Thin–Walled Structures 146 (2020) 106470
Available online 9 November 2019
0263-8231/© 2019 Elsevier Ltd. All rights reserved.
Full length article
Collapse mechanism and shaking table test validation of a 3D mid-rise CFS composite shear wall building
Liqiang Jiang
a, Jihong Ye Dean, Dr., Professor
b,c,*aSchool of Civil Engineering, Central South University, Changsha, 410075, China
bJiangsu Key Laboratory Environmental Impact and Structural Safety in Engineering, China University of Mining and Technology, Xuzhou, 211116, China
cXuzhou Key Laboratory for Fire Safety of Engineering Structures, China University of Mining and Technology, Xuzhou, 221116, China
A R T I C L E I N F O Keywords:
Cold-formed steel structure Mid-rise
Collapse
Structural vulnerability theory Shaking-table test
Earthquake engineering
A B S T R A C T
Structural vulnerability theory (SVT) is a method that identifying the weakness and potential collapse modes of building structures in the finite element (FE) aspect, and the determinant of the global stiffness matrix |K|�0 is proposed as the judgment of collapse of building structures. Due to the limitation of the SVT in identifying plastic hinge failure modes of the framed structures, this paper proposes the improved SVT (SVT), in which the transformation processes of the connections from rigid ones to pinned ones are considered. Based on the foun- dations of the ISVT using in planar framed structures, which was conducted by the authors, this paper develops the ISVT from 2-dimension to 3-dimension, and a 5-story cold-formed steel (CFS) building was tested to verify the ISVT. The results show that: the predicted collapse mode with maximum vulnerability index from the ISVT is same with the failure patterns of the test building, the first unzipped elements predicted by the ISVT are same with the most serious damaged components observed form the test. Such two aspects demonstrate the feasibility of the ISVT in revealing the collapse mechanism of the 3D CFS structures. Besides, this paper recommends the expected collapse mode for mid-rise CFS structures, and proposes “strong-frame weak-wall” concept to realize seismic design of the expected collapse mode. The findings can be used for collapse resistance design for mid-rise CFS buildings subjected to earthquakes.
1. Introduction
Cold-formed steel (CFS) structure is a composite structural system, the CFS components and the wall sheathings are used to resist gravity loads and lateral loads, respectively. By comparing with the hot-rolled steel and reinforced concrete (RC) structures, CFS structures exhibit many advantages, such as less-weight, easy for construction and cost- efficiently. The CFS structures are commonly used in North America, Australia and East Asia [1]. Due to the earthquake is one of the most important natural hazards in these regions, the seismic performance as well as seismic failure mechanism of the CFS structures were investi- gated by researchers, including experimental studies [2–14], numerical modelling methods [15–21] and seismic design methodologies [22–32].
But most of them were focusing on the low-rise CFS buildings. Ye’s team proposed a novel mid-rise CFS composite shear wall system, where continuous CFS concrete-filled tube (CFRST) columns were used as the end studs of the CFS shear walls instead of the CFS build-up section, and the CFS-ALC with concrete covering composite slabs were used as the
composite floors. A series of experimental studies were performed to examine the proposed system, including quasi-static loading tests on the proposed CFS shear walls [33–35], fire tests on the proposed composite floors and CFS shear walls [36–38], shaking table tests on the proposed system [39,40]. Though large amount of experimental and numerical studies were conducted to analyze the seismic performance of CFS structures, very few studies have been performed for collapse mecha- nism analysis of the CFS structures [41], especially for the studies on theoretical methodologies.
The collapse mechanism analysis has become a hot topic since the collapse of the Ronan Point apartment in London at 1968, the World Trade Center in New York at 2001 and many buildings due to Wenchuan earthquake at 2008 in China. A large amount of investigations on collapse analyzing methodologies were conducted by the researchers [42–54], including the aspects of strength, stiffness, energy, probability, consequence, vulnerability, robustness, etc. The structural vulnerability theory (SVT) is one of them, which was proposed by Blockley’s team since 1993 [55–57], and it was used for collapse analyses of truss
* Jiangsu Key Laboratory Environmental Impact and Structural Safety in Engineering, China University of Mining and Technology, Xuzhou, 211116, China E-mail addresses: [email protected], [email protected] (L. Jiang), [email protected] (J. Ye).
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Thin-Walled Structures
journal homepage: http://www.elsevier.com/locate/tws
https://doi.org/10.1016/j.tws.2019.106470
Received 4 June 2019; Received in revised form 10 October 2019; Accepted 23 October 2019
structures [58–60]. Besides, Murta et al. [61] analyzed the collapse modes of the truss roof in traditional Portuguese wooden buildings, and the predicted collapse modes were compared with the actual collapse modes. Ye’ team developed the SVT to space structures [62,63], and shaking table tests on three K-6 space domes were used to verify the predicted collapse modes by the SVT. Besides, Ye’s team found the correlative relationship between the distribution of joint well-formedness and the stability resistance of space domes, and per- formed optimization of single-layer space domes based on this finding [64,65]. A component was used as the basic element for the SVT, thus the predicted collapse mode is consisted of a collection of components.
