Structural Dynamics, Dynamic Force and Dynamic System Although much less used by practicing engineers than conventional structural analysis, the use of. Special Issue: Earthquake Engineering and Structural Dynamics The 9th biennial Structural Engineering Convention (SEC) was hosted by. PDF | On Jul 31, , Arash Rostami and others published Structural Dynamics of Earthquake Engineering (Volume I).
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Numerical modelling of reinforced‐concrete structures under seismic loading based on the finite element method with discrete inter‐element cracks. Comparative assessment of nonlinear static and dynamic methods for analysing building response under sequential earthquake and tsunami · Tiziana Rossetto. Integrating visual damage simulation, virtual inspection, and collapse capacity to evaluate post‐earthquake structural safety of buildings · Henry V. Burton.
Series-A are to highlight the current research trends in earthquake engineering, and to report the scientific contributions made therein focusing on newly constructed steel, reinforced concrete RC , and composite structures, retrofitting and upgrading issues of the existing structures. Subsequently, we present the MPA procedure for unsymmetric-plan buildings, demonstrate its equivalence to standard response spectrum analysis RSA for elastic systems, and identify its underlying assumptions and approximations for inelastic buildings. The principal objective of this paper is to extend MPA to estimate seismic demands for unsymmetric-plan buildings. Remember me on this computer. Join with us. Statistics of single-degree-of-freedom estimate of displacements for pushover analysis of buildings.
It is one of six symmetric-plan buildings used as examples to determine the bias and dispersion in the MPA procedure . Natural periods and modes of vibration of 9-story unsymmetric-plan systems: Unsymmetric-Plan 1 U1. Figure 3 a shows the natural vibration periods and modes of Copyright? GOEL system U1. Unsymmetric-Plan 2 U2. The IOj value for every oor was increased by a factor of 2. Unsymmetric-Plan 3 U3. The IOj value for every oor was increased by a factor of 6.
Figure 3 c shows the natural vibration periods and modes of system U3. These three unsymmetric-plan systems will undergo coupled y-lateral and torsional motions due to the y-component of ground motion, which is the focus of this paper. The purely lateral response along the x-axis due to the x-component of excitation is not considered as it has been the subject of previous investigations [14, 15].
Ground motion The ground motion selected for this investigation is the LA25 ground motion shown in Figure 4. LA25 ground motion, one of twenty ground motions assembled for the SAC project. Recorded at a distance of 7: This intense ground motion enables testing of the approximate procedures developed herein under severe conditions.
This force distribution can be expanded as a summation of modal inertia force distributions sn Reference , section The lateral forces are in the positive Copyright? GOEL 0. Basic concept Two procedures for approximate analysis of inelastic buildings will be described next: It is governed by: However, these standard equations have been derived in an unconventional way.
In contrast to the classical derivation found in textbooks e. This concept will provide a rational basis for the modal pushover analysis procedure to be developed later. Inelastic systems Although modal analysis is not valid for an inelastic system, its response can be usefully discussed in terms of the modal coordinates of the corresponding linearly elastic system.
Both systems have the same Copyright? GOEL mass and damping. Other modes start responding as soon as the 30 10 2 Mode 1 Mode 1 Mode 1 Mode 1 ur1 cm 0.
Modal decomposition of roof displacement at the CM of the symmetric building: Modal decomposition of roof displacement at the right frame of the torsion- ally- exible system U3: This procedure will be described later.
Introducing the n-th mode inelastic SDF system permitted extension of the well-established concepts for elastic systems to inelastic systems; compare Equations 10 to 17 , Equations 12 to 19 , and note that Equation 11 applies to both systems. Solution of the non-linear Equation 19 provides Dn t , which when substituted into Equa- tions 13 and 14 gives oor displacements and story drifts.
Equations 13 and 14 Copyright? When specialized for linearly elastic systems, it becomes identical to the rigorous classical modal RHA described earlier. The striking similarity between the equations for the elastic and inelastic systems is apparent.
Such comparison for roof-displacement and top-story drift is presented in Figures 10 and 11, respectively. The errors in UMRHA results are slightly larger in drift than in displacement, but the errors in either response quantity seem acceptable for approximate methods to estimate seismic demands for unsymmetric-plan buildings.
This is exactly valid for linear elastic systems but is an approximation for inelastic systems. This approximation is avoided in the MPA procedure, which is presented next, but a modal combination approximation must be introduced as will be seen later.
Static analysis of the structure subjected to forces Copyright? Therefore, the three components of roof displacement of an elastic system will simultaneously reach the values given by Equation The peak modal response rn , each determined by one modal pushover analysis, can be combined by the Complete Quadratic Combination CQC Rule Reference , section Alternatively, Dn can be determined from the inelastic response or design spectrum Reference , sections 7.
