Asia-Pacific Conference on FRP in Structures (APFIS 2007) S.T. Smith (ed) 2007 International Institute for FRP in Construction FLEXURAL RESPONSE OF FIBER RENFORCED PLASTIC DECKS USING HIGHER-ORDER SHEAR DEFORMABLE PLATE THEORY Y. Kim 1 and J. Lee 2* 1 Currently at Mirae ISE Consultants 2 Department of Architectural Engineering, Sejong University 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Korea. Email: jhlee@sejong.ac.kr ABSTRACT An analytical model was developed to investigate the flexural behavior of a pultruded fiber-reinforced plastic deck of rectangular unit module. The model is based on Higher-Order Shear Deformable Plate Theory (HSDT), and capable of predicting deflection of the deck with arbitrary laminate stacking sequences. To formulate the problem, two-dimensional plate finite element method is employed. Numerical results are obtained for FRP decks under uniformly-distributed loading, addressing the effects of fiber angle and span-to-height ratio and for GFRP decks under the given loading respectively, addressing the effects of span-to-width of GFRP decks. It is found that the present analytical model is accurate and efficient for solving flexural behavior of FRP decks. KEYWORDS FRP deck, pultrusion, classical lamination theory, Higher-order shear deformable plate theory. INTRODUCTION During the past decade, fiber reinforced plastics (FRP) have been increasingly used in civil engineering structures. This is because these materials offer significant advantages over conventional materials due to their chemical and corrosion resistance, and high strength-to-weight and stiffness-to-weight ratios. Among many applications of FRP in civil infrastructures, bridge decks have received a great deal of attention. The FRP decks commercially available at the present time can be classified into two types; construction-sandwich and adhesively bonded pultruded shapes. Sandwich structures have been widely used for applications in the aerospace, marine, and automotive industries, where stiffness and strength requirements must be met with minimum weight. In contrary, adhesively bonded pultruded shapes can be economically produced in continuous lengths using well-established methods such as pultrusion process (Bakis, 2002). In recent years, a number of researchers have investigated the behavior of FRP decks experimentally and analytically. Davalos and his collaborators(davalos et al. 1997) developed a simplified design procedure for cellular FRP bridge decks based on a first-order shear deformation plate solution Aref and Parsons(Aref et al. 1999) developed modular type FRP deck systems, and also proposed an optimum design procedure for FRP decks. (Aref et al. 2000) Recently, Lee et al.(2007) developed pultruded GFRP decks for light-weight vehicles, and investigated flexural performance of the decks. Due to the complexity of FRP materials, the analysis and design tools developed for conventional materials are sometimes not applicable for FRP decks. In this reason, most analysis and design of FRP decks have been carried out using general-purpose finite element programs such as ABAQUS.(2002) These programs are often difficult to use, require specialized training, a considerable amount of computational time and cost and degrees of freedom for structural analysis. In this paper, an analytical plate model based on the Higher-Order Shear Deformable Plate Theory (HSDT) is developed for FRP decks of rectangular unit module. The proposed analytical model can predict the structural behavior of the deck with arbitrary laminate stacking sequences and boundary conditions. THEORETICAL MODEL Kinematics In this section, a FRP deck is modeled as a homogeneous anisotropic plate by applying HSDT assumptions. In order to derive the analytical model for FRP decks, the following assumptions are made in the analysis: 1119
