Many researchers9),10),11) have undertaken the nonlinear finite element (FE) analysis of composite girders to investigate the load carrying capacity of composite girders. Basker et al.9) employed a threedimensional FE model to simulate the experimental tests of composite girders under combined shear and bending. Recently, Egashira et al.10) and Yakel and Azizinamini11) used nonlinear FE model for investi854

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Table 1 Definition and web slenderness limits of section classifications in AASHTO and Eurocode Design code

AASHTO

Eurocode

Note:

Section class

Definition

Compact

Mmax ≥ Mp

Noncompact Slender

Mp > Mmax ≥ My Mmax < My

Class 1

Mmax ≥ Mp

Class 2

Mmax ≥ Mp

Class 3

Mmax ≥ My

Class 4 Mmax < My Mmax : Maximum bending moment Mp : Plastic moment My : Yield moment bw : Web width tw : Web thickness

Web slenderness limits √ 2Dcp Es tw ≤ 3.76 √ fy 2Dc Es tw < 5.7 fy Other than those above { 36ε/α for α ≤ 0.5 bw tw ≤ 396ε/(13α − 1) for α > 0.5 { 41.5ε/α for α ≤ 0.5 bw tw ≤ 456ε/(13α − 1) for α > 0.5 { 42ε/(0.67 + 0.33ψ) for ψ > −1.0 bw √ tw ≤ 62ε(1 − ψ) −ψ for ψ ≤ −1.0 Other than those above Es : Young’s modulus of steel fy : Yield strength of steel Dcp √ = αbw : Depth of web in compression ε = 235/fy [N/mm2 ]: Reduction factor ψ: Stress gradient

ite bridges, the cross sections employed in the present parametric study are restricted to web plates without horizontal stiffeners and whose depth and aspect ratio are 3m and 3, respectively. On the basis of results obtained from the numerical study, section classification criteria are proposed, in which the initial bending moment effects are taken into account.

gating composite girders under positive bending moment. They proposed prediction equations for estimating the ultimate strength of composite girders. However, they did not consider the initial bending moment applied only on the steel girder in their numerical analyses. In unshored construction of composite girders, the steel element is required to support the weight of steel girder and wet concrete until the concrete has hardened. It is well known that there is almost no effect of initial bending moment on the flexural resistance of compact sections12) . However, since the initial bending moment serves as residual stress in elasto-plastic buckling, the flexural resistance of noncompact sections is significantly affected by the initial bending moment. Nevertheless, the initial bending moment effect on the flexural capacity and accordingly the classification of sections is not considered even in experimental studies13),14) . Only a limited number of published information15),16) are available for accounting the initial bending moment effect due to the difficulty in applying initial bending moment only on the steel girder. Bradford et al.15) studied the behavior of unshored curved composite girder under initial bending moment. However, their main concern is the effect of the lateral curvature on the load carrying capacity, and not about general classification of composite sections as discussed in the present paper. In this paper, the ultimate flexural strengths of composite girders under pure positive bending are investigated by using the finite element method. A threedimensional FE model is developed to perform elastoplastic large displacement analysis of composite girders. In consideration of the recent 2-I girder compos-

2. CURRENT CLASSIFICATION COMPOSITE SECTIONS

OF

In this chapter, the classifications of composite sections in the AASHTO and the Eurocode are reviewed briefly. Only the slenderness for web plates is discussed in this paper, because of the possibility of local buckling of compression flange plates is eliminated by the constraining effect of concrete slabs in composite girders. Table 1 gives a comparison for the definitions for section classes and the web slenderness limits under positive flexure loading. Different design codes use different slenderness parameters to define the section classification. In the AASHTO, the web slenderness refers to the ratio of twice the depth between the neutral axis and the compression flange to the web thickness: 2Dcp /tw or 2Dc /tw , while in the Eurocode it is the width-thickness ratio bw /tw ; see Fig. 1 for the definitions of symbols. The definition of compact sections in AASHTO corresponds to class 2 sections in Eurocode. In reality, the slenderness limit in the AASHTO is based on the provisions of American Institute of Steel Construction17) (AISC), where compact sections are proposed for plastic analysis with a rotation capacity of at least 313),18) for bending moment redistribution to take place. However, this rotation capacity 855

