# Blast and Impact Analyses of RC Beams Considering Bond-Slip Effect and Loading History of Constituent Materials

- 305 Downloads

## Abstract

An improved numerical model that can simulate the nonlinear behavior of reinforced concrete beams subjected to blast and impact loadings is introduced in this paper. The layered section approach is based in the formulation, and the dynamic material behaviors of concrete and steel are defined with the use of the dynamic increase factor. Unlike the classical layered section approach that usually gives conservative structural responses because of no consideration of the bond-slip effect, the introduced numerical model takes into account the bond-slip between reinforcing steel and surrounding concrete by changing the bending stiffness EI of elements placed within the plastic hinge length. Since the bond-slip developed after yielding of reinforcing steel is dominant and accompanies fixed-end rotation, the equivalent bending stiffness to be used in the critical region can be evaluated on the basis of the compatibility condition. In advance, the consideration of the unloading and reloading histories of reinforcing steel and concrete makes it possible to exactly trace the structural behavior even after reaching the maximum structural response. Finally, correlation studies between analytical results and experimental data are conducted to establish the validity of the introduced numerical model, and the obtained results show that it is important to consider the bond-slip effect and the loading history of constituent materials.

## Keywords

blast impact layered section approach high strain rate bond-slip loading history of constituent material## 1 Introduction

Due to an increase of accidental explosions and terrorist acts of bombing worldwide, a lot of researches to reserve the safety of structures under blast and impact loadings, which were originally limited to military facilities, have been conducted with considerable attention to civil structures (Kwak and Gang 2015; Lin et al. 2014; Luccioni et al. 2004). Since explosions usually cause severe damage to structures with loss of human lives, blast protection of structures is strongly required and can be achieved through accurate prediction of structural responses. In particular, recent increases in the size and height of structures have accelerated the need to secure structural resistance to blast and impact loadings.

Reinforced concrete (RC) structures subjected to blast and impact loadings exhibit remarkably different structural behavior from that observed under a quasi-static loading condition due to the change in the material properties of concrete and reinforcing steel at a high strain rate condition (Carta and Stochino 2013; Qu et al. 2016; Valipour et al. 2009). Thus, the accurate prediction of structural behaviors in RC structures subjected to blast or impact loadings will be possible through an exact implementation of the high strain rate deformations in concrete and reinforcing steel that occur during a short period of time and the nonlinearity such as the bond-slip effect that dominantly affects the resisting capacity of RC structures. Furthermore, consideration of the loading history of constituent materials, which is expressed through the description of loading, unloading, and reloading, is also required to precisely trace the structural response even after the application of blast and impact loadings.

In this context, many studies, from experiments to identify the strain rate dependent material properties (Cadoni et al. 2009; Cusatis 2011) to numerical analyses of RC structures under impact loading (Fujikake et al. 2009; Tachibana et al. 2010), have been conducted. Nevertheless, it is also true that few experimental results for RC structural members subjected to blast and impact loadings can be found in the literature due not only to national security reasons but also to the fact that experiments are costly, time consuming and difficult to carry out. To overcome these difficulties in dynamic experiments, numerous analytical approaches from a simple approach such as a single degree of freedom model (Biggs 1964) to a rigorous finite element analysis using numerous solid elements (Chen et al. 2012; Jones et al. 2009; Ožbolt and Sharma 2011) have been proposed to verify the structural responses of RC structures. The obtained research results have been used in developing design codes such as the CEB-FIP model code (Comite Euro-International 1993) and ACI 318-08 (ACI Committee and ACI 2008), and also have been implemented into many commercialized programs including LS-DYNA (LSTC 2007) and ABAQUS (Hibbitt and Karlsson and Sorenson Inc. 2001) to be used in tracing the nonlinear response of RC structures subjected to blast and impact loadings.

