Flexural behavior of prestressed composite beams with corrugated web: Part I. Development and analysis

Composites: Part B 42 (2011) 1603–1616

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Flexural behavior of prestressed composite beams with corrugated web: Part I. Development and analysis K.S. Kim a,1, D.H. Lee a,⇑, S.M. Choi a,2, Y.H. Choi a,3, S.H. Jung b,4 a b

Department of Architectural Engineering, University of Seoul, 90 Jeonnong-Dong, Dongdaemun-Gu, Seoul 130-743, Republic of Korea Department of Architectural Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon 402-751, Republic of Korea

a r t i c l e

i n f o

Article history: Received 7 September 2010 Received in revised form 14 February 2011 Accepted 8 April 2011 Available online 23 April 2011 Keywords: A. Hybrid B. Strength C. Analytical modelling C. Computational modelling Accordion effect

a b s t r a c t In this study, the prestressed composite beam with corrugated web has been developed, which is suitable for the long-span structure that enables a reduction in story height as well. Utilizing the corrugated steel web improves the efficiency of prestress introduced into the top and bottom flanges due to the accordion effect, which also provides better deflection control and greater resistance against local and out-of-plane buckling. Additionally, when the corrugated steel web becomes composite with concrete, the slanting plate of the corrugated web induces bearing stress that can improve composite action between the steel beam and the surrounding concrete. In this study, a parametric study on the accordion effect has been performed by a total of 24 corrugated webbed steel beams utilizing the finite element analysis. Based on the analysis result, a simple approach for the estimation of the accordion effect has been proposed. In addition, this study introduces a flexural behavior model for the prestressed composite beams with corrugated web. The verification of the proposed methods in this study is presented in a separated paper. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction For civil engineering structures such as bridges, continuous efforts are being carried out on the development of longer-span horizontal members. Also, in large buildings, there is an increasing demand for horizontal members that realize not only longer span, but also story height reduction. Although steel structures can generally offer high strength, large members used in the construction of steel structures may cause buckling instability, excessive deflection, vibration, deterioration of fatigue strength, and the need for excessive stiffeners due to the high width-to-thickness ratio [1]. In reinforced concrete structures whose span is greater than 8 m, they may have problems such as excessive cracking and deflection, making it difficult to achieve long spans while at the same time reducing story height [2]. To overcome such limitations with existing horizontal structural members, recent studies are undergoing on the development of steel–concrete composite members that can sufficiently offer the advantages of both reinforced concrete and steel [3–5]. Partic⇑ Corresponding author. Tel.: +82 2 2210 5354; fax: +82 2 2248 0382. E-mail addresses: [email protected] (K.S. Kim), [email protected] (D.H. Lee), [email protected] (S.M. Choi), [email protected] (Y.H. Choi), [email protected] (S.H. Jung). 1 Tel.: +82 2 2210 5707; fax: +82 2 2248 0382. 2 Tel.: +82 2 2210 2396; fax: +82 2 2248 0382. 3 Tel.: +82 2 2210 2194; fax: +82 2 2248 0382. 4 Tel.: +82 32 860 7585; fax: +82 32 866 4624. 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.04.020

ularly, many of these studies focus more on prestressed composite beams that offer more active serviceability than passive composite members that simply combine reinforced concrete and steel members. The prestressed composite members that combine the prestressing method used in reinforced concrete into composite steel beams are very effective in controlling deflection and strengthening capacity of member, and therefore there is increasing interest in this member whose field applications are also expanding [6–13]. Typical I-type steel beams have huge axial rigidity on their section, which result in low effective prestress when prestress introduced to these beams. As shown in Fig. 1a, however, prestressing on a steel beam with corrugated web can maximize the effective prestress introduced into the top and bottom flanges due to the accordion effect [14–17] caused by the low web axial rigidity, which also provides lager initial camber and better deflection control. Additionally, fabricating composite members by pouring concrete to the corrugated web of steel beam and upper slabs improves composite behavior as shown in Fig. 1b due to the bond between the steel beam and surrounding concrete, as well as the additional bearing area from the corrugated web. Therefore, the prestressed composite beams with corrugated web can be an effective alternative for realizing long span members and reducing story height. However, there is still a lack of understanding on the accordion effect of corrugated web, and little research has been done on the behavior of prestressed composite beams with corrugated web. Therefore, this research proposed a prestressed composite beam

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Nomenclature Aeff Aflange Ag Ap A0s Ast,b Ast,t Ast,web,i Aweb a b bf Cc C 0s Cst,t Cst,web,i Dx Dy d dp 0 ds dst,b dst,t dweb,layer,i Ep Es Est e Fs Fy fc fc0 fps fpe fpu fs fy hw Ibottom Itop Ieff Ig kg kL Md,steel Mi Mn Ni Pe r Tps

effective sectional area considering accordion effect sectional area of top and bottom flange total sectional area of wide flange steel beam with corrugated web area of prestressing tendons area of compression steel bar bottom flange area of corrugated web steel beam top flange area of corrugated web steel beam area of each web layer element area of corrugated web web panel width inclined web panel width bottom flange width compression force of concrete compression force of reinforcing bars compression force of top steel flange compression force of ith layer of steel web flexural stiffness of corrugated web about an axis parallel to girder length flexural stiffness of corrugated web about an axis parallel to girder depth wave height of corrugate web distance from extreme top fiber to tendon centroid distance from extreme top fiber to the centroid of compression reinforcement distance from extreme top fiber to the centroid of bottom flange distance from extreme top fiber to the centroid of top flange distance from extreme top fiber to the centroid of the ith web layer element elastic modulus of tendon elastic modulus of steel reinforcement elastic modulus of steel plate eccentricity of tendon about centroid of gross section at maximum moment region stress in steel plate yield stress of steel plate stress in concrete ultimate compressive strength of concrete stress in tendon effective stress of tendon ultimate strength of tendon stress in reinforcing bar yield stress of reinforcing bar height of web moment of inertia calculated from bottom stress based on FE analysis moment of inertia calculated from top stress based on FE analysis effective moment of inertia moment of inertia based on gross section coefficient for elastic global buckling coefficient for elastic local buckling dead load moment before composition moment resistance by the ith layer element moment resistance at a section sectional force of the ith layer element effective force of tendon inclined panel width resistance force by tendon

with corrugated web, an effective structural system, in which concrete resists compression, prevents buckling of corrugated web,

