Prediction of average debris launch velocity from a reinforced concrete structure based on SDOF system

Abstract

This study presents a physics-based model for debris launch velocity prediction of a reinforced concrete (RC) structure subjected to a blast load. The model is basically derived from energy conservation equation. Especially, a resistance-deflection relationship for the structural single degree of freedom (SDOF) system is newly considered to evaluate the energy consumed by the damage and fragmentation of the RC structure. By applying the resistance-deflection relationship, the proposed model can consider the interactions between reinforcing bars and concrete. Moreover, since the resistance-deflection curve is evaluated considering various structural properties as well as boundary conditions, the proposed model can be flexibly utilized compared to conventional approaches. In order to confirm the performance of the proposed model, a comparative study was carried out against benchmark experiments on closed concrete box structures under an internal blast. From the comparative study, it was shown that the debris launch velocities estimated from the proposed model had a good agreement with the test results compared with the other models.

Keywords

SDOF system debris launch velocity energy conservation resistance-deflection curve RC structure

Introduction

An unexpected blast load such as terrorist attacks and various accidental explosions is able to cause significant damages to structures. The damages can induce structural partial or overall collapse. Therefore, it is essential to perform a protective design for military facilities and facilities exposed to explosions to prevent their catastrophic failure.

An equivalent single degree of freedom (SDOF) system for a structural component has been intensively developed (Al-Thiry, 2016; Dragos and Wu, 2015; El-Dakhakhni et al., 2010; Feldgun et al., 2016; Fischer and Haring, 2009; Morison, 2006; Rigby et al., 2012; Stochino and Carta, 2014) to predict structural damages due to a blast load. The SDOF system has advantages in that the system can simply construct structural computational models and rapidly estimate structural damages. The SDOF system can also consider a variety of structural responses such as flexural response, tension membrane response, compression membrane response, and axial load arching response (UFC-3-340-02, 2008; U.S. Army Corps Engineers, 2008a). Recently, studies to evaluate structural responses exposed to various external loading conditions have been performed (Rigby and Tyas, 2014; Wang et al., 2013).

The debris dispersion resulting from an accidental explosion as well as a direct blast load is a governing factor in the determination of the structural damages and the personal injuries. A few empirical or semi-theoretical approaches under an internal blast load to predict the debris launch velocity were proposed (Dorr et al., 2002; Lu and Xu, 2007). The basic equation of the conventional semi-theoretical approach is based on energy conservation. However, the approaches independently deal with the effects of reinforcing bars and concrete to calculate the specific failure energy in a nominal fragment. The specific energy was defined as the energy per unit area in the previous study (Lu and Xu, 2007). For this reason, the effect of interactions between reinforcing bars and concrete could not be reflected for a specific failure energy calculation. In addition, the approach can not consider structural boundary conditions and various structural properties.

In order to overcome the limitations, this study proposes a new debris launch velocity prediction model based on a resistance-deflection relationship for the structural SDOF system. By applying the resistance-deflection relationship for the calculation of a specific failure energy, the proposed model can consider the interactions between reinforcing bars and concrete. Moreover, since the resistance-deflection curve is evaluated considering various structural properties as well as structural boundary conditions (UFC-3-340-02, 2008; U.S. Army Corps Engineers, 2008a), the proposed model can be more flexibly utilized than other approaches.

In this paper, the proposed model is validated against benchmark experiments (Lu and Xu, 2007) on closed concrete box structures under an internal blast to confirm its performance enhancement.

Theory

In the previous study (Lu and Xu, 2007), the authors proposed the semi-theoretical approach for the debris velocity prediction under an internal blast load. The debris velocity was derived from energy conservation as expressed in equation (1). In other words, considering the fragments in the critical region where the fragments exhibit highest velocity, the work done by the blast load is transformed into the fragmentation of structural component and the fragment kinetic energy. This study basically assumes that there is no energy loss in debris occurrence mechanism.

W = U_{c} + U_{k}

(1)

where W is the work done by the blast load, U_c is the energy consumed by the damage and fragmentation of the concrete structure due to fragment motions, and U_k is the kinetic energy in the system, respectively.

Estimation of the energy consumed by the damage and fragmentation

For a nominal fragment unit i, the U_ci can be expressed as

U_{ci} = u_{ci} \times S_{i}

(2)

where u_ci is the specific failure energy, and S_i is the front face area of the fragment.

