2. Nonlinear Modelling of the High Damping Rubber Bearings
The elastomeric bearing can experiment different phenomena influencing the seismic response of the base-isolated buildings: (1) nonlinear decrease of the nominal horizontal and vertical stiffnesses due to the increase of axial load and lateral displacement, respectively; (2) decrease of the vertical stiffness due to axial shortening or lengthening which result from second order geometric effects; (3) change of the horizontal stiffness corresponding to the reduction of the critical buckling load which occur when the lateral deformation increases; (4) decrease of the vertical stiffness due to the onset of voids and microcracks inside the rubber (i.e., cavitation phenomenon).
This work presents a model which provides a simultaneous description of all the phenomena is explained. The model, henceforth named CM2CR (Coupled Model 2 with Cavitation and Rotation), uses two (horizontal and vertical) axial elastic springs, one rotational elastic spring, and two (horizontal and vertical) dashpots with viscous damping coefficients (
Figure 1) with the following stiffness and damping factors:
and
being the fundamental vibration period and the equivalent viscous damping ratio in the horizontal (vertical) direction. The first three analytical expressions are obtained by modifying the original stiffnesses at shear zero strain (i.e.,
,
) and zero rotation (i.e.,
)
using the additional parameters
by [
13],
and
by [
15] and
[
32]
In the above expressions
represents the total thickness of the elastomeric layers,
is the thickness of the elastomeric layer with diameter
area
, moment of inertia
and total height
To take into account that the critical load buckling enclosed in the Equation (9) slacks off when lateral deformation increases, this load must be replaced by a reduced load obtained by one of the following formulas proposed in [
14]
where
being
the angle corresponding to the superposition area between the top and bottom plates of the bearing and
the reduced effective area whose value changes according to the ratio between the design horizontal displacement and the diameter of the bearing presented in [
1,
37,
38]. In the above formulas, the primary and secondary shape factors are defined as
and
, respectively. Following the aforementioned approach, forces
and
corresponding to the horizontal and vertical springs and to the dashpots and moment
corresponding to the rotational spring are obtained by the formulas
Here
(
are horizontal (vertical) displacement and velocity,
is the vertical displacement of the elastomeric bearing at the onset of cavitation inside the rubber and
represents the modified vertical displacement corresponding to second order geometric effects. In the same formulas
describes the post-cavitation variation of the tensile stiffness depending on the rubber mixture and construction quality of the HDRB,
and
denote shear and volumetric compressibility modulus of the rubber,
is the compression modulus defined by
while the modulus
is equal to
To include in the CM2CR model the damage (i.e., microcracks) affecting the rubber when loading and unloading take place many times beyond the cavitation threshold, the original expression of the cavitation load
must be replaced by the formula proposed in [
32]
where
represents the bonder rubber area before cavitation.
In this work the performances of the above CM2CR model are compared with those provided by three more simple HDRB models, denoted by the acronyms VEL (ViscoELastic Model), CM1 (Coupled Model 1) and CM1C (Coupled Model 1 with Cavitation). The first one simulates the behavior of the HDRBs through viscoelastic linear laws proposed by the Italian technical regulations [
6] as an alternative to complex hysteretic laws. It uses the expression of the horizontal and vertical stiffness at zero shear strain and the expression of the rotational stiffness at zero rotation presented in [
15,
34]. The two remaining models change the stiffnesses adopted in the VEL model by additional parameters to take into account for the phenomena summarized in the first part of this section. In particular, the CM1 model uses the same damping factors and rotational stiffness as the VEL model while correcting the remaining stiffnesses by two additional parameters
and
The latter can take into account the nonlinear decrease of the nominal horizontal and vertical stiffnesses due to the increase in axial load and lateral displacement [
7,
13], the decrease in vertical stiffness due to axial shortening or lengthening resulting from second geometric effects [
13,
34], the change of horizontal stiffness corresponding to the reduction of the critical buckling load which occurs when the lateral deformation increases [
19,
32]. The additional parameter
is obtained from the formula
where the nominal stiffness ratio
and the geometric ratio
are defined as follows
As for the CM1C model, it assumes damping factors and rotational stiffness of the VEL model and the horizontal stiffness of the CM1 model while it corrects the vertical stiffness by means of the additional parameter
to consider the decrease of the vertical stiffness due to the onset of voids and microcracks inside the rubber (i.e., cavitation phenomenon) [
22,
32].
Table 1 provides a comparison of the stiffnesses of the different models obtained after updating the stiffnesses at zero shear strain and at zero rotation of the VEL model by the above-mentioned additional parameters.
