1 Introduction
Blast loading response is an essential aspect of civilian and military structures that are likely to be subjected to accidental explosions and terrorist attacks (Chen et al.,
2011; Jones,
1990). These extreme events demand a better understanding of structural behavior especially for appraising their safety (Biju et al.,
2017; Jahami et al.,
2019; Lee & Kwak,
2018; Li et al.,
2018; Liu et al.,
2018; Magnusson & Hallgren,
2004; Magnusson et al.,
2010; Temsah et al.,
2018; Young Lee et al.,
2018; Zhang et al.,
2013). Numerous experimental and numerical studies have been carried out in which damage of blast-loaded structural components is appraised (Guzas & Earls,
2011; Kirsch & Bogomolni,
2007; Nawar et al.,
2021; Portioli,
2020; Tian et al.,
2020; Wu et al.,
2020; Yu et al.,
2019). Most of the numerical models used in research require a high level of expertise and are computationally expensive (Jones,
1976). This demands a simple numerical method that can predict the blast loading response with reasonable accuracy.
The simplicity of rigid–plastic approximation has been known for several decades as an analytical tool defining and crystallizing concepts that can be extended to a wide class of problems (Cennamo et al.,
2017; Jones,
1986; Lee & Symonds,
1952a; Taylor,
1948). The rigid–plastic theory was first applied to dynamic problems in 1949 (Taylor,
1948) and systematically studied in 1952 (Lee & Symonds,
1952b). These studies subsequently yielded vast literature on the investigations of structures submitted to extreme dynamic loading (Bleich & Shaw,
1960; Jones & Shen,
1993; Ling et al.,
2017; Lowe et al.,
1972; Mehreganian et al.,
2019; Menkes & Opat,
1973; Parkes,
1955,
1958; Symonds,
1967; Symonds & Frye,
1988). Yet, it is noteworthy that each closed-form theoretical solution requires postulating a kinematically admissible velocity profile for the evolution of displaced configuration. Therefore, most procedures incorporating the simple rigid–plastic theory are oriented towards obtaining specific results for specific problems and demand personal judgment for getting the best results.
The numerical investigation of the dynamic response of rigid–plastic structures appears not to have been examined extensively. The commercially available finite element softwares, such as ABAQUS, ANSYS, etc., experience severe mathematical complications when a purely rigid–plastic model is adopted. From this precedent, mathematical programming provides an approach to incorporate the rigid–plastic theory into a unified formalism that is not specific and problem-oriented. Furthermore, mathematical programming (Dantzig,
1998; Dantzig et al.,
1955; Guzas & Earls,
2011; Lemke,
1978; Murty,
1983) has a broader application in various specialized fields of engineering, such as robotics (Nuseirat & Stavroulakis,
2000), fluid simulation (Andersen et al.,
2017), and agriculture (Garrido et al.,
2020). It has the potential to proffer a finite element-based numerical formulation (Maier,
1984; Martin,
1964) that facilitates any distribution of mass, spatial placement of applied loading, and temporal variation of the associated load pulses. Moreover, once the physical modeling decisions have been taken, a complete algorithmic or completely automatic solution procedure can be obtained. The benefits of mathematical programming have been recognized for more than 60 years. Extensive surveys of the use of the mathematical programming application in engineering plasticity have been reported by Maier (Maier,
1984), Maier and Munro (Maier & Munro,
1982) and Maier and Lloyd Smith (Maier & Munro,
1982; Smith,
1974).
Tamuzh (Tamuzh,
1962) first offered a procedure to determine the response of rigid, perfectly plastic continua using kinematic minimum principle. Later Capurso (Capurso,
1972) used this principle to formulate a mathematical programming problem capable of tracing the response of rigid–plastic framed structures submitted to short-duration pulse loads. The capability of such mathematical programming problem can be enhanced by incorporating the effects of strain rate and large displacements in rigid–plastic material model (Cennamo et al.,
2017; Jones,
1986; Lee & Symonds,
1952a; Taylor,
1948). In recent times, Patsios and Spiliopoulos (Patsios & Spiliopoulos,
2018) have analyzed structural frames using the mathematical programming method. Moreover, Milani and Tralli (Milani et al.,
2009) have employed mathematical programming to model the behavior of masonry walls. Later, Portioli has used a similar model for the dynamic and pushover analysis of masonry structures (Portioli,
2020). Although mathematical programming has the potential to solve problems in dynamic plasticity (Khan et al.,
2013; Rodigari et al.,
2019), this tool has not been exploited to any great extent (Milani et al.,
2009; Wu et al.,
2020).
