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Numerical analysis and experimental validation for static loads of a composite bridge structure
Composite Structures 62 (2003) 235–243 www.elsevier.com/locate/compstruct
Numerical analysis and experimental validation for static loads of a composite bridge structure G. Giannopoulos
a,*
, J. Vantomme a, J. Wastiels b, L. Taerwe
c
a Civil Engineering Department, Royal Military Academy, Avenue de la Renaissance 30, B-1000 Brussels, Belgium Department of Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Department of Structural Engineering, University of Ghent, Magnel Laboratory for Concrete Research, Technologiepark-Zwijnaarde 9, B-9052 Ghent, Belgium b
c
Abstract The development of a new structure is performed in many different phases. The first phase is the conceptual design, which is necessary to give a preliminary shape to the structure. Afterwards the structural analysis phase follows which is the most important part of the development of a new structure. Analytical methods in complex structures are very demanding and sometimes even impossible to be applied. For this reason the numerical methods are entering the stage and give solutions concerning the structural integrity of the structure that is being examined. In the present work the numerical analysis of an IPC pedestrian bridge and the corresponding experimental results are presented. The aim of this work is to provide the scheme of a numerical modeling procedure that will be applied in the future for the dimensioning of similar structures. In addition different types of elements were investigated as well as different sizes of mesh in order to conclude to a model that gives an exact solution taking into account the computational cost. The model is validated by using real test data extracted from the experimental analysis performed on a real prototype under static loading. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Experimental analysis; FEM analysis; IPC; Pedestrian bridge
1. Introduction The use of composites in modern structures is steadily increased. The excellent mechanical properties in combination with the low weight makes these materials very attractive for structural applications. The use of these materials was encountered in first place in the aerospace applications. However, in the last years the application of these materials has been expanded to structural applications where traditional materials have been used. The construction of bridges has been the least years one more domain of structural applications that the use of composites takes place. The type of composites used for such applications are based on a thermoplastic matrix which is reinforced with fibers of carbon or glass leading to a material with su-
*
Corresponding author. Tel.: +32-2-737-64-22; fax: +32-2-737-64-
12. E-mail address: [email protected] (G. Giannopoulos). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00118-1
perior characteristics compared with traditional materials. Such structures are already in service in many places around the world. In UK a FRP bridge is constructed for the passage of the Stroudwater Canal [1] and it has the ability to carry heavy load traffic (trucks). Another example of such application can be found in US (Lockheed Martin Corporation Bridge) [2], as well as in Greece [3]. The common characteristic of the above mentioned structures is that are constructed from FRP pultruded beams. However, apart from the evident advantages of these structures, it is important to mention here the fact that these materials are not fire resistant and in addition this kind of design requires the construction each time of the elements in the right dimensions. This is in contradiction with the requirements for the pedestrian bridge presented here, which should have all the advantages of composite stuctures and in the same time to be fire resistant and modular in order to be applied in different cases with the minimum cost. The work presented here is performed within the framework of a research project for the development of
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a pedestrian bridge using a new material. This material named Inorganic Phosphate Cement (IPC), was developed in Vrije Universiteit Brussel (VUB) and it has similar behavior with traditional concrete. However, it has some advantages compared with traditional concrete. It can be reinforced with glass fibers resulting to a material with superior tensile stiffness, which is the main disadvantage of traditional concrete. More information about this material can be found in another work [4].
2. Description of the structure For the realization of the structure, a truss system from IPC sandwich panels was designed [4,5]; see Fig. 1. The idea behind this design is that the structure should be modular, using the least possible number of different elements. On top of this truss system is a concrete deck. However, the full IPC truss solution is not feasible for many reasons. The buckling of the upper chord panels, the low bending stiffness of the bridge structure and the vibration problems due to the low weight are the reasons that restrict the application of a full IPC truss system. In view of these problems for the top chord of the bridge, a reinforced concrete slab is chosen [6]. This material is able to take up the compressive stresses and to reduce the vibration problems due to the increase of the own weight of the structure. In addition the bending stiffness of the whole structure is increased and the stresses due to concentrated loads can be distributed all over the structure [7,8]. The structure is composed of 3 truss girders. Each girder has a width of 1 meter and a length of 13.344 m. These panels are identical in dimensions (1.05 m of length, 0.5 m of width) but different in composition. The faces of the diagonal panels are constructed using 10 layers of glass mat fibers, while the horizontal panels are constructed using 29 layers of UD fibers. The connection of the panels in order to create the truss system is performed using steel inserts and connection plates.
Fig. 2. Connection system.
Finally the connection of the truss system with the concrete deck is performed using a special system with connection plates that can take the shear stresses that are developed in this part of the structure. More information about the design and construction of the sandwich panels and these connection systems are presented in another work [9,10]. However, in Fig. 2 the system used for the connection of the panels that form the truss system is presented. The design of the IPC-concrete bridge is based on the requirements for pedestrian bridges. The design has been performed according to the requirements of the Belgian standard NBN B 03-101 [11–13]. The requirements for the design are the ability to carry a distributed load of 5 kN/m2 and the ability to carry a concentrated load of 10 kN. This is the Serviceability Limit State loading case. In order to overcome uncertainties for the IPC material, a safety factor of 3 is used for the ultimate strength of this material. In the ultimate limit state loading, the own weight of the structure is multiplied by 1.35, while the pressure and the concentrated load are multiplied by a factor of 1.5. In this loading case the deformation is not of particular importance. The admissible deflection for such a structure is between 1/200 and 1/300 of its length.
3. Finite element modeling of the structure 3.1. General description of the model
Fig. 1. IPC pedestrian bridge.
