Structures 74 (2025) 108489
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Structures
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Experimental and numerical investigations on progressive collapse
resistance of planar reinforced concrete frames with different storeys
Jian Yuan a, Kun Liu b,c , Shan Wang b,c , Liang Tan d, Shao-Bo Kang b,c,*
a
Academy of Combat Support, Rocket Force University of Engineering, Xi’an 710025, China
Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China
c
School of Civil Engineering, Chongqing University, Chongqing 400045, China
d
China Nerin Engineering Co, Ltd., Nanchang 330031, China
b
A R T I C L E I N F O
A B S T R A C T
Keywords:
Progressive collapse
Reinforced concrete frames
Vierendeel action
Column flexural failure
This study aims to investigate the effect of storey number on the progressive collapse resistance of reinforced
concrete planar frames subjected to the removal of a penultimate column. To investigate the influence of storeys
on the collapse resistance of frames, a single-storey frame and a double-storey frame were tested under a
concentrated point load. The resistance, failure modes, and reinforcement strain development were obtained the
test. Test results indicated that compressive arch action (CAA) and load capacities of the double-storey frame
were approximately 0.9 and 1.0 times greater than those of the single-storey frame, respectively. Besides, in the
double-storey frame, the first-storey beam experienced CAA and catenary action, while the second-storey beam
was under axial compression due to the presence of Vierendeel action. A simplified method was developed to
evaluate the Vierendeel action and load capacity of double-storey frames. Subsequently, a numerical model for
multi-storey frames was established using Diana software to study Vierendeel action. Experimental results
indicated that the contribution of Vierendeel action to the total load reduced as displacement increased. Frames
with varying numbers of storeys exhibited different force transfer mechanisms, and the load capacities increased
with increasing number of storeys. When the number of storeys of the plane frame increased from two to five, the
CAA and load-bearing capacity increased by 144.8 % and 153.8 %, respectively. This study provided a basis for
subsequent research on the progressive collapse resistance of multi-storey asymmetric planar frames.
1. Introduction
Progressive collapse of structures is characterized by the local failure
of critical components under accidental loads, causing the collapse of
adjacent members, and resulting in the failure of a significantly larger
portion of the structure. Such collapses are frequently triggered by accidents such as explosions and earthquakes [1], resulting in extensive
casualties and significant economic losses. These incidents highlight the
importance of investigating the progressive collapse resistance of
structures.
Numerous experimental studies on progressive collapse resistance
have been performed, mainly focusing on four types of structures,
namely, beam-column substructures [2–6], beam-slab substructures
[7–9], planar frames [10–12], and spatial frames [13–15]. These studies
identify three key mechanisms for resisting collapse, namely, flexural
action (FA), compressive arch action (CAA) and catenary action [3,16,
17]. Su et al. [2] performed static collapse tests on twelve beam-column
structures following the removal of the middle column and revealed that
the CAA capacity increased with the reinforcement ratio and the beam
span-to-height ratio. Kang et al. [3,4] carried out experimental tests on
precast concrete beam-slab substructures, and found that considering
axial forces in beams under CAA and catenary action significantly
increased the resistance of beams to progressive collapse. Besides, suggestions were also provided for improving the resistance of precast
concrete members. Gu et al. [9] investigated the influence of floor flange
widths on T-beam substructures and revealed that floor flanges substantially improved the load capacity and promoted earlier mobilisation
of catenary action. Stinger et al. [10] investigated the collapse resistance
of a two-span, double-storey frame featuring either continuous or
discontinuous longitudinal rebars, as well as partially filled walls.
Experimental results showed that the continuous longitudinal rebars
could significantly enhance the CAA capacity of the frame, whereas the
* Corresponding author at: School of Civil Engineering, Chongqing University, Chongqing 400045, China.
E-mail address: kang0119@cqu.edu.cn (S.-B. Kang).
https://doi.org/10.1016/j.istruc.2025.108489
Received 11 October 2024; Received in revised form 25 December 2024; Accepted 15 February 2025
Available online 22 February 2025
2352-0124/© 2025 Institution of Structural Engineers. Published by Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and
similar technologies.
J. Yuan et al.
Structures 74 (2025) 108489
effect on the catenary action capacity could be ignored. A four-span,
three-storey concrete structure was studied for the progressive
collapse resistance by Sagiroglu et al. [11], and it was found that Vierendeel action significantly increased its resistance to progressive
collapse. Xiao et al. [13] conducted tests on a three-storey space frame
with a removed column and found that the frame resisted collapse
through catenary action in beams and yield-line mechanism in floors. A
plastic hinge was observed to form at the end of the bottom column at
failure. The studies indicated that for multi-storey frames, particularly
when boundary restraints were weak, the force transfer mechanisms
differed significantly from those in single-storey frames. In multi-storey
frames, deflections of side columns caused Vierendeel action, which led
to different axial force distributions in the beams at each storey and
made it difficult to accurately assess the collapse resistance of the frame.
Fabio et al. [18] conducted a comparative analysis of the seismic performance of a 10-storey reinforced concrete frame subjected to a sudden
loss of central and corner columns and considered scenarios with and
without the contribution of masonry-infilled walls. The results indicated
that masonry infill walls interacted with the main structure during the
sudden column loss, thereby enhancing the resistance and stiffness of
the frame. Notably, the research on the collapse resistance of
multi-storey frames was still limited.
Currently, many design methods for frames to mitigate progressive
collapse have been proposed. Both direct and indirect methods have
been integrated into design standards [19,20] and guidelines [21,22].
Kang et al. [23] proposed a method to calculate the CAA capacity based
on the equilibrium and deformation compatibility of beams. Long et al.
[24] developed a simplified model to estimate the catenary action capacity by employing the moment capacity of column sections and the
axial force-bending moment interaction diagram of beams. Izzuddin
et al. [25] simplified a frame into basic beam elements based on the
assumption that the force transfer paths in each storey were the same,
and analysed each beam element separately. Then, the overall collapse
resistance of the frame was calculated using linear combinations.
