Abstract
Vibrations in buildings can cause occupant discomfort in the form of annoyance, headache, or sickness. While occupant comfort is considered in international standards regarding the design of high rise buildings against wind loading, it is neglected in the design of buildings with seismic protective base isolation systems. Nevertheless, due to their low flexibility, base isolated buildings can be prone to wind-induced vibrations, which makes occupant discomfort a potentially significant serviceability limit state. This paper presents a study on occupant comfort conditions in wind-excited base isolated buildings. A numerical simplified parametric procedure is proposed in order to evaluate the return period of wind events causing human discomfort. A parametric investigation is then presented to evaluate the effects of salient parameters on comfort conditions. The procedure is based on (i) the nonlinear dynamic analysis of the structure modeled as a single-degree-of-freedom oscillator with hysteretic base isolators, (ii) the digital generation of time histories of turbulent wind velocity, and (iii) comfort evaluations based on international standards. Results demonstrate that discomfort conditions can occur a few times in a year, depending upon the wind-exposure of the site, what suggests considering this serviceability limit state in the design of base isolated buildings.
1. Introduction
Occupant discomfort caused by wind-induced vibrations is a major serviceability limit state in the performance-based design of high rise buildings [1], whereby humans have subjective sensitivity to floor acceleration that, in turn, can cause discomfort in the form of annoyance, headache, or sickness.
While it is almost accepted that discomfort is caused by intense floor acceleration, there is no agreement in the scientific literature and in international standard codes, on how to carry out comfort evaluations, as the problem involves aspects that are difficult to model, including physiological and psychological aspects [2]. In this regard, Kwok et al. [3] reviewed existing studies on human perception and associated tolerance thresholds of vibrations in tall buildings, highlighting the need for the development of internationally accepted practical occupant comfort serviceability criteria, accounting for the subjective nature of the problem. Kwon and Kareem [4] compared major international standards in regard to wind effects on buildings and highlighted that most of the technical standards use comfort criteria based on along-wind acceleration. Lamb et al. [5] investigated motion sickness in tall buildings highlighting the importance of considering differences in individual perceptions when examining the effects of building motion on personal comfort, which could also affect work productivity. The same authors reported that the effects of building motions on individual comfort are difficult to assess, because most of the occupants do not complain when they feel discomfort. Bernardini et al. [6] proposed a probabilistic framework for the performance-based design of tall buildings for occupant comfort, providing a computational tool that allows the designer to meet a certain acceptable probability of exceedance of comfort conditions by a certain number of occupants.
While the importance of occupant comfort is well-understood in the design of high rise buildings, specific design criteria are considered to limit discomfort caused by wind-induced vibrations.
The same issue is typically neglected in the design of buildings with base isolation devices, with the exception of some sporadic attempts that were made at the very beginning of the development of the technology [7]. The comfort conditions assume lesser relevance in earthquake resistant systems such as walls structures [8], while assuming increased relevance for irregular structures [9].
This is motivated by the general understanding that vibration effects caused by wind loading in the case of base isolated buildings are negligible. Nevertheless, base isolated buildings [10], even though often low-rise, have low natural frequencies of vibration, thus being potentially wind-sensitive and prone to wind-induced vibrations [11, 12]. Thus, the lack of international criteria and standards that account for occupant comfort in base isolated buildings under wind loading does not appear to be fully justified at the present state of the knowledge and the investigation of this problem is a worth research effort, as addressed in this study.
The present paper is a contribution towards a more aware understanding of the role that human comfort conditions may play in the performance-based design of base isolated buildings under wind loading. To this aim, a simple general methodology to evaluate the short term return period of wind events determining discomfort in wind-excited base isolated buildings is proposed, at first. This methodology, presented in Section 2, considers a simple single-degree-of-freedom nonlinear hysteretic oscillator representing a base isolated building under turbulent wind loading and is based on comfort analysis carried out according to widely accepted international standards [13–15]. The model is then used for carrying out some parametric investigations in Section 3, highlighting the major design parameters affecting the critical wind velocity above which discomfort is expected to occur. These results demonstrate that discomfort conditions may occur a few times in a year depending on the wind characteristics of the site and on the characteristics of the base isolated structures. Finally, the paper is ended with proper conclusions in Section 4.
2. Simplified Analytical Model
In this section we present the simplified parametric numerical model finalized at rapidly evaluating the return period of wind events determining occupants’ discomfort in buildings with base isolation devices. The procedure comprises the following steps:(1)definition of input parameters regarding both the site and the building;(2)digital generation of a time history of turbulent wind velocity;(3)nonlinear dynamic analysis by time integration;(4)evaluation of the response of the structure in terms of horizontal acceleration;(5)occupants’ comfort analysis according to acknowledged international standards;(6)assessment of the critical gust intensity determining human discomfort in the building;(7)evaluation of the return period of wind events causing occupant discomfort.Steps from (2) to (5) are iterated by considering increasing intensities of the wind gust, so as to determine, in step (6), the critical condition determining discomfort and its corresponding return period in step (7).