During the design phase of the bridge, both numerical and analytical studies have been performed; initially, simplified 2D models have been developed, together with analytical studies [4,10]. The analytical studies were performed using simple 2D-truss theory. This kind of analyses, are very effective in terms of computational time and cost in order to perform the basic dimensioning of the structure. However, such methods cannot give
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detailed results considering the stress field of the structure. For this reason it was judged that a more detailed analysis taking into account more parameters had to be elaborated. The aim of this work is to give a more detailed overview of the strain field on the different parts of the structure. In addition a more realistic prediction of the overall displacement of the structure can be derived, since more parameters are incorporated than in previous models [5,10]. As a consequence the aim of the modeling procedure presented here is to provide a numerical model for the evaluation of the whole structure and also to be used as the basis for designing of future structures with similar characteristics. Due to the geometric and loading symmetry of this type of structure, it was decided that the modeling of the 1/3rd of the structure is enough. The finite element model that was created for the numerical analysis is presented in Fig. 3. The numerical analysis was performed in ANSYS 5.7 finite element software. In order to reduce as much as possible the computational time, the number of elements introduced was as small as possible, taking into account the convergence of the analysis and the accuracy of the results. In order to reassure that the number of elements is sufficient a convergence analysis was performed. Different element sizes were implemented in all parts of the structure. The element size presented here is the optimum in order to have the maximum accuracy with the minimal computational cost. In the following paragraphs a detailed analysis of each part of the structure is presented, concerning the finite element formulation. 3.2. Concrete deck description The concrete deck of the bridge has a length of 13344 mm and a thickness of 120 mm. It is obvious that the length to thickness ratio is high enough and permits the thin plate formulation to be implemented. For this reason, shell elements with bending stiffness can be used. For the analysis of the concrete deck 960 shell 63 [14] elements were implemented. Shell 63 is a 4-node (when it is used in quadrilateral shape) shell element with linear shape functions. The nodes are placed in the middle of
Fig. 3. 3D Model used for the numerical analysis of the bridge.
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the thickness of the element and the stress distribution is varying linearly from the bottom to the top surface of the element. It was decided that this kind of element is well suited in order to model the bending behavior of the concrete deck. However, in the framework of the convergence analysis different solutions were implemented. Apart from the evident solution of increasing the number of elements, an element of higher order was used for the modeling of the concrete deck. It was considered that Shell 93 [14] element was suitable. This is an 8-node element with quadrilateral shape functions. Using this type of element, a more detailed model was elaborated; however, the difference in the results of both models appeared to be negligible. The mesh size shown here and the type of element (Shell 63) are thus sufficient for this analysis. The material modeling of the deck was performed using a linear isotropic material model with modulus of elasticity 30 GPa and Poisson ratio of 0.2. The density of this material was assumed to be 2500 kg/m3 . Since the analysis is not focused on the behavior of the concrete, its behavior when cracks occur is not of particular interest and thus no special formulation is required. 3.3. IPC sandwich panel truss system For the modeling of the diagonal members of the truss system, 864 shell 91 [14] elements were used. Shell 91 is an 8-node shell element. This element was selected because it incorporates some special characteristics. A very important characteristic of this element is that the position of the nodes can be changed. It is possible to put the nodes in the upper or bottom surface of the structure. This is useful when there is a connection of two surfaces in one point in order to have correct geometrical modeling of the structure. This element supports the formulation of thick sandwich structures like the ones that are used for the construction of the truss system. However, the only difference occurs only in the bending behavior of such structures. In the bridge presented here the panels carry only compressive and tensile loads, and thus such a parameter does not have any impact in the results. Since the diagonal panels are subjected only to tensile or compressive loading it is decided that the selected element size is adequate in order to have accurate results. This type of element is accurate in case of tensile or compressive loading, but when the shear stresses through the thickness are very high, as it happens in thick sandwich structures in bending, the accuracy of this element is rather limited. [15]. The thickness of the faces of the sandwich panel should always be less than 20% of its overall thickness. In this case a convergence analysis was also performed. The order of the element remained the same but the number of the elements was altered. Elements of half
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dimensions (four time more elements) were used. As it was expected the results remained the same. This is due to the fact that the panels are axially forced and thus no artificial stiffening exists due to the mesh size as it would be in the case that these panels were forced in bending. 3.4. Steel plates and connection system For the modeling of the connection plates of the connection system, shell elements have been used. These plates are subjected to tensile and compressive loading. The main advantage of modeling the connection plates is that the rotation of the sandwich panels is preserved and as a consequence no bending moments are present in these panels. Even in the real structure, it is necessary to avoid artificial bending moments in this part of the structure since the existence of the metal inserts increases the danger of failure in the connection area of the metal insert and the IPC sandwich panel (see Fig. 2). Special attention was given to the correct modeling of the connection area. The metal insert that is fixed in the edge of the panels is responsible for the distribution of the stresses over the whole width of the panel. For this reason, the edge nodes of the panels are obliged to have the same deformation in order to avoid stress concentration in one node. As shown in Fig. 2, a straight link element connects the metal insert with the connection plates. It is not possible to have a precise modeling of this region, since the sandwich panel is modeled using shell elements. The number of elements required to model precisely this region is very high and it is thus inconvenient to have such a model in terms of computation time. For these reasons the modeling shown in Fig. 4 was adopted. In the model presented here the connection plate was modeled as a rectangular plate without holes. The four panels are connected at the edges of the plate, which leads to a much different stress field from the real one. In order to have the real field a detailed modeling of the
Fig. 4. Detail of the connection system modeling.
