Dynamic response of a two - story steel structure subjected to earthquake excitation by using deterministic and nondeterministic approaches

: An earthquake is a random phenomenon in its intensity and frequency content. Since the earthquake is a signal that contains a band of frequencies, each frequency has a di ﬀ erent energy. This means that the response of build - ings to earthquakes depends not only on the intensity of the earthquake but on its frequency content as well. In this study, two di ﬀ erent approaches have been used: deterministic approach which is the time history analysis to show how the intensity of earthquakes a ﬀ ects the building response, and the nondeterministic random vibration approach, which is to clarify the response in the frequency domain and to show the e ﬀ ect of dominant frequencies of the earthquake. Both a prototype and a 1:6 scaled model was used to simu - late a two - story steel building. In the experiential part, a shaking table was used to simulate a 1:6 scaled El - Centro 1940 NS earthquake as a base excitation with di ﬀ erent inten - sities ( 0.05, 0.15, and 0.32 g ) . In the theoretical part, Abaqus software was adopted to simulate the numerical model of the building. The results showed that the deterministic approach may be a non - conservative approach.


Introduction
Since earthquakes are completely random excitation processes, the analysis of a structure's seismic response must utilize the random vibration technique. However, random vibration methodologies have not been widely used in the research and design of structures due to the complexity of the analysis and low processing efficiency. As an alternative, time history analysis and the response spectrum approach have generally been used to study how buildings and other structures respond to seismic forces.
The randomness of ground motions requires the use of the random vibration theory in seismic response analysis. As a general rule, random vibrations can be divided into stationary and non-stationary types. If the cumulative averages for a random stimulation are not dependent on time, it is referred to as non-stationary. Due to simplicity, the designers adopted a stationary random vibration approach as a nondeterministic one [1]. It is essential for engineers to conduct a dynamic time history analysis of structures during the entire seismic excitation process in order to thoroughly investigate the structural energy dissipation characteristics and failure processes [2].
Multi-source random dynamic excitations can be identified by using the power spectral density (PSD) of dynamic responses and structural features, which is known as the socalled multi-input-multi-output problem for engineering structures subjected to numerous stationary random dynamic loads [3]. The second type of inverse problem has been extensively investigated and developed over the past few decades, especially focusing on the identification of dynamic loads in the frequency and time domains, as well as the identification of dynamic loads.
Depending on the randomness of the dynamic loading, the approach of load identification can be categorized as: (i) deterministic approach and (ii) nondeterministic or stochastic approach. In this study, the two approaches were used. The first one, deterministic approach, which is the time history analysis, was used to show how the intensity of earthquakes affects the building response. On the other hand, the second approach, which is the nondeterministic random vibration approach, was used to clarify the response in the frequency domain and to show the effect of dominant frequencies of the earthquake.
The analysis of multi-degree-of-freedom (MDOF) systems and buildings by using the nondeterministic approach had been studied by many researchers over the past 20 years. In 2002, Al-Baghdadi [4] studied the response of MDOF system subjected to a nonstationary stochastic ground motion. In this study, the formulation of the evolutionary correlation and PSD matrices was developed by using classical-complex model analysis approach and the effect of multiple-support excitation was considered. Li et al. [5] developed a pseudo-excitation method (PEM) to study the response of tall buildings subjected to seismic excitation by using the random vibration analysis. Fei et al. [6] investigated the structural systems subjected to stationary excitation with structural topology optimization orientation. He proposed an approach to transform the acceleration excitations in the base of the large mass system into force excitations. Rezayibana [1] adopted a PEM to analyze a MDOF system for different conditions of soil. On the other hand, several studies adopted the shaking table approach to evaluate the behavior and response of structures subjected to seismic excitations. Al-Baghdadi [7] studied the behavior of a 1:6 scaled two-story RC building under skew seismic excitations. A deterministic approach was adopted in the theoretical part, while a shaking table was developed by the author to cover the experimental part. A very good agreement between the theoretical and shaking table results, for both elastic and inelastic responses, were achieved. Liu et al. [2] studied the dynamic response and dynamic reliability assessment of a multi-story building under seismic excitations by using a shaking table in conjunction with a stochastic approach through a probability density evolution method. In 2020, a 1:4 scaled three-story steel structure was studied by Jing et al. [8] experimentally through a shaking table under biaxial base excitation. They also developed a numerical model to simulate the dynamic behavior of the system. The developed numerical model was verified depending on the experimental results.
In this study, three different approaches were adopted to study the dynamic behavior of a 1:6 scaled two-story building. The three approaches are: deterministic time history analysis approach, stationary stochastic analysis approach, and experimental shaking table approach.
2 Experimental work 2.1 One-sixth scale steel frame model A square-shaped steel building in plan was designed under the influence of gravitational and earthquake loadings. The prototype is two-floor building, each floor has four columns and four beams, on which the slab layer rests. The building was designed according to the American Institute of Steel Construction [9] requirements. The columns were chosen from HEA 320, the beams from IPE 400, the roof was made of concrete with thickness of 10 cm, with four floor beams of IPE 200. The steel modulus of elasticity of E = 200,000 MPa, Poisson's ratio of v 0.3 = , yield stress of 306 MPa, damping ratio ξ 2% = (this study considers damping as a constant damping), and a density of 7,850 kg m 3 / were adopted as physical parameters. As for the actual dimensions of the prototype building, each floor is 4.5 m high and 3.6 m between the columns from center to center, and with fixed support, as shown in Figure 1.
Then, the prototype building and the earthquake were modeled with a scaling factor S 6 L = (1:6), according to Harris and Sabnis 1999 [10] by the pi theory, as listed in Table 1. All dimensions and details of the model steel structure is shown in Figure 2.