2.1. Structural Model
Assuming, without any loss of generality, that the base isolated building has an approximately rectangular plan, with one dimension that is much greater than the other, and that the wind blows along the shorter building dimension, , at a first level of approximation the dynamic behavior of the structure can be modeled by a single-degree-of-freedom nonlinear oscillator (Figure 1(a)), whose equation of motion is where is the lateral displacement, denoting time, is the mass per unit length of the building, is the damping coefficient, and is the wind force per unit length. In (1), the nonlinear restoring force of the base isolation devices is written aswhich is formally similar to an elastic restoring force, but with a nonlinear stiffness coefficient . This last is computed by modeling the constitutive behavior of the base isolators through an elastic hardening plastic hysteretic model, also called bilinear hysteretic model [18], as illustrated in Figure 1(b), where is the yielding displacement, is the ultimate displacement, is the initial preyielding stiffness, and is the postyielding stiffness. These quantities are calculated aswhere is the secant stiffness and is a characteristic parameter of the isolators that for elastomeric ones assumes the typical value of 4 [19]. The secant stiffness of the isolators is chosen to meet a certain value of the fundamental period, , of the isolated mode of the equivalent linear system as follows:However, it should be noted that, in preyield conditions, occurring in the case of low wind intensities, the system behaves as a relatively rigid linear one, with a fundamental period of the isolated mode, , given by
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2.2. Wind Loading Force
The wind loading force is expressed aswhere is the air density, is the total height of the building from the ground, is the pressure coefficient of the building, is the ten-minute mean wind velocity at a height from the ground, and is the turbulent wind velocity. Considering that , the term , in (6), can be neglected with a good approximation. According to Eurocode 1 [16], the ten-minute mean wind velocity is written as a function of the height from the ground through the classic logarithmic profile that models the atmospheric boundary layer aswhere is the basic wind velocity, which depends upon the geographical location of the site and its altitude above the sea level, is the terrain factor, is the height below which the logarithmic profile loses its significance and the mean wind velocity is considered as constant, and is the roughness length of the site.
These quantities assume the values summarized in Table 1, which are taken from the Italian technical standard according the definitions of terrain categories reported in Eurocode 1 [16]. The basic wind velocity, , is instead defined as the ten-minute mean wind velocity with a return period of 50 years, irrespective of wind direction, at a height of 10 m above flat open country terrain (terrain category II).
Substituting (7) into (6) and assuming wind turbulence to be perfectly correlated along the height of the building () yieldUnder the classical assumption of modeling wind turbulence as a zero-mean Gaussian stochastic process [20, 21], time realizations of the turbulent wind velocity are digitally generated through the widely adopted formula based on waves superposition [22], as follows: where is the power spectral density (PSD) of , with being the circular frequency, is the number of points in which the PSD is discretized in the interval , with being the cutoff frequency, is the step increment in frequency, and are independent random phase angles having uniform distribution in the range.
According to Eurocode 1 [16], the PSD of the turbulent wind velocity is modeled aswhere is the root mean square of modeled as follows: is the integral length scale of turbulence assumed to be equal to
2.3. Response Simulation and Comfort Analysis
The response of the nonlinear system modeled by (1) is obtained by direct time integration using a Runge-Kutta algorithm of order 4, after transformation in state-space first order form. The response is evaluated at discrete time steps, using a total number of time steps and a sampling time dt.
The comfort analysis is carried out by accounting for the provisions of the most widely acknowledged international standards in the field, namely, ISO 6897 [13], UNI 9614 [14], and ISO 2631-2 [15]. In particular, ISO 6897 [13] refers to the interval from 0.063 to 1 Hz, while UNI 9614 [14] and ISO 2631-2 [15] provide essentially similar provisions, focusing on the interval from 1 to 80 Hz.
According to the aforementioned international standards, the one-third-octave spectral analysis of the acceleration is carried out in the interval from 0.063 to 80 Hz and the acceleration levels, , expressed in dB, where denotes the th octave band, are computed aswhere is the effective value (root mean square) of the acceleration in the th octave band and = 10−6 m/s2 is the reference effective acceleration.
In order to carry out the comfort evaluation, it has to be considered that the lower thresholds of human perception of building acceleration depend upon the propagation axis of the vibration referred to the posture of the exposed person. In this regard, axes z and x-y defined in UNI 9614 [14] are considered, as shown in Figure 2. The case of an unknown posture, as considered in UNI 9614 [14], is instead neglected because it yields essentially similar results to the case of the propagation along the x-y-axes.
