Damping effect on seismic input energy and its verification by shake table tests

Abstract

Seismic input energy per unit mass (E_I/m) imparted into a structure is a function of earthquake (duration, frequency content, amplitude etc.), soil (shear velocity, dominant period etc.) and the structural (vibrational periods etc.) characteristics. Generally, the damping properties of the structure is assumed negligible for seismic input energy. Most of the existing spectral equations derived for SDOF systems generally use a constant damping ratio of 5%. In this study, the damping effect on E_I/m is investigated experimentally and numerically on SDOF systems with distinct damping ratios. Experimental investigation and numerical computations proved that seismic input energy is very sensitive to variation of damping within the vicinity of fundamental frequencies. Specifically, up to 50% increment was observed in the plateau region of the input energy spectrum, where maximum E_I/m values occur, by variation of damping from 2% to 10%. Hence, a novel damping modification factor (DMF), which could be utilized for existing energy spectra, is proposed in this paper. Validation studies of the proposed DMF are achieved through the various energy spectra found in the literature.

Keywords

damping effect energy balance equation energy-based design input energy spectrum

Introduction

Energy-based methodologies combine not only the features of the conventional design procedures, but also includes some critical parameters of earthquake (duration, frequency content etc.) and the behavior of structural systems (hysteretic behavior of the members etc.). Akiyama (1985) derived energy equations in the time domain, as described in equation (1), where M, C, and F(u) correspond to mass, damping, and restoring force characteristics, respectively. The variables $u, \overset{\cdot}{u}, \overset{\cdot\cdot}{u}$ and ${\overset{\cdot\cdot}{u}}_{g}$ are relative displacement, velocity and acceleration of the system, as well as ground acceleration, respectively. The terms on the left-hand side of the equation correspond to kinetic (E_k), damping (E_d), and strain (E_s) energies, respectively. The rightmost term is the total seismic input energy (E_I).

M \int^{} \overset{\cdot\cdot}{u} \overset{\cdot}{u} dt + C \int^{} {\overset{\cdot}{u}}^{2} dt + \int^{} F (u) \overset{\cdot}{u} dt = - M \int^{} {\overset{\cdot\cdot}{u}}_{g} \overset{\cdot}{u} dt

(1)

Several prediction equations and attenuation relations were also suggested in the literature to obtain seismic input energy (Akiyama, 1985; Alıcı and Sucuoğlu, 2016; Benavent-Climent et al., 2002; Chai et al., 1998; Chou and Uang, 2000; Decanini and Mollaioli, 2001; Dindar et al., 2015; Fajfar et al., 1989; Güllü et al., 2019; Kuwamura and Galambos, 1989; Kuwamura et al., 1994; Lopez-Almansa et al., 2013; Mezgebo and Lui, 2016; Samimifar et al., 2019, 2020; Uang and Bertero, 1990; Wang et al., 2016).

Although the equation of motion of a SDOF system, that is the base for the derivation of the energy equations, could be sensitive to structural damping, most studies consider a constant damping ratio to calculate the seismic energy terms. In order to show the variations in the spectral velocity (SV) and spectral acceleration (SA) curves, which are the effective parameters for seismic input energy, 1992 Erzincan Earthquake (Erz-EW) ground acceleration record was plotted for 0%, 2%, 5%, and 10% damping ratios, as shown in Figure 1. The spectra (SA and SV) generated for diverse damping ratios have dissimilar shapes especially for some specific period bands (around 0.30, 0.66, and 1.67 s) those may correspond to low- and mid-rise structures (Güler et al., 2008).

Figure 1.

(a) Velocity and (b) acceleration spectra of Erz-EW record.

Zahrah and Hall (1984) calculated the input energies of eight earthquake records by considering the various ductility levels and damping ratios. The study resulted in damping ratio having insignificant effect on the overall input and hysteretic energies imparted into the structure. Akiyama (1985) proposed a ratio between energy equivalent velocity contributing to damage (V_D) to energy equivalent velocity (V_E), as shown in equation (2) with ξ being the damping ratio. The study declares that V_E does not change by damping whereas amount of V_D in V_E quite sensitive to the damping.

