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High performance steel bridge girder compactness
Journal of Constructional Steel Research 58 (2002) 859–880 www.elsevier.com/locate/jcsr
High performance steel bridge girder compactness C.J. Earls ∗, B.J. Shah Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA, USA Received 20 November 2000; received in revised form 14 August 2001; accepted 9 October 2001
Abstract The present study employs experimentally verified nonlinear finite element modeling techniques for the study of high performance steel (HPS) I-shaped bridge girders. An evaluation as to the appropriateness of using the current American bridge specification (AASHTO 1998) provisions for cross-sectional compactness and adequate bracing is carried out within the context of applications involving A709 Gr. HPS483W high performance steel. It appears that the current AASHTO bracing provisions are inadequate when applied to A709 Gr. HPS483W bridge girders; intense interactions between local and global buckling manifestations occur in the HPS bridge girders studied herein despite satisfaction of the AASHTO requirements. As a result of the interaction between local and global instabilities, the application of the separately defined AASHTO cross-sectional compactness criteria becomes invalid. An alternate bracing requirement is proposed for use with HPS bridge girders as part of this research. This new bracing scenario does not require any additional costs in fabrication or materials since only the placement of the required bracing is changed with respect to current practice and not the total number of braces. Once the new bracing scenario is implemented, the notion of crosssection compactness once again has validity and thus correlates with HPS I-shaped girder flexural ductility. New flange and web compactness criteria are proposed for use with A709 Gr. HPS483W bridge girders braced in the manner specified herein. 2002 Elsevier Science Ltd. All rights reserved. Keywords: High performance steel; HPS; Compactness; Local buckling; Composite girders
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Nomenclature bf d Fy Fb Fu Dcp L Lb Lb1 Lb2 Lp My Mp R ry tf tw ey eb est eu qp
Flange width Depth of cross-section Steel yield stress Intermediate stress value Steel ultimate stress Depth of the web portion in compression Span length of simply supported beam Distance between braces Larger distance between braces in the alternate bracing configuration Smaller distance between braces in the alternate bracing configuration Length of yielded region in beam at maximum moment Moment causing the extreme cross-sectional fiber to yield Full plastic capacity of cross-section Rotation capacity of cross-section Radius of gyration about the out-of-plane axis Flange thickness Web thickness Yield strain Intermediate strain vlaue Strain hardening strain Ultimate strain Cross-sectional rotation resulting in the attainment of the theoretical plastic moment
1. Introduction It is known that in the flexural response of I-shaped beams and girders, local and global buckling phenomena are coupled in some manner. In its guide and commentary on plastic design in steel, ASCE [1] states: “Even though local and lateral– torsional buckling in the inelastic range are manifestations of the same phenomena, namely, the development of large cross-sectional distortions at large strains, they have been treated as independent problems in the literature dealing with these subjects. This is mainly due to the complexity of the problem.” Despite this fact, a great deal of success has been achieved in notionally de-coupling local buckling from lateral–torsional buckling for the purposes of design. This technique has been promulgated in current American steel building and bridge specifications [2, 3]. The strategy of independent consideration of local buckling and lateral-torsional buckling in steel I-shaped beam and girder design has been supported by much laboratory testing carried out on specimens made mostly from common structural grades
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of steel. While there appears to be little doubt of the utility in applying such notional de-coupling to applications with current low, to medium, strength steel grades, there is evidence that the practice of separate consideration of cross-sectional slenderness and beam slenderness is not prudent for applications involving new high performance steels [4–6]. This evidence comes from the results of experimentally verified nonlinear finite element studies of compact I-shaped beams, of building type proportions, made from HSLA80 and other high performance steel grades (HSLA80 is a type of high performance steel, having a yield strength of approximately 586 Mpa, used by the US Navy in double-hull ship construction). From the numerical tests reported in these earlier studies it was observed that geometric factors such as flange slenderness, web slenderness, beam unbraced length, etc., when taken alone or in combinations, did not seem to accurately predict structural ductility as quantified by plastic hinge rotation capacity. Such observations raise concern about the applicability of current flexural design provisions in the American steel building and bridge specifications [2, 3] for use in designs involving high performance steels. The current specifications adopt a philosophy wherein it is implied that the satisfaction of certain cross-sectional compactness criteria result in some minimum plastic hinge rotation capacity; likewise, the satisfaction of separately prescribed bracing requirements also result in the attainment of some minimum degree of structural ductility as measured by plastic hinge rotation capacity. The ductility measure called rotation capacity is defined by ASCE [1] as R={(qu/qp)⫺1}, where qu is the rotation when the moment capacity drops below Mp on the unloading branch of the M⫺q plot, and qp is the theoretical rotation at which the full plastic capacity is achieved based on elastic beam stiffness. This definition is described graphically in Fig. 1. In this figure, q1 corresponds to qp, and q2 corresponds to quin the ASCE definition.
Fig. 1.
Definition of rotation capacity.
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1.1. Background Historically, the compactness requirements specified by the American Association of State Highway and Transportation Officials (AASHTO) for bridge design have been essentially the same as those contained in the contemporary American Institute of Steel Construction (AISC) steel building specification [7–9, 3]. While very slight differences existed between the AISC and AASHTO web compactness provisions in earlier editions of the AASHTO Specification, as a result of differing conventions as to what constitutes the appropriate plate width (i.e. for the web: the full crosssectional depth, or the clear distance between the flanges, etc.), the most recent AASHTO Load and Resistance Factor Design (LRFD) Specification [3] prescribes compactness requirements that are identical to those outlined in the third edition of the AISC LRFD Specification [2]. It has been tacitly assumed in these specifications that differences in steel grades can be accounted for within compactness criteria through the inclusions of a scaling factor related to the inverse of the square root of the yield stress associated with either the web or the flange in the case of the AISC LRFD building specification [2], or the compression flange in the case of the AASHTO LRFD [3]. It seems unlikely that such a single scaling factor, based on yield stress alone, could account for all of the behavioral changes that accompany the significant deviations in uniaxial material responses that are characteristic of changes in high performance steel grades. While most currently available constructional steel grades have somewhat similar general uniaxial stress-strain responses, save for differences in yield strength and strain hardening slope, the same is not true for new High Performance Steel (HPS) grades. The material properties of the high performance steels tend to be different from that of mild carbon steel, or its equivalent, in several key areas of the uniaxial stress–strain relationship; HPS grades may often display neither a well defined yield plateau nor a substantial strain-hardening modulus as compared with more commonly used grades. Some HPS grades may also exhibit somewhat less ductility than their more commonly used counterparts. While the frequently used HPS grade for bridges, A709 HPS483W, displays a somewhat significant strain hardening slope and good ductility, it only exhibits a rather abbreviated yield plateau as compared with other grades. All of the material response characteristics mentioned above can have a significant effect on observed structural ductility of I-shaped girders [6] and hence it is believed that the existing practice of accounting for differing steel grades in design provisions through the use of some scaling factor related to the square root of the minimum specified yield stress may be inadequate for applications involving high performance steel. 1.2. Scope As a result of the concerns associated with applying existing compactness criteria to the case of bridge girders made from HPS, a research program has been initiated at the University of Pittsburgh in cooperation with the New York State Thruway Authority and the Federal Highway Administration. The study is focused on
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assessing the validity of employing current compactness criteria to highway bridge designs utilizing A709 Gr. HPS483W steel. As part of this study, a subject bridge was selected from the New York State Thruway Authority’s high performance steel bridge inventory. While the bridge was not designed with plastic analysis and design techniques, the flanges of the girder sections at the pier were quite stocky and easily satisfied the current AASHTO [3] flange compactness criterion. In addition, despite the fact that the girder webs of the bridge did not quite satisfy the AASHTO web compactness criterion [3], they were somewhat close to the compactness limit (18% more slender than the criterion allows) and as a result the Thruway bridge girders represented a reasonable starting point for this study in terms of overall geometry. It will be observed later in this paper that, in actuality, the AASHTO web compactness requirements could be relaxed slightly when applied to designs incorporating A709 Gr. HPS483W steel, and as a result the web slenderness of the Thruway girders is not a concern. The current work focuses on finite element modeling that starts with the New York State Thruway Authority bridge as a point of departure from which parametric studies aimed at investigating the influence of flange and web compactness on bridge girder rotation capacity are conducted. Nonlinear finite element modeling is the vehicle used in the present work. The modeling techniques employed are experimentally validated through a verification study carried out as part of the current work (using experimental data from the laboratory testing of a series of bridge girders at the University of Pittsburgh [10]). A detailed description of the New York State Thruway Authority HPS bridge is given in Part 2 of the current paper while Part 3 discusses the nonlinear finite element modeling techniques used in the parametric study reported as part of this paper. The finite element modeling techniques introduced in Part 3 have shown good agreement with the experimental tests carried out by Morcos [10]; a comparison study focusing on the present finite element modeling techniques and the experimental results of Morcos is outlined and discussed in Part 4. The experimentally verified nonlinear finite element techniques are then employed in a parametric study of the influence that flange and web compactness have on bridge girder rotation capacity. The results of this parametric study are presented and discussed in Part 5. Conclusions related to the present study are contained in Part 6.
