Finite element modeling and capacity analysis of post-tensioned steel frames against progressive collapse

Engineering Structures 126 (2016) 446–456

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Finite element modeling and capacity analysis of post-tensioned steel frames against progressive collapse Akbar Pirmoz, Min (Max) Liu ⇑ Department of Civil Engineering, The Catholic University of America, 620 Michigan Ave., N.E., Washington, DC 20064, USA

a r t i c l e

i n f o

Article history: Received 19 April 2016 Revised 13 June 2016 Accepted 3 August 2016

Keywords: Progressive collapse Steel frame Post-tensioned connection Finite element Column removal Load redistribution Structural capacity Arching action Catenary action Failure mode

a b s t r a c t Post-tensioned (PT) steel frames, in which beams are connected to columns through high-strength PT strands, have been successfully developed in the past decade as a novel earthquake-resistant structural system. Compared to conventional steel moment frames, a PT steel frame demonstrates superior seismic performance, notably the minimum damages in main structural components and the self-centering capability under a design basis earthquake. Despite the abundant research on the seismic behavior of PT steel frames, there is an apparent lack of study on their load-redistribution behavior upon the notional removal of critical load-bearing columns, a commonly used threat-independent local structural damage scenario that potentially triggers the progressive collapse of the column-removed frame. This paper presents a first-of-its-kind numerical investigation on the unique structural behavior of PT steel connections and frames in redistributing the unbalanced gravity loads due to column removal. High-fidelity finite element structural models are constructed and validated using the available experimental data in the literature. The capacity of PT steel frames subjected to a gradually increasing vertical displacement along the removed column line is systematically studied. It is found that, besides the resistance of energydissipating elements, beam arching action and strand catenary action are the major sources of structural capacity of a PT steel frame against progressive collapse. The corresponding failure modes are identified and the design implications are suggested. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The recently emerging post-tensioned (PT) steel frames are potentially a novel alternative to conventional steel momentresisting frames [1–3]. As schematically shown in Fig. 1, at the beam-to-column connection of a PT steel frame, PT strands are installed along the beam web and anchored on the column flanges, and the beam is vertically supported by beam-column friction and/ or passive energy-dissipating elements (e.g., angles). Seismic design of PT steel frames typically requires that under a design basis earthquake, beams, columns, and PT strands provide the primary sources of strength and stiffness and essentially remain elastic, while the inelasticity and seismic damages be concentrated within the replaceable energy-dissipating elements [4]. Thus, after the design basis earthquake, a PT steel frame is expected to regain its original plumb position (i.e., self-center) without incurring residual deformation in its major structural components. Experimental studies [1,2,5,6] have demonstrated that, compared to conventional welded moment connections, the PT connection ⇑ Corresponding author. E-mail address: [email protected] (M.M. Liu). http://dx.doi.org/10.1016/j.engstruct.2016.08.005 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

possesses a similar level of rotational strength, stiffness, and ductility (realized via energy-dissipating elements), while the costly repair of cracks in welded structural elements can be favorably minimized or even eliminated. In addition to experimental investigation, both high-fidelity finite element (FE) models [7] and reduced-order macroelement models [8] have been developed to study the seismic response of PT steel frames. In particular, using the ANSYS software [9], a parametric FE model can be conveniently built to generate through computer simulation a vast amount of information on the deformation and internal forces of all components within a PT steel structure, thereby avoiding the physical installation of numerous sensors or any restriction owing to various experimental conditions, as often encountered in actual structural tests. Parallel to the fruitful research on seismic performance of PT steel frames [10–12], investigation of the progressive collapse (also known as disproportionate collapse) behavior and capacity of such a novel structural system is also very much needed. Progressive collapse refers to a catastrophic chain reaction of structural failures, propagating from the initial damages within a local portion of a building to the widespread global damages or even total collapse [13–15]. Because the final damage severity can be highly

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could lead to gravity-induced progressive collapse. Toward this goal, we develop high-fidelity FE models for both PT steel beamto-column connections and PT steel frames that contain such connections, and validate these FE models using relevant benchmark experimental data available in the literature. We then use these models to systematically investigate the capacity of a columnremoved PT steel frame subject to statically increasing displacement along the removed column line, elucidate the role of different structural components in facilitating the load redistribution upon column removal, and identify the relevant failure modes and highlight the implications for progressive collapse design.