Tst,b Tst,web,b tw v y yb yt

a a1 b b1

ebottom eb ec e0c ecu epe epe,st,b epe,st,t epe,web eps es e0s est est,b est,t est,web,i et etop Deps De0s Dest,b Dest,t Dest,web,i ga gf ra,elastic ra,FEM rb,elastic rb,FEM rt,elastic rt,FEM

scr,L scr,inelastic scr,G scr,I sy

resistance force by bottom flange of steel beam resistance force by web of steel beam below neutral axis web thickness Poisson’s ratio distance from top fiber of section to the centoid of equivalent concrete stress block distance from bottom fiber of section to the centroid of gross section distance from top fiber of section to the centroid of gross section angle of corrugation coefficient of concrete stress block ratio of panel width to inclined width coefficient of equivalent rectangular compressive stress block to neutral axis depth strain of extreme bottom flange of steel beam strain in extreme bottom fiber of section after composition strain of concrete compressive strain at ultimate concrete strength maximum compressive strain of concrete effective prestrain in tendon effective prestrain in bottom flange of steel beam effective prestrain in top flange of steel beam effective prestrain in top flange of steel beam strain of tendon strain in steel bar strain in compression steel bar strain in steel plate strain in bottom flange of steel beam strain in top flange of steel beam strain in the ith web layer element of steel beam strain in extreme top fiber of section after composition strain in top flange of steel beam strain increases in tendon strain increases in compression reinforcement strain increases in bottom fiber of steel beam strain increases in top fiber of steel beam strain increases in the ith web layer element of steel beam coefficient of effective web area coefficient of effective moment of inertia stress of corrugated steel web subjected to axial force from elastic analysis stress of corrugated steel web subjected to axial force from FE analysis stress in bottom flange of corrugated steel beam subjected to flexural moment obtained from elastic analysis stress in bottom flange of corrugated steel beam subjected to flexural moment obtained from FE analysis stress in top flange of corrugated steel beam subjected to flexural moment obtained from elastic analysis stress in top flange of corrugated steel beam subjected to flexural moment obtained from FE analysis elastic local buckling strength of web inelastic local buckling strength of web global buckling strength of web interactive buckling strength of web Von Misses yield stress

and enhances composite behavior. Also, steel beams resist tension forces, and tendons play an active role in controlling deflection. An

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A-A Section Strain in I-type beam ε pe, st ,t f pe, st ,t

A

Accordion effect

top flange

A

ε pe bottom flange

Steel beam with corrugated web

f pe

ε pe, st ,b

f pe, st ,b

Strains

stresses

(a) Accordion effect

(b) Bearing action Fig. 1. Structural characteristics of corrugated steel web.

analysis model also has been proposed for flexural behavior of the developed prestressed composite beams with corrugated web. The verification of the proposed model will be presented in a separate paper [18].

2. Review on previous studies Fig. 2 shows examples of existing composite beams. The steel beam system with concrete slab, shown in Fig. 2a, has been widely used, in which horizontal shear force is resisted by the shear connectors, compression force is resisted by the upper concrete, and tension force is resisted by the lower steel members. Such beams, however, require additional coating for fireproofing and are unfavorable for reduction of story height [3]. Encased composite beams, shown in Fig. 2b, have improved on such problems. Mullet [4] reports that, whereas in the external composite beams all the horizontal shear force is resisted by the shear connectors, in the encased composite beams about 0.6 MPa of bond stress (fsb) occurs in the shear bond perimeter between steel beam and surrounding concrete, which results in a huge improvement of composite action. Heo et al. [5] also reported that the composite beams with

deep deck-plate had the superior composite behavior from their test results on profiled steel beams. However, the effective depth of beam in such encased composite beams is too small to control deflection, and consequently their maximum span is relatively short. The externally prestressed composite beams [6–10], shown in Fig. 2c, can actively improve the deflection and strength issues of the afore-mentioned composite beam systems, as they can control downward deflection by implementing prestressing on steel beams and improve flexural strength by high-strength tendons. Lorenc and Kubica [6] reported that a superior composite behavior can result from adequate shear connectors by conducting experimental tests on three push-out specimens with shear connectors. They also conducted flexural test on six externally prestressed composite beams with the prestressing forces as the key variables, thereby confirming that the externally prestressed composite beams are more appropriate in long-span constructions than non-prestressed composite beams as the flexural strengths of the externally prestressed composite beams increased by about 35%. Chen and Gu [7] proposed a prediction model for the ultimate tendon stress and the ultimate load–displacement relationship of externally prestressed composite beams by integrating the plastic

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Shear connector

(a) Composite beam (steel beam exposed)

(b) Composite beam (steel beam enclosed)

Shear connector Steel

Concrete

External tendon

(c) External PSC composite beam External steel plate

(d) Preflex beam Concrete filled tube

Concrete Tendon External tendons

(e) SPC composite beam

(f) CFTA composite girder Fig. 2. Examples of existing composite beams.

rotation angles within the plastic hinge length based on the assumption that the elastic deformation of the members is negligibly small. Ayyub et al. [8] proposed an analysis model using the concept of the transformed area of the externally prestressed composite beams subjected to positive bending moment, and obtained very accurate results compared to the experiments. Subsequently, Ayyub et al. [9] extended their research to the externally prestressed composite beams subjected to negative bending moment. They conducted flexural tests of five specimens, and reported that the use of prestressing tendons can greatly reduce the amount of the top flange (tensile flange) steel and delay the cracking of the concrete at negative moment region. In a related paper [10], they also proposed an analysis method for the flexural behavior of their externally prestressed composite beams, called an incremental deformation method. The preflex beam [11] in Fig. 2d is a composite beam in which the design load is reversely applied in advance on the steel beam and the prestressing is introduced by removing the pre-load after pouring high-strength concrete on the bottom flange of the steel beam. Although the preflex beam can drastically improve constructability and make good use of the structural advantages of steel members and high-strength concrete, it gives a huge loss of prestressing and low reliability of prestressing control, which results in limiting its application.

