1 Introduction

Progressive collapse has become a popular research topic due to its history of causing catastrophic damage to structures and people. The terminology of “progressive collapse” is defined as the spread of an initial local failure from element to element, eventually resulting in the collapse of an entire structure or a disproportionally large part of it. Many numerical investigations were conducted in an attempt to completely understand this phenomena and improve structural resistance to progressive collapse.

The effect of blast loads on critical bridge components and bridge global response, as well as proposed protection strategies for mitigating blast hazards, were studied by (Yahia, 2009). Various blast scenarios were applied to a typical prestressed concrete girder, either above or below the bridge deck. A fine mesh has been used to simulate the variable cross sections of the bridge girder. The computer simulation results were utilized to identify the susceptible bridge components during a blast danger, as well as to estimate the magnitudes and locations of maximum shear forces and bending moments.

Moreover, progressive collapse analysis for post-tensioned box girder bridges was conducted under blast loads by (Ibarhim et al., 2012). The mish was chosen to be fine enough to simulate box section accurately by using 31,600 elements and 2,860,388 springs. A large vehicle bomb (LVB) placed beneath the bridge was studied in two different locations. The severity and mechanisms of damage in the bridge pier and deck were investigated.

In addition, the progressive collapse of a multi-span prestressed continuous bridge due to vessel collision was studied by (Zhao et al., 2014; Jiang et al., 2017). Finer meshes were utilized to adequately capture the internal contact between various components, local deformation or buckling, and material failure as best as possible. The results demonstrated that the bridge pier directly impacted by a vessel was collapsed in the lateral direction of the bridge span, whereas non-impacted piers were collapsed in the longitudinal direction of the bridge span.

The progressive collapse of Hongqi Bridge, a multi-span simply-supported bridge in Zhuzhou city, was studied by (Bi et al., 2015; Seyed Khoei et al., 2020). The results indicated that the multi-span simply-supported bridge with wall-type piers might be vulnerable to domino-type progressive collapse due to the low shear strength of the supporting piers. Furthermore, because insufficient seating length in the abutment was identified as a key effective factor in the onset of collapse, the use of restrainers as a deck-to-abutment connection may reduce the likelihood of progressive collapse.

The procedure of progressive collapse of prestressed voided slab bridges under earthquakes, as well as the effects of other parameters on the propagation of collapse of regular, semi-regular, and irregular bridges, were investigated by (Seyedkhoei et al., 2019). The findings revealed that domino-type progressive collapse occurred in bridges with voided slabs after the first failure of the deck at the bridge abutment seating. In addition, it was found that, the deck type, piers’ height, and ground slope had a significant impact on the progressive collapse procedure on both regular and irregular bridges with voided slab decks.

The behavior of bridge decks under blast loads was studied using the finite element method by (Hassan et al., 2021). Several explosive charges were deployed at various sites to ascertain the impact of explosion magnitude and placement on the bridge. The results showed that for minor charges, the bridge was almost completely unaffected, however for big charges, the deck completely failed, causing the bridge to fall.

The collapse of bridges during extreme loading conditions has prompted several concerns about their ability to withstand partial or whole collapses. It was reported that the progressive collapse could be initiated due to two broad categories, natural factors such as earthquakes, tsunamis, floods, mudslides, and hurricanes; and human factors such as design error and construction method, vehicle overloading or collision, fire, and terrorist attack (Deng et al., 2016; Zhang et al., 2022). Since the collapse process is strongly dependent on the entire structural system, non-linear numerical simulation with a reasonable solution time and good accuracy becomes a targeted method of investigation.

Recent studies have extended the Improved Applied Element Method (IAEM) to facilitate modeling prestressed concrete structures with bonded and unbonded tendons (Abdelaziz et al., 2021a; Abdelaziz et al., 2021b). The models were validated under different types of loadings, highlighting the reliability of the proposed model. However, the capability of the models to perform failure analysis was not examined. Therefore, the current work focuses on carrying out failure modeling of prestressed bridges under extreme loading circumstances using the IAEM to show the precision and validity of the numerical method and to demonstrate its capacity and efficiency in modelling the failure of the prestressed bridges.