\frac{V_{D}}{V_{E}} = \frac{1}{1 + 3 ξ + 1.2 \sqrt{ξ}}

(2)

Bruneau and Wang (1996) numerically investigated the contributing factors in the energy balance for simple rectangular pulse and sine-wave ground excitations. The values of energy terms at 0.5, 1, and 10 s after the pulse loading were compared for 0% and 2% damping ratios. They concluded that damping ratios had minor effect on the input energy. Sütçü et al. (2006) proposed a damping device design methodology for buildings with soft story based on energy balance equations. Proposed methodology assumed a uniform inter-story displacement distribution and 7% Rayleigh damping, which is considered ineffective for seismic input energy. Benavent-Climent and Zahran (2010) developed a methodology to evaluate the existing frames in terms of their seismic energy components. Seismic input energy was considered to be slightly affected by the strength of the structure, insignificantly affected by the configuration of restoring force characteristics and scarcely effected by the fraction of damping ratio. Li et al. (2014) evaluated dynamic amplification factor for progressive collapse design of RC frame structures using energy concept. In the study, contribution of structural damping energy is neglected.

Bertero and Uang (1988) constructed input energy spectra from eight different earthquake records for three damping ratios (0%, 2%, and 5%) while considering a constant displacement ductility of 5. The study revealed that greatest effect of damping is obtained for Mexico City 1985 record due to its long duration and harmonic nature. Khashaee et al. (2003) performed numerical analyses for the inelastic systems with constant displacement ductility and variable stiffness ratios. Different acceleration records as well as damping ratios were evaluated in the study. It was concluded that although the effect of damping ratio less than 5% is minor, damping ratios greater than 5% had significant effects on the input energy. Additionally, it was stated that, for longer periods, as the damping ratio increases, higher E_I were obtained. Ye et al. (2009) proposed an energy-based seismic design framework by considering the damping ratio. Their study concluded that decreasing seismic input energy was obtained by the increment of damping ratio and ductility. Moreover, a damping modification factor based on 2% damping was also proposed in the study, as shown in equations (3) and (4), where μ is the displacement ductility and NE is the normalized seismic energy.

\begin{matrix} ζ = \frac{E_{Im ax, ξ, μ}}{E_{Im ax, ξ = 0.02, μ = 1}} \\ = (0.6845 - \frac{0.6393}{μ} - 0.0882 \ln ξ - \frac{0.1517}{μ} \ln ξ) μ^{- 0.57} \end{matrix}

(3)

N E_{Im ax, ξ, μ} = (ζ + 0.05) \times N E_{Im ax, ξ = 0.02, μ = 1}

(4)

Alıcı and Sucuoğlu (2016) predicted the seismic input energy by scaling relation between pseudo spectral velocity (PSV) and energy equivalent velocity (V_E) by considering 2%, 5%, and 10% damping ratios. For the other damping ratios, using scaling factors given in current seismic codes was also suggested in this study. Güllü et al. (2019) performed experimental and numerical studies on SDOF specimens with 0.3% damping ratio. The experimental results were compared with the existing seismic energy equations in the literature. The comparisons showed that there was considerable amount of disagreements. It was considered to be related with the structural damping. Based on these studies, an equation was proposed to estimate seismic input energy per unit mass by accounting the damping properties, as shown in equation (5). Where, SV and SA are spectral velocity and acceleration, T_c is corner period, I_e is intensity, and t_e is effective duration of earthquake as well as ξ is damping ratio of structure. The terms B and k are related with soil properties.

\frac{E_{I}}{m} = B \sqrt{SV {(ξ)}_{max} SA {(ξ)}_{@ S V_{max}} T_{c} I_{e} t_{e}} {(T_{c} / T)}^{k}

(5)

Zhou et al. (2019) proposed a design input energy spectrum for self-centering single degree of freedom systems by means of a typical flag-shape response behavior. Influence of ground motion characteristics, hysteretic model, damping ratio, and ductility factor were considered as the main parameters of the study. It was found that the damping ratio and ductility factor had a significant effect on the resulted spectra.

The above-mentioned recent studies implied that damping is effective on seismic energy contrary to common understanding in the early literature. Since most existing E_I/m spectra were generated for constant damping ratio (classically 5%), there is a need for modifying for various damping ratios other than 5% as well.

