AN EXPERIMENTAL CONFIRMATION OF THE EQUAL DISPLACEMENT RULE FOR RC STRUCTURAL WALLS

Transcription

1 / AN EXPERIMENTAL CONFIRMATION OF THE EQUAL DISPLACEMENT RULE FOR RC STRUCTURAL WALLS Pierino LESTUZZI Marc BADOUX Dr., IS-Institute of Structural Engineering, Swiss Federal Institute of Technology Lausanne Post: ENAC-IS-IMAC, EPFL, CH- Lausanne. Phone: , Fax: / Prof. Dr., IS-Institute of Structural Engineering, Swiss Federal Institute of Technology Lausanne Post: ENAC-IS-BETON, EPFL, CH- Lausanne. Phone: marc.badoux@epfl.ch Keywords: dynamic tests, equal displacement rule, reinforced concrete, structural wall, seismic nonlinear behaviour, energy, local ductility, inelastic displacement INTRODUCTION The equal displacement rule is a well known empirical rule for the assessment of the non-linear behaviour of structures subjected to earthquake ground motion. As illustrated in Figure, the equal displacement rule states that inelastic peak displacements (u p ) remain almost the same as elastic peak displacement (u el ) whatever the selected yield strength (F y =F el /R or yield displacement u y =u el /R) of the structure. Note that assuming that stiffness is independent of strength, the equal displacement rule leads to strength reduction factor (R) equal to the global displacement ductility (µ =u p /u y ). A competing rule is the equal energy rule, which states that the area under the curve remain almost constant. The equal displacement rule plays a significant role in current seismic design since it constitutes the basic assumption for the definition of the strength reduction factors (e. g. behaviour factor q in EC8) for the majority of the building codes around the world. The equal displacement rule was intensively investigated numerically for recorded earthquakes as well as for synthetic earthquakes (significant investigations are reviewed in []). The investigations were mostly concentrated on non-linear Single-Degree-Of-Freedom-Systems (SDOF) defined by different hysteretic models (elastoplastic, Clough, Takeda, etc.). The equal displacement rule was found to be generally correct and almost independent of the hysteretic model, for both real and synthetic earthquakes, and for structures with natural frequencies under a frequency limit (generally between. Hz and Hz). force F el elastic F y = F el /R inelastic u y = u el /R u p = u el displacement Fig.. Elastic and inelastic force-displacement relationships relating the equal displacement rule. Dynamic tests on reinforced concrete structural walls using the ETH earthquake simulator have provided the opportunity for an experimental confirmation of the equal displacement rule. The objective of the tests was to analyse the dynamic non-linear behaviour of reinforced concrete walls under seismic excitations. Although the test results were intended to calibrate input parameters for numerical models and to check existing design rules of structural walls, the results may be used to confirm the equal displacement rule. The results of the entire dynamic tests are published in [] and [3], related interpretations are published in [4]. All test data are available on the web site of the Institute of Structural Engineering of ETHZ []. The tests were conducted on RC structural walls designed with different strength reduction factors; therefore, they only differentiated in their reinforcement ratios. Walls with different flexural strengths

