# Energy-Based Design Criterion of Dissipative Bracing Systems for the Seismic Retrofit of Frame Structures

## Abstract

**:**

## Featured Application

**An energy-based sizing criterion is proposed to help designing dissipative bracing systems incorporating fluid viscous spring–dampers for the seismic retrofit of frame structures.**

## Abstract

_{eq}, are proposed. These formulas allow calculating the ξ

_{eq}values that guarantee the achievement of the target factors. Finally, the energy dissipation capacity of the devices is deduced from ξ

_{eq}, finalizing their sizing process. A detailed description of the procedure is presented in the article, by distinguishing the cases where the prevailing structural deficiencies are represented by poor strength of the constituting members, from the cases having excessive horizontal displacements. A demonstrative application to the retrofit design of a reinforced concrete gym building is then offered to explicate the steps of the sizing criterion in practice, as well as to evaluate the enhancement of the seismic response capacities generated by the installation of the dissipative system.

## 1. Introduction

_{I}computed on each story [31,32] or the entire structure [12] was proposed by the author and co-authors. To facilitate the choice of β, preferable ranges were provided for several different structural types and checked in relation to the assumed design targets [12,31,32]. However, as the method requires a preliminary evaluation of the seismic input energy demand on the original structure, a finite element time-history analysis must be carried out first, and E

_{I}post-calculated from the results. Although an energy calculation can be performed with the help of commercial finite element programs by means of simple input instructions, professional engineers are not always familiar with this design approach and may be discouraged from using it.

_{eq}, are proposed. These formulas allow calculating the ξ

_{eq}values guaranteeing the achievement of the target α factors. Finally, the energy dissipation capacity of the devices is deduced from ξ

_{eq}, finalizing their sizing process.

## 2. Design Procedure

_{1}, the simple relation provided by Eurocode 8 [46] and adopted by several national Standards, among which the Italian Technical Standards [47], can be used: T

_{1}= C

_{t}·H

^{¾}, where the C

_{t}coefficient is set as equal to 0.075 for steel structures and 0.085 for RC ones, and H is the height of the structure measured from the foundation, expressed in meters. For H = 20 m, T

_{1}= 0.71 s (steel) and T

_{1}= 0.8 s (RC) is obtained. By jointly considering the two values, T

_{1}= 0.8 s is assumed in the following as the approximate upper limit of the fundamental period for relatively stiff frame building structures.

_{s}in the following—and the cases having excessive horizontal displacements, where α is intended as “deformation-related” reduction factor, α

_{d}. Although the two types of lacks normally coexist, the former is frequently prevailing in structures with undersized member sections, albeit stiffened by structural elements (e.g., RC shear walls) or non-structural components (e.g., masonry infills interacting with the frame members), whereas the latter is likely to prevail when these elements are not present—and thus the structural system highlights high deformability to lateral loads—but the frame members have only moderate deficiencies in terms of sizes and structural details.

#### 2.1. Structures with Poor Shear or Bending Moment Strength of Constituting Members

_{s}.

_{A}= d

_{max}, s

_{A}= s

_{max}), where d

_{max}, s

_{max}are the maximum d and s computed values—below point B with coordinates (d

_{B}= d

_{e}, s

_{B}= s

_{e}), where d

_{e}, s

_{e}are the elastic limit deformation-related and stress-related parameters for the critical member(s), respectively. On the contrary, in a traditional design approach, the reduction of the force-related parameter is pursued by exploiting the ductile plastic response of the structural members, symbolically represented by the dashed line K–B, by moving the maximum response point from A to K, with coordinates (d

_{A}, s

_{B}).

_{s}, targeted to reach an elastic response of the critical member (and thus of all remaining members) when passing from original to retrofitted conditions, is given by:

_{A}and s

_{B}parameters in Figure 1 are detailed below according to the specific lack of strength affecting the structural members in original conditions.