area should be performed. However, in such a generalized 3D model this is not possible. Experiments in an actual connection system with the application of the Digital Image Correlation method [16] made possible to have the full deformation field of this area. It was discovered that the whole system of the connection system undergoes deformations that cannot be neglected. In addition it was also obvious that between the bolts and the skins of the sandwich panels a positioning is taking place. A link element from the Ansys Finite Element Library may be used to model this part of the structure, in order to incorporate all these paremeters. However, this is not possible to be done in this model due to instability in the solution. There is at least one insufficiently restricted degree of freedom in the whole truss system and the solution cannot be performed. Another solution could be the use of a spring with an equivalent stiffness. However, the same problem raises. Since these elements have no bending stiffness the structure performs as a mechanism and thus the solution is not possible. However, an element with bending stiffness should not be used, in order to avoid bending in the panels. It is very important to mention here that in the real structure there is also one redundant degree of freedom in the truss system and thus it can behave as a mechanism. However, this does not take place due to the friction that exists in area of contact between the hole and the bolt. This of course imposes a bending moment in the panels but only in some of them. If the rotation is restricted in one connection plate due to friction, then the redundant degree of freedom does not exist any more and thus the truss system is no longer a mechanism. It is obvious that in a generalized 3D model this behavior cannot be modeled, since it is totally stochastic. Due to the above mentioned reasons it was chosen that the sandwich panel and the connection plate had to be in contact. In order to incorporate the behavior of the connection system, the Combin 7 [14] element was chosen. With this type of element it is possible to model the revolute joint that exists between the connection plate and the panel. The sandwich panel can rotate free and no parasite bending moments exist. However, the big advantage of this element is that the flexural behavior of the revolute joint can be modeled. This is performed by introducing the stiffness of a spring that connects the two nodes that form the revolute joint. In order to incorporate the deformation of the whole connection system data from DIC [16] tests were used. Due to the fact that the input of the Combin 7 element can only be linear, a best-fit procedure on the data of the DIC analysis was performed, in order to extract one value for the stiffness of the connection system. This kind of formulation gives good results for the overall deformation of the structure until the service load. From this point further, the deformations due to the posi-
G. Giannopoulos et al. / Composite Structures 62 (2003) 235–243 Table 1 Elements used for the formulation of the numerical model Part
No. of elements
Type
Deck Diagonal Members Horizontal Members Connection Plates
872 960 432 234
Shell Shell Shell Shell
63 91 91 63
tioning of the bolts in the skins of the sandwich panels, is increasing and becomes a very important deformation mechanism. Table 1 presents the number of the different types of elements that were used for the modeling of the bridge.
4. Material modeling of IPC One of the most demanding parameters during the finite element modeling of a structure is the material modeling. This is even more critical when a new material is introduced and that exhibits a special behavior. The behavior of the IPC material in tension is not linear, and for this reason a bilinear material model was used. However, this is not enough. The glass fiber reinforced IPC material exhibits a different behavior in compression. For the diagonal panels of the truss system that are subjected to compressive forces, a linear material model should be adopted. Due to the small amount of fibers in the IPC/mat reinforced material the modulus of elasticity in the compressive area was assumed to be the modulus of elasticity of the IPC matrix, which is 18 GPa. Experiments performed in specimens showed that this is true. The Poisson ratio is 0.3. The Young moduli that were used for the modeling of the IPC/mat reinforced material in tension, were extracted from numerous experiments on the actual material that was used in the construction of the bridge and from experiments that had been already presented in the past [15]. In Table 2 the material properties used for the IPC/mat reinforced panels are presented. Using this procedure, it was possible to have more representative results for the properties of the panels. Although the fabrication procedure of the panels was carried out following a strict procedure, it is sure that the properties may vary from panel to panel, due to the fact that the fabrication of the panels was based on the hand lay-up method. It is very important to mention here, that for the case of the UD reinforced panels, an anisotropic material model had to be used since the properties are not the
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Table 3 Material properties for the UD reinforced panels (GPa) Material
Ey
Ez
Gxy
Gyz
Gxz
UD
18
18
6
6
6
Table 4 Modulus of elasticity for the UD reinforced panels in the fibers direction (GPA) Material
E1
E2
UD
24
9.5
same in all directions. Due to the Poisson effect, the UD reinforced panels, which are loaded in tension, are subjected to compressive stresses in the transverse direction. In this direction the effect of the fibers is negligible and it can be assumed that the matrix is carrying all the stresses. For this reason the modulus of elasticity of the UD panel in this direction is assumed to be the one of the matrix, which behaves linearly in compression [15]. Due to the loading of the structure, all the members of the truss structure are loaded in tension and compression; the importance of the shear properties of the material are thus limited. From experiments it has also been shown that the difference in the shear properties between the pre-cracking and the post-cracking zone is very limited [15]. The properties that were used for the analysis of the UD reinforced IPC panels, are presented in Table 3. In Table 2 the two modulus of elasticity are the modulus of the two zones for the mat reinforced IPC material and in the Table 4 the corresponding modulus for the UD reinforced IPC material are presented. In pre-cracking zone there is a correspondence between the shear properties from experiments [15] and from HookÕs law. However, this is not the case for the post cracking zone: the shear modulus is reduced but not as much as the modulus of elasticity. In this case there is a difference between the properties extracted from experiments and the shear modulus extracted from HookÕs law. This problem is the result of the failure procedure that takes place in the IPC matrix. The small cracks that exist in the post-cracking zone play a significant role to the shear behavior of the IPC/glass fiber reinforced material. Until now only experimental values exist. No particular theoretical model for the prediction and explanation of these values exists.
5. Validation of the numerical modeling Table 2 Material properties for the IPC/mat reinforced material Material
E1 (GPa)
E2 (GPa)
V
IPC/mat
10
2.1
0.3
5.1. Construction and loading of the prototype In order to validate the finite element model, a prototype was realized and tested.
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According to the design guidelines, the structure should be tested with a concentrated load in the middle and a distributed load all over the concrete deck. However, the requirement for distributed loading is very difficult to be accomplished in the environment of the lab. For this reason the loading was simulated using four-point loading. This was imposed by the available experimental equipment. The load on each point was identical. 5.2. Experimental procedure and comparison with finite element analysis results 5.2.1. Overall displacement The most important parameter that had to be measured during the experimental analysis was the vertical displacement of the structure, especially until service load taking under consideration the requirement of the guidelines for a maximum permissible deformation. This measurement also was necessary in order to arrive to some conclusions about the validity of the numerical model. In Fig. 5 the vertical displacement of the structure is presented. It is obvious that the displacement of the real structure is diverging from the one calculated from the finite element analysis. This is due to the increased plasticity that is introduced in the area of the connection between the metal insert and the skins of the sandwich panels. 5.2.2. Deformation of the horizontal sandwich panels The displacement of the structure is of critical importance in order to comply with the design guidelines. The value of the overall displacement gives an overview of the effectiveness of the design, but in order to perform a detailed analysis, the evaluation of the strain field for
each sandwich panel of the structure is necessary. It is obvious that the most important part of the structure is the horizontal panels of the truss system. The failure of one of these panels will force the structure to collapse. In the Fig. 6 a comparison between the finite element and the experimental results concerning the strain in the direction of the force action is presented for the most loaded horizontal panel. As shown Fig. 6 the agreement in the deformation values between the finite element and experimental analysis proves that the modeling of the structure does not lead to artificial stiffening. In addition the stress for which the transition from the pre-cracking to the post-cracking zone is taking place is the same for both experiment and analysis. 5.2.3. Deformation of the diagonal panels of the truss system Another important parameter of the truss system is the behavior of the diagonal panels that are forced in compression. The two end panels are forced in compression while the next panels are forced in tension. This alternating type of loading is preserved throughout the whole structure. In Fig. 7 a comparison between the finite element and the experimental results is presented for the most loaded panel. From this figure it is obvious that in this case there is also a very good agreement between the finite element calculations and the experimental analysis. It is important to mention here that the influence of the deformation of the diagonal panels in the overall displacement of the structure is very limited. However, the correct modeling of this part of the structure is necessary in order to be able to predict a failure due to excessive deformation in some parts of the panels. The deformation of the diagonal panels in tension was also examined since the resistance and the modulus
70000 Finite Element Experiment
60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
10
20
30
40
50
60
70
80
Displacement (mm) Fig. 5. Displacement of the bridge: comparison between the experimental and finite element analysis.
