Torsion in tridimensional composite truss bridge decks André B. Almeida Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal e-mail: branco.almeida.a@gmail.com Abstract Torsion stiffness is an important issue in design practice of tridimensional (3D) composite truss bridge decks. Analytical methods providing a simple approach to determine the torsional stiffness factor (J) for 3D steel and composite trusses are presented. The dynamic behavior of this type of bridges is considered adopting the simplified method proposed in [1] which allows classifying the dynamic behavior of a bridge from its vibration level by evaluating its vibration frequency and deformability under traffic loading. Keywords bridge, truss, steel, composite, torsion, stiffness I. INTRODUCTION Trusses are well known structural forms and their application is wide in the field of civil engineering. Today, more than ever the economy has an important role in the design and construction of bridges. Technological advances in the manufacture of structural materials, connection methods between elements and new construction methods have made composite structures interesting and competitive alternatives for road bridge decks, compared to typical pre-stressed concrete solutions. Tubular composite tridimensional (3D) trusses with triangular cross section are an interesting solution for road bridge decks. The stiffness of such bridges is an important issue, specifically regarding the torsional stiffness, due to eccentrically positioned loads. Due to the geometrical complexity of these structures, its quantification is not obvious or easy to obtain without a numerical analysis of complex finite elements models. So was born the idea of creating a practical tool to determine the torsion stiffness factor J of steel and composite 3D trusses. The study of 3D truss torsional stiffness was based on energy equilibrium. The strain energy associated to the longitudinal deformation of the bars of the truss must be equal to the strain energy of an equivalent bar under torsion. The mathematical equation of this equality yields two analytical methods to calculate the factor J of 3D steel and composite trusses. Concerning composite trusses, some extra adjustments had to be done to include the concrete slab contribution. Both analytical methods were submitted to numerical validation. The work developed in the present dissertation brings different advantages to engineers. It provides a simple way to estimate the torsion stiffness of 3D steel and composite trusses without the need to create complex finite element models. In predesigning stages it is useful to have an approach to determine the torsion stiffness of the bridge cross section. There are also advantages in terms of numerical modeling. Finite element modeling of 3D trusses may become simpler. Once the factor J is known, they can be modeled using a beam element with generic cross-section to which is assigned the value of factor J and the other mechanical quantities. This alternative drastically reduces the time to model such structures. This paper is organized in six sections. In the following two sections the analytical methods that provide the torsion stiffness of 3D steel and composite trusses are presented. Section IV presents the results obtained from a numerical analysis of steel trusses. Section V presents the results obtained from a numerical analysis of composite trusses. The dynamic behavior of composite 3D truss bridge decks is analyzed in Section VI. At last, the results are analyzed in Section VII and conclusions are drawn therefrom. II. GEOMETRICAL FEATURES OF 3D TRUSSES The geometrical shape of the trusses can be defined by the geometrical parameters, e. These are illustrated in Figures 1 a), b), c) e d). a) b)
The strain energy [3], of a beam under torsion can be defined as c) d) Figure 1. Geometrical parameters of steel trusses Composite 3D trusses are geometrically defined by the same parameters. However, the green member in Figure 1 a) and b) has horizontal bracing functions and is replaced by a concrete slab in composite trusses, which has the same structural function. An example of a composite truss is illustrated in Figure 2. is the torsional moment is the Distortional Modulus which is, is the Poisson s ratio The strain energy of a bar under axial force is, [3]. (4) (5) (6) in which, axial force length of the bar Modulus of Young of the bar cross section area of the bar III. Figure 2. Composite truss TORSION STIFFNESS OF 3D STEEL TRUSSES When a 3D steel truss is under torsion, the axial force in truss bars can be determined if the torsional moment is in the form of a statically equivalent torque. The strain energy of a beam under torsion is a function of its rotational angle as[3], Figure (3) a) and b) shows a beam under torsion. (1) The strain energy associated to a set of bars submitted only