_{A}is the shear force in the critical member ${V}_{j,c}^{a}$, according with the notation introduced in step 1. Said ${V}_{j,c}^{e}$ = s

_{B}the elastic limit shear of this member, α

_{s}—specified in this hypothesis as ${\mathsf{\alpha}}_{Vj}$—is evaluated as follows:

_{A}is the bending moment ${M}_{j,c}^{a}$ (associated with the concurrent axial force N

_{c}if the critical member is a column), said ${M}_{j,c}^{e}$ = s

_{B}the corresponding elastic limit moment, the reduction factor α

_{s}—denoted with symbol ${\mathsf{\alpha}}_{Mj}$ in this case—is given by:

_{s}→α

_{F}).

_{A}

_{′}= F

_{max}, ID

_{A}

_{′}= ID

_{max}), and B′, with coordinates (F

_{B}

_{′}= F

_{e}, ID

_{B}

_{′}= ID

_{e}), where the indexes “max” and “e” denote the maximum response value and the corresponding elastic limit in this case too. Based on this correlation, the storey response points A′, B′ in Figure 1b are reached when the critical member attains points A, B in Figure 1a. Therefore, the reduction factor at the storey level, α

_{F}:

_{F}to the dissipated energy E

_{D}and the equivalent viscous damping ratio ξ

_{eq}of the spring–dampers.

_{eq}, using the general expression:

_{D}is the energy dissipated by the set of FV spring–dampers installed on the storey containing the critical member, and E

_{e}is the strain energy of the system estimated by referring to the global response cycle of the set of devices, schematically drawn in Figure 2. Therein, d

_{dmax}is the maximum displacement, F

_{D}is the damping force component, F

_{e}is the elastic reaction force, and ${F}_{ed}={F}_{e}+{F}_{D}$ is the total reaction force.

_{D}is set as equal to the difference between F

_{A′}and F

_{B′}:

_{D}expression is obtained:

_{e}elastic force component is set as equal to the elastic limit value of the storey shear, F

_{B′}. Therefore, F

_{ed}can be alternatively expressed as a function either of F

_{D}:

_{e}, by substituting (8) in (9):

_{D}and E

_{e}are derived:

_{eq}expression is deduced:

_{F}. Moreover, by solving (5) to express E

_{D}as a function of ξ

_{eq}and considering (10), the following relation is obtained:

_{D}, and selection of the devices with the nearest mechanical characteristics.

_{max}, is constrained below the corresponding elastic limit ID

_{B′}= ID

_{e}displayed in Figure 1b. ID

_{max}is given by the sum of d

_{dmax}and the interstorey drift contribution provided by the braced structure. The latter is normally small, because relatively stiff structures—i.e., with T

_{1}below about 0.8 s—are dealt with, as mentioned above, and also because the bracing system produces in any case a stiffening effect, although limited by the special installation layout of the FV spring–dampers. In view of this, in order to quickly pre-estimate the energy dissipation capacity of the devices, which is a function of d

_{dmax}—other than of F

_{D}—d

_{dmax}is set as equal to ID

_{e}at this stage. Based on this assumption, the energy dissipation capacity to be assigned to the FV devices on the considered storey is drawn from (11):

_{D}as a function of structure-related terms only, (8) is substituted in (16), and gives:

_{D}value estimated by (17) or (18) and a stroke approximately equal to ID

_{e}.

#### 2.2. Structures with Excessive InterStorey Drifts

_{1}next to the 0.8 s value approximately fixed as the upper limit for the application of the procedure. The design objective is reached by reducing the maximum interstorey drift computed in current conditions, identified by ID

_{A}

_{′}= ID

_{max}in Figure 1b, to the corresponding elastic limit ID

_{B}

_{′}= ID

_{e}, i.e., by scaling the drift response by a deformation-related reduction factor α

_{d}:

_{d}is proportional to α

_{F}. For cases where an inelastic evaluation analysis is developed, assuming a typical degrading strength and degrading stiffness post-elastic behaviour—like the one qualitatively schematized by curve B′–K′ in Figure 1b—α

_{d}significantly differs from α

_{F}. Therefore, the expressions of E

_{D}and ξ

_{eq}must be reformulated as a function of α

_{d}to allow quickly estimating both quantities also for the structures where the interstorey drift is the critical response parameter.

