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

materials

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

Haitao Zhou 1,2,3, Yigang Zhang 2,3, Feng Fu 4,* and Jinzhi Wu 2,3

1 School of Civil and Transportation, Henan University of Urban Construction, Pingdingshan 467001, China; [email protected] 2 Spatial Structure Research Center, Beijing University of Technology, Beijing 100124, China; [email protected] (Y.Z.); [email protected] (J.W.) 3 Key Lab of Urban Security and disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China 4 School of Mathematics, Computer Science & Engineering, Northampton Square, City, University of London, London EC1V0HB, UK * Correspondence: [email protected]

Received: 30 April 2020; Accepted: 27 May 2020; Published: 1 June 2020

Abstract: 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.

Keywords: cylindrical latticed shell; damage accumulation; progressive collapse; ﬁnite element; earthquake

1. Introduction After the events of 11 September 2001, more and more researchers have carried out research work on the investigation of the mechanism of progressive collapse in buildings, trying to find possible mitigating methods. However, few studies have been performed for space structures. As a typical long-span space structure, single-layer cylindrical reticulated domes are widely used in public buildings, such as terminals, gymnasiums, large factories and so on due to their graceful appearance. However, if a dome collapses due to a severe earthquake, heavy casualties and economic 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 processes of assessment on the necessity of anti-collapse design are recommended. According to the American Society of Civil Engineers (ASCE) [3] ductility and the sufficient connection performance of the structure are required. In Eurocode 1 [4], it is stipulated that the structure must be able to resist a certain accidental load without a large-scale collapse. Those specifications aimed at mitigating the collapse of structures due to 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,

Materials 2020, 13, 2519; doi:10.3390/ma13112519 www.mdpi.com/journal/materials Materials 2020, 13, 2519 2 of 18 which require steel-framed structural systems to have enough redundancy and resilience. However, as most of these design codes focused on buildings, no detailed design procedures to prevent the progressive collapse of reticulated shells are available. To date, some numerical investigations and experimental tests have been carried out on building structures. In 1974, McGuire [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 two-story and 1 1 span steel frame × model in 2007. Song [10] conducted a test on a steel frame building by the sudden physical removal of four columns at ground level to investigate the load redistribution within the building. Chen, et al. [11] conducted a progressive collapse resistance experiment on a two-story steel frame in which a composite concrete slab was adopted. Tsitos [12] performed 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 the progressive collapse of 2D steel-framed structures with different connections and found that horizontal column buckling propagation control was the only solution. Yamazaki [15] clarified the frame conditions that enable stories to resist progressive collapse through comparing the gravity potential energy released by the story collapse with the energy columns absorb before they completely collapse due to the compressive load. O’Dwyer [16] presented details of an algorithm for modeling the progressive collapse of framed multi-story buildings. Xingxing Chen et al. [17] presents an experiment and analysis on a 1/5-scaled 10-story SRC frame- structural model. Fu [18,19] made numerical simulation of a multi-story building under consecutive column removal scenarios and Gao et al. [20] performed experimental investigation on semi-rigid composite joints to accommodate column loss. Qian et al. [21] investigated the behavior of precast concrete frames with high-performance dry connections subjected to the loss of a penultimate column. However, most of the above research is related to building structures, while little has been carried out on long-span space structures. Fu et al. [22] performed a numerical simulation for the progressive resistance of a double-layer grid space structure. Zheng et al. [23] 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. [24] recommended that method of strengthening the compression layer members along with weakening the tension layer members of double-layer space trusses 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. [25] carried out nonlinear analyses of reinforced concrete bridges by the Applied Element Method (AEM). Miyachi et al. [26] 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. [27] proposed post-fracture analysis methods for truss structures composed of tubular members with large diameter-to-thickness ratios, and investigated the collapse mechanism of such truss towers after the buckling and fracture of the main columns and diagonals. Ponter et al. [28] discussed the programming method based on the Elastic Compensation method used for limit and shakedown analyses of steel structures. Skordeli et al. [29] studied examples of limit and push-down analyses of spatial frames under the aforementioned ellipsoidal approximations, with several aspects discussed, starting from a computational equation 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. [30] derived a full evolutive piecewise linear response from a bridge for different loading configurations. Xia et al. [31] studied the dynamic behavior of snap-through buckling in single-layer reticulated domes, based on nonlinear equilibrium equations. Kato et al. [32] performed an analysis on buckling and developed an analytical method for steel reticulated domes with semi-rigid ball joints, on the basis of a nonlinear elastic–plastic hinge analysis equation for three-dimensional beam-columns with elastic–plastic hinges perfectly located at both ends and at the mid-span for each member. Materials 2020, 13, 2519 3 of 18

However, while those studies focused on the response of the space structure under static load, little research has been carried out on 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 the stiffness of the dome, the strength of materials, and the fracture of members. In other words, an accurate prediction of the dynamical response of structures require serious attention to be paid to damage accumulation. Some researchers carried out similar studies on Glass Fiber Reinforced Polymer (GFRP) composites. Allah et al. [33] studied the effect of mean stress on the behavior of the S-N diagram of GFRP composites through constant deflection flexural fatigue tests on standard unidirectional glass fiber-reinforced polyester pultruded rods. Wang et al. [34] carried out an experimental study on the high-cycle (500,000 times) fatigue performance of a new type of GFRP composite material cross arm. Hyeong, et al. [35] investigated the degradation of GFRP composites using an accelerated aging method in which two types of E-glass/vinyl ester rods were exposed to moisture, chloride, alkali, and freeze–thaw cycling conditions for up to 132 days and found that the tensile properties of the GFRP rods were significantly reduced due to the degradation of GFR. However, in terms of the structure of buildings, in most of the numerical models, damage accumulation is ignored, and 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 [36] are used to simulate the progressive collapse mechanism of a single-layer cylindrical latticed shell. Both a corresponding constitutive relationship for beam elements and a 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 the application of an explicit dynamic algorithm, which is used in this paper. Based on the above studies, in the paper by Zhou et al. [36], 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 was proven 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 shells are performed and the collapse mechanism of this type of structure is studied.

2. The Numerical Model The collapse analysis is performed by the general-purpose program ABAQUS 6.12 by Dassault SIMULIA, through the further development of the subroutine VUMAT program in Abaqus. The numerical method introduced in Zhou et al. [36] 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 an appropriate constitutive model defined for each fiber, as shown in Figure1. In the dynamic analysis, the below damage criterion is used:

1 D dD = − dσ11 (1) 2σ11 where σ11 is normal stress existing in a beam element, D is the cumulative damage. The failure criterion of each fiber is determined based on [36]. When Dnew in a fiber develops into a certain value Dlimt, the fiber is determined to have failed. Dlimt is determined using the below Equation:

! 1 fu − 2 Dnew = 1 (2) − fy Materials 2020, 13, 2519 4 of 18

where fu is the ultimate tensile strength, fy is the yield stress. Based on the strain equivalence hypothesis, the following constitutive model was used in the simulation. In the elastic loading or unloading stage:

εee = eσ/E (3)

In the plastic loading stage, the strain equivalence hypothesis is still valid and it is

ε = eσ/Et = σ/Eet (4) where eσ = σ/(1 D) is the effective Cauchy stress tensor and Ee = E (1 D) is the effective elastic − × − tensor. 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 judged as having failed; thus, this beam element will be deleted in the program. Therefore, the process of the collapse of the dome can be simulated. After an element is deleted due to failure, the explicit dynamic analysis based on the central difference method is performed, which enables the accurate capture of the response of the dome. For a detailed explanation of the numerical algorithm, please refer to Zhou et al. [36]. Materials 2020, 13, x FOR PEER REVIEW 5 of 18

(a)

(b)

Figure 1. Single-layerSingle-layer cylindrical cylindrical latticed latticed shell shell with structural zone denoted: ( (aa)) Structural Structural zone zone in in the the model; ( b)) Structural Structural zone in the dome.

Figure 2. The location and numbering of the fibers in a single 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 (Figure 3) was applied at the support, in X, Y, and Z directions, with the peak ground acceleration (PGA) of 0.5 g in the X direction, 0.85 × 0.5 g in the Y direction, and 0.65 × 0.5 g in the Z direction, respectively According to the Chinese earthquake design code [37], the scale factors of 3.211304, 2.370329 and 3.161574 were applied to the original time history in three directions, respectively. The time duration was 20 s, which is greater than 10 times the natural period of the dome. The time step was 0.02 s. The constitutive model of the material uses the hybrid hardening model and the modulus considered the cumulative effect of damage to simulate the performance deterioration of the material, which occurred during plastic tension in the hysteresis process under seismic loading.

