Progressive Collapse Resistance of Reinforced Concrete Frame Structures

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Nov 26, 2013 (3 years and 9 months ago)

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Progressive Collapse Resistance of
Reinforced Concrete Frame Structures

Pro
f
. 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 extetior
column failed

an intetior
column failed

Previous Experimental Work

At present, e
xperimental studies have mainly focused on RC
beam
-
column subassemblies, each of which consists of two end
-
column 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 sub
-
assemblage”,
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 collapse
-
resistant 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 two
-
bay 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

Testing Specimens


420
0


780
0

780
0

78
00

780
0

1

2

7

4

3

6

5

A

B

C

D

E

600
×
600

400
×
600

400
×
400

Region to be tested

7800

7800

420
0


420
0


420
0


Plane layout of a prototype structure


Plan View of XB7 (B1A, B1, B2, TB3, TB4, TB5 and XB6)

B2 (XB6)

150


125


125


150


0.5w

0.5w

l
n

b

125


125


150


150

the longitudinal direction

the transverse direction

b

l
n

100


2
C
8

100

2
C
6


100

150

2
C
8

2
C
8

B1A and B1 (XB6)

Elevation View

of B1A, B1, B2 and XB6


Splice of B1A

TB3, TB4 and TB5 (XB7)


Elevation View

of TB3, TB4, TB5 and XB7


h

A
6@65

l
n

100

100

150

350

250


2
C
8 or
2
C
6

2
C
8

250

50

50

gauge
s

l
n

A
6@65

A
10@80

#12@100

150

100


350

2
C
8

250


250


2
C
10

2
C
10

100


h

t

50

50

100

100

1
5
0

1
0
0

gauge
s

40

#12@100

100


2
C
8

2
C
8

h


50

gauge
s

Test subassemblies and reinforcement layout

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).

Testing Specimens

Table 1 Specimen properties

Specimen No.

Section,
b
×
h

for beam
and
w
×
t

for flange, mm

*
l
n
,

mm

Top bars and
reinforcement ratios

Bottom bars and
reinforcement ratios

*
f
c

,

kN/mm
2


*
E
c
(
×
10
4
),

kN/mm
2


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

TB3

Beam

100
×
150

1800

2

8(ρ=0.86%)

2

8(ρ=0.85%)

28.9

2.86

Flange

450
×
40

#12@100(ρ=0.31%)

TB4

Beam

100
×
150

1800

2

8(ρ=0.84%)

2

8(ρ=0.84%)

26.5

2.86

Flange

450
×
40

#12@100(ρ=0.31%)

TB5

Beam

100
×
150

1800

2

8(ρ=0.85%)

2

8(ρ=0.86%)

29.8

2.86

Flange

450
×
40

#12@100(ρ=0.31%)

XB6

Longitudinal beam

100
×
150

1800

2

8(ρ=0.85%)

2

8(ρ=0.86%)

32.3

2.87

Transverse beam

100
×
100

900

2

8(ρ=1.41%)

2

6(ρ=0.80%)

XB7

Longitudinal

Beam

100
×
150

1800

2

8(ρ=0.86%)

2

8(ρ=0.85%)

31.5

3.02

Floor

1950
×
40

#12@100(ρ=0.31%)

Transverse

Beam

100
×
100

900

2

8(ρ=1.45%)

2

6(ρ=0.83%)

Floor

3750
×
40

#12@100(ρ=0.31%)

*

l
n

represents the net span of a beam.
f
c

represents the compressive strength of concrete.
E
c

represents the Young's modulus of concrete.

Testing Specimens

Table 2 Reinforcement properties

type

Diameter,

mm

Yield strength

f
y

,

kN/mm
2


Ultimate strength
f
u

,

kN/mm
2

Elongation,

(%)

Young's modulus
E
s
(
×
10
5
),

kN/mm
2



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

Boundary conditions and test setup of specimens


*
BE represent
s

the beam
end near the
end column stub

and

BM represent
s

the beam
end near the

middle joint.

