#### Abstract

The hysteretic energy () dissipated in posttensioned steel frames (PTSF) with hysteretic dampers is calculated by using equivalent single-degree-of-freedom systems (ESDOFS), where the nonlinearity of both the steel bars (beam and columns) and the connections of the structural frame is separately considered. Five multi-degree-of-freedom (MDOFS) PTSF and their corresponding ESDOFS are studied under the action of 30 seismic motions recorded in soft ground, scaled in terms of the spectral pseudoacceleration evaluated at the fundamental period of the structures. Several simple mathematical expressions are proposed. The first expression is useful to calculate modifying factors () to relate of the ESDOFS with that of the MDOFS; the second is to determine the interstory drift (*γ*) as a function of the seismic intensity. The third equation is to calculate the factor of the relative participation of the energy that the connections dissipate with respect to the total energy and the fourth equation, which is function of the *γ*, is to obtain the distribution factors of through the height of the structure. The methodology proposed can be used for the design or the structural revision of PTSF with dampers.

#### 1. Introduction

Seismic design spectra corresponding to single-degree-of-freedom systems (SDOFS) are commonly used in structural engineering for the design of multi-degree-of-freedom structures (MDOFS). Seismic codes around the world normally specify design spectra corresponding to a certain return interval for this purpose; however, these spectra and the seismic design criteria recommended by the codes do not explicitly consider the hysteretic energy () dissipated by the structure which may be related to the cumulative structural damage. New tendencies in structural design are oriented to take into account [1–4] particularly for structures subjected to long duration seismic ground motions like those occurring in the soft soil of the valley of Mexico. The energy concept for design purposes was first discussed by Housner [5]. Energy-based methodologies are aimed to provide the structures with an energy dissipating capacity that should be larger or equal than the expected energy demand [6].

There are approaches in the literature [7–9] for the seismic design of regular structural buildings that verify that the structure has the capability to control and accommodate simultaneously the maximum demands of (a) global ductility, (b) interstory drift, and (c) dissipated hysteretic energy. In particular, [9] makes use of constant maximum ductility strength spectra and constant normalized dissipated hysteretic energy strength spectra and transformation factors that take into account the differences between the response of MDOFS and ESDOFS. The approach is based on the understanding that spectra can be used to capture the global dynamic response of MDOFS; in other words, it is assumed that through the use of spectra and appropriate ESDOFS to MDOFS transformation factors, the dynamic response of a regular steel building can be estimated for structural revision purposes.

The approaches mentioned in the paragraph above are applicable for the design of regular steel frames, but not for regular posttensioned steel frames (PTSF) with energy dissipating elements, like the structures analyzed here. In the present study, a transformation factor to estimate the seismic response of multi-degree-of-freedom with dissipating elements based on the response of ESDOFS is proposed. Furthermore, the criterion is extended to find the distribution of through the height of the PTSF. The tools developed in the present study are useful to formulate design criteria of PTSF with dissipating elements that take into account the hysteretic energy dissipated by the structural elements.

The motivation behind the study of PTSF with dissipating elements is the increment of the use of the posttensioning technology in buildings around the world during recent years. Numerical and experimental results [10–15] show that buildings posttensioned connections are capable of achieving stiffness and stress comparable to those of traditional moment-resisting connections, as well as permitting the dissipation of seismic energy. This behavior can be achieved without the occurrence of inelastic deformations in beams and columns or residual structural drifts. This occurs because the connections with posttensioned elements and energy dissipating devices (PTED) include the high resistance of the posttensioned steel (which remains elastic during the seismic response), while the seismic dissipation of energy is confined to elements (i.e., steel angles) designed to develop large deformations in the inelastic range [16].

#### 2. Model of a Single-Degree-of-Freedom System with PTED Connection

In this section, it is assumed that the global dissipated energy of multiple-degree-of-freedom PTSF with dissipating elements can be estimated from the response of ESDOFS, by means of adequate transformation factors.

To calculate the total energy dissipated by the structure, an ESDOFS that separately takes into account the nonlinearity of the structural frame (beams and columns) and the connections is proposed. The model used to represent the structure and the beam-column connection is formed by a simple oscillator with two parallel springs plus a dashpot, as shown in Figure 1. The contribution of each spring is independently considered because the hysteretic behavior of each element has different behavior under cyclic loads.

The linear equation of motion of the model under seismic load is
where , , and are the acceleration, velocity, and displacement of the main system, respectively,* m* is the mass,* c* is the damping, and and are the stiffness of the structural system (beams and columns) and of the connection, respectively.