2 Improved applied element method

The IAEM has recently been developed as an efficient analysis technique for modeling large scale framed buildings with homogeneous and non-homogeneous cross sections, such as steel sections, reinforced concrete (RC) elements, retrofitting sections, and unbonded and bonded prestressed structures, up to complete failure under various hazard loads (Elkholy & Meguro, 2004; El-Kholy et al., 2012; Abdelaziz et al., 2020; Abdelaziz et al., 2021a; Abdelaziz et al., 2021b). In this technique, each structural member is divided into an appropriate number of multi-layered rigid elements composed of several layers representing the unconfined and confined concrete, the bonded non-prestressing reinforcing steel, the bonded prestressing tendon, and the retrofitting material, if any; as illustrated in Fig. 1 (Abdelaziz et al., 2021a). Different rectangular or non-rectangular sections can be simulated using the advantage of the multi-layered element type without any complications. All identical layers in the neighboring elements are connected together by shear and normal springs representing the material characteristics of each layer, as shown in Fig. 2 (El-Kholy et al., 2012).

Fig. 1
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The multi-layered element type (Abdelaziz et al., 2021a)

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Fig. 2
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Modeling RC framed structures with IAEM (Abdelaziz et al., 2021b)

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The main procedure of this approach is illustrated in the flowchart shown in Fig. 3. The structural equilibrium equation can be solved using both incremental methods, load control and displacement control. In addition, this approach takes into consideration P-∆ effects and large displacement during the structural analysis. The main distinction between the IAEM and the conventional Applied Element Method (AEM) is that the IAEM employs only one element to model the nonrectangular cross sections with high efficiency and reliable accuracy, resulting in a significant reduction in the solving time.

Fig. 3
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Flow chart of the IAEM (El-Kholy et al., 2012)

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The Okamura and Maekawa compression model (Okamoto & Maekawa, 1991), shown in Fig. 4a, is employed for both confined and unconfined concrete. The tangent modulus of each concrete spring is computed at each load step, whether the concrete spring is in the loading or unloading stage. In tension, the stress-strain curve is assumed to be linear up to the tensile strength. After this point, the stiffness of springs under tension is assumed to be zero.

Fig. 4
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Material model for (a) concrete, (b) non-prestressing reinforcement, and (c) prestressing reinforcement (Abdelaziz et al., 2021a).

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A classical elastic plastic material with strain hardening using a bilinear stress–strain relationship in both tension and compression loading conditions, shown in Fig. 4b, is employed for non-prestressing reinforcement. On the other hand, the Menegotto and Pinto stress-strain relationship (Menegotto & Pinto, 1973), shown in Fig. 4c, is employed to represent the prestressing tendons.

3 Simulation of the bridge

To understand the global behavior of bridges subjected to extreme loading conditions, a typical prestressed concrete girder superstructure bridge has been analyzed using the IAEM method. In this example, the bridge design, geometry, materials, dimensions, and details are in accordance with the Egyptian Code of Practice for Design and Construction of Concrete Structures and the Egyptian Loading Code (ECP-203, 2018; ECP-201, 2012). The bridge consisted of three simple spans; each of 30 m long. Each span is supported by prestressed girders 2685 mm apart and a 200 mm concrete deck. Three prestressing tendons are used for each girder with a nominal diameter of 15.24 mm, a yield strength of 1674 MPa, and an ultimate strength of 1860 MPa; conforming to the British Standard BS-5896 (BS-5896 TBS, 2012). Each tendon has a total prestressing force of 2930 kN. The girders are supported on 200 mm elastomeric bearing pads. The supporting structure consists of a moment-resisting frame with two columns of 1400 × 1400 mm2 and a RC girder of 1500 × 2000 mm2. The utilized concrete has a compressive strength of 55 MPa, whereas reinforcing bars have yield and ultimate strengths of 400 MPa and 600 MPa, respectively. The bridge geometry, dimensions, and reinforcement details are illustrated in Figs. 5 and 6.

