# Chapter 18 Seismic Design Requirements for Structures with Damping Systems

### 18.1.1 Scope

Every structure with a damping system and every portion thereof shall be designed and constructed in accordance with the requirements of this standard as modified by this section. Where damping devices are used across the isolation interface of a seismically isolated structure, displacements, velocities, and accelerations shall be determined in accordance with Chapter 17.

### 18.1.2 Definitions

The following definitions apply to the provisions of Chapter 18:
DAMPING DEVICE: A flexible structural element of the damping system that dissipates energy due to relative motion of each end of the device. Damping devices include all pins, bolts, gusset plates, brace extensions, and other components reqmred to connect damping devices to the other elements of the structure. Damping devices may be classified as either displacement-dependent or velocity-dependent, or a combination thereof, and may be configured to act in either a linear or nonlinear manner.
DAMPING SYSTEM: The collection of structural elements that includes all the individual damping devices, all structural elements or bracing required to transfer forces from damping devices to the base of the structure, and the structural elements required to transfer forces from damping devices to the seismic force-resisting system.
DISPLACEMENT-DEPENDENT DAMPING DEVICE: The force response of a displacement-dependent damping device is primarily a function of the relative displacement between each end of the device. The response is substantially independent of the relative velocity between each of the devices and/or the excitation frequency.
VELOCITY-DEPENDENT DAMPING DEVICE: The force-displacement relation for a velocity-dependent damping device is primarily a function of the relative velocity between each end of the device and could also be a function of the relative displacement between each end of the device.

### 18.1.3 Notation

The following notations apply to the provisions of this chapter:
 B1D = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βml (m = 1) and period of structure equal to T1D B1E = numerical coefficient as set forth in Table 18.6-1 for the effective damping equal to βI + βV1 and period equal to T1 B1M = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmM (m = 1) and period of structure equal to T1M BmD = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βml and period of structure equal to Tm BmM = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmM and period of structure equal to Tm BR = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βR and period of structure equal to TR BV+I = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to the sum of viscous damping m the fundamental mode of vibration of the structure in the direction of interest, βVm, (m = 1), plus inherent damping, βI,and period of structure equal to T1 CmFD = force coefficient as set forth in Table 18.7-1 CmFV = force coefficient as set forth in Table 18.7-2 CS1 = seismic response coefficient of the fundamental mode of vibration of the structure in the direction of interest Section 18.4.2.4 or 18.5.2.4 (m = 1) CSm = seismic response coefficient of the mth mode of vibration of the structure in the direction of interest, Section 18.4.2.4 (m = 1) or Section 18.4.2.6 (m > 1) CSR = seismic response coefficient of the residual mode of vibration of the structure in the direction of interest Section 18.5.2.8 DID = fundamental mode design displacement at the center rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.2 D1M = fundamental mode maximum displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.5 DmD = design displacement at the center of rigidity of the roof level of the structure due to the mth mode of vibration in the direction under consideration, Section 18.4.3.2 DmM = maximum displacement at the center of rigidity of the roof level of the structure due to the mth mode of vibration in the direction under consideration, Section 18.4.3.5 DRD = residual mode design displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.2 DRM = residual mode maximum displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.5 DY = displacement at the center of rigidity of the roof level of the structure at the effective yield point of the seismic force-resisting system, Section 18.6.3 fi = lateral force at level i of the structure distributed approximately in accordance with Section 12.8.3 and Section 18.5.2.3 Fil = inertial force at level i (or mass point i) in the fundamental mode of vibration of the structure in the direction of interest, Section 18.5.2.9 Fim = inertial force at level i ( or mass point i) in the mth mode of vibration of the structure in the direction of interest Section 18.4.2.7 FiR = inertial force at level i ( or mass point i) in the residual mode of vibration of the structure in the direction of interest, Section 18.5.2.9 qH = hysteresis loop adjustment factor as determined in Section 18.6.2.2.1 QDSD = force in an element of the damping system required to resist design seismic forces of displacement-dependent damping devices, Section 18.7.2.5 QmDSV = forces in an element of the damping system required to resist design seismic forces of velocity-dependent damping devices due to the mth mode of vibration of the structure in the direction of interest, Section 18.7.2.5 QmSFRS = force in an element of the damping system equal to the design seismic force of the mth mode of vibration of the structure in the direction of interest, Section 18.7.2.5 T1 = the fundamental period of the structure in the direction under consideration T1D = effective period, in seconds, of the fundamental mode of vibration of the structure at the design displacement in the direction under consideration, as prescribed by Section 18.4.2.5 or 18.5.2.5 T1M = effective period, in seconds, of the fundamental mode of vibration of the structure at the maximum displacement in the direction under consideration, as prescribed by Section 18.4.2.5 or 18.5.2.5 TR = period, in seconds, of the residual mode of vibration of the structure in the direction under consideration, Section 18.5.2.7 Vm = design value of the seismic base shear of the mth mode of vibration of the structure in the direction of interest, Section 18.4.2.2 Vmin = minimum allowable value of base shear permitted for design of the seismic force-resisting system of the structure in the direction of interest, Section 18.2.2.1 VR = design value of the seismic base shear of the residual mode of vibration of the structure in a given direction, as determined in Section 18.5.2.6 W̅1 = effective fundamental mode seismic weight determined in accordance with Eq. 18.4-2b for m = 1 W̅xR = effective residual mode seismic weight determined in accordance with Eq. 18.5-13 α = velocity exponent relating damping device force to damping device velocity βmD = total effective damping of the mth mode of vibration of the structure in the direction of interest at the design displacement, Section 18.6.2 βmM = total effective damping of the mth mode of vibration of the structure in the direction of interest at the maximum displacement, Section 18.6.2 βHD = component of effective damping of the structure in the direction of interest due to post-yield hysteretic behavior of the seismic force-resisting system and elements of the damping system at effective ductility demand μD, Section 18.6.2.2 βHM = component of effective damping of the structure in the direction of interest due to post-yield hysteretic behavior of the seismic force-resisting system and elements of the damping system at effective ductility demand, μM, Section 18.6.2.2 βI = component of effective damping of the structure due to the inherent dissipation of energy by elements of the structure, at or just below the effective yield displacement of the seismic force-resisting system, Section 18.6.2.1 βR = total effective damping in the residual mode of vibration of the structure in the direction of interest, calculated in accordance with Section 18.6.2 (using μD = 1.0 and μM = 1.0) βVm = component of effective damping of the mth mode of vibration of the structure in the direction of interest due to viscous dissipation of energy by the damping system, at or just below the effective yield displacement of the seismic force-resisting system, Section 18.6.2.3 δi = elastic deflection of level i of the structure due to applied lateral force, fi, Section 18.5.2.3 δi1D = fundamental mode design deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3.1 δiD = total design deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3 δiM = total maximum deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3 δiRD = residual mode design deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3.1 δim = deflection of level i in the mth mode of vibration at the center of rigidity of the structure in the direction under consideration, Section 18.6.2.3 △1D = design story drift due to the fundamental mode of vibration of the structure in the direction of interest, Section 18.5.3.3 △D = total design story drift of the structure in the direction of interest, Section 18.5.3.3 △M = total maximum story drift of the structure in the direction of interest, Section 18.5.3 △mD = design story drift due to the mth mode of vibration of the structure in the direction of interest, Section 18.4.3.3 △RD = design story drift due to the residual mode of vibration of the structure in the direction of interest, Section 18.5.3.3 μ = effective ductility demand on the seismic force-resisting system in the direction of interest μD = effective ductility demand on the seismic force-resisting system in the direction of interest due to the design earthquake ground motions, Section 18.6.3 μM = effective ductility demand on the seismic force-resisting system in the direction of interest due to the maximum considered earthquake ground motions, Section 18.6.3 μmax = maximum allowable effective ductility demand on the seismic force-resisting system due to the design earthquake ground motions, Section 18.6.4 ϕil = displacement amplitude at level i of the fundamental mode of vibration of the structure in the direction of interest, normalized to unity at the roof level, Section 18.5.2.3 ϕiR = displacement amplitude at level i of the residual mode of vibration of the structure in the direction of interest normalized to unity at the roof level, Section 18.5.2.7 Γ1 = participation factor of the fundamental mode of vibration of the structure in the direction of interest, Section 18.4.2.3 or 18.5.2.3 (m = 1) Γm = participation factor in the mth mode of vibration of the structure in the direction of interest, Section 18.4.2.3 ΓR = participation factor of the residual mode of vibration of the structure in the direction of interest, Section 18.5.2.7 ∇1D = design story velocity due to the fundamental mode of vibration of the structure in the direction of interest, Section 18.5.3.4 ∇D = total design story velocity of the structure in the direction of interest, Section 18.4.3.4 ∇M = total maximum story velocity of the structure in the direction of interest, Section 18.5.3 ∇mD = design story velocity due to the mth mode of vibration of the structure in the direction of interest, Section 18.4.3.4

