# G.17.4.1.3.2 Displacement Coefficient Method

The objective of the Displacement Coefficient Method is to find the target displacement which is the maximum displacement that the structure is likely to be experienced during the design earthquake. This is equivalent to the performance point in the Capacity Spectrum method. It provides a numerical process for estimating the displacement demand on the structure, by using a bilinear representation of capacity curve and a series of modification factors, or coefficients, to calculate a target displacement.

The structure, directly incorporating the nonlinear load-deformation characteristics of individual components and elements of the building, is subjected to monotonically increasing lateral loads representing inertia forces in an earthquake until a target displacement is exceeded. The damage state comprises deformations for all elements in the structure. Comparison with acceptability criteria for the desired performance goal leads to the identification of deficiencies for individual elements. Performance check at the expected maximum displacement is done to verify whether the lateral force resistance has not degraded by more than a desired percentage (generally 20%) of the peak resistance and the lateral drifts satisfy limits are per standard code.

## Target Displacement, δt

The target displacement is calculated as per procedure described in Section 3.3.3.3.2 of FEMA 356 : 2000.

It is given by the following expression:

 δt = C0C1C2C3Sa[Te2/(4π2)]g

where
 C0 = Modification factor to relate spectral displacement to building roof displacement, as determined by Table 3-2 of FEMA 356. C1 = Modification factor to relate expected maximum inelastic displacements to displacements calculated for linear elastic response = 1.5 for T e < 0.1 sec= 1.0 for T e ≥ T s= [ 1.0 + ( R - 1 ) T s / T e ] / R for T e < T sValue of C1 should not be less than 1.0. Ts = Characteristic period of the response spectrum , defined as period associated with transition from const accleration segment of the spectrum to the constant velocity segment of the spectrum (to be calculated from demand spectrum) Te = Effective fundamental time period = Ti(Ki/Ke)1/2 Ti = Elastic fundamental period Ki = Elastic lateral stiffness of the building Ke = Effective lateral stiffness of the building. Taken as equal to the secant stiffness calculated at a base shear force equal to 60% of the effective yield strength of the Structure obtained from bilinear representation of Capacity Curve. R = Ratio of elastic strength demand to calculated yield strength coefficient = Sa/(Vy/W)Cm Vy = Effective yield strength calculated using the capacity curve . For larger elements or entire structural systems composed of many components, the effective yield point represents the point at which a sufficient number of individual components or elements have yielded and the global structure begins to experience inelastic deformation. Sa = Response spectrum acceleration, at the effective fundamental period and damping ratio of the building (to be calculated from demand spectrum) W = Effective seismic weight Cm = Effective mass factor as determined by Table 3-1 of FEMA 356. C2 = Modification factor to represent the effect of pinched hysteretic shape , stiffness degradation and strength deterioration on maximum displacement response. Taken from Table 3-3 of FEMA 356 for different framing systems and structural performance levels. Alternatively, C2 may be taken as 1.0 for a nonlinear procedure. C3 = Modification factor to represent increased displacement due to dynamic P-D effects = 1.0 for buildings with positive post - yield stiffness = 1.0 + |α|(R - 1)3/2/Te for buildings with negative post - yield stiffness α = Ratio of post - yield stiffness to effective elastic stiffness , where the non-linear force-displacement relation shall be characterized by a bilinear relation

Refer to Figure 3-1 of FEMA 356 for idealized force-displacement curves.