nordicsteel2012_hmehri

Nordic Steel Construction Conference 2012Hotel Bristol, Oslo, Norway

5-7 September 2012

BRACING OF STEEL-CONCRETE COMPOSITE BRIDGE DURING CASTING OF THE DECK

Hassan Mehri, Roberto Crocetti

Division of Structural Engineering, Lund University, Sweden

Abstract: Trapezoidal cross sections are often used as main longitudinal load-bearing systems in steel-concrete composite bridges. A critical design stage for these girders occurs during casting of the bridge deck, when the non-composite steel section must support the entire construction load, including the wet concrete. A research work was undertaken to study lateral torsional buckling (LTB) capacity and stiffness requirements of U-shaped girders with focus on discrete torsional bracings and top lateral truss bracing, under uniform loading condition due to self-weight of structural system and wet concrete. Findings are then compared with the results obtained by previous research works and common code specifications.

Keywords: bracing, lateral torsional buckling, stiffness, composite bridges.

1 Introduction

Generally the top flanges of the girders are connected to the concrete deck as a continuous lateral bracing and the finished work has high torsional stiffness but during erection and construction period when the deck has not hardened or been attached, compression flanges of built-up steel girders are susceptible to instability as the girder is as open cross section and therefore relatively flexible in torsion. Different types of bracing systems are normally used to stabilize bridge girders against LTB. Typical bracing systems are i) discrete torsional bracing systems and ii) lateral bracing, in the form of a horizontal truss system. Minimizing the numbers of bracing used, will lead to a more efficient design since this bracing makes up a significant amount of the total costs. Moreover, top flange bracing has generally no function once the concrete deck has cured.

2 Background and literature review

Lateral torsional buckling capacity [1] of a simply supported beam subjected to uniform bending moment about the strong axis is:

M 0=πLb √ E I z GJ+

π2 E2 I zCw

Lb2 (1)

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Where Lb=the distance between lateral twisting supports; E=modulus of elasticity; G=shear modulus; J=torsional constant; I z=lateral moment of inertia; Cw=warping constant. It should be noted that the Eq. (1) was derived assuming the mentioned load and boundary conditions for a single, doubly-symmetric beam which is restricted to twist and free to warp at both ends. To account for the effect of variable moment gradients and load conditions, a modification factor,Cb, is typically applied to Eq. (1). When the lateral and the warping end restraints are unequal or are not free, the following general approximate method [2] may be used:

M cr=Cb M o=Cb π √ E I z GJ

K z Lb √1+π 2 E Cw

GJ ( K y Lb )2(2)

Where K z and K y are buckling length coefficients about weak axis, z, and strong axis, y, respectively. DecreasingLband providing appropriate bracing system, M cr will increase. Thus yielding or local buckling-not global buckling- will control the strength of the girder. Different types of cross frames or diaphragms are generally used to appropriately decrease the distortion of the girders by preventing relative twist between the two wings (Fig. 1). For both twin I-shaped and trapezoidal girders, with an adequate lateral cross-bracing system the warping stiffness and lateral moment of inertia of the cross section will be high enough to force two “wings” work together against torsion and lateral bending (Fig. 1a), otherwise the buckling capacity of the system will be dropped and global lateral buckling of half of the system about its own warping center is a possibility.

Fig. 1: effectiveness of intermediate cross-bracing stiffness during casting the deck.

The warping constant, Cw ,for a doubly symmetric I-shaped cross section is I z (h /2 )2and for an open top box girder with sloping or vertical webs can be calculated by equations given in handbooks or other approaches [3]. For both cross sections, the torsion constant is J=Σ(bi ti

3/3) where h, b i and t i are distance between top and bottom flange centroids, width and thickness of plates that make up the girder cross section, respectively. Assuming that the connections between the cross frames and the girder are hinged (so that there is no “Vierendeel effect”) and the cross frames are stiff enough in their plane, the total torsional rigidity of twin doubly symmetric, simply supported twin I girders is:

KT=KT ,st+KT ,w=2G J 1+2 E ( h2

4I z , 1+

S2

4I y ,1)(3)

Where KT ,st=St. Venant rigidity, KT ,w= total warping rigidity, S= distance between web centroids of I girders and index 1 indicates related parameter to one girder about the centroid of each wing, C1. SubstitutingKT ,st, KT ,wand 2 E I z ,1 for theGJ , E Cw and E I z respectively, in Eq. (2) gives the global buckling moment of doubly symmetric twin I girders [4], M g:

M g=2 πCb

K z Lg √E I z , 1G J 1+( πE2 K y Lb

)2

(I z ,12 h2+ I z , 1 I y ,1 S2)(4)

