Cyclic behaviour of a reinforced concrete braced frame with high damping rubber devices

Transcription

1 Cyclic behaviour of a reinforced concrete braced frame with high damping rubber devices F. Bartera, L. Dezi & R. Giacchetti Dept. of Struct. Engrg., Polytechnic University of the Marches, Italy Abstract High damping rubber devices were implemented atop steel braces and connected to the girders of an R/C one bay-by-one bay single storey space frame in such a manner that the relative displacement between the floor and the braces gives a shear deformation to the rubber pads. The static as well as the dynamic behaviours of the bare frame and the supplementary damped braced frame have been thoroughly investigated under numerous types of tests in order to assess the stiffness and damping characteristics of the whole system and establish a comparison with the expected theoretical results. After a very large amount of sinusoidal cycles at increasing levels of frequency, force and displacement, no apparent stiffness and damping modifications were detected. A proper choice of damping ratio and secant stiffness has been proposed which leads to a better numerical estimate of the experimental test results than it is obtainable by simply assuming the values currently provided by the factory based upon the tests on the single dissipative devices. Keywords: energy dissipation - high damping rubber devices, dynamic test, numerical model, seismic protection. 1 Introduction The latest techniques for seismic protection of buildings are based on the energy approach, in which the integration of the equation of motion (1) of a SDOF with mass m, damping coefficient c, stiffness k, in the time domain in seismic conditions m x( t ) + cx( t ) + kx( t ) = mx ( t ) (1) g

2 44 Structures Under Shock and Impact VIII leads to an expression (2) of energy equilibrium E + E + E = E (2) k in which it can be observed that the input energy E i is balanced at each instant by the total energy of the system, the sum of the elastic potential energy E s, of the kinetic energy E k and the dissipated energy due to viscous damping E d. The energy approach to the seismic problem has led to two design techniques: the base isolation, which decreases the input energy, and the energy dissipation, which increases the dissipated energy. Both techniques aim substantially at decreasing the total energy of the system in order to reduce the response in terms of displacement and acceleration. In this research dissipating braces are used to increase the energy dissipation. For the main frame to remain elastic, the fundamental parameter for this type of approach is the equivalent viscous damping ratio ξ. Designing in terms of increasing the dissipating capacity is synonymous of increasing damping and consequently reducing the response spectra. Figure 1 shows two PSA spectra of the 1976 Friuli earthquake at different damping ratios: attention is drawn to the fact that passing from a damping of 2% to 1% produces a significant reduction of the design acceleration (from point A to point B ). In view of the fact that the introduction of braces increases the stiffness of the structure, the natural period decreases shifting into the range of larger accelerations (from point B to point C ): this causes a larger design acceleration with respect to the acceleration that would be obtained simply by increasing damping, which is, however, less than the initial acceleration. d S i Figure 1: Effect of increased damping ratio on a response spectrum (Friuli 1976). 2 Description of the dissipating braces This paper concerns the experimental study of the cyclic behaviour of a reinforced concrete frame equipped with steel braces carrying high damping rubber devices (HDRDs). The mechanical model used to represent this equipped system shows the dissipating brace in parallel to the r.c. structure: it is quite clear that when the stiffness of the brace is such as to render it rigid with respect to the frame, the overall behaviour is conditioned only by the characteristics of stiffness and

