Anti-progressive collapse behavior of beam-to-column assemblies with bolted-angle connections under different span ratios

Abstract

This article discusses the anti-progressive collapse performance of three specimens, with top-seat angles with double web-angle connections, under different span ratios. Double full-span assemblies extracted from steel frames with a typical column-removal scenario were designed and applied in a monotonic static loading experiment. The failure modes/sequences, strain, internal force conditions, and the transformed law of the resistance mechanism are examined. The results obtained show that specimens with equal spans can provide greater resistance due to full development of flexural and catenary actions. For specimens with unequal spans, the catenary resistance from the short beam portion is larger than that from the long beam portion, and the asymmetry of the span is not beneficial to the structure’s bearing capacity in preventing progressive collapse. The three specimens’ dynamic increase factors are also discussed in terms of the energy balance principle and their development trend for nonlinear static analysis related to catenary action.

Keywords

experimental investigation progressive collapse span ratio steel frame top-seat angle connection with double web-angle connection

Introduction

With the development of building structures toward maximization and increasing complexity, the occurrence of unexpected events (such as gas explosions and vehicle and plane collisions) can trigger disastrous progressive collapses, which have been the focus of many practicing engineers and researchers in recent years. Progressive collapse is a phenomenon in which local damage to one or more critical members leads to the failure of adjacent members, causing partial or even complete collapse of the system (ASCE 7-10, 2010). This has non-proportional collapse characteristics and can result in serious loss of property and life. Since the noted collapse of Ronan Point in London, a series of design specifications and guidelines (such as DOD, 2013; European Committee for Standardization (ECS), 2006; U.S. General Services Administration (GSA), 2013) to prevent progressive collapse have been presented instead of continuing to use only the traditional design approach. Among these specifications and guidelines, the alternative load path method (APM) is widely used to analyze and assess the anti-progressive collapse performance of a structure subjected to removal of a key column. Following damage to a vital vertical load-carrying member, the internal forces of the remaining structure are redistributed by alternative load-transferring paths to bridge over the failed member. In this process, catenary action plays an important role, depending on the rotation capacity of the connections and the tie force between the beam and the column (Yang and Tan, 2013a).

In recent decades, numerous increasingly in-depth studies, both experimental and computational, have been conducted in order to prevent progressive collapse. Lee et al. (2010) proposed a parallel axial-flexural hinge model considering the interaction between axial forces and moments for nonlinear dynamic progressive collapse analysis of welded steel moment frames. Liu (2010a, 2010b) used computational simulation to demonstrate that bending moments at beam ends decrease significantly with the occurrence of catenary action and proposed novel retrofitting schemes for strengthening simple joints in order to prevent progressive collapse. Lew et al. (2013) conducted a monotonic static experiment to study progressive collapse behavior using two specimens with welded–bolted beam-to-column connections and reduced beam-section connections. The test results showed that the failure of the connections was due to a combination of flexure, shear, and tension. Yang and Tan (2013a, 2013b) comparatively analyzed the anti-progressive collapse behavior of seven types of connections experimentally and concluded that the failure modes of the connections usually controlled the development of the resistance of the structure. Stylianidis and Nethercot (2015) and Stylianidis et al. (2015, 2016) proposed a complete calculation model to predict connection deformations under the effects of bending moments and axial forces and developed a complete, simplified approach to the design of structures resistant to progressive collapse. Li et al. (2013, 2015) conducted experiments to investigate the behavior of beam–tubular column moment connections and the effect of beam web bolt arrangements on catenary action under a column-removal scenario. Rezvani et al. (2015) investigated the influence of span length on the progressive collapse performance of seismically designed steel moment frames under a column in the first-story loss condition. The above studies primarily focused on symmetrical structures; the effects of asymmetrical spans on the progressive collapse behavior of steel frames have not been given sufficient attention in previous studies. However, for certain ordinary/newly built structures, anisotropic structures, and reinforced or modified structures, the span ratios of the steel beams differ, which can easily trigger progressive collapse; hence, it is necessary to study this topic more deeply.

In this study, three specimens with different span ratios (1:0.6, 1:1.0, and 1:1.4) were designed and configured in a double full-span assembly in accordance with a prototype steel frame to investigate the influence of asymmetrical spans on progressive collapse of structures. A top-seat angle with double web-angle connection was used to connect each beam to the column. The failure modes/sequences, strain and internal force conditions, and the transformed law of the resistance mechanism were examined in detail and discussed with reference to the monotonic static experiments. In addition, the pseudo-static responses and dynamic increase factors for three specimens are discussed in terms of the principle of energy balance.

