A method for evaluating earthquake-induced structural damage based on displacement and hysteretic energy

Abstract

Rather than viewing earthquake-induced structural damage as a single index of maximal deformation, it can be viewed as a combination of two indices, that is, the deformation demand and dissipation of energy. A method for evaluating the earthquake-induced damage of multi-storey buildings is presented that considers the maximal storey drift and the storey hysteretic energy. In this method, the maximal storey drift is estimated by means of the strength reduction factor spectrum, pushover analysis and distribution formula of deformation along structural storeys, which is based on the modal decomposition hypothesis in the nonlinear response stage of structures. For estimating the storey hysteretic energy demand, the normalized hysteretic energy spectrum of constant ductility factors is established, where the normalized hysteretic energy is defined as the ratio of the hysteretic energy to the square of the peak ground acceleration, and the formula for the distribution of hysteretic energy along structural storeys is derived based on the relation of the simplified internal forces of structures. For demonstrating the process of the proposed method and verifying its accuracy, an example is implemented, and the analysis results of the example indicate the following: (1) the method proposed in this article is a simple and easily implemented method for evaluating the structural damage induced by earthquakes and (2) the distribution formulas of the maximal drift and hysteretic energy along structural storeys derived in this article are relatively accurate.

Keywords

damage index displacement demand hysteretic energy nonlinear response history analysis pushover analysis storey drift

During the past few years, displacement-based seismic design has been widely developed (Antoniou and Pinho, 2004; Li et al., 2012; Medhekar and Kennedy, 2000) because current seismic design allows the structure to undergo inelastic deformation, which requires inelastic analysis for estimating the displacement demand of structures. Generally, the displacement demand of structures can be solved by means of the inelastic response spectrum such as the strength reduction factor spectrum (Ordaz and Perez-rocha, 1998; Qu et al., 2011) and the inelastic displacement ratio spectrum (Zhai et al., 2007).

In addition, the evaluation of earthquake-induced damage and corresponding research indicate that structural damage can be interpreted not only in terms of deformation but also as a combination of deformation and the dissipation of energy, where the latter reflects the cumulative effects of repeated cycles of inelastic structural deformation, which is the basis of energy-based seismic design (Benavent-Climent, 2011; Li et al., 2010, 2014). If various energies are plotted as a function of the natural period, the energy spectrum can be established, and this has developed into a wide range of research such as the earthquake input energy spectrum (Amiri et al., 2008; Benavent-Climent et al., 2010), hysteretic energy spectrum (Fajfar and Vidic, 1994; Riddell and Garcia, 2001), absorbed energy spectrum (Chou and Uang, 2000), momentary absorbed energy spectrum (Hagiwara, 2000) and inelastic cyclic demand spectrum (Kunnanth and Chai, 2004).

Structural performance and damage limit states can be quantified by a suitable damage index. A general damage index is based on a set of structural response parameters such as force, deformation and dissipation of energy. In the past few years, some damage indices have been proposed (Fajfar, 1992; Ghobarah et al., 1999). In particular, Park and Ang (1985) proposed a damage index based on a linear combination of the maximal deformation and hysteretic energy dissipation, and this damage index has been calibrated against through numerous experimental results and field observations obtained during earthquakes (Park and Ang, 1987). This study employs this damage index for estimating the cumulative damage of structures under earthquake ground motion and proposes a method for evaluating the structural damage induced by an earthquake.

An outline of the remainder of this article can be expressed as follows. First, the dynamic balance equation and energy balance equation are established based on a hypothesis of the equivalent single degree of freedom (SDOF) system. The relationship between the maximal deformation of a considered structure and the displacement demand of its equivalent SDOF system is then given based on the previous research. Next, the normalized hysteretic energy spectrum (NHES) of constant ductility factors is presented, and the distribution formula of hysteretic energy along structural storeys is derived for determining the storey hysteretic energy of structures. Finally, a step-by-step procedure is proposed, and an example is implemented to demonstrate the process and verify the accuracy of the proposed method.

Equivalent SDOF system of multi-storey buildings

Balance equation of dynamic responses

Consider an n-storey plane-symmetric building. Each floor diaphragm is rigid in its own plane. The equation of motion governing the responses of the n-storey building subjected to earthquake excitation along the direction of the horizontal principal axis is given as follows

M \overset{\cdot\cdot}{x} (t) + C \overset{\cdot}{x} (t) + F (t) = - MI {\overset{\cdot\cdot}{x}}_{g} (t)

(1)

where M is the diagonal mass matrix, C is the damping matrix, F(t) is the resisting forces vector, x(t) is the displacement response vector, given as [x₁(t), x₂(t), …, x_n(t)], and the vector I represents the 1 × n identity matrix [1 1 … 1]. In a static procedure, such as pushover analysis, a major hypothesis is that the responses of buildings can be expressed as a superposition of the responses of appropriate SDOF systems just like in the linear range. As such, x(t) for an inelastic system can be expanded in terms of the natural vibration modes of the corresponding linear elastic system as follows

x (t) = Φ \cdot d = \sum_{i = 1}^{n} φ_{i} d_{i} (t)

(2)

where φ_i is the ith mode shape vector, given as[φ_i,1, φ_i,2, …, φ_i,n], and d_i(t) is the linear displacement response vector, given as [d₁(t), d₂(t), …, d_n(t)]. Substituting equation (2) into equation (1), and pre-multiplying both sides of equation (1) by the transpose of φ_i, given as $φ_{i}^{T}$ , yields the following equations

\begin{matrix} φ_{i}^{T} M φ_{i} {\overset{\cdot\cdot}{d}}_{i} (t) + φ_{i}^{T} C φ_{i} {\overset{\cdot}{d}}_{i} (t) + F_{i} (t) \\ = - φ_{i}^{T} {M φ}_{i} γ_{i} {\overset{\cdot\cdot}{x}}_{g} (t), i = 1, \dots, n \end{matrix}

