Progressive Collapse Resistance of
Reinforced Concrete Frame Structures
Prof. Xianglin Gu
College of Civil Engineering, Tongji University
28/12/2012
Acknowledgements
This research project is sponsored by the National
Natural Science Foundation of China (No. 90715004)
and the Shanghai Pujiang Program (No. 07pj14084).
Outline
 Introduction
 Experimental Investigation
Testing Specimens
Test Setup and Measurements
Test Results
 Simplified Models for Nonlinear Static
Analysis of RC Two-bay Beams
 Conclusions
Introduction
Ronan Point (1968 )
Alfred P. Murrah (1995)
World Trade Center (2001)
Important buildings may be subjected to accidental loads, such
as explosions and impacts, during their service lives. It is,
therefore, necessary not only to evaluate their safety under
traditional loads and seismic action (in earthquake areas), but
also the structural performance related to resisting progressive
collapse.
Introduction
For a reinforced concrete (RC) frame structure, columns on the first
floor are more prone to failure under an explosion or impact load,
compared with other components. The performance of a newly
formed two-bay beam above the failed column determines the
resistance capacity against progressive collapse of the structure.
an extetior
column failed
an intetior
column failed
an extetior
column failed
Previous Experimental Work
At present, experimental studies have mainly focused on RC
beam-column subassemblies, each of which consists of two endcolumn stubs, a two-bay beam and a middle joint, representing
the element above the removed or failed column.
Su, Y. P.; Tian, Y.; and Song, X.S., “Progressive collapse resistance of axially-restrained frame
beams”, ACI Structural Journal, Vol. 106, No.5, September-October 2009, pp. 600-607.
Yu, J., and Tan, K.H., “Experimental and numerical investigation on progressive collapse
resistance of reinforced concrete beam column sub-assemblages”, Engineering Structures,
2011, http://dx.doi.org/10.1016/j.engstruct.2011.08.040.
Choi H., and Kim J., “Progressive collapse-resisting capacity of RC beam–column subassemblage”, Magazine of Concrete Research, Vol.63, No.4, 2011, pp.297-310.
Yi, W.J., He, Q.F., Xiao, Y, and Kunnath, S.K., “Experimental study on progressive collapseresistant behavior of reinforced concrete frame structures”. ACI Structural Journal. Vol.105,
No.4, 2008, pp. 433-439.1
Previous Experimental Work
From the above experiments, it can be concluded that the
compressive arch and catenary actions were activated under
sufficient axial constraint and that the vertical capacities of twobay beams were improved due to the compressive arch action.
However, the mechanism of the onset of catenary action was
not sufficiently clear and needed to be further studied.
Meanwhile, the contribution of the floor slab in resisting
progressive collapse has largely been ignored, and the
influence of the space effect on the performance of a frame
structure is not studied deeply.
What does this study examine
• This study has investigated the mechanisms of progressive
collapse of RC frame structures with experiments on two-bay
beams, where the space and floor slab effects were
considered.
• Based on the compressive arch and catenary actions and
the failure characteristics of the key sections of the beam
observed in the test, simplified models of the nonlinear static