Test setup and specimens

Test setup

To investigate the performance of the structure with different span ratios in an internal-column-removal scenario, a double full-span assembly was extracted from the steel frame based on the APM, as shown in Figure 1. The assembly included a failure column in the middle, two adjacent full-span beams, and two connected side columns, assuming that the inflection points of the bending moment of the two side columns were located at the midpoint of the upper and lower columns. In general, the surrounding horizontal components (such as beams b1 and b2 in Figure 1) of the assembly may provide some axial and rotational constraints. However, some study results (Lew et al., 2013; Rasool et al., 2016) and numerical analysis by the authors show that if the stiffness of the side columns is large enough, the boundary constraints provided by both beams b1 and b2 can be neglected. Additionally, after sudden removal of an internal column of a steel frame, any floors above the failure column zone would undergo downward rigid-body movement, and the axial force in the column can then be ignored because it would be very small (Yang and Tan, 2013a). Therefore, the directly affected area can be replaced by the simplified test model.

Figure 1.

A beam-to-column assembly extracted from a steel frame.

Figure 2 shows the test loading setup, comprising a 1000-kN hydraulic actuator, an 8-m-high reaction wall, two groups of gantry mountings, and other parts. The seating plane of the hydraulic actuator was horizontally attached to the reaction wall and the other side of the hydraulic actuator connected to the top of the failure column. To achieve hinged constraints for both side column ends and to facilitate loading, the test specimen was vertically fixed between two gantry mountings, ignoring its weight. The east and west ends of the side columns were connected with horizontal hinged components to attain the horizontal hinge constraint. In order to realize the vertical hinge constraints, four tension-compression load cells were employed. The two tension–compression load cells, connected to the upper side column, were fixed on the girder of the gantry mounting. Meanwhile, the others, linked to the lower side column, were fixed to the ground beam.

Figure 2.

Test loading setup: (a) loading device and (b) photograph of the setup.

To simulate gravity loads above the double full-span assembly and to avoid out-of-plane instability, the hydraulic actuator applied a horizontal westward load with displacement control on the top of the failure column. This simulation method was slightly different from the general APM, but did not change the mechanical behavior of the assembly. During the test, the applied horizontal load was set at a rate of less than 6 mm/min until the specimen was completely damaged. Note that the upper end of the middle column (the failure column) would be horizontally and rotationally constrained in an actual steel frame building. However, the results of a numerical study (Li et al., 2015) showed that constraining the tested failure column had little influence on the bearing capacity of the structure in preventing progressive collapse. Therefore, the end of the stiffer failure column was not constrained and horizontal movement was generally maintained during the loading process.

Test specimens

Three one-third-scale specimens with different span ratios were designed and constructed following the strong column–weak beam design requirement in Chinese Codes (GB50011-2010, 2010). A top-seat angle with double web-angle connection (TSDWA) was used to connect the beam and the column. The detailed geometric dimensions and beam-to-column connection of the specimens are shown in Figure 3.

Figure 3.

Geometric dimensions and connection details of the specimens (mm): (a) geometric dimensions and (b) details of TSDWA.

The span ratios of the three specimens are 0.6, 1.0, and 1.4, identified as specimens TSDWA-0.6, TSDWA-1.0, and TSDWA-1.4, respectively. The length of left beam L₁ is 1500 mm and the length of right beam L₂ is determined by the different span ratios. The cross sections of the beam and the column were H-150×100×6×9 (mm) and H-150×150×8×10 (mm), respectively. The section of the top-seat angle was L-140×90×10 (mm) and the section of the web angle was L-70×8 (mm). In the three specimens with different span ratios, the length of the side column was 1100 mm.

Material properties

For all the specimens, the material used for beams, columns, and angles was Grade Q235 steel. The mechanical properties of each steel component as obtained from three coupon uniaxial tests are listed in Table 1. High-strength bolts with M16 Grade 10.9 were adopted. The nominal yield strength and nominal ultimate tensile strength of the bolts were 900 and 1000 MPa, respectively.

Table 1.

Material properties of the steel components.