In a companion paper by Khan et al. (Khan et al.,
2013), a mathematical programming formulation, called the Linear Complementarity Problem LCP (Lloyd Smith & Sahlit,
1991; Smith,
1990), was developed. In that study, the LCP solution was employed as a simplified design procedure for the assessment of structures under impact loading. This type of loading required an initial velocity profile for starting the Lemke Algorithm (Khan et al.,
2021). However, it was shown (Khan et al.,
2013) that such problems posed considerable numerical difficulties in finding solutions for simple bending-only models. These numerical problems were avoided by considering the bending–shear interaction in the model. In a recent research (Khan et al.,
2021), the Lemke algorithmic solution was tested by investigating several engineering problems; these included a simply supported beam subjected to rectangular pulse load and a portal frame subjected to triangular pulse load. A comparison of the LCP approach with the theoretical and the numerical solutions using lumped mass discretization showed excellent agreement. Nevertheless, there were occasions, particularly in the analysis of the portal frame problem, when the algorithm would become unstable producing illogical results. One remedy to reduce this instability was to use a smaller time step that is specified at the outset of the evolutionary process. In the current study, this numerical instability of the Lemke solver is investigated, and remedial measures are proposed. Specifically, the instability becomes more pronounced with the continuous mass discretization and mesh refinement. It turns out that this instability can disappear if the time step is allowed to adjust automatically, especially in highly nonlinear dynamic problems. A newly developed time-step controller subroutine is incorporated within a MATLAB program that reduces the increment size repeatedly until stability is achieved. It also overcomes the instability problems encountered when refining the finite element mesh (Khan et al.,
2013). Three illustrative examples are carefully selected to test the accuracy and efficiency of the improved LCP solver. The first two examples are comparable to that of the recent investigation (Khan et al.,
2021), but with the increased non-linearity to the dynamic response. The third verification example investigates the experimental study of Zhang et al. (Zhang et al.,
2013) involving the dynamic response of a semi-clamped RCC beam.
2 Research Significance and Assumptions
Computation of structural response under blast loading is important for reducing the risk and ensuring the safety of people. This demands a fast algorithm that can efficiently handle material nonlinearities and give accurate results. There exist limited numerical methods that can be applied to capture the complex dynamic behavior of structures subjected to blast loading. Commonly used methods of determining the response of structures to blast loading are simulations in commercial finite element softwares, such as ABAQUS, ANSYS, etc. However, due to their higher computational cost and requirement of skills to simulate complex models, an efficient formulation is needed that has the important practical benefit of speed and simplicity. Inspired by the promise shown by the previous investigations, rigid–plastic structures under impact (Khan et al.,
2013) and Lemke Algorithm for the rigid–plastic response (Khan et al.,
2021), the present study is a more systematic exploration of this powerful approach to efficiently predict the response of skeletal structures under blast loading. A predicament encountered previously (Khan et al.,
2021) was a tendency of the LCP algorithm to crash when a large number of continuous mass finite elements were employed in the modeling. Such problems are resolved in the current study when a newly developed automatic time-step controller is incorporated. Three computational examples are presented to prove the accuracy and efficiency of the updated LCP solver with low-order finite elements.
The following assumptions are made in the proposed LCP formulation:
1.
The initial energy is so large that the elastic response can be neglected
2.
A skeletal structure is envisaged as an assemblage of elements connected with other elements at stations called nodes.
3.
The development of plasticity is confined to discrete points or nodes on the structural element. No account is taken of the spreading of plasticity, either in the direction of the locus of cross section or within the structural element between the discrete points.
4.
The mass properties of a structure are defined at discrete points. Thus, it is idealized that massless elements connect these lumped masses.
5.
The structural system is considered as rigid–plastic, incompressible, and inextensible. Therefore, the effect of strain rate and strain hardening are ignored.
6.
The deflection of the structural system remains small during the dynamic response.