2 / were tested with the same synthetic earthquake. More specifically, two sets of test walls differing by 3% in their flexural strength are investigated in this paper. First, the measured relative peak displacements are compared. Despite their different strength, they should remain almost constant to confirm the equal displacement rule. With the dynamic tests it is possible to check the rule with different levels of excitation and to check if the rule is also valid for a pre-damaged wall. Afterwards, the input energy and the energy dissipated by plastic deformations (hysteretic energy) are investigated for both sets of test walls to compare their dynamic behaviour and to understand why the weaker walls did not exhibit greater relative peak displacements than the stronger ones. Finally, the measured curvatures at the base of the walls are compared to check if a difference appeared in the distribution of the plastic deformations. DYNAMIC TESTS The dynamic tests were performed using the ETH earthquake simulator (shake table) of the Institute of Structural Engineering (IBK) at the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland. Six walls, labelled WDH to WDH6, were tested. As shown in Figure, the test walls were designed to be representative at :3 scale of a structural wall of a three-storey reinforced concrete reference building. The reference building is a typical structural wall system consisting of flat slabs, slender columns designed for gravity loads only and a few relatively slender structural walls of rectangular cross-section. Storey masses 3 x tons RC wall in scale :3 h = 3 x.36 m ETH shake table with RC wall Reality 3-storey building with RC structural walls Fig.. Reference building with structural wall system and the corresponding reinforced concrete test wall in scale :3 with test set-up.. Test set-up The earthquake simulator consists of the shake table, the jack moving it, the electronic command and three storey masses consisting of rolling steel carts loaded with t of steel bars (Figure, right). Following initial tests with smaller storey masses directly attached to the walls [], the test set-up was extended with the three horizontally moveable masses attached to the wall by pinned steel struts. They were placed on rails, mounted on a separate structure, on which they could move freely except for slight friction damping due to their rolling. In this way, the desired relation between tributary areas of gravity loads and horizontal inertial forces could be obtained easily. This test set-up, therefore, permitted a realistic simulation of the reference building. Only the favourable and relatively small stiffness resulting from the frame effect of the slabs and columns with the structural walls in the real building was neglected. The axial force due to gravity loads at the base of the wall (plastic region) was applied by external post-tensioning bars, which were attached to the shake table and at the top of the wall. The wall footing was rigidly connected to the shake table, which operates in one horizontal direction and may move up to mm in each direction [6]. The resulting time-histories of dynamic bending moments and shear forces together with the axial force simulate well the stress situation in the wall during a real earthquake.

3 3/. Test walls Among the six tested walls, four of them, WDH3 to WDH6, are of interest for this paper. They were rectangular in cross-section with the following dimensions: horizontal wall length l w =.9 m, wall width b w =. m and total height including footing h tot =4.6 m. The related aspect ratio was 4.:, resulting in a wall behaviour dominated by flexure. The scale :3 enables the use of ordinary reinforcing steel with diameters of ø4. to ø8 mm and cement paste concrete with a maximum aggregate diameter of 6 mm. The main structural characteristics of the walls WDH3 to WDH6 are summarized in Table. Table Selected structural characteristics of the considered test walls. wall d e [mm] d w [mm] ρ tot [%] s [mm] s/d e [-] M y [knm] u y [mm] WDH3&WDH WDH WDH d e : diameter of the vertical reinforcement in the boundary region of the wall d w : diameter of the vertical reinforcement in the web region of the wall ρ tot : total vertical reinforcement ratio s: vertical spacing of the stabilizing horizontal reinforcement in the boundary region of the wall M y : nominal flexural strength u y : yield displacement The test walls were designed according to the capacity design method [7] with the exception of WDH, which was designed conventionally according to the swiss building code. It was not possible to test walls with high ductility because of limitations of the test set-up. Therefore, capacity design for middle or limited ductility with a displacement ductility of µ =3 was used. The vertical reinforcement consists mainly of bars with diameters of ø. mm for which a special high ductility steel was used. The reinforcement at the base of the walls WDH3 to WDH6 is shown in Figure 3. WDH3 and WDH4 were identical. This was done to make possible a comparison of the loading sequence on the wall behaviour. The vertical reinforcement of WDH3 and WDH4 consisted of bars with diameters of ø. mm (see Figure 3 left), leading to a total vertical reinforcement ratio of ρ tot =.47%. For WDH and WDH6 the diameter of the 4 edge vertical bars (d e ) were increased up to ø8 mm while the diameter of the 6 web bars (d w ) remained ø. mm (see Figure 3 right), leading to a total vertical reinforcement ratio of ρ tot =.6%. Considering a constant axial force of 7 kn, corresponding to 3% of the resistance of the gross section, the nominal flexural strength (M y ) of WDH3 and WDH4 was M y =4 knm and the one of WDH and WDH6 was M y =46 knm (see Table ). As a consequence, the nominal flexural strength of WDH and WDH6 was approximately 3% higher than the one of WDH3 and WDH4 (46/4=.8). WDH and WDH6 were identical except for the stirrups spacing (s) in the boundary region of the wall (i. e. s= mm for WDH and s=6 mm for WDH6). Unlike for the other test walls, the normalized stirrups spacing (s/d e ) of wall WDH did not satisfy capacity design requirements for stabilization of the vertical reinforcement bars in the plastic region (i. e. s/d e <8 according to [8]). As the test results showed, this however did not affect the seismic response leading to the conclusion that the considered limit appeared to be too large and have to be modified to incorporate the unfavorable ductility properties of European reinforcing steel []. WDH and WDH6 may be considered to be identical and are therefore referred to as twin walls in the following. The yield displacements (u y ) were directly derived from measurements [4]. Despite the different strengths, the obtained values, u y = mm for WDH3 and WDH4 and u y =3 mm for WDH and WDH6 are very close together. These results are in accordance with the conclusions of Priestley and Kowalsky [9], which showed that the yield displacement of a rectangular cantilever wall depends principally on the wall length and on the steel yield strain. By contrast, the equal displacement rule assumes that the yield displacement depends on the strength (see Figure ).

