Numerical study of dynamic responses of reinforced concrete infilled frames subjected to progressive collapse

Abstract

Recently, the contribution of infill walls on progressive collapse resistance of reinforced concrete (RC) structures attracts a great many research attentions, but the research interests are mainly concentrated on the static resistance and the macro-modeling approaches, which require predefined one-dimensional load paths through two-dimensional walls. However, the load transfer paths in dynamic loading regime are still not fully understood. To this end, high-fidelity finite element (FE) models of multi-story RC infilled frames are built and validated through quasi-static experimental results. Then the FE models are used to investigate the dynamic responses of infilled frames under different single and double CRS as well as the effect of the number of stories on the load transfer paths of full-height infill walls (FHIW) and infill walls having opening (IWHO). The results indicate that the load paths along the infill walls in static and dynamic loading regimes are similar prior to the peak resistance but different in post-peak resistance for single infilled story frames. Such difference results from the loading distribution pattern, in which the static loading is typically represented by a concentrated load whereas the dynamic loading involves the uniformly distributed load. Moreover, increasing the number of infilled stories with FHIW, trans-story load paths due to composite effect always exist to enhance resistance and such paths are scenario-dependent. In comparison, the load paths for multi-story frames with IWHO are relatively scenario-independent with minor composite effect. Therefore, to generalize the macro-modeling, it is conservative to ignore the trans-story load paths.

Keywords

reinforced concrete infilled frames infill walls progressive collapse dynamic response load transfer path

Introduction

Disproportionate collapse (or progressive collapse) of buildings is an unacceptable catastrophic event triggered by local damage of load bearing elements, such as columns or walls, due to accidental loading. The severe aftermaths of progressive collapse albeit low probability still raise the concerns of government agencies and research interests over past decades. As the accidental loading is hard to be predicted, threat-independent approaches are employed in the current design guidelines or codes (CEN, 2006; DOD, 2016; GSA, 2016).

Alternate load path method is one of threat-independent approaches to the check the ability of the remaining structures to bridge over a removed column or wall. Consequently, a series of experimental and numerical studies have been carried out to evaluate the load-redistribution capability of reinforced concrete (RC) beam-column assemblies or frames (Deng et al., 2020; Feng et al., 2016; Kang and Tan, 2015; Qian et al., 2020; Yi et al., 2008; Yu and Tan, 2013a, 2013b, 2017) and beam-slab systems (Lim et al., 2017; Lu et al., 2017; Qian et al., 2015, 2016; Yu et al., 2018, 2020) under column removal scenarios (CRS). The priority of research given to the frames and beam-slab systems is because the former is regarded as the primary structural members to redistribute the gravity loads and the large deformation of the latter is regarded as the last line of defense against progressive collapse. The results suggested that with adequate lateral and rotational restraints, RC beams are able to develop compressive arch action and catenary action on top of flexural action, and when the spatial effect of structures is considered, slabs are able to greatly improve structural resistance, in particular, through mobilizing tensile membrane action at large deformation stage.

Recently, the contribution of the secondary structural elements, such as infill walls, to the progressive collapse resistance has been concerned. Li et al. (2016) quasi-statically tested two 1/3 scaled four-bay two-story RC frames under a middle CRS, in which one was a bare frame and the other was a frame with full-height infill walls (FHIW). Qian and Li (2017) experimentally investigated structural behavior of three 1/4 scaled two-bay three-story frames with FHIW under a penultimate CRS. Brodsky and Yankelevsky (2017) tested static resistance of seven 1/2 scaled infilled frame units under a corner CRS, and explored the effects of column stiffness, reinforcement detailing, and masonry block type etc. The results of the above three tests indicate that the presence of FHIW significantly improved the structural resistance and stiffness, changed the failure modes and thus the contribution of infill walls cannot be ignored. In addition, the effect of infill walls having opening (IWHO) on progressive collapse resistance and failure modes of RC frames has also been studied. Stinger and Orton (2013) tested two-bay two-story RC frames with partial infill walls under a middle CRS. Shan et al. (2016) tested a four-bay two-story RC frame with IWHO at the center under a middle CRS. In such cases, the contribution of infill walls very much depends on the area and the location of the opening. Yu et al. (2019b) built up structural resistance of frames with IWHO as a function of opening area under a middle CRS through systematic high-fidelity numerical studies.

