stringer bridges

Composites: Part B 31 (2000) 593±609

www.elsevier.com/locate/compositesb

A systematic analysis and design approach for single-span FRP deck/stringer bridges Pizhong Qiao*, Julio F. Davalos, Brian Brown Department of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26506-6103, USA

Abstract There is a concern with worldwide deterioration of highway bridges, particularly reinforced concrete. The advantages of ®ber reinforced plastic (FRP) composites over conventional materials motivate their use in highway bridges for rehabilitation and replacement of structures. In this paper, a systematic approach for analysis and design of all FRP deck/stringer bridges is presented. The analyses of structural components cover: (1) constituent materials and ply properties, (2) laminated panel engineering properties, (3) stringer stiffness properties, and (4) apparent stiffnesses for composite cellular decks and their equivalent orthotropic material properties. To verify the accuracy of orthotropic material properties, an actual deck is experimentally tested and analyzed by a ®nite element model. For design analysis of FRP deck/stringer bridge systems, an approximate series solution for orthotropic plates, including ®rst-order shear deformation, is applied to develop simpli®ed design equations, which account for load distribution factors under various loading cases. An FRP deck fabricated by bonding side-by-side box beams is transversely attached to FRP wide-¯ange beams and tested as a deck/stringer bridge system. The bridge systems are tested under static loads for various load conditions, and the experimental results are correlated with those by an approximate series solution and a ®nite element model. The present simpli®ed design analysis procedures can be used to develop new ef®cient FRP sections and to design FRP highway bridge decks and deck/stringer systems, as shown by an illustrative design example. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Approximate series solution

1. Introduction Fiber-reinforced plastic (FRP) composites are increasingly used in civil infrastructure. In recent years, attention has been focused on FRP shapes as alternative bridge deck materials because of their high speci®c stiffness and strength, corrosion resistance, lightweight, and potential modular fabrication and installation that can lead to decreased ®eld assembly time and traf®c routing costs. Despite the overall bene®ts of using FRP sections, they are as yet not widely used. Besides concerns with material costs, in comparison with conventional materials, the complexity of material properties and structural behaviors has deterred the rapid development of design codes for practicing engineers. A critical obstacle to widespread use and applications of FRP structures in construction is the lack of simpli®ed and practical design guidelines. Unlike standard materials, FRP composites are typically orthotropic or anisotropic, and their analyses are much more dif®cult. For example, while

* Corresponding author. Present address: Department of Civil Engineering, The University of Akron, Akron, OH 44325-3905, USA.

changes in the geometry of FRP shapes can be easily related to changes in stiffness, changes in the material constituents do not lead to such obvious results. In addition, shear deformations in pultruded FRP composite materials are usually signi®cant and, therefore, the modeling of FRP structural components should account for shear effects. For applications to pedestrian and vehicular FRP bridges, there is a need to develop simpli®ed design equations and procedures which should provide relatively accurate predictions of bridge behavior and be easily implemented by practicing engineers. A number of theoretical and experimental investigations have been conducted to study stiffness, strength, and stability characteristics of FRP composite bridge decks. Henry [1] and Ahmad and Plecnik [2] examined the performance of several glass reinforced polymer bridge deck con®gurations using ®nite element analysis. In their study, an FRP deck was modeled as a truss system in the direction perpendicular to the traf®c and as a beam system in the direction parallel to the traf®c; their results indicated that the design was always controlled by de¯ection limit state rather than strength limit states. Later, FRP decks were fabricated using a combination of ®lament winding and hand lay-up processes, and the static and fatigue behaviors

1359-8368/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 1359-836 8(99)00044-X

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Fig. 1. Systematic analysis protocol for FRP bridge systems.

of decks were determined experimentally [3]; from the experimental observation, it was evident that damage under fatigue loading consisted primarily of interfacial delamination initiation and local buckling of thin delaminated layers under compressive service loads. Bakeri and Sunder [4] used balanced symmetrical laminate to study the feasibility of two FRP bridge deck systems: a deck solely made of a glass-reinforced polymer and a deck with hybrid materials of a glass-reinforced polymer, a carbon-reinforced polymer and a light-weight concrete; they concluded that the hybrid system concept achieved less de¯ection and had a promising future in infrastructure applications. With the objective of developing an FRP bridge deck system, Mongi [5] tested three full-scale ¯oor systems which differ in their size, joint type, and loading conditions. GangaRao and Sotiropoulos [6] assembled and tested two FRP bridge superstructure systems consisting of bridge decks and stringers; a simpli®ed ®nite-element model, in which equivalent plates were used as a substitute for the stringers and the cellular deck, was used to correlate results with experimental data. Burside et al. [7] presented a design optimization of an all-composite bridge deck; cellular-box and stiffened-box geometries were optimized with consideration of de¯ection and buckling. Most recently, two highway bridges were constructed with a modular FRP composite deck in West Virginia [8]: one was built as an all-composite short-span deck/stringer system and another one was constructed with a modular FRP deck supported by steel stingers. Fatigue and failure characteristics of a modular deck were investigated by Lopez-Anido et al. [9] and a satisfactory performance was observed. Several projects [10] also demonstrated the potential of FRP composites as

