Probabilistic performance analysis of structural elements subjected to blast load using equivalent static force model

Abstract

This study presents a probabilistic analysis of a single degree of freedom (SDOF) system subjected to drift-controlled distant blast loading employing Monte-Carlo simulation using MATLAB. The simulations are achieved using an equivalent static force (ESF)–based model as the deterministic model. The loading and structural parameters are treated as random variables in the parametric sensitivity analysis. ESF factor and the resistance of the SDOF system are observed as the response parameters. ESF factor is found to be highly sensitive to positive pulse duration, whilst the resistance is found to be more sensitive to both positive pulse duration and the peak blast force. With the log-normal distribution of input parameters, the ESF factor and the resistance of the SDOF system follow the log-normal distribution. The present study suggests that the probabilistic analysis is more conservative than the deterministic analysis. The uncertainty can be incorporated in a deterministic approach for both analysis and design purposes by opting suitable factor of safety (FOS) based on probabilistic analysis.

Introduction

Blast is defined as a rapid phase of pressure created by sudden release of energy, generating blast waves in different shapes, causing both external and internal catastrophic damage to structures. Considerable importance has been given to issues caused by blast and earthquake loads in past few decades. While designing engineering structures, exceptional loads play a crucial role, especially in case of public facilities. Structures of public importance such as hospitals, parliament buildings, school buildings, stadiums, bridges, and dams should be designed to resist exceptional loads, such as fire and blast, along with permanent and variable loads (Biggs, 1964; May and Smith, 1995). The negligence of unusual loads may result in severe damage to the structures in the event of such loads. With the rise in the number of blasts caused by terrorist acts and accidental explosions worldwide, a thorough investigation should be conducted to analyze the effect of these exceptional loads. Proper guidelines should be incorporated in the design procedures of important structures for the same.

In general, designers assume that probability of occurrence of a blast is negligible. However, existing structures, as well as those under construction, are highly vulnerable to explosives. The extent of the structural collapse is governed by the amount of explosive material detonated and the location of detonation from the structural element. Hence, to understand the characteristics of blast loads, Goel et al. (2012) defined and explained various associated parameters such as explosion, peak over-pressure, positive phase duration of pulse, arrival time of pulse, waveform parameter defining the decay of wave, and wave incident angle. The energy released by explosive charge can broadly be classified into two categories, viz. thermal energy and pressure energy. Thermal energy is released in the form of a bright flash. However, pressure energy is released in the form of shock waves expanding radially and transmitted either through the air or through the ground as a medium. These pressure waves are categorized as high-intensity and short-duration pulses. As the pressure wave propagates, the wave’s intensity and speed of propagation reduce, as shown in Figure 1 (Ngo et al., 2007).

Figure 1.

Variation of blast pressure with distance (after, Ngo et al., 2007).

The shock wave propagation is represented by the pressure-time profile which is categorized into two main phases: the phase above ambient atmospheric pressure is known as the positive phase of duration $t_d$ , while the phase below ambient atmospheric pressure is known as the negative phase of duration $t_d^{-}$ . After reaching the peak over-pressure, the pressure reduces exponentially.

Blast loads can be broadly classified as near blast or distant blast based on the stand-off distance of the structure exposed to an explosion. The near-blast loading produces non-uniform lateral pressure. While the distant-blast loading produces a uniform lateral pressure on a structure, causing a significant side sway, and thus severely damaging the structure. Various deterministic performance-based or probabilistic performance-based approaches can be used to obtain the response as well as the design of a structure subjected to blast loading which are discussed in subsequent paragraphs.

For a performance-based design, a performance-indicating parameter is required to monitor a pre-defined level of damage. As the distant-blast scenario is similar to earthquake load, drift can be used as a performance indicator for the design of the structure. A considerable drift can be observed for framed structures when subjected to blast loading (May and Smith, 1995; Li et al., 2007). Li et al. (2007) proposed a deterministic design methodology for reinforced concrete frame structures subjected to distant-blast conditions using drift as a controlling parameter and presented distant blast loading as an equivalent static force (ESF). ESF is calculated in such a way that this static force reproduces the same drift as produced by blast loading. A lateral deflection, support rotation, and storey drift are well-known performance indicators in the performance-based design of structures subjected to seismic and blast loads. Various deterministic performance-based approaches for structures subjected to blast load are used for preliminary design, using a single degree of freedom approximation without considering strain rate effect (Biggs, 1964; Defence, 2008; El-Dakhakhni et al., 2010; Krauthammer and Astarlioglu, 2017) or considering strain rate effect (Stochino and Carta, 2014; Al-Thairy, 2016), assuming mid-span deflection and support rotation as performance indicators. Pressure impulse curves (or iso-damage curves) were also proposed using a single degree of freedom approximation to aid the design of structural elements subjected to blast (Fallah and Louca, 2007; El-Dakhakhni et al., 2009; Dragos and Wu, 2014; Xu et al., 2014). Detailed finite element analysis using various FEM software such as Abaqus, Ansys, and LS-DYNA were used to simulate blast load and observe the corresponding dynamic response of structure (Jones, 2008; Jayasooriya, 2010; Biju and Ramesh, 2017).