To cater for the modeling of infill walls in a large scale computation, such as entire three-dimensional multi-story buildings, infill walls are simplified as one-dimensional macro-members that are able to demonstrate the load transfer paths in two-dimensional walls. However, the current macro-modeling of infill walls for progressive collapse analysis are proposed only in accordance with the load transfer paths in single infilled story. For instance, Shan et al. (2016) proposed a four-strut model for IWHO based on their experimental results to simulate the structural resistance. Li et al. (2019) proposed a three-strut model for FHIW based on their test results as well, in which the diagonal strut is used to describe the load path prior to the failure of infill walls and the two off-diagonal struts are used to represent the load paths of the remaining infill walls. Eren et al. (2019) employed a two-strut model for FHIW under a middle CRS to account for the actual contact between the infill walls and the surrounding frames.

The above-mentioned experimental studies of RC infilled frames were conducted with a quasi-statically concentrated load under a middle CRS. However, the nature of progressive collapse is a dynamic process, during which a uniformly distributed service load always exists. So far, only a few site tests for dynamic responses of infilled frames induced by explosively or manually removing columns prior to building demolition were reported. Sasani and Sagiroglu (2008) tested the dynamic responses of a six-story RC frame with a corner and a penultimate exterior column imploded, and found that the bidirectional Vierendeel action is the primary load transfer mechanism to rebalance the gravity load. Song and Sezen (2013) and Song et al. (2014) conducted on-site tests of two steel frames with infill walls, and recorded the dynamic responses of the remaining structure under sequential removal of columns in a ground story. However, the afore-mentioned studies did not clearly report the contribution of infill walls as well as the load transfer paths in infill walls.

In summary, the load transfer paths of infill walls under CRS are mainly studied through quasi-static tests and analysis, and the dynamic tests were very limited. To account for dynamic effect, the quasi-static resistance can be converted to dynamic resistance through energy-based approach (Izzuddin et al., 2008) provided that the load transfer paths in both static and dynamic loading regimes are similar. However, such similarity is not known yet. Moreover, the current macro-modeling of infill walls are proposed in accordance with the load transfer path of single infilled story under middle CRS (Eren et al., 2019; Li et al., 2019; Shan et al., 2016) or by analogy with single infilled unit under horizontal cyclic loading (Crisafulli, 1997; Eldakhakhni et al., 2003; Uva et al., 2012). The validity of those macro-modeling for multi-story infilled frames in dynamic loading regime is not checked yet, in particular for implementing the macro-models in multi-story frames.

Besides the experimental study, high-fidelity finite element (FE) modeling is an alternate effective solution to explore the load transfer paths of infill walls (Yu et al., 2019b). Accordingly, in this paper FE models of RC frames with infill walls are built and validated through quasi-static results. Then the same numerical models are analyzed in dynamic loading regime to investigate the dynamic behavior of RC infilled frames under a middle CRS, as well as the similarity and the difference of load transfer paths in static and dynamic loading regimes. Thereafter, the two-story models are extended to four-story ones, to explore the effect of the number of infilled stories on the load transfer paths under single and double CRS. Finally, reasonable macro-models of infill walls for dynamic analysis are suggested.

Finite element models of RC infilled frames

The FE models of the RC infilled frames are built and validated in a quasi-static manner firstly through commercial software LS-DYNA. Then, the dynamic models considering the effect of sudden removal of columns are developed based on the validated static models, to investigate the dynamic response of RC infilled frames under progressive collapse scenarios.

Brief of the reference experiments

Li et al. (2016) and Shan et al. (2016) quasi-statically tested two 1/3 scaled RC infilled frames, including a frame with full-height infill walls (F-FHIW) and the other with infill walls having opening (F-IWHO) under a middle CRS. The specimens contained two stories and four bays with the middle column removed at the first story. Every bay in the second story had the same type of infill walls. The geometric dimensions of half F-FHIW and F-IWHO are respectively shown in Figure 1 due to symmetry. The two specimens had identical dimensions, that is, 100 × 150 × 1700 mm for the beams, 200 × 200 × 1400 mm and 200 × 200 × 1100 mm for the columns in the first story and second story, respectively. Each infill wall was made of solid concrete masonry units (CMU) and connected to the columns through tie bars. The loading process was simulated by slowly increasing the vertical displacement of the remaining middle column using a hydraulic jack. More detailed information of the tests were reported in the papers (Li et al., 2016; Shan et al., 2016).

Figure 1.

Geometric dimensions and reinforcement detailing of RC infilled frames in reference experiments.

Numerical models for static analysis

Introduction of modeling details

To save computational time, only half of each specimen is modeled due to symmetry, as illustrated in Figure 2. All nodes in the symmetry plane are constrained in y-direction to limit the translational movement. The bottom ends of the columns at the first story is assumed to be fixed. Meanwhile, a rigid plate on the top of middle column, specified with a velocity-time curve, is used to accomplish the quasi-static loading with displacement control.

Figure 2.

Numerical modeling details of RC infilled frames.