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70000 Experiment Finite Element 60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
200
400
600
800
1000
1200
1400
1600
1800
Strain (µm)
Fig. 6. Deformation of the most loaded horizontal panel.
70000
60000 Finite Element Experiment
Force (Nt)
50000
40000
30000
20000
10000
0 -800
-700
-600
-500
-400
-300
-200
-100
0
Strain (µm)
Fig. 7. Deformation of the most loaded diagonal panel in compression.
of elasticity for the IPC in tension is limited as it is already shown in the previous chapters. In Fig. 8 the deformation of the diagonal panel in tension for both finite element and experimental analysis is presented. As shown in Fig. 8 there is a mismatch between the deformation values predicted by the finite element model and the values achieved by the experimental analysis. However, the deformations are higher in the finite element model. It is obvious that in the modeling of the sandwich panels no artificial stiffening is introduced. As it has been mentioned the behavior of the IPC was derived from numerous experiments performed in specimens. Examining the graphs in detail it is easy to notice that in both cases there is a transition point where
the IPC passes from the pre-cracking to the postcracking zone. However, this is taking place in lower force values in the finite element than in the real case, which means that the improved techniques introduced for the construction of these sandwich panels leaded to a higher transition stress level. In addition the stiffness of these panels in the post-cracking zone seems to be higher in the real case. The issue that is raised directly is what is the influence of this behavior on the stress levels of the other panels. In order to give an answer to this issue more analyses were performed changing the modulus of elasticity of these panels in the pre-cracking zone. As it was expected the strain level on tensile loaded diagonal panels was influenced while in the same time the strain levels on the diagonal panels loaded in compression and
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G. Giannopoulos et al. / Composite Structures 62 (2003) 235–243 70000 Finite Element Experiment 60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
200
400
600
800
1000
1200
Strain (µm)
Fig. 8. Deformation of a diagonal panel in tension.
on the horizontal panels remained almost the same. This is due to the fact that the diagonal panels in tension are very weak compared with the other panels and as a consequence any redistribution of forces due to the change of their properties has small influence on the other parts of the structure. It is thus concluded that a more detailed analysis of the IPC using new specimens is necessary in order to introduce these values in a new model.
6. Conclusions In the present work the numerical modeling and its evaluation using the results derived from experiments to a real prototype is presented. A finite element model has been created in order to predict the behavior of the structure and use it in the future for the design of similar structures. The model does not experience artificial stiffening and thus the strains predicted in the different parts of the structure are not smaller than in the real case. This is usually the danger of the results of a numerical model that leads to wrong conclusions and failure of parts that are designed to take a specific load. However, the overall displacement of the structure seems to be derived by a model that is artificially stiffened. This is due to the fact that parameters, like the positioning of the bolts inside the sandwich skins, are not incorporated in the finite element model. A thorough examination of the results shows that the modeling of the elements of the structure is adequate for this type of analysis. The deformations predicted are in good agreement with the experiments and this type of modeling can thus be used in the future for the prediction
of the behavior of similar structures. The assumption of the thin plate for the concrete deck is valid as expected due to the high length/thickness ratio. The size and the type of the elements used for the modeling of the panels is also adequate and this is proved by the agreement that exists in the values of the deformation between the finite element analysis and the experimental procedure. As far as the design of the structure is concerned, according to the guidelines, this structure fulfils all the necessary criteria in order to be put in service. In addition the results of this work show clearly that use of IPC as a structural material can be extended to such applications, leading to structures with significant advantages over the traditional concrete structures. However, its long-term behavior under severe environmental conditions is still to be investigated.
References [1] Head PR. Advanced composites in civil engineering a critical overview at this high interest, low use stage of development. Adv Compos Mater Bridges Struct 1996:3–13. [2] Dumlao C, Lauraitis K, Abrahamson E, Hurlbut B, Jacoby M, Miller A, et al. Demonstration low-cost modular composite highway bridge. Proceedings of the First International Conference on Composites in Infrastructure 1996. p. 1141–55. [3] Kostopoulos V, Markopoulos YP, Vlachos DE, Katerelos D, Galiotis C. A motorway composite bridge made of glass/polyester pultruded elements. Tenth European Conference on Composite Materials (ECCM-10), 2002, Brugge, Belgium. [4] De Roover C, Vantomme J, Wastiels J, Kroes K, Taerwe L, Blontrock H. A new non alkaline cement material reinforced by glass fibers (IPC) for the construction of bridges, Third International Conference on Advanced Composite Materials in Bridge and Structures ACMBS III, Otawa, Canada, 2000. p. 429–36. [5] De Roover C, Vantomme J, Wastiels J, Kroes K, Cuypers H, Taerwe L, Blontrock H. Modeling of a modular pedestrian bridge
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composed of a concrete deck and a truss girder with IPC sandwich panels, Fifth International Conference on Computational Structures Technology CST2000, Leuven, Belgium, 2000. p. 327–36. Triantafillou TC, Meier U. Innovative design of FRP combined with concrete. In: Neale KW, Labossiere P (Eds.), Advanced composite materials in bridges and structures (ACMBS III), Milan, 1994. p. 491–500. Genders C. Eerste composietbrug in Nederland, Cement en Beton, 6 June 1997. p. 16–9. Composite cooperation. Bridge design and engineering. 1997. p. 76. Blontrock H, Taerwe L, Nurchi A, Vantomme J, De Roover C, Wastiels J, et al. Testing of a dowel connection for a bridge with a concrete deck and a sandwich panel truss structure. Connections between concrete and steel, Rilem Symposium, Stuttgart, September 2001.
