Rakesh K. Goel

Evaluation of a Modified MPA Procedure Assuming Higher Modes as Elastic to Estimate Seismic Demands

The modal pushover analysis (MPA) procedure, which includes the contributions of all significant modes of vibration, estimates seismic demands much more accurately than current pushover procedures used in structural engineering practice. Outlined in this paper is a modified MPA (MMPA) procedure wherein the response contributions of higher vibration modes are computed by assuming the building to be linearly elastic, thus reducing the computational effort. After outlining such a modified procedure, its accuracy is evaluated for a variety of frame buildings and ground motion ensembles. Although it is not necessarily more accurate than the MPA procedure, the MMPA procedure is an attractive alternative for practical application because it leads to a larger estimate of seismic demands, improving the accuracy of the MPA results in some cases (relative to nonlinear response history analysis) and increasing their conservatism in others. However, such conservatism is unacceptably large for lightly damped systems, with damping significantly less than 5%. Thus the MMPA procedure is not recommended for such systems.

Capacity‐Demand‐Diagram Methods Based on Inelastic Design Spectrum

An improved capacity-demand-diagram method that uses the well-known constant-ductility design spectrum for the demand diagram is developed and illustrated by examples. This method estimates the deformation of inelastic SDF systems consistent with the selected inelastic design spectrum, while retaining the attraction of graphical implementation of the ATC-40 Nonlinear Static Procedure. One version of the improved method is graphically similar to ATC-40 Procedure A whereas the second version is graphically similar to ATC-40 Procedure B. However, the improved procedures differ from ATC-40 procedures in one important sense. The demand diagram used is different: the constant-ductility demand diagram for inelastic systems in the improved procedure versus the elastic demand diagram in ATC-40 for equivalent linear systems. The improved method can be conveniently implemented numerically if its graphical features are not important to the user. Such a procedure, based on equations relating the yield strength reduction factor, R y , and ductility factor, μ, for different period, T n , ranges, has been presented, and illustrated by examples using three different R y - μ - T n relations.

Evaluation of Modal and FEMA Pushover Analyses: SAC Buildings

This paper comprehensively evaluates the Modal Pushover Analysis (MPA) procedure against the “exact” nonlinear response history analysis (RHA) and investigates the accuracy of seismic demands determined by pushover analysis using FEMA-356 force distributions; the MPA procedure in this paper contains several improvements over the original version presented in Chopra and Goel (2002). Seismic demands are computed for six buildings, each analyzed for 20 ground motions. It is demonstrated that with increasing number of “modes” included, the height-wise distribution of story drifts and plastic rotations estimated by MPA becomes generally similar to trends noted from nonlinear RHA. The additional bias and dispersion introduced by neglecting “modal” coupling and P-Δ effects due to gravity loads in MPA procedure is small unless the building is deformed far into the inelastic range with significant degradation in lateral capacity. A comparison of the seismic demands computed by FEMA-356 NSP and nonlinear RHA showed that FEMA-356 lateral force distributions lead to gross underestimation of story drifts and completely fail to identify plastic rotations in upper stories compared to the values from the nonlinear RHA. The “Uniform” force distribution in FEMA-356 NSP seems unnecessary because it grossly overestimates drifts and plastic rotations in lower stories and grossly underestimates them in upper stories. The MPA procedure resulted in estimates of demand that were much better than from FEMA force distributions over a wide range of responses—from essentially elastic response of Boston buildings to strongly inelastic response of Los Angeles buildings. However, pushover analysis procedures cannot be expected to provide satisfactory estimates of seismic demands for buildings deforming far into the inelastic range with significant degradation of the lateral capacity; for such cases, nonlinear RHA becomes necessary.

Direct Displacement‐Based Design: Use of Inelastic vs. Elastic Design Spectra

Direct displacement-based design requires a simplified procedure to estimate the seismic deformation of an inelastic SDF system, representing the first (elastic) mode of vibration of the structure. This step is usually accomplished by analysis of an “equivalent” linear system using elastic design spectra. In this paper, an equally simple procedure is developed that is based on the well-known concepts of inelastic design spectra. We demonstrate that the procedure provides the following: (1) accurate values of displacement and ductility demands, and (2) a structural design that satisfies the design criteria for allowable plastic rotation. In contrast, the existing procedure using elastic design spectra for equivalent linear systems in shown to underestimate significantly the displacement and ductility demands. The existing procedure is shown to be deficient in yet another sense; the acceptable value of the plastic rotation, leaving an erroneous impression that the allowable plastic rotation constraint has been satisfied.