The concrete and CMU are modeled using 8-node solid elements with only one integration point in each element. The original continuous surface cap model (CSCM), which considers a set of parameters about elastic properties, cap location and strength surface, etc., is adopted for concrete and CMU materials. Rebars and stirrups are represented by 2-node Hughes-Liu beam elements, and material properties are defined by a symmetric bilinear elastic-plastic model using the keyword *Mat_Plastic_Kinematic. The connection between steel reinforcement and concrete is assumed to be perfect bond using the keyword ∗CONSTRAINED_LAGRANGE_IN_SOLID in LS-DYNA. The critical material properties of the infilled frames are listed in Table 1.

Table 1.

Material properties of the infilled frames.

Materials	Mechanical properties
Concrete in first story	Compressive strength	41.3 MPa
Concrete in second story	Compressive strength	31.8 MPa
Concrete masonry unit (CMU)	Compressive strength	18 MPa
Contact surfaces between CMUs	Shear strength	1.5 MPa
	Tensile strength	1.08 MPa
Rebars with diameter of 8 mm	Yield strength	415 MPa
	Ultimate strength	588 MPa
	Ultimate strain	0.18
Rebars with diameter of 4 mm	Yield strength	235 MPa
	Ultimate strength	322 MPa
	Ultimate strain	0.31

For masonry walls, as indicated in Figure 2, the tiebreak-based contact model (Bala, 2019) is adopted for representing the interface between CMU, in which the interface damage is controlled by the normal and the shear failure strengths of mortar joint. Note that the mortar thickness is halved and reallocated to each neighboring units. More detailed information of the static models can be found in the paper (Yu et al., 2019b).

Validation of the static models

Figure 3 shows the comparisons of the numerical and experimental progressive collapse resistance of the two specimens under quasi-static loading, and Figure 4 demonstrates the comparisons of corresponding failure modes. It is observed that although the randomness of the quality and the craftsmanship of mortar joints in practice made numerical models hard to capture sudden drop of the resistance caused by wall cracking, the primary characteristic of the structural resistances and failure modes are in good agreement with experimental results, indicating that the static FE models are reliable and reasonable to study the structural behavior of infilled frames against progressive collapse. Therefore, in the next section, the static models are used to develop dynamic models.

Figure 3.

Comparison of experimental and numerical structural resistance of RC infilled frames (Yu et al., 2019b).

Figure 4.

Comparisons of experimental and numerical failure modes of RC infilled frames (Yu et al., 2019b).

Numerical models for dynamic analysis

Dynamic modeling methodology

The nonlinear dynamic analysis of infilled frames is achieved in two steps: (1) Apply the service gravity load to the structure using the keyword *LOAD_BODY_Z prior to column removal, followed by a wait time to eliminate dynamic vibration; (2) Instantaneously remove the steel support which replaces the column to be removed by setting the material erosion time of the steel support via the keyword *MAT_ADD_EROSION. The gravity load (Q_g) is calculated as 1.2 DL+0.5 LL according to (DOD, 2016; GSA, 2016), where DL and LL are dead load and live load, respectively. Base on the Chinese standard (MOHURD, 2009), except the self-weight, the extra imposed dead load and live load are 2.5 kN/m² and 2.0 kN/m², respectively. All slab surface loads are transformed into line loads which are directly imposed on the beams by increasing the density of beams to consider the effect of mass on inertial force. The damping is applied using the keyword *DAMPING_GLOBAL. Moreover, except the above loading scheme, other model details are the same as the static models, as shown in Figure 2. Note that according to Yu et al. (2014), during progressive collapse phase the strain rate varied in the range of 10^–2/s to 10^–1/s. The corresponding dynamic increase factor (DIF) for concrete ranged from 1.035 to 1.053 and steel strength from 1.112 to 1.172. Neglecting such strength increase due to DIF will result in slight conservative prediction. As a result, the limited effect of strain rate is not considered in the dynamic model.

Comparison of the dynamic and static analysis results

Figure 5(a) and (b) demonstrate the dynamic responses of F-FHIW and F-IWHO without considering damping under a middle CRS, respectively. It is seen that under the service load (i.e. 1×Q_g) the maximum vertical displacement of F-FHIW and F-IWHO is merely 0.9 mm and 4.1 mm, respectively. This means that both RC infilled frames are still at elastic stage. Through increasing the applied gravity load until 6.0×Q_g and 3.5×Q_g, F-FHIW and F-IWHO fail to capture the downward movement, respectively, corresponding to the occurrence of collapse. Evidently, the presence of infill walls significantly enhances the progressive collapse resistance of RC frames, and the improvement contributed by FHIW is much larger than that by IWHO. This finding also confirms the observation in accordance with the comparisons of quasi-static resistance as shown in Figure 3. Moreover, Figure 5(c) and (d) show that incorporating the damping, the vertical displacement under a given gravity load become less.