Though it is appropriated for the truss and space structures which are dominated by axial force, it can be not used to identify the failure modes that plastic hinge formed at the ends of the components. However, such failure modes are usually occurred on steel frames and CFS structures subjected to earthquake loads, and such failure modes can not be expressed in the hierarchical model and unzipping procedures of the classical SVT.
Therefore, an improved SVT (ISVT) was proposed by the authors to outcome the limitation of the SVT. In the ISVT, the rigid connections were proposed as important as the components, and they were treated as basic elements in the ISVT, thus the plastic hinge failure modes could be identified by the proposed ISVT. The authors conducted failure mech- anism analyses of a planar steel frame [66], planar CFS shear walls [67]
and the FIU bridge [68]. This paper aims to develop the proposed ISVT from 2-dimension to 3-dimension, and new computing codes are developed. A 5-story 3D CFS building was tested by the authors to verify the ISVT on collapse mechanism analysis of the 3D CFS structures.
Section 2 introduces the classical structural vulnerability theory (SVT), and the improvements of the ISVT are presented in Section 3. The in- formation of the 5-story CFS shaking table test model are described in Section 4. The detailed analyzing procedures and results of the predic- tion by the ISVT as well as the test validations are presented in Section 5.
2. Methodology of structural vulnerability theory (SVT)
The collapse mechanism analysis of building structures could be viewed on different aspects, many types of methodologies have been proposed in literature, and three categories can be concluded:
(1) Engineering aspect. A structure could be judged as collapse if the structure could not resist the loads, including the static loads and dynamic loads.
(2) Mechanical aspect. A structure could be judged as collapse if the structure is changed from statically determinate or statically indeterminate system to geometrically unstable system.
(3) Finite element (FE) aspect. A structure could be judged as collapse if the determinant of the global stiffness matrix |K|�0.
The structural vulnerability theory (SVT) is one of the most typical theory in the FE aspect. The SVT was proposed by Blockley’s team at 1993 [55–60], and it was validated, applied and developed by Murta et al. [61] and Ye’s team [62–68]. The SVT identifies the most vulner- able positions of the structural systems based on the importance level of the elements in the global stiffness matrix K. Then it weakens the K of the structure by inflicting a set of damage events at these vulnerable positions until |K|�0. The failure elements due to these damage events are the key elements of the structure corresponding to a typical collapse mode.
The analytical procedures of the SVT method include following three parts.
(1) Clustering procedure. A structure can be divided as several initial structural clusters, and it can be expressed by a hierarchy model with rigorous hierarchical relationship according to the connec- tion characteristics of the structural clusters. The clustering
procedure starts from the bottom of the hierarchical model, and it is accomplished once all the elements are already clustered into the hierarchical model.
(2) Unzipping procedure. The unzipping procedure starts from the top of the hierarchical model by inflicting a damage event. Then a new hierarchical model is built for the residual structure, and a new damage event is used to unzip the new model. This pro- cedure repeats until |K|�0, and final new residual structure is judged as collapse.
(3) Vulnerability evaluation. The vulnerability index is proposed to assess the disproportionate level of the identified failure/collapse modes, it is calculated by the ratio of failure consequence to damage demand, and it is used to identify the worst collapse mode. According to the previous findings [62–68], the identified worst collapse modes of structures were similar with the final collapse modes observed in the tests or engineering practices.
By comparing with the other collapse analyzing methodologies, the SVT exhibits the following advantages.
(1) The SVT could identify the various collapse modes because it is not limited by the types of loads, thus some dangerous but not easily be found failure modes could be identified by the SVT.
Generally, the collapse tests and numerical analyses of building structures are always accomplished subjected to some typical load conditions, however, the collapse modes of the building structures might be changed in their servicing life. The reasons are: not only the load conditions might be changed due to envi- ronmental and man-made interference, but also the material properties as well as the connection performance might be changed in their servicing life. Thus it is important to identify the potential and dangerous collapse modes which are difficult to find.
(2) The vulnerability index of the SVT could evaluate the dispro- portion consequence levels of different failure/collapse modes, thus it can be used to find the worst collapse modes of existing or new buildings. Though the model tests and numerical analyzing methods could simulate the collapse scenarios and collapse risks of building structures, the disproportionate consequence level of these collapse scenarios/modes are difficult to evaluate, and relative few studies were focusing on this issue. However, the disproportion level is an important factor for risk-based robust- ness assessment of building structures [69,70], and the impor- tance of the factor is as important as the failure probability factor.
The authors assessed the structural robustness of steel frame structures due to progressive collapse based on the dispropor- tionate consequence quantification by the SVT [71].
2.1. Structural ring, structural cluster and well-formedness
A structural ring is the basic element to resist loads from arbitrary directions, and some structural rings consist a structural cluster. A structural ring with minimal amount of elements is named as an initial structural ring. Several initial structural rings consist a leaf structural cluster, and the foundation is named as reference cluster.
Initial structural rings with pin connections and with rigid connec- tion are shown in Fig. 1. A structural ring would transform to a mech- anism if it encountered a damage event, such as pin or fracture at an element or pin at the joint. The SVT changes the structural form of a structure through a set of damage events and weakens the structure to a mechanism, and then identifies the collapse modes and key elements of the structure.
