Simplified Life Cycle Cost Estimation of Low-Rise Steel Buildings Using Fundamental Period

Abstract

In the current study, a simplified seismic life cycle cost (LCC) estimation procedure is proposed utilizing the mean values of the structure’s main input variables. The main input variables of the building are used for constructing a relationship between the structural fundamental period (T) and an average estimation of the LCC (LCCavg). Using the actual building properties related to damage probability, the T–LCCavg relationship is used to obtain the final LCC (LCCfin). The equivalent single degree of freedom (ESDOF) model and SAC-FEMA framework are utilized for damage probability calculation. The dispersion measure in demand is approximately calculated based on the mean plus one standard deviation of the seismic hazard response spectrum, and, then, verified through nonlinear time history (NLTH) analyses of the original structure. Five and three-story steel buildings are used as case studies for verification of the proposed method. The analysis results indicate that the proposed procedure provides reasonable LCC estimations for low-rise buildings dominated by the fundamental mode of vibration.

1. Introduction

The advancement in PBEE (performance-based earthquake engineering) in the last two decades has made great leaps in terms of quantifying and assessing the performance of buildings in a manner suitable for decision-makers. This can be attributed to the robust, yet complicated, probabilistic models, and the advancements in structural analysis and computational capabilities. On the other hand, the role of the structural engineer is challenged because of the complexity inherited in the process itself [1]. To this end, the need for simple and reliable methods suitable for loss assessment and LCC estimation is justified. Limited studies have been conducted on simplifying the LCC of building structures, which makes it an attractive field to investigate for researchers and engineers. Many of the recent studies in LCC are dedicated to the assessment and optimization of LCC. For example, Mahmoud and Cheng [2] provided a probabilistic framework for evaluating different design approaches when considering LCC. Two steel buildings with different heights subjected to different seismic and wind intensities were used as case studies. Cutfield et al. [3] presented a life cycle cost analysis of a conventional and base-isolated steel braced frame office building. The study uses the FEMA P-58 approach to assess the impact of moat wall pounding and business interruption. Matta [4] proposed a multi-hazard LCC approach as an efficient alternative for Cost-effective assessment of a TMD. Noureldin et al. [5] evaluated the cost-effectiveness of retrofitting steel structures using steel slit dampers and shape memory alloy-based hybrid dampers. Recently, Torti et al. [6] proposed an approach based on life cycle cost analysis to assess the effectiveness of equipping bridges with seismic structural health monitoring systems. Gidaris and Taflanidis [7] described a probabilistic, simulation-based framework for estimating life cycle cost and developed a stochastic search approach to support an efficient optimization under different design scenarios. Esteva et al. [8] provided an overview of the general framework supporting the optimum life cycle-based engineering decisions to examine the relationship between the target safety level and the required maintenance of the structure. Vitiello et al. [9] developed a semi-probabilistic approach to assess structures economically. The approach addresses the cost-effective retrofitting strategies of structures during their lifetime. Noureldin and Kim [10] proposed a parameterized LCC evaluation method of structures considering stiffness, strength, and ductility.

Risk assessment is considered an important step in any LCC methodology and simplifying this step will reduce the computational cost required for LCC calculations. Recently, some simplified methods have been proposed for risk assessment. For example, Fragiadakis and Vamvatsikos [11] used pushover analysis and approximate incremental dynamic analysis to propose a fast performance uncertainty-estimation method. Another important study was conducted by Dolsek [12], who proposed an approximate method for seismic risk assessment taking into account various uncertainties. The method was demonstrated using an example of a four-story reinforced concrete frame. Fajfar and Dolsek [13] proposed a practice-oriented probabilistic approach for the seismic performance assessment of building structures using the SAC-FEMA approach and the pushover-based N2 method. Sullivan et al. [14] discussed some simplified tools for risk assessment, such as the simple definition of collapse fragility, based on FEMA P-58 guidelines [15], and applied it to a benchmark 4-story building. Welch et al. [1] proposed a simplified probabilistic loss assessment methodology that builds on a direct displacement-based framework. The methodology was tested via an examination of two RC frame buildings. Kosič et al. [16] utilized a deterministic structural model with a probabilistic equivalent SDOF model to introduce an approximate seismic risk assessment framework. Kosič et al. [17] proposed dispersion values for a practice-oriented pushover-based method for the estimation of failure probability for eight selected examples. Noureldin and Kim [18] developed a simple methodology for seismic life cycle cost (LCC) estimation for a steel jacket offshore platform structure. In their study, a localized incremental dynamic analysis and an approximate fragility curve are used. In their study, Liu and Mi [19] proposed a new approach for analyzing the costs of energy-efficient buildings over their lifespan, called LCC analysis. The method probabilistically predicts the levels of seismic damage in energy-efficiency building features and includes the costs of repairing or replacing those features in the analysis. Recent studies in earthquake engineering [e.g., [20,21,22,23,24,25]] places an emphasis on the importance on the seismic assessment of different types of structures. This necessitates proposing new approaches of seismic assessment.