highway bridge materials. An overview of the current status on research and applications of ®ber-reinforced polymeric bridge decks has been presented by Zureick [11]. Although several experimental and numerical efforts have been conducted, there is no simpli®ed design protocol available for FRP composite bridges. The design procedures developed for bridges composed of isotropic materials [12] cannot be directly applied; they need to be modi®ed to incorporate the anisotropy of composite materials. Thus a simple but accurate solution for the analysis and design of FRP composite bridges is needed. This solution should account for the geometry and material properties of FRP decks and stringers; and it should provide reasonably accurate predictions for performance and load distribution of the system. In this paper, a systematic approach for analysis and design of all FRP deck/stringer bridges is presented. This approach (Fig. 1) is based on analyses at micro-level (material), macro-level (structural component) and system level (structure) to design all FRP deck/stringer bridge systems. First, based on manufacturer's information and material lay-up, ply properties are predicted by micromechanics. Once the ply stiffnesses are obtained, macromechanics is applied to compute the panel mechanical properties. Beam or stringer stiffness properties are then evaluated from mechanics of thin-walled laminated beams (MLB). Using elastic equivalence, apparent stiffnesses for composite cellular decks are formulated in terms of panel and single-cell beam stiffness properties, and their equivalent orthotropic material properties are further obtained. For design analysis of FRP deck/stringer bridge systems, an approximate series solution for ®rstorder shear deformation orthotropic plate theory is applied to develop simpli®ed design equations, which account for load distribution factors for various load cases. As illustrated in Fig. 1, the present systemic approach, which accounts for the microstructure of composite materials and geometric orthotropy of a deck system, can be used to design and optimize ef®cient FRP deck and deck/stringer systems. To introduce the systematic approach shown in Fig. 1, this paper is organized in the following three main sections: (1) panel and beam analyses by micro/macromechanics and mechanics of thin-walled laminated beams, (2) FRP cellular decks by elastic equivalence analysis, and (3) analysis of deck/stringer system by an approximate series solution technique. To verify the accuracy of the equivalent orthotropic material properties, a multi-box-beam deck fabricated by bonding side-by-side box FRP beams is experimentally tested and analyzed by a ®nite element model. To validate the approximate series solution, the multi-box-beam deck is attached to FRP wide-¯ange (WF) beams and tested and analyzed as a deck/stringer bridge system. The box beams for decks and WF beams for stringers were both produced by pultrusion. Both deck and bridge systems are tested under static loads for various load conditions. To

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Fig. 2. Microstructure and dimensions of a FRP box beam section.

illustrate the systematic design procedures developed in this paper, an example of an FRP deck/stringer bridge is presented. 2. Panel and beam analyses Extensive research has been conducted in the area of analysis and design of composite materials at micro- and macro-levels. The analysis of FRP beams from micro/ macromechanics to beam response has been presented in Ref. [13]. In this section, the analyses of micro/macrostructure and beam component are brie¯y reviewed and include: (1) constituent materials and prediction of ply properties, (2) laminated panel engineering properties, and (3) beam or stringer stiffness properties. 2.1. Panel analysis by micro/macromechanics Although pultruded FRP shapes are not laminated structures in a rigorous sense, they are pultruded with

material architectures that can be simulated as laminated con®gurations. A typical pultruded section mainly includes the following three types of layer [13] (see Figs. 2 and 3): (1) continuous strand mats (CSM); (2) stitched fabrics (SF); and (3) rovings or unidirectional ®ber bundles. Each layer is modeled as an homogeneous, linearly elastic, and generally orthotropic material. Based on information provided by the manufacturer, the ®ber volume fraction (Vf) can be evaluated and used to compute the ply stiffnesses from micromechanics models [14]. For the box section of Fig. 2 and the wide-¯ange section of Fig. 3, the predicted ply properties are given in Tables 1 and 2, respectively. Once the ply stiffnesses for each laminate (panel) of a FRP beam are computed, the stiffnesses of a laminated panel can be computed from macromechanics [15]. For example, for the box beam shown in Fig. 2, the panel properties (Ex, Ey, nxy and Gxy) predicted by the micro/macromechanics models [14] correlate well with experimental results for coupon samples (Table 3) tested in tension and torsion.

Fig. 3. Panel ®ber architecture of a wide ¯ange beam.

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Table 1 Ply material properties of a box section (Fig. 2) computed by a micromechanics model [14] Lamina

E1 ( £ 10 6 psi)

E2 ( £ 10 6 psi)

v12

G12 ( £ 10 6 psi)

1/2 oz CSM 1 oz CSM 15.5 oz 908 SF 12 oz 1/2458 SF 61 yield roving

2.093 1.710 4.118 3.505 8.469

2.093 1.710 1.183 1.056 3.374

0.407 0.402 0.389 0.396 0.343

0.744 0.610 0.457 0.405 1.429

2.2. Beam analysis by mechanics of laminated beams The response of FRP shapes in bending is evaluated using the mechanics of thin-walled laminated beams (MLB) [16]. In MLB, the stiffness coef®cients (axial, A; bending, D; axial-bending coupling, B; and shear, F) of a beam are computed by adding the contributions of the stiffnesses of the component panels, which in turn are obtained from the effective beam moduli. Based on MLB, engineering design equations for FRP beams under bending have been formulated [17], and they can be easily adopted by practicing engineers and composite manufacturers for the analysis, design, and optimization of structural FRP beams or bridge stringers. MLB is suitable for straight FRP beams or columns with at least one axis of geometric symmetry and can be used to evaluate the stiffness properties and response of bridge stringers. As an example, the bending (D) and shear (F) stiffnesses of a box beam (Fig. 2) and a wide-¯ange beam (Fig. 3) by MLB are listed in Table 4, and experimental results for de¯ections and strains compared favorably with MLB predictions [13,18,19]. The panel and beam properties obtained above by micro/ macromechanics and MLB can be ef®ciently implemented in deck and deck/stringer system designs, as described in Sections 3 and 4. 3. FRP cellular decks: elastic equivalence A multicellular FRP composite bridge deck can be modeled as an orthotropic plate, with equivalent stiffnesses that account for the size, shape and constituent materials of the cellular deck. Thus, the complexity of material anisotropy of the panels and structural orthotropy of the deck system can be reduced to an equivalent orthotropic plate with global elastic properties in two orthogonal directions: parallel and transverse to the longitudinal axis of the deck cell. These equivalent orthotropic plate properties can be directly used in the design and analysis of deck/stringer