FC

represent
s
the interface of
flange
and c
onstraint

beam and FT

represent
s

the

interface of
flange and
transverse beam in the middle of the
specimen.

Bolts

RC reaction wall

End column stab

Roller

Torque angle indicator

Bolts

Hydraulic actuator

l
n

Strain

gauges

LDVT

BE
,FC

*
B
M,FT

*
BE
,FC

B
M,FT

Constraint

beam

P

h

150


l
n

N

N

350

1
50

40
0

Test Setup and Measurements

Test Results

Failure modes of specimens

(a) Specimen B1A

(b) Specimen B1

(c) Specimen B2

(d) Specimen TB4

(e) Specimen TB5

(f) Specimen XB6

(h) Specimen XB7: Bottom view

(g) Specimen XB7: Top view

Test Results

Compressive arch action and catenary action


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

B2

B1A,

B1

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

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.

-
60
-
40
-
20
0
20
40
60
0
50
100
150
200
250
300
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Specimen B1A
Specimen B1
Horizontal
Reaction
N (kN)
Elestic Stage
Compressive Arch Stage
Catenary Stage
Yielding at BE and BM
Peak Load

Vertical load
P

and horizontal reaction force
N

versus
middle joint deflection
Δ

for B1A and B1


Vertical load
P

and horizontal reaction force
N

versus middle joint deflection
Δ

for B2

-
30
-
20
-
10
0
10
20
30
40
0
50
100
150
200
250
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Horizontal
Reaction
N (kN)
Compressive Arch
Stage
Catenary Stage
Elestic Stage
Peak Load
Yielding at BE and BM
(a) Specimen B1A

(b) Specimen B1

(c) Specimen B2

RC Beam
-
column Subassemblies

Strain of rebars in B1A

(a) at BM for B1A

-
4000
-
2000
0
2000
4000
6000
8000
0
50
100
150
200
250
300
Middle Joint Deflection
Δ
(mm)
Top rebar 1
Top rebar 2
Bottom rebar 1
Bottom rebar 2
Yield Strain
Yield Strain
Strain of
Bars at BM
Middle Joint Deflection
Δ
(in.)
0
9.8
7.9
5.9
3.9
2.0
11.8
(
×
10
-
6
)
(b) at BE for B1A

-
6000
-
4000
-
2000
0
2000
4000
6000
0
50
100
150
200
250
300
Middle Joint Deflection
Δ
(mm)
Top rebar 1
Top rebar 2
Bottom rebar 1
Bottom rebar 2
Yield Strain
Yield Strain
Strain of
Bars BE
Middle Joint Deflection
Δ
(in.)
0
9.8
7.9
5.9
3.9
2.0
11.8
(
×
10
-
6
)
RC Beam
-
column Subassemblies

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.

-
60
-
40
-
20
0
20
40
60
0
50
100
150
200
250
300
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Specimen B1A
Specimen B1
Horizontal
Reaction
N (kN)
Elestic Stage
Compressive Arch Stage
Catenary Stage
Yielding at BE and BM
Peak Load

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)

RC T
-
Beam
-
column Subassemblies

-
10
0
10
20
30
0
50
100
150
200
250
300
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Specimen B1
Specimen TB3
Specimen TB4
Specimen TB5
Elestic Stage
Compressive Arch Stage
Catenary Stage
Peak Load
Yielding at BE and BM
Vertical load
P

versus middle joint deflection
Δ

for B1, TB3, TB4 and TB5

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.


Specimen TB4


Specimen TB5

RC T
-
Beam
-
column Subassemblies

P
s

N

N

N

N

P
s

TB3,

TB4,

TB5

(a) Compressive arch action

(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.