load-displacement responses for RC two-bay beams were
proposed.
Experimental Investigation
 Testing Specimens
 Test Setup and Measurements
 Test Results
D
C
B
A
2
3
350
250
50
50
7
6
2C 8
ln
A 6@65
150
100
B2 (XB6)
Splice of B1A
gauges
#12@100
2C 8
2C 8
50
100
TB3, TB4 and TB5 (XB7)
Test subassemblies and reinforcement layout
150
2C 8
Plan View of XB7 (B1A, B1, B2, TB3, TB4, TB5 and XB6)
Elevation View of TB3, TB4, TB5 and XB7
2C 6
100
40
#12@100
100
2C 10
A 10@80
0.5w
7800
2C 8
h
150
gauges
100
100150 100
100 h 100
2C 8 or 2C 6 50
ln
2C 10
7800
5
4
gauges
100 t
h
b 125
7800
7800
Elevation View of B1A, B1, B2 and XB6
150
125
0.5w
ln
150
7800
7800
2C 8 A 6@65
350
ln
Region to be tested
Plane layout of a prototype structure
250
150
400×600
400×400
1
250
150
50
600×600
4200
E
250
the longitudinal direction
125 b
the transverse direction
125
The test specimens contained 1/4-scaled RC
structures: rectangle beam-column
subassemblies (named B1A, B1, B2
respectively), T-beam-column subassemblies
(named TB3, TB4 and TB5 respectively), a
substructure with cross beams (named XB6)
and a substructure with cross beams and a
floor slab (named XB7).
4200 4200 4200
Testing Specimens
2C 8
100
B1A and B1 (XB6)
Testing Specimens
Table 1 Specimen properties
TB3
TB4
TB5
Specimen No.
Section, b×h for beam
and w×t for flange, mm
mm
B1A
100×150
1800
28(ρ=0.86%)
28(ρ=0.86%)
25.6
2.82
B1
100×150
1800
28(ρ=0.86%)
28(ρ=0.86%)
21.8
2.73
B2
100×100
900
28(ρ=1.49%)
26(ρ=0.83%)
25.8
2.80
28(ρ=0.86%)
28(ρ=0.85%)
28.9
2.86
26.5
2.86
29.8
2.86
32.3
2.87
31.5
3.02
*l ,
n
*f ,
Top bars and
Bottom bars and
c
reinforcement ratios reinforcement ratios kN/mm2
Beam
100×150
Flange
450×40
Beam
100×150
Flange
450×40
Beam
100×150
Flange
450×40
Longitudinal beam
100×150
1800
28(ρ=0.85%)
28(ρ=0.86%)
Transverse beam
100×100
900
28(ρ=1.41%)
26(ρ=0.80%)
28(ρ=0.86%)
28(ρ=0.85%)
1800
XB7
Transverse
*
Beam
100×150
Floor
1950×40
Beam
100×100
Floor
3750×40
c(×10
1800
28(ρ=0.84%)
28(ρ=0.84%)
#12@100(ρ=0.31%)
1800
28(ρ=0.85%)
28(ρ=0.86%)
#12@100(ρ=0.31%)
1800
#12@100(ρ=0.31%)
900
28(ρ=1.45%)
26(ρ=0.83%)
#12@100(ρ=0.31%)
ln represents the net span of a beam. fc represents the compressive strength of concrete. Ec represents the Young's modulus of concrete.
4),
kN/mm2
#12@100(ρ=0.31%)
XB6
Longitudinal
*E
Testing Specimens
Table 2 Reinforcement properties
type
Diameter,
mm
Yield strength fy ,
kN/mm2
Ultimate strength fu ,
kN/mm2
Elongation,
(%)
Young's modulus Es(×105),
kN/mm2
6
5.75
569
714
14.9
2.34)
8
7.60
537
670
14.0
1.92
#12
2.80
238
319
24.0
1.46
6
6.60
329
523
22.1
2.18
bolts
19.63
561
671
-
1.86
Test Setup and Measurements
RC reaction wall Constraint beamP
BE,FC
Hydraulic actuator
*BM,FT
BM,FT
LDVT
Strain
gauges
*BE,FC
h
N
350 150 400
N
Bolts
End column stab
Roller
ln
150
Torque angle indicator
ln
Boundary conditions and test setup of specimens
Bolts
*BE represents the beam end n
end column stub and BM rep
the beam end near the middl
FC represents the interface of
and constraint beam an
represents the interface of flan
transverse beam in the middle
specimen.
Test Setup and Measurements
Test Results
(a) Specimen B1A
(b) Specimen B1
(d) Specimen TB4
(e) Specimen TB5
(g) Specimen XB7: Top view
(c) Specimen B2
(f) Specimen XB6
(h) Specimen XB7: Bottom view
Failure modes of specimens
Test Results
P
N
BE
BE
N
BE
N
B2
N
BM
BM
Failed column
BM
