doi:10.1016/j.engstruct.2005.05.019www.elsevier.com/locate/engstruct
es
formed by nts. Finally,
Wonseok Chunga, Elisa D. Sotelinob,∗
aTrack and Civil Engineering Research Department, Korea Railroad Research Institute, Republic of Korea bDepartment of Civil and Environmental Engineering, 214 Patton Hall, Virginia Tech, Blacksburg, VA 24061, USA
Received 19 February 2004; received in revised form 4 January 2005; accepted 10 May 2005 Available online 26 August 2005
Abstract
This paper investigates finite element (FE) modeling techniques of composite steel girder bridges focusing on the overall flexural of the system. In particular, four three-dimensional FE bridge models are examined. Various modeling techniques, which are e to overcome displacement incompatibility and geometric modeling errors, are studied and issues related to the selection of ele discussed. A technique that uses the concept of work equivalent nodal loads to accurately represent the applied tire pressure is also The accuracy of each model is verified against the results acquired from full-scale laboratory test experiments and a field test per other researchers. Furthermore, the results are also compared with those of a detailed finite element model that uses solid eleme the efficiency of each model, based on a comparison of computer resources usage, is also presented. © 2005 Elsevier Ltd. All rights reserved.
Keywords:Finite element method; Three-dimensional analysis; Composite bridge; Compatibility
s re na e
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1. Introduction
With the development of high-powered computer together with state-of-the-art finite element (FE) softwa and user-friendly graphical interfaces, three-dimensio (3-D) FE analysis has become a popular choice ev for straightforward bridge analysis. More specificall the design bending moment for steel girders can determined more accurately using FE analysis of a brid superstructure rather than using the lateral load distribut factor specified in the AASHTO (American Association o State Highway and Transportation Officials) specification However, accurate finite element models must be used reliable bridge analysis.
This study is implemented to evaluate the curren adopted 3-D FE models used in the analysis of a popu type of bridge, composite steel girder bridges. It is know that some FE bridge models available in the literatu introduce geometric errors and compatibility errors whi can result in incorrect predictions of flexural behavio
∗ Corresponding author. Tel.: +1 540 231 3174; fax: +1 540 231 7532. E-mail address:sotelino@vt.edu (E.D. Sotelino).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2005.05.019
ab. ur- the
In this study, techniques to minimize modeling errors a introduced and implemented. These techniques include use of displacement transformations and the proper selec of finite elements.
The simplest 3-D FE model utilizes shell elements f the deck slab with eccentrically stiffened beam elements the girders [1–4]. The eccentricity of the girders is take into account by using rigid links between the centroid of t concrete slab and the centroid of the steel girders.
Brockenbrough [5] and Tabsh and Tabatabai [6] modeled deck slabs using four-node shell elements that includ membrane and bending effects. Each steel girder w divided into flange and web parts. Each flange of the gird was idealized by beam elements, and the web was mod by the four-node shell elements. Bishara et al. [7] adopted the same modeling technique to represent the girder, they used three-node thin plate triangular elements to mo the slab. The eccentricity between the concrete deck and steel girder flange was modeled by a rigid link.
Mabsout et al. [8] used three-dimensional solid element which have linear shape functions, to model the deck sl The steel girder flanges and web were modeled by fo node shell elements. Imposing no releases between
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shell elements and beam elements simulated the comp behavior between the concrete deck and steel girder. Ea and Nowak [9] used solid elements to represent the de while beam elements were used to represent the br girders. The beam elements were attached directly to bottom of the solid deck. In these works, no details w provided concerning the lack of compatibility at the interfa between the slab solid element and the elements use model the girders.
In the published literature, various FE modelin techniques have been used to idealize bridge superstruct However, there is a lack of information about th compatibility at interfaces when different element typ are adopted. The geometric errors also observed in s FE models which overlap two shell elements by shar the same node have not yet been fully explored. T objective of this study is, thus, to identify the source the incompatibilities and geometric errors, and to prov guidelines to help analysts avoid these types of mode errors in the FE analysis of bridge superstructures.
2. Finite element models
A bridge is a hybrid structure that combines seve structural components. Since the proper selection of fi elements is key in a FE model, different types of fin elements have been used in an attempt to model br superstructures. First, the modeling of the bridge d and girders is discussed separately. This is followed a discussion of the techniques employed to model composite behavior of the structure.
