Equal Displacement Rule Research Articles

Comparison between Design Methods for Seismic Retrofit of Reinforced Concrete Frames Using Dissipative Bracing Systems

Braces equipped with dissipative devices are among the most widespread methods for the seismic strengthening of seismically prone reinforced concrete (RC) frames. It allows for high reductions in seismic vulnerability with inexpensive, quickly executed interventions. They can often be carried out mainly at the exterior, resulting in interruptions of use that are limited both in time and to only small portions of the building. The design methods of dissipative devices are based on the extensive use of pushover analyses (POA). POA is capable of highlighting the structural deficiencies of the building and comparing the performances of design performed according to different methods and sizing criteria. In the present work, with reference to a case study represented by a four-story spatial frame having characteristics representative of design and construction common practice of the 1970s in Southern European countries, the performances of three different design methods were evaluated and compared. The examined procedures differ, including the following: (i) methods for estimating the peak displacement response of the nonlinear systems, namely (i1) the well-known equal displacement rule and (i2) the equivalent (secant) stiffness and damping rule, and (ii) criteria for distributing stiffness and strength of the braces along the height, namely (ii1) the distribution of stiffness and strength proportionally to those of the frame and (ii2) methods that vary the stiffness and strength along the height in order to minimize the eventual irregularity in elevation of the bare frame. The effectiveness of the procedures was checked by both POA and nonlinear response history analysis, the latter performed assuming both unidirectional and bidirectional input. The stiffness was found to increase by about 10 times and the strength between 7.5 to 3.7 times depending on the design method, and reduction in the displacements ranged between 31% and 42% compared to the values of the original frame. The pros and cons of each procedure are summarized, as all procedures are able to provide brace designs that meet the performance requirements set during the design phase.

Displacement‐based design procedures for rigid block isolation

AbstractWhen subjected to earthquakes, many objects or structural elements behave like rocking rigid blocks. Computer servers, medical shelves, art objects, statues, and electrical transformers are frequently included in this category. Protection of these objects is an important task, considering that their value could be inestimable or their operation crucial during earthquakes; base isolation technology has been proven to be a viable option for this purpose. Initially, the dynamic model of a rocking rigid block placed on a base isolation device is reviewed. Then, two equivalent‐static displacement‐based procedures for designing the isolators for these types of objects are proposed, and the main steps are illustrated. The first procedure aims to determine isolator characteristics to prevent the initiation of rocking motion during the code‐level earthquake event. The second procedure is aimed at designing isolators that allow a specified maximum rotation of the block during seismic events. The proposed procedures are validated by means of time‐history analyses for a suite of spectrum‐compatible accelerograms. The first displacement‐based procedure appears particularly suitable for objects of small to medium size. The validation of the second procedure demonstrates that the equal displacement rule can be applied for this kind of systems, despite their softening. The results also indicate that the approach is particularly effective for medium to large structures/objects, if small oscillations are acceptable. The controlled rocking procedure offers a significant advantage by allowing for a reduction in the maximum displacement and period of the isolator, compared to situations where rocking motion must be prevented entirely.

Assessment of seismic inelastic displacement profiles of steel MRFs by means of modal behavior factors

A new method for the estimation of inelastic displacements of steel moment resisting frames (MRFs) under strong seismic ground motions is presented. The method comes as an extension of previous work of the authors from reinforced concrete to steel MRFs. Key-elements of the present method are the modal behavior (strength reduction) factors, accounting for the first four modes of significance. These factors are expressed in terms of natural period, interstorey drift ratio and/or plastic rotation, soil type and performance level. These factors are multiplied by the corresponding elastic modal displacements to generate the inelastic modal displacement profile. Eventually, the inelastic displacements can be calculated after combining the modal responses through any combination rule (e.g. SRSS). The results of the presented method are compared with that of the well-known equal displacement rule and three others and attested through nonlinear time history analyses.

Peak Inelastic Displacement of Bilinear Systems in Support of Performance-Based Wind Design

A fundamental notion in building engineering is the equal displacement rule, which posits that the peak inelastic displacement of a system subjected to a ground motion excitation is approximately equal to the displacement of the same system responding elastically. The purpose of this study is to determine if the equal displacement rule can additionally be applied to wind excitations. To achieve this purpose, bilinear single-degree-of-freedom systems were subjected to B-spline wavelet excitations, Fejér–Korovkin wavelet excitations, and wind excitations derived from wind tunnel tests. The results showed the equal displacement rule generally held for excitations with neutral polarity. The frequency content of the excitation had a significant effect on the response because it shifted the location of the displacement-controlled region of the response spectrum. Duration had a mild effect for excitations with neutral polarity. The effect of stiffness and strength degradation due to gravity loads on the response was more pronounced for short-period structures. For regularly shaped buildings subjected to wind forces, the findings suggest that the equal displacement rule applies in the cross-wind direction however not in the along-wind direction.