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Yura [4] has presented a simplified equation retaining only the term I y ,1 S2, which is the dominating term of the Eq. (4). Thus the global lateral buckling capacity is directly proportional to the girder spacing “S”. In this study, Yura also mentions that a substitution of I z , 1with effective moment of inertia,I z 1 , eff , could be directly applied to a mono-symmetric cross sections and open box girder system with reasonable accuracy:

M g=Cbπ 2 SE

Lg2 √ I y ,1 I z 1, eff (5)

I z 1 , eff=I z 1 , c+tc

I z 1 , t(6)

where t /c is the distance ratio of tension and compression flanges from the centroid of cross; I z 1 , c and I z 1 , t are the lateral moment of inertia of the compression and the tension flanges about weakest axis, respectively. In the current work, the validity of Eq. (5) for LTB of Marcy Bridge will be also investigated.

3 Case study

The Marcy Bridge in New York was a pedestrian bridge and collapsed during construction, resulted in nine severely injured workers and one fatality. The bridge, which spanned 51m, consisted of a trapezoidal girder with concrete deck acting compositely as a top flange. The concrete deck was 4200mm wide and approximately 200mm thick. The bridge was straight in plan and arched in elevation and collapsed during casting of the concrete deck. Casting begun at the abutments and moved to the mid-span. As the pouring reached the mid-span, global lateral buckling occurred and the girder collapsed twisting off of its supports at the ends, Fig. 2. The cross sectional dimensions at mid-span and near the supports are shown in Fig. 3. The web thickness is constant over the entire length, while bottom flange and top flanges thicknesses varies between 22mm and 28mm. Eight internal k-frames were used in the Marcy Bridge with no top lateral truss bracing. All cross frame bracing members were L 3× 3×3/8 (¿) angles with a cross sectional area of 1361mm2.

Fig. 2: The Marcy bridge after collapse [5]. Fig. 3: Steel cross sectional dimensions of Marcy Bridge (CS-e: at the ends, CS-m: at the mid span)

4 Mono-symmetric cross sections

The effect of having unsymmetrical cross section (i.e. with different flanges) should be taken into account in stability analysis especially when the compression flange is the smaller flange. Eq. (1), (2) and (4) are derived assuming doubly symmetric cross section. The lateral torsional

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buckling of mono-symmetric beams can have complicated behaviour depending on boundary conditions, type of loading and degree of asymmetry. It must be also noted that for mono-symmetric beams, where bending is in plane of symmetry, the shear center and the centroid do not coincide. Most of previous approaches for buckling of mono-symmetric cross sections can lead to incorrect results for particular cases [6]. An exact formula was presented by Vlasov [7] expressing the elastic buckling moment of simply supported mono-symmetric beams under uniform moment. Since then, equations for the elastic critical moment have been developed by various authors [6] with slight modifications on coefficients and expressions depending on various boundary and loading conditions. A general formula for the critical bending moment is given by Galambos [2]:

M cr=Cb π2 E I z β y

2 ( K z Lb )2 {1±√1+4

β y2 [Cw K z

2

I z K y2 +

GJ ( K z Lb )2

π2 E I z]}(7)

Where β y=¿the coefficient of asymmetry can be approximated [8] as follow:

β y=0.9 d (2 ρ−1) [1−( I z

I y)

2](8)

Where ρ=I zc¿ I z and d is the depth of cross section. To account the effects of mono-symmetry of the cross section, other approach given in AASHTO [9] is substituting 2 I zc , 1 for I z , 1 in which Eq. (1) is used. The result of an exact solution [8] with approximate coefficient of asymmetry and also an approximate equation given in AISC LRFD [10] are used in current work to verify the model.

5 Linear eigenvalue buckling analyses

Generally, two types of analysis can be performed to study buckling problems using the finite element method: 1) Linear eigen-value buckling analysis, and 2) non-linear incremental buckling analysis. The eigen-value buckling analysis is used in the current study which is limited to problems where the pre-buckling displacements are relatively small and any changes in material properties do not significantly affect the assumption of linearity.