4 4/ WDH3 and WDH4 WDH and WDH6 Cross-section stirrup Ø 4. s = 8 stirrup Ø 4. s = 8 Ø. stirrup Ø 4. s = or 6 Ø 8 Ø 8 6 Ø Elevation s = WDH WDH s = s = Fig. 3. Reinforcement at the wall s base for the two sets of twin specimens (dimensions in mm). displacement [m] velocity [m/s].. d min = -8.4 mm d max = 8.4 mm v max =.88 m/s v min = -.3 m/s S a [m/s ] 4 3 Eurocode 8 calculated. frequency [Hz] 4 3 acceleration [m/s ].. -. a max =.4 m/s a min = -.7 m/s S a [m/s ] time [s].. period [s] Fig. 4. Synthetic earthquake compatible with Eurocode 8 s design spectrum for soft soils.

5 /.3 Dynamic excitations A synthetic spectrum-compatible earthquake was used for the tests in order to facilitate the comparison of the results with the code assumptions. The synthetic earthquake was generated using a stationary simulation. Figure 4 shows the ground motion used in the tests. It is compatible with the design spectrum of Eurocode 8 (ENV to 3) [] for soft soils and a peak ground acceleration of.6 m/s. The time-histories of the displacement, the velocity and the acceleration (left) and the related acceleration response spectra (right) are plotted. The corner frequencies of. Hz and Hz define the plateau with a constant spectral acceleration of 3.6 m/s. The earthquake lasts approximately 4 seconds. Each wall was subjected to several tests. The sequence of the tests is indicated in Table. An 8% earthquake means that the accelerations of the test excitation reached 8% of the accelerations of the reference synthetic earthquake shown in Figure 4. 3 MEASURED RELATIVE DISPLACEMENTS Key test results are summarized in Table for the tests in which no failure occurred. The test walls exhibited a stable ductile behaviour as they were subjected to significant non-linear deformations, reaching in all these cases global ductilities (µ ) above 3. and average storey drifts (δ m ) of more than.8%. Despite the different strengths, the measured relative peak displacements at the 3 rd floor for the % earthquake tests remained remarkably constant (between u 3 floor =7.3 mm and u 3 floor =73. mm). This confirms the equal displacement rule. Table Summary of key test results wall test earthquake u 3 floor [mm] δ m [%] µ [-] E tot [kj] E hyst [kj] WDH3 % WDH4 8% % % % % WDH 8% % WDH6 8% % u 3 floor : peak relative displacement at the third floor δ m : average storey drift (u 3 floor / wall height) µ : displacement ductility (u 3 floor / u y ) E tot : total input energy E hyst : energy dissipated by plastic deformations The time-histories of the measured relative displacements for the first two tests of WDH3 to WDH6 are plotted in Figure. From the comparison of the relative displacements of WDH and WDH6, it is clear that the variability in the seismic behaviour is small enough to ensure the reproductibility of the tests. The comparison of the relative displacements of WDH3 and WDH4 for % earthquakes is still more instructive. The peak values are quite identical; however, they did not occur at the same time nor in the same direction. It should be noted that the twin specimens WDH3 and WDH4 were not tested with the same sequence of dynamic excitations. WDH3 failed at the second excitation of 8% after an initial excitation of %. In the inverse sequence of excitation, WDH4 survived an 8% and a % excitation, as well as three further 6% excitations. This difference showed the influence of the sequence of excitations []. However, regarding peak displacements, these results show that the equal displacement rule is also valid for a pre-damaged wall (WDH4, % after an initial test of 8%). Initial cracking influences only the first small cycles of the seismic response.

