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| Due to initial local failures in accident events |
| 作者：本站 来源：转载 发布时间：2008-7-24 7:56:51 发布人：guo8130 |
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Due to initial local failures in accident events, domino phenomena or even progressive
collapse might arise in some vulnerable structures, and consequently, robustness or
collapse resistance becomes one important requirement in structural design. Because
of its low redundancy, truss roof used in large-scale public buildings is one system
different from that of frame and bearing wall, of which plentiful literatures can be
achieved. In order to discover its potential risk, progressive collapse analysis is
carried out in this paper using sophisticated tool LS-DYNA, and its redistribution
mechanism of internal forces, dynamic response and failure modes are investigated.
The conceptions of sensitivity element and key element are put forward to identify
different responses of members due to initial failures. Finally, a Safety Assessment
Method (SAM) based on static linear analysis is proposed to evaluate collapse
resistance performance of truss structures.
Keywords: steel truss roof, progressive collapse, safety assessment, sensitive element,
key element
1 Introduction
Since several catastrophic events in the past years, it becomes a significant consensus that progressive
collapse starting from initial failure of local elements should be avoided or mitigated in accident events
[1-2]. Tie force, alternate load path and specific local resistance were proposed as primary design
methods to improve structural robustness and integrity [3]. Because of a high possibility of intended
attacks for federal or commercial buildings, frame and bearing wall systems had been paid more
attentions to explore its collapse resistance performance [4-5]. For public buildings using plane or
space system with low redundancy, however, its weak point and vulnerability have been overlooked for
a long time. Focusing on steel truss roof, numerical analysis and mechanism discussion are carried out
in this paper and a Safety Assessment Method (SAM) is established to identify sensitivity elements and
key elements and to improve structural robustness.
1 Supported by Specialized Research Fund for the Doctoral Program of Higher Education, SRFDP, China
(20060247034)
-2-
2 Progressive collapse analysis for truss roof
Nowadays, progressive collapse analysis for real structures is still one of complicated numerical
techniques, which involves troublesome problems, such as large deflection and large rotation, rupture
and discontinuous deformation, contact and impact, and so on. However, when attentions are closely
paid to mechanics behavior of structure system, some complex simulation problems can be neglected in
special situations. For example, contact and impact is not considered in this paper since load path could
not be effectively established by contacts of members and impact effects would be of little importance.
Sophisticated tool LS-DYNA with explicit time-integration is adopted in this paper for progressive
collapse analysis. Hughes-Liu Beam based on co-rotational formulation is used with improved section
integration precision and refined line meshing. Elastoplastic model is used for steel material. When
effective plastic strain reaches a specified value, 0.15 in this study, the section integration point is
supposed disable to account for fictitious fiber rupture, and full section shall lose functions if the
process continues.
Figure 1: Dimensions and sections for truss roof
The steel truss roof analyzed in this paper is sketched as Fig.1, with a span of 33 meter and a total height
of 27.63 meter. In this system, T-type sections and double angle sections are used as main members,
and H-type sections as columns. The semi-rigid behavior of gusset connection and no secondary
members, such as purlins and roof bracings, are included. As dead load, uniform roof load of 0.9 kN/m2
and walling load of 0.15 kN/m2 are adopted in accidental event.
The results show that, if top chord is supposed to fail initially as Figure 2, the original equilibrium
condition would break up and a new load path would be effect soon by the redistribution of internal
force for members. Static plastic analyses show that weakened bending resistance of truss could lead to
a stress concentration at local area and then load path would consequently change. The dynamic
collapse analysis reveals the similar behaviors as well as its dynamic effect. As depicted in Figure 3 and
Figure 4 for nodal deflection and member axial force, it can be found that the dynamic response is of a
vibration from initial position toward a new static equilibrium position if exists, otherwise, progressive
collapse would come forth [6].
1
1
2400 2400
33000
13054 12571 2005
1386
5230 4760 4760 3500 4760 4760 5230
27630
1
2 2
H900S
斜腹杆2e 斜腹杆2d
d 2c d 2c 2c
TW250
TW300
TW250
TW300
TW300 T150×300×10×15
编号截 面牌号
TW250 T125×250×9×14
2d 2L90×7 Q235B
2e 2L100×7
2c 2L75×6
d L90×7
BH700 H700×450×16×28 Q345B
HM700 H700×300×13×24
H900S H900×520×20×30
2TM244 2T122×175×7×11 Q235B
HM900 H900×300×16×28
2
Web Chord Web Chord
Name Section Material
-3-
a) Bending moments for members b) Axial forces for members
Figure 2: Redistribution of internal forces after initial failure of top chord
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-90
-75
-60
-45
-30 upper bound
lower bound
Displacement/mm
Time/s
by dynamic
by static
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-100
0
100
200
300
lower bound
upper bound
by dynamic
by static
Axial Force/kN
Time/s
Figure 3: Deflection at node ND8 Figure 4: Axial force of bottom chord BT1
The similar results can be found when initial failure of web chord occurs as Figure 5, however, with
more serious stress concentration and remarkable redistribution of internal forces. Figure 6 illustrates
that multiple integration points of bottom chord BT1 are ruptured and potential collapse resistance of
truss might reach its ultimate capacity. So, the moment capacity of columns is crucial to make a new
load path, as Figure 7.