However, this model was only suitable for frames with symmetric or
strong boundary restraints, as it ignored inter-storey interactions within
the frame. In addition, numerous finite element analyses have been
conducted to quantify the progressive collapse resistance of frames.
Feng et al. [26] employed a finite element model to systematically
evaluate the effects of multi-span continuous beams and various column
removal scenarios on reinforced concrete frame structures with and
without slabs. The results showed that continuous span beams increased
the internal compressive force within the beam during the CAA stage
and significantly enhanced the resistance of the substructure. Fu et al.
[27] demonstrated through numerical simulations that in multi-storey
frames, CAA and catenary action occurred only in the first-storey
beam, while the axial forces in the beams on other storeys were quite
limited. Currently, there is no simplified method available for predicting
the Vierendeel action capacity and collapse resistance of multi-storey
frames characterized by weak boundary restraints.
In this study, two reinforced concrete plane frames with a removed
penultimate column were tested subjected to a quasi-static pushdown
loading regime to explore their resistance, failure modes and reinforcement strain development. The effect of storey numbers on the
resistance of frames was investigated. Subsequently, a simplified
calculation method was proposed to predict the collapse resistance of
double-storey frames. Besides, a numerical model was established for
multi-storey frames using Diana software to study the mechanism of
Vierendeel action and the relationship between the storey number and
the load capacity of multi-storey frames.
accordance with the Chinese standards [28–30]. Fig. 1 shows the dimensions of the prototype structure and the specimen locations. The
height of the first storey was 4.2 m, but the height of the remaining
storeys was 3.6 m. The centre-to-centre spacing between columns in two
perpendicular directions was 6.6 m. The live and dead loads were
designed as 2.5 kN/m² and 4.5 kN/m² per storey, respectively. The dimensions of beams were 300 mm × 600 mm, and the column dimensions
were 600 mm square. In this experimental study, double-span planar
frames in the prototype structure were extracted, as shown in Fig. 1. The
scenario of column removal, a realistic condition in engineering applications, was considered on the first storey. To fit the physical constraints
of the laboratory, 1/3 scaled models of the planar frames were used
instead of the prototype frame. Two planar frames, named F-1 and F-2,
were designed for testing. Table 1 provides the dimensions and reinforcement details of the two planar frames. In the designated notation, F
represents the frame specimen, and the second numeral indicates the
number of storeys of each specimen.
Fig. 2 shows the dimensions and reinforcement details of the two
frames. F-1 was designed as a single-storey, two-bay planar frame subjected to the removal of a penultimate column, as shown in Fig. 2(a). The
beam had a clear span of 2000 mm, and the storey height was 1400 mm.
To maintain the boundary conditions of the extracted frame, the beam
was extended by one-fourth of its span beyond the right column. Besides,
four bolts were embedded in the beam extension to connect it to the
horizontal restraint.
The cross-sectional dimensions of columns and beams were 200 mm
× 200 mm and 100 mm × 200 mm, respectively. The beams were
reinforced with two 12 mm diameter steel rebars at the top and two
10 mm diameter rebars at the bottom, and the overall reinforcement
ratio was 1.22 %. The columns contained eight longitudinal steel rebars,
each with a diameter of 8 mm. Transverse reinforcement, consisting of
6 mm diameter steel rebars, was placed at 100 mm spacings in the
beams and columns. In F-2, the reinforcement ratios of beams and columns remained the same at the first and second storeys. However, the
heights of the first and second storeys were different, with 1400 mm for
the first storey and 1200 mm for the second storey, as shown in Fig. 2(b).
To facilitate the installation of frames, the longitudinal reinforcement at
the top of the columns was welded to steel plates with a thickness of
10 mm.
2.2. Material properties
Deformed rebars HRB400 with diameters of 10 mm, 12 mm, and
14 mm were utilized for the longitudinal reinforcement in the beams
and columns, and round rebars HPB300 of 6 mm diameter were utilized
for the transverse reinforcement. The mechanical properties of the steel
rebars were determined through uniaxial tension tests in accordance
with GB/T 228–2010 [31], as shown in Fig. 3. Table 2 summarizes the
tensile properties of steel rebars. The yield strength of stirrups was
367 MPa, and those of longitudinal reinforcement with diameters of
10 mm, 12 mm, and 14 mm were 461 MPa, 434 MPa, and 415 MPa,
respectively. A series of uniaxial compression tests were conducted on
cubic concrete samples with dimensions of 150 mm to measure the
compressive strength of concrete. The compressive strengths of concrete
for frames F-1 and F-2 were 47.2 MPa and 46.9 MPa, respectively.
2.3. Test setup and instrumentation
The test setup for F-1 and F-2 consisted of the loading system, reaction frame and restraint systems, as shown in Fig. 4. To simulate a
progressive collapse scenario, the top of the main column was loaded
with a quasi-static force from a hydraulic jack with a load cell, and
compression forces were also applied to side columns using hydraulic
jacks. The hydraulic jacks were connected to the reaction frame using
rollers to ensure that the applied compression remained vertical. To
replicate the boundary conditions, the right beam extension was
2. Experimental programme
2.1. Test specimens
A five-storey prototype reinforced concrete frame was designed in
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Structures 74 (2025) 108489
Fig. 1. Geometric dimensions of the prototype structure and the location of specimens.
Table 1
Details of test frames.
Specimen
Beam
Cross-section (mm)
F− 1
F− 2
100 × 250
100 × 250
Column
Reinforcement
Top
Bottom
2T12
2T12
2T10
2T10
Stirrups
Cross-section (mm)
Stirrups
Reinforcement
R6@100
R6@100
200 × 200
200 × 200
R6@100
R6@100
8T10
8T10
connected to the reaction wall using hinges to consider the horizontal
restraints of the remaining frame on the extracted frame. Two steel
beams were used to fix the frame to the testing floor with four anchor
bolts. Besides, to increase the stiffness of the reaction frame, the steel
beam was fastened to the ground using four vertical tension rods. Due to
the different lengths of the hydraulic jacks, connecting rods were used to
link the hydraulic jack to the top of the side columns. Before testing, a
constant vertical axial compression force of 152.8 kN was applied to
each side column to simulate the load transferred from the upper storey,
corresponding to an axial compression ratio of 0.2. Thereafter, a
displacement-control loading was applied to the middle column at a rate
of 2 mm/min using a hydraulic jack until final failure occurred.