The conventional semi-theoretical approach (Lu and Xu, 2007), however, independently considered the effects of reinforcing bars and concrete to calculate the specific failure energy in a nominal fragment, as expressed in equation (3).

u_{ci} = \frac{1}{2} \times t \times [f_{yd} {\bar{ε}}_{yd} R_{d} + f_{cd} {\bar{ε}}_{cd} R_{c}]

(3)

where ${\bar{ε}}_{yd}$ is the dynamic yield strain of the reinforcing bars, ${\bar{ε}}_{cd}$ is the dynamic failure string of the concrete, f_yd is the dynamic yield strength of the reinforcing bars, f_cd is the dynamic strength of the concrete, t is the thickness of reinforced concrete structure, and R_c and R_d are the volumetric ratios of concrete and reinforcement, respectively.

For more precise predictions, the effect of interactions between reinforcing bars and concrete needs to be considered for the specific failure energy estimation. To overcome the limitations, this study estimates u_ci in equation (3) based on a resistance-deflection relationship of the structural SDOF system.

The resistance of the SDOF system is the spring force that develops in the structural component due to internal stresses as it deflects (U.S. Army Corps of Engineers, 2008a). In addition, the SDOF system is derived on the assumption that it is subjected to uniformly distributed loads (U.S. Army Corps of Engineers, 2008a; UFC-3-340-02, 2008). Thus, the u_ci can be calculated based on the resistance-deflection relationship of the SDOF system as:

u_{ci} = \int_{0}^{FP} R (x) dx

(4)

where R(x) is the resistance of the structural SDOF system (See Figure 1). The FP is the fracture point (FP) of the structure.

Figure 1.

Typical example of a resistance-deflection curve for the structural SDOF system.

This study defines the FP as the blow out threshold based on the structural damage criteria (U.S. Army Corps of Engineers, 2008b). The damage criteria are estimated as the rotation angle, as shown in Figure 2. Thus, the deflections at the criteria can be calculated based on the relationship between the deflection (Δ_Threshold) at the center of the structure and the rotation angle (θ_Threshold). The FP may be adjusted by users according to engineering judgement based on experimental data. The bigger rotation angle between the L-direction rotation angle and the H-direction rotation angle is utilized as the threshold for the RC slab damages.

Figure 2.

Estimation of the deflection corresponding to the rotation angle for the damage criterion.

In order to construct the resistance-deflection curve for a reinforced concrete (RC) structure, an ultimate resistance and a stiffness of the structure are required. The equations for the ultimate resistance and the stiffness are described in UFC-3-340-02 (2008). For a one-way structural component, the values can be evaluated according to moment capacity, and boundary conditions, etc. In addition, the ultimate resistance and the stiffness in a two-way component can be obtained utilizing the equations that are described in accordance with moment capacity, yield lines, and boundary conditions, etc. In other words, the proposed model can consider the structural type (i.e. one-way or two-way) as well as the structural boundary conditions.

Estimation of the kinetic energy in the system

After the process of fragmentation, the debris will continue accelerating until the pressure can be no longer impart energy on a fragment, and the debris reaches the maximum velocity (Lu and Xu, 2007). Lu and Xu (2007) considered the flexural failure of the RC slab and the average fragment size. In addition, the direction of debris was considered in one direction (i.e. perpendicular to the RC slab). The assumption of this study is identical to the methodology presented by Lu and Xu (2007). The kinetic energy of a fragment at the maximum velocity can be expressed as

U_{ki} = \frac{1}{2} \times M_{i} \times v_{i}^{2} = u_{ki} \times S_{i}

(5)

u_{ki} = \frac{1}{2} \times m_{i} \times v_{i}^{2}

(5a)

where M_i is the mass of the fragment, v_i is its velocity, m_i is the specific mass, m_i = M_i/S_i, and u_ki is the specific kinetic energy

Estimation of the work done by the blast load

The specific work done by the shock waves and the gas overpressures can be calculated based on the impulse of the blast load and the specific mass, as expressed in equation (6a) (Lu and Xu, 2007). The impulse of the blast load can be calculated using the explosive weight, the distance from the target location to the explosive location, the volume of the room, the opening size of the room, etc (Anderson, 1983). According to the detonation environment, a user can utilize a proper blast propagation model (Anderson, 1983; Needham, 2018). Thus, this study can simultaneously consider the shock waves and the gas overpressures delivered to the fragment i by the blast load as

W_{i} = (w_{s} + w_{g}) \times S_{i}

(6)

w_{s} + w_{g} = \frac{i_{r}^{2}}{2 \times m_{i}}

(6a)

where w_s the specific work by the shock wave, w_g the specific work by the gas overpressure, and i_r is the impulse per unit area, respectively.