Nevertheless, most numerical methods still accompany some problems when applied to RC beams. The use of solid elements gives mesh-dependent numerical results (Kwak and Gang 2015) and necessitates choosing one of the three dimensional failure criteria for concrete even though these criteria cannot simulate the concrete cracking behavior accurately (Tu and Lu 2009; Wu et al. 2012). A single degree of freedom (SDOF) model also has some limitations in simulating the nonlinear response of RC beams, even though it has been popularly adopted in design practice because of ease-of-use. Since the SDOF method not only adopts a lot of approximations but also cannot exactly take into account the nonlinear behavior in a RC section induced from the cracking of concrete and yielding of reinforcing steel, the use of the SDOF method may be inappropriate when a precise evaluation of the structural response is required.

The use of beam elements is also not exceptional. The beam model cannot consider the bond-slip effect between reinforcing steel and surrounding concrete because the strain compatibility has been based upon the perfect bond assumption (Bicanic et al. 2011). This restriction makes it more difficult to take into account the fixed-end rotation which occurs after yielding of the main reinforcement. The nonlinear analyses of RC beams consequently may give different results by ignoring the bond-slip effect, and in advance, the accuracy of the simulation results may not be guaranteed. Nevertheless, the bond-slip effect is still excluded in the numerical formulation of RC beams subjected to blast and impact loadings (Yao et al. 2016).

To address these limitations in the numerical analyses of RC beams, this paper presents a numerical model developed to consider the bond-slip effect in a beam element. The layered section method is based in the formulation, and the dynamic material behaviors of concrete and steel are defined with the use of the dynamic increase factor (DIF). The very different feature of the introduced numerical model is the implementation of the bond-slip effect, which cannot be considered in the classical layered section approach because of the difficulty in defining the relative slip along the reinforcing steel. In order to take into account the influence of bond-slip, the proposed numerical model suggests using the equivalent bending stiffness \( EI_{eq} \) within the plastic hinge length, upon the assumption that a large portion of the bond-slip will be concentrated, due to the anchorage slip, within the plastic hinge length where the yielding of reinforcing steel is subjected. Based on the compatibility condition, the equivalent bending stiffness \( EI_{eq} \) to be used in the critical region is evaluated.

In advance, the consideration of the loading histories of reinforcing steel and concrete makes it possible to exactly trace the structural behavior even after reaching the maximum structural response. Since the residual structural response beyond the maximum structural response is due to the unloading and reloading behavior of constituent materials, its exactness will be directly related to the consideration of the loading history for reinforcing steel and concrete. Finally, the validity of the proposed numerical model is confirmed by the comparison of analytical predictions with experimental data. Furthermore, the effects of bond-slip and loading history of the constituent materials are discussed through parametric studies, and the obtained results show the importance of considering both effects in the nonlinear dynamic behavior of RC beams subjected to blast and impact loadings.

## 2 Material Properties

### 2.1 Concrete

For the tensile region, concrete is linearly elastic up to the tensile strength. Beyond that, the tensile stress decreases along a linear softening branch with increasing principal tensile strain. It is assumed that ultimate failure take places by cracking, when the strain exceeds the value of \( \varepsilon_{t0} \) where \( b \) is the length of elements and \( G_{f} \) denotes the fracture toughness of concrete, as shown in Fig. 1b (Kwak and Hwang 2010).

After defining the monotonic envelope curve, it is necessary to exactly define the unloading–reloading behavior in order to describe the hysteretic response of concrete. However, the monotonic envelope curves are obtained on the basis of experimental studies, whereas the definition of an accurate cyclic stress–strain relation is very limited since it is difficult to carry out experiments for concrete subjected to cyclic loadings. Only a few of cyclic constitutive models have been proposed through experimental results (Konstantinidis et al. 2007).

