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

Mustafa Qasim Dows; Hayder A. Al-Baghdadi

doi:10.1515/jmbm-2022-0261

Article Open Access

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

Mustafa Qasim Dows and Hayder A. Al-Baghdadi

Published/Copyright: July 3, 2023

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From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

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 different energy. This means that the response of buildings to earthquakes depends not only on the intensity of the earthquake but on its frequency content as well. In this study, two different approaches have been used: deterministic approach which is the time history analysis to show how the intensity of earthquakes affects the building response, and the nondeterministic random vibration approach, which is to clarify the response in the frequency domain and to show the effect of dominant frequencies of the earthquake. Both a prototype and a 1:6 scaled model was used to simulate 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 different intensities (0.05, 0.15, and 0.32g). 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.

Keywords: stationary random vibration analysis; time history analysis; earthquake excitation

1 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 so-called 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.

Figure 1

Prototype structure details.

Then, the prototype building and the earthquake were modeled with a scaling factor S L = 6 (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.

Table 1

Similitude requirements for the model structure [10]

Quantity	Symbol	Dimension	Scale factor
Geometry
Linear dimension	L	L	S L
Displacement	δ	L	S L
Frequency	ω	T − 1	S L − 1 / 2
Material properties
Modulus of elasticity	E	F L − 2	S E
Stress	σ	F L − 2	S E
Strain	ε	−	1
Poisson’s ratio	ν	−	1
Mass density	ρ	F L 4 T 2	S E / S L
Energy	EN	FL	S E S L 3
Loading
Force	Q	F	S E S L 2
Pressure	q	F L − 2	S E
Gravitational acceleration	g	L T − 2	1
Acceleration	a	L T − 2	1
Velocity	v	L T − 1	S L 1 / 2
Time	t	T	S L 1 / 2

Figure 2

Model structure details.

2.2 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].

(1) M m i = M p i S L 2 − M om i ,

where M m i is the additional mass to be added to the model’s i th story, M p i is the prototype’s i th story dead load, and M om i is the own weight of the model’s i th floor.

M m i = 253 kg .

The mass added to the model for every story is shown in Figure 3.

Figure 3

The model on the shaking table with added mass.

2.3 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).

Figure 4

The shaking table, accelerometer, and strain gages.

Figure 5

The LVDT of each story.

Figure 6

The model on the shaking table with the instrument.

Table 2

Specifications and characteristics properties of the shaking table

Specifications	Value
Weight of table (kg)	−1,000
Max. payload (kg)	2,000
Max. displacement (mm)	±120
Max. velocity (m/s)	1.0
Max. acceleration (g)	∼1
Frequency range up to	20 Hz

3 Analysis of MDOF systems

3.1 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]:

(2) M X ̈ ( t ) + C X ̇ ( t ) + KX ( t ) = − M U ̈ g ( t ) ,

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 U ̈ g 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:

(3) X ( x , t ) = ∑ i = 1 N ϕ i ( x ) u i ( t ) ,

where ϕ ( x ) is the mode-shape vector and u ( t ) is the modal amplitude. Substituting Eq. (3) in Eq. (2) yields the following:

(4) M ϕ u ̈ ( t ) + C ϕ u ̇ ( t ) + K ϕ u ( t ) = − M { 1 } U ̈ g ( t ) .

To make the damping and stiffness matrices as a diagonal matrix, one can multiply Eq. (4) by ϕ T

(5) ϕ T M ϕ u ̈ ( t ) + ϕ T C ϕ u ̇ ( t ) + ϕ T K ϕ u ( t ) = − ϕ T M { 1 } U ̈ g ( t ) .

Then, by using the orthogonality conditions [11] and dividing Eq. (5) by the generalized mass ( ϕ i T M ϕ i ), the following modal equation of motion may be expressed in an alternative form:

(6) u ̈ i ( t ) + 2 ξ i ω i u ̇ i ( t ) + ω i 2 u i ( t ) = − ϕ i T M { 1 } U ̈ g ( t ) ,

where u i is an SDOF response which can be defined by using Duhamel’s integral [12] as follows:

(7) u i ( t ) = ∫ 0 t h i ( t − τ ) F i ( τ ) d τ ,

where

(8) h i ( t − τ ) = 1 ϕ i T M ϕ i ω D i e − ξ i ω i ( t − τ ) sin ω D i ( t − τ ) ,

and

(9) ω D i = ω i 1 − ξ 2 .

