Collapse Mechanism of Single-Layer Cylindrical Latticed Shell under Severe Earthquake

In this paper, the results of finite element analyses of a single-layer cylindrical latticed shell under severe earthquake are presented. A 3D Finite Element model using fiber beam elements is used to investigate the collapse mechanism of this type of shell. The failure criteria of structural members are simulated based on the theory of damage accumulation. Severe earthquakes with peak ground acceleration (PGA) values of 0.5 g are applied to the shell. The stress and deformation of the shell are studied in detail. A three-stage collapse mechanism “double-diagonal -members-failure-belt” of this type of structure is discovered. Based on the analysis results, measures to mitigate the collapse of this type of structure are recommended.


Introduction
After the events of September 11, 2001, more and more researchers have carried out research work on investigation of the mechanism of progressive collapse in buildings, trying to find possible mitigating methods. But few studies have been performed for space structures. As a typical long-span space structure, single-layer cylindrical reticulated dooms are widely used in public buildings, such as terminals, gymnasiums, large factories and so on for its graceful appearance. However, if the dome collapses under severe earthquake, heavy casualties and economical losses would be caused.
Therefore, research in this area is imperative.
In the current design practice, some design procedures are available in Europe and the United States on mitigating the progressive collapse of buildings，such as the design guidance of the Department of the American Concrete Institute (ACI) [1] and the General Services Administration (GSA) [2] in which specific process of assessment on the necessity of anti-collapse design is recommanded. In the American Society of Civil Engineers (ASCE) [3] ductility and sufficient connection performance of the structure is required. In Eurocode 1 [4], it is stipulated that the structure must be able to resist a certain accidental load without a large scale of collapse. Those specifications aimed at mitigating of the collapse of structures under specific accidents. In Eurocode 2 [5], special requirements are given to the construction of reinforced concrete, especially about the anchorage and connection of bars. Some references such as FEMA [6] and NIST [7] also provide general design recommendations, which require steel-framed structural systems to have enough redundancy and resilience. However, most of these design codes focused on buildings, no detailed design procedure to prevent progressive collapse of reticulated shells is available.
Up to now, some numerical investigations and experimental tests have been carried out on building structures. In 1974 Mc Guire [8] reviewed the problem of progressive collapse of multi-story masonry buildings under abnormal loading and presented some recommendations to prevent progressive collapse. Shimada [9] conducted a shaking table test of a full-scale 2-story and 1×1 span steel frame model in 2007. Song [10] conducted a test on a steel frame building by physically sudden removal of four 3 columns at ground level to investigate the load redistribution within the building. Chen, J.L [11] conducted a progressive collapse resistance experiment on a two-storey steel frame in which composite concrete slab was adopted. Tsitos, [12] performed a quasi-static "push-down" experiments on a 1:3 scale three-story steel frame considering multi-hazard extreme loading. Starossek [13] suggested a pragmatic approach for designing against progressive collapse and a set of design criteria. Lim, [14] investigated progressive collapse of 2D steel-framed structures with different connection and found that horizontal column buckling propagation control was the only solution. Yamazaki [15] clarified the frame conditions which enable stories to resist progressive collapse through comparing the gravity potential energy released by the story collapse with the energy which columns absorbe before they completely collapse due to the compressive load. O'Dwyer [16] presented details of an algorithm for modelling the progressive collapse of framed multi-story buildings.
However, as it can be seen that, most of above research is related to building structures; little has been carried out on long-span space structures. Zheng et al. [17] developed a force-displacement hysteresis model for the collapse simulation of a power transmission pylon considering the buckling/softening-based fracture criterion.
Rashidyan et al. [18] recomended the method of strengthening the compression layer members along with weakening the tension layer members of double-layer space trussesis was an effective method for increasing the structure's ductility and load-bearing capacity against progressive collapse. To study the progressive collapse phenomenon of structures during earthquakes, Lau et al. [19] carried out an nonlinear analyses of reinforced concrete bridges by the Applied Element Method (AEM).
Miyachi et al. [20] carried out a progressive collapse analysis for three continuous steel truss bridge models with a total length of 230.0 m using large deformation and elastic plastic analysis. Takeuchi et al. [21] proposeed the post-fracture analysis methods for truss structures composed with tubular members of large diameter-to-thickness ratios, and investigated the collapse mechanism of such truss towers after the buckling and fracture of main columns and diagonals. Ponter et al. [22] discussd of the programming method based on the Elastic Compensation method used for limit and shakedown analysis of steel structures. Skordeli et al. [23] studied examples of limit and push down analysis of spatial frames under the aforementioned ellipsoidal approximations with several aspects discussed. Starting from a computational formulation for the elastoplastic limit analysis of 3D truss-frame systems, apt to provide the exact limit load multiplier and attached collapse mechanism. Ferrari et al. [24] derived a full evolutive piece-wise-linear response of the bridge, for different try-out loading configurations. Xia et al. [25] studied the dynamic behavior of snap-through buckling in single-layer reticulated domes, based on the nonlinear equilibrium equations. Kato et al. [26] peformed an analysis on buckling collapse and its analytical method of steel reticulated domes with semi-rigid ball joints, on the basis of a nonlinear elastic-plastic hinge analysis formulated for three-dimensional beam-columns with elastic perfectly plastic hinges located at both ends and the mid-span for each member.
However, those researches focused on the response of the space structure under static load, little research has been done for the collapse mechanism of a space structure under seismic load. Under seismic load, damage accumulation induced by cyclic loading is a very important factor that needs to be considered for its effect on the numerical simulation of the deterioration of both the stiffenss of the dome and strength of materials, and the fracture of members. In another word, an accurate prediction of the dynamical response of structures needs serious attention to be paid on damage 5 accumulation. However, in most of the numerical models used in above research, damage accumulation is ignored, little work has been done in this area. In this paper, based on thermodynamic theory, the damage evolution equations for fiber beam elements derived by the authors [27] is used to simulate the progressive collapse mechanism of single-layer cylindrical latticed shell. The corresponding constitutive relationship for beam elements and the relevant numerical analysis method are also developed.
Due to the redundancy and indeterminacy of single layer space structures, a dynamic instability criterion based on an implicit algorithm can capture neither the collapse mechanism of the reticulated shell nor the failure mode of individual components.
Therefore, the simulation of the whole collapse process requires application of an explicit dynamic algorithm, which is used in this paper. Based on the above studies, in Zhou et al. [27], the authors developed a subroutine program based on an explicit dynamic algorithm to analyze the response of the single-layer reticulated shell under a severe earthquake. The program was validated against experimental tests. It is proved that the proposed numerical method can accurately simulate the members' failure, the redistribution of internal forces, and the collapse mechanism of the whole structure.
Based on this numerical method, parametric studies on single-layer reticulated shell are performed and the collapse mechanism of this type of structure is studied.