Fig. 5
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The geometric configuration of the typical prestressed girder

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Fig. 6
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Cross section of bridge

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3.1 Structural modeling

A typical girder has been selected to perform the analysis. Using the improved approach, the studied bridge has been modelled taking into account the constituent material properties. With the multi-layered element feature, each bridge girder has been modeled using 25 elements, with a total number of 256 elements. The girders’ dimensions and the tendons’ varying eccentricities along the girders’ length have been considered in the model. The model and the elements’ configurations are shown in Fig. 7. The characteristic parameters for each material are listed in Table 1. The bridge model has been analyzed under two extreme loading scenarios. The first one is applying a detonation charge located in the middle of the central span. The second scenario is a sudden failure of a column representing a truck or a vessel colliding with the bridge pier. At first, the design load of the structure is statically applied as an initial load in 50 load increments, while the blast loads or the sudden failure of the column are applied as loading in the time domain. The time step for blast load should be different from the time step for sudden column removal because the blast load reaches its highest value and decays to atmospheric pressure in milliseconds (Amr Ramadan Ibrahim, 2018). The chosen time steps are 0.0001 sec. and 0.01 sec. For the blast load condition and the column removal condition, respectively.

Fig. 7
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IAEM model of the analyzed bridge

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Table 1 Material properties of the concrete, the reinforcing steel, and prestressing tendon
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3.2 Blast load

In general, an explosion is a phenomena caused by the quick and unexpected release of a massive quantity of energy (Beshara, 1991). When an explosion occurs, the hot gases released by the explosion source forcefully push the atmosphere surrounding the explosion and generate a blast wave. This wave exerts significant pressure on exposed surfaces and penetrates the structure through apertures. In a fraction of a second after the explosion, the blast wave spreads outward from the explosion site. As the shock propagates outward, the peak of the overpressure caused by this hemispherical explosion decays rapidly as a function of distance from the source, as shown in Fig. 8. The reflected wave’s pressure amplitude is at least twice that of the original shock wave and is proportional to the initial shock’s strength, which is proportional to the charge weight. When a blast wave reflects, the pressure amplitude can be substantially larger than the pressure caused by the original shock alone (Beshara, 1991). Figure 9 shows how the blast pressure decays exponentially and finally becomes negative.

Fig. 8
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Variation of pressure with distance (UFC-3-340-02, 2008)

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Fig. 9
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Blast wave pressure- time history (UFC-3-340-02, 2008)

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Unlike static and seismic loading, blast loading is a high-magnitude pressure that affects just a small portion of the building. The fundamental distinction between blast loading and other forms of loading is the quick load rate, which can excite higher structural modes that are typically ignored for other types of hazards, such as earthquakes.

The factors influencing a structure’s behavior during explosion scenarios are classified as internal and external factors. Internal factors are those that are dependent on the properties of the structural element subjected to the blast load, such as stiffness, mass, ductility, redundancy, and overall structure continuity, while external factors are those that are affected by the explosion situation. They include the explosive material, standoff distance, charge weight, and angle of incidence. In explosion situations, the standoff distance is the most important element. Simply extending the standoff distance by a few meters reduces the pressure significantly. Therefore, a scaled distance is used to weight the energy of the blast load as follows:

$$Z=\frac{R}{\sqrt[3]{W}}$$
(1)

where Z is the scaled distance, R is the stand-off distance, and W is the equivalent TNT charge weight. This relationship illustrates that the energy delivered to different targets at the same scale distance is similar. For example, the energy given to a target by 8 ton TNT at a stand-off distance of 10 m from a specified point, is the same as the energy delivered by 64 kg of TNT at a stand-off distance of 2 m from the point (both having the same scaled distance of 0.50 m/kg1/3) (Amr Ramadan Ibrahim, 2018).