It is shown that the dynamic compressive and tensile strengths of concrete under rapid loading increase significantly due to the lateral inertia confinement effect and the change of the crack pattern (Yan and Lin 2006). Previous experimental studies provide a more detailed description of strain rate effects on concrete (Cusatis 2011). Moreover, many mathematical models have also been proposed, which express an increase in strength and critical strain depending on the strain rate (Hao et al. 2012; Shkolnik 2008). In spite of many accurate numerical models for the consideration of the strain rate effect, however, the simple relations introduced by Saatcioglu et al. (2011) are adopted in this paper for computational convenience.

This model introduced a dynamic increase factor (DIF) to take into account the strain rate effect, and the compressive and tensile strength under a high strain rate condition can be determined by multiplying the DIF corresponding to the developed strain rate. Nevertheless, the strain at peak stress and the shape of the descending branch were assumed to be constant regardless of the change in the strain rate (see Fig. 1). This model requires only the strain rate \( \dot{\varepsilon } \) to compute the dynamic strength increase in concrete and can be expressed as follows (Saatcioglu et al. 2011): \( DIF = 0.03\ln \dot{\varepsilon } + 1.30 \ge 1.0\,\,\,{\text{for}}\,\,\dot{\varepsilon } < 30\;{\text{s}}^{ - 1} \), and \( DIF = 0.55\ln \dot{\varepsilon } - 0.47\,\,{\text{for}}\,\,\dot{\varepsilon } \ge 30\;{\text{s}}^{ - 1} \). The same equations are used in compression and tension for the computational convenience because DIF values for compression and tension do not show a large difference in a structure subjected to general blast loading which accompanies the relatively small strain rate.

### 2.2 Steel

The strain rate effect is considered by introducing a DIF (see Fig. 2), as mentioned in connection to Fig. 1. The relation of DIF as used by Saatcioglu et al. (2011) is adopted under the same assumption to define the dynamic stress–strain relation of reinforcing steel, and the corresponding equation is presented as follows: \( DIF = 0.034\ln \dot{\varepsilon } + 1.30 \ge 1.0 \).

*b*is the strain-hardening ratio between slope \( E_{S1} \) and \( E_{S2} \), and

*R*is a parameter that influences the shape of the transition curve and represents the Bauschinger effect. More details related to this model can be found elsewhere (Kwak and Kim 2006).

## 3 Consideration of Bond-Slip Effect

The perfect bond assumption usually adopted in the analysis of RC beams is reasonable only in uncracked regions where bond stress transferred along the interface between reinforcing steel and surrounding concrete is negligibly small (Monti and Spacone 2000). The influence of bond-slip, however, is particularly noticeable in a cracked region, and the bond-slip will be remarkably enlarged with the yielding of reinforcing steel (Kwak and Kim 2010). Therefore, consideration of the bond-slip effect is required to simulate the structural behavior more exactly.

In this regard, many studies have been conducted to consider this effect (Oliveira et al. 2008; Santos and Henriques 2015), and the bond-slip models such as the bond-link element and the bond-zone element have been introduced to take into account the bond-slip effect (Lowes et al. 2004). In these models, the relative slip between concrete and reinforcement is evaluated by using a double node. However, in a beam element defined by both end nodes along the length direction, it is impossible to use the double node at each end node. To address this limitation in adopting the bond model, a numerical algorithm that includes the bond-slip effect is proposed in this paper.

*x*is an experimentally determined parameter ranging from 0.9 to 1.0 and

*h*is the section depth. As shown in Fig. 5a, if a point load P is applied at the mid-span of an RC beam, the maximum deflection \( \Delta_{1} = P \times (EI_{eq} L_{1}^{3} + EIL_{p} (3L_{p} L_{1} + 3L_{1}^{2} ))/(3EI_{eq} EI) \) can be obtained by the moment area method.

In addition, the beam can also be idealized by using the equivalent rotational stiffness \( K_{\theta } \), as shown in Fig. 5b, because additional rigid body rotation, which causes a reduction of the bending stiffness, will be accompanied by slippage of the main reinforcement and can be simulated by introducing the end rotational stiffness. In this case, \( K_{\theta } \) can be obtained by the ratio of the moment to the fixed-end rotation (\( K_{\theta } = M_{y} /\theta_{fe} \)) and, the fixed-end rotation is determined by the relation of \( \theta_{fe} = \delta /(d - c) \), where \( \delta \) denotes the bond-slip of the reinforcing steel (see Fig. 4), and *d* and *c* are the effective depth in an RC section and the distance from the extreme compression fiber to the neutral axis, respectively.