_{dmax}= ID

_{A′}− ID

_{B′}= ID

_{max}− ID

_{e}, and F

_{ed}= F

_{K′}= F

_{D}. The first assumption corresponds to assigning to the dampers the capacity of absorbing the post-elastic displacement demand of the structure computed in original conditions, so as to meet the design objective of limiting its response to the elastic field after retrofit. The second assumption derives from the fact that the displacement performance enhancement must be achieved essentially by means of the dissipative capacity of the FV devices, by neglecting at the sizing stage the slight stiffening effects related to their elastic spring function. Then, according to (19), the following ξ

_{eq}expression is obtained:

_{D}, estimated by (20), and a stroke not less than (ID

_{max}− ID

_{e}).

## 3. Geometrical and Structural Characteristics of the Case-Study Building

^{2}; yield stress and limit stress of steel equal to 417 MPa and 594 MPa, respectively. The total seismic weight of the building is equal to 2960 kN.

## 4. Verification Analysis in Current Conditions (Step 1 of the Design Procedure)

#### 4.1. Modal Analysis

#### 4.2. Time-History Verification and Performance Assessment Analysis

_{R}); Serviceability Design Earthquake (SDE, with 50%/V

_{R}probability); Basic Design Earthquake (BDE, with 10%/V

_{R}probability); and Maximum Considered Earthquake (MCE, with 5%/V

_{R}probability). The V

_{R}period is fixed at 75 years, which is obtained by multiplying the nominal structural life V

_{N}of 50 years by a coefficient of use C

_{u}equal to 1.5, imposed to structures whose seismic resistance is of importance in view of the consequences associated with their possible collapse, like the case-study school gym building. By referring to topographic category T1 (flat surface) and B-type soil, the resulting peak ground accelerations for the four seismic levels referred to the city of Florence are as follows: 0.065 g (FDE), 0.078 g (SDE), 0.181 g (BDE), and 0.227 g (MCE). The relevant pseudo-acceleration elastic response spectra at linear viscous damping ratio ξ = 5% are plotted in Figure 8.

_{r}, which is equivalent to the interstorey drift ratio in the presence of a system of continuous intermediate beams, although without a floor. The maximum ILD

_{r}values induced by the most severe among the seven groups of input motions, ILD

_{r,max}, are as follows: 0.07% (FDE), 0.09% (SDE) in X, and 0.06% (FDE), 0.07% (SDE) in Y, on the first level; 0.13% (FDE), 0.16% (SDE) in X, and 0.53% (FDE), 0.64% (SDE) in Y, on the second level. The drift ratios in X are far below the 0.33% limitation adopted by [47] at the Operational (OP) performance level for frame structures interacting with drift-sensitive non-structural elements, like the masonry infills on the first level and the curtain wall-type windows on the second level, for the main façades of the building, and the infills situated on both levels, for the side façades. The ILD

_{r,max}values obtained on the second level in Y are 1.6 times (FDE) and about twice (SDE) the OP-related limit, and also greater than the drift threshold adopted by [47] for the Immediate Occupancy (IO) performance level, equal to 0.5%.

