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.
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.
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.
The following notations apply to the provisions
of this chapter:
B_{1}_{D} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{ml} (m = 1) and period of structure equal to T_{1}_{D} | |
B_{1}_{E} | = | numerical coefficient as set forth in Table 18.6-1 for the effective damping equal to β_{I} + β_{V1} and period equal to T_{1} | |
B_{1}_{M} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mM} (m = 1) and period of structure equal to T_{1}_{M} | |
B_{m}_{D} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{ml} and period of structure equal to T_{m} | |
B_{m}_{M} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mM} and period of structure equal to T_{m} | |
B_{R} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{R} and period of structure equal to T_{R} | |
B_{V+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, β_{Vm}, (m = 1), plus inherent damping, β_{I},and period of structure equal to T_{1} | |
C_{m}_{FD} | = | force coefficient as set forth in Table 18.7-1 | |
C_{m}_{FV} | = | force coefficient as set forth in Table 18.7-2 | |
C_{S}_{1} | = | 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) | |
C_{S}_{m} | = | seismic response coefficient of the m^{th} 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) | |
C_{S}_{R} | = | seismic response coefficient of the residual mode of vibration of the structure in the direction of interest Section 18.5.2.8 | |
D_{I}_{D} | = | 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 | |
D_{1}_{M} | = | 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 | |
D_{m}_{D} | = | design displacement at the center of rigidity of the roof level of the structure due to the m^{th} mode of vibration in the direction under consideration, Section 18.4.3.2 | |
D_{m}_{M} | = | maximum displacement at the center of rigidity of the roof level of the structure due to the m^{th} mode of vibration in the direction under consideration, Section 18.4.3.5 | |
D_{R}_{D} | = | 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 | |
D_{R}_{M} | = | 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 | |
D_{Y} | = | 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 | |
f_{i} | = | lateral force at level i of the structure distributed approximately in accordance with Section 12.8.3 and Section 18.5.2.3 | |
F_{i}_{l} | = | 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 | |
F_{i}_{m} | = | inertial force at level i ( or mass point i) in the m^{th} mode of vibration of the structure in the direction of interest Section 18.4.2.7 | |
F_{i}_{R} | = | 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 | |
q_{H} | = | hysteresis loop adjustment factor as determined in Section 18.6.2.2.1 | |
Q_{DSD} | = | force in an element of the damping system required to resist design seismic forces of displacement-dependent damping devices, Section 18.7.2.5 | |
Q_{mDSV} | = | forces in an element of the damping system required to resist design seismic forces of velocity-dependent damping devices due to the m^{th} mode of vibration of the structure in the direction of interest, Section 18.7.2.5 | |
Q_{mSFRS} | = | force in an element of the damping system equal to the design seismic force of the m^{th} mode of vibration of the structure in the direction of interest, Section 18.7.2.5 | |
T_{1} | = | the fundamental period of the structure in the direction under consideration | |
T_{1}_{D} | = | 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 | |
T_{1}_{M} | = | 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 | |
T_{R} | = | period, in seconds, of the residual mode of vibration of the structure in the direction under consideration, Section 18.5.2.7 | |
V_{m} | = | design value of the seismic base shear of the m^{th} mode of vibration of the structure in the direction of interest, Section 18.4.2.2 | |
V_{min} | = | 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 | |
V_{R} | = | 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 | |
W̅_{1} | = | effective fundamental mode seismic weight determined in accordance with Eq. 18.4-2b for m = 1 | |
W̅x_{R} | = | effective residual mode seismic weight determined in accordance with Eq. 18.5-13 | |
α | = | velocity exponent relating damping device force to damping device velocity | |
β_{mD} | = | total effective damping of the m^{th} mode of vibration of the structure in the direction of interest at the design displacement, Section 18.6.2 | |
β_{mM} | = | total effective damping of the m^{th} mode of vibration of the structure in the direction of interest at the maximum displacement, Section 18.6.2 | |
β_{HD} | = | 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 μ_{D}, Section 18.6.2.2 | |
β_{HM} | = | 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, μ_{M}, Section 18.6.2.2 | |
β_{I} | = | 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 | |
β_{R} | = | 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 μ_{D} = 1.0 and μ_{M} = 1.0) | |
β_{Vm} | = | component of effective damping of the m^{th} 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 | |