Since the bond-slip \( \delta \) can be evaluated on the basis of the assumption that the crack width \( \omega \) caused by the bending behavior is equivalent to two times the bond-slip (\( 0.5\omega = \delta \)) at the considered position, and the crack width can directly be evaluated by the formula introduced by Gergely and Lutz (1968), the rotational stiffness \( K_{\theta } \) can be evaluated from \( K_{\theta } = M_{y} /\theta_{fe} = M_{y} (d - c)/\delta \). Moreover, Eq. (4) not only gives the equivalent bending stiffness \( EI_{eq} \) but also takes into account the strain rate dependent bond-slip effect indirectly, and the modification of the bending stiffness of the elements within the plastic hinge length will be followed.

## 4 Solution Procedure

To analyze RC beams subjected to blast and impact loadings, the construction of an element stiffness is based on the layered Timoshenko beam theory, which takes the shear deformation into consideration, and this paper adopts the Newmark method, in which a constant average acceleration with Newmark coefficients of \( \beta = 0.25 \) and \( \gamma = 0.5 \) is based. More details related to the construction of the element stiffness and the numerical evaluation of the dynamic response can be found elsewhere (Ayoub and Filippou 1999).

In advance, to minimize the difference in numerical results according to the finite element mesh size, as was mentioned in a previous study (Kwak and Gang 2015), all the RC beams considered in this paper have been idealized by ten elements through a convergence test. On the other hand, the critical regions within the plastic hinge length are discretized by the use of two elements to accurately estimate the plastic deformation especially after yielding of reinforcing steel because such a separate consideration of the critical region is required to avoid overestimation of the ultimate resisting capacity and underestimation of the developed lateral deformation.

## 5 Numerical Applications

Material properties of each specimen.

Specimens | \( f_{c} \) (MPa) | \( f_{y} \) (MPa) | \( E_{c} \) (GPa) | \( E_{s} \) (GPa) | Reinforcement |
---|---|---|---|---|---|

B40_D1 | 43 | 595 | 31 | 205 | 5 \( \phi \) 16 mm |

B40_D2 | 43 | 595 | 31 | 205 | 5 \( \phi \) 16 mm |

WE2 | 27 | 457 | 27.3 | 199.8 | 2 No. 7/2 No. 9 |

WE5 | 27 | 462 | 27.3 | 199.8 | 2 No. 7/2 No. 9 |

WE6 | 28.4 | 455 | 27.3 | 199.8 | 2 No. 7/2 No. 9 |

SS3a | 46.7 | 464 | 33 | 195 | 2 No. 30/2 No. 30 |

SS3b | 46.7 | 464 | 33 | 195 | 2 No. 30/2 No. 30 |

The material properties of each specimen are summarized in Table 1, and more details of the experimental setup can be found elsewhere (Magnusson and Hallgren 2000; Saatci 2007; Seabold 1967). Moreover, three beams B40_D1, B40_D2, and SS3a, among these specimens are also analyzed by using the equivalent SDOF model on the basis of the approach introduced at UFC-3-340-02 (US DoD 2008), to compare the accuracy of the proposed numerical model. To define the dynamic material properties of constituent materials in the SDOF, differently from the introduced numerical model, which considers the change of DIF according to the strain rate, a constant value of DIF = 1.2, which is corresponding to the minimum value among the mainly used DIF values ranged from 1.2 to 1.4 (Magnusson 2007; Fujikura and Bruneau 2011), is used, because the test specimens considered in this paper are subjected to relatively small blast loadings. To trace the post-peak nonlinear behavior of RC beams after the maximum loading history, the damping coefficient *c* is assumed to be about 3% of the critical damping (\( C = 0.03C_{cr} \)) because the experiments considered in this paper did not give any damping value. This value is usually considered in the nonlinear dynamic analyses of RC structures.