Joint well-formedness qk was proposed in the SVT to describe the capacity of the joint jk resisting loads from different directions [55–60], and the calculation is
qk¼Yc
1
λi¼delðKkkÞ (1)
detðKkk λIÞ ¼0 (2)
where, Kkk is the substiffness matrix associated with the joint jk; c is the dimensions of the joint jk, and it is equal to the degree of freedom of the joint; λi is the principal stiffness coefficient for the ith eigenvector; λ and I are the eigenvector associated with the joint jk and the unit stiffness with c orders, respectively.
The well-formedness of a structure was proposed as the average value of the joint well-formedness of all joints in the structural system [55–60].
QðnÞ ¼Xn
i¼1
qi
n (3)
where Q(n), n and qi are the well-formedness of a structure, amount of the joints in the structure (excluding the joints connected with the foundation) and the well-formedness of the ith joint, respectively.
A planar pined structure is selected as an example, as shown in Fig. 2, the well-formedness of the structure with different angles is calculated and listed in Table 1. It is estimated that the line stiffness of the com- ponents is equal, k ¼E1A1/l1 ¼E2A2/l2 ¼E3A3/l3.
According to the results listed in Table 1, the structure with form of regular triangle exhibits the largest well-formedness, which result is complied with the basic concept of structural mechanic theory “regular triangle is the most stable form of a structure”. The finding indicates that the quality of the structural form of a structure could be expressed by the value of well-formedness.
2.2. Clustering and unzipping rules
The detailed clustering and unzipping rules can be found in previous studies [55–60], and a brief introduction is presented in this paper. The clustering rules include the following.
(1) Maximum well-formedness of the structure (Q);
(2) Minimal damage demand of the structure (Dmin);
(3) Maximum nodal connectivity (N);
(4) Maximum distance from the reference (Dis);
(5) Free choice if the above rules can not be used (FC).
Damage demand Dj is the required damage events that resulting jth failure mode of the structure, and it is associated with the lost principal stiffness coefficient in the joint stiffness matrix. The nodal connectivity N is the amount of the components at the joints of the structure, it represents the clustering ability in next step. The distance from the reference Dis is the distance between the structural cluster after a certain of clustering steps and the reference cluster, and it is equal to the amount of the initial structural rings from the structural cluster to the reference cluster. Thus the smaller the Dis, the more serious failure consequence the structure if the structural cluster is separate from the reference cluster.
The followings are the unzipping rules.
(1) It is not the reference cluster (NR);
(2) It can not be formed as a structural ring with the reference cluster (FR);
(3) It can not be formed as a structural ring with the reference cluster, even though it connects with the reference cluster (CD);
(4) It is not a leaf cluster but an initial cluster (L);
(5) It has the lowest well-formedness (SQ);
(6) It has the lowest damage demand (SD);
(7) It is located at the highest hierarchy (CL);
(8) Free choice if the above rules can not be used (FC).
2.3. Vulnerability index
The vulnerability index ϕ was proposed by the SVT [55–60].
ϕ¼ γ Dr
¼½QðSÞ QðS’Þ�Dmax
QðSÞD (4)
where Dr and γ are the damage demand and the separateness of a Fig. 1. Failure patterns of the initial structural rings.
Fig. 2.Three-bar pined planar structures.
structure, respectively; Q(S) and Q(S0) are the well-formedness of the original structure and the damaged structure, respectively; D and Dmax
are the damage demand of the damage components in a typical failure mode and the damage demand of all the components in the structure, respectively.
3. Improved SVT (ISVT)
A component was proposed as the basic element of the hierarchical model in the SVT, and it is appropriate for the truss and space structures which are dominated by axial force. For the steel-moment frames and cold-formed steel (CFS) framed structures, the plastic hinges formed at the ends of the components due to bending moment are the typical failure modes of them subjected to earthquakes. However, the SVT did not consider the connections transformed from rigid to pine in the hi- erarchical model, thus the plastic-hinge failure modes can not be iden- tified based on the SVT. Therefore, this paper proposed improved SVT (ISVT) for collapse mechanism analyses of planar structures [66–68], and the aim of this paper is promoting the ISVT to 3D structures, and reveals the collapse mechanism of 3D CFS structures.
3.1. Effects of the rigid connections
Fig. 3 shows three 2-bar planar structures with different types of boundary conditions, the geometry and material performance of the components in these structures are: the length of the components is
l1 ¼l2 ¼0.33 m, the elastic modulus is E1 ¼E2 ¼2 �105 MPa, the sec- tion area is A1 ¼A2 ¼42.74 mm2, the inertia moment is I1 ¼I2¼135.77 mm4. In Fig. 3, α1 ¼β and α2 ¼αþβ. To distinguish the rigid and pined connections, the capital letters represent rigid connec- tions (J1, R1 and R2) and the lower-case letters represent pined con- nections (j1, r1 and r2). Such expression is also used in the following sections.