### 18.2.1 Seismic Design Category A

Seismic Design Category A structures with a damping system shall be designed using the design spectral response acceleration determined in accordance with Section 11.4.4 and the analysis methods and design requirements for Seismic Design Category B structures.

### 18.2.2 System Requirements

Design of the structure shall consider the basic requirements for the seismic force-resisting system and the damping system as defined in the following sections. The seismic force-resisting system shall have the required strength to meet the forces defined in Section 18.2.2.1. The combination of the seismic force-resisting system and the damping system is permitted to be used to meet the drift requirement.

### 18.2.2.1 Seismic Force-Resisting System

Structures that contain a damping system are required to have a seismic force-resisting system that, in each lateral direction, conforms to one of the types indicated in Table 12.2-1.
The design of the seismic force-resisting system in each direction shall satisfy the requirements of Section 18.7 and the following:
1. The seismic base shear used for design of the seismic force-resisting system shall not be less than Vmin, where Vmin is determined as the greater of the values computed using Eqs. 18.2-1 and 18.2-2:
2.           (18.2-1)
(18.2 -2)
where
 V = seismic base shear in the direction of interest, deter mined in accordance with Section 12.8 Bv+1 = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to the sum of viscous damping in the fundamental mode of vibration of the structure in the direction of interest, βvm (m = 1), plus inherent damping, βI, and period of structure equal to T1

EXCEPTION: The seismic base shear used for design of the seismic force-resisting system shall not be taken as less than 1.0V, if either of the following conditions apply :

1. In the direction of interest, the damping system has less than two damping devices on each floor level, configured to resist torsion.
2. The seismic force-resisting system has horizontal irregularity Type 1b (Table 12.3-1) or vertical irregularity Type lb (Table 12.3-2).
3. Minimum strength requirements for elements of the seismic force-resisting system that are also elements of the damping system or are otherwise required to resist forces from damping devices shall meet the additional requirements of Section 18.7.2.

### 18.2.2.2 Damping System

Elements of the damping system shall be designed to remain elastic for design loads including unreduced seismic forces of damping devices as required in Section 18.7.2.1, unless it is shown by analysis or test that inelastic response of elements would not adversely affect damping system function and inelastic response is limited in accordance with the requirements of Section 18.7.2.6.

### 18.2.3.1 Design Spectra

Spectra for the design earthquake ground motions and maximum considered earthquake ground motions developed in accordance with Section 17.3.1 shall be used for the design and analysis of a structure with a damping system. Site-specific design spectra shall be developed and used for design of a structure with a damping system if either of the following conditions apply:
1. The structure is located on a Class F site.
2. The structure is located at a site with S1 greater than or equal to 0.6.

### 18.2.3.2 Ground Motion Histories

Ground motion histories for the design earthquake and the maximum considered earthquake developed in accordance with Section 17.3.2 shall be used for design and analysis of all structures with a damping system if either of the following conditions apply:
1. The structure is located at a site with S1 greater than or equal to 0.6.
2. The damping system is explicitly modeled and analyzed using the response-history analysis method.

### 18.2.4 Procedure Selection

A structure with a damping system shall be designed using linear procedures, nonlinear procedures, or a combination of linear and nonlinear procedures, as permitted in this section.
Regardless of the analysis method used, the peak dynamic response of the structure and elements of the damping system shall be confirmed by using the nonlinear response-history procedure if the structure is located at a site with S1 greater than or equal to 0.6.

### 18.2.4.1 Nonlinear Procedures

The nonlinear procedures of Section 18.3 are permitted to be used for design of all structures with damping systems.

### 18.2.4.2 Response-Spectrum Procedure

The response spectrum procedure of Section 18.4 is permitted to be used for design of a structure with a damping system provided that
1. In the direction of interest, the damping system has at least two damping devices in each story, configured to resist torsion.
2. The total effective damping of the fundamental mode, βmD (m = 1), of the structure in the direction of interest is not greater than 35% of critical.

### 18.2.4.3 Equivalent Lateral Force Procedure

The equivalent lateral force procedure of Section 18.5 is permitted to be used for design of a structure with a damping system provided that
1. In the direction of interest, the damping system has at least two damping devices in each story, configured to resist torsion.
2. The total effective damping of the fundamental mode, βmD (m = 1), of the structure in the direction of interest is not greater than 35% of critical.
3. The seismic force-resisting system does not have horizontal irregularity Type 1a or 1b (Table 12.3-1) or vertical irregularity Type 1a, 1b, 2, or 3 (Table 12.3-2).
4. Floor diaphragms are rigid as defined in Section 12.3.1.
5. The height of the structure above the base does not exceed 100 ft (30 m).

### 18.2.5.1 Device Design

The design, construction, and installation of damping devices shall be based on response to maximum considered earthquake ground motions and consideration of the following:
1. Low-cycle, large-displacement degradation due to seismic loads;
2. High-cycle, small-displacement degradation due to wind, thermal, or other cyclic loads;
3. Forces or displacements due to gravity loads;
4. Adhesion of device parts due to corrosion or abrasion, biodegradation, moisture, or chemical exposure; and
5. Exposure to environmental conditions, including, but not limited to, temperature, humidity, moisture, radiation (e.g., ultraviolet light), and reactive or corrosive substances (e.g., salt water).
Damping devices subject to failure by low-cycle fatigue shall resist wind forces without slip, movement, or inelastic cycling.
The design of damping devices shall incorporate the range of thermal conditions, device wear, manufacturing tolerances, and other effects that cause device properties to vary during the design life of the device.