P
N
N
BM
Failed column
(c) catenary action for top and bottom bars in beam
(a) compressive arch action
B1A,
B1
P
BE
BE
P
N
N
BM
Failed column
(b) catenary action for top bars in beam
Failed column
(d) catenary action for bottom bars in beam
Compressive arch action and catenary action
Test Results
The details of the test results are presented by
dividing the specimens into 4 types:
• RC Beam-column subassemblies
• RC T-beam-column subassemblies
• RC Cross-beam systems without floor slab
• RC Cross-beam systems with a floor slab
RC Beam-column Subassemblies
60
Compressive Arch Stage
40
Catenary Stage
Elestic Stage
20
0
-20
Specimen B1A
Specimen B1
Yielding at BE and BM
Peak Load
-40
-60
0
50
100
150
200
Middle Joint Deflection Δ (mm)
250
300
Vertical load P and horizontal reaction force N versus
middle joint deflection Δ for B1A and B1
(a) Specimen B1A
Horizontal Reaction N (kN) Vertical Load P (kN)
Horizontal Reaction N (kN) Vertical Load P (kN)
It can be seen that the failure process can be divided into three stages:
an elastic stage, a compressive arch stage and a catenary stage.
40
Compressive Arch
Stage
30
Catenary Stage
Elestic Stage
20
10
0
-10
Yielding at BE and BM
Peak Load
-20
-30
0
50
100
150
Middle Joint Deflection Δ (mm)
200
250
Vertical load P and horizontal reaction force N
versus middle joint deflection Δ for B2
(b) Specimen B1
(c) Specimen B2
RC Beam-column Subassemblies
0
Middle Joint Deflection Δ (in.)
3.9
2.0
5.9
7.9
9.8
11.8
0
6000
4000
Yield Strain
Top rebar 1
Top rebar 2
Bottom rebar 1
Bottom rebar 2
2000
0
-2000
2.0
11.8
6000
-6
Strain of Bars BE (×10 )
Strain of Bars at BM (×10 -6)
8000
Middle Joint Deflection Δ (in.)
3.9
5.9
7.9
9.8
Yield Strain
4000
Yield Strain
2000
Top rebar 1
Top rebar 2
Bottom rebar 1
Bottom rebar 2
0
-2000
Yield Strain
-4000
-6000
-4000
0
50
100
150
200
Middle Joint Deflection Δ (mm)
(a) at BM for B1A
250
300
0
50
Strain of rebars in B1A
100
150
200
Middle Joint Deflection Δ (mm)
(b) at BE for B1A
250
300
Horizontal Reaction N (kN) Vertical Load P (kN)
RC Beam-column Subassemblies
60
Compressive Arch Stage
40
Catenary Stage
Elestic Stage
20
0
-20
Specimen B1A
Specimen B1
Yielding at BE and BM
Peak Load
-40
-60
0
50
100
150
200
Middle Joint Deflection Δ (mm)
250
300
Vertical load P and horizontal reaction force N versus middle
joint deflection Δ for B1A and B1
(Definitions of BE and BM are given in Table 3)
The shapes of the curves for
B1A and B1 were similar, and no
indication of splice failure was
observed in B1A, implying that
the lap splice according to
GB50010-2010 can meet the
continuity requirements in
progressive collapse resistant
design.
RC T-Beam-column Subassemblies
It can be seen that the failure process can be divided into three
stages: an elastic stage, a compressive arch stage and a catenary
stage.
Vertical Load P (kN)
30
20
Compressive Arch Stage
Elestic Stage
Catenary Stage
Yielding at BE and BM
Peak Load
10
Specimen B1
Specimen TB3
Specimen TB4
Specimen TB5
0
-10
0
50
100
150
200
Middle Joint Deflection Δ (mm)
250
300
Vertical load P versus middle joint deflection Δ for B1, TB3, TB4 and TB5
Specimen TB4
Specimen TB5
RC T-Beam-column Subassemblies
Ps
N
N
(a) Compressive arch action
N
Ps
(b) catenary action
Considering the effect of floor slabs, there was only one
mechanism that activated the catenary action, that is, the