Fig. 4(a) and Fig. 4(b) present the nodal stiffness matrix of the joint J1 for the planar 2-bar structures in Fig. 3(a) and (c), respectively. It can be found that the nodal stiffness matrix of the joint is changed along with the changing of the boundary conditions. The well-formedness of the joint J1 in these structures is shown in Fig. 5, the maximum difference on the well-formedness of the joint in these structures is about 40% when the angle α is 90�. The angle between columns and beams is 90�in many cases for the steel frames and CFS framed structures.
It can be found that the effects of the rigid connections on well- formedness of structures are similar with the effects of the Fig. 3. Planar 2-bar structures with different boundary conditions.
Table 1
Well-formedness of joints and the 3-bar pined planar structures.
Structure (
Fig. 3) Joint well-formedness qi
(i ¼1,2,3) Well-formednss of the structure Q (n)
q1 q2 q3
I 0.85
k2 0.5 k2 0.85
k2 0.73 k2
II 0.75
k2 0.75k2 0.75
k2 0.75 k2
III 0.50
k2 k2 0.50
k2 0.67 k2
Fig. 4. Nodal stiffness matrix of the joint J1 in the planar 2-bar structures.
Fig. 5.Well-formedness of joint J1 in the 2-bar structure with different boundary conditions.
components. Besides, there is a requirement on damage demands if the rigid connections are transformed to pined ones. Therefore, this paper proposes that the rigid connections should also be treated as basic ele- ments as same as the components in the hierarchical model. Further- more, it is recommended that the transforming process from rigid to pin of the connections should be expressed in the unzipping procedures.
This recommendation is one of the most important idea of the ISVT proposed by the authors. Due to the rigid connections are considered into the hierarchical model, clustering and unzipping procedures, thus the ISVT could identify the plastic-hinge failure modes which could not be identified by the SVT. The advantages of the ISVT in identifying the plastic-hinge failure modes are approved according to the comparisons for planar steel frames in the authors’ previous work [66]. Based on the previous findings, this paper aims to develop the ISVT from 2-dimension to 3-dimension, and explore the feasibility of the ISVT in collapse mechanism analysis of 3D CFS structures.
3.2. Improvements on the clustering and unzipping rules
Due to the rigid connections are considered into the clustering and unzipping procedures, the traditional rules of the SVT are suggested to be improved, including the followings.
(1) New clustering rule. The rigid connection (labeled as R) is used to distinguish the difference of the rigid connections and the pined connections, it is used as a basic element in the clustering pro- cedures as like as a component, and it also follows the traditional rules defined by the SVT.
(2) New 4th unzipping rule. In the ISVT, the 4th unzipping rule is improved as:
(4–1) It is not a leaf cluster but an initial cluster, and the initial cluster is a rigid connection R (L1).
(4–2) It is not a leaf cluster but an initial cluster, and the initial cluster is not a rigid connection R but a component (L1).
Furthermore, this paper suggests that the ISVT should be used for failure mechanism analysis of framed structures (including steel frames and CFS framed structures) subjected to earthquake loads; the SVT could be used for the truss and space structures, because the components of these structures are dominated by axial force.
3.3. Application of the ISVT in 3D CFS structures
Due to the high nonlinear behaviors of the CFS structures, including fracture of wallboards, buckling of CFS components, failure of screw connections, etc., the collapse modelling of the CFS structures is complicated. The contributions of wall sheathings, CFS components, connections, boundary restraints and composite floors should be accu- rately modeled for collapse analysis. Thus very few works have been reported on the area of collapse analysis of CFS structures.
In this paper, a simplified macroscopic model is used to perform the collapse mechanism analysis of the CFS structures instead of the complicated microscopic models, which is presented in Section 4. The collapse mechanism of the CFS structures is conducted by the simplified model taking the basis of the ISVT methodology. The simplified model was also recommended for seismic responses analysis for CFS structures by many researchers [15–21]. It should be noted that the simplified model is suggested for collapse mechanism analysis of CFS structures, but not for preliminary structural design of them, because the stability analysis of CFS components as well as the uncertainty factors are not considered in this work.
As stated in Section 2, the ISVT is a FE-based numerical methodol- ogy, and it identifies the potential collapse modes according to a crite- rion that if the global stiffness matrix |K|�0, and it finds the most vulnerable one by the vulnerability indexes. Thus the global stiffness matrix are the important factors in identifying the collapsed modes.
Besides, the damage demand is an important factor to calculate the vulnerability indexes of the identified collapse modes. The ISVT is applicable to the CFS structures due to the following reasons:
(1) The contributions of wall sheathings, CFS components, boundary conditions could be considered into the global stiffness matrix, and thus be considered in the ISVT analysis. The lateral stiffness contribution as well as the yield strength of the CFRST end studs are considered. The cross-section properties and the length of the CFS components (CFRST end studs and the CFS steel beams) and sheathing walls are considered.
(2) The connection type as well as the connection performance could also be considered in the ISVT, and different joint well- formedness could be obtained to different types of connections.
Thus the plastic hinge failure modes could be considered in the ISVT.
(3) The failure of the wall sheathings and CFS components (including buckling of the CFS components) could be considered in deter- mining the damage demand of these components. The damage demand is defined as the loss of the principal stiffness coefficient.