Doboku Gakkai Ronbunshuu A Vol.62 No.4, 854-864, 2006. 10 0.85 fc fy D cp = αb w

bw

σc σ1

Mp

M1

M2

+

M

Dc

=

tw

σ2=ψσ1

fy

(a) Compact section

(b) Noncompact section

Fig.1 (a) Stress distribution in compact section and definition of plastic moment Mp ; (b) Stress history of noncompact section in unshored construction. M1 is the initial bending moment for steel section only and M2 is bending moment for composite section. buf bc

tc

bw = 3m

m

z y

x

a/

2=

4.5

blf

(a) Phase 1 - Erection Phase

(b) Phase 2 - Service Phase

Fig.2 Different Construction Phases

in concrete slabs has been neglected in FE models. In the following parametric study, a two-phase analysis is carried out to simulate the unshored construction of composite girders. In the first phase shown in Fig. 2(a), there is no composite action, and only steel girder carries the initial bending moment. In actual unshored composite bridges, this initial bending moment results from the self-weight of the steel girder, the weight of the wet concrete, and construction loads. In the second phase shown in Fig. 2(b), the incremental applied moment is resisted by a composite section with concrete slab. The superimposed dead and live load in unshored composite bridges cause the incremental moment. The magnitude of the initial bending moment, M1 were assigned to 0, 20, 30 and 40% of the yield moment of the steel section Mys and acts as an external moment in phase 1 (Fig. 3). Fig. 2(b) shows the boundary condition of the present FE model. In this figure at right hand side edge of the girder, which is the center of 9m girders, the displacement in the girder axis direction is pre-

requirement for the definition of compact sections is not given explicitly in the current AASHTO specifications. Hence, in this paper, the AASHTO’s definition of “compact”, “noncompact” and “slender” sections will be adopted, where the compact sections are defined as one that can develop a plastic moment Mp as shown in Fig. 1 without any rotation capacity requirement.

3. NUMERICAL STUDY (1) Nonlinear FE Model A three-dimensional finite element model was used in the present study to analyze the composite girders under positive bending. In the analysis, only half of the girder is considered because of symmetry of their structures and loading conditions. The flanges and web plates of the steel I-girder were modeled with 4-node thin shell elements. The concrete slab was modeled with 8-node solid elements. As is usually done in positive bending, longitudinal reinforcement 856

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σ1 0

bw

tw

M1 =

20% M ys 30% M ys 40% M ys

Fig.3 Stress distribution for different magnitudes of initial bending moment M1 on steel girder; Mys stands for yield bending moment for steel girder alone (σ1 = fy , when M1 = Mys )

Fig.5 Stress–strain curve for steel

(2) Constitutive Models In this study, two material models have been used to predict the ultimate behavior of the steel- concrete composite girders. The material non-linearity was considered by applying elasto-plastic material models. a) Structural Steel Steel is modeled as an elasto-plastic strain hardening material and von Mises yield criterion 20) was applied. The assumed uniaxial multi-linear stress-strain curve is shown in Fig. 5. For steel grade SM490Y, the yield strength of web and flange materials for the analysis has been taken as 355 MPa and its Young’s modulus as 2.0 × 105 MPa. The ratio of the strain when strain–hardening starts to the yield strain εst /εy employed was 3. The following stress-strain equation 21) has been assumed for steel. { [ ( )]} σs εst 1 Est εs = 1 − exp −ξ − + 1 (2) fy ξ Es εy εy where σs and εs are the corresponding stress and strain of the steel respectively, fy and εy are the yield strength and strain of steel respectively, Est = 1730 MPa is the strain-hardening slope and ξ = 0.02. b) Concrete Concrete in compression is modeled as a linearelastic, plastic hardening material. The MohrCoulomb criterion is employed as a yield criterion. The yield function for this criterion is expressed in terms of the principal stresses σi , (i = 1, 2, 3) as follows 1 1 F = (σ1 − σ3 ) + (σ1 + σ3 ) sin φ − c cos φ (3) 2 2 where φ and c are the angle of internal friction and cohesion, respectively. The internal friction angle is assumed to be constant φ = 20deg., while the cohe-

