©Civil-Comp Press, 2012

Proceedings of the Eighth International Conference

Paper 24 on Engineering Computational Technology,
B.H.V. Topping, (Editor),
Civil-Comp Press, Stirlingshire, Scotland

The Design of Special Truss Moment Frames Against

Progressive Collapse
H.K. Kang, J.Y. Park and J.K. Kim
Department of Architectural Engineering
Sungkyunkwan University, Suwon, Korea

Abstract
In this study the progressive collapse resisting capacity of the special truss moment
frames (STMF) was investigated. As analysis models, STMF with various span
lengths, numbers of storeys, and lengths of special segment were designed and their
performance compared. It was observed that all the model structures designed as per
the ATSC seismic provision collapsed as a result of plastic hinge formation at the
special segment when a column was suddenly removed. A design procedure was
developed to prevent progressive collapse based on the energy balance concept. The
model structures redesigned using the developed design procedure turned out to
remain stable after a column was suddenly removed and satisfies the acceptance
criteria of the GSA guidelines.

Keywords: special truss moment frames, progressive collapse, nonlinear analysis,

energy based design.

1 Introduction
This study investigated the progressive collapse resisting capacity of the special
truss moment frames (STMF) structures. To this end analysis model structures with
vierendeel special segment were designed per the AISC (American Institute of Steel
Construction) Seismic Provisions. The design parameters such as the length of
special segment, depth of panels, span length, and number of stories were
considered in the investigation. The progressive collapse potential of the structures
was evaluated based on the arbitrary column loss scenario recommended in the GSA
(General Service Administration) guidelines. A design procedure was proposed
based on energy-balance principle to prevent progressive collapse of the STMF
structures, and the validity of the proposed procedure was evaluated by nonlinear
static and dynamic analyses of four analysis model structures.

1
2 Design of STMF systems

According to the AISC Seismic Provisions (2008), STMF are required to be

designed to maintain elastic behavior of the truss members, columns, and
connections, except for the members of the special segment that are involved in the
formation of the yield mechanism. All members outside the special segment are to
be designed for calculated loads by applying the combination of gravity and lateral
loads that are necessary to develop the maximum expected nominal shear strength of
the special segment.
The AISC Seismic provisions 2009 (draft) presents the expected vertical shear
strength of the special segment at mid-length, Vne, as follows:

3.6 R y M nc ( L − Ls )
Vne = + 0.036 Es I 3
+ R y ( Pnt + 0.3Pnc ) sin α (1)
Ls Ls

where Ry = yield stress modification factor, Mnc = nominal flexural strength of the
chord members of the special segment, EsI = flexural elastic stiffness of the chord
members of the special segment, L = span length of the truss, Ls = length of the
special segment, center-to-center of supports, Pnt = nominal axial tension strength of
diagonal members of the special segment, Pnc = nominal axial compression strength
of diagonal members of the special segment, α = angle of diagonal members with
the horizontal members.

3 Design of analysis model structures

In this study a redesign procedure was proposed to enhance the progressive collapse
resisting capacity of the STMF structures above the acceptance criterion of the GSA
guidelines. Fig. 1 illustrates the failure mechanism of the STMF structures subjected
to loss of a column, where large plastic deformation occurs in the members located
in the special segment. For a STMF structure to remain stable after a column is
removed, the internal work of the members subjected to plastic deformation needs to
be in equilibrium with the external work done by the removed column. As plastic
hinges occur only in the members of the special segment, the required size of the
members in the special segment can be obtained from the equilibrium of the internal
and external works. Fig. 3 shows the moment-rotation relationship of the members
in the special segment idealized for design purpose, and the plastic moment of the
cord members, Mpc, can be obtained as follows:

M pc = Fyc Z c = αFyc Sc (4)

where Fyc is the yield stress of the chord members, Sc and Zc are the elastic and the
plastic section moduli of the chord members, respectively. The shape factor α is the
ratio of the plastic and the elastic section moduli.