The objective of the current study is to estimate the life cycle cost of a building structure using its fundamental period as the main input parameter. The probability assessment proposed by the SAC-FEMA method in a closed-form is utilized. The proposed LCC procedure is used for estimating the seismic LCC of three- and five-story steel buildings. The accuracy of the proposed method is assessed by comparison with the results obtained from the conventional method based on NLTHA. Establishing a relationship between the fundamental structural period and the expected LCC will be useful both in new and existing structures. For new structures, during the preliminary structural design stage, where different lateral resisting systems are examined, it is required to estimate LCC quickly to finalize the structure system. The simplest parameter that can be used in this case is the building fundamental period (T). In this case, using an LCC–T relationship will be useful for judging the cost-effectiveness of each structural system. For existing structures, the natural period of the structure can be changed during its service life; for example, in the case of the installation of new non-structural elements, experiencing inelastic deformation, retrofitting of the existing structural system, or considering the effect of soil-structure interaction. Mwafy and Elnashai [26] showed that the natural period can be doubled when regular frames experience inelastic deformation. Soil-structure interaction may reduce the global system stiffness [27] of the building, which may cause period elongation of the superstructure. The change in the fundamental period of the structure will, consequently, lead to a change in the total seismic LCC. The proposed procedure will provide a simple and reliable estimation tool for the seismic life cycle cost of the structures that will be readily available for designers and decision-makers in the preliminary design of structures.

The current study addresses a crucial need in the field of LCC estimations of structures by focusing on simplification rather than assessment. This study stands out from previous studies by proposing a novel approach of using only one factor, the fundamental period, to estimate LCC approximately, instead of the multiple factors used in previous studies. Furthermore, this study establishes a graphical relationship between LCC and the fundamental period of the structure, providing a valuable tool for structural designers, particularly at the early stages of design where information about the structure is limited. This makes the current study a significant contribution to the field of LCC and sets it apart from previous studies on LCC.

2. Framework for Constructing LCC Curve

The proposed framework is based primarily on establishing a relationship between the fundamental structural period (T) and the expected LCC for a certain hazard area. This relationship will be configured as a curve relating the T and LCC (LCC curve). This LCC curve is similar to the response spectrum curve because it gives an average estimate of the LCC based on the average values of the main parameters affecting the LCC.

2.1. The Steps of the Proposed Procedure

Figure 1 shows the proposed procedure, which is summarized as follows:

• Modal analysis of the original MDOF structure is conducted and the basic dynamic properties are extracted. Three different hazard levels corresponding to 75, 500, and 2500 years return periods are selected based on the site-specific hazard curve along with their response spectra. The inelastic responses of different SDOF systems with different periods (T = 0.5, 1.0, 1.5, 2.5, etc.) are estimated using a nonlinear static technique such as the capacity spectrum method [28] or the N2 method [29,30].
• Damage state probability is computed to estimate the LCC. Cornell et al. [31] provide the following equation to compute the damage state probability $P_{Ls}$:

$$P_{Ls}=H\left(S_a^{\widehat{c}}\right)exp\left[\frac{1}{2}\frac{k^2}{b^2} \left({\beta }_{D{s}_a}^2+{\beta }_C^2\right)\right]$$

(1)

where $S_a^{\widehat{c}}$ is median drift capacity spectral acceleration (which can be obtained from the approximate fragility curve); $H\left(S_a^{\widehat{c}}\right)$ is the probability that the intensity in a site will exceed or equal to $S_a^{\widehat{c}}$ on an annual basis; k is the slope of the hazard curve on a log–log plot; b is the slope of the demand–intensity relationship on a log–log plot; β_D|s is the standard deviation of the natural logarithm of the drift demands for a given S_a; and β_c is the standard deviation of natural logarithm for the median capacity of drift C (assumed 0.3 considering previous studies, e.g., [31]).

After obtaining the probabilities of damage states, the estimated LCC can be given as [32]

$$E\left[C_{Lc}\right]=C_o+{{\displaystyle \int }}_0^{L}E\left[C_{SD}\right]{\left(\frac{1}{1+\lambda }\right)}^{t}dt=C_o+\propto L E\left[C_{SD}\right]$$