to axial forces can be defined as [3], (7) refers to bar is the strain energy of the bar is the axial force in bar is the length of the bar is the Modulus of Young of the bar is the area of the cross section of the bar is the number of bars As said, the strain energy associated to torsion of the truss must be equal to the sum of strain energy associated to the longitudinal deformation of the truss bars, i.e. As the rotational angle per unit length as, in which where is defined as, a) b) Figure 3. Beam under torsion is the beam length J is the torsional stiffness factor. is a function of the torsional rotation (2) (3) Solving equation (8) in order to J yields, J Equation (9) determines the factor J of 3D steel trusses. However if the axial force and length of each member are defined as a function of the truss geometrical parameters, this last equation can be subjected to mathematical transformations in order to be expressed in terms of geometrical parameters, the Poisson ratio and the area of the cross sections of each bar [2]. Table I shows the factor J equation for 3D steel trusses. (8) (9)
=1=Upper Chord =2=Diagonal bar =3=Bracing bar =4=Horizontal bar IV. TABLE I. J OF 3D STEEL TRUSSES J (10) TORSION STIFFNESS OF 3D COMPOSITE TRUSSES Due to the difference in mechanical characteristics of materials in the composite structure, the analysis of a composite cross section can be simplified through its homogenization. This technique is based on the principle that each part of the cross section has the same deformation of the global section, meaning, the rotation angle of global section is equal to the rotation angle of the concrete and steel parts as, Hence it is possible to define the torsional modular ratio The factor J of the composite cross section can be defined as (11) as, (12) (13) The concrete slab was replaced by two equivalent bars in order to simplify the structural analysis as shown in Figure 4. Hence it is possible to define the area of the cross section of each fictitious bar simulating the slab capacity to resist distortion. A two dimensional truss was idealized in order to determine the energy associated to the fictitious bars under distortion, as shown in Figure 5. According to [2] the strain energy in this structure is given by the following expression, (14) in which, is the fictitious bar length is the area of the cross section of each fictitious bar is the truss width is the Young s Modulus of steel is a transverse force Figure 5. Fictitious bars The analysis of the concrete slab under distortion, as shown in Figure 6, yields expression (15) which represents the strain energy of the slab [2], (15) in which, is the length of the concrete slab is the Distortional Modulus of concrete is a transverse force is the shear area of the cross section of the concrete slab and is given by, (16) Figure 4. Composite truss with fictitious bars In this way it was possible to determine the axial force in each member of the truss when it is submitted to torsion. In order to correctly replace a concrete slab by two steel bars, an equality of strain energy was established. Once the slab influence on truss torsional stiffness is basically due to its capacity to absorb distortional loading we can say the energy absorbed by the two equivalent fictitious bars when submitted to distortion has to be equal to the energy absorbed by the slab also when it is submitted to distortion. Figure 6. Concrete slab under distortion Hence equalizing the strain energies of the concrete slab and the fictitious bars both under distortion leads to the following expression, (17)
Solving the expression (17) in order to and after some mathematical transformations we get to expression (18). (18) In conclusion, expression (18) represents the area that should be applied to the cross section of each one of the fictitious bars to simulate the slab influence on the torsional stiffness of the global truss. Once again replacing a torsional moment by a torque, as shown in Figure 7, due to the fictitious bars, it is possible to determine the axial force in each member of a composite truss under torque by a simple static analysis, with the assumption of neglecting the axial force in the upper chords. Figure 7. Fictitious truss Like in steel trusses, it is possible to establish the equality between the strain energy associated to an equivalent beam under torque and the sum of strain energy associated to the longitudinal deformation of each one of the truss bars. The equality of Energies yields, (19) As demonstrated in [2] the coefficients, and can be defined in terms of truss geometrical parameters. After some mathematical transformations and defining as, where refers to the truss diagonal bars and refers to the fictitious bars, the following expression was obtained for the factor J of the fictitious truss, in which, Jfict.truss (20) torsional stiffness. The