Figure 5.

Time history of vertical displacement of RC infilled frames with different dampings: (a) F-FHIW without damping, (b) F-IWHO without damping, (c) F-FHIW with damping ratio of 5%, and (d) F-IWHO with damping ratio of 5%.

The nature of progressive collapse is a dynamic process. As a result, nonlinear dynamic (ND) analysis is the most straightforward approach to conduct progressive collapse analysis, such as the dynamic modeling methodology presented herein. In comparison, nonlinear static (NS) analysis is not a direct method. To evaluate structural capacity against progressive collapse, NS results should be modified through taking account of dynamic effect. One of the approaches is converting a static resistance curve into a pseudo-static resistance curve by equating external work and strain energy, which was proposed by Izzuddin et al. (2008). Note that the energy-based approach neglects damping. However, it is necessary to check the applicability of the energy-based approach for RC infilled frames over the entire deformation history.

Figure 6 demonstrates the quasi-static and dynamic progressive collapse resistance normalized by service load Q_g, in which the former is achieved through dividing the static resistance by the axial force of the column to be removed under service load (1×Q_g), and the latter is obtained from incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) as shown in Figure 5(a) and (b) or the pseudo-static resistance. Consequently, the normalized resistance is presented as load factors. It is observed that load factor curves obtained from the IDA agree well with those from the energy-based approach prior to the collapse of both F-FHIW and F-IWHO. This suggests that if the dynamic response of RC infilled frames under a middle CRS is concerned only prior to the collapse, energy-based approach is a good start. However, when the applied load exceeds the critical load (corresponding to the maximum load prior to imminent collapse), say, 5.5×Q_g for F-FHIW and 3.0×Q_g for F-IWHO, a small increment of the applied load results in a tremendous increase of the vertical displacement as illustrated by IDA curves shown in Figure 6, corresponding to the collapse of the RC infilled frames. In such a situation, the softening pseudo-static resistance determined by the energy-based approach is not realistic.

Figure 6.

Comparisons of dynamic progressive collapse resistance: (a) F-FHIW and (b) F-IWHO.

One of critical pre-assumptions of using energy-based approach is that the load transfer path in both quasi-static and dynamic loading regimes should be identical. For simplicity, Figure 7 just compares the von-Mises stress contours in the infill walls directly connected to the removed column in two loading regimes. Figure 7(a) shows that the load is transferred through the diagonal of FHIW in both static and dynamic loading prior to the failure of the infill wall. With further statically increasing middle column displacement, cracks initiate along the primary diagonal and eventually cause the failure of infill walls in the diagonal direction, but the load is still able to transfer in two new small diagonal paths, which are separately located in the remaining infill walls within a single-span panel. In comparison, for F-FHIW in dynamic loading regime, after the peak resistance the load is only transferred in a diagonal direction of the upper-right remaining infill wall. The difference mainly results from the distribution pattern of the applied gravity load. A concentrated load is more conveniently used to determine static resistance, as shown in Figure 2, whereas a uniformly distributed load has to be eventually balanced in the dynamic response induced by a CRS. With the concentrated load, the interaction between the 2nd story middle column and the lower-left remaining infill wall is still active and thus that wall is still able to transfer load.

Figure 7.

Load transfer path in the infill walls under static and dynamic loading regime: (a) F-FHIW and (b) F-IWHO.

Likewise, Figure 7(b) demonstrates the load transfer paths through IWHO in static and dynamic loading. The presence of opening weakens the stiffness of the infill wall and makes the load transferred through the perimeter of the opening, corresponding to four diagonal struts. With increasing the vertical displacement, the upper and lower diagonal paths gradually lose function due to the shift of transferring compression to tension. In static loading regime, the left and right diagonal paths keep working until the end, whereas in dynamic loading regime, only the right diagonal path plays a role.

In a word, the load transfer paths through infill wall in static and dynamic loading regimes are similar prior to the peak resistance and different at failure stage due to the loading pattern. Therefore, if more concerns are focused on dynamic response before the peak resistance, the macro-modeling of FHIW under a middle CRS also could be represented by a diagonal strut, whereas for IWHO the load path can be simulated as a four-strut model.

Influence of the damping

To investigate the dynamic responses of structures against progressive collapse, energy-based approach ignores damping, but damping always exists and can be easily considered in ND analysis. Figure 5 shows that increasing damping ratio from 0 to 5%, the peak displacement becomes smaller and the vibration of the infilled frame disappears faster for a given load. Moreover, increasing applied load results in a larger mass of the structure, and the contribution of damping to resist progressive collapse becomes more evident.

