1. College of Civil Engineering, Tongji University
Progressive Collapse Resistance of Reinforced Concrete Frame Structures
Prof. Xianglin Gu
College of Civil Engineering, Tongji University
28/12/2012

2. Acknowledgements
This research project is sponsored by the National Natural Science Foundation of China (No ) and the Shanghai Pujiang Program (No. 07pj14084).

3. Outline Introduction Experimental Investigation Testing Specimens
Test Setup and Measurements
Test Results
Simplified Models for Nonlinear Static Analysis of RC Two-bay Beams
Conclusions

4. 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.

5. 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

6. Previous Experimental Work
At present, experimental 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
Yu, J., and Tan, K.H., “Experimental and numerical investigation on progressive collapse resistance of reinforced concrete beam column sub-assemblages”, Engineering Structures, 2011,
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
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

7. 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.

8. 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.

9. Experimental Investigation
Testing Specimens
Test Setup and Measurements
Test Results

10. Testing Specimens
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).
Plane layout of a prototype structure
Test subassemblies and reinforcement layout

11. Testing Specimens Table 1 Specimen properties Specimen No.
Section, b×h for beam and w×t for flange, mm
*ln, mm
Top bars and reinforcement ratios
Bottom bars and reinforcement ratios
*fc ,
kN/mm2
*Ec(×104),
B1A
100×150
1800
28(ρ=0.86%)
25.6
2.82 B1
21.8
2.73 B2
100×100
900
28(ρ=1.49%)
26(ρ=0.83%)
25.8
2.80
TB3
Beam
28(ρ=0.85%)
28.9
2.86
Flange
450×40
TB4
28(ρ=0.84%)
26.5
TB5
29.8
XB6
Longitudinal beam
32.3
2.87
Transverse beam
28(ρ=1.41%)
26(ρ=0.80%)
XB7
Longitudinal
31.5
3.02
Floor
1950×40
Transverse
28(ρ=1.45%)
3750×40
* ln represents the net span of a beam. fc represents the compressive strength of concrete. Ec represents the Young's modulus of concrete.

12. Testing Specimens Table 2 Reinforcement properties type Diameter, mm
Yield strength fy ,
kN/mm2
Ultimate strength fu ,
Elongation,
(%)
Young's modulus Es(×105), 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
6.60
329
523
22.1
2.18
bolts
19.63
561
671 -
1.86

13. Test Setup and Measurements
Boundary conditions and test setup of specimens

14. Test Setup and Measurements

15. 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
Failure modes of specimens

16. Compressive arch action and catenary action
Test Results B2
B1A, B1
Compressive arch action and catenary action

17. 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

18. 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.
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
(a) Specimen B1A
(b) Specimen B1
(c) Specimen B2

19. RC Beam-column Subassemblies
(a) at BM for B1A
(b) at BE for B1A
Strain of rebars in B1A

20. (Definitions of BE and BM are given in Table 3)
RC Beam-column Subassemblies
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 GB can meet the continuity requirements in progressive collapse resistant design.

21. 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 versus middle joint deflection Δ for B1, TB3, TB4 and TB5
Specimen TB4
Specimen TB5

22. RC T-Beam-column SubassembliesPs N N Ps
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.

23. RC T-Beam-column Subassemblies
Strain of steel bars at FT for TB5
Strain of steel bars at FC for TB5

24. RC Cross-beam system without a floor slab
(a) longitudinal direction
(b) transverse direction
Vertical load P versus middle joint deflection Δ for B1, B2 and XB6
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.

25. Vertical load P versus middle joint deflection Δ for XB6 and XB7
RC Cross-beam system with floor slab
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.

26. 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.
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)
Areas and yielding and ultimate strengths for the top and bottom bars in beams

27. 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.
the yielding moment of two beam ends, respectively
Static load-deflection response for two-bay beams (the catenary action was activated by the concrete crushing)
the yielding moment of the left and right sections of beams near the middle column, respectively
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
the stiffness of the most unfavorable section of beam (BM)
Areas and yielding and ultimate strengths for the top and bottom bars in beams

28. Simplified Models for Nonlinear Static Analysis
Deformation mode of two-bay beams under ultimate state considering the compressive arch action
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.
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)

29. 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.
the drift of BE
Static load-deflection response for two-bay beams (the catenary action was activated by the concrete crushing)
Geometrical relationship of BE and BM for the left bay of two-bay beams
For the left bay:
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
For the right bay:
For the two- bay beam:
Deformation made of two-bay beams under ultimate state considering the compressive arch action

30. 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.
Static load-deflection response for two-bay beams (the catenary action was activated by the concrete crushing)
Stress and strain distribution of BE in the left bay
The stress of bottom bars can be derived.
simplified
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)

31. 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.
According to the equilibrium condition of the internal forces at BE, x1L can be determined.
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
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.

32. Simplified Models for Nonlinear Static Analysis
N and are quadratic functions of Δs and there is always a Δs that makes become the maximum and the corresponding vertical deflection 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.
Δs＝Δs +dΔs
Determine N.
Calculate x1L, x0L, x1R and x0R.
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 , x0R and the bending moment s Mu1L, Mu0L, Mu1R and Mu0R .
Calculate
become the maximum , and ，

33. 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.
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
When the catenary action is activated by the concrete crushing, the relationship of Ps and Δs at the catenary stage can be expressed as
Concrete is deactivated at the transition point,
So,
Loadings working on the two-bay beam at the catenary stage

34. Simplified Models for Nonlinear Static Analysis
The vertical deflection at the ending of the catenary stage is depended on the elongations of the top and bottom bars.
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°.
So, or
Loadings working on the two-bay beam at the catenary stage

35. 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
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
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
Loadings working on the two-bay beam at the catenary stage

36. 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
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
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
Loadings working on the two-bay beam at the catenary stage
So,

37. 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.
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
Loadings working on the two-bay beam at the catenary stage
Static load-deflection responses for two-bay beams
The top value of the vertical load at the transition point can be determined as

38. 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
Static load-deflection response for two-bay beams (the catenary action was activated by the beam bars fracture)
The carrying capacity at the catenary stage is depended on the ultimate strength of the top bars.
Loadings working on the two-bay beam at the catenary stage

39. 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
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.
Loadings working on the two-bay beam at the catenary stage

40. Simplified Models for Nonlinear Static Analysis
(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.

41. 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.

42. 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 GB can meet the continuity requirements in progressive collapse resistant design.

43. 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.

44. Thank You !