Fig.4 Initial imperfection in web plate of the steel I-girder

vented by the use of rollers. At left side edge, where rigid plane is set using tyings19) , forced rotational angle is increased at the support, which is located at the mid-depth of the web plate and supported only in the vertical direction. The out-of-plane displacement for the web uy is considered as the initial imperfection. The maximum out-of-plane displacement is assigned to bw /250 according to JSHB, and the following equation is assumed: ( ) ( ) bw π π uy = sin z cos x (1) 250 bw bw where bw is the depth of web plate; x and z are the global nodal coordinates of the web plate as shown in Fig. 4. For built-up steel sections, in general, tensile residual stresses are introduced into both upper and lower parts of web plates, while the compressive residual stresses are introduced into the center of the web plate. Since the neutral axis of recent composite girder sections is located at the upper part of the web plate, the tensile residual stress prevents from local buckling of the web plate. The residual stresses are therefore neglected in the present parametric study, because the neglect corresponds to the safe side assumption in identifying the web slenderness limits. 857

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Fig.6 Stress–strain curve for concrete

Fig.7 Bending moment and curvature relationship

Table 2 Cross–section properties of girder

Table 3 Results of different section class

Width-thickness ratio bw /tw U. flange width buf [mm] U. flange thickness tuf [mm] L. flange width blf [mm] L. flange thickness tlf [mm] Slab thickness tc [mm]

143 550 25 1200 59 300

Section class Compact Noncompact Slender

Non-dimensionalized moment Mu /My 1.19 1.09 0.97

Proportion-of-web depth in comp. α or α0 0.29 0.42 0.50

tionships obtained from the numerical analysis accounting for the initial bending moment. Table 2 shows the dimensions of the steel cross–section and thickness of the concrete slab. The concrete slab width was taken as 2.5m, 2.0m and 1.5m whose behavior can be classified into compact, noncompact and slender sections, respectively. For phase 1, rotation corresponding to M1 = 30% of Mys was applied at the left support and a linear relationship was observed. The concrete slab was then added to the steel I-girder and the results obtained from phase 1 were used as initial values at the beginning of phase 2 along with additionally applied rotations. The moment– curvature plot obtained from phase 2 analysis shows a linear relationship initially but becomes nonlinear with increasing curvature for composite girders with same steel I-girder cross-section but different concrete slab widths. The kink in the moment–curvature plot shows the change from one phase to another. Compact, noncompact and slender behavior was observed depending on the dimensions of sections for M1 = 30% of Mys . The results of two-phase numerical analysis are shown in Table 3. At the ultimate state, the greatly enhanced ductility in compact sections compared to that in noncompact sections is due to the reduced depth of web in compression as shown in Table 3. For noncompact

sion is considered as a function of the equivalent plastic strain adopting strain hardening hypothesis. The equivalent plastic strain ε¯p is defined by ∫ √ 2 p p ε¯p = dε dε (4) 3 ij ij by integrating plastic strain increments dεpij where dεpij stands for the incremental plastic strain tensor, and the summation convention is used over all increments. The cohesion is calculated from 1 − sin φ c = σc (5) 2 cos φ with the uniaxial stress-strain relationship22) under compression shown in Fig. 6. In Eq. (5), σc is the compressive stress in concrete, and the stress–strain relationship in Fig. 6 is assumed to be linear up to 0.7fc , where fc is the compressive strength of concrete. In the present analysis, the associated flow rule and isotropic hardening were used. The compressive strength of concrete is assigned to 40 MPa, and the influence of creep and shrinkage of concrete are neglected throughout this paper. (3) Example of numerical results Fig. 7 shows the bending moment–curvature rela858

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Table 4 Parameter study variables

Z Y X

Fig.8 Failure mode for noncompact section; Maximum lateral displacement of the web plate is 9mm

Variable

Min.

Max.

Width-thickness ratio bw /tw U. flange width buf [mm] U. flange thickness tuf [mm] L. flange width blf [mm] L. flange thickness tlf [mm] Slab width bc [mm] Ini. bending moment M1 /Mys