The blast loads acting on the bridge have been determined using a developed software, named VecTor-Blast (Miller, 2004) which calculates the pressure-time history at specified points on a three-dimensional cuboid structure for known values of charge weight and standoff distance. This numerical tool was validated using experimental data, which demonstrated that VecTor-Blast is capable of reliably estimating pressure time histories on the structure’s front and back sides.

In order to model a vehicle explosion over the bridge in the current study, several charge weights have been explored at a height of 1.5 m above the middle bridge’s mid-span until the girder collapsed. The pressure-time histories for each element in the present model have been calculated based on their stand-off distance. In addition, the pressures have been converted into nodal forces by multiplying the pressures by the tributary areas of the elements. The generated load-time histories have been used on the bridge model with a time step of 0.0001 sec. It should be noted that the negative phase’s effect has ignored during the analysis. The procedure for applying the time history of the blast load is illustrated in Fig. 10.

Fig. 10
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Procedure of applying the blast load time history

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The general differential equation of motion governing the response of the bridge is described as (Tagel-Din & Meguro, 2000):

$$\left[M\right]\left\{\Delta \overset{..}{U}\right\}+\left[C\right]\left\{\Delta \overset{.}{U}\right\}+\left[K\right]\left\{\Delta U\right\}=\left\{\Delta f(t)\right\}+\left\{{R}_m\right\}+\left\{{R}_G\right\}$$
(2)

where: [M] is the mass matrix; [C] is the damping matrix; [K] is the nonlinear stiffness matrix; {Δf(t)} is the incremental applied load vector; \(\left\{\Delta \overset{..}{U}\right\}\), \(\left\{\Delta \overset{.}{U}\right\}\), and {ΔU} are the incremental acceleration, velocity, and acceleration vectors, Rm is the residual force vector due to cracking and incompatibility between strain and stress of each spring; and RG is the residual force vector due to geometrical changes in the structure during loading.

3.3 Collapse mechanism due to blast loads

When an explosion occurs over the bridge’s mid-span, the hemispherical blast waves flow towards the bridge structure. The structural elements affected by these blast waves are the girders. The failure history of the bridge due to the blast load is illustrated in Fig. 11. Collapse of a bridge due to blast in reality is shown Fig. 12. The history of failure can be summarized as follows:

  1. 1-

    The middle span has experienced severe vertical displacement, resulting in additional straining actions on the girder.

  2. 2-

    Tensile cracks have occurred at the mid-span followed by yielding of the top and bottom reinforcement and the tendon, as well as, crushing of the top fiber of the concrete.

  3. 3-

    With the increase of the additional loads caused by the propagation of the blast wave, the bridge girder has been exposed to more severe damage until the mid-span section failed resulting in downward movement of the two failed parts of the girder behaving as two cantilevers connected to the elastomeric bearing.

  4. 4-

    The subsequent mode of failure depends on the type of bearing between the girder and the supporting column. Separation between the girder and the column has happened when the girders are simply resting on the elastomeric bearing (free bearing), as in the current model. On the other hand, for a typical fixed bearing connection, another mode of failure might occur depending on the tensile strength of the anchor bolts which connect the sole plate to the pier.

  5. 5-

    The girder has fallen down and collided with the ground, while the other side has collided with the adjacent column, resulting in a shear load acting on the column. If this shear load is more than the shear capacity of the column, it will collapse, resulting in the collapse of the adjacent span, and this procedure will continue progressively until the complete collapse of the bridge takes place. However, in the current model, the shear capacity of the column was sufficient to sustain the shear force resulting from the girder impact on the column.

