in comparison with that of wall 1 This would occur during allocation of the total base shear between the different elements (Section 3.5.6) Note that the approach

outlined in the previous section indicates that there will still be a torsional component of response despite the zero strength eccentricity, which needs to be considered in the design process outlined in (c) below. Torsional response of systems with stiffness eccentricity, but no strength eccentricity is confirmed by inelastic time-history results^4).

(b) Design to Minimize Strength Eccentricity: When the plan layout of lateral force-

resisting elements is such that strength eccentricity is unavoidable, the design objective will generally be to minimize the strength eccentricity, so that the inelastic twist will be minimized. This will be the case when the drift of the most flexible element governs displacement-based design, but may not be the optimum solution when ductility capacity of the stiffest element governs (see Section 6.4.5).

(c) M odification o f Design Displacement to Account for Torsion: The most

common design situation, particularly for frame buildings, but also for many wall buildings of more than four storeys, will be that design displacements are governed by code drift limits. In these cases, the code drift will apply to the element with greatest displacement, including torsional effects, meaning that the design displacement at the building centre of mass, used in the SDOF design, will need to be reduced in proportion to the torsional displacements. The design displacement for the centre of mass will thus be found, reorganizing Eq.(3.53) to give:

where A^cr is the drift-controlled displacement of the critical element. With reference again to Fig.3.28, and assuming that drift limits apply to wall 2, the design displacement for the SDOF substitute structure will be

^ZM ~ ^i,cr @(Xi,cr evx) (3.58)

Chapter 3. Direct D isplacem ent-B ased D esign: Fundam ental C onsiderations 125

It is also possible, particularly for low-rise wall buildings, or buildings containing walls with low height/length aspect ratios, that the displacement capacity of the stiffest wall corresponding to material strain limits may govern design. In this case the design displacement at the centre of mass will be larger than the displacement of the critical element. Equation (3.56) still applies, with due consideration of signs, and with reference to Fig.3.28, the design displacement for the substitute structure will be

~ (3.60)

In general it will be necessary to adopt an iterative approach to determine the design displacement when torsional effects are significant, since 6 depends on eR and ey

which in turn depend on the relative strengths assigned to the lateral force-resisting elements in both orthogonal directions, and the system ductility. However, adequate simplifying assumptions can often be made to avoid the necessity for iteration. Since this is mainly relevant to the behaviour of wall structures, it is discussed in further detail in Chapter 6.

3.9 CAPACITY DESIGN FOR DIRECT DISPLACEMENT-BASED DESIGN

Direct displacement-based design is a procedure for determining the required strength ot different structural systems to ensure that a given performance state, defined by flexural strain or drift limits, is achieved under a specified level of seismic intensity. From this design strength, the required moment capacity at intended locations of plastic hinges or shear capacity of seismic isolation devices, with seismic isolated structures) can be determined. As with force-based design, it is essential to ensure that inelastic action occurs only in these intended locations, and only in the desired inelastic mode. For example, a cantilever wall building will have intended plastic hinges at the bases of the various walls, where inelastic action will be required to occur by inelastic flexural rotation. Special measures are required to ensure that unintended plastic hinges do not occur at other locations up the wall height, where adequate detailing for ductility has not been provided, and to ensure that inelastic shear displacements, which are accompanied by rapid strength degradation, do not occur.

Moments and shears throughout the structure resulting from the distribution of the base shear in accordance with Sections 3.5.6 and 3.5.7 include only the effects of the first inelastic mode of vibration. This is adequate for determining the required strength at plastic hinge locations. However, actual response of the structure will include effects of higher modes. These will not affect the moments at the plastic hinge locations, as these are defined by, and limited to, the first inelastic mode values, but will influence moments and shears at other locations.