### 18.2.5.2 Multiaxis Movement

Connection points of damping devices shall provide sufficient articulation to accommodate simultaneous longitudinal, lateral, and vertical displacements of the damping system.

### 18.2.5.3 Inspection and Periodic Testing

Means of access for inspection and removal of all damping devices shall be provided.
The registered design professional responsible for design of the structure shall establish an appropriate inspection and testing schedule for each type of damping device to ensure that the devices respond in a dependable manner throughout their design life. The degree of inspection and testing shall reflect the established in-service history of the damping devices and the likelihood of change in properties over the design life of the devices.

### 18.2.5.4 Quality Control

As part of the quality assurance plan developed in accordance with Section 11A.1.2, the registered design professional responsible for the structural design shall establish a quality control plan for the manufacture of damping devices. As a minimum, this plan shall include the testing requirements of Section 18.9.2.

### 18.3 Nonlinear Procedures

The stiffness and damping properties of the damping devices used in the models shall be based on or verified by testing of the damping devices as specified in Section 18.9. The nonlinear force-deflection characteristics of damping devices shall be modeled, as required, to explicitly account for device dependence on frequency, amplitude, and duration of seismic loading.

### 18.3.1 Nonlinear Response-History Procedure

A nonlinear response-history analysis shall utilize a mathematical model of the structure and the damping system as provided in Section 16.2.2 and this section. The model shall directly account for the nonlinear hysteretic behavior of elements of the structure and the damping devices to determine its response.
The analysis shall be performed in accordance with Section 16.2 together with the requirements of this section. Inherent damping of the structure shall not be taken as greater than 5% of critical unless test data consistent with levels of deformation at or just below the effective yield displacement of the seismic force-resisting system support higher values.
If the calculated force in an element of the seismic force-resisting system does not exceed 1.5 times its nominal strength, that element is permitted to be modeled as linear.

### 18.3.1.1 Damping Device Modeling

Mathematical models of displacement-dependent damping devices shall include the hysteretic behavior of the devices consistent with test data and accounting for all significant changes in strength, stiffness, and hysteretic loop shape. Mathematical models of velocity-dependent damping devices shall include the velocity coefficient consistent with test data. If this coefficient changes with time and/or temperature, such behavior shall be modeled explicitly. The elements of damping devices connecting damper units to the structure shall be included in the model.
EXCEPTION: If the properties of the damping devices are expected to change during the duration of the time history analysis, the dynamic response is permitted to be enveloped by the upper and lower limits of device properties. All these limit cases for variable device properties must satisfy the same conditions as if the time-dependent behavior of the devices were explicitly modeled.

### 18.3.1.2 Response Parameters

In addition to the response parameters given in Section 16.2.4, for each ground motion used for response-history analysis, individual response parameters consisting of the maximum value of the discrete damping device forces, displacements, and velocities, in the case of velocity-dependent devices, shall be determined.
If at least seven pairs of ground motions are used for response-history analysis, the design values of the damping device forces, displacements, and velocities are permitted to be taken as the average of the values determined by the analyses. If less than seven pairs of ground motions are used for response-history analysis, the design damping device forces, displacements, and velocities shall be taken as the maximum value determined by the analyses. A minimum of three pairs of ground motions shall be used.

### 18.3.2 Nonlinear Static Procedure

The nonlinear modeling described in Section 16.2.2 and the lateral loads described in Section 16.2 shall be applied to the seismic force-resisting system. The resulting force-displacement curve shall be used in lieu of the assumed effective yield displacement, DY, of Eq. 18.6-10 to calculate the effective ductility demand due to the design earthquake ground motions, μD, and due to the maximum considered earthquake ground motions, μM, in Eqs. 18.6-8 and 18.6-9, respectively. The value of (R/Cd) shall be taken as 1.0 in Eqs. 18.4-4, 18.4-5, 18.4-8, and 18.4-9 for the response-spectrum procedure, and in Eqs. 18.5-6, 18.5-7, and 18.5-15 for the equivalent lateral force procedure.

### 18.4 Response-Spectrum Procedure

Where the response-spectrum procedure is used to analyze a structure with a damping system, the requirements of this section shall apply.

### 18.4.1 Modeling

A mathematical model of the seismic force-resisting system and damping system shall be constructed that represents the spatial distribution of mass, stiffness, and damping throughout the structure. The model and analysis shall comply with the requirements of Section 12.9 for the seismic force-resisting system and to the requirements of this section for the damping system. The stiffness and damping properties of the damping devices used in the models shall be based on or verified by testing of the damping devices as specified in Section 18.9.
The elastic stiffness of elements of the damping system other than damping devices shall be explicitly modeled. Stiffness of damping devices shall be modeled depending on damping device type as follows:
1. Displacement-dependent damping devices: Displacement-dependent damping devices shall be modeled with an effective stiffness that represents damping device force at the response displacement of interest (e.g., design story drift). Alternatively, the stiffness of hysteretic and friction damping devices is permitted to be excluded from response spectrum analysis provided design forces in displacement-dependent damping devices, QDSD, are applied to the model as external loads (Section 18.7.2.5).
2. Velocity-dependent damping devices: Velocity-dependent damping devices that have a stiffness component (e.g., viscoelastic damping devices) shall be modeled with an effective stiffness corresponding to the amplitude and frequency of interest.

### 18.4.2.1 Seismic Base Shear

The seismic base shear, V, of the structure in a given direction shall be determined as the combination of modal components, Vm, subject to the limits of Eq. 18.4-1:

VVmin         (18.4-1)

The seismic base shear, V, of the structure shall be determined by the sum of the square root method (SRSS) or complete quadratic combination of modal base shear components, Vm.

### 18.4.2.2 Modal Base Shear

Modal base shear of the mth mode of vibration, Vm, of the structure in the direction of interest shall be determined in accordance with Eqs. 18.4-2:

(18.4-2a)
(18.4-2b)
where
 Csm = seismic response coefficient of the mth mode of vibration of the structure in the direction of interest as determined from Section 18.4.2.4 (m = 1) or Section 18.4.2.6 (m > 1) W̅m = effective seismic weight of the mth mode of vibration of the structure

### 18.4.2.3 Modal Participation Factor

The modal participation factor of the mth mode of vibration, Γm, of the structure in the direction of interest shall be determined in accordance with Eq. 18.4-3:
(18.4-3)
where
 ϕim = displacement amplitude at the ith level of the structure in the mth mode of vibration in the direction of interest, normalized to unity at the roof level.