2. Description of New York State Thruway Authority bridge The New York State Thruway Authority HPS bridge studied consists of a system of five, equally spaced, two-span continuous girders tied together with transverse diaphragm members and a reinforced concrete deck slab of 240 mm thickness. While the top girder flanges are continuously tied to each other by the deck system, the girders are also connected through diaphragm members occurring at the supports and along the length of girders at variable spacing. Composite action is achieved between the deck and girders by way of shear studs (22 mm in diameter). Bearing and transverse stiffeners are present along the bridge longitudinal axis. The bearing
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Fig. 2.
Elevation view of New York State Thruway Authority HPS bridge girder.
stiffeners are provided at the end supports and at interior support, whereas intermediate stiffeners are provided in each span at equal spacings of 7.625 m, except in the portion of the span region where there is some adjustment made due to the bolted flange splice. An elevation and section view of the girder are given in Figs. 2 and 3 respectively.
Fig. 3.
Section view of New York State Thruway Authority HPS bridge girder.
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The behavior investigated in this work focuses on bare steel compactness and hence is most relevant to the design of the hogging moment regions of continuous span girders due to the composite action occurring in the positive moment region. To represent a portion of continuous bridge girders between adjacent inflection points at an interior support, a simply supported beam with a overhang of L/2 is considered as shown in Fig. 4. The mid span concentrated load represents the pier reaction and the simple supports represent the point of contraflexure in the prototype continuous span member. The flanges of the HPS beam are 406 mm wide and 45 mm thick in the negative moment region with a slenderness ratio (bf/2tf) of 4.5 and 38 mm thick in the positive moment region. The web is 16 mm thick and 736 mm deep, resulting in a slenderness ratio (2Dcp/tw) of 90 at the plastic cross-sectional capacity (including the effect of negative moment reinforcing bars in the deck). All stiffeners are provided for the entire height of the web and are continuously welded to the web on both sides of the stiffener. Bearing stiffeners at the supports and at mid-span are provided on either side of the web as 25 mm thick and 200 mm wide steel plates. Similarly, intermediate stiffeners are also provided on both sides of the web, but with a 12 mm thickness and 125 mm width as shown in Fig. 3. Limiting slenderness ratios for the flange (bf/2tf) and web (2Dcp/tw) compactness are 7.75 and 76.5 respectively according to AASHTO provisions [3]. While the girder cross-section is essentially compact (the web is not quite compact), and thus able to develop its full plastic capacity without the threat of local buckling interfering with the required plastic hinge rotation needed to develop a collapse mechanism in the girder as a whole, the same is not true for the bracing. Since the girder was not proportioned using plastic analysis and design methodologies, the New York State Thruway Authority designed the girder with a bracing
Fig. 4.
Schematic of model geometry and boundary conditions used in parametric study.
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spacing that is considered inadequate for plastic moment redistribution to take place (Lb=7.625 m). According to AASHTO Section 6.10.4.1.7, Lb must be 4.68 m or less in order to ensure that adequate structural ductility will be displayed at the full cross-sectional girder capacity without lateral–torsional buckling interfering with the formation of a collapse mechanism. Hence the finite element models used in the compactness parametric study incorporate two additional braced stiffeners present in the girder regions on either side of the pier adjacent to the first stiffener location (i.e. within the first 7.625 m on either side of the mid-span bearing stiffener shown in Fig. 4). While in the first part of the parametric study the stiffeners and bracing are placed at a distance of 3.8125 m from mid-span, this was subsequently changed as discussed in Part 5 of the current paper due to the strong coupling that exists between local and global buckling modes in HPS I-shaped girders at the ultimate load. 3. Finite element modeling techniques The commercial multipurpose finite element software package ABAQUS [11] is employed in this research. ABAQUS has the ability to consider both geometric and material nonlinearity in a given model. Advantage has been taken of this capability and thus all modeling herein considers both nonlinear geometric and material influences. Incremental solution strategies are required to trace the proper nonlinear equilibrium path in analyses such as these; hence the modified Riks–Wempner solution strategy [11–14] is chosen for this work. In modeling studies where inelastic buckling is studied it is important that the evolution of the modeling solution be carefully monitored so that any indication of bifurcation in the equilibrium path is carefully assessed so as to guarantee that the equilibrium branch being followed corresponds to the lowest energy state of the system. While different strategies exist for guaranteeing that the lowest energy path is taken [15], the strategy of seeding the finite element mesh with a initial displacement field is employed in this study [11]. In this technique, the finite element mesh is subjected to a linearized-eigenvalue buckling analysis from which an approximation to the first buckling mode of the girder is obtained. The displacement field associated with this lowest mode is then superimposed on the finite element model as a seed imperfection for use in the incremental nonlinear analysis. This seed imperfection displacement field is scaled so that the maximum initial displacement anywhere in the mesh is equal to one-one-thousandth of the entire span length of the girder (L/1000). While it is recognized that the technique of seeding a finite element mesh with an initial imperfection has shortcomings [15], this technique is nonetheless employed in the current study due to the fact that results obtained from this method have agreed quite well with experimental tests as discussed in Part 4 herein. 3.1. Finite element mesh The models of the bridge girders considered in this study are constructed from a dense mesh of four node nonlinear shell finite elements. The planes of the mesh
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surfaces correspond with the middle surfaces of the constituent cross-sectional plate components of the given I-section (see Fig. 5). A moment gradient loading is used in this study since it represents the loading condition experienced by the girder at pier locations. It is the pier location where the most plastic hinge rotation capacity is needed in order to accommodate formation of a collapse mechanism in a twospan continuous highway bridge. The moment gradient loading is achieved in the finite element modeling by imposing a concentrated load at the mid-span of a simply supported I-shaped girder assembly. The concentrated force simulates the pier reaction of the subject bridge and the simple supports are placed at the approximate points of inflection on either side of the pier. In the subject bridge, diaphragm members are present at the pier and at the point of inflection and so the finite element models are braced against out-of-plane deflection at the load point and supports. In the finite element models an additional length of girder is present beyond the support locations to help simulate the torsional-warping restraint provided by the adjacent beam segments in the actual bridge. The length of the additional beam segments is chosen to be 7.625 m which corresponds to the distance to the next diaphragm member occurring after the point of inflection, as measured along the longitudinal axis of the girder. Since this additional beam segment represents an additional unbraced length, the ends of the segments are braced against out-of-plane translation in an idealized way so as to simulate the presence of the associated diaphragm members in the real bridge. Fig. 4 provides a schematic of the loading configuration just described. It is noted that in Fig. 4, the top flange is in compression and hence has dimensions of: 406 mm width and 45 mm thickness. The bottom flange is in tension and hence has a modified thickness of 84 mm in conjunction with the original flange width of 406 mm. The exaggerated tension flange thickness is used to account for the presence of negative moment steel in the bridge deck. The S4R nonlinear, finite strain, shell element from the ABAQUS element library
Fig. 5.
Representative bucked finite element mesh.
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is employed in this research. The S4R shell element is shear deformable and subsequently both reduced integration and the assumed strain method [16] are employed to improve the overall thin-shell behavior of this element. A single integration point is used in this particular element hence stabilization of spurious zero energy modes is provided for by ABAQUS. This element was selected for use in the parametric study based on its superior performance in the verification study described in Part 4 of the current paper. A uniaxial representation of the A709 Gr. HPS483W steel constitutive law employed in this study appears in Fig. 6 as a plot of true stress versus logarithmic strain (the steel grade HSLA80 mentioned in Section 1 is also shown). ABAQUS uses the von Mises yield criterion to extrapolate a yield surface in three-dimensional principal stress space from the uniaxial material response given in Fig. 6. The corresponding ABAQUS metal plasticity model is characterized as an associated flow plasticity model with isotropic hardening being used as the default hardening rule.
4. Validation of modeling techniques Experimental verification of the modeling techniques used in the current study are carried out as a preliminary step in the present research. An experimental test program conducted by Morcos [10] at the University of Pittsburgh serves as the vehicle for the model verification phase of the current study.
Fig. 6.
Steel constitutive laws—true stress vs logarithmic strain.
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4.1. Experimental tests Morcos tested three full-scale bridge girders in the Structural Engineering Laboratory at the University of Pittsburgh. The proportions of the three girders varied in such a way that insight into the influence of web compactness on rotation capacity could be studied within the context of grade 345 steel. Morcos’ specimen “S” [10] is selected for the present validation study since its cross-sectional proportions permit the girder to be considered compact with respect to current AASHTO [3] design provisions. This overall cross-sectional compactness is seen as a favorable feature for a validation study since the present work has at its goal the identification of appropriate compactness criteria for applications involving new high performance steel grades. Specimen “S” was fabricated from ASTM A572 Gr. 345 plate steel all originating from the same heat. The gas metal arc process was used to join three plates together forming an I-shaped cross-section. Actual girder cross-sectional dimensions are provided in Fig. 7. AWS A5.18 ER70S-X filler material was used for the welding. The girder was 3.96 m in overall length with bearing stiffeners present at the supports and mid-point in addition to intermediate stiffeners spaced at varying intervals along the girder longitudinal axis (see Fig. 7). The overall span length was set at 3.66 m with a concentrated load being imposed at mid-span thus simulating the momentgradient loading condition present in girder section adjacent to the pier of a twospan continuous bridge. Out-of-plane bracing was provided at the load point and supports in the form of threaded bars with eye-bar ends and turnbuckles. The eye-
Fig. 7.