bottom beam bars fractured at BM.
N
TB3,
TB4,
TB5
RC T-Beam-column Subassemblies
0
0.4
Middle Joint Deflection Δ (in.)
1.6
0.8
1.2
2.0
0
2.4
Middle Joint Deflection Δ (in.)
1.6
0.8
1.2
2.0
2.4
6000
Flange rebar 1
Flange rebar 2
Flange rebar 3
Flange rebar 4
4000
from inside to outside:1,2,3,4
2000
Yield strain
0
Strain of Bars at FC (×10 -6)
6000
Strain of Bars at FT (×10 -6)
0.4
Flange rebar 1
Flange rebar 2
Flange rebar 3
Flange rebar 4
4000
from inside to outside:1,2,3,4
2000
Yield strain
0
-2000
-2000
0
10
20
30
40
Middle Joint Deflection Δ (mm)
50
Strain of steel bars at FT for TB5
60
0
10
20
30
40
Middle Joint Deflection Δ (mm)
50
Strain of steel bars at FC for TB5
60
RC Cross-beam system without a floor slab
Elestic Stage
40
20
0
Yielding at BE and BM
Peak Load
50
100
150
200
Middle Joint Deflection Δ (mm)
250
Vertical load P versus middle joint
deflection Δ for B1, B2 and XB6
45
Longitudinal of XB6
30
15
0
-15 0
50
100
150
200
-30
300
-60
250
300
Transverse of XB6
30
15
0
0
50
100
150
200
-15
-45
-20
0
Specimen B2
Specimen B1
45
Specimen B1
Specimen B2
Specimen XB6
Horizontzl reaction N (kN)
60
Catenary Stage
Horizontzl reaction N (kN)
Compressive Arch Stage
Vertical Load P (kN)
60
60
80
Middle joint deflection Δ (mm)
(a) longitudinal direction
-30
Middle joint deflection Δ (mm)
(b) transverse direction
Horizontal reaction N versus middle joint deflection Δ for two directions of XB6
It can be seen that the failure process can be divided into three
stages: an elastic stage, a compressive arch stage and a catenary
stage.
250
RC Cross-beam system with floor slab
Vertical Load P (kN)
100
Compressive Arch Stage
80
Catenary Stage
Elestic Stage
60
40
20
0
Specimen XB7
Specimen XB6
Yielding at BE and BM
Peak Load
-20
0
50
100
150
200
Middle Joint Deflection Δ (mm)
250
Vertical load P versus middle joint deflection Δ for XB6 and XB7
It can be seen that the failure process can be divided into three
stages: an elastic stage, a compressive arch stage and a catenary
stage.
Simplified Models for Nonlinear Static Analysis
For the simplicity, the models of the nonlinear static analysis of RC two-bay
beams were derived by linking the critical points.
Ps
Elastic stage
Catenary stage
Ps
Compressive arch stage
Catenary stage
Compressive arch stage
Pua
Puc
Pua
c
u
P '
Ptr
Py
0
Elastic stage
Ptrt
Py
Ptrb
a
y
 tr
u
Static load-deflection response for two-bay beams (the
catenary action was activated by the concrete crushing)
Ast1L
Ast 0 L
f yt1L
fut1L
Ast1R
f yt0 L
f yt0 R
f yt1R
fut0L
fut0R
fut1R
BM
f
b
y1L
fub1L
BM
Asb0 L
f
b
y0L
fub0L
失效柱
BE
Asb0 R
Asb1R
b
y0R
b
y1R
f
fub0R
y
a
tr'
u'
a
Static load-deflection response for two-bay beams (the
catenary action was activated by the beam bars fracture)
Ast 0 R
BE
Asb1L
Ps
0
s
f
The areas and the yielding and ultimate
strengths for the continuous top beam bars
were Ast , f yt and fut , and for the continuous
bottom beam bars were Asb ,
fub1R
Areas and yielding and ultimate strengths for the top and bottom bars in beams
f yb
and
fub
Simplified Models for Nonlinear Static Analysis
The yielding load, which was the load of the ending
of the elastic stage, could be determined not
considering the influence of the axial constraint.
Elastic stage
Ps
Catenary stage
Compressive arch stage
Pua
Puc
Ptr
Py
the yielding moment of two beam ends, respectively
Py 
M y1L  M y 0 L
ln1