bridge systems, as presented in Section 4, and they can also serve to simplify modeling procedures either in numerical or explicit formulations. The design equations necessary for such a model are presented in this section, along with numerical and experimental veri®cation of the results. In this section, the development of equivalent stiffness for cellular decks consisting of multiple FRP box beams is presented. Multicell box sections are commonly used in deck construction because of their light-weight, ef®cient geometry, and inherent stiffness in ¯exure and torsion. Also, this type of deck has the advantage of being relatively easy to build. It can be either assembled from individual box-beams or manufactured as a complete section by pultrusion or a vacuum-assisted resin transfer molding process. The elastic equivalence approach [20] used in this paper accounts for out-of-plane shear effects, and the results for a multicell box section are veri®ed experimentally and by ®nite element analyses. 3.1. Equivalent stiffness for cellular FRP decks As an illustrative example, we derive the bending, shear and torsional equivalent stiffnesses for a deck composed of multiple box sections (Fig. 4). 3.1.1. Longitudinal stiffnesses of cellular FRP deck The bending stiffness of the deck in the longitudinal direction, or x-axis in Fig. 4, is expressed as the sum of the bending stiffness of individual box beams (Db, see Table 4): Dx ˆ nc Db

…1†

where nc ˆ number of cells. For the section shown in Fig. 4, b ˆ width of a cell, h ˆ height of a cell, tf ˆ thickness of the ¯ange, and tw ˆ thickness of the web. If all panels have identical material lay-up and tf ˆ tw ˆ t, Eq. (1) becomes   ht …2† Dx ˆ nc Ex h2 1 3bh 6

Table 2 Ply material properties of a wide-¯ange section (Fig. 3) computed by a micromechanics model [14] Lamina

E1 ( £ 10 6 psi)

E2 ( £ 10 6 psi)

v12

G12 ( £ 10 6 psi)

3/4 oz CSM 17.7 oz 1/2458 SF 62 yield roving

1.710 4.157 6.732

1.710 1.191 2.077

0.402 0.294 0.278

0.610 0.460 0.826

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Table 3 Panel properties of a box section (4 00 £ 8 00 £ 1/4 00 ) Ex Experimental Micro/macromechanics %Difference a b

Ey 6

3.293 £ 10 psi 3.377 £ 10 6 psi 12.6%

a

vxy 6

2.491 £ 10 psi 2.620 £ 10 6 psi 15.2%

a

0.269 0.285 15.9%

Gxy a

8.599 £ 10 5 psi b 8.760 £ 10 5 psi 11.9%

From tension tests. From torsion tests.

where Ex ˆ modulus of elasticity of a panel in the x-direction computed by micro/macromechanics or obtained experimentally (Table 3). The out-of-plane shear stiffness of the deck in the longitudinal direction, Fx, is expressed as a function of the stiffness for the individual beams (Fb): F x ˆ nc F b

…3†

where Fb is given in Table 4, and nc ˆ number of cells. This expression can be further approximated in terms of the inplane shear modulus of the panel, Gxy (see Table 3) and cross-sectional area of the beam webs: Fx ˆ nc Gxy …2t†h:

…4†

3.1.2. Transverse stiffnesses of cellular FRP deck An approximate value for the deck bending stiffness in the transverse direction, Dy, may be obtained by neglecting the effect of the transverse diaphragms and the second moment of area of the ¯anges about their own centroids. For a deck as shown in Fig. 4 with tf ˆ t: Dy ˆ

1 E …w†…t†h2 2 y

…5†

where w is the length of the deck in the longitudinal direction and Ey is the modulus of elasticity of the panel in the y-direction (Table 3). For multiple box sections, the simplest way to obtain the deck's out-of-plane transverse shear stiffness is to treat the structure as a Vierendeel frame in the transverse direction [21]. For the Vierendeel frame (Fig. 5), the in¯ection points are assumed at the midway of top and bottom ¯anges between the webs. The shear stiffness in the transverse direction, Fy, for the cross-section shown in Fig. 4 may be written as Fy ˆ

12Ey V  ˆ  h b u b 1 Iw 2If

…6†

Table 4 Strong-axis beam bending and shear stiffness coef®cients by MLB Beam stiffness

Db (lb/in 2 2 in 4)

Fb (lb/in 2 2 in 2)

Box 4 00 £ 8 00 £ 1=4 00 WF 12 00 £ 12 00 £ 1=2 00

1:795 £ 108 1:706 £ 109

3:474 £ 106 5:026 £ 106

where the moments of inertia I are de®ned as: ÿ
w 2tw 3 wtf3 If ˆ ; Iw ˆ 12 12

…7†

For tf ˆ tw ˆ t, Eq. (6) can be simpli®ed as Fy ˆ

2E wt3  y  h b b1 4

…8†

where Ey is the modulus of elasticity of a panel in the y-direction (Table 3). 3.1.3. Torsional stiffness of cellular FRP deck The torsional rigidity of a multi-cell section, GJ, is evaluated by considering the shear ¯ow around the crosssection of a multi-cell deck. For a structure where the webs and ¯anges are small compared with the overall dimensions of the section, Cusens and Pama [21] have shown that the torsional rigidity may be written as GJ ˆ

X 4A2 Gxy t3 1 Gxy …ds† P ds 3 t

…9†

where A ˆ area of the deck section including the void area P ds=t represents the and is de®ned as A ˆ nc bh, and summation of the length-to-thickness ratio taken around the median line of the outside contour of the deck crosssection. For a constant panel thickness t, the torsional rigidity can be simpli®ed as ÿ

2 nc bh 2 Gxy t 2ÿ
1 nc b 1 h Gxy t3 …10† GJ ˆ ÿ 3 nc b 1 h The above approximate equation is justi®ed by the fact that for a multi-cell deck, the net shear ¯ows through interior webs are negligible and only the shear ¯ows around the outer webs and top and bottom ¯anges are signi®cant. The second term in Eq. (10) is relatively small compared with the ®rst term and can be ignored. If the deck is treated as an equivalent orthotropic plate, its torsional rigidities depend upon the twist in two orthogonal directions. Thus, torsional stiffness Dxy may be taken as onehalf of the total torsional rigidity given by Eq. (10) divided by the total width of the deck: Dxy ˆ

GJ 2nc b

…11†

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Fig. 4. Geometric parameters of a multi-cell box deck.