120

231

Nos. of values 7

310

590

9

20

37

12

800

1200

2

58

65

6

1200

2900

8

0

0.4

4

4. PARAMETRIC STUDY Parametric studies using a finite element numerical model present an attractive alternative to physical tests when formulating or checking design rules for members exhibiting complex behavior. The present parametric investigation was carried out by extending existing studies 10),18),23),24) to include parameters that were held constant in the previous researches and focused on the variation of the seven main parameters listed in Table 4. The range of values for each parameter was chosen as being slightly higher than what would be considered practical. Table 4 indicates the maximum and minimum values for each parameter along with the number of values used in the parametric study. The aspect ratio and depth of the web plates are fixed to 3 and 3 m, respectively. The concrete slab thickness is assigned to 300 mm in all numerical calculations. For each bw /tw ratio, the concrete slab width bc is varied from 1.2 m to 2.9 m. In addition, the upper flange width buf is varied from 310 mm to 590 mm for each concrete slab width bc . When arriving at individual web slenderness limits for compact or noncompact web response, the ratio bw /tw is varied with a fixed combination of the other five parameters until compact, noncompact or slender response is achieved. For each bw /tw group there will be at least 28 finite element model runs. The resulting number of hypothetical girders obtained by all combinations provided in Table 4 is 290304. In order to minimize the total number of hypothetical girders, the following criteria were adopted. While selecting sections for analysis, the first criterion is taken from the Eurocode, which states that compression flange thickness should at least satisfy the requirements for class 3 sections i.e. buf /2tuf ≤

α’b w or αb w bw

tw

Fig.9 Strain distribution for compact and noncompact sections, and definition of α and α0

sections, the increased depth of web in compression weakens the composite girder and there is greater tendency for web buckling and reduced inelastic curvatures and rotations. The reduction in the ultimate bending moment for noncompact sections is due to local buckling of the web plate as shown in Fig. 8. Since a portion of the noncompact section exhibits yielding in the steel part at the ultimate state, the neutral axis is calculated from the strain distribution. The definition of α0 is shown in Fig. 9, and α = Dcp /bw for compact sections and α0 = Dc /bw for noncompact sections. In this paper, the compact section is defined as those sections whose ultimate bending moment Mu is larger than its plastic moment Mp defined in Fig. 1(a). Since the simplified stress-strain relationship without a strain-softening regime is used for concrete modeling in the present numerical analysis, the strain in the concrete slab may reach εcu = 0.0035 shown as the example moment-curvature curve for a compact section in Fig. 7. If this is the case, the ultimate bending moment Mu is defined as the bending moment at which the concrete strain attains εcu = 0.0035. 859

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14ε. Other criteria were assumed mostly based on practical considerations, as follows: (a) The thickness of the flange is not less than the thickness of the web. This limit is simply a practical limit, which would not be violated in an economical design10) . (b) For economical design, the area of tension flange is greater than the area of compression flange is employed. (c) An area ratio Aw /A0uf = 2 was assumed for most sections, while 1.5 was used for sections with higher thickness of the upper flange, where Aw is the area of web plate and √ A0uf = k/kuf Auf (6)

1.2 b /t =130 w w

b /t =143 w w

b /t =158

1.1

w w

b /t =176 w w

u

M /M

y

b /t =200 w w

1

0.9

0.8 0

0.1

0.2

0.3

M /M 1

0.4

0.5

ys

26)

where k = 2.31 is the buckling coefficient for flange plates with three sides fixed, kuf = 0.425 is the buckling coefficient for unstiffened upper flange plates and Auf is the area of the upper flange. (d) The width of the concrete slab bc > 2buf is adopted. Apart from the slab width limitation, all criteria consider the steel section alone. When all the criteria are applied, the number of remaining sections for the parametric study is reduced to 28426) . Since the bw /tw ratios of the selected parametric sections is rather high, the web plates of these sections are classified as class 4 sections in Eurocode and the compression flanges into class 3 sections. On the other hand, from the viewpoint of AASHTO specifications, the web plates and compression flange of the remaining sections fall in slender and noncompact category, respectively. In the subsequent sections, the effects of the initial bending moment on the web slenderness limit is first reported to demonstrate the effect of the initial bending moment. The maximum value of the initial bending moment ratio M1 /Mys is selected on the basis of the allowable bending stress reaching maximum of 0.66fy at the top of the steel girder specified in AISC Specifications.

