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Progressive collapse assessment of friction damped post-tensioned steel frames based on a simplified model
Structures 23 (2020) 447–458
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Progressive collapse assessment of friction damped post-tensioned steel frames based on a simplified model
T
⁎
Zhongwei Zhao , Xiangyang Jian, Bing Liang, Haiqing Liu School of Civil Engineering, Liaoning Technical University, Fuxin 123000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: 3D-FDPT connections Friction element Simplified numerical model Progressive collapse Dynamic analysis
Post-tensioned energy-dissipating (PTED) connections for steel frames have drawn great research attention on account of their good seismic performance. Few studies focus on the progressive collapse of friction damped post-tensioned (FDPT) frame structures. The progressive collapse of FDPT frames exhibits a complex threedimensional mechanical behavior that is quite different from that of traditional frame structures. In this work, a simplified 3D PTED connection model was first established and then used to investigate the progressive collapse behavior of FDPT frames. The influences of material strength, initial PT force, friction force, and axial and bending stiffness of beams on the collapse behavior of FDPT were studied. Finally, the dynamic response of FDPT frames was compared with that of traditional frame structures (TF).
1. Introduction Progressive collapse of building structures can cause massive economic losses and many casualties [1]. Many studies [2–4] on the progressive collapse of MRF have been conducted, and these studies are usually based on different component levels, such as beam-to-column connections, 2D frames, 3D skeletal frames, and 3D frames with slab systems. Numerical and experimental methods are mainly used for analysis [5–8]. Dynamic effects are key factors influencing progressive collapse analysis. The existing literature indicates that rate effects could change the failure mechanism of joints and influence their collapse capacity. Many procedures and explicit expressions have been proposed to approximately compensate for dynamic effects when a static procedure is used [9,10]. Stoddart et al. [11], for example, introduced rate-dependent springs to component-based joint models. McKay [12] concluded that dynamic factors in existing guidelines tend to yield overly conservative results, which often translate to expensive designs and retrofit. Liu [13] presented an empirical method to calculate the dynamic increase factor used to amplify gravity loads on the affected bays of a building frame when nonlinear static alternate path analysis is carried out to predict the peak dynamic responses to sudden column removal. Ferraioli [14] proposed a modal pushdown analytical to assess the progressive collapse of multi-story steel-frame buildings under sudden removal of a column due to catastrophic events. The catenary action of the beams of traditional steel frames involves a load-carrying
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mechanism that can potentially arrest the progressive collapse of steel structures [15–17]. However, the catenary action in posttensioned (PT) self-centering frames is considerably different from that in traditional frame structures due to self-centering connections. Several researchers [18–22] have confirmed the satisfactory seismic performance of PT self-centering beam–column connections. A method for estimating the influence of PT loss was previously proposed by Granello [23]. Garlock et al. [24] conducted experimental studies on full-scale PT steel connections and evaluated the effects of several parameters, including the initial posttensioning force, the number of posttensioning strands, and the length of the reinforcing plates. The results of this work demonstrated that PT connections allow good energy dissipation and ductility, which indicates that the connection strength in upper floors improves the seismic response of the frame. Kim and Christopoulos [25] proposed a new connection for steel moment-resisting frames incorporating posttensioning elements to provide self-centering capacity, along with friction mechanisms to dissipate energy. Lin et al. [26] investigated the seismic performance of a largescale steel self-centering moment-resisting frame. PT steel connections have been proposed to eliminate permanent deformations following severe earthquakes. Friction devices and viscous dampers [27] can be used to improve energy dissipation capacity. Christopoulos et al. [28] experimentally investigated PT energy-dissipating connections that dissipate input energy through the axial yielding of high-strength threaded bars with couplers welded to the top and bottom flanges of the beams. The seat angles were designed to yield
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.istruc.2019.09.005 Received 10 June 2019; Received in revised form 11 August 2019; Accepted 11 September 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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0.5
Force(KN)
to provide energy dissipation capacity in some PT connections [29]. The seismic performance of PT steel moment-resisting frames with friction devices was studied by Rojas et al. [20], who eventually confirmed that the variability of maximum friction forces developing in the friction devices does not have a significant effect on MRF performance. Although PT energy-dissipating (ED) connections have been proven to be a proper alternative to semi-rigid connections, no guidelines in the current seismic design codes [30] are yet available. Thus, further research studies are required to thoroughly investigate the application of these connections to buildings. Finite element (FE) analysis is a method that allows analysis of the behavior of PTED connections from different aspects that are difficult to capture in an experimental test. The entire FE model of PT steel frames is very limited. The progressive collapse of FDPT frames exhibits a complex 3D mechanical behavior that differs from that of traditional frame structures. The high-strength strands utilized in PTED connections is a key component of FTED. Progressive collapse may occur if the strands fail. However, research on the progressive collapse of FDPT frame cannot be found in the existing literature. Research on PT connections mainly focuses on seismic performance, and studies relevant to the progressive collapse of FDPT frame structures are scarce. The main purpose of this study is to provide a detailed FE modeling procedure that can accurately capture the progressive collapse behavior of FDPT connections under seismic loading. As the progressive collapse behavior of FDPT frame is very different from that of traditional frame structures, a simplified numerical model for 3D FDPT frames is presented.
0.4 0.3 0.2 0.1
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0.0 0.0 -0.1
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Fig. 2. Force–deflection curve of the contact spring element.
was adopted to prohibit invasion between steel columns and beams (contact element) and model a friction device (friction element). The PT strand was modeled using the Link180 element. A negative temperature load was applied to Link180 to generate an initial tensioning force. Combin39 is a unidirectional element with nonlinear generalized force-deflection capability that can be used in any analysis. The element is defined by two (preferably coincident) node points and the force–deflection curve. This element has longitudinal or torsional capability in 1D, 2D, and 3D applications. The longitudinal option is a uniaxial tension-compression element with up to three degrees of freedom at each node, i.e., translations in the nodal x, y, and z directions. No bending or torsion was considered in the analysis [31,32]. As the function of the contact spring is to prohibit invasion between steel columns and beams, the stiffness under compression was set to a value large enough to make the invasion negligible. The stiffness under tension was set to 0. The force–deflection curve of Combin39 is shown in Fig. 2. The parameter required for the friction spring element is the maximum static friction, i.e., Fmax. The nonlinear friction element adopted should be able to capture the mechanical characteristics of friction, i.e., the friction value is
2. Establishing of simplified numerical model Prior research has indicated that the columns and beams of PT connection are elastic under the action of a bending moment. The energy input is mainly dissipated by the friction device. Therefore, the beam element (Beam188) in ANSYS was used to model steel beams and columns in a PT connection. The simplified mechanical model of a PT connection is shown in Fig. 1. The nonlinear spring element Combin39
Fig. 1. Simplified mechanical model of a 2D PT connection. 448
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F 4''
I 4 II
4'
H 1''
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Rigid beam
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5''
IV 5 III
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Fig. 5. Constraint relation among internal nodes (2D). Fig. 3. Mathematical model of sliding friction.
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relevant to the frictional coefficient and normal force, and the direction of friction depends on the relative motion tendency. The static friction force can be determined through the equilibrium condition of forces. If the internal force is lower than Fmax, two nodes of the friction element remain relatively static. The internal force can be derived from the unbalanced forces at the two nodes. A mathematical model of sliding friction force is shown in Fig. 3. Sliding occurs when the static friction force reaches Fmax (or −Fmax), which is a constant value under a certain normal force. When the static friction force reaches Fmax, the cable will slide in the positive direction, such as point 1. By contrast, the cable will slide in the negative direction when the static friction force reaches −Fmax, such as point 6. The static friction force varies between Fmax and −Fmax when the cable is static relative to the joints. ANSYS was adopted to simulate the friction force, and the nonlinear spring element was utilized to model the friction force between the cable and the joint. The force–displacement curves of the COMBIN39 element are shown in Fig. 4. The angle θ is set to approximately 90° to allow the mechanical behavior of the nonlinear spring element to precisely model the sliding friction. The KEYOPT (1) of COMBIN39 was set to 1, so that the spring will unload along the line parallel to the slope at the origin of the loading curve. The mechanical model of the 2D-FDPT connection and its deformation under a bending moment are shown in Figs. 5 and 6, respectively. The distance between the friction device (or friction force) and the beam edge is indicated by H. The rigid beam element MPC184 was adopted to connect. Nodes 3′, 7, 3″, and 6 indicate nodes establishing the beam element. Node 3 is attached to the steel column and shares the same location as 3′and 3″. Nodes 8, 9, 10, and 11, which are located at the end of the high-strength strand, are also connected to the side end of the beam, i.e., 7, by a rigid beam element. A 3D-FDPT connection was established based on the 2D-FDPT connection. The constraint relation among internal nodes of the simplified 3D-FDPT connection are shown in Fig. 7. All of the nodes are in the vertical middle section of the column in the x or y directions. Nodes 1′-x, 1″-x, 1′-y, 1″-y, 2′-x, 2″-x, 2′-y, and 2″-y indicate the top and
4'
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2(2') 5
5' M
Fig. 6. Deformation under a bending moment.