In this paper, the damping effect on the seismic input energy was investigated by experimental and numerical studies using SDOF systems having different damping properties. A damping modification factor (DMF) for input energy per unit mass spectrum was proposed since damping, especially within the fundamental frequency, was found to be sensitive to the variation of damping. The calibration of the proposed DMF was carried out with spectral equations of Güllü et al. (2019) and Alıcı and Sucuoğlu (2016).

Shake table experiments

Effects of damping ratios (0.3% and 3.0%) are discussed herein for three discrete SDOF systems having vibrational periods of 0.385 s (SDOF #1), 0.526 s (SDOF #2), and 0.667 s (SDOF #3). The specimens were tested on the uni-axial shake table test setup using SS-CIG acceleration record, (Bertero and Uang, 1988). The record was selected after the preliminary evaluation of command and feedback properties of the shake table. Since the record has a relatively smoother frequency content, command/feedback ratio was close to unity. It was scaled by a factor of 0.25 to make sure that the system remains in the elastic range since elastic E_I/m spectrum is the envelope for the inelastic one (Akiyama 1985). Therefore, PGA of the applied acceleration data became close to 0.2g.

The tested steel cantilever columns consisted of three parts. The identical bottom and top parts were utilized to connect the column (middle piece) to the shake table and the supplementary mass, as shown in Figure 2(a). The middle part was the test specimen that had a rectangular box cross section with 40 mm in depth, 80 mm in width, and 2 mm in thickness. Height of the specimen was 1500 mm. Total heights of the specimens became different due to the additional weights (Güllü et al., 2019). They were determined as 1.725, 1.775, and 1.795 m. The pieces were connected to each other with steel plates having 10 mm of thickness and 150 × 150 mm of plan dimensions. The specimens were fixed to the shake table.

Figure 2.

Experimental study: (a) the test specimens on the shake table and (b) measurement system.

Two groups of specimens were tested on the shake table. The first group consisted of free cantilever and they were called as non-instrumented. The second group had a special type of spring damping device that was attached to the specimens and they were called as instrumented.

The measuring system consisted of accelerometers, strain gauges and linear potentiometric transducers (LPTs), Figure 2(b). Tri-axial accelerometers with ±2g capacity were utilized on the shake table and top of the specimens. The received raw acceleration data was filtered by using a Butterworth band pass filter with cut-off frequencies of 0.1 and 18 Hz. LPTs were utilized to measure the displacements of the shake table. Two strain gauges were attached on the bottommost section and both faces of the columns. Top displacement of the specimens was computed through the procedure described in Güllü et al. (2019).

In the standard coupon tests, yielding strain and strength, and ultimate strain of the steel were obtained as 0.16%, 350 MPa and 22%, respectively, as shown in Figure 3.

Figure 3.

The coupon test results.

Free vibration characteristics of the specimens were determined by using alternative test methods such as white noise, step function and hammer tests. Fourier amplitude spectra of non-instrumented and instrumented specimens were plotted for the top acceleration, as shown in Figure 4. Fourier amplitudes were normalized with its own maximum value to reach comparable graphs. It could be stated that the lateral stiffness of the specimens was almost unaffected by the existence of the damping device since dominant frequencies were quite close to each other. Maximum relative difference between the dominant frequencies was 2.32% for SDOF #2.

Figure 4.

Free vibrational characteristics of the specimens: (a) SDOF #1, (b) SDOF #2, and (c) (a) SDOF #3.

Contrary, the test results exposed that damping properties of the specimens were significantly affected from the existence of the damping device, as shown in Figure 5. Although the damping ratio of the non-instrumented specimens was determined as ξ = 0.3%, the damping ratio of the instrumented specimens was about ξ = 3%. They were determined by both half band width and logarithmic decrement methods.

Figure 5.

Free vibration responses of the specimens.

The effect of damping ratio on the energy response is presented in Figure 6 based on the experimental results. E_I/m and its components were determined to be judiciously different for SDOF #2 and SDOF #3 specimens. On the other hand, the comparable seismic input energies were obtained for SDOF #1 specimen. For the instrumented specimen, E_d was almost equal to the seismic input energy. Specifically, all the input energy was dissipated by damping at the end of the record. However, the input energy became higher than E_d for non-instrumented specimens because of inevitable long-term free vibration response after the execution of the record. Hence, some part of elastic energy (E_k/m, E_s/m) still existed at the end of the record.

