Anil K. Chopra

Perfectly matched layers for time-harmonic elastodynamics of unbounded domains : Theory and finite-element implementation

Abstract One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.

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.

Critical response of structures to multicomponent earthquake excitation

This paper aims to develop an improved understanding of the critical response of structures to multicomponent seismic motion characterized by three uncorrelated components that are defined along its principal axes: two horizontal and the vertical component. An explicit formula, convenient for code applications, has been derived to calculate the critical value of structural response to the two principal horizontal components acting along any incident angle with respect to the structural axes, and the vertical component of ground motion. The critical response is defined as the largest value of response for all possible incident angles. The ratio rcr/rsrss between the critical value of response and the SRSS response—corresponding to the principal components of ground acceleration applied along the structure axes—is shown to depend on three dimensionless parameters: the spectrum intensity ratio γ between the two principal components of horizontal ground motion characterized by design spectra A(Tn) and γA(Tn); the correlation coefficient α of responses rx and ry due to design spectrum A(Tn) applied in the x- and y-directions, respectively; and β = ry/rx. It is demonstrated that the ratio rcr/rsrss is bounded by 1 and . Thus the largest value of the ratio is , 1.26, 1.13 and 1.08 for γ = 0, 0.5, 0.75 and 0.85, respectively. This implies that the critical response never exceeds times the result of the SRSS analysis, and this ratio is about 1.13 for typical values of γ, say 0.75. The correlation coefficient α depends on the structural properties but is always bounded between −1 and 1. For a fixed value of γ, the ratio rcr/rsrss is largest if β = 1 and α = ±1. The parametric variations presented for one-storey buildings indicate that this condition can be satisfied by axial forces in columns of symmetric-plan buildings or can be approximated by lateral displacements in resisting elements of unsymmetrical-plan buildings. Copyright

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.

Modal-Pushover-Based Ground-Motion Scaling Procedure

Earthquake engineering is increasingly using nonlinear response history analysis (RHA) to demonstrate the performance of structures. This rigorous method of analysis requires selection and scaling of ground motions appropriate to design hazard levels. This paper presents a modal-pushover-based scaling (MPS) procedure to scale ground motions for use in a nonlinear RHA of buildings. In the MPS method, the ground motions are scaled to match to a specified tolerance, a target value of the inelastic deformation of the first-mode inelastic single-degree-of-freedom (SDF) system whose properties are determined by the first-mode pushover analysis. Appropriate for first-mode dominated structures, this approach is extended for structures with significant contributions of higher modes by considering elastic deformation of second-mode SDF systems in selecting a subset of the scaled ground motions. Based on results presented for three actual buildings—4, 6, and 13-story—the accuracy and efficiency of the MPS procedure ...

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.