7 Results Discussion and Future Work
The important feature of the numerical implementation developed in this work is its provision for the automatic change of the magnitude of the time increment. This feature is highly advantageous for tracing the true behavior of the concrete and steel structures. Three examples are reported to demonstrate the accuracy and computational efficiency of the LCP formulation, considering the time increment algorithm. Interestingly, the numerical complications identified in the previous study (Khan et al.,
2013) of impact problems are nonexistent in the current case of pulse loading. This is because the impact problems required adjustment of the initial velocity profile for starting the LCP formulation; however, the current loading case requires an initial load to initiate the LCP. Moreover, it is found that the Lemke Algorithm (Khan et al.,
2021) can break down in the case of blast loading problems if finely discretized continuous mass elements are adopted. Again, this numerical difficulty can be resolved by allowing automatic control of the time increment. However, the error in the results for the continuous element procedure is marginally greater than in the lumped mass discretization, which is contrary to expectation. There is a possibility that the large size of the LCP matrix for the continuous mass model may have contributed to this in the form of round-off errors. The following three examples illustrate the effectiveness of the automatic time control algorithm.
The first investigation involves a theoretical study to explore the underlying mechanics of a rigid–plastic propped cantilever beam acted on by a linearly decaying triangular pulse. The nonlinear effects resulting from the asymmetry in the support condition and the decaying pulse load provide a better opportunity to test the improved LCP solver, compared to the relatively simple problem examined recently (Khan et al.,
2021). It is seen that the time-step controller subroutine automatically identifies the instant where instability occurs, reduces the time-step size until the instability is resolved, and thereafter continues the evolutionary process. With either the masses being lumped at the element ends or uniformly distributed along the element, the LCP solution traces the dynamic response that matches the derived continuum solution remarkably well.
A second investigation of a pulse-loaded steel portal frame is borrowed from the recent study (Khan et al.,
2021) showing that the LCP formulation can effectively capture the complex dynamic response through lumped mass discretization of the frame. In the current study, the portal frame is discretized into continuous mass elements and then the Lemke solver is examined for its computational merit. It is found that mesh refinement using continuous mass elements is a source of algorithmic instability. However, the LCP solver supplemented by the automatic time-stepping algorithm is computationally advantageous. The previous ABAQUS model (Khan et al.,
2021) is further improved by the addition of stiffeners at various locations of beam and columns to make the assembly rigid, unlike previously reported analysis (Khan et al.,
2021) where the modulus of elasticity was significantly enhanced to ensure the rigidity of the structure. The comparison of LCP results with the ABAQUS solution indicates that the sequence of mechanism movements is difficult to identify in the ABAQUS output because of the interspersed elastic and plastic deformations. In addition, the dynamic response of the ABAQUS frame shows the influence of bending and shear deformations, which is absent in the LCP model since that is based on the idealization of infinite resistance to shear deformation. However, the final deformation mode of the ABAQUS frame model is broadly similar to that of the LCP model. It also turned out that the problems encountered by continuous mass discretization (Khan et al.,
2013), such as spurious oscillations in stress-resultants, are eliminated in solving the blast loading problems.
The third benchmark test is the experimental results (Zhang et al.,
2013) against which is compared the LCP mid-span displacement for a blast-loaded RC beam with both ends semi-clamped. This experimental result is borrowed from the literature (Zhang et al.,
2013). ABAQUS model of the beam is also developed and calibrated on the experimental results. The concrete damage plasticity model is utilized for simulating concrete behavior in ABAQUS. The calibration is carried out until an acceptable level of error is reached in the mid-span displacement. After this calibration, the plastic hinge distribution, evolution of displacement, and strain time history are compared with the LCP solution. It is shown that the LCP solution for both the lumped mass and continuous mass discretization of 50 elements can capture the mid-span deflection satisfactorily. In the case of continuous mass discretization, the time-stepping algorithm again shows exceptional promise in ensuring numerical stability.
Finally, it is worth asserting that the proposed automatic time-stepping scheme has made the LCP formulation an efficient computer-based tool, which can be applied to size the skeletal structures subjected to blast loads. This simple finite element tool can predict the main features of behavior for a particular structural design rather quickly for both lumped mass and continuous mass discretization, thereby avoiding the use of advanced finite element analysis methods that impose considerable demand on computational resources. For problems where increased accuracy or further details of the structural behavior is required, then the present formulation may be straightforwardly extended to incorporate additional aspects, such as bending–shear interaction, strain rate, strain hardening, and large displacements. The LCP model involving large displacement of skeletal structures is in progress and the analysis will be reported in due course.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.