(2)

where C_o is the initial cost of the construction, L is the expected life span (service life) of the building, λ is the rate of return (or annual discount rate), and E[C_SD] is the damage cost of the building on an annual basis. A and E[C_SD] can be formulated as:

$$\alpha =1-\frac{exp\left(-qL\right)}{qL}$$

(3)

$$q=ln\left(1+\lambda \right)$$

(4)

$$\left[C_{SD}\right]={\displaystyle \sum }_{i=1}^N{C}_i{P}_i$$

(5)

where N represents the considered limit states, P_i is the i^th damage state probability of the building during its service life, and C_i is the i^th damage state cost including damage and its repair. Three different limit states are used, which are IO, LS, and CP [33]. C_i is taken as a fraction of the initial cost (0.3, 0.7, and 1.0, respectively, for IO, LS, and CP limit states; [34,35]). P_i is given by:

$$P_i=P\left({\Delta }_D>{\Delta }_{C,i} \right)-P\left({\Delta }_D>{\Delta }_{C,i+1} \right)$$

(6)

where Δ_D is the maximum inter-story drift that a building experiences under an earthquake excitation (demand), and Δ_C,i is the maximum inter-story drift that corresponds to the capacity of the structure at a the i^th damage state (capacity). The probability of demand is greater than capacity, ${\Delta }_D>{\Delta }_{C,i}$, is evaluated as discussed in the previous step.

After that, an average LCC graph is constructed, which relates T (the fundamental period of the structure) with the corresponding LCC. T represents the stiffness of the structure and is usually used for the design or retrofit of structures [36]. The LCC graph is based on calculating the mean inter-story drift ratio (MIDR) of the multi-degree-of-freedom (MDOF) system using the building characteristics (structural variables) such as height, modal participation factor, and modal amplitude at the roof. All the parameters are adjusted to the most common mean values of building structures. Each point in the graph represents the average LCC for a particular fundamental period. This graph can be adjusted for any other building characteristics using a simple algorithm or electronic spreadsheet tool. The following equation is used for adjusting the MIDR of the equivalent single degree of freedom (ESDOF) system:

$$D_{MDOF}=s_{SDOF\left(inel\right)}\frac{\Gamma .{\phi }_r}{\mathsf{H}}$$

(7)

where $\Gamma$ is the modal participation factor; ${\phi }_r$ is the modal amplitude at the roof of the fundamental mode shape; and H is the height of the structure.

Some parameters such as drift, seismic hazard regression coefficients, and damping ratio, have a lognormal distribution considering previous studies [37,38]. However, other parameters such as initial cost and limit state repair costs have ranges and coefficients of variations mentioned in previous studies [37,39,40], but no specific statistical distribution is found in literature for them.

A probabilistic relationship between the response of the MDOF system and its corresponding ESDOF system needs to be established to quantify the uncertainty in obtaining the system response using the SDOF system. This probabilistic relationship is investigated by Jeong and Elnashai [41] for different structural systems and can be expressed as

$${\overline{D}}_{max}^{MDOF}=1.13 {\overline{D}}_{max}^{ESDOF}$$

(8)

where ${\overline{D}}_{max}^{MDOF}$ and ${\overline{D}}_{max}^{ESDOF}$ are the mean maximum seismic drift demand of the MDOF and ESDOF systems, respectively.

Next, similar graphs (T–LCC) for the main cost and damage probability variables, e.g., dispersion in demand, dispersion in capacity, hazard-curve slope factor, service time, etc., are constructed. In each graph, the mean, the upper bound (mean plus one standard deviation) and lower bound (mean minus one standard deviation) of a variable are considered, while the other input variables remain at their mean values. Some variables can be assumed based on previous studies such as b and β_c. For moment frames, b and β_c can be assumed as 1.0 and 0.3, respectively [31], and k and β_D|s can be obtained from the hazard curve and the hazard response spectra, respectively. Based on these parameter values, the average LCC (LCC_avg) is estimated using the fundamental period of the structure and the average LCC graph. Similarly, from the (T–LCC) curves of the main variables, the difference in LCC between the LCC_avg and the corresponding LCC of the value of the variable can be obtained at the same T. If the LCC is larger than LCC_avg, the difference is added to the LCC_avg.. If the difference is smaller than LCC_avg, then it is subtracted from LCC_avg. This difference will be used for estimating the final LCC of the building.

After obtaining the difference in LCC (LCC_dif) from the (T–LCC) graphs of the main variables, the final estimate of the LCC is determined by

LCC_final = LCC_avg ± LCC_dif.

(9)

2.2. Median Drift Demand and the Drift Capacity Intensity

The relationship between the drift demand (D) and the hazard intensity (S_a) can be obtained accurately using NLTHA. Based on that, the responses of the building are obtained for different levels of intensity measured and a regression relationship is established to consider record-to-record variability. Using power-law distribution [42], the relationship can be expressed as,

$$\widehat{D}=a{(S_a)}^b$$

(10)