Steel component	Elastic modulus E (GPa)	Yield strength f_y (MPa)	Tensile strength f_u (MPa)	Fracture strain^a
Column flange (t = 10 mm)	2.10	262	395	0.30
Column web (t = 8 mm)	1.98	293	405	0.27
Beam flange (t = 9 mm)	1.95	266	397	0.26
Beam web (t = 6 mm)	1.90	312	422	0.27
Angle (L70 × 8 mm)	2.09	268	405	0.29
Angle (L140 × 90 × 10 mm)	2.01	265	402	0.30

Fracture strain is based on the proportional coupon gauge length of 5.65 $\sqrt{S_{0}}$ , where S₀ is the original cross-sectional area of the coupon.

Instrumentation

The instrumentation layout for the three specimens was similar; for brevity, only that of specimen TSDWA-0.6 is described here and is shown schematically in Figure 4. Six displacement transducers (D1–D6) were placed along the length of the beam to record the deflection of the left/right beams. In addition, two other displacement transducers (D7 and D8) were positioned to measure the deflection at the middle of the side column. More than 46 strain gauges were placed at critical sections of the beam (sections B1–B6), along which strain gauges attached at sections B2 and B5 were used to calculate axial forces in the beams while other strain gauges were arranged to monitor the strain condition of the connection region, as described in Figure 4. In order to obtain the vertical support reactions at the ends of the side columns, four 100-kN tension–compression load cells were installed, as shown in Figure 4(a). Additionally, the horizontal support reactions at the end of the side columns could be calculated according to the strain gauges measuring results at sections C1–C4. The horizontal displacement of the failure column (D0, shown in Figure 4(a)) was determined by the automatically recorded hydraulic actuator.

Figure 4.

Arrangement of (a) displacement transducers and (b) strain gauges.

Test results

Deflection and failure modes/sequences

This section describes the load–deflection responses and the observed test phenomena for the three specimens. Figure 5 shows the relationship curve between the applied load and displacement at the failure column; the coordinates of the associated key points during the test are marked on the curves. Furthermore, the failure modes/sequences of assemblies with different span ratios throughout the loading process are shown in Figure 6.

Figure 5.

Load–displacement curves of specimens and associated key points.

Figure 6.

Failure phenomena of specimens: (a) failure phenomena of specimen WUF-0.6, (b) failure phenomena of specimen WUF-1.0, and (c) failure phenomena of specimen WUF-1.4.

For specimen TSDWA-0.6, when the loading displacement was less than 124 mm (point a1 in Figure 5), the specimen exhibited almost flexural bending performance. With increasing loading displacement at 225 mm (point a2 in Figure 5), the west angle in tension connecting the failure column and the upper beam (short beam) fractured at the bolt hole (see Figure 6 (A1)), leading to a load drop from 151 to 139 kN. This indicated that the flexural bending generated by the top and seat angles (east and west angles) at the connection zone decreased to almost zero. Subsequently, the internal forces were redistributed and a new balanced system was formed in the assembly. Additionally, the internal forces of the local failure connection were transferred mainly by the web bolts. Subsequently, fracture at the east angle connecting the upper side column and the upper beam was observed at the displacement of 245 mm (point a3 in Figure 5; the fracture is shown in Figure 6 (A2)), at which moment the angles connecting the lower beam (long beam) and the side column exhibited larger plastic deflection. With further increase of the loading displacement, the web bolt holes at the lower end of the upper beam incurred severe bearing damage (see Figure 6 (A3)). Meanwhile, the short beam stopped functioning before the long beam, and the test was then terminated.

For specimen TSDWA-1.0, flexural bending behavior was observed before the loading displacement reached 194 mm (point b1 in Figure 5) and the load slowly continued to increase. Subsequently, the load began to increase rapidly until the loading displacement reached 358 mm (point b2 in Figure 5). In this process, the rapid development of the axial forces in the beams was conducive to resisting the external load. Subsequently, a fracture at the west angle in tension connecting the failure column and the upper beam was observed (see Figure 7 (B1)), causing a dramatic drop in the load from 172 to 124 kN. With further increase of the loading displacement, the stress transfer path was redistributed, and a new balanced system was formed in the assembly. The internal forces of the local failure connection were then transferred primarily by the web bolts, and the load was somewhat regained. This trend was maintained until the east angle connecting the lower side column and the lower beam fractured at the displacement of 397 mm (point “b3” in Figure 5, phenomenon shown in Figure 7 (B2)), and as a result, the load decreased from 146 to 76 kN. Thereafter, the load began to pick up again until the loading displacement reached 492 mm (point “b4” in Figure 5). Meanwhile, the web bolt holes of the lower of the upper beam occurred severe damage and the rest of other angles exhibited obvious plastic deflection. Then the specimen lost its all resistance and had to be interrupted. It is worth mentioning that the several destructions of the connection zones of the specimen were not completely symmetrical due to the initial defects of material and deviation of installation.

