Shear Beam Model Research Articles

SHEAR SPRING MODEL FOR TIME HISTORY ANALYSIS OF MULTI-STORY PASSIVE CONTROLLED BUILDINGS

This paper proposes a method to create a shear beam model consisting of elements that represent hysteretic characteristics of the dampers, their support members/connections, and frames in a passively controlled muluti-story and multi-bay building. Four types of dampers, steel, oil, viscous, and viscoelastic dampers, are considered. The model enables efficient and practical time history analyses of the building subjected to ground motions. Degrees of freedom of the building are reduced to a few per floor, by conducting two or three static analyses. Then, a shear beam model of a few degrees of freedom per floor level is constructed by matching its behavior exactly with the building. The accuracy of the shear beam model is demonstrated via static analyses as well as dynamic analyses using a variety of ground motions.

Near-Field Ground Motion Modal versus Wave Propagation Analysis

The response spectrum generally provides a good estimate of the global displacement and acceleration demand of far-field ground motion on a structure. However, it does not provide accurate information on the local shape or internal deformation of the response of the structure. Near-field pulse-like ground motion will propagate through the structure as waves, causing large, localized deformation. Therefore, the response spectrum alone is not a sufficient representation of near-field ground motion features. Results show that the drift-response technique based on a continuous shear-beam model has to be employed here to estimate structure-demand parameters when structure is exposed to the pulse like ground motion. Conduced modeling shows limited applicability of the drift spectrum based on the SDOF approximation. The SDOF drift spectrum approximation can only be applied to structures with smaller natural periods than the dominant period of the ground motion. For periods larger than the dominant period of ground motion the SDOF drift spectra model significantly underestimates maximum deformation. Strong pulse-type motions are observed in the near-source region of large earthquakes; however, there is a lack of waveforms collected from small earthquakes at very close distances that were recorded underground in mines. The results presented in this paper are relevant for structures with a height of a few meters, placed in an underground excavation. The strong ground motion sensors recorded mine-induced earthquakes in a deep gold mine, South Africa. The strongest monitored horizontal ground motion was caused by an event of magnitude 2 at a distance of 90 m with PGA 123 m/s2, causing drifts of 0.25%–0.35%. The weak underground motion has spectral characteristics similar to the strong ground motion observed on the earth's surface; the drift spectrum has a maximum value less than 0.02%.

Near-Field Ground Motion Modal versus Wave Propagation Analysis

The response spectrum generally provides a good estimate of the global displacement and acceleration demand of far-field ground motion on a structure. However, it does not provide accurate information on the local shape or internal deformation of the response of the structure. Near-field pulse-like ground motion will propagate through the structure as waves, causing large, localized deformation. Therefore, the response spectrum alone is not a sufficient representation of near-field ground motion features. Results show that the drift-response technique based on a continuous shear-beam model has to be employed here to estimate structure-demand parameters when structure is exposed to the pulse like ground motion. Conduced modeling shows limited applicability of the drift spectrum based on the SDOF approximation. The SDOF drift spectrum approximation can only be applied to structures with smaller natural periods than the dominant period of the ground motion. For periods larger than the dominant period of ground motion the SDOF drift spectra model significantly underestimates maximum deformation. Strong pulse-type motions are observed in the near-source region of large earthquakes; however, there is a lack of waveforms collected from small earthquakes at very close distances that were recorded underground in mines. The results presented in this paper are relevant for structures with a height of a few meters, placed in an underground excavation. The strong ground motion sensors recorded mine-induced earthquakes in a deep gold mine, South Africa. The strongest monitored horizontal ground motion was caused by an event of magnitude 2 at a distance of 90 m with PGA 123 m/s2, causing drifts of 0.25%–0.35%. The weak underground motion has spectral characteristics similar to the strong ground motion observed on the earth's surface; the drift spectrum has a maximum value less than 0.02%.