a) Bending moments for members b) Axial forces for members
Figure 5: Redistribution of internal forces after initial failure of web chord
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-400
-200
0
200
400
Pnt5 Pnt6 Pnt7 Pnt8
Pnt1 Pnt2 Pnt3 Pnt4
Pnt8
... ...
Pnt1
Stress / MPa
Time/s
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-3000
-2000
-1000
0
1000
Bending moment / kN.m
Time/s
CL1
CL2
Figure 6: Fiber stresses of integration points for BT1 Figure 7: Bending moments at bottom of columns
BT1
ND8
BT1
-4-
3 Mechanisms of internal force redistribution
Because of notable stress concentrations after initial local failure, plastic hinges are commonly
inevitable at ends of members, which consequently results in the redistribution of internal forces in
total structure. The effect of plastic hinges to total structure is called local redistribution mechanism
and the change of load path in total structure called as total redistribution mechanism in this paper.
Table 1 lists two typical local redistribution mechanisms due to different initial failures: rotating pin
and sliding surface, which represent no moment or shear resistance capacity, respectively. Since
unbracing length of top chord is increased due to initial failure of web chord, rotating pin mechanism
might happen if top chord could not keep stable.
Table 1: Local redistribution mechanisms
INITIAL
FAILURE
LOCAL
MECHANISMS SCHEMATIC
Top chord Rotating pin - down
Bottom chord Rotating pin - up
Web chord Sliding surface
Web chord Sliding surface
Web chord Unbracing length is
increased
A simplified beam-column system is conceived instead of truss-column structure, so that the local
redistribution mechanisms can be introduced to roughly explore the effect of different initial failures. If
pin mechanism occurs at a distance of αL from right column as shown in Figure 8, the moment and
shear at controlling point B can be derived as following (other points are similar to point B):
( )
2
B 6 2
12
qL
M =α α+ γ × , ( ) B 2
2
qL
V = α+γ × (1a,b)
Where, q is uniform load and L is the span of truss, γ is a factor about sectional parameters of truss [6].
Then effect factors can be obtained representing the ratio of moment or shear to one before initial
failure. It can be judged from Figure 9 that initial failure of top or bottom chords at mid-span would
result in a remarkable increase of axial forces for chords at end-span.
1 0.8 0.6 0.4 0.2 0
0.0
0.5
1.0
1.5
2.0
2.5
MB
MC
MM
Reff for Moment
Location-a
Figure 8: Rotating pin for total redistribution Figure 9: Effect factors for MB, MC & MM
H
D
C
(1-α)L αL A
M
X1
X2
B
-5-
Figure 10 and Figure 11 discover the effect of sliding surface mechanism. It is concluded that sliding
surface mechanism is very adverse to top and bottom chords at end of truss and the initial failure of web
chord is sensitive to progressive collapse. The moment and shear at controlling point B are written as:
( ) 2
B
6 2
12
qL
M = α -γ × , B 2
2
qL
V = α× (2a,b)
1 0.8 0.6 0.4 0.2 0
-4
-2
0
2
4
6
Reff for Moment
Location-a
MB
MC
MM
Figure 10: Sliding surface for total redistribution Figure 11: Effect factors for MB, MC & MM
4 Safety assessment for steel truss roof
In order to identify the particular function of members, the concepts on sensitivity element and key
element are proposed. The sensitivity element represents that initial failure of which would tend to
induce progressive failure or even collapse, and consequently it is a weak point in original load path of
structural system. The key element represents one member which contributes to resist progressive
collapse within residual element system. Thus, key elements are essential to achieve a robust structure
with indispensable redundancies. It is necessary to point out, however, that the concept of key element
by HMSO [2] or JSSC [7] is factually equal to sensitivity element in this paper. Stressing two different
concepts in this paper is helpful for more deep understanding.
Based on the identification of sensitivity element and key element, a Safety Assessment Method (SAM)
for truss structure is established using linear static analyses and member internal force checks. The flow
of this assessment method is illustrated in Figure 13, including four phases: a) Threat assessment,
which estimates the possibility of external threat and possible initial failure in incident events, b)
Initialization analysis, which investigates original structural bearing capacity in common loads, c)
Sensitivity analysis, which calculates sensitivity index for initial failure members and key index for
residual members, and d) Safety assessment, which investigates the distribution of sensitivity elements
and key elements and then makes a general evaluation of structural robustness as well as suggestion for
design or retrofitting.