δ_{i} | = | elastic deflection of level i of the structure due to applied lateral force, f_{i}, Section 18.5.2.3 | |
δ_{i}_{1}_{D} | = | 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 | |
δ_{i}_{D} | = | total design deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3 | |
δ_{i}_{M} | = | total maximum deflection of level i at the center of rigidity of the structure in the direction under consideration, Section 18.5.3 | |
δ_{i}_{RD} | = | 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 | |
δ_{i}_{m} | = | deflection of level i in the m^{th} mode of vibration at the center of rigidity of the structure in the direction under consideration, Section 18.6.2.3 | |
△_{1}_{D} | = | 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 m^{th} 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 | |
μ_{D} | = | 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 | |
μ_{M} | = | 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 | |
μ_{max} | = | maximum allowable effective ductility demand on the seismic force-resisting system due to the design earthquake ground motions, Section 18.6.4 | |
ϕ_{i}_{l} | = | 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 | |
ϕ_{iR} | = | 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 | |
Γ_{1} | = | 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) | |
Γ_{m} | = | participation factor in the m^{th} mode of vibration of the structure in the direction of interest, Section 18.4.2.3 | |
Γ_{R} | = | participation factor of the residual mode of vibration of the structure in the direction of interest, Section 18.5.2.7 | |
∇_{1}_{D} | = | 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 m^{th} mode of vibration of the structure in the direction of interest, Section 18.4.3.4 |
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.
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.
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:
The design of the seismic force-resisting system in each direction shall satisfy the requirements of Section 18.7 and the following:
- The seismic base shear used for design of the seismic force-resisting system shall not be less than V_{min}, where V_{min} is determined as the greater of the values computed using Eqs. 18.2-1 and 18.2-2:
- In the direction of interest, the damping system has less than two damping devices on each floor level, configured to resist torsion.
- The seismic force-resisting system has horizontal irregularity Type 1b (Table 12.3-1) or vertical irregularity Type lb (Table 12.3-2).
- 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.
(18.2-1)
(18.2 -2)
where
V | = | seismic base shear in the direction of interest, deter mined in accordance with Section 12.8 |
B_{v+}_{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, β_{vm} (m = 1), plus inherent damping, β_{I}, and period of structure equal to T_{1} |
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 :
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.
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:
- The structure is located on a Class F site.
- The structure is located at a site with S_{1} greater than or equal to 0.6.
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:
- The structure is located at a site with S_{1} greater than or equal to 0.6.
- The damping system is explicitly modeled and analyzed using the response-history analysis method.
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 S_{1} greater than or equal to 0.6.
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 S_{1} greater than or equal to 0.6.
The nonlinear procedures of
Section 18.3 are permitted to be used for design of all structures
with damping systems.
The response spectrum
procedure of Section 18.4 is permitted to be used for
design of a structure with a damping system provided that
- In the direction of interest, the damping system has at least two damping devices in each story, configured to resist torsion.
- The total effective damping of the fundamental mode, β_{mD} (m = 1), of the structure in the direction of interest is not greater than 35% of critical.
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
- In the direction of interest, the damping system has at least two damping devices in each story, configured to resist torsion.
- The total effective damping of the fundamental mode, β_{mD} (m = 1), of the structure in the direction of interest is not greater than 35% of critical.
- 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).
- Floor diaphragms are rigid as defined in Section 12.3.1.
- The height of the structure above the base does not exceed 100 ft (30 m).
The design, construction, and installation
of damping devices shall be based on response to maximum
considered earthquake ground motions and consideration of the
following:
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.
- Low-cycle, large-displacement degradation due to seismic loads;
- High-cycle, small-displacement degradation due to wind, thermal, or other cyclic loads;
- Forces or displacements due to gravity loads;
- Adhesion of device parts due to corrosion or abrasion, biodegradation, moisture, or chemical exposure; and
- 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).