3 Structures Under Shock and Impact VIII 441 dissipating capacity of the rubber element. The devices tested in this study are made of vulcanised rubber with the addition of black carbon filler die-cast on steel plates (Figure 2). This type of devices is designed to operate by slipping on its own plane. The force-displacement relationship is shown in Figure 3. The fundamental parameters of the devices are the secant stiffness (determined experimentally in proximity of a shear deformation γ=1% obtained from quasi static tests) and the loss factor η D (determined experimentally with harmonic tests as the tangent of the phase angle θ between the shear strain τ and the shear deformation γ ), whose link with the damping ratio is: ηd = tanθ = 2ξ (3) The application of the modal deformation energy model, under the simplifying hypothesis on the nature of the vibrations mode of a structure characterised by many degrees of freedom equipped with HDRDs, leads to the formulation of the damping ratio for the i-th mode ξi in the form T ( ηd 2ξC ) φi K BDφi ξ i = ξc + (4) 2 2 ωi in which the importance of the parameters of the dissipating capacity of the r.c. frame (damping ratio ξ c ) and of the material (loss factor η D ) and the matrix stiffness of the dissipating brace system (K BD ), is evident. From the mechanical model proposed, in which the brace and the devices are in series, it can be assumed that if the stiffness of the steel brace K B is such as to consider it rigid with respect to the dissipating device, the behaviour of the system will depend only upon the characteristics of the device. The expression of the dissipating brace stiffness K BD is: 2 K B K D [ K B + K D( 1 ηd )] K BD = (5) K B + 2K B K D + ( 1 ηd )K D which, for K B, can be simplified to the term of the device stiffness K D. With the theory proposed, it is possible to obtain, according to the damping target and the frame storey stiffness K S, the total storey stiffness K D of the dissipating devices necessary and it is therefore possible to design them dimensionally 2( ξ ξc ) G' A K D = K S = n (6) ηd 2ξ t where n is the number of dissipating devices, A the effective shear area and G the shear strain modulus of the single device. Experimental state-of-the-art involving dissipating braces is characterised by the design of devices according to the damping target established, and continues therefore with the characterisation of the devices in quasi static conditions and the implementation of the same on the structure. Tests on shake tables or pseudodynamic tests are then performed. This method is subject to criticism in that the shake table and the pseudodynamic tests aim at investigating the overall behaviour of the system subjected

4 442 Structures Under Shock and Impact VIII to earthquakes, without permitting an in-depth analysis of its basic dynamic characteristics. It is in fact not possible to obtain the stiffnesses in question and thus explain the cause of eventual behavioural differences with respect to the theory. This paper therefore aimed at: 1. studying the behaviour of dissipating braces by creating a simple damped oscillator subject to sinusoidal force and displacement cycles so as to obtain results which are simple and clear to interpret; 2. determining the F-d relationship of the dual system (bare frame plus dissipating braces), the system damping and dynamic stiffness; 3. comparing the resulting behaviours with different configurations of the devices; 4. optimising a simplified numerical model by matching the experimental observations that may be extended to the study of the system response subject to real earthquakes; 5. comparing the results obtained and theoretical conclusions. The scope of the implementation is a one bay by one bay single storey space frame mounted on a 4.2 by 4.2 m mesh (see Figure 4). The column cross sections are,2x,3 m, the beams,3x,35 and,2x,35 m and the slab has a thickness of,1 m. The total height of the frame is 3,15 m. The r.c. frame has been statically and dynamically characterised in order to obtain the values of stiffness K ca and damping ξ ca with which to compare the equipped situation, and are: K ca =3,277 kn/mm, ξ ca =2,52%. The steel braces used are designed in such way as to be considered rigid with respect to the r.c. frame, providing to the same a stiffness K B equal to 6,61 kn/mm, equal to approx. 2 times that of the frame. These braces were realised according to a K layout with the decussated vertex stiffened by two bars arranged cross-wise. The arms of the K are constructed with steel pipes 133 mm in diameter and 4 mm thick and the stiffening with steel bars φ24 mm with threaded ends. The bars were added to prevent even the slightest movement that might compromise the study of the expected response of pure shear. The bars were pre-tensioned in order to avoid instability. The two dissipating braces, mounted on the r.c. frame symmetrically with respect to the direction of motion, were realised with the following configurations (Figure 5): a) 2 dissipating devices on each brace placed vertically b) 1 dissipating device on each brace obtained from the previous configuration by removing the inner device; c) 1 dissipating device on each brace placed horizontally F d t γ = d /t Figure 2: Details of the high damping rubber devices.