The steel-confined prestressed composite beam (SCP composite beam) shown in Fig. 2e is a composite member in which concrete is filled in the I-shaped steel tube and prestress is introduced to composite member by the prearranged tendon. Kim et al. [12] tested a large SCP composite beam, 49 m long. They reported that its constructability and flexural performance were greatly improved compared to the existing external PSC composite beams, which is because it require neither auxiliary steel reinforcement nor additional forms and the external steel tube can confine the concrete inside. Because the SCP composite beam has large axial stiffness, however, it is considered that the efficiency of prestress can be degraded, resulting in an increase of the required amount of tendon. Furthermore, the external steel tube is disadvantageous against fire and requires regular maintenance work, such as anti-rust work. The concrete-filled tubular arch (CTFA) girder, a composite beam filled with concrete in the tapered steel tubes with prestressing, has been recently developed and applied in a construction site [13]. The CFTA girder is favorable to long-span bridge constructions, but it may not be suitable for building systems due to its high depth. The steel tubes and tendons that are externally exposed also possibly reduce its durability. As observed in the case study on the existing composite beam systems, in order to achieve both story height reduction and

K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

long-span structural member, it is considered that introduction of prestress is advantageous and that increase of the composite efficiency is also necessary. 3. Development of prestressed composite beams with corrugated web This study aimed at developing a flexural member that can be suitable for both long-span construction and story height reduction. Fig. 3 shows the summary of the characteristics of the proposed prestressed composite beams with corrugated web. As has been summarized in Fig. 3, deflection control is very important in a long-span member, and therefore in this study, the prestressing method was applied to steel beams with high-strength tendon, which increases flexural rigidity as well as its ultimate strength. However, the introduction of prestressing to steel beams with high rigidity results in low introduction efficiency of the prestressing force. To solve out this problem, the steel beam with corrugated web has been utilized, which reduces axial rigidity and thereby increases the introduction efficiency of prestress. The steel beam with corrugated web had been applied to structures in Europe as early as 1960s [1], and then was also applied to bridge constructions in Japan as well as Europe in the 1980s [19] and recently in the USA [20]. The flexural members with corrugated web have many advantages in addition to the afore-

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mentioned. As shown conceptually in Fig. 4, the flexural members with corrugated web have improved structural stability against non-symmetrical load because their shear buckling strength by the in-plane and out-of-plane load are relatively greater than typical wide flange beams [19–29]. Furthermore, the web can be considerably thinner than that of existing wide flange beam and the amount of stiffener can be drastically reduced so that it allows a more economical use of expensive steel members than previously possible [1]. The shear buckling strength of corrugated web can be determined by local buckling, global buckling, and interactive buckling. Local shear buckling strength (scr,L) of the web panel based on the well-known theory of elasticity is given by [30]

scr;L ¼ kL

 2 tw 12ð1  t2 Þ a

pE s

ð1Þ

where kL is the buckling factor based on the shape ratio of panel and boundary condition, which is 5.34 for simple support, and 8.98 for the fixed support, Es is the elastic modulus of steel plate, m is a Poisson’s ratio, tw is the plate thickness, and a is the panel width. In Eq. (1), the panel width of corrugated web (a) becomes smaller than that of a typical I-shaped steel beam, which results in the improved local shear buckling strength. Meanwhile, Easley [21], under the condition that the rigidity ratio of the orthotropic plate on the strong axis to the weak axis (Dy/Dx) is larger than 200, proposed the global shear buckling strength (scr,G):

*concrete is along the beam

Fig. 3. Concept of proposed PSC composite beam with corrugated web.

Fig. 4. Resistance to unsymmetrical load.

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scr;G ¼ kg

3=4 D1=4 y Dx 2 tw hw

ð2Þ

where kg is the buckling factor reflecting the boundary conditions of web panel and flange, which is 36 for the simple support, and 64.8 for the fixed support, and hw is the height of web. Dx and Dy are defined as:

Es t3w 12ð1  t2 Þ      Es t 3w d d þ3 a 3 þ1 þr Dy ¼ tw tw 6

Dx ¼

ð3aÞ ð3bÞ

where r is the panel width of inclined web, b is the cosine length of inclined web panel, and d is the wave height (refer to Fig. 7a). Moreover, Driver et al. [1] proposed, by introducing the non-dimensional geometric coefficient of the corrugated web, global shear buckling strength (scr,G) as:

scr;G ¼ k

3=2 Et1=2 w a

12hw

Fða; bÞ

ð4Þ

where F(a, b) is

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #3=4 3 ð1 þ bÞ sin a 3b þ 1 Fða; bÞ ¼  2 b þ cos b b ðb þ 1Þ

ð5Þ

where a is the angle of the wave height, and b is the a/b ratio. Additionally, Elgaaly [21] proposed that if the elastic buckling strength, evaluated by Eqs. (1) and (2), excesses 80% of the shear yielding strength (sy), the following inelastic strength (scr,inelastic) can be used for the check of both the local and global buckling strengths:

scr;inelastic ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:8sy ðscr Þ 6 sy

ð6Þ

where scr is smaller value of the elastic local buckling strength and the elastic global buckling strength, evaluated by Eqs. (1) and (2), respectively. The shear yielding strength (sy) can be expressed by using the Von Mises yield criterion:

F

sy ¼ pyffiffiffi

3

ð7Þ

where Fy is the yield strength of steel plate. Abbas et al. [27] and El-Meltway and Loov [16] performed studies on the interactive buckling behavior, combined local and global buckling, and they proposed simple equations for the estimation of the interactive buckling strength (scr,I), respectively, as:

1 1 1 ¼ þ ðscr;I Þn ðscr;L Þn ðscr;G Þn 1 1 1 1 ¼ þ þ ðscr;I Þn ðscr;L Þn ðscr;G Þn ðsy Þn

ð8Þ

member with corrugated web, the web behaves like an accordion and folds easily to the longitudinal direction. By the accordion effect, the axial rigidity of web decreases and the stresses introduced to the top and bottom flanges increases. The existing studies on flexural behavior of beams with corrugated web [14–17], however, either did simple calculations to obtain the flexural strengths of their specimens, just ignoring the sectional area of web, or performed the finite element analysis on the certain types of section for the verification of their behavior, showing limitations in quantitative estimation of the accordion effect for design and analysis in general. In this study, therefore, a wide range of parametric study on the accordion effect has been performed using the finite element (FE) analysis for a various geometric characteristics of the beams with corrugated web. In order to quantify the results of FE analysis in detail, this study introduced the concept of effective moment of inertia (Ieff) and effective sectional area (Aeff);