Fig. 5 shows the arrangement of the linear variable displacement
transducers (LVDTs) and strain gauges. The vertical load applied was
recorded by the load cell within the hydraulic jack, and the horizontal
reaction force was monitored by load cells placed in the horizontal
hinges. The vertical displacement of the middle column was monitored
using LVDT R-3. Additionally, the vertical deflection of the beams in F-1
and at the first storey of F-2 was recorded by four LVDTs installed
beneath the beams. Horizontal deformations of the columns were
measured with LVDTs arranged along the column height. Strain gauges
were employed to measure the strains in the longitudinal reinforcement
of beams and columns.
Beam clear span
(mm)
Number of storeys
Horizontal
restraint
2000
2000
1
2
Right side
Right side
stage. F-1 and F-2 maintained load capacities of 30 kN and 57 kN,
respectively. As the displacement increased further, the vertical load
increased gradually due to the onset of catenary action. F-1 achieved its
catenary action capacity of 31.7 kN at 358 mm displacement (point D),
only 5 % higher than its CAA capacity. Then, a sudden drop in the load
of F-1 occurred due to the fracture of the bottom reinforcement on the
left side of the middle joint, indicating the failure of the frame. Frame F2 eventually reached a load capacity of 63.7 kN under catenary action,
11.6 % higher than its CAA capacity, at a vertical displacement of
331 mm. Two sudden drops in the curve of F-2 corresponded to the
fracture of the bottom reinforcement on the right side of the secondstorey middle joint at a displacement of 340 mm and that of the firststorey middle joint at 362 mm.
Fig. 7 presents the horizontal reaction force-middle point displacement curves for the frames, with positive values in tension and negative
values in compression. Initially, the horizontal compression force in F-1
gradually increased to the first peak at 115 mm displacement. As the
vertical displacement increased, the axial compression force gradually
decreased and was transformed into tension between 200 mm and
320 mm of vertical displacement. The plateau stage might be caused by
connection gaps in the lateral restraint. Under catenary action, the
horizontal force increased sharply once the vertical displacement
exceeded 320 mm. Finally, the horizontal force was reduced due to the
failure of the frame. For frame F-2, the horizontal force-displacement
curve of the first storey was similar to that of F-1, but the peak
compression of the former was larger than that of the latter. However,
the horizontal force in the second storey remained in compression
throughout the loading process. It initially increased rapidly to 5.0 kN at
a 29 mm displacement, and then gradually dropped to 2.9 kN when the
displacement reached 323 mm. Thereafter, a sharp increase in the
horizontal force occurred at 340 mm displacement when steel rebars in
the beam on the right of the second-storey middle joint fractured.
Notably, the axial force in the second storey continued to increase after
the steel rebars fractured, due to Vierendeel action in the frame under
large deformation conditions.
Table 3 summarizes the load resistances and the associated
3. Test results and discussions
3.1. Load-displacement curves
Fig. 6 shows the vertical load-middle joint displacement curve of
frames F-1 and F-2. Initially, the curve increased linearly, and steel rebars and concrete were in the elastic stage. Once the vertical displacements reached 20 mm for F-1 and F-2, respectively, the load started
increasing slowly, as the longitudinal reinforcement in the beams near
the middle joints began to yield. Then, F-1 reached its CAA capacity of
30.1 kN at a displacement of 52 mm (point C), while F-2 reached
57.1 kN at 67 mm (point A). Thereafter, the frames entered a plateau
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Structures 74 (2025) 108489
Fig. 2. Dimensions and reinforcement details of specimens.
displacements of F-1 and F-2 at different stages. The CAA capacity of F-2
was 57.1 kN, 0.90 times greater than that of F-1, and the displacement at
this capacity of F-2 was also larger than that of F-1. The load capacity of
F-2 was 1.0 times greater than that of F-1, whereas the associated
displacement of F-1 was 27 mm larger than that of F-2. Therefore, the
load capacity of frames increased significantly as the number of storeys
increased from one to two. However, the deformation capacity of F-2
was less than that of F-1, which might be caused by the larger bending
moment at the middle joint at the second storey of F-2.
two bottom steel rebars fractured at the middle joint, and crushing and
spalling of the concrete were also observed in the compression zone at
the beam end.
Fig. 9 illustrates the failure modes of the beams in F-2. Fig. 9(a)
shows that the beam in the first storey exhibited failure modes similar to
those of F-1. However, the cracks in the second-storey beam were primarily located at the beam ends, without apparent tension cracks along
the beam length, as shown in Fig. 9(b). It indicated that catenary action
was not mobilized at the second storey. The bottom reinforcement
adjacent to the middle joints at the first and second storeys fractured at
340 mm and 390 mm displacements, respectively. Moreover, at the
middle joints, obvious crushing of concrete occurred at the top of the
beam, while crushing of concrete appeared at the bottom of the beam at
the side joints. Notably, the crushing of concrete at the beam ends at the
second storey was more severe than at the first storey due to the presence of compressive forces in the beam.
Fig. 10 illustrates the failure modes of side columns in F-1 and F-2. As
depicted in Fig. 10(a and b), the left and right columns of F-1 exhibited
different failure modes due to differences in the boundary condition. On
the outside face of the column sections beneath the side joint, flexural
3.2. Crack patterns and failure modes
Fig. 8 presents the failure modes of beams in F-1. Bending cracks
occurred in the tension zone at the beam ends when the vertical
displacement reached 20 mm. The cracks then propagated towards the
compression zone, reducing the neutral axis depth and activating CAA.
As the displacement increased, diagonal cracks developed at the beam
end due to the combined effect of bending moment and shear force.
Thereafter, catenary action induced axial tension in the beam, resulting
in tension cracks along its length. At a vertical displacement of 370 mm,
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Structures 74 (2025) 108489
Fig. 3. Uniaxial tension tests of steel rebars.
Table 2
Mechanical characteristics of steel reinforcement.