5 Structures Under Shock and Impact VIII 443 Figure 3: Force-displacement loop for a single damper at 1% γ. Figure 4: Test structure incorporating the dissipating bracing. RC beam pre-existantplate anchor plate connectio n s to th e anchor plate RC beam pre-existantplate anchor plate HDRD HDRDs ste e lbrace conne cting plates steelbrace T shap e d connecting plate Figure 5: Configurations (a), (b) and (c). The following tests were performed: 1. static and quasi static displacement-controlled tests: monotone and cyclic tests to determine the constitutive law and the hysteretic behaviour; 2. Free vibration tests: Snap-back tests to determine the damped free vibration characteristics (frequencies and damping) 3. Forced vibration tests: - Sine sweep with constant displacement amplitude and variable frequency to investigate the dependence of the damping ratio on the displacement amplitude in a wide frequency range; - Sine sweep with constant force amplitude and variable frequency to determine the critical frequencies, stiffness and damping ratio.

6 444 Structures Under Shock and Impact VIII The damping ratio has been calculated as the logarithmic decrement in free vibration conditions and, according to the expression ξ=w D /4πW S, in resonance condition for the forced vibration tests, where W S is the maximum elastic potential energy and W D is the dissipated energy per cycle. For the load application, a hydraulic jack was used in the monotonic test, while a servo-controlled hydraulic actuator was used for both the constant displacement amplitude cyclic test and the snap-back test, both mounted on a reacting element. Transmission of the load of the cyclic tests was achieved by means of a connection consisting of a special joint that hooks the contrasting steel plate to the r.c. beam connected to another symmetric plate, on the beam opposed, by four tie rods. The constant force amplitude cyclic test was performed by means of an electric-hydraulic shaker mounted directly on the frame. 3 Experimental results on the dissipating braces 3.1 Stiffness and constitutive laws The rigidity of the brace with respect to the frame was confirmed by the comparison between the displacement time history of the floor and the relative displacement time history measured between the brace plate and the beam, coinciding with the shear deformation of the device. In these conditions, an increment of the system stiffness with respect to that of the bare frame, equal to the sum of the devices stiffnesses, was measured, and the constitutive law of the equipped structure turned out to be very similar to that of the single dissipating device (Figure 6). Figure 6: Global force-displacement relationship for all the tested layouts.

7 Structures Under Shock and Impact VIII Free vibration tests The tests were performed by imposing an initial displacement to the floor by means of a servo-controlled actuator and instantaneously removing the perturbation. Snap-back tests were carried out by imposing the storey displacement equal to 2,5 5, 7,5 1, 12,5 mm (respectively equal to.8, 1.6, 2.4, 3.2 and 4. interstorey drift). Three tests at the same conditions were performed for each displacement. The displacement time histories show an evident reduction of the number of oscillations for the equipped configuration, that never performs more than two cycles before stopping, while the motion occurs around a position different to the starting (zero) position. It was ascertained that this deformation coincides with the residual displacement obtained from the constitutive law of the system when the force returns to zero. The residue was monitored in time, and it was ascertained that it is reabsorbed completely within 24 hours, which confirms the re-centring capacity of the material used. The average values of the damping ratio calculated according to logarithmic decrement method in the two configurations are as follows: ξ(a) = 39,2 %, ξ(b) = 27,4 %, ξ(c) = 26,5 %. Figure 7: Free vibration tests results. 3.3 Forced vibration tests with constant displacement amplitude and variable frequency Tests were performed imposing displacement sinusoidal cycles whose amplitude was equal to 2,5 5, mm with a frequency sweep from,1 to 3,5 Hz. Figure 8: Constant displacement amplitude test results.