0
 tr
a
y
u
s
Static load-deflection response for twobay beams (the catenary action was
activated by the concrete crushing)
M y1 R  M y 0 R
ln 2
Ps
Elastic stage
Catenary stage
Compressive arch stage
Pua
the yielding moment of the left and right sections of beams near
the middle column, respectively
Puc '
Ptrt
Py
Ptrb
0
m  ln1  0.5b
'
n  ln 2  0.5b
'
3
3
2
3
2
Py  n 2  l  2m  3 mn 2 2  n  l  3ml  2m  3m n  

y 
m  2 m 
m
Bs 
6 1  n 
6l 3
2l


the stiffness of the most unfavorable section of beam
(BM)
y
a
u'
tr'
a
Static load-deflection response for two-bay
beams (the catenary action was activated
by the beam bars fracture)
Ast1L
Ast 0 L
Ast 0 R
Ast1R
f yt1L
f yt0 L
f yt0 R
f yt1R
t
u0L
t
u 0R
fut1R
fut1L
BE
Ps
f
BM
f
BM
BE
Asb1L
Asb0 L
Asb0 R
Asb1R
f yb1L
f yb0 L
f yb0 R
f yb1R
fub1L
fub0L
fub0R
fub1R
失效柱
Areas and yielding and ultimate strengths for
the top and bottom bars in beams
The
stren
were
botto
Simplified Models for Nonlinear Static Analysis
Ps
BE
Pu
Δs
BM
BM
BE
Elastic stage
Ps
Catenary stage
Compressive arch stage
t
ln1
b'
ln2
Pua
Puc
t
Ptr
Py
Deformation mode of two-bay beams under ultimate state considering the compressive arch action
0
To determine the ultimate bearing capacity of the
RC two-bay beam considering the compressive arch
action, it was assumed that:
1) the beam between the plastic hinges is elastic;
2) the stress distribution block of concrete in
compressive zone at BE and BM can be equivalent
to the rectangular block;
3) the axial reactions N applied on BE and BM have
the same value and the applied points of N are all
on the middle of the sections;
4) the tensile strength of concrete is neglected.
 tr
a
y
u
s
Static load-deflection response for twobay beams (the catenary action was
activated by the concrete crushing)
Ps
Elastic stage
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
0
y
a
u'
tr'
a
Static load-deflection response for two-bay
beams (the catenary action was activated
by the beam bars fracture)
P
N
BE
BE
N
N
BM
BM
Failed column
(a) compressive arch action
P
N
Simplified Models for Nonlinear Static Analysis
From the Deformation made of two-bay beams under ultimate
state considering the compressive arch action, the deformation
compatibility of the RC two- bay beams can be derived. P
Elastic stage
Catenary stage
s
ln1+t
Ptr
the drift of BE
θ1
Δs
N
z0L
xn1L
Py
e  N ln1 /  EA
xn0L
z1L
BE
Compressive arch stage
Pua
Puc
ln1  t
 ln1  e  z1L tg1  z0 L tg1
cos 1
N
ln1-e
relationship of BE and
1  s /  ln1  e   s / ln1
Fig.17Geometrical
Geometrical
relationship of BE and
BM for the left bay of two-bay
BM for
the left bay of two-bay beams
beams
For the right bay:
s
1
 x1L  x0 L    s h 
s
1
Elastic stage
Ps
u
s
Catenary stage
Compressive arch stage
Pua
ln12 ln1
 2s P
BL 


EA k s 2 EA
c
u
Ptrt
'
 2s
 BL N
2
 2s
 x  x0 R    s h   BR N
1 1R
2
BR 
Py
Ptrb
y
0
u'
tr'
a
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
s
For the two- bay beam:
 tr
a
y
Static load-deflection response for two-bay
beams (the catenary action was activated by
the concrete crushing)
z0 L  0.5h  xn 0 L  0.5h  x0 L / 1
BM
For the left bay:
0
Ps
ln 2 2 ln 2
2

 s
EA ks 2 EA
BM
t
 x1L  x0 L  x1R  x0 R   2 s h   2s   BL  BR  N
Δs
Pu
BE
ln1
BE
BM
b'
ln2
t
Deformation made of two-bay beams under
ultimate state considering the compressive arch
action
Simplified Models for Nonlinear Static Analysis
x1L, x0L, x1R and x0R can be determined by the equilibrium
conditions of the internal forces at BE and BM.
Elastic stage
Ps
Catenary stage
Compressive arch stage
Pua
Puc
Ast1L f yt1L
N
Ptr
Py
Mu1L
Asb1Lsb1L
 sb1L
 cu
0
 tr
a
y
x1L
u
s
Static load-deflection response for two-bay
beams (the catenary action was activated by
the concrete crushing)
1 fc
Stress and strain distribution of BE in the left bay
Ps
The stress of bottom bars can be derived.
 b
 1asb 
b
b
 s1L  Es  cu 1 
  s1 L  f y 1 L
x1L 


b
b
  f
 sb1L  f yb1L
y1 L
 s1L
 sb1L  f yb1L
1
f yb1L
Es  cu
D1L 
f yb1L
 1L asb  1asb
C1L 
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
0
simplified
y
a
u'
tr'
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
x1L  1asb
 D1L x1L  C1L
 1L asb  1asb
1
 1L 
Elastic stage
P
1 f
1   1L
b
y1L
N
BE
BE
N
N
BM
BM
Failed column
(a) compressive arch action
P
N
Simplified Models for Nonlinear Static Analysis
x1L, x0L, x1R and x0R can be determined by the equilibrium
conditions of the internal forces at BE and BM.
Elastic stage
Ps
Catenary stage
Compressive arch stage
Pua
Puc
According to the equilibrium condition of the internal
forces at BE, x1L can be determined.
Ptr
Py
f yt1L Ast1L  N  1 fc bx1L   sb1L Asb1L
0
 tr
a
y
u
s
Static load-deflection response for two-bay
beams (the catenary action was activated by
the concrete crushing)
Accordingly, the bending moment of BE can be determined
as
h x
M u1L  1 f c bx1L   1L
2 2
h