2. Benchmark PT structures Fig. 1. Schematic of a PT steel beam-to-column connection.

disproportionate to its triggering event, the associated economic and societal consequences are often enormous [16]. Therefore, a structural system must be adequately designed/retrofitted to mitigate the risk of such a devastating event. Although the existing research has offered a comprehensive understanding of the seismic behavior of PT steel frames, their structural characteristics against progressive collapse can be quite different and are hence not readily inferred from the available seismic findings. This is because, as clearly revealed by previous studies on other types of structural systems [17–21], although structures that are properly designed against lateral seismic loads favorably possess some reserve capacity beyond what is needed to resist the normal gravity loads, such a reserve capacity can be far from adequate to safely redistribute the unbalanced gravity loads upon the notional removal of critical load-bearing columns from the original intact building, a commonly used scenario that simulates the sudden column loss caused by extreme loading and is often implemented with the widely adopted threat-independent alternate path method [22,23]. Indeed, the desirable behavior and corresponding design philosophy of a PT steel frame under unbalanced gravity loads are very different from those under seismic loads. For seismic design, the PT frame is expected to return to its initial position after a design basis earthquake so that the building can resume its normal functionality. To this end, the main structural components (i.e., strands, beams, and columns) of a PT steel frame must remain elastic while the energy-dissipating elements (e.g., top-seat angles) function as a fuse to bear all structural damages. For gravity-induced progressive collapse design, however, it is neither necessary nor economical to make the major structural components of a PT steel frame stay elastic under postulated column removal scenarios, considering the extremely rare, highly unpredictable nature of a triggering local damage. Instead, a controlled level of plastic deformation and local failure of main structural components is acceptable, as long as the path for damage propagation is blocked and thus the overall system integrity is maintained to avoid the total frame collapse. So far, research on the performance of PT steel frames in redistributing the unbalanced gravity loads upon column removal is very scarce [24–26]. In particular, numerical modeling of PT steel frames specifically for progressive collapse analysis and corresponding investigation of their structural capacity against progressive collapse do not exist. As a result, the capability of a seismically sound PT steel frame to arrest the progressive collapse risk is essentially unknown. Lack of such fundamental knowledge can severely hinder the adoption of PT steel frames in a building system that could experience the loss of critical load-bearing columns during its service life. In response to this concern, the study reported in this paper aims to understand the behavior and quantify the capacity of PT steel frames under column removal that

In order to validate and/or calibrate the present nonlinear FE structural models, we consider two sets of benchmark experimental data available in the literature. The first set of data was generated from the testing of a series of PT beam-to-column connections under quasi-static cyclic lateral loads [5]. In the present study, we use the test data associated with a particular connection specimen designated as PC2A (Fig. 2). This connection included two beams (with a total length of 6096 mm or 240 in.) with an identical PT connection on each side of an H-shaped column (with a total length of 3658 mm or 144 in.) and was post-tensioned by four strands along each side of the beam web. Each seven-wire, 0.6in. (15 mm)-diameter strand was initially pre-tensioned to onethird of its ultimate stress of 1864 MPa (270 ksi). Although not directly produced from a progressive collapse test, such experimental data are used here to confirm the general effectiveness of the FE model for simulating the nonlinear behavior of a PT connection. Table 1 lists the actual (measured) yield stresses of individual steel components in the tested connection. The second set of data was generated directly from a progressive collapse experiment on a reduced-scale three-story-two-bay PT steel frame [24] (Fig. 3), which was fixed at the base and seismically designed as a lateral load-resisting frame within a prototype building. The tested frame had a uniform bay width of 1219 mm (49.75 in.) and a height of 1264 mm (49.75 in.) for the first story and 1289 mm (50.75 in.) for both second and third stories. At the mid-height of each column, restraining steel plates were mounted as a lateral support to prevent the out-of-plane instability. The frame was tested by applying a monotonically increasing displacement along the middle column line, where the first-story middle column was intentionally unsupported to simulate its removal. The beams and columns were W-shape steel sections made of ASTM A572 Grade 50 steel. At each beam-to-column connection, both beam and column flanges were reinforced with steel plates

Fig. 2. Experimental setup of a PT steel connection under a lateral load (adapted from Ricles et al. [5]).

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Table 1 Material properties of a tested PT steel connection. Structural component

Beam flange

Beam web

Reinforcing plate

Shim plate

Strand

Actual yield stress [MPa]

230

266

843

843

1305

Language capability [9] to create parametric FE models for PT steel connections and frames, and then validate/calibrate these models against the afore-mentioned experimental data. The specific modeling strategies are described as follows. 3.1. Modeling of the PT connection

Fig. 3. Experimental setup of a PT steel frame under a vertical load (adapted from Tsitos et al. [24]).

made of ASTM A514 steel; four dog-bone-shaped energydissipating rods, which had a cross-sectional area of 50 mm2 at the second floor and 40 mm2 at the third floor and roof, were installed and allowed to yield in tension and compression without buckling. Two PT strands were installed for each beam, and the initial PT forces per strand were approximately 88.9 kN, 66.7 kN, and 55.6 kN at the second-floor, third-floor, and roof levels, respectively. The PT strands were of DYWIDAG type and each had a cross-sectional area of 140 mm2 at the second and third floors and 99 mm2 at the roof. The nominal yield stresses of different components in the tested PT frame are listed in Table 2 based on data in the literature [27], where more detailed information about the tested frame can be found. 3. FE modeling High-fidelity FE modeling enables us to understand the detailed structural responses and thus helps identify the potential failure modes, a salient advantage over the simplified structural models [28]. In the present study, we exploit the ANSYS Parametric Design