Though a number of deterministic design methodologies for structures were proposed by researchers, the response predictions and respective design methodologies based on this approach are onerous due to high uncertainties related to blasts and earthquakes. The reliability of the structure is quite sensitive to loads and structural parameters. A probabilistic performance-based approach can yield more realistic and conservative results. As a result of the uncertainty associated with loading parameters such as external load, load duration, and structural parameters such as mass and stiffness, the sensitivity analysis becomes an important aspect of the study for structures subjected to blast. Several research studies were conducted on the probabilistic design aspect of structures, and probabilistic blast load models were proposed for sensitivity analysis. Twisdale et al. (1994) conducted statistical analysis for the blast, recommending a coefficient of variance (COV) of 0.3 for peak over-pressure and 0.25 for impulse. Ahmad et al. (2015) performed a probabilistic analysis for blast load and proposed a COV of 0.55 for peak over-pressure and 0.69 for impulse. Stewart (2021) proposed simplified approach to predict the probabilistic model of air blast variability and associated reliability-based design load factors (or RBDFs) for all combinations of range, explosive mass, and model errors. Some works on probabilistic analysis for structural parameters of masonry and concrete structures subjected to blast load were conducted by a few researchers (Low and Hao, 2002; Eamon, 2007; Netherton and Stewart, 2010).

The main focus of these studies was on variability associated with type, the weight of explosives, stand-off distance, and their impact on blast load parameters viz., peak over-pressure and impulse. It was observed that the response of a structure is more sensitive to peak over-pressure, impulse, and the positive duration of a blast load (Smith and Hetherington, 1994). Borenstein and Benaroya (2009) performed a study to observe the effect of uncertainties of blast loading parameters on the maximum deflection of a clamped aluminum plate and concluded the response to be most sensitive to the duration of loading. Low and Hao (2002) developed probabilistic distribution function for the resistance parameters of a structure, taking into account the uncertainties associated with blast loading. Netherton and Stewart (2010) investigated and established the variability of blast loading parameters. Hao et al. (2010) investigated the failure probability of reinforced columns subjected to blast while also emphasizing the significance of taking random variations in structural parameters and blast loadings into account for evaluating the blast load effects on RC columns. By varying blast pressure and soil/rock parameters in a parametric sensitivity analysis for cast iron-lined tunnels, Khan et al. (2016) discovered that the response is more sensitive to tunnel lining thickness, peak blast pressure in the tunnel, and the elastic moduli of soil and rock, while less sensitive to dilation angle of soil and rock. Kumar et al. (2019) noticed the influence of uncertainty in input parameters, along with the effect of pulse shape on the response of the SDOF system; concluding the response to be highly sensitive to a triangular pulse with a sudden rise in force. Kumar et al. (2019) further concluded that stiffer structures are more susceptible to input parameter uncertainties.

Acito et al. (2011) and Olmati et al. (2017) developed probabilistic design methodologies to estimate the probability of damage and failure of a structure subjected to a blast load, and a corresponding factor of safety was proposed. Although few research works have been undertaken on the sensitivity of the response parameters of a structure subjected to blast load, it is obvious from the foregoing literature review that a detailed study is required to establish the framework for the probabilistic performance based design approach.

In view of this, the present study examines performance parameters, drift, and the resistance for a single degree of freedom system for the structure subjected to blast load. The parameter sensitivity analysis is performed by varying loading parameters such as peak over-pressure, equivalent force ( $F_1$ ), positive pulse duration ( $t_d$ ), and structural parameters, viz., the equivalent mass of SDOF ( $m$ ) and equivalent stiffness of SDOF ( $K_s$ ). The present work aims to develop the factor of safety (FOS) in respect of the specified resistance of the SDOF system for a specified target displacement based on the uncertainty of the parameters (loading and structural). The uncertainty in performance parameters is quantified by altering all parameters while considering the log-normal distribution of parameters and the FOS is obtained, using a numerical simulation via Monte-Carlo (MC) approach explained in the subsequent section. Applying such an uncertainty-driven factor of safety to ordinary deterministic designs is expected to add significant contribution to the more rational design of structures.

Methodology

Monte-Carlo simulations

The present work employs Monte-Carlo simulations for probabilistic analysis of system under consideration. Monte-Carlo simulation (Harrison, 2009; Taylor, 2009; Taylor et al., 2011) is a numerical study to estimate the probabilistic mathematical model formed by random sampling of variables. Monte-Carlo simulation process aids in explaining the impact of risk and uncertainty in estimation and forecasting models. The Monte-Carlo simulation is based on the assignment of several random values to an uncertain variable. These values are then fed into a deterministic model to generate state variable values, and the process is repeated several times. After the simulations are finished, the data are averaged to obtain an estimate. The Monte-Carlo simulation, which uses numerous values, may prove to be a superior alternative for producing a forecast or estimation, rather than just replacing the uncertain variable with a single average figure. Monte-Carlo simulation helps in the generation of probability density/distribution functions that describes the possibility of occurrence of a certain step in the simulation. These functions are created using a known mean and the standard deviation of the random process. The relevant mean and standard deviation necessary for simulation are achieved by employing the deterministic models. These simulations depend on spatial coordinates and the number of iterations adopted.

