Ambient Vibration Tests Research Articles

Seismic analysis of a full scaled self-supported tower using ambient vibration testing

Seismic analysis of a full scaled self-supported tower using ambient vibration testing

Accelerating convergence in Bayesian operational modal analysis with Fisher information matrix

Bayesian operational modal analysis (BAYOMA) has been increasingly applied to ambient vibration tests, providing a fundamental means for estimating modal properties as well as quantifying their identification uncertainty consistent with probability rules and modelling assumptions. Computing the most probable value of modal parameters in BAYOMA involves a high-dimensional numerical optimisation of the negative log-likelihood function (NLLF). Brute-force optimisation using generic algorithms treating the NLLF as a black box is computationally prohibitive and often non-converging. Efficient iterative algorithms have been developed over the past decade, but challenges still exist, e.g., for very close modes that may require significantly more iterations or (in some cases) may not be converging. Leveraging on recent advance in the mathematics and understanding of the Fisher Information Matrix (FIM) of modal parameters, an efficient method based on Newton-type iterations is proposed in this work to improve computational efficiency and convergence robustness. The MPV estimate in each iteration is updated based on two characteristic types of principal directions of the FIM. Move of the first type involves mode shapes only and is orthogonal to the subspace spanned by the current mode shape estimates. The second type involves updates of all modal parameters, where the mode shapes move within the subspace. Compact analytical expressions are derived for the gradient of NLLF to ensure accurate and efficient use in determining each iteration move. The performance of the proposed method is investigated with a comprehensive study based on synthetic, laboratory and field data. Results reveal that the proposed algorithm generally outperforms existing ones in terms of computational time by an order of magnitude, and it has better convergence robustness especially for challenging cases with very close modes and high modal force coherence.

Boundary element model for the analysis of the dynamic response of the Soria arch dam and experimental validation from ambient vibration tests

This paper presents a 3D boundary element time–harmonic model for the analysis of the dynamic behavior, under low-level vibrations, of the Soria arch dam (Gran Canaria, Spain). The model takes into account dynamic Soil–Structure Interaction (DSSI), Fluid–Structure Interaction (FSI) and the actual geometry of dam wall and reservoir. The rock and the dam wall are considered as viscoelastic regions, and the water in the reservoir is modeled as an inviscid fluid. The influence of DSSI, FSI, height of the water in the reservoir, accuracy of the geometric modeling of the canyon, and type of transmitting boundary conditions at the truncated end of the reservoir, are evaluated. DSSI and FSI have a significant influence on the dynamic response of the dam, and the accurate representation of the geometry of the canyon is relevant for the correct estimation of the modal shapes. Ambient vibrations tests were also performed, from which the first and third modes could be clearly identified. The comparison between the experimental and numerical natural frequencies and mode shapes suggests that the proposed complete numerical model is able to capture the dynamic response and can be used for the structural health monitoring of the wall of this dam.

Influence of Hankel matrix dimension on system identification of structures using stochastic subspace algorithms

In stochastic subspace methods, the modal analysis results are highly dependent on the dimensions of the Hankel matrix. Increasing the dimensions of the Hankel matrix (especially the number of rows) improves the estimation of the modal features by decreasing the impact of noise effects. Due to processing time and memory use, it is impossible to adjust the size of the Hankel matrix to the maximum compatible value. Therefore, this study uses a sensitivity analysis of the dimensions of the Hankel matrix to pick models with the slightest estimate error in the data-driven (DD-SSI) method. First, using the condition number criteria, the desirable models in which the effects of system errors are not predominating are determined. Second, the modal properties of desired models are validated by clustering the damping ratios and modal frequencies using the Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm and assessing the complexity of shape modes with the Modal Complexity Factor (MCF). Finally, the optimum models are identified by analyzing the estimated error of modal properties (particularly damping ratio) and the coefficient of variation (CV) of damping of validated clusters. The proposed method was investigated for a two-dimensional simulated concrete building frame, a three-dimensional experimental model, and the Namin city Overpass Bridge ambient vibration tests. Sensitivity analysis was performed for canonical correlation analysis (SSI-CCA) and canonical variate analysis (SSI-CVA) methods. Analyses of computational and laboratory models revealed that the likelihood of non-physical modes arising outside the system's maximum order is relatively high. Also, the MCF is valuable for identifying computational and noise modes. These criteria accurately detected the flexural mode of the foundation set, which was clustered as the stable mode of the slab deck in the practical bridge. Furthermore, the CV analysis revealed that the desired dimension obtained from the condition number could be a reasonable estimate of the optimal system dimension.

