# 12.8.2 Determination of the Building's Fundamental Period

The fundamental period of the building is determined automatically by modal analysis.

The standard requires that the fundamental period, T, used to determine the design base shear, V, does not exceed the approximate fundamental period, T_{a}, times the upper limit coefficient, C_{u}, provided in Table 12.8-1.

**Symbols**

* A_{b }=* area of base of structure [ft2 (m2)]

*= web area of shear wall i [ft2 (m2)]*

**A**_{i }*Coefficient used in the empirical natural vibration period calculation*

**C**_{t }=*Coefficient calculated with eq. 12.8-10*

**C**_{w }^{ }=*0.0019 ft (0.00058 m)*

**C**_{q }^{ }=*= Total height of the upper part of the building above the basement floors [m]*

**h**_{N}*number of stories above the base*

**N =***length of shear wall i [ft (m)]*

**D**_{i}=*The fundamental period of the building [s]*

**T =***The approximate fundamental period calculated empirically [s]*

**T**_{A}=The standard requires that the fundamental period, T, used to determine the design base shear, V, does not exceed the approximate fundamental period, T_{a}, times the upper limit coefficient, C_{u}, provided in Table 12.8-1.

This period limit prevents the use of an unusually low base shear for design of a structure that is, analytically, overly flexible because of mass and stiffness inaccuracies in the analytical model. If T from a properly substantiated analysis (Section 12.8.2) is less than C_{u}T_{a}, then the lower value of T and C_{u}T_{a} should be used for the design of the structure.

It is calculated * automatically* for comparison and compared with the fundamental period obtained as a result of modal analysis. The dynamic analysis report is given in detail in the parameters used in the calculation in the earthquake parameters section.

Historically, the exponent, x, in Eq. (12.8-7) has been taken as 0.75 and was based on the assumption of a linearly varying mode shape while using Rayleigh’s method. Because the empirical expression is based on the lower bound of the data, it produces a lower bound estimate of the period for a building structure of a given height. This lower bound period, when used in Eqs. (12.8-3) and (12.8-4) to compute the seismic response coefficient, C_{s}, provides a conservative estimate of the seismic base shear, V.