The nonlinear equation corresponding to the combined system is

The terms represent the restoring force of the main structural system; the first and second terms represent nonlinear nonhysteretic and nonlinear hysteretic components, respectively, of the structural restoring force; is the ratio between the postyielding and the initial stiffness of the main structural system (elastic behavior); and represents the hysteretic component having units of displacement [17].

The term in (2) describes the flag-shaped hysteretic behavior of the connection in terms of the force-displacement relationship, and it is given by (3) and (4) for the unloading and loading cycles, respectively [18, 19]: where is the postyielding stiffness of the connection, defines the transition zone from elastic to inelastic behavior, defines the width of the flag, is the decompressing force (exactly when the connection opens), ( is the yield force in the PTSF), and and are the maximum displacement and maximum force reached in each load cycle, respectively. Equation (3) is used for either positive or negative load cycles, while (4) is for unloading. Figure 2 shows a hysteretic cycle of the semirigid posttensioned connection; the parameters previously mentioned are indicated in the figure, as well as the parameter which is the force when the connection closes. Parameter values of (3) and (4) were obtained by means of incremental static analyses (push-over) and are given in Table 1, where F4 represents a 4-story frame; this notation is also applied for the other models with number of stories ranging from 6 to 14.

Dividing (2) by and expressing it by the following system of differential equations, the following is obtained: where , and are the parameters of the model proposed by Bouc [20] and modified by Baber and Wen [21], that control the amplitude, the shape of the hysteretic cycle, and the smoothness of the transition from the elastic to the inelastic zone; and are parameters that control the deterioration of both stiffness and strength. In this study, no structural deterioration was considered. The parameter values of and for steel structures are equal to [22] , where represents the yield force of the structural system and is the structural stiffness. The values for and for were assumed to be equal to 1 and 15, respectively.

By introducing the variables ;; and , the system of (5) can be written as the following system of first-order differential equations:

By solving (6), the displacement, velocity, and acceleration of the main structure are calculated, as well as the hysteresis cycles of both the main system and the PTED connections. The corresponding is obtained by calculating the area enclosed by the hysteretic curves. The sum of the energies of both the main system and the connections is the total energy that the combined system dissipates.

#### 3. Algorithm for Obtaining the Transformation Factors between ESDOFS and MDOSFS

The steps proposed to calculate the transformation factors between the response of the ESDOFS and MDOFS are as follows.(1)The dynamic characteristics of the ESDOFS are assumed to be equal to those of the MDOFS to be analyzed. The properties are as follows: the fundamental structural vibration period (), the damping (), the yield force of the main system (), and the elastic and plastic stiffness of the main system and connections (, , ). The values of both and stiffness are obtained by means of “push-over,” as was mentioned in Section 2, for the MDOFS with and without PTED connections.(2)The mass of the ESDOFS is calculated as , where .(3)The yield displacement of the structural system is obtained by the ratio .(4)The parameter corresponding to the hysteretic model of the structure is calculated.(5)The earthquakes recorded in soft ground of Mexico City are scaled [23]. More details about the scaling criterion are given later.(6)The response of the combined ESDOFS is calculated by using a “step-by-step” time history method (the fourth-order Runge-Kutta method was used herein). The dissipated hysteretic energy, the maximum restoring force, and the maximum displacement of the system are obtained by solving (6).(7)The response of the MDOFS with dissipating elements is calculated by means of a “step-by-step” dynamic analysis in time domain.(8)Once the responses of the MDOFS and their corresponding ESDOFS are calculated, the transformation factors that relate the responses of both structural systems are obtained as follows: where is the transformation factor of the response parameter of the ESDOFS to obtain that of the MDOFS and is the response parameter. Dissipated hysteretic energy was the parameter selected in this study, but it could be any other parameter, such as base shear, maximum distortion, or ductility demand.

#### 4. Multi-Degree-of-Freedom PTSF Models

Five regular PTSF with semirigid connections designed according to the requirements of Mexico City Building Code [24] are considered in this study. It is assumed that the buildings which are used as offices have 3 bays and the number of stories ranges from 4 to 14. Their dimensions are shown in Figure 3(a), and in Figure 3(b) a detail of the posttension strands and the steel angles which represent the energy dissipating elements is shown. A36 steel and W sections are used in beams and columns. The assumed damping is 3% of the critical damping. The relevant characteristics of each frame, such as , the shear force and the yield displacement ( and ), and the plastic stiffness of the main system with connections () are shown in Table 2. It can be observed that the structural frames present an ample interval of vibration periods (0.89 s < < 2.10 s). In the last two columns of Table 2, values of the postyielding and the initial stiffness of the structural system are given ().