Appraising the Seismic Response of a Retrofitted Adobe Historic Structure, the Role of Modal Updating and Advanced Computations

The concepts of structural assessment and retrofit of historical constructions are of particular complexity and require advanced knowledge in material science, conservation techniques and structural analysis. In particular, adobe constructions, given their low mechanical properties and brittle failure modes, are in immense need of comprehensive assessment and retrofitting plans. The current work focuses on the adobe Church of Kuñotambo in Peru, having experienced long periods of deterioration and earthquake-related damage. Under the ongoing Seismic Retrofitting Project (SRP) of the Getty Conservation Institute (GCI), the structural assessment of the church initiated in 2015 confirmed the low lateral capacity of the building and the poor connectivity between the structural parts. Based on the existing cracks and damage, a strengthening scheme was optimized and validated. After the implementation of the retrofitting plan, the quality of its execution and efficiency were assessed in 2019 with a new in situ campaign, which included ambient vibration testing (AVT) and sonic testing. From the acquired field data, the FE model of the retrofitted church was optimized, by updating the stiffness properties of masonry and discontinuities. Moreover, nonlinear static analyses were performed on the updated model in all in-plan directions. Finally, a displacement-based performance assessment was undertaken, under different earthquake limit states, demonstrating the adequacy of the retrofitting.

Post-earthquake damage assessments of unreinforced masonry (URM) buildings by shake table test and numerical visualization

Masonry structures are one of the structures most affected by earthquakes. In addition, since masonry structures constitute a large part of the building inventory, it is important to determine the dynamic characteristics of these structures. The present study is aimed to evaluate the seismic performance of a half-scale hollow brick unreinforced masonry (URM) building considering different damage states. Shake table test was conducted with seven loading conditions. In order to obtain different damage states, seismic excitations were applied incrementally to structure. After each loading, the damage conditions and the corresponding damage patterns were examined comparatively, and the structure collapsed in the last loading condition. To experimentally observe the damage, the modal parameters for un-damaged and damaged situations of the building were identified by ambient vibration tests performed before and after the seismic loadings. The experimental results obtained were then compared with the analyses made in numerical softwares, and similar results were obtained.

Using a rhythmic human shaker to identify modal properties of a stationary human body on a footbridge

Dynamic properties of the human body are basic parameters in biomechanics and many related areas. Experimental test is the most reliable manner to obtain dynamic parameters of the human body. Traditional method usually employs shaking table test or well-designed test rig driven by a mechanical shaker, which is always expensive and sometimes not available. This paper presents a simple but efficient transmissibility-based procedure for testing the dynamic properties of a human body. The excitation is generated by a human shaker with rhythmic movement acting on a flexible structure (either a footbridge or a long-span floor). Wide band excitation frequency can be generated even excite in a narrow-band owing to the multi-harmonics of the footfall force. The procedure consists of two main steps, namely, to identify the modal properties of the supporting structure and to establish the transmissibility of the human body. The human excitation force is found to be highly correlated with the body motion based on simultaneous footfall measurement and 3D motion tracking. To this end, the human excitation force is reconstructed and calibrated using the human body motion. The structural frequency response function is established using the footfall force and the bridge acceleration. Then the modal properties of the supporting structure, i.e. the vibration frequencies, mode shapes, and damping ratios and modal mass, are identified. The performance of human excitation method is checked by conventional ambient vibration testing and impact testing methods. Transmissibility of the human body is obtained by motion measurement at the supporting point and the stationary human body. The identified resonance frequency of the human body is further modified to obtain its natural frequency since the body is heavily damped. The feasibility of the procedure is demonstrated in identification of the dynamic properties of a footbridge with a stationary occupant having different postures.