Deformation-based pushover analysis method for transverse seismic assessment of inverted Y-shaped pylons in kilometer-span cable-stayed bridges: Formulation and application to a case study

Kilometer-span cable-stayed bridges usually adopt tall inverted Y-shaped pylons. Such pylons may behave inelastically under strong transverse seismic excitations. Therefore, efficient analysis methods (e.g., pushover analysis) for transverse elastoplastic behavior assessments of inverted Y-shaped pylons are of great interest for academic and industrial communities. Based on equal displacement rule, this study proposes an applied deformation-based pushover analysis (DPA) incorporating a novel multi-node load pattern for the efficient transverse seismic behavior assessments of inverted Y-shaped pylons in kilometer-span cable-stayed bridges. A case study of the inverted Y-shaped pylon of Sutong Bridge is adopted to demonstrate the proposed DPA. The proposed multi-node load pattern for DPA is verified through the validation of the loaded-node selection and equal displacement rule for the studied pylon. The ability of DPA in seismic behavior assessments is then systematically validated by comparative studies with incremental dynamic analysis (IDA), conventional pushover analysis (CPA), and modal pushover analysis (MPA) in terms of displacement, bending moment and curvature responses of pylon as well as computational costs.

Seismic displacements and behaviour factors assessment of an innovative steel and concrete hybrid coupled shear wall system

The objective of this research is to facilitate the design process of innovative steel and concrete hybrid coupled wall system, which is obtained through the connection of a reinforced concrete (RC) wall to two steel side columns by means of steel links. To this end, a group of 120 innovative hybrid coupled wall systems subjected to a set of 100 far-field ground motions are selected. Thousands of nonlinear dynamic time histories on the basis of incremental dynamic analysis (IDA) are carried out in order to generate a databank of specified response quantities. Subsequently, nonlinear regression analyses are conducted in order to derive simple formulae which offer a direct estimation of: fundamental periods of vibration, seismic displacements, ductility demands and behaviour factors q for both designed and undesigned structures at different limit states; involving steel first yielding of internal wall, various predefined maximum inter-storey drifts and links rotations values. The effect of the following parameters: number of storeys, coupling ratio value, steel area ratio of boundary elements, uniformity status of shear links, and length of links is thoroughly investigated. The results indicate that storeys number and uniformity status parameters have the largest influence on the most of response quantities. The current q behaviour factor value is approximately suitable for low-rise building, but it is highly underestimated in medium-rise buildings. Furthermore, the equal-displacement rule clearly overestimates the roof ductility. Overall, this study bridges the gaps about the innovative hybrid coupled wall system design methodology and enables a rapid seismic assessment of such structures.

Prediction of seismic inelastic displacement profiles of R/C moment resisting frames

A new method for predicting inelastic displacements of reinforced concrete moment resisting frames under seismic motions is presented. The method uses modal behavior (or strength reduction) factors for the first few significant modes, which are functions of period, interstorey drift ratio, plastic rotation, performance level and soil type. Explicit empirical expressions for these factors have been derived by the first author in the past for design purposes. These factors are used here for inelastic displacement prediction. Thus, these factors are multiplied by the corresponding elastic modal displacements to produce the inelastic modal displacements, which can provide the inelastic displacements via a combination rule like the SRSS rule. The proposed method is compared against the equal displacement rule and three other well-known methods on the basis of nonlinear time history analyses and its merits are demonstrated.

Seismic response of yielding structures equipped with inerters

This paper investigates the seismic response of one- and two-degree-of-freedom yielding structures equipped with inerters at the first story. Inerters are mechanical devices that their resisting force is proportional to the relative acceleration at their end-nodes. This class of response modification devices complements the traditional supplemental damping devices that their resisting force is proportional to the relative velocity at their end-nodes, also examined in this study for comparison. The paper develops a stable nonlinear response analysis methodology that implements a state-space formulation. Given that the engagement with an inerter lengthens the apparent preyielding period of the inelastic structure, the paper shows that when a yielding structure is equipped with supplemental rotational inertia, the equal-displacement rule is valid starting from lower values of the preyielding period. The effectiveness of a single inerter versus the use of a pair of clutching inerters that can only resist the motion of the yielding structure is examined, and the paper concludes that a single inerter suppresses effectively the displacement response of inelastic structures by outperforming the response modification with supplemental damping in particular when the supporting frame of the response modification devices is compliant.