Material
Stirrup
Longitudinal reinforcement
R6
T10
T12
T18
Diameter
(mm)
Yield strength
(MPa)
Ultimate strength (MPa)
Elastic modulus (GPa)
Yield strain
(με)
6
10
12
18
367
461
434
415
518
600
608
593
166
206
203
190
2713
2360
2090
2290
Fig. 5. Arrangement of LVDTs and strain gauges in F-2.
the side joints at the first and second storeys due to CAA. Besides,
flexural cracks were also observed on the outer face of the first-storey
column near the footing due to catenary action. However, due to the
presence of the horizontal restraint at the right joint of the first storey,
CAA did not induce flexural cracks at the right column at the first storey.
Under catenary action, bending cracks developed at the right joint at the
first storey. Finally, evident crushing of concrete appeared at the footing
of the left column. Notably, the right column exhibited smaller lateral
deformations and fewer cracks than the left column due to the horizontal
support.
Fig. 4. Experiments setup for frames.
cracks first emerged in the CAA stage. Multiple flexural cracks appeared
close to the footing during the catenary action stage as a result of inward
bending of the side columns under axial tension in the beam. As the
vertical displacement increased, diagonal cracks developed at the side
joint. Notably, the right column showed fewer bending cracks and a
shorter diagonal crack than the left column, due to the restraint provided by the horizontal support. Finally, flexural failure of the column
occurred due to the crushing of concrete at the base of the left column.
Fig. 10(c and d) show the failure modes of the side columns in F-2.
For the left column, flexural cracks appeared on the outer surface below
3.3. Deformation capacity
3.3.1. Deformation capacity of beams
Fig. 11 illustrates the vertical deformation profiles of the beam at
various load levels. Initially, each beam segment remained nearly
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Structures 74 (2025) 108489
Vertical load (kN)
80
F-1
F-2
A (67 mm,57.1 kN)
60
observed to be smaller than that of F-1. This was mainly due to that the
beam at the second storey of F-2 provided an additional force redistribution path.
B (331 mm,63.7 kN)
θc =
40
C (52 mm,30.1 kN)
3.3.2. Deformation capacity of columns
The lateral deflection of columns in F-2 was measured and analysed,
as shown in Fig. 12. Positive values signify deflections towards the
middle joint, while negative values denote deflections away from the
joint. As seen in Fig. 12(a), at the CAA stage, the axial compression force
pushed the side columns away from the joint, and then the side columns
were pulled towards the joint by the axial tension force under catenary
action. Catenary action induced greater deformations in the side column
compared to CAA. Finally, the inward displacement of the column
increased rapidly due to cracking on the outer side of the side column.
The displacement of the second-storey column exceeded 62 mm at
failure and beyond the range of the LVDTs. From the results of 2-L6 and
2-L8, it was found that the lateral deflection of the right column was
nearly zero under CAA, and it was also noticeably smaller than that of
the left column when catenary action started. Fig. 12(b) illustrates the
deflections of the left column. Under CAA, the outward deflection at the
first-storey joint was almost equal to that at the second-storey joint,
indicating that the relative deformation between them could be ignored.
However, under catenary action, the lateral deflection at the secondstorey joint exceeded that at the first-storey joint. The maximum
lateral deflection of the left column reached 62.5 mm.
Fracture of bars
0
100
200
300
400
Middle joint displacement (mm)
500
Fig. 6. Vertical load-displacement curves of frames.
Horizontal force (kN)
20
10
0
-10
-20
F-1
1st storey of F-2
2nd storey of F-2
-30
-40
0
100
200
300
400
Middle joint displacement (mm)
500
3.4. Strain development
Fig. 7. Horizontal reaction force-displacement curves of frames.
Fig. 13 presents the strain development of steel reinforcement at the
beam mid-span of F-2. The analysis focused on the left beam segments,
due to the consistent strain variations observed between the left and
right beam segments. The strain at 1-BT3-L and 1-BB4-L were consistently tensile, while the strain at 1-BB3-L was kept compressive. Interestingly, the strain at 1-BT4-L changed from compressive to tensile at a
displacement of 280 mm, signifying the change from CAA to catenary
action in the first storey. However, as illustrated in Fig. 13(b), the axial
force at 2-BT4-L at the equivalent position 1-BT4-L on the second storey
remained compressive throughout testing, indicating that catenary action did not occur in the second storey.
Fig. 14 presents the strain development of steel rebars in columns of
F-2. Fig. 14(a) displays that the reinforcement strain in the left column of
F-2 was similar to that of F-1. Notably, the strain at 2-CB3-L, located at
the bottom of the second-storey column, did not change from tensile to
compressive. This might be due to the inward movement of the firststorey column and the downward rotation of the first-storey beam.
Moreover, at the CAA stage, the strain of the first-storey column was
larger than that of the second-storey column, suggesting more deformations in the first-storey column under CAA.
Fig. 14(b) displays the strain development of the steel rebars in the
right column. At the CAA stage, the strains in the first-storey column
were close to zero due to the presence of horizontal restraints connected
to the column. After a displacement of 340 mm, the strains at 1-CT1-R,
1-CT3-R and 2-CB1-R increased significantly. It was due to that the steel
rebar at the second-storey middle joint fractured, which increased the
axial tension of the first-storey beam, thereby increasing the bending
moment at the top of the first-storey column and the bottom of the
second-storey column. Besides, the fracture of the reinforcement also
caused a decrease in the reinforcement strain in the second-storey beam,
significantly reducing the strains at 2-CT1-R and 2-CT3-R at the top of
the second-storey column.
Table 3
Load capacities and associated displacements of frames.
Specimen
F− 1
F− 2
Compressive arch action
Catenary action/ the second peak
capacity
CAA
capacity
(kN)
Displacement
(mm)
Load
capacity
(kN)
Displacement
(mm)
30.1
57.1
52
67
31.7
63.7
358
331
(1)
where θc represents the chord rotation of beams, δm represents the
ultimate displacement at the middle joint, and ln represents the net span
of the beam.