(1) Each unit of the deck is composed of rectangular tubular section. Local deformations of the FRP deck are neglected. (3) Transverse shear deformation is considered as cubic function of the thickness coordinate. Figure 1. Definition of coordinates in thin-walled open sections Based on the HSDT, the displacement fields of the FRP deck can be written as (Reddy, 1984): U x y z u x y z x y z x y z x y 2 3 (,, ) = (, ) + ψ x(, ) + ξx(, ) + ζ x(, ) Vxyz vxy z xy z xy z xy 2 3 (,, ) = (, ) + ψ y(, ) + ξy(, ) + ζ y(, ) W( x, y, z) = w( x, y) The strains associated with small-displacement theory of elasticity are given by: 2 ε = ε + z( κ + z κ ) x x x x ε = ε + z( κ + z κ ) 2 y y y y γ = γ + z( κ + z κ ) 2 xy xy xy xy γ = γ + z κ 2 yz yz yz γ = γ + z κ 2 xz xz xz (1) Constitute Equations for the FRP Deck The representative element is subdivided into two parts: the top and bottom flanges (segment I) and the web plate (segment II). The constitutive equations for the composite deck can now be written by adding the stress resultants in segment I and segment II together as: II N A11 A12 A B11 B12 B E11 E12 E 2 II x 0 A 0 h 12A ε x N A22 A26 B22 B26 E22 E26 0 0 0 0 ε II 2 II N sym. A66 sym. B66 sym. E66 A45 0 x h 12 A45 0 γ x M x D11 D12 D F11 F12 F 0 0 0 0 kx M D 22 D26 F22 F26 0 0 0 0 k M sym. D xy 66 sym. F66 0 0 0 0 kxy Px = H11H12 H 0 0 0 0 kx P H22 H26 0 0 0 0 k y P sym. H xy 66 0 0 0 0 kxy Q yz A44 A45 D44 D γ 45 yz Q xz A45 A55 D45 D γ 55 xz R yz k F44 F 45 yz sym. R ( 2) xz F k 45 F 55 xz (3) where N x, N y, and N xy are axial and in-plane shear forces, M x, M y, and M xy are bending and twisting moments, P x, P y, and P xy are higher-order stress resultants, Q yz, and Q xz are transverse shear forces, R yz and R xz are higherorder shear forces as defined in Reddy (1984). APFIS 2007 1120
NUMERICAL EXAMPLES Effects of Span-to-height and Height-to-thickness Ratios A four-edge simply-supported FRP deck under uniformly distributed loading is considered. The central deflection is compared with the results by general purpose finite element program ABAQUS (2002). In ABAQUS modeling, eight-noded shell elements (Figure 2) are used to represent the surfaces made of laminated composite material. Figure 2. ABAQUS finite element mesh of the FRP deck Figure 3. Sectional profile of the FRP deck The geometry of the FRP deck is given as a/b=1 ; B/H=1 (Figure 3). The following engineering constants are used in the analysis: E 1 /E 2 =26, p 12 =0.3, G 12 /E 2 =0.5, G 13 /E 2 =0.5, G 23 /E 2 =0.5 For convenience, the following nondimensional deflection is used: 3 E2 H 2 w = w 10 4 q0 b In Table 1, the maximum nondimensional central deflection of the FRP decks are calculated by using various plate theories including Classical Lamination Theory (CLT), First-Order Shear Deformable Plate Theory (FSDT), HSDT, and ABAQUS for various span-to-height ratios (a/h). In calculating the deflection, the height-tothickness ratio of the deck is fixed to 8 (H/t=8). The stacking sequences of the flange and the web of the FRP deck are assumed to be unidirectional fiber orientation. As given in Table 1, it is found that the HSDT gives closer result than FSDT or CLT to ABAQUS. For a relatively high span-to-height ratio (a/h=50), the result by HSDT rather overestimates the deflection. For moderate span-to-height ratio (a/h=25), however, the result by HSDT shows excellent agreement with ABAQUS, whereas the FSDT and CLT solutions remarkably underestimate the deflection. For lower span-to-height ratio (a/h=10), HSDT and ABAQUS results show discrepancy up to 35% error. In addition, FSDT and CLT solutions become inadequate in this case. That is, CLT and FSDT become inadequate for modeling of the deck with low span-to-height ratio. In order to investigate the local effects on the flexural behavior of the deck, the deflection of the deck with respect to the height-to-thickness ratio (H/t) is investigated in Table 2. In all computation, a/t is assumed to be 400. Once again, the trend is similar to the case with span-to-height ratio variation. HSDT