slab width must be divided in overhangs width and the width between truss upper chords. The slab width between truss upper chords forms a closed section with the truss triangular cross section and contributes for the truss torsional stiffness by its capacity to resist distortion and transmit horizontal transverse forces for the steel truss members. The slab overhangs have a contribution to the truss torsional stiffness due to the torsional stiffness of their cross sections. So it is necessary to add the factor J of slab overhangs. According to [2] the factor J of the overhangs must be homogenized dividing the overhangs width by the torsional modular ratio as, J slab (21) is the overhangs total width is the thickness of the slab overhangs Concluding, in order to calculate 3D composite trusses torsional stiffness factor, the cross section area of the fictitious bars is defined according to expression (18). Then the factor J of the steel fictitious truss can be determined applying expression (20). A more accurate result may be obtained if the factor J of concrete slab overhangs is added. Hence, the factor J for a 3D composite truss is given by, J = Jfict.truss + J slab V. ANALYSIS OF RESULTS FOR STEEL TRUSSES (22) In order to verify the accuracy of the factor J equation for 3D steel trusses (10), obtained in Section III, a comparison between analytical and numerical results was made. Different finite elements models of 3D steel trusses were created in order to numerically determine the torsional stiffness factor. The factor J could be numerically determined from joints displacement data [2]. The first model is called Steel Truss I and is defined by the parameters shown in Table II. TABLE II. GEOMETRICAL PARAMETERS OF STEEL TRUSS I Outside diameter Thickness Upper Chord 323,9 16 Bottom Chord 508 40 Expression (20) represents the factor J of a composite truss with its concrete slab replaced by two steel bars. According to [2], it is necessary to include the contribution of the slab cross section torsional stiffness to the global truss Diagonal 244,5 16 Bracing bar 244,5 12,5 Horizontal bar 244,5 12,5
Numerical analysis and data treatment led to the following value [2], JTruss I 14615244260 (mm 4 ) The factor J of Steel Truss I was calculated with (10). The following result was obtained [2], Analytical results JTruss I 14615244260 (mm 4 ) Comparison between analytical and numerical results of Steel Truss I shows there is no difference between values. The second model is called Steel Truss II and is defined by the parameters shown in Table III. The cross section dimensions of the steel bars are equal to the ones used on Steel Truss I. TABLE III. GEOMETRICAL PARAMETERS OF STEEL TRUSS II Numerical analysis and data treatment led to the following value [2], Numerical analysis JTruss II 32019181968 (mm 4 ) The J of Steel Truss II was calculated with (10). The following result was obtained [2], Analytical analysis JTruss II 32019181968 (mm 4 ) Comparison between analytical and numerical results of Steel Truss II shows that there is no difference between values. JTruss III 28998194792 (mm 4 ) The J of Steel Truss III was calculated with (10). The following result was obtained [2], Analytical analysis JTruss III 28998194792 (mm 4 ) Comparison between analytical and numerical results of Steel Truss III shows that there is no difference between values. VI. ANALYSIS OF RESULTS FOR COMPOSITE TRUSSES Three different 3D composite trusses were modeled with finite element software. These models were created with a combination of column elements and shell elements connected with rigid bars. The truss bars were modeled with biarticulated column elements and the concrete slab was modeled with thin shell elements. The steel bars of all three models have the same cross section dimensions. These are presented in the following table, TABLE V. CROSS SECTION DIMENSIONS OF STEEL BARS Outside diameter Thickness Upper Chord 323,9 16 Bottom Chord 508 40 Diagonal 244,5 16 Composite Truss I geometrical characteristics are shown in the next table. TABLE VI. GEOMETRICAL PARAMETERS OF COMPOSITE TRUSS I At last, Truss III is defined by the following parameters, TABLE IV. GEOMETRICAL PARAMETERS OF TRUSS III Slab width Slab thickness Total: Overhangs: each one Outside diameter Thickness Upper Chord 508 16 Bottom Chord 508 40 Diagonal 244,5 16 Bracing bar 244,5 12,5 Horizontal bar 244,5 16 Numerical analysis and data treatment led to the following value [2], Node displacements obtained by the computational analysis allowed the necessary information to determine the torsional rotation per unit length J. The factor J could be determined with (3). For Composite Truss I the following results were obtained [2], Composite Truss I - Composite Truss I J rad/m Figure 8 shows the finite element model of Truss I. Its deformed shape is illustrated in Figure 9.