The uncoupled force can be defined in Eq. (10) as follows:

(10) F i ( τ ) = ϕ i T M { 1 } U ̈ g ( t ) .

In which, Eq. (11) represents the participation factor

(11) Γ i = ϕ i T M { 1 } ϕ i T M ϕ i .

Substituting Eq. (10) in Eq. (7) yields the following:

(12) u i ( t ) = Γ i ω D i ∫ 0 t e − ξ i ω i ( t − τ ) sin ω D i ( t − τ ) U ̈ g ( t ) d τ .

Finally, the response of each story for all modes, X ( x , t ) , can be defined by substituting Eq. (12) in Eq. (3)

(13) X ( x , t ) = ∑ i = 1 N ϕ i ( x ) Γ i ω D i ∫ 0 t e − ξ i ω i ( t − τ ) sin ω D i ( t − τ ) U ̈ g ( t ) d τ .

3.2 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]:

(14) − ω 2 u i ( ω ) + 2 i ξ i ω i ω u i ( ω ) + ω i 2 u i ( ω ) = Γ i U ̈ g ( ω ) ,

where u i ( ω ) is an SDOF response in the frequency domain when the system is subjected to random ground excitation. The response can be expressed as follows:

(15) u i ( ω ) = Γ i U ̈ ( ω ) ( ω i 2 − ω 2 ) + ( 2 i ξ i ω i ω ) = H i ( ω ) Γ i U ̈ g ( ω ) ,

where H i ( ω ) is the transfer function (frequency response function) given as follows:

(16) H i ( ω ) = 1 ( ω i 2 − ω 2 ) + ( 2 i ξ i ω i ω ) .

The total response for each floor can be written in Eq. (17).

(17) X i ( ω ) = ∑ i = 1 N ϕ i H i ( ω ) Γ i U ̈ g ( ω ) .

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:

(18) S XX ( ω ) = lim T → ∞ 1 T 〈 | X i ( ω ) | 2 〉 ,

where the notation < > denotes mathematical expectation of a value. Therefore, the PSD output will be

(19) S XX ( ω ) = ∑ i = 1 N ∑ j = 1 M ϕ i ϕ m T H i ( ω ) H j * ( ω ) Γ i Γ j S U ̈ U ̈ ( ω ) ,

where S U ̈ U ̈ ( ω ) 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. (19) can be rewritten as Eq. (20), and it is known as the square root of the sum of squares [1].

(20) S XX ( ω ) = ∑ p = 1 q | H structure ( ω ) | 2 Γ p 2 ϕ p ϕ p T S U ̈ U ̈ ( ω ) .

4 Modeling of earthquake excitation

4.1 Time domain model

Three earthquakes were used, with a change in their intensity, where the original earthquake was 0.32g as 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.

Figure 7

El-Centro Earthquake 0.32g without scale.

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.

Figure 8

Scale down El-Centro Earthquake 0.32, 0.15, and 0.05g.

4.2 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)

(21) S U ̈ U ̈ ( ω ) = ω 4 + ( 2 ξ g ω g ω ) 2 ( ω g 2 − ω 2 ) 2 + ( 2 ξ g ω g ω ) 2 S o .

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 ̈ g ( t ) . Finally, Kanai-Tajimi double filter model can be expressed in Eq. (22) as follows [1]:

(22) S U ̈ U ̈ ( ω ) = 1 + 4 ξ g 2 ( ω / ω g ) 2 [ 1 − ( ω / ω g ) 2 ] 2 + 4 ξ g 2 ( ω / ω g ) 2 × ( ω / ω f ) 4 [ 1 − ( ω / ω f ) 2 ] 2 + 4 ξ g 2 ( ω / ω f ) 2 S o ,

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 g = 15.6 rad / s , ξ g = 0.6 , ω f / ω g = 0.1 , and ξ f / ξ g = 1 , 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 is 0.3539 m 2 / s 4 (area under the PSD curve). And S o = 0.0078 m 2 / s 3 [16].