D (358 mm,31.7 kN)
20
0
δm
ln
straight, and the rotation primarily occurred near the joint. F-1 exhibited
symmetrical deformation, as shown in the Fig. 11(a). The difference in
final deformation was caused by the fracture of the bottom reinforcement at the beam end close to the middle joint. In F-2, the deformation
of the first-storey beam remained symmetrical even after the bottom
reinforcement fractured at the beam end at the second-storey middle
joint, as shown in Fig. 11(b). However, the deformation of the firststorey beams in F-2 became different after the bottom steel reinforcement fractured at the first-storey joint.
Table 4 includes the plastic hinges at the middle joint and the side
column rotating. These rotations were determined by dividing the ultimate displacement at R3 by the net span of beams and by dividing the
ultimate displacement at R1 by the distance from R1 to the left column.
The chord rotation of the beams was determined by the ratio of the ultimate displacement at the middle joint to the net span of the beams, as
expressed in Eq. (1) [32]. It was found that the chord rotation was more
significant than the rotation of the plastic hinges at the middle joints.
These rotations surpassed the rotation threshold of 0.20 rad prescribed
in UFC 4–023–03, which is required for beams to develop alternate load
paths. Notably, in the first storey of F-2, the plastic hinge rotation was
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Fig. 8. Failure modes of beams in F-1.
Fig. 9. Failure modes of beams in F-2.
Fig. 10. Failure modes of side columns in F-1 and F-2.
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Vertical displacement (mm)
Vertical displacement (mm)
J. Yuan et al.
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Structures 74 (2025) 108489
Left side column
0
R3
R2
catenary action. The relationship between the axial tension force (Tb1 ) in
the beam, the bending moment (Mb1 ) at the end of the first-storey beam,
and the bending moment (MA1 ) at the bottom of the left column can be
obtained from Eq. (2). Moreover, the effect of shear (Vb1 ) is ignored as
shear diminishes under catenary action.
Right side column
Penultimate column
R1
R4
R5
-100
-200
-300
-400
Original position
CAA
CA
Bar fracture
Final
The beam as
a rigid segement
MA1 + Vc (l1 + l2 ) = Mb1 + Tb1 ⋅l1
where MA1 is the bending moment at the bottom of the left column,
Vc is the shear force at the inflection point, l1 and l2 are the distances
measured from the centre of the first-storey beam to the bottom of the
first-storey column and the inflection point of the second-storey column,
Mb1 is the bending moment at the end of the first-storey beam, and Tb1 is
the axial tension force in the first-storey beam.
Fig. 16 displays the relationship between the axial force and the
bending moment [24]. Between the point (M0 , 0) corresponding to the
pure flexure of the beam section and the point (M1 , T1 ) corresponding to
the tensile yielding of the top and bottom reinforcement, it is assumed
that the axial force and bending moment followed a linear relationship.
When all reinforcement yields, the negative moment was not zero as the
cross-sectional area of the top reinforcement is larger than that of the
bottom reinforcement. The correlation between the negative bending
moment and the axial tension can be established as follows:
-500
-2500 -2000 -1500 -1000 -500
0
500 1000 1500 2000 2500
Monitor point position (mm)
100
(a) F-1
Left side column
2-R1
0
Right side column
Penultimate column
2-R3
2-R2
2-R4
2-R5
-100
-200
-300
-400
Original position
CAA
CA
2nd bar fracture
1st bar fracture
Final
The beam as
a rigid segement
-500
-2500 -2000 -1500 -1000 -500
0
500 1000 1500 2000 2500
Monitor point position (mm)
(b) F-2
Tb1 =
Fig. 11. Vertical deformation profiles of beams in frames.
F− 1
The first
storey
of F− 2
Rotation of plastic hinges
at middle joint (rad)
Rotation of plastic hinges
at side column (rad)
Left
Right
Mean
value
Left
Right
Mean
value
0.22
0.20
0.22
0.22
0.22
0.21
0.24
0.23
0.22
0.21
0.23
0.22
Chord
Rotation θc
(rad)
0.24
0.24
4. Simplified calculation method
In the scenario of penultimate column removal at the first storey, as
the first storey entered the catenary action stage, Vierendeel action
commenced in the frame, with the second-storey beam in compression
and the first-storey beam in tension. Assuming that the inflection point
of the second-storey column was located at the mid-height of the column, simplified models of the first and second storeys were established
to calculate the load capacity of the double-storey frame.
Fig. 15 presents the force equilibrium of the left column under
2-L1
2-L2
2-L3
2-L4
50
40
Pca =
2-L6
2-L8
30
20
10
0
-10
0
100
200
300
400
Middle joint displacement (mm)
(3)
2(Mb + Mm +
βl
Tb δ)
(4)
where βl is the net span length of the single-span beam, Mm is the
bending moment of the beam at the middle joint, δ is the vertical
displacement of the middle joint, taken as the vertical displacement
when the chord rotation of the beam is 0.2 rad [22]. The sagging
moment (Mm ) at the middle joint can be obtained from the interaction
Distance to bottom support (mm)
Side column deflection (mm)
70
60
M0 − Mb1
T1
M0 − M1
where M0 is the moment capacity of the beam in pure flexure, M1 and
T1 are the bending moment and axial force when the top and bottom
reinforcement yields simultaneously.
The axial compression force (Nva ) in the second-storey beam is
estimated to be 25.6 kN based on the model Wang [33], and it is also
assumed that this force is equal to the peak axial compression force at
the CAA stage when frame F-2 attains its load capacity. Therefore, the
shear force at the inflection point (Vc ) is equal to 25.6 kN. Moreover, the
reinforcement at the bottom of the left column yields. Using the
cross-sectional area of the reinforcement and mechanical properties, the
bending moment (MA1 ) is calculated to be 24.2 kN⋅m. Eqs. (2) and (3)
can be used to calculate the bending moment (Mb ) and the associated
axial tension force (Tb ).
Fig. 17 shows the force equilibrium of the first-storey beam under
catenary action. Eq. (4) expresses the catenary action capacity (Pca ).
Table 4
Plastic hinge and chord rotations under catenary action.