Substituting Eq. (10) into Eq. (11) and neglecting the second term in Eq. (10), we get:

3.2. Veri®cation of deck stiffness equations by ®nite element analysis

nc Gxy bh2 t
Dxy ˆ ÿ nc b 1 h

The formulae for bending and torsional stiffnesses obtained in Section 3.1 are based on the assumption that the deck system behaves as a beam and does not account for the Poisson effects of the deck. To verify the accuracy of the above deck stiffness equations, a ®nite element analysis of the deck system is performed. The model is shown in Fig. 4 and consists of box beams (Fig. 2) bonded side-by-side to form an integral deck. The computer program NISA [22] is used, and the panels are modeled with 8-node isoparametric layered shell elements. The cellular decks subjected to lineloading for longitudinally and transversely supported conditions are shown in Figs. 6 and 7, and the model for torsional loading is given in Fig. 8.

…12†

where D xy is the torsional stiffness per unit width (lb 2 in 3 /in 2 ).

3.2.1. Veri®cation of bending and shear stiffnesses The deck bending and shear stiffnesses in the longitudinal and transverse directions are used to evaluate midspan de¯ections from the following

Fig. 5. Vierendeel distortion in a multi-cell box-beam [21].

d3 ˆ

PL3 PL 1 …3-point bending† 48Di 4 k Fi

…13†

d4 ˆ

23PL3 PL 1 …4-point bending† 6kFi 1296Di

…14†

where P ˆ total applied load, L ˆ span length, k ˆ shear correction factor (k ù 1:0 is assumed in the analysis), and Di and Fi ˆ bending and shear stiffness (i ˆ x for longitudinal or y for transverse directions). The de¯ections by Eqs. (13) and (14) in terms of stiffness properties are compared with results from the ®nite element model for actual cellular systems under line loading (Figs. 6 and 7). For the longitudinal stiffness veri®cation, the length of the decks is kept constant (L ˆ 108 in), and the de¯ection in terms of bending and shear stiffnesses is a function of the number of cells. Each deck is simply supported and

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Fig. 7. FE model for veri®cation of transverse bending and shear stiffnesses equations.

Fig. 6. FE model for veri®cation of longitudinal bending and shear stiffnesses equations.

subjected to either 3-point or 4-point bending due to uniformly distributed line loads. The comparisons between the predictions of Eqs. (13) and (14) based on simpli®ed stiffness formulae and the ®nite element results for actual decks are presented graphically in Fig. 9. Similarly, the midspan de¯ections in the transverse direction are found by modeling several multicellular decks comprised of 4 00 £ 8 00 £ 1/4 00 box sections (Fig. 7). For these models, the width (w) is kept constant (w ˆ 12 in), and the de¯ection is a function of the number of cells. The model is simply supported and subjected to either 3-point or 4-point bending due to uniformly distributed line loads. The results of the ®nite element models and theoretical predictions are shown in Fig. 10. 3.2.2. Veri®cation of torsional stiffness of the deck The simpli®ed formula for the torsional rigidity, GJ, of the deck system was also veri®ed using ®nite element analyses, which indirectly serve to verify the torsional stiffness of the deck (Dxy). The model shown in Fig. 8 consisted of a multicellular deck with one end ®xed, by constraining

displacements and rotations in all three principal directions and all three rotations, and the other end subjected to a uniform torque. The longitudinal torsional rigidity of a deck is expressed in terms of the angle of twist f and the torque applied at the end of the section as GJ ˆ

TL f

…15†

where T ˆ 2qnc bh (as shown in Fig. 8) is the applied torque. The specimen length L is held constant (L ˆ 108 in), and the number of cells is used as the design variable. The ®nite element results are compared with the theoretical predictions of Eq. (10), and the results are presented in Fig. 11. 3.2.3. Comparisons and remarks As shown in Figs. 9±11, a good correlation is obtained between the theoretical predictions based on the simpli®ed stiffness formulas and the ®nite element analyses of an actual deck. For the de¯ection in terms of longitudinal stiffnesses (Dx and Fx), the maximum percent difference is 4%, and for the de¯ection in terms of transverse stiffnesses (Dy and Fy), the maximum difference is about 10%. For the longitudinal torsional stiffness, the discrepancy of results increases steadily from 6% for one cell to 22% for 15 cells. Some limited experimental data available for one and two cells [23] match closely the analytical results. The favorable de¯ection comparisons between beam

600

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Fig. 10. Transverse central de¯ection of a multi-cell deck (Fig. 7). Fig. 8. FE Model for veri®cation of the torsional rigidity equation.

equations and ®nite element results indirectly verify the accuracy of the deck bending stiffness equations. Similarly, the torsion results indicate that the simpli®ed torsional stiffness equations are acceptable for practical applications. Therefore, the proposed relatively simple stiffness equations account for both shape and material anisotropy of the deck and can be used with relative con®dence in design analysis of cellular bridge deck systems. 3.3. Equivalent orthotropic material properties

deck can further simplify the design analysis of deck and deck/stringer bridge systems. To calculate the moduli of elasticity …Ex †p and …Ey †p for the equivalent orthotropic plate, the relationship D ˆ EI is used, leading to  ÿ
D  Ex p ˆ 12 3 x 1 2 nxy nyx t p bp

…16†

   Dy  Ey ˆ 12 3 1 2 nxy nyx p tp lp

…17†

Once the stiffness properties of an actual deck are obtained, it is a simple matter to calculate effective material properties for an equivalent orthotropic plate. To obtain the equivalent orthotropic plate material properties for an actual

where the subscript ªpº indicates property related to the equivalent orthotropic plate; tp ˆ thickness of the plate ( ˆ h for the actual deck, Fig. 4), bp ˆ width of the plate ( ˆ ncb for the actual deck), and lp ˆ length of the plate ( ˆ w for the actual deck). The Poisson's

Fig. 9. Longitudinal central de¯ection of a multi-cell deck (Fig. 6).