Motion Planning and Tracking for Tip Displacement and Deflection Angle for Flexible Beams

Abstract Explicit motion-planning reference solutions are presented for flexible beams with Kelvin–Voigt (KV) damping. The goal is to generate periodic reference signals for the displacement and deflection angle at the free-end of the beam using only actuation at the base. The explicit deflection angle reference solution is found as a result of writing the shear beam model in a strict-feedback form. Special “partial differential equation (PDE) backstepping” transformations relate the strict-feedback model to a “target system,” governed by an exponentially stable wave equation with KV damping, whose displacement reference solution is relatively easy to find. The explicit beam displacement reference solution is found using the target system solution and an inverse backstepping transformation. The explicit reference solutions for the wave equation and shear beam with KV damping are novel results. State-feedback tracking boundary controllers are found by extending previous PDE backstepping stabilization results. Application of the shear beam results to the more complicated Timoshenko beam is discussed.

MODAL PROPERTIES AND FREE VIBRATION OF UPLIFTING BEHAVIOR OF MULTISTORY BUILDINGS MODELED AS UNIFORM SHEAR-BEAM

Dynamic behavior of buildings allowed to uplift is investigated by means of classical modal analysis of a uniform shear-beam model on a rigid ground. The equations of motion are derived and eigenproblem is solved to investigate the modal properties during uplift. Free vibration under gravity during the first excursion of uplift is analyzed to show the responses of transient uplift behavior. In addition to the confirmation of significant reduction effect, it is pointed out from the results that the relatively complicated uplift responses at a glance can be recognized as natural consequences of higher modes' properties and their behaviors.

Rotations in a shear-beam model of a seven-story building caused by nonlinear waves during earthquake excitation

SUMMARY We model a seven-story, reinforced-concrete building in Van Nuys, CA, which was damaged during the 1994 Northridge earthquake. We use a one-dimensional, layered, shear-beam model with bi-linear material properties, and we examine how the rotations (local strains and drifts) in this model depend upon the distribution of the stiffness along the building height, the nonlinear properties of the reinforced concrete, and the nature of strong motion. We show how, following the powerful waves propagating up and down the building, point rotations take place that differ from the corresponding average drift angles. These point rotations are larger near the rigid floor slabs (near the top and bottom ends of the columns, shear walls, and nonstructural members found at all floors) and propagate as slow waves up and down the building. These slow rotational waves occur only while the material is experiencing large nonlinear deformations, and their amplitudes grow with ductility. We note that recording these rotational waves in real time may provide a powerful new vehicle for health monitoring of full-scale structures excited by large transient forces when their structural members enter large nonlinear deformations. Copyright r 2008 John Wiley & Sons, Ltd.

Discussion of “On the Comparison of Timoshenko and Shear Models in Beam Dynamics” by Noël Challamel

This technical note questions the validity of a general approach developed by Aristizabal-Ochoa 2004 for the dynamic analysis of a Timoshenko beam-column with generalized boundary conditions and the nonclassical modes of vibration of shear beams, in particular the theoretical relationship between both models for large values of bending to shear stiffness parameter. A simply supported beam is analytically studied for both models by the author of the discussed paper, and asymptotic solutions are obtained for large values of bending to shear stiffness parameter. Using the particular case of a simply supported beam with zero rotational inertia i.e., r =0 , the author claims that, “In the general case, it is proven that the shear beam model cannot be deduced from the Timoshenko model, by considering large values of bending to shear stiffness parameter. This is only achieved for specific geometrical parameter in the present example.” Finally, the author concludes based on his asymptotic solutions of the simply supported beam with zero rotational inertia that “the capability of the shear model to approximate Timoshenko model for large values of bending to shear stiffness parameter is firmly dependent on the material and geometrical characteristics of the beam section and on the boundary conditions.” The author is congratulated for studying the free vibration of beams. However, it is unfortunate that the author uses a degenerate beam model to prove that the general approach developed by Aristizabal-Ochoa 2004 for the dynamic analysis of a Timoshenko beam-column with generalized boundary conditions and the nonclassical modes of vibration of shear beams is in some way incorrect. The main objective of this discussion is to show that 1 it is incorrect to assume a Timoshenko beam with r = 0 and still expect that it is capable of predicting the free vibration of the Timoshenko shear model for large values of bending to shear stiffness parameter; and 2 the conclusions based on the asymptotic solutions presented by the author are not correct because they are based on a free-vibration analysis of a degenerate Timoshenko beam. What follows is the proof that it is incorrect to assume that the rotational inertia can be simply ignored in the free-vibration analysis of shear beams and to expect that the natural frequencies and modal shapes are correct. Kausel 2002 elegantly proved this point in shear beams with unrestrained or partially restrained ends against rotation of the cross sections, showing that the classical solutions which are based on r =0 violate the principle of angular moment.