Probabilistic design using ESF model for SDOF system

The present study concerns the probabilistic design of the SDOF system that is subjected to distant blast load (Figure 2). The loading parameters ( $F_1$ and $t_d$ ) and the structural parameters ( $m$ and $K_s$ ) for the system are treated as probabilistic processes. These variables are assumed to be random variables with a known variance and mean (based on the results of past research). Specifics of the adopted approach are given in following sections.

Figure 2.

Equivalent static force (ESF) for SDOF system (after, Li et al., 2007).

Deterministic model

A deterministic model is required to conduct Monte-Carlo simulations. A computational model using the equivalent static force (ESF) technique for blast-resistant design of structures to achieve the specified target displacement given by Li et al. (2007) is used as a deterministic model in the present study and is discussed subsequently.

The response is found for an SDOF system subjected to a distant blast load of amplitude F(t) and duration $t_d$ . To achieve the required response, dynamic blast load is converted into equivalent static force. In the present study, the triangular pulse shape is adopted as a blast loading model for simplification (Biggs, 1964).

Equivalent blast load for SDOF system can be given as follows (Biggs, 1964):

$F\left(t\right)=F_1\left(1-\frac{t}{t_d}\right),\hspace{0.17em}0\le t\le t_d$

(1a)

$F\left(t\right)=0,t_d\le t$

(1b)

Figure 2 provides the conceptual background to obtain equivalent static force for an SDOF system as adopted by Li et al. (2007). It was assumed that once the blast load ceases at $t_d$ , no more external energy will be delivered and hence total energy of the SDOF system will remain constant. It was also assumed that maximum displacement for the SDOF system occurs at $t_d$ or after $t_d$ An equivalent static force (ESF) factor (λ) was introduced to reduce the static force (P) for producing the equivalent static force (ESF) (P _st ) to create same maximum response as would have been produced by dynamic analysis of the SDOF system (Li et al., 2007) as given below:

$P_{st}=\lambda \hspace{0.17em}P$

(2)

For an elastic-perfectly plastic SDOF system, a closed-form solution of equivalent static force (ESF) factor (λ) was derived for three cases based on the response of the system and is mentioned as follows:

Case I: Elastic state at $t=t_d$ and $t=t_m$

$P=K_s{y}_m\left(\frac{y_m}{y_{t_d}}\right)$

(3a)

$P_{st}=K_s{y}_m$

(3b)

$\lambda =y_{t_d}/y_m$

(3c)

Case II: Elastic state at $t=t_d$ and plastic state at $t=t_m$

$P=\frac{K_s{y}_e\left(2y_m-y_e\right)}{y_{t_d}}$

(4a)

$P_{st}=K_s{y}_e$

(4b)

$\lambda =\frac{y_{t_d}}{\left(2y_m-y_e\right)}$

(4c)

Case III: Plastic state at $t=t_d$ and $t=t_m$

$P=K_s{y}_e\frac{\left(2y_m-y_{t_d}\right)}{y_{t_d}}$

(5a)

$P_{st}=K_s{y}_e$

(5b)

$\lambda =\frac{y_{t_d}}{\left(2y_m-y_{t_d}\right)}$

(5c)

For designing an SDOF system with the ESF, the procedure given by Li et al. (2007) can be followed which included the determination of ultimate strength of the SDOF system such that specific target displacement is achieved. For this purpose, initial resistance $R_{mo}$ and $y_{mref}$ were obtained by linear dynamic analysis for input parameters such as equivalent stiffness of the SDOF system (K _s ), equivalent mass of the SDOF system (m), maximum value of blast load (F ₁ ), and the duration of blast load (t _d ). λ and P can then be obtained using equations (Biggs, 1964; Biju and Remesh, 2017; Borenstein and Benaroya, 2009), and thus a new $R_m$ was obtained based on conditions for various cases as mentioned above. The process is repeated until the solution converges; thus, giving us the required resistance $R_m$ for pre-defined value of displacement or drift $y_t$ .

Details of simulations

Aforementioned deterministic model is adopted for conducting the Monte-Carlo simulation, with $K_s$ , $m$ , $F_1$ , and $t_d$ treated as random variables. These random variables are considered log-normally distributed (Low and Hao, 2002; Netherton and Stewart, 2010; Khan et al., 2016; Kumar et al., 2019) with a mean (μ) and standard deviation (σ). The mean and standard deviation must be known to form a probability density/distribution function of the selected parameters. The mean is assumed to be a deterministic value of the selected parameter. The adopted standard deviations (σ) suggested by Gupta and Manohar (2006) are given in Table 1. Monte-Carlo simulations are performed by developing a code in MATLAB to analyze the probabilistic attributes of random variables ( $K_s$ , $m$ , $F_1$ , and $t_d$ ). The resistance of the SDOF system (R _m ) and the equivalent static load factor (λ) for the specified blast-load scenario are the major state variables of importance from the perspective of design of the SDOF system.

Table 1.

Coefficient of variance (COV) for various input parameters (Gupta and Manohar, 2006).

Parameter	Mass (m) (kg)	Stiffness, % (K s ) (N/m)	Peak force (F 1 ), % (N)	Positive pulse duration (t d ), % (s)
COV, %	5	5	10	5

The present study for design considerations purposefully investigates cases II (elastic state at $t=t_d$ and plastic state at $t=t_m$ ) and III (plastic state at $t=t_d$ and $t=t_m$ ) in which target displacement occurs at plastic stage. The formulation described in section 2 is demonstrated using three examples of the SDOF system covering different cases II and III, based on the response of the system investigated by earlier studies. Table 2 summarizes the features of the SDOF system as well as the blast-load information. Results from the analysis are explored in depth in Section 3 of this paper.