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.
Connection points of damping
devices shall provide sufficient articulation to accommodate
simultaneous longitudinal, lateral, and vertical displacements of
the damping system.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, D_{Y}, of Eq.
18.6-10 to calculate the effective ductility demand due to the
design earthquake ground motions, μ_{D}, and due to the maximum
considered earthquake ground motions, μ_{M}, in Eqs. 18.6-8 and
18.6-9, respectively. The value of (R/C_{d}) 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.
Where the response-spectrum procedure is used to analyze a
structure with a damping system, the requirements of this section
shall apply.
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:
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:
- 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, Q_{DSD}, are applied to the model as external loads (Section 18.7.2.5).
- 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.
The seismic base shear, V, of the
structure in a given direction shall be determined as the combination
of modal components, V_{m}, subject to the limits of Eq. 18.4-1:
V ≥ V_{min} (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, V_{m}.
V ≥ V_{min} (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, V_{m}.
Modal base shear of the m^{th} mode
of vibration, V_{m}, 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
C_{sm} | = | seismic response coefficient of the m^{th} 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) |
W̅_{m} | = | effective seismic weight of the m^{th} mode of vibration of the structure |
The modal participation
factor of the m^{th} mode of vibration, Γ_{m}, of the structure in the
direction of interest shall be determined in accordance with Eq.
18.4-3:
(18.4-3)
where
ϕ_{im} | = | displacement amplitude at the i^{th} level of the structure in the m^{th} mode of vibration in the direction of interest, normalized to unity at the roof level. |
The fundamental mode (m = 1) seismic response coefficient, C_{S}_{1}, in the direction of interest shall be determined in
accordance with Eqs. 18.4-4 and 18.4-5:
For T_{1}_{D} < T_{S},
For T_{1}_{D} ≥ T_{S},
For T_{1}_{D} < T_{S},
(18.4-4)
For T_{1}_{D} ≥ T_{S},
(18.4-5)
The effective fundamental mode (m = 1) period at the
design earthquake ground motion, T_{1}_{D}, and at the MCE_{R} ground
motion, T_{1}_{M}, 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)
Higher
mode (m > 1) seismic response coefficient, C_{Sm}, of the m^{th} 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 T_{m} < T_{S},
For T_{m} ≥ T_{S},
For T_{m} < T_{S},
(18.4-8)
For T_{m} ≥ T_{S},
(18.4-9)
where
T_{m} | = | period, in seconds, of the m^{th} mode of vibration of the structure in the direction under consideration |
B_{mD} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mD} and period of the structure equal to T_{m} |
Design lateral force at level i
due to the m^{th} mode of vibration, F_{im}, 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.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, δ_{iD} and δ_{iM}; story drifts, Δ_{D} 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.
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, δ_{iD} and δ_{iM}; story drifts, Δ_{D} 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.
The deflection
of structure due to the design earthquake ground motions at level
i in the m^{th} mode of vibration, δ_{imD}, of the structure in the direction
of interest shall be determined in accordance with Eq.
18.4-11:
δ_{imD} = D_{mD}ϕ_{im} (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.
δ_{imD} = D_{mD}ϕ_{im} (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.
Fundamental
(m = 1) and higher mode (m > 1) roof displacements due to
the design earthquake ground motions, D_{1}_{D} and D_{mD}, 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,
For m > l,
For m = 1,
(18.4-12a)
(18.4-12b)
For m > l,
(18.4-13)
Design story drift in
the fundamental mode, Δ_{1}_{D}, and higher modes, Δ_{mD} (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, Δ_{D}, shall be determined by the SRSS or complete quadratic combination of modal design earthquake drifts.
Total design story drift, Δ_{D}, shall be determined by the SRSS or complete quadratic combination of modal design earthquake drifts.
Design story
velocity in the fundamental mode, ∇_{1}_{D}, 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, Δ_{D}, shall be determined by the
SRSS or complete quadratic combination of modal design
velocities.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,
For m > 1,
For m = 1,
(18.4-16a)
(18.4-16b)
For m > 1,
(18.4-17)
where
B_{mM} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mM} and period of the structure equal to T_{m} |
---|
Where the equivalent lateral force procedure is used to design
structures with a damping system, the requirements of this
section shall apply.
