The Dynamic Properties of the Bridge Deck Model Reinforced With FRP Bars

Significant application of composites can be observed in numerous fields over the last decades. Composite materials mainly used in the aeronautics industry are gaining more and more application in various areas such as the construction of bridges. The article presents a set of dynamic research carried out on a plate made of lightweight concrete reinforced with composite rods. The tested plate with dimensions 514 x 190 x 18 cm was a model of the actual bridge deck. A multi-channel signal recorder with specialized software was used for performing the measurements and estimating modal parameters of the plate. Dynamic response of the plate on a modal hammer impact was measured with numerous of piezoelectric acceleration sensors. A range of FEM models was created, started with 1-D beam model. Next, 2-D plate and 3-D volume, more complex models were developed. Only in the 3-D model the composite rods were modeled. The basic frequencies and mode shapes obtained for physical model were compared with those for numerical models. There were significant differences. The conducted analysis indicated more complicated than theoretical boundary conditions of the tested plate.


Introduction
Composites which are relatively new in civil engineering have been used for many years in the aircraft industry.The availability and low price are the biggest advantages of traditional materials in comparison to the price of those new ones, like Fiberglass Reinforced Plastic composites.However, competitive advantages of FRP composites are low weight, high strength and total corrosion resistance which means low maintenance costs.Because of the fact that the exploitation is more expensive, composites are cost effective compared to conventional materials.
FRP composites have been widely used in bridge structures to strengthen the existing bridges since 1997 [1].However, still no bridge or overpass with the use of composite reinforcement bars or carrying elements has been built in Poland.
The research conducted by consortium Mostostal Warszawa SA and Rzeszow University of Technology has been the first attempt to change the existing situation [2].This paper presents the research which has been a part of the so-called "Combridge" project.This project is financed by The National Centre for Research and Development in "Demonstrator+" program.The aim of this program is to build the first Polish bridge made of FRP composites [3].

Description of the tested plate
A double span plate has been studied.A theoretical length of each span has been 240 cm.The discussed plate has a e-mail: gpiatkowski@prz.edu.plbeen a model of the real bridge deck made of a FRP gridreinforced concrete LC35/38.Bars used in the grids has been diameter of 12 mm.Static scheme and model dimensions have been presented in Fig. 1.
The plate has been pinned in the centre and supported in the sliding manner near the edges.Support has been implemented in the form of cylindrical roller bearings and steel panels.Plate has been attached to steel panels by means of cement mortar.Theoretical length of each support has been equal 190 cm which is the width of the plate.
Steel profiles and rubber spacers have been used to transmit the load generated by the actuators of the research station on the deck.The load has been transferred to four 40 cm x 40 cm areas.The total additional mass has been 440 kg, Fig. 1.The classical modal analysis has been used, which is based on the measurement of the existing force and accelerations at various points of the structure as a response to this force.
The studies have been conducted in several steps, which has been motivated by parallel static research.The first two stages of the test have been as follows: • stage 1 -uncracked state, mass of a plate, • stage 2 -uncracked state, mass of a plate + additional mass.
Pulse excitation has been generated with PCB Piezotronics modal hammer weighing 5,5 kg and with a sensitivity of 0.23 mV/N.Soft tip has been used.Pulse excitation has been generated in four points shown in Fig. 2 in the vertical direction.
Dynamic response of the plate has been measured by piezoelectric accelerometers.The vertical accelerations have been measured by 27 accelerometer sensors.All the measurement points have been located on the upper surface of the plate and located as in Fig. 2. The placement of measurement points has corresponded to support scheme of the plate.Sensor offset from the longer edges of the plate has been 5 cm and from the shorter ones 17 cm.A measurement mesh of a total size 480 cm x 180 cm has been received.
Multi-channel signal recorder Scadas Mobile's LMS International has been used in measurements.Signals in the frequency range 0-256 Hz with a resolution of 0.125 Hz have been recorded during acceleration measurements.Recorder software allows for the classical experimental modal analysis including estimation of poles using Polymax method and visualization of the results.

Modal test results
The vibration of the plate has been measured in stage 1.In second stage the additional mass of metal and rubber spacers located on the plate has been included.Dynamic characteristics which are the results of modal analysis for stage 1 have been shown in Table 1.Mode shapes for the first two modes have been presented in Fig. 3 and Fig. 4. The range of the values of the first and second frequencies have been presented in the min-max variability because excitation for modal analysis has been made in different points.