Water–structure interaction analysis of a segmental bridge using ambient vibration testing at different water levels

Updating a finite-element (FE) model using dynamic parameters obtained from field testing requires an adequate understanding of the variation of such parameters due to changes in environmental conditions. This paper presents an experimental program to study the influence of the water–structure interaction in the dynamic response of a prestressed concrete segmental bridge with partially submerged piers in an artificial reservoir. The bridge, located in Colombia, has a total length of 558 m. Ambient Vibration Tests (AVTs) consisted of four experimental campaigns at different water levels to perform modal identification. In addition, existing empirical formulations accounted for the effect of the hydrodynamic masses concentrated in the piers. This paper shows that updating FE models from identified dynamic properties in partially submerged bridges involves percentage reductions in natural frequencies related to the reservoir's water level. Thus, providing a better insight into the incidence of water–structure interaction on partially submerged bridges' dynamic response. Finally, considerations during dynamic characterization tests are discussed.

A holistic methodology for the non-destructive experimental characterization and reliability-based structural assessment of historical steel bridges

Nowadays, several historical steel structures present damage and an advanced deterioration state induced by human or natural actions, causing fluctuations in geometrical, physical, and mechanical properties that dramatically affect their mechanical behavior. Due to the economic, cultural, and heritage value, these constructions must be comprehensively assessed to verify their current condition state. This work presents a holistic methodology aimed at the non-destructive experimental characterization and reliability-based structural assessment of historical steel bridges. It comprehends from the experimental data acquisition to the finite element model updating and the probabilistic-based structural assessment to obtain the reliability indexes of serviceability and ultimate limit states. Several sources of information are considered in the evaluation process, thus, results are more realistic and accurate and can be used for optimal decision-making related to maintenance and retrofitting actions. The feasibility of the methodology has been tested on O Barqueiro Bridge, an aging riveted bridge located in Galicia, Spain. The study first involved a comprehensive experimental campaign to characterize the bridge effectively at multiple levels: geometry, material, and structural system by the synergetic combination of different tools and methods: in-depth visual inspection, terrestrial laser scanner survey, ultrasonic testing, and ambient vibration test. Subsequently, a detailed FE model was developed and calibrated with an average relative error in frequencies of 2.04% and an average MAC value of 0.94. Finally, the reliability-based structural assessment was performed, yielding reliability indexes of 1.80 and 1.99 for the serviceability and ultimate limit states, respectively. Thus, the bridge could not withstand traffic loads with satisfactory structural performance in its current condition.

Vibration testing and performance evaluation of Hagia Sophia bell tower after recent restoration

Historical structures are very valuable in terms of culture and history because they are the legacy of the past and the trust of the future. These structures require structural investigation and assessment to survive their life. By developing technology, the continuous and/or partial monitoring of these structures has become widespread in many parts of world for last decades. This study presents the structural dynamic parameter identification and the seismic performance evaluation of Hagia Sophia Bell Tower. This historical tower was built by using masonry cut stones in 1204 in Trabzon, Türkiye. Many restorations were made on the bell tower in different times in the past and the last one was executed in 2020. After this restoration, ambient vibration tests were performed on the tower by using developed measurement systems. The dynamic characteristics were attained by analyzing the collected data and were compared the results of previous studies to determine the effects of the last restoration. Also, an initial finite element model was generated by using solid elements in ANSYS Workbench software and the model was calibrated according to the experimental natural frequencies attained from the last restoration. The structural performance was evaluated by performing seismic analyses according to the current Türkiye Building Earthquake Code by considering the local site effects and seismicity. These analyses were performed on the weak direction of the tower in the initial and calibrated finite element models. The maximum top displacement, drift ratios, principal stress values and their distributions were determined and the seismic vulnerability was evaluated by considering these values.

In situ seismic testing for experimental modal analysis of civil structures

This paper presents a novel excitation method for in situ experimental modal analysis (EMA) and compares its performance to ambient vibration analysis, also known as operational modal analysis (OMA). The method consists of base excitation through ground accelerations generated through a large mobile shaker, a unique form of building excitation which has not been studied to date. The controllable and repeatable nature of the excitation complements some limitations of OMA, including inconsistent excitation conditions and low energy at high frequencies. The method is applied to an instrumented eleven-story building in Austin, Texas, resulting in observations of significant energy input up to 50 Hz. The high input bandwidth is essential to characterizing high order modes which ambient sources do not excite. Using a five-minute set of multiple 20-s chirp excitations from 5 to 50 Hz in two directions (vertical and transverse), it is shown that the in situ seismic EMA yields modes which are not identified through standard ambient OMA. The differences in identification methods were most noticeable above 7 Hz, where vertical contributions accounted for a significant portion of the modal response. Above 11 Hz, ambient excitation was too weak for proper OMA, but EMA enabled the identification of high-order “local” modes, although their complete identification was limited by low sensor density. The primary benefit of OMA comes from processing time series of much longer duration (20 min), which reduced variation in most of the modal estimates, given the fairly consistent ambient excitation conditions during testing. Overall, in situ seismic testing is shown to be a viable option for forced vibration testing of very large civil structures. With refinements in the testing methods, the technique could be used as a complement to ambient vibration testing in contexts where the ambient excitation bandwidth is limited or quick, short-term tests are desired.