The numerical simulations were performed usi the general-purpose finite element software pack ABAQUS [10]. This software provides an extensive libra of elements that can model virtually any geometry and s eral multi-point-constraint (MPC) options that can be us to avoid displacement incompatibility between elements.
2.1. Modeling of bridge deck
In FE modeling, the bridge deck is typically modele either by solid or shell elements. Shell elements are wid used to idealize the bridge deck since behavior of t structural component is governed by flexure and in t case a mesh of shell elements is computationally m efficient when compared to one of solid elements. In bridge application, the finite element that accounts for transverse shear flexibility, the Mindlin type shell element preferred for an accurate analysis, even though the transv shear deformations are not usually significant. It should noted that Mindlin type elements are susceptible to lock with full integration when the thickness of the bridge de becomes thinner. This phenomenon is due to the fact tha shear strain energy term tends to dominate the total pote energy in these cases. This leads to the deterioration o element bending stiffness, thus producing over-stiff resu
ite on , e e
Fig. 1. Maximum deflection of ABAQUS shell elements.
Several ABAQUS-provided shell elements were tested evaluate their applicability for bridge deck modeling. Th tested plate was a 2.54 m (100 in.) by 1.27 m (50 in rectangular plate under a distributed load. All supports we clamped. The modulus of elasticity was 20.7 MPa (3000 p and Poisson’s ratio was 0.3. The exact solution for a th plate is given by [11]
wmax = 0.00254qL4
D (1)
whereD = Eh3
12(1−ν2) . The central deflections of ABAQUS
shell elements were compared to the exact solution of t thin plate, as shown inFig. 1. It should be noted that the mesh density of each model presented inFig. 1 is identical. It was found that a quadrilateral nine-node (or eight-nod shell element with reduced integration (S9R5/S8R5) and quadrilateral eight-node thick shell element with reduce integration (S8R) predicted the same response up to a ra of span length to depth(L/h) of approximately 150 which is a reasonable upper bound for bridge analyses. Thus transverse shear deformation may be neglected in typi bridge analyses. In this study, the shear flexible shell elem (S8R) was selected to model the concrete bridge deck.
Some models proposed in the literature utilize sol elements to model a concrete bridge deck. The ma drawback of solid deck models is the computational cost predict the correct flexural behavior of the bridge. Multipl layers are required through the thickness direction in order model the deck with linear solid elements (e.g., eight-no brick elements), since the strain variation of these eleme is constant through the thickness. An alternative option the use of higher order solid elements, but this may entail even higher computational cost.
2.2. Modeling of bridge girders
In this study, four different modeling techniques fo steel girders, named G1, G2, G3, and G4, respective are investigated. The element selection for each model technique is shown inTable 1.
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71 65
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Model name Girder part Web Flanges
G1 Shell element Shell element G2 Shell element Beam element G3 Beam element Shell element G4 Beam element
Fig. 2. Bridge model 1 and girder model G1.
The G1 model is a detailed model of a steel girde The flanges and the web are modeled by shell eleme as shown inFig. 2. It should be noted that only the girde part of the bridge model is considered in this section. Sh elements must be placed along the mid-surface of struct components. Numerical tests revealed that overlapp flange elements with web elements by sharing the same n results in significant modeling errors because of an incorr moment of inertia about the primary bending axis. ABAQU has the capability to input an offset distance for sh elements from their node locations. The moment of iner can be matched with the actual I-beam moment of iner by adjusting the offsety, shown inFig. 2. Alternatively to the offset option, rigid links or constraints can be used create an offset. However, this creates additional model complexity and is therefore not used in this study.
The next model, G2, is similar to the G1 model exce that the flange is modeled by beam elements instead of s elements, as shown inFig. 3. As a result, this model requires less computing resources to represent the three-dimensi nature of the girder structures. The beam elements have same properties as a girder flange with the centroid of
ts
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Fig. 3. Bridge model 2 and girder model G2.
flange offset from the node location by one-half of the flan thickness.