RC T
-
Beam
-
column Subassemblies

-
2000
0
2000
4000
6000
0
10
20
30
40
50
60
Middle Joint Deflection
Δ
(mm)
Flange rebar 1
Flange rebar 2
Flange rebar 3
Flange rebar 4
Yield strain
Strain of
Bars at FC
from
inside to outside:1,2,3,4
(
×
10
-
6
)
Middle Joint Deflection
Δ
(in.)
0
2.4
2.0
1.6
1.2
0.8
0.4
-
2000
0
2000
4000
6000
0
10
20
30
40
50
60
Middle Joint Deflection
Δ
(mm)
Flange rebar 1
Flange rebar 2
Flange rebar 3
Flange rebar 4
Yield strain
Strain of
Bars at FT
from
inside to outside:1,2,3,4
(
×
10
-
6
)
Middle Joint Deflection
Δ
(in.)
0
2.4
2.0
1.6
1.2
0.8
0.4
Strain of steel bars at FT for TB5

Strain of steel bars at FC for TB5

RC
Cross
-
beam system without a floor slab

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

versus middle joint
deflection
Δ
for B1, B2 and XB6

-
20
0
20
40
60
80
0
50
100
150
200
250
300
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Specimen B1
Specimen B2
Specimen XB6
Compressive Arch Stage
Elestic Stage
Catenary Stage
Peak Load
Yielding at BE and BM
-
30
-
15
0
15
30
45
60
0
50
100
150
200
250
Horizontzl reaction N (kN)
Middle joint deflection
Δ
(mm)
Specimen B2
Transverse of XB6
(a) longitudinal direction

(b) transverse direction

-
60
-
45
-
30
-
15
0
15
30
45
60
0
50
100
150
200
250
300
Horizontzl reaction N (kN)
Middle joint deflection
Δ
(mm)
Specimen B1
Longitudinal of XB6
Horizontal reaction
N

versus middle joint deflection
Δ

for two directions of XB6

RC
Cross
-
beam system with floor slab

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

versus middle joint deflection
Δ

for XB6 and XB7

-
20
0
20
40
60
80
100
0
50
100
150
200
250
Vertical Load P (kN)
Middle Joint Deflection
Δ
(mm)
Specimen XB7
Specimen XB6
Elestic Stage
Compressive Arch Stage
Catenary Stage
Peak Load
Yielding at BE and BM
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.



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y




0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay beams (the
catenary action was activated by the concrete crushing)

Static load
-
deflection response for two
-
bay beams (the
catenary action was activated by the beam bars fracture)


失效柱

P
s

1
t
s L
A

1
t
y L
f

1
t
u L
f

1
b
s L
A

1
b
y L
f

1
b
u L
f

0
t
s L
A

0
t
y L
f

0
t
u L
f

0
b
s L
A

0
b
y L
f

0
b
u L
f

1
t
s R
A

1
t
y R
f

1
t
u R
f

1
b
s R
A

1
b
y R
f

1
b
u R
f

0
b
s R
A

0
b
y R
f

0
b
u R
f

0
t
s R
A

0
t
y R
f

0
t
u R
f

BE

BE

BM

BM

The areas and the yielding and ultimate
strengths for the
continuous

top beam bars
were
t
s
A
,
t
y
f
and

t
u
f
, and for the
continuous

bottom beam bars were
b
s
A
,
b
y
f
and
b
u
f

Areas and yielding and ultimate strengths for the top and bottom bars in beams

Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y




0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay beams (the catenary action was
activated by the concrete crushing)

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated
by the beam bars fracture)

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.