Geometry of Morcos’experimental specimen “S”.
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bars allowed for the bracing to be attached at discrete points on the top and bottom of each bearing stiffener on either side of the web. The turnbuckles at the center of the bracing provided some degree of bracing flexibility not easily quantified; more will be said about this flexibility in the sequel. A single 900 kN actuator provided the concentrated force needed to form a plastic hinge in the specimen. 4.2. Finite element models of experiments The finite element techniques employed in the model validation study, and described in this section, incorporate variations on the same finite element modeling techniques as discussed in Section 3 for use in the parametric study described in Part 5 of the present paper. The validation study served as a test-bed from which to evaluate differing modeling strategies in order that the most realistic model behavior is obtained thus yielding a representative response of a bridge girder in the field. In every case within the present validation study nonlinear shell finite elements are specified at the middle surfaces of the constituent plate elements of the I-girder cross-sections. Both S9R5 and S4R shell elements are considered in the validation study. Also, varying degrees of initial imperfection and bracing stiffness are considered. The influences of these individual variations are now discussed. 4.2.1. Comparison of shell finite elements Nonlinear shell elements are chosen for this study so as to be able to explicitly model local buckling deformations and the spread of plasticity effects. ABAQUS has a wide variety of shell elements in its element library; theses include shells that are suitable for “thick” or “thin” shell applications utilizing reduced integration. The ABAQUS S4R and S9R5 shell elements are considered in this verification study and the results of these element types are compared in Fig. 8. The results from the model incorporating the S4R element show a better agreement with Morcos’experimental results as compared to the model incorporating the S9R5 element type. Hence, all
Fig. 8.
Influence of shell element formulation on predicted flexural response.
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subsequent girder models considered herein are constructed from a mesh of S4R shell finite elements. 4.2.2. Effects of initial imperfections The primary role of geometric imperfections is to provide a perturbation to the perfect model geometry so as to initiate a response that is asymptotic to the ideal equilibrium path of the structure. This asymptotic response will be close to the ideal response so long as the imperfections are small enough so as not to affect gross cross sectional properties. As can be seen from Fig. 9, the results of a model with perfect geometry produces a rather gradual unloading response that does not agree well with the behavior exhibited in the experimental tests of Morcos. Improvements in the modeling response related to the effects of geometric imperfections are investigated by conducting several analyses with varying levels of geometric imperfections. In the study, small (i.e. L/2000) to very large (i.e. L/200) imperfections are added to the model and the results studied. As expected, the magnitude of the seed imperfection has a marked influence on the rotation capacity of the girder studied. At a relatively large imperfection (i.e. L/200), the rotation capacity is greatly reduced (Fig. 9). It is noted that differences in response associated with differing degrees of small imperfections (i.e. L/500 and L/1000) have very little impact on observed rotation capacity, and based on the results presented in Fig. 9, it appears that an imperfection of L/1000 provides for reasonably accurate results. The magnitude of such an imperfection is also a reasonable estimate for the initial out-of-straightness (or sweep) that might be encountered in a fabricated girder. Based on the results obtained in this phase of the verification study, an imperfection of L/1000 is used in the parametric studies carried out in the sequel related to A709 HPS483W girders. 4.2.3. Influence of bracing stiffness on flexural response Considering the fact that the bracing forces can be large at points in and around the plastic hinge regions of a beam, the effect of lateral bracing flexibility on the
Fig. 9.
Influence of seed imperfections on predicted flexural response.
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flexural behavior of the girder is investigated in the current verification study. The SPRING1 element from the ABAQUS element library is used to simulate flexible bracing behavior. SPRING1 elements are attached to the I-girder at the specified bracing locations on the tension and compression flange tips on either side of the girder web (see Fig. 10 inset). When defining the spring stiffness for the SPRING1 elements, a displacement baseline of L/1000 is used. The spring stiffness, for each of the four springs at a given cross-section, is then arrived at through consideration of variations in bracing force only. The spring forces are specified as percentages of the force necessary to yield the compression flange of the braced cross-section (i.e. 0.5, 1 and 2%). When considering validation results obtained using ideal lateral restraint, as well as the three other spring stiffnesses studied, it appears that the finite element models incorporating a SPRING1 element whose stiffness is based on 1% of the force necessary to yield the compression flange at a displacement of L/1000 works well (Fig. 10). It is noted that the models considered in this portion of the verification study possess an initial geometric imperfection of the type described in Section 4.2.2 above with a maximum amplitude of L/1000. This initial imperfection maximum value of L/1000 is unrelated to the L/1000 baseline deflection used in the spring stiffness calculations. While the bracing stiffness mentioned has worked well in the verification study described, there is evidence that optimal bracing stiffness may, to some degree, be problem dependent [4]. As a result of this situation, rigid braces are used in the parametric study to follow in order that additional complexities may be reduced. 4.3. Discussion of validation results Owing to the fact that the current validation study is used to help identify the most efficient and accurate modeling strategies to be employed in the parametric studies associated with high performance steel bridge girders considered later (Part
Fig. 10.
Influence of bracing force on predicted flexural response.
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5), the favorable agreement between the results of the experimental tests and the finite element analogs (as demonstrated by the comparison of load-deflection response displayed in Fig. 10) is encouraging. It is further noted that while the agreement in load-deflection response between the finite element models and the experiments is good, so too is the agreement between the modal manifestations exhibited by the finite element models and those observed by Morcos to be present in the experimental test beam (see Fig. 11). The fact that the finite element models display a more distinct leveling-off of the load-deflection behavior displayed in Fig. 10, as compared with the experimental tests, is related to the presence of residual stresses in the cross-section. While the actual girder cross-sections undoubtedly possess residual stresses, the finite element models did not incorporate any since residual stresses do not have a significant influence on plastic hinge rotation capacity: the focus of the present study.
5. Results A finite element parametric study is carried out using the verified modeling techniques previously discussed. At issue is whether the current AASHTO compactness criteria are suitable for use in applications involving the new A709 HPS483W steel. While these provisions have worked well in the past for structural applications involving conventional grades of steel, there has been some recent research that would indicate that this is not the case for several of the new high performance steels [4–6]. 5.1. Influence of cross-sectional compactness on structural ductility It has been tacitly assumed in current bridge specifications that localized buckling influences may be considered independently of global buckling influences in girders that possess a compact cross-section (as related to the plate slenderness of the web and flange) in conjunction with adequate bracing. Satisfaction of both the com-
Fig. 11.
Comparison of experimental and finite element buckling modes.
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pactness and bracing requirements is intended to result in a girder design that can not only attain the full plastic cross-sectional capacity, but also exhibited sufficient plastic hinge rotation capacity for moment redistribution to occur; thus ultimately allowing for the formation of a collapse mechanism. The validity of the current American bridge specification (AASHTO 1998) is tested in this regard for its applicability to A709 HPS483W steel. 5.1.1. Beam braced according to AASHTO LRFD requirements As indicated earlier in this paper, a typical HPS girder from a bridge in the New York State Thruway inventory is selected as a point of departure for the parametric study now discussed. While the cross-section of this girder comes close to satisfying AASHTO compactness criteria, the bracing spacing, as originally specified, is considered by AASHTO to be inadequate for plastic moment redistribution to take place. Hence in this first portion of the parametric study, two additional braces (one on either side of the pier) are added to the original girder configuration in the region between the mid-span bearing stiffener and the first original intermediate stiffener location (i.e. Lb was originally 7.625 m near the pier, but for the purposes of the parametric study it is reduced to 3.8125 m). The AASHTO compactness limits for this application are bf/2tf=7.75 and 2Dcp/tw=76.5 for the flange and web respectively. The New York State Thruway girder has a bf/2tf=4.5 and 2Dcp/tw=90 and hence comes close to satisfying the compactness requirements. However, as can be seen in Table 1 and Figs. 12 and 13, despite the satisfaction of the AASHTO flange compactness and bracing provisions (and near satisfaction of the AASHTO web compactness requirements), the girders were not able to attain the desired rotation capacity of three as needed for moment redistribution. While it is alarming that a girder which is considered by AASHTO to be compact and adequately braced displays less than half of the rotation capacity needed for mechanism formation, what is more of a concern is the apparent lack of correlation between cross-sectional rotation capacity and flange and web slenderness. Table 1 Results from the parametric study involving A709 HPS 483W bridge girders braced according to AASHTO Sec. 6.10.4.1.7 bf/2tf
2Dcp/tw
Lb/ry
R
3 3.5 4 4.5 5 4.5
69.9 85.6 89.2 89.6 89.8 42.6 55.5 68.1 77.9 89.6
37.4
1.14 1.38 1.45 1.32 1.18 2.23 2.03 1.79 2.26 1.32
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Fig. 12.