The G3 model is proposed here to investigate the poss incompatibility at the element connection between web a flanges found in the previous two models. A typical flat sh element is formulated by superimposing plate bending a membrane action. The resulting shell element has, thus, degrees of freedom (DOFs); three translational DOFs two in-plane rotational DOFs at each node. A sixth DO known as the drilling DOF, is often added to these eleme to avoid singularity. This DOF is associated with the sh normal rotation. However, if two neighboring elements a not coplanar, compatibility between the in-plane rotation a drilling rotation is generally violated [12]. The G1 and G2 models share the drilling rotation of the shell element in t web with the in-plane bending rotation of the shell or bea element in the flanges. Thus, displacement is not compat along the element boundary of these models.
The G3 model places shell elements at the centroid girder flanges. Beam elements are placed at the cent of the girder web. Rigid links, through the constraints DOFs, are applied to ensure composite action. This mod illustrated inFig. 4. The shape functions of beam and sh elements should be identical to avoid incompatibility alo the element boundary.
The G4 model is the simplest model and utilizes be elements with the geometric properties of girder sectio This model is shown inFig. 5. It should be noted that the G4 model is not able to represent different material propertie web and flanges.
The performance of each girder model is evaluated working through a numerical example. A simply support
66 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
o ti ar
ty.
ust ions wo on igid he ips
o a
Fig. 4. Bridge model 3 and girder model G3.
Fig. 5. Bridge model 4 and girder model G4.
beam having an I-shape cross section is subjected t concentrated load at the center span. Because an analy solution of the maximum deflection considering she flexibility is known from the theory of elasticity [13], a direct comparison with the FE result is possible. The giv single girder structure is modeled by models G1, G2, and using either linear elements or quadratic elements. The fi element selected for the G4 model was either an Euler be element or a shear flexible Timoshenko beam element.
The convergence trends of the FE solutions are sho in Fig. 6. The G1 and G2 models required significant me refinement to converge to the analytical solution, while t G3 and G4 models produced less than 1% error compare
a cal
Fig. 6. Convergence of finite element girder models.
the analytical solution, regardless of the mesh refineme In the G1 and G2 models, the prescribed incompatibil between the drilling rotation and the transverse rotat tends to diminish as the mesh is refined. It is also obser that the G1 and G2 models using quadratic eleme converged more quickly than those using linear elements
It is concluded from the previous observation that t G3 and G4 models are simple yet produce accurate res regardless of the mesh density. Models G1 and G2, howe are able to model the local behavior of a bridge girder b require a refined mesh due to displacement incompatibili
2.3. Modeling of composite action

1 0 0 0 e 0


(2)
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71 67
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d de he th ng c m e e o
of ap t i
h
e uch of d ed
Fig. 7. DOFs for solid and shell/beam elements.
wheree is the eccentricity between the solid element no and the shell/beam element node.
In the case where shell elements are used to mo the bridge deck, the nodes of the girder do not coinci with the nodes of the shell elements in the deck. T shell elements in the bridge deck are connected with prescribed girder models through an MPC. Typical bendi elements, such as the Kirchhoff shell element (for the de and for girder models G1 and G3) and the Bernoulli bea element (for the girder model G2 and G4), should be avoid for the modeling of the composite girder bridge sinc displacement incompatibility occurs at the interface of tw bending elements [14]. These bending elements make use a linear shape for the axial displacement and a cubic sh for the transverse displacement. The axial displacemen the girder is given as
ug 1 = ud
1 − e · θd 2 . (3)
This incompatibility is noticeable since the axial displac ments of the deck and the girder (ud
1 and ug 1) are linear,
but the rotation of the deck(θd 2 ) is quadratic in the axial
direction. Even though this incompatibility error completely
disappears as the mesh is refined, many methods have b proposed to eliminate this nonconforming error [15–17]. In the ABAQUS implementation, the use of S8R elements f the concrete deck and B32 elements for the girder giv full compatibility between the boundary of two differen elements.