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[10] De Roover C, Vantomme J, Wastiels J, Croes K, Taerwe L, Blontrock H. Modular pedestrian bridge with concrete deck and IPC truss girder. Engineering structures, in press. [11] Eurocode 1: Basis of design and actions on structures. Brussels: Institut Belge de Normalisation (IBN). [12] Eurocode 2: Design of concrete structure. Brussels: Institut Belge de Normalisation (IBN). [13] Eurocode 3: Design of steel structures. Brussels: Institut Belge de Normalisation (IBN). [14] Ansys Theory Reference. [15] Cuypers H. Analysis and design of sandwich panels with brittle matrix composite faces for building applications. Ph D Thesis, November 2001. [16] De Roover C, Vantomme J, Wastiels J. Deformation analysis of a modular connection system by digital image correlation. Experimental Techniques, 2002.
Numerical analysis and experimental validation for static loads of a composite bridge structure G. Giannopoulos
a,*
, J. Vantomme a, J. Wastiels b, L. Taerwe
c
a Civil Engineering Department, Royal Military Academy, Avenue de la Renaissance 30, B-1000 Brussels, Belgium Department of Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Department of Structural Engineering, University of Ghent, Magnel Laboratory for Concrete Research, Technologiepark-Zwijnaarde 9, B-9052 Ghent, Belgium b
c
Abstract The development of a new structure is performed in many different phases. The first phase is the conceptual design, which is necessary to give a preliminary shape to the structure. Afterwards the structural analysis phase follows which is the most important part of the development of a new structure. Analytical methods in complex structures are very demanding and sometimes even impossible to be applied. For this reason the numerical methods are entering the stage and give solutions concerning the structural integrity of the structure that is being examined. In the present work the numerical analysis of an IPC pedestrian bridge and the corresponding experimental results are presented. The aim of this work is to provide the scheme of a numerical modeling procedure that will be applied in the future for the dimensioning of similar structures. In addition different types of elements were investigated as well as different sizes of mesh in order to conclude to a model that gives an exact solution taking into account the computational cost. The model is validated by using real test data extracted from the experimental analysis performed on a real prototype under static loading. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Experimental analysis; FEM analysis; IPC; Pedestrian bridge
1. Introduction The use of composites in modern structures is steadily increased. The excellent mechanical properties in combination with the low weight makes these materials very attractive for structural applications. The use of these materials was encountered in first place in the aerospace applications. However, in the last years the application of these materials has been expanded to structural applications where traditional materials have been used. The construction of bridges has been the least years one more domain of structural applications that the use of composites takes place. The type of composites used for such applications are based on a thermoplastic matrix which is reinforced with fibers of carbon or glass leading to a material with su-
*
Corresponding author. Tel.: +32-2-737-64-22; fax: +32-2-737-64-
12. E-mail address: [email protected] (G. Giannopoulos). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00118-1
perior characteristics compared with traditional materials. Such structures are already in service in many places around the world. In UK a FRP bridge is constructed for the passage of the Stroudwater Canal [1] and it has the ability to carry heavy load traffic (trucks). Another example of such application can be found in US (Lockheed Martin Corporation Bridge) [2], as well as in Greece [3]. The common characteristic of the above mentioned structures is that are constructed from FRP pultruded beams. However, apart from the evident advantages of these structures, it is important to mention here the fact that these materials are not fire resistant and in addition this kind of design requires the construction each time of the elements in the right dimensions. This is in contradiction with the requirements for the pedestrian bridge presented here, which should have all the advantages of composite stuctures and in the same time to be fire resistant and modular in order to be applied in different cases with the minimum cost. The work presented here is performed within the framework of a research project for the development of
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a pedestrian bridge using a new material. This material named Inorganic Phosphate Cement (IPC), was developed in Vrije Universiteit Brussel (VUB) and it has similar behavior with traditional concrete. However, it has some advantages compared with traditional concrete. It can be reinforced with glass fibers resulting to a material with superior tensile stiffness, which is the main disadvantage of traditional concrete. More information about this material can be found in another work [4].
2. Description of the structure For the realization of the structure, a truss system from IPC sandwich panels was designed [4,5]; see Fig. 1. The idea behind this design is that the structure should be modular, using the least possible number of different elements. On top of this truss system is a concrete deck. However, the full IPC truss solution is not feasible for many reasons. The buckling of the upper chord panels, the low bending stiffness of the bridge structure and the vibration problems due to the low weight are the reasons that restrict the application of a full IPC truss system. In view of these problems for the top chord of the bridge, a reinforced concrete slab is chosen [6]. This material is able to take up the compressive stresses and to reduce the vibration problems due to the increase of the own weight of the structure. In addition the bending stiffness of the whole structure is increased and the stresses due to concentrated loads can be distributed all over the structure [7,8]. The structure is composed of 3 truss girders. Each girder has a width of 1 meter and a length of 13.344 m. These panels are identical in dimensions (1.05 m of length, 0.5 m of width) but different in composition. The faces of the diagonal panels are constructed using 10 layers of glass mat fibers, while the horizontal panels are constructed using 29 layers of UD fibers. The connection of the panels in order to create the truss system is performed using steel inserts and connection plates.
Fig. 2. Connection system.