Parameter values for equivalent SDOF models.

Specimens | \( P_{l} \) (kN) | \( t_{d} \) (ms) | \( R_{m} \) (kN) | M (kg) | K (MN/m) |
---|---|---|---|---|---|

B40_D1 | 175 | 12 | 259 | 80 | 29 |

B40_D2 | 256 | 8.3 | 259 | 80 | 29.5 |

SS3a | 1410 | 1.1 | 428 | 668 | 31 |

The details of blast loadings.

Specimens | Q (kg) | \( P_{r} \) (kPa) | T (s) |
---|---|---|---|

B40_D1 | 1.1 | 650 ± 31 | 0.023 |

B40_D2 | 2.0 | 1060 ± 116 | 0.026 |

Moreover, the displacement time history for both specimens is also obtained by the SODF analysis. As shown in Figs. 7b and 8b, the SDOF model also shows a similar displacement history to the experimental data in the case of B40_D1 but a large difference from the experimental data in the case of B40_D2. This appears to be induced by the classical SDOF model not taking into account the bond-slip effect, which suddenly increased with the yielding of reinforcing steel. Accordingly, if the bond-slip effect can be implemented in the equivalent stiffness of the SDOF model, additional improvement in the numerical results can be expected.

On the other hand, since the blast loading for the beam B40_D1 is not large enough to develop yielding of tensile reinforcements, the results with and without consideration of the loading history of the constituent materials are almost the same even at the unloading stage (see Fig. 7a). A slight difference in numerical results according to the consideration of the loading history of materials can be found in beam B40_D2, which accompanies the yielding of tensile reinforcements (see Fig. 8). However, comparison of the displacement history limited to the first unloading behavior appears to be insufficient to exactly evaluate the influence of considering the loading history of the constituent materials.

Moreover, the displacement history in Fig. 10 also shows that ignoring the loading history of the constituent materials lengthens the returning period in the displacement cycle. Since ignoring the loading history of materials leads the constituent materials to behavior along the loading path even at the unloading stage after developing large deformation, the stiffness of the structure will be underestimated compared to the real stiffness. This underestimation of the stiffness causes an increase of the returning period because the period is inversely proportional to the stiffness of the structure. This result is induced by the fact that concrete and reinforcing steel under the unloading path represent larger stiffness, which is not substantially different from the initial stiffness. These differences in following the unloading path cause a large difference in the displacement history at the post-peak structural response, as shown in Fig. 10.

Besides, differently from experimental data in the beam SS3b, which represents a rapid decrease of the returning period from the second half of the first fluctuation in the displacement history, the numerical model still sustains almost the same returning period regardless of the number of displacement cycles (see Fig. 13b). This means that the beam model may have a limitation in simulating the local effect induced by repetition of the crack opening and closing after experiencing the large deformation because the formulation of the beam element is basically based on the continuum displacement field, which does not allow discontinuity in the deformation.

## 6 Conclusion

This paper introduces an improved layered section model that can simulate the nonlinear dynamic analysis of RC beams subjected to blast and impact loadings. Unlike the classical layered section approach that usually gives conservative structural responses because of the absence of consideration of the bond-slip effect, the introduced numerical model takes into account the bond-slip between reinforcing steel and surrounding concrete by changing the bending stiffness EI of elements placed within the plastic hinge length, and the consideration of the bond-slip effect leads to a remarkable improvement in the accuracy of the numerical results. Moreover, the consideration of the loading history in the constituent materials makes it possible to trace the structural response after reaching the maximum response. Through correlation studies between the numerical results and experimental data, the following conclusions are reached: (1) to improve the accuracy of the simulation results in RC beams, the bond-slip effect and the loading history of the constituent materials must be considered. In particular, the importance of considering the bond-slip effect needs to be emphasized; (2) the SDOF method has a limitation in application to RC beams with large deformation induced by the yielding of reinforcing steel; and (3) the proposed model can be used effectively in predicting the structural response of entire RC structures subjected to blast and impact loadings; nevertheless, (4) it is also true that the proposed model has a limitation in simulating the shear-dominant structural behavior.