where $\widehat{D }$ is the median drift demand; $S_a$ is the elastic spectral acceleration (a measure of ground motion intensity); and a and b are the regression coefficients for linear regression of drift demand D at the intensity $S_a$ in logarithmic space. The most accurate method to determine the regression coefficients a and b for an ESDOF system is by conducting a nonlinear dynamic time history response analysis (NLTHA) using all available ground motion records. This analysis should be used to plot the relationship between the drift demand and the intensity, and, then, a regression analysis should be performed on the natural logarithm of the drift demand versus the natural logarithm of the intensity. A simpler alternative may be used such that the mean response spectrum for each set of records is used to calculate the drift demand by using a non-linear static approach (e.g., the N2 method or the capacity spectrum method) based on the equivalent SDOF system. This means that three drift demands will be available to draw the relationship between the drift demand (D) and hazard intensity (S_a) in logarithmic space. From this relationship, a and b are obtained. It needs to be mentioned that two conditions are required to guarantee the accuracy of the procedure at this stage. The first is having an elastic-perfectly plastic force-deformation relationship for the equivalent SDOF system (i.e., bilinear idealization of the capacity curve of the original MDOF system). The second is the applicability of the equal displacement rule (i.e., the seismic demand of the inelastic SDOF system is equal to the seismic demand of the corresponding elastic SDOF system with the same period). It is worth mentioning that the precision of determining regression coefficients by utilizing the mean response spectrum based on a nonlinear static method may be low due to the following reasons:

• Limited applicability: Nonlinear static analysis is typically only applied to regular structures and may not be suitable for irregular or complex structures;
• Simplistic input motion: Nonlinear static analysis uses a simplified input motion, typically a lateral load pattern that simulates the expected ground motion. This simplification may not accurately represent the actual ground motion that the structure will experience during an earthquake;
• The actual drift of the structure during an earthquake may be different from the performance-based drift obtained from non-linear static analysis, as the actual ground motion may not be as severe as the load pattern used in the analysis, and the structure may not reach its maximum capacity. Additionally, the actual ground motion may be more complex than the load pattern used in the analysis, and the structure may experience multiple load cycles.

The median drift capacity intensity $S_a^{\widehat{c}}$ corresponding to each limit state can be calculated from the relationship between the drift demand (D) and the hazard intensity (S_a), explained above. Using the assumed limit state values (for IO, LS, and CP) and the slope of the (D–S_a) relationship, the corresponding median drift capacity intensity $S_a^{\widehat{c}}$ can be obtained for each limit state. Assuming that the hazard curve is linear on a log–log space, the relationship between $s_a$ and $H\left(s_a\right)$ can be approximated by the following form [31]:

$$H\left(s_a\right)=P\left[S_a\ge s_a\right]=k_o{s}_a^{-k}$$

(11)

where $s_a$ is the elastic spectral acceleration (the intensity measure); $H\left(s_a\right)$ is the corresponding annual probability of exceeding $s_a$, $k_o$ and $k$ are the linear regression coefficients on a log–log graph.

2.3. Dispersion in the Seismic Demand

In the proposed method, two alternatives are proposed for calculating the dispersion in demand ${\beta }_{D|s_a}$. One is based on generating the elastic mean and the mean plus one standard deviation response spectra, which can be generated from the smooth elastic response spectrum for the case of a firm soil profile using amplification factors recommended by Newmark and Hall [43] and illustrated by Chopra [44]. Using the N2 method [18,45] and the elastic spectral accelerations from both the elastic mean and the mean plus one standard deviation response spectra, the inelastic responses of the SDOF system can be calculated. The difference between the two responses indicates the standard deviation in demand, which is β_D|Sa. This alternative is used in the proposed method because of its simplicity, where only the natural period of the structure is required. The second alternative is based on conducting a response spectrum analysis on the MDOF structure using the elastic mean and mean plus one standard deviation response spectra. The difference between the two responses defines β_D|Sa.

In the current study, a deterministic SDOF model is utilized and the other epistemic and aleatoric uncertainties are treated simply without using sampling techniques such as Monte Carlo simulations and Latin hypercube sampling. This adds an advantage to the proposed framework in the current study compared, for example, with other studies such as work from Kosič et al. [16,17], which requires many Monte Carlo simulations with Latin hypercube sampling at the level of the SDOF model. Another advantage of the proposed framework is through using the main input variables of the building and implementing them to adjust the mean LCC to obtain the final LCC. This is different than the study conducted by Fajfar and Dolsek [13], where all the variables required for the probabilistic loss estimation calculation are assumed.

3. Case Study Structures and Seismic Hazards

3.1. Design and Modelling of the Case Study Structures

To validate the proposed simplified LCC procedure, three- and five-story steel buildings were used as the case study structures. The steel structures were special moment resisting frames, having a typical story height of 4000 mm and a first story height of 5500 mm. A span of 6000 mm is considered for each bay in both directions. The configuration of the case-study models is shown in Figure 2 (one of the exterior frames was selected for the analysis). The gravity loads used for design were 4.1 kN/m² and 2.5 kN/m², respectively, for dead and live loads. The beams and columns of both structures were W-shaped sections. The steel yield stresses used for beams and columns were 250 and 345 MPa, respectively.