A further factor to be considered is that conservative estimates of material strengths will normally be adopted when determining the size of members, and (for reinforced concrete design) the amount of reinforcing steel. If the material strengths exceed the design values, as will normally be the case, then the moments developed at the plastic

126 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

hinge locations will exceed the design values. Since response is inelastic, it is the actual strength, not the theoretical design strength that will be developed under the design level of seismic intensity. AW moments throughout the structure corresponding to the first inelastic mode will then increase in proportion.

Required strengths at these locations, or for actions other than flexure, are found from capacity design considerations^1!. Basic strengths Se for these locations and actions corresponding to the first-mode force distribution are thus amplified by an overstrength factor (fP to account for maximum feasible flexural overcapacity at the plastic hinge locations, and by a dynamic amplification factor 0) to represent the potential increase in design actions due to higher mode effects. The relationship between design strength Sd and basic strength Se is thus

~ Sr — CtiSE (3.61)

where Sr is the required dependable strength of the design action S, and </>s is the corresponding strength reduction factor. As is discussed in Chapter 4, a value of (f>s - 1 should be adopted for flexural design of plastic hinges, but values of (j>$ < 1 are

appropriate for other actions and locations.

The conventional approach currently adopted in force-based design is explained with reference to design of a cantilever wall building. For the required moment capacity of cantilever walls, the base moment is amplified to account for material overstrength, and a linear distribution of moments is generally adopted up the wall height to account for higher mode effects. As is apparent from Fig.3.29(a), this implies higher amplification of moments at mid-height than at the base or top of the wall. Reinforcement cut-off is determined by consideration of tension shift effects. This is achieved by vertical offset of the moment profile.

Shear forces corresponding to the design force distribution are amplified by the flexural overstrength factor, and the dynamic amplification factor 0^ directly in accordance with Eq.(3.61), as shown in Fig.3.29(b). The factor adopted in this figure has been obtained from previous research, related to force-based design and is presented elsewhere^1] in the following form:

0)y = 0.9 + n/10 for n<6 ^ ^

cov = 1.3 + 77 / 30 for 6 < n < 15

where n is the number of storeys in the wall, and need not be taken greater than 15. It is shown in Section 6 . 6 that this equation is generally non-conservative, and alternative

recommendations are made for displacement-based design.

^ For frame structures, beam shear forces, and moments at locations other than potential plastic hinges are amplified by the flexural overstrength factor. Since higher modes are not normally considered for beam design, the dynamic amplification factor is not normally included. However, it should be noted that vertical response is essentially a

Chapter 3. Direct D isplacem ent-B ased D esign: F undam ental Considerations 127

higher mode, and may amplify the gravity moments considerably. A strict formulation of capacity design would take this into account.

Column end moments and shear forces are amplified for both beam plastic hinge overstrength and dynamic amplification. For one-way frames, upper limits for dynamic amplification of column moments of 1.80 have been recommended, with 1.3 for column shear forces. Further amplification for beam flexural overstrength is required^1!.

In this section we have discussed conventional capacity design, as currently applied to force-based design of structures. We show in Section 4.5 and the design chapters related to specific structural types that modifications to the capacity factors for both flexural overstrength, and dynamic amplification are appropriate for direct displacement-based design. 1 ° ' 8 jz .3° * 0.6_C/5 X JU Q_O g 0.4 £ s

0.2

Fig. 3.29 Recent Recommendations for Dynamic Amplification of Design Forces for Equivalent Lateral Force Design of Cantilever Walls^P1l 3.10 SOME IMPLICATIONS OF DDBD

3.10.1 Influence of Seismic Intensity on Design Base Shear Strength

Direct displacement-based seismic design implies significantly different structural sensitivity to seismic intensity than found from current codified force-based design procedures. This can be illustrated with reference to Fig.3.30, where acceleration spectra (Fig.3.30(a)), and displacement spectra (Fig.3.30(b)) are shown for two seismic zones. It

0.8 - .SP 'C A o "9

_o

Dimensionless Moment

(a) Moment Profiles

0.6 - 0.4 -

0.2

Dimensionless Shear Force