8 446 Structures Under Shock and Impact VIII The results showed few dispersed values of damping and of the quantity of energy dissipated as the frequency varies for the different configurations; the average values are respectively as follows: ξ (a) = 12,2 %, ξ (b) = 1 %, ξ (c) = 14 %. Figure 8 (a) and (b) show the results obtained in terms of dissipated energy per cycle and damping ratios at the different test frequencies. 3.4 Forced vibration tests with constant force amplitude Tests were performed at different force levels and the behaviour of the system at critical conditions was recorded. The dissipating capacity obtained is fully compatible with the results of the constant displacement amplitude tests. Figure 9: Constant force amplitude test results. As the force ranges vary (and consequently the motion amplitudes) a variation of the resonance frequencies was reported due to a variation of the system stiffness. The stiffnesses in question were therefore calculated analytically according to the resonance frequencies and it was ascertained that these values coincide with the secant stiffness obtained from the constitutive law. Considering that the quasi static tests demonstrated that the force-deformation relationship is a sort of envelope of the cycles having lower amplitude, an analytical law was identified that, starting from the constitutive law, provides the secant stiffness at each displacement range. Since the load branches of the constitutive law may be well approximated by straight lines represented by the equation F = s tanγ + c (Figure 1a), the following secant stiffness expression, according to the displacement, was obtained c + stan γ c K () s = = tan γ + s s (7) where c represents the interception on the force axis and γ the slope with respect to the displacement axis of the straight line interpolating the load branch of the constitutive law. Expression (7) is represented by a hyperbola (Figure 1b). Experimental observation, for values of displacement larger than 1 mm (range in which seismic applications fall), confirmed the expression obtained.

9 Structures Under Shock and Impact VIII 447 Integrating the proposed equation, a curve was obtained that can be considered as the force-deformation relationship of the equipped system, whose point by point tangent represents the system secant stiffness. F () s = stan γ + cln s (8) Observing the curve (Figure 11), it can be noted that it is possible to interpolate, by means of a regression straight line having a correlation coefficient equal to 99%, the values for displacement ranges larger than 3 mm, in order to obtain an average (interpolated) stiffness which describes the system behaviour in a wide range of displacements. It was decided to use this stiffness value to study the dynamic response of the system, which is equivalent to the HDRD s secant stiffness at a shear strain γ equal to 7%, in that this was considered to better represent the real behaviour obtained experimentally. F c γ s (a) (b) Figure 1: (a) Approximate constitutive law (b) Secant stiffness diagram. Figure 11: Secant stiffness integration curve. 4 Comparison between theoretical conclusions and experimental results The following table shows the stiffness and damping values obtained analytically according to the real stiffness and damping values of the frame and steel brace with dissipating devices.

10 448 Structures Under Shock and Impact VIII The table shows that the results obtained from the theoretical formulation underestimate the stiffness and overestimate damping, with respect to the average values obtained from the experimental investigation. This discrepancy means that the structural design based on the theoretical formulation is characterised by an underestimation of the actions on the brace and consequently of the additional shear on the joints of the real structure. Table 1: Comparison of expected damping ratio and stiffnesses. Configuration (a) Configuration (b) 5 Numerical model Theoretical value Experimental value Stiffness 1.7 kn/mm kn/mm Damping 15.4 % 12 % Stiffness 7.9 kn/mm 8.84 kn/mm Damping 12.4 % 1 % A two-dimensional numerical model was created with Sap2NL where the HDRD was implemented as a chevron for which a suitable value of the elastic module of the rubber was selected in such a way as to obtain the experimental stiffness (reduced secant stiffness) of the braced frame tested experimentally. The scheme adopted is that shown in Figure 12. The dissipating devices with configurations (a) and (b) were simulated with the average stiffness values. Firstly, the model was verified by subjecting it to constant load sinusoidal cycles identical to those imposed during the test, assigning both the stiffness and damping values obtained and the average values. The numerical results coincide with those obtained experimentally, no matter what the load amplitude is: the difference is always less than.3mm of peak displacement. The response of the numerical model was then studied under real seismic conditions, imposing the real acceleration time histories of the 194 El Centro, the 1976 Friuli, the 1972 Ancona and the 1981 Campania shakings in order to cover a wide range of frequencies and thus evidence the critical behaviours of each configuration. The original RC frame was also tested and the theoretical damping and stiffness values forecast were implemented in the system. K D K F /2 K B K F /2 Figure 12: Structural scheme adopted in the numerical model.