shows best results among all plate theories for all range of height-to-thickness ratio. As shown in Tables 1 and 2, the solution error by the various theories depends only on the span-to-height ratio of the deck, and seems almost insensitive to the height-to-thickness ratio. For example, the errors between CLT and ABAQUS for a/h=10 become 76% and 77% for H/t=8,40, respectively. In order to investigate the span-to-height ratios of the FRP deck further, nondimensional maximum deflection of the deck with unidirectional fiber direction is shown with respect to the span-to-height ratio of the deck (Figure 4). The results by HSDT agree well with those of ABAQUS for a/h>20. It should be noted here that the FRP deck with very low span-to-height ratio (say, a/h<15) is not practical. Accordingly, proposed HSDT model gives good approximation for the flexural response of FRP deck. Table 1. Comparison of the nondimensional central deflections with respect to span-to-height ratio for a four-edge simply-supported FRP deck under uniformly distributed loading Present a / H H / t ABAQUS CLT FSDT HSDT 50 8 0.99 1.03 1.10 1.05 25 8 0.99 1.13 1.32 1.38 10 8 0.99 1.83 2.74 4.20 APFIS 2007 1121
Table 2. Comparison of the nondimensional central deflections with respect to height-to-thickness ratio for a four-edge simply-supported FRP deck under uniformly distributed loading Present a / H H / t ABAQUS CLT FSDT HSDT 50 8 0.99 1.03 1.10 1.05 25 1.68 1.94 2.37 2.48 10 40 3.82 7.70 12.88 17.10 Effects of Fiber Angle Variation The next example shows the effects of fiber angle variation on the flexural behavior of the deck. The top and bottom flanges of the deck are assumed to be unidirectional fiber orientation, whereas the stacking sequence of the web plate is anti-symmetric angle-ply [ θ / - θ]. The maximum nondimensional deflections with respect to the fiber angle are shown in Figures 5, 6, 7, 8 and 9 for three different span-to-height ratios and two different heightto-thickness ratios. It is seen that the deflection is not much sensitive to the fiber angle change for a high span-to-height ratio. For all the range of fiber angle, CLT, FSDT and HSDT results agree well with ABAQUS solution (Figure 5). For a/h=25 (Figures 6, 7), however, the ABAQUS and the HSDT solutions show some discrepancy as fiber angle becomes off-axis. CLT and FSDT solutions show remarkable error for all the range of fiber orientation. Figures 6 and 7 show similar trend for all the range of fiber angle. For a lower span-to-height ratio (a/h=10), the error becomes larger, yet still HSDT yields better results (Figures 8, 9). ABAQUS solution shows almost symmetric distribution of deflection with respect to 45º, whereas the plate theories show minimum values near θ =20º. Figure 4. Nondimensional deflection with respect to span-to-height ratio(a/h) change for a four-edge simply-supported FRP deck under uniformly distributed loading Figure 5. a/h = 50, H/t=8 Figure 6. a/h = 25, H/t= Nondimensional deflection with respect to the fiber angle change of web for a four-edge simply-supported FRP deck under uniformly distributed loading Figure 7. a/h = 25, H/t = 8 Figure 8. a/h = 10, H/t=40 Figure 9. a/h = 10, H/t = 8 Nondimensional deflection with respect to the fiber angle change of web for a four-edge simply-supported FRP deck under uniformly distributed loading APFIS 2007 1122
Experimental Investigation In this example, the fiber system of the decks was designed by two types of DBT [45/90/-45] series and LT [0/90] series.