Materials 2020, 13, x FOR PEER REVIEW 5 of 18

Materials 2020, 13, 2519 5 of 18

3. Progressive Collapse Analysis of Single-Layer Cylindrical Latticed Shell Based on the aforementioned analysis method,(a) a single-layer monoclinic cylindrical reticulated dome with a span of 20 m, 30 m in length and 7.5 m in height was modeled using the general-purpose program Abaqus (Figure1). The dome was supported with pins along the two longitudinal edges, with bottom nodes hinged to the ground. All the structural members are Chinese steel tubular sections, where, the lateral and diagonal members are Chinese Ø166 5, and the longitudinal member are × Chinese Ø114 5. The grade of the steel is Chinese Q235-B), with an initial yield stress of 235 MPa. × Von Mises theory is adopted as yield criteria. The Young’s modulus was 2.06 105 MPa. The Poisson × ratio was 0.3. The live load of the roof was taken as 2.5 KN/m2. Figure1 also shows the structural zone divided by the authors, which is shown to clearly demonstrate the response of the dome under earthquake loading. In the simulation, all the structural members were modeled using beam elements. Each beam element was further divided into four segments along its length to model the buckling behavior. As is explained Zhou et al. [36], this is an effective way to model the member buckling with sufficient accuracy. In addition, each beam element was subdivided(b) into eight fiber beams across its cross-section, as shown in Figure2. The fibers are numbered from one to eight along the circumference of the beam Figure 1. Single-layer cylindrical latticed shell with structural zone denoted: (a) Structural zone in the elementmodel; (Figure (b) Structural2). zone in the dome.

Figure 2. The location and numbering of the ﬁbersfibers in a single beam element.

The analysisanalysis waswas divided divided into into two two steps. steps. The The first first step step was was static static analysis, analysis, where where the gravitythe gravity load wasload appliedwas applied to the to dome. the dome. The secondThe second step wasstep dynamicwas dynamic analysis, analysis, where where the time the time history history of the ofTaft the earthquakeTaft earthquake (Figure (Figure3) was 3) applied was applied at the support,at the su inpport, X, Y, in and X, ZY, directions, and Z directions, with the with peak the ground peak accelerationground acceleration (PGA) of(PGA) 0.5 g of in 0.5 the g X in direction, the X direction, 0.85 0.850.5 g × in0.5 the g in Y the direction, Y direction, and 0.65and 0.650.5 × g0.5 in g the in × × Zthe direction, Z direction, respectively respectively According According to theto the Chinese Chinese earthquake earthquake design design code code [37 ],[37], the the scale scale factors factors of 3.211304,of 3.211304, 2.370329 2.370329 and 3.161574and 3.161574 were appliedwere applied to the original to the timeoriginal history time in threehistory directions, in three respectively. directions, Therespectively. time duration The time was 20duration s, which was is greater20 s, which than is 10 greater times the than natural 10 times period the of natural the dome. period The of time the stepdome. was The 0.02 time s. step The constitutivewas 0.02 s. The model constitutive of the material model usesof the the material hybrid uses hardening the hybrid model hardening and the modulusmodel and considered the modulus the cumulative considered e fftheect cumulative of damage toeffect simulate of damage the performance to simulate deterioration the performance of the material,deterioration which of occurredthe material, during which plastic occurred tension during in the plastic hysteresis tension process in the under hysteresis seismic process loading. under seismicFigure loading.4 depicts the deformation and damage distribution during the collapse process, where the distribution of the D value can be checked. It demonstrates the process where the D value was zero, which indicates that no damage occurred until the rupture of the member when D > Dlimt. According to the analysis results, the dome entered the plastic stage at 1.88 s for the first time. Damage distribution spread to zones 2-2 and 3-3 and the symmetric zone of 6-6/7-7, but the D value was extremely small in this stage. With the ground motion continued, at 6.16 s, the D value of many diagonal and lateral members in zones 2-2 and 3-3, at adjacent ends, had obviously exceeded that of members in other zones (Figure4b). By 11.78 s, the damage state of the whole dome had furtherly deteriorated. The D value of damaged structural members kept rising and more diagonal and lateral members were damaged. At 11.80 s, three diagonal bars in zone 2-2 failed at the ends of the joints with a threshold value of 0.209. Materials 2020, 13, x FOR PEER REVIEW 6 of 18

Materials 2020, 13, 2519 6 of 18 Materials 2020, 13, x FOR PEER REVIEW 6 of 18

(a) (b)

(c)

Figure 3. Amplified Taft earthquake wave: (a) X direction, (b) Y direction, (c) Z direction.

Figure 4 depicts the deformation and damage distribution during the collapse process, where (c) the distribution of the D value can be checked. It demonstrates the process where the D value was zero, whichFigure indicates 3. AmplifiedAmpliﬁed that no Taft damage earthquake occurred wave: until ( a) Xthe direction, rupture ( b of) Y the direction, member (c) when Z direction. D > D limt.

Figure 4 depicts the deformation and damage distribution during the collapse process, where the distribution of the D value can be checked. It demonstrates the process where the D value was zero, which indicates that no damage occurred until the rupture of the member when D > Dlimt.

(a) (b)

Figure 4. Cont.

Materials 2020, 13, 2519 7 of 18 Materials 2020, 13, x FOR PEER REVIEW 7 of 18

(e) (f)

(g) (h)

(i) (j)

Figure 4. Deformation and and D D value distribution of of structural members members during during the the collapse collapse process: process: (a) t t == 4.564.56 s, s, (b (b) )t t= =6.166.16 s, s,(c) ( ct )= t 11.78= 11.78 s, (d s,) (td =) 11.80 t = 11.80 s, (e s,) t (=e )12.03 t = 12.03 s, (f) s,t = ( f14.38) t = s,14.38 (g) t s, = (15.40g) t = s,15.40 (h) t = s, 15.83(h) t = s,15.83 (i) t = s, 16.38 (i) t =s, 16.38(j) t = s,16.78 (j) t =s. 16.78 s.

AccordingAt 12.03 s, allto the diagonalanalysis results, members the in dome zone 2-2entere failed,d the while plastic the stage damage at 1.88 range s for spread the first to the time. left Damageof the zone, distribution with the damagespread to area zones increased 2-2 and dramatically, 3-3 and the symmetric although no zone member of 6-6/7-7, facture but was the observed. D value wasAt 14.38 extremely s, all the small diagonal in this membersstage. With in zonethe ground 2-2 failed motion because continued, of continuous at 6.16 s, damage the D value accumulation, of many diagonaland the failed and lateral zone wasmembers like a in continuous zones 2-2 andand regular3-3, at adjacent “diagonal ends, member had obviously failure belt”. exceeded Lateral that bars of memberswere left unbrokenin other zones and the (Figure damage 4b). area By continue11.78 s, the to propagate.damage state The of deflection the whole of dome the dome had furtherly had been deteriorated.significant. Furthermore, The D value some of damaged lateralmembers structural and members diagonal kept members rising and in zone more 6-6 diagonal/7-7 were and damaged. lateral membersAfterwards, were damaged. the damage At 11.80 area wass, three found diagonal between bars zones in zone 6-6 and2-2 failed 7-7. At at 15.40the ends s, most of the diagonal joints withmembers a threshold in the zone value were of 0.209. damaged and a few at two longitudinal ends failed. By 15.83 s, most diagonal membersAt 12.03 in zone s, all 6-6 the and diagonal two diagonal members members in zone at2-2 the failed, right while of zone the 7-7 damage had failed. range The spread position to the of leftstructural of the memberszone, with failed the damage in a similar area manner increased to the dramatically, “diagonal member although failure no member belt “with facture its width was observed.becoming At slightly 14.38s, wider all the than diagonal when it members first came in into zone being 2-2 (Figurefailed 4becauseh). No lateralof continuous members damage failed, accumulation,though they were and obviouslythe failed zone damaged. was like Furthermore, a continuous the and overall regular downward “diagonal deflection member failure of the domebelt”. Lateralbecame bars significant were left during unbroken this process. and the With damage the continuous area continue action to of propagate. the earthquake, The deflection the dome of began the dometo collapse had been downward significant. rapidly. Furthermore, Horizontally, some the lateral dome members collapsed and towards diagonal the position members of in the zone first “diagonal6-6/7-7 were member damaged. failure belt”, the whole dome now looked like an italic letter “M”. Finally, 1.13 s later,Afterwards, with the formation the damage of the area second was found “diagonal between member zones failure 6-6 and belt”, 7-7. the At dome15.40 s, collapsed most diagonal totally. membersThe D value in ofthe longitudinal zone were barsdamaged of the and dome a few had at not tw increasedo longitudinal much throughout.ends failed. By 15.83 s, most diagonal members in zone 6-6 and two diagonal members at the right of zone 7-7 had failed. The position of structural members failed in a similar manner to the “diagonal member failure belt “with

Materials 2020,, 13,, xx FORFOR PEERPEER REVIEWREVIEW 8 of 18 its width becoming slightly wider than when it first came into being (Figure 4h). No lateral members failed, though they were obviously damaged. Furthermore, the overall downward deflection of the dome became significant during this process. With the continuous action of the earthquake, the dome began to collapse downward rapidly. Horizontally, the dome collapsed towards the position of the first “diagonal member failure belt”, the whole dome now looked like an italic letter "M". Finally,Materials 20201.13, 13 s, 2519later, with the formation of the second “diagonal member failure belt”, the dome8 of 18 collapsed totally. The D value of longitudinal bars of the dome had not increased much throughout.