3. Layout and Design of the Test Structures
The seismic response of r.c. framed buildings can be notably influenced by the nonlinear behavior of the bearings used in the base-isolation system. These devices must retain their ability to function also when subjected to nonlinear effects including those due to the tensile axial loads resulting from the combined action of vertical and seismic loads (i.e., cavitation, post-cavitation and buckling). To identify and study the consequences of such phenomena and estimate their importance in comparison with other nonlinear effects, three office buildings with three-(B3), seven-(B7) and ten-storeys (B10) are considered below. Their r.c. framed structures include masonry infill walls located along the perimeter, regularly distributed in elevation. In particular, the frames along X direction have two lateral bays of 4 m and two central bays of 5 m, those along Y direction contain two bays of 5 m (
Figure 2a) while the first floor and upper floors are of 4 m and 3 m in height, respectively (
Figure 2b). The superstructures of the above buildings are designed in line with the Italian seismic code NTC18 [
6]. To this end, a live load equal to 2 kN/m
2 for all the floors and a gravity load equal to 4.98 kN/m
2, 6.28 kN/m
2 and 6.68 kN/m
2 for the top, intermediate and base-isolated floors, respectively, are considered; an average weight of about 2.7 kN/m
2 for the masonry infill walls located along the exterior perimeter is also assumed. Seismic loads are evaluated on the basis of the masses summarized in
Table 2 and assuming: elastic response of the superstructure (behaviour q
H = 1.0); medium-dense subsoil (type class C); high-risk seismic zone, with peak ground acceleration in the horizontal direction
PGAH = 0.201 g, at the life-safety (LS) limit state, and
PGAH = 0.264 g, at the collapse prevention (CP) limit state. Moreover, a yield strength of 375 MPa for the steel and a cylindrical compressive strength of 25 MPa for the concrete are assumed to design both the beams and columns of the test structures, whose geometric dimensions are summarized in
Table 3,
Table 4 and
Table 5. The base-isolation system of the test structures is made up of fifteen (i.e.,
nb = 15) identical high-damping-rubber-bearings (HDRBs). The first step in their design consists of evaluating the horizontal and vertical stiffnesses (
KH0,
KV0) of the single elastomeric device considering the same fundamental vibration period in the horizontal direction (i.e.,
T1H = 2.5 s) and selecting a nominal stiffness ratio α
K 0=
KH0/
KV0 (i.e., 400, 1200, 2400). To this end the following formulas can be used
The damping coefficients (
CH0,
CV0) of the single HDRB are then obtained assuming different equivalent viscous damping ratios in the horizontal (
ξH = 10%) and vertical (
ξV = 5%) direction by
Next, the effective geometric dimensions of the HDRB, like diameter
D, total thickness
te of the elastomeric layers and thickness
ti of the single elastomeric layers, are evaluated by an iterative procedure including: a design displacement
Sd = 27.8 cm, at the collapse prevention (CP) limit state of the Italian seismic code [
6]; a shear modulus
G = 0.035 kN/cm
2; a volumetric compression modulus
Eb = 200 kN/cm
2. This procedure is repeated until the effective horizontal and vertical stiffnesses (
KH,
KV) match the initial stiffnesses (
KH0,
KV0), and the ultimate limit state verifications regarding the maximum shear strains are fulfilled
Here
γtot,
γs,
γc and
γα represent the total shear strain of the elastomer and its shear strains due to the seismic displacement, axial compression, and angular rotation, respectively. In particular, the thickness of the single elastomeric layer (
ti), the total thickness (
te) of the elastomeric layers and the diameter
D of the bearing can be obtained by using the expressions
Two further verifications are added to complete the design of the bearings. The first obviates buckling phenomena by satisfying the following condition
In this expression
P is the maximum axial load on the single HDRB while
Pcr is its critical load
with reduced area
Ar. The second verification ensures prevention of rollout phenomena checking that
where
t1 (
t2) is the thickness of the contiguous rubber layer contiguous to the upper (lower) steel plates each having thickness
ts while
is the yield strength. Main properties of the base-isolation systems are presented in
Table 5,
Table 6 and
Table 7 for the B3, B7, and B10 test structures.
