Modal base forces in structures having a straight‐line mode of vibration (original) (raw)
Related papers
Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium
Journal of Engineering Mechanics, 2007
The dynamic modal analysis ͑i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors͒ and static stability ͑i.e., critical loads and buckling modes͒ of two-dimensional ͑2D͒ cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base ͑i.e., zero rotation of the cross sections͒, the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium "ignoring" the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: ͑1͒ the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness ͑L / r͒; and ͑2͒ the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer ͑2005͒ in columns made of elastomeric materials.
Modal Parameters for the Analysis of Inelastic Asymmetric-Plan Structures
Earthquake Spectra, 2009
The conventional inelastic response spectra constructed by using the single-degree-of-freedom (SDOF) modal systems representing the relationship of roof translation versus base shear are short of the capability to accurately estimate rotational seismic demands for asymmetric-plan structures. This paper illustrates the inelastic response spectra for one-way asymmetric-plan structures constructed from the two-degrees-of-freedom (2DOF) modal systems. The 2DOF modal system can simultaneously represent the roof translation versus base shear and the roof rotation versus base torque relationships for each vibration mode of any one-way asymmetricplan structure. Thus, the proposed inelastic response spectra are able to estimate not only the translational but also the rotational seismic demands. This new type of response spectra is named as the T-R response spectra. In order to construct the T-R inelastic response spectra, this research identifies three independent elastic 2DOF modal parameters: vibration period, T n , frequency ratio, ⍀ n , and modal eccentricity, e n. The relationships between the inelastic 2DOF modal parameters and the strength ratio are established in this study. The ranges of the 2DOF modal parameter values are investigated for the general one-story, one-way, asymmetric-plan buildings. The translational and rotational constant-strength response spectra constructed from the 2DOF modal systems are illustrated. This paper demonstrates that the proposed inelastic response spectra are more suitable for computing the seismic demands of one-way asymmetric-plan structures than the conventional SDOF inelastic response spectra.
Static versus modal analysis: influence on inelastic response of multi-storey asymmetric buildings
The paper investigates the influence of design procedures on the seismic response of multi-storey asymmetric buildings. To this end, some structures are designed according to methods based on either static or modal analysis, with or without design eccentricities. The seismic response of these systems is determined by means of inelastic dynamic analyses and the design is thoroughly examined in order to explain the results of the dynamic analyses. Attention is basically focused on the ability of design methods to prevent asymmetric buildings from experiencing ductility demands much larger than those of the corresponding torsionally balanced systems. Numerical analyses underline that while design procedures based on either static or modal analysis are suitable for the design of torsionally rigid structures only those based on modal analysis lead to the satisfactory performance of torsionally flexible buildings. Furthermore, the study highlights the qualities of a design method proposed by the Authors. Its application does not require any explicit calculation of design eccentricities and leads to proper seismic response of both torsionally rigid and flexible asymmetric buildings.
Effect of higher modes on the seismic response and design of moment-resisting RC frame structures
Engineering Structures, 2013
Recently, extensive research has been conducted regarding higher-mode effects on the response of multi degree-of-freedom (MDOF) systems. The research has been focused mainly on structures with a lateral force resisting system consisting of slender walls, since these types of buildings are expected to be mostly affected by higher-mode phenomena according to structural dynamics, and simplified expressions have been proposed for slender-wall structures to account for higher-mode response in estimating shear forces. Current seismic design practice assumes the same reduction factor for all modes, even though there is strong evidence that inelasticity affects higher modes of vibration unequally. Additionally, simplified design methods are based only on the fundamental mode of vibration neglecting the effect of higher modes or considering them as elastic. In this paper, higher-mode contributions on the overall response of a nine-storey moment resisting frame (MRF), for which a domination of the first mode is expected, are investigated. The accuracy of a modified Modal Response Spectrum Analysis (mMRSA) method and other available methods is evaluated by comparing the results with the ones of the nonlinear response history analysis. Modal behaviour (reduction) factors are directly calculated for the first three modes and the validity of common assumptions is examined. The assessment of the methods is not restricted to deformations, but is extended to storey inertial forces and shears as well, which have attracted less interest from structural engineers, even though they are considered responsible for numerous structural and non-structural failures during major recent earthquakes and are critical for the design of several structures, such as precast buildings. The results suggest that the storey inertial forces and accelerations at all storeys and shear forces at higher storeys are significantly underestimated by methods neglecting or non-properly accounting for higher modes, even for first-mode dominated structures. The contribution of higher modes depends on the ground motion characteristics, the overstrength associated with each mode and the response quantity examined.