2.4. Discretization of applied loading
AASHTO specifications [18] specify the use of tire contact area for a more exact analysis of bridge structur The applied loading on a bridge deck consists of press loads applied through a tire patch. In the finite eleme modeling, this requirement imposes the need for a fine m in the deck, so that the element is fitted with the pat size. As a part of this research, the equivalent nodal lo algorithm is employed in order to uncouple the patch lo
el
e
k
d
Fig. 8. Discretization error of patch load.
from the mesh size. The equivalent load of the patch loa can be calculated by the surface integral as
Re = ∫
S NTt dS (4)
whereN is the shape function matrix andt is the surface traction. In this formal way, one must identify the nodes an elements that lie on the patch load. For practical purpose in this work, the patch load is discretized as a numbe of uniformly distributed concentrated loads which will be called “sub-point loads”. Each sub-point load is considere as a single concentrated load. If there areK sub-point loads applied to the tire patch on an element(p), then equivalent nodal forces are computed as
Re = K∑
i=1
NT i pi . (5)
The saved equivalent nodal forces are assembled to loaded nodes. The major advantage of the discretized pa load algorithm is that it eliminates the cumbersome loa boundary search problem and numerical integration whi the accuracy of the FE solution is retained with the sufficien refinement of tire patch. The discretization error of patc load is illustrated inFig. 8 when an eight-node Mindlin shell element is loaded under distributed load represent by sub-point loads. Different levels of discretization ar considered by increasing the number of sub-point load The exact equivalent nodal forces are calculated b Eq. (4). It is clear that equivalent nodal forces for both corne nodes and interior nodes using the proposed discretiz algorithm converge to the exact value of equivalent nod forces as discretization level increases. It is observed th approximately 100 sub-point loads results in less than 0.5 error for both corner and interior nodes.
2.5. Boundary conditions
Since the main purpose of this study is to analyz bridge superstructures, it is assumed that substructures, s as piers and abutments, do not influence the behavior the superstructure. Although bearings are typically locate below the beam element, many previous models neglect
68 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
Table 2 Material properties of Nebraska bridge test
Material properties Concrete Steel
Slab Parapet Top Bottom Bottom web Reinforcing flange flange (int) flange (ext) bar
Young’s modulus, GPa 193.7 180.6 191.0 200.6 190 Yield stress, MPa – – 283 268 279 323 500 Ultimate stress, MPa – – 467 445 447 415 827 Compressive strength, MPa 43.0 41.8 – – – – –
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this fact and assumed bearings to be located at the cent of the beam element or at the bottom flange of th beam. In this study, bearings are modeled by assign boundary conditions to the zero-dimensional elements their real location. For simply supported beams, rotatio in all directions are allowed in order to simulate th simply supported structure. Minimum restraints are assign for longitudinal and transverse movement while vertic restraint is placed at the supports. Kinematic constraints also supplied to nodes between the girders and the deck.
3. Numerical comparisons
The objective of this section is to compare the resu of finite element analyses to those of a full-scale laborato test and an actual field test. The comparison is restricted elastic load levels to investigate the validity of the variou bridge models.
3.1. Full-scale laboratory test
A full-scale steel girder bridge was designed, constructe and tested in the structural laboratory of the University Nebraska at Lincoln [19]. The bridge is 21.4 m (70 ft) long and 7.9 m (26 ft) wide. The superstructure consists of thr 137 cm (54 in.) deep welded plate girders built composite with a 19 cm (7.5 in.) thick reinforced concrete deck. The are three girders with girder spacing of 3 m (10 ft). Th reinforced concrete deck was reinforced in both the top a bottom of the slab. The details of the bridge configuratio and loading can be found in the original report. The mater properties are summarized inTable 2.
The test loading set-up consisted of 12 post-tension rods simulating approximately two side-by-side AASHTO HS-20 design trucks. The four rods simulating front axle are placed at 6.1 m from the left support. The rods we spaced at 3.7 m (12 ft) and 4.6 m (15 ft) instead of th typical AAHSTO HS-20 spacing of 4.3 m (14 ft) and 4.3 m (14 ft). Two sets of loads simulating two trucks ar placed symmetrically with respect to the center girder the transverse direction. The tire contact area was simula using steel plates with dimensions of 50 cm (20 in.) by 20 c (8 in.) for rear and center wheels and 25 cm (10 in.) by 10 c (4 in.) for front wheels. For the elastic test, 2.5 times HS2
id
l
d
truck load was applied on the rod plates. This load consist of 362.5 kN (40 kips) for the center and rear wheels an 87.5 kN (10 kips) for the front wheels. The applied loadin on the bridge deck is discretized using the equivalent nod force algorithms as discussed in the previous section.