Table 2 Material properties of a tested PT steel frame. Structural component

Beam & column

Reinforcing plate

Rod

Strand

Nominal yield stress [MPa]

345

690

490

1675

The three-dimensional eight-node solid element SOLID185 provided in the ANSYS element library is used to mesh most structural components of the tested PT steel connection (Fig. 2). The full integration option is selected for calculating the element stiffness matrix. Note that the compressive force at the beam-to-column interface gives rise to a frictional force that resists the shear in the PT connection. To model such surface interaction, we use the contact elements CONTA174 and TARGE170, which are paired to prevent the penetration of the nodes from one surface into the other [9], to couple the end contact surfaces of both the beam and its flange reinforcing plates with that of the flange reinforcing plates of the column. A friction coefficient of 0.35 is assigned for such contact surfaces, a normal penalty stiffness factor of 1.0 with a penetration tolerance factor of 0.05 is chosen when defining the pair of contact elements, and the close gap option is used to enable the automatic contact adjustment. In order to prevent the column stress concentration under a point load and ensure a simple connection condition, we place fictitious rigid elements (i.e., cap plates with a significantly high, say ten times as high, modulus of elasticity compared to that of steel) at both ends of the column and beam. We model the PT strands (assuming all wires within a strand behave identically) by using the one-dimensional two-node element LINK180, each node having three degrees of translational freedom. This element is able to consider the large deformation and large strain effects, and thus it can be conveniently used to simulate the necking (i.e., shrinking in the cross section) of a PT strand. To pre-tension the strands in the FE model, we apply a fictitious negative thermal gradient to the whole structure but assign a fictitious thermal expansion coefficient only to the strands. As the strands tend to shrink under the internal thermal load, they are restrained from both ends and thus pre-tensioned. Values of the two fictitious parameters are adjusted by trial and error to obtain the target initial PT forces. In order to trigger the potential beam flange local buckling, the initial imperfections in beams are modeled by applying a nominal set of transverse perturbation forces, equivalent to a line load of 0.5 kN/m, at the flange edge nodes after the strands have been pre-tensioned and thus the load-carrying capacity of the connection has been established. We account for the potential inelastic deformation of different structural components by defining their individual material nonlinearities with appropriate bilinear stress-strain relationships — the actual steel yield stresses for different components [5] are used (Table 1), and the strain-hardening ratios are assumed to be 0.02 and 0.05 for strands and all other steel components, respectively [7], along with the isotropic hardening rule and von Mises yield criterion. Note that the elastic modulus of steel is taken as 200 GPa. The simple supports at the column ends of the PT connection are modeled by restraining the middle nodes of the rigid elements at the column ends in all translational directions. The roller supports at the beam ends of the PT connection are modeled by restraining the bottom row of the beam end nodes in both vertical and out-of-plane directions (i.e., along y and x axes). The displacement-controlled loading is applied at the upper column tip after pre-tensioning the strands and imposing the initial imperfections. Fig. 4 shows the FE model of the PT steel connection built in ANSYS using a total of 7813 nodes and 4720 elements. The following options and parameters are further considered when carrying out the numerical procedure in ANSYS. The sparse

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Fig. 4. The FE model of the PT connection.

solution method is selected to solve the system of equations for the FE model, the structural force and displacement convergence criteria are adopted with a tolerance level of 0.001, the minimum and maximum displacement increments are 0.01 mm and 5 mm, respectively, and the automatic time stepping and line search options are activated. 3.2. Modeling of the PT frame Fig. 5. The FE model of the three-story frame (symmetry considered).

4. Results and discussion 4.1. Validation of the FE model for the PT connection We compare in Fig. 6 the test data, which were associated with the loading portion of the first hysteretic cycle [5], and the simulation results using the FE model (Fig. 4). Note that the vertical axis represents the lateral force (denoted as H in the literature [5]) that is applied at the upper column tip in the z direction (Fig. 4), and the horizontal axis represents the corresponding lateral displacement in the same direction. It is clearly seen that the FE model

250

FE results 200

Lateral Force [kN]