As one of main subjects in Threat-Vulnerability-Risk Assessment system [8], threat assessment has not
been stipulated well in most nations and regions. But it is a starting point for Safety Assessment
Method.
In phase of initialization analysis, bearing capacity factors can be defined for truss members and
columns as:
k
bc
0
R A f
N
×
=j× , p k
bc
0
W f
R
M
×
= (3a,b)
Where, N0 is axial force of truss chords and M0 is maximum moment of columns in initial condition, A
is area and Wp is plastic section modulus of cross section, φ is stable factor for axial compression and fk
is the characteristic value of steel yielding strength. Weak points in structural system can be
preliminarily judged by Checking bearing capacity factors.
B
A
H
D (1-α)L αL
M
C X1 X3
-6-
Sensitivity analysis is a series of linear analyses in which different initial failures are supposed in turns.
In this phase, effect factors can be defined for truss members and columns as:
Reff=N/N0, eff 0 R =M/M (4a,b)
Where, N is axial force of truss chords and M is moment of column in condition with initial failures.
The dynamic effect for static linear analysis can be modified by:
( ) d 0
0
1+
/
N N NN N
N N
æ ö
= + - =ç - ÷×
è ø
h
h h (5a)
( ) d 0
0
1+
/
M M MM M
M M
æ ö
= + - =ç - ÷×
è ø
h
h h (5b)
Where, η is a corrective factor as multiple DOF systems. A value of 1.05~1.10 is suggested [6].
During checking internal force for members, sensitivity index of initial failure members represents the
maximum value of residual elements’ demand-capacity ratios, and key index of residual element means
the maximum value of its demand-capacity ratios during different initial failures.
j j
i i
sen j i sen j i
k p k
R Max N or R Max M
¹ A f ¹W f
æ ö æ ö
= çèj××÷ø = ççè ×÷÷ø
(6a,b)
j j
j j
key i j key i j
k p k
R Max N or R Max M
¹ A f ¹W f
æ ö æ ö
= çèj××÷ø = ççè ×÷÷ø
(7a,b)
Where, i
sen R represents sensitivity index for initial failure member i, and j
key R represents key index for
residual member j.
Theoretically, it is reasonable to identify all members of Rsen>1.0 as sensitivity elements. Considering
the actual existence of energy dissipation capacity of primary chords and other assistant members,
members of Rsen >1.5 are only deemed as real sensitivity elements (also very sensitive elements) in this
paper. Similarly, key elements are only ones of Rkey>1.5.
In fact, sensitivity index and key index can also be uniformly defined as:
i j
i eff
sen j i j
bc
Max R R
R
®
¹
æ ö
= ç ÷
è ø
,
( i j)
j i j eff
key j
bc
Max R
R
R
®
= ¹ (8a,b)
Where, j
bc R means bearing capacity factor of residual member j, and i j
eff R ® denotes effect factor of
member j due to initial failure of member i. If array expression is used, two arrays for bearing capacity
factors and effect factors can be written as:
[ ]
1
bc
2
bc
bc n n
n
bc
0 0
0 0
0 0
R
R
R
R
´
é ù
ê ú
=êê úú
ê ú
êë úû
(9)
[ ]
1 2 1 m 1 n
eff eff eff
2 1 2 m 2 n
eff eff eff
eff m n
m 1 m 2 m n
eff eff eff
0
0
0
R R R
R R R
R
R R R
® ® ®
® ® ®
´
® ® ®
é ù
ê ú
=êê úú
ê ú
êë úû
(10)
Therefore, the array for safety factors can be calculated:
-7-
[ ] [ ] [ ] 1
sf m n eff bc
1 2 1 m 1 n
eff eff eff
2 m n
bc bc bc
2 1 2 m 2 n
eff eff eff
1 m n
bc bc bc
m 1 m 2 m n
eff eff eff
1 2 n
bc bc bc
0
0
0
R R R
R R R
R R R
R R R
R R R
R R R
R R R
-
´
® ® ®
® ® ®
® ® ®
= ×
é ù
ê ú
ê ú
ê ú
ê ú
=ê ú
ê ú
ê ú
ê ú
ê ú
ë û
(11)
It is obvious that the value of sensitivity index for member i is equal to the maximum of all entries in
low i of array [ ] sf R and the value of key index for member j equal to the maximum of all entries in
column j of array [ ] sf R . It is necessary to point out that, since the bearing capacities of members for
tension or compression differs, such array expression can be only used as a literal explanation and be
disabled for calculations.
When shortage of structural robustness requires modification of design or retrofitting, it is not advised
to take some measures for reinforcement involving excessive change of structural system and the idea
of “protecting sensitivity elements and strengthening key elements” is suggested. Although protecting
sensitivity elements is good to improve structural reliability, strengthening key elements is still
encouraged to reflect the robustness ideology.