**(a) Dimension of the frames**

**(b) Angles and posttensioned strands**

#### 5. Seismic Records

The structural models mentioned above are subjected to the action of several narrowband long duration earthquakes. Narrowband seismic motions have the special characteristic of considerably affecting specific structures in a short interval of periods, especially to structures that undergo “softening” [25, 26] or to structures with vibration periods close to the period of the ground. This type of records demands large amounts of energy to structures, compared to that demanded by broadband motions [2]. A set of 30 seismic motions recorded on soft ground of Mexico City are used in this study. The interval of magnitudes of the events goes from 6.9 to 8.1 (including the record obtained in the station “Secretaría de Comunicaciones y Transportes” (SCT) on September 19, 1985). The final seismic motions were obtained by rotating both horizontal components of the ground motion recorded in the station and maximizing their Arias intensity [27]. Details of the records are described in Table 3, where the last two columns represent the maximum acceleration and maximum velocity of the ground motion, respectively.

#### 6. Transformation Factors of Hysteretic Energy

Transformation factors of hysteretic energy () according to (7) are presented in this section. Each of the frames described in Section 2 is analyzed under the action of the 30 accelerograms listed in Table 3. The ground motions are scaled in terms of the pseudoacceleration corresponding to the fundamental period of the frame (()); the scaling ranges from 0.1 g to 1.7 g with increments of 0.1 g (where g is the gravity acceleration), in such a way that 510 records for each frame are obtained (a total of 2550 records for the 5 frames). A “step-by-step” nonlinear dynamic analysis in time domain is performed by using the RUAUMOKO computer program [28], and the response parameters in terms of moment rotation are obtained. The corresponding E_{H} is obtained by calculating the area enclosed by the hysteretic curve corresponding to each connection. For the analysis, it is assumed that the elements in each story dissipate equal amount of energy.

Figures 4(a), 4(b), 4(c), 4(d), and 4(e) show transformation factors for frames F4, F6, F8, F10, and F14, respectively. It is important to note that for the first values of , there are no results of transformation factors because there is no dissipated energy ; on the other side, some graphs are truncated for big values of because they correspond to structural failure. For example, Figure 4(a) shows that frame F4 failed for four ground motions (Ea17, Ea18, Ea19, and Ea21) before reaching the maximum intensity of 1.0 g. From Figures 4(a)–4(e), the following observations can be made: the magnitude of depends on the seismic intensity , when increases, so does and the dispersion increases as increases.

(a) = 0.89 s |

(b) = 1.03 s |

(c) = 1.25 s |

(d) = 1.37 s |

(e) = 2.10 s |

Figures 4(a)–4(e) also show that the mean value of (), indicated by a black circles, can be reasonably fitted with a linear equation. The parameter is used for that purpose which is expressed by the equation

where is given as a fraction of the gravity acceleration and and are fitted in terms of the fundamental period of the frame under consideration, with the following equations:

Figure 5 shows with continuous lines the graphs of the mean values of obtained from the results of the dynamic analyses and with dotted lines the values calculated with (8a), for all the structural models. It can be observed that the fitting obtained with (8a) is adequate.

#### 7. Maximum Story Drift for the MDOFS

The maximum story drift (*γ*) is one of the main parameters used in the design codes to measure the damage and the performance of buildings under seismic actions; it is therefore useful to propose a simple procedure to determine the maximum story drift of posttensioned frames as a function of its fundamental structural period and of the seismic intensity. Figure 6 shows with continuous line the mean values of maximum drifts of F4, F6, F8, F10, and F14 for different intensity values of . It is observed that, except for frame F4, there is practically a linear variation of *γ* with . Using a regression analysis, the following equation to calculate *γ* is proposed:
where is the maximum story drift, fitted to the mean values and expressed in radians.

Figure 6 shows with dotted lines the values of (obtained with (10)) for each of the frames studied and with continuous lines the values obtained from dynamic analyses. A good approximation is observed in all cases, although it is better for the frames with longer periods (greater height); the reason for this is that frames with lower height present nonlinear behavior for lower seismic intensities due to the structural “softening” phenomenon.