Figure 9

Actual and theoretical PSD function of El Centro north-south component, 1940.

Figure 10

The three types of structures (parametric study).

5 Numerical work

In this study, three types of structure as shown in Figure 10 are used as a parametric study as follows:

Two-story steel structure with fixed support.
Four-story steel structure with fixed support.
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/m³ 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 3

Predominant frequencies for the prototype in both strong and weak axes

Structure	Mode	Frequency in strong axis (Hz)	Frequency in weak axis (Hz)
Two-story structure	1st	3.3072	2.3992
	2nd	10.419	6.9846
Four-story structure	1st	1.6487	1.0906
	2nd	5.2935	3.4295
	3rd	9.5868	5.8523
	4th	13.404	7.8814
Two-story L-shape structure	1st	3.0712	1.9225
Two-story L-shape structure	2nd	9.4584	5.619

Figure 11

The first mode shape for all types of structures.

6 Results and discussion

After making the model and testing it on the shaking table and taking the results practically, the results were compared by modeling the model in the Abaqus/2019 program for the strong and weak directions as shown in Figures 13, 15 and 17 for the weak direction under three time histories 0.32, 0.15, and 0.05g, and Figures 12, 14 and 16 for the strong direction and under the same time histories. The dynamic response of the THA approach (time domain) of the model was obtained as shown in Figures 12–17.

The building was analyzed on Abaqus/2019 by (SRVA) approach (frequency domain), the response was obtained as shown in Figures 18–21 for strong direction and for weak direction (Table 4).

Table 4

Results of the RMS comparison between deterministic and nondeterministic methods

Structure	Actual response				Kanai-Tajimi (Theoretical response)
	Variance of dis. (mm²)		RMS of dis. (mm)		Variance of dis. (mm²)		RMS of dis. (mm)
	F 1	F 2	F 1	F 2	F 1	F 2	F 1	F 2
Two-story/strong	5 . 4715	32 . 213	2 . 339	5 . 6756	10 . 673	63 . 0436	3 . 27	7 . 94
Two-story/weak	26 . 554	124 . 46	5 . 1532	11 . 1562	50 . 41	242 . 1136	7 . 1	15 . 56
Irregularity two-story	7 . 5963	38 . 737	2 . 756	6 . 224	15 . 887	81 . 18	3 . 986	9 . 01
Four-story	18 . 167	128 . 45	4 . 2623	11 . 33	31 . 95	227	5 . 652	15 . 066
	F 3	F 4	F 3	F 4	F 3	F 4	F 3	F 4
	302 . 2	456 . 45	17 . 384	21 . 364	535 . 43	810	23 . 14	28 . 46

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 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.

Figure 12

Dynamic response of El-Centro 0.32g for 1st and 2nd floors in Z-direction.

Figure 13

Dynamic response of El-Centro 0.32g for 1st and 2nd floors in X-direction.

Figure 14

Dynamic response of El-Centro 0.15g for 1st and 2nd floors in Z-direction.

Figure 15

Dynamic response of El-Centro 0.15g for 1st and 2nd floors in X-direction.

Figure 16

Dynamic response of El-Centro 0.05g for 1st and 2nd floors in Z-direction.

Figure 17

Dynamic response of El-Centro 0.05g for 1st and 2nd floors in X-direction.

Figure 18

PSD of displacement and acceleration for 1st and 2nd floors for 2-story structure.

Figure 19

PSD of displacement and acceleration for 1st and 2nd floors for 2-story structure.

Figure 20

PSD of displacement and acceleration for 1st and 2nd floors for irregular structure.

Figure 21

PSD of displacement and acceleration for the 1st, 2nd, 3rd, and 4th floors.

7 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:

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

Acknowledgments

We wish to express our gratitude to the Staff of the Structural Laboratory at the Department of Civil Engineering, University of Baghdad, for their valuable support.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2022-04-23

Revised: 2022-08-22

Accepted: 2022-09-24

Published Online: 2023-07-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/jmbm-2022-0261

Keywords for this article

stationary random vibration analysis; time history analysis; earthquake excitation

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