### 18.4.2.4 Fundamental Mode Seismic Response Coefficient

The fundamental mode (m = 1) seismic response coefficient, CS1, in the direction of interest shall be determined in accordance with Eqs. 18.4-4 and 18.4-5:

For T1D < TS,
(18.4-4)

For T1DTS,
(18.4-5)

### 18.4.2.5 Effective Fundamental Mode Period Determination

The effective fundamental mode (m = 1) period at the design earthquake ground motion, T1D, and at the MCER ground motion, T1M, shall be based on either explicit consideration of the post-yield force deflection characteristics of the structure or determined in accordance with Eqs. 18.4-6 and 18.4-7:
(18.4-6)
(18.4-7)

### 18.4.2.6 Higher Mode Seismic Response Coefficient

Higher mode (m > 1) seismic response coefficient, CSm, of the mth mode of vibration (m > 1) of the structure in the direction of interest shall be determined in accordance with Eqs. 18.4-8 and 18.4-9:

For Tm < TS,
(18.4-8)

For TmTS,
(18.4-9)
where
 Tm = period, in seconds, of the mth mode of vibration of the structure in the direction under consideration BmD = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmD and period of the structure equal to Tm

### 18.4.2.7 Design Lateral Force

Design lateral force at level i due to the mth mode of vibration, Fim, of the structure in the direction of interest shall be determined in accordance with Eq. 18.4-10:
(18.4-10)
Design forces in elements of the seismic force-resisting system shall be determined by the SRSS or complete quadratic combination of modal design forces.

### 18.4.3 Damping System

Design forces in damping devices and other elements of the damping system shall be determined on the basis of the floor deflection, story drift, and story velocity response parameters described in the following sections.
Displacements and velocities used to determine maximum forces in damping devices at each story shall account for the angle of orientation of each device from the horizontal and consider the effects of increased response due to torsion required for design of the seismic force-resisting system.
Floor deflections at level i, δiD and δiM; story drifts, ΔD and ∆M; and story velocities, ∇D and ∇M, shall be calculated for both the design earthquake ground motions and the maximum considered earthquake ground motions, respectively, in accordance with this section.

### 18.4.3.1 Design Earthquake Floor Deflection

The deflection of structure due to the design earthquake ground motions at level i in the mth mode of vibration, δimD, of the structure in the direction of interest shall be determined in accordance with Eq. 18.4-11:

δimD = DmDϕim              (18.4-11)

The total design deflection at each floor of the structure shall be calculated by the SRSS or complete quadratic combination of modal design earthquake deflections.

### 18.4.3.2 Design Earthquake Roof Displacement

Fundamental (m = 1) and higher mode (m > 1) roof displacements due to the design earthquake ground motions, D1D and DmD, of the structure in the direction of interest shall be determined in accordance with Eqs. 18.4-12 and to 18.4-13:

For m = 1,
(18.4-12a)

(18.4-12b)

For m > l,

(18.4-13)

### 18.4.3.3 Design Earthquake Story Drift

Design story drift in the fundamental mode, Δ1D, and higher modes, ΔmD (m > 1), of the structure in the direction of interest shall be calculated in accordance with Section 12.8.6 using modal roof displacements of Section 18.4.3.2.
Total design story drift, ΔD, shall be determined by the SRSS or complete quadratic combination of modal design earthquake drifts.

### 18.4.3.4 Design Earthquake Story Velocity

Design story velocity in the fundamental mode, ∇1D, and higher modes, ∇mD (m > 1), of the structure in the direction of interest shall be calculated in accordance with Eqs. 18.4-14 and 18.4-15:
(18.4-14)

(18.4-15)
Total design story velocity, ΔD, shall be determined by the SRSS or complete quadratic combination of modal design velocities.

### 18.4.3.5 Maximum Considered Earthquake Response

Total modal maximum floor deflection at level i, design story drift values, and design story velocity values shall be based on Sections 18.4.3.1, 18.4.3.3, and 18.4.3.4, respectively, except design roof displacement shall be replaced by maximum roof displacement. Maximum roof displacement of the structure in the direction of interest shall be calculated in accordance with Eqs. 18.4-16 and 18.4-17:

For m = 1,
(18.4-16a)
(18.4-16b)

For m > 1,
(18.4-17)
where
BmM = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmM and period of the structure equal to Tm

### 18.5 Equivalent Lateral Force Procedure

Where the equivalent lateral force procedure is used to design structures with a damping system, the requirements of this section shall apply.

### 18.5.1 Modeling

Elements of the seismic force-resisting system shall be modeled in a manner consistent with the requirements of Section 12.8. For purposes of analysis, the structure shall be considered to be fixed at the base.
Elements of the damping system shall be modeled as required to determine design forces transferred from damping devices to both the ground and the seismic force-resisting system. The effective stiffness of velocity-dependent damping devices shall be modeled.
Damping devices need not be explicitly modeled provided effective damping is calculated in accordance with the procedures of Section 18.6 and used to modify response as required in Sections 18.5.2 and 18.5.3.
The stiffness and damping properties of the damping devices used in the models shall be based on or verified by testing of the damping devices as specified in Section 18.9.

### 18.5.2.1 Seismic Base Shear

The seismic base shear, V, of the seismic force-resisting system in a given direction shall be determined as the combination of the two modal components, V1 and VR, in accordance with Eq. 18.5-1:

(18.5-1)
where
 V1 = design value of the seismic base shear of the fundamental mode in a given direction of response, as determined in Section 18.5.2.2 VR = design value of the seismic base shear of the residual mode in a given direction, as determined in Section 18.5.2.6 Vmin = minimum allowable value of base shear permitted for design of the seismic force-resisting system of the structure in direction of the interest, as determined in Section 18.2.2.1

### 18.5.2.2 Fundamental Mode Base Shear

The fundamental mode base shear, V1, shall be determined in accordance with Eq. 18.5-2:
(18.5-2)
where
 CS1 = the fundamental mode seismic response coefficient, as determined in Section 18.5.2.4 W̅1 = the effective fundamental mode seismic weight including portions of the live load as defined by Eq. 18.4-2b for m = 1

### 18.5.2.3 Fundamental Mode Properties

The fundamental mode shape, ϕil, and participation factor, Γ1, shall be determined by either dynamic analysis using the elastic structural properties and deformational characteristics of the resisting elements or using Eqs. 18.5-3 and 18.5-4:

(18.5-3)
(18.5-4)
where
 hi = the height above the base to level i hn = the structural height as defined in Section 11.2 wi = the portion of the total effective seismic weight, W, located at or assigned to level i
The fundamental period, T1, shall be determined either by dynamic analysis using the elastic structural properties and deformational characteristics of the resisting elements, or using Eq. 18.5-5 as follows:

(18.5-5)
where
 fi = lateral force at level i of the structure distributed in accordance with Section 12.8.3 δi = elastic deflection at level i of the structure due to applied lateral forces fi

### 18.5.2.4 Fundamental Mode Seismic Response Coefficient

The fundamental mode seismic response coefficient, CS1, shall be determined using Eq. 18.5-6 or 18.5-7:

For T1D < TS,
(18.5 -6)
For T1DTS,

(18.5 -7)
where
 SDS = the design spectral response acceleration parameter in the short period range SD1 = the design spectral response acceleration parameter at a period of 1 s B1D = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmD (m = 1) and period of the structure equal to T1D