_{r,max}values computed for the second level in Y are equal to 1.36% at the BDE, and 1.69% at the MCE, assessing moderate (BDE) to high (MCE) potential plastic demands on the columns—should an inelastic finite element analysis be carried out—and severe (BDE) to very severe (MCE) damage of infills and curtain-wall windows. Consequently, the performance level attained in terms of displacement response is Life Safety (LS), both for the BDE and the MCE. At the same time, ILD

_{r,max}is no greater than 0.18% (BDE) and 0.22% (MCE) on the first level in Y, i.e., only 13% of the second level values. This identifies a cantilever-like response of the structure along Y, with structural and non-structural damage located on the second level. As discussed in the following Sections, this suggests incorporating the dissipaters on the upper level only, in order to adequately exploit their damping capacity and limit the cost of the retrofit intervention. In X direction, ILD

_{r,max}is equal to 0.2% (BDE), 0.25% (MCE) on the first level, and 0.36% (BDE), 0.45% (MCE) on the second level. The interlevel drift profile depicts a frame-like layout along this axis, which approaches a shear-type shape on the second level, as a consequence of the high flexural stiffness of the roof beams in the X–Z vertical plan (which determines nearly a sliding-clamped constraint condition on the top section of the columns).

_{X,c}–M

_{Y,c}biaxial moment interaction curves (being M

_{X,c}, M

_{Y,c}the bending moments around the X and Y axes) graphed by jointly plotting the two bending moment response histories obtained from the most demanding among the seven groups of MCE-scaled accelerograms, are plotted in Figure 9 for a corner column, namely C17. The boundary of the M

_{X,c}–M

_{Y,c}elastic interaction domain—which is a function of the time-history variation of the axial force in the columns—is also traced out in the two graphs for the value of the axial force conventionally referred to the basic combination of gravity loads, i.e., N

_{c}= 104 kN.

_{X,c}–M

_{Y,c}combined values slightly exceeding the safe domain boundary at the BDE, and 1.77 times greater than the corresponding values situated on the boundary, with prevailing contribution of M

_{Y,c}, at the MCE. The curves traced out for the second level show more marked unsafe conditions at the BDE, as compared to the first-level ones, and exceed the boundary by a factor equal to 2.07 at the MCE, but with inverted role of the moments (i.e., with prevailing contribution of M

_{X,c}for the second level).

## 5. Dissipative Bracing Retrofit Solution

#### 5.1. Characteristics of the Protective System

_{D}damping and F

_{ne}non-linear elastic reaction forces corresponding to the damper and spring functions are effectively simulated by the following expressions [54,56]:

_{0}= static pre-load force; k

_{1}, k

_{2}= stiffness of the response branches situated below and beyond F

_{0}; and x(t) = device displacement. For the development of the numerical analyses, the finite element model of a FV spring–damper is obtained by combining in parallel a non-linear dashpot and a non-linear spring with reaction forces given by expressions (22) and (23), respectively. Both types of elements are currently incorporated in commercial structural analysis programs, such as the SAP2000NL code used in this study.

#### 5.2. Application of the Design Method to the Case-Study Building

#### 5.2.1. X Direction—Lack of Bending Moment Strength in the Columns

_{Y}) in the first-level columns, with the highest unsafe conditions checked in the four corner columns. By referring to the nomenclature in Section 2.1, the maximum moment ${M}_{Y,c}^{a}$ corresponding to the peak response point in Figure 9c and associated with the concurrent axial force N

_{c}= 104 kN mentioned above, is equal to 398.7 kNm. The elastic limit moment ${M}_{Y,c}^{e}$ of the corner columns around the Y axis is equal to 224.8 kNm. Thus, the stress reduction factor ${\mathsf{\alpha}}_{s}={\mathsf{\alpha}}_{MY}$ of the critical members in X direction results as follows:

_{F}ratio coincides with α

_{MY}:

_{D}energy dissipation capacity of the spring–dampers is calculated by expression (18). The elastic limit values of the level shear F

_{e}(i.e., the sum of the elastic limit shear forces of the columns) and the first interlevel drift ILD

_{e}(replacing ID

_{e}in this case) computed in X direction, named F

_{e,X}and ILD

_{e,}

_{1L,X}, are equal to 969 kN and 22 mm, respectively. Introducing these values, as well as α