Fig. 13.
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Influence of flange slenderness on rotation capacity using AASHTO bracing configuration.
Influence of web slenderness on rotation capacity using AASHTO bracing configuration.
It has long been thought that as either flange or web slenderness (or both) decrease, cross-sectional ductility and rotation capacity should increase. What is apparent from Table 1 and Figs. 12 and 13 is that a decrease in these plate slenderness values can lead to either an increase or a decrease in rotation capacity. This counter-intuitive situation seems to arise out of a highly interactive buckling mode that evolves in the HPS girders in which local and global influences are very much coupled. Based on an examination of the results from graphical post processing of the finite element models, it becomes clear that substantial interaction between local and global buckling is occurring despite the satisfaction of the AASHTO bracing provisions. Thus the AASHTO bracing provisions are at issue in HPS applications as well the com-
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pactness criteria. It will be seen in the next section that once adequate bracing has been achieved, the notion of cross-sectional compactness will once again become valid. 5.1.2. Beam with alternate bracing geometry In earlier work by Earls [4] building size girders, subjected to a moment gradient loading much like that of the present research, and made from the high performance steel grade HSLA80 were studied. Two generally different inelastic mode shapes were identified in this earlier study: one yielding a much more ductile response than the other. After careful examination of the geometric characteristics of the less ductile mode, a bracing scheme was devised whereby the braces were located at points along the beam longitudinal axis where excessive out-of-plane cross-sectional deflections were noted. The new bracing scheme effectively precluded the formation of the less favorable mode and hence drastically improved the observed structural ductility (see Fig. 14). The new bracing configuration consisted of simply restricting the compression flange-web junction from moving out-of-plane at two discrete points at a distance of d/2 on either side of the point where the concentrated load was applied (“d” is the total depth of the cross-section). A very similar bracing strategy has been adopted in the present study with two notable exceptions. Firstly, in the present study the out-of-plane bracing at the bearing stiffener location over the pier is retained in contrast to the earlier work by Earls [4] in which the bracing at the load point was removed. Secondly, in the earlier study by Earls [4] the d/2 bracing members were attached to the compression flangeweb junction with a pin (thus providing no direct torsional restraint to the crosssection) while in the current work the d/2 bracing occurs at four cross-sectional locations at each bracing location along the longitudinal axis (i.e. the bracing is
Fig. 14. Schematic of model incorporating proposed bracing configuration.
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Table 2 Results from the parametric study involving A709 HPS 483W bridge girders braced according to the proposed method described in Part 5 of this paper bf/2tf
2Dcp/tw
4 4.5 4.5 (AASHTO) 5 4.5
89.2 89.6 89.6 89.8 77.9 89.6 89.6 (AASHTO) 97.4
R
Lb/ry
Lb1/ry
Lb2/ry
3.3 3.18 1.32 2.72 3.3 3.18 1.32 2.73
N.A. N.A. 37.4 N.A. N.A. N.A. 37.4 N.A.
3.6 3.6 N.A. 3.6 3.6 3.6 N.A. 3.6
71.2 71.2 N.A. 71.2 71.2 71.2 N.A. 71.2
imposed at the four flange tips of the cross-section at each bracing location) and hence torsional restraint of the cross-section is directly provided. It is further noted that the total number of braces and stiffeners have not been changed as a result of the new bracing scheme. The intermediate stiffeners and diaphragm members, formerly placed at a distance of 3.8125 m on either side of the bearing stiffener at the pier location, have simply been shifted to a distance of 0.375 m on either side of the pier location. As can be observed from Table 2 and Figs. 15 and 16, the effect of this simple shift in bracing location can be quite pronounced with the observed rotation capacity being more than doubled. Furthermore, now that the interactive buckling mode has been mitigated to some extent, there once again is an observed correlation between flange and web slenderness and the observed cross-sectional rotation capacity.
Fig. 15.
Influence of flange slenderness on rotation capacity using the proposed bracing configuration.
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Fig. 16.
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Influence of web slenderness on rotation capacity using the proposed bracing configuration.
6. Discussion of results Table 2 clearly shows that in conjunction with the new bracing configuration proposed in this paper, flange and web plate slenderness parameters once again correlate with cross-sectional rotation capacity. From Table 2 it is noted that flange slenderness (bf/2tf) of 4.5 results in a rotation capacity of 3.18 while a flange slenderness of 5 results in a rotation capacity of 2.72. Based on the compactness requirement that a cross-sectional rotation capacity of three be achieved, the flange slenderness value of 4.5 is proposed as the compactness limit for bridge applications involving A709 HPS483W. It is noted that this flange slenderness of 4.5 is much less than the value of 7.75 which is prescribed by the current AASHTO flange compactness criteria. Also in Table 2 it is observed that that in conjunction with the new bracing configuration proposed in this paper, a web slenderness value (2Dcp/tw) of 90 seems to be an appropriate compactness limit for the girder web in bridge applications involving A709 HPS483W. It is noted that this web slenderness of 90 is slightly larger than the value of 76.5 which is prescribed by the current AASHTO web compactness criteria.
7. Conclusion It appears that the current American bridge specification provisions related to flange compactness are un-conservative and not presently applicable to bridges made from A709 HPS483W steel. This situation is partially a result of the interactive buckling mode that develops in the negative moment region at the pier of HPS bridge girders. This interactive buckling mode involves a very close coupling of local and global buckling manifestations thus effectively rendering as useless currently held notions related to the role of flange and web compactness on flexural ductility. Based
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on the present research, it appears that the AASHTO bracing provisions contained in section 6.10.4.1.7 [3] are un-conservative for applications involving A709 HPS483W steel. This paper prescribes a new bracing configuration for use in the negative moment region of high performance steel bridges. The new bracing scheme does not require any additional stiffeners or diaphragm members. Rather a repositioning of the two stiffeners and diaphragm members on either side of the bridge pier, so as to be closer together, is all that is required (i.e. moving the existing stiffeners and diaphragm members to a new location d/2 away from the pier on either side). The result of the repositioning of the braces is to more than double the observed cross-sectional rotation capacity. Furthermore, now that the interactive buckling mode has been mitigated to some extent, there once again is an observed correlation between flange and web slenderness and the observed cross-sectional rotation capacity. Thus in addition to providing new bracing requirements for bridge applications requiring moment redistribution and involving A709 HPS483W steel, the current research also provides for a corrected flange compactness limit of bf/2tf=4.5 and a corrected web compactness limit of 2Dcp/tw=90 to be used in bridge applications involving A709 HPS483W. The latter compactness limit is a slightly more liberal value than that currently recommended by AASHTO. Acknowledgements The Authors wish to express their sincere thanks to the Federal Highway Administration for the funding needed to pursue this research under the Technology Deployment: Innovative Bridge Construction Program. The Authors also wish to thank the New York State Thruway Authority for its interest and cooperation in the current research. References [1] American Society of Civil Engineers. In: Plastic design in steel, a guide and commentary. New York: American Society of Civil Engineers; 1971, p. 80. [2] AISC. Load and resistance factor design specification for structural steel buildings, LRFD vol. I, 3rd ed. Chicago (IL): American Institute of Steel Construction Inc, 1999. [3] AASHTO. LRFD Bridge Specification. Washington DC: American Association of State Highway and Transportation Officials, Inc, 1998. [4] Earls CJ. On the inelastic failure of high strength steel I-shaped beams. Journal of Constructional Steel Research 1999;49(1):1–24. [5] Earls CJ. On geometric factors influencing the structural ductility of compact I-shaped beams. Journal of Structural Engineering 2000;126(7):780–9. [6] Earls CJ. The influence of material effects on the structural ductility of Compact I-shaped beams. Journal of Structural Engineering 2000;126(11):1268–78. [7] Taly N. In: Design of modern highway bridges. New York: McGraw Hill; 1998. p. 75-9. [8] Haaijer G. Objectives and early research of autostress design. Transportation Research Record No. 1380, Transportation Research Board—National Research Council, National Academy Press, Washington DC, 1993.
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[9] AASHTO. Guide specification for alternate load and resistance factor design procedures for steel beam bridges using braced compact sections. American Association of State Highway and Transportation Officials, Inc, Washington DC, 1991. [10] Morcos SS. Moment–rotation tests of steel bridge girders for autostress design. M.S. Thesis, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA, 1988. [11] ABAQUS. ABAQUS Theory Manual. Pawtucket, RI, USA: Hibbitt, Karlsson & Sorensen, Inc, 1999. [12] Riks E. The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics 1972;39:1060–6. [13] Riks E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures 1979;15:529–51. [14] Crisfield MA. A fast incremental/iterative solution procedure that handles ‘snap-through’. Computers & Structures 1981;13:55–62. [15] Teh LH, Clarke MJ. Tracing secondary equilibrium paths of elastic framed structures. Journal of Engineering Mechanics 1999;125(12):1358–64. [16] MacNeal RH. A simple quadrilateral shell element. Computers and Structures 1978;8:175–83.