Finally the connection of the truss system with the concrete deck is performed using a special system with connection plates that can take the shear stresses that are developed in this part of the structure. More information about the design and construction of the sandwich panels and these connection systems are presented in another work [9,10]. However, in Fig. 2 the system used for the connection of the panels that form the truss system is presented. The design of the IPC-concrete bridge is based on the requirements for pedestrian bridges. The design has been performed according to the requirements of the Belgian standard NBN B 03-101 [11–13]. The requirements for the design are the ability to carry a distributed load of 5 kN/m2 and the ability to carry a concentrated load of 10 kN. This is the Serviceability Limit State loading case. In order to overcome uncertainties for the IPC material, a safety factor of 3 is used for the ultimate strength of this material. In the ultimate limit state loading, the own weight of the structure is multiplied by 1.35, while the pressure and the concentrated load are multiplied by a factor of 1.5. In this loading case the deformation is not of particular importance. The admissible deflection for such a structure is between 1/200 and 1/300 of its length.
3. Finite element modeling of the structure 3.1. General description of the model
Fig. 1. IPC pedestrian bridge.
During the design phase of the bridge, both numerical and analytical studies have been performed; initially, simplified 2D models have been developed, together with analytical studies [4,10]. The analytical studies were performed using simple 2D-truss theory. This kind of analyses, are very effective in terms of computational time and cost in order to perform the basic dimensioning of the structure. However, such methods cannot give
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detailed results considering the stress field of the structure. For this reason it was judged that a more detailed analysis taking into account more parameters had to be elaborated. The aim of this work is to give a more detailed overview of the strain field on the different parts of the structure. In addition a more realistic prediction of the overall displacement of the structure can be derived, since more parameters are incorporated than in previous models [5,10]. As a consequence the aim of the modeling procedure presented here is to provide a numerical model for the evaluation of the whole structure and also to be used as the basis for designing of future structures with similar characteristics. Due to the geometric and loading symmetry of this type of structure, it was decided that the modeling of the 1/3rd of the structure is enough. The finite element model that was created for the numerical analysis is presented in Fig. 3. The numerical analysis was performed in ANSYS 5.7 finite element software. In order to reduce as much as possible the computational time, the number of elements introduced was as small as possible, taking into account the convergence of the analysis and the accuracy of the results. In order to reassure that the number of elements is sufficient a convergence analysis was performed. Different element sizes were implemented in all parts of the structure. The element size presented here is the optimum in order to have the maximum accuracy with the minimal computational cost. In the following paragraphs a detailed analysis of each part of the structure is presented, concerning the finite element formulation. 3.2. Concrete deck description The concrete deck of the bridge has a length of 13344 mm and a thickness of 120 mm. It is obvious that the length to thickness ratio is high enough and permits the thin plate formulation to be implemented. For this reason, shell elements with bending stiffness can be used. For the analysis of the concrete deck 960 shell 63 [14] elements were implemented. Shell 63 is a 4-node (when it is used in quadrilateral shape) shell element with linear shape functions. The nodes are placed in the middle of
Fig. 3. 3D Model used for the numerical analysis of the bridge.
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the thickness of the element and the stress distribution is varying linearly from the bottom to the top surface of the element. It was decided that this kind of element is well suited in order to model the bending behavior of the concrete deck. However, in the framework of the convergence analysis different solutions were implemented. Apart from the evident solution of increasing the number of elements, an element of higher order was used for the modeling of the concrete deck. It was considered that Shell 93 [14] element was suitable. This is an 8-node element with quadrilateral shape functions. Using this type of element, a more detailed model was elaborated; however, the difference in the results of both models appeared to be negligible. The mesh size shown here and the type of element (Shell 63) are thus sufficient for this analysis. The material modeling of the deck was performed using a linear isotropic material model with modulus of elasticity 30 GPa and Poisson ratio of 0.2. The density of this material was assumed to be 2500 kg/m3 . Since the analysis is not focused on the behavior of the concrete, its behavior when cracks occur is not of particular interest and thus no special formulation is required. 3.3. IPC sandwich panel truss system For the modeling of the diagonal members of the truss system, 864 shell 91 [14] elements were used. Shell 91 is an 8-node shell element. This element was selected because it incorporates some special characteristics. A very important characteristic of this element is that the position of the nodes can be changed. It is possible to put the nodes in the upper or bottom surface of the structure. This is useful when there is a connection of two surfaces in one point in order to have correct geometrical modeling of the structure. This element supports the formulation of thick sandwich structures like the ones that are used for the construction of the truss system. However, the only difference occurs only in the bending behavior of such structures. In the bridge presented here the panels carry only compressive and tensile loads, and thus such a parameter does not have any impact in the results. Since the diagonal panels are subjected only to tensile or compressive loading it is decided that the selected element size is adequate in order to have accurate results. This type of element is accurate in case of tensile or compressive loading, but when the shear stresses through the thickness are very high, as it happens in thick sandwich structures in bending, the accuracy of this element is rather limited. [15]. The thickness of the faces of the sandwich panel should always be less than 20% of its overall thickness. In this case a convergence analysis was also performed. The order of the element remained the same but the number of the elements was altered. Elements of half
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dimensions (four time more elements) were used. As it was expected the results remained the same. This is due to the fact that the panels are axially forced and thus no artificial stiffening exists due to the mesh size as it would be in the case that these panels were forced in bending. 3.4. Steel plates and connection system For the modeling of the connection plates of the connection system, shell elements have been used. These plates are subjected to tensile and compressive loading. The main advantage of modeling the connection plates is that the rotation of the sandwich panels is preserved and as a consequence no bending moments are present in these panels. Even in the real structure, it is necessary to avoid artificial bending moments in this part of the structure since the existence of the metal inserts increases the danger of failure in the connection area of the metal insert and the IPC sandwich panel (see Fig. 2). Special attention was given to the correct modeling of the connection area. The metal insert that is fixed in the edge of the panels is responsible for the distribution of the stresses over the whole width of the panel. For this reason, the edge nodes of the panels are obliged to have the same deformation in order to avoid stress concentration in one node. As shown in Fig. 2, a straight link element connects the metal insert with the connection plates. It is not possible to have a precise modeling of this region, since the sandwich panel is modeled using shell elements. The number of elements required to model precisely this region is very high and it is thus inconvenient to have such a model in terms of computation time. For these reasons the modeling shown in Fig. 4 was adopted. In the model presented here the connection plate was modeled as a rectangular plate without holes. The four panels are connected at the edges of the plate, which leads to a much different stress field from the real one. In order to have the real field a detailed modeling of the
Fig. 4. Detail of the connection system modeling.