Plastic hinges used for conducting NLTHAs were introduced at the end of the columns and beams to account for the inelastic behavior of the members according to FEMA-356 [46]. Figure 3a shows the bending moment vs. rotation angle relationship of the flexural members. The level of deformations corresponding to each limit state according to FEMA-356 [46] is shown in Figure 3b. The figure shows that between the first yield (point B) and collapse (point C) there were three different levels, which were immediate occupancy (IO), life safety (LS), and collapse prevention (CP). SAP2000 [47] was used for conducting NLTHAs of the model structures including the automatic definition of the limit states related to each cross section of the structure based on ‘Tables 5 and 6’ in FEMA-356 [46]. A modal damping ratio of 2% of the critical damping was used for nonlinear static and dynamic analyses.

3.2. Seismic Hazards and Earthquake Ground Motions

A site-specific hazard curve with various peak ground accelerations was used. A location in Los Angeles with S_DS = 1.46 and S_D1 = 0.737 with soil type D was used in the study. The response spectra and the hazard curve of the site were obtained from the USGS [48] and shown in Figure 4.

Thirty earthquake records, three sets of ground motions corresponding to 75, 500, and 2500 years return periods as shown in

Table 1. Selected ground motions for different hazard levels.

Return Period	No.	NGA #	Earthquake Event	Year	Station	Mag.	Vs (m/Sec)
75 YR	1	31	Parkfield	1952	Cholame—Shandon	6.19	385.43
	2	20	Northern California	1954	Ferndale City Hall	6.5	219.31
	3	68	San Fernando	1971	LA—Hollywood Stor FF	6.61	316.46
	4	138	Tabas, Iran	1978	Boshrooyeh	7.35	324.57
	5	161	Imperial Valley	1979	Brawley Airport	6.53	208.71
	6	163	Imperial Valley	1979	Calipatria Fire Station	6.53	205.78
	7	173	Imperial Valley	1979	El Centro Array #10	6.53	471.53
	8	172	Imperial Valley	1979	El Centro Array #1	6.53	237.33
	9	175	Imperial Valley	1979	El Centro Array #12	6.53	196.88
	10	176	Imperial Valley	1979	El Centro Array #13	6.53	249.92
500 YR	11	6	Imperial Valley	1940	El Centro Array #9	6.95	213.44
	12	158	Imperial Valley	1979	Aeropuerto Mexicali	6.53	259.86
	13	161	Imperial Valley	1979	Brawley Airport	6.53	208.71
	14	165	Imperial Valley	1979	Chihuahua	6.53	242.05
	15	169	Imperial Valley	1979	Delta	6.53	242.05
	16	170	Imperial Valley	1979	EC County Center FF	6.53	192.05
	17	171	Imperial Valley	1979	El Centro—Meloland Geot	6.53	264.57
	18	164	Imperial Valley	1979	Cerro Prieto”	6.53	202.85
	19	174	Imperial Valley	1979	El Centro Array #11	6.53	196.25
	20	178	Imperial Valley	1979	El Centro Array #3	6.53	162.94
2500 YR	21	6	Imperial Valley	1940	El Centro Array #9	6.95	213.44
	22	30	Parkfield	1966	Cholame—Shandon	6.19	289.56
	23	15	Kern County	1966	Taft Lincoln School	7.36	256.82
	24	33	Parkfield	1966	Temblor pre-1969	6.19	527.92
	25	57	San Fernando	1971	Castaic—Old Ridge Route	6.61	450.28
	26	95	Managua Nicar	1972	Managua_ESSO	6.24	288.77
	27	125	Friuli, Italy	1976	Tolmezzo	6.5	505.23
	28	126	Gazli, USSR	1976	Karakyr	6.8	259.59
	29	139	Tabas, Iran	1978	Dayhook	7.35	471.53
	30	158	Imperial Valley	1979	Aeropuerto Mexicali	6.53	259.86

, were selected from the Pacific Earthquake Engineering Research NGA database [49] and used in previous studies [50,51,52,53,54]. The earthquake records were scaled such that the peak ground accelerations of response spectra geometric mean and the target spectra are the same (Figure 5). Scale factors less than four were used to maintain the original record characteristics [55]. Some earthquake events were repeated for different return periods in Table 1 because the same earthquake event in the NGA data base can be used again by multiplying it by a different scale factor to obtain similar peak ground accelerations of both the earthquake response spectrum and the target spectrum for a specific return period.

