Evaluation of building period formulas for seismic design

The paper investigates the fundamental period of buildings, a key parameter in equivalent lateral force seismic design. Because fundamental periods cannot be known before design, practitioners rely on empirical formulas in building codes or preliminary analytical models. These empirical formulas also constrain analytical periods via an upper-bound factor (Cu) in NEHRP 2003 and ASCE 7-05. Historically, codes used simple height- or story-based formulas, later refined and calibrated (e.g., ATC 3-06; Goel and Chopra) primarily for MRFs and, more recently, shear walls. However, calibration datasets were limited in size and scope, some structural systems (e.g., braced frames, “other” types) lack robust calibration, and all instrumented buildings are from strong seismic regions, raising questions about applicability in lower seismicity. The research question is to evaluate the accuracy and conservatism of ASCE 7-05 approximate period formulas across multiple structural systems using a substantially expanded set of measured (apparent) building periods, and to assess factors such as occupancy importance that may influence periods and, consequently, seismic demands.

The authors summarize the evolution of code-based approximate period formulas since the 1970s across UBC, BOCA, NEHRP, ASCE 7, Eurocode 8, and ATC 3-06. Early codes had two broad formulas (for MRFs and others). ATC 3-06 calibrated RC and steel MRF formulas using periods from the 1971 San Fernando Earthquake, assuming linear lateral force distribution and drift-controlled deflection profiles. Goel and Chopra later recalibrated MRF formulas and proposed a new shear wall formula dependent on equivalent shear area and height, adopted in NEHRP 2000/2003 and ASCE 7, while noting code shear-wall height-only formulas can be non-conservative. Housner and Brady reported wide dispersion in shear wall periods due to stiffness and mass variability. Tremblay developed an empirical relation for concentrically braced frames (CBFs) largely from analytical studies, highlighting gaps in measured data for braced systems. Prior datasets were modest (e.g., 42 steel MRFs, 27 RC MRFs, 9 shear wall buildings), leaving several system types under-evaluated and possibly biased toward strong-seismic-region designs; the Cu factor addressing regional design-level effects was judgment-based rather than data-driven.

• Data compilation and selection:

  • 191 buildings total: 141 from California Geological Survey’s CSMIP/CGS stations and 50 from prior sources (ATC 3-06, NOAA, USGS, and others).
  • Over 800 apparent fundamental periods identified from 411 recordings of 67 earthquake events (1970 Lytle Creek to 2008 Yucaipa).
  • Buildings in California only; excluded highly irregular, base-isolated, or energy-dissipation-system buildings from general comparisons; treated directions separately, yielding 382 lateral systems evaluated.
  • Structural systems represented include steel MRFs (125 systems), RC MRFs (58), RC shear walls (56), CBFs (34), EBFs (8), RM/URM shear walls (23), precast tilt-up shear walls (12), and 66 “other” systems (dual/multiple systems, plywood shear walls, composite frames, etc.).
  • Occupancy categories noted, particularly Category IV (essential facilities); 34 essential facilities among CGS stations.

• Apparent period identification:

  • Used transfer function method with base acceleration as input and multiple outputs along height (roof and intermediate floors) per direction (longitudinal/transverse).
  • Excluded channels at locations not representative of global response (e.g., penthouses, irregular wings).
  • Defined building’s apparent period for a direction as the average of peaks across output channels.
  • Selected low-intensity events to limit nonlinear effects: PGA < 0.15g (0.20g threshold for CBFs due to limited data).
  • Recognized that “apparent periods” may differ from true elastic fundamental periods because of inelasticity, SSI, instrumentation layout, and noise; using base input neglects SSI effects in comparisons.

• Code formulas evaluated:

  • ASCE 7-05 generic form Ta = Cr h^x with system-specific Cr and x; and alternative Ta = 0.10 N for up to 12 stories with ≥10 ft story heights (also compared beyond nominal limits to assess conservatism).
  • Steel MRFs: Cr = 0.028, x = 0.8; also Ta = 0.10 N.
  • RC MRFs: Cr = 0.016, x = 0.9; also Ta = 0.10 N.
  • Shear walls (RC, RM/URM, PC): Cr = 0.020, x = 0.75; also noted alternative Ta = 0.0019 hn / Cw but could not evaluate it due to lack of detailed wall data.
  • Braced frames: CBFs Cr = 0.020, x = 0.75; EBFs Cr = 0.030, x = 0.75; compared CBF results also to Tremblay’s Ta = 0.0076 h.
  • Other structural types: Cr = 0.020, x = 0.75 (code), assessed suitability.
  • Considered code upper-bound factor Cu (e.g., Cu = 1.4 for SD1 ≥ 0.4g) in illustrating upper limits.

• Analyses performed:

  • Plotted measured periods vs height and number of stories by system type and compared to code curves and Cu-limited upper bounds.
  • For steel MRFs, segregated essential vs non-essential buildings (especially for heights < 100 ft) and ran regressions constrained to x = 0.8 to quantify period reductions associated with importance factors.

• Dataset and identification:

  • 800 apparent periods from 191 buildings and 67 earthquakes; 382 lateral systems analyzed.

  • Periods identified via transfer functions using base input; low-PGA events to limit nonlinearity.