area should be performed. However, in such a generalized 3D model this is not possible. Experiments in an actual connection system with the application of the Digital Image Correlation method [16] made possible to have the full deformation field of this area. It was discovered that the whole system of the connection system undergoes deformations that cannot be neglected. In addition it was also obvious that between the bolts and the skins of the sandwich panels a positioning is taking place. A link element from the Ansys Finite Element Library may be used to model this part of the structure, in order to incorporate all these paremeters. However, this is not possible to be done in this model due to instability in the solution. There is at least one insufficiently restricted degree of freedom in the whole truss system and the solution cannot be performed. Another solution could be the use of a spring with an equivalent stiffness. However, the same problem raises. Since these elements have no bending stiffness the structure performs as a mechanism and thus the solution is not possible. However, an element with bending stiffness should not be used, in order to avoid bending in the panels. It is very important to mention here that in the real structure there is also one redundant degree of freedom in the truss system and thus it can behave as a mechanism. However, this does not take place due to the friction that exists in area of contact between the hole and the bolt. This of course imposes a bending moment in the panels but only in some of them. If the rotation is restricted in one connection plate due to friction, then the redundant degree of freedom does not exist any more and thus the truss system is no longer a mechanism. It is obvious that in a generalized 3D model this behavior cannot be modeled, since it is totally stochastic. Due to the above mentioned reasons it was chosen that the sandwich panel and the connection plate had to be in contact. In order to incorporate the behavior of the connection system, the Combin 7 [14] element was chosen. With this type of element it is possible to model the revolute joint that exists between the connection plate and the panel. The sandwich panel can rotate free and no parasite bending moments exist. However, the big advantage of this element is that the flexural behavior of the revolute joint can be modeled. This is performed by introducing the stiffness of a spring that connects the two nodes that form the revolute joint. In order to incorporate the deformation of the whole connection system data from DIC [16] tests were used. Due to the fact that the input of the Combin 7 element can only be linear, a best-fit procedure on the data of the DIC analysis was performed, in order to extract one value for the stiffness of the connection system. This kind of formulation gives good results for the overall deformation of the structure until the service load. From this point further, the deformations due to the posi-
G. Giannopoulos et al. / Composite Structures 62 (2003) 235–243 Table 1 Elements used for the formulation of the numerical model Part
No. of elements
Type
Deck Diagonal Members Horizontal Members Connection Plates
872 960 432 234
Shell Shell Shell Shell
63 91 91 63
tioning of the bolts in the skins of the sandwich panels, is increasing and becomes a very important deformation mechanism. Table 1 presents the number of the different types of elements that were used for the modeling of the bridge.
4. Material modeling of IPC One of the most demanding parameters during the finite element modeling of a structure is the material modeling. This is even more critical when a new material is introduced and that exhibits a special behavior. The behavior of the IPC material in tension is not linear, and for this reason a bilinear material model was used. However, this is not enough. The glass fiber reinforced IPC material exhibits a different behavior in compression. For the diagonal panels of the truss system that are subjected to compressive forces, a linear material model should be adopted. Due to the small amount of fibers in the IPC/mat reinforced material the modulus of elasticity in the compressive area was assumed to be the modulus of elasticity of the IPC matrix, which is 18 GPa. Experiments performed in specimens showed that this is true. The Poisson ratio is 0.3. The Young moduli that were used for the modeling of the IPC/mat reinforced material in tension, were extracted from numerous experiments on the actual material that was used in the construction of the bridge and from experiments that had been already presented in the past [15]. In Table 2 the material properties used for the IPC/mat reinforced panels are presented. Using this procedure, it was possible to have more representative results for the properties of the panels. Although the fabrication procedure of the panels was carried out following a strict procedure, it is sure that the properties may vary from panel to panel, due to the fact that the fabrication of the panels was based on the hand lay-up method. It is very important to mention here, that for the case of the UD reinforced panels, an anisotropic material model had to be used since the properties are not the
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Table 3 Material properties for the UD reinforced panels (GPa) Material
Ey
Ez
Gxy
Gyz
Gxz
UD
18
18
6
6
6
Table 4 Modulus of elasticity for the UD reinforced panels in the fibers direction (GPA) Material
E1
E2
UD
24
9.5
same in all directions. Due to the Poisson effect, the UD reinforced panels, which are loaded in tension, are subjected to compressive stresses in the transverse direction. In this direction the effect of the fibers is negligible and it can be assumed that the matrix is carrying all the stresses. For this reason the modulus of elasticity of the UD panel in this direction is assumed to be the one of the matrix, which behaves linearly in compression [15]. Due to the loading of the structure, all the members of the truss structure are loaded in tension and compression; the importance of the shear properties of the material are thus limited. From experiments it has also been shown that the difference in the shear properties between the pre-cracking and the post-cracking zone is very limited [15]. The properties that were used for the analysis of the UD reinforced IPC panels, are presented in Table 3. In Table 2 the two modulus of elasticity are the modulus of the two zones for the mat reinforced IPC material and in the Table 4 the corresponding modulus for the UD reinforced IPC material are presented. In pre-cracking zone there is a correspondence between the shear properties from experiments [15] and from HookÕs law. However, this is not the case for the post cracking zone: the shear modulus is reduced but not as much as the modulus of elasticity. In this case there is a difference between the properties extracted from experiments and the shear modulus extracted from HookÕs law. This problem is the result of the failure procedure that takes place in the IPC matrix. The small cracks that exist in the post-cracking zone play a significant role to the shear behavior of the IPC/glass fiber reinforced material. Until now only experimental values exist. No particular theoretical model for the prediction and explanation of these values exists.
5. Validation of the numerical modeling Table 2 Material properties for the IPC/mat reinforced material Material
E1 (GPa)
E2 (GPa)
V
IPC/mat
10
2.1
0.3
5.1. Construction and loading of the prototype In order to validate the finite element model, a prototype was realized and tested.