Figure 7.

Deflection development of the beams: (a) specimen TSDWA-0.6, (b) specimen TSDWA-1.0, and (c) specimen TSDWA-1.4.

For specimen TSDWA-1.4, the failure modes/sequences were similar to those for specimen TSDWA-0.6. Before the displacement increased to 204 mm (point c1 in Figure 5), the flexural bending generated by the top and seat angles resisted the external load. Beyond this point, axial tension started to develop. At a displacement of 404 mm (point c2 in Figure 5), the trend was rapidly interrupted due to fracture at the east angle connecting the lower beam (short beam) and the lower side column (see Figure 8 (C1)), where the load abruptly dropped from 133 to 110 kN. Then, a new balanced system was formed in the specimen due to the redistribution of the stress transfer path. The internal forces of the local failure connection were transferred mainly by the web bolts and the load slightly increased. With further increase of the loading displacement, the west angle in tension connecting the failure column and the lower beam fractured (see Figure 8 (C2)). At the same time, the angles connecting the upper beam (long beam) and the side column exhibited obvious plastic deflection. This was followed by a slight increase of the load again, and the web bolt holes at the lower part of the lower beam then showed severe bearing damage (see Figure 8 (C3)). Subsequently, the short beam rapidly failed before the long beam, and then the specimen completely lost its bearing capacity.

Figure 8.

Strain development of the mid-span beam sections of specimens: (a) Specimen TSDWA-0.6, (b) Specimen TSDWA-1.0, and (c) Specimen TSDWA-1.4.

The test results indicate that local damage of all the specimens occurred at the beam-to-column connection zone. The failure sequences of the assemblies showed that the angle in tension connecting the beam and the column sustained fractures at the bolt holes, and then the web bolt holes at the beam end exhibited severe bearing damage. Note that for specimens with unequal spans, the short beam with larger linear stiffness supported more horizontal load than the long beam when the assembly conducted redistribution of internal forces.

In addition, a comparison of the resisting load for three specimens with different span ratios is shown in Figure 5. The resisting loads of the specimens decreased with increasing span ratio at the initial loading stages. This was because specimens with smaller span ratios possess a larger overall flexural rigidity than those with larger ratios. At the later loading stage, the specimen with equal spans can obtain maximum resisting loads compared with the other two specimens with unequal spans under larger deformation due to the cooperative effect of the upper and lower beams. In contrast, for specimens with unequal spans, the short beam was damaged before the long beam without giving full play to its performance, causing it not to attain high resisting loads. This means that span asymmetry is not conducive to the strength reserve of a structure against progressive collapse, whereas specimens with equal spans possess better resistance reserves in the final stage of loading.

Deflection development

Figure 7 depicts the deflection profiles of the beams for the three specimens at different levels of loading displacement based on the displacement transducers (D1–D6) positioned along the length of the beam. It can be seen that the deflection profile turns from a quadratic parabola to two approximately straight lines with increasing loading displacement. This means that the anti-collapse mechanism of the assembly converts from a typical flexural pattern to a catenary action pattern. It should be noted that the deformation process of the upper and lower beams was closely related to the location of local damage to connections and to the length of the beam. In particular, the fracture of tension angles makes readings at measuring points in the vicinity more complex.

Strain evolution of beam sections

According to the strain measurement results of sections B1, B3, B4, and B6, when the tension angle connecting the failure column and the beam experienced fracture, it was found that the compressive flange of the beam had yielded, and local buckling deformation could be observed clearly. Meanwhile, virtually all strain measurements at sections B2 (S10–S14) and B5 (S33–S37) were significantly less than the nominal yield strain of 2,000 µε, indicating that the mid-span sections of the beams were within the elastic range throughout the test, as expected. Therefore, these mid-span strain gauge measurements could be used to analyze and calculate the axial forces in the beams. The evolution of strains at the beams’ mid-span (sections B2 and B5) versus the loading displacement for the three specimens is shown in Figure 8. In addition, the strain development at these sections may exhibit different responses due to different span ratios as follows.