## Notes

### Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (No. 2017R1A5A1014883), and this research was supported by a Construction Technology Research Project (17SCIP-B128706-01) funded by the Ministry of Land, Infrastructure and Transport.

## References

- ACI Committee, & ACI. (2008). Building code requirements for structural concrete (ACI 318-08) and commentary.Google Scholar
- Ayoub, A., & Filippou, F. C. (1999). Mixed formulation of bond-slip problems under cyclic loads.
*Journal of Structural Engineering,**125*(6), 661–671.CrossRefGoogle Scholar - Bayrak, O., & Sheikkh, S. (1997). High-strength concrete columns under simulated earthquake loading.
*ACI Structural Journal,**94*(6), 708–722.Google Scholar - Belarbi, A., & Hsu, T. T. C. (1994). Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete.
*ACI Structural Journal,**91*(4), 465–474.Google Scholar - Bicanic, N., de Borst, R., Mang, H., & Meschke, G. (2011).
*Computational modelling of concrete structures*. Boca Raton: CRC Press.Google Scholar - Biggs, J. M. (1964).
*Introduction to structural dynamics*. New York: McGraw-Hill.Google Scholar - Cadoni, E., Solomos, G., & Albertini, C. (2009). Mechanical characterisation of concrete in tension and compression at high strain rate using a modified Hopkinson bar.
*Magazine of Concrete Research,**61*(3), 221–230.CrossRefGoogle Scholar - Carta, G., & Stochino, F. (2013). Theoretical models to predict the flexural failure of reinforced concrete beams under blast loads.
*Engineering Structures,**49,*306–315.CrossRefGoogle Scholar - Chen, G. M., Chen, J. F., & Teng, J. G. (2012). On the finite element modelling of RC beams shear-strengthened with FRP.
*Construction and Building Materials,**32,*13–26.CrossRefGoogle Scholar - Comite Euro-International. (1993).
*Ceb-fip model code 1990: Design code du beton*. Wiltshire: Redwood Books.CrossRefGoogle Scholar - Craig, R. R., Jr., & Kurdila, A. J. (2006).
*Fundamentals of structural dynamics*(2nd ed.). Hoboken: Wiley.MATHGoogle Scholar - Cusatis, G. (2011). Strain-rate effects on concrete behavior.
*International Journal of Impact Engineering,**38*(4), 162–170.CrossRefGoogle Scholar - Filippou, F. C., Popov, E. P., & Bertero, V. V. (1983).
*Effects of bond deterioration on hysteretic behaviour of reinforced concrete joints*. Berkeley, CA: Earthquake Engineering Research Center.Google Scholar - Fujikake, K., Li, B., & Soeun, S. (2009). Impact response of reinforced concrete beam and its analytical evaluation.
*Journal of Structural Engineering,**135*(8), 938–950.CrossRefGoogle Scholar - Fujikura, S., & Bruneau, M. (2011). Dynamic analysis of multihazard-resistant bridge piers having concrete-filled steel tube under blast loading.
*Journal of Bridge Engineering,**17*(2), 249–258.CrossRefGoogle Scholar - Gergely, P., & Lutz, L. A. (1968). Maximum crack width in reinforced concrete flexural members,
*Sp 20*-*6*, 87–117.Google Scholar - Guner, S. (2016). Simplified modeling of frame elements subjected to blast loads.
*ACI Structural Journal,**113*(2), 363–372.MathSciNetCrossRefGoogle Scholar - Hao, Y., Hao, H., & Zhang, X. H. (2012). Numerical analysis of concrete material properties at high strain rate under direct tension.
*International Journal of Impact Engineering,**39*(1), 51–62.CrossRefGoogle Scholar - Hibbitt, & Karlsson and Sorenson Inc. (2001).