Storey
(2)
500
(a) Side column deflection-middle joint
displacement curve
3000
Original position
2500
2000
1500
1000
CAA-67 mm
CA-331 mm
1st fracture-340mm
2nd fracture-390mm
Final-480 mm
500
0
-20
0
20
40
60
Lateral deflection (mm)
80
(b) Lateral deflections of the left side column
Fig. 12. Lateral deflections of column in F-2.
8
J. Yuan et al.
Structures 74 (2025) 108489
3000
3000
Yield strain
1000
0
-1000
-2000
-3000
0
1-BT3-L
1-BT4-L
Yield strain
2000
1000
Strain (µe)
Strain (µe)
2000
-1000
1-BB3-L
1-BB4-L
Yield strain
100
200
300
400
Middle joint displacement (mm)
0
2-BT3-L
2-BT4-L
-2000
-3000
0
500
2-BB3-L
2-BB4-L
Yield strain
100
200
300
400
Middle joint displacement (mm)
500
(b) Left beam segment in the second storey
(a) Left beam segment in the first storey
Fig. 13. Strain development of steel rebars at beam mid-span of F-2.
3000
Yield strain
2000
1000
Strain (µe)
Strain (µe)
2000
3000
0
-1000
-2000
-3000
0
1-CT4-L
2-CT3-L
1-CT1-L
1-CB1-L
2-CB1-L
1000
0
-1000
2-CT1-L
1-CB4-L
2-CB3-L
Yield strain
100
200
300
400
Middle joint displacement (mm)
Yield strain
1-CT1-R
2-CT1-R
2-CB1-R
-2000
-3000
0
500
(a) Left column
1-CT3-R
2-CT3-R
1-CB3-R
2-CB3-R
Yield strain
100
200
300
400
Middle joint displacement (mm)
500
(b) Right column
Fig. 14. Strain development of steel rebars at column sections of F-2.
400
200
Axial force (kN)
Negative moment
Assumed relationship
Positive moment
T1
0
M0
M1
-200
-400
-600
0
10
20
30
40
Moment (kN·m)
50
60
Fig. 16. Relationship between bending moment and axial force.
Fig. 17. Force equilibrium of first-storey beam under catenary action.
the sagging moment of the beam at the middle joint (Mva ) is 14.2 kN⋅m.
The load capacity of the second-storey beam under Vierendeel action
can be obtained from Eq. (5).
Fig. 15. Force equilibrium of left side column under catenary action.
diagram of the axial force and bending moment, as depicted in Fig. 16.
Fig. 18 shows the force equilibrium of the second-storey beam under
Vierendeel action. According to Fig. 16, with an axial force of 25.6 kN,
2
Pva = (Nva (l3 − δ) + Mva )
βl
(5)
where Pva is the second-storey beam load capacity under Vierendeel
9
Structures 74 (2025) 108489
J. Yuan et al.
5.3. Discussion of numerical results
5.3.1. Force in each storey
Based on the numerical results, the vertical load sustained by each
storey of the frame can be obtained from the force equilibrium of the
middle joints at each storey. Fig. 22 shows the force equilibration of the
middle joint in the first storey of F-2.
The vertical load carried by the first-storey frame (P1 ) can be obtained from Eq. (6).
Fig. 18. Force equilibrium of second-storey beam under Vierendeel action in
F-2.
P1 = V1L cosθ + N1L sinθ + V1R cosθ + N1R sinθ
action, l3 is the distance from the centroid of the second-storey beam to
the inflection point of the second-storey column.
The total load capacity of frame F-2 is determined by superposing the
catenary action capacity of the first-storey beam and the resistance of
the second-storey beam under Vierendeel action. As shown in Table 5,
the analytical and experimental load capacity of F-2 are determined to
be 58.1 kN and 63.7 kN, respectively. The ratio of analytical to experimental results is 0.91, indicating good agreement between the analytical
and test results.
(6)
where V1R and V1L are the shear forces in the right and left beams
near the middle joint of the first storey, respectively, N1R and N1L are the
axial forces in the right and left beams at the first storey, and θ is the
relative rotation angle of the first-storey beam.
Fig. 23(a) shows that the load sustained by the first and second
storeys are almost the same during the flexural and CAA stages. Once the
displacement reaches about 240 mm, the load on the first storey increases due to the commencement of catenary action, but the load on the
second storey gradually decreases. The curves of the beam at the first
and second storeys experience a sudden drop at 325 mm and 360 mm,
respectively, corresponding to the fracture of the bottom reinforcement
of the beam at the middle joint in each storey. Before a displacement of
240 mm is reached, the loads on the first and the second storeys account
for about 50 % of the total load, respectively, as displayed in Fig. 23(b).
However, once the frame reaches its load capacity, the proportion of the
load sustained by the beam at the first storey increases to 61.6 %, but the
portion carried by that at the second storey decreases to 38.4 %.
5. Numerical simulations of frames
5.1. Finite element model and material models
Considering the negligible out-of-plane deformation of frames
observed during testing, a two-dimensional plane model is established
by DIANA for the frame with a removed first-storey middle column, as
shown in Fig. 19. The plane-stress element CQ16M is utilized for concrete, and the embedded reinforcement rebar element is used for the
reinforcement in DIANA software. The bond between concrete and
reinforcement is ignored due to the use of embedded reinforcement elements [27]. The model mesh size is determined to be 30 mm. Notably,
the horizontal stiffness of the frame is simulated by spring elements at
the beam extension, with the axial stiffness for the first and
second-storey beams set as 15 kN/mm and 20 kN/mm, respectively.
Fig. 20(a) shows that the stress-strain curve of steel reinforcement is
simplified as a piece-wise linear model, with the mechanical properties
obtained from tension tests. The rotating crack model depend on total
strain is used for the concrete. The concrete models in tension and
compression are the nonlinear softening model developed by Hordijk
and the model developed by Maekawa, respectively, as displayed in
Fig. 20(b and c).
5.3.2. Internal force in beams
Fig. 24 shows the variation of forces in beams of F-2, with the force
acting at the beam mid-span. During the CAA stage, the axial force in the
first-storey beam is in compression, and is shifted to tension at 240 mm
due to the commencement of catenary action, as displayed in Fig. 24(a).
