Fundamental period and natural frequency of buildings

A building’s natural frequency (n) is simply the inverse of the fundamental period (T), and, for flexible and dynamically sensitive buildings and other structures, this natural frequency is used in the calculation of the Gust Factor. The ASCE 7 standard specifies that buildings are considered “flexible” if they have a fundamental period greater than 1 second. Tall building or moment-frame buildings tend to be more flexible than others, so these code criteria can have a significant impact on their design and their construction costs. And engineering judgement is required when the natural frequency/fundamental period of the structure is near 1.0 and note that the nonbuilding structure or components with the fundamental period less than 0.06 s shall be considered a rigid element.

Recently I heard Bhavin Shah discuss this topic on the weekly session and it reminds me of the article I read over a decade ago in Structure magazine written by Will Jacobs titled “Building Periods: Moving Forward (and Backward)”.

Equation form 1

Equation form 2

For Indian scenario, (with equation form 2) 0.09*h/SQRT (d), seems reasonable for all buildings except moment resisting frames, while the equations with form 1 is used in ASCE7 and seems appropriate for regular rectangular buildings. For additional information on the IS 1893 (Part 1):2016, empirical formula for calculation of time period as defined in cl no. 7.6.2 was discussed in detail by Bhavin Shah.

The article by Will Jacobs published in 2008 explained in detail that the research on the topic has been done worldwide for seismic considerations, and the formulae in the building codes contain an inherent bias towards shorter-period structures. The comparison of the periods based on ASCE 7-05 was well explained in the article and the wind based approximate natural frequency (na) are rewritten in the same form as the approximate fundamental period (Ta) for easier comparison. The ratio of this upper bound "wind" period to "seismic" period is approximately 1.4 to 1.7, thus mirroring the coefficient for the upper limit on the calculated period for seismic design! 

For the preliminary design stages, some empirical relationships for building period Ta (Ta = 1∕n1) are available in the earthquake-related chapters of every building code and design standard. However, these expressions are based on recommendations for earthquake design with inherent bias toward higher estimates of fundamental frequencies, because shorter periods result in higher seismic loading and are therefore conservative. For wind design applications and drift calculations, these values may be unconservative because an estimated frequency higher than the actual frequency would yield lower values of the gust-effect factor and thus a lower design wind pressure. However, the estimated period based on the analytical model is limited by the upper bound (Cu*Ta) to avoid overestimation of the period and 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. So, in general, if the fundamental period, T, from a properly substantiated analysis (per Section12.8.2) is less than upper bound of the period, then T should be used for the design of the structure.

For wind design (n < 1), the general recommendation is to use the lower bound approximate natural frequency based on the equations mentioned in the code for wind design or (in a rare case) higher frequency based on the stiffer analytical model with realistic lateral stiffness. As an alternative to using the approximate period formulae, we can use structural software to perform a dynamic analysis to determine the building periods as the software calculates only the stiffness of the bare lateral frame, without any stiffness contribution from the gravity framing, interior partitions, infill walls and cladding, etc. Also note that higher mass and lower stiffness assigned in the model, will increase the estimated fundamental period. So, these calculated fundamental periods are generally too long (lower frequency), that would yield higher values of the gust factor, and considered conservative. This is especially true for short-term wind loading and elastic-level forces such as those considered for serviceability.

Most engineers are now aware of these, since this information moved into the body of ASCE 7-10, rather than the code commentary in prior codes. In general, buildings with a mean roof height < 60’ (18 m) can be considered rigid, but for those that are flexible the following text from ASCE 7-16 should be reviewed carefully.

ASCE 7-16

Section 12.8.2.1 Approximate Fundamental Period.

The approximate fundamental period (Ta), in seconds, shall be determined from the following equation:

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. The exponents provided in the standard, however, are based on actual response data from building structures, thus more accurately reflecting the influence of mode shape on the exponent. 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.

Section 26.11.2.1 Limitations for Approximate Natural Frequency.

As an alternative to performing an analysis to determine n1, the approximate building natural frequency, na, shall be permitted to be calculated in accordance with Section 26.11.3 for structural steel, concrete, or masonry buildings meeting the following requirements:

1. The building height is less than or equal to 300ft (91m), and

2. The building height is less than 4 times its effective length, Leff, based on Eq. (26.11-1)

Section 26.11.3 Approximate Natural Frequency.