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According to the design guidelines, the structure should be tested with a concentrated load in the middle and a distributed load all over the concrete deck. However, the requirement for distributed loading is very difficult to be accomplished in the environment of the lab. For this reason the loading was simulated using four-point loading. This was imposed by the available experimental equipment. The load on each point was identical. 5.2. Experimental procedure and comparison with finite element analysis results 5.2.1. Overall displacement The most important parameter that had to be measured during the experimental analysis was the vertical displacement of the structure, especially until service load taking under consideration the requirement of the guidelines for a maximum permissible deformation. This measurement also was necessary in order to arrive to some conclusions about the validity of the numerical model. In Fig. 5 the vertical displacement of the structure is presented. It is obvious that the displacement of the real structure is diverging from the one calculated from the finite element analysis. This is due to the increased plasticity that is introduced in the area of the connection between the metal insert and the skins of the sandwich panels. 5.2.2. Deformation of the horizontal sandwich panels The displacement of the structure is of critical importance in order to comply with the design guidelines. The value of the overall displacement gives an overview of the effectiveness of the design, but in order to perform a detailed analysis, the evaluation of the strain field for
each sandwich panel of the structure is necessary. It is obvious that the most important part of the structure is the horizontal panels of the truss system. The failure of one of these panels will force the structure to collapse. In the Fig. 6 a comparison between the finite element and the experimental results concerning the strain in the direction of the force action is presented for the most loaded horizontal panel. As shown Fig. 6 the agreement in the deformation values between the finite element and experimental analysis proves that the modeling of the structure does not lead to artificial stiffening. In addition the stress for which the transition from the pre-cracking to the post-cracking zone is taking place is the same for both experiment and analysis. 5.2.3. Deformation of the diagonal panels of the truss system Another important parameter of the truss system is the behavior of the diagonal panels that are forced in compression. The two end panels are forced in compression while the next panels are forced in tension. This alternating type of loading is preserved throughout the whole structure. In Fig. 7 a comparison between the finite element and the experimental results is presented for the most loaded panel. From this figure it is obvious that in this case there is also a very good agreement between the finite element calculations and the experimental analysis. It is important to mention here that the influence of the deformation of the diagonal panels in the overall displacement of the structure is very limited. However, the correct modeling of this part of the structure is necessary in order to be able to predict a failure due to excessive deformation in some parts of the panels. The deformation of the diagonal panels in tension was also examined since the resistance and the modulus
70000 Finite Element Experiment
60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
10
20
30
40
50
60
70
80
Displacement (mm) Fig. 5. Displacement of the bridge: comparison between the experimental and finite element analysis.
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70000 Experiment Finite Element 60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
200
400
600
800
1000
1200
1400
1600
1800
Strain (µm)
Fig. 6. Deformation of the most loaded horizontal panel.
70000
60000 Finite Element Experiment
Force (Nt)
50000
40000
30000
20000
10000
0 -800
-700
-600
-500
-400
-300
-200
-100
0
Strain (µm)
Fig. 7. Deformation of the most loaded diagonal panel in compression.
of elasticity for the IPC in tension is limited as it is already shown in the previous chapters. In Fig. 8 the deformation of the diagonal panel in tension for both finite element and experimental analysis is presented. As shown in Fig. 8 there is a mismatch between the deformation values predicted by the finite element model and the values achieved by the experimental analysis. However, the deformations are higher in the finite element model. It is obvious that in the modeling of the sandwich panels no artificial stiffening is introduced. As it has been mentioned the behavior of the IPC was derived from numerous experiments performed in specimens. Examining the graphs in detail it is easy to notice that in both cases there is a transition point where
the IPC passes from the pre-cracking to the postcracking zone. However, this is taking place in lower force values in the finite element than in the real case, which means that the improved techniques introduced for the construction of these sandwich panels leaded to a higher transition stress level. In addition the stiffness of these panels in the post-cracking zone seems to be higher in the real case. The issue that is raised directly is what is the influence of this behavior on the stress levels of the other panels. In order to give an answer to this issue more analyses were performed changing the modulus of elasticity of these panels in the pre-cracking zone. As it was expected the strain level on tensile loaded diagonal panels was influenced while in the same time the strain levels on the diagonal panels loaded in compression and
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G. Giannopoulos et al. / Composite Structures 62 (2003) 235–243 70000 Finite Element Experiment 60000
Force (Nt)
50000
40000
30000
20000
10000
0 0
200
400
600
800
1000
1200
Strain (µm)
Fig. 8. Deformation of a diagonal panel in tension.
on the horizontal panels remained almost the same. This is due to the fact that the diagonal panels in tension are very weak compared with the other panels and as a consequence any redistribution of forces due to the change of their properties has small influence on the other parts of the structure. It is thus concluded that a more detailed analysis of the IPC using new specimens is necessary in order to introduce these values in a new model.
6. Conclusions In the present work the numerical modeling and its evaluation using the results derived from experiments to a real prototype is presented. A finite element model has been created in order to predict the behavior of the structure and use it in the future for the design of similar structures. The model does not experience artificial stiffening and thus the strains predicted in the different parts of the structure are not smaller than in the real case. This is usually the danger of the results of a numerical model that leads to wrong conclusions and failure of parts that are designed to take a specific load. However, the overall displacement of the structure seems to be derived by a model that is artificially stiffened. This is due to the fact that parameters, like the positioning of the bolts inside the sandwich skins, are not incorporated in the finite element model. A thorough examination of the results shows that the modeling of the elements of the structure is adequate for this type of analysis. The deformations predicted are in good agreement with the experiments and this type of modeling can thus be used in the future for the prediction
of the behavior of similar structures. The assumption of the thin plate for the concrete deck is valid as expected due to the high length/thickness ratio. The size and the type of the elements used for the modeling of the panels is also adequate and this is proved by the agreement that exists in the values of the deformation between the finite element analysis and the experimental procedure. As far as the design of the structure is concerned, according to the guidelines, this structure fulfils all the necessary criteria in order to be put in service. In addition the results of this work show clearly that use of IPC as a structural material can be extended to such applications, leading to structures with significant advantages over the traditional concrete structures. However, its long-term behavior under severe environmental conditions is still to be investigated.
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