1 0 1 0
1 2
y L y L y R y R
y
n n
M M M M
P
l l
 
 

失效柱

P
s

1
t
s L
A

1
t
y L
f

1
t
u L
f

1
b
s L
A

1
b
y L
f

1
b
u L
f

0
t
s L
A

0
t
y L
f

0
t
u L
f

0
b
s L
A

0
b
y L
f

0
b
u L
f

1
t
s R
A

1
t
y R
f

1
t
u R
f

1
b
s R
A

1
b
y R
f

1
b
u R
f

0
b
s R
A

0
b
y R
f

0
b
u R
f

0
t
s R
A

0
t
y R
f

0
t
u R
f

BE

BE

BM

BM

The areas and the yielding and ultimate
strengths for the
continuous

top beam bars
were
t
s
A
,
t
y
f
and

t
u
f
, and for the
continuous

bottom beam bars were
b
s
A
,
b
y
f
and
b
u
f

Areas and yielding and ultimate strengths for
the top and bottom bars in beams







3 3 2 3 2
2
2
3 2
3 2
3 2 3
2
6 1
6 2
y
y
s
n l ml m m n
P
n l m
mn
m m m
B n
l l
 
   

 
   

 
 
the stiffness of the most unfavorable section of beam
(BM)

the yielding moment of two beam ends, respectively

the yielding moment of the left and right sections of beams near
the middle column, respectively

'
1
0.5
n
m l b
 
'
2
0.5
n
n l b
 
Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y




0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay beams (the catenary action was
activated by the concrete crushing)

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated
by the beam bars fracture)

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.


Δ
s

P
s

b'

l
n1

l
n2

P
u

t

t

BE

BM

BM

BE


Deformation mode of two
-
bay beams under ultimate state considering the compressive arch action


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

Simplified Models for Nonlinear Static Analysis

1
1 1 1 0 1
1
cos
n
n L L
l t
l e z tg z tg
 


   


0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the concrete crushing)

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

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.

Deformation made of two
-
bay beams under
ultimate state considering the compressive arch
action

the drift of BE


Δ
s

l
n1
+
t

l
n1
-
e

z
1
L

z
0
L

BE

N

N

BM

θ
1

x
n1L

x
n0L

Fig.1
7

G
eometrical

relationship

of BE and
BM for the left bay of two
-
bay beam
s

Geometrical relationship of BE and
BM for the left bay of two
-
bay
beams


Δ
s

P
s

b'

l
n1

l
n2

P
u

t

t

BE

BM

BM

BE




1
/
n
e Nl EA

0 0 0 1
0.5 0.5/
L n L L
z h x h x

   


1 1 1
//
s n s n
l e l

    
For the left bay:



2
1 0
1
2
s s
L L s L
x x h B N

 
    


2
1 0
1
2
s s
R R s R
x x h B N

 
    




2
1 0 1 0
1
2
s
L L R R s s L R
x x x x h B B N


       
For the right bay:

For the two
-

bay beam:

2 2
2 2
2
n n s
R
s
l l
B
EA k EA

  
2 2
1 1
2
n n s
L
s
l l
B
EA k EA

  


0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y


Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y




0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the concrete crushing)

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

x
1L
,
x
0L
,
x
1R

and
x
0R
can be determined by the equilibrium
conditions of the internal forces at BE and BM.

Stress and strain distribution of BE in the left bay


N

1 1
b b
s L s L
A

1 1
t t
s L y L
A f
cu

1
b
s L

1
c
f

1
L
x
1
u L
M
1
1 1 1
1
1 1 1 1
1
b
b b b
s
s L s cu s L y L
L
b b b b
s L y L s L y L
a
E f
x
f f

  
 

 
  

 

 

 

The stress of bottom bars can be derived.

1 1
1 1 1 1 1
1 1
b
b b
L s
s L y L L L L
b b
L s s
x a
f D x C
a a


 

  

1
1
1 1
b
y L
L
b b
L s s
f
D
a a
 


1 1
1
1 1
b
y L
L
L
f
C

 


1
1
1
1
L
b
y L
s cu
f
E





simplified


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y




0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the concrete crushing)

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

x
1L
,
x
0L
,
x
1R

and
x
0R
can be determined by the equilibrium
conditions of the internal forces at BE and BM.