High performance steel bridge girder compactness C.J. Earls ∗, B.J. Shah Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA, USA Received 20 November 2000; received in revised form 14 August 2001; accepted 9 October 2001
Abstract The present study employs experimentally verified nonlinear finite element modeling techniques for the study of high performance steel (HPS) I-shaped bridge girders. An evaluation as to the appropriateness of using the current American bridge specification (AASHTO 1998) provisions for cross-sectional compactness and adequate bracing is carried out within the context of applications involving A709 Gr. HPS483W high performance steel. It appears that the current AASHTO bracing provisions are inadequate when applied to A709 Gr. HPS483W bridge girders; intense interactions between local and global buckling manifestations occur in the HPS bridge girders studied herein despite satisfaction of the AASHTO requirements. As a result of the interaction between local and global instabilities, the application of the separately defined AASHTO cross-sectional compactness criteria becomes invalid. An alternate bracing requirement is proposed for use with HPS bridge girders as part of this research. This new bracing scenario does not require any additional costs in fabrication or materials since only the placement of the required bracing is changed with respect to current practice and not the total number of braces. Once the new bracing scenario is implemented, the notion of crosssection compactness once again has validity and thus correlates with HPS I-shaped girder flexural ductility. New flange and web compactness criteria are proposed for use with A709 Gr. HPS483W bridge girders braced in the manner specified herein. 2002 Elsevier Science Ltd. All rights reserved. Keywords: High performance steel; HPS; Compactness; Local buckling; Composite girders
∗
Corresponding author. Tel.: +1-412-624-9575; fax: +1-412-624-0135. E-mail address: [email protected] (C.J. Earls).
0143-974X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 9 7 4 X ( 0 1 ) 0 0 0 8 6 - 4
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Nomenclature bf d Fy Fb Fu Dcp L Lb Lb1 Lb2 Lp My Mp R ry tf tw ey eb est eu qp
Flange width Depth of cross-section Steel yield stress Intermediate stress value Steel ultimate stress Depth of the web portion in compression Span length of simply supported beam Distance between braces Larger distance between braces in the alternate bracing configuration Smaller distance between braces in the alternate bracing configuration Length of yielded region in beam at maximum moment Moment causing the extreme cross-sectional fiber to yield Full plastic capacity of cross-section Rotation capacity of cross-section Radius of gyration about the out-of-plane axis Flange thickness Web thickness Yield strain Intermediate strain vlaue Strain hardening strain Ultimate strain Cross-sectional rotation resulting in the attainment of the theoretical plastic moment
1. Introduction It is known that in the flexural response of I-shaped beams and girders, local and global buckling phenomena are coupled in some manner. In its guide and commentary on plastic design in steel, ASCE [1] states: “Even though local and lateral– torsional buckling in the inelastic range are manifestations of the same phenomena, namely, the development of large cross-sectional distortions at large strains, they have been treated as independent problems in the literature dealing with these subjects. This is mainly due to the complexity of the problem.” Despite this fact, a great deal of success has been achieved in notionally de-coupling local buckling from lateral–torsional buckling for the purposes of design. This technique has been promulgated in current American steel building and bridge specifications [2, 3]. The strategy of independent consideration of local buckling and lateral-torsional buckling in steel I-shaped beam and girder design has been supported by much laboratory testing carried out on specimens made mostly from common structural grades
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of steel. While there appears to be little doubt of the utility in applying such notional de-coupling to applications with current low, to medium, strength steel grades, there is evidence that the practice of separate consideration of cross-sectional slenderness and beam slenderness is not prudent for applications involving new high performance steels [4–6]. This evidence comes from the results of experimentally verified nonlinear finite element studies of compact I-shaped beams, of building type proportions, made from HSLA80 and other high performance steel grades (HSLA80 is a type of high performance steel, having a yield strength of approximately 586 Mpa, used by the US Navy in double-hull ship construction). From the numerical tests reported in these earlier studies it was observed that geometric factors such as flange slenderness, web slenderness, beam unbraced length, etc., when taken alone or in combinations, did not seem to accurately predict structural ductility as quantified by plastic hinge rotation capacity. Such observations raise concern about the applicability of current flexural design provisions in the American steel building and bridge specifications [2, 3] for use in designs involving high performance steels. The current specifications adopt a philosophy wherein it is implied that the satisfaction of certain cross-sectional compactness criteria result in some minimum plastic hinge rotation capacity; likewise, the satisfaction of separately prescribed bracing requirements also result in the attainment of some minimum degree of structural ductility as measured by plastic hinge rotation capacity. The ductility measure called rotation capacity is defined by ASCE [1] as R={(qu/qp)⫺1}, where qu is the rotation when the moment capacity drops below Mp on the unloading branch of the M⫺q plot, and qp is the theoretical rotation at which the full plastic capacity is achieved based on elastic beam stiffness. This definition is described graphically in Fig. 1. In this figure, q1 corresponds to qp, and q2 corresponds to quin the ASCE definition.
Fig. 1.
Definition of rotation capacity.
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1.1. Background Historically, the compactness requirements specified by the American Association of State Highway and Transportation Officials (AASHTO) for bridge design have been essentially the same as those contained in the contemporary American Institute of Steel Construction (AISC) steel building specification [7–9, 3]. While very slight differences existed between the AISC and AASHTO web compactness provisions in earlier editions of the AASHTO Specification, as a result of differing conventions as to what constitutes the appropriate plate width (i.e. for the web: the full crosssectional depth, or the clear distance between the flanges, etc.), the most recent AASHTO Load and Resistance Factor Design (LRFD) Specification [3] prescribes compactness requirements that are identical to those outlined in the third edition of the AISC LRFD Specification [2]. It has been tacitly assumed in these specifications that differences in steel grades can be accounted for within compactness criteria through the inclusions of a scaling factor related to the inverse of the square root of the yield stress associated with either the web or the flange in the case of the AISC LRFD building specification [2], or the compression flange in the case of the AASHTO LRFD [3]. It seems unlikely that such a single scaling factor, based on yield stress alone, could account for all of the behavioral changes that accompany the significant deviations in uniaxial material responses that are characteristic of changes in high performance steel grades. While most currently available constructional steel grades have somewhat similar general uniaxial stress-strain responses, save for differences in yield strength and strain hardening slope, the same is not true for new High Performance Steel (HPS) grades. The material properties of the high performance steels tend to be different from that of mild carbon steel, or its equivalent, in several key areas of the uniaxial stress–strain relationship; HPS grades may often display neither a well defined yield plateau nor a substantial strain-hardening modulus as compared with more commonly used grades. Some HPS grades may also exhibit somewhat less ductility than their more commonly used counterparts. While the frequently used HPS grade for bridges, A709 HPS483W, displays a somewhat significant strain hardening slope and good ductility, it only exhibits a rather abbreviated yield plateau as compared with other grades. All of the material response characteristics mentioned above can have a significant effect on observed structural ductility of I-shaped girders [6] and hence it is believed that the existing practice of accounting for differing steel grades in design provisions through the use of some scaling factor related to the square root of the minimum specified yield stress may be inadequate for applications involving high performance steel. 1.2. Scope As a result of the concerns associated with applying existing compactness criteria to the case of bridge girders made from HPS, a research program has been initiated at the University of Pittsburgh in cooperation with the New York State Thruway Authority and the Federal Highway Administration. The study is focused on
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assessing the validity of employing current compactness criteria to highway bridge designs utilizing A709 Gr. HPS483W steel. As part of this study, a subject bridge was selected from the New York State Thruway Authority’s high performance steel bridge inventory. While the bridge was not designed with plastic analysis and design techniques, the flanges of the girder sections at the pier were quite stocky and easily satisfied the current AASHTO [3] flange compactness criterion. In addition, despite the fact that the girder webs of the bridge did not quite satisfy the AASHTO web compactness criterion [3], they were somewhat close to the compactness limit (18% more slender than the criterion allows) and as a result the Thruway bridge girders represented a reasonable starting point for this study in terms of overall geometry. It will be observed later in this paper that, in actuality, the AASHTO web compactness requirements could be relaxed slightly when applied to designs incorporating A709 Gr. HPS483W steel, and as a result the web slenderness of the Thruway girders is not a concern. The current work focuses on finite element modeling that starts with the New York State Thruway Authority bridge as a point of departure from which parametric studies aimed at investigating the influence of flange and web compactness on bridge girder rotation capacity are conducted. Nonlinear finite element modeling is the vehicle used in the present work. The modeling techniques employed are experimentally validated through a verification study carried out as part of the current work (using experimental data from the laboratory testing of a series of bridge girders at the University of Pittsburgh [10]). A detailed description of the New York State Thruway Authority HPS bridge is given in Part 2 of the current paper while Part 3 discusses the nonlinear finite element modeling techniques used in the parametric study reported as part of this paper. The finite element modeling techniques introduced in Part 3 have shown good agreement with the experimental tests carried out by Morcos [10]; a comparison study focusing on the present finite element modeling techniques and the experimental results of Morcos is outlined and discussed in Part 4. The experimentally verified nonlinear finite element techniques are then employed in a parametric study of the influence that flange and web compactness have on bridge girder rotation capacity. The results of this parametric study are presented and discussed in Part 5. Conclusions related to the present study are contained in Part 6.