Table 2.

Parameters and their mean values of various examples under consideration.

S. No.	Parameter	Case II	Case III	Case III
		Example 1 (Li et al., 2007)	Example 2 (Li et al., 2007)	Example 3 (Biggs, 1964)
1	Peak blast force ( $F_1$ ) (N)	2000	1000	4225 × 103
2	Positive pulse duration ( $t_d$ ) (s)	0.25	0.04	0.058
3	Equivalent mass of the SDOF system (m) (kg)	200	100	29,846.35
4	Equivalent stiffness of the SDOF system ( $K_s$ ) (kN/m)	20	10	142.12 × 103
5	Target displacement ( $y_t$ ) (mm)	500	100	200

Results and discussion

Parameter sensitivity analysis

For the SDOF systems having parameters as mentioned in Table 2, a sensitivity analysis is performed by varying peak blast force ( $F_1$ ), positive pulse duration ( $t_d$ ), equivalent mass of the SDOF system (m), and equivalent stiffness of the SDOF system ( $K_s$ ), individually at a time while keeping other parameters constant. All parameters are varied ±5%, ±10%, and ±20% from their mean values as mentioned in Table 2 (values of the deterministic model are assumed as mean values). The effect of variation of considered parameters is presented in subsequent paragraphs and figures.

Example 1

Example 1 represents case II (elastic state at $t=t_d$ and plastic state at $t=t_m$ ) of the deterministic model based on the response of the SDOF system. The equivalent SDOF system properties can be referred from Table 2. For 2000 N, peak blast force at mean positive pulse duration ( $t_d$ ) of 0.25 sec, ESF factor (λ) of 0.2141, and resistance ( $R_m$ ) of 279.6967 N is observed for the SDOF system for targeted displacement, y _t of 500 mm (Li et al., 2007). Figures 3 and 4 depict the fluctuation of random variables, ESF factor (λ) and resistance ( $R_m$ ), respectively, for the SDOF system under study after being subjected to blast load. The mean value of initial peak blast force ( $F_1$ ) is set as 2000 N with the variations of ±5%, ±10%, and ±20% of the peak value. For the same positive pulse duration, the ESF factor (λ) and resistance ( $R_m$ ) drop as the peak force decreases. When the positive pulse duration is taken as 0.25 sec, with the variations of ±5%, ±10%, and ±20% of the mean value at peak blast load ( $F_1$ ), a similar pattern is observed.

Figure 3.

ESF factor (λ) for different parametric variations for the SDOF system: Example 1.

Figure 4.

Resistance ( $R_m$ ) (N) for different parametric variations for the SDOF system: Example 1.

The ESF factor (λ) and resistance ( $R_m$ ) of SDOF decrease as the equivalent mass (m) of the SDOF system increases. Though the same trend is observed when an equivalent stiffness of the SDOF system ( $K_s$ ) is increased, however, any significant increase is not observed when this parameter is varied.

The variation of examined parameters does not display similar response patterns on the state variables, ESF factor (λ), and resistance ( $R_m$ ) of the SDOF system, according to the findings of parametric sensitivity analysis. For greater insight into the observed structural response trend of the SDOF system corresponding to the parametric variation in the present analysis, a function to calculate error percentage has been introduced (Khan et al., 2016). An error percentage function has been defined as a ratio of variation in response obtained using varying input parameters from the response obtained using mean values of input parameters to the response obtained for the input parameters’ mean values. The error in response of the SDOF system for variation in each specified parameter is determined as follows:

$Error\hspace{0.17em}(\%)=\frac{P_{mean}-P_{var}}{P_{mean}}\hspace{0.17em}*\hspace{0.17em}100$

(6)

where $P_{mean}$ is the response of the SDOF system for mean values of the parameters and $P_{var}$ is the response of the SDOF system for variation in the 201 value of the parameters.

For variations of ±5%, ±10%, and ±20% in the selected parameters, the error has been calculated. The comparison of the variations for a given case has been shown in Figures 5 and 6 for ESF factor (λ) and resistance ( $R_m$ ), respectively. It can be observed from respective figures that positive pulse duration of the load ( $t_d$ ) has a significant effect on the ESF factor (λ) while peak blast force ( $F_1$ ) has a substantial impact on the resistance ( $R_m$ ) of the SDOF system. A maximum of 44.56% variation in ESF factor is observed for a 20% increase in positive pulse duration ( $t_d$ ), followed by 36.43% for a 20% reduction in positive pulse duration ( $t_d$ ). A maximum of 42.3% variation in resistance ( $R_m$ ) is observed for a 20% increase in peak blast force ( $F_1$ ), followed by a 37.34% for a 20% increase in positive pulse duration ( $t_d$ ). Both ESF factor (λ) and resistance ( $R_m$ ) are least sensitive to the equivalent stiffness of the SDOF system, implying stiffness has significantly less role to play for given targeted displacement to achieve required resistance. A significantly higher value of equivalent stiffness of the SDOF system ( $K_s$ ) is selected for the design model (case II) for attaining a prescribed target displacement/drift at the plastic state. Slight variations opted for study in the equivalent stiffness do not influence the required resistance to a greater extent as target displacement is achieved at plastic state (at $t=t_m)$ . Thus, it can be concluded that the required resistance to attain the pre-defined target displacement/drift response obtained using the proposed design method is greatly influenced by loading parameters, peak blast force ( $F_1$ ), and blast load duration ( $t_d$ ).

