Determination of dynamic behavior of structures

For the determination of the dynamic behavior of structures under earthquake loading, four main seismic analysis methods can be distinguished:

1. Lateral force method (LFM);

2. Modal response spectrum analysis (MRS);

3. Nonlinear pushover analysis (NLPO);

4. Nonlinear time history analysis (NLTH).

The lateral force method and the modal response spectrum analysis perform a linear analysis, whereas the latter two methods perform a nonlinear analysis. In sections 2.4.1 to 2.4.4, the four main seismic analysis methods are elaborated. Afterwards, section 2.4.5 introduces different approaches to numerically model a structure to be analyzed.

2.4.1 Lateral force method

The lateral force method (LFM) is a linear analysis method, which can be regarded as the most simple seismic analysis method. The method may only be applied for buildings whose response is not significantly affected by contributions from higher eigenmodes, i.e. the response of the building should be dominated by the fundamental mode [21]. The different steps of the LFM are illustrated in Figure 2.17.

a) b) c)

Figure 2.17: Illustration of LFM.

The first step of the LFM is defining the seismic base shear force Fb, which depends on the struc-ture’s total mass, as well as on the spectral acceleration that corresponds to the natural period of the structure. This latter value can be read from the design spectrum. Then, the seismic base shear force is applied to the idealized structure as an equivalent static load. In case the structure is idealized as a SDOF system, the total base shear force Fb is acting on the single degree of freedom.

In case of a MDOF system, the total base shear force F_bis distributed over the different DOFs. The distribution of Fbis based on the masses of the different DOFs and their horizontal displacements in

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the fundamental mode. Although a MDOF structure has multiple eigenmodes, only the first eigen-mode is taken into account. This is a characteristic of the lateral force method. The displacement demand of the degrees of freedom, as a result of the seismic base shear force Fb that represents the earthquake load, can simply be determined using a static analysis.

By means of hand calculations, the lateral force method can be applied to quickly give an indication of the displacement demand of a structure. However, the lateral force method may not be used on its own, since it concerns a linear analysis and due to the fact that only the first eigenmode is considered. The analysis method may only be used to check the order of magnitude of results that are derived using a more sophisticated analysis method.

2.4.2 Modal response spectrum analysis

The modal response spectrum (MRS) analysis is a linear analysis method. This method must be used in case a structure does not meet the requirements of the lateral force method in terms of regularity. For such more complex structures, the contribution from higher modes can be significant and is therefore incorporated in the analysis. The different steps of the MRS analysis are illustrated in Figure 2.18.

a) b) c)

Figure 2.18: Illustration of MRS analysis.

Again, the seismic base shear force Fbon the structure is subjected to the different DOFs as equiv-alent static loads. In the MRS analysis, however, the equivequiv-alent static loads on the different DOFs are determined for all modes that are taken into account, based on the masses of the DOFs and their horizontal displacements in the included eigenmodes. The equivalent static loads that correspond to the different eigenmodes are then combined. The most common methods to combine such modal information are the Square-Root-of-Sum-of-Squares (SRSS) method and the Complete Quadratic Combination (CQC) method [39]. The displacement demand of the lumped masses, as a result of the application of the combined horizontal static loads that represent the earthquake load, can be determined using a static analysis.

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2.4.3 Nonlinear pushover analysis

The nonlinear pushover analysis (NLPO) is, as its name implies, a nonlinear analysis method. The NLPO analysis applies a similar static analysis as applied in both the LFM and the MRS analysis, however, in contrast to these two analysis methods, the NLPO analysis does take into account the nonlinear behavior of the structure. The different steps of the NLPO analysis are illustrated in Figure 2.19.

a) b) c) d)

Figure 2.19: Illustration of NLPO analysis.

In the NLPO analysis, an increasing seismic base shear force F_b is applied to the structure step-by-step. Similar as for the LFM and the MRS analysis methods, this force is distributed over the different DOFs, based on the masses of the different DOFs and their horizontal displacements in the fundamental mode. After each step, the displacements of the lumped masses are calculated.

This static calculation takes into account the formation of plastic hinges and the NL behavior of the structure. A capacity- or NLPO-curve is constructed for all DOFs. Also, the curve for the corresponding equivalent SDOF system is derived using the modal participation factor Γ.

Afterwards, the NLPO curve of the equivalent SDOF system is compared to the Acceleration Dis-placement Response Spectrum (ADRS). This ADRS-curve is constructed using the elastic response spectrum that corresponds to the earthquake that is expected at the location of the analyzed struc-ture. The intersection of the NLPO- and the ADRS-curve, which is referred to as the performance point, describes the seismic acceleration and displacement demand of the equivalent SDOF system as a result of the analyzed earthquake load.

2.4.4 Nonlinear time history analysis

The nonlinear time history analysis (NLTH) is, as its name implies, a nonlinear analysis method.

Besides the implementation of the nonlinear behavior of the analyzed structure, dynamic loading is applied in the analysis. The different steps of the NLTH analysis are illustrated in Figure 2.20.

Often, an extensive 3D model is made of the analyzed structure, however, less complex models or idealized systems may also be used as a representation of the structure. A transient analysis is then performed, which means that the modelled structure is subjected to loading that varies over time.

Hence, scaled accelerograms are used as a representation of earthquake loading. Due to the usage of accelerograms, the changes in magnitude and direction of the ground acceleration are included in

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a) b) Figure 2.20: Illustration of NLTH analysis.

the seismic analysis. The NLTH analysis method is therefore the most accurate method of the four mention methods, since the LFM, MRS and NLPO analysis methods make use of data from the response spectrum. Hence, in these methods the structure’s response is only based on the maximum value of the ground acceleration that is expected to occur at the specific location. An NLTH analysis yields the dynamic response of a structure over time.

2.4.5 Numerical modelling

For each seismic analysis method, a model is set up that represents the evaluated structure. The models can be constructed using different levels of detail and varying complexities. Often, numerical models are applied in order to obtain the dynamic response of a structure. For numerical modelling, two approaches are generally used: micro-modelling and macro-modelling [40]. In the first approach, the structure is discretized into small finite elements, as shown in Figure 2.21a. To all these micro elements, detailed nonlinear hysteretic constitutive properties are assigned. The second approach represents the structure by the fewest number of finite elements possible, as shown in Figure 2.21b.

All macro elements represent a structural component, to which material properties are assigned.

These material properties, such as the backbone curve, are typically derived through calibration with experimental data or they are constructed using codes or guidelines.

a) micro-modelling b) macro-modelling

Figure 2.21: Numerical modelling approaches [41].

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