#### 8. Participation of the Connections in the Total Energy Dissipation ()

As previously mentioned, the PTED are designed to dissipate most of the . When the columns of the base are fixed at the foundation, plastic hinges are formed and some amount of is dissipated. A similar situation may happen in other locations of the structure when the seismic demands increase considerably. Table 4 shows the factors of relative participation of the connections in the dissipation of the total for the PTDE studied herein, calculated for different distortion demands. It is observed from the table that as the number of levels increases (thus increasing the number of connections) the participation of the energy dissipated by the connections also increases.

Table 4 also shows that taller frames (F10, F14) dissipate more energy (because they have more stories implying a large number of connections) than the smaller frames (F4). To calculate the relative participation of the dissipated energy of PTDE connections, (11a) is proposed, which is a function of the interstory drift demand (listed in the first column of Table 4). Coefficients and are calculated with (11b) and (11c) and both depend on . The parameters were obtained through a regression analysis of the data presented in Table 4:

Figure 7 shows with continuous lines the values of the factors of participation of connections (FPC) obtained by means of the dynamic analysis of the PTDE and with dotted lines those calculated with (11a); a good approximation is observed for all frames. It is important to highlight that the PTSF studied herein are fixed at their base, which makes an important contribution to the base columns in the total dissipated energy . In the case of columns with hinged bases, the participation is null.

#### 9. Distribution of the Dissipated Energy through the Height of the Structure

It is known that the demands of are not uniformly distributed in the structure; moreover, one of the main objectives in the design of PTSF with dissipating elements is that the is dissipated at the connections, while beams and columns remain essentially elastic [29–31]. As previously mentioned, this is true except when the columns are fixed at the base; in this case, even for moderate seismic demands, plastic hinges are formed in the base, dissipating important amounts of .

After calculating the total in the ESDOFS, as shown previously, it is necessary to determine how the is distributed through the height of the structure. Considering that the connections with PTED devices in each story dissipate equal amount of energy, López-Barraza et al. [32] proposed (12a) to calculate distribution factors of (). To obtain the energy dissipated in each story, the factors should be multiplied by the energy () dissipated by the connections:

where is the total height and is the height of the th story measured from the base; the functions ,, and , given by (12b), (12c), and (12d), respectively, are obtained by using regression analysis.

#### 10. Calculation of the Energy Dissipated in PTSF Based on the Results of ESDOFS

Using the mathematical expressions proposed in this study, an algorithm to find the distribution of the hysteretic energy dissipated through the height of the PTSF is presented in the following.(1)Calculate the total by using ESDOFS.(2)Calculate the transformation factors , substituting in (8a).(3)Calculate the total of the PTSF.(4)Calculate the interstory drift value () with (10), corresponding to and .(5)Calculate the relative participation of the connections (FPC), substituting *γ* in (11a).(6)Calculate the energy dissipated by the connections.(7)Calculate the energy distribution factors corresponding to the th story of the PTSF, substituting in (12a).(8)Calculate the demand of in the th story ().

It is important to note that the difference between and is the hysteretic energy that the base columns dissipate when they are fixed.

#### 11. Conclusions

(i)The study proposes a methodology to estimate the hysteretic energy () dissipated through the height of regular posttensioned steel frames with energy dissipating elements, based on the dissipated by equivalent single-degree-of-freedom systems (ESDOFS) and by some simple mathematical expressions proposed here. These relate the interstory drifts of the PTSF with the seismic intensity and the fundamental structural vibration period (10), the relative participation of the connections with respect to the total dissipated energy (11a), (11b), and (11c), and the distribution of the dissipated energy as a function of the interstory drift (12a), (12b), (12c), and (12d).(ii)The expressions are simple and can be applied to fixed PTSF with dissipating elements, with fundamental structural vibration periods ranging between 0.89 and 2.1s, located in soft soils of the valley of Mexico; however, the general methodology proposed here is applicable to PTSF with different structural conditions.(iii)Furthermore, spectra of to be included in the methods of seismic design can be developed. It is observed that for frames with few connections, such as F4, the efficiency in the dissipation of is small due to the reduced number of PTED connections. For this case, other energy dissipating mechanisms may be considered.(iv)The methodology and the expressions proposed here are useful tools for the seismic design or structural revision of PTSF with energy dissipating elements. For this purpose, it would be necessary to verify that the energy dissipating structural capacity is larger or equal to the energy demand. In addition, it would be also necessary to verify the requirements related to ductility capacity and maximum interstory drifts.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The support given by PROMEP under Grant UAS-PTC-060 and the Universidad Autónoma de Sinaloa under Grant PROFAPI2013/160 is appreciated. Financial support also was received from the Universidad Nacional Autónoma de México under Project PAPIIT-IN102114.