At this roof displacement, non-linear static analysis provides an estimate of the peak value rn of response quantity rn t: For an inelastic system, no invariant distribution of forces will produce displacements pro- portional to the n-th elastic mode. Therefore, the three components of roof displacement of an inelastic system will not simultaneously reach the values given by Equation One of the two lateral components will be selected as the controlling displacement; the choice of the component would be the same as the dominant motion in the mode being considered.
At It would be natural to use the x or y pushover curve for a mode in which the x or y component of displacements is dominant compared to their y or x component. As mentioned earlier, rn determined by pushover analysis of an elastic system is the exact peak value of rn t , the n-th mode contribution to response r t.
As a result, the oor displacements are no longer proportional to the mode shape, as implied by Equation This application of modal combination rules to inelastic systems obviously lacks a rigorous theoretical basis, but seems reasonable because the modes are weakly coupled.
Summary of MPA A step-by-step summary of the MPA procedure to estimate the seismic demands for an unsymmetric-plan multistorey building is presented as a sequence of steps: Compute the natural frequencies,! Between the two pushover curves obtained corresponding to two lateral directions, x and y, preferably choose the pushover curve in the dominant direction of motion of the mode.
Gravity loads, including those present on the interior gravity frames, are applied before pushover analysis. Note the value of the lateral roof displacement due to gravity loads, urg. Idealize the pushover curve as a bilinear curve. The MPA procedure summarized in this paper is an extension to unsymmetric-plan buildings of the procedure originally developed for symmetric buildings  with three improvements . They will be computed from the total story drifts; such a procedure to determine beam plastic rotations has been published , but the one for element forces remains to be reported.
Figure 14 a shows the oor displacements and story drift demands at the CM for the symmetric building together with the exact value determined by non-linear RHA of the system. Figures 14 b , c , and d show similar results for the three unsymmetric systems, but the demands are now for the frame at the right edge of the plan.
For the excitation considered, the MPA results are accurate for two unsymmetric systems, U1 and U3, to a similar degree as they were for the symmetric building, which is apparent by comparing Figures 14 b and d with Figure 14 a ; however, the results are less accurate for system U2. This loss of accuracy could be due to two reasons: However, in spite of the resulting stronger modal coupling Figure 8 , the approximate UMRHA procedure was shown to be valid for this system Figure Thus, strong lateral—torsional coupling does not seem to be the source of the entire discrepancy.
Another plausible reason is that the roof displacement of system U2 due to the selected ground motion is considerably under-estimated in the MPA procedure Figure 14 c. The ABSSUM rule provides a conservative estimate of the roof displacement, as it should, and over-estimates displacements at most oors and drifts in most stories. The preceding scenario points to the need for evaluating the MPA procedure considering an ensemble of ground motions and documenting the bias and dispersion in this procedure applied to unsymmetric buildings, as has been accomplished for symmetric buildings .
Such a statistical investigation is necessary for two reasons. First, the SRSS and CQC modal combination rules are based on random vibration theory and the combined peak response should be interpreted as the mean of the peak values of response to an ensemble of earthquake excitations. Thus, the modal combination rules are intended for use when the excitation is characterized by a smooth response or design spectrum.
Although modal combination rules can also approximate the peak response to a single ground motion characterized by a jagged response spectrum, the errors are known to be much larger.
Second, accurate estimation of roof displacement is necessary for the success of any pushover procedure and this usually is not assured for individual ground motions, as has been observed for the six SAC buildings . Based on structural dynamics theory, the MPA pro- cedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice.
The MPA estimate of seismic demand due to an intense ground motion including a for- ward directivity pulse has been shown to be generally accurate for unsymmetric systems to a similar degree as it was for a symmetric building. This conclusion is based on a comparison of the MPA estimate of demand and its exact value determined by non-linear RHA for four structural systems: For the excitation considered, the MPA estimates for two unsymmetric systems, U1 and U3, are sim- ilarly accurate as they were for the symmetric-plan building; however, the results deteriorated for system U2 because of a stronger coupling of elastic modes and b under-estimation of roof displacement by the CQC modal combination rule, which occurs because the individual modal responses attain their peaks almost simultaneously.
This implies that for system U2 and the selected ground motion the CQC modal combination rule would not give an accurate estimate of the peak response even if the system were linearly elastic. This points to the need for evaluating the MPA procedure considering an ensemble of ground motions and documenting the bias and dispersion in the procedure applied to un- symmetric buildings, as has been accomplished for symmetric buildings .
Simultaneous action of two horizontal components of ground motion and structural plans unsymmetric about both axes also remain to be inves- tigated.
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