4. Numerical Results
Although traditional linear models of HDRBs (e.g., VEL model) are usually considered sufficient to predict the seismic response of base-isolated buildings, more complex models can become necessary in particular circumstances. Indeed, various developments can crop up, such as coupling of horizontal and vertical motions, change of critical buckling load due to large horizontal displacement, tension buckling, cavitation, and post-cavitation phenomena affecting the rubber layers in tension and strength degradation in cyclic tensile loading. Advanced nonlinear models of shear and axial laws of HDRBs, like the CM1, CM1C and CM2CR models mentioned in
Section 2, take into account these phenomena, providing more accurate expressions of their horizontal and vertical stiffnesses. Herein particular attention is paid to the study of the CM2CR model used to include the consequences of tension buckling and cavitation and post-cavitation phenomena. To assess the reliability of this model when the seismic response of base-isolated buildings subjected to strong EQs is simulated, the incremental dynamic analysis (IDA) of the test structures described in
Section 3 is carried out with reference to different values of the nominal stiffness ratio, comparing results provided by linear (i.e., VEL) and nonlinear (i.e., CM1, CM1C, CM2CR) models. For this purpose, a homemade computer code based on a lumped plasticity approach [
34,
35] has been developed, using a bending moment-axial load interaction domain for the columns and adopting a bilinear moment-curvature law with a hardening ratio of 5% for the sections of all frame members where inelastic deformations occur. Moreover, each beam is discretized in four elements of equal length and each column in a single element, since the inelastic deformations can appear at the end, quarter-span and mid-span sections, or at the end sections, respectively. An equivalent viscous damping equal to 2% is used for the superstructure following the Rayleigh approach, with reference to the fundamental vibration modes in the horizontal (Y) and vertical (Z) directions. As for the seismic input considered in the IDAs, three near-fault EQs are selected from the PEER database [
36,
37] ensuring that each one has long-duration horizontal velocity pulses and agrees with main design hypotheses (i.e., high-risk seismic region and medium subsoil class).
Table 8 lists names and dates of the EQs, the recording station, magnitude (
Mw), epicentral distance (Δ), horizontal (
PGAH), and vertical (
PGAV) peak ground accelerations of the selected near-fault ground motions whose horizontal and vertical response spectra are shown in
Figure 3a,b, respectively.
After some preliminary results presented in [
38], the accuracy of the CM2CR model is tested by examining the seismic response of superstructures and isolators of the test buildings. As first step, the ductility demands of beams and columns provided by the CM2CR model are compared with those obtained by less sophisticated HDRBs models (i.e., VEL, CM1, CM1C). To this end, the results are plotted as a function of the submultiple
of the peak ground acceleration of the selected EQ (i.e.,
), for each test structure, storey, nominal stiffness ratio and HDRB model. More specifically, three different values of the nominal stiffness ratio
(i.e., 400, 1200, 2400) and three different near-fault EQs, one with prevailing horizontal component (i.e., Chi-Chi EQ), one with prevailing vertical component (i.e., Imperial Valley EQ), and one with comparable horizontal and vertical components (i.e., Northridge EQ), are considered in the IDA of three—(i.e., B3), seven—(i.e., B7) and ten—(i.e., B10) storey buildings. For the sake of brevity, ductility demands of the B3 structure corresponding to nominal stiffness ratio
= 1200 are plotted in
Figure 4 with reference to the Imperial Valley EQ. These figures show that the different HDRB models provide overlapping results until the coefficient
attains a specific value, different for each storey. Once exceeded such threshold, the different models continue to provide similar results except for the first storey (
Figure 4a–c) where the ductility demands corresponding to the CM2CR model are lower than those provided by the other models, with a percentage reduction compared to the VEL model reaching −110.9% and −156.4% at the end sections and mid-span sections of the beams, respectively, when the maximum value of the acceleration ratio (i.e., α = 0.71) is attained (
Figure 4a,b). Moreover, the highest ductility demands in the mid-span section of beams are recorded at the third storey (
Figure 4h), where a significant contribution of the vertical component of the selected earthquake is expected. In order to better understand the response of the superstructure, main parameters of the base-isolation systems are shown in
Figure 5a–c with reference to the most stressed isolator (see isolator n.9 in
Figure 2a): i.e., the maximum shear strain of the elastomer due to seismic displacements (
), the maximum total shear strain (
and the ratio (
) between the maximum and critical axial loads. As can be observed, the base-isolation system presents a behavior opposite to that of the superstructure, with the CM2CR model highlighting the most conservative results, contrary to the other models which correspond to the lower-bound values of
(i.e., an increase of about +25% with reference to the VEL model in
Figure 5a) and
(i.e., an increase of about +32% with reference to the VEL model in
Figure 5b) and
(i.e., an increase of about +52.8% with reference to the VEL model in
Figure 5c).