3.1. Collapse Collapse Pattern Pattern of “Double- “Double-DiagonalDiagonal Member Failure Belt” FigureFigures 55 and 6Figure depict 6 the depict deformation the deformation and distribution and distribution of the damage of the of damage the structural of the structural member memberduring the during collapse the process collapse for process the other for twothe domes,other tw witho domes, span:depth with span:depth ratios of 4:1 ratios and 2:1, of respectively.4:1 and 2:1, Theirrespectively. spans are Their kept spans at 20 m.are Figureskept at7 20 and m.8 ,Figure respectively, 7 and depictFigure the8, respectively, other two 20-m-span depict the domes, other one two with20-m-span a length:width domes, one ratio with of 1:1, a length:w anotheridth with ratio a ratio of 1:1, of 2:1. another With thewith increase a ratio inof the2:1. length:widthWith the increase ratio, inthe the position length:width of the “diagonal ratio, the member position failure of the belt” “diagonal moved frommember zones failure 2-2/6-6 belt” to 3-3 moved/7-7. The from decrease zones 2-2/6-6in the length:width to 3-3/7-7. The ratio decrease caused a contrastingin the length:w trend.idth With ratio the exceptioncaused a ofcontrasting the above, thetrend. basic With process the exceptionof collapse of and the ﬁnal above, collapse the basic forms process are similar. of collapse and final collapse forms are similar. Figure9 9 depicts depicts the the failure failure mode mode of double-diagonal of double-diagonal member member cylindrical cylindrical reticulated reticulated domes, domes, which whichare similar are similar to single-diagonal to single-diagonal member member cylindrical cylindrical reticulated reticulated domes. domes.

(a) (b) (c)

Figure 5. Collapse process of cylindrical reticulated dome domess of different diﬀerent span/depth span/depth ratios at typical times (span/depth (span/depth = 4:1):4:1): ( (aa)) belt belt 1 1 formed, formed, ( (bb)) belt belt 2 2 formed, formed, ( (cc)) dome dome collapses. collapses.

(a) (b) (c)

Figure 6. Collapse process of cylindrical reticulated dome domess of different diﬀerent span/depth span/depth ratios at typical times (span/depth (span/depth = 2:1):2:1): ( (aa)) belt belt 1 1 formed, formed, ( (bb)) belt belt 2 2 formed, formed, ( (cc)) dome dome collapses. collapses.

(a) (b) (c)

Figure 7. Collapse process of cylindrical reticulated domes of different diﬀerent length/width length/width ratios at typical Materialstimes 2020 (length/width (length, 13, x FOR/width PEER = 1:1):1:1):REVIEW ( (aa)) belt belt 1 1 formed, formed, ( (bb)) belt belt 2 2 formed, formed, ( (cc)) dome dome collapses. collapses. 9 of 18

(a) (b) (c)

Figure 8. Collapse process of cylindrical reticulated domes of of different diﬀerent length/width length/width ratios at typical times (length/width (length/width = 2:1):2:1): ( (aa)) belt belt 1 1 formed, formed, ( (bb)) belt belt 2 2 formed, formed, ( (cc)) dome dome collapses. collapses.

(a) (b) (c)

Figure 9. Double diagonal member failure belt: (a) belt 1 formed, (b) belt 2 formed, (c) dome collapses.

Apart from the above analysis, extensive parametric studies were performed by the authors with different spans, different span:depth ratios, different length:width ratios and different roof loadings. We have proven that similar collapse processes were observed. In addition to the above case studies, all the analysis cases were applied with a peak ground acceleration (PGA) of 0.5 g in the Y direction, 0.85 × 0.5 g in the X direction, which is different to the first loading regime. The results showed that nothing, apart from the time when the dome collapsed, changed. Therefore, a consistent failure pattern for the collapse process of single-layer cylindrical reticulated domes can be concluded. Under severe earthquakes, serious damaged first occurs at 1/4 height of the dome. Diagonal members failed continuously from the center to the boundary lengthwise and the first “diagonal member failure belt” came into being (Figure 10). The stiffness of the dome also greatly weakened. Then, another damaged zone appeared at the middle of the dome and, in this zone, diagonal members failed continuously lengthwise from the ends to the center. A second “diagonal member failure belt” was formed. The stiffness of the dome was reduced further. The whole deflection became significant. The position of the “belt” varied slightly with the change in geometry. In the final stage, the dome deflected rapidly at the second “belt”, dropping towards the first “belt”. The dome collapsed. We call this collapse process and phenomenon the “double-diagonal -members-failure-belt” pattern.

Figure 10. The demonstration of the collapse mechanism of the shell.

Materials 2020, 13, x FOR PEER REVIEW 9 of 18

(a) (b) (c)

Figure 8. Collapse process(a) of cylindrical reticulated(b domes) of different length/width(c) ratios at typical times (length/width = 2:1): (a) belt 1 formed, (b) belt 2 formed, (c) dome collapses. MaterialsFigure2020 ,8.13 Collapse, 2519 process of cylindrical reticulated domes of different length/width ratios at typical9 of 18 times (length/width = 2:1): (a) belt 1 formed, (b) belt 2 formed, (c) dome collapses.

(a) (b) (c)