2. Description of New York State Thruway Authority bridge The New York State Thruway Authority HPS bridge studied consists of a system of five, equally spaced, two-span continuous girders tied together with transverse diaphragm members and a reinforced concrete deck slab of 240 mm thickness. While the top girder flanges are continuously tied to each other by the deck system, the girders are also connected through diaphragm members occurring at the supports and along the length of girders at variable spacing. Composite action is achieved between the deck and girders by way of shear studs (22 mm in diameter). Bearing and transverse stiffeners are present along the bridge longitudinal axis. The bearing
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Fig. 2.
Elevation view of New York State Thruway Authority HPS bridge girder.
stiffeners are provided at the end supports and at interior support, whereas intermediate stiffeners are provided in each span at equal spacings of 7.625 m, except in the portion of the span region where there is some adjustment made due to the bolted flange splice. An elevation and section view of the girder are given in Figs. 2 and 3 respectively.
Fig. 3.
Section view of New York State Thruway Authority HPS bridge girder.
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The behavior investigated in this work focuses on bare steel compactness and hence is most relevant to the design of the hogging moment regions of continuous span girders due to the composite action occurring in the positive moment region. To represent a portion of continuous bridge girders between adjacent inflection points at an interior support, a simply supported beam with a overhang of L/2 is considered as shown in Fig. 4. The mid span concentrated load represents the pier reaction and the simple supports represent the point of contraflexure in the prototype continuous span member. The flanges of the HPS beam are 406 mm wide and 45 mm thick in the negative moment region with a slenderness ratio (bf/2tf) of 4.5 and 38 mm thick in the positive moment region. The web is 16 mm thick and 736 mm deep, resulting in a slenderness ratio (2Dcp/tw) of 90 at the plastic cross-sectional capacity (including the effect of negative moment reinforcing bars in the deck). All stiffeners are provided for the entire height of the web and are continuously welded to the web on both sides of the stiffener. Bearing stiffeners at the supports and at mid-span are provided on either side of the web as 25 mm thick and 200 mm wide steel plates. Similarly, intermediate stiffeners are also provided on both sides of the web, but with a 12 mm thickness and 125 mm width as shown in Fig. 3. Limiting slenderness ratios for the flange (bf/2tf) and web (2Dcp/tw) compactness are 7.75 and 76.5 respectively according to AASHTO provisions [3]. While the girder cross-section is essentially compact (the web is not quite compact), and thus able to develop its full plastic capacity without the threat of local buckling interfering with the required plastic hinge rotation needed to develop a collapse mechanism in the girder as a whole, the same is not true for the bracing. Since the girder was not proportioned using plastic analysis and design methodologies, the New York State Thruway Authority designed the girder with a bracing
Fig. 4.
Schematic of model geometry and boundary conditions used in parametric study.
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spacing that is considered inadequate for plastic moment redistribution to take place (Lb=7.625 m). According to AASHTO Section 6.10.4.1.7, Lb must be 4.68 m or less in order to ensure that adequate structural ductility will be displayed at the full cross-sectional girder capacity without lateral–torsional buckling interfering with the formation of a collapse mechanism. Hence the finite element models used in the compactness parametric study incorporate two additional braced stiffeners present in the girder regions on either side of the pier adjacent to the first stiffener location (i.e. within the first 7.625 m on either side of the mid-span bearing stiffener shown in Fig. 4). While in the first part of the parametric study the stiffeners and bracing are placed at a distance of 3.8125 m from mid-span, this was subsequently changed as discussed in Part 5 of the current paper due to the strong coupling that exists between local and global buckling modes in HPS I-shaped girders at the ultimate load. 3. Finite element modeling techniques The commercial multipurpose finite element software package ABAQUS [11] is employed in this research. ABAQUS has the ability to consider both geometric and material nonlinearity in a given model. Advantage has been taken of this capability and thus all modeling herein considers both nonlinear geometric and material influences. Incremental solution strategies are required to trace the proper nonlinear equilibrium path in analyses such as these; hence the modified Riks–Wempner solution strategy [11–14] is chosen for this work. In modeling studies where inelastic buckling is studied it is important that the evolution of the modeling solution be carefully monitored so that any indication of bifurcation in the equilibrium path is carefully assessed so as to guarantee that the equilibrium branch being followed corresponds to the lowest energy state of the system. While different strategies exist for guaranteeing that the lowest energy path is taken [15], the strategy of seeding the finite element mesh with a initial displacement field is employed in this study [11]. In this technique, the finite element mesh is subjected to a linearized-eigenvalue buckling analysis from which an approximation to the first buckling mode of the girder is obtained. The displacement field associated with this lowest mode is then superimposed on the finite element model as a seed imperfection for use in the incremental nonlinear analysis. This seed imperfection displacement field is scaled so that the maximum initial displacement anywhere in the mesh is equal to one-one-thousandth of the entire span length of the girder (L/1000). While it is recognized that the technique of seeding a finite element mesh with an initial imperfection has shortcomings [15], this technique is nonetheless employed in the current study due to the fact that results obtained from this method have agreed quite well with experimental tests as discussed in Part 4 herein. 3.1. Finite element mesh The models of the bridge girders considered in this study are constructed from a dense mesh of four node nonlinear shell finite elements. The planes of the mesh
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surfaces correspond with the middle surfaces of the constituent cross-sectional plate components of the given I-section (see Fig. 5). A moment gradient loading is used in this study since it represents the loading condition experienced by the girder at pier locations. It is the pier location where the most plastic hinge rotation capacity is needed in order to accommodate formation of a collapse mechanism in a twospan continuous highway bridge. The moment gradient loading is achieved in the finite element modeling by imposing a concentrated load at the mid-span of a simply supported I-shaped girder assembly. The concentrated force simulates the pier reaction of the subject bridge and the simple supports are placed at the approximate points of inflection on either side of the pier. In the subject bridge, diaphragm members are present at the pier and at the point of inflection and so the finite element models are braced against out-of-plane deflection at the load point and supports. In the finite element models an additional length of girder is present beyond the support locations to help simulate the torsional-warping restraint provided by the adjacent beam segments in the actual bridge. The length of the additional beam segments is chosen to be 7.625 m which corresponds to the distance to the next diaphragm member occurring after the point of inflection, as measured along the longitudinal axis of the girder. Since this additional beam segment represents an additional unbraced length, the ends of the segments are braced against out-of-plane translation in an idealized way so as to simulate the presence of the associated diaphragm members in the real bridge. Fig. 4 provides a schematic of the loading configuration just described. It is noted that in Fig. 4, the top flange is in compression and hence has dimensions of: 406 mm width and 45 mm thickness. The bottom flange is in tension and hence has a modified thickness of 84 mm in conjunction with the original flange width of 406 mm. The exaggerated tension flange thickness is used to account for the presence of negative moment steel in the bridge deck. The S4R nonlinear, finite strain, shell element from the ABAQUS element library
Fig. 5.
Representative bucked finite element mesh.
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is employed in this research. The S4R shell element is shear deformable and subsequently both reduced integration and the assumed strain method [16] are employed to improve the overall thin-shell behavior of this element. A single integration point is used in this particular element hence stabilization of spurious zero energy modes is provided for by ABAQUS. This element was selected for use in the parametric study based on its superior performance in the verification study described in Part 4 of the current paper. A uniaxial representation of the A709 Gr. HPS483W steel constitutive law employed in this study appears in Fig. 6 as a plot of true stress versus logarithmic strain (the steel grade HSLA80 mentioned in Section 1 is also shown). ABAQUS uses the von Mises yield criterion to extrapolate a yield surface in three-dimensional principal stress space from the uniaxial material response given in Fig. 6. The corresponding ABAQUS metal plasticity model is characterized as an associated flow plasticity model with isotropic hardening being used as the default hardening rule.
4. Validation of modeling techniques Experimental verification of the modeling techniques used in the current study are carried out as a preliminary step in the present research. An experimental test program conducted by Morcos [10] at the University of Pittsburgh serves as the vehicle for the model verification phase of the current study.
Fig. 6.
Steel constitutive laws—true stress vs logarithmic strain.
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4.1. Experimental tests Morcos tested three full-scale bridge girders in the Structural Engineering Laboratory at the University of Pittsburgh. The proportions of the three girders varied in such a way that insight into the influence of web compactness on rotation capacity could be studied within the context of grade 345 steel. Morcos’ specimen “S” [10] is selected for the present validation study since its cross-sectional proportions permit the girder to be considered compact with respect to current AASHTO [3] design provisions. This overall cross-sectional compactness is seen as a favorable feature for a validation study since the present work has at its goal the identification of appropriate compactness criteria for applications involving new high performance steel grades. Specimen “S” was fabricated from ASTM A572 Gr. 345 plate steel all originating from the same heat. The gas metal arc process was used to join three plates together forming an I-shaped cross-section. Actual girder cross-sectional dimensions are provided in Fig. 7. AWS A5.18 ER70S-X filler material was used for the welding. The girder was 3.96 m in overall length with bearing stiffeners present at the supports and mid-point in addition to intermediate stiffeners spaced at varying intervals along the girder longitudinal axis (see Fig. 7). The overall span length was set at 3.66 m with a concentrated load being imposed at mid-span thus simulating the momentgradient loading condition present in girder section adjacent to the pier of a twospan continuous bridge. Out-of-plane bracing was provided at the load point and supports in the form of threaded bars with eye-bar ends and turnbuckles. The eye-
Fig. 7.