_{F}and ${\mathsf{\xi}}_{eq,\mathsf{\alpha}F}$ values given by (25) and (26), in (18), the following E

_{D}estimate is derived:

_{D}by the number of spring–dampers placed in X, the minimum energy dissipation capacity E

_{D,X}

_{,d}to be assigned to each of the eight devices in order to reach the target performance at the MCE results as follows: E

_{D,X}

_{,d}= 8.2 kJ. The spring–damper type with the nearest nominal energy dissipation capacity, E

_{n}, to E

_{D,X}

_{,d}has the following mechanical properties, drawn from the manufacturer’s catalogue [48]: E

_{n}= 9 kJ; stroke s

_{max}= ±30 mm; damping coefficient c = 9.9 kN(s/mm)

^{γ}, with γ = 0.15; F

_{0}= 17 kN; and k

_{2}= 1.74 kN/mm.

#### 5.2.2. Y Direction—Lack of Bending Moment Strength in the Columns and Excessive Inter Level Drift

_{X}) in the second-level columns, with the highest unsafe conditions checked in the four corner columns too, and the second interlevel drifts. The maximum moment ${M}_{X,c}^{a}$, corresponding in this case to the peak response point in Figure 9d, is equal to 174.2 kNm, whereas the elastic limit moment ${M}_{X,c}^{e}$ is equal to 84.2 kNm. Therefore, the stress reduction factor ${\mathsf{\alpha}}_{s}={\mathsf{\alpha}}_{MX}$ of the critical members in Y direction is:

_{d}given by (19) is calculated for the ILD

_{max}(replacing ID

_{max}) and ILD

_{e}(replacing ID

_{e}) values computed on the second level in Y direction, named ILD

_{max,}

_{2L,Y}, ILD

_{e,}

_{2L,Y}in the following. ILD

_{max,}

_{2L,Y}—corresponding to the ILD

_{r,max}value of 1.69% mentioned in Section 4.2—is equal to 72.7 mm, and ILD

_{e,}

_{2L,Y}to 36.8 mm, yielding:

_{F}and to α

_{d}, using expressions (14) and (20), respectively:

_{e,Y}the elastic limit level shear in Y direction, by applying the E

_{D}energy dissipation capacity expressions (18) and (21), the following E

_{D}estimates are obtained:

_{e,Y}= 638 kN.

_{eq}expression (5), the damping coefficient depends on E

_{e}, and thus on the elastic properties of the device, which are a function of the maximum displacement and force reached in the time-history response, in addition to the hysteretic response. On the other hand, the dissipated energy E

_{D}is only determined by the area covered by the response cycles, which identifies it as a more stable and reliable parameter for the design of the FV devices. ξ

_{eq}is only a useful synthetic measure of their limit damping capacity.

_{D,Y}

_{,d}, in order to achieve the target performance at the MCE, is obtained by dividing ${E}_{D,\mathsf{\alpha}F}$ by the number of spring–dampers: E

_{D,Y}

_{,d}= 12.6 kJ. The device with the nearest nominal energy dissipation capacity to E

_{D,Y}

_{,d}has the following mechanical properties: E

_{n}= 14 kJ; stroke s

_{max}= ±40 mm; damping coefficient c = 14.16 kN(s/mm)

^{γ}, with γ = 0.15; F

_{0}= 28 kN; and k

_{2}= 2.1 kN/mm.

#### 5.3. Numerical Verification of the Retrofit Solution

_{Xc}–M

_{Yc}interaction curves of the first and second level base sections of column C17, plotted in Figure 9c,d above for the original structure, are duplicated in Figure 13 in retrofitted conditions. The two graphs show that the dissipative action of the protective system allows confining the interaction curves within the biaxial moment safe domain, reducing the maximum M

_{Yc}(Figure 13a) and M

_{Xc}(Figure 13b) moments nearly by the targeted α

_{MY}and α

_{MX}factors of 1.77 and 2.07, as given by (24) and (29).