(3)

where γ_i is the modal participation factor. The ith modal resisting force F_i(t) can be represented as $φ_{i}^{T} K_{p} (t) φ_{i} d_{i} (t)$ , where K_p(t) is the elastic–plastic instantaneous stiffness matrices. In addition, $φ_{i}^{T} M φ_{i}$ can be represented by m_eq,i, which is interpreted as the ith modal equivalent mass. Both sides of equation (3) are divided by m_eq,i, and, employing the orthogonality of modes, the uncoupled equations can be derived as

{\overset{\cdot\cdot}{q}}_{i} (t) + 2 ξ ω_{i} {\overset{\cdot}{q}}_{i} (t) + f_{s, i} (t) = - {\overset{\cdot\cdot}{x}}_{g} (t), i = 1, \dots, n

(4)

where ω_i is the natural vibration frequency, given in terms of the natural vibration period T_i as 2π/T_i, ξ is the damping ratio, the displacement response q_i(t) represents d_i(t)/γ_i and the resisting force f_s,i(t) represents F_i(t)/γ_im_eq,i. Equation (4) represents the motion equations of equivalent SDOF systems from the first to the nth mode of the considered building. The above deduction indicates that the nonlinear response of a building with n degrees of freedom subjected to earthquake excitation can be expressed as the sum of the responses of n equivalent SDOF systems, each of which corresponds to a vibration mode, under equivalent earthquake excitation. The equivalent displacement responses, q₁(t), …, q_i(t), …, q_n(t), can be solved by equation (4) and then x(t) can be solved by equation (2) using the relationship d_i(t) = γ_i·q_i(t).

Energy balance equation

According to equation (1), the energy balance equation of the considered n-storey building can be derived as

\begin{matrix} \int_{0}^{t_{0}} \overset{\cdot}{x} {(t)}^{T} M \overset{\cdot\cdot}{x} (t) dt + \int_{0}^{t_{0}} \overset{\cdot}{x} {(t)}^{T} C \overset{\cdot}{x} (t) dt + \int_{0}^{t_{0}} \overset{\cdot}{x} {(t)}^{T} F (t) dt \\ = - \int_{0}^{t_{0}} \overset{\cdot}{x} {(t)}^{T} MI {\overset{\cdot\cdot}{x}}_{g} (t) dt \end{matrix}

(5a)

Here, t is the duration of earthquake motion, and $\overset{\cdot}{x} (t) dt$ is equal to dx(t), where $\overset{\cdot}{x} (t)$ is the velocity vector. Equation (5a) can be simplified as

E_{k} (t) + E_{d} (t) + E_{h} (t) = E_{I} (t)

(5b)

where E_k(t), E_d(t), E_h(t) and E_I(t) are the kinetic energy, viscous damping energy, hysteretic energy and input energy, respectively, of the considered building. Equation (2) is substituted into equation (5a), and, after pre-multiplying both sides of the adjusted equation by $φ_{i}^{T}$ , the following equations can be derived

\begin{matrix} \int_{0}^{t_{0}} φ_{i}^{T} M φ_{i} {\overset{\cdot\cdot}{d}}_{i} (t) {\overset{\cdot}{d}}_{i} (t) dt + \int_{0}^{t_{0}} φ_{i}^{T} C φ_{i} {\overset{\cdot}{d}}_{i} (t) {\overset{\cdot}{d}}_{i} (t) dt + \int_{0}^{t_{0}} F_{i} (t) {\overset{\cdot}{d}}_{i} (t) dt \\ = - \int_{0}^{t_{0}} φ_{i}^{T} M φ_{i} γ_{i} {\overset{\cdot\cdot}{x}}_{g} (t) {\overset{\cdot}{d}}_{i} (t) dt, i = 1, \dots, n \end{matrix}

(6a)

Equation (6a) can be simplified as

E_{k, i} (t) + E_{d, i} (t) + E_{h, i} (t) = E_{I, i} (t)

(6b)

Both sides of equation (6a) are divided by $φ_{i}^{T} M φ_{i} γ_{i}^{2}$ and the following equations are obtained

\begin{matrix} \int_{0}^{t_{0}} {\overset{\cdot\cdot}{q}}_{i} (t) {\overset{\cdot}{q}}_{i} (t) dt + 2 ξ ω_{i} \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{i} (t) {\overset{\cdot}{q}}_{i} (t) dt + \int_{0}^{t_{0}} f_{s, i} (t) {\overset{\cdot}{q}}_{i} (t) dt \\ = - \int_{0}^{t_{0}} {\overset{\cdot\cdot}{x}}_{g} (t) {\overset{\cdot}{q}}_{i} (t) dt, i = 1, \dots, n \end{matrix}

(7a)

Equation (7a) can be simplified as

e_{k, i} (t) + e_{d, i} (t) + e_{h, i} (t) = e_{I, i} (t), i = 1, \dots, n

(7b)

where e_k,i(t), e_d,i(t), e_h,i(t) and e_I,i(t) are the kinetic energy, viscous damping energy, hysteretic energy and earthquake input energy of the ith modal equivalent SDOF system, respectively. Comparing e_h,i, E_h,i and E_h, the following relationship is obtained

E_{h} (t) = \sum_{i = 1}^{n} E_{h, i} (t) = \sum_{i = 1}^{n} m_{eq, i} \cdot γ_{i}^{2} \cdot e_{h, i} (t)