Fig.10 Effect of initial bending moment M1 /Mys on ultimate bending moment Mu /My

0.6 0.55

'

0.5 0.45 0.4 b /t =130

b /t =176

w w

w w

b /t =143

0.35

b /t =200 w w

w w

b /t =158 w w

0.3 0

0.1

0.2

0.3

M /M 1

0.4

0.5

ys

Fig.11 Effect of initial bending moment M1 /Mys on proportion-of-depth in compression α0

causes first yielding in either of the flanges. When M1 /Mys < 0.3, Mu occurs before yielding due to premature local buckling of the web plate. Accordingly, all sections except for the section with M1 /Mys = 0.3 and bw /tw = 130 are classified into slender. On the other hand, for M1 /Mys > 0.3, the ultimate bending moment exceeds the yield moment as the initial bending moment increases, and all sections with M1 /Mys = 0.4 exhibit noncompact behavior. From the analyses it is found that the effect of bw /tw on Mu /My is larger for lower values of M1 /Mys . It is to be also noted that as the initial bending moment increases, both yield and ultimate moments decreases however the change in Mu itself is not large26) , only around 0.5%.

5. NUMERICAL RESULTS AND PROPOSED LIMITS FOR FLEXURE (1) Effect of initial bending moments In order to demonstrate the effects of initial bending moment on the section classification, the results of numerical analyses for noncompact sections with different width-thickness ratios ranging from 130 to 200 for bc =1.5 m is selected. Fig. 10 shows the plot between the non-dimensionalized ultimate bending moments Mu /My and M1 /Mys ratios for different bw /tw ratios where My is the yield moment that 860

Doboku Gakkai Ronbunshuu A Vol.62 No.4, 854-864, 2006. 10

(a) M1 = 0

(b) M1 = 0.2Mys

(c) M1 = 0.3Mys

(d) M1 = 0.4Mys

Fig.12 Web slenderness vs α or α0 : Proposal for web slenderness limit for compact sections

with AAHSTO’s compact and Eurocode’s class 1 and class 2 limits in Fig. 12.

One reason for increase in Mu /My is due to the position of the neutral axis. Fig. 11 shows the position of the neutral axis at the ultimate state calculated from the strain distribution obtained from the numerical analysis. It is observed that with increasing M1 /Mys , the neutral axis moves up increasing the tension region, which results in an increase in Mu /My . Moreover, composite girders with higher bw /tw show more area in tension zone than girders with lower bw /tw .

Comparing AASHTO with Eurocode, since compact sections in AASHTO corresponds to class 2 in Eurocode, AASHTO allows thinner web plates than those in Eurocode as compact sections. From the plotted results, it was found that our numerical results support AASHTO’s limit, and the web slenderness limit in Eurocode’s class 2 seems to be conservative. Moreover, our numerical results show that sections with further thinner web plates than AASHTO’s limit can behave as compact sections. Contrary to web slenderness limit for noncompact sections, sections with any bw /tw ratio was not influenced by the change in the initial bending moment and hence no change in sectional class was observed. As a result, the web slenderness limit, which provides a suitable fit for developing the plastic moment, becomes independent of the initial bending moment and is ex-

(2) Proposed limits for compact sections In this section, the web slenderness limit for compact sections is presented. The width-thickness ratios bw /tw of web are plotted against parameter α obtained from numerical analysis. The numerical results are expressed by square symbols for compact sections and open circle symbols for noncompact sections as shown in Fig. 12 for different magnitude of the initial bending moment. For each magnitude of the initial bending moment, the plotted results are compared 861

Doboku Gakkai Ronbunshuu A Vol.62 No.4, 854-864, 2006. 10

(a) M1 = 0

(b) M1 = 0.2Mys

(c) M1 = 0.3Mys

(d) M1 = 0.4Mys

Fig.13 Web slenderness vs α0 : Proposal for web slenderness limit for noncompact sections

in Fig. 13. Similar to α for compact sections, the parameter α0 is also calculated from strain distribution corresponding to the maximum bending moment Mmax . In Fig. 13, noncompact and slender sections are denoted by open circle and solid triangle symbols, respectively. It is apparent from the results in Fig. 13 that the increase of M1 is to change the classification of sections from noncompact to slender. This change in section classification due to increasing initial bending moment is not adequately considered in the provisions of either Eurocode or AASHTO specifications.

pressed as bw 59ε ≤ (7) tw α where the definition of ε is identical with that given by Eurocode. The above proposed web slenderness limit for compact sections represents the boundary between compact and noncompact sections, and plotted as the dotted curve in Fig. 12. Eq.(7) of the proposed limit is similar to that proposed by Eurocode. It is to be noted that the term on the right-hand side of Eq.(7) is dependent only on steel’s yield stress unlike AASHTO’s limits, which is dependent on both yield stress and Young’s modulus of steel.