Graphs similar to the previous ones are plotted in
Figure 6, referring to the B7 structure (
. subjected to the Imperial Valley EQ. Specifically, ductility demands at the first level confirm that CM2CR model is the least conservative among those examined (
Figure 6a–c), with percentage reduction of about −95.7% and −165.5%, similar to those observed for the B3 structure. Effects of the vertical component of the Imperial Valley EQ, characterized by high values of spectral acceleration in the range of vibration periods of interest for the selected structure (see
Figure 3b and
T1V values in
Table 6) are more evident at the mid-span sections at the seventh floor (
Figure 6e), where bending moments due to the vertical seismic loads are more important than those due to the horizontal ones.
More evident differences among the four HDRB models can be observed by examining the ductility demands of beams and columns provided by the IDAs of the B7 structure with nominal stiffness
. As shown in
Figure 7 for the Northridge EQ, after an initial overlapping of results the ductility demands take different values depending on the HDRB model used in the IDA. In particular, at the lower storeys the highest and lowest values of ductility demands are obtained by the VEL and CM2CR models, respectively, with a percentage reduction of about −223.2% and −528.3% at the end and mid-span section of the beams respectively, while at the top storey the different models provide very close values. To complete the comparison, the results provided the IDA of the B10 structure.
Figure 8 shows the ductility demands of beams and columns obtained carrying out the IDA of the test structure with nominal stiffness ratio
subjected to Chi-Chi EQ. These results prove that all the HDRB models lead to comparable values of the ductility demand for all the columns and for all the mid-span sections of the beams while differences between the CM2CR model and the other models concern the end sections of the beams. Specifically, ductility demands for beams under the Chi-Chi ground motion, characterized by high values of the corresponding pseudo-acceleration (see
Figure 3a), are more evident, especially at the lower storeys (
Figure 8a,d), while the addition of the vertical component of Chi-Chi ground motion is negligible at the upper storeys (
Figure 8g,j). After having analyzed how the CM2CR model can influence the evaluation of the ductility demands of beams and columns, the influence of this numerical model on the predictions of the seismic response of the HDRBs is investigated in
Figure 9.
For this purpose, the maximum shear strain of the elastomer due to seismic displacements (
), the maximum total shear strain (
, the ratio between the maximum axial load and the critical axial load
and the ratio (
) between the maximum axial load and the cavitation load provided by the CM2CR model are compared with those provided by the other HDRB models. As with beams and columns, the above quantities are plotted as a function of the submultiple
of the peak ground acceleration of the selected EQs, for each test structure, nominal stiffness ratio, isolator, and HDRB model. For the sake of brevity, only the results for the most stressed isolators are presented below, denoting each device by the reference number in
Figure 2a. In particular, the plots of
Figure 9 show the shear strain
of the corner isolator 1 when the IDA of the test structures B3, B7, and B10 are carried out considering the nominal stiffness
and the Chi-Chi EQ. These results prove that as the value of the submultiple α of the peak ground acceleration increases, the shear strains
obtained by the CM2CR model and other models diverge progressively although their starting values, corresponding to
, are close. Specifically, percentage variations of
range from +29.1% for the B3 (
Figure 9a) to +33% for the B10 (
Figure 9c). The same remark can be made for the Imperial Valley EQ while for the Northridge EQ the different HDRB models lead to similar evaluations of the shear strains affecting the elastomeric bearings. A similar situation can be observed when comparing the total shear strains
, as shown by
Figure 9 where the plots corresponding to the perimetral isolator 7 and to the Chi-Chi EQ are presented. Specifically, percentage variations of
, range from +40.7% for the B3 (
Figure 9d) to +21.3% for the B10 (
Figure 9f). Certain differences can be noted in the plots of
Figure 9 where the ratio
, provided by the IDAs for the Imperial Valley EQ, is presented for isolators 8 and 9. They highlight that the CM2CR and VEL models provide the highest and lowest values of the ratio
all the test structures, with a percentage variation of about +38.1% and +19.2% for the B3 and B10 test structures respectively, while the CM1 and CM1C models lead to intermediate results. These differences are less evident for the Chi-Chi and Northridge EQs. Finally, the plots where the ratios
provided by the different HDRB models are presented for the Imperial Valley and Northridge EQs. They show that substantial differences are obtained when using different nonlinear models for HDRBs, also depending on the nominal stiffness ratio of the isolation system. Further results omitted for the sake of brevity highlight similar behavior for the Chi-Chi and Northridge EQs.