Figure 8 demonstrates the load factor and the corresponding maximum vertical displacement obtained from the time history curves of displacement as shown in Figure 5. It is seen that for a service load (i.e. load factor = 1), the maximum vertical displacement of F-FHIW is 0.92, 0.77 and 0.76 mm for damping ratio of 0, 3% and 5%, respectively. However, when the load factor reaches 5.5, the maximum vertical displacement of F-FHIW is 11.3, 5.2 and 4.5 mm for damping ratio of 0, 3% and 5%, respectively. That is, the damping ratio of 5% is able to reduce the peak displacement by 60%. For F-IWHO, the contribution of damping ratio is more significant. For example, at load factor = 3.0, the damping ratio of 5% decreases the peak displacement by 76%. This indicates that when the structural resistance is weak, the damping force can play a more important role to resist progressive collapse. Therefore, it is suggested that global damping ratio should be considered for infilled frames due to their large mass, and the damping ratio of 5% will be used in the following ND analysis.

Figure 8.

Effect of damping on the load factor curves of RC infilled frames: (a) F-FHIW and (b) F-IWHO.

Dynamic responses of RC infilled frames under single sudden column loss

Section 2 focuses on the dynamic responses and corresponding load transfer paths of 2-story RC infilled frames. In this Section the numerical models of 4-story infilled frames are built on top of the ones of the 2-story frame to illustrate the effects of the number of stories on the dynamic responses and load transfer path of the infilled frames. Except the first story, all the rest three story of the infilled frame share the same geometric and material properties.

Middle (C3) column removal scenario

Figure 9 demonstrates the displacement responses of 4-story infilled frame under a middle CRS. It is seen that the failure load is 8.0×Q_g and 4.5×Q_g for F-FHIW and F-IWHO, respectively. Compared with 2-story infilled frame, the failure load increases by 33.3% and 28.5% for F-FHIW and F-IWHO, respectively. This can be attributed to two potential reasons. One is that increasing the number of infilled stories, the increase of overall structural resistance due to involving more structural members is faster than the increase of the applied load. The other is that increasing the number of infilled stories could affect the load transfer paths, which further improves the structural resistance (Yu et al., 2019b). In addition, Figure 10 presents that the initial stiffness of F-FHIW-2FL (2FL denotes two stories) and F-IWHO-2FL is 1.33×Q_g and 0.41×Q_g N/mm, respectively, and the one of F-FHIW-4FL and F-IWHO-4FL is 1.60×Q_g and 0.87×Q_g N/mm, respectively, indicating that increasing the number of infilled stories enhances not only the peak load factor of frames but also the stiffness. In a word, increasing the number of infilled stories is more beneficial to enhance structural resistance for F-FHIW, and to improve structural stiffness for F-IWHO.

Figure 9.

Dynamic responses of 4-story RC infilled frames under a middle CRS: (a) F-FHIW and (b) F-IWHO.

Figure 10.

Load factor-displacement curves of RC infilled frames with different stories under middle CRS.

Figure 11 displays the von-Mises stress contours of 4-story RC infilled frame at different stages under a middle CRS. For F-FHIW-4FL, the change of the load transfer path through infill walls is categorized into three stages. At stage 1 (i.e. vertical displacement less than 5 mm), the load at each story is transferred through the diagonal of each wall panel. At stage 2 (i.e. vertical displacement between 5 and 7 mm), as shown in Figure 11(a), the load paths through the infill wall at each story tend to form a composite trans-story compression arch and the vertical component of compression used to resist gravity load becomes larger. Accordingly, more infilled stories result in a greater peak load factor due to such composite effect. At stage 3 (i.e. vertical displacement greater than 7 mm), the composite trans-story arch starts to fail in a brittle manner accompanied with the dysfunction of trans-story struts, and imminent collapse is about to occur.

Figure 11.

Load transfer path of 4-story RC infilled frames under a middle CRS: (a) F-FHIW-4FL and (b) F-IWHO-4FL.

Figure 11(b) illustrates that the load transfer path through IWHO at each story is basically similar and independent, regardless of the number of infilled stories. At each wall, the load is transferred through the equivalent compression struts at the perimeter of the opening, in which the struts farthest from the middle column are the most critical whereas the ones nearest the middle column basically transfer the least load. Moreover, the presence of opening obstructs the trans-story load path, but the local composite effect is still able to form in neighboring stories. For example, at right-side span, the lower compression strut in the third and fourth story grouped to the right compression strut in the second and third story, respectively.