5.1 Finite element modelling

The commercial finite element software, SAP 2000 [11], was used for the current research. Four-node shell elements with sufficient fine meshing were utilized for modelling the cross section. Shell elements were utilized since a significant portion of the total strain energy of the deformed state of these structural systems is due to in-plane behaviour of the elements. Beam elements with 6 degree of freedoms were used to simulate lateral top flange and cross-frame bracing throughout the girder. In order to avoid “Vierendeel behaviour”, beam elements were considered as hinged at the both ends. At the ends of the unbraced length of the girder, the beam is free to warp but prevented to twist laterally. The girder is free to slide in longitudinal direction at only one end. A linear elastic isotropic material with E=200GPa and n=0.3 is used for all elements. Linear eigenvalue buckling analysis has been carried out and the material non-linearity is neglected. Marcy Bridge was designed as a vertically curved beam with a camber at the mid-span. In this work the model is assumed initially straight and the benefits of presence of deck formwork close to the top flanges are also neglected conservatively.

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5.2 Model verification

The results obtained by numerical analysis should always be checked by means of traditional approaches, engineering judgments and laboratory tests. Eq. (1) for LTB capacity is conservative in predicting the buckling strength for the unbraced case due to the more favourable moment gradient, load and support height in practice; factors that are all neglected by the equation. Other factors which are not considered in general equations for LTB capacity of a beam are the magnitude and distribution residual stresses and effect of initial imperfections (loading and geometry), discontinuities in the cross section and slope of the webs in trapezoidal cross sections. The analytical aspects of determining the critical moment strength of a beam are quite complex and closed-form solutions exist only for most simple cases. Global LTB values of Marcy Bridge, assuming constant cross section CS-e, (Fig. 3), is quantitatively presented with exact and approximate solutions in Table 1.

Table 1: Result of FEM analysis and theoretical solutions (Loads at N.A.)Analysis type M cr (MNm) Diff. (%)FEM 10.4 -

Exact solution, with approximate βz [8] 10.0 -3.9

Approximate, AISC LRFD [10] 8.7 -16.9

Eq. (5) 10.3 -1.3

The theoretical solutions are derived for loading with uniform distributed moment while FEM simulations are derived for girders subjected to uniform distributed load. Therefore theoretical equations should be adjusted [4] by gradient moment factor, Cb=1.12, which is suggested by other approaches such as [8]. Comparisons between theoretical and FEM results show good agreement despite that LTB of trapezoidal cross sections is more complicate than buckling of twin I girders as the shear center is outside of the cross section and the effects of inclined webs in LTB capacity is ignored in all theoretical solutions. The similar way used for twin I girders showed more consistent results with general solutions. Supports height in theoretical formulas is assumed at the Neutral axis level (N.A.) but in reality and in FEM simulations, supports are placed at bottom layer of the cross section and global buckling capacity of the girder is slightly increased in this case.

5.3 Non-prismatic cross sections

The flange thickness of the cross-section of the Marcy Bridge varies between 22mm and 28mm at the top flange and between 20mm and 22.5mm at the bottom flange. The effect of changing the cross section along the girder is normally not considered in traditional lateral torsional buckling solutions.

Table 2: Effect of Non-prismatic cross sections (Load at N.A.)Section type M cr (MNm) Diff. (%)Constant Cs-e (Fig. 3) 10.4 -0.1

Non-prismatic (Marcy Bridge) 10.4 -

Constant Cs-m (Fig. 3) 10.7 +3.1

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Although, there is no high change in cross sectional dimensions in Marcy Bridge, the results shown in Table 2 indicate that this effect should be considered in buckling capacity of the girders when cross-sectional properties are significantly changed. Next investigations are based on real cross sections of Marcy Bridge.

5.4 Load and support height

The effect of load height must be taken into account for determining the LTB capacity of a beam. The case where the load acts on the compression flange of simply supported beam is the most detrimental as it increases the torque arm. Top flange loading reduces buckling capacity of a single I-shaped beam for mid-height loading by an approximate factor of 1/1.4 whereas tension flange loading improves the buckling capacity by a 1.4 factor. Yura [4] has achieved that the load height effect can be ignored for doubly symmetric twin I girders, with single-curvature bending moment gradient and adequate intermediate torsional bracing. The effect of load height on buckling capacity with respect to different cross frame stiffness for Marcy Bridge is shown in Fig. 4. It is evident that load height effect cannot be ignored for trapezoidal girders even though the girder is torsionally braced by means of cross frames with large equivalent torsional stiffness,βb(¿ Nβ /Lb), where N=number of intermediate cross frames and βb=¿torsional stiffness of each cross frame. Top flange loading reduces LTB capacity of the Marcy Bridge by 0.806 compared to loading at neutral axis (N.A.).

Fig. 4: Effect of load height for Marcy Bridge where M cr ,br and M N . A . ,ubrare global LTB capacity for braced and unbraced case (Load at N.A.); bracing by means of intermediate cross frames

The effect of load height for different web slopes is also investigated where the vertical depth was held constant while top width was varied and the results are shown in Table 3.

