Modal response spectrum analysis

The Modal Response Spectrum Analysis (also known as a dynamic analysis) more accurately depicts the response of the structure to the earthquake. This is done by considering the response of multiple modes instead of just the first one (as is done in the ELF). Whereas the use of a static analysis assumes that a load is applied relatively slowly, a dynamic analysis should be used if the application of the load is faster than the response of the structure. For this reason a dynamic analysis is mainly... 

For non symmetric buildings, two different pushover analysis should be performed for each seismic input direction, in the positive and negative directions. For structures that are torsionally flexible, ECS suggests to account for the torsional effects by following a simplified procedure where the frame displacements are increases based on a modal response spectrum analysis. This approach is based on the modified N2 procedure (Fajfar et al, 2005). 

In many seismic codes, the equivalent static lateral force V is used as a reference value of the total seismic design base shear. For instance, the base shear calculated using the modal response spectrum analysis should not be less than 80 % or 90 % of the equivalent lateral base shear. This minimum base shear is provided because the computed period of vibration may be the result of an overly flexible (incorrect) analytical model. 

Lateral force method Modal analysis Modal response spectrum analysis Modal superposition Response spectrum Seismic performance assessment... 

Eurocode 8, linear elastic analysis can be performed either with (a) the lateral force method of analysis for buildings meeting specific conditions or (b) with the modal response spectrum analysis, applicable to all types of builduigs and being the reference method for obtaining seismic response estimates. 

According to the lateral force method of analysis, a lateral load pattern that follows the first mode is applied (e.g., Eq. 35). In Eurocode 8, the method is applicable only if the fundamental period of vibration is less than 2s or 4Tc, where Tc is the comer period of the design spectrum. A second condition that also has to be satisfied is that the stiucture should meet the criteria for regularity in plan. Different criteria may be found in other codes. The modal response spectrum analysis is reconunended for all other cases, with the only exception of buildings with seismic isolation provided by highly nmilinear devices. Practically every design code in the world uses these two methods for linear elastic analysis and recommends criteria to determine if their applicable. 

An alternative nonlinear static procedure was proposed by Aydinoglu (2003). The procedure, called Incremental Response Spectrum Analysis (IRSA) does not properly belong to the pushover methods. The procedure is displacement-based, it uses the equal displacement rule and the structure nonlinear behavior is modeled as piece-wise linear. The initial modes are computed and the elastic response spectrum for the initial structure is carried out. In the first stop the spectral ordinates are scaled to the formation of the first plastic hinge, which corresponds in the piece-wise linear capacity curve, to the first change of stiffness. The updated modal quantities are computed and additional spectral ordinates are computed up to the formation of the second plastic hinge. The procedure continues until the entire spectral ordinates are applied. The method does not require any transformation to an equivalent SDOE system. 

Multi-mode pushover analysis procedure IRSA (Incremental Response Spectrum Analysis) has been introduced by the first author to enable the two and three dimensional nonlinear analyses of buildings and bridges (Aydmoglu 2003). The practical version of the procedure (Aydmoglu 2004, 2007) works directly with smoothed elastic response spectrum and makes use of the well-known equal displacement rule to scale modal displacement increments at each piecewise linear step of an incremental application of linear Response Spectrum Analysis (RSA). In this paper, main steps of IRSA are summarized and its performance is evaluated on two example bridges under three different ground motions. 

Incremental Response Spectrum Analysis (IRSA) is a multi-mode pushover procedure, in which the incremental response is assumed piecewise linear at each pushover step in between the formation of two consecutive plastic hinges. The key point in IRSA is modal scaling, which is applied at each step to modal displacement increments, to identify their proportions in different modes (Aydmoglu 2003, 2007). 

A standard linear response spectrum analysis (RSA) is performed at each (i) th incremental pushover step for the unit value of the unknown incremental scale factor = 1) by considering instantaneous mode shapes that are compatible with the current distribution of plastic hinges and the initial elastic spectral displacements taken as seismic input. Such a linear response spectrum analysis (RSA) effectively corresponds to adaptive pushover analyses, which are simultaneously performed in each mode followed by the application of an appropriate modal combination rule. Thus, any response quantity of interest, which is represented by a generic response quantity, r, is obtained for the unit value of the unknown incremental scale factor. Now, the increment of the generic response quantity, Ar, is expressed as... 