Figure 3(a) depicts the acceleration perception curves as reported in ISO 6897 [13] and UNI 9614 [14]. The acceleration perception curves in the interval 1–80 Hz, represented with continuous lines in Figure 3(a), are taken from the latter standard, while the curve referring to the x-y-axes in the interval 0.063–1.0 Hz is taken from ISO 6897 [13]. It should be noticed that, at 1 Hz, the x-y curves of UNI 9614 and ISO 6897 assume the same value. The curve referring to the z-axis in the interval 0.063–1.0 Hz is instead not reported in any of the considered standards and is here assumed by extending the curve of UNI 9614 below 1 Hz with a constant slope. It is noted that this branch of the curve is parallel to the curve referring to the x-y-axes.
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The acceleration threshold curves of Figure 3(a) are transformed into equivalent acceleration level curves, depicted in Figure 3(b), by means of (13). The curves in Figure 3(b) are used to weight the one-third-octave spectral analysis acceleration levels in such a way to make all acceleration levels equivalent in terms of perception and therefore of disturbance. To this aim, the generic acceleration level corresponding to the th octave, , is corrected aswhere is the frequency-dependent human threshold level at the th octave (Figure 3(b)), while is a constant threshold.
After application of (14), the level of the total frequency weighted acceleration, , is computed asThe comfort analysis consists of verifying the following condition:where represents a critical limit. According to UNI 9614 [14], values of the critical limit are differentiated for so-called “critical areas,” residences, offices, and factories, where critical areas are, for instance, operating rooms in hospitals, laboratories, or places where delicate manual works are carried out. The same limit values are also differentiated on the basis of the periods of the day to which the comfort analysis is referred and, in particular, day-time, from 7:00 a.m. to 10:00 p.m., and night-time, from 10:00 p.m. to 7:00 a.m. Values of the critical limits and of the constant threshold, in (14), are summarized in Table 2. Considering that the building acceleration considered in this work are horizontal, propagation along the -axis is only taken into account for residences at night.
2.4. Evaluation of the Return Period of Discomfort
The measure of the gust intensity adopted in this work to evaluate the return period of wind events causing discomfort is the ten-minute mean wind velocity at 10 m height, denoted as . The critical value of , is the one corresponding to the case where . The return period, , of , expressed in years, can be evaluated through the following equation, derived from the indications contained in Eurocode 1 [16], as a function of the basic wind velocity, , associated with a return period of 50 years: In the case of a return period shorter than one year, (17) cannot be applied. In this case, is approximated by using a spline interpolating curve passing through (, ) and through the curve versus , (17), with assuming integer values from 1 to 50. Figure 4 depicts the curves showing versus , for smaller than 1 year and for different values of , considering terrain categories from I to V. It is noted that, for fixed values of and , the return period monotonically increases with increasing , that is, with increasing terrain category, due to the effect of on the atmospheric boundary layer, (7).
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3. Simulation Results and Discussion
The simplified model presented in the previous section is adopted to perform some parametric investigations. A specific building example is considered at first and, then, salient parameters of the model are changed in order to evaluate their effects over comfort conditions.
3.1. Analysis for a Building Example
The parameters assumed in the first analysis case are summarized in Table 3. The analysis allows to obtain the curves representing the ratios between the levels of the total frequency weighted acceleration, , and the critical limits, , versus the ten-minute mean wind velocity at 10 m height, . The results of the analysis are presented in Figure 5(a). It is noted that when the wind velocity exceeds a certain critical value, discomfort conditions occur. This critical wind velocity value decreases with decreasing (see Table 2), ranging from about 3 m/s for critical areas and vibration propagating along x-y-axes up to about 8 m/s for factories. Values of the return period associated with these critical values of ten-minute mean wind velocity at 10 m height are plotted in Figure 5(b) as functions of the basic wind velocity. It is noted that, for the chosen case study parameters, discomfort conditions can occur about once every two months in critical areas. Considering that wind storms have typical durations of two-three days, it can be concluded that discomfort conditions may occur a few days in a year.
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3.2. Parametric Analysis by Varying Building and Site Characteristics
In order to gain more insight into the investigated problem, it is important to assess how variations in salient model parameters affect the return period of critical wind events. To this aim, the comfort analysis is carried out by varying building and site parameters, one at a time, using the values reported in Table 3 as references.