Fig. 11. Torsional rigidity vs number of cells (Fig. 8).

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Fig. 12. Experimental setup of a multi-cell box deck (5 0 £ 10 0 £ 8 00 ).

ratios n ij are de®ned as

nij ˆ

2 1j 1i

…18†

where 1 is the strain in the i or j direction. For orthotropic materials, the Poisson's ratio must obey the following relationship:

nij nji nxy nyx ˆ or ˆ Ei Ej Dx Dy

…19†

In this study, we use the approximation nxy ˆ 0:3, which is typically used for pultruded composites. To calculate the out-of-plane shear moduli …Gxz †p and …Gyz †p , the relationship F ˆ GA is used, leading to ÿ


F Gxz p ˆ x t p bp



 Fy Gyz ˆ p tp lp

…20†

…21†

Finally, to calculate the in-plane shear modulus …Gxy †p , we use   Dxy …22† Gxy ˆ 6 3 p tp With these equivalent material properties, it is now easy to use explicit plate solutions (see Section 4) for analysis and design of cellular decks. 3.4. Experimental and numerical veri®cation of equivalent orthotropic material properties To indirectly verify the accuracy of the equivalent orthotropic material properties given in Eqs. (16)±(22) (see Section 3.3), a multi-box-beam deck of 5 0 £ 9 0 £ 8 00 (Fig. 12a) subjected to a patch load is tested and analyzed for three load conditions: (1) at the center of the deck, (2) at 16 00 to one side from the center along the line AA 0 , and (3) at 16 00 to the other side from the center along the line AA 0 .

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Fig. 13. FE simulation and de¯ection contour of a multi-cell box deck under symmetric loading.

The ®nite element program NISA [22] is used to conduct two distinct analyses: (1) the actual deck (Fig. 13) is modeled using 8-node isoparametric layered shell elements and the material properties of Table 1; (2) an equivalent solid orthotropic plate of the same global dimensions as the actual deck is modeled using the same elements and the equivalent material properties computed from Eqs. (16)±(22) and given in Table 5. The experimental results and correlations with ®nite element analyses are presented next.

3.4.1. Experimental details The test sample was fabricated by bonding FRP box beams side-by-side with epoxy [24]. For each load condition, the displacements are recorded at several locations with LVDTs (see Fig. 12b), and the strains in the longitudinal and transverse directions are obtained at three locations by bonding 350 ohm strain gages at the bottom of the deck (Fig. 12c). Note that from Fig. 12, for the asymmetric load cases 2 and 3, the following displacement values should be approximately equal: d 1 and d 5, and d 2 and

Table 5 Equivalent deck stiffness properties and orthotropic material properties for cellular deck 5 0 £ 9 0 £ 8 00 Dx (lb 2 in 4/in 2)

Dy (lb 2 in 4/in 2)

n xy

Dxy (lb 2 in 4/in 2)

Fx (lb 2 in 2/in 2)

Fy (lb 2 in 2/in 2)

2:689 £ 109

2:250 £ 109

0.3

1:153 £ 107

4:896 £ 107

3:662 £ 105

Ex (psi)

Ey (psi)

n yx

Gxy (psi)

Gxz (psi)

Gyz (psi)

9:713 £ 10

5

5

4:515 £ 10

0.25

1:351 £ 10

5

5

1:020 £ 10

4:238 £ 102

P. Qiao et al. / Composites: Part B 31 (2000) 593±609 Table 6 Experimental and ®nite element comparison for a multi-cell box deck under load case 1 (centric) Parameter

Experiment

FE (actual deck)

FE equivalent plate

d 1 (in/kips) d 2 (in/kips) d 3 (in/kips) d 4 (in/kips) d 5 (in/kips) d 6 (in/kips) d 7 (in/kips) 1 1 (m1 /kips) 1 2 (m1 /kips) 1 3 (m1 /kips) 1 4 (m1 /kips) 1 5 (m1 /kips) 1 6 (m1 /kips)

0.00721 0.00971 0.02036 0.00964 0.00710 0.01544 0.01421 28.111 214.701 68.584 215.266 28.361 212.966

0.00627 0.00861 0.01644 0.00861 0.00625 0.01033 0.01033 28.463 25.832 61.043 226.522 28.463 25.832

0.00602 0.00947 0.01900 0.00947 0.00603 0.01242 0.01242 31.752 27.813 73.539 223.643 31.752 27.813

d 4. Similarly, the following strains should correspond to each other: 1 1 and 1 5, and 1 2 and 1 6. 3.4.2. Experimental results and correlation Comparisons of experimental results and ®nite element analyses for FRP deck under centric loading (load case 1) and asymmetric loading (load cases 2 and 3) are shown in Tables 6 and 7, respectively. Tables 6 and 7 indicate that the measured displacements and strains compare relatively well with FE models of actual deck and equivalent plate for both the symmetric loading (case 1) and asymmetric loading (cases 2 and 3). For the symmetric load case (case 1), the difference of de¯ection (d 3) under load point between the experiment and the FE equivalent plate model is about 6.5%; whereas there are 6.7% difference for longitudinal strain (1 3) at the center of the deck (Fig. 12). As noted in Table 7, values for the asymmetric loading (cases 2 and 3) also compare favorably between the average experimental data and FE equivalent plate model when the measurements were close to the applied load; the differences are about 0.9% for de¯ection and 5.0% for longitudinal strain under applied load. The good correlation between experimental data and FE models validates the orthotropic plate material properties obtained by elastic equivalence analysis, which can be then be used in the analysis of FRP deck/stringer systems. 4. Analysis of FRP deck/stringer bridge system The equivalent properties for cellular decks and stiffnesses for FRP beams can be ef®ciently used to analyze and design deck/stringer systems. In this section, we present an overview of a series solution for stiffened orthotropic plates based on ®rst-order shear deformation theory and transverse interaction of forces between the deck and the stringers. The solutions for symmetric and antisymmetric load cases are used to obtain the solution for asymmetric loading. Based on deck-stringer transverse interaction force