However, the axial force in the second-storey beams remains in
compression throughout the loading process. Fig. 24(b) presents the
shear force-displacement curve in beams. The curves in the first-storey
beam are consistent with those in the second-storey beam before a
displacement of 240 mm is reached. Then, the shear force in the firststorey beam begins to increase due to the onset of catenary action,
whereas the shear force in the second-storey beam decreases gradually.
5.3.3. Mechanism under Vierendeel action
Vierendeel action in double-storey reinforced concrete frames is
characterized by axial tension in the first-storey beam and axial
compression in the second-storey beam. According to the abovementioned analysis, the second-storey resistance is mainly provided by
the bending moment at the beam end and the axial force in the beam.
Thus, a parameter α is defined to represent the contribution of Vierendeel action to the resistance of the second storey, β represents the
contribution of the bending moment at the second-storey beam end to
the resistance of the second storey, γ represents the contribution of
Vierendeel action to the overall resistance of F-2, as expressed in Eqs.
(7–9).
5.2. Model validation
Fig. 21 presents a comparison between the test and numerical results
of F-2. As illustrated in Fig. 21(a), the numerical vertical loaddisplacement curve shows good agreement with the test curve, and
the numerical model accurately predicts the load capacity of F-2 at each
stage. Moreover, the numerical horizontal displacement-displacement
curve agrees well with the experimental results, and the numerical
model effectively predicts the outward and inward deflections of the left
column relative to the middle joint, as displayed in Fig. 21(b). Fig. 21(c)
shows that the failure mode and crack distribution in the numerical
results are consistent with those observed in the experimental results.
Table 5
The resistance of F-2 under catenary action and Vierendeel action.
Frame
Simplify calculation model
Resistance of single-storey under
catenary action
Pc (kN)
F− 2
The first storey
The second
storey
​
38.8
Experimental result
Resistance of single-storey under
Vierendeel action
Pva (kN)
19.32
10
Load capacity of frame Pca
(kN)
Load capacity of the
frame
Peca (kN)
58.1
63.7
Pca /Peca
0.91
J. Yuan et al.
Structures 74 (2025) 108489
s
ft
fu
fy
0 ?y ?s,h
?s,u
Tensile strain
(a) Piece-wise linear model for
reinforcement
s
Tensile stress
Tensile stress
Fig. 19. Numerical modelling for F-2.
?
GIf /h
0
?
Tensile strain
(b) Hordijk model for concrete
tension
fc'
(c) Maekawa model for concrete
compression
Fig. 20. Constitutive models for concrete and steel reinforcement.
Horizontal displacement (mm)
Vertical load (kN)
80
60
40
20
0
Test result
FEA result
0
100
200
300
400
Displacement (mm)
500
(a) Vertical load-displacement curves
80
L-1/Test
L-2/Test
L-3/Test
L-4/Test
60
40
L-1/FEM
L-2/FEM
L-3/FEM
L-4/FEM
20
0
-20
0
100
200
300
400
500
Middle point displacement (mm)
(b) Horizontal displacement- displacement curves
(c) Failure modes
Fig. 21. Comparisons of experimental and numerical results of F-2.
11
J. Yuan et al.
Structures 74 (2025) 108489
of the second-storey beam increases from 51.1 % to 64.0 %. Besides, the
contribution of Vierendeel action to the overall resistance of F-2 decreases from 23.0 % to 13.8 % with increasing vertical displacement due
to the larger catenary action in the first-storey beam and the bending
moment at the second-storey beam end.
Fig. 22. Force equilibration of the middle joint at the first storey of F-2.
Nva (l3 − δ)
Pva βl/2
(7)
β=
Mva
Pva βl/2
(8)
γ=
Nva (l3 − δ)
Pβl/2
(9)
To accurately simulate the boundary restraints of the frame, a numerical model of the double-storey, four-span frame (F2B4) is
Contribution to resistance (%)
α=
5.4. Influence of the number of storeys
where P is the overall vertical load carried by frame F-2, Pva is the
resistance obtained by the second-storey beam under Vierendeel action,
Nva is the axial force in the second-storey beam under Vierendeel action,
Mva is the bending moment at the beam end near the middle joint at the
second storey, and l3 is the distance between the second-storey beam and
the inflection point of the second-storey column.
The contribution of Vierendeel action and the bending moment at
the second-storey beam end to the resistance of the second storey is
calculated based on numerical results, as shown in Fig. 25. With the
vertical displacement increases, the contribution of Vierendeel action to
the resistance of the second-storey beam decreases from 48.9 % to
36.0 %, while the contribution of the bending moment to the resistance
Total load
First storey
(330 mm,55.3 kN)
Second storey
60
35 mm,56.8 kN)
(360 mm,41.9 kN)
40
(35 mm,28.9 kN)
(35 mm,27.9 kN)
20
0
(325 mm,24.8 kN)
0
80
60
64.0
51.1
40 48.9
20
0
36.0
23.0
13.8
240
260
280
300
320
Middle joint displacement (mm)
100
80
60
40
20
0
100
200
300
400
Middle joint displacement (mm)
(a) Vertical load carried by single-storey
frame
a
ß
?
Fig. 25. Contribution of Vierendeel action and bending moment to
the resistance.
Percent of total load (%)
Vertical load (kN)
80
100
Second storey
First storey
0
100
200
300
400
Middle joint displacement (mm)
(b) Contribution of the load carried by each layer of
the frame to the total load
Fig. 23. Load resistance of beams at each storey in frame F-2.
40
1st-AB beam
2nd-AB beam
1st-BC beam
2nd-BC beam
40
20
Shear force in beam (kN)
Axial force in beam (kN)
60
0
-20
-40
0
20
0
-20
-40
100
200
300
400
Middle joint displacement (mm)
(a) Axial force
1st-AB beam
1st-BC beam
0
100
200
300
400
Middle joint displacement (mm)
(b) Shear force
Fig. 24. Variation of forces in beams of F-2.