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.
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.
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, V_{1} and
V_{R}, in accordance with Eq. 18.5-1:
(18.5-1)
where
V_{1} | = | 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 |
V_{R} | = | design value of the seismic base shear of the residual mode in a given direction, as determined in Section 18.5.2.6 |
V_{min} | = | 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 |
The fundamental
mode base shear, V_{1}, shall be determined in accordance with Eq.
18.5-2:
(18.5-2)
where
C_{S}_{1} | = | the fundamental mode seismic response coefficient, as determined in Section 18.5.2.4 |
W̅_{1} | = | the effective fundamental mode seismic weight including portions of the live load as defined by Eq. 18.4-2b for m = 1 |
The fundamental
mode shape, ϕ_{i}_{l}, and participation factor, Γ_{1}, 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:
The fundamental period, T_{1}, 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:
(18.5-3)
(18.5-4)
where
h_{i} | = | the height above the base to level i |
h_{n} | = | the structural height as defined in Section 11.2 |
w_{i} | = | the portion of the total effective seismic weight, W, located at or assigned to level i |
(18.5-5)
where
f_{i} | = | lateral force at level i of the structure distributed in accordance with Section 12.8.3 |
δ_{i} | = | elastic deflection at level i of the structure due to applied lateral forces f_{i} |
The fundamental mode seismic response coefficient, C_{S}_{1}, shall be
determined using Eq. 18.5-6 or 18.5-7:
For T_{1}_{D} < T_{S},
(18.5 -7)
where
For T_{1}_{D} < T_{S},
(18.5 -6)
For T_{1}_{D} ≥ T_{S},
(18.5 -7)
S_{DS} | = | the design spectral response acceleration parameter in the short period range |
S_{D}_{1} | = | the design spectral response acceleration parameter at a period of 1 s |
B_{1}_{D} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mD} (m = 1) and period of the structure equal to T_{1}_{D} |
The effective fundamental mode period at the design earthquake,
T_{1}_{D}, and at the maximum considered earthquake, T_{1}_{M}, 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)
(18.5-8)
(18.5-9)
Residual mode base
shear, V_{R}, shall be determined in accordance with Eq. 18.5-10:
V_{R} = C_{SR}W̅_{R}
(18.5-10)
where
V_{R} = C_{SR}W̅_{R}
(18.5-10)
C_{SR} | = | the residual mode seismic response coefficient as determined in Section 18.5.2.8 |
W̅_{R} | = | the effective residual mode effective weight of the structure determined using Eq. 18.5-13 |
Residual mode shape, ϕ_{iR},
participation factor, Γ_{R}, effective residual mode seismic weight
of the structure, W̅_{R}, and effective period, T_{R}, shall be determined
using Eqs. 18.5-11 through 18.5-14:
(18.5-11)
Γ_{R} = 1 − Γ_{1}
(18.5-12)W̅_{R} = W − W̅_{1}
(18.5 -13)T_{R} = 0.4T_{1}
(18.5-14)
The
residual mode seismic response coefficient, C_{SR}, shall be determined
in accordance with Eq. 18.5-15:
(18.5-15)
where
(18.5-15)
B_{R} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{R}, and period of the structure equal to T_{R} |
The design lateral force in elements
of the seismic force-resisting system at level i due to
fundamental mode response, F_{i}_{l}, and residual mode response,
F_{iR}, 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.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, δ_{iD} and δ_{iM}; story drifts, Δ_{D} and Δ_{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 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, δ_{iD} and δ_{iM}; story drifts, Δ_{D} and Δ_{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 the following sections .