Analytical model
Even in case of complex structures preliminary estimates of the dynamic properties can be based on the simplest numerical model.The first model of a studied plate has been a two-spanned beam, for which two first frequencies have been calculated using closed formulas.The formulas (1) and ( 2) have been determined for half schemes of symmetric beam and appropriate boundary conditions.The first mode shape has been shown in Fig. 5 and frequency for this mode has been described by formula (1).
where: l -span length, μ -linear density, EI -stiffness, k -number of particular type of mode.The second mode shape has been shown in Fig. 6 and frequency for this mode has been described by formula (2).The data used in calculations: the Young's modulus equal to 21,562 GPa, moment of inertia 92340 cm4 and linear density μ = 673, 74 kg/m.The length of each span has been equal to 2,4 m.The cantilevers have been omitted.
For second stage, homogeneous mean density with mass of additional elements located on the deck have been adopted and the value of linear density has been increased to 765,41 kg/m.

FEM 1-D and 2-D models
The ADINA program has been used for building of numerical models.
The Young's modulus in numerical models has been the same as in analytical model and volume density has been equal to 1970 kg/m3, which is also corresponding with the above mentioned data.
Two numerical BEAM [4] models have been created to compare obtained frequencies with analytical results.The cantilevers have been considered in all numerical models.In the first model only the mass of the plate has been considered.In the second model additional mass has been included.The results have been shown in Table 3.
Received in the ADINA results differ from those obtained analytically by less than 0.5% in the case without the additional mass.The difference is due to the fact that the analytical calculation does not include overhangs.Consideration of additional masses on appropriate sections of the beam reduces the numerically calculated vibration frequency compared to the frequency obtained with the analytical model of a mean weight of the whole span length, but the difference in frequency values is less than 5%.
Comparing these results with the results of the measurements given in Tables 1 and 2, the calculated frequencies seem to be too high.The analytical model and numerical model 1-D are limited in terms of defining the actual conditions of the support of the plate.It is impossible therefore to consider the lack of contact between the plate and the support.
In order to consider deviation from the theoretical support the 2-D model has been created.It has been formed with the use of eight-node type SHELL elements [4].The plate has been analysed as non-cracked, and the stiffness of reinforcement has been omitted.Four models in which the ratio of the length of the left support to width of the plate was as follows: 1/1, 1/2, 1/4, 1/8, in order to test the effect of these boundary conditions on the basic frequen- cies.In Fig. 7 the model with entire supported width has been shown.

FEM 3-D model
The 3-D model plate has been made of eight-node SOLID elements [4].The plate has been analysed as uncracked, again.The composite reinforcement has been included with the use of two-node TRUSS element [4].The Young's modulus of this composite rods has been equal 57.8 GPa.A fragment of the model is presented in Fig. 8.  5.
The mode shapes of 3-D model with 1/1 support ratio have been shown in Fig. 9.They have been such like these obtained for 1-D and 2-D models, compare to Fig. 5 and Fig. 6.
The frequency of the plate has increased of 2-3% by included reinforcement.The frequency has decreased in case of reducing the length of the contact between the plate and one of the support of the edges.The difference between the results for the total support and support 1/8 of the width reaches 22% for second mode.
The Modal Assurance Criterion (MAC) was used to compare the test and numerical mode shapes.The correlation between test eigenvector and numerical eigenvectors where: x T -test eigenvector x N -numerical eigenvector The first mode shape is anti-symmetrical like in tested plate.Numerical forms differ from each other only with length of stabilized segment and are comparable to these obtained in tests.
The second mode shape for 1/2 support ratio was still symmetrical like for a beam model.For 1/4 and 1/8 support ratio the nature of the second mode shape was rather anti-symmetrical, what can be observed in Fig. 11.That mode shape is similar to this obtained in test, Fig. 4. Calculated MAC values confirm this observation.The results presented above allow for the conclusion that the actual length of support has been close to 1/4 of the width of the plate.

Conclusions
The way of support implementation has an significant impact on the gap between measurements and models concerning natural frequencies, because one line support was implemented differently than expected.The most similar to results obtained in modal test are frequencies calculated

Figure 1 .
Figure 1.Static scheme and dimensions of the plate

Figure 2 .
Figure 2. Location of measurement points

Figure 5 .
Figure 5. Antisymmetric eigen mode shape of a beam for k=1

Figure 6 .
Figure 6.Symmetric eigen mode shape of a beam for k=1

Figure 8 .
Figure 8. Fragment of the 3-D model with reinforcement

Table 2 .
Modal test results -stage 2 Dynamic characteristics which are the results of modal analysis for stage 2 have been shown in Table2.As expected, additional mass resulted in decreasing of the frequencies and increasing of damping coefficient values.The obtained in both stages mode shapes of the plate have suggested problem with one of the supports.

Table 3 .
Frequencies of two first mode shapes for 1-D model

Table 4 .
Frequencies of two first mode shapes for 2-D model

Table 5 .
Frequencies of two first mode shapes for 3-D model