Consequently, the collapse process of F-IWHO-4FL can be divided into two stages. At stage 1 (i.e. vertical displacement less than 10 mm), the load is transferred through the equivalent compression paths around the opening, as shown in Figure 11(b). After reaching the peak load factor, the infill walls start to damage. This is categorized as stage 2, in which the load is mainly transferred through the equivalent compression struts far from the removed column.

Corner (C1) column removal scenario

Progressive collapse of a building can be induced by column failure at different locations. Previous researches (Li et al., 2016; Shan et al., 2016) on the progressive collapse resistance of RC infilled frames are mainly concentrated on single middle CRS, and macro-based numerical models of infill walls are proposed based on such a scenario as well. Whether those macro-models apply to the other CRS is still unclear. As a result, a corner CRS is additionally concerned in this Section.

Figure 12 demonstrates the dynamic responses of 2- and 4-story RC infilled frames subjected to a corner CRS. It is seen that for both F-FHIW and F-IWHO, the failure load of the 4-story infilled frame exceeds that of the 2-story infilled frame, suggesting that trans-story composite effect exists even in a corner CRS. In addition, the failure load is 3.5×Q_g and 5.5×Q_g for F-FHIW-2FL and F-FHIW-4FL, respectively, under a corner CRS, which is only 58.3% and 68.8% that of the counterparts under a middle CRS, respectively. Meanwhile, compared with the middle CRS, the failure load of F-IWHO-2FL and -4FL under a corner CRS is reduced by 28.6% and 22.2%, respectively. In summary, the RC infilled frame under a corner CRS is more prone to collapse than the one under a middle CRS. This is mainly ascribed to the fact that the boundary restraints to the affected spans under a corner CRS are much weaker than those under a middle CRS. Moreover, F-FHIW is more sensitive to the variation of the boundary restraints, and thus the reduction of peak load factor under a corner CRS is more evident than that of F-IWHO.

Figure 12.

Dynamic responses of RC infilled frames under a corner CRS: (a) F-FHIW-2FL, (b) F-FHIW-4FL, (c) F-IWHO -2FL, and (d) F-IWHO -4FL.

Figure 13 show the von-Mises stress contours of 2- and 4-story RC infilled frames under a corner CRS, respectively. As shown in Figure 13(a), for F-FHIW-2FL, the load path is initially along the diagonal of the wall panel and then shifted to the diagonal of the upper-right remaining infill wall far from the removed column. This is quite similar to the load paths in a middle CRS. For F-FHIW-4FL, the transfer path keeps evolving with three stages. At stage 1, the load is transferred along the wall diagonal at each infilled story. At stage 2, trans-story load paths appear, but those paths only link the neighboring two stories, such as between the fourth and third story, as shown in Figure 13(b). Moreover, the lowest infilled story still has a path along the wall diagonal. Consequently, the load paths in a corner CRS is more complex than those in a middle CRS. At stage 3, in which the applied load exceeds the structural resistance, F-FHIW-4FL is about to collapse and part of the paths at stage 2 is damaged. In comparison, for F-IWHO-2FL and F-IWHO-4FL, as shown in Figure 13(c) and 13(d), the load is basically transferred through the perimeter of the opening, still complying with the four-strut models before the collapse occurs, in which the right-most strut is the most critical path.

Figure 13.

Load transfer path of RC infilled frames under a corner CRS: (a) F-FHIW-2FL, (b) F-FHIW-4FL, (c) F-IWHO-2FL, and (d) F-IWHO-4FL.

Dynamic response of RC infilled frames under double columns loss

In reality, the accidental load could result in multiple column failure, further triggering the progressive collapse of a building, such as Murrah Building Collapse, Oklahoma City, 1995. Therefore, in this Section, the dynamic behavior of RC infilled frames under double CRS are investigated.

Corner (C1) and penultimate (C2) column removal scenario

Under a C1-C2 CRS, the displacement responses measured at remaining column C1 and C2 of infilled frames are shown in Figure 14. Under service load (1×Qg), the concerned vertical displacements of F-IWHO-2FL and -4FL keep increasing, as shown in Figure 14(c) and (d), respectively, suggesting that the two infilled frames are not able to rebalance the applied gravity load and about to collapse. Figure 14(a) and (b) demonstrate that the applied load of 2.0×Qg and 2.5×Qg results in the collapse of F-FHIW-2FL and -4FL, respectively. Consequently, the failure load of F-FHIW-2FL and -4FL is 1.5×Qg and 2.0×Qg, much less than the counterparts under a single CRS.

Figure 14.

Dynamic responses of RC infilled frames under a corner-penultimate CRS: (a) F-FHIW-2FL, (b) F-FHIW-4FL, (c) F-IWHO-2FL, and (d) F-IWHO-4FL.