603

functions, wheel load distribution factors are derived, which are used later to provide design guidelines for deck/stringer bridge systems. Finally, the approximate series solution is veri®ed by testing a 10 0 £ 10 0 £ 8 00 multi-box-beam deck supported by WF 12 00 £ 12 00 £ 1=2 00 FRP beams; this system is also analyzed by ®nite element model [22]. 4.1. First-order shear deformation theory for FRP composite deck A ®rst-order shear deformation theory [25] is applied to analyze the behavior of a geometrically orthotropic FRP composite deck. Instead of direct modeling of the actual deck geometry, an equivalent orthotropic plate, as discussed in Section 3, is used to simplify the analysis. The formulae for equivalent orthotropic material properties accounting for deck geometry and panel laminated material properties are given in Section 3. The equilibrium equations accounting for ®rst-order shear deformation of an orthotropic plate are:     2 2w 0 2 2w 0 cx 1 cy 1 1 A44 1 q…x; y† ˆ 0 A55 2x 2y 2x 2y     2cy 2cy 2 2c 2 2cx D11 x 1 D12 1 1 D66 2x 2y 2x 2x 2y 2y   2w0 ˆ0 2 A55 cx 1 2x

…23†

    2cy 2cy 2 2cx 2 2c 1 D12 x 1 D22 1 2x 2y 2y 2x 2x 2y   2w0 ˆ0 2 A44 cy 1 2y

D66

where Aij (i; j ˆ 4; 5) are the intralaminar shear stiffnesses, and Dij (i; j ˆ 1; 2; 6) are the bending stiffnesses for an orthotropic material. The deck/stringer bridge system can be ®rst analyzed as an orthotropic plate stiffened by edge stringers (or beams) [26]. Then, the contributions of interior stringers are accounted for in the formulation by considering the interaction forces and the comparability conditions along rib lines between the deck and stringers. The analysis is general with respect to: (1) size and stiffness of the deck, and (2) type of loading (uniform and/or concentrated). The formulation is concerned ®rst with symmetric and antisymmetric loading conditions. 4.1.1. System under symmetric loading case A Fourier polynomial series is employed to obtain the solutions for the equilibrium equations [Eq. (23)]. The solution for a symmetric loading is w0 …x; y† ˆ

1 X i;jˆ1

Wij sin ax…sin by 1 W0 †

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

20.00117 0.00252 0.00901 0.02255 0.02104 0.00632 0.00632 8.327 20.692 30.010 26.142 84.803 226.792

cx …x; y† ˆ

0.00094 0.00352 0.00860 0.01940 0.01729 0.00610 0.00566 12.826 22.176 28.528 25.086 70.076 229.444

FE actual deck

FE equivalent plate

604

cy …x; y† ˆ

1 X i;jˆ1 1 X i;jˆ1

q…x; y† ˆ

1 X

20.02801 0.00356 0.00993 0.02235 0.01883 0.00399 0.00815 24.201 219.956 28.579 226.379 81.666 230.770 20.05410 0.00412 0.01091 0.02330 0.01901 0.00996 0.00872 23.750 223.297 29.289 231.206 83.287 234.467

Average of load cases 2 and 3 Experiment Load Case 3

20.00191 0.00299 0.00895 0.02139 0.01865 0.00798 0.00757 24.652 216.614 27.868 221.552 80.044 227.072

d 1 and d 5 (in/kips) d 2 and d 4 (in/kips) d 3 and d 3 (in/kips) d 4 and d 2 (in/kips) d 5 and d 1 (in/kips) d 6 and d 6 (in/kips) d 7 and d 7 (in/kips) 1 1 and 1 5 (m1 /kips) 1 2 and 1 6 (m1 /kips) 1 3 and 1 3 (m1 /kips) 1 4 and 1 4 (m1 /kips) 1 5 and 1 1 (m1 /kips) 1 6 and 1 2 (m1 /kips)

Qij sin ax sin by:

…25†

where Kij are the deck stiffness coef®cients (for a symmetric loading) [24]. For a one-term approximation, the constants W0 and X0 are obtained by satisfying the boundary conditions of the edge-stiffened orthotropic plate [Fig. 14(b)]:   Y W0 ˆ A44 c 11 1 b W11

where

Experiment load case 2

Yij sin ax cos by

Qij are the Fourier coef®cients in the representation of the load q(x,y). By substituting the general solution Eqs. (24) and (25) into Eq. (23) and reducing by orthogonality conditions [12,23], we obtain the following system of equations for any number of terms (i,j): 9 9 8 2 38 Wij > K11 K12 K13 > Qij > > > > > > = = < 6 7< 6 K21 K22 K23 7 Xij ˆ 0 …26† 4 5> > > > > > > : ; : ; > Yij K13 K23 K33 0

X0 ˆ 2

Parameter load cases 2 and 3

…24†

where a ˆ ip=a and b ˆ jp=b, and Wij, Xij, and Yij are the coef®cients to be determined to complete the solution. Note that these series approximations satisfy the essential boundary conditions. The generalized loading can be written as the following in®nite double series

i;jˆ1

Table 7 Experimental and ®nite element comparison for a multi-cell box deck under load cases 2 and 3 (asymmetric)

Xij cos ax…sin by 1 X0 †

1 cˆ 2 a

  A44 Y11 1 bW11 X11 a3 D 

…27†

 1 1 1 2 ; kF aD

k ˆ the stringer shear correction factor, and F and D are, respectively, the shear and bending stiffnesses of the stringer and are obtained based on mechanics of laminated beams (MLB) (Table 4) [16]. For any interior stringer at any location r (r ˆ 0; 1; ¼; n) [see Fig. 14(c)], the generalized de¯ection function for any symmetric loading is [24]     1 1 1 px pr 1 2 sin 1 W0 sin wR …x; r† ˆ R11 2 a n a kF aD …28† where R11 ˆ

1 a2



Q  11  : 1 1 Q11 n 4W0 11 1 2 1 b kF p a D W11

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

605

Fig. 14. Deck/stringer bridge system.