b
b h
b 
t
t
   s1L As1L  2  as   f y1L As1L  h0  2 





Ps
Elastic stage
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
0
y
a
u'
tr'
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
In a similar way, x0L, x1R and x0R can be determined and
Mu0L, Mu1R and Mu0R can also be calculated accordingly.
P
N
BE
BE
N
N
BM
BM
Failed column
(a) compressive arch action
P
N
Simplified Models for Nonlinear Static Analysis
a
P
N and Pua are quadratic functions
of
Δ
and
there
is
always
a
Δ
that
makes
u
s
s
a
'
P
u
max
become the maximum.
and the corresponding vertical deflection s can
be determined by trial and error method.
Given the stiffness of the axial constraint and the properties
of the two-bay beam .
Set a starting value for Δs.
Assume all rebars at BE and BM are yielded and the expressions of x1L, x0L, x1R and x0R can be determined.
Judge whether the rebars are yielded or not.
Choose the appropriate expressions of
x1L, x0L, x1R and x0R , determine N again.
Calculate x1L, x0L, x1R and x0R.
Determine N.
Calculate x1L, x0L, x1R , x0R and the bending
moment s Mu1L, Mu0L, Mu1R and Mu0R .
Calculate Pua .
Δs＝Δs +dΔs
0
Pu a  Pu a
n
n1
0
Pua become the maximum Puamax , and Pua  Puamax ， a  's   y .
Simplified Models for Nonlinear Static Analysis
The load and deflection at the transition point of the
compressive arch stage and the catenary stage can
be determined on the base of the mechanism
activated the catenary action.
Pua
Puc
Ptr
Py
0
u
s
P
BE
N
BE
N
N
 1
1  arch action
Ps   f yt Ast  f yb Asb   s  (a)compressive

 ln1 ln 2  P
N
Concrete is deactivated at the transition point,
So, tr 
 tr
a
y
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
BM
BM
Failed column
Ptr  Py
Catenary stage
Compressive arch stage
When the catenary action is activated by the concrete
crushing, the relationship of Ps and Δs at thePcatenary
stage can be expressed as
BE
BE
N
Elastic stage
Ps
BM
Failed column
(b) catenary action for top bars in beam
Ptr ln1ln 2
 f yt Ast  f yb Asb  ln1  ln2 
BM
BM
Failed column
(c) catenary action for top and bottom bars in beam
N
P
BE
N
Ps
T
θ1
Ps
T
Δs
θ2
Failed column ln2
ln1
b'
(d) catenary action for bottom bars in beam
图 22 梁悬索阶段受力分析
Loadings working on the two-bay beam at
the catenary stage
N
Simplified Models for Nonlinear Static Analysis
Elastic stage
Ps
The vertical deflection at the ending of the catenary
stage is depended on the elongations of the top and
bottom bars.
Compressive arch stage
Pua
Puc
Ptr
Py
0
P
BE
ln  min(ln1 , ln 2 )
t
s
b
u
t
s
b
y
N
b
s
BM
Failed column
(c) catenary action for top and bottom bars in beam
N
1 1 
or P   f A  f A  u  BM 
ln 2 column
 ln1 Failed