Figure 6.

Comparison of the experimentally obtained energy components: (a) non-instrumented and (b) instrumented.

Seismic input energy intensities calculated from the experimental data were normalized with masses of the specimens in order to be able to compare spectra constructed through piece-wise method (Güllü and Yüksel, 2019), as shown in Figure 7. Power spectrum of the applied record was also depicted in this figure with black dotted line.

Figure 7.

Comparison of input energy spectra with experimental results and power spectrum.

The figure revealed that: (i) experimental results satisfactorily matched with the constructed spectra and (ii) damping had an important effect on the dominant periods of the records. Relative differences in resulted energy response for varying damping cases were found to be 5.28%, 33.84%, and 39.64% for the specimens SDOF #1, SDOF #2, and SDOF #3, respectively.

Numerical modeling and analyses

Since the experimental test results on specimens with distinct damping ratios have proved that the damping could be a dominant parameter on the energy response, numerical validations were performed to investigate the effects of higher damping values and longer period ranges. The tests and their corresponding numerical analyses were performed in the elastic range. Frame type elements in the software was utilized to represent the specimens.

Reproduction of the experimental results

The experimentally obtained top acceleration responses of SDOF #1 were compared with the numerical results for the non-instrumented (ξ = 0.3%) and the instrumented (ξ = 3.0%) cases as shown in Figure 8(a) and (b), respectively. Similarly, the experimental and numerical seismic energy graphics of SDOF #3 are presented for two discrete damping cases, as shown in Figure 8(c) and (d).

Figure 8.

The experimental and numerical responses of the SDOF systems: (a) acceleration response for ξ = 0.3%, (b) acceleration response for ξ = 3%, (c) energy response for ξ = 0.3%, and (d) energy response for ξ = 3%.

It is seen from the above figures that the numerical results are in good agreement with experimental work.

Response history analyses

After assessment of the numerical models, 0%, 2%, 5%, and 10% damped SDOF systems with ten specific natural periods of 0.159, 0.277, 0.333, 0.385, 0.454, 0.500, 0.526, 0.667, 1.000, 1.250, 1.667, and 2.050 s (Güllü et al. 2019) were analyzed for the earthquake records (taken from PEER NGA database) given in Table 1 where M_w is moment magnitude.

Table 1.

Earthquake records utilized in the numerical study.

Record	Symbol	M _w	d (km)	PGA (m/s²)	PGV (m/s)
Erzincan 03/13/1992 Erzincan S.	ERZ-EW	6.90	8.97	4.866	0.643
Duzce_Turkey 1999 Bolu	BOL090	7.14	12.04	7.903	0.658
Chi-Chi, Taiwan 09/20/1999 Cwb 99999 Tcu067 S.	TCU067	7.62	28.70	3.188	0.666
Darfield_ New Zealand 2010 GDLC	GDLCN55W	7.00	1.22	7.500	1.160
Kobe_ Japan 1995 Takatori	TAK090	6.90	1.47	6.583	1.229
Bam_Iran 2003 Bam	BAM-L	6.60	1.70	7.923	1.241
Northridge-01 1994 Sylmar-Converter Sta East	SYL360	6.69	5.30	8.273	1.293

The E_I/m intensities (10 periods × 4 damping ratios = 40 points) that were computed by means of numerical models were illustrated in Figure 9 together with power amplitude spectra of the records. The power amplitude was included into the figures in order to show the dependency of the damping effect to the frequency content of the record. Left ordinates of the graphs represent E_I/m intensity while the right one corresponds to power amplitude.

Figure 9.

The effect of damping on the E_I/m intensities.

The following results could be driven from Figure 9.

Damping had a significant effect on the seismic input energy (E_I/m) for the systems with fundamental periods of the earthquake record.

Seismic input energy was barely affected by damping except for the vicinity of dominant periods of earthquake record.

Although the seismic input energy decreased by the increment of damping ratio within main peaks of power spectrum of the record, there was no direct correlation for the other periods which were close to fundamental periods. Hence, the damping effect should be considered in order to determine the accurate input energy (EI/m) spectrum.

Hence, based on the experimental and numerical studies presented in this paper, there is a need for a damping modification factor (DMF) to convert input energy spectrum generated for a specific damping ratio to another form with a different damping ratio.