2
 Mp
d δ
θ
Lp
Ls P

L
Fig. 1 Failure mechanism of STMF subjected to column removal

Parametric study results for the parameters α with respect to the varying h/b (depth /
width) of the angle section with three different width thickness ratios (b/t) showed
that α decreases almost monotonically from 1.8 to 1.65 as h/d varies from 1 to 3. In
this study lower bound value of 1.65 was used for α to derive conservative solution
for the required section modulus of the special segment members to prevent
progressive collapse. The plastic moment of the vertical members in the special
segment can be computed as follows:

M pv = Fyv Z v = αFyv S v = γM pc (5)

where Fyv is the yield stress of the vertical members, Sv and Zv are the elastic and the
plastic section moduli of the vertical members in the special segment, respectively,
and γ is the ratio of the plastic moment of the vertical and the chord members as
follows:

M pv
γ= (6)
M pc

In the bi-linearly idealized moment-rotation relationship of the members in the

special segment, shown in Fig. 12, the yield rotations of the chord and the vertical
elements are obtained as follows:

M pc L p M pv d
θ ec = , θ ev = (7)
6 Es I c 6 Es I v

where Es is the elastic modulus, Ic and Iv are the second moments of inertia of the
chord and the vertical members, respectively, Lp is the length of the special segment,
and d is the depth of the special segment panel. The limit state for member rotation
was set to be 0.035rad following the GSA guidelines. The moments of inertia of the
chord and the vertical members in the special segment are represented as follows:

I c = S c βhc , I v = S v βhv (8)

3
where βh is the depth of the centroid. The variation of the parameter β as a function
of h/b of the angle section showed that β decreases monotonically from about 0.7 to
0.6 as h/b increases from 1.0 to 3.5. In this study the lower bound value of 0.55 was
used for β to induce conservative results. Based on the above simplification, the
energy balance equation of the internal and the external work is formulated as
follows:

⎛ M pcθ ec (2M pc + ηkcθ pc )θ pc ⎞ ⎛M θ (2M pv + ηkvθ pv )θ pv ⎞

N c × ⎜⎜ + ⎟⎟ + N v × ⎜⎜ pv ev + ⎟⎟ = P × δ (9)
⎝ 2 2 ⎠ ⎝ 2 2 ⎠

6 Es I c 6 Es I v
kc = , kv = (10)
Lp d

The left hand side of Eq. 9 represents the internal work done by the member force
and the deformation of the elements in the special segment, and the right hand side
corresponds to the external work done by the force supported by the removed
column, P, and the vertical displacement, d, at the beam-column joint from which
the column was removed. Nc and Nv are the number of plastic hinges formed in the
chord and the vertical members in the special segment, and θpc and θpv are the plastic
rotation at the chord and the vertical members, respectively. The post yield stiffness
η was assumed to be 10% of the initial stiffness. Based on the above equations the
section moduli of the chord and the vertical members in the special segment required
to satisfy the energy balance equation, Sc(req) and Sv(req), respectively, are derived as
follows for the given depths of the chord and the vertical members, hc and hv,
respectively:

2 PLsθ u
S c ( req ) =
⎡ 2
⎡ ⎛ 0.15αFyc L p ⎞ 0.6 βhc E sθ u2⎤
⎛ 0.15αFyv d ⎞ 0.6 βhv E sθ u Fyc ⎤
N c ⎢αFyc ⎜ 1.8θ u − ⎟⎟ + ⎥ + γN v ⎢αFyv ⎜⎜ 1.8θ u − ⎟⎟ + ⎥
⎜
⎣⎢ ⎝ βhc E s ⎠ L p ⎥⎦ ⎢
⎣ ⎝ βhv E s ⎠ dFyv ⎥⎦

Fyc
S v ( req ) = γS c ( req ) (12)
Fyv

To prevent progressive collapse of the STMF structures caused by sudden column

loss, the sectional moduli of the members in the special segment need to be larger
than those derived above. Therefore after a STMF is designed based on the current
design code, the above procedure needs to be applied before finalization of design.
Once the member sizes of the special segment are increased, the other members also
need to be redesigned so that plastic hinges form only at the special segment.

4
 2@4.2m

2@4.2m
1.35m 1.35m
5 .2 m

5 .2 m
1m 2m

3 @ 6m 3 @ 6m
(a) Ls/L=0.16 (b) Ls/L=0.33

Fig. 2 3-story 6m span model

As analysis model structures three-story STMF structures with different span

lengths were designed following the guidelines of the Seismic Provisions. Fig. 2
shows the side view of the three-story analysis model structure with two different
lengths of the special segment. The design dead and live loads of 4.9kN/m2 and
2.5kN/m2, respectively, were used as vertical load, and the seismic load was
evaluated based on the spectral acceleration coefficients of SDS=0.43 and SD1=0.23
with the response modification factor of 7 in the ASCE 7-10 format. The columns
were designed with wide flange sections with ultimate strength of 490 MPa, and the
truss members outside of the special segment were designed with double angle
sections with the same ultimate strength. The double angle sections in the special
segment were designed to have the ultimate strength of 400 MPa. Fig. 3 shows the
force-deformation relationship of structural members (IO: Immediate Occupancy, LS: Life
Safety, CP: Collapse Prevention).