Table 3: Effect of Load height for different web slopes (cross frames as Marcy Bridge, M.B.)

Slope of websM cr ,ubr (MNm) M cr ,br(MNm) M cr ,Top/ M cr , N . A .

Top N.A. Top N.A. Unbraced M.B. braced0 °(Vertical webs) 6.1 7.6 12.3 15.2 0.806 0.8066.5 ° 5.7 7.1 10.0 12.5 0.795 0.806

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13 °(Marcy Bridge) 5.1 6.5 8.4 10.4 0.779 0.806

The results indicate that for trapezoidal girders interconnected with cross frames the load height effect is not affected with increasing slope of webs if the depth of girder is kept constant while top flange loading effect increases with sloping of the web for trapezoidal girders which are not interconnected by means of cross frames. It is also shown that critical bending moment of these girders decreases with increasing slope of the webs for both cases: unbraced, M cr ,unbr, and braced , M cr ,br even though I zis increasing. Bracing is achieved by means of intermediate cross frames.

5.5 Equivalent stiffness of internal cross frames as torsional bracing system

For low stiffness of internal cross frames, the wings of the girder system (Fig. 1b) behave independently, governed by general LTB equations considering half part of the girder. Increasing the cross-frames stiffness, the girder will still buckle in a single half wave until buckling occurs between the cross frames. Yielding can also control the capacity of the girder [12]. It can be derived from Fig. 4 or Fig. 5 that for torsional bracing stiffness higher than the ideal stiffness, the effectiveness of the cross-frames stiffness is dramatically reduced. The global buckling of a girder system can also be predicted by considering the cross frames as continuous torsional bracing along the length of the girder [12]:

M g=√Cubr2 M ubr

2 +Cbr

2 E I z βT

CT

≤ Min(M s , M p)(9)

WhereβT=N βT /Lg, N=number of intermediate cross frames; M ubr=¿bucking capacity of unbraced beam; Cubr , Cbr are limiting factors corresponding to an unbraced and an effectively braced beam, respectively;CTis top flange loading modification factor; M s and M p are the critical moment corresponding to buckling between the brace points and the plastic strength of the section, respectively; and βT=¿the effective torsional brace stiffness including the stiffness of the cross frames, βb, the web stiffness, βsec, and girder system stiffness, βg, as follow [12]:

1βT

= 1βb

+ 1β sec

+ 1β g

(10)

For k-brace systems [12] such as cross-frames which are used in Marcy Bridge, βb is:

βb=2 E bbot

2 h02

8 Lc3+bb

3 Ab(11)

Where bbot=¿width of bottom flange; h0=¿height of cross frame; Lc=¿length of diagonal members; and Ab=¿cross area section of cross-frame members. Although it is shown that the sum of intermediate cross frames stiffness is the dominating variable and not the actual number or spacing of the cross frames [4], the moment capacity of girder system drops when the cross-frame spacing exceeds 0.25Lb [13]. It should be considered that in some cases, fewer cross-frames are normally used under erection rather than service conditions. The GLB capacity of Marcy bridge with different number of intermediate cross frames,M cr ,N −br, compared to unbraced buckling capacity,M cr ,ubr, with respect to equivalent torsional stiffness, βb, is shown in Fig. 5. The result indicates the cross frame bracing system used in Marcy Bridge has a stiffness which is several times larger than the required torsional stiffness corresponding to full bracing. It is also shown that providing only one intermediate cross frame at mid span is more efficient than two cross frames, each at one third of the span, due to

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maximum lateral deformation at the mid span. For Marcy Bridge,I y /I z>1.0 where I z and I y are respectively in-plane and out-of-plane moments of inertia, if only the steel cross section is present. Marcy Bridge is one example proving that, the concept “lateral-torsional buckling of the girder cannot occur if the beam is bending about the weaker bending axis” is not valid for trapezoidal girders without top flange bracing system.

Fig. 5: Equivalent cross frame stiffness of M.B. (Load at N.A.). N=Number of cross frames.