Ieff ¼ gf Ig

ð10aÞ

Aeff ¼ Aflange þ ga Aweb

ð10bÞ

where gf is the coefficient of effective moment of inertia, Ig is the gross moment of inertia, Aflange is the sectional area of the upper and bottom flanges, ga is the coefficient of effective web area, and Aweb is the sectional area of web. The effective moment of inertia (Ieff) and the effective sectional area (Aeff) reflects the decrease in flexural rigidity and axial rigidity of the beam with corrugated web, respectively. As previously discussed, the steel beam with corrugated web has a higher local buckling strength compared to the typical Ishaped steel beam. As reported by Driver et al. [1], however, local buckling possibly occurs from the lower part of the corrugated web near ultimate load. Because the local buckling of the corrugated web, as shown in Fig. 5, lowers the resistance and ductility of the member, it is, of course, desirable to prevent local buckling. In this study, therefore, the corrugated web was encased by concrete after the introduction of prestressing force so that the local buckling of corrugated web can be avoided. As shown in Fig. 1b, the protruding area of the height of corrugated web also functions as a bearing plate when loaded, which improves the composite action between web plate and concrete in addition to the surface bond stress between them. The surrounding concrete also improves fire-resistance performance and durability of the member [4]. On the other hand, as the prestressed steel beam is placed, no form or supporting posts are necessary under the beam, which leads to save construction cost. In summary, the prestressed composite beam with corrugated web developed in this study is comprised of the corrugated steel webbed beam, concrete, and prestressing tendon, which can

ð9Þ

where the value of n was suggested as 2 for Eqs. (8) and (3) or smaller for Eq. (9). Recently, Abbas et al. [20], Sayed-Ahmed [31], and Moon et al. [23] extended their studies to the evaluation of lateral-torsional buckling strength of steel beams with corrugated web subjected to in-plane load. There have been, of course, many other researches on the shear buckling strength of steel beams with corrugated web [19–29], and they reported that the in-plane and out-of-plane shear buckling strengths of the beams with corrugated web were superior to the typical wide flange beam due to the geometric advantages of corrugated web. Huang et al. [14], Egaaly et al. [15], El-Metwally and Loov [16], and Khalid et al. [17] conducted experiments on the accordion effect of steel beams with corrugated web and performed finite element analysis as well. As shown in Fig. 1a, the accordion effect is a phenomenon in which, when prestressing is introduced to a

Fig. 5. Local buckling of corrugated web.

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shows the sectional stress distribution in linear elastic state, in which the concrete and the corrugated web are perfectly bonded. As shown in stage 4, the increase of the applied load results in a non-linear behavior of the concrete in compression. When the strains on the tension side greatly increases due to the applied load, the bottom flange and the lower part of web may partially reach their yielding stresses. Different from typical composite beams with I-shaped section, the lower part of the web of the prestressed composite beams with corrugated web may yield earlier than the bottom flange. This is because only the top and bottom flanges have large pre-compressive stresses due to the accordion effect, whereas the web plates do not have it. Consequently, the top and bottom flanges experience the decompression process before they gain tensile stresses, but the lower part of the web after composition with concrete starts to get tensile stresses right after or soon after loading. As the applied load further increases, the prestressed composite beam with corrugated web reaches its ultimate strength. At ultimate, as shown in stage 5, the concrete on the compression side shows fully non-linear behavior with maximum compressive strain on top fiber and the most parts of steel beam in tension side reach their yield stress.

maximize the prestressing efficiency utilizing the accordion effect and can improve the composite action between steel beams and concrete. Furthermore, the steel beams with corrugated web prestressed by tendons, before pouring concrete, provide enough flexural stiffness and strength during construction, which also enable to shorten the construction period by eliminating the form work.

4. Analysis model For the complete estimation of the flexural behavior of the proposed composite beam, its installation and construction process should be considered. The behavior of the proposed composite beam can be divided into the behavior before and after composite with concrete, which should be examined in detail according to construction phase. In the first construction phase of the proposed composite beam, prestressing is introduced to the steel beam with corrugated web. As shown in stage 1 of Fig. 6a, the prestress is mostly introduced to the top and bottom flanges due to the accordion effect, and little or no stress to the corrugated web. In this stage, the steel beam with corrugated web is fully in elastic state, and the upward camber, varying with the self-weight of the steel beam, would normally occur due to the prestress. As aforementioned, the accordion effect of the prestressed composite beams would be quantitatively estimated by the proposed analysis model, explained in detail in Section 4.1, which would enable us to estimate the initial camber and member deflection more properly. In the second construction phase, the reinforcing bars are placed on the prestressed steel beam, and concrete is poured. As shown in stage 2 of Fig. 6a, the self-weight of concrete is added to the steel beam in this stage, which causes the compressive stress in the top flange and the tensile stress in the bottom flange, changing the effective prestress of the steel flange. Before the hardening of the concrete, the steel beam and concrete do not act as a composite member. Fig. 6b shows the behavioral characteristics of the prestressed composite beams after composition as loading increases. Stage 3

4.1. The accordion effect When the prestress is introduced to a normal I-shaped steel beam (without the accordion effect), the axial strain distributions on the top and bottom fibers of the section (etop and ebottom) are expressed by:

etop ¼ ebottom

rt

Pe Pe e Md;steel þ y  y E s Ag E s I g t Es Ig t rt Pe Pe e Md;steel ¼ ¼  y þ y Es E s Ag E s I g b Es Ig b

Es

¼

ð11aÞ ð11bÞ

where rt and rb are the stresses on the top and the bottom fibers of section, Pe is the effective prestressing force, Es is the elastic modulus of steel plate (2.0  105 MPa), Ag is the area of gross section, Ig is

F pe, st ,t

F pe, st ,t

Corrugated web

Fweb ≈ 0

f pe

Tendon

f ps

F pe, st ,b

F pe, st ,b

Stage 1 : After release

Stage 2 : After pouring concrete

(a) Before composition Comp. rebar

fc < fc '

f c = Ec ε t f s'

fc ' f s'

Fst ,t

Fst , web = Fy

f s'

Fst ,t Corrugated web

Fst,t

Fst , web = Fy

Tendon

f ps

f ps

f ps Fst ,b = Fy

Fst ,b Stage 3 : Linear elastic state

Stage 4: Non-linear state (Bottom flange and web yielding)

(b) After composition Fig. 6. Stress distribution at various loading stages.