6 6/ relative displacement [mm] relative displacement [mm] relative displacement [mm] relative displacement [mm] - WDH3,. EC8 soft soils % WDH3,. EC8 soft soils 8% -7 mm 7.39 s mm 7.36 s time [s] 4 mm 9.9 s 4 mm.3 s mm.68 s - WDH4,. EC8 soft soils 8% WDH4,. EC8 soft soils % WDH6,. EC8 soft soils 8% WDH6,. EC8 soft soils % 34 mm 6.7 s -48 mm 7.3 s 4 mm 9.7 s mm 7.39 s mm.3 s 7 mm 9. s WDH,. EC8 soft soils 8% WDH,. EC8 soft soils % 34 mm 6.7 s -47 mm 7.3 s mm.3 s failure -73 mm 7.36 s - -7 mm 7.36 s - time [s] u / wall height [%] u / wall height [%] u / wall height [%] u / wall height [%] st floor nd floor 3 rd floor Fig.. Time-histories of the measured relative displacements of walls WDH3 to WDH6 for the first two tests. It is noteworthy that the equal displacement rule is also confirmed for the smaller dynamic excitations of WDH4. The measured relative peak displacement of approximately 9 mm for the 8% earthquake (Table, WDH4 test ) corresponds well to 8% of the mean value of the relative peak displacement for the % earthquakes (8% of 7 mm is 7 mm). Moreover, the measured relative peak displacements of approximately mm for the three 6% earthquakes (Table, WDH4 tests 3, 4 and ) correspond also approximately to 6% of the mean value of the relative peak displacement for the % earthquakes (6% of 7 mm is 44 mm). Hysteretic loops give a good insight of the overall seismic response during the tests. Figure 6 shows the hysteretic loops of walls WDH3 and WDH measured during the % earthquake tests. The bending moment at the wall s base is plotted against the measured relative displacement at the

7 7/ 3 rd floor. The curves confirm that significant plastic deformations occurred during the tests. Despite the difference of the flexural strength, the hysteretic curves are similar in shape and have identical peak relative displacements in both directions. The curves for the weaker wall WDH3 are only scaled down in the vertical direction by a factor corresponding approximately to the flexural strength ratio. WDH3,. EC8 soft soils % WDH,. EC8 soft soils % M base [knm] mm 73 mm relative displacement at the 3 rd floor [mm] Fig. 6. Hysteretic loops of walls WDH3 and WDH for the % earthquake tests. 4 FUNDAMENTAL FREQUENCY To be complete, a discussion on experimental confirmation of the equal displacement rule should include consideration of the stiffness. The equal displacement rule assumes equal stiffness (see Figure ), whereas one would expect different stiffness for the weaker and the stronger wall sets. The following two observations apply: Analysis of the dynamic response of the test walls showed that the effective fundamental frequency is almost the same for all tests, except for the initial 8% earthquake tests of the stronger walls (WDH and WDH6). Note that WDH and WDH6 were pre-damaged for the next % earthquake tests. As reported in a companion paper [], numerical simulations estimated the effective fundamental frequency to be.37 Hz for walls WDH and WDH6 for the initial 8% earthquake test and. Hz for all other tests. This means that the effective stiffness for both the weaker and the stronger wall sets are very similar for the % earthquake tests. As performed in the previous section, the direct comparison of their peak displacement can be conducted without consideration of stiffness difference. In the case of the smaller initial dynamic excitations of WDH and WDH6, the difference in effective stiffness explains why the equal displacement rule does not apply linearly. The measured relative peak displacements of approximately 47 mm for the 8% earthquake test (Table, WDH and WDH6 test ) do not correspond to 8% of the mean value of the relative peak displacement for the % earthquakes (8% of 7 mm is 7 mm). However, if one considers a modification factor accounting for the frequency shift, the equal displacement rule is confirmed again. Assuming constant spectral acceleration (plateau in Figure 4), the modification factor corresponding to the ratio of spectral displacements can be calculated with the square of the ratio of the fundamental frequencies, in this case (.37 Hz/. Hz) =.. The modified value of peak displacement (. x 47 mm=6 mm) corresponds then well to 8% of the mean value of the relative peak displacement for the % earthquakes, confirming also in this case the equal displacement rule. ENERGY Energy considerations are useful in understanding the observed constant peak relative displacements. The determination of the energy, more specifically the total input energy and the energy dissipated by plastic deformations (hysteretic energy) at the end of the tests, shows the