Figure 5.

Percentage error in the ESF factor (λ) for the SDOF system: Example 1.

Figure 6.

Percentage error in resistance ( $R_m$ ) (N) for the SDOF system: Example 1.

Example 2

Example 2 represents case III (plastic state at $t=t_d$ and plastic state at $t=t_m$ ) based on the response of the SDOF system. The equivalent SDOF system properties can be referred from Table 2. For 1000 N, peak blast force at mean positive pulse duration ( $t_d$ ) of 0.04 sec, ESF factor (λ) of 0.027, and resistance ( $R_m$ ) of 20.042 N is observed for the SDOF system for targeted displacement ( $y_t=100\hspace{0.17em}mm$ ). Figures 7 and 8 show the variation of ESF factor (λ) and resistance ( $R_m$ ), respectively, for the SDOF system, corresponding to variation in input parameters. All observed trends are similar to the trends observed in the case of Example 1.

Figure 7.

ESF factor (λ) for different parametric variations for the SDOF system: Example 2.

Figure 8.

Resistance ( $R_m$ ) (N) for different parametric variations for the SDOF system: Example 2.

For variations of ±5%, ±10%, and ±20% in the selected parameters, the error is calculated. The comparison of the variations for the given case is shown in Figures 9 and 10 for ESF factor (λ) and resistance ( $R_m$ ), respectively. A maximum of 45.18% variation in ESF factor (λ) is observed for a 20% increase in positive pulse duration ( $t_d$ ), followed by 36.29% for a 20% decrease in positive pulse duration ( $t_d$ ). A maximum of 44.59% variation in resistance ( $R_m$ ) is observed for a 20% increase in peak blast force ( $F_1$ ), followed by a 44.31% for a 20% increase in positive pulse duration ( $t_d$ ).

Figure 9.

Percentage error in the ESF factor (λ) for the SDOF system: Example 2.

Figure 10.

Percentage error in resistance ( $R_m$ ) (N) for the SDOF system: Example 2.

Example 3

In this example, a frame is idealized as an equivalent SDOF system, as given in Figure 11 (Biggs, 1964). The equivalent SDOF system properties can be referred from Table 2 for analysis of the frame. Example 3 represents case III (plastic state at $t=t_d$ and plastic state at $t=t_m$ ) based on the response of the SDOF system.

Figure 11.

Frame subjected to lateral blast load (after, Biggs, 1964).

An ESF model is applied for targeted displacement ( $y_t$ ) of 200 mm. The required resistance for the pre-defined target displacement obtained by the analysis is 1001.4 kN. In comparison, resistance for the same target displacement was reported 995.952 kN according to graphs used by Biggs (1964).

Figures 12 and 13 show the variation of ESF factor (λ) and resistance ( $R_m$ ), respectively, for the SDOF system corresponding to variation in input parameters. All observed trends are similar to the trends observed in the case of Example 1.

Figure 12.

ESF factor (λ) for different parametric variations for the SDOF system: Example 3.

Figure 13.

Resistance ( $R_m$ ) (N) for different parametric variations for the SDOF system: Example 3.

For variations of ±5%, ±10%, and ±20% in the selected parameters, the error is calculated. The comparison of the variations for the given case is shown in Figures 14 and 15 for ESF factor (λ) and resistance (R _m ), respectively. A maximum of 40.6% variation in ESF factor is observed for a 20% increase in positive pulse duration ( $t_d$ ), followed by 35.64% for a 20% decrease in positive pulse duration ( $t_d$ ). A maximum of 39.85% variation in resistance ( $R_m$ ) is observed for a 20% increase in peak blast force ( $F_1$ ), followed by a 33.86% for a 20% decrease in peak blast force ( $F_1$ ).

Figure 14.

Percentage error in the ESF factor (λ) for the SDOF system: Example 3.

Figure 15.

Percentage error in resistance ( $R_m$ ) (N) for the SDOF system: Example 3.

Uncertainty quantification

Selection of an optimal number of realizations for Monte-Carlo simulation

The total number of iterations to be used does not have any upper limit for performing simulations using Monte-Carlo method. With the increase in the number of iterations, the sample approaches the population. Driels and Shin (2004) proposed an equation, mentioned as equation Driels and Shin (2004), to determine an optimum number of iterations that is required to attain a pre-defined percentage error with a particular confidence level. For the number of iterations obtained using equation Driels and Shin (2004), it has been observed that the solution converges quickly and is relatively stable.

