1 Introduction

As most terrorist attacks may aim to demolish important reinforced concrete structures (e.g. embassies and government departments) many researches in recent years investigate the interaction of blast wave with concrete structures and various failure mechanisms and damage of concrete structures under blast loadings [1,2,3,4,6]. Damage analysis is important in the decision-making process not only in post blast rehabilitation but also immediately after the blast [7]. Damaged buildings may undergo further collapse due to a lack of strength of lower level elements, sometimes after the incident, and causing rescue workers to serve with extensive risk. Damage evaluation is carried out to identify the post blast performance of the RC structures when subjected to explosive loads. A damage index numerically indicates the level of damage of a particular structure or a component [8, 9]. This is a valuable tool in evaluating the damage and design to resist disasters such as an earthquake or blast [10–12].

However, limited numerical derivations identify damage indices when subjected to explosion loads, and most have focused on a single structural component investigation to calculate the indices [9, 13–15]. Damage indices based on the residual capacity of the damaged structure may offer the most reliable estimate of damage. That is the remaining life of the structure after the blast event would depend on post blast residual capacity of the structural elements. Load carrying capacities of the columns and beams are measured before and after the explosion to calculate the damage indices of the beams and columns with respect to each blast load scenario. The cross sectional properties of the concrete elements are used to determine residual load capacities [10]. In this study variations of the damage index are presented with respect to blast load scenarios. Damage criterion and mode of failure in reinforced concrete structures subjected to explosive loads was overviewed in this research study. Conclusions and future work recommendations are presented to develop of reinforced concrete structures under blast loads.

2 Blast Load Induced Damage

There are three types of localized concrete damage under direct impact of blast loads. They are named compression failure, spalling and scabbing. As the blast pressure in contact with the surface of an obstacle (i.e. a building structure) there will be an instantaneous rise in the compressive stress on the external face of the structure. These stresses can be sufficiently large to result in cratering surface (compressive failure). Compressive stresses will then propagate through the concrete in a form of a compressive shock wave which will continue to travel into the interior face of the concrete structure until it hits the boundary. Reflection of shock wave takes place at the boundary interface and initiates the tensile stress. Since tensile strength of concrete is much weaker than its compressive strength, tensile fracturing usually occurs on the interior face, resulting in debris being projected into the building structure [16]. Another commonly occurred damage is known as scabbing. This happens when large and rapid plastic deformation of structure causing lumps of concrete to be detached from the interior face. The lumps might then be projected into the interior in a similar manner to spalling. Figure 1 illustrates the damage of a concrete slab under blast load impact.

Fig. 1
figure 1

Contact detonation on a concrete slab [17]

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Besides the immediate and localized blast effects, progressive collapse must also be considered as it not only imposes threats to the building structural integrity but also public safety. Progressive collapse of building is triggered by the alteration in either boundary conditions or loading pattern (or combination of the two) such that structural elements are loaded beyond the design ultimate strength and then collapse. Under these circumstances, the residual structure will search alternative paths for redistribution of excess load. This can be accomplished either by finding a stable alternative load path or by shedding load (i.e. as by-products of failing elements).

Many guidelines are proposed (e.g. Balance Survivability Assessment, [18] and Structural Design for Physical Security—State of the Practical Report, [19]) and incorporated in subsequent British Standards (e.g. [20]; [21] and [22]) in constructing civilian structures such that the structural damage severity can be reduced when exposed to either accidental or hostile explosion threats. Although the effectiveness of these guidelines have been attributed to the good performance of numerous buildings that were subjected to actual blast loading, it might not always be possible to quantify the risk of progressive collapse of the structure as there are various crucial factors that can influence the failure process of the building [23].

3 Damage Criterion

In order to define damage level in RC structures, the damage criteria used should be suitable for evaluation of RC structures related to the member global and material damage, easy to use in assessing the element conditions and easily obtained from numerical or experimental test [24, 25]. Damage assessment plays an important role in the evaluation of the stability and strength of structures [26]. In an extreme condition, such as a blast load scenario that is normally not considered in the original structural design of civilian structures, structural elements may experience damage in different degrees. An effective damage assessment method of a structure is essential in order to apply protective measures when there exist potential blast load risks. For a RC structural element, the analysis becomes more complicated because the reinforced concrete always deforms in a nonlinear way, especially in the post-failure stage.