Figure 9. Double(a )diagonal member failure belt:(b )(a ) belt 1 formed, (b) belt 2 formed,(c) (c) dome collapses. Figure 9.9. DoubleDouble diagonal diagonal member member failure failure belt: belt: (a) belt(a) 1belt formed, 1 formed, (b) belt (b 2) formed, belt 2 (formed,c) dome collapses.(c) dome Apartcollapses. from the above analysis, extensive parametric studies were performed by the authors with differentApart from spans, the abovedifferent analysis, span:depth extensive ratios, parametric different studies length:width were performed ratios and by thedifferent authors roof with loadings.differentApart We spans, from have di theprovenfferent above span:depththat analysis, similar ratios,collapseextensive di ffprocesses erentparame length:widthtric were studies observed. ratioswere andperformed different by roof the loadings. authors WewithIn have different addition proven spans,to that the similar abovedifferent collapsecase span:depth studies, processes all ratios, the were analysis different observed. cases length:width were applied ratios with and a peakdifferent ground roof accelerationloadings.In addition We (PGA) have to of proven the 0.5 aboveg in that the case simila Y direction, studies,r collapse all 0.85 theprocesses × analysis0.5 g in were the cases Xobserved. direction, were applied which with is different a peak groundto the accelerationIn addition (PGA) to the of 0.5above g in case the studies, Y direction, all the 0.85 analysis0.5 cases g in thewere X direction,applied with which a peak is di groundfferent first loading regime. The results showed that nothing, apart× from the time when the dome collapsed, changed.accelerationto the first loading (PGA) of regime. 0.5 g in The the resultsY direction, showed 0.85 that × 0.5 nothing, g in the apartX direction, from the which time is when different the dometo the firstcollapsed,Therefore, loading changed. regime. a consistent The results failure showed pattern that fornothin theg, collapseapart from process the time of whensingle-layer the dome cylindrical collapsed, reticulatedchanged.Therefore, domes a consistentcan be concluded. failure pattern Under for severe the collapse earthquakes, process serious of single-layer damaged cylindrical first occurs reticulated at 1/4 heightdomesTherefore, of can the be dome. concluded. a consistent Diagonal Under failure members severe pattern earthquakes, failed for co thentinuously serious collapse damaged fromprocess the first ofcenter occurs single-layer to at 1the/4 height boundarycylindrical of the lengthwisereticulateddome. Diagonal and domes the members firstcan be“diagonal concluded. failed member continuously Under failure severe from belt” earthquakes, the came center into to seriousbeing the boundary (Figure damaged 10). lengthwise first The occursstiffness and at of the1/4 theheightfirst dome “diagonal of also the greatly memberdome. weakened. Diagonal failure belt” Then,members came another into failed being damaged co (Figurentinuously zone 10). appeared Thefrom sti fftheness at thecenter of middle the to dome the of alsotheboundary dome greatly and,weakened.lengthwise in this zone, Then,and the diagonal another first “diagonal damaged members zonemember failed appeared continuously failure at belt” the middle camelengthwise into of the being from dome (Figure the and, ends in 10). this to The the zone, stiffnesscenter. diagonal A of secondthemembers dome “diagonal failedalso greatly continuouslymember weakened. failure lengthwise belt” Then, was another from formed. thedamaged endsThe stiffness to zone the center.appeared of the Adome secondat the was middle “diagonal reduced of thefurther. member dome Theand,failure whole in belt”this deflection zone, was formed. diagonal became The members significant. stiffness failed of The the continuouslyposition dome was of reducedthe lengthwise “belt” further. varied from The slightly the whole ends with deflection to the the change center. became in A geometry.secondsignificant. “diagonal In Thethe final position member stage, of failurethe dome “belt” belt” deflected varied was formed. slightly rapidly The with at stiffnessthe the second change of the“belt”, in geometry.dome dropping was Inreduced the towards final further. stage, the firstThethe domewhole“belt”. deflected deflection The dome rapidly became collapsed. at thesignificant. second We “belt”, The call position dropping this ofcollapse the towards “belt” process the varied first “belt”.slightlyand phenomenon The with dome the change collapsed. the in “double-diagonalWegeometry. call this In collapse the -members-failure-belt”final process stage, andthe dome phenomenon deflected pattern. the ra “double-diagonalpidly at the second -members-failure-belt” “belt”, dropping towards pattern. the first “belt”. The dome collapsed. We call this collapse process and phenomenon the “double-diagonal -members-failure-belt” pattern.

FigureFigure 10. 10. TheThe demonstration demonstration of of the the collapse collapse mechanism mechanism of of the the shell. shell.

3.2. Mechanism of ProgressiveFigure 10. CollapseThe demonstration Process of the collapse mechanism of the shell. Earthquake loading causes deformations and material damage. Deformation and damage

cause a variety of mechanical characteristics of dome. The variety deeply aﬀect the deformation and development of material damage in turn (“failure” is the extreme state). This kind of complex relationship exists and works during the whole process. According to the analysis results in the earlier section, it can be concluded that, of the deformation and mechanical characteristics of dome changed along the whole process, different of deformation characteristics and mechanism properties were observed, the collapse process can be divided into three stages. Materials 2020, 13, x FOR PEER REVIEW 10 of 18

3.2. Mechanism of Progressive Collapse Process Earthquake loading causes deformations and material damage. Deformation and damage cause a variety of mechanical characteristics of dome. The variety deeply affect the deformation and development of material damage in turn (“failure” is the extreme state). This kind of complex relationship exists and works during the whole process. According to the analysis results in the earlier section, it can be concluded that, of the deformation and mechanical characteristics of dome changed along the whole process, different of Materials 2020, 13, 2519 10 of 18 deformation characteristics and mechanism properties were observed, the collapse process can be divided into three stages. StageStage 11:: ThisThis stagestage refersrefers toto thethe processprocess fromfrom thethe earthquakeearthquake occurringoccurring toto thethe formationformation ofof thethe firstfirst “belt”“belt” (0–14.38(0-14.38s). s). During During this this stage, stage, the the ph phenomenonenomenon of material dama damagege and diagonal members failingfailing mainlymainly occuroccur inin zonezone 2-2.2-2. TheThe cylindricalcylindrical reticulatedreticulated dome,dome, withwith longitudinallongitudinal edges supported with pins, is oneone typetype ofof spacespace structurestructure thatthat exhibitsexhibits planaryplanary archarch features.features. TheThe responseresponse ofof domedome underunder severesevere earthquakeearthquake isis veryvery similarsimilar toto thatthat ofof aa planaryplanary arch.arch. TheThe diagonaldiagonal membersmembers areare forfor bracing,bracing, andand thethe longitudinallongitudinal membermember worksworks as as a tiea bar,tie whichbar, which coordinates coordinates the continuity the continuity of the vertical of the and vertical horizontal and deformation. horizontal Therefore,deformation. during Therefore, earthquakes, during the earthquakes, axial force, the bending axial momentforce, bending and torsion moment of each and type torsion of bar of varyeach withtype time,of bar although vary with the time, overall although characteristics the overall are characteristics relatively stable. are relatively stable. Thus,Thus, atat thethe beginningbeginning ofof anan earthquake,earthquake, thethe maximummaximum internalinternal forceforce cancan bebe observedobserved atat thethe 11/4/4 spanspan ofof thethe domedome fromfrom thethe boundary,boundary, andand thethe membermember thatthat firstfirst developeddeveloped intointo aa plasticplastic statestate andand causedcaused thethe materialmaterial damagedamage mustmust comecome fromfrom thisthis zone.zone. However,However, EarthquakeEarthquake loading loading is is asymmetric, asymmetric, so so plastic plastic deformation deformation and and damage damage accumulation accumulation in thein the dome dome are are also also inevitably inevitably asymmetric. asymmetric. This This means means the theD valueD value and and the the increasing increasing speed speed of theof theD valueD value of theof the bars bars in this in zonethis zone are larger are larger than thosethan ofthose bars of in bars other in zones. other Withzones. the With continuing the continuing ground motionground andmotion gradually and gradually increasing increasing amplitude, amplitude, 5.13 s later, 5.13 thes later, center the of center gravity of ofgravity the dome of the moved dome towardsmoved towards the X direction the X (Figuredirection 11 ).(Figure This is because11). This plastic is because deformation plastic occurreddeformation due tooccurred vibrations, due and to thevibrations, overall geometricand the overall configuration geometric shifted configuratio towards zonen shifted 2-2. This towards phenomenon zone 2-2. conversely This phenomenon caused the internalconversely forces caused of members the internal in zone forces 2-2 of/3-3 members to become in zone significantly 2-2/3-3 to larger become than significantly those in the larger symmetric than zonethose of in 11-11 the symmetric/10-10. Consequently, zone of 11-11/10-10. the speed Consequently, of damage of the bars speed in zone of 2-2damage was higherof barsthan in zone that 2-2 of barswas inhigher the symmetric than that of zone bars (Figure in the 4symmetricb,c). zone (Figure 4b, c).

FigureFigure 11.11. The horizontal translation of gravitygravity centercenter inin XX direction.direction.

AsAs shownshown inin FigureFigure 1212,, throughthrough aa comparisoncomparison ofof thethe timetime historyhistory curvescurves ofof bendingbending momentsmoments MMx’x’ andand My’y’, ,torsion torsion MMx’y’x’y’ andand axial axial force force FF ofof members members 242/254/278 242/254/278 in in zone zone 2-2, 2-2, it it can be found that, althoughalthough thethe axialaxial forceforce FF ofof 254254 waswas lessless thanthan thatthat ofof 242, 242, the the bending bending moments momentsM Mx’x’and and MMyy andand torsiontorsion MMx’y’x’y’ ofof 254 254 are obviously larger than those of 242. The axial force andand bendingbending momentsmoments ofof 278278 areare approximate to those of of 254, 254, but but the the torsion torsion of of 278 278 is is less less than than that that of of 242. 242. So, So, the the D Dvalues values of ofthe the slant slant bar bar ofof 254 254 and and neighboring neighboring bars bars in in zone zone 2-2 2-2 are are obviously obviously larger larger th thanan those of other barsbars (Figure(Figure4 4b,b,c). c). According to Figure 13, which shows the variety of internal forces of traverse bars 14, 78 and 74 at the center of zone 2-2, the traverse bars are mainly subjected to bending moment My, while axial force F, bending moment Mx’ and torsion Mx’y’ at this stage are minimal, so My’ contributed mainly to the damage increment. This is why, in Figure 14, the normal stresses of upper section fiber 3 and lower section fiber 7 were much larger than those of middle section fibers 1 and 5, which hardly exceeded the yield point and caused corresponding material damage. Thus, damage development concentrated in fiber 3 and 7 and traverse bars hardly failed. This is the common response observed in all the transverse bars of the monoclinic and double-skew cylindrical reticulated domes under the same ground motion. Materials 2020, 13, x FOR PEER REVIEW 11 of 18

Materials 2020, 13, 2519 11 of 18

Based on the above analysis, bar 254 and neighboring bars failed at first. Furthermore, once they lost their bearing capacity, the internal force redistribution effect imposed a force on the neighboring bars that had not yet failed (as shown in Figure 14a), and the D value developed rapidly until the bar failed (Figure 15). Then, the internal force redistribution effect worked again. With the failure propagating, the force redistribution to the neighboring bars continued. Finally, all the bars in zone 2-2 failed. The collapse process then entered the second stage. Materials 2020, 13, x FOR PEER REVIEW(a) (b) 11 of 18

((ca)) ((bd))

Figure 12. Time history of the internal force of the diagonal members: (a) Mx’, (b) My’, (c) Mx’y’, (d) F.