Development of new damping modification factor for seismic input energy

A novel damping modification factor (DMF) for seismic input energy spectrum is proposed based on extensive response history analyses of SDOF systems. Non-pulse like and pulse-like records (totally 69) were selected from PEER NGA database and grouped according to their V_s30 values, as shown in Table 2.

Table 2.

Earthquake records utilized for DMF proposal.

Set #	Non-pulse like record	V_s30	Pulse-like record	V_s30
1 0 < V_s30 < 179	IMPVALL.A_A-E03140	162.94	IMPVALL.H_H-E03140	162.94
	SUPER.A_A-IVW360	179.00	TOTTORI_TTR008NS	139.21
	LOMAP_LKS360	169.72	PARK2004_C02090	173.02
	CHICHI_CHY076-E	169.84	PARK2004_COW360	178.27
	DUZCE_ATS300	175.00	DARFIELD_REHSS88E	141.00
	WHITTIER.B_B-WAT180	160.58	CCHURCH_REHSN02E	141.00
	TOTTORI_SMN002NS	138.76	CCHURCH_REHSS88E	141.00
2 180 < V_s30 < 359	HUMBOLT_FRN225	219.31	COYOTELK_G02050	270.84
	IMPVALL.BG_B	213.44	IMPVALL.H_H-ECC002	192.05
	ELALAMO_ELC270	213.44	WESMORL_PTS315	348.69
	NCALIF.AG_B-FRN314	219.31	SUPER.B_B-KRN360	266.01
	BORREGO_A-PEL180	316.46	LOMAP_G03000	349.85
	FRIULI.A_A-COD270	249.28	NORTHR_RRS228	282.25
	TABAS_FER-T1	302.64	KOBE_TAK000	256.00
3 360 < V_s30 < 759	HELENA.B_B-FEB090	551.82	SANSALV_GIC090	489.34
	KERN_PAS270	415.13	LOMAP_STG090	380.89
	SCALIF_SLO324	493.50	NORTHR_SYL360	440.54
	SFERN_L01021	425.34	KOCAELI_ARE000	523.00
	DURSUN.BEY_DUR–L	585.04	CHICHI_TCU076-E	614.98
	IMPVALL.H_H-CPE237	471.53	DARFIELD_LPCCN80E	649.67
	ANZA_PFT135	724.89	DUZCE_487-NS	690.00
4 760 < V_s30 < 1499	SANFRAN_GGP100	874.72	TABAS_TAB-L1	766.77
	HOLLISTR_A-G01157	1428.14	LANDERS_LCN260	1369.00
	LOMAP_PTB297	1315.92	KOCAELI_GBZ000	792.00
	NORTHR_VAS000	996.43	KOCAELI_IZT090	811.00
	DUZCE_1060-E	782.00	KOCAELI_IZT180	811.00
	TOTTORI_HYG007NS	760.54	LOMAP_LEX000	1070.34
	CHUETSU_GIFH11NS	904.15	LOMAP_LEX090	1070.34
5 1500 < V_s30	CHICHI_HWA003-N	1525.85	SFERN_PUL164	2016.13
	CHICHI_HWA003-W	1525.85	SFERN_PUL254	2016.13
	CHICHI.05_HWA003N	1525.85	NORTHR_PAC175	2016.13
	CHICHI.06_HWA003N	1525.85	NORTHR_PAC265	2016.13
	CHICHI.06_HWA003W	1525.85	NORTHR_PUL104	2016.13
	TOTTORI.1_YMGH06EW	2100.00	NORTHR_PUL194	2016.13
	TOTTORI.1_YMGH06NS	2100.00

Seismic energy per unit mass spectra were constructed for SDOF systems having natural vibrational periods between 0 and 10 s and damping ratios of 0% to 20%. Increments for period and damping ratio were 0.01 s and 0.1%, respectively. Hence, totally 69 × 1001 × 201 ≈ 13.8 × 10⁶ response history analyses were performed owing to the integrated algorithm PW-SPECTs (Aydınoğlu and Fahjan, 2003; Güllü, 2018) that construct response and energy spectra using piece-wise methods.