(figure 10) The dimensions of unit module and fiber architectures are shown in Figure 11. Geometry of Glass Fiber Reinforced Plastic(GFRP) decks, load and boundary condition are performed as Figures 12 and 13. On this study, analytical model, which developed for evaluating the capacity on flexural behavior of GFRP decks and commercial finite element program, ABAQUS analysis, and experiment are performed. As a result, deflection on the deck center is compared. The following lamina properties are used in the analysis: E GPa) = 32.599 E ( GPa) = 5.663 G ( GPa) = 2.306 ν 0.312 1 ( 2 12 12 = Fabric Series Orientation Biaxial & Double Bias LT series 0º 45º Tiaxial DBT series +45º 90º -45º Figure 10. Layer construction(www.iparamaxc.de) Figure 11. Dimensions and stacking sequences of a GFRP deck Analysis case Section(Unit module number) Width(m) Length(m) DBT - 1M LT - 1M 0.3 3 DBT - 2M LT - 2M DBT - 5M LT - 5M Figure 12. Geometry of GFRP decks 0.54 3 1.26 3 a) The GFRP 1 Module and 2 Module deck b) The GFRP 5 Module deck Figure 13. Load and boundary condition of the GFRP decks Table 3 shows the central deflection of a GFRP deck loaded on 50kN. The results of 1 and 2 module on the experiment, ABAQUS and analytical model are almost same. Structural behavior of 1 and 2 module, which length-width rates of deck are big, happen one-action, while 5 module is considered to have errors as behavior on two-action. Also, there are two reasons for happening the errors: firstly, load of 1 and 2 module is the same as figure 13a, and 5 module isn't the same as the errors of 1 and 2 module as loaded on the two positions, which are APFIS 2007 1123
the same load as the automobile wheel on GFRP deck. Secondly, as the length and width of deck are bigger, the errors according to the prior results on the length-height of deck are considered to happen. Figures 14 shows load-deflection curve on the GFRP deck. In case of 1 and 2 module, deflection form is almost same. In 5 module, deflection form is different according to load condition and two-action structural behavior. Table 3. Deflection of GFRP-Decks from 4-point bending experiment, ABAQUS and analytical model Unit Module Fiber system Experiment ABAQUS HSDT 1 Module DBT 57.15 56.70 60. LT - 48.01 47.25 2 Module DBT 29.29 30.67 33.30 LT 26.12 25.00 26.20 5 Module DBT 8.21 9.12 7.87 (DECK) LT 8.15 7.84 6.38 (mm) CONCLUSIONS a) 1 Module b) 2 Module c) 5 Module Figure 14. Load-deflection curve of GFRP A simplified analytical model for a FRP deck was proposed based on Higher-order Shear-Deformable Plate Theory (HSDT). A two-dimensional plate finite element model was developed to formulate the problem. The model is capable of predicting deflection of the deck of arbitrary laminate stacking sequences. The effects of fiber angle, span-to-height ratio and height-to-thickness ratio on the flexural behavior of the FRP decks were investigated. Based on the above analytical developments and numerical results, the proposed analytical model is found to be efficient and accurate for solving flexural behavior of the FRP decks. ACKNOWLEDGMENTS The support of the research reported here by Korea Ministry of Construction and Transportation through Grant KMCT-2003-C103A1040001-00110 is gratefully acknowledged. REFERENCES Bakis, C. E. (2002), Fiber-Reinforced Polymer Composites for Construction-State of the Art Review. Journal of Composite for Construction, ASCE, 6. Salim, H. A., Davalos, J. F., Qiao, P. and Kiger, S. A.(1997) Analysis and design of fiber reinforced plastic composite deck-and-stringer bridges. Composite Structures, 38, 295~307. Aref., A. J., Parsons., I. D. (1999), Design optimization procedures for fiber reinforced plastic bridges. Journal of Engineering Mechanics, ASCE, 125(9), 1040~1047. Aref., A. J., Parsons., I. D. (2000), Design and performance of a modular fiber reinforced plastic bridge. Composites, Part B, 31, 619~628. Lee, J., Kim, Y., Jung, J., and Kosmatka, J. (2007) Experimental characterization of a pultruded GFRP bridge deck for light-weight vehicles, Composite Structures, 80, 141-151. ABAQUS (2002), User's Manual version 6.3, Karlsson and Sorensen, Inc. Pawtucket, R.I. Reddy, J. N. (1993) An Introduction to the Finite Element Method, McGRAW-Hill International Editions. Reddy, J.N., (1984) "A Simple Higher-Order Theory for Laminated Composite Plates, Journal of Applied Mechanics, 51, 745-752. APFIS 2007 1124