The parameter that mostly affects the critical wind velocity is the roughness length, which is associated with the terrain category (see Table 1). Figure 6 shows the results obtained by varying such a terrain category, both in terms of critical wind velocity and in terms of return period of discomfort conditions as a function of the basic wind velocity. These results show that the critical wind velocity decreases with increasing roughness length, , due to an increase in turbulence intensity, according to (11). Because, however, an increase in also determines a modification of the mean wind velocity profile, (7), and, consequently, an increase in the return period for a fixed value of the ten-minute mean wind velocity at 10 m height (Figure 4), a lower value of the critical value of such wind velocity does not necessarily entail a lower value of the return period of discomfort, . This is apparent by looking at the results of Figure 6, where the smallest value of , equal to 64 days, is obtained for a terrain category III, = 0.1 m, and for a basic wind velocity, , equal to 35 m/s.
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The results of the investigation considering variations in all other building and site parameters are summarized in Figure 7, where the basic wind velocity is assumed to be equal to 27 m/s and a terrain category V is considered. These results confirm that the lowest values of the critical wind velocity are obtained, again, for critical areas and vibrations propagating along the x-y-axes. The same results also show that the return period, , of discomfort conditions significantly decreases with decreasing building width, B, due to a reduction in mass for a constant wind exposed area. For the same reason, increases with increasing aerial mass of the floors, . A slight reduction of with increasing is observed, which is associated with a reduction of the stiffness of the isolators. On the contrary, is marginally affected by the height of the building, H, because an increased height, corresponding to an increased wind loading force, is compensated by an increased mass. The effect of an increased damping is, of course, that of reducing the structural vibrations and hence increasing . As a final remark, some variation of with the natural period, , of the isolated mode of the equivalent linear system is also observed that, however, is not monotonic. In particular, the worst conditions are obtained for almost equal to 2 s and 4 s, while different values of result in less critical situations.
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Figure 8 shows the results of some time history analyses carried out in order to investigate, in more detail, the role played by the base isolation period, , of the linearized system on comfort conditions. These results show that the behavior of the system is linear up to relatively large values of the wind velocity, whereby the only case in Figure 8 where the isolators reach the postyield conditions is the one corresponding to = 4 s and = 15 m/s. This means that comfort conditions are governed by the fundamental period of the isolated mode, , in preyield conditions, (5), and by the flexibility of the system. The worst values of are those corresponding to natural frequencies comprised in the interval 1-2 Hz ( = 0.5–1 s) for vibrations along x-y-axes (see Figure 3). However, a reduction in flexibility is also associated with an increase in vibration amplitudes, which make the variation of with not monotonic.
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4. Conclusions
In this paper we have investigated, for the first time in the literature, occupants’ comfort conditions in base isolated buildings subjected to turbulent wind loading.
A simple parametric numerical procedure has been developed to estimate the minimum gust intensity that determines occupants’ discomfort in the building and the mean recurrence intervals of discomfort conditions. As gust intensity measure, we have considered the ten-minute mean wind velocity at 10 m height, which is a standard reference for evaluating wind effects on structures and for which historical statistical data are available, as considered in most international codes. The model has been purposely formulated in the simplest way possible, with one single degree of freedom, in such a way to control the effects of the different parameters governing the problem. It considers elastomeric base isolation devices having an elastic hardening plastic constitutive behavior and is based on a classic digital generation formula for wind velocity time histories and on human comfort analysis using international standard provisions based on third-octave band spectral analysis of horizontal acceleration and on the application of appropriate filters to account for human perception thresholds.
A parametric study demonstrates that base isolated buildings can undergo discomfort conditions a few times in a year, depending on the site exposure and its geographical location and depending upon the type of building, whereby the worst conditions are observed for critical areas and for residences at night. The minimum gust intensity determining discomfort is seen to be very sensitive to the terrain category, to the along-wind dimension of the building, and to the aerial mass of the floors, while it is comparatively less sensitive to variations in the base isolation period, the constitutive behavior of the base isolation devices, the height of the building, and the damping ratio of base isolated mode. Furthermore, it has been observed that discomfort occurs at dynamic excitation levels where the behavior of the elastomeric base isolators is still in preyield conditions, so governed by the linear response of the isolators in the range of displacement where their stiffness is maximum.
In conclusions, the analysis presented in this paper demonstrates that human comfort conditions should be considered in the design of base isolation systems, especially for buildings containing critical areas such as hospitals. However, if evaluated with appropriate methodologies accounting for the frequency content of the motion and for frequency-dependent human perception thresholds, as addressed in this study, comfort conditions are not expected to represent a significant design constraint in practice. On the contrary, comfort analysis methods merely based on peak acceleration in the time domain might be too conservative for the considered problem and might significantly underestimate the critical wind velocity causing discomfort.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper. The authors confirm that the mentioned received funding in “Acknowledgments" did not lead to any conflict of interests regarding the publication of this manuscript.
Acknowledgments
This work has been partially supported by ReLUIS project, line 6, “Base Isolation and Dissipation.”