Y
4 1 3
X 2 5
Fig. 7. Mechanical model of a 3D FDPT connection.
bottom contact points of the beam to the column. Symbols “-x” and “-y” indicate nodes attached to beams in the x and y directions, respectively. Nodes 1 and 2 indicate the corresponding locations on the steel column. Nodes 4 (4′-x, 4″-x, 4′-y, and 4″-y) and 5 (5′-x, 5″-x, 5′-y, and 5″-y) indicate the location of the friction device. MPC184 was adopted to
F
F
90°
θ
Displacement
Displacement
Fig. 4. Force–displacement curves of COMBIN39. 449
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3. Validation of the numerical model
4''-x 4''-y 1''-x 1''-y 3''-x
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3.1. Validation of the friction spring element
1'-y 1'-x 3'-y 3'-x 2'-y
The friction spring element is a key component in the simplified FDPT connection model, and its accuracy directly determines the seismic response of the overall structure. Thus, the accuracy and applicability of the friction spring element are validated in this section. A numerical test was conducted based on a numerical beam model to validate the accuracy of the friction spring element, as depicted in Fig. 11. The beam model was established by a planar beam element in ANSYS. The horizontal degree of the beam element was constrained, and a cyclic displacement was applied in the vertical direction. The curves of displacement versus load step are displayed in Fig. 12. The spring element was established at mid-span to simulate the influence of friction force, and the translational degree of the free end of the friction spring element was constrained. The force versus displacement curves of the friction spring element are shown in Fig. 13. The maximum displacement of the spring was set to a value sufficiently large (i.e., 1 m) to cover the real displacement, which occurs during the pre-stressing process of practical engineering. Numerical analysis was conducted on the basis of the premise above. The value and direction of the friction force during the loading process were derived. The change process of the friction force during the different load steps is depicted in Fig. 14. In this figure, the magnitude of the friction force is equal to Fmax, and its direction is consistently opposite the direction of movement. Therefore, the numerical method proposed in this work can exactly predict the value and direction of the friction force.
2'-x 5'-y 5'-x Z
Y X
Fig. 8. Internal nodes of connections.
connect nodes 1′-x, 1″-x, 1′-y, 1″-y, 2′-x, 2″-x, 2′-y, and 2″-y to the beam end nodes 3′-x, 3″-x, 3′-y, and 3″-y and allow them to deform with the rotation of the beam. The rigid beam elements are shown as bold lines in black and blue in Figs. 7 and 8. The friction spring element described in previous sections was established between nodes 4 and 4′-x, 4″-x, 4′y, and 4″-y, as well as between nodes 5 and 5′-x, 5″-x, 5′-y, and 5″-y to model the friction device. The friction device comes into play after gap opening occurs. The nodal coordinates of nodes with the same starting number are identical in the numerical model; for example, the coordinates of nodes 5 and 5′-x, 5″-x, 5′-y, and 5″-y are identical. The translational degree of freedom in the x direction and the rotational degree of the beam in the y direction are coupled together with the column nodes at the same location. Similarly, the translational degree of freedom in the y direction and the rotational degree of the beam in the x direction are coupled together with column nodes at the same location, as shown in Fig. 9. As the main objective of this study is to investigate the seismic performance of FDPT connections, the influence of shear force is ignored and the vertical displacements of nodes 3, 3′, and 3″ are coupled together. Dotted lines in Figs. 5 and 7 represent the outline of the actual section of a beam, and a mechanical model of the connections in one story is shown in Fig. 10.
3.2. Validation of PT connection model The author has investigated PT steel connections with bolted angles. This type of PT connection mainly dissipates seismic energy through angles. The friction spring element in this PT connection is replaced by angle as shown in Fig. 15. An FE specimen model (PC4) was extensively tested by Ricles et al. [33] and analyzed by Moradi and Alam [29] on the basis of the method presented in the previous section. The results derived in the present paper were compared with those of Moradi and Alam [29], as shown in Fig. 16. The simplified FE model of the PT connection can accurately predict the mechanical behavior of PT steel connections with bolted angles. 3.3. Validation of the FDPT connection An FE model of 2D FDPT connection specimens was established
Couple: Translational degree in y direction Rotational degree in x direction
4'-y 4''-x
4
4'-x
Couple: Translational degree in x direction Rotational degree in y direction
4''-y Planform Z
Y X
Fig. 9. Constraint relations among internal nodes of connections. 450
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Fig. 10. Mechanical model of connections in one story.
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Fig. 14. Change rule of the friction force during different load steps.
4''
Fig. 11. Schematic of the beam model. 1''
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Fig. 15. FE model of a PT connection with bolted angles.
-0.02
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Fig. 12. Displacement–load step curve.
I II
Moradi and Alam 2016 Presented method
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1m Displacement
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100
Fig. 16. Comparison of results derived in the present paper and those derived by Moradi and Alam [29].
-80N -100N
respectively. The cross-sectional area of each PT strand is reported to be 140 mm2. A bilinear elastoplastic behavior was selected as the stress–strain relationship for steel in the PT strand, beam, and column. The tangent modulus of steel used by the PT strand and that of the beam and column were set to 0.05E and 0.02E, respectively, as shown in Table 1. To predict yielding in the steel material, the von Mises yield
Fig. 13. Force–displacement curve of the spring element.
according to the PC4 system analyzed by Ricles [33] and Moradi [29], and the angles were replaced by the friction spring element. The sections of beam and columns were W24 × 62 and W14 × 311, 451
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1000
Table 1 Material properties of the PT steel frame.
800
Beam and column
Strand
600
Nominal yield stress [MPa] Elastic modulus [GPa] Tangent modulus
276 202 0.02E
1305 199 0.05E
400 Lateral load (KN)
Structural component
*Note: E indicates the modulus of elasticity.
Kim 2008 Presented method
200 0 -200 -400 -600 -800
-1000 -100 Fig. 17. Numerical model of the connection adopted by Kim [25].
3.4. Simulation of column failure The gravity load should be applied before dynamic analysis is performed, and the dynamic response of the structure must be kept to a minimum until it reaches a steady state. This effect is usually achieved by ensuring that the rise time is high enough to obtain the quasi-static response. Three loading steps were considered in the column failure analyses, as shown in Fig. 21. First, a gravity load and PT force were applied for 0.2 s, and the time integral effect was switched off by “timint, off;” this process can be regarded as a static analysis. During this time, the internal column, which is about to be removed, remains intact. Second, the time integral effect was switched on and the selected failure column was removed by “ekill.” This process was completed within 0.001 s, i.e., the reaction force of the failure column was removed within 0.001 s. Then, the remaining structure was analyzed for 1.2 s. The detailed location of the failure column is shown in Fig. 23. To predict and control the load step size increments, the automatic time stepping option is switched on in this study. The model stiffness is updated in ANSYS by Newton–Raphson equilibrium iterations, which default to 30 equilibrium equations. Geometric nonlinearity is considered by “nlgeom, 1.” The damping ratio is set to 0.02 in accordance with the existing literature. The convergence criteria in this study were based on the force, displacement, and tolerance limits; here, convergence criteria default values of 0.1% for force and moment checking and 5% for displacement checking were initially selected. Convergence was difficult to achieve using the default values due to the highly nonlinear behavior of the friction spring element. Thus, the convergence tolerance limits were increased to 1% for the moment checking criterion to achieve convergence of the equilibrium iterations.
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Displacement (mm)
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presented in this paper was compared with that observed in the experimental work conducted by Kim and Christopoulos [25], as shown in Fig. 20. The gap opening behavior is derived in the hysteresis analysis described above. This behavior could be exactly modeled by the simplified numerical model. The contact spring element can prevent invasion between beams and columns correctly.
150 100
0 50 Displacement (mm)
Fig. 19. Lateral load–displacement responses. Here, the FE results are compared with results derived by Kim [25].
criterion was employed. The behavior after the yield point follows kinematic strain hardening and the associated flow rule. Therefore, the Bauschinger effect was considered in the material models. As the column and beam are relatively elastic, the materials of the beam and column adopted in Sections 3 and 4 were assumed to be ideally elastic. One FDPT beam-column connection among those specimens tested by Kim and Christopoulos [25] was modeled to assess the accuracy of the results derived by the numerical model proposed in this paper. The numerical model of the adopted connection is shown in Fig. 17. The cyclic loading history is shown in Fig. 18. Fig. 19 shows the lateral load–displacement response derived by using the simplified numerical model alongside the corresponding FE result derived by Kim. The lateral displacement is the displacement at the point of load application on the column flange. The column and beams were W360 × 509 and W610 × 113 sections, respectively. The diameter of each PT strand is reported to be 32 mm. The initial PT force in an individual PT strand, Ti, is 200 kN, and the Fmax of the friction spring element is set to 280 kN. Other material properties were set accordingly. The time needed for one cycle of analysis is about 2 min. Thus, the computational cost is reduced significantly. The results shown in Fig. 19 reveal that the simplified numerical model agrees well with Kim’s model. The ultimate load predicted by the simplified numerical model showed marked similarity to the experimental results. The area of the hysteresis loop derived in this paper was identical to that derived by Kim. As local buckling or yielding cannot be exactly reflected by the simplified numerical model, the unloading stiffness derived by the proposed method exhibited some errors relative to that in experimental work. However, this parameter has a minimal influence on the seismic performance of connections. The gap opening behavior derived by the numerical model
100
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0
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3.5. Introduction and validation of the double element method
-80
-100
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-108
The double element method was adopted to estimate the influence of axial and bending stiffness on the dynamic response caused by column failure. This method assumes that every beam in the frame structure consists of two elements: a beam element with only bending stiffness and a beam element without bending stiffness (Fig. 22). The beam element without bending stiffness is realized by releasing the
-150 0
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Loading step Fig. 18. Cyclic loading history. 452
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a) Experiment (Kim and Christopoulos[25])
b) Numerical results of this paper
Fig. 20. Gap opening behavior.
Load Gravity load
0.2 0.201
1.2
t(s)
Reaction force of failure column Fig. 21. Loading scheme. Fig. 23. Location of the failure column.