4. Seismic LCC Curves of the Case Study Structures

Figure 6 shows the relationship between the fundamental period and the average LCC for structures having a different number of stories with the same number of bays, story height, and mass per floor as the case study buildings. This graph was obtained after conducting a small-scale parametric study on these structures. To construct these graphs, the fundamental mode shapes were normalized to have a modal mass equal to 1.0 for all buildings. The modal participation factor and the fundamental modal amplitude on the roof were calculated from the modal analysis. Each building curve on the graph was adjusted based on the aforementioned characteristics. The figure shows that if the natural period of the structure increases, the LCC increases exponentially. In other words, if the frame system experiences more ductility demand, this causes the LCC to increase. This is an interesting observation because, commonly, allowing the system to be more ductile is considered an advantage for the system since it allows for more energy dissipation. However, on the other hand, it causes larger damage to the structure and increases overall. This necessitates the structural designer to make a preliminary LCC estimation in the early design stage using the basic structural information available (such as the fundamental period), as proposed in the current study. This requires designers to make an early estimate of the LCC using basic structural information, such as the fundamental period, as proposed in the current study.

Figure 7 shows the average LCC graph for the three-story and five-story buildings with an indication of the building’s fundamental period and the corresponding average LCC. Figure 8 and Figure 9 show the mean and the upper and lower bounds of the LCC curves corresponding to the variation in the selected variables for the three- and five-story buildings. As shown in these figures, the upper and lower bounds of the LCC curves are, sometimes, inversely proportional to the upper and lower bounds of the parameter value itself. This is true because some parameters have an inverse relationship with LCCs. For example, increasing the damping ratio will result in more energy dissipation in the system, which causes smaller lateral displacement and, consequently, leads to less LCC estimation.

Table 2. Mean and covariance of the parameters used for the LCC estimation [56].

	k	b	BD|s (75, 500, 2500)yr	βc	Co	t	λ	Ci	η	Limit State (IO, LS, CP)
Mean	2.5	1.0	(0.4,0.5,0.6)	0.3	-	40	0.03	0.3,0.7,1.0	0.03	(0.0,0.02,0.03)
COV	0.1	0.05	0.3	0.3	0.3	0.3	0.3	0.3	0.5	0.3

shows the mean value and coefficient of variation (COV) used for all variables required to estimate the LCC. All references for these values can be found in [56].

Table 3. Summary of the parameters used in LCC estimation of the five-story structure.

Parameter	LCCavg Using an Average Value for All Parameters (1)			LCCfinal Proposed Method Adjusted to 5 Story Building (2)			LCCdif The Difference in LCC (3)–(1)	LCC Adjusted to a Particular Parameter (3)
Limit state	LS1	LS2	LS3	LS1	LS2	LS3	-	-
MIDR (%)	1	2	4	1	1.5	2	+353	2158
t (years)	40			30			−122	1683
${\mathrm{S}}_{\mathrm{A}}^{\widehat{\mathrm{c}}}$ (g)	0.107	0.215	0.430	-			-	-
βD|Sa	0.4	0.5	0.6	0.53	0.37	0.27	+255	2060
H (Sa )	3.82c 10−2	6.75 × 10−3	1.19 × 10−3	-			-	-
ko	0.0001444						-	-
k	2.50			2.26			−381	1424
b	1.0			1.0			-	-
βc	0.3			0.3
P(LS |Sa ) %	8.35%	1.95%	0.49%	-
Pi (%)	6.39%	1.47%	0.49%
Co ($)	1000 (Assumed reference value)
Ci ($)	300	700	1000				Assuming 0.3, 0.7 and 1.0 of Co, respectively.
λ	0.03			0.03
q	0.03			0.03			=ln(1 + λ)
α	0.66			0.66			=1 − exp(−ql)/ql
(LCC − Co) ($)	805			810			additional cost
LCC ($)	1805			1910 [=1805 + 353 – 122 + 255 − 381]			-
LCC from NLTH and MDOF (LCCMDOF)	1630

and

Table 4. Summary of the parameters used in LCC estimation of the three-story structure.

Parameter	LCCavg Using an Average Value for all Parameters (1)			LCCfinal Proposed Method Adjusted to 5 Story Building (2)			LCCdif The Difference in LCC (3)–(1)	LCC Adjusted to a Particular Parameter (3)
Limit state	LS1	LS2	LS3	LS1	LS2	LS3	Difference in LCC	Calculate LCC
MIDR (%)	1	2	4	1	1.5	2	+396	2299
t (years)	40			30			−137	1766
${\mathrm{S}}_{\mathrm{A}}^{\widehat{\mathrm{c}}}$ (g)	0.128	0.256	0.512	-			-	-
βD|Sa	0.4	0.5	0.6	0.54	0.54	0.23	+340	2243
H (Sa)	4.29 × 10−2	7.58 × 10−3	1.34 × 10−3	-			-	-
ko	0.000252						-	-
k	2.50			2.38			−235	1668
b	1.0			1.0			-	-
βc	0.3			0.3
P(LS|Sa) %	9.36%	2.19%	0.55%	-
Pi (%)	7.17%	1.65%	0.55%
Co ($)	1000 (Assumed reference value)
Ci ($)	300	700	1000				Assuming 0.3, 0.7 and 1.0 of Co, respectively.
λ	0.03			0.03
q	0.03			0.03			= ln(1 + λ)
α	0.66			0.66			= 1 − exp(− ql)/ql
(LCC − Co) ($)	903			1267			additional cost
LCC ($)	1903			2267 [=1903 + 396 – 137 + 340 − 235]			-
LCC from NLTH and MDOF	2147