An approximate procedure for estimating the member demands in mid-rise reinforced concrete buildings

Seismic assessment of mid-rise reinforced concrete buildings is important for urban seismic risk reduction. Nonlinear time history analysis is the state-of-the-art analysis tool for this purpose. However, the number of such buildings may reach to several tens of thousands in metropolitan cities of earthquake prone countries. This necessitates faster yet sufficiently accurate assessment methods to mitigate the seismic risk. In this study, a simple and efficient approach is developed and validated for the assessment of such buildings. The proposed method relies on estimating the reduced secant stiffness values for structural members based on elastic analysis. Afterwards, subsequent elastic analyses with the reduced stiffness values is conducted employing the equal displacement rule. The procedure is tested for twelve buildings with and without shear walls, and torsional irregularity. The member end chord rotation results obtained with the proposed procedure are compared with the mean results of eleven nonlinear time history analyses. It is found that the proposed procedure can estimate column and beam chord rotation demands with reasonable accuracy, significantly reducing the computational time.

A simplified risk-targeted decision model for the verification of the seismic performance of critical infrastructure components to the operational limit state

A simplified risk-targeted decision model for the verification of the seismic performance of infrastructure components to the operational limit state is presented. It assumes linearity between the intensity measure and the engineering demand parameter and requires the evaluation of the displacement demand for a hazard-targeted design earthquake and of the displacement capacity. The latter is defined as the ratio between the operational limit-state displacement and a risk-targeted safety factor, which depends on the seismicity of the relevant location, on the randomness of the intensity measures causing the operational limit-state, and on the acceptable (target) probability of exceeding this limit state. In the paper, the theoretical background of risk-targeted safety factor is first explained, followed by a discussion on the sensitivity of the risk-targeted safety factor to the input parameters. The design procedure involving the simplified risk-targeted decision model is then introduced and demonstrated by means of an example of a simple pipe rack located at a petrochemical plant. Within the scope of the presented example, the operational limit-state displacement of the pipe rack is estimated on the basis of the limit-state strains of a pipe attached to the frame. The risk-targeted safety factor is then evaluated, and the frame is designed. It is shown that the application of the proposed procedure is straightforward once the operational limit-state displacement has been determined. The advantage of the procedure is that the target risk is accounted for by the risk-targeted safety factor, which can be calculated with respect to the importance of the equipment of the critical infrastructure. Additionally, the proposed decision model can be easily adopted by engineers, because the seismic demand is calculated by analogy to the Eurocode. However, at this stage of the research, the application of the proposed decision model is limited to structures that do not exhibit any significant nonlinear seismic response at the operational limit state, and to structures where the equal displacement rule applies.

Cyclic shear resistance for seismic design, based on monotonic shear models in fib Model Code 2010 and in the 2018 draft of Eurocode 2

AbstractShear resistance decays during cycling of inelastic deformations, causing flexural plastic hinges to fail in cyclic shear before reaching their ultimate flexural deformation. The monotonic shear model in MC2010 does not work for shear failure after flexural yielding. Accurate, unbiased, lack‐of‐fit‐free prediction of cyclic shear resistance, before or after flexural yielding, is possible, if: (a) inelasticity is considered in the calculation of longitudinal strains at section mid‐depth, through the “equal displacement rule” of Earthquake Engineering; (b) axial forces are taken to contribute to shear resistance with the transverse component of the diagonal strut between member ends; (c) the Eurocode 2 shear model for loads near supports is used for squat members; and (d) minor changes are made to the minimum compression field angle and the concrete strength reduction along it in MC2010, Levels II or III of approximation.

Evaluating the code approaches for estimating the seismic drifts of steel frame buildings designed under variable levels of seismicity

Two different approaches are considered by seismic codes for estimating the seismic drifts of moment resisting steel frames (MRSFs). The first approach which is adopted by the European and the Canadian codes is driven by the equal displacement rule and defines the deflection amplification factor (DAF) to be equal to the response modification factor R. The second approach which is motivated by the opinion of professionals who believe that the equal displacement rule often over-estimates displacements is employed by the ASCE 7-10 and the Egyptian seismic code, where the DAF is considered equal to a fraction of R. This study evaluates the two approaches considered for estimating the seismic drifts of MRSFs. Four MRSF buildings having 2-, 4-, 8- and 12-stories have been designed under variable levels of seismic intensity. Five lateral-stiffness scenarios are considered for each building to account for the stiffness increase due to the nonstructural components as well as the modeling approximations. The seismic drifts of the MRSFs are calculated under the effect of twenty-two pairs of far-field ground motion records by time history analysis and with using the code simplified approaches. The results obtained indicate that the current code approaches of amplifying the elastic drifts significantly overestimate the seismic drifts of buildings designed under low level of seismicity. In contrast, the ASCE 7-10 and the Egyptian code approaches of estimating the seismic drifts underestimate the maximum story drift ratios under earthquake loading for long period buildings designed under high level of seismicity.