### 18.5.2.5 Effective Fundamental Mode Period Determination

The effective fundamental mode period at the design earthquake, T1D, and at the maximum considered earthquake, T1M, shall be based on explicit consideration of the post-yield force deflection characteristics of the structure or shall be calculated using Eqs. 18.5-8 and 18.5-9:

(18.5-8)
(18.5-9)

### 18.5.2.6 Residual Mode Base Shear

Residual mode base shear, VR, shall be determined in accordance with Eq. 18.5-10:

VR = CSRR

(18.5-10)
where
 CSR = the residual mode seismic response coefficient as determined in Section 18.5.2.8 W̅R = the effective residual mode effective weight of the structure determined using Eq. 18.5-13

### 18.5.2.7 Residual Mode Properties

Residual mode shape, ϕiR, participation factor, ΓR, effective residual mode seismic weight of the structure, R, and effective period, TR, shall be determined using Eqs. 18.5-11 through 18.5-14:
(18.5-11)

ΓR = 1 − Γ1

(18.5-12)

W̅R = W − 1

(18.5 -13)

TR = 0.4T1

(18.5-14)

### 18.5.2.8 Residual Mode Seismic Response Coefficient

The residual mode seismic response coefficient, CSR, shall be determined in accordance with Eq. 18.5-15:

(18.5-15)
where
 BR = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βR, and period of the structure equal to TR

### 18.5.2.9 Design Lateral Force

The design lateral force in elements of the seismic force-resisting system at level i due to fundamental mode response, Fil, and residual mode response, FiR, of the structure in the direction of interest shall be determined in accordance with Eqs. 18.5-16 and 18.5-17:

(18.5-16)

(18.5-17)
Design forces in elements of the seismic force-resisting system shall be determined by taking the SRSS of the forces due to fundamental and residual modes.

### 18.5.3 Damping System

Design forces in damping devices and other elements of the damping system shall be determined on the basis of the floor deflection, story drift, and story velocity response parameters described in the following sections.
Displacements and velocities used to determine maximum forces in damping devices at each story shall account for the angle of orientation of each device from the horizontal and consider the effects of increased response due to torsion required for design of the seismic force-resisting system.
Floor deflections at levels i, δiD and δiM; story drifts, ΔD and ΔM; and story velocities, ∇D and ∇M, shall be calculated for both the design earthquake ground motions and the maximum considered earthquake ground motions, respectively, in accordance with the following sections .

### 18.5.3.1 Design Earthquake Floor Deflection

The total design deflection at each floor of the structure in the direction of interest shall be calculated as the SRSS of the fundamental and residual mode floor deflections. The fundamental and residual mode deflections due to the design earthquake ground motions, δi1D and δiRD, at the center of rigidity of level i of the structure in the direction of interest shall be determined using Eqs. 18.5-18 and 18.5-19:

δi1D = D1Dϕil

(18.5-18)

δiRD = DRDϕiR (18.5-19)
where
 D1D = fundamental mode design displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.2 DRD = residual mode design displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.5.3.2

### 18.5.3.2 Design Earthquake Roof Displacement

Fundamental and residual mode displacements due to the design earthquake ground motions, D1D and D1R, at the center of rigidity of the roof level of the structure in the direction of interest shall be determined using Eqs. 18.5-20 and 18.5-21:

(18.5-20a)

(18.5-20b)

(18.5-21)

### 18.5.3.3 Design Earthquake Story Drift

Design story drifts, ΔD, in the direction of interest shall be calculated using Eq. 18.5-22:
(18.5-22)
where
 Δ1D = design story drift due to the fundamental mode of vibration of the structure in the direction of interest ΔRD = design story drift due to the residual mode of vibration of the structure in the direction of interest

Modal design story drifts, Δ1D and ΔRD, shall be determined as the difference between the deflections at the top and bottom of the story under consideration using the floor deflections of Section 18.5.3.1.

### 18.5.3.4 Design Earthquake Story Velocity

Design story velocities, ∇D, in the direction of interest shall be calculated in accordance with Eqs. 18.5-23 through 18.5-25:

(18.5-23)

(18.5-24)

(18.5-25)
where
 ∇1D = design story velocity due to the fundamental mode of vibration of the structure in the direction of interest ∇RD = design story velocity due to the residual mode of vibration of the structure in the direction of interest

### 18.5.3.5 Maximum Considered Earthquake Response

Total and modal maximum floor deflections at level i, design story drifts, and design story velocities shall be based on the equations in Sections 18.5.3.1, 18.5.3.3, and 18.5.3.4, respectively, except that design roof displacements shall be replaced by maximum roof displacements. Maximum roof displacements shall be calculated in accordance with Eqs. 18.5-26 and 18.5-27:

(18.5-26a)

(18.5-26b)

(18.5-27)

where
 SM1 = the MCER, 5% damped, spectral response acceleration parameter at a period of 1 s adjusted for site class effects as defined in Section 11.4.3 SMS = the MCER, 5% damped, spectral response acceleration parameter at short periods adjusted for site class effects as defined in Section 11.4.3 B1M = numerical coefficient as set forth in Table 18.6-1 for effective damping equal to βmM (m = 1) and period of structure equal to T1M

### 18.6 Damped Response Modification

As required in Sections 18.4 and 18.5, response of the structure shall be modified for the effects of the damping system.

### 18.6.1 Damping Coefficient

Where the period of the structure is greater than or equal to T0, the damping coefficient shall be as prescribed in Table 18.6-1. Where the period of the structure is less than T0, the damping coefficient shall be linearly interpolated between a value of 1.0 at a 0-second period for all values of effective damping and the value at period T0 as indicated in Table 18.6-1.

### 18.6.2 Effective Damping

The effective damping at the design displacement, βmD, and at the maximum displacement, βmM, of the mth mode of vibration of the structure in the direction under consideration shall be calculated using Eqs. 18.6-1 and 18.6-2:
(18.6-1)
(18.6-2)
where
 βHD = component of effective damping of the structure in the direction of interest due to post-yield hysteretic behavior

Table 18.6-1 Damping Coefficient, BV+I, B1D, BR, B1M, BmD, BmM

(Where Period of the Structure ≥ T0)

 Effective Damping, β (percentage of critical) Bv+l, B1D, BR, B1M, BMD, BmM (where period of the structure ≥ T0) ≤25102030405060708090≥100 0.81.01.21.51.82.12.42.73.03.33.64.0
 of the seismic force-resisting system and elements of the damping system at effective ductility demand, μD βHM = component of effective damping of the structure in the direction of interest due to post-yield hysteretic behavior of the seismic force-resisting system and elements of the damping system at effective ductility demand, μM βI = component of effective damping of the structure due to the inherent dissipation of energy by elements of the structure, at or just below the effective yield displacement of the seismic force-resisting system βVm = component of effective damping of the mth mode of vibration of the structure in the direction of interest due to viscous dissipation of energy by the damping system, at or just below the effective yield displacement of the seismic force-resisting system μD = effective ductility demand on the seismic force-resisting system in the direction of interest due to the design earth quake ground motions μM = effective ductility demand on the seismic force-resisting system in the direction of interest due to the maximum considered earthquake ground motions
Unless analysis or test data supports other values, the effective ductility demand of higher modes of vibration in the direction of interest shall be taken as 1.0.