3.1 Damage Criterion for RC Columns

Fallah and Louca evaluated a damage criterion based on the mid-height deflection of the column for the SDOF system [27]. The yC values and the corresponding damage degrees are given in Table 1. In their study damage criterion is based on flexural failure of the column that it may not give reliable prediction of column capacity with other failure modes. Fallah and Louca [27] neglects the axial load effects on column load carrying capacity. Since columns are primarily designed to carry the axial load, the residual axial load carrying capacity should be a better damage criterion of a column.

Table 1 Damage criterion based on Fallah and Louca study [27]
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Shi [24] and Mutalib and Hao [3] proposed a new method to estimate the damage index in RC columns. To observe the damage level of the target column, three steps of loading are introduced into the model as given in the following;

3.1.1 Stage 1: Initial Stress Analysis

Herein, 20% an axial force is applied initially to the column prior to the blast loads to simulate the stress state arising from the existing imposed and dead loads.

3.1.2 Stage 2: Blast Loading Analysis

At the beginning of this loading stage, the velocity of the model is reset to zero. During this step, the target column is evaluated for its response when subjected to blast loads face of the column. The dynamic analysis is executed for a specified duration of time, depending on the type of column applied, to enable capture of the peak and steady state response. The axial load which was applied on the top of the column during step 1 is maintained. In this case, the top of column is also free to move vertically downward.

3.1.3 Stage 3: Residual Capacity Check

Post-blast analysis is carried out in the third stage to evaluate the residual axial load carrying capacity of the column. There are two ways to check the post-blast analysis:

  1. i.

    Displacement Control Method (DC)

A prescribed motion control is applied at the tip of the target column. This prescribed motion control must be applied gradually due to explicit dynamic analysis used.

  1. ii.

    Load Control Method (LC)

An axial load is applied on the top of the target column. Similar to that of displacement control, this axial force is applied gradually within the time duration of analysis. In stage 3, a prescribed motion control is applied at the tip of the target column using a specified displacement rate to obtain the load deflection curve and the residual axial load carrying capacity of the target column after being subjected to blast loads. Procedure of the three stages loading that is used to assess the damage level of RC columns is representing in the Fig. 2. This is achieved by applying a static axial load to the top of damaged column until the column fails.

Fig. 2
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Form of load history applied to a column to compute its residual capacity [24, 28]

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Shi et al. [24] and Mutalib and Hao [28] for assessment the RC column damage determined damage index based on the residual axial load carrying capacity. It is defined as

$$D = 1 - \frac{Presidual }{ Pdesign }$$
(1)

where D is level of damage, Presidual is the residual axial load-carrying capacity of the damaged column, and Pd is the maximum axial load-carrying capacity of the undamaged column [29]. Residual capacity of the RC column is estimated from finite element modeling [30]. The axial load-carrying capacity degradation is suitable for evaluating the RC column shear damage and flexural damage, as well as local damage. The axial load-carrying capacity of an undamaged RC column depends on the longitudinal reinforcement and concrete. According to Macgregor [31] and ACI Code the following equation is used to assess the maximum axial load-carrying capacity of an undamaged RC column:

$$P_{design} = 0.85f_{c}^{{}} \left( {A_{c} - A_{s} } \right) + f_{y} A_{s}$$
(2)

where fc is the compressive strength of concrete, fy is the yield strength of the longitudinal reinforcement, Ac gross area of the column cross-section, AS the area of the longitudinal reinforcement. The different values of D are correlated to different damage degrees show in Table 2 [32].

Table 2 Damage Index classification in the Shi study [24]
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The advantages of the damage index proposed by Shi are that it has direct physical meanings that are independent of the failure modes and also it is easy to use in evaluating column conditions because the primary function of columns is to carry axial load [24]. In addition, it is easy to use in numerical simulation or experiment tests. Damage Index classification shown in Table 2 and Fig. 3 respectively.