Concurrent effects of inertial and kinematic soil-structure interactions on strength-ductility-period relationship

As a key stage of performance-based seismic design of structures, strength-ductility-period relationship is used to determine either inelastic strength or ductility ratio of the structure with a specific fundamental vibration period. On the other hand, period, strength, and ductility demands of structures are significantly affected by inertial and kinematic soil-structure interactions. Hence, this study addresses strength-ductility-period relationship of structures with embedded foundations subjected to non-pulse earthquakes. To achieve this goal, a soil-foundation-structure interaction system well suited for parametric study was used: the soil beneath the rigid foundation was considered as a half-space, the foundation was modelled using a stack of discretised embedded disks, and the structure was modelled as an equivalent single-degree-of-freedom oscillator. A number of soil-foundation-structure interaction systems defined according to key interaction dimensionless parameters were analysed subjected to a suit of 40 non-pulse ground motions. The results show that ductility ratio spectra are significantly influenced by inertial and kinematic interaction effects. It is also found that equal displacement rule can be either valid or invalid under different soil-foundation-structure interaction effects. Further, a practice-oriented mathematical expression was recommended for ductility ratio of soil-foundation-structure interaction systems, which can find use in performance evaluation of existing structures with embedded foundation.

An adaptive modal pushover analysis procedure (VMPA-A) for buildings subjected to bi-directional ground motions

A new modal pushover analysis procedure (VMPA-A) is developed and implemented in MATLAB code for three-dimensional buildings subjected to bidirectional ground motions. VMPA-A uses stepwise force patterns to represent changes in the dynamic characteristics because of the accumulated structural damages. The hybrid-spectrum concept is introduced to account for the bidirectional ground motion effects. Due to enactments of the equal displacement rule and the secant stiffness-based linearization process, nonlinear analysis is performed for specific displacement targets without stipulation of full modal capacity curves for each mode. Horizontal components of an earthquake record are considered simultaneously, and the consistency between the force and displacement vectors for each mode is provided. These are the main advantages of the proposed procedure against modal pushover analysis (MPA). An existing 21-story reinforced concrete building is analyzed to exemplify VMPA-A. The response parameters such as displacements, story drifts, internal forces, strains, etc. are discussed by comparing the results of VMPA-A with nonlinear time history analyses, which is accepted as the “exact solution”. Though consistent demand estimations are obtained for story drifts, displacements and deformations, some conservative results are obtained for story shears.

Seismic Protection of the Piers of Integral Bridges using Sliding Bearings

Seismic resilience and continued operation of bridges after earthquakes are important seismic design criteria. A new seismic protection concept for integral bridge piers is explored that uses sliding bearings to separate the superstructure from the piers. The influence of sliding bearings on the seismic response of a representative 3-span integral highway bridge is investigated. With sliding bearings, the pier column shear force was limited to the bearing design friction force. Furthermore, the abutment ductility demands were found to be insensitive to the friction forces in the sliding bearings because the bridge displacement demands were controlled by the equal displacement rule.

Empirical Correlation between Inelastic and Elastic Spectral Displacement Demands

Inelastic spectral displacement demand is arguably one of the most effective, simplified means of relating earthquake intensity to building damage. However, seismic hazard assessment is typically conducted using empirical ground-motion prediction equations (GMPEs) that only provide indications of elastic spectral response quantities, which an engineer subsequently relates to inelastic demands using empirical relationships such as the equal-displacement rule. An alternative approach is to utilize relationships for the inelastic spectral displacement demand directly within the seismic hazard assessment process. Such empirical relationships are developed in this work, as a function of magnitude, distance, building period, and yield strength coefficient, for four different hysteretic models that are representative of a wide range of possible structural typologies found in practice. The new relationships are likely to be particularly useful for performance-based seismic design and assessment.