*ABAQUS/standard user’s manual*. Providence, RI: Hibbitt, & Karlsson and Sorenson Inc.Google Scholar - Jones, J., Wu, C., Oehlers, D. J., Whittaker, A. S., Sun, W., Marks, S., et al. (2009). Finite difference analysis of simply supported RC slabs for blast loadings.
*Engineering Structures,**31*(12), 2825–2832.CrossRefGoogle Scholar - Karsan, I. D., & Jirsa, J. O. (1969). Behavior of concrete under compressive loadings.
*Journal of the Structural Division,**95*(12), 2543–2563.Google Scholar - Kent, D. C., & Park, R. (1971). Flexural members with confined concrete. In
*Journal of the Structural Division*(Vol. 97(ST7), pp. 1969–1990). Reston: American Society of Civil Engineers.Google Scholar - Konstantinidis, D. K., Kappos, A. J., & Izzuddin, B. A. (2007). Analytical stress-strain model for high-strength concrete members under cyclic loading.
*Journal of Structural Engineering,**133*(4), 484–494.CrossRefGoogle Scholar - Kwak, H. G., & Gang, H. G. (2015). An improved criterion to minimize FE mesh-dependency in concrete structures under high strain rate conditions.
*International Journal of Impact Engineering,**86,*84–95.CrossRefGoogle Scholar - Kwak, H. G., & Hwang, J. W. (2010). FE model to simulate bond-slip behavior in composite concrete beam bridges.
*Computers & Structures,**88*(17), 973–984.CrossRefGoogle Scholar - Kwak, H. G., & Kim, J. K. (2006). Implementation of bond-slip effect in analyses of RC frames under cyclic loads using layered section method.
*Engineering Structures,**28*(12), 1715–1727.CrossRefGoogle Scholar - Kwak, H. G., & Kim, S. P. (2010). Simplified monotonic moment-curvature relation considering fixed-end rotation and axial force effect.
*Engineering Structures,**32*(1), 69–79.CrossRefGoogle Scholar - Lin, X., Zhang, Y. X., & Hazell, P. J. (2014). Modelling the response of reinforced concrete panels under blast loading.
*Materials and Design,**56,*620–628.CrossRefGoogle Scholar - Livermore Software Technology Corporation (LSTC). (2007).
*LS-DYNA Keyword User’s Manual Version 97*, Livermore, CA.Google Scholar - Lowes, L. N., Moehle, J. P., & Govindjee, S. (2004). Concrete-steel bond model for use in finite element modeling of reinforced concrete structures.
*ACI Structural Journal,**101,*501–511.Google Scholar - Luccioni, B., Ambrosini, R., & Danesi, R. (2004). Analysis of building collapse under blast loads.
*Engineering Structures,**26*(1), 63–71.CrossRefGoogle Scholar - Magnusson, J. (2007). Structural concrete elements subjected to air blast loading, Dissertation.Google Scholar
- Magnusson, J., & Hallgren, M. (2000).
*High performance concrete beams subjected to shock waves from air blast*. Tumba, Sweden.Google Scholar - Mashaly, E.-S., El-Heweity, M., Abou-Elfath, H., & Ramadan, M. (2011). A new beam-column model for seismic analysis of RC frames—Part I: Model derivation.
*Alexandria Engineering Journal,**50*(4), 313–320.CrossRefGoogle Scholar - Menegotto, M., & Pinto, P. E. (1973). Method of analysis for cyclically loaded R.C. plane frames including changes in geometry and non-elastic behaviour of elements under combined normal force and bending. In
*IABSE reports of the working commissions*(Vol. 13).Google Scholar - Monti, G., & Spacone, E. (2000). Reinforced concrete fiber beam element with bond-slip.
*Journal of Structural Engineering,**126,*654–661.CrossRefGoogle Scholar - Oliveira, R. S., Ramalho, M. A., & Corrêa, M. R. S. (2008). A layered finite element for reinforced concrete beams with bond-slip effects.