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,
δ_{i}_{1}_{D} and δ_{i}_{RD}, 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:
δ_{i}_{1}_{D} = D_{1}_{D}ϕ_{i}_{l}
(18.5-18)
δ_{i}_{1}_{D} = D_{1}_{D}ϕ_{i}_{l}
(18.5-18)
δ_{i}_{RD} = D_{RD}ϕ_{i}_{R}
(18.5-19)
where
D_{1}_{D} | = | 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 |
D_{RD} | = | 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 |
Fundamental
and residual mode displacements due to the design earthquake
ground motions, D_{1D} and D_{1R}, 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)
(18.5-20a)
(18.5-20b)
(18.5-21)
Design story drifts, Δ_{D}, in the direction of interest shall be calculated using Eq.
18.5-22:
Modal design story drifts, Δ_{1D} and Δ_{RD}, 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.
(18.5-22)
where
Δ_{1D} | = | design story drift due to the fundamental mode of vibration of the structure in the direction of interest |
Δ_{RD} | = | design story drift due to the residual mode of vibration of the structure in the direction of interest |
Modal design story drifts, Δ_{1D} and Δ_{RD}, 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.
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
(18.5-23)
(18.5-24)
(18.5-25)
∇_{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 |
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)
where
(18.5-26a)
(18.5-26b)
(18.5-27)
S_{M}_{1} | = | the MCE_{R}, 5% damped, spectral response acceleration parameter at a period of 1 s adjusted for site class effects as defined in Section 11.4.3 |
S_{M}_{S} | = | the MCE_{R}, 5% damped, spectral response acceleration parameter at short periods adjusted for site class effects as defined in Section 11.4.3 |
B_{1M} | = | numerical coefficient as set forth in Table 18.6-1 for effective damping equal to β_{mM} (m = 1) and period of structure equal to T_{1M} |
Where the period of the structure
is greater than or equal to T_{0}, the damping coefficient shall be as
prescribed in Table 18.6-1. Where the period of the structure is
less than T_{0}, 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 T_{0} as indicated in
Table 18.6-1.
The effective damping at the design
displacement, β_{mD}, and at the maximum displacement, β_{mM}, of
the m^{th} mode of vibration of the structure in the direction under
consideration shall be calculated using Eqs. 18.6-1 and 18.6-2:
Table 18.6-1 Damping Coefficient, B_{V+I}, B_{1D}, B_{R}, B_{1M}, B_{mD}, B_{mM}
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.
(18.6-1)
(18.6-2)
where
β_{HD} | = | component of effective damping of the structure in the direction of interest due to post-yield hysteretic behavior |
Table 18.6-1 Damping Coefficient, B_{V+I}, B_{1D}, B_{R}, B_{1M}, B_{mD}, B_{mM}
(Where Period of the Structure ≥ T_{0})
Effective Damping, β (percentage of critical) |
B_{v+l}, B_{1D}, B_{R}, B_{1M}, B_{MD}, B_{mM} (where period of the structure ≥ T_{0}) |
≤2 5 10 20 30 40 50 60 70 80 90 ≥100 |
0.8 1.0 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 4.0 |
of the seismic force-resisting system and elements of the damping system at effective ductility demand, μ_{D} |
||
β_{HM} | = | 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, μ_{M} |
β_{I} | = | 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 |
β_{Vm} | = | component of effective damping of the m^{th} 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 |
μ_{D} | = | effective ductility demand on the seismic force-resisting system in the direction of interest due to the design earth quake ground motions |
μ_{M} | = | effective ductility demand on the seismic force-resisting system in the direction of interest due to the maximum considered earthquake ground motions |
Inherent damping, β_{I}, 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.
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:
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.
(18.6-3)
(18.6-4)
where
q_{H} | = | hysteresis loop adjustment factor, as defined in Section 18.6.2.2.1 |
μ_{D} | = | effective ductility demand on the seismic force-resisting system in the direction of interest due to the design earthquake ground motions |
μ_{M} | = | effective ductility demand on the seismic force-resisting system in the direction of interest due to the maximum considered earthquake ground motions |
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, q_{H}, calculated using Eq. 18.6-5:
The value of q_{H} shall not be taken as greater than 1.0 and need
not be taken as less than 0.5.