Thus different failure patterns in different sheathing walls or CFS components cause different damage demands, and the damage demands are used to calculate the vulnerability indexes.
Unlike the planar structures, the rigid joints in 3D structures have six freedoms, including three translation freedoms and three rotation free- doms, thus the nodal stiffness matrix of a rigid joint is 6-order matrix.
Therefore, this paper redesigned the computational codes for the ISVT, and stiffness matrix of 3D structures is used for calculating the well- formedness of the joints and the structure.
Besides, the 3D CFS structures use composite floors to resist gravity loads, and the axial stiffness of these composite floors is much larger than the stiffness of other components. In the simplified numerical model, the composite floors are commonly modeled by rigid plane ele- ments, and the assumption of the rigid diaphragm behavior of the floor was validated in the authors’ previous study [39]. The nodal stiffness matrix of a typical 3D joint is selected for example, the matrix is diag- onalized, as shown in Fig. 6, where E is estimated as the contribution of the composite floors. In the numerical model of a 3D CFS structures, the contribution of E is ∞, and the determinant of the nodal stiffness matrix is ∞. Thus the contributions of the other components (A, B, C, D, F and G) are “concealed” by the contribution of the composite floors E, which results in futility of the ISVT in identifying the vulnerable parts of the structure. Therefore, this paper suggests that the contributions of the composite floors could be wiped out in the nodal stiffness matrix, as shown in Fig. 6, and only the contributions of the other components are considered in the procedures of the ISVT, because the composite floors are not the vulnerable parts of the CFS structures according to the shaking table test [39].
Fig. 6.Diagonalization nodal stiffness matrix of a 3D joint.
4. Simplified numerical model of the shaking-table test building To verify the proposed ISVT for collapse mechanism analysis of the CFS structures, a 5-story CFS building was tested and used as the vali- dation case, as shown in Fig. 7. The story height of the building was 1.5 m, and the total height was 7.5 m. The building was designed as two- bay two-span, and the length of each bay and each span was 1.8 m. Only single-direction earthquake was input on the building along the east- west diction. The CFS shear walls in 1-axis ~ 3-axis were used to resist earthquake loads, gypsum wallboard (GWB) with 12 mm in thickness was sheathed at the both sides of the walls, and no openings were design for these CFS shear walls. The construction details of the 1- axis ~ 3-axis CFS shear walls is depicted in Fig. 7(b), and these shear walls were designed with novel mid-rise construction, which was pro- posed by the authors. The concrete-filled CFS steel tube (CFRST) is used as the end stud instead of the commonly used build-up sections, thus both the compression strength and the flexural capacity of the end studs are strengthened to resist the earthquake loads, the details of the novel mid-rise construction of the CFS shear walls were reported in the authors previous study [35,39]. The shear walls in A-axis ~ B-axis were not subjected to earthquake loads, and 12 mm-thickness GWB with 0.9 m �0.6 m door openings was used as the wall sheathings, and other construction details of these shear walls were same with the shear walls in the 1-axis ~ 3-axis. The test building was designed as 1/2-scale due to the limitation of the test equipment and laboratory, and it was designed according to the similarity relation, which listed in Table 2. The detailed construction, loading pattern and test results were presented in the au- thors’ previous study [39].
Due to large amount of stiffness matrix calculations are involved in the ISVT, a numerical model of the test building is required, and the global stiffness matrix as well as the nodal stiffness matrix of the nu- merical model are the basic data for vulnerability analyses. The nu- merical model is established by OpenSees platform, as shown in Fig. 7 (b). The cold-formed steel tube (CFDST) end studs are modeled by nonlinear beam-column elements, and the CFS shear walls are modeled by two-node link elements with Pingching04 material. The details of the numerical model of the test building can be found in the authors’ pre- vious study [39].
The time-history drift responses measured from tests are used to verify the numerical model, comparisons on time-history drift curves at
the 300 gal and the 800 gal loading cases are presented in Fig. 8, the detailed validation can be found in the authors’ previous study [39]. The numerical model captures the dynamic responses of the test building, and predicts the changing process of the lateral stiffness of the test building subjected to earthquake loads. Such results indicate that the proposed numerical model has relative high accuracy, and it can be used for the following vulnerability analyses.
The numerical model includes cold-formed steel tube (CFRST) col- umns, CFS beams, equivalent springs for the CFS shear walls and the rigid connections, the first-story of the building is selected to present the numbering rules, as shown in Fig. 9. The rigid connections at the column bases are labeled as R1 ~ R9. The CFDST columns are labeled as 1–9, the equivalent springs are labeled as 10–45. Such numbers are increased according to the number of the located story n with increment of 45 �(n-1), thus the maximum number is 225 and it is located at the roof story. The rigid connections of the continuous CFRST columns are labeled as J1 ~ J36, and J1 ~ J9 are used in the first story. The section properties as well as the damage demands of the elements in the test model are listed in Table 3.
Fig. 7. Shaking table test on a 5-story cold-formed steel composite shear wall building.
Table 2
Similarity relation of the test building.