(8)

Estimation of the maximal displacement and hysteretic energy

Estimation of the maximal displacement

The mass of the ith modal equivalent SDOF system is equal to 1; therefore, the peak force of this system in the linear elastic stage can be expressed as

f_{s, i, e, \max} = β (ω_{i}, ξ) \cdot \max (| {\overset{\cdot\cdot}{x}}_{g} (t) |)

(9)

where β(ω_i, ξ) is the amplification coefficient spectrum. The strength reduction factors can be defined as follows (Ordaz and Perez-rocha, 1998)

R = \frac{f_{s, i, e, \max}}{f_{s, i, yie}}

(10)

Here, f_s,i,yie are the yield forces of the system, which are given by f_s,i,yie = k_s,i q_i,yie, where k_s,i and q_i,yie are the elastic stiffness and yield displacement, respectively. Substituting equation (9) into equation (10) and solving for q_i,yie yields

q_{i, yie} = \frac{β (ω_{i}, ξ) \cdot \max | {\overset{\cdot\cdot}{x}}_{g} |}{ω_{i}^{2} \cdot R}

(11)

After dividing both sides of equation (4) by q_i,yie, and substituting equation (11), the following equation is obtained based on the relationship u_i(t) = q_i(t)/q_i,yie

{\begin{matrix} {\overset{\cdot\cdot}{u}}_{i} (t) + 2 ξ ω_{i} {\overset{\cdot}{u}}_{i} (t) + ω_{i}^{2} \frac{f_{s, i} (t)}{f_{s, i, yie}} = - \frac{ω_{i}^{2} R}{β (ω_{i}, ξ)} \frac{{\overset{\cdot\cdot}{x}}_{g} (t)}{\max (| {\overset{\cdot\cdot}{x}}_{g} (t) |)} \\ μ = \max | u_{i} (t) | \end{matrix}

(12)

Here, µ is the ductility factor of the equivalent SDOF system. On the other hand, the definitions of R and µ yield the following equation

D_{i} = ω_{i}^{2} \cdot \frac{μ}{R} \cdot A_{i}

(13)

where A_i is the pseudo acceleration demand, given as $A_{i} \approx β (ω_{i}, ξ) \cdot \max | {\overset{\cdot\cdot}{x}}_{g} (t) |$ , and D_i is the displacement demand (i.e. maximal displacement), which is given as q_i,max. The ratio of µ/R for different frequencies (or periods) can be obtained through equation (12). The relation between A_yie and D, that is, the A_yie–D spectrum, can be established, where A_yie = A/R, by substituting µ/R, different vibration frequencies ω and $A \approx β (ω, ξ) \cdot \max | {\overset{\cdot\cdot}{x}}_{g} (t) |$ into equation (13). According to the improved capacity spectrum method (Fajfar, 2003), µ and D_i can be obtained by substituting the bilinear A_i–D_i format capacity curve into the A_yie–D demand spectrum.

The R spectrum and β(ω_i, ξ) are illustrated in Figures 1 and 2, respectively, for three sites with different types of soils based on previously collected earthquake acceleration records (Wang, 2007).

Figure 1.

Strength reduction factor R spectra: (a) hard soil site, (b) intermediate soil site and (c) soft soil site.

Figure 2.

Amplification coefficient β spectra: (a) hard soil site, (b) intermediate soil site and (c) soft soil site.

Estimation of hysteretic energy

Dividing equation (7a) by $q_{yie}^{2}$ and substituting equations (11), (10) and (9) into the adjusted equation, the following equation yields

\begin{matrix} η_{i} \int_{0}^{t} {\overset{\cdot\cdot}{u}}_{i} (t) {\overset{\cdot}{u}}_{i} (t) dt + 2 ξ ω_{i} η_{i} \int_{0}^{t} {\overset{\cdot}{u}}_{i} (t) {\overset{\cdot}{u}}_{i} (t) dt + ω_{i}^{2} η_{i} \int_{0}^{t} \frac{f_{s, i} (t)}{f_{s, i, yie}} {\overset{\cdot}{u}}_{i} (t) dt \\ = - \frac{ω_{i}^{2} R η_{i}}{β (ω_{i}, ξ)} \int_{0}^{t} \frac{{\overset{\cdot\cdot}{x}}_{g} (t)}{\max (| {\overset{\cdot\cdot}{x}}_{g} (t) |)} {\overset{\cdot}{u}}_{i} (t) dt \end{matrix}

(14a)

where $η_{i} = β^{2} (ω_{i}, ξ) / ω_{i}^{4} R_{i}^{2}$ . Equation (14a) can be simplified as

N e_{k, i} (t) + N e_{d, i} (t) + N e_{h, i} (t) = N e_{I, i} (t)

(14b)

where Ne_k,i(t), Ne_d,i(t), Ne_h,i(t) and Ne_I,i(t) are the normalized kinetic energy, normalized viscous damping energy, normalized hysteretic energy and normalized earthquake input energy, respectively. The relationship between the normalized energy terms in equation (14a) and the corresponding energy terms of the equivalent SDOF system in equation (7a) is given as

N e_{h, i} (t) = \frac{e_{h, i} (t)}{PG A^{2}}

(15)

where PGA is the abbreviation of peak ground acceleration and $PGA = \max | {\overset{\cdot\cdot}{x}}_{g} (t) |$ . Equation (14a) is regarded as the basis for establishing the hysteretic energy spectrum. The form of the spectrum is defined as the relationship between Ne_h,i(t) and the natural period T_i, where Ne_h,i(t) is a function of T_i. The values of µ for each spectral curve are equal. Therefore, the spectrum is denoted as the NHES of constant ductility factors, which can be expressed as NHES(T_i), where NHES(T_i) = Ne_h,i(t_max). The relationship between NHES(T_i) and e_h,i(t_max) is therefore given as

NHES (T_{i}) = \frac{e_{h, i} (t_{\max})}{PG A^{2}}

(16)

Mean normalized hysteretic spectra are illustrated in Figure 3 for three sites with different types of soils based on previously collected earthquake motion records (Wang, 2007).