Moment
(M/My)
b
a

C
1 B CP
IO LS

D E

c
A
θy Rotation

(a) Bending members (b) Bracing

Fig. 3 Force-deformation relationship of structural members (IO: Immediate Occupancy,

LS: Life Safety, CP: Collapse Prevention)

4 Analysis of model structures

The validity of the proposed design procedure to prevent progressive collapse was
investigated by analyzing STMF structures. Fig. 4 shows the nonlinear static and

5
dynamic analysis results of the single story STMF structure with 6m span length
(Ls/L=0.33) subjected to sudden loss of one of the interior columns. The results of
the structure designed with conventional (Original) and the proposed method
(Redesigned) were compared. The pushdown analysis results show that the
maximum load factor of the original structure designed per the AISC Seismic
Provision is less than 0.5, well below the required value of 1.0. The nonlinear time
history analysis results show that the vertical displacement is unbounded when the
column is suddenly removed. The maximum load factor of the structure with only
the member sizes of the special segment redesigned considering progressive collapse
reached about 0.75 and the structure remained stable around the vertical
displacement specified as limit state in the GSA guidelines after sudden removal of
the column. The structure with complete redesign showed maximum load factor
higher than 1.0 and remained stable at the vertical displacement above the limit
state. Fig. 5 shows the plastic hinge formation in the 1-story model structure
obtained from pushdown and pushover analyses. It can be observed that in the
structure with all members redesigned following the proposed procedure, plastic
hinges formed only in the special segment, which conforms to the basic philosophy
of STMF structures. Fig. 6 depicts the Plastic hinge formation in the 3-story 9m span
model. The structures designed following the proposed procedure turned out to
remain stable at vertical displacements smaller than the limit states specified in the
GSA guidelines. It was also observed that plastic hinges formed only in the special
segment as required by the Seismic Provisions either when they were subjected to
seismic load or exposed to sudden column loss.
1.5 20
Vertical displacement(cm)

1 0
Load factor

0.5 -20

0 -40

0 10 20 30 40 0 4 8 12 16 20
Vertical displacement(cm) Time(sec)

(a) Pushdown curves (b) Time histories of vertical displacement

Fig. 4 Analysis results of 1-story 6m span model (Ls/L=0.33)

(a) Pushdown analysis (b) Pushover analysis

Fig. 5 Plastic hinge formation in the 1-story 6m span model with all elements redesigned
with proposed method

6
 (a) Pushdown analysis (b) Pushover analysis

Fig. 6 Plastic hinge formation in the 3-story 9m span model

5 Summary
In this study the progressive collapse resisting capacity of special truss moment
frames (STMF) was investigated based on the arbitrary column removing scenario.
As analysis models, STMF with various span lengths, numbers of storeys, and
lengths of special segment were designed and their performances were compared
using nonlinear static and dynamic analyses.
A closed form formula was derived to obtain the required section moduli of the
members in the special segment to prevent progressive collapse based on the energy
balance concept. The remaining elements were resized based on the AISC seismic
provisions to ensure plastic hinge formation only in the special segment. The model
structures redesigned using the developed design procedure turned out to satisfy the
acceptance criteria of the GSA guidelines to prevent progressive collapse. The
nonlinear static pushover analysis of the redesigned structures showed that plastic
hinges formed only in the special segment as required by the seismic provisions.

Acknowledgement
This research was financially supported by a grant (Code# ’09 R&D A01) funded by
the Ministry of Land, Transport and Maritime Affairs of Korean government.

References
[1] AISC. Seismic Provisions for Structural Steel Building (Draft). AISC-341-10,
American Institute of Steel Construction, Chicago, Illinois, 2010
[2] ASCE 7-10, "Minimum Design Loads for Buildings and Other Structures",
American Society of Civil Engineers, New York, 2010.
[3] GSA, Progressive Collapse Analysis and Design Guidelines for New Federal
Office Buildings and Major Modernization Project, The U.S. General Service
Administrations, 2003
[4] SAP2000, Structural Analysis Program, Computers and Structures, Berkeley,
2004