In the present study, a total of five three-dimension finite element models are implemented for the tested bridg Four models make use of eight-node Mindlin shell elemen (ABAQUS S8R) as the deck model with different girde models (girder models G1, G2, G3, and G4) as discuss in the previous section. These models are illustrated Figs. 2–5. Model 1 uses shell elements to represent th bridge deck and the quadratic G1 model to idealiz the bridge girders. The parapet of the tested bridge modeled by three-node beam elements. The compos action between the deck and the girder is modeled by rig links. The second model (Model 2) is the same as Mod 1, but the girders are modeled by the quadratic G2 mod In Model 3, the girder model G3 is used for modeling the girders. Model G4 is the simplest model, which use three-node beam elements (G4) to represent bridge girde This model is known as the eccentric beam model. Th last model is denoted as “Solid Model”, and uses the mo detailed girder model, G1, while quadratic solid elemen (ABAQUS C3D20) are used to model the deck and parap Full composite action and displacement compatibility ar achieved by imposing kinematic constraints between th solid elements and shell elements. The FE mesh of the So Model is shown inFig. 9.
Figs. 10 and 11 present the bottom flange deflections at the interior girder and exterior girders, respectively. A finite element models generally produce deflections simil to the measured deflections. The maximum error betwe predicted deflection and measured deflection is 5% for t Solid Model and 6% for Model 4. The predicted bottom flange strains are compared to the measured strains Fig. 12. The maximum error is 5% for the interior girder at mid-span in Model 1 and 9% for the exterior girder a the quarter-span in the Solid Model. Model 3 and Mode 4 predict values that are closer to the measured strains both cases.
The lateral load distributions of each bridge mode are compared through the AASHTO specified loa distribution factor (LDF).Fig. 13shows the LDF from the
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71 69
Fig. 9. Finite element model of Nebraska bridge.
s. tly ar
Fig. 10. Deflection of interior girder.
Fig. 11. Deflection of exterior girders.
AASHTO Standard specifications [18], AASHTO LRFD specifications [20], experiments, and finite element model The LDFs calculated by finite element models are sligh larger (up to 6%) than the measured LDF. However, it is cle that the LDF values from the AASHTO-Standard and t ASSHTO-LRFD are more conservative than both measu
Fig. 12. Strains at the bottom flanges.
Fig. 13. Load distribution factors.
and predicted LDF values. All finite element models a capable of predicting the load distribution mechanism of composite bridge.
The amount of computational resources used by various FE models is compared inFig. 14. The total number of DOFs required for the Solid Model is three times larg than the number required for Model 4. This indicates th the eccentric beam model (Model 4) is as accurate as other models yet it is the simplest model. The eccentric be
70 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
t
re an d
ed
l 4) ong tely g re, s a ge
A &
Fig. 14. Total number of DOFs.
model is thus further verified with a field test in the nex section.
3.2. Michigan field test
The eccentric beam model (Model 4) is verified with the results from a field test conducted at the University Michigan [21]. The tested bridge is a simple span located o Stanley Road over I-75 in Flint, Michigan. The span length 38.4 m (126 ft). There are seven girders with girder spacin of 2.2 m (7.25 ft) and an overhanging width of 74.7 cm (2.45 ft). The slab thickness is 20 cm (8 in.). Strain gaug were installed at the bottom flanges of the girders. All strain were measured along the centerline of the bridge span. T test load was the Michigan three-unit, 11-axle truck. Th load test was performed with the truck at crawl speed produce the maximum static strain at the steel girders.
The predicted strains and load distribution factors a compared to those obtained from the test results. As c be seen inFig. 15, good agreement between measured an calculated values is observed in all girders. The maximu error in the finite element model is within 6% of the measured strain value at girder 4. The model is also able predict the lateral load distribution accurately. Therefore, can be concluded that the eccentric beam model (Model used in this study, is capable of accurately predicting th actual behavior of steel girder bridges.
4. Summary and conclusions
This paper discusses two key issues in the thre dimensional finite element modeling of composite girde bridges: element compatibility and geometric error. Sinc the proper selection of finite elements is key to avoi compatibility errors, different types of finite elements wer used to model bridge superstructures. In addition, techniqu to reduce geometric errors and to discretize the appli pressure load are also presented.