(18.6 -5)
where
T_{S} | = | period defined by the ratio, S_{D1}/S_{DS} |
T_{1} | = | period of the fundamental mode of vibration of the structure in the direction of the interest |
Viscous damping of the m^{th} mode
of vibration of the structure, β_{vm}, shall be calculated using Eqs.
18.6-6 and 18.6-7:
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.
(18.6-6)
(18.6-7)
where
W_{mj} |
= |
work done by j^{th} damping device in one complete cycle of dynamic response corresponding to the m^{th} mode of vibration of the structure in the direction of interest at modal displacements, δ_{im} |
W_{m} |
= |
maximum strain energy in the m^{th} mode of vibration of the structure in the direction of interest at modal displacements, δ_{im} |
F_{im} | = | m^{th} mode inertial force at level i |
δ_{im} | = | deflection of level i in the m^{th} mode of vibration at the center of rigidity of the structure in the direction under consideration |
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.
The effective ductility
demand on the seismic force-resisting system due to the design earthquake ground motions, μ_{D}, and due to the maximum considered
earthquake ground motions, μ_{M}, shall be calculated using
Eqs. 18.6-8, 18.6-9, and 18.6-10:
The design ductility demand, μ_{D}, shall not exceed the maximum value of effective ductility demand, μ_{max}, given in Section 18.6.4.
(18.6-8)
(18.6-9)
(18.6-10)
where
D_{1D} |
= |
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 |
D_{1M} |
= |
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 |
D_{Y} |
= |
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 |
C_{d} | = | deflection amplification factor from Table 12.2-1 |
Ω_{0} | = | overstrength factor from Table 12.2-1 |
Γ_{1} |
= |
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) |
C_{S1} |
= |
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) |
T_{1} | = | period of the fundamental mode of vibration of the structure in the direction of interest |
The design ductility demand, μ_{D}, shall not exceed the maximum value of effective ductility demand, μ_{max}, given in Section 18.6.4.
For determination
of the hysteresis loop adjustment factor, hysteretic
damping, and other parameters, the maximum value of effective
ductility demand, μ_{max}, shall be calculated using Eqs. 18.6-11
and 18.6-12:
For T_{1D} ≤ T_{S},
For T_{1} < T_{s} < T_{1D}, μ_{max} shall be determined by linear interpolation between the values of Eqs. 18.6-11 and 18.6-12.
For T_{1D} ≤ T_{S},
µ_{max} = 0.5{[R/(Ω_{0}I_{e})]^{2} + 1}
(18.6-11)
For T_{1} ≥ T_{S},
µ_{max} = R/(Ω_{0}I_{e})
(18.6-12)
where
I_{e} | = | the importance factor determined in accordance with Section 11.5.1 |
T_{}1D |
= |
effective period of the fundamental mode of vibration of the structure at the design displacement in the direction under consideration |
For T_{1} < T_{s} < T_{1D}, μ_{max} shall be determined by linear interpolation between the values of Eqs. 18.6-11 and 18.6-12.
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.
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.
The damping devices and their
connections shall be sized to resist the forces, displacements, and
velocities from the maximum considered earthquake ground
motions.
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, Q_{E}, 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.
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.
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.
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, Δ_{D}, 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.
The design story drift, Δ_{D}, 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.
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.
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.
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.
- 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, Δ_{D}.
- 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.
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, Q_{E}, 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.