Identification of nonlinear elastic structures using empirical mode decomposition and nonlinear normal modes

The empirical mode decomposition (EMD) method is well-known for its ability to decompose a multi-component signal into a set of intrinsic mode functions (IMFs). The method uses a sifting process in which local extrema of a signal are identified and followed by a spline fitting approximation for decomposition. This method provides an effective and robust approach for decomposing nonlinear and non-stationary signals. On the other hand, the IMF components do not automatically guarantee a well-defined physical meaning hence it is necessary to validate the IMF components carefully prior to any further processing and interpretation. In this paper, an attempt to use the EMD method to identify properties of nonlinear elastic multi-degree-of-freedom structures is explored. It is first shown that the IMF components of the displacement and velocity responses of a nonlinear elastic structure are numerically close to the nonlinear normal mode (NNM) responses obtained from two-dimensional invariant manifolds. The IMF components can then be used in the context of the NNM method to estimate the properties of the nonlinear elastic structure. A two-degree-of-freedom shear-beam building model is used as an example to illustrate the proposed technique. Numerical results show that combining the EMD and the NNM method provides a possible means for obtaining nonlinear properties in a structure.

On the Comparison of Timoshenko and Shear Models in Beam Dynamics

The classical Timoshenko beam model and the shear beam model are often used to model shear building behavior both for stability or dynamic analysis. This technical note questions the theoretical relationship between both models for large values of bending to shear stiffness parameter. The simply supported beam is analytically studied for both models. Asymptotic solutions are obtained for large values of bending to shear stiffness parameter. In the general case, it is proven that the shear beam model cannot be deduced from the Timoshenko model, by considering large values of bending to shear stiffness parameter. This is only achieved for specific geometrical parameter in the present example. As a conclusion, the capability of the shear model to approximate Timoshenko model for large values of bending to shear stiffness parameter is firmly dependent on the material and geometrical characteristics of the beam section and on the boundary conditions.

Application of Internally Damped Shearbeam Model to Analysis of Buildings under Earthquakes: Robust Procedure for Quick Evaluation of Seismic Performance

A step-by-step procedure for linear analysis of building structures with rigid floor beams is developed and presented. The method outputs the maximum story drift envelope, and consists of a series of explicit formulas that do not require iteration. The basis of the method is an internally damped shearbeam model. Drift demand spectra are used to provide information on the ground motion hazard of the site. In addition to the drift demand spectra, two additional spectra are incorporated, which describe the higher mode behavior. To test accuracy, three sample buildings structures (3-, 9-, and 20-story buildings at Los Angeles) are analyzed for five sample earthquake records using the proposed procedure, and conventional time history analysis. A comparison of the results shows that the proposed procedure is accurate enough for quick evaluation of seismic performance.

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.