Mass similitude
For accurate modeling of dynamic behavior, the model's mass similitude must be satisfied. Using constant acceleration scaling and the same material for the model, more mass must be added to compensate for the difference between the needed and given material densities, according to the similitude requirements listed in previous literature [10].
where Mm i is the additional mass to be added to the model's ith story, M pi is the prototype's ith story dead load, and Mom i is the own weight of the model's ith floor.
The mass added to the model for every story is shown in Figure 3.

Shake table
There is no other experimental equipment that tries to recreate the true nature of the earthquake input like shaking tables, hence they are important in earthquake engineering. Using a ground motion at the structure's base, they simulate realistic inertia forces over the entire mass of the structure. The displacements and strains that result from the response are caused by these forces. The shaking table motion has a one-directional horizontal excitation only with the ability of model skewness (Figures 4-6) ( Table 2).

Deterministic time-domain analysis
The Time-history analysis (THA) is a dynamic analytical technique in which structures can be linearly and nonlinearly analyzed. In this approach, the earthquake recording is a signal that varies in intensity over time. The complete dynamic equilibrium of MDOF systems can be defined by the following equation of motion [11][12][13]: where M is the mass matrix of the structure, C is the damping matrix, K is the stiffness matrix, X is the relative response vector, and Ug is the earthquake excitation vector. In order to make the MDOF system as a series of SDOF systems, Eq. (3) can be developed by superposing suitable amplitudes of the normal modes as follows: where ϕ x ( ) is the mode-shape vector and u t ( ) is the modal amplitude. Substituting Eq. (3) in Eq. (2) yields the following: To make the damping and stiffness matrices as a diagonal matrix, one can multiply Eq. (4) by ϕ T Then, by using the orthogonality conditions [11] and dividing Eq. (5) by the generalized mass (ϕ Mϕ i T i ), the following modal equation of motion may be expressed in an alternative form: where u i is an SDOF response which can be defined by using Duhamel's integral [12] as follows: where h t τ ϕ Mϕ ω e ω t τ 1 sin , and ω ω ξ 1 .    Response of a two-story steel structure subjected to earthquake excitation  5 The uncoupled force can be defined in Eq. (10) as follows: In which, Eq. (11) represents the participation factor Substituting Eq. (10) in Eq. (7) yields the following: Finally, the response of each story for all modes, X x t , ( ), can be defined by substituting Eq. (12) in Eq. (3)

Stationary random vibration analysis (SRVA)
A statistical description of the loading is the only way to describe it because it is nondeterministic. To make this characterization possible, one must make several assumptions. It is important that the statistical qualities do not change over time, even while the excitation does. The Fourier transform can be used to convert the domain of the equation of motion of a MDOF system form time domain to frequency domain (Eq. (6)), as follows [11]: where u ω i ( ) is an SDOF response in the frequency domain when the system is subjected to random ground excitation. The response can be expressed as follows: where H ω i ( ) is the transfer function (frequency response function) given as follows: The total response for each floor can be written in Eq. (17).
On the other hand, in order to represent the response as a PSD function in the frequency domain, the response can be written as follows: where the notation <> denotes mathematical expectation of a value. Therefore, the PSD output will be

S ω ϕ ϕ H ω H ω ΓΓ S ω ,
where S ω UÜ̈( ) is the PSD of the earthquake excitation which is defined in Eq. (21). Eq. (19) is known as the complete quadratic combination method. If the correlation between the parameters is eliminated, Eq.  shown in (Figure 7), and it was reduced to 0.15g, and then it was reduced to 0.05g, to show what happened if the intensity of the earthquake changed. The acceleration recording is modeled through time modeling, and the original time is divided by the square root of the scaling factor. As for the acceleration, it remains the same value recording to Table 1. Figure 8 shows the shape of the earthquake after modeling it.