Geometry of Morcos’experimental specimen “S”.
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bars allowed for the bracing to be attached at discrete points on the top and bottom of each bearing stiffener on either side of the web. The turnbuckles at the center of the bracing provided some degree of bracing flexibility not easily quantified; more will be said about this flexibility in the sequel. A single 900 kN actuator provided the concentrated force needed to form a plastic hinge in the specimen. 4.2. Finite element models of experiments The finite element techniques employed in the model validation study, and described in this section, incorporate variations on the same finite element modeling techniques as discussed in Section 3 for use in the parametric study described in Part 5 of the present paper. The validation study served as a test-bed from which to evaluate differing modeling strategies in order that the most realistic model behavior is obtained thus yielding a representative response of a bridge girder in the field. In every case within the present validation study nonlinear shell finite elements are specified at the middle surfaces of the constituent plate elements of the I-girder cross-sections. Both S9R5 and S4R shell elements are considered in the validation study. Also, varying degrees of initial imperfection and bracing stiffness are considered. The influences of these individual variations are now discussed. 4.2.1. Comparison of shell finite elements Nonlinear shell elements are chosen for this study so as to be able to explicitly model local buckling deformations and the spread of plasticity effects. ABAQUS has a wide variety of shell elements in its element library; theses include shells that are suitable for “thick” or “thin” shell applications utilizing reduced integration. The ABAQUS S4R and S9R5 shell elements are considered in this verification study and the results of these element types are compared in Fig. 8. The results from the model incorporating the S4R element show a better agreement with Morcos’experimental results as compared to the model incorporating the S9R5 element type. Hence, all
Fig. 8.
Influence of shell element formulation on predicted flexural response.
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subsequent girder models considered herein are constructed from a mesh of S4R shell finite elements. 4.2.2. Effects of initial imperfections The primary role of geometric imperfections is to provide a perturbation to the perfect model geometry so as to initiate a response that is asymptotic to the ideal equilibrium path of the structure. This asymptotic response will be close to the ideal response so long as the imperfections are small enough so as not to affect gross cross sectional properties. As can be seen from Fig. 9, the results of a model with perfect geometry produces a rather gradual unloading response that does not agree well with the behavior exhibited in the experimental tests of Morcos. Improvements in the modeling response related to the effects of geometric imperfections are investigated by conducting several analyses with varying levels of geometric imperfections. In the study, small (i.e. L/2000) to very large (i.e. L/200) imperfections are added to the model and the results studied. As expected, the magnitude of the seed imperfection has a marked influence on the rotation capacity of the girder studied. At a relatively large imperfection (i.e. L/200), the rotation capacity is greatly reduced (Fig. 9). It is noted that differences in response associated with differing degrees of small imperfections (i.e. L/500 and L/1000) have very little impact on observed rotation capacity, and based on the results presented in Fig. 9, it appears that an imperfection of L/1000 provides for reasonably accurate results. The magnitude of such an imperfection is also a reasonable estimate for the initial out-of-straightness (or sweep) that might be encountered in a fabricated girder. Based on the results obtained in this phase of the verification study, an imperfection of L/1000 is used in the parametric studies carried out in the sequel related to A709 HPS483W girders. 4.2.3. Influence of bracing stiffness on flexural response Considering the fact that the bracing forces can be large at points in and around the plastic hinge regions of a beam, the effect of lateral bracing flexibility on the
Fig. 9.
Influence of seed imperfections on predicted flexural response.
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flexural behavior of the girder is investigated in the current verification study. The SPRING1 element from the ABAQUS element library is used to simulate flexible bracing behavior. SPRING1 elements are attached to the I-girder at the specified bracing locations on the tension and compression flange tips on either side of the girder web (see Fig. 10 inset). When defining the spring stiffness for the SPRING1 elements, a displacement baseline of L/1000 is used. The spring stiffness, for each of the four springs at a given cross-section, is then arrived at through consideration of variations in bracing force only. The spring forces are specified as percentages of the force necessary to yield the compression flange of the braced cross-section (i.e. 0.5, 1 and 2%). When considering validation results obtained using ideal lateral restraint, as well as the three other spring stiffnesses studied, it appears that the finite element models incorporating a SPRING1 element whose stiffness is based on 1% of the force necessary to yield the compression flange at a displacement of L/1000 works well (Fig. 10). It is noted that the models considered in this portion of the verification study possess an initial geometric imperfection of the type described in Section 4.2.2 above with a maximum amplitude of L/1000. This initial imperfection maximum value of L/1000 is unrelated to the L/1000 baseline deflection used in the spring stiffness calculations. While the bracing stiffness mentioned has worked well in the verification study described, there is evidence that optimal bracing stiffness may, to some degree, be problem dependent [4]. As a result of this situation, rigid braces are used in the parametric study to follow in order that additional complexities may be reduced. 4.3. Discussion of validation results Owing to the fact that the current validation study is used to help identify the most efficient and accurate modeling strategies to be employed in the parametric studies associated with high performance steel bridge girders considered later (Part
Fig. 10.
Influence of bracing force on predicted flexural response.
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5), the favorable agreement between the results of the experimental tests and the finite element analogs (as demonstrated by the comparison of load-deflection response displayed in Fig. 10) is encouraging. It is further noted that while the agreement in load-deflection response between the finite element models and the experiments is good, so too is the agreement between the modal manifestations exhibited by the finite element models and those observed by Morcos to be present in the experimental test beam (see Fig. 11). The fact that the finite element models display a more distinct leveling-off of the load-deflection behavior displayed in Fig. 10, as compared with the experimental tests, is related to the presence of residual stresses in the cross-section. While the actual girder cross-sections undoubtedly possess residual stresses, the finite element models did not incorporate any since residual stresses do not have a significant influence on plastic hinge rotation capacity: the focus of the present study.
5. Results A finite element parametric study is carried out using the verified modeling techniques previously discussed. At issue is whether the current AASHTO compactness criteria are suitable for use in applications involving the new A709 HPS483W steel. While these provisions have worked well in the past for structural applications involving conventional grades of steel, there has been some recent research that would indicate that this is not the case for several of the new high performance steels [4–6]. 5.1. Influence of cross-sectional compactness on structural ductility It has been tacitly assumed in current bridge specifications that localized buckling influences may be considered independently of global buckling influences in girders that possess a compact cross-section (as related to the plate slenderness of the web and flange) in conjunction with adequate bracing. Satisfaction of both the com-
Fig. 11.
Comparison of experimental and finite element buckling modes.
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pactness and bracing requirements is intended to result in a girder design that can not only attain the full plastic cross-sectional capacity, but also exhibited sufficient plastic hinge rotation capacity for moment redistribution to occur; thus ultimately allowing for the formation of a collapse mechanism. The validity of the current American bridge specification (AASHTO 1998) is tested in this regard for its applicability to A709 HPS483W steel. 5.1.1. Beam braced according to AASHTO LRFD requirements As indicated earlier in this paper, a typical HPS girder from a bridge in the New York State Thruway inventory is selected as a point of departure for the parametric study now discussed. While the cross-section of this girder comes close to satisfying AASHTO compactness criteria, the bracing spacing, as originally specified, is considered by AASHTO to be inadequate for plastic moment redistribution to take place. Hence in this first portion of the parametric study, two additional braces (one on either side of the pier) are added to the original girder configuration in the region between the mid-span bearing stiffener and the first original intermediate stiffener location (i.e. Lb was originally 7.625 m near the pier, but for the purposes of the parametric study it is reduced to 3.8125 m). The AASHTO compactness limits for this application are bf/2tf=7.75 and 2Dcp/tw=76.5 for the flange and web respectively. The New York State Thruway girder has a bf/2tf=4.5 and 2Dcp/tw=90 and hence comes close to satisfying the compactness requirements. However, as can be seen in Table 1 and Figs. 12 and 13, despite the satisfaction of the AASHTO flange compactness and bracing provisions (and near satisfaction of the AASHTO web compactness requirements), the girders were not able to attain the desired rotation capacity of three as needed for moment redistribution. While it is alarming that a girder which is considered by AASHTO to be compact and adequately braced displays less than half of the rotation capacity needed for mechanism formation, what is more of a concern is the apparent lack of correlation between cross-sectional rotation capacity and flange and web slenderness. Table 1 Results from the parametric study involving A709 HPS 483W bridge girders braced according to AASHTO Sec. 6.10.4.1.7 bf/2tf
2Dcp/tw
Lb/ry
R
3 3.5 4 4.5 5 4.5
69.9 85.6 89.2 89.6 89.8 42.6 55.5 68.1 77.9 89.6
37.4
1.14 1.38 1.45 1.32 1.18 2.23 2.03 1.79 2.26 1.32
37.4
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Fig. 12.