_{I}and E

_{D}denote the total input and dissipated energies, and E

_{I,X}, E

_{D,X}, E

_{I,Y}, E

_{D,Y}the relevant portions in X and Y, show E

_{D,X}, E

_{D,Y}values of 66.9 kJ and 101.9 kJ, which differ only by 2% and 1% from the corresponding ${E}_{D,\mathsf{\alpha}F}$ estimates (27) and (33), respectively.

_{D}/E

_{I}ratio are found in X and Y, namely: E

_{D,X}/E

_{I,X}= 0.855; E

_{D,Y}/E

_{I,Y}= 0.85, identifying a well-balanced energy dissipation demand in the two directions.

_{e,}

_{2L,Y}= 36.8 mm, as targeted in the design.

## 6. Conclusions

_{F}, α

_{d}relevant to stress states and drifts, using formulas (14) and (20), respectively, resulted to be notably different. On the other hand, a slight difference was found between the corresponding energy dissipation measures, ${E}_{D,\mathsf{\alpha}F}$ and ${E}_{D,\mathsf{\alpha}d}$. This identified E

_{D}as a more stable and reliable design parameter, as compared to ξ

_{eq}, consistently with the fact that E

_{D}is only determined by the area covered by the response cycles of the dissipaters. This was also confirmed by the fact that the E

_{D}values in X and Y computed from the time-history verification analysis were very similar to the ${E}_{D,\mathsf{\alpha}F}$ estimates.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic response of the critical structural member (

**a**) and the frame structure storey to which it belongs (

**b**).

**Figure 2.**Schematic response cycle of the set of fluid viscous (FV) spring–dampers installed on a storey, and parameters for evaluating ξ

_{eq}.

**Figure 6.**Redrawn cross sections of: (

**a**) roof beams–half-span and ends; (

**b**) top longitudinal beams TB—half-span and ends; (

**c**) intermediate longitudinal beams IB; (

**d**) columns (dimensions in millimeters).

**Figure 9.**Current state (CS). M

_{X,c}–M

_{Y,c}biaxial moment interaction curves at the base section of column C17 on the first level (

**a**,

**c**) and second level (

**b**,

**d**) obtained from the most demanding BDE-scaled (

**a**,

**b**) and MCE-scaled (

**c**,

**d**) group of accelerograms.

**Figure 13.**Retrofitted structure (RS). M

_{X,c}–M

_{Y,c}biaxial moment interaction curves at the base section of column C17 on the first level (

**a**) and second level (

**b**) obtained from the most demanding MCE-scaled group of accelerograms.

**Figure 14.**Retrofitted structure (RS). Response cycles of the spring–damper pairs installed in Al. X1 (

**a**), Al. X3 (

**b**), Al. Y2 (

**c**) and Al. Y4 (

**d**) vertical alignments obtained from the most demanding MCE-scaled group of accelerograms.

**Figure 15.**Retrofitted structure (RS). Energy time-histories obtained from the most demanding MCE-scaled group of accelerograms.

**Figure 16.**Rooftop displacement time-histories in X and Y direction obtained from the most demanding MCE-scaled group of accelerograms.

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**MDPI and ACS Style**

Terenzi, G. Energy-Based Design Criterion of Dissipative Bracing Systems for the Seismic Retrofit of Frame Structures. *Appl. Sci.* **2018**, *8*, 268.
https://doi.org/10.3390/app8020268

**AMA Style**

Terenzi G. Energy-Based Design Criterion of Dissipative Bracing Systems for the Seismic Retrofit of Frame Structures. *Applied Sciences*. 2018; 8(2):268.
https://doi.org/10.3390/app8020268

**Chicago/Turabian Style**

Terenzi, Gloria. 2018. "Energy-Based Design Criterion of Dissipative Bracing Systems for the Seismic Retrofit of Frame Structures" *Applied Sciences* 8, no. 2: 268.
https://doi.org/10.3390/app8020268