4.1.2. System under antisymmetric loading case Analogous to the symmetric case, Eqs. (24) and (25) are modi®ed for a ®rst-term approximation of an antisymmetric loading as    2y w0 …x; y† ˆ W12 sin ax sin 2by 1 W1 1 2 b

cx …x; y† ˆ X12

   2y cos ax sin 2by 1 X1 1 2 b

…29†

q…x; y† ˆ Q12 sin ax sin 2by By substituting Eq. (29) into Eq. (23), we obtain the stiffness matrix for an orthotropic deck under antisymmetric loading [24]. The constants W1 and X1 are determined as 1 0   B C Y 1 C W1 ˆ A44 c 12 1 2b B @ A 2 W12 1 1 A44 c b   A44 Y12 1 2bW12 X12 a3 D

…30†

where c is the same as for Eq. (27). The generalized de¯ection function for antisymmetric loading is   1 1 1 1 2 wR …x; r† ˆ R12 2 a kF aD  sin

  px 2p r pr sin 1 W1 cos a n n

where R12 ˆ

1 a2



Q  12  : 1 1 Q12 4n 2W1 11 1 2 1 b kF 3p a D W12

wR …x; r† ˆ " R11

cy …x; y† ˆ Y12 sin ax cos 2by

X1 ˆ 2

4.1.3. System under asymmetric loading case The asymmetric case is obtained by superposition of the symmetric and antisymmetric load conditions. By simply adding the symmetric and antisymmetric responses, the generalized de¯ection function for an interior stringer under an asymmetric load is written as

…31†

!# ! pr 2pr pr 1 W0 1 R12 sin 1 W1 cos sin n n n

1  2 a

! 1 1 px 1 2 sin kF a aD

(32)

4.2. Wheel load distribution factors The above solution is used to de®ne wheel-load distribution factors for any of the stringers. The load distribution factor for any interior stringer ith is de®ned as the ratio of the interaction forces R(x,r) for the ith stringer to the sum of interaction forces for all stringers. The general expressions of load distribution factors in terms of the number of stringers m (where, m ˆ n 1 1) for symmetric and asymmetric loads [24] are, respectively WfSym …r†

ˆ

sin

r21 p 1 W0 m21

2 …m 2 1† 1 mW0 p

WfAsym …r† ˆ 

     r21 r21 r21 R11 sinp 1 W0 1 R12 sin2p 1 W1 1 2 2 m21 m21 m21      m  X r21 r21 r21 R11 sinp 1 W0 1 R12 sin2p 1 W1 1 2 2 m21 m21 m21 rˆ1

…33†

606

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

Fig. 15. FE simulation and de¯ection contour of a deck/stringer system under symmetric load.

4.3. Design guidelines Based on the wheel distribution factors obtained above, the number of stringers necessary for a given bridge deck can be determined. The dimensions of the deck are used to evaluate the maximum allowable moment per lane (Mmax) according to AASHTO [27]. Then an equivalent concentrated load (Pe) is calculated as [23,24] Pe ˆ

4Mmax L

…34†

where L is the length of a stringer (span of the bridge). The equivalent deck properties (Section 3) and the bending and shear stiffnesses (D and F) for a given type of stringer (Section 2) are then used to calculate the edge de¯ection coef®cient W0 and/or W1. Next, a design load (Pd) is de®ned either for the symmetric or asymmetric

load case as ÿ
Pd ˆ Pe NL Wf max

…35†

where NL is the number of lanes and (Wf)max is found from Eq. (33) as a function of number of stringers m. Two design criteria based on the performance of stringers and deck can be used to design the system. 4.3.1. Design criterion based on performance of stringer The midspan de¯ection d LL of a stringer is evaluated as ! L3 L …1 1 DLA† dLL ˆ Pd 1 …36† 4k F 48D where DLA is the dynamic load allowance factor, and for short-span bridges DLA ù 0:2 [26]. Eq. (36) is then set equal to the maximum allowable de¯ection (from AASHTO [27]) to determine the number of stringers required for

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

607

Fig. 16. Comparison for a deck/stringer bridge system.

the bridge deck. Once a suitable system is chosen, the maximum moment due to live load (MLL) is calculated from MLL ˆ

Pd L …1 1 DLA† 4

…37†

Finally, the approximate maximum extreme ®ber normal stress (s c) in the stringer can be found from

sc ˆ

MLL y 0 I

…38†

where y 0 is the distance from the neutral axis of the stringer to the top surface of the stringer and I is the moment of inertia of the stringer. This stress can then be compared with the material compressive or tensile strength to con®rm that the system will be effective. Also, as an approximation, shear stress in the stringer can be estimated as:

tˆ

Pd …1 1 DLA† 2Aw

…39†

where Aw is the area of the web panels. The shear stress in Eq. (39) should be less than the shear strength of the stringer.