The Numerical model
The collapse analysis is performed by general purpose program ABAQUS through the further development of the subroutine VUMAT program in Abaqus. The numerical method introduced in Zhou et al. [27] is used in this research. All the structural members of the reticulated shell are simulated using the beam elements. Each beam element is further discretized into eight longitudinal fibers across their cross-section with appropriate constitutive model defined to each fiber as is shown in Fig 2. In the dynamic analysis, below damage critrien is used: 11  is normal stress existing in a beam element, D is cumulative damage The failure criterion of each fiber is determined based on the studies by the authors in (2018). When new D in a fiber develops into certain value Dlimt, the fiber is determined as failure. Dlimt is determined using below formula: In the designed VUMAT subroutine program, when the failure of one fiber is triggered, the elastic modulus of this fiber will be set as zero. When all the fibers in a beam element fail, that beam element is determined as failure, thus this beam element will be deleted by the program. Therefore, the process of collapse of the dome can be simulated. After an element is deleted due to failure, the explicit dynamic analysis based on Central difference method, which enabling the accurate capture of the response of the dome. For detailed explanation of the numerical algorithm, please refer to Zhou et al. [27].

Progressive collapse analysis of single-layer cylindrical latticed shell
Base on afrometioned analysis method, a single-layer monoclinic cylindrical reticulated dome with span of 20m, 30 metres in length and 7.5 metres in height was modeled using general purpose program Abaqus (Fig. 1) Figure 1 also shows the structural zone divided by the authers, which is to clearly demonstate the response of the dome under earthquake loading.
In the simulation, all the structural members were modeled using beam elements. Each beam element was divided into four segments along its length to model the buckling behavior. as it is explained Zhou et al. [27], this is an effective way to model the member buckling with sufficient accuracy. In addition, each beam element was subdivided into eight fiber beams across its cross-section as shown in Fig.2. The fibers are numbered from 1 to 8 accordinly along the circumference of the beam element The analysis was divided into two steps. The first step was static analysis, where the gravity load was applied to the dome. The second step was dynamic analysis, where the time history of the Taft earthquake was applied at the support, in X, Y, and Z directions, with the peak ground acceleration (PGA) of 500 gal in the X direction, 0.85 times 500 gal in the Y direction, and 0.65 times 500 gal in the Z direction, respectively.
The time duration was 20s, which is greater than 10 times the natural period of the dome. The constitutive model of the material is using the hybrid hardening model and the modulus considered the cumulative effect of damage to simulate the performance deterioration of material occured during plastic tension in hysteresis process under seismic loading.     Apart from above analysis, extensive parametric studies were performed by the authors with diferent spans, diferent span:depth ratios, diferent length: width ratios, diferent roof loadings. It has proved that the similar collapse process were observed.

