Empirically Derived Design Spectra
Newmark and Hall's Method

N. M. Newmark and W. J. Hall's procedure for developing elastic design spectra starts with the peak values of ground acceleration, velocity, and displacement. These values are used to generate a baseline curve that the spectrum will be generated from. The values of peak ground acceleration and velocity should be obtained from a deterministic or probabilistic seismic hazard analysis. The value of peak ground displacement is a bit more difficult to obtain due to the lack of reliable attenuation relationships. Some empirical functions utilizing the PGA are available to provide additional estimates of the peak ground displacement. A typical baseline curve plotted on tripartite axes is shown below.

Structural Response Amplification Factors

Structural response amplification factors are then applied to the different period-dependent regions of the baseline curve. These factors differ for acceleration, velocity, and displacement, especially at low values of damping. The factors decrease rapidly with increasing damping, especially at small damping values. These factors are shown in the table below.

 Damping (% critical) Structural response amplification factors Median + One Sigma aa av ad aa av ad 1 3.21 2.31 1.82 4.38 3.38 2.73 2 2.74 2.03 1.63 3.66 2.92 2.42 3 2.46 1.86 1.52 3.24 2.64 2.24 5 2.12 1.65 1.39 2.71 2.3 2.01 7 1.89 1.51 1.29 2.36 2.08 1.85 10 1.64 1.37 1.2 1.99 1.84 1.69 20 1.17 1.08 1.01 1.26 1.37 1.38

Newmark and Hall's structural response amplification factors can also be used to change the damping value of other spectra, such as those generated using attenuation relationships. This modification technique is presented in the viscous damping section of the notes.

Tripartite Plots

Newmark and Hall's spectra are plotted on a four-way log plot called a tripartite plot. This is made possible by the simple relation between spectral acceleration, velocity, and displacement:

Sa/w = Sv = Sdw

A tripartite plot begins as a log-log plot of spectral velocity versus period as shown.

Then spectral acceleration and spectral displacement axes are superimposed on the plot at 45 degree angles.

All three types of spectrum (Sa vs. T, Sv vs. T, and Sd vs. T) can be plotted as a single graph, and three spectral values for a particular period can easily be determined. The Sa, Sv, and Sd values for a period of 1 second are shown below.