Items Parameters Symbols Similarity factors
Geometric parameters Length SL 1/2
Area SA 1/4
Drift ratio Sα 1
Material parameters Strain Sε 1
Elastic modulus SE 1
Stress Sσ 1
Poisson’s ratio Sμ 1
Mass density Sρ 2
Load parameters Concentrated force SP 1/4
Area load Sq 1
Dynamic parameters Period ST 1= ffiffiffi
p2
Frequency Sf ffiffiffi
p2
Acceleration Sa 1
Gravity acceleration Sg 1
Damping ratio SC 1/21.5
5. Analyzing procedures and results of the shaking table test model based on the ISVT
5.1. Clustering procedures
Based on the clustering rules of the ISVT, the clustering procedures of the test model include the following.
(1) Initial clustering stage. An initial structural ring {m5, R5, R} is selected as the seed cluster C0 based on the clustering rules, as show in Fig. 10(a). The other elements are successively clustered with the C0, and the clustering procedures are shown in Fig. 10. It is terminated as it processes to the stage in Fig. 11(a), because the well-formedness of the leaf cluster C5 cannot be increased in the next step, and the hierarchical model at this stage is depicted in Fig. 12(a). Then, new seed clusters are selected in the residual elements which have not been participated in the clustering procedures, and new leaf clusters can be obtained, such as the leaf clusters of the side column 2 and the corner column 1, as shown in Figs. 11 and 12. The initial clustering stage is finished when no initial structural rings can be found in the residual elements of the structure.
(2) Second and final clustering stages. According to the rules of in- crease or cause the least reduction in well-formedness of the Fig. 8. Comparisons on displacement time-history curves of test and numerical results.
Fig. 9.Numbering rule of the first story in the numerical model.
Table 3
Section properties and damage demands of the elements in the test model (Unit: N, m).
Element type Section type Length EA EI Damage demand
Corner column “L” CFRST 1.5 1.78 �108 2.74 �105 6.40 �106
Side column (weak-axis) “T” CFRST 1.5 2.07 �108 4.08 �105 1.10 �107
Side column (strong-axis) “T” CFRST 1.5 2.07 �108 5.10 �105 1.36 �107
Central column “þ” CFRST 1.5 2.41 �108 5.41 �105 1.50 �107
CFS beams Buildup H section 1.8 8.03 �107 2.58 �106 2.29 �107
CFS shear walls in 1-axis ~ 3-axis 12 mm GWB sheathing 2.343 5.57 �106 – 9.51 �106
CFS shear walls in A-axis ~ C-axis 12 mm GWB sheathing with 0.9 m �0.45 m opening 2.343 3.48 �106 – 5.94 �106
Rigid connections at the column bases Damage demand: (1) Central column is 2.41 �106; (2) Side column is 2.19 �106 and 1.88 �106 in strong-axis and weak-axis, respectively;
(3) Corner column is1.02 �106
Note: the details of the CFRST columns, CFS beams and CFS shear walls can be found in Ref. [39].
structure, the leaf clusters and the residual elements are clustered each other. The second clustering state is finished if no more clusters can be clustered except of the reference cluster, and the finial clustering state is accomplished when the reference cluster is clustered into the hierarchical model. The final hierarchical model of the test building is shown in Fig. 13.
5.2. Unzipping procedures
The unzipping procedure starts from the highest level of cluster C102, as shown in Fig. 13, it proceeds to the C101 according to the unzipping rule of NR. After that, it proceeds to the C30 based on the rule of FR, and then unzips the rigid connection R1 for the rule of L1, as shown in Fig. 14(a). Due to the determinant of the global stiffness matrix |K|>0,
thus a new hierarchical model is rebuilt taking basis of the residual el- ements. The elements R3, R7, R9, R2, R4, R6, R8 and R5 are then be unzipped successively according the unzipping rules, and the first-story of the test building is shown in Fig. 15(a), but the determinant of the global stiffness matrix |K|>0 at this stage. Thus a new hierarchical model is rebuilt, and further unzipping steps are needed. At this stage, the leaf cluster C35 changes to the cluster in Fig. 14(b), and the com- ponents 31, 30, 32, 33, 22, 23, 24, 25, 26, 27, 28 and 29 are unzipped successively. A newer hierarchical model is rebuilt due to the |K|>0 at the end of this stage, and the new leaf cluster C35 is depicted in Fig. 14 (c). The rigid connections R at the ends of the CFDST columns 1, 3, 7, 9, 2, 4, 6, 8 and 5 are unzipped successively, the |K| ¼0 can be calculated at the end of this stage, and the structure is judged as collapse. The first story of the finial residual structure is shown in Fig. 15(b), and the it is considered as the finial collapse mode predicted by the ISVT, an eleva- tion view of the predicted collapse mode is depicted in Fig. 16(a).
The unzipping procedures of the unzipped elements can be sequenced as: the rigid connections at the column bases are firstly unzipped, the CFS shear walls in the first-story are then unzipped, the rigid connections at the ends of the CFDST columns in the first-story are unzipped finally. Besides, the unzipping sequences of the rigid connec- tions in the CFDST columns are the corner columns, the side columns and the central column.