Most modeling strategies for the PT connection are applicable for the FE modeling of the tested three-story PT steel frame (Fig. 3). So only the unique aspects of the frame modeling are presented here. We only consider half of the frame by taking advantage of the symmetry about a plane passing through the centerline of the middle column and being perpendicular to the frame plane. All nodes on this symmetry plane are restrained along the beam longitudinal direction. Note that another symmetry about a plane passing through the middle of beam webs is not considered in the FE model — otherwise, the nodes within this plane would be prevented from moving transversely and thus the potential beam web local buckling be eliminated. Besides, the reinforcing plates are treated as being fully integrated into the corresponding steel members through the adjacent nodes in the FE model. As a result, the effect of fillet welds that connect the plates to beams and columns is ignored, a reasonable approximation considering the minor impact of such welded joints on frame behavior. Each energy-dissipating rod is modeled as a single two-node LINK180 element with a constant cross-sectional area associated with the middle portion of the actual rod. Selected nodes at the middle portion of each column are restrained against the out-of-plane displacement to simulate the existence of restraining steel plates mounted on the tested frame. Consistent with the displacementcontrolled test, the FE model is loaded by statically increasing the displacement along the middle column line. Fig. 5 presents the FE model of the tested three-story frame using a total of 16,771 nodes and 10,606 elements in ANSYS. In order to accurately study the capacity of a frame structure, the realistic stress-strain relationships of material based on the measured (actual) strength of coupon should best be used in the FE modeling. Because such data were not reported for the tested three-story PT frame [24], the expected steel yield stresses are calculated based on their nominal values (Table 2) per AISC provisions [29] and are used in the FE model. For example, the yield stress of steel for beams and columns used in the FE model is taken equal to an expected value of 375 MPa, as opposed to a nominal value of 345 MPa.

150

Test data [5] 100

50

0

0

20

40

60

80

100

120

Lateral Displacement [mm] Fig. 6. Comparison of test and FE results for the PT connection under a lateral load.

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satisfactorily predicts the response of the tested connection in terms of the initial elastic stiffness, decompression load, and post-gap-opening stiffness of the PT connection. It appears that the FE model slightly overestimates the response in the nonlinear range, likely due to ignoring the inevitable uncertainties associated with, for example, material properties, residual stresses, and/or experimental conditions of the tested PT connection. 4.2. Study of a hypothetical one-story PT frame Because the outer ends of beams in the tested connection (Fig. 2) were not restrained in their beam longitudinal direction, the beneficial beam arching action [21,30] (to be defined next) and column flexural resistance, which typically exist in the response of an actual PT steel frame, are unable to be achieved. As a result, the FE model in Fig. 4 should not be directly used to study the progressive collapse behavior of a PT steel connection. To resolve this issue, we hypothetically create a one-story, twobay PT frame (Fig. 7) based on the PT connection (Fig. 2) and then consider a middle column removal scenario. Taking advantage of the structural symmetry, we only model half of the frame in ANSYS leading to 9646 nodes and 5621 elements (Fig. 8). An incrementally increasing vertical displacement is applied along the middle column to investigate the vertical force capacity of the frame upon the middle column loss. Before discussing the progressive collapse behavior of the onestory PT frame, we hereby introduce the concept of beam arching action caused by the rotation of an axially restrained beam with respect to its original horizontal position. Illustrated in Fig. 9, as the right-hand-side column is pushed downward by D, the beam rotates by h. Accordingly, the horizontally projected length of the beam tends to increase. However, the restraining columns at both ends of the beam, along with the PT strands, prevent such a length increase, leading to equal and opposite horizontal forces F that act at the beam ends and are vertically offset by e. This force couple is equilibrated by the vertical forces V that also act at the beam ends. Mathematically,

V ¼ Fe=L

ð1Þ

e ¼ d cosðhÞ  D  d  D

ð2Þ

where L and d are the original clear length and depth of the beam, respectively. Accordingly, the vertical force Varch acting on the righthand-side column by the beam is upward and equal in value to the afore-mentioned V exerted by the column. This ‘‘vertical beam force” Varch contributes to the frame capacity against the applied vertical force G along the removed column line, as long as the line of application of the horizontal force F at the right end of the beam is above that of the horizontal force at the left end of the beam

Fig. 8. The FE model of the hypothetical one-story frame (symmetry considered).

Fig. 9. Illustration of the beam arching action in a PT steel frame.

Fig. 7. Schematic of a hypothetical one-story PT steel frame under a vertical load.