Figure 3.

Normalized hysteretic energy spectra: (a) hard soil site, (b) intermediate soil site and (c) soft soil site.

The hysteretic energy of the jth storey of the considered building can be expressed as

E_{h}^{j} = \sum_{i = 1}^{n} E_{h, i}^{j} = \sum_{i = 1}^{n} \int_{0}^{t_{\max}} V_{i, j} (t) Δ u_{i, j} (t) dt, j = 1, \dots, n

(17)

where V_i,j(t) and Δu_i,j(t) are the momentary shear force and drift of the jth storey for the ith mode, respectively, which may be expanded as follows

\begin{matrix} V_{i, j} (t) = \sum_{k = j}^{n} f_{k, i} (t) = \sum_{k = j}^{n} m_{k} ϕ_{i, k} A_{i} (t); \\ Δ u_{i, j} (t) = Δ ϕ_{i, j} q_{i} (t) γ_{i} \end{matrix}

(18)

Here, A_i(t) is the pseudo acceleration response of the ith mode, φ_i,k is the modal value of the kth storey of the ith mode and Δϕ_i,j equals ϕ_i,n − ϕ_i,n−1. Substituting equations (18) and (16) into equation (17), the hysteretic energy of the jth storey can be rewritten as

\begin{matrix} E_{h}^{j} = \sum_{i = 1}^{n} ((\sum_{k = j}^{n} m_{k} ϕ_{i, k}) \frac{Δ ϕ_{i, j}}{γ_{i}} NHES (T_{i}) \cdot PG A^{2}), \\ j = 1, \dots, n \end{matrix}

(19)

Considering variations of mode shape in the plastic response stages of structures, the modes in equation (19) can be replaced by φ_p,i, where φ_p,i = [φ_p,i,1, φ_p,i,2, …, φ_p,i,n], which can be obtained by modal pushover analysis (Manoukas et al., 2012; Miao et al., 2011; Poursha et al., 2011).

Damage index

For considering the influences of maximal deformation and cumulative energy, the Park–Ang damage index is used to evaluate the earthquake-induced damage in the structural storeys of the considered building. Combining the above discussed fundamentals, the Park–Ang damage index DM_j of the jth storey can be expressed as

D M_{j} = \frac{Δ x_{j}}{Δ x_{j, cu}} + η \cdot \frac{E_{h}^{j}}{V_{yie}^{j} \cdot Δ x_{j, cu}}

(20)

Here, $V_{yie}^{j}$ is the yield shear force of the jth storey, η is the energy dissipation factor, which is in the range 0–0.85, Δx_j and Δx_j,cu are the maximal storey drift and the storey drift limit of the jth storey, respectively, where Δx_j = x_j,max − x_j_−1,max and Δx_j,cu = x_j,cu − x_j_−1,cu. Because serious damage to any given storey represents a risk to the entire building, the earthquake-induced structural damage index DM for the entire building can be determined conservatively as

DM = \max (D M_{j})

(21)

Summary of the procedure

The proposed method is referred to herein as the simplified method. The procedure of the simplified method consists of three phases: estimation of the maximal storey drift, estimation of the storey hysteretic energy and evaluation of structural damage, which are given according to individual steps 1–12 as follows.

Estimation of the maximal storey drift

The A_yie–D demand spectrum of constant ductility factors µ is established according to section ‘Estimation of the maximal displacement’.

The ith mode shape vector φ_i of the considered building is determined by modal analysis, where φ_i = [φ_i,1, φ_i,2, …, φ_i,n], and φ_i is normalized such that φ_i,1 = 1.

The equivalent SDOF system of the building is established according to the ith mode, and the equivalent mass m_eq,i and the modal participation factors γ_i are then determined.

The V_b,i–x_r,i (base shear–roof displacement) capacity curve of the building is determined by the ith modal pushover analysis, and the V_b,i–x_r,i capacity curve is converted to the bilinear A_i–D_i (pseudo acceleration–displacement demand) format capacity curve of the ith modal equivalent SDOF system.

The equivalent period T_eq,i is determined by

T_{eq, i} = 2 π \sqrt{\frac{D_{i, yie}}{A_{i, yie}}}

(22)

where A_i,yie and D_i,yie are the pseudo yield acceleration and yield displacement, respectively, which are determined from the bilinear A_i–D_i format capacity curve.

6. The values of µ and D_i of the ith modal equivalent SDOF system are determined by the intersection of the A_i–D_i format capacity curve and the A_yie–D demand spectrum.

7. According to D_i, the maximal responses of the ith mode, such as the maximal storey drift Δx_j,i,max of the jth storey, can be determined by equation (2) and the relation q_i(t) = d_i(t)/γ_i.

8. Steps 2–7 are repeated. The maximal responses of different modes (i = 1, 2, …, n) solved in Step 7 can be combined to the responses of the building by square root of sum of square (SRSS) rule.

Estimation of the storey hysteretic energy

9. The NHES of constant ductility factors µ is established according to equations (14a) and (16).

10. The values of T_eq,i and µ determined in Steps 5 and 6 are substituted into NHES, and NHES(T_eq,i) is determined.

11. Step 10 is repeated for different modes (i = 1, 2, …, n). The jth storey hysteretic energy $E_{h}^{j}$ of the building is solved by equation (19).