Fig. 3
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Damage Index classification [24]

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There is no comprehensive definition given for each term. However the physical meaning is that the supporting building is increasingly at high risk at each stage and the column has to be replaced when the damage index is in the 0.8–1.0 range. The advantage of this index is none of the commonly used damage criteria, such as residual deflection, maximum stress and strain conditions, satisfy the above requirements. On the other hand, the axial load capacity degradation will reflect the shear damage, flexural damage or local damage conditions due to impact while expressing the global behaviour of the impact, and these factors are easily obtain from numerical simulation techniques.

Cui proposed new formulae to analysis and damage assessment of RC columns under close-in explosions [33]. They show that the maximum residual plastic deflection of the RC column often occurs at the height of the centre of the blast loaded area. The residual deflection along the RC column is a combination of the overall deflection and the local deflection directly induced by the concentrated blast load. Based on the numerical results, the blast load concentrated area is defined, which is shown in Fig. 4a. And the damage index λ is defined as the ratio of the relative residual deflection and the column depth;

$$\uplambda = \frac{\delta }{h}$$
(3)

where δ is relative residual deflection of the blast load concentrated area, which is calculated as the difference between the maximum residual deflection and the residual deflection at the edge of the blast load concentrated area as shown in Fig. 4b; h is the column depth.

Fig. 4
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a Area of concentrated blast load and b relative residual deflection, in the Jian Cui study [33]

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Cui [33] evaluated the relationship between damage index λ and axial load carrying capacity degradation-based damage index D (based on the Shi [24] study) that it is independent of column depth. The damage D can be derived as a function of λ;

$$D = 1.9 + 0.25ln\uplambda \quad \left( {\uplambda \le 0.015} \right)$$
(4)

Abedini defined damage criterion based on the residual axial load carrying capacity [30]. Damage analysis for structural components such as columns, is usually carried out by a separate analysis of the damaged structure. In his study the residual capacity of a structure or structural components is based on a separate analysis for the blast damaged structure applying loads or displacements. Damage indices are defined using the residual capacities of the structural elements in a separate analysis. Residual capacities are evaluated by applying additional loads or displacements to the blast damaged structure until complete failure occurs. The maximum load which the damaged structure can sustain is measured and identified as the maximum residual capacity as presented in Fig. 5.

Fig. 5
figure 5

Depiction of the procedure that is used to assess the residual capacity of RC columns

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A new method based on damaged area of components is proposed in the Abedini study as an appropriate tool to evaluate post blast residual capacity of structural components [34]. The damaged area of the concrete is identified from the development of the plastic strain and the level of damage is then evaluated based on the damaged area. This enables the calculation of the true residual capacity which remains in the partially damaged structural component, in order to accurately determine the damage indices. Post blast performance capacities of the RC building structure can then be evaluated and appropriate engineering decisions can be made for the structural adequacy of the blast damaged building. The residual capacity of the damaged concrete can be calculated by evaluating the minimum cross sectional area of damaged concrete at the weakest point of the column.

$$C_{r} = P_{d} - \left( {\frac{{P_{d} \times A_{i} }}{{A_{c} }}} \right)$$
(5)

where Cr is the residual axial capacity of RC columns, Pdesign is the maximum axial load-carrying capacity of the undamaged column, \(A_{i}\) is the damaged cross-sectional area of concrete and \(A_{c}\) is the cross-section area of RC columns.

Cross section behavior at the critically damaged locations in the columns is investigated to identify the level of damage and residual capacity following the blast [35]. Damaged indices based on cross section area of the RC columns quantify the actual damage to the structure. Therefore, a study is required to identify the true response for the damaged structure to identify the level of damage and then to calculate a damage index. The damage index is a ratio used for quantifying the level of damage in a particular structure. The damage index η is defined as the damaged cross-sectional area of concrete and the column cross-section area of RC column [35].

$$\upeta = \frac{{A_{i} }}{{A_{G} }}$$
(6)

where η is the damage index of RC columns based on the actual yielding of the materials at post blast conditions, \(A_{i}\) is the damaged cross-sectional area of concrete and \(A_{c}\) is the cross-section area of RC column. Plastic strain diagrams at selected key elements can be used to determine the damage mechanism and the damage extent for assessment of residual capacity in RC columns as represent in Fig. 6.