Estimation of the debris launch velocity

According to equation (1), the kinetic energy of the fragment can be expressed as

U_{ki} = W_{i} - U_{ci} = (w_{s} + w_{g} - u_{ci}) \times S_{i}

(7)

Substituting equations (4)–(6) into equation (7), the debris launch velocity can be obtained as

v_{i} = \sqrt{2 \times {\frac{\frac{i_{r}^{2}}{2 \times m_{i}} - \int_{0}^{FP} R (x) dx}{m_{i}}}}

(8)

In general, the failure of the RC slab occurs later than the build-up time of the gas overpressure as well as the shock propagation time. Thus, the proposed model can consider the spalling debris and whole fragment simultaneously.

Validation against benchmark experiments

Experimental setup

The experiments for the RC structure were carried out by Dorr et al. (2002). The fragmentation process was recorded with a high-speed video camera and a photo pole. The frame rate of the high-speed video camera is 200 frames per second. The test arrangement was a subsurface cubicle chamber of 1 m by 1 m by 1 m dimension (i.e. inner volume V = 1 m³), as shown in Figure 3. The RC structure was mounted on top of the cubicle chamber. Hemispherical HE Pentolite charges (Plastic Nitropenta) were detonated at the center of the chamber. Distance of cover to center of bars was not described in the literature. The value was assumed as 20 mm which is reasonable in 100 mm thickness RC structure. Detailed information on structural properties is described in Table 1.

Figure 3.

Configuration of the test box for clamped concrete structure tests (Lu and Xu, 2007).

Table 1.

Properties of the benchmark structure (Lu and Xu, 2007).

Parameter		Value
Concrete information	Thickness (mm)	100
	Density (kg/m³)	2550
	Compressive strength, f_c (MPa)	35
Reinforcement information	Yield strength (MPa)	337
	Yield strain	0.002
	Young’s modulus, E_s (GPa)	200
Geometry information	Long span length, L (mm)	1000
	Short span length, H (mm)	1000
	Reinforcing bars spanning parallel to L (mm)	100
	Reinforcing bars spanning parallel to H (mm)	100
	Reinforcing bar area (mm²) →positive & negative moment steel parallel to L(inbound & rebound →identical)	50.27
	Reinforcing bar area (mm²) →positive & negative moment steel parallel toH(inbound & rebound →identical)	50.27
	Distance of cover to center of bars (mm)(Loaded & non-loaded side spanning parallel to L)	20
	Distance of cover to center of bars (mm)(Loaded & non-loaded side spanning parallel to H)	20

Analytical results

The estimated resistance-deflection curve for the target structure considering a variety of structural properties as well as the boundary condition (See Table.1) is depicted in Figure 4. The rotation angle for the blow out threshold of a two way RC structure is 10° (U.S. Army Corps of Engineers, 2008b) Thus, the estimated fracture point of the target structure is about 87.3 mm as center deflection, and the total failure energy is about 48,328 J.

Figure 4.

Estimated resistance-deflection curve for the target structure

Equations (9) and (10) represent the simple equations of the conventional models to predict the debris launch velocity which are the empirical formula (Dorr et al., 2002) derived from the test data and the semi-theoretical approach (Lu and Xu, 2007), respectively. Figure 5 plots the analytical results compared with the values calculated from the empirical formula, the semi-theoretical approach results, the numerical simulation results (Wu et al., 2019), and the test results against the benchmark experiments. For direct comparison, this study utilized the identical blast loads calculated from the equations described in Lu and Xu (2007). It was found that the present analytical results agree well with the test results compared to other predictions, as depicted in Figure 5.

v_{i} = 525 \times \sqrt{\frac{γ L}{M_{a}}} (m / s)

(9)

v_{i} = 575 \times \sqrt{\frac{γ L}{M_{a}}} (m / s)

(10)

where v_i is debris velocity, $γ$ is the charge loading density, L is the cubicle volume divided by the cross-section area, and M_a is the mass per unit area.

Figure 5.

Variation of the debris launch velocity with charge loading density.

The expressions for the equations (9) and (10) were simplified by curve fitting and rearranging terms in the previous study (Lu and Xu, 2007). Especially, equation (10) represents the nominal peak debris launch velocity for the test cases.

Since the information on the cover value was not described in the open literature (Lu and Xu, 2007), the value was assumed as 20 mm. In order to confirm the effect of the cover value, this study performed an analytical investigation by changing the value from 10 to 30 mm with 5 mm interval. From the investigation, it was found that the effect of the cover value on the debris launch velocity was insignificant in the test data region, as shown in Figure 6.