show the parameters used in the estimation of the LCC. One case is for estimation of the LCC while all parameters are maintained at their mean values. The other case is for estimation of the LCC with some parameters adjusted to the building under investigation (three or five-story building). For each adjusted parameter, the LCC is re-calculated, and the difference is shown in the table. If the difference has a positive sign, as in the case of the limit state, the calculated LCC resulting from this change gives a larger value than the LCC_avg, and the opposite is true. The parameters different from the mean are indicated in the table in bold letters. These parameters have been adjusted to the original building case.

The dispersion in demand parameter β_D|Sa has been calculated by conducting full NLTH analyses on the original structure using the 30 time history records. The two alternative calculations for β_D|Sa, which were explained earlier, were compared with the case of the NLTH analysis of the MDOF system as shown in

Table 5. Comparison of β_D|Sa obtained from NLTHA, RSA, and the proposed method.

Model	Limit State (Return Period)	βD|Sa Based on NLTHA Using the MDOF	βD|Sa Based on RSA Using the MDOF	βD|Sa Based on the Proposed Method Using the ESDOF
	IO (75 yr)	0.58	0.54	0.54
3 story	LS (500 yr)	0.56	0.53	0.54
	CP (2500 yr)	0.56	0.21	0.23
	IO (75 yr)	0.61	0.55	0.53
5 story	LS (500 yr)	0.57	0.37	0.37
	CP (2500 yr)	0.57	0.27	0.27

for the three different return period spectra. As can be observed in the table, the alternative method provides acceptable results, especially for the 75 and 500-year return period earthquakes. For the three-story building, the percentage differences between the proposed method and the NLTH case using the MDOF system turned out to be 7.4% and 3.7% for the case of the 75 and 500-year return period earthquakes, respectively. The differences were 15.0% and 54.0%, respectively, for the five-story building. For the case of the 2500-year return period earthquake, the difference was found to be considerably large for the two buildings. It is worth mentioning that the probability of exceedance of the CP limit state is generally small, as shown in Table 3 and Table 4; the values being 0.49% and 0.55 for the five and three-story buildings, respectively. These small values will overshadow the inaccuracy in calculating the dispersion demand associated with the 2500-year return period.

Figure 10 shows the mean and the mean plus one standard deviation response spectra used for calculating β_D|Sa. In this figure, the mean plus one standard deviation response spectra were obtained by multiplying the mean response spectra by amplification factors obtained from previous studies [43,44]. These factors were 2.7, 2.3, and 2.1, respectively, for acceleration, velocity, and displacement sensitive regions of the elastic response spectra with a 5% damping ratio assuming lognormal probability distribution for the spectral ordinates. Figure 11 shows the difference in the probability of exceeding the limit state P(LS|_Sa) for both the proposed method and the NLTH reference method. It can be observed that the proposed method provides generally reasonable results for both buildings for the limit states one and two (i.e., IO and LS, respectively). The difference in the five-story building was found to be 0.96%, 0.25%, and 0.38%, respectively, for the IO, LS, and CP limit states. For the three-story building, the differences were 3.84%, 1.28%, and 1.19%, respectively, for the IO, LS, and CP limit states.

It can be observed in Table 3 and Table 4 that the effect of some parameters on the LCC was more significant than others. For example, in the five-story building, the LS and k had more effect on the LCC compared to β_D|Sa and t, while in the three-story building the LS and β_D|Sa had more effect on the LCC compared to the k and t. In the five-story building, the percentage difference in LCC between the proposed method and the reference method was 17.2%, while in the three-story building this difference reduced to 5.6% (refer to Figure 12). This indicates that the proposed method provides an upper bound for the LCC for both cases with less margin in the case of the three-story building and that the method is applicable for low-rise buildings. This may be attributed to transforming the MDOF to the ESDOF system, wherein accuracy depends on the dominance of the fundamental mode. In addition, it was observed that the change in a lower limit state, such as IO, has more effect on the LCC compared to the change in a higher limit state, such as CP. On the other hand, the structure service life (t) was found to have a limited effect on the LCC; 7.2% and 6.8% of the LCC for the three and five-story buildings, respectively.

It is worth mentioning that the main purpose of the proposed framework is to easily predict the LCC of any building with acceptable accuracy while considering some limitations. For example, it is suitable for low-to-medium-rise regular buildings in which dynamic responses are dominated by the first vibration mode. Additionally, it is suitable for buildings where the equal displacement rule can be applied. Moreover, the framework assumes common values for some variables required for calculating damage probability and cost items. If more specific values of these variables are known, then they can be easily implemented in the framework.