Evaluation of the structural damage

12. The maximal drift and hysteretic energy of each storey of the building are substituted into equations (20) and (21), and the damage index DM is determined, which can be used to evaluate the earthquake-induced damage of the entire building.

Numerical example

Description of the example building

To clarify how the simplified method should be applied, a simple example building is designed. The example building is the 10-storey symmetric reinforced concrete frame structure illustrated in Figure 4. Ground motion is considered to act along the principal axis. Each floor diaphragm is rigid in its own plane. All storeys are 3.6 m in height and in the plane, as shown in Figure 4. The sectional sizes of the beams are 300 mm × 700 mm for all storeys; the sectional sizes of the columns are 700 mm × 700 mm for the first through third storeys, 600 mm × 600 mm for the fourth through seventh storeys and 500 mm × 500 mm for the eighth through tenth storeys. Steel ratios are approximately 1.5% for beam sections and 2% for column sections. The concrete compression strength is designated as 30 MPa for all columns and beams. The designated dead load and live load of each floor (roof) are 6.6 kN/m² (4.7 kN/m²) and 1.0 kN/m² (2.0 kN/m²), respectively. The damping of the example building is modelled by Rayleigh damping, and the damping ratio ξ = 5%.

Figure 4.

Schematic of the example building.

Earthquake acceleration records

For verifying the accuracy of the method, the results of the simplified method are compared with the results of the nonlinear response history analysis (NL-RHA) method, based on the acceleration records of actual earthquake motion. Thirty earthquake acceleration records for hard soil site (V_s = 360–750 m/s), corresponding to soil type B of the US Geological Survey (USGS), are selected and listed in Table 1. The rules for selecting these records are defined as (1) magnitude = 6–8, (2) fault distance = 15–45 km and (3) peak acceleration ≥0.1 g, approximately. The PGA values of the acceleration records are all adjusted to 5.1 m/s².

Table 1.

Information obtained from earthquake acceleration records.

Stations	Earthquakes	Fault distance (km)	Components	PGA (g)	PGV (cm/s)	PGD (cm)	Duration (s)
1095 Taft Lin-Coln School	Kern County (21 July 1952, Ms7.7)	41	TAF021	0.16	15.3	9.2	54
1095 Taft Lin-Coln School	Kern County (21 July 1952, Ms7.7)	41	TAF111	0.18	17.3	9.0	54
1652 Anderson Dam	Loma Prieta (18 October 1989, Ms7.1)	20	AND270	0.24	20.3	7.7	39
1652 Anderson Dam	Loma Prieta (18 October 1989, Ms7.1)	20	AND360	0.24	18.4	6.7	39
24157 LA-Baldwin Hills	Northridge (17 January 1994, Ms6.7)	31	BLD090	0.24	14.9	6.2	40
24157 LA-Baldwin Hills	Northridge (17 January 1994, Ms6.7)	31	BLD360	0.17	17.6	4.8	40
14403 LA-116th St School	Northridge (17 January 1994, Ms6.7)	42	116090	0.21	10.3	2.67	40
14403 LA-116th St School	Northridge (17 January 1994, Ms6.7)	42	116360	0.13	13.5	2.83	40
23 Coolwater	Landers (28 June 1992, Ms7.4)	21	CLW-LN	0.28	25.6	13.7	28
23 Coolwater	Landers (28 June 1992, Ms7.4)	21	CLW-TR	0.42	42.3	13.8	28
57504 Coyote Lake Dam	Loma Prieta (18 October 1989, Ms7.1)	22	CLD195	0.16	13.0	6.11	40
57504 Coyote Lake Dam	Loma Prieta (18 October 1989, Ms7.1)	22	CLD285	0.179	22.6	13.2	40
Tcu045	ChiChi (20 September 1999, Ms7.6)	24	TCU045-N	0.50	39.0	14.3	90
Tcu045	ChiChi (20 September 1999, Ms7.6)	24	TCU045-W	0.47	36.7	50.7	90
Tcu047	ChiChi (20 September 1999, Ms7.6)	33	TCU047-N	0.41	40.2	22.2	90
Tcu047	ChiChi (20 September 1999, Ms7.6)	33	TCU047-W	0.30	41.6	51.1	90
24389LA-Century City CC North	Northridge (17 January 1994, Ms6.7)	26	CCN090	0.26	21.1	6.68	40
24389LA-Century City CC North	Northridge (17 January 1994, Ms6.7)	26	CCN360	0.22	25.2	5.7	40
89324 Rio Dell Overpass-FF	Cape Mendocino (25 April 1992, Ms7.1)	19	RIO270	0.39	43.9	22.0	36
89324 Rio Dell Overpass-FF	Cape Mendocino (25 April 1992, Ms7.1)	19	RIO360	0.55	42.1	18.6	36
1061 Lamont	Duzce (12 November 1999, Ms7.3)	15.6	1061-N	0.11	11.5	8.21	42
1061 Lamont	Duzce (12 November 1999, Ms7.3)	15.6	1061-E	0.13	13.7	8.19	42
24577 Fort Irwin	Landers (28 June 1992, Ms7.4)	65	FTI000	0.11	9.7	3.66	40
24577 Fort Irwin	Landers (28 June 1992, Ms7.4)	65	FTI090	0.12	16.4	21.8	40
57383 Gilroy Array#6	Loma Prieta (18 October 1989, Ms7.1)	20	G06000	0.13	12.8	4.74	39
57383 Gilroy Array#6	Loma Prieta (18 October 1989, Ms7.1)	20	G06090	0.17	14.2	3.79	39
58378 APEEL 7-Pulgas	Loma Prieta (18 October 1989, Ms7.1)	47	A07000	0.16	16.1	7.75	39
58378 APEEL 7-Pulgas	Loma Prieta (18 October 1989, Ms7.1)	47	A07090	0.09	15.7	8.41	39
24611 LA-Temple & Hope	Northridge (17 January 1994, Ms6.7)	32	TEM090	0.13	13.9	3.15	40
24611 LA-Temple & Hope	Northridge (17 January 1994, Ms6.7)	32	TEM180	0.18	20.0	2.74	40

PGA: peak ground acceleration; PGV: peak ground velocity; PGD: peak ground displacement.