The approximate lower bound natural frequency (na), in hertz, of concrete or structural steel buildings meeting the conditions of Section 26.11.2.1 is permitted to be determined from one of the following equations:

Explicit calculation of the gust-effect factor per section 26.11

These equations for gust factor and natural frequency are based on studies for regular rectangular building models, with side ratio (D∕B, where D is the depth of the building section along the oncoming wind direction) from 1∕3 to 3. So wind tunnel studies should be considered for slender and irregular tall buildings or dynamically sensitive structures.

Commentary:

BUILDING OR OTHER STRUCTURE, FLEXIBLE: A building or other structure is considered “flexible” if it contains a significant dynamic resonant response. The gust effects and the natural frequency of buildings or other structures greater than 60 ft (18.3 m) in height is determined in accordance with Sections 26.11. When buildings or other structures have a height exceeding 4 times the least horizontal dimension or when there is reason to believe that the natural frequency is less than 1 Hz (natural period greater than 1 s), the natural frequency of the structure should be investigated. Approximate equations for natural frequency or period for various building and structure types in addition to those given in Section 26.11.2 for buildings are contained in Commentary Section C26.11.

Gust energy and the resonant response of most buildings and structures with lowest natural frequency above 1 Hz ("rigid structure") are sufficiently small that resonant response can often be ignored. A general guidance is that most rigid buildings and structures have height to-minimum-width less than 4.

Approximate fundamental frequency:

Lower bound estimates of frequency that are more suited for use in wind applications and are now given in Section 26.11.2; graphs of these expressions are shown in Fig. C26.11-1.

Because these expressions are based on regular buildings, limitations based on height and slenderness are required. The effective length, Leff, based on Eq. (26.11-1), uses a height-weighted average of the along-wind length of the building for slenderness evaluation. The top portion of the building is most important; hence, the height-weighted average is appropriate. This method is an appropriate first-order equation for addressing buildings with setbacks. Observations from wind tunnel testing of buildings where frequency is calculated using analysis software show that the following expression for frequency can be used for steel and concrete buildings less than about 400 ft (122 m) in height:

n1 = 100∕H(ft) average value (C26.11-6)

na = 75∕H(ft) lower bound value (C26.11-7)

Based on full-scale measurements of buildings under the action of wind, the following expression has been proposed for wind applications:

f n1 = 150∕h (ft) (C26.11-8)

This frequency expression is based on older buildings and overestimates the frequency common in U.S. construction for smaller buildings less than 400 ft (122 m) in height, but it becomes more accurate for tall buildings greater than 400 ft (122 m) in height. The Australian and New Zealand Standard AS/NZS 1170.2, Eurocode ENV1991-2-4, Hong Kong Code of Practice on Wind Effects (2004), and others have adopted Eq. C26.11-8 for all building types and all heights.

Studies in Japan involving a suite of buildings under low amplitude excitations have led to the following expressions for natural frequencies of buildings:

n1 = 220∕h (ft) (concrete buildings) (C26.11-9)

n1 = 164∕h (ft) (steel buildings) (C26.11-10)

These expressions result in higher frequency estimates than those obtained from the general expression given in Eqs. (C26.11-6) through (C26.11-8), particularly since the Japanese data set has limited observations for the more flexible buildings sensitive to wind effects, and Japanese construction tends to be stiffer.

The construction in India is similar to Japan and buildings tend to be stiffer, so the lower bound 75∕H(ft) may be too conservative.

Structural Damping:

Because the level of structural response in the strength and serviceability limit states is different, the damping values associated with these states may differ.

In wind applications, damping ratios of 1% and 2% are typically used in the United States for steel and concrete buildings at serviceability levels, respectively, while ISO (1997) suggests 1% and 1.5% for steel and concrete, respectively. Damping ratios for buildings under ultimate strength design conditions may be significantly higher, and 2.5% to 3% is commonly assumed.

Damping values for steel support structures for signs, chimneys, and towers may be much lower than buildings and may fall in the range of 0.15–0.5%. Damping values of special structures like steel stacks can be as low as 0.2–0.6% and 0.3–1.0% for unlined and lined steel chimneys, respectively. These values may provide some guidance for design.

Damping levels used in wind load applications are smaller than the 5% damping ratios common in seismic applications because buildings subjected to wind loads respond essentially elastically, whereas buildings subjected to design-level earthquakes respond inelastically at higher damping levels.

Damping and stiffness modifiers appropriate for seismic and wind design will be further explained in near future. While wind drifts and wind speeds based on various mean return intervals were covered in the previous article.