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Frame capacity G

(a)

300 250

Vertical strand force Tv

Force [kN]

200

Varch + Tv

150 100 Vertical beam force Varch

50 0 -50

0

200

400

600

800

1000

Vertical Displacement [mm] 2000

(b ) Horizontal beam force F

1500

Force [kN]

(Fig. 9). Such a beneficial beam arching action continues to exist until the beam rotates significantly so that the horizontal force F at the right end is below the force at the left end. Under this situation, the resulting vertical beam force along the removed column line starts to point downward and thus reduce the vertical force capacity of the frame. That is, the beneficial beam arching action is no longer available. Note that in the above introduction to the beam arching action, an infinitely rigid beam is assumed to simplify the equation development and improve the concept understanding. Under this assumption, the gap between the beam and column opens immediately when a vertical displacement is applied, and all forces apply only at the top/bottom edges of the beam ends. In reality, however, the gap starts to open (i.e., the beam end separates from the column face on the tension flange side) only when a decompression moment capacity (provided by the initial PT forces) is exhausted by the vertical force-induced flexural demand. Therefore, before gap opens (i.e., when the vertical displacement is very small), the vertical beam force is in the form of beam end shear associated with the beam flexural demand (up to the decompression moment) and thus should not be calculated by Eqs. (1) and (2). After the gap opens, the vertical beam force is contributed by both the shear resistance of energy-dissipating elements (if existing) and the beam arching action. At this stage, the vertical beam force Varch due to the arching action can be approximately calculated by Eqs. (1) and (2) while keeping in mind that the horizontal reaction force is distributed over a finite contact surface at each beam end. Fig. 10 presents the changes in the overall vertical force capacity of the one-story PT frame as well as the vertical and horizontal forces of the beam (acting on the right-hand-side column) and the total axial force of strands, as the vertical displacement along the removed column increases. Note that both the overall frame capacity and the strand forces are directly obtained from the ANSYS outputs, while the vertical beam force is calculated by Eqs. (1) and (2). It is clearly observed in Fig. 10(a) that the algebraic sum of the vertical beam force Varch and the vertical strand force Tv (i.e., the vertical component of axial strand force in Fig. 9) essentially matches the overall vertical force capacity of the frame. This observation reveals that two major sources — beam arching action Varch and strand catenary action Tv — contribute to the vertical force capacity G of the frame. Such a relationship is expressed as G = Varch + Tv in Fig. 9. Note that the gap between the beam and column opens at a vertical displacement of about 10 mm, when the moment capacity due to the initial PT forces is exhausted. As discussed above, the vertical beam force prior to gap opening is offered by the beam end shear corresponding to moment demand and should not be calculated per Eqs. (1) and (2), and it is equal in value to the difference between the frame vertical force capacity and the vertical strand force. As the vertical displacement goes beyond 10 mm, it appears in Fig. 10(a) that the accuracy of using Eqs. (1) and (2) to calculate the vertical beam force is quite satisfactory, as evidenced by the proximity between the curve for the overall frame capacity and the curve for the sum of vertical beam and strands forces. Note that for the particular PT connection (Fig. 2), no energy-dissipating elements were installed. As a result, the vertical beam force is entirely from the beam arching action after the gap opens. It is observed in Fig. 10(a) that the beam arching action dominates the vertical force capacity of the frame as long as the vertical displacement is moderate (less than 346 mm). When the vertical displacement reaches 69 mm, the beam starts to yield under axial compression, as verified by the von Mises stress distribution in the beam (Fig. 11). At a vertical displacement of 117 mm, the vertical beam force Varch attains its peak value and then decreases afterwards, due to, based on Eq. (1), the combined effects of reduced

1000 Strand axial force T

500

0

0

200

400

600

800

1000

Vertical Displacement [mm] Fig. 10. Responses of (a) the vertical forces and (b) horizontal beam force and axial strand force of the one-story frame.

vertical offset e (Fig. 9) and almost constant horizontal beam force F (Fig. 10(b)). As the vertical displacement further increases to 257 mm, the horizontal beam force F becomes 1325 kN, a value very close to half of 2700 kN, which is the axial yield strength of beam W24X62. After that, the horizontal beam force vs. displacement curve is almost flattened (Fig. 10(b)), indicating that roughly half of the beam cross-section yields. This is because only a portion of the beam cross section (i.e., that along the diagonal line connecting the localized contact areas at the two ends of the inclined beam) is fully engaged to resist the compression by PT strands. Note that unlike the vertical beam force Varch, the vertical strand force Tv ( = Tsinh) increases with an increasing vertical displacement (Fig. 10(a)). This is quite obvious because as the vertical displacement increases, not only the strands are kept stretched (so the strand axial force T increases, as seen in Fig. 10(b)) but also the beam rotation angle h (Fig. 9) increases. After a vertical displacement reaches 346 mm, the vertical strand force Tv starts to exceed the vertical beam force Varch, indicating that the strand catenary action begins to dominate the vertical force capacity of the frame.

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Fig. 11. Distribution of von Mises stresses in the beam of the one-story frame at a vertical displacement of 69 mm.