5 Application Demonstration of safety assessment method
According to Safety Assessment Method, the robustness of truss roof in Figure 1 can be evaluated. The
redistribution of sensitivity elements and key elements is depicted as Figure 12. The results show that
the truss roof is very sensitive to the initial failure of three web chords at end-span and two top chords
at mid-span, and columns and several bottom chords at end-span is key to resist progressive collapse.
This conclusion is with a good similarity to progressive collapse analysis in section 2.
Figure 12: Redistribution of sensitivity elements and key elements
WB1
0.83
WB1
0.68
WB1
0.68
BT6
1.08
BT6
1.07
BT6
1.67
BT6
1.67
0.74
WB1
0.74
WB1
0.67
WB1
0.67
WB1
0.64
WB1
0.92
BT6
2.74
BT6
2.17
BT6
1.60
BT6
1 .24
WB1
1.05
WB1
0.88
WB1
0.76
WB1
2.01
WB1
2.0 1
WB1
1.84
WB1
1.84
WB1
2.7 4
WB1
0.99
0.89
WB1
TP1
0.44
TP1
0.50
TP1
0.73
1.08
W B1
1.38
WB1
WB1
0. 74
1.00
WB1
1.22
WB1
WB1
1.20
WB1
1.32
0.6 9
WB1
1.79
WB1
WB1
1. 88
a)敏感性系数分布
关键构件
非关键构件
1.90
WB1
2.23
WB1
非常敏感
较敏感
不敏感
a) Sensitivity Elements b)关键性系数分布
Very Sensitive
Sensitive
Insensitive
b) Key Elements
Not Key Elements
Key Elements
-8-
Figure 13. Flow for Safety Assessment Method
0 N ( 0 M )
k p k
bc bc
0 0
R A f or R W f
N M
j
× ×
= × =
Initialization
analysis
i j ( )j i j ( )j
eff 0 eff 0 R®=N/N or R®=M/M
i j
i eff
sen j i j
bc
Max
R R
R
®
¹
æ ö
= ç ÷
è ø
( i j)
j i j eff
key j
bc
Max R
R
R
®
= ¹
Sensitive
Initial failure
member i
N,M
Static
( )
( )
d 0
d 0
N N N N
M M M M
h
h
= + - ìï
í
= + - ïî
Dynamic
Threat
Start
Safe
Dangerou
s
Threat
assessment
Sensitivity
analysis
Safety
assessment
Modifying
project design
Sensitivity
elements i i
sen sen 1.0£R <1.5 or R ³1.5
Key elements
j j
key key R ³1.0 or R ³1.5
Insensitiv
Safety
Very sensitive
Threat
High dangerous
Dangerous
End
-9-
6 Conclusion
The mechanisms of internal forces redistribution and dynamic effects are really revealed for truss roof
after initial failures and program LS-DYNA is testified to be an ideal tool for progressive collapse
analysis. Two local redistribution mechanisms of rotating pin and sliding surface is introduced into a
simplified beam-column system for studying the total redistribution of internal forces. It is roughly
indicated that the web chords at end-span are mostly sensitivity elements and top and bottom chords at
end-span are key elements. For common truss systems, a Safety Assessment Method (SAM) is
establishing based on the identification of sensitivity elements and key elements using linear static
analysis. This method is clear in conception and convenient for application, and can also be a good tool
for relevant structural design and retrofitting.
References
[1] American Society of Civil Engineers. ASCE 7-05 Minimum Design Loads for Buildings and Other Structures
[S]. 2005.
[2] HMSO. The Building Regulations 2000: Approved document A – Structure [S]. Norwich: The Stationery
Office, 2004.
[3] Leyendechker EV, Ellingwood BR. Design methods for reducing the risk of progressive collapse in buildings
[R]. Washington: National Bureau of Standards, 1977.
[4] U.S. General Services Administration. Progressive Collapse Analysis and Design Guidelines for New Federal
Office Buildings and Major Modernization Projects [S]. 2003.
[5] U.S. Department of Defense. UFC 4-023-03 Design of Buildings to Resist Progressive Collapse [S]. 2005.
[6] Jiang XF. Study on mechanism analysis and resistant design of progressive collapse for truss/beam-type
structural systems in large-span [D]. Shanghai. 2008.
[7] Japanese Society of Steel Construction & Council on Tall Buildings and Urban Habitat. Guidelines for
Collapse Control Design-Construction of Steel Buildings with High Redundancy [S]. 2004.
[8] Renfroe N A, Smith J L. Threat/Vulnerability Assessments and Risk Analysis [EB/OL]. 2007.
Author Brief Introduction：
Jiang Xiao-Feng, Man, Birth in December 26, 1979. Ph.D. Research on structural progressive collapse
and structural design.
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