Simplified seismic analysis of asymmetric building systems
Earthquake Engineering & Structural Dynamics, 2007
The paper reviews the uncoupled modal response history analysis (UMRHA) and modal pushover analysis (MPA) procedure in the analysis of asymmetric structures. From the pushover curves in ADRS format, showing the relationships of base shear versus roof translation and base torque versus roof rotation, a bifurcating characteristic of the pushover curves of an asymmetric structure is observed. A two-degreeof-freedom (2DOF) modal stick is constructed using lump mass eccentrically placed at the end of beam which is connected with the column by a rotational spring. By converting the equation of motion of a whole structure into 2DOF modal equations, all of the elastic properties in the 2DOF modal sticks can be determined accurately. A mathematical proof is carried out to demonstrate that the 2DOF modal stick is consistent with the single-degree-of-freedom (SDOF) modal stick at elastic state. The bifurcating characteristic of modal pushover curves and the interaction of modal translation and rotation can be considered rationally by this 2DOF modal stick. In order to verify the effectiveness of this proposed 2DOF modal stick, a two-storey asymmetric building structure was analysed by the UMRHA procedure incorporating this novel 2DOF modal sticks (2DMPA) and conventional SDOF modal sticks (SDMPA), respectively. The analytical results are compared with those obtained by nonlinear response history analysis (RHA). It is illustrated that the accuracy of the rotational response histories obtained by 2DMPA is much better than those obtained by SDMPA. Consequently, the estimations of translational response histories on flexible side (FS) and stiff side (SS) of the building structure are also improved. Figure 3. Seismic ground acceleration records: (a) sine wave; and (b) 1940 El Centro earthquake EW component.
dist.unina.it
The vertical elastic axis and the horizontal principal axes of a building constitute the necessary presupposition in order to apply the documented equivalent static method (simplified spectrum method) of seismic design. However, on the contrary to single-story building, in the general case of asymmetric multistory buildings real vertical elastic axis does not exist. To confront this problem, Hellenic Seismic Code (EAK/2003) introduces the meaning of vertical fictitious elastic axis of an irregular asymmetric multistory building utilizing the optimum torsion (around z-axis) criterion. Using the fictitious elastic center, namely the point (in the plan) from which the vertical fictitious elastic axis and the horizontal fictitious principal axes pass through, the definition of the static eccentricities and the torsional stiffness radii of gyration are achieved, along the two horizontal real/fictitious principal axes of the building. So, the documented simplified spectrum method of seismic design can be applied on irregular in the plan asymmetric multistory building. In the present paper, the dynamic properties of the real elastic center of a single-story building are proved that hold good approximately to the fictitious elastic center of irregular in the plan asymmetric multistory buildings.
Bulletin of the New Zealand Society for Earthquake Engineering
The primary objective of this paper is to evaluate the effectiveness of the 3-dimensional modal response spectrum method (MRSM), as specified in the seismic design regulations of NZS 4203:1992, in accounting for torsional effects arising in stiffness-asymmetric buildings. Such buildings are assumed to be excited well into the inelastic range of response, where ultimate limit state design criteria are applicable. Also included is a study of the horizontal regularity conditions of NZS 4203:1992. This preliminary study focuses on the influence that these regularity conditions have on the selection of an appropriate codified approach for the design of torsionally asymmetric buildings. In particular, restrictions on the use of the equivalent static method of design (ESM), leading to a requirement to employ the MRSM to account for dynamic torsional effects, are discussed in some detail. The dynamic response studies indicate that in systems designed by the MRSM, no significant increase in ...
On the modal damping ratios of shear-type structures equipped with Rayleigh damping systems
Journal of Sound and Vibration, 2006
The effects of added manufactured viscous dampers upon shear-type structures are analytically investigated here for the class of Rayleigh damping systems. The definitions of mass proportional damping (MPD) and stiffness proportional damping (SPD) systems are briefly recalled and their physical counterpart is derived. From basic physics, a detailed mathematical demonstration that the first modal damping ratio of a structure equipped with the MPD system is always larger than the first modal damping ratio of a structure equipped with the SPD system is provided here. All results are derived for the class of structures characterised by constant values of lateral stiffness and storey mass, under the equal ''total size'' constraint. The paper also provides closed form demonstrations of other properties of modal damping ratios which further indicate that the MPD and the SPD systems are respectively characterised by the largest and the smallest damping efficiency among Rayleigh damping systems subjected to base excitation. A numerical application with realistic data corresponding to an actual seven-storey building structure is presented to illustrate and verify the theoretical findings.
On the dynamic response of regular structures exhibiting tilt
Earthquake Engineering & Structural Dynamics, 2000
The mechanical characteristics of earthquake-resistant structures and the ground motion properties that have an important in#uence on the seismic performance of structures with asymmetric load}deformation behaviour, are identi"ed from the results obtained from a parametric study of the dynamic response of single-degree-of-freedom systems. The dynamic response of tilted single and multi-degree-of-freedom systems, designed according to Mexican requirements for the design of asymmetric structures is studied. After a general understanding of this phenomenon is o!ered, shortcomings in current design requirements for tilted structures are discussed. Research needs to establish a rational design approach of this type of structures are identi"ed.