The effects of non-structural components on the dynamic characteristics and vulnerability of concrete structures using ambient vibration tests and Nakamura's criterion

From the past up to now, selecting a reliable structural system as well as analyzing and designing it accurately have considerably attracted the attention of researchers and engineers. An appropriate evaluation of the dynamic characteristics of the structures such as natural frequency and damping ratio has a considerable effect on the accurate analysis and design of structures. In addition, the vulnerability assessment of structures plays a significant role in selecting a resilient structural system against the seismicity of the area. In the light of these facts, one of the most important challenges is that the effects of non-structural components (NSCs) are commonly considered as a concentrated/distributed mass in most FE models and their stiffness and damping ratio are ignored. Therefore, this research aims to figure out the effects of NSCs on the dynamic characteristics and the vulnerability indices of concrete structures using microtremor measurements. For this purpose, the ambient vibrations of four concrete buildings (with two bending frame and two shear wall structural systems) during the construction process are measured in three independent stages: (1) after the completion of the structural system; (2) after the completion of the interior and exterior partition walls; (3) after the completion of the flooring, facade, and parapet elements (when the building is completely constructed). Afterward, to extract the dynamic characteristics and to compute the vulnerability indices of the structures, the measured microtremors are subjected to two signal processing techniques, i.e., floor spectral ratio (FSR) and random decrement method (RDM). It is observed that, by taking the effects of NSCs into account, the values of both dynamic characteristics of the concrete structures (i.e., natural frequency and damping ratio) are increased when the buildings are under the erection process. Furthermore, the obtained results for the concrete structures with the bending frame structural system demonstrate that the vulnerability index increases in the second stage compared with the first one, whereas it remarkably decreases in the third stage compared with the second one. Reciprocally, the obtained vulnerability indices for the concrete structures with the shear wall structural system show a similar behavior.

Multiple Tests for Dynamic Identification of a Reinforced Concrete Multi-Span Arch Bridge

This paper presents the results of an experimental dynamic campaign carried out on a reinforced concrete multi-span arch bridge. Five expeditious ambient vibration tests were conducted separately on five spans (one test in each span) of the bridge using only six piezoelectric uniaxial accelerometers. Modal parameters were identified through the well-known Enhanced Frequency Domain Decomposition (EFDD) procedure developed using Matlab R2021b software. At the same time, a finite element model was accurately implemented through a commercial software (Midas Civil) to evaluate the main modal features. A manual model update was successively pursued varying the elastic modulus of the reinforced concrete to make the identified and numerical modes as close as possible. A complete and suitable instrumentation to perform global experimental dynamic tests is not always available. Recursive/Multiple tests have different advantages: handy, easily executable, and could provide a more robust identification thanks to a statical characterization. The paper aims to highlight the peculiarities of recursive/multiple dynamic tests on multi-span arch bridges. The procedure also provides useful suggestions for designing a permanent and continuous vibration-based monitoring system.

Dynamic Testing in Support of the Seismic Assessment of a Century Old Masonry Building Complex

The vulnerability assessment of existing masonry buildings is a largely investigated research topic with some aspects still to be faced. In historic towns, masonry buildings are aggregated and together confined, and their final appearance is derived from interventions and additions during their lives in different times and with different masonry textures or different construction materials. Demolitions and reconstructions of some parts were frequent, with the difficulty of now understanding the effectiveness of the mutual constraints. The seismic assessment of a case study of a 175-year-old building complex in Udine (Italy) provides an opportunity to use the results of ambient vibration tests to face the problem of modelling aggregate buildings for their seismic assessment. The “Padiglione Lodi” building complex was built in 1847 and extended and renovated several times afterwards. It was built mostly using URM with limited use of reinforced concrete. It consists of a main building and three wings (western, central and eastern). The inspections, experimental survey and analysis of the available documentation are used to suitably calibrate a Finite Element Model of the whole complex. Moreover, this allows the singling out of the central wing, as the unit needs more careful investigation. Non-destructive dynamic testing is then applied to the central wing in order to further validate the model and improve the knowledge of the interaction of the unit with the rest of the building. General remarks on the effective application of non-destructive dynamic analysis in conjunction with other methods to the seismic assessment of large URM building complexes are drawn.