Beam
element
structures. The load value should be adjusted to investigate the level of vertical displacements that develop. As such, the load factor (LF) should be increased in the load combination LF × (DL + 0.25LL), where DL is the dead load and LL is the live load. However, the floor slab is not established in this simplified model, and the live load cannot be directly applied. The dead and live loads of the floor slab will eventually transform into the line load on the beam. The present work is a qualitative research on the progressive collapse of FDPT frames. The magnitude of the vertical load was adjusted by the steel density of the columns. The symbol “R×” indicates the ratio of one parameter to its actual value. For example, the symbol “RL” indicates that the steel density of columns was enlarged by RL times. The steel density of the beams was kept constant at 7850 kg/m3. A comparison of results is shown in Fig. 24. The displacement–time curves derived by the double element method were in absolute agreement with those derived by Beam188. The vertical displacement of the collapsed columns increased with increasing vertical load. The structure is kept stable when RL is equal to 6 and 7, i.e., progressive collapse will not occur after bottom column failure. A critical state is observed when RL is equal to 8. Progressive collapse occurs when RL is equal to 9. Note that elastic analysis was conducted in this section as the double element in this paper cannot consider plasticity. The influence of material strength, initial PT force, friction force, and axial and bending stiffness of the beams on the dynamic response was investigated. The actual value of each parameter is shown in Table 2. When one parameter was adjusted in the parametrical analysis,
with
bending stiffness
Beam element without bending stiffness Fig. 22. Numerical model of a double element.
rotational degree at both ends and behave similar to a truss element. The sectional area of the beam element with bending stiffness is set to a small-enough value such that its contribution to the axial stiffness of a double element is negligible. The two elements share the same nodes at both ends; thus, their displacement vectors are equal. The beam element in the double element method exhibits bending stiffness, which can be easily adjusted to consider its influence. Similarly, the axial stiffness of the beam element without bending stiffness can be easily adjusted. More information about double elements can be found in Zhao et al.’s work [32,34,35]. The double element can only be adopted in elastic analysis. Results derived by the double element method and Beam188 in ANSYS were compared to validate the reliability of the double element in estimating the progressive collapse behavior of FDPT frame
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dynamic response. The FDPT frame structure was somewhat elastic when its yield strength was set to 500 MPa, and the structural response was nearly identical in both conditions. The vertical displacement reached 350 mm within 1 s, the PT force increased from 156 kN to 240 kN, and the bending moment of the beam increased to 2250 kN × m within 1 s after column failure. The vertical displacement of the FDPT frame was large when the yield strength of the beam was set to 276 MPa. The response of the PT force and bending moment decreased. The PT force was influenced to a larger degree compared with the bending moment. However, the yield strength did not influence the dynamic response (including displacement, PT force, and bending moment of the beam) at the initial state of column failure and exerted only a slight influence in the later period. The contour of stress of the FDPT frame approximately 1 s after column failure is shown in Fig. 27. Here, most of the top flange of the beam yielded, and the stress level of the bottom flange was low. A traditional frame structure was also analyzed to compare its dynamic response with that of the FDPT frame. A traditional frame structure is one where the steel columns and beams are rigidly connected and the PT strand and friction spring are removed. Other parameters, such as section and material, were kept identical to those in the FDPT frame. The vertical displacement and bending moment of the beam are shown in Fig. 28. Progressive collapse does not occur when RL is equal to 9. The dynamic response derived when the yield strength is 500 MPa is identical to that derived in the elastic analysis. A plastic hinge occurs at the end of the beam when the yield strength is 276 MPa, as shown in Fig. 28b, and the corresponding maximum displacement is 89.2 mm, which is far smaller than that obtained from the FDPT structure. The response of the TF frame showed remarkable vibrations due to its lack of an energy dissipation device. The ability of FDPT frame structures to resist progressive collapse could also be concluded to be far weaker than that of TF.
0 -50
-150 -200 -250 -300 -350 0.0
RL=9(Double element) RL=9(Beam188) RL=8(Double element) RL=8(Beam188) RL=7(Double element) RL=7(Beam188) RL=6(Double element) RL=6(Beam188)
0.2
0.4
0.6 Time (s)
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1.2
Fig. 24. Comparison of results derived by different models (Elastic analysis). Table 2 Actual value of each parameter. Item
Yield strength
Initial PT force
Friction force
Cross section area
Moment of inertia
Value
276 MPa
156 kN
223 kN
0.030503 m2
0.001877 m4
the other parameters were kept unchanged. Results derived from the double element (elastic analysis) and Beam188 (elastoplastic analysis) models were compared to investigate the application of the double element method since this method can only be adopted in elastic analysis, as shown in Fig. 25. The yield strength of steel was set to 276, 345, and 500 MPa. Results derived from both types of elements were in absolute agreement when RL is equal to 6 and 7. Results derived by the double element were slightly smaller than those obtained by Beam188 in the elastoplastic analysis when RL is equal to 8 and 9, likely because of the yield of the beam. This difference was reduced when the yield strength of steel was set to 345 and 500 MPa. The results here indicate that yielding of the beam has a slight influence on the dynamic response of the FDPT frame during column failure. Thus, the double element can be adopted to analyze FDPT frame structures and is adopted in Section 7 to investigate the influence of the axial and bending stiffness of beam on progressive collapse behavior of FDPT frame structures.
5. Influence of initial PT force The influence of initial PT force on the dynamic response caused by column failure was investigated. The yield force of the PT strand was 370 kN according to the parameters given in Tables 1 and 2. The initial PT force was set to 150, 300, and 370 kN. Dynamic responses corresponding to different initial PT forces are shown in Fig. 29. The initial PT force had a large influence on the vertical displacement response. As shown in Fig. 29c, the PT strand remained elastic during column failure when the initial PT force was set to 150 and 300 kN. The PT force changed from 151.5 kN to 168 kN and from 304 kN to 314 kN when the initial PT force was set to 150 and 300 kN, respectively. The PT strand came into the yield station at the initial stage when the initial PT force was set to 370 kN. In general, the peak bending moment decreased with increasing initial PT force. The initial PT force had a large influence on the vertical displacement, which increased with decreasing initial PT force.
4. Influence of material strength The influences of material strength on the dynamic response of the FDPT frame, including its vertical displacement, PT force, and bending moment of the beam, are investigated in this section. The dynamic responses corresponding to different material strengths are shown in Fig. 26. The yield strength of the material had a slight influence on the 50
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b) fy = 345 MPa Fig. 25. Comparison of results derived with different load grades. 454
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Fig. 26. Influence of material strength on dynamic responses (RL = 9, FDPT).
Fig. 27. Contour of stress with different material strengths (RL = 9, FDPT).
vertical displacement and PT force of the beam. As shown in Fig. 30, progressive collapse did not occur when Fmax = 200 kN; however, when Fmax = 150 kN, progressive collapse was observed. The peak bending moment was not influenced by Fmax. However, Fmax influenced the shock behavior of the frame during later periods. The PT force increased to the ultimate loading capacity within a very short time after collapse. For example, the PT force exceeded 270 kN approximately 0.82 s after column failure when RL = 9 and Fmax = 100 kN. The results thus far indicate that Fmax has a significant influence on the ability of the FDPT frame to resist continuous collapse. Thus, more attention should be given to this parameter during the structural design phase. Deformation at FDPT connections is shown as in Fig. 32. In addition, the gap opening behavior of the PT connection can be accurately captured.
Changes in PT force did not change very much because the axial deformation of the beam was limited. The deformation of the PT strand was also limited. As the vertical displacements of the beam end and column were coupled, the shear capacity of the FDPT connection could be assumed to be sufficiently large.
6. Influence of friction force The friction device in the FDPT frame has a great influence on the dynamic response because it is the main energy dissipation device. Thus, the influence of this device is investigated in this section. The maximum friction force was set to 100, 200, 223, 500, and 2230kN, and the RL was set to 7 and 9. Changes in the dynamic response with time are shown in Figs. 30 and 31. Fmax had a significant influence on the 50
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Fig. 28. Influence of material strength on dynamic responses (RL = 9, TF). 455
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Fig. 29. Influence of initial PT force on dynamic responses (RL = 7). 50
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Fig. 31. Influence of friction force on dynamic responses (RL = 9).
Fig. 32. Deformation at FDPT connections (RL = 7, Fmax = 100 kN).
7. Influence of axial and bending stiffness of beam
factor, which is used to adjust the axial stiffness of the steel beam, and is defined as the axial stiffness ratio of the double element and the beam element. Here, RA was set to 1.0, 0.8, and 0.6. Results derived from different conditions were compared, as shown in Figs. 33 and 34. As seen in the figures, the axial stiffness of the beam has a large influence on its displacement and PT force but only a minimal influence on its bending moment. The PT force in the stable state increased with
7.1. Influence of axial stiffness The axial stiffness of the double element was adjusted, and the bending stiffness was kept unchanged to investigate the influence of axial stiffness on the dynamic response. RA indicates the axial stiffness 456
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Fig. 33. Influence of axial stiffness on dynamic responses (RL = 7). 2500
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7.2. Influence of bending stiffness
decreasing RA, and the action of the PT strand was enhanced with the weakening of the beam. The influence of the axial stiffness of the beam on the collapse behavior of FDPT frame was not significant compared with that of Fmax as the axial force of the beam was transferred to the friction device. This behavior is very different from that in the TF structure.