Fig. 13.
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Influence of flange slenderness on rotation capacity using AASHTO bracing configuration.
Influence of web slenderness on rotation capacity using AASHTO bracing configuration.
It has long been thought that as either flange or web slenderness (or both) decrease, cross-sectional ductility and rotation capacity should increase. What is apparent from Table 1 and Figs. 12 and 13 is that a decrease in these plate slenderness values can lead to either an increase or a decrease in rotation capacity. This counter-intuitive situation seems to arise out of a highly interactive buckling mode that evolves in the HPS girders in which local and global influences are very much coupled. Based on an examination of the results from graphical post processing of the finite element models, it becomes clear that substantial interaction between local and global buckling is occurring despite the satisfaction of the AASHTO bracing provisions. Thus the AASHTO bracing provisions are at issue in HPS applications as well the com-
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pactness criteria. It will be seen in the next section that once adequate bracing has been achieved, the notion of cross-sectional compactness will once again become valid. 5.1.2. Beam with alternate bracing geometry In earlier work by Earls [4] building size girders, subjected to a moment gradient loading much like that of the present research, and made from the high performance steel grade HSLA80 were studied. Two generally different inelastic mode shapes were identified in this earlier study: one yielding a much more ductile response than the other. After careful examination of the geometric characteristics of the less ductile mode, a bracing scheme was devised whereby the braces were located at points along the beam longitudinal axis where excessive out-of-plane cross-sectional deflections were noted. The new bracing scheme effectively precluded the formation of the less favorable mode and hence drastically improved the observed structural ductility (see Fig. 14). The new bracing configuration consisted of simply restricting the compression flange-web junction from moving out-of-plane at two discrete points at a distance of d/2 on either side of the point where the concentrated load was applied (“d” is the total depth of the cross-section). A very similar bracing strategy has been adopted in the present study with two notable exceptions. Firstly, in the present study the out-of-plane bracing at the bearing stiffener location over the pier is retained in contrast to the earlier work by Earls [4] in which the bracing at the load point was removed. Secondly, in the earlier study by Earls [4] the d/2 bracing members were attached to the compression flangeweb junction with a pin (thus providing no direct torsional restraint to the crosssection) while in the current work the d/2 bracing occurs at four cross-sectional locations at each bracing location along the longitudinal axis (i.e. the bracing is
Fig. 14. Schematic of model incorporating proposed bracing configuration.
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Table 2 Results from the parametric study involving A709 HPS 483W bridge girders braced according to the proposed method described in Part 5 of this paper bf/2tf
2Dcp/tw
4 4.5 4.5 (AASHTO) 5 4.5
89.2 89.6 89.6 89.8 77.9 89.6 89.6 (AASHTO) 97.4
R
Lb/ry
Lb1/ry
Lb2/ry
3.3 3.18 1.32 2.72 3.3 3.18 1.32 2.73
N.A. N.A. 37.4 N.A. N.A. N.A. 37.4 N.A.
3.6 3.6 N.A. 3.6 3.6 3.6 N.A. 3.6
71.2 71.2 N.A. 71.2 71.2 71.2 N.A. 71.2
imposed at the four flange tips of the cross-section at each bracing location) and hence torsional restraint of the cross-section is directly provided. It is further noted that the total number of braces and stiffeners have not been changed as a result of the new bracing scheme. The intermediate stiffeners and diaphragm members, formerly placed at a distance of 3.8125 m on either side of the bearing stiffener at the pier location, have simply been shifted to a distance of 0.375 m on either side of the pier location. As can be observed from Table 2 and Figs. 15 and 16, the effect of this simple shift in bracing location can be quite pronounced with the observed rotation capacity being more than doubled. Furthermore, now that the interactive buckling mode has been mitigated to some extent, there once again is an observed correlation between flange and web slenderness and the observed cross-sectional rotation capacity.
Fig. 15.
Influence of flange slenderness on rotation capacity using the proposed bracing configuration.
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Fig. 16.
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Influence of web slenderness on rotation capacity using the proposed bracing configuration.
6. Discussion of results Table 2 clearly shows that in conjunction with the new bracing configuration proposed in this paper, flange and web plate slenderness parameters once again correlate with cross-sectional rotation capacity. From Table 2 it is noted that flange slenderness (bf/2tf) of 4.5 results in a rotation capacity of 3.18 while a flange slenderness of 5 results in a rotation capacity of 2.72. Based on the compactness requirement that a cross-sectional rotation capacity of three be achieved, the flange slenderness value of 4.5 is proposed as the compactness limit for bridge applications involving A709 HPS483W. It is noted that this flange slenderness of 4.5 is much less than the value of 7.75 which is prescribed by the current AASHTO flange compactness criteria. Also in Table 2 it is observed that that in conjunction with the new bracing configuration proposed in this paper, a web slenderness value (2Dcp/tw) of 90 seems to be an appropriate compactness limit for the girder web in bridge applications involving A709 HPS483W. It is noted that this web slenderness of 90 is slightly larger than the value of 76.5 which is prescribed by the current AASHTO web compactness criteria.
7. Conclusion It appears that the current American bridge specification provisions related to flange compactness are un-conservative and not presently applicable to bridges made from A709 HPS483W steel. This situation is partially a result of the interactive buckling mode that develops in the negative moment region at the pier of HPS bridge girders. This interactive buckling mode involves a very close coupling of local and global buckling manifestations thus effectively rendering as useless currently held notions related to the role of flange and web compactness on flexural ductility. Based
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on the present research, it appears that the AASHTO bracing provisions contained in section 6.10.4.1.7 [3] are un-conservative for applications involving A709 HPS483W steel. This paper prescribes a new bracing configuration for use in the negative moment region of high performance steel bridges. The new bracing scheme does not require any additional stiffeners or diaphragm members. Rather a repositioning of the two stiffeners and diaphragm members on either side of the bridge pier, so as to be closer together, is all that is required (i.e. moving the existing stiffeners and diaphragm members to a new location d/2 away from the pier on either side). The result of the repositioning of the braces is to more than double the observed cross-sectional rotation capacity. Furthermore, now that the interactive buckling mode has been mitigated to some extent, there once again is an observed correlation between flange and web slenderness and the observed cross-sectional rotation capacity. Thus in addition to providing new bracing requirements for bridge applications requiring moment redistribution and involving A709 HPS483W steel, the current research also provides for a corrected flange compactness limit of bf/2tf=4.5 and a corrected web compactness limit of 2Dcp/tw=90 to be used in bridge applications involving A709 HPS483W. The latter compactness limit is a slightly more liberal value than that currently recommended by AASHTO. Acknowledgements The Authors wish to express their sincere thanks to the Federal Highway Administration for the funding needed to pursue this research under the Technology Deployment: Innovative Bridge Construction Program. The Authors also wish to thank the New York State Thruway Authority for its interest and cooperation in the current research. References [1] American Society of Civil Engineers. In: Plastic design in steel, a guide and commentary. New York: American Society of Civil Engineers; 1971, p. 80. [2] AISC. Load and resistance factor design specification for structural steel buildings, LRFD vol. I, 3rd ed. Chicago (IL): American Institute of Steel Construction Inc, 1999. [3] AASHTO. LRFD Bridge Specification. Washington DC: American Association of State Highway and Transportation Officials, Inc, 1998. [4] Earls CJ. On the inelastic failure of high strength steel I-shaped beams. Journal of Constructional Steel Research 1999;49(1):1–24. [5] Earls CJ. On geometric factors influencing the structural ductility of compact I-shaped beams. Journal of Structural Engineering 2000;126(7):780–9. [6] Earls CJ. The influence of material effects on the structural ductility of Compact I-shaped beams. Journal of Structural Engineering 2000;126(11):1268–78. [7] Taly N. In: Design of modern highway bridges. New York: McGraw Hill; 1998. p. 75-9. [8] Haaijer G. Objectives and early research of autostress design. Transportation Research Record No. 1380, Transportation Research Board—National Research Council, National Academy Press, Washington DC, 1993.
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[9] AASHTO. Guide specification for alternate load and resistance factor design procedures for steel beam bridges using braced compact sections. American Association of State Highway and Transportation Officials, Inc, Washington DC, 1991. [10] Morcos SS. Moment–rotation tests of steel bridge girders for autostress design. M.S. Thesis, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA, 1988. [11] ABAQUS. ABAQUS Theory Manual. Pawtucket, RI, USA: Hibbitt, Karlsson & Sorensen, Inc, 1999. [12] Riks E. The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics 1972;39:1060–6. [13] Riks E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures 1979;15:529–51. [14] Crisfield MA. A fast incremental/iterative solution procedure that handles ‘snap-through’. Computers & Structures 1981;13:55–62. [15] Teh LH, Clarke MJ. Tracing secondary equilibrium paths of elastic framed structures. Journal of Engineering Mechanics 1999;125(12):1358–64. [16] MacNeal RH. A simple quadrilateral shell element. Computers and Structures 1978;8:175–83.