Since the records were not scaled to avoid any deterioration in the frequency and energy content, each record had different form and amplitude in the energy spectra. Therefore, spectral values obtained from the varied damping ratios were normalized with the seismic energy calculated for 5% damping ratio. The resultant spectrum is nominated as “normalized spectrum” throughout the paper. Hereafter, maximum values of the normalized spectra at the fundamental period, that corresponds to the highest peak in the power spectrum, for each damping ratios were determined. It was realized that they had a concave form with the increment of damping ratio, as shown in Figure 10. It was seen that normalized maximum input energy decrease by the increment of damping ratio.

Figure 10.

General representation of damping versus normalized input energy relation.

The graph representing damping vs. normalized input energy relation had two discrete parts: a quadratic part between 0% and 5% damping ratio and linear part beyond 5% damping ratio. Given the disparity of the two distinct curve functions, it would be more appropriate to define one unique equation for the whole curve. For this purpose, a new DMF is proposed in equation (6), where λ is a scalar value to be determined from the statistical analyses. Since most of the existing E_I/m spectra in the literature were generated for 5% damping ratio, the proposed DMF was also designed for the ratio.

DMF = \frac{λ + 5}{λ + ξ}

(6)

The following procedure is applied to determine λ values for each group of earthquakes (2 types × 5 sets) given in Table 2.

Step 1: Compute “real” normalized mean spectra of the selected records, see Figure 10.

Step 2: Assign a value to λ between 0 and 10 with the increments of 0.01.

Step 3: Calculate DMF by equation (6).

Step 4: Multiply DMF by “real” normalized mean spectra at 5% damping ratio.

Step 5: Calculate “converted” normalized mean spectra for the other damping ratios.

Step 6: Calculate residuals between “real” and “converted” normalized mean spectra.

Step 7: Calculate standard deviations of the residuals.

Step 8: Select λ value that yields minimum standard deviation.

Variation of standard deviations against λ are depicted in Figure 11(a) and (b) for specific cases, and the other λ values are given in Figure 11c. Average values for non-pulse like and pulse like records were 3.26 and 4.90, respectively.

Figure 11.

(a) and (b) Selection of λ values and (c) resultant λ values: (a) set 1: non-pulse like records, (b) set 2: pulse like records, and (c) λ value for each record group.

Based on the statistical analyses, global λ values are proposed as 3 for non-pulse like records and 5 for pulse-like records. Thus, equation (6) converts to the following forms, as shown in equations (7a) and (7b). Since, it was aimed to reach a simplest equation representing the whole group, some amount of discrepancy is inevitable. One may easily produce separate DMFs for distinct soil and record types.

DMF = \frac{8}{3 + ξ}

(7a)

DMF = \frac{10}{5 + ξ}

(7b)

To evaluate validity of the proposed equations (7a) and (7b), “real” and “converted” normalized input energy spectra were compared in Figure 12 for each group of earthquakes (2 types × 5 sets) given in Table 2.

Figure 12.

Application of the proposed DMF to the assigned record sets.

The “converted” normalized mean spectra attained by the proposed DMF had a good agreement with the “real” case. The divergence had tendency to increase for higher damping ratios and V_s30 values.

Application of the new DMF to the existing input energy spectra

The effectiveness of the proposed DMF was evaluated through limited data existing in the literature. Two independent comparative procedures were applied for this purpose. They were;

Comparison #1: It is a comparison between the results of the target spectra by Güllü et al. (2019) and the proposed DMF, as well as the one suggested by Ye et al. (2009).

Comparison #2: It is a comparison between the target spectra by Alıcı and Sucuoğlu (2016) and the proposed DMF.

The essential steps of Comparison #1 are listed in the following paragraphs.

Step 1: The E_I/m spectrum was generated for 5% damping ratio by using the spectral equation of Güllü et al. (2019). The spectrum was converted to other spectra with 2% and 10% damping ratios by means of the proposed DMF. The results were compared with the corresponding target spectra.

Step 2: The E_I/m spectrum was generated for 2% damping ratio by using the spectral equation of Güllü et al. (2019). The spectrum was converted to the spectra with 5% and 10% damping ratios by means of the DMF suggested by Ye et al. (2009). The results were compared with the corresponding target spectra.