8 8/ difference in the seismic behaviour and provides a justification for the almost constant value of the peak relative displacements. The energy values are computed from the measured response. The total input energy is calculated by integrating numerically the jack forces with respect to the shaking table displacements. The hysteretic energy at the end of the tests may be calculated by integrating the storey-forces with respect to the related storey-relative displacements [4]. The total input energy and the hysteretic energy at the end of the tests are given in Table for the considered test walls. The cumulative energy time-histories for the test walls WDH3 and WDH during the % earthquake tests are plotted in Figure 7. The different energy parts are plotted together. The total input energy (E tot ) consists of the sum of dissipated energy and recoverable energies (kinetic, E kin and strain, not plotted). The energy can be dissipated by friction (E frict ), by viscous damping (E D ) or by plastic deformations (E hyst ). In this particular test set-up, friction energy appears in the rolling of the moveable storey masses. The cumulative energy time-histories for the two test walls show that: The energy dissipated by plastic deformations is lower for the weaker wall WDH3, by a factor corresponding to the flexural strength ratio (8.7/6.9=.7). The total input energy was also smaller for the weaker wall WDH3, again by a factor corresponding to the flexural strength ratio (./9.8=.4). In other words, the weaker wall compensated for its smaller hysteretic dissipation capacity by absorbing less seismic energy. This confirms that the weaker wall does not have to have larger peak relative displacement to dissipate the absorbed seismic energy. WDH3,. EC8 soft soils % E tot 9.8 E frict energy [kj] E kin E D 6.9 E hyst time [s] WDH,. EC8 soft soils %. energy [kj] E frict E tot E D 8.7 E kin E hyst time [s] Fig. 7. Cumulative energy time-histories for the walls WDH3 and WDH for the % earthquake tests. 6 LOCAL DUCTILITY Since the estimated yield displacements and the peak relative displacements were similar for both sets of walls, the global ductility (displacement ductility) was similar for both sets of test walls (see Table ). Differences were, however, observed in terms of the local ductility (curvature ductility) reached at the wall s base. Figure 8 shows the distribution of the peak curvatures at the wall s base of

9 9/ test walls WDH3 and WDH for the % earthquake. During the tests, the vertical deformations of both sides of the wall at the wall s base were measured at different heights of the plastic region. The measurements enabled the determination of the mean curvatures for four portions at the base of the walls []. The lower portion is extended into the wall footing, to a depth corresponding to times the diameter of the greater vertical reinforcement bar (d e in Table ). This extension takes into account the pull-out effect of the vertical reinforcement bars. The measured curvatures of the stronger wall WDH are better distributed. The curvature of the weaker wall WDH3 is more concentrated at the base of the wall, reaching a peak of 6-3 m - (vs. 43 for the stronger wall). In other words, the local ductility demand (curvature ductility) was significantly higher (6 vs. ) for the weaker wall. WDH3,. EC8 soft soils % 9 curvature ductility [ ] curvature [-3 m-] WDH,. EC8 soft soils % 9 curvature ductility [ ] curvature [-3 m-] Fig. 8. Distribution of the curvatures at the wall s base for the walls WDH3 and WDH for the % earthquake tests. 7 DESIGN IMPLICATIONS As the test walls provide a realistic simulation of a building subjected to earthquake loading, the exposed experimental confirmation of the equal displacement rule strengthens the classic design assumptions for ductile structures. More specifically, the behaviour factors (q in EC8) corresponding to the selected level of global ductility according to the equal displacement rule are well adapted for design purpose in case of reinforced concrete. The experimental results confirm the well known consequence of the equal displacement rule, which is that the selection of a weaker wall in the design does not lead to increased relative displacements but to increased local ductility demand. In other words, increasing behaviour factors increase local ductility demand and more demanding design and detailing requirements are needed in order to ensure that the available local ductility is sufficient. All other things equal, this means that a possible advantage of a stronger wall is not to reduce displacements in a strong earthquake but to reduce local damage level.