Seismic design force, Q_{E}, in each element
of the damping system shall be taken as the maximum force of
the following three loading conditions:
- 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)whereQ_{mSFRS} = force in an element of the damping system equal to the design seismic force of the m^{th} mode of vibration of the structure in the direction of interest Q_{DSD} = force in an element of the damping system required to resist design seismic forces of displacement-dependent damping devices - 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)whereQ_{mDSV} = force in an element of the damping system required to resist design seismic forces of velocity-dependent damping devices due to the m^{th} mode of vibration of the structure in the direction of interest - 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, C_{mFD} and C_{mFV}, 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, C_{1FD} and C_{1FV}, shall be based on the velocity exponent, α, that relates device force
Table 18.7-1 Force Coefficient, C_{mFD}^{a,b}μ ≤ 1.0 Effective Damping α ≤ 0.25 α = 0.5 α = 0.75 α ≥ 1.0 C_{mFD}^{a,b} = 1.0_{}^{c} ≤0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
≥1.01.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.001.00
1.00
0.95
0.92
0.88
0.84
0.79
0.75
0.70
0.66
0.621.00
1.00
0.94
0.88
0.81
0.73
0.64
0.55
0.50
0.50
0.501.00
1.00
0.93
0.86
0.78
0.71
0.64
0.58
0.53
0.50
0.50μ ≥ 1.0
μ ≥ 1.0
μ ≥ 1.1
μ ≥ 1.2
μ ≥ 1.3
μ ≥ 1.4
μ ≥ 1.6
μ ≥ 1.7
μ ≥ 1.9
μ ≥ 2.1
μ ≥ 2.2
^{b}Interpolation shall be used for intermediate values of velocity exponent, α, and ductility demand, μ.
^{c}C_{mFD} 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, C_{mFV}^{a,b}Effective Damping α ≤ 0.25 α = 0.5 α = 0.75 α ≥ 1.0 ≤0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
≥1.01.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.000.35
0.44
0.56
0.64
0.70
0.75
0.80
0.83
0.90
1.00
1.000.20
0.31
0.46
0.58
0.69
0.77
0.84
0.90
0.94
1.00
1.000.10
0.20
0.37
0.51
0.62
0.71
0.77
0.81
0.90
1.00
1.00
^{b}Interpolation 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 (β_{1D} − β_{HD} or β_{1M} − β_{HM}) at the response level of interest(μ = μ_{D} or μ = μ_{M})
Elements of the damping
system are permitted to exceed strength limits for design loads
provided it is shown by analysis or test that
- Inelastic response does not adversely affect damping system function and
- Element forces calculated in accordance with Section 18.7.2.5, using a value of Ω_{0} taken as equal to 1.0, do not exceed the strength required to satisfy the load combinations of Section 12.4.
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:
The design review shall include, but need not be limited to, the following:
- 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;
- Review of the preliminary design of the seismic force-resisting system and the damping system, including design parameters of damping devices;
- Review of the final design of the seismic force-resisting system and the damping system and all supporting analyses; and
- Review of damping device test requirements, device manufacturing quality control and assurance, and scheduled maintenance and inspection requirements.
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.
The fabrication and quality control procedures used for all prototype and production damping devices shall be identical.
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:
Representative sizes of each type of device are permitted to be used for prototype testing, provided both of the following conditions are met:
- Fabrication and quality control procedures are identical for each type and size of device used in the structure.
- Prototype testing of representative sizes is accepted by the registered design professional responsible for design of the structure.
The force-deflection relationship for
each cycle of each test shall be recorded.
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.
- 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 (f_{1} = 1/T_{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/T_{1M} 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.
- Alternative methods of testing are equivalent to the cyclic testing requirements of this section.
- Alternative methods capture the dependence of the damping device response on ambient temperature, frequency of loading, and temperature rise during testing.
- Alternative methods are accepted by the registered design professional responsible for the design of the structure.
- 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/T_{1M} to 2.5/T_{1}, then the preceding
tests shall also be performed at frequencies equal to
1/T_{1M} and 2.5/T_{1}.
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.
Damping devices need not
be prototype tested provided that both of the following conditions
are met:
- 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.
- The registered design professional substantiates the similarity of the damping device to previously tested devices.
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.
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.
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:
- For Test 1, no signs of damage including leakage, yielding, or breakage.
- 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.
- 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.
- For Tests 2 and 3, the area of hysteresis loop (E_{loop}) 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.
- The average maximum and minimum forces at zero displacement and maximum displacement, and the average area of the hysteresis loop (E_{loop}), 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.
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:
- For Test 1, no signs of damage including leakage, yielding, or breakage.
- 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.
- 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.
- For Tests 2 and 3, the area of hysteresis loop (E_{loop}) 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.
- The average maximum and minimum forces at zero displacement, effective stiffness (for damping devices with stiffness only), and average area of the hysteresis loop (E_{loop}) 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.
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.