The bending stiffness of the double element was adjusted, and the axial stiffness was kept unchanged to investigate the influence of bending stiffness on dynamic responses, as shown in Figs. 35 and 36. RB indicates the bending stiffness factor, which is used to adjust the bending stiffness of the steel beam, and is defined as the bending 457
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stiffness ratio of the double element and the beam element. RB was set to 1.0, 0.8 and 0.6. The results obtained show that the bending stiffness of the beam has a large influence on its displacement and PT force but only a slight influence on its bending. The bending stiffness of the beam would have a minimal influence on the collapse of the frame as a plastic hinge forms with increasing vertical load. Thus, the axial loading capacity of the beam was more determinant compared with bending capacity. The bending stiffness also influences the dynamic response during collapse.
[3] Izzuddin BA, Vlassis AG, Elghazouli AY, Nethercot DA. Progressive collapse of multi-storey buildings due to sudden column loss-part I: simplified assessment framework. Eng Struct 2008;30(5):1308–13. [4] Johnson ES, Meissner JE, Fahnestock LA. Experimental behaviour of a half-scale steel concrete composite floor system subjected to column removal scenarios. J Struct Eng 2016;142:04015133. [5] Donahue S, Hadjioannou M, Williamson EB, Engelhardt MD, Izzudin BA, Nethercot D, et al. Experimental evaluation of floor slab contribution in mitigating progressive collapse of steel structures. WIT Trans Built Environ 2013;67:547–55. [6] Yang B, Tan KH. Experimental tests of different types of bolted steel beam–column joints under a central-column-removal scenario. Eng Struct 2013;54(9):112–30. [7] Zhao Zhongwei, Jinjia Wu, Liu Haiqing, Liang Bing, Sun Qingwei. Numerical investigations on mechanical behavior of friction damped posttensioned steel connections. Arch Appl Mech 2018;88(12):2247–60. [8] Chen Zhihua, Li Hongxing, Zhao Zhongwei, Liu Haiqing. A simplified numerical model for posttensioned steel connections with bolted angles. Int J Steel Struct 2017;17(4):1487–2149. [9] Mohamed OA. Calculation of load increase factors for assessment of progressive collapse potential in framed steel structures. Case Stud Struct Eng 2015;3:11–8. [10] Mashhadi JJ, Saffari H. Dynamic increase factor based on residual strength to assess progressive collapse. Steel Comp Struct 2017;25(5):617–24. [11] Stoddart E, Byfield M, Davison B, Tyas A. Strain rate dependent component based connection modelling for use in non-linear dynamic progressive collapse analysis. Eng Struct 2013;55:35–43. [12] McKay A, Marchand K, Diaz M. Alternate path method in progressive collapse analysis: variation of dynamic and nonlinear load increase factors. Practice periodical on structural design and construction. ASCE 2012;17(4):152–60. [13] Liu M. A new dynamic increase factor for nonlinear static alternate path analysis of building frames against progressive collapse. Eng Struct 2013;48:666–73. [14] Ferraioli M. A modal pushdown procedure for progressive collapse analysis of steel frame structures. J Constr Steel Res 2019;156:227–41. [15] Kim T, Kim J, Park J. Investigation of progressive collapse-resisting capability of steel moment frames using push-down analysis. J Perform Constr Facil 2009;23(5):327–35. [16] Ferraioli M. Evaluation of dynamic increase factor in progressive collapse analysis of steel frame structures considering catenary action. Steel Comp Struct 2019;30(3):253–69. [17] Yang B, Tan KH. Experimental tests of different types of bolted steel beam–column joints under a central-column-removal scenario. Eng Struct 2013;54:112–30. [18] Zhang Ai-lin, Zhang Yan-xia, Li Rui, Wang Zong-yang. Cyclic behavior of a prefabricated self-centering beam–column connection with a bolted web friction device. Eng Struct 2016;111:185–98. [19] Saber Moradi, Shahria Alam M. Finite-element simulation of posttensioned steel connections with bolted angles under cyclic loading. J Struct Eng 2016;142(1). 04015075-1-04015075-15. [20] Rojas P, Ricles JM, Sause R. Seismic performance of post-tensioned steel moment resisting frames with friction devices. J Struct Eng 2005. [21] Herning G, Garlock MEM, Vanmarcke E. Reliability based evaluation of design and performance of steel self-centering moment frames. J Constr Steel Res 2011;67(10):1495–505. [22] Kim H-J, Christopoulos C. Numerical models and ductile ultimate deformation response of post-tensioned self-centering moment connections. Earthquake Eng Struct Dyn 2009;38(1):1–21. [23] Granello G, Leyder C, Palermo A, Frangi A, Pampanin S. Design approach to predict post-tensioning losses in post-tensioned timber frames. J Struct Eng 2018;144(8):04018115. [24] Garlock M, Ricles J, Sause R. Experimental studies of full-scale posttensioned steel connections. J Struct Eng 2005;131(3):438–48. [25] Kim H, Christopoulos C. Friction damped posttensioned self-centering steel moment-resisting frames. J Struct Eng 2008;134(11):1768–79. [26] Lin Y-C, Sause R, Ricles J. Seismic performance of a large-scale steel self-centering moment-resisting frame: MCE hybrid simulations and quasi-static pushover tests. J Struct Eng 2013;139(7):1227–36. [27] Angelos ST, Athanasios ID, Theodore LK. EC8-based seismic design and assessment of self-centering post-tensioned steel frames with viscous dampers. J Constr Steel Res 2015;105:60–73. [28] Christopoulos C, Filiatrault A, Uang CM, Folz B. Posttensioned energy dissipating connections for moment-resisting steel frames. J Struct Eng 2002;128(9):1111–20. [29] Moradi S, Alam MS. Finite-element simulation of posttensioned steel connections with bolted angles under cyclic loading. J Struct Eng 2016;142(1). 04015075-104015075-15. [30] Kajbaf AA, Fanaie N, Najarkolaie KF. Numerical simulation of failure in steel posttensioned connections under cyclic loading. Eng Fail Anal 2018;91:35–57. [31] ANSYS, ANSYS Customer Support, https://www.ANSYS.com, accessed 26 June 2017. [32] Zhao Z, Liu H, Liang B. Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity. Adv Eng Softw 2017;114:208–14. [33] Ricles JM, Sause R, Peng SW, Lu LW. Experimental evaluation of earthquake resistant posttensioned steel connections. J Struct Eng 2002;128(7):850–9. [34] Zhao Z, Chen Z, Yan X, Xu H, Zhao B. Simplified numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints. Adv Struct Eng 2016;19(4):689–702. [35] Zhao Z, Liu H, Liang B, Sun Q. Semi-rigid beam element model for progressive collapse analysis of steel frame structures. Proc Inst Civil Eng – Struct Build 2019;172(2):113–26.
8. Conclusions In this paper, a simplified numerical model for analyzing 3D friction damped PT steel connections was presented, and the progressive collapse of FDPT frame structures was systematically investigated using a 3D finite-element model of the FDPT frame structure. The influence of material strength, initial PT force, friction force, and axial and bending stiffness of beams was studied, and the dynamic responses observed were compared with those of TF. The conclusions can be summarized as follows: (1). The results derived by the simplified numerical model agree with those derived through experimental work. The accuracy of the proposed numerical mode was validated, and the parameter Fmax was found to determine the exact sliding behavior of the friction device. This parameter can be readily changed according to the real conditions of a project. (2). The yield strength of the material had a minimal influence on the dynamic response during column failure, which is very different from the behavior observed in TF. The response of TF showed large vibrations due to the lack of an energy dissipation device. Moreover, the capacity of the FDPT frame structures to resist progressive collapse was far lower than that of TF. (3). The maximum bending moment decreased with increasing initial PT force in general, and vertical displacement increased with decreasing initial PT force. The change in range of PT force was limited due the restriction of axial deformation of beams. (4). Fmax has a significant influence on the resistance of FDPT frames to continuous collapse. More attention should be given to this property during the structural design phase. (5). The axial loading capacity of beams was more determinant compared with their bending capacity. The influence of the axial stiffness of beams on the collapse behavior was not significant when compared with that of Fmax as the axial force of the beam was transferred to the friction device. This behavior is very different from that of TF. Declaration of Competing Interest The authors declare that they have no kNown competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was financially supported by the project funded by the China Postdoctoral Science Foundation (No. 2017M621156) and the Natural Science Foundation of Liaoning Province (No. 20180540144). References [1] Fu Qiu Ni, Tan Kang Hai, Zhou Xu Hong, Yang Bo. Numerical simulations on threedimensional composite structural systems against progressive collapse. J Constr Steel Res 2017;135:125–36. [2] Fu QN, Yang B, Xiong G, Zhang WF, Nie SD, Dai GX. Dynamic analyses of boltedangle steel joints against progressive collapse based on component-based model. J Constr Steel Res 2016;1172:161–74.
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Progressive collapse assessment of friction damped post-tensioned steel frames based on a simplified model
T
⁎
Zhongwei Zhao , Xiangyang Jian, Bing Liang, Haiqing Liu School of Civil Engineering, Liaoning Technical University, Fuxin 123000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: 3D-FDPT connections Friction element Simplified numerical model Progressive collapse Dynamic analysis
Post-tensioned energy-dissipating (PTED) connections for steel frames have drawn great research attention on account of their good seismic performance. Few studies focus on the progressive collapse of friction damped post-tensioned (FDPT) frame structures. The progressive collapse of FDPT frames exhibits a complex threedimensional mechanical behavior that is quite different from that of traditional frame structures. In this work, a simplified 3D PTED connection model was first established and then used to investigate the progressive collapse behavior of FDPT frames. The influences of material strength, initial PT force, friction force, and axial and bending stiffness of beams on the collapse behavior of FDPT were studied. Finally, the dynamic response of FDPT frames was compared with that of traditional frame structures (TF).