The outcome of this study has the potential to greatly benefit both researchers and end users in the field of structural design and LCC analysis. For researchers, the proposed simplification of LCC estimations using only one factor, the fundamental period, opens up new opportunities for further investigation and advancements in the field. Additionally, the graphical representation of the relationship between LCC and the fundamental period of the structure provides a clear and intuitive tool for researchers to use in future studies. For end users such as structural designers, the outcome of this study can be applied in the early design stages of a structure, where information about the structure is limited, to make informed decisions about the design and ultimately reduce the overall LCC of the building. This can result in cost savings and improved energy efficiency for the building. Additionally, the graphical representation of the relationship between LCC and the fundamental period of the structure can be useful for designers to make trade-off decisions between different design options.

5. Limitations of this Study and Recommendations for Future Research

As this study is focused on the simplification of the seismic LCC estimation process of building structures, it has some limitations. One limitation of this study is that it is based on a set of assumptions, such as the seismic hazard level and the failure probability of structural systems, which may not be accurate for all regions and buildings. Additionally, the study only considers a limited number of structural systems for simplification, and it may not be applicable to other types of systems such as high-rise buildings. Another limitation is that the proposed method is based on SAC-FEMA framework for damage probability calculation, which may not be accurate for all structures. This framework needs to be validated for different types of structures and for different seismic hazard levels. It is also worth mentioning that the proposed method is based on a deterministic approach, which might not be suitable for some situations where a probabilistic approach is needed. These limitations should be considered when interpreting the results and when applying the proposed method to real-world structures.

It also should be mentioned that more research still needs to be carried out for the validation of the proposed method in different types of structures. It would also be valuable to investigate the effect of different seismic hazard levels on the proposed method. The study results can be used to develop design guidelines for cost-effective buildings, and the authors recommend that future research should investigate the possibility of incorporating the proposed method in building design codes and software. This will enable designers to have an automated LCC analysis tool, which will help them to make better decisions in terms of cost and sustainability.

6. Conclusions

In this study, a simplified procedure for estimating the seismic LCC of building structures was presented. The procedure used the fundamental period of the structure as the main input to obtain an average seismic LCC estimation. This average LCC was adjusted using the actual value of the variables related to damage probability and LCC to obtain the final LCC of the building. In the present study, the dispersion measure in demand β_D|Sa was calculated using the mean plus one standard deviation of the response spectrum, which eliminates the need for performing nonlinear dynamic time history analyses. The SAC-FEMA framework was utilized along with the transformation of the MDOF system to the ESDOF system. The procedure was applied to three- and five-story buildings located in LA. The results were verified through a comparison with the results obtained from NLTH analysis-based procedure. The main findings can be summarized as follows:

• The analysis results showed that the proposed procedure provides an upper bound for the seismic LCC of the low-rise buildings, which makes it suitable for preliminary LCC estimations. The percentage difference in LCC between the proposed method and the reference method is 17.2% and 5.6%, respectively, for the five- and three-story buildings.
• The structure service life (t) is found to have a limited effect on the LCC; 7.2% and 6.8% of the LCC for the three and five-story buildings, respectively. The contributions of the limit state and β_D|Sa were found to have a significant impact on LCC estimation.
• The percentage difference in seismic demand dispersion measure (β_D|Sa) compared with the reference method for the 75 and 500 years return period spectra was found to be 7.4% and 3.7% for the three-story building, respectively. For the five-story building, these values were 15.0% and 54.0%, respectively.
• The differences between the proposed and the reference method in the probability of exceeding a limit state P(LS|_Sa) for the IO, LS, and CP limit states were found to be 0.96%, 0.25%, and 0.38%, respectively, for the five-story building. For the three-story building, the differences turned out to be 3.84%, 1.28%, and 1.19%, respectively, for the IO, LS, and CP limit states.

The proposed method eliminates the use of nonlinear dynamic analysis; instead, a simple nonlinear static method such as N2 or capacity spectrum method can be used. The method does not need the construction of fragility curves as the capacity spectral acceleration is calculated from the relationship between the spectral acceleration and the MIDR. The proposed method utilizes the response spectrum approach to calculate the dispersion measure in demand without the need for constructing the IDA curves. The results of the method provide a good approximation for the expected LCC of low-rise buildings and the framework can be generalized for other buildings, which can be used for preliminary LCC estimation needed for decision-makers and early design stage. Moreover, the framework can be easily customized for any particular structure dominated by the first mode shape. Additionally, the framework provides an insight into the effect of important variables on LCCs through a set of graphs showing the relationship between the LCC and the period of the structure for a different variation in the the main input variables of the LCC.