The last model, Composite Truss III, has the following geometrical parameters, TABLE VIII. GEOMETRICAL PARAMETERS OF COMPOSITE TRUSS III Figure 8. Finite element model of Truss I Slab width Slab thickness Total: Overhangs: each one The numerical analysis yields the following result [2], Composite Truss III - Composite Truss III J Figure 9. Deformed shape The analytical approach, using (22), for the factor J of Composite Truss I yields [2], Analytical results Fictitious C. Truss I J Concrete slab J Composite Truss I total J Comparison between analytical and numerical factor J shows no relevant difference. The approach for Composite Truss I has a relative error. Composite Truss II is similar to Composite Truss I but has wider overhangs, as shown in the next table. TABLE VII. GEOMETRICAL PARAMETERS OF COMPOSITE TRUSS II The analytical approach, using (22), for the factor J of Composite Truss III yields, Analytical results Fictitious C. Truss III J Concrete slab J Composite Truss III total J There is a relative error between the numerical and analytical results of Composite Truss III. Concluding, the error in second and third truss model is bigger than the error obtained in the first model. This difference may be associated to the enlargement of the slab width. The concrete slab was analyzed according to simplified hypothesis, and so, as the slab width increases the error in the solution of the analytical approach grows. However, the biggest difference does not exceed 10% and so it is an acceptable error. Slab width Slab thickness Total: Overhangs: each one The following result was obtained from numerical analysis[2], Composite Truss II - Composite Truss II J rad/m The analytical approach, using (22), for the factor J of Composite Truss II yields, Analytical results Fictitious C. Truss II J Concrete slab J Composite Truss II total J Hence, there is an 8,85% relative error in the analytical determination of factor J of Composite Truss II. VII. DYNAMIC BEHAVIOR OF COMPOSITE 3D TRUSS BRIDGE DECKS An important issue to consider when designing a 3D truss bridge deck is its behavior under dynamic loads, particularly referring to oscillatory behavior. Bridge decks vibrations are an important cause for discomfort. There is a suggested methodology in [1] to classify the dynamic behavior of a bridge from its vibration level evaluating its vibration frequency and deformation under traffic loading. The basic principles of this methodology are summarized below. This methodology is based on a limit for the vertical acceleration of the bridge deck, i.e., (23) is the bridge first vertical vibration mode frequency in Hz
Based on expression (23) the document [1] establishes a limit for the static displacement from a uniform load moving at a speed of 130Km/h. This limit guarantees an acceptable level of vibrations. As shown in [2]. TABLE X. DIMENSIONS OF THE STEEL TRUSS Outside diameter Thickness (24) is the main span length of the bridge, in meters(m). This displacement should be calculated using a uniform load of 10 kn/m 2 distributed along the deck width (m) and the longitudinal length (m) given by, (25) Upper Chord 323,9 16 Bottom Chord 508 40 Diagonal 244,5 16 The finite element model of the bridge is illustrated below. There is also a suggested way for classifying the bridge comfort level. This is given by the coefficient KB, (26) Figure 10. Bridge finite element model A modal analysis was performed, from which the first vertical vibration mode frequency was obtained, This coefficient has an associated scale shown in Table VII which indicates the comfort level provided by the bridge. TABLE IX. COMFORT LEVEL CLASSIFICATION KB Class Perception level 0,1 A Imperceptible 0,25 B Low perception 0,63 C Perceptible 1,6 D Quite perceptible 4 E Very perceptible 10 F High Perception 25 G High Perception H e I High Perception It is suggested the value 12 for KB as the maximum limit from which the bridge starts to generate an intolerable vibration level. This value allowed to determine the static maximum displacement associated with the 10kN/m 2 load already referred with (24), It was also possible to determine the coefficient KB associated with the bridge, Hence the bridge