Adaptive pushover analysis is an alternative to conventional pushover analysis. In force-based adaptive pushover analysis, the load vectors are gradually updated due to the damage occurring to the structure, which affects the mode shapes. One of the first procedures for the adaptive lateral load for pushover analysis was proposed by Bracci et al. (1997). Many different procedures for force-based adaptive pushover analysis followed. Most approaches involve response spectrum analysis which is performed for the different modal characters of the structure which are associated with selected stages of the pushover analysis. However, Aydinoglu (2003) has argued that the load patterns should be based on an inelastic rather than an elastic response spectrum. It was soon realized that force-based adaptive pushover analysis offers only a relatively minor advantage in comparison with conventional pushover analysis. As a result of such an observation, a displacement-based adaptive pushover procedure was proposed (Antoniou and Pinho 2004), which. 

Response spectrum analysis (RSA) is a method widely used for the design of buildings. Conceptually the method is a simplification of modal analysis, i.e., response history (or time history) analysis (RHA) using modal decomposition, that benefits from the properties of the response spectrum concept. The purpose of the method is to provide quick estimates of the peak response without the need to carry out response history analysis. This is very important because response spectrum analysis (RSA) is based on a series of quick and simple calculations, while time history analysis requires the solution of the differential equation of motion over time. Despite its approximate nature, the method is very useful since it allows the use of response spectrum, a very convenient way to describe seismic hazard. 

Example 2 Response Spectrum Modal Analysis of a Plane Frame For the building of Fig. 1 a, estimate the storey displacements using response spectrum analysis. The structure is again subjected to the Imperial Valley grotmd motion record of Fig. lb. 

Regarding the method of analysis, according to ASCE/AWEA RP2011 a fully coupled time-domain analysis and decoupled analyses based on equivalent lateral force method or modal response spectrum method are acceptable, as permitted by the local building code. For the specific implementation of each method of analysis, the local building code or ASCE/SEl 7-05 is referred to. In particular, if the equivalent lateral force procedure is used, the vertical distribution of seismic forces should be calculated based on the procedure given in ASCE/SEl 7-05,... 

When distribution systems are connected to two or more points having different movements and applicable response spectra, a single response spectrum of a particular support point should be applied with care. To account for inertial effects, either an envelope spectrum or multiple spectra should be applied. However, the results are not always conservative and engineering judgement should be used in their evaluation. In the event that results are unreliable, methods should be used in which multisupport excitation is considered in conjunction with modal analysis. 

When the output of the numerical analysis is requested in terms of floor response spectra, maximum relative displacements, relative velocities, absolute accelerations and maximum stresses during an earthquake, linear dynamic analysis (e.g. direct time integration, modal analysis, frequency integration and response spectrum) is generally adequate for most models. Alternatively, non-hnear dynamic analysis should be used where appropriate or necessary (e.g. structural hft-off, non-linear load dependent support, properties of foundation materials in soil-structure interaction problems or interactions between solid parts). 

In the response spectrum method, the maximum response of each mode should be calculated by direct use of the design response spectrum. The maximum response in each principal direction should be determined by an appropriate combination of the modal maxima, such as the square root of the sum of the squares of each modal response, or by the complete quadratic combination procedure. For closely spaced modal frequencies, a conservative procedure should be applied by taking the sum of the absolute values of each closely spaced modal and rigid response. The missing mass as a function of the modelling detail, cut-off frequencies and modal participation factors used in the analysis should also be carefully assessed and documented. 

Reddy, D.V., Cheema, P.S., Swamidas, A.S.J., Ice Force Response Spectrum Modal Analysis of Offshore Towers , Proc. Third Int. Conf. on Port and Ocean Engg. under Arctic Conditions (POAC), Fairbanks, Alaska, 887-910, 1975. 

ABSTRACT The paper deals with the seismic linear response spectrum modification based on a probabilistic approach. The modification improves the correctness of the deterministic seismic analysis of building structures using the response spectrum method. In that case, the structure response depends strongly on computed natural frequencies. The proposed response spectrum modification accounts for uncertainties in response due to uncertainties in frequencies caused by variances of computation model parameters. The response spectrum is modified so, that with a selected probability the given spectral values should not be exceeded. The variance of natural frequency values is obtained by modal analysis of a stochastic model of the structure. 

Respons Spectrum Analysis of Structures Subjected to Seismic Actions, Fig. 2 Response histories of modal displacements (t) (Eq. 3)... 

Response Spectrum Modal Analysis in Seismic Design Codes... 

Response spectrum modal analysis has been presented, discussing that it is a simplified version of modal analysis and appropriate for stmc-tural design. The method allows the use of smooth design spectra for the assessment and the design of stractures. All concepts discussed are presented in a numerical example, while the adaptation of the method by modem design codes has been conceptually explained. 

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