• Steel MRFs:

  • Ta = 0.028 h^0.8 generally predicts the lower bound across heights, but exhibits large discrepancies for low-to-medium rise buildings (< ~100 ft; roughly 6–8 stories).
  • Story-based formula Ta = 0.10 N is conservative and captures lower bounds well for ≤ ~5 stories; underestimates periods for >12 stories but remains conservative; may be usable beyond nominal limits if economy is not critical.
  • Essential facilities (Category IV) show about 40% shorter periods than comparable non-essential buildings for low-to-medium rises; suggested applying a factor of ~0.6 to Ta for essential steel MRFs in this height range.

• RC MRFs:

  • Ta = 0.016 h^0.9 captures the lower bound of measured periods well.
  • Ta = 0.10 N also follows lower bounds even up to ~25 stories, though dispersion is larger when using story count instead of height.

• Braced steel frames:

  • CBFs: Code Ta = 0.020 h^0.75 can substantially underestimate periods of high-rise CBFs (>200 ft), risking uneconomical design; for low-to-medium rises it tracks lower bounds reasonably. Tremblay’s Ta = 0.0076 h matches measured periods across heights better than the code formula.
  • EBFs: Limited measured data; code Ta = 0.030 h^0.75 aligns with observed trend for available cases, but more data are needed, especially for high-rises.

• Shear wall buildings (RC, RM/URM, PC tilt-up):

  • Code Ta = 0.020 h^0.75 often overestimates the lower bound of measured periods (i.e., predicts longer periods), which is non-conservative; consistent with Goel and Chopra.
  • Suggested revising Cr from 0.020 to 0.015 (keeping x = 0.75) for shear walls; RM/URM and PC walls exhibit similar trends to RC walls.
  • Could not evaluate Ta = 0.0019 hn / Cw due to insufficient wall property data.

• Other structural types (dual/multiple systems, plywood SW, composite frames, etc.):

  • Code Ta = 0.020 h^0.75 largely overestimates measured periods; many dual/multiple systems are stiffer/stronger than single-system assumptions imply.
  • Suggested using Cr = 0.015 (x = 0.75) as a better lower-bound estimator for initiating design in absence of detailed modeling.

• Influence of design level:

  • Importance factor (higher design base shear) correlates with shorter periods (e.g., ~40% shorter for essential steel MRFs). This implies that base design level affects fundamental period; buildings in low seismic regions may have longer periods than those in high seismic regions. Current Cu partially addresses this variability but lacks direct calibration with low-seismic-region data.

The study directly addresses whether ASCE 7-05 approximate period formulas provide conservative and reliable lower-bound estimates across structural systems. Using a substantially larger measured dataset than prior calibrations, the findings show that MRF formulas are broadly conservative for lower bounds (especially RC MRFs) but that the shear wall height-only formula can be non-conservative. For braced frames, the code relation is acceptable for low-to-medium rises but less accurate for taller CBFs; Tremblay’s height-linear relation performs better overall. The pronounced reduction in periods for essential buildings suggests that code prescriptions using importance factors alter stiffness and period in ways that may inadvertently increase seismic demand (due to shorter periods) and influence fragility, challenging assumptions about the safety margin provided by importance factors. Additionally, the prevalence of dual/multiple systems in practice means “other” structures often have shorter periods than code estimates; a reduced Cr improves lower-bound prediction. These insights reinforce the need to refine empirical formulas by system type, height range, and design level, and to acquire data from low-seismic regions to calibrate Cu and regional applicability, thereby improving both safety and economy in design.

• A large database of measured apparent periods from 191 buildings was assembled and used to evaluate ASCE 7-05 approximate period formulas across major lateral systems.
• Steel MRFs: The formula Ta = 0.028 h^0.8 captures lower bounds but shows considerable spread for low-to-medium rises. Essential steel MRFs exhibit ~40% shorter periods; applying a 0.6 factor to Ta is suggested for low-to-medium rise essential buildings.
• RC MRFs: Ta = 0.016 h^0.9 adequately predicts lower bounds.
• Braced frames: For CBFs, code Ta = 0.020 h^0.75 underestimates high-rise periods; Tremblay’s Ta = 0.0076 h better matches data; EBFs appear reasonably described by code but require more measurements.
• Shear walls (RC, RM/URM, PC): Code Ta = 0.020 h^0.75 is often non-conservative; using Cr = 0.015 (x = 0.75) is recommended when a simple height-based relation is desired.
• Other structural types: Code periods are overestimated; recommend Cr = 0.015 to better reflect lower bounds at the preliminary stage.
• The dependence of period on design base shear (e.g., via importance factor) implies code period limits and Cu factors should be re-examined and potentially recalibrated, especially for low-seismic regions. Future work should gather and analyze data from low-to-moderate seismicity areas and obtain detailed wall properties to evaluate shear wall area-based formulas.

• Apparent periods were derived from recorded responses using transfer functions; true elastic fundamental periods may differ due to inelasticity, soil-structure interaction, instrumentation layout, and measurement noise.
• Base acceleration was used as input; SSI and soil effects were not explicitly considered.
• Low-intensity events (PGA < 0.15g; 0.20g for CBFs) were used to limit nonlinear effects, but some nonlinearity may persist.
• Detailed design and wall property data were unavailable for many buildings, preventing evaluation of the shear wall area-based formula (Ta = 0.0019 hn / Cw).
• The dataset is limited to California (strong seismic region), limiting direct inference to low-seismic regions; essential-building comparisons included a small sample (8 steel MRF Category IV), limiting generalization to high-rises.
• Braced frame datasets were limited; many instrumented CBF buildings have isolation/dampers and were excluded from comparisons.