The outcomes of Comparison #1 are given in Figure 13. In the figure, blue solid lines represent the target spectra by Güllü et al. (2019) while black dashed lines denote results of the DMF suggested by Ye et al. (2009). Finally, red dashed dot lines are results of the proposed DMF.

Figure 13.

Evaluation of the proposed DMF against the spectra and the DMF by Ye et al. (2009): (a) ξ = 2%, (b) ξ = 5%, and (c) ξ = 10%.

It is clearly seen that the plateau of the seismic input energy spectra decreases by the increment of damping ratio for all the earthquake records.

Another evaluation process of the two DMFs was achieved by the comparison of peaks (plateau region) of “target” and “converted” spectra. The obtained maximum relative differences respect to the “target” are 19% and 36% for the proposed and the literature DMFs, respectively. Averages of the differences for all earthquakes were 11.1% and 13.7%, respectively. For 10% damping ratio, DMF proposed by Ye et al. (2009) generally estimates the seismic energy conservatively, while the proposed DMF (equation (7)) resulted in underestimated predictions for some records.

Comparison #2: The proposed DMF was applied to the input energy spectra suggested by Alıcı and Sucuoğlu (2016) that was based on PSV and V_eq. The equation used for the calculation of seismic input energy is given in equation (8). The parameters of the equation were originally presented in the paper.

V_{eq} / PSV = a e^{- bT} + c

(8)

The pseudo spectral velocity (PSV) of the records given in Table 1 were computed. Hereafter, V_eq was extracted from equation (8). Then E_I/m of the earthquake records were computed. Hence, three independent spectra having 2%, 5%, and 10% damping ratios were generated for each earthquake record.

The energy spectra corresponding to 2% and 10% damping ratios were estimated by the proposed DMF. In this process, 5% damped spectrum of Alıcı and Sucuoğlu (2016) was taken as a base, as shown in Figure 14.

Figure 14.

Comparison of the proposed DMF with Alıcı and Sucuoğlu (2016) spectra.

The spectra attained by using the proposed DMF shown reasonably good agreement with the “target” spectra. The possible reason for the divergence between target and converted spectra at some period bands could be attributed to damping effect on PSV.

The comparative works yield that the DMF presented in equation (7) are reasonably effective to predict input energy spectra for diverse damping ratios. It would be utilized as an efficient tool in the estimation of seismic input energy spectrum for damping ratios different from 5%.

Conclusions

The damping effect on seismic input energy was investigated through the experimental and numerical studies. Similar to the recent studies about the topic (Güllü et al., 2019; Zhou et al., 2019), damping was found to be effective on the imparted seismic energy. Hence, an easy applicable damping modification factor (DMF) was proposed to predict seismic input energy spectra having damping ratios rather than 5%. The following conclusions has been driven:

Seismic input energy is independent or scarcely affected from the variation of damping for some period bands corresponding to low power amplitudes. However, there are severe differences in the seismic input energy around the fundamental periods.

It was not observed a proportional relation between damping and seismic input energy. Based on the power spectrum of the records, seismic input energy is decreased by the increment of damping on the fundamental periods. However, the reverse relation is also experienced for the periods having lower power amplitudes and in the vicinity of fundamental period.

Since frequency content of the record is the decisive parameter for the damping effect, damping characteristics of the structural system should be considered in the true determination of seismic input energy.

The ultimate values (plateau region) of seismic input energy computed by the equation of Güllü et al. (2019) decreased about 50% by the increment of damping ratio from 2% to 10%.

New damping modification factors (DMFs) are proposed for non-pulse like and pulse-like records. The mean relative difference between the calculated and converted spectra is about 5% in the damping range of 0%–10%. It might be utilized as an efficient tool owing to its easy formulation and adaptability to the existing seismic input energy equations.

Discrepancies between “real” and “converted” normalized mean spectra that observed for relatively high damping and V_s30 values can be diminished by defining two discrete functions for the segments of the spectra.

It was obtained acceptable results in the comparisons made between the products of new DMFs and two spectral equations given in the literature (Alıcı and Sucuoğlu, 2016; Güllü et al., 2019).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was conducted at the Structural and Earthquake Engineering Laboratory of Istanbul Technical University (STEELab). It was financed by the research project of ITU BAP 39305. Their support is gratefully acknowledged.

ORCID iD

Ahmet Gullu

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