(b) catenary action for top bars in beam
t
u
BE
BM
b
s
N
c
u
s
N
BM
 1 Failed
BM
1 column
So, P   f A  f A  u  (a) compressive
 arch action
l
 n1 ln 2 P
t
y
u
P
BE
N
BE
N
c
u
 tr
a
y
Static load-deflection response for two-bay
beams (the catenary action was activated by
the concrete crushing)
According to GSA2003, the acceptance criterion of the
rotation degree for beam is 12°.
u  ln tg12  0.2ln
Catenary stage
P
BE
N
Ps
T
θ1
Ps
T
Δs
θ2
Failed column ln2
ln1
b'
(d) catenary action for bottom bars in beam
图 22 梁悬索阶段受力分析
Loadings working on the two-bay beam at
the catenary stage
N
Simplified Models for Nonlinear Static Analysis
P
P
N
BE
BE
N
N
BE
BE
N N
BE
N
BM
BM
BM
Failed
BM column
BE
N
N
BM
Failed
(c) catenary action
forcolumn
top and bottom bars in beam
(c) catenary action for top and bottom bars in beam
P
N
N
N
BE
P
BM
(a) compressive arch action
P
P
BMFailedBM
column
Failed column
(a) compressive
arch action
N
BE
BE
N
BE
P
P
N
N
N
BM
BM
Failed column
Failed column
(b)(b)catenary
action
bars in
in beam
beam
catenary action for
for top
top bars
Failed column
Failed column
(d)
catenary
action
for bottom
in beam
(d) catenary action for bottom
bars bars
in beam
Ps
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of Ps and Δs at the
catenary stage can be expressed as
Pua
Puc '
Ptrt
Py
Ptrb
 1
1 
Ps  f yt Ast  s   
 ln1 ln 2 
 1
1 
Ps  f yb Asb  s   
 ln1 ln 2 
Catenary stage
Compressive arch stage
0
When the mechanism is the fracture of the top bars
at BE, the relationship of Ps and Δs at the catenary
stage can be expressed as
Elastic stage
y
a
u'
tr'
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
Ps
T
θ1
Ps
ln1
b'
T
θ2
Δs
ln2
图 22 梁悬索阶段受力分析
Loadings working on the two-bay beam at
the catenary stage
a
Simplified Models for Nonlinear Static Analysis
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of Ps and Δs at the
catenary stage can be expressed as
Ps
Pua
Puc '
Ptrt
Py
Ptrb
0
y
u'
tr'
a
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
The vertical load was carried by the top beam bars
after the bottom bars fracture. So the bottom value of
the vertical load at the transition point can be
determined as
y
y
M u0
M u0

'
ln1  b ln 2  b '
Catenary stage
Compressive arch stage
 1
1 
Ps  f yt Ast  s   
 ln1 ln 2 
Ptrb 
Elastic stage
Ps
T
θ1
Ps
P
ln1
BE
N
T
θ2
Δs
b'
ln2
N
BE
图 22 梁悬索阶段受力分析
N
Loadings working on the two-bay beam at
BM
BM
the catenary stage
Failed column
Ptrb ln1ln 2
So,   f b Ab  l  l 
y
s
n1
n2
'
tr
(a) compressive arch action
(c
P
N
N
BM
Failed column
(b) catenary action for top bars in beam
N
Simplified Models for Nonlinear Static Analysis
Due to the load-deflection response for the mechanisms
of concrete crushing and rebars fracture being
coincident before fracture of bars , the top value of the
vertical load at the transition point can be determined by
the descending branch in the compressive arch stage
for the mechanism of concrete crushing.
Ps
Elastic stage
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
0
y
u'
tr'
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
Ps
Ps
T
Pua
θ1
Ps
Ptrt
Ptr
T
θ2
Δs
P
ln1
b'
ln2
a
 tr'
 tr
s
N
N
BE
BE
0
a
图 22 梁悬索阶段受力分析
Loadings working on the two-bay beam at
N
BM
Failed column
BM
the catenary stage
Static load-deflection
responses for two-bay beams
图 25 双跨梁荷载－位移曲线
(a) compressive arch action
(c
P
N
N
The top value of the vertical load at the transition
point can be determined as
Pua  Ptr
P P 
 tr'   a 

 tr   a
t
tr
a
u
BM
Failed column
(b) catenary action for top bars in beam
N
Simplified Models for Nonlinear Static Analysis
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of Ps and Δs at the
catenary stage can be expressed as
Ps
Elastic stage
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
0
 1
1 
Ps  f yt Ast  s   
 ln1 ln 2 
y
a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
Ps
T
θ1
The carrying capacity at the catenary stage is depended
on the ultimate strength of the top bars.
u  ln tg12  0.2ln
u'
tr'
a
ln  min(ln1 , ln 2 )
Ps
T
Δs
P
ln1
θ2
b'
ln2
N
BE
BE
N
图 22 梁悬索阶段受力分析
Loadings working on the two-bay beam at
N
BM
Failed column
BM
the catenary stage
1 1 
Puc  fut Ast u   
 ln1 ln 2 
(a) compressive arch action
(c
P
N
N
BM
Failed column
(b) catenary action for top bars in beam
N
Simplified Models for Nonlinear Static Analysis
When the mechanism is the fracture of the top bars at
BE, the relationship of Ps and Δs at the catenary stage
can be expressed as
Catenary stage
Compressive arch stage
Pua
Puc '
Ptrt
Py
Ptrb
 1
1 
Ps  f A  s   
 ln1 ln 2 
b
y
Elastic stage
Ps
0
b
s
y
a
Ps
T
θ1
Ps
P
M uy0
M uy0 BE
b
Ptr 