Building Period Formulas for Estimating Seismic Displacements

Traditionally, empirical formulas for building period recommended for code applications are intentionally calibrated to underestimate the period in order to estimate base shear conservatively. At a shorter period, the seismic displacements are smaller, however, and hence underestimated. This note discusses this issue and recommends formulas for estimating seismic displacements of buildings.

Evaluation of Bridge Abutment Capacity and Stiffness during Earthquakes

The “actual” capacity and stiffness values of the abutment-soil systems at the US 101/Painter Street Overpass, determined from its earthquake motions, are used to investigate how abutment stiffness varies during earthquakes and to evaluate current modeling procedures. It is found that the “actual” abutment stiffness may be significantly different during different phases of the shaking and decreases significantly as the abutment deformation increases. The CALTRANS modeling procedure leads to a good estimate of the transverse abutment stiffness and capacity. However, this procedure may overestimate the normal abutment stiffness and capacity by a factor of over two, indicating that the assumed value of 7.7 ksf for the ultimate passive resistance of the soil, used in the CALTRANS procedure, may be too high. The AASHTO-83 and ATC-6 procedures lead to an initial estimate of the abutment stiffness that is too high in both directions.

Extension of Modal Pushover Analysis to Compute Member Forces

This paper extends the modal pushover analysis (MPA) procedure for estimating seismic deformation demands for buildings to compute member forces. Seismic demands are computed for six buildings, each analyzed for 20 ground motions. A comparison of seismic demands computed by the MPA and nonlinear response history analysis (RHA) demonstrates that the MPA procedure provides good estimates of the member forces. The bias (or error) in forces is generally less than that noted in earlier investigations of story drifts and is comparable to the error in the standard response spectrum analysis (RSA) for elastic buildings. The four FEMA-356 force distributions, on the other hand, provide estimates of member forces that may be one-half to one-fourth of the value from nonlinear RHA.

Effects of supplemental viscous damping on seismic response of asymmetric-plan systems

Coupling between lateral and torsional motions may lead to much larger edge deformations in asymmetric-plan systems compared to systems with a symmetric plan. Supplemental viscous damping has been found to be effective in reducing deformations in the symmetric-plan system. This investigation examined how supplemental damping affects the edge deformations in asymmetric-plan systems. First, the parameters that characterize supplemental viscous damping and its plan-wise distribution were identified, and then the effects of these parameters on edge deformations were investigated. It was found that supplemental damping reduces edge deformations and that reductions by a factor of up three are feasible with proper selection of system parameters. Furthermore, viscous damping may be used to reduce edge deformations in asymmetric-plan systems to levels equal to or smaller than those in the corresponding symmetric-plan system.

Seismic behaviour of asymmetric buildings with supplemental damping

This paper investigates the response of asymmetric-plan buildings with supplemental viscous damping to harmonic ground motion using modal analysis techniques. It is shown that most modal parameters, except dynamic amplification factors (DAFs), are affected very little by the plan-wise distribution of supplemental damping in the practical range of system parameters. Plan-wise distribution of supplemental damping significantly influences the DAFs, which, in turn, influence the modal deformations. These trends are directly related to the apparent modal damping ratios; the first modal damping ratio increases while the second decreases as CSD moves from right to left of the system plan, and their values increase with larger plan-wise spread of the supplemental damping. The largest reduction in the flexible edge deformation occurs when damping in the first mode is maximized by distributing the supplemental damping such that the damping eccentricity takes on the largest value with algebraic sign opposite to the structural eccentricity. Copyright

Role of Higher-"Mode" Pushover Analyses in Seismic Analysis of Buildings

The role of higher-“mode” pushover analyses in seismic analysis of buildings is examined in this paper. It is demonstrated that the higher-“mode” pushover curves reveal plastic hinge mechanisms that are not detected by the first-“mode” or other FEMA-356 force distributions, but these purely local mechanisms are not likely to develop during realistic ground motions in an otherwise regular building without a soft and/or weak story. Furthermore, the conditions necessary for “reversal” of a higher-“mode” pushover curve are examined. It is shown that “reversal” in a higher-“mode” pushover curve occurs after formation of a mechanism if the resultant force above the bottom of the mechanism is in the direction that moves the roof in a direction opposite to that prior to formation of the mechanism. Such “reversal” can occur only in higher-“mode” pushover analyses but not in the pushover analyses for the first-“mode” or other FEMA-356 force distributions. However, the “reversal” in higher-“mode” pushover curves was found to be very rare in several recent investigations that examined behavior of many moment-resisting frame buildings. Included are guidelines for implementing the Modal Pushover Analysis for buildings that display “reversal” in a higher-“mode” pushover curve.