PSD function model
The ground acceleration Kanai-Tajimi model has been widely employed in engineering structures under earthquake excitation analysis. A spectral density of the ground acceleration was idealized as a stationary randomness process [14,15] as defined in Eq. (21) The three parameters, namely S , o ω , g and ξ g represent the broad-band excitation level at the base, the system's natural frequency, and the damping to critical damping ratio, all normalized to one mass unit. The magnitude, frequency, and attenuation of seismic waves on the ground can all be taken into account while adjusting these parameters [14]. There are some difficulties with the Kanai-Tajimi model, and this problem appears in the low frequencies, especially in the displacement and velocity. A second filter, known as high pass filter function (HPFF), has been suggested to overcome the problem of low frequencies [11]. HPFF is necessary to correct possible drifting in the time of the first and second integral functions of U ẗ g ( ). where S o is the amplitude of the white-noise bedrock acceleration, ω g and ξ g are the frequency and damping ratio of the first filter related to the soil type, respectively; ω f and ξ f are the frequency and damping ratio of the second filter, respectively, which are applied to consider the ground acceleration. The double-filter function given in Eq. (22) has been used for ground acceleration. The optimal parameters, ω 15.6 rad s g = /, ξ 0.6 g = , ω ω 0.1 f g / = , and ξ ξ 1 f g / = , have been used, consistent with Kanai's suggestion for firm soil conditions. The white-noise intensity S o for the ground acceleration has been adjusted to be comparable with the intensity of the north-south component of acceleration of the 1940 El Centro earthquake, Figure 9. In other words, the variance of El-Centro ground acceleration shown in Figure 10   Response of a two-story steel structure subjected to earthquake excitation  7 In this study, three types of structure as shown in Figure  10 are used as a parametric study as follows: 1. Two-story steel structure with fixed support. 2. Four-story steel structure with fixed support. 3. Two-story L-shape steel structure (structure with irregularity in plan) with fixed support.
These three structures were modeled by using ABAQUS/ standard 2019 software.
The behavior of the steel structures under dynamic loadings for both approaches, deterministic (THA) and nondeterministic (SRVA), can be simulated. The models consist of five parts which are column, beam, added mass, stiffeners, and base plate. The models were designed to be within the elastic range. The steel modulus of elasticity of E = 200,000 MPa, Poisson's ratio of v 0.3 = , yield stress of 306 MPa, damping ratio ξ 2% = (this study considers damping of all structures as constant damping), and a density of 7,850 kg/m 3 were adopted as physical parameters. The solid element (C3D8R) 8-node linear brick, reduced integration, hourglass control was used to model the steel columns, beams, stiffeners, and the added mass [17]. The mesh size for columns, beams, and stiffeners is 12, 16, and 6 mm, respectively.
Two approaches were adopted in the analysis: deterministic dynamic modal analysis superposition method approach and dynamic SRVA. The predominant frequencies in both strong and weak axes for each structure are listed in Table 3. The corresponding mode shapes for the first and second predominant frequencies are shown in Figure 11.    Table 4).
After using three time-domain records and changing with different intensities (0.32, 0.15, and 0.05g), along weak direction (Figures 13, 15 and 17), it was noticed that the behavior of the response of the structure does  Response of a two-story steel structure subjected to earthquake excitation  9        not change significantly, but only the intensity of the response changes. This was also noticed for strong direction (Figures 12, 14 and 16). By comparing the response for strong and weak directions ( Figures  12-17), it was noticed that the response in both behavior and intensity changed significantly as the PSD function of the earthquake (shown in Figure 9) has different intensities (different energies) corresponding to the frequency content. SRVA approach can give a clear view on the effect of frequency content on the response as shown in Figures 18-21. This means that the change in the natural frequency of the structure from one direction to another, as well as the intensity of the response because the energy distribution on the frequencies content in the earthquake, is not stationary with frequency domain.

Conclusion
In this study, a 1:6 scaled down model has been adopted to idealize a two-story steel building prototype. Three types of earthquakes were used to study their impact on the steel structure with different intensities. From the previous results: 1. Analyzing structures in the frequency domain is more clear than in the time domain, giving an idea of which frequencies of the structure are involved in forming the response. 2. Random vibration analysis transforms the stochastic phenomenon into a scalar function by stochastic model, and also gives the concept of energy distribution on the frequencies contained in the earthquake by PSD function, and thus gives clarification when designing buildings around the frequencies that dominate the area, in order to avoid them when designing. 3. Changing the intensity of the earthquake does not affect the behavior of the response of the model, just the intensity of the response. Also, changing the natural frequency of the model affects the behavior of the model's response. 4. Due to different distribution of energy intensity on the frequency content in the earthquake, changing the natural frequency of the building from one direction to another will make the response of the structure differ in behavior as well as the intensity.

5.
The results showed that the deterministic approach may be a non-conservative approach.