5.3. Vulnerability results
According the ISVT, the other collapse modes of the test model can also be identified by unzipping other leaf cluster firstly except for the C102, and some of them are presented in Table 3 and Fig. 16. The vulnerability indexes of these collapse modes are calculated taking basis of Eq. (4) and the damage demands in Table 2, and the results are listed in Table 4. Due to the failure modes are same for the CFS shear walls in 1-axis ~ 3-axis, thus the elevation view of the 3-axis CFS shear walls is used to express the collapse modes, as shown in Fig. 16.
Fig. 10.Initial clustering processes of the test model.
Fig. 11.Diagrams of the leaf clusters after initial clustering processes.
Fig. 12.Hierarchical models of the leaf clusters in Fig. 11.
It can be found that the collapse mode A (Fig. 16(a)) exhibits the maximum vulnerability index, indicating that the minimal damage de- mand is required for this collapse mode, and it is determined as the worst collapse mode. The vulnerability index of the collapse mode D (Fig. 16(d)) is the minimal one in the group of the global collapse modes, indicating that this collapse mode requires the maximum damage de- mands and it is not easy to happen.
5.4. Validation by shaking table test
The failure modes of the test building at the last loading case (earthquake intensity is 970 gal) are used to verify the predicted collapse mode of the ISVT. The finial failure modes of the CFS shear walls are presented in Fig. 17, and the finial failure modes of the CFRST columns are shown in Fig. 18 and Fig. 19.
The CFS shear walls were serious damaged and many wallboards were separated from the CFS frames, as shown in Fig. 17. In Fig. 18, obvious buckling deformation was observed at the ends of the CFRST Fig. 13.Hierarchical model of the test building.
Fig. 14.Partial unzipping processes of the leaf cluster C35 in the hierarchical model (corner column m1).
columns in the first story, and plastic hinges were formed at the columns bases (R1, R3, R7 and R9). Local buckling was observed at the ends of the CFRST columns nearby the beam-to-column joints of the first story, as shown in Figs. 18 and 19. Such failure patterns of the test building indicate that plastic hinges were formed at the column bases, the CFS shear walls at the first story were damaged and plastic hinges would be formed at the ends of the CFRST columns nearby the beam-to-column
joints of the first story. In order to ensure the safety of the test equip- ment and measurements, the test was terminated until the structural collapse. However, the observed failure patterns of the CFS shear walls and the CFRST columns demonstrate that the test building would collapse with the weaken first-story collapse mode. This collapse mode concluded form the test observations is same with the predicted collapse mode A with the maximum vulnerability index.
Fig. 15.Diagram of the unzipping processes for the test building. (only the first story is depicted, no components are unzipped at the other stories).
Fig. 16.Possible collapse modes of the test building.
According to the test observations, the damages of the corner col- umns were more serious than the side columns, and the damages of the side columns were more serious than the central columns; the damages of the rigid connections at the column bases were more serious than the damages of the rigid connections nearby the beam-to-columns joints at the first story. Thus it can be found that the first unzipped elements based on the ISVT are same with the most serious damaged elements observed in the test.
Therefore, the finial failure patterns observed form the test demon- strate the predicted collapse mode and the predicted unzipping pro- cedures of the proposed ISVT, and then demonstrate the feasibility of the ISVT for collapse mechanism analysis of 3D CFS structures.
Besides, the collapse mode D in Fig. 16(d) is quite similar with the
“beam-hinge” collapse mode proposed in the standards of many coun- tries. Therefore, this paper proposes the collapse mode D as the expected (or design) collapse mode of the mid-rise CFS structures, and proposes
“strong-frame weak-wall” designing concept for seismic design of the structures. Because the CFS shear walls could not be fully used if the CFRST columns damage early, and the similar finding was also reported
in the authors’ previous study on failure mechanism analyses of planar CFS shear walls [67].
The maximum vulnerability index of the test mid-rise CFS building (18.59) is smaller than the value of a mid-rise steel frame structure (31.44) [66], indicating that the disproportionate damage consequence of the mid-rise CFS building is smaller than the mid-rise steel frame, because the CFS building has two defense lines to resist earthquake loads (CFS frame and CFS shear walls). The disproportion level in damage consequence is also an important aspect for safety evaluation of building structures [69]. Though it is just a unique example, this paper thinks that a well-designed mid-rise CFS building has not only good seismic per- formance, but also has less seismic vulnerability. The results obtained from this paper indicate that the investigated mid-rise CFS building with the special structural typology under the specific conditions exhibits good seismic performance and relative low seismic vulnerability, thus it might be a good choice for the seismicity regions. More investigations on different types of structural typologies and ground motions would be conducted in the future to demonstrate the applicability of the proposed mid-rise CFS building system.
5.5. Limitation of the current work
Based on the above validations, the proposed ISVT could be a good choice for collapse mechanism analysis of the CFS structures subjected to earthquake loads. The basic concept that using the global stiffness matrix to judge the collapse of CFS structures is also reasonable, and the identified collapse mode is also validated by the test observations.