$n={\left(\frac{100z_c{\sigma }_x}{E\mu }\right)}^2$

(7)

where $z_c$ is the confidence value for the corresponding confidence interval, ${\sigma }_x$ is the standard deviation, $\mu$ is the mean, and $E$ is the percentage error. The ESF factor data of Example 1 is used for finalizing the number of realizations for the Monte-Carlo simulation. The confidence level of 95% is aimed, and the corresponding values of $z_c$ , $\mu$ , and ${\sigma }_x$ are 1.96, 0.2141, and 0.0373, respectively (the deterministic model is assumed as mean value and data obtained performing sensitivity analysis is used to obtain standard deviation). The value of error (E) is assumed as 1. The number of iterations computed using equation Driels and Shin (2004) is 1166. Any number of iterations greater than 1166 may be adopted for the simulation. Hence, in the present study, probability distribution function is plotted for 1500, 1750, 2000, and 2250 number of iterations and the results are shown in Figures 16 and 17 with respect to ESF factor, λ, and R _m , respectively. Accordingly, 2000 iterations are adopted for further analysis.

Figure 16.

Probability distribution curves for ESF factor (λ).

Figure 17.

Probability distribution curves for resistance ( $R_m$ ) (N).

Monte-Carlo simulation study

After fixing the number of iterations for the Monte-Carlo process, the simulation is conducted on the considered deterministic SDOF system models (Section 2.2.1 with parameters mentioned in Table 2). This helps in analyzing the effect of uncertainties associated with input parameters ( $K_s$ , $m$ , $F_1$ , and $t_d$ ) on the response of the SDOF system in terms of the ESF factor (λ) and resistance (R _m ) of the SDOF system.

Ensuing to generations of n (= 2000) realizations of the response parameters, ESF factor (λ) and resistance (R _m ) of the SDOF system, the attributes of these two random state variables are produced as follows:

I. Distribution functions—The realizations of random variables (λ and R _m ) as obtained from simulations are employed to generate the distribution functions of random variables λ _i and $R_{m_i}$ (i = 1, 2, ..., N) as follows:

$F_{D_i}\left(\lambda \right)=P\left({\lambda }_i\le \lambda \right)=\frac{k_1}{n+1}$

(8a)

$F_{D_i}\left(R_m\right)=P\left(R_{m_i}\le R_m\right)=\frac{k_2}{n+1}$

(8b)

where k ₁ and k ₂ are the ranks of λ and R _m , respectively, among arrays of n (= 2000) realizations.

• ii. Mean and standard deviation—Invoking “n” corresponding realizations, the mean and standard deviation of random state variables (λ and R _m ) are obtained (i.e., λ _i , $R_{m_i}$ and i = 1, 2, .., n).

• iii. Factor of safety (FOS)—With the aforementioned context, the deterministic design requirement (restricting R _m to $R_{m_1}$ ) may be phrased as follows in a probabilistic form (Maheshwari and Kashyap, 2009, 2011):

$E\left(R_m\right)=R_{m_1}$

(9)

In the deterministic design, the risk of failure (i.e., R _m below $R_{m_m}$ ) may be represented as follows:

$\epsilon =P\left(R_m\le R_{m_m}\right)$

(10a)

$or$

$\epsilon =F_D\left({\mu }_{R_m}/FOS\right)$

(10b)

where $\epsilon$ is the risk of failure connected with the deterministic design characterized by FOS. $F_D$ (R _m ) is the probability distribution function (pdf) of R _m . As a result, the required FOS for a given risk level may be calculated as follows:

$FOS={\mu }_{R_m}{/F}_D^{-}\left(\epsilon \right)$

(11)

It can be stated from above that a lower recommended risk level will always result in a higher factor of safety. This means that by appropriately raising the FOS, the risk of failure can be minimized significantly.

The present study assumes that the input parameters (say r— $K_s$ , $m$ , $F_1$ , and $t_d$ ) follow a log-normal distribution (Low and Hao, 2002; Netherton and Stewart, 2010; Khan et al., 2016; Kumar et al., 2019). This indicates that ln(r) is normally distributed. These realizations, along with the stipulated values of input parameters, are entered into the previously defined deterministic SDOF model for the ESF factor and resistance to compute the realizations of λ and R _m . These realizations are involved in building the probability distribution function ( $F_D$ ) of R _m, which yields an array of $\epsilon$ and the appropriate FOS according to equations (10b) and Fallah and Louca (2007). Results obtained using the Monte-Carlo simulations are presented in depth for each of the three examples (Table 2) subsequently.

Example 1

Histograms of the ESF factor (λ) and resistance (R _m ) for log-normal distribution of input parameters for Example 1 are shown in Figures 18 and 19, respectively. After considering that the input parameters follow the log-normal distribution, the distributions of response parameters, ESF factor and resistance of the SDOF system, are obtained with the mean value of 0.2142 and 281.7 N, respectively.

Figure 18.

Distribution of the ESF factor (λ) considering the log-normal distribution of input parameters: Example 1.

Figure 19.

Distribution of resistance ( $R_m$ ) (N) considering the log-normal distribution of input parameters: Example 1.

The visual shape of the histograms indicates a log-normal distribution; thus, the ESF factor and resistance distributions are examined for being log-normally distributed. Both ESF factor distribution and resistance distribution are subjected to a Chi-square value test for a 95% confidence level. For ESF factor distribution, the tabular value is 132.144, while the calculated value obtained is 118.5642. Hence, the given ESF factor distribution can be treated as log-normally distributed with a 95% confidence level. Similarly, the tabular value for resistance distribution is 80.232, while the calculated value obtained is 78.16896. Hence, the given resistance distribution can also be treated as log-normally distributed with a 95% confidence level.