5.6 Top flange bracing of partial or entire length of the girder

Top flange bracing of the U-shaped girder is significantly effective in increasing GLB strength as it creates a ‘pseudo box’ cross section which has much higher torsional stiffness to resist system buckling. LTB can be avoided by properly positioned and designed lateral truss bracing, before dropping off due to combination of lateral deflection and twisting. Partial or entire top flange lateral bracing is commonly used to ensure that the design moment does not exceed the LTB capacity. It should be noted that setting a top lateral bracing system for the entire length of the girder may be too expensive. It has been shown that installing partial top flange bracing near the supports is more efficient than bracing at the middle of the girder [14] as it converts lateral supports from hinged to semi-clamped. In this study it is also indicated that using single diagonal bracing system (Fig. 6) improves GLB strength of the girder in the same extent as using X-type system if the cross section area of the diagonal member of the single brace system is equal to the sum of the cross section areas of the X-type bracing. Having high amount of Lbr / Lb at the ends will increase the GLB of the girder, where Lbr=¿top flange braced length of the girder. For three different trapezoidal cross sections with different length between 45m to 62.5m it is shown [14] that top bracing of those bridges with more than 20% of the entire length at each end does not have a large impact on the GLB strength. Local buckling of compression flange or webs can occur rather than global lateral torsional buckling [15], as the unbraced length decreases.

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Fig. 6: Common top flange bracings Fig. 7: Decreasing the unbraced global length of the girder by providing partial bracing at the ends

The effect of entire X-type top flange bracing for Marcy Bridge is shown in Fig. 8.

Fig. 8: Effect of entire top flange bracing on GLB capacity of Marcy Bridge, M cr ,br.M cr ,ubr=¿GLB capacity of Marcy Bridge without top flange bracing; M Ed=¿Maximum bending

moment of M.B. due to self-weight of the steel girder and wet concrete deck; and M Rd ,br=¿design moment resistance of M.B. with top flange bracing, with respect to Ab, According to Eurocode 3.

With such a bracing, the moment capacity of top flange braced girder, M cr ,br, increases linearly with respect to increasing cross area of top flange bracing bars, Ab. It can also be derived that for Marcy Bridge providing top flange truss bracing was a necessity to carry applied moment, M ED, caused by self-weight of steel, formworks and wet concrete. It must be noted that the effect of imperfections should also be taken into account and more bracing are needed to reach the resistance moment, M Rd ,br, according to Eurocode 3 or other codes. It is shown in Fig. 9 that with a partial top flange bracing, applied over a relatively small part of span length at each end, LTB capacity of Marcy Bridge significantly increases due to providing warping end restraint similar to the laterally fixed ends. Maximum global LTB capacity of the adequately partial top flange braced bridge at the ends can be predicted considering K z ≈ 0.5in theoretical equations such as Eq. (7). It is also shown that partial top flange bracing at the ends is more efficient than at the mid-span.

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Fig. 9: comparison between entire, partial ends and partial mid-span top flange bracing. M cr ,top−br=¿GLB capacity of Marcy Bridge with top flange bracing; M cr ,top−ubr=¿GLB capacity of Marcy Bridge

without top flange bracing

6 Conclusions

The main conclusions are:1- Load height effect must be considered in design for both braced and unbraced cases.

This effect is constant for efficiently braced trapezoidal cross sections by intermediate cross frames and decreases with increasing the slopes for unbraced girders. For trapezoidal girders subjected to uniform vertical load at the top flanges, the modification factor,Cb, is suggested as 0.9(¿0.806×1.12) to take the effect of load height and moment gradient into consideration.

2- Buckling capacity of non-prismatic cross sections cannot be predicted by current general solutions and the effect of changing properties of the cross section should be properly taken into account.

3- Intermediate cross frames stiffness for Marcy Bridge was several times higher than required for full bracing against distortion. It should be noted that these values for stiffness and partial bracing length are achieved conducting linear buckling analysis of Marcy Bridge and must be adequately enhanced to account the effects of imperfections.

4- If the results of an analysis indicate inadequate bending moment capacity, the strength of the girder can be improved by adding top flange bracing at the 10-20% of the span near the supports. Providing X-type bracing with relatively small area cross section (≈ 8 mm2) moment capacity of Marcy Bridge increases about 28% which was needed to prevent the girder against lateral torsional buckling during casting the deck.

References

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Journal of Structural Engineering, vol. 134, pp. 1487-1494, 2008.[5] Pedestrian Bridge Collapse. Available: http://www.exponent.com/pedestrian_bridge_collapse/

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[6] I. Balaz and Y. Kolekova, Stability of monosymmetric beams. Amsterdam: Elsevier Science Bv, 1999.[7] V. Z. Vlasov, Thin-walled elastic beams 2. ed., rev. and augmen. ed. Jerusalem: Israel Program for

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[15] B. H. Choi, Y.-S. Park, and T.-y. Yoon, "Experimental study on the ultimate bending resistance of steel tub girders with top lateral bracing," Engineering Structures, vol. 30, pp. 3095-3104, 2008.