Analysis model and structural parameters

The example building is simplified as a nonlinear multi-storey frame model. The seismic concepts, including strong columns and weak beams, strong shear and weak bending, and strong joints and weak components, are used for designing the example building. As such, the restoring force relation of a column and a joint is assumed to be a linear elastic model, and the restoring force relation for the bending of two beam ends is assumed to be a bilinear stiffness degradation model, where the yield stiffness coefficient is 0.03 and the stiffness degradation coefficient is 0.4. The storey mass matrix M is given with corresponding diagonal terms 400 t, 450 t, 450 t, 450 t, 450 t, 450 t, 450 t, 450 t, 450 t and 500 t. The first three natural vibration periods of the example building are 1.067, 0.364 and 0.211 s, respectively. The equivalent masses m_eq, the modal participation factors γ and the equivalent periods T_eq of the first three modal equivalent SDOF systems of the example building are listed in Table 2.

Table 2.

The employed parameters and some calculation results.

Mode	m_eq,i	γ_i	Pushover capacity curve		A–q capacity curve		T_eq,i	μ	Maximal deformation		Hysteretic energy
Mode	m_eq,i	γ_i	V_b,i,yie	x_r,i,yie	A_i,yie	q_i,yie	T_eq,i	μ	q _r,i,max	x _r,i,max	e_h,i	E_h,i
1st	2461.7	1.197	4194.1	0.0586	1.423	0.049	1.165	2.88	0.0587	0.169	0.0182	1668.3
2nd	2314.9	0.4815	4556.9	0.0296	4.088	0.0614	0.77	1.6	0.0982	0.0474	0.0144	200.8
3rd	2769.4	0.2661	8396.0	0.0343	11.395	0.1288	0.668	<1	0.0588	0.0156	–	–

Estimation of deformation and hysteretic energy

Maximal displacement and hysteretic energy

The V_b–x_r capacity curves of the different modes are determined by the modal pushover analysis method. As shown in Figure 5, the bilinear A–D format capacity curves (see the red lines in the figure) of different modes are respectively substituted into the A_yie–D demand spectrum of constant ductility factors to determine displacement demands D and ductility factors µ. According to Figure 5, D can be determined through the intersection of bilinear A–D format capacity curves and the A_yie–D demand spectrum, which is the fundamental principle of the capacity spectrum method. After that, the values of D for the different modes can be converted to the maximal roof displacements x_r,max of the corresponding modes of the example building. On the other hand, the values of µ and T_eq for the different modes are respectively substituted into the NHES and equation (16) to determine hysteretic energies e_h of the corresponding modal equivalent SDOF systems, as shown in Figure 6. After that, e_h of multi modes can be converted to the structural entire hysteretic energy E_h of the example building by equation (8). The employed parameters and some calculation results are listed in Table 2.

Figure 5.

Estimation of the displacement demand and ductility factor: (a) first mode and (b) second mode.

Figure 6.

Estimation of the normalized hysteretic energy: (a) first mode and (b) second mode.

To verify the estimation accuracy of the simplified method, the mean responses calculated by the NL-RHA method and the responses estimated by the simplified method are compared. The mean value of the entire hysteretic energy E_h calculated by the NL-RHA method is 2005.7 kN m and that of the simplified method is 1869.1 kN m (from Table 2, only the first three modes are considered, where E_h = E_h₁+E_h₂+E_h₃). Comparing the results of the two methods, the latter is lower, and the deviation is 6.8%. Moreover, for the maximal roof displacement x_r,max, the mean x_r,max obtained by the NL-RHA method is 0.187 m and that estimated by the simplified method is 0.176 m (from Table 2, only the first three modes are considered, where x_r,max = (x²_r,1,max+x²_r,2,max+x²_r,3,max)^1/2). Comparing the results of the two methods, the value obtained by the simplified method is lower, and the deviation is 5.9%.

Distributions of maximal storey drift and storey hysteretic energy

To explore the characteristics of storey response distributions, including the maximal storey drift and storey hysteretic energy, it is necessary to solve the statistical mean response distributions based on the given earthquake records by the NL-RHA method. The responses of structures induced by different earthquake excitations are different, so the statistical method cannot be used directly to determine mean response distributions. To solve this problem, the distribution coefficients of storey response demands, including that of the maximal storey drift α and that of the storey hysteretic energy ε, are defined as the ratio of the jth storey response to the sum of the responses of all storeys, that is, α = Δx_j/ΣΔx_j and $ε = E_{h}^{j} / Σ E_{h}^{j}$ . Based on the NL-RHA method, the values of α and ε for each earthquake excitation are illustrated in Figure 7(a) and (b), respectively, where the mean distributions of α and ε are given.

Figure 7.

Storey response demands for each earthquake excitation obtained by the NL-RHA method: (a) maximal storey drift and (b) storey hysteretic energy.