Relating Factor of safety (FOS) with risk of failure ( $\epsilon )$ :

The design challenge of achieving optimum resistance R _m of the SDOF system for a pre-defined target displacement may be tackled using a probabilistic or deterministic approach using the aforementioned model. This problem of finding optimum resistance is greatly influenced by the uncertainties associated with the structural and loading parameters.

All the realizations generated for the resistance R _m of the SDOF system in the probabilistic approach meet the requirement of attaining the prescribed target displacement of 500 mm (for Example 1). The obtained resistance R _m can be characterized by a probability distribution function or by its mean and standard deviation. For the SDOF under consideration, the probability of undershooting (P (R _m ≤ $R_{m_m}$ )) of the resistance parameter ${\hspace{0.17em}R}_m$ with respect to the postulated threshold $R_{m_m}$ indicates the probability of failure or risk of failure ( $\epsilon )$ Figure 20 illustrates the probability distribution function $F_D\left(R_m\right)$ for the resistance of the SDOF system under consideration, describing the probability of undershooting (P (R _m ≤ $R_{m_m}$ ) = 10%) of the resistance parameter $R_m$ with respect to the threshold $R_{m_m}=213.35\hspace{0.17em}\mathrm{N}$ (the shaded area under the curve shows failure).

Figure 20.

Probability distribution curve for resistance (R _m ) (N) of the SDOF system: Example 1.

In the deterministic approach, the resistance parameter $R_m$ is computed using the deterministic model (Section 2.2.1 of present work) by assigning the mean values to the uncertain input parameters. If a value of resistance lower than the deterministic resistance $R_{m_1}$ is adopted, a structure developed using a deterministic technique is said to fail. Further, this will cause the structure to collapse as the displacement exceeds 500 mm (prescribed target displacement).

The computed resistance parameter $R_m$ estimated using the deterministic approach inherits the inherent uncertainties. Thus, the uncertainties in a deterministic analysis model can be addressed by adopting a suitable factor of safety (FOS) based on the probabilistic analysis. Keeping this in view, the resistance of the SDOF system should be subjected to a lower limit $R_{m_m}$ described previously, such that values lower than $R_{m_m}$ will lead to the failure of the structure. The lower limit $R_{m_m}$ is controlled or imposed on $R_{m_1}$ (deterministic value) by the use of an arbitrarily assigned factor of safety, that is, FOS = $R_{m_1}/R_{m_m}$ . Hence, by constraining variable R _m to deterministic resistance, $R_{m_1}$ as per equation El-Dakhakhni et al. (2009), the deterministic design may be expressed in a probabilistic mode.

In the present example, the variable resistance (R _m ) calculated using a probabilistic approach is limited to a predetermined lower limit, say $R_{m_m}=213.35\hspace{0.17em}\mathrm{N}$ with an acceptable margin of failure $\epsilon =P\left(R_m\le R_{m_m}\right)=10\hspace{0.17em}\%$ and the system is said to fail if it falls below this limit. The means identified as deterministic resistance $R_{m_1}$ , that is, 279.69 N, obtained via deterministic analysis. The lower limit on $R_{m_1}$ (deterministic value) is controlled or imposed by an arbitrarily assigned factor of safety (FOS) = 1.311 ( $FOS=R_{m_1}/R_{m_m}$ ).

Using equations El-Dakhakhni et al. (2010) and Fallah and Louca (2007), the generated probability distribution function (F _D ) for the SDOF system’s resistance (R _m ) is used to build an array of the associated risk of failure ( $\epsilon$ = (P (R _m ≤ $R_{m_m}$ ))) and the corresponding FOS, as depicted in Figure 21. The lower prescribed risk of failure level shall always lead to a higher factor of safety. This scenario can be compared with the design strength opted in the limit state design method by incorporating a suitable factor of safety (FOS) to address uncertainties.

Figure 21.

Factor of safety (FOS) vs. probability of failure or risk of failure (ε %): Example 1.

Example 2

Figures 22 and 23 display histograms of the ESF factor (λ) and resistance ( $R_m$ ) for log-normal distribution of input parameters for Example 2. After considering that the input parameters follow the log-normal distribution, the value of the mean ESF factor is 0.02715, and the mean resistance is 20.3 N. The Chi-square test tabular value for ESF factor distribution is 132.144, whereas the computed value achieved is 118.5642. Similarly, the tabular value for resistance distribution is 80.73, whereas the computed value obtained is 78.16. The probability distribution function $F_D\left(R_m\right)$ for resistance of the SDOF system under investigation is shown in Figure 24. Figure 25 presents the discrete FOS generated with respect to the resistance of the SDOF system corresponding to the associated probability of failure or risk of failure ( $\epsilon$ = (P (R _m ≤ $R_{m_m}$ ))).

Figure 22.

Distribution of the ESF factor (λ) considering the log-normal distribution of input parameters: Example 2.

Figure 23.

Distribution of resistance ( $R_m$ ) (N) considering the log-normal distribution of input parameters: Example 2.

Figure 24.

Probability distribution curve for resistance (R _m ) (N) of the SDOF system: Example 2.

Figure 25.

Factor of safety (FOS) vs. probability of failure or risk of failure (ε %): Example 2.