Fig. 10(a) shows that at a vertical displacement of 600 mm, the positive (i.e., upward) vertical beam force Varch reduces to zero, suggesting the ending of beneficial beam arching action. In fact, Eq. (2) suggests that, when the vertical displacement D is equal to the beam depth d, the vertical offset e is close to zero and hence V (which is equal in value to Varch) calculated by Eq. (1) is zero. Note that the depth of the beam W24X62 used in the one-story PT frame is about 610 mm, a value practically equal to 600 mm considering the approximation made in Eq. (2). As the vertical displacement goes beyond 600 mm, the vertical beam force switches its direction by acting downward along the removed column line (i.e., its value becomes negative in Fig. 10(a)), thereby reducing the vertical force capacity of the frame. At the same time, the horizontal beam force (acting on the column) decreases due to the significantly inclined position of the beam. At a vertical displacement of 950 mm, both horizontal and vertical beam forces drop to zero (Fig. 10), indicating that the beam totally separates from the column face (Fig. 12). At this stage, the beam completely loses its gravity load-carrying capacity. The corresponding strain of the strands is calculated to be 0.018, a value slightly more than the yield strain of 0.012 but

far less than the typical fracture strain of 0.05 [31], indicating that the strands still possess considerable reserve capacity although they have already yielded at a vertical displacement of 760 mm (Fig. 10(b)). 4.3. Study of the tested three-story PT frame We compare in Fig. 13 the vertical force vs. displacement curves generated from the tested three-story frame (Fig. 3) and its FE model (Fig. 5), respectively. In general, the two sets of data agree well. For example, the gap opening forces are 176 kN and 167 kN from the test and FE simulation, respectively. The peak vertical force capacities of the frame are 497 kN and 502 kN from the test and FE simulation, respectively. Note that the FE model exhibits a slightly higher initial stiffness than that of the tested frame. As will be discussed in detail later, the actual lower-than-reported axial stiffness of strands in the tested frame, along with other

500

FE results

Loading

Force [kN]

400

Test data [24] 300

nd

2 - floor strand

200

rd

3 - floor strand

100

Roof strand 0

0

50

100

150

200

Vertical Displacement [mm] Fig. 12. Deformation of the PT connection of the one-story frame at a vertical displacement of 950 mm.

Fig. 13. Comparison of tested and simulated vertical force vs. displacement curves of the three-story frame. Simulated axial forces of individual strands also shown.

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various experimental uncertainties, can be a plausible reason for such a relatively stiffer FE model, which is originally built by assuming a defect-free strand assembly. It is observed in Fig. 13 that the peak capacity of the FE model occurs shortly after the second-floor strands yield but slightly before the vertical displacement reaches 89 mm, when the tested frame was reported to gradually lose PT forces due to the successive fracture (failure) of wires in its second-floor strands [24]. We calculate the strain of the second-floor strands at this displacement to be 0.014, a value far below a typical ultimate (i.e., fracture) strain of 0.05 [31]. Such a considerable strain difference implies that the strand fracture was unlikely to be the true cause of PT force loss in the tested frame. Accordingly, it is not possible to determine whether a tested strand failed or not based on its strain level. In order to simulate the successive wire failures, we decide to model each wire of the second-floor strand using a LINK180 element, and sequentially remove each element using the KILL ELEMENT option, leading to a stepped decrease in the second-floor strand force (Fig. 13). It was also reported [24] that the gradual failure of second-floor strands, along with the beam local buckling and energy-dissipating rod failure, caused the third-floor strands of the tested frame to yield, a phenomenon clearly captured by the FE model, as observed in Fig. 13. Note that as the second-floor strands gradually fail, an increase in the vertical displacement is accompanied by a decrease in the overall frame capacity (Fig. 13). This decrease is contributed not only by the reduced cross-sectional area of strands due to successive wire failure but also by the weakened beam arching action, the latter similarly observed in Fig. 10 (a) for the one-story frame case. Besides, it was reported [24] that at a vertical displacement of 239 mm, the third-floor strands of the tested frame lost their PT forces, causing another major decrease in the vertical force capacity of the frame. The corresponding deformed shapes of the threestory frame obtained from the test and predicted by the FE model

are compared in Fig. 14. It is clearly seen that both the overall deflection patterns and the member deformation details (e.g., beam local buckling locations and gap opening sizes) match satisfactorily, confirming the high accuracy of the FE model in predicting the nonlinear behavior of a column-removed PT frame. The overall discrepancy between the tested and simulated responses of the three-story PT frame could be attributed to ignoring in the FE model miscellaneous experimental errors, for example, the inherent imperfection and residual stresses in structural members, out-of-plumbness of columns, deviation of beam axes with respect to one another, inherent flexibility at column bases, and non-uniform beam-to-column contact surfaces, as well as potential defects of the steel materials and/or strand anchorage. Indeed, because the tested three-story frame is more complex than the tested connection, more experimental uncertainty and errors can be expected. As a side note, such factors could also be responsible for the irregular fluctuation in the test data (Fig. 13). In the following subsection, the effect of possible strand defects (as an illustration of such experimental errors) on the performance of the three-story PT frame is investigated.