Vibration Serviceability Assessment of a Historic Suspension Footbridge

Experimental and numerical studies for the structural and vibration serviceability assessment of a historic suspension footbridge adopting non-invasive surveys and low-cost equipment are presented. Field surveys have been carried out to determine geometric properties, ambient vibration tests have been performed to estimate the dynamic properties, and the dynamic response of the footbridge under the action of a single crossing pedestrian has been recorded. Based on field surveys, a 3D Finite Element model was built and was then calibrated against ambient vibration test results. The experimentally-measured maximum acceleration under the action of one crossing pedestrian is compared with the ones obtained numerically and analytically. Furthermore, vibration serviceability assessment under multi-pedestrian loading is carried out, adopting the simplified procedure recommended by a recent guideline. Results show that low-cost non-invasive dynamic testing is suitable to correctly identify the footbridge vertical natural frequencies and mode shapes, including higher-order ones, and to draw considerations about the state of degradation of the structure. Moreover, the level of vibration under the action of a single pedestrian can be estimated with sufficient accuracy using a simplified loading model, provided that the modal damping ratio is properly tuned.

Experimental Validation of a Double-Deck Track-Bridge System under Railway Traffic

This article describes the experimental and numerical evaluation of the dynamic behaviour of the Cascalheira bridge, located on the Northern Line of the Portuguese railway network. The bridge has a short span formed by two filler-beam half-decks, each one accommodating a railway track. The study includes the development of a finite element numerical model in ANSYS® software, as well as in situ dynamic characterization tests of the structure, namely ambient vibration tests, for the estimation of natural frequencies, modes shapes and damping coefficients, and a dynamic test under railway traffic, particularly for the passage of the Alfa Pendular train. The damping coefficients’ estimation was performed based on the Prony method, which proved effective in situations where the classical methods (e.g., decrement logarithm) tend to fail, particularly in the case of mode shapes with closed natural frequencies, as typically happens with the first vertical bending and torsion modes. The updating of the numerical model of the bridge was carried out using an iterative methodology based on a genetic algorithm, allowing an upgrade of the agreement between the numerical and experimental modal parameters. Particular attention was given to the characterization of the ballast degradation over the longitudinal joint between the two half-decks, given its influence in the global dynamic behavior of this type of double-deck bridges. Finally, the validation of the numerical model was performed by comparing the acceleration response of the structure under traffic actions, by means of numerical dynamic analyses considering vehicle-bridge interaction and including track irregularities, with the ones obtained by the dynamic test under traffic actions. The results of the calibrated numerical model showed a better agreement with the experimental results based on the accelerations evaluated in several measurement points located in both half-decks. In the validation process the vertical stiffness of the supports, as well as the degradation of the ballast located over the longitudinal joint between half-decks, was demonstrated to be relevant for the accuracy and effectiveness of the numerical models.

Operational modal analysis and finite element model updating of ultra-high-performance concrete bridge based on ambient vibration test

The advancement of technology for accurate and dependable monitoring and evaluation of current bridge infrastructure conditions has become increasingly important in ensuring that bridge structures operate safely. Operational modal analysis (OMA) has been used to solve various engineering problems, including civil engineering structures, where OMA has been used to monitor systems on a global scale. Structural health monitoring (SHM) is a method of observing and assessing a process that involves sampling response measurements regularly to track changes in the material and geometric properties of structures. However, knowledge of massive structural health monitoring (SHM) data is poorly interpreted due to computational constraints and a lack of data analysis methodologies. By determining the structure's modal parameters, including the natural frequency and mode shape, this study applied ambient vibration testing to acquire vibration response measurements and thus determine the structure's dynamic characteristics. A comparison of three modal detection approaches (Frequency Domain Decomposition (FDD), Enhanced Frequency Domain Decomposition (EFDD), and Stochastic Subspace Identification (SSI)) was performed, and the mode shapes of each method were validated using the Modal Assurance Criterion (MAC) value to verify the accuracy of the results. On the basis of experimental data, a sensitivity-based updating method was used to justify and update the numerical finite element (FE) model of bridge structures. The difference between the updated FE and that measured for the first five dominant natural frequencies in this study was less than 5%. The updated model's minimum MAC value of 90% indicates that it was updated successfully. The updated dynamic parameter of the first dominant natural frequency (3.348 Hz) was used to figure out the structure's serviceability vibration limit state in accordance with EN 1991–2. The result was in the range that indicates that the bridge is safe for people to drive on.