Example 3

Histograms of the ESF factor (λ) and resistance ( $R_m$ ) for log-normal distribution of input parameters for Example 3 are shown in Figures 26 and 27, respectively. After considering that the input parameters follow the log-normal distribution, the value of the mean ESF factor is 0.4037, and the mean resistance is 1008 kN. For ESF factor distribution, the Chi-square test tabular value is 93.945, while the calculated value obtained is 84.66. Similarly, the tabular value for the resistance distribution is 74.468, while the calculated value obtained is 63.57. Figure 28 displays the probability distribution function $F_D\left(R_m\right)$ for resistance of the SDOF system under consideration. Figure 29 presents the generated discrete FOS with respect to the resistance of the SDOF system corresponding to the associated probability of failure or risk of failure ( $\epsilon$ = (P (R _m ≤ $R_{m_m}$ ))).

Figure 26.

Distribution of the ESF factor (λ) considering the log-normal distribution of input parameters: Example 3.

Figure 27.

Distribution of resistance (R _m ) (N) considering the log-normal distribution of input parameters: Example 3.

Figure 28.

Probability distribution curve for resistance (R _m ) (N) of the SDOF system: Example 3.

Figure 29.

Factor of safety (FOS) versus probability of failure or risk of failure (ε %): Example 3.

Factor of safety (FOS)

Figure 30 presents the generated discrete FOS with respect to the resistance of the SDOF system corresponding to the associated probability of failure or risk of failure ( $\epsilon$ = (P (R _m ≤ $R_{m_m}$ ))) for all the three aforementioned examples (Table 2). It can be observed from the figure that variable parameters for all three cases do not affect the factor of safety (FOS) significantly. Taking this under consideration, an envelope on the FOS data is fitted onto all the three considered examples. The fit is found to be good with the correlation coefficient $R^2=0.9388$ . Thus, in order to integrate the impacts of uncertainties in the deterministic design technique of getting optimal resistance of the SDOF system for pre-requisite target displacement, an appropriate factor of safety (FOS) can be adopted for the specified risk of failure, as presented below.

Figure 30.

Factor of safety (FOS) versus probability of failure or risk of failure (ε %).

The closed-form expression for the factor of safety (FOS) in terms of the probability of failure or risk of failure ( $\epsilon$ = (P (R _m ≤ $R_{m_m}$ ))) is given as follows:

$FOS=1.6083\hspace{0.17em}{\left(\epsilon \right)}^{-0.081}$

(12)

Table 3 shows the computed factor of safety (FOS) for different risk levels ( $\epsilon$ ). It can be concluded from above that the lower prescribed risk of failure ( $\epsilon$ ) will result in a higher factor of safety (FOS).

Table 3.

Factor of safety (FOS) corresponding to the associated probability of failure or risk of failure ( $\epsilon$ ).

Probability of failure or risk of failure (ε) (%)	1	5	10	20	50
Factor of safety (FOS)	1.6083	1.4117	1.3346	1.2617	1.1715

Conclusion

An equivalent static force (ESF) model for the preliminary design of the SDOF system subjected to triangular pulse shaped load was considered for the probabilistic analysis. The loading parameters such as peak force ( $F_1$ ) and positive pulse duration $\left(t_d\right)$ and structural parameters such as equivalent mass ( $m$ ) and equivalent stiffness $(K_s$ ) of the system under consideration were treated as probabilistic processes. The resistance of the SDOF system (R _m ) and the equivalent static force factor (ESF) for the specified blast load scenario are the major state variables of importance from the perspective of the SDOF system design, hence considered as response parameters for the study. The formulation is demonstrated using three SDOF system examples covering different cases based on the response of the system.

Parametric sensitivity analysis was performed by varying input parameters ( $F_1$ , $t_d$ , $m$ , and $K_s$ ) by a value of ± 5%, ± 10%, and ± 20% from their deterministic values assuming them as means based on the results of past research. It is observed that as the peak blast force and positive pulse duration rise, so do the ESF factor and the resistance of the SDOF system. The ESF factor is very sensitive to positive pulse duration, whereas resistance is highly sensitive to the peak blast force and positive pulse duration. Furthermore, when the equivalent mass of the SDOF system increases, the ESF factor and the resistance of the system drop. In comparison, the variations in the equivalent stiffness of the SDOF system have less impact on the ESF factor and the required resistance of the SDOF system for pre-defined targeted displacement, because once chosen for achieving a prescribed target displacement/drift (which is generally attained at plastic state), the required resistance is not affected.

Probabilistic analysis was performed using Monte-Carlo simulation after selecting an optimal number of iterations. Assuming a log-normal distribution of input parameters, the derived ESF factor and the resistance distributions also follow log-normal distribution which is validated using the Chi-square test. The present study proposes that the structural elements exposed to blast loading should be designed in a probabilistic mode because of the inherent uncertainty associated with the loading as well as structural parameters. However, deterministic analysis is the most often utilized approach for designing SDOF systems exposed to blast load because of its simplicity. The present work amalgamates the two approaches by associating the appropriate factor of safety (FOS) (in terms of minimum resistance) to the permitted risk of failure, which is in essence a characteristic of the probabilistic design. A closed-form equation for the factor of safety (FOS) in terms of risk of failure was proposed, which may be used to calculate the requisite factor of safety for limiting the risk of failure to any specified level.