1. Introduction Progressive collapse of building structures can cause massive economic losses and many casualties [1]. Many studies [2–4] on the progressive collapse of MRF have been conducted, and these studies are usually based on different component levels, such as beam-to-column connections, 2D frames, 3D skeletal frames, and 3D frames with slab systems. Numerical and experimental methods are mainly used for analysis [5–8]. Dynamic effects are key factors influencing progressive collapse analysis. The existing literature indicates that rate effects could change the failure mechanism of joints and influence their collapse capacity. Many procedures and explicit expressions have been proposed to approximately compensate for dynamic effects when a static procedure is used [9,10]. Stoddart et al. [11], for example, introduced rate-dependent springs to component-based joint models. McKay [12] concluded that dynamic factors in existing guidelines tend to yield overly conservative results, which often translate to expensive designs and retrofit. Liu [13] presented an empirical method to calculate the dynamic increase factor used to amplify gravity loads on the affected bays of a building frame when nonlinear static alternate path analysis is carried out to predict the peak dynamic responses to sudden column removal. Ferraioli [14] proposed a modal pushdown analytical to assess the progressive collapse of multi-story steel-frame buildings under sudden removal of a column due to catastrophic events. The catenary action of the beams of traditional steel frames involves a load-carrying
⁎
mechanism that can potentially arrest the progressive collapse of steel structures [15–17]. However, the catenary action in posttensioned (PT) self-centering frames is considerably different from that in traditional frame structures due to self-centering connections. Several researchers [18–22] have confirmed the satisfactory seismic performance of PT self-centering beam–column connections. A method for estimating the influence of PT loss was previously proposed by Granello [23]. Garlock et al. [24] conducted experimental studies on full-scale PT steel connections and evaluated the effects of several parameters, including the initial posttensioning force, the number of posttensioning strands, and the length of the reinforcing plates. The results of this work demonstrated that PT connections allow good energy dissipation and ductility, which indicates that the connection strength in upper floors improves the seismic response of the frame. Kim and Christopoulos [25] proposed a new connection for steel moment-resisting frames incorporating posttensioning elements to provide self-centering capacity, along with friction mechanisms to dissipate energy. Lin et al. [26] investigated the seismic performance of a largescale steel self-centering moment-resisting frame. PT steel connections have been proposed to eliminate permanent deformations following severe earthquakes. Friction devices and viscous dampers [27] can be used to improve energy dissipation capacity. Christopoulos et al. [28] experimentally investigated PT energy-dissipating connections that dissipate input energy through the axial yielding of high-strength threaded bars with couplers welded to the top and bottom flanges of the beams. The seat angles were designed to yield
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.istruc.2019.09.005 Received 10 June 2019; Received in revised form 11 August 2019; Accepted 11 September 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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0.5
Force(KN)
to provide energy dissipation capacity in some PT connections [29]. The seismic performance of PT steel moment-resisting frames with friction devices was studied by Rojas et al. [20], who eventually confirmed that the variability of maximum friction forces developing in the friction devices does not have a significant effect on MRF performance. Although PT energy-dissipating (ED) connections have been proven to be a proper alternative to semi-rigid connections, no guidelines in the current seismic design codes [30] are yet available. Thus, further research studies are required to thoroughly investigate the application of these connections to buildings. Finite element (FE) analysis is a method that allows analysis of the behavior of PTED connections from different aspects that are difficult to capture in an experimental test. The entire FE model of PT steel frames is very limited. The progressive collapse of FDPT frames exhibits a complex 3D mechanical behavior that differs from that of traditional frame structures. The high-strength strands utilized in PTED connections is a key component of FTED. Progressive collapse may occur if the strands fail. However, research on the progressive collapse of FDPT frame cannot be found in the existing literature. Research on PT connections mainly focuses on seismic performance, and studies relevant to the progressive collapse of FDPT frame structures are scarce. The main purpose of this study is to provide a detailed FE modeling procedure that can accurately capture the progressive collapse behavior of FDPT connections under seismic loading. As the progressive collapse behavior of FDPT frame is very different from that of traditional frame structures, a simplified numerical model for 3D FDPT frames is presented.
0.4 0.3 0.2 0.1
-0.4
-0.2
0.0 0.0 -0.1
Deflection(m) 0.2
0.4
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Fig. 2. Force–deflection curve of the contact spring element.
was adopted to prohibit invasion between steel columns and beams (contact element) and model a friction device (friction element). The PT strand was modeled using the Link180 element. A negative temperature load was applied to Link180 to generate an initial tensioning force. Combin39 is a unidirectional element with nonlinear generalized force-deflection capability that can be used in any analysis. The element is defined by two (preferably coincident) node points and the force–deflection curve. This element has longitudinal or torsional capability in 1D, 2D, and 3D applications. The longitudinal option is a uniaxial tension-compression element with up to three degrees of freedom at each node, i.e., translations in the nodal x, y, and z directions. No bending or torsion was considered in the analysis [31,32]. As the function of the contact spring is to prohibit invasion between steel columns and beams, the stiffness under compression was set to a value large enough to make the invasion negligible. The stiffness under tension was set to 0. The force–deflection curve of Combin39 is shown in Fig. 2. The parameter required for the friction spring element is the maximum static friction, i.e., Fmax. The nonlinear friction element adopted should be able to capture the mechanical characteristics of friction, i.e., the friction value is
2. Establishing of simplified numerical model Prior research has indicated that the columns and beams of PT connection are elastic under the action of a bending moment. The energy input is mainly dissipated by the friction device. Therefore, the beam element (Beam188) in ANSYS was used to model steel beams and columns in a PT connection. The simplified mechanical model of a PT connection is shown in Fig. 1. The nonlinear spring element Combin39
Fig. 1. Simplified mechanical model of a 2D PT connection. 448
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F 4''
I 4 II
4'
H 1''
Fmax
1
1'
3'' 3 3'
6 2''
Rigid beam
Rigid beam
9 7
7 8
2'
2
11
10
Displacement
5''
IV 5 III
5'
-Fmax
Fig. 5. Constraint relation among internal nodes (2D). Fig. 3. Mathematical model of sliding friction.
4
relevant to the frictional coefficient and normal force, and the direction of friction depends on the relative motion tendency. The static friction force can be determined through the equilibrium condition of forces. If the internal force is lower than Fmax, two nodes of the friction element remain relatively static. The internal force can be derived from the unbalanced forces at the two nodes. A mathematical model of sliding friction force is shown in Fig. 3. Sliding occurs when the static friction force reaches Fmax (or −Fmax), which is a constant value under a certain normal force. When the static friction force reaches Fmax, the cable will slide in the positive direction, such as point 1. By contrast, the cable will slide in the negative direction when the static friction force reaches −Fmax, such as point 6. The static friction force varies between Fmax and −Fmax when the cable is static relative to the joints. ANSYS was adopted to simulate the friction force, and the nonlinear spring element was utilized to model the friction force between the cable and the joint. The force–displacement curves of the COMBIN39 element are shown in Fig. 4. The angle θ is set to approximately 90° to allow the mechanical behavior of the nonlinear spring element to precisely model the sliding friction. The KEYOPT (1) of COMBIN39 was set to 1, so that the spring will unload along the line parallel to the slope at the origin of the loading curve. The mechanical model of the 2D-FDPT connection and its deformation under a bending moment are shown in Figs. 5 and 6, respectively. The distance between the friction device (or friction force) and the beam edge is indicated by H. The rigid beam element MPC184 was adopted to connect. Nodes 3′, 7, 3″, and 6 indicate nodes establishing the beam element. Node 3 is attached to the steel column and shares the same location as 3′and 3″. Nodes 8, 9, 10, and 11, which are located at the end of the high-strength strand, are also connected to the side end of the beam, i.e., 7, by a rigid beam element. A 3D-FDPT connection was established based on the 2D-FDPT connection. The constraint relation among internal nodes of the simplified 3D-FDPT connection are shown in Fig. 7. All of the nodes are in the vertical middle section of the column in the x or y directions. Nodes 1′-x, 1″-x, 1′-y, 1″-y, 2′-x, 2″-x, 2′-y, and 2″-y indicate the top and
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Fig. 6. Deformation under a bending moment.
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Fig. 7. Mechanical model of a 3D FDPT connection.
bottom contact points of the beam to the column. Symbols “-x” and “-y” indicate nodes attached to beams in the x and y directions, respectively. Nodes 1 and 2 indicate the corresponding locations on the steel column. Nodes 4 (4′-x, 4″-x, 4′-y, and 4″-y) and 5 (5′-x, 5″-x, 5′-y, and 5″-y) indicate the location of the friction device. MPC184 was adopted to
F
F
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Displacement
Displacement
Fig. 4. Force–displacement curves of COMBIN39. 449
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3.1. Validation of the friction spring element
1'-y 1'-x 3'-y 3'-x 2'-y
The friction spring element is a key component in the simplified FDPT connection model, and its accuracy directly determines the seismic response of the overall structure. Thus, the accuracy and applicability of the friction spring element are validated in this section. A numerical test was conducted based on a numerical beam model to validate the accuracy of the friction spring element, as depicted in Fig. 11. The beam model was established by a planar beam element in ANSYS. The horizontal degree of the beam element was constrained, and a cyclic displacement was applied in the vertical direction. The curves of displacement versus load step are displayed in Fig. 12. The spring element was established at mid-span to simulate the influence of friction force, and the translational degree of the free end of the friction spring element was constrained. The force versus displacement curves of the friction spring element are shown in Fig. 13. The maximum displacement of the spring was set to a value sufficiently large (i.e., 1 m) to cover the real displacement, which occurs during the pre-stressing process of practical engineering. Numerical analysis was conducted on the basis of the premise above. The value and direction of the friction force during the loading process were derived. The change process of the friction force during the different load steps is depicted in Fig. 14. In this figure, the magnitude of the friction force is equal to Fmax, and its direction is consistently opposite the direction of movement. Therefore, the numerical method proposed in this work can exactly predict the value and direction of the friction force.