Based on comparisons between the results obtained us several finite element models and available experimen results or analytical solutions, the following conclusion
e
(b) Load distribution factor.
Fig. 15. Results of Michigan field tests.
can be drawn. First, girder models, which utilize sh elements for their girder modeling (girder model G1 a G2), require a higher level of mesh refinement to conve due to the displacement incompatibility between the drilli DOF of the web element and the rotational DOF of t flange element. In general, for the same level of me density, quadratic elements are more accurate than lin elements. Secondly, the eccentric beam model (Mode has been identified as the most economical model am all studied models, since this model is capable of accura predicting the flexural behavior of girder bridges, includin deflection, strain, and lateral load distribution. Furthermo the finite element model for slab on girder bridges provide rational tool for the understanding of the behavior of brid superstructures.
References
[1] Chen Y. Distribution of vehicular loads on bridge girders by the FE using ADINA: modeling, simulation, and comparison. Computers Structures 1999;72:127–39.
[2] Chan THT, Chan JHF. The use of eccentric beam elements the analysis of slab-on-girder bridges. Structural Engineering a Mechanics 1999;8(1):85–102.
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71 71
in rin
al ral
): ls;
s.
[3] Barr PJ, Eberhard MO, Stanton JF. Live-load distribution factors prestressed concrete girder bridges. Journal of Bridge Enginee ASCE 2001;6(5):298–306.
[4] Shahawy M, Huang D. Analytical and field investigation of later load distribution in concrete slab-on-girder bridges. ACI Structu Journal 2001;98(4):590–9.
[5] Brockenbrough RL. Distribution factors for curved I-girder bridge Journal of Structural Engineering, ASCE 1986;112(10):2200–15.
[6] Tabsh SW, Tabatabai M. Live load distribution in girder bridg subject to oversized trucks. Journal of Bridge Engineering, AS 2001;6(1):9–16.
[7] Bishara AG, Liu MC, Nasser DE. Wheel load distribution on simp supported skew I-beam composite bridges. Journal of Struct Engineering, ASCE 1993;119(2):399–419.
[8] Mabsout ME, Tarhini KM, Frederick GR, Tayar C. Finite eleme analysis of steel girder highway bridges. Journal of Brid Engineering, ASCE 1997;2(3):83–7.
[9] Eamon CD, Nowak AS. Effect of edge-stiffening elements a diaphragms on bridge resistance and load distribution. Journa Bridge Engineering, ASCE 2002;7(5):258–66.
[10] ABAQUS, Inc. ABAQUS/Standard user’s manual—version 6 Pawtucket, RI; 2001.
[11] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and she 2nd ed. McGraw-Hill; 1959.
[12] Zienkiewicz OC, Taylor RL. The finite element method, vol. 2. 4th e
g,
l
f
.
McGraw-Hill; 1989. [13] Timoshenko S, Goodier JN. Theory of elasticity. 3rd ed. McGra
Hill; 1970. [14] Gupta AK, Ma PS. Error in eccentric beam formulation. Internation
Journal for Numerical Methods in Engineering 1977;11:1473–7. [15] Miller RE. Reduction of the error in eccentric beam modelin
International Journal for Numerical Methods in Engineering 1980;1 575–82.
[16] Khaleel AK, Itani RY. Live-load moments for continuous skew bridges. Journal of Structural Engineering, ASCE 1990;116( 2361–73.
[17] Sadek EA, Tawfik SA. A finite element model for the analys of stiffened laminated plates. Computers & Structures 2000;75( 369–83.
[18] American Association of State Highway and Transportation Officia (AASHTO). Standard specifications for highway bridges. 16th e Washington, DC; 1996.
[19] Kathol S, Azizinamini A, Luedke J. Strength capacity of ste girder bridges. NDOR Research Project No. RES1 90099. Nebra Department of Roads; 1995.
[20] AASHTO LRFD bridge specifications. 2nd ed. Washington (DC American Association of State Highway and Transportation Officia 1998.
[21] Eom J, Nowak AS. Live load distribution for steel girder bridge Journal of Bridge Engineering, ASCE 2001;6(6):489–97.
Three-dimensional finite element modeling of composite girder bridges
Introduction