### 18.6.2.1 Inherent Damping

Inherent damping, βI, shall be based on the material type, configuration, and behavior of the structure and nonstructural components responding dynamically at or just below yield of the seismic force-resisting system. Unless analysis or test data supports other values, inherent damping shall be taken as not greater than 5% of critical for all modes of vibration.

### 18.6.2.2 Hysteretic Damping

Hysteretic damping of the seismic force-resisting system and elements of the damping system shall be based either on test or analysis or shall be calculated using Eqs. 18.6-3 and 18.6-4:

(18.6-3)
(18.6-4)
where
 qH = hysteresis loop adjustment factor, as defined in Section 18.6.2.2.1 μD = effective ductility demand on the seismic force-resisting system in the direction of interest due to the design earthquake ground motions μM = effective ductility demand on the seismic force-resisting system in the direction of interest due to the maximum considered earthquake ground motions
Unless analysis or test data supports other values, the hysteretic damping of higher modes of vibration in the direction of interest shall be taken as zero.

### 18.6.2.2.1 Hysteresis Loop Adjustment Factor

The calculation of hysteretic damping of the seismic force-resisting system and elements of the damping system shall consider pinching and other effects that reduce the area of the hysteresis loop during repeated cycles of earthquake demand. Unless analysis or test data support other values, the fraction of full hysteretic loop area of the seismic force-resisting system used for design shall be taken as equal to the factor, qH, calculated using Eq. 18.6-5:

(18.6 -5)
where
 TS = period defined by the ratio, SD1/SDS T1 = period of the fundamental mode of vibration of the structure in the direction of the interest
The value of qH shall not be taken as greater than 1.0 and need not be taken as less than 0.5.

### 18.6.2.3 Viscous Damping

Viscous damping of the mth mode of vibration of the structure, βvm, shall be calculated using Eqs. 18.6-6 and 18.6-7:
(18.6-6)
(18.6-7)
where
 Wmj = work done by jth damping device in one complete cycle of dynamic response corresponding to the mth mode of vibration of the structure in the direction of interest at modal displacements, δim Wm = maximum strain energy in the mth mode of vibration of the structure in the direction of interest at modal displacements, δim Fim = mth mode inertial force at level i δim = deflection of level i in the mth mode of vibration at the center of rigidity of the structure in the direction under consideration
Viscous modal damping of displacement-dependent damping devices shall be based on a response amplitude equal to the effective yield displacement of the structure.
The calculation of the work done by individual damping devices shall consider orientation and participation of each device with respect to the mode of vibration of interest. The work done by individual damping devices shall be reduced as required to account for the flexibility of elements, including pins, bolts, gusset plates, brace extensions, and other components that connect damping devices to other elements of the structure.

### 18.6.3 Ductility Demand

The effective ductility demand on the seismic force-resisting system due to the design earthquake ground motions, μD, and due to the maximum considered earthquake ground motions, μM, shall be calculated using Eqs. 18.6-8, 18.6-9, and 18.6-10:

(18.6-8)
(18.6-9)
(18.6-10)
where
 D1D = fundamental mode design displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.4.3.2 or 18.5.3.2 D1M = fundamental mode maximum displacement at the center of rigidity of the roof level of the structure in the direction under consideration, Section 18.4.3.5 or 18.5.3.5 DY = displacement at the center of rigidity of the roof level of the structure at the effective yield point of the seismic force-resisting system R = response modification coefficient from Table 12.2-1 Cd = deflection amplification factor from Table 12.2-1 Ω0 = overstrength factor from Table 12.2-1 Γ1 = participation factor of the fundamental mode of vibration of the structure in the direction of interest, Section 18.4.2.3 or 18.5 .2.3 (m = 1) CS1 = seismic response coefficient of the fundamental mode of vibration of the structure in the direction of interest, Section 18.4.2.4 or 18.5.2.4 (m = 1) T1 = period of the fundamental mode of vibration of the structure in the direction of interest

The design ductility demand, μD, shall not exceed the maximum value of effective ductility demand, μmax, given in Section 18.6.4.

### 18.6.4 Maximum Effective Ductility Demand

For determination of the hysteresis loop adjustment factor, hysteretic damping, and other parameters, the maximum value of effective ductility demand, μmax, shall be calculated using Eqs. 18.6-11 and 18.6-12:

For T1DTS,

µmax = 0.5{[R/(Ω0Ie)]2 + 1} (18.6-11)
For T1TS,

µmax = R/(Ω0Ie)
(18.6-12)
where
 Ie = the importance factor determined in accordance with Section 11.5.1 T1D = effective period of the fundamental mode of vibration of the structure at the design displacement in the direction under consideration

For T1 < Ts < T1D, μmax shall be determined by linear interpolation between the values of Eqs. 18.6-11 and 18.6-12.

### 18.7 Seismic Load Conditions and Acceptance Criteria

For the nonlinear procedures of Section 18.3, the seismic force-resisting system, damping system, loading conditions, and acceptance criteria for response parameters of interest shall conform with Section 18.7.1. Design forces and displacements determined in accordance with the response-spectrum procedure of Section 18.4 or the equivalent lateral force procedure of Section 18.5 shall be checked using the strength design criteria of this standard and the seismic loading conditions of Section 18.7.1 and 18.7.2.

### 18.7.1 Nonlinear Procedures

Where nonlinear procedures are used in analysis, the seismic force-resisting system, damping system, seismic loading conditions, and acceptance criteria shall conform to the following subsections.

### 18.7.1.1 Seismic Force-Resisting System

The seismic force-resisting system shall satisfy the strength requirements of Section 12.2.1 using the seismic base shear, Vmin, as given by Section 18.2.2.1. The story drift shall be determined using the design earthquake ground motions.

### 18.7.1.2 Damping Systems

The damping devices and their connections shall be sized to resist the forces, displacements, and velocities from the maximum considered earthquake ground motions.

### 18.7.1.3 Combination of Load Effects

The effects on the damping system due to gravity loads and seismic forces shall be combined in accordance with Section 12.4 using the effect of horizontal seismic forces, QE, determined in accordance with the analysis. The redundancy factor, ρ, shall be taken equal to 1.0 in all cases, and the seismic load effect with overstrength factor of Section 12.4.3 need not apply to the design of the damping system.

### 18.7.1.4 Acceptance Criteria for the Response Parameters of Interest

The damping system components shall be evaluated using the strength design criteria of this standard using the seismic forces and seismic loading conditions determined from the nonlinear procedures and ϕ = 1.0. The members of the seismic force-resisting system need not be evaluated where using the nonlinear procedure forces.

### 18.7.2 Response-Spectrum and Equivalent Lateral Force Procedures

Where response-spectrum or equivalent lateral force procedures are used in analysis, the seismic force-resisting system, damping system, seismic loading conditions, and acceptance criteria shall conform to the following subsections.