Fig. 6
figure 6

Plastic strain of RC column to assessment blast damage [35]

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Wijesundara M.G. Lakshitha performed the damage criterion adopted for developing column assessment charts considers two main aspects of the column response and consequential damage behaviour [15]:

  • Damage progression and overall failure mechanism (qualitative assessment)

  • Residual axial capacity of post-damaged columns (quantitative assessment)

Table 3 provides the defined damage classification system comprising five damage levels based on the extent of overall damage and residual axial load capacity of columns.

Table 3 Damage classification system
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The extent of damage inflicted on internal RC columns by blast loading was quantified both qualitatively and quantitatively using two parameters: (1) Damage index (η) and, (2) Residual axial capacity index (α). The residual axial capacity index (α) was defined as Nd/Nud, where Nd and Nud are the axial compression capacity of a column respectively in damaged and undamaged conditions. The numerical procedure used for estimating column residual axial capacity is illustrated in Fig. 7. The damage index parameter (η) has a specific range for each damage level defined in the damage classification system. Based on the extent of overall damage, particularly considering damage progression, ultimate failure mechanism and the integrity of reinforcement, a value of η was carefully assigned for each column. Table 4 provides typical ranges of α and η corresponding to defined damage classes [15].

Fig. 7
figure 7

Estimation of residual axial capacity of post-damaged RC columns [15]

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Table 4 Quantification of the extent of column damage
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Figure 8 shows a set of numerical modeling outputs grouped according to the defined damage classification system. The η versus α graph corresponding to each damage class provides the approximate relationship of column residual axial capacity with the extent of damage [15].

Fig. 8
figure 8

Overall failure mechanism and residual axial capacity of columns [15]

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The P–I diagram is generally used to numerically describe an exact damage level to joint blast pressures and impulses applied on a specific structural member [36, 37]. Each P–I curve represents a damage level that a structure experiences due to the various blast loading conditions. Biggs, John, M. and Mendis observed an example of a P–I diagram to illustrate levels of damage of a structural element that is shown in Fig. 9 [38, 39]. Region (I) describes significant structural damage and region (II) displays no or minor damage. Three categories of blast-induced injury concern with human response to blast for develop P–I diagram, namely: primary, secondary, and tertiary injury [40, 41].

Fig. 9
figure 9

Typical pressure–impulse (P–I) diagram

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3.2 Damage Criterion for RC Wall and Panels

Yufeng Shi and Mark G. Stewart conducted the damage and risk assessment for RC wall panels subjected to explosive blast loading [42]. They used the maximum support rotation (θ) obtained from the LS-DYNA analysis to estimate the degree of damage of RC wall panels. Three damage limit states based on test data and UFC 3-340-02 [43] are used that shown in Table 5 [39]:

Table 5 Damage criterion based on test data and UFC 3-340-02 [39, 43]
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Support rotation \(\frac{1}{4}\) Arctan (mid height deflection/mid-wall height). Repairable damage means that the component has some permanent deflection. It is generally repairable, if necessary, although replacement may be more economical and aesthetic. Heavy damage defines that the component has not failed, but it has significant permanent deflections causing it to be unrepairable. Hazardous failure means the component has failed, and debris velocities range from insignificant to very significant.