4.4. Effect of strand defects on the capacity of the three-story PT frame As already discussed, the strain associated with the reported failure of the second-floor strand wires is 0.014, which is much less than the typical ultimate strand strain of 0.05. Besides, it was reported that at a vertical displacement of 239 mm, the thirdfloor strands of the tested frame began to lose their PT forces [24]. We calculate the corresponding strain of the third-floor strands to be 0.024, again far below a typical ultimate strain of 0.05. The calculated strains at which the second and third-floor strands reportedly lost their PT forces would have been similar and both close to the ultimate strain of typical strand materials, if the strand fracture were indeed the true cause of the PT force losses during the test. In other words, the existence of such a considerable difference in the strains of failed strands implies that the loss of PT forces in the tested three-story frame might not necessarily be contributed by PT strand rupture. Therefore, it is interesting to explore probable reasons for the strands to lose their PT forces at such a low strain level.

250

Load cell capacity [24]

Axial Force [kN]

200

Test data [24]

150

FE results

100

50

0

0

20

40

60

80

100

120

Vertical Displacement [mm] Fig. 14. (a) Tested [24] and (b) simulated deformed shapes of the three-story frame.

Fig. 15. Tested and simulated second-floor strand forces of the three-story frame.

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4.5. Capacity analysis of the three-story PT frame with defect-free strands So far we have demonstrated that an impaired strand assembly could noticeably reduce the PT frame capacity against progressive collapse due to the premature loss of PT forces. Therefore, it is important to ensure that every component of a strand assembly works properly in order to prevent the gradual or sudden loss of PT forces. In the following discussion, we report the results from a re-analysis of the three-story frame while assuming that the strands would possess their ideal stiffness as the vertical displacement increases. Under this assumption, a strand would rigorously follow the assigned bilinear stress-strain relationship, that is, it would yield at a strain of 0.008 (which is calculated by dividing the strand yield stress by its elastic modulus) and fracture (and thus completely loses its PT force) at an ultimate strain of 0.05. We statically push the FE frame model down along the middle column until reaching a vertical displacement of 100 mm. The resulting forces (including the vertical force capacity of the frame and the axial forces of individual strands) vs. displacement curves are plotted in Fig. 17. It is observed that at a vertical displacement of 42 mm the second-floor strands yield first. Then, as the middle column continues to be pushed down, the third-floor strands yield at a vertical displacement of 66 mm, and almost simultaneously the overall force capacity of the frame reaches a peak value of 497 kN. Then, there are slight drops in the forces of the frame and third-floor strands, likely due to the initiation of local buckling at the bottom flange of the third-floor beam (Fig. 18). Note that the roof strands remain elastic during this process. Similarly, the decrease in the axial force of the roof strands after a vertical displacement of 88 mm is likely due to the initiation of local buckling at the bottom flange of the roof beam and the inward flexural deflection of columns, both factors relaxing the PT strands at the roof level. Because the strands at the second and third-floor levels have already yielded or nearly so, the gradual decrease in the PT force of roof strands causes the vertical force capacity of the frame to drop accordingly. We also perform a parametric analysis to examine the effect of strand numbers on the response of the three-story PT frame against column removal, using the same FE model except that the number of defect-free strands is doubled (i.e., from the original

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Fig. 15 compares the tested and simulated PT forces of a secondfloor strand in the three-story frame. It is observed that at the beginning, both strand forces are equal to the initial pretensioning force of 88.9 kN. As the vertical displacement increases, the axial stiffness of the tested strand is lower than that of the simulated strand. At a vertical displacement of 52 mm, the simulated strand yields at an axial force of 233 kN, which well exceeds the load cell capacity of 204.6 kN in the test [24]. For this reason, it is unclear whether the tested strand indeed yielded or not. Since it is unlikely that major mechanical properties (e.g., elastic modulus and cross-sectional area) of the tested strands were considerably less than the reported values [24], we hypothesize that the following two issues cause the relatively low axial stiffness of a strand in the actual test. First, because the wires of a tested strand were reported to rupture one by one as the vertical displacement increased gradually [24], it is reasonable to infer that the wires of the tested strand also prematurely yielded one by one at a low strain level. Thus, such successive yielding of strand wires gradually reduced the overall axial stiffness of the tested strand. Second, it is also possible that the strand anchorage gradually slipped, thus relaxing the strand and decreasing the overall axial stiffness of the strand assembly that can be viewed as a series system. To test this hypothesis, we numerically adjust the axial stiffness of strands to reflect the effects of one or both of the above strand defect issues (i.e., premature wire yielding and anchorage slippage). A trial-and-error approach is used to reduce the elastic moduli of individual strands in the FE model to equivalently albeit approximately account for such discrepancy in strand stiffness, until the simulated axial force vs. displacement curves of individual strands satisfactorily match those from the test, respectively, up to a vertical displacement of 89 mm. The curves obtained from the test and the FE model using the properly reduced strand moduli are compared in Fig. 16. It is observed that such curves of all individual strands match very well, respectively. However, the vertical force vs. displacement curve of the overall frame obtained from the FE model exhibits a somehow lower post-gap-opening stiffness than that from the test. Such discrepancy could be attributed, at least in part, to the miscellaneous sources of structural restraining (e.g., friction between the column end and the sliding support plates) in the actual assembly of the tested frame.