Figure 6.

Effect of the cover value about the debris launch velocity.

Conclusion

This study presented a debris launch velocity prediction model of a RC structure under an internal blast load based on energy conservation equation. This study basically assumes that there is no energy loss in debris occurrence mechanism. Thus, the work done by the internal blast load is transformed into the fragmentation of structural component and the fragment kinetic energy by considering the fragments in the critical region where the fragments exhibit highest velocity.

A resistance-deflection relationship for the structural SDOF system was newly considered to evaluate the energy consumed by the damage and fragmentation of the RC structure.

By applying the resistance-deflection relationship for the specific failure energy estimation, the proposed model can reflect the effect of the interactions between reinforcing bars and concrete for the specific failure energy calculation in a nominal fragment.

Moreover, the model can consider various structural properties as well as boundary conditions.

In order to confirm the performance of the proposed model, a comparative study was carried out against benchmark experiments on closed concrete box structures under an internal blast. For a comparative study, the semi-theoretical approaches which were simplified by curve fitting and rearranging terms were introduced. From the comparison, it was found that the present analytical results agree well with the test results compared to other predictions.

In conclusion, it was confirmed that the proposed model has crucial advantages in terms of its accuracy and flexible usage compared to the other models.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Seung-Hun Sung

References

Al-Thiry

(2016) A modified single degree of freedom method for the analysis of building steel columns subjected to explosion induced blast load. International Journal of Impact Engineering 94: 120–133.

Anderson

Jr Baker

Wauters

(1983) Quasi-static pressure, duration, and impulse for explosions (e.g. HE) in structures. International Journal of Mechanical Sciences 25(6): 455–464.

Dorr

Michael

Gurke

(2002) Experimental investigations of the debris launch velocity from internally overloaded concrete structures. Final Report DLV 4-2002, Institut Kurzzeitdynamik Ernst-Mach-Institut.

Dragos

(2015) Single-degree-of-freedom approach to incorporate axial load effects on pressure impulse curves for steel columns. Journal of Engineering Mechanics 141(1): 1–10.

El-Dakhakhni

Mekky

Changiz Rezaei

(2010) Validity of SDOF models for analyzing two-way reinforced concrete panels under blast loading. Journal of Performance of Constructed Facilities 24(4): 311–325.

Feldgun

Yankelevsky

Karinski

(2016) A nonlinear SDOF model for blast response simulation of elastic thin rectangular plates. International Journal of Impact Engineering 88: 172–188.

Fischer

Haring

(2009) SDOF response model parameters from dynamic blast loading experiments. Engineering Structures 31:1677–1686.

(2007) Prediction of debris launch velocity of vented concrete structures under internal blast. International Journal of Impact Engineering 34(11): 1753–1767.

Morison

(2006) Dynamic response of walls and slabs by single-degree-of-freedom analysis-a critical review and revision. International Journal of Impact Engineering 32(8):1214–1247.

10.

Needham

(2018) Blast Waves. 2nd edn. Heidelberg: Springer.

11.

Rigby

Tyas

Bennett

(2012) Single-degree-of-freedom response of finite targets subjected to blast loading – The influence of clearing. Engineering Structures 45: 396–404.

12.

Rigby

Tyas

Bennett

(2014) Elastic-plastic response of plates subjected to cleared blast loads. International Journal of Impact Engineering 66: 37–47.

13.

Stochino

Carta

(2014) SDOF models for reinforced concrete beams under impulsive loads accounting for strain rate effects. Nuclear Engineering and Design 276:74–86.

14.

UFC-3-340-02 (2008) Structures to resist the effect of accidental explosions. Washington, DC: US Department of the Army, Navy and the Air Force.

15.

U.S. Army Corps of Engineers (2008a) Methodology manual for the single-degree-of freedom blast effects design spreadsheets (SBEDS), pp. 1.1–10.4. PDC-TR-06-01. Omaha, Nebraska, USA.

16.

U.S. Army Corps of Engineers (2008b) Single degree of freedom structural response limits for antiterrorism design, pp. 1.1–5.4. PDC TR-06-08. Omaha, Nebraska, USA.

17.

Wang

Zhang

Fangyun Lu

, et al. (2013) A new SDOF method of one-way reinforced concrete slab under non-uniform blast loading. Structural Engineering & Mechanics 46(5): 595–613.

18.

Zhang

Fan

Liu

(2019) Debris characteristics and scattering pattern analysis of reinforced concrete slabs subjected to internal blast loads-a numerical study. International Journal of Impact Engineering 131: 1–16.