In accordance with the comparison of Figure 14(a) and (b), the failure load of F-FHIW-4FL is 25% larger than that of F-FHIW-2FL. Likewise, the comparison of Figure 14(c) and (d) shows that under service load 1×Q_g, the vertical displacement of each column of F-IWHO-4FL is always smaller than that of the corresponding column of F-IWHO-4FL. The above two findings suggest that even under double CRS, increasing the number of infilled stories is able to enhance peak load factor, and thus the trans-story load transfer path still exists.

Figure 15 show that the load transfer paths of F-FHIW under a C1-C2 CRS are mainly concentrated at the span C2-C3, and the evolution of load paths is very similar to that under a single corner CRS as shown in Figure 13. For example, in the four-story infilled frames, the path changes from the individual wall diagonal at each infilled story to the trans-story paths. Moreover, the infill walls at the span C1-C2 only increase the applied load with marginal resistance contribution. Therefore, instead of the resistance contributed by infill walls, the gravity load resulting from infill walls in this scenario is more detrimental to the structure against collapse.

Figure 15.

Load transfer path of RC infilled frames under a corner-penultimate CRS: (a) F-FHIW-2FL and (b) F-FHIW-4FL.

Penultimate (C2) and Middle (C3) column removal scenario

Under a C2-C3 CRS, the time histories of displacements measured at remaining column C2 and C3 of F-FHIW and F-IWHO are shown in Figure 16. It is observed the peak load factor of this scenario is much larger than that under C1-C2 CRS. For example, under service load, both F-IWHO-2FL and -4FL survive without collapse. The failure load of F-FHIW-4FL reaches 6×Q_g, which is 1.4 times the failure load under C1-C2 CRS, suggesting that the progressive collapse resistance depends on the scenario very much, and the affected spans with stronger boundary restraints can more easily mobilize the resistance of infill walls.

Figure 16.

Dynamic responses of RC infilled frames under a penultimate-middle CRS: (a) F-FHIW-2FL, (b) F-FHIW-4FL, (c) F-IWHO-2FL, and (d) F-IWHO-4FL.

Figure 17 demonstrate stress contours of F-FHIW at different stages. The gravity load is mainly transferred through the infill walls at the affected span of C1-C2 and C3-C4. The evolution of load transfer paths at the above-mentioned two spans is similar to the one under a single corner CRS. For simplicity, it is not to show the load paths of F-IWHO under double CRS, which are still the same as that under single CRS.

Figure 17.

Load transfer path of 4-story RC infilled frames under a penultimate-middle CRS: (a) F- FHIW-2FL, and (b) F-FHIW-4FL.

Discussions

In Section 3 and 4, the dynamic responses of infilled frames and corresponding load transfer paths are investigated under different CRS. The comprehensive results are very helpful to understand how infill walls affect load factors of RC frames, and to provide suggestions for macro-modeling of infill walls in a large scale computation.

Table 2 lists the critical load and the associated primary load transfer paths of different types of infilled frames under each CRS. Except that C1-C2 CRS results in the collapse of F-IWHO-2FL and F-IWHO-4FL under service load (1.0×Q_g), the infilled frames under other scenarios are all able to rebalance the service load, indicating adequate progressive collapse resistance of the infilled frames. Basically, the critical load under double CRS is smaller than that under single CRS, and the case involving corner column removal is more threatening.

Table 2.

Summary of progressive collapse resistance and load path.

RC infilled frame^*	Column removal scenario	Critical load	Displacement corresponding to critical load/mm	Load transfer path stage2
F-FHIW-2FL	C1	3.0 × Q_g	4.2	–
	C3	5.5 × Q_g	4.5
	C1+C2	1.5 × Q_g	23.6
	C2+C3	3.0 × Q_g	5.6
F-FHIW-4FL	C1	5.0 × Q_g	4.4
	C3	7.5 × Q_g	5.9
	C1+C2	2.0 × Q_g	14.5
	C2+C3	5.0 × Q_g	11.0
F-IWHO-2FL	C1	2.0 × Q_g	10.2	–
	C1+C2	<1.0 × Q_g	–
	C3	3.0 × Q_g	11
	C2+C3	1.0 × Q_g	7.9
F-IWHO-4FL	C1	3.0 × Q_g	11.1	–
	C3	4.0 × Q_g	9.1
	C1+C2	<1.0 × Q_g	–
	C2+C3	1.5 × Q_g	8.5

The first story of the frame contains no infill walls.

Except F-FHIW-2FL under C1-C2 CRS, the monitored displacement corresponding to the critical load for other cases is all less than 17 mm (0.1% beam span), suggesting that prior to the critical load the displacement response is very small, but after the critical load the infilled frames quickly fail in a brittle manner, resulting in imminent collapse. Therefore, engineers should be concerned with the resistance of infilled frames prior to critical load only.