### 18.7.2.1 Seismic Force-Resisting System

The seismic force-resisting system shall satisfy the requirements of Section 12.2.1 using seismic base shear and design forces determined in accordance with Section 18.4.2 or 18.5.2.
The design story drift, ΔD, as determined in either Section 18.4.3.3 or 18.5.3.3 shall not exceed (R/Cd) times the allowable story drift, as obtained from Table 12.12-1, considering the effects of torsion as required in Section 12.12.1.

### 18.7.2.2 Damping System

The damping system shall satisfy the requirements of Section 12.2.1 for seismic design forces and seismic loading conditions determined in accordance with this section.

### 18.7.2.4 Modal Damping System Design Forces

Modal damping system design forces shall be calculated on the basis of the type of damping devices and the modal design story displacements and velocities determined in accordance with either Section 18.4.3 or 18.5.3.
Modal design story displacements and velocities shall be increased as required to envelop the total design story displacements and velocities determined in accordance with Section 18.3, where peak response is required to be confirmed by response-history analysis.
1. Displacement-dependent damping devices: Design seismic force in displacement-dependent damping devices shall be based on the maximum force in the device at displacements up to and including the design story drift, ΔD.
2. Velocity-dependent damping devices: Design seismic force in each mode of vibration in velocity-dependent damping devices shall be based on the maximum force in the device at velocities up to and including the design story velocity for the mode of interest.
Displacements and velocities used to determine design forces in damping devices at each story shall account for the angle of orientation of the damping device from the horizontal and consider the effects of increased floor response due to torsional motions.

### 18.7.2.3 Combination of Load Effects

The effects on the damping system and its components due to gravity loads and seismic forces shall be combined in accordance with Section 12.4 using the effect of horizontal seismic forces, QE, determined in accordance with Section 18.7.2.5. The redundancy factor, ρ, shall be taken equal to 1.0 in all cases, and the seismic load effect with overstrength factor of Section 12.4.3 need not apply to the design of the damping system.

### 18.7.2.5 Seismic Load Conditions and Combination of Modal Responses

Seismic design force, QE, in each element of the damping system shall be taken as the maximum force of the following three loading conditions:
1. Stage of maximum displacement: Seismic design force at the stage of maximum displacement shall be calculated in accordance with Eq. 18.7-1:

(18.7-1)
where
 QmSFRS = force in an element of the damping system equal to the design seismic force of the mth mode of vibration of the structure in the direction of interest QDSD = force in an element of the damping system required to resist design seismic forces of displacement-dependent damping devices
Seismic forces in elements of the damping system, QDSD, shall be calculated by imposing design forces of displacement-dependent damping devices on the damping system as pseudostatic forces. Design seismic forces of displacement-dependent damping devices shall be applied in both positive and negative directions at peak displacement of the structure.
2. Stage of maximum velocity: Seismic design force at the stage of maximum velocity shall be calculated in accordance with Eq. 18.7-2:

(18.7-2)
where
 QmDSV = force in an element of the damping system required to resist design seismic forces of velocity-dependent damping devices due to the mth mode of vibration of the structure in the direction of interest
Modal seismic design forces in elements of the damping system, QmDSV, shall be calculated by imposing modal design forces of velocity-dependent devices on the nondeformed damping system as pseudostatic forces. Modal seismic design forces shall be applied in directions consistent with the deformed shape of the mode of interest. Horizontal restraint forces shall be applied at each floor level i of the nondeformed damping system concurrent with the design forces in velocity-dependent damping devices such that the horizontal displacement at each level of the structure is zero. At each floor level i, restraint forces shall be proportional to and applied at the location of each mass point.
3. Stage of maximum acceleration: Seismic design force at the stage of maximum acceleration shall be calculated in accordance with Eq. 18.7-3:

(18.7-3)

The force coefficients, CmFD and CmFV, shall be determined from Tables 18.7-1 and 18.7-2, respectively, using values of effective damping determined in accordance with the following requirements:
For fundamental-mode response (m = 1) in the direction of interest, the coefficients, C1FD and C1FV, shall be based on the velocity exponent, α, that relates device force

Table 18.7-1 Force Coefficient, CmFDa,b
 μ ≤ 1.0 Effective Damping α ≤ 0.25 α = 0.5 α = 0.75 α ≥ 1.0 CmFDa,b = 1.0c ≤0.050.10.20.30.40.50.60.70.80.9≥1.0 1.001.001.001.001.001.001.001.001.001.001.00 1.001.000.950.920.880.840.790.750.700.660.62 1.001.000.940.880.810.730.640.550.500.500.50 1.001.000.930.860.780.710.640.580.530.500.50 μ ≥ 1.0μ ≥ 1.0μ ≥ 1.1μ ≥ 1.2μ ≥ 1.3μ ≥ 1.4μ ≥ 1.6μ ≥ 1.7μ ≥ 1.9μ ≥ 2.1μ ≥ 2.2
aUnless analysis or test data support other values, the force coefficient CmFD for viscoelastic systems shall be taken as 1.0.
bInterpolation shall be used for intermediate values of velocity exponent, α, and ductility demand, μ.
cCmFD shall be taken as equal to 1.0 for values of ductility demand, μ, greater than or equal to the values shown.

Table 18.7-2 Force Coefficient, CmFVa,b
 Effective Damping α ≤ 0.25 α = 0.5 α = 0.75 α ≥ 1.0 ≤0.050.10.20.30.40.50.60.70.80.9≥1.0 1.001.001.001.001.001.001.001.001.001.001.00 0.350.440.560.640.700.750.800.830.901.001.00 0.200.310.460.580.690.770.840.900.941.001.00 0.100.200.370.510.620.710.770.810.901.001.00
aUnless analysis or test data support other values, the force coefficient CmFV for viscoelastic systems shall be taken as 1.0.
bInterpolation shall be used for intermediate values of velocity exponent, α.

to damping device velocity. The effective fundamental-mode damping shall be taken as equal to the total effective damping of the fundamental mode less the hysteretic component of damping (β1D − βHD or β1M − βHM) at the response level of interest(μ = μD or μ = μM)
For higher-mode (m > 1) or residual-mode response in the direction of interest, the coefficients, CmFD and CmFV shall be based on a value of a equal to 1.0. The effective modal damping shall be taken as equal to the total effective damping of the mode of interest (βmD or βmM). For determination of the coefficient CmFD, the ductility demand shall be taken as equal to that of the fundamental mode (μ = μD or μ = μM).

### 18.7.2.6 Inelastic Response Limits

Elements of the damping system are permitted to exceed strength limits for design loads provided it is shown by analysis or test that
1. Inelastic response does not adversely affect damping system function and
2. Element forces calculated in accordance with Section 18.7.2.5, using a value of Ω0 taken as equal to 1.0, do not exceed the strength required to satisfy the load combinations of Section 12.4.