doesn t exceed the suggested limit of 12. In order to verify the static limit requirement two load cases were created. In the first place, the bridge was submitted to a 10KN/m 2 uniform load distributed along the total deck width and the longitudinal length of, A bridge with a 3D composite truss deck was modeled in finite element software to analyze the dynamic behavior of this type of bridges according to this method. This load corresponds to a concentrated force of, The bridge has a total length of 100m divided in three spans of 30m, 40m and 30m. The longitudinal structural system is like a continuous beam. Its cross section is a 3D triangular composite truss. The steel truss is made of steel and the slab is made of concrete. The concrete slab has a 0,2m thickness and a total width of 9,5m. The geometrical parameters of the steel truss and the dimensions of the bars cross sections are shown in the next table. One half of this force was applied at each of the two upper nodes of the truss cross section. This load case yields a displacement at the mid-span section of, This displacement is lower than the limit, The deformed shape of the bridge under this load case is illustrated in Figure 11.
According to expression (3) we have, Figure 11. Bridge deformed shape According to the method in [1] the bridge doesn t generate intolerable vibrations. A second load case was applied to generate bending and torsion of the bridge deck. This load case represents the effect of eccentric traffic loads. The uniform load is distributed on half of the deck width. The load length is the same as before. This load case is equivalent to a torsional moment applied at the mid-span section plus a vertical force. The torsional moment is given by, This moment was simulated by an equivalent torque. The value of each force is given by, is the truss width between upper chords These forces generated a mid-span rotation angle of, The vertical load generated a vertical displacement of, The maximum displacement occurs at the edge of the concrete slab overhang and is given by, The result is, Therefore, This value is very close to the one numerically determined, so once again it is shown that the method to calculate the torsional stiffness factor of a 3D composite truss provides sufficient accuracy. VIII. CONCLUSIONS To conclude, the comparison of numerical and analytical results in Section V and Section VI shows that the analytical methods developed in [2] and presented in Section III and IV yields to an accurate determination of the torsional stiffness factor of 3D steel trusses. The determination of factor J of composite trusses is not exact, but an approach with an acceptable error. The hypotheses considered in the analysis of composite trusses were applied with the purpose of developing a simple and practical method to estimate the torsion stiffness of 3D composite trusses. These methods were developed in order to create tools which can contribute to a perception of the torsion stiffness of such trusses without the need to develop complex finite element models It can be concluded that the methods developed provide an acceptable approximation for the factor J of 3D trusses and therefore they can be useful in pre-design stages of structures. At last, the dynamic behavior of 3D composite truss bridges was analyzed with a simplified approach. The bridge model was analyzed and classified with respect to acceptable comfort levels. Concluding, the proposed objectives for this dissertation were successfully achieved. The rotation angle of the cross section was also analytically determined using the method presented in Section IV as indicated below. The cross section of the bridge model is equal to Composite Truss I cross section, so the factor J was already defined in Section VI and is, J The rotation angle can be determined as, (27) REFERENCES [1] Ministerio de Fomento: RPM-95 Recomendaciones para el proyecto de puentes metálicos para carreteras. Espanha 1996 [2] Almeida, A., Torção em tabuleiros de pontes em treliça tridimensional mista aço-betão, M. Sc. Thesis, IST-UTL, Lisbon, Portugal, 2010 [3] Beer, F., Johnston, E., DeWolf, J. Resistência dos materiais, McGraw-Hill, Espanha, 2006