A variant of modal pushover analyses (VMPA) based on a non-incremental procedure

A variant of modal pushover analysis (VMPA) is presented to evaluate the seismic performance of the structures. The suggested procedure is based on an iterative process in which secant stiffnesses are used both at the element level and in the modal response. VMPA diverges from the existing modal pushover analyses for the following reasons: (1) mode compatible adaptive forces are applied to the structure at each iteration step, (2) the application of the equal displacement rule in combination with secant stiffness based linearization in the spectral displacement (Sd)—spectral acceleration (Sa) relation eliminates the necessity to produce a modal capacity diagram for each mode. The displacement controlled algorithm determines the single ordinate of the modal capacity diagram corresponding to the target displacement demand for the nth mode (Sdn_p, San_p) by reducing elastic spectral acceleration (San_e) to converge to the plastic acceleration (San_p). A Matlab based computer program known as DOC3D-v2 is developed to implement the proposed procedure. To verify the success of the suggested procedure, 9- and 20-storey LA SAC buildings are analysed, and the resulting demands are compared with several existing procedures, such as the extended N2, MPA and MMPA, as well as nonlinear time history analyses performed for two different set of acceleration records. VMPA yields enhanced results in terms of storey drifts, especially for the 20-storey LA building, compared with the other methods. Although the storey displacements and drifts are largely consistent with nonlinear time history analysis results, conservative estimates are obtained for the storey shear forces.

Effects of inertial soil–structure interaction on inelastic displacement ratios of SDOF oscillators subjected to pulse-like ground motions

This study is devoted to investigate the effects of inertial soil-structure interaction (SSI) on the constant-strength inelastic displacement ratios of elastoplastic single-degree-of- freedomsystemsusingasuitof91pulse-likegroundmotions.Thesoilbeneaththefoundation is simulated based on the cone model. A local minimum for the inelastic displacement ratios around the interacting system-to-pulse period ratio of one is demonstrated. Moreover, the soil flexibility increases the inelastic displacement ratios at all interacting system-to-pulse period ratios. However, the aspect ratio has decreasing and increasing effects on the inelastic displacement ratios before and after a threshold interacting system-to-pulse period ratio, approximately very close to one. It is confirmed that for slender structures, the SSI effects are the lowest at small interacting system-to-pulse period ratios and as this ratio increases, the SSI effects on the inelastic displacement ratios increase. However, for squat structures, the SSI approximately has more significant effects on the inelastic displacement ratios at lower interacting system-to-pulse period ratios and the effects decrease for higher interacting system-to-pulse period ratios. It is noted that the equal displacement rule is not valid as the SSI effects are taken into account. In addition, a formula is proposed to estimate the inelastic displacement ratios of soil-structure systems using nonlinear regression analysis, which is desirable for the displacement assessment of existing structures. Besides, the mean ratios of approximate-to-analyticalvalues,veryclosetoone,emphasizewellaccuracyoftheproposed formula.

Simplified non-linear seismic displacement demand prediction for low period structures

The prediction of non-linear seismic demand using linear elastic behavior for the determination of peak non-linear response is widely used for seismic design as well as for vulnerability assessment. Existing methods use either linear response based on initial period and damping ratio, eventually corrected with factors, or linear response based on increased equivalent period and damping ratio. Improvements to the original EC8 procedure for displacement demand prediction are proposed in this study. Both propositions may be graphically approximated, which is a significant advantage for practical application. A comparison with several other methods (equal displacement rule, EC8 procedure, secant stiffness and empirical equivalent period methods) is performed. The study is based on non-linear SDOF systems subjected to recorded earthquakes, modified to match design response spectra of different ground types, and focuses on the low frequency range that is of interest for most European buildings. All results are represented in the spectral displacement/fundamental period plane that highlights the predominant effect of the fundamental period on the displacement demand. This study shows that linearized methods perform well at low strength reduction factors but may strongly underestimate the displacement demand at strength reduction factors greater than 2. This underestimation is an important issue, especially for assessment of existing buildings, which are often related with low lateral strength. In such cases, the corresponding strength reduction factors are therefore much larger than 2. The new proposals significantly improve the reliability of displacement demand prediction for values of strength reduction factors greater than 2 compared to the original EC8 procedure. As a consequence, for the seismic assessment of existing structures, such as unreinforced masonry low-rise buildings, the current procedure of EC8 should be modified in order to provide accurate predictions of the displacement demand in the domain of the response spectrum plateau.

Lateral vibration analysis of continuous bridges utilizing equal displacement rule

The application of equal displacement rule simplifies the evaluation of lateral displacement demand forSDOF system. For complex multi-degree-of-freedom (MDOF) structures such as continuous bridge systems, however, it requires more investigations. In this paper, a comprehensive parametric study of the ratio of maximum inelastic displacement to maximum elastic displacement for typical continuous bridges is performedto advance the application of equal displacement rule to MDOF systems. Particurlarly for the bridges with long periods, this adapted methodlogy is further simplified. It is concluded that equal displacement rule of MDOF is applicable to continuous bridges when the periods of the main modes are no less than the limiting period, which usually serves as an indication to the level of inelastic deformation for a bridge subjected to an earthquake.