### 18.8 Design Review

A design review of the damping system and related test programs shall be performed by an independent team of registered design professionals in the appropriate disciplines and others experienced in seismic analysis methods and the theory and application of energy dissipation systems.
The design review shall include, but need not be limited to, the following:
1. Review of site-specific seismic criteria including the development of the site-specific spectra and ground motion histories and all other project-specific design criteria;
2. Review of the preliminary design of the seismic force-resisting system and the damping system, including design parameters of damping devices;
3. Review of the final design of the seismic force-resisting system and the damping system and all supporting analyses; and
4. Review of damping device test requirements, device manufacturing quality control and assurance, and scheduled maintenance and inspection requirements.

### 18.9 Testing

The force-velocity displacement and damping properties used for the design of the damping system shall be based on the prototype tests specified in this section.
The fabrication and quality control procedures used for all prototype and production damping devices shall be identical.

### 18.9.1 Prototype Tests

The following tests shall be performed separately on two full-size damping devices of each type and size used in the design, in the order listed as follows.
Representative sizes of each type of device are permitted to be used for prototype testing, provided both of the following conditions are met:
1. Fabrication and quality control procedures are identical for each type and size of device used in the structure.
2. Prototype testing of representative sizes is accepted by the registered design professional responsible for design of the structure.
Test specimens shall not be used for construction, unless they are accepted by the registered design professional responsible for design of the structure and meet the requirements for prototype and production tests.

### 18.9.1.1 Data Recording

The force-deflection relationship for each cycle of each test shall be recorded.

### 18.9.1.2 Sequence and Cycles of Testing

For the following test sequences, each damping device shall be subjected to gravity load effects and thermal environments representative of the installed condition. For seismic testing, the displacement in the devices calculated for the maximum considered earthquake ground motions, termed herein as the maximum device displacement, shall be used.
1. Each damping device shall be subjected to the number of cycles expected in the design windstorm, but not less than 2,000 continuous fully reversed cycles of wind load. Wind load shall be at amplitudes expected in the design windstorm and shall be applied at a frequency equal to the inverse of the fundamental period of the structure (f1 = 1/T1)
EXCEPTION: Damping devices need not be subjected to these tests if they are not subject to wind-induced forces or displacements or if the design wind force is less than the device yield or slip force.
1. Each damping device shall be loaded with five fully reversed, sinusoidal cycles at the maximum earthquake device displacement at a frequency equal to 1/T1M as calculated in Section 18.4.2.5. Where the damping device characteristics vary with operating temperature, these tests shall be conducted at a minimum of three temperatures (minimum, ambient, and maximum) that bracket the range of operating temperatures.
EXCEPTION: Damping devices are permitted to be tested by alternative methods provided all of the following conditions are met:
1. Alternative methods of testing are equivalent to the cyclic testing requirements of this section.
2. Alternative methods capture the dependence of the damping device response on ambient temperature, frequency of loading, and temperature rise during testing.
3. Alternative methods are accepted by the registered design professional responsible for the design of the structure.
1. If the force-deformation properties of the damping device at any displacement less than or equal to the maximum device displacement change by more than 15% for changes in testing frequency from 1/T1M to 2.5/T1, then the preceding tests shall also be performed at frequencies equal to 1/T1M and 2.5/T1.
If reduced-scale prototypes are used to qualify the rate-dependent properties of damping devices, the reduced-scale prototypes should be of the same type and materials and manufactured with the same processes and quality control procedures as full-scale prototypes and tested at a similitude-scaled frequency that represents the full-scale loading rates.

### 18.9.1.3 Testing Similar Devices

Damping devices need not be prototype tested provided that both of the following conditions are met:
1. All pertinent testing and other damping device data are made available to and are accepted by the registered design professional responsible for the design of the structure.
2. The registered design professional substantiates the similarity of the damping device to previously tested devices.

### 18.9.1.4 Determination of Force-Velocity-Displacement Characteristics

The force-velocity-displacement characteristics of a damping device shall be based on the cyclic load and displacement tests of prototype devices specified in the preceding text. Effective stiffness of a damping device shall be calculated for each cycle of deformation using Eq. 17.8-1.

### 18.9.1.5 Device Adequacy

The performance of a prototype damping device shall be deemed adequate if all of the conditions listed below are satisfied. The 15% limits specified in the following text are permitted to be increased by the registered design professional responsible for the design of the structure provided that the increased limit has been demonstrated by analysis not to have a deleterious effect on the response of the structure.

### 18.9.1.5.1 Displacement-Dependent Damping Devices

The performance of the prototype displacement-dependent damping devices shall be deemed adequate if the following conditions, based on tests specified in Section 18.9.1.2, are satisfied:
1. For Test 1, no signs of damage including leakage, yielding, or breakage.
2. For Tests 2 and 3, the maximum force and minimum force at zero displacement for a damping device for any one cycle does not differ by more than 15% from the average maximum and minimum forces at zero displacement as calculated from all cycles in that test at a specific frequency and temperature.
3. For Tests 2 and 3, the maximum force and minimum force at maximum device displacement for a damping device for any one cycle does not differ by more than 15% from the average maximum and minimum forces at the maximum device displacement as calculated from all cycles in that test at a specific frequency and temperature.
4. For Tests 2 and 3, the area of hysteresis loop (Eloop) of a damping device for any one cycle does not differ by more than 15% from the average area of the hysteresis loop as calculated from all cycles in that test at a specific frequency and temperature.
5. The average maximum and minimum forces at zero displacement and maximum displacement, and the average area of the hysteresis loop (Eloop), calculated for each test in the sequence of Tests 2 and 3, shall not differ by more than 15% from the target values specified by the registered design professional responsible for the design of the structure.

### 18.9.1.5.2 Velocity-Dependent Damping Devices

The performance of the prototype velocity-dependent damping devices shall be deemed adequate if the following conditions, based on tests specified in Section 18.9.1.2, are satisfied:
1. For Test 1, no signs of damage including leakage, yielding, or breakage.
2. For velocity-dependent damping devices with stiffness, the effective stiffness of a damping device in any one cycle of Tests 2 and 3 does not differ by more than 15% from the average effective stiffness as calculated from all cycles in that test at a specific frequency and temperature.
3. For Tests 2 and 3, the maximum force and minimum force at zero displacement for a damping device for any one cycle do not differ by more than 15% from the average maximum and minimum forces at zero displacement as calculated from all cycles in that test at a specific frequency and temperature.
4. For Tests 2 and 3, the area of hysteresis loop (Eloop) of a damping device for any one cycle does not differ by more than 15% from the average area of the hysteresis loop as calculated from all cycles in that test at a specific frequency and temperature.
5. The average maximum and minimum forces at zero displacement, effective stiffness (for damping devices with stiffness only), and average area of the hysteresis loop (Eloop) calculated for each test in the sequence of Tests 2 and 3, do not differ by more than 15% from the target values specified by the registered design professional responsible for the design of the structure.

### 18.9.2 Production Testing

Prior to installation in a building, damping devices shall be tested to ensure that their force-velocity-displacement characteristics fall within the limits set by the registered design professional responsible for the design of the structure. The scope and frequency of the production-testing program shall be determined by the registered design professional responsible for the design of the structure.
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