According to Figure 13, which shows the variety of internal forces of traverse bars 14, 78 and 74 at the center of zone 2-2，the traverse bars are mainly subjected to bending moment My, while axial force F, bending moment Mx’ and torsion Mx’y’ at this stage are minimal, so My’ contributed mainly to the damage increment. This is why, in Figure 14, the normal stresses of upper section fiber 3 and lower section fiber 7 were much larger than those of middle section fibers 1 and 5, which hardly exceeded the yield point and caused corresponding material damage. Thus, damage development concentrated in fiber 3 and 7 and traverse bars hardly failed. This is the common response observed (c) (d) in all the transverse bars of the monoclinic and double-skew cylindrical reticulated domes under the sameFigure groundFigure 12. 12. motion.Time Time history history of of the the internal internal force force of of the the diagonal diagonal members: members: ((aa)) MMx’x’,(, (b) My’,,( (cc)) MMx’y’x’y’, (,(dd) F) .F .

According to Figure 13, which shows the variety of internal forces of traverse bars 14, 78 and 74 at the center of zone 2-2，the traverse bars are mainly subjected to bending moment My, while axial force F, bending moment Mx’ and torsion Mx’y’ at this stage are minimal, so My’ contributed mainly to the damage increment. This is why, in Figure 14, the normal stresses of upper section fiber 3 and lower section fiber 7 were much larger than those of middle section fibers 1 and 5, which hardly exceeded the yield point and caused corresponding material damage. Thus, damage development concentrated in fiber 3 and 7 and traverse bars hardly failed. This is the common response observed in all the transverse bars of the monoclinic and double-skew cylindrical reticulated domes under the same ground motion. (a) (b)

(a) (b) (c) (d)

FigureFigure 13.13. TimeTime historyhistory ofof internal internal forces forces of of traverse traverse bar bar 14, 14, 78 78 and and 74: 74: ( a()a)M Mx’x’,(, b(b))M My’y’,(, (cc)) MMx’y’x’y’,,( (d) F.

Figure 13. Time history of internal forces of traverse bar 14, 78 and 74: (a) Mx’, (b) My’, (c) Mx’y’, (d) F.

Materials 2020, 13, x FOR PEER REVIEW 12 of 18 Materials 2020, 13, x FOR PEER REVIEW 12 of 18

Materials 2020, 13, 2519 12 of 18 Materials 2020, 13, x FOR PEER REVIEW 12 of 18

(a(a) ) ((bb)) ((cc))

FigureFigure 14. 14. Time Time history history of of normal normal stress stress in in fiberfiber 1,3,5,71,3,5,7 of traverse bar 14, 78 and 74:74: ((aa)) barbar 74,74, ((bb)) bar bar 14,14, ( c()c )bar bar 78 78. .

BasedBased on on the the above above analysis,analysis, barbar 254254 andand neighborneighboringing bars failed atat firsfirst.t. Furthermore,Furthermore, onceonce theythey lost lost their their bearingbearing capacity,capacity, thethe internalinternal foforcerce redistributionredistribution effect imposedimposed aa forceforce onon thethe neighboringneighboring bars bars that that had had not not yet yet failed failed (as(as shownshown inin FigureFigure 14a), and the D value developeddeveloped rapidly rapidly (a) (b) (c) untiluntil the the bar bar failed failed (Figure (Figure 15). 15). Then, Then, thethe internalinternal forceforce redistributionredistribution effect workedworked again.again. WithWith thethe failurefailureFigure propagating, propagating, 14. Time history the the force force ofof normalnormal redistribution redistribution stress stress in in ﬁber fiber toto the1,3,5,7the 1,3,5,7 neighboringneighboring of of traverse traverse bar bars bar 14, 14,continued. 78 78 and and 74: 74: ( a Finally,Finally, )( bara) bar 74, 74, all (allb) ( the barbthe) bar bars 14,bars in in zonezone(14, c2-2 )2-2 bar( cfailed.) failed.bar 78. 78 The. The collapse collapse process process then then enteredentered thethe secondsecond stage.

Based on the above analysis, bar 254 and neighboring bars failed at first. Furthermore, once they lost their bearing capacity, the internal force redistribution effect imposed a force on the neighboring bars that had not yet failed (as shown in Figure 14a), and the D value developed rapidly until the bar failed (Figure 15). Then, the internal force redistribution effect worked again. With the failure propagating, the force redistribution to the neighboring bars continued. Finally, all the bars in zone 2-2 failed. The collapse process then entered the second stage.

(a(a) ) ((bb)) ((cc))

FigureFigure 15.15. 15. Time Time history history of of D D value value in in ﬁberfiber fiber 1,3,5,71,3,5,71,3,5,7 ofofof barbarbar 14,14, 7878 andand 74:74: (a) bar 74, 74, ( (bb)) bar bar 14, 14,14, ( (cc))) bar barbar 78 78.78. .