2'-x 5'-y 5'-x Z
Y X
Fig. 8. Internal nodes of connections.
connect nodes 1′-x, 1″-x, 1′-y, 1″-y, 2′-x, 2″-x, 2′-y, and 2″-y to the beam end nodes 3′-x, 3″-x, 3′-y, and 3″-y and allow them to deform with the rotation of the beam. The rigid beam elements are shown as bold lines in black and blue in Figs. 7 and 8. The friction spring element described in previous sections was established between nodes 4 and 4′-x, 4″-x, 4′y, and 4″-y, as well as between nodes 5 and 5′-x, 5″-x, 5′-y, and 5″-y to model the friction device. The friction device comes into play after gap opening occurs. The nodal coordinates of nodes with the same starting number are identical in the numerical model; for example, the coordinates of nodes 5 and 5′-x, 5″-x, 5′-y, and 5″-y are identical. The translational degree of freedom in the x direction and the rotational degree of the beam in the y direction are coupled together with the column nodes at the same location. Similarly, the translational degree of freedom in the y direction and the rotational degree of the beam in the x direction are coupled together with column nodes at the same location, as shown in Fig. 9. As the main objective of this study is to investigate the seismic performance of FDPT connections, the influence of shear force is ignored and the vertical displacements of nodes 3, 3′, and 3″ are coupled together. Dotted lines in Figs. 5 and 7 represent the outline of the actual section of a beam, and a mechanical model of the connections in one story is shown in Fig. 10.
3.2. Validation of PT connection model The author has investigated PT steel connections with bolted angles. This type of PT connection mainly dissipates seismic energy through angles. The friction spring element in this PT connection is replaced by angle as shown in Fig. 15. An FE specimen model (PC4) was extensively tested by Ricles et al. [33] and analyzed by Moradi and Alam [29] on the basis of the method presented in the previous section. The results derived in the present paper were compared with those of Moradi and Alam [29], as shown in Fig. 16. The simplified FE model of the PT connection can accurately predict the mechanical behavior of PT steel connections with bolted angles. 3.3. Validation of the FDPT connection An FE model of 2D FDPT connection specimens was established
Couple: Translational degree in y direction Rotational degree in x direction
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Fig. 9. Constraint relations among internal nodes of connections. 450
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Fig. 14. Change rule of the friction force during different load steps.
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Fig. 11. Schematic of the beam model. 1''
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Fig. 15. FE model of a PT connection with bolted angles.
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Fig. 12. Displacement–load step curve.
I II
Moradi and Alam 2016 Presented method
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Fig. 16. Comparison of results derived in the present paper and those derived by Moradi and Alam [29].
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respectively. The cross-sectional area of each PT strand is reported to be 140 mm2. A bilinear elastoplastic behavior was selected as the stress–strain relationship for steel in the PT strand, beam, and column. The tangent modulus of steel used by the PT strand and that of the beam and column were set to 0.05E and 0.02E, respectively, as shown in Table 1. To predict yielding in the steel material, the von Mises yield
Fig. 13. Force–displacement curve of the spring element.
according to the PC4 system analyzed by Ricles [33] and Moradi [29], and the angles were replaced by the friction spring element. The sections of beam and columns were W24 × 62 and W14 × 311, 451
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Table 1 Material properties of the PT steel frame.
800
Beam and column
Strand
600
Nominal yield stress [MPa] Elastic modulus [GPa] Tangent modulus
276 202 0.02E
1305 199 0.05E
400 Lateral load (KN)
Structural component
*Note: E indicates the modulus of elasticity.
Kim 2008 Presented method
200 0 -200 -400 -600 -800
-1000 -100 Fig. 17. Numerical model of the connection adopted by Kim [25].
3.4. Simulation of column failure The gravity load should be applied before dynamic analysis is performed, and the dynamic response of the structure must be kept to a minimum until it reaches a steady state. This effect is usually achieved by ensuring that the rise time is high enough to obtain the quasi-static response. Three loading steps were considered in the column failure analyses, as shown in Fig. 21. First, a gravity load and PT force were applied for 0.2 s, and the time integral effect was switched off by “timint, off;” this process can be regarded as a static analysis. During this time, the internal column, which is about to be removed, remains intact. Second, the time integral effect was switched on and the selected failure column was removed by “ekill.” This process was completed within 0.001 s, i.e., the reaction force of the failure column was removed within 0.001 s. Then, the remaining structure was analyzed for 1.2 s. The detailed location of the failure column is shown in Fig. 23. To predict and control the load step size increments, the automatic time stepping option is switched on in this study. The model stiffness is updated in ANSYS by Newton–Raphson equilibrium iterations, which default to 30 equilibrium equations. Geometric nonlinearity is considered by “nlgeom, 1.” The damping ratio is set to 0.02 in accordance with the existing literature. The convergence criteria in this study were based on the force, displacement, and tolerance limits; here, convergence criteria default values of 0.1% for force and moment checking and 5% for displacement checking were initially selected. Convergence was difficult to achieve using the default values due to the highly nonlinear behavior of the friction spring element. Thus, the convergence tolerance limits were increased to 1% for the moment checking criterion to achieve convergence of the equilibrium iterations.
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presented in this paper was compared with that observed in the experimental work conducted by Kim and Christopoulos [25], as shown in Fig. 20. The gap opening behavior is derived in the hysteresis analysis described above. This behavior could be exactly modeled by the simplified numerical model. The contact spring element can prevent invasion between beams and columns correctly.
150 100
0 50 Displacement (mm)
Fig. 19. Lateral load–displacement responses. Here, the FE results are compared with results derived by Kim [25].
criterion was employed. The behavior after the yield point follows kinematic strain hardening and the associated flow rule. Therefore, the Bauschinger effect was considered in the material models. As the column and beam are relatively elastic, the materials of the beam and column adopted in Sections 3 and 4 were assumed to be ideally elastic. One FDPT beam-column connection among those specimens tested by Kim and Christopoulos [25] was modeled to assess the accuracy of the results derived by the numerical model proposed in this paper. The numerical model of the adopted connection is shown in Fig. 17. The cyclic loading history is shown in Fig. 18. Fig. 19 shows the lateral load–displacement response derived by using the simplified numerical model alongside the corresponding FE result derived by Kim. The lateral displacement is the displacement at the point of load application on the column flange. The column and beams were W360 × 509 and W610 × 113 sections, respectively. The diameter of each PT strand is reported to be 32 mm. The initial PT force in an individual PT strand, Ti, is 200 kN, and the Fmax of the friction spring element is set to 280 kN. Other material properties were set accordingly. The time needed for one cycle of analysis is about 2 min. Thus, the computational cost is reduced significantly. The results shown in Fig. 19 reveal that the simplified numerical model agrees well with Kim’s model. The ultimate load predicted by the simplified numerical model showed marked similarity to the experimental results. The area of the hysteresis loop derived in this paper was identical to that derived by Kim. As local buckling or yielding cannot be exactly reflected by the simplified numerical model, the unloading stiffness derived by the proposed method exhibited some errors relative to that in experimental work. However, this parameter has a minimal influence on the seismic performance of connections. The gap opening behavior derived by the numerical model
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3.5. Introduction and validation of the double element method
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The double element method was adopted to estimate the influence of axial and bending stiffness on the dynamic response caused by column failure. This method assumes that every beam in the frame structure consists of two elements: a beam element with only bending stiffness and a beam element without bending stiffness (Fig. 22). The beam element without bending stiffness is realized by releasing the
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a) Experiment (Kim and Christopoulos[25])
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Fig. 20. Gap opening behavior.
Load Gravity load
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Reaction force of failure column Fig. 21. Loading scheme. Fig. 23. Location of the failure column.
Beam
element
structures. The load value should be adjusted to investigate the level of vertical displacements that develop. As such, the load factor (LF) should be increased in the load combination LF × (DL + 0.25LL), where DL is the dead load and LL is the live load. However, the floor slab is not established in this simplified model, and the live load cannot be directly applied. The dead and live loads of the floor slab will eventually transform into the line load on the beam. The present work is a qualitative research on the progressive collapse of FDPT frames. The magnitude of the vertical load was adjusted by the steel density of the columns. The symbol “R×” indicates the ratio of one parameter to its actual value. For example, the symbol “RL” indicates that the steel density of columns was enlarged by RL times. The steel density of the beams was kept constant at 7850 kg/m3. A comparison of results is shown in Fig. 24. The displacement–time curves derived by the double element method were in absolute agreement with those derived by Beam188. The vertical displacement of the collapsed columns increased with increasing vertical load. The structure is kept stable when RL is equal to 6 and 7, i.e., progressive collapse will not occur after bottom column failure. A critical state is observed when RL is equal to 8. Progressive collapse occurs when RL is equal to 9. Note that elastic analysis was conducted in this section as the double element in this paper cannot consider plasticity. The influence of material strength, initial PT force, friction force, and axial and bending stiffness of the beams on the dynamic response was investigated. The actual value of each parameter is shown in Table 2. When one parameter was adjusted in the parametrical analysis,
with
bending stiffness
Beam element without bending stiffness Fig. 22. Numerical model of a double element.