Fst ,b = Fy Stage 5: Ultimate state

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(a) Detail of corrugated web

(b) Finite element detail Fig. 7. Description of finite element analysis model.

the moment of inertia of gross section, e is the eccentricity from the centroid of section to the centroid of tendon, Md,steel is the bending moment by the self-weight of steel beam, and yt and yb are the distances from the centroid of section to the top and the bottom fiber of section. The axial and flexural rigidity of the corrugated webbed beams, however, is smaller than the typical beams with I-shaped section because of the accordion effect. This means that the strains of etop and ebottom are greater than the values calculated by Eq. (11) [14]. Therefore, as afore-mentioned, the sectional stresses and strains of the prestressed composite beams with corrugated web were estimated by utilizing the concept of the effective moment of inertia (Ieff) and the effective sectional area (Aeff). As defined in Eq. (10), the values of Ieff and Aeff can be obtained by determining the coefficient of effective moment of inertia (gf) and the coefficient of effective web area (ga). In order to determine the coefficients gf and ga, a parametric study has been conducted by a finite element analysis (FEA) using a commercial program [32]. The accordion effect depends on the geometric characteristics of the corrugated web such as panel width (a), wave height (d), web thickness (tw), and web height (hw), as shown in Fig. 7a. As the most influencing factors on the accordion effect are considered to be the relative ratios of the

geometric dimensions, the parametric study has been conducted for the key variables of the wave height to web thickness ratio (d/tw) and the panel width to web height ratio (a/hw). As shown in Table 1, a total of 24 FE models were selected, and the values of the web height (hw) were 350 mm, 600 mm, 1000 mm, and 1500 mm. The d/tw ratios were 10.8, 15, and 20, and the a/hw ratios were 0.2 and 0.51. The distance between the supports was 6750 mm, and the widths of top and bottom flanges were 200 mm and 300 mm, respectively. The elastic modulus and the yield strength of steel plates were 2.0  105 MPa and 235 MPa, respectively. As shown in Fig. 7b, the eight-node 3D solid element was used in the FE analysis. In the FE analysis to determine the coefficient of effective moment of inertia (gf), two-point loading was applied to the member with a magnitude of 100 kN. It should be noted that the coefficient gf considers only the reduction of the moment of inertia, and thus, only flexural moments need to be applied. Table 2 shows the analysis results of the I-shaped section by the flexural stress formulas, which can be obtained from Eq. (11), and those of the steel beam with corrugated web obtained from FE analysis. In Table 2, rt,FEM and rb,FEM are the flexural stresses on top and bottom flange of corrugated steel beam obtained from FE analysis, and rt,Eq.11 and rb,Eq.11 are those obtained from elastic

Table 1 Details of finite element analysis models. No.

hw (mm)

d/tw ()

a/hw ()

a (mm)

tw (mm)

No.

hw (mm)

d/t ()

a/h ()

a (mm)

tw (mm)

1 2 3 4 5 6

350

10.8 10.8 15.0 15.0 20.0 20.0

0.20 0.51 0.20 0.51 0.20 0.51

70.0 178.5 70.0 178.5 70.0 178.5

6.00 6.00 4.33 4.33 3.25 3.25

13 14 15 16 17 18

1000

10.8 10.8 15.0 15.0 20.0 20.0

0.20 0.51 0.20 0.51 0.20 0.51

200.0 510.0 200.0 510.0 200.0 510.0

6.00 6.00 4.33 4.33 3.25 3.25

7 8 9 10 11 12

600

10.8 10.8 15.0 15.0 20.0 20.0

0.20 0.51 0.20 0.51 0.20 0.51

120.0 306.0 120.0 306.0 120.0 306.0

6.00 6.00 4.30 4.30 3.25 3.25

19 20 21 22 23 24

1500

10.8 10.8 15.0 15.0 20.0 20.0

0.20 0.51 0.20 0.51 0.20 0.51

300.0 765.0 300.0 765.0 300.0 765.0

6.00 6.00 4.30 4.30 3.25 3.25

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K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616 Table 2 Analysis results for the coefficient of effective flexural stiffness, gf.

gf

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.81 0.87 0.83 0.90 0.86 0.91 0.73 0.81 0.77 0.84 0.81 0.86 0.62 0.77 0.70 0.80 0.74 0.84 0.56 0.71 0.62 0.77 0.66 0.79

rt,FEM

rb,FEM

rt,Eq.11

rb,Eq.11

(MPa)

(MPa)

(MPa)

(MPa)

62.7 92.5 64.1 95.6 64.9 97.4 45.3 73.4 46.3 77.6 46.8 80.1 35.1 56.5 35.7 60.8 36.5 63.8 28.6 46.2 30.2 51.4 31.2 55.0

43.5 65.2 43.9 66.6 44.0 67.4 31.7 52.2 32.1 54.2 32.3 55.4 24.9 42.2 25.1 44.3 25.4 45.8 20.7 35.9 21.5 38.6 21.9 40.4

47.80 76.30 51.00 81.70 53.20 85.30 30.38 54.95 33.57 60.72 35.85 64.83 20.09 40.18 23.00 46.00 25.19 50.38 14.48 30.91 17.10 36.49 19.16 40.89

38.00 60.09 39.50 63.41 40.50 64.82 25.51 46.14 27.30 49.37 28.42 51.40 17.66 35.31 19.61 39.23 20.94 41.88 13.14 28.05 15.11 32.25 16.55 35.31 Average

rt;FEM rt;Eq:11

rb;FEM rb;Eq:11

1.32 1.22 1.27 1.18 1.22 1.14 1.49 1.33 1.37 1.28 1.30 1.23 1.75 1.41 1.56 1.32 1.45 1.27 1.96 1.49 1.75 1.41 1.64 1.35 1.40

1.15 1.06 1.11 1.05 1.09 1.04 1.25 1.14 1.18 1.10 1.14 1.08 1.41 1.19 1.28 1.12 1.22 1.10 1.59 1.28 1.43 1.19 1.32 1.15 1.19

analysis based on Eq. (11). Due to the decrease in flexural stiffness of the steel beam with corrugated web, the stresses on the top and bottom flanges increased by an average of 40% and 19%, respectively, compared with the case without the accordion effect. The different increase of the stress ratios between top and bottom is because the width of the bottom flange is wider than top flange. Based on the top and bottom flange stresses (rt,FEM and rb,FEM) resulting from FE analysis, two values of the moment of inertia (Itop and Ibottom) can be calculated:

Itop ¼

M

rt;FEM

Ibottom ¼

yt

M

rb;FEM

ð12aÞ yb

ð12bÞ

Then, the effective moment of inertia (Ieff) can be obtained approximately as:

Ieff 

Itop þ Ibottom 2

ð13Þ

Therefore, the coefficient of effective moment of inertia, gf, is given by:

gf ¼

Ieff ðItop þ Ibottom Þ=2  Ig Ig

ð14Þ

where Ig is the effective moment of inertia of gross section of steel beam. Fig. 8a presents the effect of the web height (hw) on gf in case the panel width to web height ratio (a/hw) is 0.2. The gf values are distributed between 0.86 and 0.56, and the gf value decreases as the web height (hw) increases. The reduction of flexural stiffness also tends to be larger as the wave height to web thickness ratio (d/tw) increases. Fig. 8b also shows the gf values when the panel width to web height ratio (a/hw) is 0.51. Although the reduction of flexural stiffness can be observed similarly as in the case when a/hw is 0.2, the decrease of gf values was relatively smaller. On the other hand, the gf values are between 0.91 and 0.71, which are larger than the case of a/hw = 0.2. This is because, as the a/hw becomes larger, the sectional shape of the web becomes close to that of a typical I-shaped beam (the case without the accordion effect),

Fig. 8. Coefficient of effective flexural stiffness, gf, based on finite element analysis.

implying the decrease of the accordion effect due to greater stiffness of web. A regression analysis was performed to reflect the observations on the gf values, resulting in the following equation:



gf ¼ 0:70

d a bf tw hw hw

0:15 ð15Þ

where bf is the width of the bottom flange and the other variables are identical to what have been described above. Fig. 8c shows the comparison of the gf values by Eq. (15) to the results of FE analysis. The ratios of the gf values by these two approaches gave a

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K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

Table 3 Analysis results for the coefficient of effective web area, ga. No.

ga

ra,FEM (MPa)

ra,Eq.11 (MPa)

ra,FEM/ra,Eq.11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.35 0.42 0.38 0.44 0.40 0.47 0.24 0.28 0.26 0.29 0.27 0.31 0.17 0.22 0.20 0.23 0.21 0.24 0.15 0.20 0.16 0.20 0.17 0.21

82.4 79.3 76.0 73.4 71.7 69.7 106.8 101.6 94.0 90.9 85.9 83.9 141.5 130.5 119.0 114.0 106.1 102.9 177.0 158.0 149.5 137.5 131.0 123.0

57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 57.18 Average

1.44 1.39 1.33 1.28 1.25 1.22 1.87 1.78 1.64 1.59 1.50 1.47 2.47 2.28 2.08 1.99 1.86 1.80 3.10 2.76 2.61 2.40 2.29 2.15 1.90

mean of 0.99 and a coefficient of variation of 0.04, implying that the Eq. (15) matches well the FE analysis. The reduction in axial stiffness of the corrugated web during prestressing due to accordion effect should be also considered, whereas the gf value reflects the reduction in flexural stiffness. As previously mentioned, in order to consider the reduction of the axial stiffness, this study introduced the coefficient of effective web area (ga) by applying the concept of effective web area. In order to consider the effect of the prestressing force in axial direction alone, separated from the reduction in flexural stiffness, four tendons with 98.2 mm2 of sectional area each (u12.7 mm) were arranged at the centroid of section so that no eccentricity occurs. Tendons were prestressed up to 0.75fpu in the model No. 1 in Table 1, and in all other FE models, the tendon force was adjusted so that the axial stress of the steel beam remains the same. Table 3 shows the comparison of axial stresses by FE analysis (ra,FEM) and by Eq. (11) (ra,Eq.11). It should be again noted that the FE analysis considered the effect of accordion effect while Eq. (11) did not. The analysis result showed that the accordion effect of the corrugated web increased the introduced stress by 1.9 times on average. Based on the axial stress obtained from the FE analysis (ra,FEM), the effective sectional area of the web (Aeff) is given by:

Aeff ¼

Pe

ra;FEM

ð16Þ

Then, the coefficient of effective web area (ga) is given by:

ga ¼

ðAeff  Aflange Þ Aweb

Fig. 9. Coefficient of effective web area, ga, based on finite element analysis.

ð17Þ

where Aflange and Aweb are the sectional area of flange and web, respectively. Fig. 9a shows the coefficient of ga versus the web height (hw) in case when the panel width to web height ratio (a/hw) is 0.2. As the web height (hw) increases, the value of ga decreases to between 0.40 and 0.15 for all d/tw ratios, resulting from the increase of the accordion effect as web plate gets thinner. By the same reason, the smaller the wave height to web thickness ratio (d/tw) is, the thicker the web becomes, resulting in the decrease of ga. Fig. 9b shows the ga values in case when the panel width to web height

ratio (a/hw) is 0.51, in which the value of ga decreases to between 0.47 and 0.20 as the web height (hw) increases. The ga values in this case, however, are still greater than the case of a/hw = 0.2, which means that the accordion effect decreases as the panel width to web height ratio (a/hw) increases. Considering the effects of d/tw, a/hw, and bf/hw, the coefficient of effective web area (ga) can be expressed as:

" 0:19    0:56 # 0:19 bf d a ga ¼ 0:30 tw hw hw

ð18Þ

Fig. 9c gives the comparison between ga from Eq. (18) and from the FE analysis shown in the Table 3. The results of Eq. (18) were very

K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

1613

Fig. 10. Sectional analysis of PSC beam with corrugated steel web.

close to the FE analysis, showing that the ga ratios by these two approaches have a mean of 1.06 and a coefficient of variation of 0.05. Once the values of gf and ga are obtained, the effective prestrains of the top and bottom fiber of the section (epe,st,t and epe,st,b) for the corrugated webbed beams can be modified to reflect the accordion effect as:

epe;st;b

rt

Pe Pe e Md;steel þ y  y Es Es Aeff Es Ieff t Es Ieff t rb Pe Pe e Md;steel ¼ ¼  y þ y Es Es Aeff Es Ieff b Es Ieff b

epe;st;t ¼

¼

ð19aÞ ð19bÞ

where Aeff and Ieff are the effective sectional area and the effective moment of inertia, respectively, obtained by substituting Eqs. (15) and (18) to Eq. (10). The effective prestrain of tendon is also given by:

epe ¼

Pe fpe ¼ Ap Ep Ep

ð20Þ

where Pe is the effective prestressing force, Ap is the sectional area of prestressing tendon, Ep is the elastic modulus of prestressing tendon, and fpe is the effective prestress. 4.2. Analysis model – behavior of composite beam Fig. 10 shows the strains and stresses of the prestressed composite beams with corrugated web for all loading stages, that is, at prestressing before composite in Fig. 10a, and under the service load and at ultimate after composite with concrete in Fig. 10b and c, respectively. The strains and stresses of top and bottom flanges right after prestressing, as afore-mentioned, can be obtained by Eq. (19). After composite with concrete, the tendon strain (eps) is calculated by:

eps ¼ epe þ Deps

ð21Þ

where the effective prestrain (epe) is given by Eq. (20), and the incremental strain due to applied load (Deps) can be calculated by:

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K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

Deps ¼ et þ

eb  et h

dp

ð22Þ

Then, the tendon stress (fps) corresponding to the tendon strain (eps) can be obtained, once the stress–strain relation of tendon is known. In this study, the modified Ramberg–Osgood Curve [33] is used, which is defined by:

(

0:975

fps ¼ Epe eps 0:03 þ

½1 þ ð118eps Þ10 0:1

) 6 fpu

ð23Þ

where fpu is the tensile strength of tendon. The strains on the top and bottom flanges of the composite beam can be expressed by:

est;t ¼ epe;st;t þ Dest;t est;b ¼ epe;st;b þ Dest;b

ð24aÞ ð24bÞ

where epe,st,t and epe,st,b are the prestrains on the top and bottom flanges of the steel beam before composite, respectively, and Dest,t and Dest,b are the increased strain by loading that can be calculated by:

Dest;t ¼ et þ Dest;b ¼ et þ

eb  et h

eb  et h

Thus, the total strain of corrugated web needs to consider the increased strain by loading (Dest,web) only as shown in Fig. 10b and c. Also, this study utilized the Layered Analysis [34], shown in Fig. 10b, and then, the web strain (est,web) at each layer is given by:

est;web ¼ epe;web þ Dest;web ¼ Dest;web ¼ et þ

eb  et h

dweb;layer;i

ð26Þ

where dweb,layer,i is the distance from the top fiber of the composite section to the centroid of each layer. The top reinforcing steel (in compression side) in the slab of the composite beam is placed after the introduction of prestressing, and thus, only the strain increase by loading (De0s ) can be considered as:

e0s ¼ De0s ¼ et þ

eb  et h

0

ds

ð27Þ

0

dst;t

ð25aÞ

where ds is the distance from the top fiber of the composite section to the centroid of the top reinforcing steel. As shown in Fig. 11a, the stress–strain relationships of rebar and steel plate applied in this study are the elastic–plastic model, which are given by:

dst;b

ð25bÞ

fs ¼ Es es 6 fy

ð28aÞ

F s ¼ Est est 6 F y

ð28bÞ

where dst,t and dst,b are the distance from the top fiber of the composite section to the centroid of the top flange and of the bottom flange, respectively. As shown in Fig. 10a, little stress occurs on the corrugated web before composite with concrete due to the accordion effect, and therefore, the effective prestrain in web (epe,web) can be ignored.

where es and est are the strain of rebar and steel plate, respectively, Es and Est are the elastic modulus of rebar and steel plates, respectively, fs and Fs are the stress of rebar and steel plates, respectively, fy and Fy are the yield strength of rebar and steel plates, respectively. The non-linear model by Collins and Mitchell [35] is adopted for the

Fig. 11. Stress–strain relationships of materials.

K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

1615

relationship between the stress (fc) and strain (ec) of concrete, which is given by:

"     # ec ec 2 fc ¼ 2 0  0 fc0

ec

ð29Þ

ec

where fc0 and e0c the compressive strength of concrete and the corresponding strain. Then, the sectional force by the concrete in compression can be obtained by integrating the stress of each layer, which is given by:

Z

c

fc bdy ¼ a1 fck b1 cd

ð30Þ

0

where b is the width of the composite section, a1 and b1 are the coefficients of concrete stress block, and c is the distance from top fiber of composite section to the neutral axis. By substituting Eqs. (29) and (30), a1b1 is expressed as:

a1 b1 ¼

 2

et 1 et  e0c 3 e0c

ð31Þ

) Also, by definition, the centroid of the concrete stress block (y should be:

Rc f bydy  ¼ R0 c c ¼ c  0:5b1 c y f bdy 0 c

ð32Þ

In addition, using the relation between Eqs. (29)–(32), the coefficient of concrete stress block (b1) is given by:

b1 ¼

4  et =e0c 6  2et =e0c

ð33Þ

As shown in Fig. 10, once all the stresses at each layer on the section are obtained, the sectional force can be calculated by multiplying the corresponding sectional area, which should satisfy the equilibrium condition below:

X

F x ¼ C c þ C 0s þ C st;t þ C st;web þ T st;web;b þ T ps þ T st;b

ð34Þ

Then, the bending moment of the composite beam (Mn) is calculated by:

Mn ¼

N X

C i ðdi  cÞ þ

i¼1

N X

T i ðdi  cÞ

ð35Þ

i¼1

Also, the curvature is obtained by:

/¼

et c

¼

eb ðh  cÞ

ð36Þ

Fig. 12 shows the flowchart of the whole analysis process, by which the moment–curvature response of the prestressed composite beams with corrugated web can be obtained in all loading stages. To verify the proposed analysis method, an experimental test on the proposed prestressed composite beams with corrugated web has been conducted, which will be discussed in a separated paper [18].

Fig. 12. Flow-chart of the proposed analysis model.

moment of inertia. Based on this study, the following results are drawn:

5. Conclusion This study developed the prestressed composite beam with corrugated web, which is advantageous for long-span structures and saving story height as well. An analysis method has also been proposed to estimate the flexural behavior of the developed prestressed composite beams. Particularly, the proposed analysis method is capable of predicting the prestress of the steel beams with corrugated web before composite with concrete, by applying the concept of the effective sectional area and the effective

1. Based on the FE analysis, it was verified that the use of corrugated web is very efficient to increase the prestress introduced to the top and bottom flanges utilizing, so called, the accordion effect. 2. The proposed method utilizing the coefficients of effective web area and of effective moment of inertia, based on the parametric study, was very useful to estimate the accordion effect, which is considered to be applicable for design and analysis of the corrugated webbed steel beam.