Considering the deviation of the structural nonlinear lateral deformation estimated with elastic modes, the first three equivalent lateral deformation vectors determined by modal pushover analysis are adopted to replace the first three elastic modes: φ_p,1 = [1 0.9875 0.9588 0.9104 0.8489 0.7377 0.5721 0.3772 0.2023 0.0655], φ_p,2 = [−1 −0.8591 −0.4786 0.01823 0.2724 0.3918 0.4144 0.3521 0.2383 0.0952] and φ_p,3 = [1 0.4072 −0.4299 −0.7559 −0.5992 −0.1078 0.4305 0.6588 0.5618 0.2605]. The above vectors are substituted into equations (2) and (19) successively, and the values of α and ε can be approximately determined, respectively. The distribution coefficients determined by the NL-RHA method and the simplified method are compared in Figure 8. The mean results of the storey response demands of the NL-RHA method are considered to be exact solutions, which can be used for verifying the accuracy of the estimation obtained by the simplified method. As shown in Figure 8(a) and (b), the deviations of the storey response demands between the two methods are relatively small, which indicates that the simplified method proposed in this article can be used for estimating the distributions of maximal storey drift and storey hysteretic energy.

Figure 8.

Comparison of the storey response demands obtained by the NL-RHA method and the simplified method: (a) maximum storey drift and (b) storey hysteretic energy.

Evaluation of structural damage

Damage indices of individual storeys

To determine the storey damage of the example building, the storey drift angle limit is set as 1/50 (i.e. the storey drift limit $Δ x_{cu}^{j}$ of the jth storey is 0.072 m). From the results listed in Table 2, the maximal storey drift Δx^j and the storey hysteretic energy $E_{h}^{j}$ can be solved. Substituting Δx^j, $E_{h}^{j}$ , $Δ x_{cu}^{j}$ and the storey yield shear force $V_{yie}^{j}$ into the damage equation (equation (20)), where η = 0.3, the damage index of each storey is calculated, and the results are listed in Table 3.

Table 3.

Comparison of the storey response demands and damage indices obtained by the NL-RHA method and the simplified method.

Storey	Maximal storey drift (m)			Storey hysteretic energy (kN m)			Damage index
Storey	Mean of NL-RHA	Simplified method	Deviation (%)	Mean of NL-RHA	Simplified method	Deviation (%)	Mean of NL-RHA	Simplified method	Deviation (%)
1st	0.01005	0.00981	−2.4	236.27	148.80	−37.0	0.367414	0.279735	−23.8
2nd	0.02141	0.01913	−10.6	374.63	307.58	−17.9	0.65861	0.562289	−14.6
3rd	0.02737	0.02345	−14.3	396.97	380.72	−4.1	0.76293	0.692816	−9
4th	0.03114	0.02595	−9.2	357.64	398.92	+11.5	0.805042	0.775958	−3.6
5th	0.02847	0.02281	−16.7	281.99	306.18	+8.6	0.689156	0.635743	−7.8
6th	0.02328	0.01639	−29.6	187.07	176.95	−5.4	0.518198	0.411962	−20.5
7th	0.01775	0.01243	−29.9	104.96	79.89	−23.9	0.355861	0.255858	−28.1
8th	0.01525	0.01953	+28.1	54.21	47.55	−12.3	0.273185	0.325088	+19.0
9th	0.00850	0.01754	+106.3	12.42	18.65	+50	0.132118	0.264728	+100.3
10th	0.00381	0.00896	+135.2	0	3.87	–	0.052917	0.128826	+143.4
Mean deviation of storey drift			+15.7%	Mean deviation of hysteretic energy		−3.05%	Mean deviation of damage index		−15.5%

NL-RHA: nonlinear response history analysis.

In Table 3, the mean response demands obtained by the NL-RHA method are considered to be exact solutions. A comparison of the response demands and the damage indices obtained by the NL-RHA method and the simplified method indicate the following: (1) for the upper storeys (9th and 10th storeys), the deviations of the two response demands obtained by the two methods are substantial and (2) for the other storeys, the deviations of the two response demands fall within the range ±30%, except for that of the hysteretic energy of the first storey. The above analysis suggests that, in a practical application, satisfactory results can be obtained through appropriate amendments to the proposed simplified method.

Damage evaluation of the entire building

Considering that the damage of individual storeys determines the safety of the entire building, the maximal damage index of all storeys can be regarded as the damage index of the entire building. As shown in Table 3, the maximal damage indices determined by the NL-RHA method and the simplified method are 0.805 and 0.776, respectively, for the fourth storey, and the deviation is −3.6%. The evaluation result indicates that, for a hard soil site, the degree of damage of the example building subjected to strong earthquake motion (PGA = 5.1 m/s²) represents serious damage.

Conclusion

A simplified method for evaluating earthquake-induced structural damage was proposed. Some novelties of the method and conclusions of the study are given as follows:

The maximal storey drift and storey hysteretic energy are considered for the evaluation of storey damage. Moreover, considering that the damage of individual storeys determines the safety of an entire multi-storey building, the maximal damage index of all structural storeys is regarded as the damage index of the entire building.

A novel approach was presented for establishing the hysteretic energy spectrum. The cumulative energy of an SDOF system subjected to earthquake excitation is normalized as the ratio of the energy to the square of the PGA. An improved energy balance equation is established based on the form of the normalized energy. In the balance equation, the excitation is simplified as a relative value, and the normalized energies are not affected by the PGA, which provides a convenient means of establishing the statistical mean energy spectrum based on selected earthquake acceleration records with different PGA values. In practical applications, the hysteretic energy of the equivalent SDOF system of a structure can be solved by the product of the normalized hysteretic energy and the square of the objective PGA.