Fig. 11
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Collapse mechanism of the prestressed girder under blast loading

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Fig. 12
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Collapse of a prestressed bridge due to blast scenario

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3.4 Sudden failure of a column

The entire bridge’s integrity is vital to prevent the horizontal spread of damage to nearby areas. Progressive collapse happens when a local failure of a key structural element causes a chain reaction of structural failures, resulting in the collapse of the entire structure or a disproportionately significant part of it. Proper design and details can greatly reduce the probability of such a collapse. This can be accomplished by providing structural elements with enough continuity, redundancy, and energy-dissipating ability to shift loads from the locally injured region to nearby regions capable of supporting these extra loads without collapsing.

In different cases, the main supporting members in the bridges are exposed to different hazardous loading conditions (e.g., bomb, earthquake, or vehicle impact). Intermediate piers in multiple span bridges should be designed to resist the loss of one or more of their supporting columns. In addition, the cap beam should be designed for different scenarios, including column removal.

To obtain full knowledge of the total response of the bridge due to the sudden failure of one supporting member, the analyzed bridge has been exposed to an internal pier removal to represent truck or vessel impacts to the bridge pier. Initially, the structure’s design load is applied as an initial load in 50 load increments followed by the unexpected failure of one supporting structural column at 0.2 sec. With a time step of 0.01 sec.

3.5 Collapse mechanism due to column removal

In this case of failure, an interior column is assumed to fail suddenly to represent different cases of pier collapse, such as vehicle or vessel impact. The failure of the column is modeled by removing the springs connecting the column elements during a period of 0.20 second. The dynamic nonlinear collapse analysis using IAEM has been performed. Based on the simulation results, the mechanism of failure due to the column collapse is shown in Fig. 13. Moreover, the collapse of a bridge due to column removal in reality is shown in Fig. 14. The history of failure can be summarized as follows:

  1. 1-

    Once the interior column has collapsed, the two adjacent girders have fallen down acting as two cantilevers connected to the elastomeric bearing. In addition, tensile cracks, as well as, yielding of the top reinforcement, bottom reinforcement, and tendon have occurred.

  2. 2-

    Due to the lack of continuity between the adjacent spans, no catenary action has been developed in the superstructure of the bridge. The two spans have continued falling dawn until one of their sides collided with the ground.

  3. 3-

    The type of the bearing between the girder and the supporting column affects the post-behavior of the girders. In the current bridge, the connection between their other sides and the elastomeric supports has failed. As a result, these sides have collided with the adjacent column, resulting in a shear load acting on the column.

  4. 4-

    As previously stated, if the shear capacity of the column is less than the action shear force, it will collapse, and the collapse of the adjacent span will take place, resulting in progressively collapse of the bridge. However, the shear capacity of the column in the current model is sufficient to resist the shear force caused by the girder impact on the column.

  5. 5-

    The final mode of in the current model is that one of the girder sides has rested on the ground while the other side has rested on the column. However, there is another possibility: the entire length of the bridge might fall down to the ground after the failure of the connection between the girder and the elastomeric bearing. This depends on the span of the bridge, the height of the piers, and the column’s mode of failure.

Fig. 13
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Collapse mechanism of the prestressed girder due to column removal

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Fig. 14
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Collapse of bridge due to column removal in practice

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4 Conclusions

In this study, the IAEM has been utilized to perform collapse analysis of prestressed bridges under various hazard loads. A typical prestressed bridge has been analyzed under two severe loading conditions; blast load and sudden failure of column. According to the presented results, progressive collapse phenomena can occur due to the lack of redundancy. Alternative load paths should be provided to avoid such catastrophic collapses and minimize the induced damage during blast hazards. Blast resistant bridge design should be developed and implemented by the national building codes to protect bridges against terrorist events. The numerical approach, IAEM, presented in this research, demonstrates a strong capability for studying the overall behavior of prestressed bridges under high loading circumstances, from the early stages of loading to final collapse. The investigation has confirmed the IAEM’s dependability in simulating prestressed bridges using a minimum number of elements and, therefore, a reasonable CPU solving time.