rotational degree at both ends and behave similar to a truss element. The sectional area of the beam element with bending stiffness is set to a small-enough value such that its contribution to the axial stiffness of a double element is negligible. The two elements share the same nodes at both ends; thus, their displacement vectors are equal. The beam element in the double element method exhibits bending stiffness, which can be easily adjusted to consider its influence. Similarly, the axial stiffness of the beam element without bending stiffness can be easily adjusted. More information about double elements can be found in Zhao et al.’s work [32,34,35]. The double element can only be adopted in elastic analysis. Results derived by the double element method and Beam188 in ANSYS were compared to validate the reliability of the double element in estimating the progressive collapse behavior of FDPT frame
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dynamic response. The FDPT frame structure was somewhat elastic when its yield strength was set to 500 MPa, and the structural response was nearly identical in both conditions. The vertical displacement reached 350 mm within 1 s, the PT force increased from 156 kN to 240 kN, and the bending moment of the beam increased to 2250 kN × m within 1 s after column failure. The vertical displacement of the FDPT frame was large when the yield strength of the beam was set to 276 MPa. The response of the PT force and bending moment decreased. The PT force was influenced to a larger degree compared with the bending moment. However, the yield strength did not influence the dynamic response (including displacement, PT force, and bending moment of the beam) at the initial state of column failure and exerted only a slight influence in the later period. The contour of stress of the FDPT frame approximately 1 s after column failure is shown in Fig. 27. Here, most of the top flange of the beam yielded, and the stress level of the bottom flange was low. A traditional frame structure was also analyzed to compare its dynamic response with that of the FDPT frame. A traditional frame structure is one where the steel columns and beams are rigidly connected and the PT strand and friction spring are removed. Other parameters, such as section and material, were kept identical to those in the FDPT frame. The vertical displacement and bending moment of the beam are shown in Fig. 28. Progressive collapse does not occur when RL is equal to 9. The dynamic response derived when the yield strength is 500 MPa is identical to that derived in the elastic analysis. A plastic hinge occurs at the end of the beam when the yield strength is 276 MPa, as shown in Fig. 28b, and the corresponding maximum displacement is 89.2 mm, which is far smaller than that obtained from the FDPT structure. The response of the TF frame showed remarkable vibrations due to its lack of an energy dissipation device. The ability of FDPT frame structures to resist progressive collapse could also be concluded to be far weaker than that of TF.
0 -50
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0.2
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1.2
Fig. 24. Comparison of results derived by different models (Elastic analysis). Table 2 Actual value of each parameter. Item
Yield strength
Initial PT force
Friction force
Cross section area
Moment of inertia
Value
276 MPa
156 kN
223 kN
0.030503 m2
0.001877 m4
the other parameters were kept unchanged. Results derived from the double element (elastic analysis) and Beam188 (elastoplastic analysis) models were compared to investigate the application of the double element method since this method can only be adopted in elastic analysis, as shown in Fig. 25. The yield strength of steel was set to 276, 345, and 500 MPa. Results derived from both types of elements were in absolute agreement when RL is equal to 6 and 7. Results derived by the double element were slightly smaller than those obtained by Beam188 in the elastoplastic analysis when RL is equal to 8 and 9, likely because of the yield of the beam. This difference was reduced when the yield strength of steel was set to 345 and 500 MPa. The results here indicate that yielding of the beam has a slight influence on the dynamic response of the FDPT frame during column failure. Thus, the double element can be adopted to analyze FDPT frame structures and is adopted in Section 7 to investigate the influence of the axial and bending stiffness of beam on progressive collapse behavior of FDPT frame structures.
5. Influence of initial PT force The influence of initial PT force on the dynamic response caused by column failure was investigated. The yield force of the PT strand was 370 kN according to the parameters given in Tables 1 and 2. The initial PT force was set to 150, 300, and 370 kN. Dynamic responses corresponding to different initial PT forces are shown in Fig. 29. The initial PT force had a large influence on the vertical displacement response. As shown in Fig. 29c, the PT strand remained elastic during column failure when the initial PT force was set to 150 and 300 kN. The PT force changed from 151.5 kN to 168 kN and from 304 kN to 314 kN when the initial PT force was set to 150 and 300 kN, respectively. The PT strand came into the yield station at the initial stage when the initial PT force was set to 370 kN. In general, the peak bending moment decreased with increasing initial PT force. The initial PT force had a large influence on the vertical displacement, which increased with decreasing initial PT force.
4. Influence of material strength The influences of material strength on the dynamic response of the FDPT frame, including its vertical displacement, PT force, and bending moment of the beam, are investigated in this section. The dynamic responses corresponding to different material strengths are shown in Fig. 26. The yield strength of the material had a slight influence on the 50
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Fig. 26. Influence of material strength on dynamic responses (RL = 9, FDPT).
Fig. 27. Contour of stress with different material strengths (RL = 9, FDPT).
vertical displacement and PT force of the beam. As shown in Fig. 30, progressive collapse did not occur when Fmax = 200 kN; however, when Fmax = 150 kN, progressive collapse was observed. The peak bending moment was not influenced by Fmax. However, Fmax influenced the shock behavior of the frame during later periods. The PT force increased to the ultimate loading capacity within a very short time after collapse. For example, the PT force exceeded 270 kN approximately 0.82 s after column failure when RL = 9 and Fmax = 100 kN. The results thus far indicate that Fmax has a significant influence on the ability of the FDPT frame to resist continuous collapse. Thus, more attention should be given to this parameter during the structural design phase. Deformation at FDPT connections is shown as in Fig. 32. In addition, the gap opening behavior of the PT connection can be accurately captured.
Changes in PT force did not change very much because the axial deformation of the beam was limited. The deformation of the PT strand was also limited. As the vertical displacements of the beam end and column were coupled, the shear capacity of the FDPT connection could be assumed to be sufficiently large.
6. Influence of friction force The friction device in the FDPT frame has a great influence on the dynamic response because it is the main energy dissipation device. Thus, the influence of this device is investigated in this section. The maximum friction force was set to 100, 200, 223, 500, and 2230kN, and the RL was set to 7 and 9. Changes in the dynamic response with time are shown in Figs. 30 and 31. Fmax had a significant influence on the 50
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Fig. 28. Influence of material strength on dynamic responses (RL = 9, TF). 455
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Fig. 31. Influence of friction force on dynamic responses (RL = 9).
Fig. 32. Deformation at FDPT connections (RL = 7, Fmax = 100 kN).
7. Influence of axial and bending stiffness of beam
factor, which is used to adjust the axial stiffness of the steel beam, and is defined as the axial stiffness ratio of the double element and the beam element. Here, RA was set to 1.0, 0.8, and 0.6. Results derived from different conditions were compared, as shown in Figs. 33 and 34. As seen in the figures, the axial stiffness of the beam has a large influence on its displacement and PT force but only a minimal influence on its bending moment. The PT force in the stable state increased with
7.1. Influence of axial stiffness The axial stiffness of the double element was adjusted, and the bending stiffness was kept unchanged to investigate the influence of axial stiffness on the dynamic response. RA indicates the axial stiffness 456
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Uz (mm)
-100 -150 RA=0.8 -200
RA=0.6
-250 -300 -350 0.0
0.2
0.4
0.6 Time (s)
0.8
1.0
180 RA=1.0
2000 170 PT force (kN)
50
1500
1000
RA=0.6 160
150
RA=1.0
500
RA=0.8
RA=0.8 0 0.0
1.2
a) Vertical displacement
RA=0.6 0.2
0.4
0.6 Time (s)
0.8
1.0
140 0.0
1.2
0.2
b) Bending moment
0.4
0.6 Time (s)
0.8
1.0
1.2
1.0
1.2
1.0
1.2
1.0
1.2
c) PT force
Fig. 33. Influence of axial stiffness on dynamic responses (RL = 7). 2500
50
240
-50
Uz (mm)
-100 -150 RA=1.0
-200 -250
RA=0.8
-300 -350 0.0
2000
1500
1000
0.4
0.6 Time (s)
0.8
1.0
RA=1.0 RA=0.8
500
RA=0.6 0.2
PT force (kN)
Bending Moment (kN×m)
0
0 0.0
1.2
a) Vertical displacement
210 RA=1.0 RA=0.8 180
RA=0.6
RA=0.6
0.2
0.4
0.6 Time (s)
0.8
1.0
150 0.0
1.2
0.2
b) Bending moment
0.4
0.6 Time (s)
0.8
c) PT force
Fig. 34. Influence of axial stiffness on dynamic responses (RL = 9). 2500
0
Bending Moment (kN×m)
RB=1.0
-50
RB=0.6
Uz (mm)
-100 -150 RB=0.8 -200 -250 -300 -350 0.0
0.2
0.4
0.6 Time (s)
0.8
1.0
180 RB=1.0
2000
RB=0.8 170 PT force (kN)
50
1500
1000
160
RB=1.0
500
RB=0.8 0 0.0
1.2
RB=0.6
a) Vertical displacement
RB=0.6 0.2
0.4
0.6 Time (s)
0.8
1.0
150 0.0
1.2
0.2
b) Bending moment
0.4
0.6 Time (s)
0.8
c) PT force
Fig. 35. Influence of bending stiffness on dynamic responses (RL = 7). 2500
50
240
Uz (mm)
-100 -150 RB=1.0
-200
RB=0.8
-250 -300 -350 0.0
RB=0.6 0.2
0.4
0.6 Time (s)
0.8
1.0
a) Vertical displacement
1.2
2000
PT force (kN)
-50
Bending Moment (kN×m)
0
1500
1000
RB=1.0 RB=0.8
RB=1.0 RB=0.8 180
RB=0.6
RB=0.6
500
0 0.0
210
0.2
0.4
0.6 Time (s)
0.8
1.0
1.2
150 0.0
b) Bending moment
0.2
0.4
0.6 Time (s)
0.8
c) PT force
Fig. 36. Influence of bending stiffness on dynamic responses (RL = 9).