The relationship between the cumulative hysteretic energies of a multi-storey building and its equivalent SDOF system was derived. The distribution of hysteretic energy along storeys was derived and then the procedure for estimating storey hysteretic energies of a building through the NHES was presented.

The maximal storey drift is estimated through the strength reduction factor spectrum, pushover analysis and the distribution formula of the maximal displacement along structural storeys, which is based on the modal decomposition hypothesis in the nonlinear response stage of structures.

To verify the accuracy of the proposed simplified method, an example building was designed, and 30 earthquake acceleration records for hard soil site were selected and applied as the horizontal excitations of the example building. The maximal storey drift, storey hysteretic energy and storey damage index were, respectively, solved by the NL-RHA method and the simplified method, and the results of comparative analysis are given as follows: (1) the simplified method proposed in this article is simple and easily implemented, and the method can be used for evaluating earthquake-induced structural damage and (2) the distribution formulas employed for the maximal storey drift and storey hysteretic energy in the procedure are relatively accurate.

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 research was supported by the National Natural Science Foundation of China (Grant No. 51478091), the Program for Liaoning Excellent Talents in University (Grant No. LJQ2014139) and the project of Dalian Nationalities University (DC201502040301).

References

Amiri

Darzi

Amiri

(2008) Design elastic input energy spectra based on Iranian earthquakes. Canadian Journal of Civil Engineering 35: 635–646.

Antoniou

Pinho

(2004) Development and verification of a displacement-based adaptive pushover procedure. Journal of Earthquake Engineering 8(5): 643–661.

Benavent-Climent

(2011) A energy-based method for seismic retrofit of existing frames using hysteretic dampers. Soil Dynamics and Earthquake Engineering 31: 1385–1396.

Benavent-Climent

Lopez-Almanse

Bravo-Gonzalez

(2010) Design energy input spectra for moderate-to-high seismicity regions based on Colombian earthquakes. Soil Dynamics and Earthquake Engineering 30: 1129–1148.

Chou

Uang

(2000) Establishing absorbed energy spectra – an attenuation approach. Earthquake Engineering & Structural Dynamics 29: 1441–1455.

Fajfar

(1992) Equivalent ductility factors, taking into account low cycle fatigue. Earthquake Engineering & Structural Dynamics 21: 837–848.

Fajfar

(2003) A nonlinear analysis method for performance-based seismic design. Earthquake Spectra 16(3): 573–592.

Fajfar

Vidic

(1994) Consistent inelastic design spectra: hysteretic and input energy. Earthquake Engineering & Structural Dynamics 23(5): 523–537.

Ghobarah

Abou-Elfath

Biddah

(1999) Response-based damage assessment of structures. Earthquake Engineering & Structural Dynamics 28: 79–104.

10.

Hagiwara

(2000) Momentary energy absorption and effective loading cycles of structures during earthquakes. In: The 12th world conference on earthquake engineering, Auckland, New Zealand, 30 January–4 February, paper no. 0457. Upper Hutt, New Zealand: New Zealand National Society for Earthquake Engineering.

11.

Kunnanth

Chai

(2004) Cumulative damage-based inelastic cyclic demand spectrum. Earthquake Engineering & Structural Dynamics 33: 499–520.

12.

Zhai

. (2010) Energy-based modal pushover procedure for asymmetric structures. Advances in Structural Engineering 13(6): 129–138.

13.

Zhai

Xie

(2012) Evaluation of displacement-based, force-based and plastic hinge elements for structural non-linear static analysis. Advances in Structural Engineering 15(3): 503–514.

14.

Guan

. (2014) An energy-based assessment on dynamic amplification factor for linear static analysis in progressive collapse design of ductile RC frame structures. Advances in Structural Engineering 17(8): 1217–1225.

15.

Manoukas

Athanatopoulou

Avramidis

(2012) Multimode pushover analysis for asymmetric buildings under biaxial seismic excitation based on a new concept of the equivalent single degree of freedom system. Soil Dynamics and Earthquake Engineering 38: 88–96.

16.

Medhekar

Kennedy

DJL

(2000) Displacement-based seismic design of buildings – theory. Engineering Structures 22: 201–209.

17.

Miao

Guan

. (2011) Evaluation of modal and traditional pushover analyses in frame-shear-wall structures. Advances in Structural Engineering 14(5): 815–836.

18.

Ordaz

Perez-rocha

(1998) Estimation of strength-reduction factors for elastoplastic systems: a new approach. Earthquake Engineering & Structural Dynamics 27: 889–901.

19.

Park

Ang

AH-S

(1987) Damage-limiting aseismic design of buildings. Earthquake Spectra 3(1): 1–26.

20.

Park

Ang

AH-S

(1985) Mechanistic seismic damage model for reinforced concrete. Journal of Structural Engineering 111(4): 722–739.

21.

Poursha

Khoshnoudian

Moghadam

(2011) A consecutive modal pushover procedure for nonlinear static analysis of one-way unsymmetric-plan tall building structures. Engineering Structures 33: 2417–2434.

22.

Zhang

Zhao

(2011) Strength reduction factors for seismic analyses of buildings exposed to near-fault ground motions. Earthquake Engineering and Engineering Vibration 10(2): 195–209.

23.

Riddell

Garcia

(2001) Hysteretic energy spectrum and damage control. Earthquake Engineering & Structural Dynamics 30: 1791–1816.

24.

Wang

(2007) Studies on performance-based seismic design methods of structures subjected to multi-dimensional earthquake excitations. PhD Thesis, Dalian University of Technology, Dalian, China.

25.

Zhai

Xie

. (2007) Study on inelastic displacement ratio spectra for near-fault pulse-type ground motions. Earthquake Engineering and Engineering Vibration 6(4): 351–355.