However, due to the simplifications are used in this paper, the effects of these simplifications on the collapse mechanism of CFS structures and the limitation of the current work are noted as followings:
(1) The middle studs of the CFS shear walls are ignored in the simplified model and a CFS wall is modeled by two nonlinear Table 4
Vulnerability indexes of the main collapse modes for the test building.
Type Collapse mode Separateness Relative damage demand
Vulnerability index
Global
collapse A in Fig. 16(a) 1.0 0.0538 18.59
B in Fig. 16(b) 1.0 0.0748 13.37
C in Fig. 16(c) 1.0 0.0846 11.82
D in Fig. 16(d)
Expected 1.0 0.2002 5.00
E in Fig. 16(e) 1.0 0.0979 10.21
Partial
collapse F in Fig. 16(f) 0.9997 0.1005 9.95 G in Fig. 16(g) 0.7136 0.1123 6.35 H in Fig. 16(h) 0.9997 0.1869 5.35
Fig. 17.Final failure patterns of the CFS shear walls in the test building (970gal loading case).
springs. Because this paper analyzes the collapse mechanism of the CFS structures subjected lateral earthquake loads. The lateral contribution of the CFS frame is relative few, and it is quite different with the progressive collapse analysis studied by Bae et al. [41]. Due to this simplification, the collapse modes due to failure of middle studs might not be identified in this paper, even though these collapse modes are not commonly happened in a proper-designed CFS structures. And the middle studs were recommend to be ignored in seismic analysis of CFS buildings in literature, and a CFS shear wall could be modeled by two crossed nonlinear springs [15–21].
(2) The CFS composite floors are simplified as rigid plane in the simplified model. The progressive collapse modes as well as some vertical collapse modes may not be identified due to this
simplification, even though it was also used for seismic analysis of CFS structures in literature [17–20,26,32].
(3) The simplifications of the boundary conditions and the connec- tions. The beam-to-column connections are estimated as pin and the boundary connections are estimated as rigid. Such simplifi- cations idealize the connection performance, a more compre- hensive investigation on the effect of the connection performance is needed to extent the collapse mechanism analysis of CFS structures.
(4) The uncertainty factors as well as the imperfections are not considered in this model, and only one test is performed to vali- date the analytical results. Thus the results of this paper are applicable for the simplified model presented in this paper, and more work are acquired to establish a more general model for collapse mechanism analysis of CFS structures.
Fig. 18.Buckling patterns of the CFRST corner columns (loading case of 970gal).
Fig. 19.Failure patterns of the CFRST columns (loading case of 970gal).
6. Conclusions
The classical structural vulnerability theory (SVT) aimed to identify the weakness and potential collapse modes of building structures on the aspect of the FE method by using the determinant of the global stiffness matrix of the structures, and it has been verified in truss and space structure by many researchers [58–65]. However, a component was proposed as a basic element in the SVT, it cannot be used for identifying the failure modes of plastic hinges formed at the rigid connections, thus it cannot be used for steel frame and CFS structures. Therefore, an improved SVT (ISVT) was proposed by the authors and was applied for the planar structures [66–68], and this paper aims to develop the ISVT from 2-dimension to 3-dimension, and new computation codes are developed. A 5-story CFS building was tested to verify the ISVT in collapse mechanism analysis of 3D CFS structures. Following conclu- sions can be made.
(1) The collapse mode with maximum vulnerability index predicted by the ISVT is same with the failure modes of the test building, and the first unzipped elements by the ISVT are similar with the most serious damaged components observed form the test. Such findings demonstrate the feasibility of the ISVT in revealing the collapse mechanism of the 3D CFS structures.
(2) The collapse mode D in Fig. 17(d) is proposed as the expected or design collapse mode of the mid-rise CFS structures, because such collapse mode has the minimal vulnerability index and it is quite similar with the “beam-hinge” collapse mode of the steel frame structures, which was proposed by the standards of many coun- tries. The lateral resistance of the CFS shear walls could be fully used if the finial collapse mode of the CFS structures is the collapse mode D, and the CFS frame could be better protected. To realize the expected collapse mode of the CFS structures, the
“strong-frame weak-wall” concept is proposed for seismic design of the CFS structures, because the CFS shear walls would be wasted if the CFRST columns damaged early.
The ISVT evaluates the structural vulnerability taking basis of the disproportion in damage consequence, which is quite different with commonly used probability-based seismic fragility analysis. The damage consequence is also another important index for safety evaluation of building structures as like as the failure probability, and it is corre- sponding to the direct and induced consequences in the risk-based robustness assessment framework. This framework was proposed by Baker et al. [69] and was adopted by JSCC [46], and it is the stat-of-the-art framework in the field of structural robustness assess- ment. Based on this finding, a previous trial on progressive collapse robustness assessment of steel frame structures has been reported by the authors [70], and the next paper about the vulnerability-fragility-based seismic robustness assessment of CFS structures will also be reported.
Finally, the limitation of this work is also presented and discussed.
Conflicts of interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Collapse Mechanism and shaking table test validation of a 3D mid-rise CFS composite shear wall building”.
Acknowledgement
This work is sponsoredby the National Key Program Foundation of China (51538002).
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