Measuring configuration of multi-setup ambient vibration test

In the ambient vibration test (AVT) of structures, when the number of desired locations for mode shapes to be measured is larger than that of available sensors, a ‘multi-setup’ strategy is often adopted. That is, sensors are ‘roved’ (placed) at different locations in different setups while ‘reference’ sensors are placed at common locations to provide information for estimating the ‘global’ mode shape covering all desired locations from ‘local’ information in individual setups. Currently, multi-setup AVTs are often planned based on experience. Naturally, questions on how to judge the performance of the test configuration are relevant, e.g., number and location of reference/rover sensors. This paper proposes metrics to quantify a multi-setup configuration by comparing the identification uncertainty of mode shapes with its best achievable one. The latter is informed by recent theoretical findings, i.e., ‘uncertainty law’ of multi-setup in operational modal analysis. Two indices are proposed to measure the overall configuration and the placement of reference sensors. They are also simplified to aid the design of multi-setup AVT. Two examples, a laboratory shear-type building model and a (full-scale) suspension footbridge, are presented to illustrate the usage of the proposed indices for performance quantification. The theory and examples reveal that the placement of reference sensors has a decisive influence. Notably, it is found that as long as the reference sensors are in ‘good’ positions, the identification uncertainty of the global mode shape is insensitive to the choice of rover sensors. Practically, and somewhat consistent with conventional wisdom, this suggests that reference sensors should be placed with due consideration to signal-to-noise ratio and the modes that can be detected, but rover sensors can be planned primarily based on logistics constraints/considerations.

Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating

Numerical model updating using the data measured from the actual structure is required in order to minimize the error between the initial numerical model and the actual structure. Field load tests, which are conducted in order to assess the condition and safety of high-speed railway bridges, are generally expensive and restricted by railway control and weather conditions. Therefore, a method for evaluating the performance of high-speed railway bridges using updated numerical models without conducting field load tests is required. In this study, numerical model updating was performed by using the data measured from the ambient vibration test in order to assess the dynamic stability of high-speed railway bridges. In the ambient vibration test, the measurement point roaming method was applied in order to accurately measure high-speed railway bridges using a limited number of sensors. For numerical model updating, the univariate search method was used, and several measured parameters were updated and converted into the properties of the target bridges in the numerical models. The vertical and torsional modes of the updated numerical models differed by less than 5% from those estimated using the data measured from the target bridges. The responses of the updated numerical models were found to be similar to those measured from the high-speed railway bridges in operation. It was also shown that the updated numerical models could be used to assess the dynamic stability of the bridges.

Experimental out-of-plane damage limits of historical stone masonry walls

In stone masonry buildings, failure under strong ground motions usually occurs in the form of out-of-plane overturning of the walls before reaching their in-plane strength, as the walls have long unsupported spans perpendicular to their plane and the lack of slabs ensuring a diaphragm effect. Post-earthquake damage observations reveal this specific weakness of the historic masonry walls. Experimental determination of the out-of-plane damage limits of these walls, emerges as a need for use, especially during numerical evaluations. This article presents the results of static and dynamic testing of double- and three-leaf large-scale U-shaped masonry wall specimens. For this purpose, nondestructive material characterization was conducted on a reference historical masonry structure. Large-scale wall specimens were designed considering these findings and material tests performed on mortar samples. Quasi-static cyclic tests and ambient vibration tests for both initial and damaged conditions were performed on these specimens to capture the damage evolution and the corresponding drift limits. Damage limits are suggested to be used in performance-based evaluations of other territorial structures.