Figure 8(a) describes the strain development at sections B2 and B5 for specimen TSDWA-0.6. It can be observed that these sections primarily show compressive strain toward a displacement of less than 124 mm (point a1 in Figure 8(a)). This reveals the increasing development of compressive forces in the beams. With the increase in the loading displacement from point a1 to point a2, the strains at these sections gradually changed from compression to tension. Then, the strains of the east flange abruptly fell and became negative due to the fracture of the tension angle connecting the failure column and the upper beam; in contrast, other strains showed obvious positive values until specimen failure. This implies that some small bending moments were formed and that, ultimately, the external load was mainly resisted via the axial tensile forces in the beams.

Figure 8(b) plots the strain evolution at the beams’ mid-span versus the loading displacement for specimen TSDWA-1.0. At the early stages, it can be observed that sections B2 and B5 showed flexural bending behavior. Beyond the loading displacement of 194 mm (point b1 in Figure 8(b)), the strains at sections B2 and B5 rapidly converted from compression to tension, indicating that the beam sections at mid-span began to be subjected to axial tension. This trend was maintained until the tension angle connecting the failure column and the upper beam experienced fracture (point b2 in Figure 12). After that point, the strains of the west flanges of sections B2 and B5 decreased substantially, but these sections were still subjected to axial tension. With further increase of the loading displacement, the strain of the web suddenly dropped and the strains of the west flanges of sections B2 and B5 shifted from tension to compression, indicating that some small bending moments were generated due to the neutral axis moving gradually from the mid-height of the section as the axial tensile force decreased. After the beam–column connection near the failure column was completely destroyed (point b4 in Figure 8(b)), the strain development of the beams was terminated.

Figure 8(c) depicts the strain development at the mid-span of the beams (at sections B2 and B5) with increasing displacement of the failure column for specimen TSDWA-1.4. When the loading displacement was less than 204 mm (point c1 in Figure 8(c)), the strains at sections B2 and B5 were basically in compression and developed slowly, indicating that a small axial compressive force in the beams had developed. As the loading displacement increased, strains at sections B2 and B5 increased rapidly, which caused axial tensile forces in the beams to develop quickly. When the loading displacement attained point c2 (in Figure 8(c)), strains at sections B2 and B5 were all in tension. Subsequently, strains of the east flanges of sections B2 and B5 dropped abruptly due to the fracture at the tension angle connecting the failure column and the lower beam. In contrast, the strains of the web and the east flange were still in tension. With further increase of in the loading displacement, the strain of the web decreased and the strains of the west flanges of sections B2 and B5 changed from tension to compression, indicating that some small bending moments were generated. This trend developed continuously until the web bolt holes at the lower part of the lower beam incurred severe bearing damage (point c4 in Figure 8(c)).

Analysis of the assembly’s resistance mechanism

Simplified analysis model and development of internal force

The calculation model for the double full-span assembly under a middle-column-removal scenario can be simplified, as shown in Figure 9. In fact, the intermittent damage characteristics of the specimens may make little difference for two-beam end rotation angles for one beam and axial force along the length of the beam. However, the beam presents a straight line under large deflection based on experimental phenomena and results. Here, two assumptions are made: (1) the axial force along the length of the beam is uniform and (2) the beam end rotation angle is the average of the upper and lower beam end rotation angles for one beam, as follows

N_{l} = N_{1} = N_{2}; N_{u} = N_{3} = N_{4}

(1)

θ_{l} = \frac{(θ_{1} + θ_{2})}{2}; θ_{u} = \frac{(θ_{3} + θ_{4})}{2}

(2)

where N_l and N_u are the axial forces of the lower and upper beams, respectively; N₁–N₄ are the beam end axial forces of the upper or lower beam; θ_l is the beam end rotation angle of the lower beam; θ_u is the beam end rotation angle of the upper beam; and θ₁–θ₄ are the beam end rotation angles of the upper or lower beam (as shown in Figure 9), which can be obtained from the measurement results of the displacement transducers.

Figure 9.

Force equilibrium diagram for the specimens.

According to above calculation model, axial forces in the upper and lower beams were calculated using the strain measurements at sections B2 and B5, which remained in the elastic range, as discussed in section “Strain evolution of beam sections.” Recombined with the measured data from the tension–compression load cells and the strain measurements at sections (C1–C4) of the side column ends, bending moments at beam ends were obtained.