Accordingly, the bending moment of BE can be determined
as

In a similar way,

x
0L
,
x
1R

and
x
0R
can be determined and
M
u0L
,
M
u1R

and
M
u0R
can also be calculated accordingly.

According to the equilibrium condition of the internal
forces at BE,

x
1L
can be determined.

1 1 1 1 1 1
t t b b
y L s L c L s L s L
f A N f bx A
 
  
1
1 1 1 1 1 1 1 0
2 2 2 2
b b b t t
L
u L c L s L s L s y L s L
x
h h h
M f bx A a f A h
 
 
   
     
   
 
   
 

BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

Simplified Models for Nonlinear Static Analysis

N

and are quadratic functions of
Δ
s

and there is always a
Δ
s
that makes
become the maximum. and the corresponding vertical deflection can
be determined by trial and error method.

a
u
P
a
u
P
max
a
u
P
'
s

Given the stiffness of the axial constraint and the properties

of the two
-
bay beam .

Assume all
rebars

at BE and BM are yielded

and

the expressions of
x
1L
,
x
0L
,
x
1R

and
x
0R

can be determined.


Set a starting value for
Δ
s
.


Determine
N.

Calculate
x
1L
,
x
0L
,
x
1R

and
x
0R
.

Judge whether the rebars are yielded

or not.

Choose the appropriate expressions of
x
1L
,
x
0L
,
x
1R

and
x
0R

, determine
N
again.

Calculate
x
1L
,
x
0L
,
x
1R

,
x
0R

and

the bending
moment s
M
u1L
,
M
u0L
,
M
u1R

and
M
u0R
.

Calculate

.

a
u
P
1
a a
u u
n n
P P


0

Δ
s

Δ
s
+d
Δ
s



become the maximum

, and



.

a
u
P
max
a a
u u
P P

max
a
u
P
0

'
a s y
   
Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y


Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

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.

Concrete is deactivated at the transition point,

So,

Loadings working on the two
-
bay beam at
the catenary stage



1 2
1 1
t t b b
s y s y s s
n n
P f A f A
l l
 
   
 
 


2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s

tr y
P P





1 2
1 2
tr n n
tr
t t b b
y s y s n n
P l l
f A f A l l
 
 

BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

When the catenary action is activated by the concrete
crushing, the relationship of
P
s

and
Δ
s

at the catenary
stage can be expressed as

Simplified Models for Nonlinear Static Analysis



0

C
ompressive arch

stage

Catenary

stage

Elastic stage

u


s


tr


a


s
P

y
P

c
u
P

a
u
P

tr
P

y


Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the concrete crushing)

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

The vertical deflection at the ending of the catenary
stage

is depended on the elongations of the top and
bottom bars.

According to GSA2003, the acceptance criterion of the
rotation degree for beam is 12
°
.

So,

1 2
12 0.2 min(,)
u n n n n n
l tg l l l l
   


1 2
1 1
c t t b b
u y s u s u
n n
P f A f A
l l
 
   
 
 


1 2
1 1
c t t b b
u u s y s u
n n
P f A f A
l l
 
   
 
 
or

Simplified Models for Nonlinear Static Analysis

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s



0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

1 2
1 1
t t
s y s s
n n
P f A
l l
 
  
 
 

BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

1 2
1 1
b b
s y s s
n n
P f A
l l
 
  
 
 
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of
P
s

and
Δ
s

at the
catenary stage can be expressed as

When the mechanism is the fracture of the top bars
at BE, the relationship of
P
s

and
Δ
s

at the catenary
stage can be expressed as

Simplified Models for Nonlinear Static Analysis

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s



0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

1 2
1 1
t t
s y s s
n n
P f A
l l
 
  
 
 
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of
P
s

and
Δ
s

at the
catenary stage can be expressed as

0 0
''
1 2
y y
b
u u
tr
n n
M M
P
l b l b
 
 


'
1 2
1 2
b
tr n n
tr
b b
y s n n
P l l
f A l l
 

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

So,

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.