11 Structures Under Shock and Impact VIII 449 From the comparison of the numerical results (Figure 13) it is evident that the equipped configuration shows a significant reduction of response in terms of displacements, while it is also evident that if only the theoretical values were considered, a non conservative result would be obtained. In particular, the underestimation of the stiffness due to theoretical conclusions leads to non conservative results in terms of storey shear and axial force on the brace, thus underestimating the added strain on the beam-column joint. Lateral Displacement Acceleration max lateral displacement [mm Main Frame DBF "a" DBF "b" DBF "a"-theory DBF "b"-theory peak acceleration [m/s^2] DBF "a" DBF "b" DBF "a"-theory DBF "b"-theory Base Shear Axial Force Base Shear [kn] DBF "a" DBF "b" DBF "a"-theory DBF "b"-theory brace axial force [kn] DBF "a" DBF "b" DBF "a"-tarrc DBF "b"-tarrc 1 5 Figure 13: Numerical results of linear time-history analyses. 6 Conclusions The force-deformation relationship of the equipped system may be forecast, in the case of a rigid brace with respect to the frame, starting from the constitutive laws of the structure which is to be equipped and of the dissipating devices, simply by summing them. The characteristics of stiffness of the equipped system are variable with respect to the range of displacements applied, in compliance with theoretical application. The experimental results show that it is not conservative to consider the HDRD s stiffness evaluated at 1% of shear strain as is currently advised to take. A critical frequency was also reported in which amplification of the displacements is much higher than that reported during the resonance phase.

12 45 Structures Under Shock and Impact VIII To better describe the real situation in a wide range of displacements, the authors conservatively propose adopting as the design stiffness the value corresponding to 7% of shear strain of the device (Average Stiffness) and to also carry out a control with a reduced value equal to 15% of strain (Reduced Average Stiffness), which is representative of the behaviour of the system at the critical frequency (Figure 14). Lateral Displacement Acceleration max lateral displacement [mm Main Frame DBF "a" AS DBF "a" RAS DBF "b" AS DBF "b" RAS peak acceleration [m/s^2] DBF "a" AS DBF "a" RAS DBF "b" AS DBF "b" RAS Base Shear Axial Force Base Shear [kn] DBF "a" AS DBF "a" RAS DBF "b" AS DBF "b" RAS brace axial force [kn] DBF "a" AS DBF "a" RAS DBF "b" AS DBF "b" RAS 1 5 Figure 14: Numerical results with average stiffness and reduced average stiffness. The two-dimensional model proposed gives an excellent approximation of the experimental behaviour of the equipped system and can be used to quantify to what extent the implementation of dissipating braces on the structure actually reduces the response in seismic conditions with extremely diversified frequencies and amplitudes. The calculation performed with the reduced average experimental values is more conservative from all points of view with respect to the use of the values given theoretically. High damping rubber show very interesting features such as re-centring ability, no evident modification in damping capacity even after a large number of cycles (at least up to 12% shear deformation), no evident modification of overall dynamic characteristics with temperature.

13 Structures Under Shock and Impact VIII 451 Acknowledgements The high damping rubber devices used in this experimental study were supplied by TARRC (Tun Abdul Razak Research Centre) whose co-operation is greatly acknowledged. References [1] Dumoulin, C., Magonette, G., Taucer, F., Fuller, K.N.G:, Goodchild, I.R., Ahmadi,H.R., Viscoelastic energy dissipaters for earthquake protection of reinforced concrete buildings, 11 th European Conference on Earthquake Engineering, 1998 Rotterdam. [2] Shen, K.L., Soong, T.T., Chang, K.C., Lai, M.L., Seismic behavior of reinforced concrete frame with added viscoelastic dampers, Engineering Structures, 17(5), [3] Soong, T.T., Dargush, G.F., Passive energy dissipation systems in structural engineering, John Wiley & son. [4] Ungar, E.E., Kerwin, E.M., Loss factors of viscoelastic systems in terms of energy concepts, The Journal of the acoustical society of America, vol.34 n.7, [5] Giacchetti R., Bartera F., Antonucci R., Earthquake upgrading of R.C. frames by steel dissipating braces, 4 th International conference on Behaviour of Steel Structures in Seismic Areas STESSA 23, Naples, Italy. [6] Bartera, F. & Giacchetti, R., Steel dissipating braces for upgrading existing building frames. 3 rd European conference on steel structures, 22, Coimbra.