7.2. Influence of bending stiffness
decreasing RA, and the action of the PT strand was enhanced with the weakening of the beam. The influence of the axial stiffness of the beam on the collapse behavior of FDPT frame was not significant compared with that of Fmax as the axial force of the beam was transferred to the friction device. This behavior is very different from that in the TF structure.
The bending stiffness of the double element was adjusted, and the axial stiffness was kept unchanged to investigate the influence of bending stiffness on dynamic responses, as shown in Figs. 35 and 36. RB indicates the bending stiffness factor, which is used to adjust the bending stiffness of the steel beam, and is defined as the bending 457
Structures 23 (2020) 447–458
Z. Zhao, et al.
stiffness ratio of the double element and the beam element. RB was set to 1.0, 0.8 and 0.6. The results obtained show that the bending stiffness of the beam has a large influence on its displacement and PT force but only a slight influence on its bending. The bending stiffness of the beam would have a minimal influence on the collapse of the frame as a plastic hinge forms with increasing vertical load. Thus, the axial loading capacity of the beam was more determinant compared with bending capacity. The bending stiffness also influences the dynamic response during collapse.
[3] Izzuddin BA, Vlassis AG, Elghazouli AY, Nethercot DA. Progressive collapse of multi-storey buildings due to sudden column loss-part I: simplified assessment framework. Eng Struct 2008;30(5):1308–13. [4] Johnson ES, Meissner JE, Fahnestock LA. Experimental behaviour of a half-scale steel concrete composite floor system subjected to column removal scenarios. J Struct Eng 2016;142:04015133. [5] Donahue S, Hadjioannou M, Williamson EB, Engelhardt MD, Izzudin BA, Nethercot D, et al. Experimental evaluation of floor slab contribution in mitigating progressive collapse of steel structures. WIT Trans Built Environ 2013;67:547–55. [6] Yang B, Tan KH. Experimental tests of different types of bolted steel beam–column joints under a central-column-removal scenario. Eng Struct 2013;54(9):112–30. [7] Zhao Zhongwei, Jinjia Wu, Liu Haiqing, Liang Bing, Sun Qingwei. Numerical investigations on mechanical behavior of friction damped posttensioned steel connections. Arch Appl Mech 2018;88(12):2247–60. [8] Chen Zhihua, Li Hongxing, Zhao Zhongwei, Liu Haiqing. A simplified numerical model for posttensioned steel connections with bolted angles. Int J Steel Struct 2017;17(4):1487–2149. [9] Mohamed OA. Calculation of load increase factors for assessment of progressive collapse potential in framed steel structures. Case Stud Struct Eng 2015;3:11–8. [10] Mashhadi JJ, Saffari H. Dynamic increase factor based on residual strength to assess progressive collapse. Steel Comp Struct 2017;25(5):617–24. [11] Stoddart E, Byfield M, Davison B, Tyas A. Strain rate dependent component based connection modelling for use in non-linear dynamic progressive collapse analysis. Eng Struct 2013;55:35–43. [12] McKay A, Marchand K, Diaz M. Alternate path method in progressive collapse analysis: variation of dynamic and nonlinear load increase factors. Practice periodical on structural design and construction. ASCE 2012;17(4):152–60. [13] Liu M. A new dynamic increase factor for nonlinear static alternate path analysis of building frames against progressive collapse. Eng Struct 2013;48:666–73. [14] Ferraioli M. A modal pushdown procedure for progressive collapse analysis of steel frame structures. J Constr Steel Res 2019;156:227–41. [15] Kim T, Kim J, Park J. Investigation of progressive collapse-resisting capability of steel moment frames using push-down analysis. J Perform Constr Facil 2009;23(5):327–35. [16] Ferraioli M. Evaluation of dynamic increase factor in progressive collapse analysis of steel frame structures considering catenary action. Steel Comp Struct 2019;30(3):253–69. [17] Yang B, Tan KH. Experimental tests of different types of bolted steel beam–column joints under a central-column-removal scenario. Eng Struct 2013;54:112–30. [18] Zhang Ai-lin, Zhang Yan-xia, Li Rui, Wang Zong-yang. Cyclic behavior of a prefabricated self-centering beam–column connection with a bolted web friction device. Eng Struct 2016;111:185–98. [19] Saber Moradi, Shahria Alam M. Finite-element simulation of posttensioned steel connections with bolted angles under cyclic loading. J Struct Eng 2016;142(1). 04015075-1-04015075-15. [20] Rojas P, Ricles JM, Sause R. Seismic performance of post-tensioned steel moment resisting frames with friction devices. J Struct Eng 2005. [21] Herning G, Garlock MEM, Vanmarcke E. Reliability based evaluation of design and performance of steel self-centering moment frames. J Constr Steel Res 2011;67(10):1495–505. [22] Kim H-J, Christopoulos C. Numerical models and ductile ultimate deformation response of post-tensioned self-centering moment connections. Earthquake Eng Struct Dyn 2009;38(1):1–21. [23] Granello G, Leyder C, Palermo A, Frangi A, Pampanin S. Design approach to predict post-tensioning losses in post-tensioned timber frames. J Struct Eng 2018;144(8):04018115. [24] Garlock M, Ricles J, Sause R. Experimental studies of full-scale posttensioned steel connections. J Struct Eng 2005;131(3):438–48. [25] Kim H, Christopoulos C. Friction damped posttensioned self-centering steel moment-resisting frames. J Struct Eng 2008;134(11):1768–79. [26] Lin Y-C, Sause R, Ricles J. Seismic performance of a large-scale steel self-centering moment-resisting frame: MCE hybrid simulations and quasi-static pushover tests. J Struct Eng 2013;139(7):1227–36. [27] Angelos ST, Athanasios ID, Theodore LK. EC8-based seismic design and assessment of self-centering post-tensioned steel frames with viscous dampers. J Constr Steel Res 2015;105:60–73. [28] Christopoulos C, Filiatrault A, Uang CM, Folz B. Posttensioned energy dissipating connections for moment-resisting steel frames. J Struct Eng 2002;128(9):1111–20. [29] Moradi S, Alam MS. Finite-element simulation of posttensioned steel connections with bolted angles under cyclic loading. J Struct Eng 2016;142(1). 04015075-104015075-15. [30] Kajbaf AA, Fanaie N, Najarkolaie KF. Numerical simulation of failure in steel posttensioned connections under cyclic loading. Eng Fail Anal 2018;91:35–57. [31] ANSYS, ANSYS Customer Support, https://www.ANSYS.com, accessed 26 June 2017. [32] Zhao Z, Liu H, Liang B. Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity. Adv Eng Softw 2017;114:208–14. [33] Ricles JM, Sause R, Peng SW, Lu LW. Experimental evaluation of earthquake resistant posttensioned steel connections. J Struct Eng 2002;128(7):850–9. [34] Zhao Z, Chen Z, Yan X, Xu H, Zhao B. Simplified numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints. Adv Struct Eng 2016;19(4):689–702. [35] Zhao Z, Liu H, Liang B, Sun Q. Semi-rigid beam element model for progressive collapse analysis of steel frame structures. Proc Inst Civil Eng – Struct Build 2019;172(2):113–26.
8. Conclusions In this paper, a simplified numerical model for analyzing 3D friction damped PT steel connections was presented, and the progressive collapse of FDPT frame structures was systematically investigated using a 3D finite-element model of the FDPT frame structure. The influence of material strength, initial PT force, friction force, and axial and bending stiffness of beams was studied, and the dynamic responses observed were compared with those of TF. The conclusions can be summarized as follows: (1). The results derived by the simplified numerical model agree with those derived through experimental work. The accuracy of the proposed numerical mode was validated, and the parameter Fmax was found to determine the exact sliding behavior of the friction device. This parameter can be readily changed according to the real conditions of a project. (2). The yield strength of the material had a minimal influence on the dynamic response during column failure, which is very different from the behavior observed in TF. The response of TF showed large vibrations due to the lack of an energy dissipation device. Moreover, the capacity of the FDPT frame structures to resist progressive collapse was far lower than that of TF. (3). The maximum bending moment decreased with increasing initial PT force in general, and vertical displacement increased with decreasing initial PT force. The change in range of PT force was limited due the restriction of axial deformation of beams. (4). Fmax has a significant influence on the resistance of FDPT frames to continuous collapse. More attention should be given to this property during the structural design phase. (5). The axial loading capacity of beams was more determinant compared with their bending capacity. The influence of the axial stiffness of beams on the collapse behavior was not significant when compared with that of Fmax as the axial force of the beam was transferred to the friction device. This behavior is very different from that of TF. Declaration of Competing Interest The authors declare that they have no kNown competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was financially supported by the project funded by the China Postdoctoral Science Foundation (No. 2017M621156) and the Natural Science Foundation of Liaoning Province (No. 20180540144). References [1] Fu Qiu Ni, Tan Kang Hai, Zhou Xu Hong, Yang Bo. Numerical simulations on threedimensional composite structural systems against progressive collapse. J Constr Steel Res 2017;135:125–36. [2] Fu QN, Yang B, Xiong G, Zhang WF, Nie SD, Dai GX. Dynamic analyses of boltedangle steel joints against progressive collapse based on component-based model. J Constr Steel Res 2016;1172:161–74.
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