Figure 10(a) shows the development of axial forces in the beams versus the displacement of the failure column for the three specimens. The trends of axial force development of the upper and lower beams are basically consistent. It can also be clearly seen that the beams were subjected to axial compression at the initial loading stages due to the compressive arching action, which may depend on the ratio of the upper beam length to the lower beam length (L₁/L₂). In comparison, the displacement corresponding to the maximum axial compressive force increased with increasing ratio, and the maximum compressive force of specimen TSDWA-0.6 was much higher than for the other two specimens. Axial forces in the beams then increased rapidly and converted from compression to tension. After fracture occurred at the tension angle connecting the failure column and the beam, axial forces in the beams were all in tension, indicating that catenary action dominated to resist the gravity load. Subsequently, axial forces in the beams generally dropped due to the intermittent damage of other connections until the axial forces transfer paths were completely interrupted. It is apparent that specimen TSDWA-1.0 had the largest axial tension while specimen TSDWA-0.6 had the smallest in the latter stage.

Figure 10.

Internal force development of specimens: (a) axial force development of specimens and (b) bending moment development of specimens.

The development of beam end bending moments for the three specimens is shown in Figure 10(b). The bending moment was mainly formed by the top and seat angles at the beginning. After the tension angle connecting the failure column and the beam fractured, bending moments at the beam ends abruptly dropped. With intermittent damage of the specimens, bending moments gradually decreased to zero. It should be noted that the beam end bending moment near the failure column was higher than that adjacent to the side columns. This was because the beam-to-column connection adjacent to the failure column is the first to bear the gravity load and incurs local damage.

Resistance mechanism of the specimens

The change law for the force transfer path of the specimens can be further analyzed by the above calculation model and the development of the internal forces. It can be found that the total resistance P of the specimen, comprising the total resistance of the lower beam portion P_l and the total resistance of the upper beam portion P_u, is carried by the beams through the horizontal component of axial forces and shear forces. Moreover, resistances P_l and P_u can be obtained by the supporting horizontal force of the lower and upper side columns, respectively, expressed as follows

P = P_{l} + P_{u}, P_{l} = R_{1} + R_{2}, P_{u} = R_{3} + R_{4}

(3)

where the supporting horizontal force of the side columns (R₁, R₂, R₃, and R₄) can be calculated from the strain gauge readings in sections C1–C4, respectively, which remained elastic according to the measurement results.

In addition, P is mainly provided by flexural and catenary actions. The total resistance from catenary action P_c comprises the catenary resistance generated by the lower beam portion P_cl and that generated by the upper beam portion P_cu, which can be calculated using equation (4). It can also be seen that the full development of the resistance from catenary action depends on reliable axial tensile forces in the beams and the rotation capacity of the beam-to-column connection

P_{c} = P_{cl} + P_{cu}, P_{cl} = N_{l} \sin θ_{l}, P_{cu} = N_{u} \sin θ_{u}

(4)

Therefore, the total flexural resistance P_w is the difference between P and P_c. In the same way, the flexural resistance formed by the lower beam portion P_wl and that formed by the upper beam portion P_wu can be obtained by the following equation

P_{w} = P - P_{c}, P_{wl} = P_{l} - P_{cl}, P_{wu} = P_{u} - P_{cu}

(5)

Figure 11(a) shows the resistance contribution of the upper and lower beam portions. For specimen TSDWA-0.6, the total resistance contributed by the short beam portion was larger than that of the long beam portion. This further demonstrates that the short beam portion sustains a larger external load when assemblies undergo redistribution of internal force. When specimen TSDWA-0.6 reaches the maximum resistance, the contributions of P_wl and P_wu are clear, whereas the resistance from catenary action is very small, indicating that the catenary action of upper and lower beam portions cannot allow full play to develop.

Figure 11.

Resistance mechanism development of the specimens: (a) resistance development of the upper and lower beams and (b) total resistance development of the specimens.

For specimen TSDWA-1.0, the development trend of resistance for the upper and lower beam portions was consistent before the fracture of the first tension angle. Meanwhile, P_u was slightly larger than P_l, and this is why the initial damage of the upper portion was observed. Afterward, some differences could be seen in resistance between the upper and lower beam portions due to the asymmetric damage of the connections.