12
2nd-AB beam
2nd-BC beam
J. Yuan et al.
Structures 74 (2025) 108489
established, and its modelling method, material parameters and loading
method are consistent with those of F-2, as shown in Fig. 26(a). Besides,
to study the effect of the storey number on the resistance of multi-storey
planar frames, models F3B4, F4B4 and F5B4 are developed based on the
F2B4 model, as shown in Fig. 26(b-d).
Fig. 27 shows the vertical load-displacement curve of multi-storey
planar frames. The CAA capacities of F3B4, F4B4 and F5B4 are 1.48
times, 1.96 times and 2.45 times as much as that of F2B4, respectively.
Moreover, the load capacities of F3B4, F4B4 and F5B4 are 1.55 times,
2.00 times and 2.54 times as much as that of F2B4. Therefore, in scenarios with the penultimate column on the first storey being removed,
the CAA and load capacities of the multi-storey planar frame show a
roughly linear increase with the storey number. However, the effect of
the number of storeys on the deformation capacity of multi-storey
frames can be neglected.
Fig. 28 illustrates the vertical load-displacement curve of each beam
in multi-storey frames. It is evident that the curves of the middle storeys
are almost the same. At the CAA capacity, the first storey contributes the
most to the total load, the middle storey contributes the least to the total
load, and the difference in the load sustained by each storey is insignificant. With the displacement increases, the load sustained by the firststorey beam increases significantly due to the mobilisation of catenary
action in the first storey. Meanwhile, the load sustained by the beam in
the middle storey gradually increases, whereas the vertical load on the
top storey remains relatively constant. Upon the load capacity is
reached, the first-storey beam still contributes the most to the total load,
but the smallest contribution to the total load comes from the top storey.
This indicates different vertical load transfer mechanisms are mobilised
at the first, middle and top storeys.
Vertical load (kN)
200
163.2
135.4
100
129.2
108.6
99.9
81.9
50
0
55.3
0
64.3
F2B4
F4B4
F3B4
F5B4
100
200
300
400
Middle joint displacement (mm)
Fig. 27. Vertical load-displacement curve of multi-storey planar frames.
storey joint of F-2, while no tensile cracks were found along the
beam span at the second storey.
(3) In F-2, the reinforcement at 1-BT-L was shifted from compression
to tension due to the development of catenary action, whereas the
steel rebars at the corresponding position in the second-storey
beam remained under compression. This highlighted the difference in the force transfer mechanisms between the first and
second storeys of F-2.
(4) Assuming that the inflection point of the column is at the
midpoint of the second-storey column, a method was developed
to obtain the load capacity of the double-storey frame. This
method assumes that the negative bending moment varies linearly with the axial force in the beam. It can accurately predict the
load capacity of the frame, with the ratio of the calculated to
experimental load capacity of 0.91.
(5) According to the verified finite element model, the contribution
of Vierendeel action to the load capacity decreases with
increasing displacement. Moreover, frames with varying numbers
of storeys exhibit different force transfer mechanisms, and the
load capacities of the frames increase nearly linearly with the
number of storeys. At the CAA capacity, the first-storey beam
contributes the most to the total load, while the middle-storey
beam contributes the least. However, at the ultimate load capacity, the contribution of the middle-storey beam to the overall
load exceeds that of the top-storey beam.
6. Conclusions
This paper explores the resistance, failure modes, and reinforcement
strain development of two reinforced concrete planar frames under a
scenario of removing the penultimate column. The frames were tested
under a quasi-static pushdown loading regime to analyse the influence
of the number of storeys on the resistance of frames. Subsequently, a
simplified model was proposed to quantify the collapse resistance of
double-storey frames. Besides, a numerical model of multi-storey frames
was developed by using Diana finite element software to study the
mechanism of Vierendeel action. The following conclusions can be
drawn:
CRediT authorship contribution statement
(1) The vertical load-middle joint displacement curves of frames F-1
and F-2 were similar in shape, consisting of an elastic stage and a
CAA stage. However, unlike F-1, the first storey of F-2 exhibited
catenary action after CAA, while the second-storey beam was
always under axial compression. The CAA and load capacities of
F-2 were 0.9 and 1.0 times greater than those of F-1, respectively.
(2) Damage in frame F-1 and the first storey of F-2 included crushing
of concrete and tensile cracking at the joints, fracture of beam
rebars at the middle joint, tensile cracks along the beam span, and
crushing of concrete at the bottom of the first-storey column.
Besides, more severe crushing of concrete appeared at the second-
(a) F2B4
150
Load capacity
CAA capacity
Liu Kun: Writing – original draft, Investigation, Formal analysis,
Data curation, Conceptualization. Wang Shan: Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Tan Liang:
Investigation, Formal analysis, Data curation. Kang Shao-Bo: Writing –
original draft, Visualization, Project administration, Funding acquisition, Formal analysis. Yuan Jian: Writing – original draft, Investigation,
Funding acquisition, Formal analysis.
(b) F3B4
(c) F4B4
Fig. 26. Geometric model of multi-storey planar frames.
13
(d) F5B4
J. Yuan et al.
Structures 74 (2025) 108489
50
40
30
20
10
0
0
100
200
300
400
Middle joint displacement (mm)
(a) F3B4
60
First storey
Second storey
Third storey
Fourth storey
50
40
30
20
10
0
0
100
200
300
400
Middle joint displacement (mm)
(b) F4B4
60
Vertical load (kN)
First storey
Second storey
Third storey
Vertical load (kN)
Vertical load (kN)
60
First storey
Second storey
Third storey
Fourth storey
Fifth storey
50
40
30
20
10
0
0
100
200
300
400
Middle joint displacement (mm)
(c) F5B4
Fig. 28. Vertical load-displacement curve of a single-storey beam in multi-storey frames.
Declaration of Competing Interest
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The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
The authors gratefully acknowledge the financial support provided
by the National Natural Science Foundation of China (No. 52278145),
by the Luzhou Science and Technology Program, China (no. 2022-GYF2), and by Scientific and Technological Research Program of Chongqing
Education
Commission
(grant
no.
KJQN202004010
and
KJZDK202204001).
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