StageStage 2: 2: This This refers refers to to the the stage stage from from thethe formationformation ofof the firstfirst “belt” in in zone zone 2-2 2-2 to to the the formation formation ofof the the second second “slant-bars-failed-belt” “slant-bars-failed-belt”“slant-bars-failed-belt” inin zonezone zone 6-6/6-6/ 6-6 /7-77-7 (14.38–15.83(14.38–15.83 (14.38–15.83 s). s). In In this this stage, stage, thethe the phenomenaphenomena phenomena of of ofmaterialmaterial material damage damage damage accumulation accumulation accumulation and and and thethe the failurefailure failure ofof thethe of thebarsbars bars mainly mainly occurred occurred in zoneszones in zones 6-66-6 andand 6-6 7-7. and7-7. The The 7-7. Theeffecteffect eff of ectof material material of material damage damage damage accumula accumula accumulationtiontion causedcaused caused notnot onlyonly not onlythethe failure the failure of all of diagonaldiagonal all diagonal membersmembers members inin zonezone in (a) (b) (c) zone2-2,2-2, but 2-2,but also butalso the alsothe significant significant the significant deterioration deterioration deterioration ofof flexuralflexural of flexural stiffnessstiffness stiffness in fibers in fibers 3 and 3 and 77 inin 7 thethe in traverse thetraverse traverse bars.bars. bars. At At Atthisthis this time,Figure time, time, the the15. the dome Timedome dome historyacted acted acted as ofas as anD an anvalue arch arch arch inwhose whose whosefiber 1,3,5,7 locallocal local fl flof flexuralexuralexural bar 14, stiffness 78 sti ffandness 74: in in ( zonea zone) bar 2-274, 2-2 weakened(weakenedb weakened) bar 14, ( c significantlysignificantly) significantly bar 78. (Figure 16). (Figure(Figure 1616).). Stage 2: This refers to the stage from the formation of the first “belt” in zone 2-2 to the formation of the second “slant-bars-failed-belt” in zone 6-6/ 7-7 (14.38–15.83 s). In this stage, the phenomena of material damage accumulation and the failure of the bars mainly occurred in zones 6-6 and 7-7. The effect of material damage accumulation caused not only the failure of all diagonal members in zone 2-2, but also the significant deterioration of flexural stiffness in fibers 3 and 7 in the traverse bars. At this time, the dome acted as an arch whose local flexural stiffness in zone 2-2 weakened significantly Figure 16. The transition between stages 1 and 2. (Figure 16). Figure 16.16. The transition between stages 1 and 2. In Figure 14a, we can see that the normal stress at upper fiber 3 and lower fiber 7 at the end InIn FigureFigure 14 14a,a, we we can can see see that that the the normal normal stress stress at upper at upper fiber fiber 3 and 3 lower and lower fiber 7 fiber at the 7 endat the section end section of traverse bar 74 was maintained at a value of 280 MPa, which was much greater than the ofsection traverse of traverse bar 74 was bar maintained 74 was maintained at a value at of a 280 value MPa, of which280 MPa, was which much greaterwas much than greater the initial than yield the initial yield stress of 235 MPa, while the normal stress of fibers 5 and 1 also increased dramatically, stressinitial ofyield 235 stress MPa, of while 235 theMPa, normal while stressthe normal of fibers stress 5 and of fibers 1 also 5 increased and 1 also dramatically, increased dramatically, acting as a acting as a plastic hinge. Fiber 3 and 7 of traverse bar 14 failed at a short time after entering this plastic hinge. Fiber 3 and 7 of traverse bar 14 failed at a short time after entering this stage, but the actingstage, asbut a theplastic stress hinge. of fibers Fiber 1 and3 and 5, whose7 of traverse stress was bar relatively14 failed small,at a short increased time afterrapidly entering to around this stress of fibers 1 and 5, whose stress was relatively small, increased rapidly to around 200 MPa. We can stage,200 MPa. but theWe stresscan conclude of fibers that 1 and the 5, normal whose stresses stress was of the relatively remaining small, fibers increased of bar 14 rapidly were larger to around than conclude that the normal stressesFigure of16. theThe remaining transition between fibers of stages bar 141 and were 2. larger than the yield point. 200the MPa. yield We point. can Thisconclude means that that the the normal end of stresses bar 14 ofacted the remainingas a plastic fibers hinge of as bar well. 14 wereFinally, larger we thancan This means that the end of bar 14 acted as a plastic hinge as well. Finally, we can conclude that all the theconclude yield point. that allThis the means traverse that bars the in end zone of 2-2bar th14at acted had notas a yet plastic failed hinge acted as as well. plastic Finally, hinges. we The can traverseIn Figure bars in 14a, zone we 2-2 can that see had that not the yet normal failed acted stress as at plastic upper hinges. fiber 3 The and formation lower fiber of plastic 7 at the hinges end concludeformation that of allplastic the traversehinges induced bars in zonesignificant 2-2 th atinternal had not force yet redistributionfailed acted as effect. plastic As hinges. shown The in inducedsectionformation of significant traverseof plastic bar internal hinges 74 was forceinduced maintained redistribution significant at a va e lueffinternalect. of As280 shownforce MPa, redistribution inwhich Figure was 17 much, theeffect. bending greater As shown momentthan the in Minitial y’ axial yield force stress F, bending of 235 momentMPa, whileMx’ theand normal torsion stressMx’y’ of diagonalfibers 5 and bars 1 161also and increased 232 (Figure dramatically,1), one at theacting end as and a plastic the other hinge. at the Fiber mid 3 of and zone 7 6-6,of traverse were less bar than 14 thosefailed of at bars a short 242, 254time or after 278 (Figureentering 12 this) in stagestage, 1.but Thus, the stress almost of no fibers damage 1 and was 5, donewhose to stress these bars.was relatively However, small, the value increased of My’ ,rapidlyMx’, torsion to aroundMx’y, began200 MPa. rapid We increasing can conclude since that the the formation normal ofstresses the first of the “belt”. remaining fibers of bar 14 were larger than the yield point. This means that the end of bar 14 acted as a plastic hinge as well. Finally, we can conclude that all the traverse bars in zone 2-2 that had not yet failed acted as plastic hinges. The formation of plastic hinges induced significant internal force redistribution effect. As shown in

Materials 2020, 13, x FOR PEER REVIEW 13 of 18

Figure 17, the bending moment My’ axial force F, bending moment Mx’ and torsion Mx’y’ of diagonal bars 161 and 232 (Figure 1), one at the end and the other at the mid of zone 6-6, were less than those of bars 242, 254 or 278 (Figure 12) in stage 1. Thus, almost no damage was done to these bars. However, the value of My’, Mx’, torsion Mx’y, began rapid increasing since the formation of the first Materials 2020, 13, x FOR PEER REVIEW 13 of 18 “belt”. Therefore, continuously increasing the internal force induced by ground motion resulted in Figure 17, the bending moment My’ axial force F, bending moment Mx’ and torsion Mx’y’ of diagonal severebars 161 damage and 232 to (Figure the diagonal 1), one bars at the 0.8s end later and when the other entering at the the mid second of zone stage, 6-6, becausewere less the than internal those forcesof bars of 242, the bars254 orat two278 ends(Figure were 12) larger in stage than 1. that Thus, of thealmost bars noin thedamage middle; was the done bars toat thethese ends bars. of Materialsthe nodes2020 of, 13 zone, 2519 6-6 failed first (Figure 4g). Then, the other bars failed successively from both13 ends of 18 However, the value of My’, Mx’, torsion Mx’y, began rapid increasing since the formation of the first to“belt”. the middle in a short time. Finally, at 15.83 s, the second belt was formed. Therefore, continuously increasing the internal force induced by ground motion resulted in severe damage to the diagonal bars 0.8s later when entering the second stage, because the internal forces of the bars at two ends were larger than that of the bars in the middle; the bars at the ends of the nodes of zone 6-6 failed first (Figure 4g). Then, the other bars failed successively from both ends to the middle in a short time. Finally, at 15.83 s, the second belt was formed.

(a) (b)

FigureFigure 17. 17.Time Time history history of of internal internal forces forces of of diagonal diagonal bar bar 161and 161and 232: 232: ( a(a))M Mx’x’,(, (bb))M My’y’,(, (cc)) Mx’y’x’y’,,( (dd)) FF..

Therefore,For traverse continuously bar 78 in zone increasing 6-6, although the internal the forceinternal induced forces by of ground the bar motion significantly resulted increased in severe damagebecause toof theforce diagonal redistribution, bars 0.8 sthe later damage when enteringaccu mulation the second mainly stage, developed because in the upper internal fiber forces 3 and of thelower bars fiber at two 7, while ends werefibers larger 1 and(c) than 5 were that not of the damaged bars in theseriously. middle; Thus, the(d bars )no bars at the failed ends in of theall fibers. nodes ofIn zoneFigure 6-6 14c, failed the firstnormal (Figure stresses4g). Then,in fibers the 3 otherand 7 bars in bar failed 78 were successively very high; from we bothcould ends conclude to the that middle this Figure 17. Time history of internal forces of diagonal bar 161and 232: (a) Mx’, (b) My’, (c) Mx’y’, (d) F. inbar a acted short time.as a plastic Finally, hinge at 15.83 as well. s, the second belt was formed. For traverse bar 78 in zone 6-6, although the internal forces of the bar significantly increased TheFor traversesuccessive bar failure 78 in zoneof diagonal 6-6, although members the and internal damage forces to oftraverse the bar bars significantly in zone 6-6 increased greatly because of force redistribution, the damage accumulation mainly developed in upper fiber 3 and lower weakenedbecause of the force vertical redistribution, stiffness of the the damage dome. Thus, accu mulationunder the mainlyearthquake developed loading, in the upper dome fiber deflected 3 and fiber 7, while fibers 1 and 5 were not damaged seriously. Thus, no bars failed in all fibers. In Figure 14c, downwardslower fiber 7, significantly. while fibers At1 and the same5 were time, not angulardamaged displacement seriously. Thus, was noobserved bars failed in the in traverse all fibers. bars In the normal stresses in fibers 3 and 7 in bar 78 were very high; we could conclude that this bar acted as inFigure zone 14c, 2-2, thewhich normal made stresses damage in infibers fibers 3 and 3 and 7 in 7 developbar 78 were rapidly very (Figure high; we 14). could conclude that this a plastic hinge as well. bar actedUp to as now, a plastic the generation hinge as well. of the second “diagonal member failure belt” and the existence of the The successive failure of diagonal members and damage to traverse bars in zone 6-6 greatly first The“diagonal successive member failure failure of diagonal belt” caused members the andformation damage of totwo traverse zones barswhose in zonestiffnesses 6-6 greatly were weakened the vertical stiffness of the dome. Thus, under the earthquake loading, the dome deflected seriouslyweakened weakened,the vertical stiffnessor two ofplastic the dome. hinges. Thus, As under shown the inearthquake Figure 18,loading, the lateralthe dome mechanism deflected downwards significantly. At the same time, angular displacement was observed in the traverse bars in characteristicsdownwards significantly. of the dome At changed the same again. time, angular displacement was observed in the traverse bars zone 2-2, which made damage in fibers 3 and 7 develop rapidly (Figure 14). in zone 2-2, which made damage in fibers 3 and 7 develop rapidly (Figure 14). Up to now, the generation of the second “diagonal member failure belt” and the existence of the Up to now, the generation of the second “diagonal member failure belt” and the existence of the first “diagonal member failure belt” caused the formation of two zones whose stiffnesses were seriously first “diagonal member failure belt” caused the formation of two zones whose stiffnesses were weakened, or two plastic hinges. As shown in Figure 18, the lateral mechanism characteristics of the seriously weakened, or two plastic hinges. As shown in Figure 18, the lateral mechanism dome changed again. characteristics of the dome changed again. Figure 18. The transition between stages 2 and 3.