10 / 8 SUMMARY AND CONCLUSIONS Dynamic tests on 3-storey reinforced concrete walls have provided the opportunity for an experimental confirmation of the equal displacement rule. Two sets of identical walls differing only by 3% in their flexural strength were investigated. The test walls exhibited a stable ductile behaviour as they were subjected to significant non-linear deformations. The measured relative displacements at the 3 rd floor confirm the equal displacement rule since: Despite the different strengths, the measured relative peak displacements are almost equal for a given earthquake level. Even with a pre-damaged wall (WDH4), no significant difference appeared in the measured relative peak displacement. The determination of the energy shows that, at the end of the tests, the total input energy and the energy dissipated by plastic deformations (hysteretic energy) are different for both sets of test walls. Energy ratios correspond well to the flexural strength ratios. The weaker walls compensated for their smaller hysteretic dissipation capacity by absorbing less seismic energy. This explains how the weaker wall does not have to have larger peak relative displacement to dissipate the seismic energy. Through the comparison of the curvature ductility demand at the base of the walls, it was found that even though the global ductility demand is similar for weaker and stronger wall, the local ductility demand (curvature ductility) is higher for the weaker walls. As a consequence, the experimental results further confirm that the selection of a weaker wall in the design does not lead to increased relative displacements but to increased local ductility demand. REFERENCES [] Miranda E., Bertero V.: Evaluation of Strength Reduction Factors for Earthquake-Resistant Design. Earthquake Spectra. Vol, No., 994. [] Lestuzzi P, Wenk T., Bachmann H.: Dynamische Versuche an Stahlbetontragwänden auf dem ETH-Erdbebensimulator. Institut für Baustatik und Konstruktion (IBK), ETH Zürich. Bericht No. 4, ISBN X. Birkhäuser Verlag, Basel, [3] Bachmann H., Dazio A., Lestuzzi P.: Developments in the Seismic Design of Buildings with RC Structural Walls. Proceedings of the th European Conference on Earthquake Engineering, September 6-, CNIT, Paris la Défense, France, 998. [4] Lestuzzi P.: Dynamisches plastisches Verhalten von Stahlbetontragwänden unter Erdbebeneinwirkung. Dissertation ETH Nr. 376, Zürich,. [] Dazio A., Wenk T., Bachmann H.: Vorversuche an einer Stahlbetontragwand auf dem ETH- Rütteltisch. Institut für Baustatik und Konstruktion (IBK), ETH Zürich. Bericht No. 3, ISBN Birkhäuser Verlag, Basel, 99. [6] Bachmann H., Wenk T., Baumann M., Lestuzzi P.: Der neue ETH-Erdbebensimulator. Schweizer Ingenieur und Architekt SI+A, Heft 4/99, Zürich, 999. [7] Paulay T., Priestley M.J.N.: Seismic Design of Reinforced Concrete and Masonry Buildings. ISBN John Wiley & Sons, New York, 99. [8] Bachmann H.: Erdbebensicherung von Bauwerken. ISBN X. Birkhäuser Verlag, Basel, 99. [9] Priestley M.J.N., Kowalsky M. J.: Aspects of Drift and Ductility Capacity of Rectangular Cantilever Structural Walls. Bulletin of the New Zealand National Society for Earthquake Engineering, 998. [] Eurocode 8: Auslegung von Bauwerken gegen Erdbeben. Europäische Vornorm ENV bis 3 (SIA V 6.8 bis 83, Ausgabe 998). Schweiz. Ingenieur- und Architekten- Verein. Zürich, 998. [] Lestuzzi P., Badoux M.: The γ-model: a Simple Hysteretic Model for Reinforced Concrete Walls. Proceedings of the fib-symposium; Concrete Structures in Seismic Regions, Athens, 3.