Collapse pattern of "Double-diagonal member failure belt"
In addition to above case studies, all the analysis cases was applied with the peak from nearly center to boundary along the lengthwise and the first "diagonal member failure belt" come into being (Fig.7). The stiffness of dome is also greatly weakened.
Then another damaged zone appear at middle of dome and in the zone diagonal members failed continuously from ends to nearly center along the lengthwise. A second "diagonal member failure belt" was formed. The stiffness of dome is reduced further.
The whole deflection become significant. The position of two "belt" varied slightly with the change of geometry. At final stage, the dome deflect rapidly at the second "belt", with dropping twords the first "belt". The dome collapse. The paper calls the collapse process and phenomen as "double-diagonal -members-failure-belt" pattern.  (Figure 8). This is because plastic deformation occured in vibration, and the overall geometric configuration shifted towards zone of 2-2. This phenomenon conversly caused that internal forces of member in zone of 2-2/3-3 were significantly larger that in symmetric zone of 11-11/10-10. Consquenly, the development speed of damage of bars in zone of 2-2 were more than that of bars in the symmetric zone( Fig.3b and 3c).
As it shown in Figure 9, At this time, the dome acted as a arch whose local flexural stiffenss at zone of 2-2 weakened significantly ( Fig.13).

Figure 13 The transision between stage 1 and 2
In Fig. 11a, we can see that the normal stress at upper fiber 3 and lower fiber 7 at end section of traverse bar 74 was maintained at a value much greater than the yield stress of 280 MPa, while normal stress of fiber 5 and 1 also increased dramtically, which acted as a plastic hinge. Fiber 3 and 7 of traverse bar 14 failed at a short time later entering the stage, but the stress of its fiber 1 and 5 whose stress is relatively small now increased rapidly near to 200MPa. We can conclude that the normal stresses of remaining fibers of bar 14 were larger than yield point. This means that the end of bar 14 acted as a plastic hinge too. Finally we can concluded that all the traverse bars in this zone of 2-2 that yet not failed acted as plastic hinges. The formation of plastic hinges induced significant internal-force-redistribution effect. As shown in Figure 14, bending moment My' axial force F, bending moment Mx', torsion Mx'y'of diagonal bar 161 and 232 (Fig.1) , one at end and another at mid of zone of 6-6, were less than those of bar 242/254/278 (Fig.8) (Fig.11).
Untill now, the generation of the second "diagonal member failure belt" and the exist of the first "diagonal member failure belt" caused the formation of two zone whose stiffness were seriously weakened, or two plastic hinges. As showen in Fig.15, the lateral mechanism characteristics of the dome changed again.  In conclusion, in the three stages, variation of mechanical characteristics, damage zone, deformation status of dome is of distinct difference respectively. But we should understand, the three stages are closely continuous rather than intermittent or absolutely strictly divided.

Displacement during the collapse process
The deformation of the dome changed along the whole collapse process, therefore, the gravity center of the structure changed as well. Figure 17 shows the change of the locations of the center of the gravity of the structure during the whole collape process.
Different peak ground acceleration was selected with the same duration of 18 s. It can be seen that the center of the gravity went down to differernt distance with differernt PGA values

Conclusion
In this paper, a detailed progressive collapse analysis of a single-layer cylindrical latticed shells under severe earthquakes is performed using the fibre beam element method with the inclusion of the accumulation of material damage. Below conclusion can be made: 1. A failure pattern called "double-diagonal -members-failure-belt" is discovered for this type of space structure.
2. The failure pattern can be divided into three consecutive stages: a. Significant variation of deformation in diiferent structural zone; b. Continuously increasing internal forces; c. Damage structural members.