Dynamic Characterisation of Base-Isolated Structures Using Analytical Shear-Beam Model

Dynamic Characterisation of Base-Isolated Structures Using Analytical Shear-Beam Model

Generalized Damped Wave Approach for Evaluating Seismic Shear Demands

This paper presents a new analytical approach for simple and explicit computation of the seismic base shear demand of structural systems that can be idealized by a uniform shear-beam model. The approach is based on a Green’s function representation for the relative displacement response that is assumed to be composed of exponentially decaying wave sequences. Explicit solutions for both the strain and displacement response are derived in terms of an effective ground velocity and displacement that can be computed incrementally from the ground acceleration. A physical interpretation for the damping mechanism is proposed. The method is further generalized to form a class of physically motivated shear-beam systems referred to as the continuous spring-dashpot (CSD) model. The response characteristics of three cases of the CSD model along with a shear beam equipped with a mass-proportional external damping are compared and discussed for the case of near-field earthquake excitation.

Ground motion analysis in seasonally frozen regions

In seasonally frozen regions, the seasonal appearance of the frozen layer may have certain influence on the ground motion features of the area. In this paper, a site in a seasonally frozen region in northwestern China is taken to investigate the ground motion features in different seasons. As an easy method to obtain the linear features of the ground, micro-tremors are measured twice, once without the frozen layer and the other during the maximum frozen depth in the year. It is shown from the micro-tremor analyses that the shape of frequency spectrum changes and the predominant period of the ground is enhanced when the frozen layer is present. To examine the nonlinear properties of soils under strong ground motions, triaxial dynamic tests were performed on undisturbed samples taken from the working site to obtain the soil's dynamic parameters. These parameters, together with the results from seismic risk analysis, are input into a one-dimensional shear beam model and strong ground motion parameters are calculated for the site in seasons with and without frozen layer. This calculation shows that the seasonal frozen layer can reduce the strong motion to some extent, and change the predominant period slightly.

Inelastic torsion of multistorey buildings under earthquake excitations

The inelastic earthquake response of eccentric, multistorey, frame-type, reinforced concrete buildings is investigated using three- and five-storey models, subjected to a set of 10, two-component, semi-artificial motions, generated to match the design spectrum. Buildings designed according to the EC8 as well as the UBC-97 code were included in the investigation. It is found that contrary to what the simplified one-storey, typical, shear-beam models predict, the so-called ‘flexible’ side frames exhibit higher ductility demands than the ‘stiff’ side frames. The substantial differences in such demands between the two sides suggest a need for reassessment of the pertinent code provisions. This investigation constitutes one of the first attempts to study the problem of inelastic torsion by means of realistic, multistorey inelastic building models. Additional studies with similar or even more refined idealizations will certainly be required to arrive at definite results and recommendations for possible code revisions. Copyright © 2005 John Wiley & Sons, Ltd.

SYSTEM IDENTIFICATION OF LANDFILL SEISMIC RESPONSE

Assessment of landfill seismic response necessitates the availability of reliable dynamic material properties. During the past decade, geophysical surveys and computational studies have been conducted to investigate the seismic response of the Operating Industries, Inc. (OII) landfill in Southern California. In this paper, a survey and summary of available research results is presented. In addition, a set of Oil input-output seismic records during six earthquakes is thoroughly analysed. Spectral analyses are conducted to shed light on the landfill dynamic response characteristics. A simple shear beam model is found to be useful in modelling the landfill resonant behaviour. System identification techniques are employed to estimate the landfill stiffness and damping properties. These properties are defined by minimising the difference between computed and recorded acceleration response spectra at the landfill top. The identified stiffness properties are found to be near the lower bound of those documented through geophysical measure-ments. Identified damping of about 5% (at resonance) is within the range of earlier investigations. Comparisons of the computed and recorded accelerations show: (I) effectiveness of a linear viscous shear beam model in simulating the landfill dynamic behaviour, for the recorded small to moderate levels of dynamic excitation (up to 0.26 g peak lateral acceleration), and (ii) potential of the employed system identification procedure for analysis of input-output seismic motions.