Syed [44] utilized the UFC-3-340-02 [43] damage criteria for damage assessment of RC panel and beam in equivalent SDOF analysis. The damages are defined based on the support rotation of the members and are classified into low, moderate and severe. When support rotation is 2°, yielding of the reinforcement is first initiated and the compression concrete crushes, the damage of the wall is termed as low (LD). When the support rotation is 4°, the element loses its structural integrity, and moderate damage (MD) occurs. At 12° support rotation, tension failure of the reinforcement occurs. This is defined as the severe damage (SD). Zhou et al. [45] used a dynamic plastic damage model to estimate responses of both an ordinary reinforced concrete slab and a high strength steel fibre concrete slab for concrete material under to explosive loading. Shope [46] used the maximum deflection, δ corresponds to the specific support rotation, θ defined in UFC-3-340-02 and illustrated in Fig. 10 to define the damage level. The respective δ value for each damage level is calculated by

$$\delta = \frac{b}{2}\tan \theta$$
(7)

where b is the shortest span of the wall. The critical values of δ are set to be the numerical maximum mid-height deflection of the RC panel. These damage criteria are used in this study to define damage levels of P–I diagrams.

Fig. 10
figure 10

Resistance-deflection curve for flexural response of concrete elements [43]

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3.3 Damage Criterion for RC Slabs

Wang established the damage criteria for different levels of damage in the square reinforced concrete slab under close-in explosion that show in the Table 6 [47]. They proposed empirical damage criterion by the support rotation angle and the support rotation is defined by the ratio of the calculated peak deflection to half a span length for one-way slabs:

$$\tan \theta = \frac{{x_{m} }}{L/2}$$
(8)

where \(x_{m}\)= the centre maximum deflection, L = length of the slab.

Table 6 Damage criterion of the slab in the Wei Wang study [47]
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4 Mode of Failure

Blast loading effects on structural members may produce both local and global responses associated with different failure modes [38]. Global failure results in the collapse of the entire structure, localised failure occurs on only some reinforced concrete members such as beams, slabs and columns. Global failure usually occurs under static and quasi-static loading [48]. The general failure modes associated with blast loading can be flexure, direct shear or punching shear [49]. In particular, failure of RC columns due to high loading rates typical of blast (102s−1–104s−1) is predominantly shear-critical. This mode of failure is influenced by strength deterioration due to concrete cracking and spallation typically occurring prior to ultimate failure as represent in Fig. 11 [50, 51].

Fig. 11
figure 11

Blast induced damage in reinforced concrete columns; a shear failure (courtesy of TPS consult, London), b concrete spalling phenomenon [15]

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Krauthammer et al. [40]. concluded that the flexural failure modes and direct shear failure modes are always independent to each other. In general, the response and failure for most structures can occur in more than one mode. Although flexure is usually the predominant mode, but under certain circumstances, failure may occur in other mode (e.g. direct shear). According to a Fig. 12, there is two failure modes that in the P–I diagram consists of two threshold curves, each representing a failure mode that the true threshold curve shown by the dotted line [52].

Fig. 12
figure 12

P–I diagram with two failure modes [52]

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According to T. Ngo, The essential characteristics of loading and building response for transient loads produced by explosions depend primarily on the relationship between the effective duration of the loading and the fundamental period of the structure on which the loading acts [38]. T. Ngo done a field blasting test of a RC wall subjected to a close-in explosion of 6000 kg TNT equivalent, the wall suffered direct shear failure as shown in Fig. 13. Direct shear failure occurs in high velocity impact and in the case of explosions close to the surface of structural members.

Fig. 13
figure 13

Breaching failure due to a close-in explosion [38]

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Mutalib and Hao [28] observed three main damage modes of RC columns in their numerical simulation as shown in Fig. 14. They observed that when the column is subjected to impulsive load the failure is inclined to be damaged by shear and in the dynamic loading region failure of the column is a combination of shear and flexural damage and finally in the quasi-static region, the column is likely damaged by flexural failure mode.

Fig. 14
figure 14

Damage modes of the non-retrofitted RC column [28]

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Fujikura and Bruneau [53] performed blast testing on 1/4 scale ductile RC columns, and non-ductile RC columns retrofitted with steel jacketing. Figure 15 show the shear failure at the base and top of the RC column. According to Fig. 15, crack patterns observed after tests are shown along the height of the RC columns. Therefore, column did not exhibit a ductile behavior under blast loading, but rather failed in shear at the base of the column. A negative moment developed at the top of the column, leading to fracture of the steel bars on the tension side at the top of the column and to spall off the cover concrete on the compression side, as shown in Fig. 15.