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s



0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)


BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

The top value of the vertical load at the transition
point can be determined as



'
a
t a
u tr
tr u tr a
tr a
P P
P P

   
 

s
P
0


2
5

双跨梁
荷载-位移
曲线

a


tr

s

t
tr
P

a
u
P

tr
P

'
tr

Static load
-
deflection responses for two
-
bay beams

Simplified Models for Nonlinear Static Analysis

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s



0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

1 2
1 1
t t
s y s s
n n
P f A
l l
 
  
 
 

BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

The carrying capacity at the catenary stage is depended
on the ultimate strength of the top bars.

1 2
1 1
c t t
u u s u
n n
P f A
l l
 
  
 
 
1 2
12 0.2 min(,)
u n n n n n
l tg l l l l
   
When the catenary action is activated by the fracture of
bottom bars at BM, the relationship of
P
s

and
Δ
s

at the
catenary stage can be expressed as

Simplified Models for Nonlinear Static Analysis

Loadings working on the two
-
bay beam at
the catenary stage



2
2

梁悬索阶段受力
分析

l
n1

b
'

Δ
s

P
s

T

l
n2

θ
1

θ
2

T

P
s



0

C
ompressive arch

stage


Catenary

stage


Elastic stage

'
u


a


'
tr


a


s
P

y
P

'
c
u
P

a
u
P

t
tr
P

y


b
tr
P

Static load
-
deflection response for two
-
bay
beams (the catenary action was activated by
the beam bars fracture)

1 2
1 1
c t t
u u s u
n n
P f A
l l
 
  
 
 
1 2
12 0.2 min(,)
u n n n n n
l tg l l l l
   

BM

Faile
d

column

P

N

N

(
b
)
catenary

action

for top bars in beam

BM

Fail
ed

column

P

N

N

BE

BE

BM

(
c
)
catenary

action

for top and bottom bars in beam

BE

Fail
ed

column

P

N

N

(
d
)
catenary

action

for bottom bars in beam

(a)
compressive arch action

B
E

BM

B
E

BM

Faile
d

column

N

N

P

1 2
1 1
b b
s y s s
n n
P f A
l l
 
  
 
 
When the mechanism is the fracture of the top bars at
BE, the relationship of
P
s

and
Δ
s

at the catenary stage
can be expressed as

0 0
''
1 2
y y
b
u u
tr
n n
M M
P
l b l b
 
 


'
1 2
1 2
b
tr n n
tr
b b
y s n n
P l l
f A l l
 

The vertical load was carried by the top beam bars after
the bottom bars fracture.



'
a
t a
u tr
tr u tr a
tr a
P P
P P

   
 
Simplified Models for Nonlinear Static Analysis

0
5
10
15
20
25
30
0
100
200
300
400
Vertical load P(kN)
Middle joint deflection
Δ
(mm)
B1A
Ps
-
ex
Ps
-
cal
0
5
10
15
20
25
30
0
100
200
300
400
Vertical load P (kN)
Middle joint deflection
Δ
(mm)
B1
Ps
-
ex
Ps
-
cal
0
5
10
15
20
25
30
0
50
100
150
200
250
Vertical load P (kN)
Middle joint deflection
Δ
(mm)
B2
Ps
-
cal
Ps
-
ex
0
50
100
150
200
0
50
100
150
200
250
300
Vertical load P (kN)
Middle jiont deflection
Δ
(mm)
A1
Ps
-
ex
Ps
-
cal
0
50
100
150
200
250
0
50
100
150
200
250
300
Vertical load P
(kN)
Middle jiont deflection
Δ
(mm)
A5
Ps
-
ex
Ps
-
cal
0
30
60
90
120
150
0
100
200
300
400
500
Vertical load P
(kN)
Middle jiont deflection
Δ
(mm)
B3
Ps
-
ex
Ps
-
cal
(a) Static load
-
deflection response for test specimens

(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 !