Table 2 also summarizes the primary load paths prior to critical load. For F-IWHO, the load paths are quite constant, equivalently as four compression struts at the perimeter of the opening. For F-FHIW, the initial load path is story-independent, along a wall diagonal at each infilled story, but trans-story paths can further form in multi-infilled stories, in which the trans-story paths form but vary depending on CRS. It is not easy to generalize the macro-models for infill walls. Therefore, for simplicity and safety, it can be conservative to ignore the trans-story paths.

In summary, FHIW and IWHO can be equivalently represented by a single compressive strut and four compressive struts, respectively, as shown in Figure 18. According to (FEMA356, 2000; Shan et al., 2016; Yu et al., 2019a), the width of the compressive strut a for each infill region is suggested as

a = 0.175 {(λ l_{b})}^{- 0.4} r_{\inf}

(1)

where in FHIW, l_b is the length of the beam; In IWHO, l_b is the length of the beam for the strut 1 and 2 and the length from column centerline to the edge of infill region for the strut 3 and 4 (for example, l_b for FHIW and the strut 4 of IWHO are shown in Figure 18); r_inf is the diagonal length of the infill region; and λ is determined based on equation (2):

λ = \sqrt[4]{\frac{E_{m} t_{inf} \sin 2 θ}{4 E_{fe} I_{beam} l_{inf}}}

(2)

Figure 18.

Macro-modeling of the equivalent compressive struts for infill walls.

where θ = arctan (l_inf/h_inf); t_inf is the thickness of the infill region; I_beam is the moment inertia of beams; l_inf and h_inf are the length and height of the infill region; E_m and E_fe are the elastic modulus of the infill region and the surrounding frame, respectively, in which E_m = 550 f’_m and E_fe =5000 $\sqrt{f_{c}^{'}}$ ; and f’_m and f’_c are the compressive strength of the infill walls and the concrete.

In addition, the compressive strength of the equivalent struts can be estimated as f’_m. Because the tensile strength of infill walls is very low, the tensile strength of the equivalent struts is not considered.

Conclusions

In this paper, high-fidelity FE models are built to investigate dynamic responses of RC infilled frames under different column removal scenarios (CRS) and the corresponding load transfer paths. To this end, the FE models of RC infilled frames are established and validated through quasi-static experimental results. Then the same numerical models are analyzed in dynamic loading regime to demonstrate the similarity and the difference of load transfer paths of infill walls in static and dynamic loading regimes. Thereafter, the four-story models are built to explore the effect of the number of infilled stories on the load transfer paths of infill walls. Finally, the suggestions for macro-modeling of infill walls are suggested. The main conclusions are shown below:

For a two-story frame with infill walls at the second story only, the load transfer path prior to the peak resistance is similar under static and dynamic middle CRS. That is, the diagonal compression strut for full-height infill walls (FHIW), and four struts for infill walls having opening (IWHO).

The load transfer paths of four-story frames with FHIW at the top three stories are divided into three stages under a middle CRS. At initial stage, the load is transferred through diagonal of FHIW at each infilled story. At stage 2, trans-story load path similar to a compressive arch forms, followed by the brittle failure at post-peak stage 3. For four-story frames with IWHO, there are only two stage, that is, the load is transferred through the equivalent compression paths around the opening at stage 1, and failure at stage 2.

Under no matter which CRS, trans-story load paths always exist for multi-story infilled frames with FHIW, and the trans-story paths are scenario-dependent. The composite effect of multiple stories for RC frames with IWHO is quite insignificant, locally exists between the two neighboring stories. Moreover, the composite effect is more beneficial to improve the peak resistance of RC frames with FHIW and more helpful to increase the stiffness of RC frames with IWHO.

The comparisons of dynamic responses of RC infilled frames indicate that double CRS is much more dangerous than single CRS, and the scenario with corner column removal is particularly threatening, in which the corner-penultimate CRS is the most critical scenario.

For engineering practice and macro-modeling of infill walls with one-dimensional predefined load paths, it can be conservative to ignore the trans-story load paths in multi-story frames with FHIW and to employ diagonal compression strut at each infilled story for general situations. For frames with IWHO, the four compression strut model around the perimeter of the opening is recommended.

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 authors would like to acknowledge the financial support by the National Natural Science Foundation of China (No. 51978246); Natural Science Foundation of Jiangsu Province of China (No. BK20180073); the Fundamental Research Funds for the Central Universities (No. 2019B12814) and Open Foundation of Engineering Research Center of Construction Technology of Precast Concrete of Zhejiang Province.

ORCID iDs

Jun Yu

Yi-Ping Gan

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