Question 13.6: Develop a spectrum between yield strength reduction factor a...

Equations (13.30) can be re-written as follows:

F_{y}=\frac{\text{Elastic strength}}{R}                (13.30a)

F_{y}=kx_{y}=m\omega ^{2}_{n}x_{y}=mA_{y}=\frac{\mathrm{w}}{g} A_{y}              (13.30b)

R=\frac{\text{Elastic strength}}{\text{Yield strength}}

The reduction factors for different regions were given by Equation (13.31). From the tripartite spectra generated in Figure 13.31, determine the periods T_{c} and {T_{c}}^{\prime}. The relation between yield strength reduction factor R and period T for different ductility ratio is shown in Table 13.3 and log-log graph Figure 13.32. The period T_{c} is 0.5 sec while {T_{c}}^{\prime} values are highlighted in Table 13.3.
Knowing the elastic spectra for µ = 1, the acceleration ordinate for a given ductility can be computed using the following equation:

A^{\prime}=\frac{A}{\sqrt{2\mu -1} }                (13.34c)

where A = elastic acceleration spectrum ordinate

R = 1             for  T < T_{A}  (13.31a)

R=(2\mu -1)^{\gamma /2}      for T_{A} < T < T_{B}              (13.31b)

R=(2\mu -1)^{0.5}      for T_{B}< T <{T_{C}}^{\prime}          (13.31c)

R= (T/T_{C})\mu      for {T_{C}}^{\prime} < T < T_{C}        (13.31d)

R = µ             for  T >T_{C}        (13.31e)

Table 13.3 Relation between R and T for a Given Ductility μ

The relation between acceleration S_{a} and period T for different ductility µ is shown in Table 13.4 and in log-log graph in Figure 13.33. The values in the table are shown only upto μ = 2. Similar calculations can be done for other values of μ.

Table 13.4 Relation between S_{a} and T for Different Ductility μ