ln1  b' N ln 2  b '
 tr' 
Pl l
f yb A  l  l
a
u
P
n1
ln2
b'
BE
N
图 22 梁悬索阶段受力分析
(c) catenary action for top and bottom bars in beam
P
Failed column
(b) catenary action for top bars in beam
1 1 
Puc  fut Ast u   
 ln1 ln 2 
Δs
BM on the
Loadings working
BMtwo-bay beam at
the catenary stage
Failed column
(a) compressive arch action
a N
u
u  ln tg12  0.2ln
BE l
T
θ2
N
P  Ptr
P P 
 tr'   a 

 tr   a
BM
t
tr
N
BE
BM
BM
Failed column

a
Static load-deflection response for two-bay
beams (the catenary action was activated by
the beam bars fracture)
The vertical load was carried by the top beam bars after
the bottom bars fracture.
b
tr n1 n 2
b
s
n1
n2
u'
tr'
ln  min(ln1 , ln 2 )
N
BE
P
N
N
Failed column
(d) catenary action for bottom bars in beam
Simplified Models for Nonlinear Static Analysis
25
Ps-cal
20
15
10
5
Ps-cal
20
15
10
5
0
100
200
300
Middle joint deflection Δ(mm)
Ps-ex
20
15
10
5
0
0
400
Ps-cal
25
0
0
B1A
30
Ps-ex
25
Vertical load P (kN)
30
Ps-ex
Vertical load P (kN)
Vertical load P(kN)
30
B1
100
200
300
Middle joint deflection Δ(mm)
400
0
B2
50
100
150
200
Middle joint deflection Δ(mm)
250
(a) Static load-deflection response for test specimens
Ps-ex
Ps-cal
200
150
150
100
100
50
A1
50
0
Vertical load P (kN)
Ps-cal
Vertical load P (kN)
Ps-ex
Vertical load P (kN)
150
250
200
Ps-ex
0
0
50 100 150 200 250
Middle jiont deflection Δ(mm)
300
A5
Ps-cal
120
90
60
30
0
0
50 100 150 200 250
Middle jiont deflection Δ(mm)
300
B3
0
100 200 300 400 500
Middle jiont deflection Δ(mm)
(b) static load-deflection response for test specimens carried by Su et al[1]
static load-deflection response for test specimens
It can be seen that the shapes of the calculated load-deflection response
curves have good match with the tested curves.
Conclusions
• Based on the test results, it can be concluded that the failure process for
the specimens can be divided into an elastic stage, a compressive arch
stage and a catenary stage, regardless of floor and/or space effects.
• The ultimate carrying capacity of a beam or cross-beam system in the
compressive arch stage increases when considering the effect of a floor
slab, and the ultimate carrying capacity for unidirectional beams increases
with increased floor slab width.
• The ultimate bearing capacity of a cross-beam system in the
compressive arch stage is enhanced by the space effect, larger than that of
the longitudinal or transverse direction, but not the sum of the ultimate
bearing capacities of these two directions.
Conclusions
• Mechanisms to activate the catenary action were discussed, which
yielded that the elongations of beam bars are an important factor in
determining the mechanism.
• When considering the effect of floor slabs for unidirectional beams, there
is only one mechanism to activate the catenary action, which is the fracture
of the bottom steel bars in beams at BM. The ultimate carrying capacity in
the catenary stage depends on the top bars.
• When considering the space effect and effect of floor slabs at the same
time, there are two probable mechanisms to activate the catenary action
fracture of the bottom bars at BM in the either longitudinal direction or the
transverse direction.
• The lap splice of the bottom bars according to GB50010-2010 can meet
the continuity requirements in progressive collapse resistant design.
Conclusions
• The simplified models of the nonlinear static load-deflection response
for RC two-bay beams were proposed based on the test results. They
were verified to be effective by comparing the calculated and test results.
Thank You !