FigureFigure 18.18. TheThe transitiontransition betweenbetween stagesstages 22 andand 3.3.

Stage 3: This refers to the process of transitioning from the formation of the second “diagonal member failure -bar-failed belt” to the ﬁnal collapse of the dome (15.83 s~ end). This process is mainly an irreversible collapse process of the whole geometric conﬁguration of the dome. In “belt” two, the loading is borne and transferred only by the seriously damaged traverse bars. With the MaterialsMaterials 2020 2020, 13, 13, x, FORx FOR PEER PEER REVIEW REVIEW 1414 of of18 18

StageStage 3 : 3This: This refers refers to to the the process process of of transitioning transitioning from from the the formation formation of of the the second second “diagonal “diagonal memberMaterialsmember 2020failure failure, 13, 2519-bar-failed -bar-failed belt” belt” to to the the final final collap collapsese of of the the dome dome (15.83s~ (15.83s~ end). end). This This process process14 ofis 18 is mainlymainly an an irreversible irreversible collapse collapse process process of of the the whole whole geometric geometric configuration configuration of of the the dome. dome. In In “belt” “belt” two,two, the the loading loading is isborne borne and and transferred transferred only only by by the the seriously seriously damaged damaged traverse traverse bars. bars. With With the the developmentdevelopmentdevelopment of ofof damage damagedamage and andand the the deterioration deterioration of of angle angle stiffness stistiffnessﬀness of of traverse traverse bars bars in in the the vertical vertical plane, plane, thethethe dome domedome acted actedacted as asas an anan almost almostalmost unstable unstableunstable system. system.system. Then, Then,Then, under underunder an anan earthquake, earthquake,earthquake, total totaltotal collapse collapsecollapse occurred, occurred,occurred, as asas shownshownshown in inin Figure FigureFigure 19. 1919. The. TheThe calculation calculationcalculation results resultsresults showed showedshowed that thatthat the thethe process processprocess is is isvery veryvery rapid rapidrapid and andand irreversible. irreversible.irreversible.

FigureFigureFigure 19. 19.19. The The collapse collapse of ofof the thethe dome domedome in inin stage stagestage 3. 3. 3.

TraverseTraverseTraverse bar bar bar 1414 14 andand and the the the neighboring neighboring neighboring bars bars inbars zone in in zone 2-2zone rapidly 2-2 2-2 rapidly failedrapidly becausefailed failed because ofbecause suddenly of of suddenly increasingsuddenly increasingtorsion.increasing Then, torsion. torsion. the bendingThen, Then, the the moment bending bendingM y’moment ofmoment bar 74 M obviouslyMy’ y’of of bar bar 74 increased 74 obviously obviously and increased made increased the damageand and made made value the the of damageﬁbersdamage 1 value and value 5 of increase. of fibers fibers 1 and1 and 5 increase.5 increase. InInIn conclusion, conclusion,conclusion, in inin the thethe three threethree stages, stages,stages, the thethe variation variationvariation in inin the thethe mechanical mechanicalmechanical characteristics, characteristics,characteristics, damage damagedamage zone zonezone andandand deformation deformationdeformation status statusstatus of ofof the thethe dome domedome show showshow distinct distinctdistinct differences. didifferences.ﬀerences. However, However,However, we wewe should shouldshould understand understandunderstand that thatthat thethethe three threethree stages stagesstages are areare closely closelyclosely related relatedrelated rather ratherrather strictly strictlystrictly divided. divided.divided.

3.3.3.3.3.3. Displacement Displacement During duringDuring the the the Collapse CollapseCollapse Process Process Process TheTheThe deformation deformationdeformation of of of the the domedome dome changedchanged changed throughout throughout throughout the the wholethe whole whole collapse collapse collapse process; process; process; therefore, therefore, therefore, the center the the centerofcenter the of structureof the the structure structure changed changed changed as well. as as Figurewell. well. Figure 20Figure shows 20 20 shows theshows change the the change change in the in locations in the the locations locations of the of center of the the center of center gravity of of gravityofgravity the of structure of the the structure structure during during theduring whole the the whole collapsewhole collapse collapse process. pr process. Diocess.ﬀerent Different Different peak peak ground peak ground ground acceleration acceleration acceleration values values values were wereselectedwere selected selected with with thewith samethe the same same duration duration duration of 18 of of s.18 18 It s. can s.It Itca be can n seenbe be seen seen that that thethat the center the center center of gravityof of gravity gravity decreased decreased decreased with with with the thedithe ﬀdifferenterent different PGA PGA PGA values. values. values.

FigureFigure 20. 20. Time Time history history of ofvertical vertical translation translation of ofof gravity gravitygravity center. center.center.

WhenWhenWhen PGA PGAPGA was waswas 0.45 0.45 0.45 g, g, g, the the the damage damage damage was was observed observed atat at 1.91.9 1.9 ss and ands and the the the rupture rupture rupture of of theof the the first first first diagonal diagonal diagonal bar baroccurredbar occurred occurred at 16.44at at 16.44s. 16.44s. s. However, However, However, the the the failure failure failure belt belt belt was was was not not not formed formed formed until until until 18 18 s. 18 s. At s.At 1.9At 1.9 s,1.9 thes, s,the locationthe location location of of the of thecenterthe center center of theof of the gravity the gravity gravity fluctuated fluctuated fluctuated and noand and significant no no signif signif accelerationicanticant acceleration acceleration in the centerin in the the of center gravitycenter of of wasgravity gravity observed. was was observed.observed.When PGA was 0.465 g, the first diagonal member failure belt formed at 16.55 s. The second belt startedWhenWhen to PGA form PGA was at was 17.63 0.465 0.465 s. g, However, g, the the first first diagonal the diagonal failure member beltmember was fa notfailureilure formed belt belt formed formed until18 at at s.16.55 16.55 It can s. s.The also The second be second observed belt belt startedthatstarted the to to slopeform form at of at 17.63 the 17.63 curve s. s.However, However, in Figure the 17the failureincreased failure belt belt dramatically. was was not not formed formed until until 18 18 s. s.It Itcan can also also be be observed observed thatthat the Whenthe slope slope PGA of of the wasthe curve curve 0.475 in in g,Figure Figure the rupture 17 17 increased increased of the dramatically. first dramatically. diagonal member failure belt formed at 16.38 s. TheWhen secondWhen PGA PGA belt was startedwas 0.475 0.475 to g, formg, the the rupture atrupture 17.48 of s.of the However,the first first di diagonal itagonal can member also member be observedfailure failure belt belt that formed formed the slope at at 16.38s. 16.38s. of the ThecurveThe second second increased belt belt started at started stage to 2to comparedform form at at 17.48 17.48 to stage s. s.Howeve 3.Howeve Whenr, r, PGAit itcan can was also also 0.5 be be g, observed theobserved rupture that that of the the the slope first slope diagonal of of the the member failure belt occurred at 14.38 s. The second belt started to form at 15.83 s. MaterialsMaterials2020 2020, 13, 13,, 2519 x FOR PEER REVIEW 1515 of of 18 18

curve increased at stage 2 compared to stage 3. When PGA was 0.5 g, the rupture of the first diagonal memberIt can failure also be belt seen occurred that when at 14.38s. PGA The was second 0.475 g,belt the started acceleration to form distance at 15.83 s. of the center reached 1286mm;It can however, also be seen for a that larger when PGA PGA of0.5 was g, 0.475 when g, the the time acceleration was shorter distance than of 16 the s, thecenter acceleration reached distance1286mm; of however, the center for was a larger smaller PGA than of 0.5 1286 g, mm,when which the time indicates was shorter the eﬀ thanect of 16 the s, the duration acceleration on the timedistance history. of the center was smaller than 1286 mm, which indicates the effect of the duration on the timeBased history. on the above analysis, the following can be concluded: Based on the above analysis, the following can be concluded: 1. At the ﬁrst stage, the descent of the center of gravity is slow; 1. At the first stage, the descent of the center of gravity is slow; 2. 2.The The descent descent of the of the center center of gravity of gravity speeds speeds up up from from stage stage 1 to 1 stageto stage 3; 3; 3. 3.The The time time to reachto reach stage stage 2 or 2 or 3 is3 determinedis determined by by the the peak peak ground ground acceleration acceleration appliedapplied to the structure—thestructure—the larger larger the the PGA, PGA, the the quicker quicker it will it will reach reach stage stage 2 or 2 3;or 3; 4. 4.The The time time duration duration will will determine determine the the collapse collapse mechanism mechanism of of thethe dome,dome, which is one one of of the the dominantdominant factors. factors.