*Cement & Concrete Composites,**30*(3), 245–252.CrossRefGoogle Scholar - Ožbolt, J., & Sharma, A. (2011). Numerical simulation of reinforced concrete beams with different shear reinforcements under dynamic impact loads.
*International Journal of Impact Engineering,**38*(12), 940–950.CrossRefGoogle Scholar - Qu, Y., Li, X., Kong, X., Zhang, W., & Wang, X. (2016). Numerical simulation on dynamic behavior of reinforced concrete beam with initial cracks subjected to air blast loading.
*Engineering Structures,**128,*96–110.CrossRefGoogle Scholar - Saatci, S. (2007).
*Behaviour and modelling of reinforced concrete structures subjected to Impact Loads.*Ph.D. Thesis. University of Toronto, Toronto.Google Scholar - Saatcioglu, M., Lloyd, A., Jacques, E., Braimah, A., & Doudak, G. (2011).
*Focused research for development of a CSA standard on design and assessment of buildings subjected to blast loads*. Ottawa: University of Ottawa.Google Scholar - Santos, J., & Henriques, A. A. (2015). New finite element to model bond–slip with steel strain effect for the analysis of reinforced concrete structures.
*Engineering Structures,**86,*72–83.CrossRefGoogle Scholar - Scott, B. D., Park, R., & Priestley, M. J. N. (1982). Stress–strain behavior of concrete confined by overlapping hoops at low and high strain rates.
*Journal Proceedings,**79*(1), 13–27.Google Scholar - Seabold, R. H. (1967).
*Dynamic shear strength of reinforced concrete beams, part 2*. Port Hueneme, CA: US Naval Civil Engineering Laboratory.CrossRefGoogle Scholar - Shkolnik, I. E. (2008). Influence of high strain rates on stress-strain relationship, strength and elastic modulus of concrete.
*Cement & Concrete Composites,**30*(10), 1000–1012.CrossRefGoogle Scholar - Tachibana, S., Masuya, H., & Nakamura, S. (2010). Performance based design of reinforced concrete beams under impact.
*Natural Hazards and Earth System Science,**10*(6), 1069–1078.CrossRefGoogle Scholar - Taucer, F. F., Spacone, E., & Filippou, F. C. (1991).
*A fiber beam*-*column element for seismic response analysis of reinforced concrete structures*. Report No. UCB/EERC-91/17. Berkeley, CA.Google Scholar - Tu, Z., & Lu, Y. (2009). Evaluation of typical concrete material models used in hydrocodes for high dynamic response simulations.
*International Journal of Impact Engineering,**36*(1), 132–146.CrossRefGoogle Scholar - US DoD. (2008).
*Unified facilities criteria (Ufc) structures to resist the effects of accidental*. Washington, DC: Department of Defence.Google Scholar - Valipour, H. R., Huynh, L., & Foster, S. J. (2009). Analysis of RC beams subjected to shock loading using a modified fibre element formulation.
*Computers and Concrete,**6*(5), 377–390.CrossRefGoogle Scholar - Wu, Y., Crawford, J. E., & Magallanes, J. M. (2012). Performance of LS-DYNA concrete constitutive models. In
*12th International LS*-*DYNA Users conference*(pp. 1–14).Google Scholar - Yan, D., & Lin, G. (2006). Dynamic properties of concrete in direct tension.
*Cement and Concrete Research,**36*(7), 1371–1378.CrossRefGoogle Scholar - Yao, S. J., Zhang, D., Lu, F. Y., Wang, W., & Chen, X. G. (2016). Damage features and dynamic response of RC beams under blast.
*Engineering Failure Analysis,**62,*103–111.CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.