0.2 0.4 0.6 0.8 Dimensionless Distance
3.5.3 Effective Mass
From consideration of the mass participating in the first inelastic mode of vibration, the effective system mass for the substitute structure is
me = ' Z i {mA i ) /Ad (333)
/'=!
where is the design displacement given by Eq.(3.26). Typically, the effective mass will range from about of the total mass for multi-storey cantilever walls to more than 85% for frame buildings of more than 20 storeys. For simple multi-span bridges the effective mass will often exceed 95% of the total mass. The remainder of the mass participates in the higher modes of vibration. Although modal combination rules, such as
100 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures
the square-root-sum-of-squ ares (SRSS) or complete quadratic combination (CQC)rulesl°J may indicate a significant increase in the elastic base shear force over that from the first inelastic mode, there is much less influence on the design base overturning moment. The effects of higher modes are inadequately represented by elastic analyses, as will be shown in the chapters devoted to specific structural forms, and are better accommodated in the capacity design phase, rather than the preliminary phase of design.
3.5.4 Equivalent Viscous Damping
(a) System Damping: The effective damping depends on the structural system and
displacement ductility demand, as illustrated in Fig.3.1(c) and Eqs.3.17. This requires determination of the displacement ductility demand of the substitute structure. This poses few problems, since the design displacement A^ has already been determined, from Eq.(3.26). The effective yield displacement Ay needs to be interpolated from the profile of displacements at yield (e.g. Eq.(3.31) for cantilever walls, or Eq.(3.8) for frames). For frames it Is adequate to assume that the yield drift is constant with height (i.e. the yield displacement profile is linear with height), and hence the yield displacement is
A y = 0 y . H e (3.34)
where 6y is given by Eq.(3.8). For walls, the yield displacement is found from Eq.(3.31) with Hi — He. In both cases this requires knowledge of the effective height of the
substitute structure, which may be taken as:
H.
= E M ' Z M (3-35)/-I /=1
The design ductility factor, for use in Eq.(3.17) is then
// = A d /Ay (3.36)
in the usual fashion.
Note that provided reasonable ductility is implied by the design displacement
A</,
Fig.3.1(c) and Eq.(3.17) indicate that the damping is not strongly dependent on the ductility, and average values may be adopted. This is also implied in Fig.3.17(b). Note also, that concrete and masonry structures are much more flexible than normally assumed by designers, and hence code drift limits, rather than displacement ductility capacity tends to govern design (see Section 5.3.1, e.g.). As a consequence, the design ductility, and the effective damping are known at the start of the design process, and no iteration is needed in determining the design base shear force.When the lateral resistance of a building in a given direction is provided by a number of walls of different length, the ductility demand of each wall will differ, since the yield displacements of the walls will be inversely proportional to the wall lengths (see Eq.(3.6c)), while the maximum displacements at design-level response will be essentially
Chapter 3. D irect D isplacem ent-B ased D esign: F undam ental C onsiderations 101
equal, subject only to small variations resulting from torsional response and floor diaphragm flexibility. This was discussed in Section 1.3.6, with reference to Fig. 1.13. Hence the system damping will need to consider the different effective damping in each wall.
In the general case, where different structural elements with different strengths and damping factors contribute to the seismic resistance, the global damping may be found by rhe weighted average based on the energy dissipated by the different structural elements. That is,
m m
£ = K
fA £ ) /I ,/A
p-37>
7=1 j=1
where Vj, Aj and ^ are the design strength at the design displacement, displacement at height of centre of seismic force, and damping, respectively, of the /h structural element. Alternatively, the energy dissipated may be related to the moment and rotation of different plastic hinges (VjA\ —MjOj). This form may be more appropriate for frame structures.
With multiple in-plane walls, the displacements of the different walls will all be the same, and hence Eq.(3.37) can be simplified to
m m
z. = u v,z, ) , 1l vj <3'38>
J= 1 7=1
where Vj and are the base shear force and damping of the m walls in a given direction. Some modification of Eq.(3.38) may be required when torsional response of a building containing more than one plane of walls in a given direction is considered. In this case, Eq.(3.37) applies. However, the error involved in using Eq.(3.38) is small, even when torsional response is expected.
A rational decision will be to apportion the total base shear force requirement between the walls in proportion to the square of the length. This will result in essentially constant reinforcement ratios between the walls. With wall strength proportional to length squared, Eq.(3.38) may be rewritten as:
m m
(3-39)
7=1 7=1
(b) Influence o f Foundation Flexibility on Effective Damping: Although the
influence of foundation flexibility on seismic design can be incorporated into force-based design, albeit with some difficulty, it is rarely considered. Foundation flexibility will increase the initial elastic period, and reduce the ductility capacity corresponding to the strain or drift limit statesiP4l It is comparatively straightforward, however, to incorporate the influence of elastic foundation compliance into Direct Displacement-Based Design. If the limit state being considered is strain-limited, then the design displacement will be increased by the elastic displacement corresponding to foundation compliance (this
102 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures
requires a knowledge of the design base moment and shear force, and hence some iteration may be required). If, however, the limit state is defined by code drift limits, there will be no change in the design displacement, thus implying reduced permissible structural deformation.
Fig.3.20 Damping Contributions of Foundation and Structure
The second influence relates to the effective damping. Both foundation and structure will contribute to the damping. Consider the force-displacement hysteresis loops of Fig.3.20, where foundation (A/) and structure (As) components of the peak response displacement A^ = As + Af have been separated for a cantilever wall building. Assuming sinusoidal displacement response, the area-based equivalent viscous damping for the foundation and for the structure can be separately expressed as
Ai
Foundation: £/>«,=-
7
-=----— (3-40a)t A
Structure: £ = — —--- — (3.40b)
where A f and As are hysteretic areas within the loops (i.e. energy absorbed per cycle) for foundadon and structure respectively. As shown in Fig.3.20, the hysteretic area of the combined structure/foundation system will be the sum of the two components, and hence the system equivalent viscous damping will be
System: q —--- r = —--- (3.40c)
C hapter 3. Direct D isplacem ent-B ased D esign: Fundam ental Considerations 103
Equation (3.40) is based on the assumption that the ratios of the effective damping of the structure and foundation is equal to the ratios of the hysteretic areas. However, it will be seen that the identical final result would be obtained from the more general form of Eq.(3.37).