4.4. New New EnhancedEnhanced SchemesSchemes toto MitigateMitigate thethe Collapse BasedBased on on the the numerical numerical simulation simulation results, results, seven seven mitigation mitigation schemes schemes were developedwere developed to prevent to theprevent collapse the ofcollapse the dome: of the in dome: Figure in 21 Figurea, the cross21a, the section cross of section all top of chords all top in chords the transverse in the transverse direction increased.direction increased. In Figure 21In b,Figure the cross 21b, sectionthe cross of section three top of chordsthree top in chords the transverse in the transverse direction increased.direction Inincreased. Figure 21 c,Inthe Figure cross 21c, section the cross of the section top chords of the at centertop chords was increased.at center was In Figure increased. 21d, theIn crossFigure section 21d, ofthe the cross diagonal section members of the diagonal as shown members in the Figure as shown increased. in the Figure In Figure increased. 21e, the In cross Figure section 21e, the of allcross the diagonalsection of members all the diagonal increased. members In Figure increased. 21f, extra In diagonal Figure 21f, and extra bottom diagonal cords were and addedbottom to cords the archwere at theadded center to ofthe the arch shell. at Inthe Figure center 21 ofg, the extra shell. diagonal In Figure and bottom21g, extra cords diagonal were added and bottom to the two cords arches were at theadded end ofto the two shell. arches All the at increasedthe end of section the shell. sizes All used the increased the ones speciﬁedsection sizes in GBT17395-1998 used the ones specified [38], with ain wall GBT17395-1998 thickness of 6[38], mm. with a wall thickness of 6 mm.

(a) (b)

(e) (f)

(g)

FigureFigure 21. 21. SevenSeven different diﬀerent schemes schemes to to enhance enhance the the dome: dome: (a)( ascheme) scheme 1, (b 1,)scheme (b)scheme 2, (c 2,) scheme (c) scheme 4, (d) 4, (dscheme) scheme 4, ( 4,e) ( escheme) scheme 5, ( 5,f) (schemef) scheme 6, ( 6,g) (schemeg) scheme 7. 7.

Materials 2020, 13, x FOR PEER REVIEW 16 of 18 Materials 2020, 13, 2519 16 of 18 Note: Red solid line represents the enhanced bars; the red dotted line indicates the plane arch truss. The height of the truss section is 450mm, and the diameter of the truss web members is Φ30 × 3.5 Note: Red solid line represents the enhanced bars; the red dotted line indicates the plane arch truss. The height of the truss section is 450mm, and the diameter of the truss web members is Φ30 3.5 A time history analysis was performed for all seven enhanced schemes. The results are shown× in FiguresA time 22 history and 23. analysis Figure was 22 shows performed the relation for all sevenship between enhanced the schemes. various Theearthquake results are inputs shown with in Figuresthe increase 22 and in the23. use Figure of the 22 showssteel material. the relationship between the various earthquake inputs with the increase in the use of the steel material. Materials 2020, 13, x FOR PEER REVIEW 16 of 18 It can be seen that, with the increase in the steel consumption, the collapse peak acceleration, which can be taken as the seismic capacity index of schemes 6 and 7, increases most rapidly. On the Note: Red solid line represents the enhanced bars; the red dotted line indicates the plane arch truss. The height other hand, for schemes 2 and 3, the capacity index increases less than in schemes 6 and 7, while in of the truss section is 450mm, and the diameter of the truss web members is Φ30 × 3.5 schemes 4 and 5, the capacity index increases the least. 2 2 A timeFigure history 23 shows analysis the relationshipwas performed between for all the seven various enhanced live loads schemes. (1.3 KN The/m resultsand 1.7 are KN shown/m ) on in theFigures roof 22 with and the 23. use Figure of the 22 steelshows material the relation aftership the analysis.between the It can various be seen earthquake that scheme inputs 7 shows with a thereasonably increase in high the use earthquake of the steel resistance material. with a small increase in steel usage.

Figure 22. Relation between collapse peak acceleration vs. steel consumption increment.

It can be seen that, with the increase in the steel consumption, the collapse peak acceleration, which can be taken as the seismic capacity index of schemes 6 and 7, increases most rapidly. On the other hand, for schemes 2 and 3, the capacity index increases less than in schemes 6 and 7, while in schemes 4 and 5, the capacity index increases the least. Figure 23 shows the relationship between the various live loads (1.3 KN/m2and 1.7 KN/m2) on the roof with the use of the steel material after the analysis. It can be seen that scheme 7 shows a reasonablyFigureFigure high 22. earthquake 22.RelationRelation between betweenresistance collapse collapse with peak peaka smallacceleration acceleration increase vs. vs.steelin steel consumption consumptionusage. increment. increment.

It can be seen that, with the increase in the steel consumption, the collapse peak acceleration, which can be taken as the seismic capacity index of schemes 6 and 7, increases most rapidly. On the other hand, for schemes 2 and 3, the capacity index increases less than in schemes 6 and 7, while in schemes 4 and 5, the capacity index increases the least. Figure 23 shows the relationship between the various live loads (1.3 KN/m2and 1.7 KN/m2) on the roof with the use of the steel material after the analysis. It can be seen that scheme 7 shows a reasonably high earthquake resistance with a small increase in steel usage.

(a) (b)

Figure 23.23. Curve of accelerationacceleration vs.vs. steel consumption increment under various livelive loads:loads: (a) proofproof livelive loadload 1.31.3 KNKN/m/m22,(, (b) proof live load 1.71.7 KNKN/m2./m2.

5. Conclusion Conclusions InIn thisthis paper,paper, aa detaileddetailed progressiveprogressive collapsecollapse analysisanalysis ofof aa single-layersingle-layer cylindricalcylindrical latticedlatticed shellsshells under severe earthquakes was performed using the ﬁberfiber beam element method with the inclusion of the accumulation of material damage. The below conclusions can be made: the accumulation of material(a) damage. The below conclusions can be made:(b)

1. 1.A failureA failure pattern pattern called called “double-diagonal “double-diagonal -members-failure-belt” -members-failure-belt” is discovered is discovered for thisfor this type type of Figure 23. Curve of acceleration vs. steel consumption increment under various live loads: (a) proof spaceof structure;space structure; live2. load The 1.3 failureKN/m2, pattern(b) proof can live be load divided 1.7 KN/m2. into three consecutive stages: 2. The failure pattern can be divided into three consecutive stages: i. Significant variation in the deformation in different structural zones; 5. Conclusion i. ii. SigniﬁcantContinuously variation increasing in the internal deformation forces; in diﬀerent structural zones; In thisii. iii. paper, ContinuouslyDamage a detailed to structural progressive increasing members. internal collapse forces; analysis of a single-layer cylindrical latticed shells under severeiii. earthquakesDamage to was structural performed members. using the fiber beam element method with the inclusion of the accumulation of material damage. The below conclusions 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: i. Significant variation in the deformation in different structural zones; ii. Continuously increasing internal forces; iii. Damage to structural members.

Materials 2020, 13, 2519 17 of 18

Author Contributions: Conceptualization, H.Z., Y.Z. and F.F.; methodology, H.Z. and F.F.; writing—original draft preparation, H.Z. and F.F.; writing—review and editing F.F.; project administration, J.W.; funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Henan Province Science and Technology Research, China, grant number 172102210181 and 172102210182. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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