4.3.2. Design criterion based on performance of deck Excessive local deck deformation and punching-shear failure may be observed in FRP bridge applications. Thus it is necessary in the design process to check the local deck de¯ection and bending and shear stresses in a deck section between two adjacent stringers [28,29]. Further research is needed to address these issues. 4.4. Experimental testing and numerical analysis of FRP deck/stringer systems To validate the approximate series solution presented above, an FRP deck (10 0 £ 10 0 £ 8 00 ) is fabricated by bonding side-by-side box beams of 4 00 £ 8 00 £ 1=4 00 (Fig. 2); the deck is attached to FRP I-beams 12 00 £ 12 00 £ 1=2 00 (Fig. 3) and tested and analyzed as a deck/stringer bridge system. The deck with either three or four stringers is subjected to various static load conditions [24]. The ®nite element model with NISA [22] is shown in Fig. 15 for a 3-stringer system under a concentrated centric loading. The comparisons among the FE, series solution and experiments for both 3-stringer and 4-stringer systems under centric loading are shown in Fig. 16, and relatively consistent trends are observed. The maximum differences of stringer de¯ections between experiments and series approximation are

608

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

Fig. 17. Dimensions and panel ®ber architectures of an optimized winged-box beam [30].

about 16% for a 3-stringer system and 23% for a 4stringer system. The detailed study on the experimental program and comparisons for various cases can be found in Ref. [24].

5. Design analysis procedures and illustrative example General guidelines for the applications of the above series approximation solution for design analysis of FRP composite deck-and-stringer bridge systems and several illustrative design examples are given in Ref. [24]. 5.1. General design procedures The following step-by-step design procedures are recommended: 1. de®ne bridge dimensions and allowable loads; 2. obtain deck panel and bridge stringer properties by micro/ macromechanics [13±15] and mechanics of laminated beams (MLB) [16]; 3. determine deck equivalent material properties by the equations derived in Section 3; 4. perform the series approximation analysis and determine the number of stringers m based on the required de¯ection limit;

5. check the stress level on the stringers; 6. check the local stresses, de¯ections, and other details. 5.2. Design example As an illustration, a single-lane short-span bridge of 15 ft width and 25 ft span is designed using side-by-side 4 00 £ 8 00 £ 1=4 00 bonded FRP box sections for the cellular deck assembly and optimized FRP winged-box beams 12 00 £ 24 00 [30] for the stringers (Fig. 17). The material properties for the stringers are: D ˆ 1:248 £ 1010 lb-in4 =in2 and F ˆ 1:940 £ 107 lb-in2 =in2 , which are computed by micro/macromechanics and MLB. The de¯ection limit of L/500 and the loading of AASHTO HS-20 [27] are considered, and the number of stringers (m) is used as a design variable. The edge de¯ection coef®cient for symmetric loading is evaluated from Eq. (27) as W0 ˆ 1:691, and the de¯ection limit is written as a function of the number of stringers m: 0 1   B C L 4 1 1 W0 C ˆ Mmax NL B dLL ˆ @ 2 A 500 L mW0 1 …m 2 1† p 

! L3 48 L …1 1 DLA† 1 D 4k F

…40†

P. Qiao et al. / Composites: Part B 31 (2000) 593±609

where NL ˆ 1:0 and k ˆ 1:0. In this example, the dynamic load allowance DLA ˆ 0:20 and AASHTO lane-moment M max ˆ 207:4 kip-ft are used. Solving for the number of winged box beams (stringers) (m) required for this singlespan bridge, we get m ˆ 6:15, and therefore, m ˆ 7 is used, which corresponds to 30 in center-to-center spacing of seven longitudinal stringers. The maximum stress in the stringer becomes sc ˆ 1:48 ksi, which is below the allowable stress of 21.2 ksi [30]. 6. Conclusions As described in this paper, a systematic approach for design analysis of FRP deck/stringer bridge systems is proposed, and the constitutive material properties and micro/macrostructure of a composite system are accounted for in the design. This design approach (Fig. 1) includes the analyses of ply (micromechanics), panel (macromechanics), beam or stringer (mechanics of laminated beam), deck (elastic equivalence model), and ®nally combined deck/ stringer system (series approximation technique). This relatively simple and systematic concept accounts for the complexity of composite materials and geometry of the bridge system. The approximate series solution, which is used to obtain wheel load distribution factors for symmetric and asymmetric loading, is an ef®cient way to analyze and design single-span FRP deck/stringer systems. The present design analysis approach can be ef®ciently used to design bridge systems and also develop new design concepts for single-span FRP deck/stringer bridges. References [1] Henry JA. Deck girders system for highway bridges using ®ber reinforced plastics. MS Thesis, North Carolina State University, 1985. [2] Ahmad SH, Plecnik JM. Transfer of composite technology to design and construction of bridges. US DOT Report, September 1989. [3] Plecnik JM, Azar WA. Structural components, highway bridge deck applications. In: Lee I, Stuart M, editors. International encyclopedia of composites, vol. 6, 1991. p. 430±45. [4] Bakeri B, Sunder SS. Concepts for hybrid FRP bridge deck system. Proceedings of 1st Materials Engineering Congress, ASCE, vol. 2. Denver, CO, 1990. p. 1006±14. [5] Mongi ANK. Theoretical and experimental behavior of FRP ¯oor system. MS Thesis, West Virginia University, Morgantown, WV, 1991. [6] GangaRao HVS, Sotiropoulos SN. Development of FRP bridge superstructural systems. US DOT Report, June 1991. [7] Burnside P, Barbero EJ, Davalos JF, GangaRao HVS. Design optimization of an all-FRP bridge. Proceedings of 38th Int SAMPE Symposium, 1993. [8] Lopez-Anido R, Troutman DL, Busel JP. Fabrication and installation of modular FRP composite bridge deck. Proceedings of Int Composites Expo 0 98, Composites Institute, 1998. p. 4-A (1±6). [9] Lopez-Anido R, Howdyshell PA, Stephenson LD, GangaRao HVS.

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