Comparison of eigenanalysis and Ritz vector approaches for response spectrum analysis of soil-pile-bridge systems

Frequency-based analysis techniques such as response spectrum analysis (RSA) are widely used for designing bridges in seismically active regions. Two well-known analysis procedures that underlie RSA are the solution of the eigenproblem and the approximation of the solution to the eigenproblem (i.e., approximation of eigenvectors and eigenvalues) through use of force-dependent Ritz vectors. While frequency-based methods have achieved widespread adoption in practice, certain simplifications remain common, such as neglecting soil-structure interaction (SSI) due to a fixed-base assumption. In the present study, frequency-based techniques packaged within a research version of a design-oriented computational tool are employed to analyze, assess, and compare results obtained from RSA with use of the eigenanalysis, and separately, Ritz vector approaches. Importantly, for the bridge configurations analyzed, SSI is taken into account. As outcomes, the potential benefits of the Ritz vector approach (as well as modeling strategies) are demonstrated. The study outcomes are intended to aid practicing engineers when the need to account for SSI is recognized as pertinent to a given bridge seismic design application.

Improved Experimental Ritz Vector Extraction With Application to Damage Detection

Load-dependent Ritz vectors, or Lanczos vectors, are alternatives to mode shapes as a set of orthogonal vectors used to describe the dynamic behavior of a structure. Experimental Ritz vectors are extracted recursively from a state-space system realization, and they are orthogonalized using the Gram–Schmidt process. In addition to the Ritz vectors themselves, the associated nonorthogonalized vectors are required for application to damage detection. First, this paper presents an improved experimental Ritz vector extraction algorithm to correctly extract the nonorthogonalized Ritz vectors. Second, this paper introduces a Ritz vector accuracy indicator for use with noisy data. This accuracy indicator is applied as a tool to guide the deflation of a state-space system realization identified from simulated noisy data. The improved experimental Ritz vector extraction algorithm produces experimental nonorthogonalized Ritz vectors that match the analytically computed vectors. The use of the accuracy indicator with simulated noisy data enables the identification of a state-space realization for Ritz vector extraction from which damage location and extent are correctly estimated. The improved Ritz vector extraction algorithm improves the application of Ritz vectors to damage detection, more accurately estimating damage location and extent. The accuracy indicator extends the application of Ritz vectors to damage detection in noisy systems as well.

On Combined Use of POD Modes and Ritz Vectors for Model Reduction in Nonlinear Structural Dynamics

An approach to develop Proper Orthogonal Decomposition (POD) based reduced order models for systems with local nonlinearities is presented in this paper. This technique is applied to multi-degree of freedom systems of coupled oscillators with isolated nonlinear elements. Typically, reduced order models are obtained using POD modes exclusively. In this work, we explore the suitability of using a combination of POD modes and other physically based “Ritz vectors” to produce the reduced model. The objectives are 1). to improve the accuracy of the reduced order differential equation model and 2). to expand the range of system parameters for which the reduced basis provides reasonably accurate approximations. The “Ritz vectors” used in this work are static displacement vectors that are calculated in one of the following three ways: 1). “Load – based Ritz vectors” [1, 2, 7, 8, 12, 16–18] – This is the static displacement vector due to a static loading that is proportional to the static version of the actual (assumed dynamic) loading to which the structure is subjected. 2). “Milman – Chu vectors” [3] – This is the static Ritz vector due to the imposition of equal and opposite static loads on the two masses to which the non-linear element is connected. The loading used to generate the first M – C vector is dictated by the location of the non-linearity. 3). “K – B (Kumar – Burton) vector” – This is a new Ritz vector defined in the spirit of the Milman – Chu vector. The K – B vector is the static displacement vector due to the imposition of a). equal and opposite static loads on the two masses to which the nonlinear element is connected (i.e. same as M – C loading) and b). equal and opposite static loads on the nearest neighbors. Thus, four masses are statically loaded. As for the M – C vector, the K – B loading is dictated by the location of the nonlinear element. The nonlinear model is numerically integrated to generate a full ODE model solution, which we call the “baseline solution”. We select a set of POD modes of the baseline nonlinear system response as basis functions. The POD modes are then augmented by various combinations of the three aforementioned Ritz vectors to generate reduced order models for system having parameters in vicinity of baseline system parameters. Our results indicate that the K – B augmentation vector combined with the Milman – Chu vector is an effective way to account for nonlinear effects for the system considered. The use of combined M – C/K – B augmentation also expands the range of system parameters for which the baseline POD modes provide accurate reduction. This is considered to be a significant result.