Experimental study of boundary effects in dynamic centrifuge modelling

Dynamic centrifuge modelling has been established as a powerful tool for studying soil–structure interaction problems under earthquake loading. Increasingly complex models are being tested in centrifuges all around the world in an attempt to understand real structure behaviour under earthquake loading. However, there is a need to model the field conditions correctly in these centrifuge models. In a geotechnical centrifuge, the space available to model real situations is not infinite, and it is necessary to enclose the model within the finite boundaries of a container. The boundary effects of the soil container are important and can lead to inaccurate simulation of a field situation that has infinite lateral extent. The equivalent shear beam (ESB) model container used in dynamic centrifuge testing is built to achieve the same dynamic response as the soil sample to minimise the boundary effects. A series of dynamic centrifuge tests involving loose and dense, dry and saturated models of homogeneous horizontal sand layers has been carried out to investigate the effects of the end walls of the ESB model container on soil behaviour. In both saturated and dry sand models it was seen that amplification of acceleration occurs at the base of the model close to the end wall. In saturated models, as excess pore pressures build up, the response of the centre of the model gets progressively more different from that near to the end wall. This is to be expected as the box stiffness was matched to that of the non-liquefied soil column. It was confirmed that when the relative density of the soil in the model was around 50%, close to that of the design soil layer, the change in behaviour towards the end wall was minimal.

Experimental study of boundary effects in dynamic centrifuge modelling

Dynamic centrifuge modelling has been established as a powerful tool for studying soil–structure interaction problems under earthquake loading. Increasingly complex models are being tested in centrifuges all around the world in an attempt to understand real structure behaviour under earthquake loading. However, there is a need to model the field conditions correctly in these centrifuge models. In a geotechnical centrifuge, the space available to model real situations is not infinite, and it is necessary to enclose the model within the finite boundaries of a container. The boundary effects of the soil container are important and can lead to inaccurate simulation of a field situation that has infinite lateral extent. The equivalent shear beam (ESB) model container used in dynamic centrifuge testing is built to achieve the same dynamic response as the soil sample to minimise the boundary effects. A series of dynamic centrifuge tests involving loose and dense, dry and saturated models of homogeneous horizontal sand layers has been carried out to investigate the effects of the end walls of the ESB model container on soil behaviour. In both saturated and dry sand models it was seen that amplification of acceleration occurs at the base of the model close to the end wall. In saturated models, as excess pore pressures build up, the response of the centre of the model gets progressively more different from that near to the end wall. This is to be expected as the box stiffness was matched to that of the non-liquefied soil column. It was confirmed that when the relative density of the soil in the model was around 50%, close to that of the design soil layer, the change in behaviour towards the end wall was minimal.

Considerations of Multimode Structural Response for Near-Field Earthquakes

This paper presents an investigation of multimode effects of tall buildings idealized as a continuous shear-beam model subjected to near-field pulse-like ground motion. The investigation is based on three analytical approaches: a damped wave solution approach, a fundamental-mode approach, and a modal summation approach. In the modal summation approach, all modal damping ratios are assumed to be equal and a set of Green’s functions for the shear strain response is explicitly derived. The multimode effects on the base-level shear strain/force demands are compared by using an effective response spectrum for shear-beam systems. The study results show that the occurrence of major spectral differences is conditioned on the ratio of the fundamental structural period to the duration of the predominant excitation pulse. Seismic analyses for a set of recorded near-field earthquake data indicate a strong correlation between the characteristics of effective response spectra and the ground pulse parameters.

Simplified procedure for seismic analysis of asymmetric buildings

A simplified procedure, which is a modification of the shear beam model, has been developed and applied to treat the seismic analysis of asymmetric buildings. This three dimensional procedure accounts for torsional coupling and the bending rotations at beam–column junctions and it can be used with a personal computer to give fast and reasonably accurate results, which compare well with those from comprehensive finite element analysis. This procedure will therefore be useful for preliminary seismic analysis and design of buildings.