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Vertical Displacement [mm] Fig. 17. Simulated force vs. displacement curves of the three-story frame (with two defect-free strands per beam) and its individual strands.

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strands in place, the strands are partially relaxed and thus unable to fully stretch to reach their desired strength. Therefore, doubling the number of strands alone does not cost-effectively increase the overall force capacity of this particular frame. In order to fully engage the strands, beams should be strengthened to increase their yield and local buckling capacities. In the meantime, columns should not only have adequate flexural capacity to avoid excessive inward deflection but also possess enough axial strength to resist the elevated axial compression due to the increased PT forces.

5. Concluding remarks

Fig. 18. Local buckling at the bottom flange of the third-floor beam of the threestory frame (assuming defect-free strands) at a vertical displacement of 66 mm.

two to four) per beam. The corresponding vertical force vs. displacement curve of the PT frame, along with the axial forces of individual strands, is shown in Fig. 19. Comparison to Fig. 17 reveals that, for this particular PT frame, doubling the number of strands increases the gap-opening load (from 119 kN to 231 kN) of the frame but does not significantly affect the peak force capacity (from 493 kN to 513 kN, i.e., only a 4% increase) of the frame within the studied vertical displacement range. This is because, as discussed in the previous paragraph, the peak force capacity of the original frame (i.e., the one with two defect-free strands per beam) is controlled by the yielding of second and third-floor beams followed by the local buckling of the third-floor and roof beams. Although the strands are doubled, such a beam failure sequence essentially remains, yet the higher PT forces cause the beams to fail at smaller vertical displacements, when all individual strands behave elastically and their axial forces are well below those of their counterparts in the original frame (Fig. 17). As beams fail while columns do not have enough flexural capacity to hold the

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We have successfully developed, calibrated, and validated highfidelity FE models to investigate the load redistribution capability of post-tensioned (PT) steel frames against progressive collapse upon notional column removal. For the studied PT connections and frames, we have found that the vertical force capacity of a column-removed PT steel frame is primarily contributed by the vertical components of both beam reaction force acting on the column (i.e., arching action) and strand axial forces (i.e., catenary action). A major failure mode of a PT steel frame subject to column removal is the beam axial yielding and/or local buckling under the large axial compressive force imposed by PT strands (along with the column restraining effect) as the frame moves downwards under the unbalanced gravity loads. After that, the role of beam as a support to hold the strands in place is weakened. Unless columns have enough flexural capacity to avoid excessive inward deflection, the strands are relaxed and thus the vertical force capacity of the frame is negatively affected. Eventually, the beams completely separate from columns and/or the strands attain an unacceptable inelastic strain (even if they have not yet fractured), signifying the imminent widespread failure of the PT steel frame. An immediate implication for the design of a PT steel frame against progressive collapse is that the beams and columns must be adequately sized so that PT strands can be fully engaged to resist the unbalanced gravity loads resulting from a missing column scenario. In other words, the strengths and stiffness of beams, columns, and PT strands within a PT steel frame should be properly balanced in order to achieve an overall cost-effective design. Once such a design consideration has been implemented, it is expected that under the unbalanced gravity loads, beams reach their axial compression capacity while PT strands yield or experience an acceptable level of plastic deformation. At this stage, the gravitycarrying capacity of the frame is safeguarded by the available shear resistance (through friction or energy-dissipating elements), transverse structural members, and/or strand catenary action. In the meantime, columns should be able to resist the large compression forces imposed by the sloped PT strands. Furthermore, it is emphasized that in order to attain the desired capacity of a PT steel frame against progressive collapse, early failure (e.g., premature wire fracture and gradual anchorage slippage) of the strand assembly should be avoided by all means. Finally, it is worth noting that a more accurate study of the progressive collapse potential of a frame structure requires that the three-dimensional (3D) effect be considered. Indeed, load redistribution upon sudden column removal is by nature a 3D phenomenon due to the existence of transverse beams and floor slabs. It is expected that considering such a 3D effect can have an important impact on the prediction of progressive collapse response and capacity of PT steel connections and frames. The present FE modeling and capacity investigation has focused on twodimensional (2D) structures solely because it is based on the 2D benchmark test data available in the literature. However, it is straightforward to extend the FE models developed in the present study to account for the 3D effect by explicitly including the trans-

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