For specimen TSDWA-1.4, similar to the case for specimen TSDWA-0.6, the total resistance from the short beam portion (P_l) was larger than for the long beam portion (P_u), whereas the total resistance of the upper and lower beam portions was contributed by flexural and catenary action upon initial fracture of the tension angle. Subsequently, the resistance contributions of P_wl and P_wu gradually decreased, and the resistance generated by P_cl was fully developed while the resistance formed by P_cu was not sufficiently evident.

Figure 11(b) depicts the development curve of the total resistance P, P_w, and P_c versus the displacement of the failure column for the three specimens. P was mainly provided by the bending moments generated by the top and seat angles at the initial loading stage. When the tension angle connecting the failure column and the beam incurred fracture, P_w and P_c together contributed the total resistance P. Thereafter, P_w decreased abruptly, while P_c increased continuously, indicating that the resistance mechanism converted from flexural action to catenary action. With intermittent damage of the connections, the resistance from P_w gradually decreased and catenary action supported the resistance until specimen failure.

Taken as a whole, the resistance process can be divided into three phases for the three specimens: a flexural action phase, a flexural-catenary mixed phase, and a catenary phase. Specimen TSDWA-1.0 can provide a greater resistance due to cooperation of the upper and lower beams in combination with flexural and catenary actions. For specimens with unequal spans, the short beam contributing more resistance failed before the long beam without allowing full play in catenary action.

Dynamic increase factors of the assemblies

Pseudo-static response of the assemblies

The analysis and discussion above for static experiments investigated the static behavior of double full-span assemblies in a middle-column-removal scenario in detail. However, in the event of an accident, a structure subjected to progressive collapse involves nonlinear dynamic effects, and the vertical load above the failure area can suddenly affect the remaining structure. In fact, the effect of sudden loss of a column is similar to the case of applying a gravity load suddenly on the remaining structure, especially when vital displacements are sustained by the system as a result (Izzuddin et al., 2008). In order to consider the dynamic response in a sudden-column-loss scenario, Izzuddin et al. (2008) proposed a simplified approach according to the static response to achieve dynamic response assessment using the principle of energy balance without considering the effect of damping. With determination of the nonlinear static response (such as the static load–displacement curve in the current study), the pseudo-static response can be obtained by considering an equivalence such that the external work done by applying the sudden load is equal to the internal absorption energy of the structure, as shown graphically in Figure 12. If an allowable suddenly applied load P_ti (i.e. load control) or displacement δ_ti (i.e. displacement control) is given, the corresponding dynamic displacement or load response can be deduced when the two shaded areas become identical. Then, the entire pseudo-static curve can be established. This approach is beneficial in achieving a simplified nonlinear dynamic assessment instead of conducting an overly complicated and expensive dynamic experiment and/or analysis.

Figure 12.

Principle of energy balance and pseudo-static response.

Based on the principle of energy balance, the pseudo-static response curves for the three specimens with different span ratios can be obtained through transformations of the static test curves, as shown in Figure 13. Although the span ratios of the specimens are different, the pseudo-static loads maintain an increasing trend before fracture at the tension angle connecting the failure column and the beam, just as in the static test results. Subsequently, the assemblies achieve the maximum pseudo-static load marked by the blue point in Figure 13. Beyond this point, the pseudo-static loads begin to decrease, indicating that this may trigger the failure of assemblies because the absorbed energy of the assemblies cannot balance the external work. In other words, in a sudden-column-loss scenario, if the suddenly applied gravity load (the maximum pseudo-static load) is less than the bearing load of the remaining structure, the structure can avoid progressive collapse; otherwise, the remaining structure may trigger collapse due to the external work done by the upper vertical load exceeding the stored strain energy of the system. It should be noted that after the fracture of the first tension angle, the failure of one connection may not immediately lead to the complete failure of an assembly due to the residual ductility and redundancy of other connections. It can be seen from Figure 13 that the residual ductility of assemblies gradually increases with increasing span ratio due to the increase of redundancy before complete damage of an assembly, which is advantageous in energy dissipation in the final stage.

Figure 13.

Pseudo-static response results for the specimens.