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K.S. Kim et al. / Composites: Part B 42 (2011) 1603–1616

3. The flexural behavior model for the composite beams with corrugated web has been proposed, which is capable of estimating the behavior of steel beam before composite and of the composite beam under service as well as at ultimate. Authors believe that the proposed approach is able to provide a good estimation of the behavior of PSC composite beams with corrugated web; however, an experimental study is required to verify the accuracy of this approach. In order to verify the proposed analysis model of PSC composite beams with corrugated web, an experimental study has been conducted, which is also to be presented separately in the next consecutive paper [18].

Acknowledgments This work (Grants No. 00041190) was supported by Business for Cooperative R&D between Industry, Academy and Research Institute funded by the Korea Small and Medium business Administration in 2010.

References [1] Driver RG, Abbas HH, Sause R. Shear behavior of corrugated web bridge girders. ASCE J Struct Div 2006;132(2):195–203. [2] Nilson AH. Design of prestressed concrete. 2nd ed. John Wiley & Sons; 1987. 592 pp. [3] Salmon CG, Johnson JE. Steel structures design and behavior. 4th ed. Berlin: Harper Collins; 1996. 1024 pp. [4] Mullett DL. Composite floor system. Blackwell Science Ltd.; 1998. 311 pp. [5] Heo BYW, Kwak MK, Bae KW, Jeong SM. Flexural capacity of profiled steel composite beams: deep deck plate. J Korean Soc Steel Constr 2007;19(3):247–58. [6] Lorenc W, Kubica E. Behavior of composite beams prestressed with external tendons: experimental study. J Constr Steel Res 2006;62(12):1353–66. [7] Chen S, Gu P. Load carrying capacity of composite beams prestressed with external tendons under positive moment. J Constr Steel Res 2005;61(4):515–30. [8] Ayyub BM, Sohn YG, Saadatmanesh H. Prestressed composite girders under positive moment. ASCE J Struct Eng 1990;116(11):2931–51. [9] Ayyub BM, Sohn YG, Saadatmanesh H. Prestressed composite girders. I: experimental study for negative moment. ASCE J Struct Eng 1992;118(10):2743–62. [10] Ayyub BM, Sohn YG, Saadatmanesh H. Prestressed composite girders. II: analytical study for negative moment. ASCE J Struct Eng 1992;118(10):2763–83. [11] Bae DB, Lee KM. Behavior of preflex beam in manufacturing process. KSCE J Civil Eng 2007;8(1):111–5.

[12] Kim JH, Park KH, Hwang YK, Choi YM, Cho HN. Experimental study for the development of steel-confined concrete girder. J Korean Soc Steel Constr 2002;14(5):593–602. [13] Kim JI, Kim DK, Lee JH, Kim JH. Static behavior of concrete-filled and tied steel tubular arch (CFTA) girder. KISM J Korea Inst Struct Maint Insp 2009;13(3):225–31. [14] Huang L, Hikosaka H, Komine K. Simulation of accordion effect in corrugated steel web with concrete flanges. Comput Struct 2004;82(23–26):2061–9. [15] Elgaaly M, Seshadri A, Hamilton RW. Bending strength of steel beams with corrugated webs. ASCE J Struct Eng 1997;123(6):772–82. [16] Metwally AE, Loov RE. Corrugated steel webs for prestressed concrete girders. RILEM Mater Struct 2003;36(3):127–34. [17] Khalid YA, Chan CL, Sahari BB, Hamouda AMS. Bending behavior of corrugated web beam. J Mater Process Technol 2004;150(3):242–54. [18] Kim KS, Lee DH. Flexural behavior of prestressed composite beams with corrugated web, part II: experiment and verification. J Compos Part B: Eng 2011;42(6):1617–29. [19] Bergfelt A, Leiva-Aravena J. Buckling of trapezoidally corrugated web and panels. IASBE Colloq 1986:67–74. [20] Abbas HH, Sause R, Driver RG. Analysis of flange transverse bending of corrugated web. I-girders under in-plane loads. ASCE J Struct Div 2007;133(3):347–55. [21] Easley JT. Buckling formulas for corrugated metal shear diaphragms. ASCE J Struct Div 1975;101(ST7):1403–17. [22] Elgaaly M, Hamilton RW, Seshadri A. Shear strength of beams with corrugated webs. ASCE J Struct Eng 1996;122(4):390–8. [23] Moon JH, Yi JW, Choi BH, Lee HE. Lateral–torsional buckling of I-girder with corrugated webs under uniform bending. Thin-Walled Struct 2009;47(1):21–30. [24] Gil HB, Lee SR, Lee JW, Lee HK. Shear buckling strength of trapezoidally corrugated steel webs for bridges, transportation research board of the national academies. J Transport Res Board, CD 2005;11-S:473–80. [25] Moon JH, Yi JW, Choi BH, Lee HE. Shear strength and design of trapezoidally corrugated steel webs. J Constr Steel Res 2009;65(5):1198–205. [26] Luo R, Edlund B. Shear capacity of plate girder with trapezoidally corrugated webs. Thin-Walled Struct 1996;26(1):19–44. [27] Abbas HH, Sause R, Driver RG. Shear strength and stability of high performance steel corrugated web girders. SSRC Conference. p. 361–87. [28] Naito T, Hattori M. Prestressed concrete bridge using corrugated steel webs – Shinkai Bridge. Proceedings of the Xll FIP congress; 1994. p. 101–4. [29] Ibrahim SA, El-Dakhakhni WW, Elgaaly M. Behavior of bridge girders with corrugated webs under monotonic and cyclic loading. Eng Struct 2006;28(14):1941–55. [30] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. McGraw-Hill Co.; 1961. 518 pp. [31] Sayed-Ahmed EY. Lateral torsion-flexure buckling of corrugated web steel girders. Proc ICE Struct Build 2005;158(1):53–69. [32] MIDAS IT, MIDAS/GEN User manual, Ver. 6.3.2, MIDAS Information Technology Co., Ltd., Korea; 2004. [33] Mattok AH. Flexural strength of prestressed concrete sections by programable calculator. PCI J 1979;24(1):32–54. [34] Bentz EC. Sectional analysis of reinforced concrete members, PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada; 2000. p. 198. [35] Collins MP, Mitchell D. Prestressed concrete structure. Prentice Hill; 1991. 766 pp.