Fig. 15
figure 15

Shear failure at the base and top of the RC column [53]

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In the Shi study, two damage modes have been observed during the numerical simulation of RC column damage to blast loads [24]. One is shear damage, and the other is flexural damage. Sometimes the failure of the column could be a combination of the above two modes. The typical results of these three damage modes derived from numerical simulation are shown in Fig. 16.

Fig. 16
figure 16

Damage modes of RC column under blast loads; a shear damage; b flexural damage; c combined shear and flexural damage [24]

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Ngo performed a blast test on one-way panels with the average reflected impulse and average reflected pressure of 2876 kPa.ms and 735 kPa, respectively [54]. The one-way panel failed due to concrete breach and mid-span crack formed vertically at the front and rear surface. Figure 17 shows the failure mode of the RC panel [54]. The similar damage mode is observed in [55] in a shock tube test of a 0.62 m × 1.75 m × 0.12 m panel as shown in Fig. 18. The panel failed in the flexural mode at the tested value of 208 kPa peak pressure and 3038 kPa.ms impulse.

Fig. 17
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After the blast in Ngo field test [54]

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Fig. 18
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Result from a shock tube test of a RC panel in [55]

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Weerheijim tested square RC panels simply supported at four sides by a blast simulator at different pressure levels [56]. Figure 19a show the extensive cracks after a blast load of 160 kPa peak pressure. The crack pattern is consistent with that observed in field blasting test carried out by Muszynski and Purcell [57] shown in Fig. 19b, where the tested wall failed due to tension failure resulted from an explosive charge of 830 kg detonated at 14.6 m standoff distance from the structure.

Fig. 19
figure 19

Crack patterns in a Weerheijim et al. field test [56] and b Muszynski and Purcell test [57]

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The failure mode categories for simply supported beams and fully clamped beams are evaluated in Ma study [58]. They assessment bending failure and shear failure in their research and the responses of the beams are analysed based on five transverse velocity profiles. The failure mode for simply supported beams and fully clamped beams are show in Fig. 20. Failure modes of simply supported and fully clamped beams are shown in Table 7.

Fig. 20
figure 20

Distribution of Failure modes for simply supported and fully clamped beams [58]

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Table 7 Failure modes of simply supported and fully clamped beams [58]
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In 1973, Menkes and Opat [59] were the first to report the three possible failure modes on fully clamped plates and beams loaded impulsively. Three clearly different damage modes are shown schematically in Fig. 21. They are described as: Mode I: large inelastic deformation; Mode II: tearing (tensile failure) in outer fibres, at or over the support; and Mode III: transverse shear failure.

Fig. 21
figure 21

Failure modes for explosively loaded plates and beams

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The same failure modes are also observed in blast load experiments on circular [60] and square steel plates [61, 62]. When the explosion centre is very close to the structure, RC panel might suffer localized crushing and spalling damage. Otherwise, damage modes of RC panels are similar to those of steel plate as observed in various field blasting tests reviewed above. The RC panel damage mode not only depends on the amplitude and duration of the blast loads but also depends on the boundary conditions and panel material properties. Ma et al. [63]. observed three failure modes in their study. Mode 1 contains the shear failure only. Mode 2 indicates the bending failure which has a plastic hinge at the centre of the element. Mode 3 can be considered as the combination of mode 1 and mode 2.

5 Conclusion

The current study performed an overview on the damage criterion and failure modes of RC structures when subjected to extreme dynamic loads. The damage index is calculated to establish a numerical value to identify the level of damage in the post event investigations. In this study different damage criterion illustrated to evaluate the damage level of RC structures when subjected to high impulsive loads. The results demonstrated that the residual axial load carrying capacity of the partially loaded RC columns is a suitable method to investigate the level of damage in the columns as it has a proven history of success against blast loads. The general failure modes associated with blast loading can be flexure, direct shear or punching shear. This research work and the conclusions drawn may be has been increased awareness about safety and behaviour of RC structures from explosive loads.