Dynamic increase factor

In the nonlinear static design approach allowed by DOD (2013), the dynamic increase factor (DIF) is a vital parameter for considering approximately the dynamic effect. Using force-controlled design, the DIF is defined as the ratio of static load P_s to dynamic load P_d at the same displacement, expressed as in equation (6). In this section, the DIFs for assemblies with different span ratios are comparatively discussed on the basis of the static and dynamic values obtained in section “Pseudo-static response of the assemblies.” Additionally, the calculation of the DIF for nonlinear static design is given for the DOD (2013) guideline by equation (7)

DI F_{P} = \frac{P_{s}}{P_{d}}

(6)

DI F_{DOD} = 1.08 + 0.76 / (θ_{pra} / θ_{y} + 0.83)

(7)

θ_{y} = \frac{(W_{p} f_{by} l_{b})}{(6 E_{b} I_{b})}

(8)

where θ_y is the yield rotation of the beam end and can be calculated by equation (8) (ASCE/SEI 41-06, 2007); W_p is the plastic section modulus of the beam; f_by is the yield strength of the beam; l_b is the length of the beam; E_b is the elastic modulus of the beam; I_b is the inertia moment of the beam section; and θ_pra is the plastic rotation of the beam end and is equal to the difference between θ (the total rotation of the beam end) and θ_y. In the current study, θ is approximately equal to the maximum experimental values of θ_u and θ_l, as discussed in section “Simplified analysis model and development of internal force.”

Figure 14 compares the results for the DIFs between the calculated values using equation (6) and DOD recommendations for the three specimens. It can be observed that the results based on equation (7) monotonically decrease with increasing displacement while the results from equation (6) first decrease and then increase for specimens TSDWA-1.0 and TSDWA-1.4 with full realization of catenary action. Moreover, although the DIFs from the calculation results using equation (6) and DOD (2013) recommendations basically vary between one and two, the former are much larger than the latter, especially in the late stages of specimens TSDWA-1.0 and TSDWA-1.4. There are two main reasons for this difference. First, the relevant provisions in DOD (2013) are based on seismic studies, which may be different from behaviors under the condition of progressive collapse. Second, the calculation formula of the DIFs for DOD (2013) cannot reflect the effect of catenary action. However, the full realization of catenary action leads to increased DIFs, which even exceed a value of two under large deformation, as shown in Figure 14. This illustrates that the increase of catenary action greatly improves the static load capacity and does not significantly promote the increase of the pseudo-static load capacity using a simplified energy balance approach. It can be preliminarily concluded that the development trend of the DIFs is also related to catenary action.

Figure 14.

Dynamic increase factors for the specimens.

Conclusion

In this article, three test specimens with different span ratios were examined under monotonic static loading to study the behavior of the assemblies with top-seat angle with double web-angle connections under a middle-column-removal scenario. The failure modes/sequences, strain and internal force condition response, laws of the resistance mechanism, and pseudo-static response were discussed and analyzed in detail. The following conclusions were reached:

Local damage of all specimens with different span ratios occurred at the beam-to-column connection zone. The failure sequence of the specimens started with the fracture at the bolt holes of the tension angle connecting the beam and the column first and then the web bolt holes at the beam end appeared to have severe bearing damage.

At the initial loading stages, the resisting loads of specimens decreased with increasing span ratio because specimens with smaller span ratios possess larger overall flexural rigidity values than specimens with larger ratios. At the later loading stages, the specimen with equal spans can attain maximum resisting loads compared with the other two specimens with unequal spans due to cooperation between the upper beam and lower beam. In contrast, for specimens with unequal spans, the short beam was damaged before the long beam without allowing its behavior to develop fully.

Specimen TSDWA-1.0 can provide a greater resistance due to the full development of flexural and catenary actions. For specimens with unequal spans, the catenary resistance from the short beam portion was larger than that for the long beam portion; when the allowable span ratio was much smaller, catenary action was not obvious. This means that the asymmetry of the span is not beneficial to the strength reserve of the anti-collapse capacity of a structure, while a specimen with equal spans provides a better resistance reserve before complete damage.

The pseudo-static responses and the dynamic increase factors of three specimens subjected to different levels of sudden gravity loading were discussed. The results demonstrate that the development trend of the dynamic increase factor is also related to catenary action.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research presented in this article is supported by the National Natural Science Youth Foundation of China through grant no. 51308442 and the National Natural Science Foundation of China through grant no. 51678476.All opinions, findings, conclusions, and recommendations expressed in this article are those of the writers and do not necessarily reflect the views of the sponsors.

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