The solid curves in Fig. 13 represent Eurocode’s class 3 and AASHTO’s noncompact limits. From Fig. 13, it is observed that the boundary between the noncompact and slender section symbols obtained from the numerical analysis shows a trend similar to Eurocode’s class 3 limit. A reason for the inadequacy in AASHTO’s limit is considered that AASHTO web slenderness limit is based on the elastic buckling theory for the pure bending case α0 = 0.5, and accordingly the effect of variation of α0 is not taken into ac-

(3) Proposed limits for noncompact sections Next, the web slenderness limit for noncompact sections will be considered in this section. The web width-thickness ratios bw /tw are plotted against the numerical values of proportion-of-depth in compression α0 for varying amount of initial bending moment 862

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Fig.14 Determination of Λ: Effect of initial bending moment.

same boundary, and that can be expressed as ( ) ( ) M1 M1 2 Λ = 1 − 0.1 + 2.3 Mys Mys

count properly. Hence, the web slenderness limit in Eurocode is used as a basis for deriving the web slenderness limit for noncompact sections, and the limit is shifted to fit the noncompact-slender boundary obtained from the numerical analysis for M1 = 0: bw 52ε ≤ for ψ > −1.0 tw (0.67 + 0.33ψ) √ bw ≤ 77ε(1 − ψ) −ψ for ψ ≤ −1.0 tw

Eventually, we propose Eqs.(10) and (11) with Eq.(12) as the web slenderness limit for noncompact sections. Comparing with Eurocode, the proposed web slenderness limit for M1 = 0 turns out to be 25% larger than Eurocode’s class 3 limits. Furthermore, the proposed web slenderness limit increases up to 33% with increasing initial bending moment up to 40% of Mys .

(8) (9)

where the stress gradient parameter ψ is defined in Fig. 1, which has a relationship with α0 through ψ = (α0 − 1)/α0 for elastic states. Next, in order to include the initial bending moment effect, we introduce a new coefficient Λ into the right hand sides of Eqs. (8) and (9): bw 52εΛ ≤ for ψ > −1.0 tw (0.67 + 0.33ψ) √ bw ≤ 77εΛ(1 − ψ) −ψ for ψ ≤ −1.0 tw

(12)

6. CONCLUDING REMARKS The slenderness limits for web plates without horizontal stiffeners in composite girders under positive bending moment are proposed. It is shown that the proposed web slenderness limits for compact and noncompact sections are significantly higher than that in Eurocode, and a little higher than that in AASHTO. The compact web slenderness limit is independent of the initial bending moments, whereas the noncompact web slenderness limit is expressed as a function of the initial bending moment and the stress gradient parameter. Most composite girder bridges are built with the unshored construction method, and hence the initial bending moment is applied to their steel sections. Comparisons were made between the proposed slenderness limits with existing AASHTO and Eurocode slenderness limits. Although the proposed limits have been developed on the basis of Eurocode’s slenderness limits, it may be modified for any other code for the classification of composite I-girder sections. The main feature of the proposed slenderness limits is to

(10) (11)

Assuming that the stress gradient effect (the effect of variation of α0 ) on the web slenderness limit is properly considered in Eqs. (8) and (9), the ratio of the right to the left hand sides of Eqs. (8) and (9) should be independent of the stress gradient, and be a function of the initial bending moment. Strictly speaking this assumption is only true for elastic buckling, since Eqs. (8) and (9) are based on Euler buckling theory. To examine this, all the numerical results for noncompact and slender sections are plotted into the ratio versus the initial bending moment diagram as shown in Fig. 14. For both ψ > −1.0 and ψ ≤ −1.0, the lower limits of slender sections gives an almost 863

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take account of the initial bending moment effect in determining the noncompact slenderness limit, which is not considered in conventional design codes explicitly. Finally, it should be noted that, since the present parametric study is focused on cross sectional dimensions for the recent composite girders with 2-I girder cross sections, further research is necessary to extend the scope of the proposed web slenderness limits.

12)

13) 14)

ACKNOWLEDGMENT: This study was carried out as a part of the activity of Technical committee for enhancement of steel bridges performance in Japanese Society of Steel Construction. Valuable comments from other committee members are gratefully acknowledged.

15)

16)

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