Advances in Steel Structures - part 9 pps

60 S.Z. Shen
p/g are shown in Fig.5. It's seen that the curves sustainedly go down, do not approach a limit even as
p/g=2, meanwhile the limit load has dropped to a rather low value of about 30% of the case of
symmetrical loading. Based upon regression analysis the coefficient K2 of considering the effect of
unsymmetrical loading can be given as Eqn.9. This formula is applicable for both net systems.
Kz = 1 / [ 1 + 0.956 p/g + 0.076 (p/g)2 ] (applicable for p/g=0-2.0) ( 9 )
"
~ moveable hinge
15 N fixed hinge
z lO
-
., .,~ .~ .~. ~'- ~
5 - f/L = 1/6 ~"-" ~~.~~~
f/L = 1/7
f/L= 1/8 p/g
0 0 015 i ' ' 1.5 2
Figure 5 : Limit loads of reticulated shallow shells with increase of p/g
In reference to the analytical formula of linear theory for domes and based upon regression analysis,
the design formula for predicting limit load of reticulated shallow shell can adopt rather simple form:
x/BD
For triangular system qcr = 1.29K2 ( 10 )
R1R2
4"D
For orthogonal system qcr = 1"07K2 ( 11 )
R1R 2
In which R1 and R 2 are the radiuses of curvature in two directions, respectively, and the coefficient K 2
is given in Eqn.9. The effect of initial imperfections has been considered in the formulas.
STABILITY OF RETICULATED SADDLE SHELLS
The complete load-deflection response of the reticulated HP shells are varied with variation of
geometrical and structural parameters such as the raise-span ratio, the net system, the rigidity of edge
beams and etc Some load-deflection curves may rise sustainedly with no critical point emerging.

Some curves may have bifurcation point appearing, but the load continues up with the rigidity matrix
of the structure keeping positive definite. And in some other curves buckling of limit-point type may
occur, but the load rises again after certain post-buckling path downwards. HoweVer, there exists a
common character for the load-deflection curves of HP shells: the load has a general tendency to
keep going up, and from practical viewpoint the load-capacity of the shells is maintained. As an
example, the load-deflection curves of HP shells of L=60m with different raises (H = 6,9,12,15 and
18m) are shown in Fig.6a. It demonstrates the specific characteristics of the shells of negative
Gaussion curvature. Further more, it can be supposed that the feature of monotonous rise would be
revealed more obviously for load-deflection curves of the practical shells with initial imperfections.
Design Formulas for Stability Analysis of Reticulated Shells
61
It seems rational to conclude that the stability problem is not significant for reticulated HP shells, and
as a necessary substitutive measure the rigidity of the shell should betaken as a main structural
property to be checked in practical design. The maximum deflections of the shells with different
raises under service load (2kN/m 2) are shown in Fig.6b. It's seen that the rigidity of HP shells with
H=9m and 6m is obviously not enough.
Figure 6 : a. Load-deflection curves of reticulated HP shells with different raises
b. Maximum deflections under service load of these shells
CONCLUSIONS
1. Based upon the complete load-deflection analysis for more than 2800 examples of reticulated
shells of prototype the varied and colorful structural behaviors developing with the loading
process, the practical mechanism of structural instability and the complex effects of different
factors were revealed rather thoroughly for different types of reticulated shells.
2. Based upon the regression analysis of the plentiful data obtained from the parametrical analysis as
described above design formulas for predicting limit loads of reticulated domes, reticulated vaults
with different supporting conditions, as well as reticulated shallow shells, rather simple for
application but obtained on the basis of accurate theoretical procedure, were proposed.
3. For reticulated saddle shells it's suggested just to carry out routine rigidity check instead of the
complicated stability analysis.
REFERENCES

Chen X. and Shen S.Z (1993). Complete Load-Deflection Response and Initial Imperfection
Analysis of Single-Layer Lattice Dome.
International Journal of Space Structures
8:4, 271-278
Wang N., Chen X. and Shen S.Z. (1993). Geometric and Material Non-linear Analysis of Latticed
Shells of Negative Gaussion Curvature.
Space Structures 4.
London. 649-655
Shen S.Z. and Chen X. (1999).
Stability of Reticulated Shells.
The Science Publisher, Beijing, China
62
S.Z. Shen
APPENDIX: Formulas for Equivalent Rigidities of Reticulated
Shells
The net systems used for reticulated shells can be classified into three basic types as shown in the
attached figure.
Attached Figure: Three typical net systems
The equivalent rigidities in two main directions can be calculated as follows:
1 .For net system ( a ) and system ( b ) with single diagonal
B11
EA1 EAc EI1 EIc
= + sin4
a
Dll = + sin4
a
A 1 A c A 1 A c
B22 EA 2 EA~ E12 EZc 4
= + cos4a 022 = + cos a
A 2 A~ A 2 Ac

2.For net system ( b ) with double diagonals
EA~
EAc E11 El c
Bll = +
2 sin 4 ct D~ = + 2 sin 4 a
A 1 Ac A1 Ac
B22
EA 2 EA c EI 2 EI
= + 2 cos4 ct D22 = + 2 cos 4 a
A 2 A~ A 2 Ac
3.For net system ( c )
EA 1 EAc
Bll + 2 sin4 a Dll
E11 Elc
= = + 2 sin 4 ct
A 1 Ac A1 A c
B22
2
EAc
cos4
a
022 = 2
EI c
= COS 4 a
Ac Ac
In the formulas A1,A 2 and A c are the cross-section areas of members in direction 1 and 2 and of
diagonals, respectively, I l, I 2 and Ic are the corresponding moments of inertia, the intervals between
members A 1
, A 2
and A c , as well as the inclination angle a are as shown as in the figure.

DUCTILITY ISSUES IN THIN-WALLED STEEL
STRUCTURES
T. Usami 1, Y. Zheng 1, and H.B. Ge 1
1Department of Civil Engineering, Nagoya University, Nagoya, 464-8603, JAPAN
ABASTRACT
The ductility of thin-walled steel box stub-columns under compression and bending is studied in this
paper through extensive parametric analyses, and empirical ductility equations are developed. The
equations for isolated plates and pipe stub-columns proposed in the previous studies are also presented.
On this basis, a simplified ductility evaluation procedure is proposed for practical steel structures with
thin-walled box or pipe sections. An inelastic pushover analysis is employed and a failure criterion is
introduced. The implementation of the proposed procedure is demonstrated by application to some
cantilever columns and a one-story frame. Moreover, the computed results are compared with the
ductility estimations through cyclic analyses reported in the literature, which leads to the validation of
the proposed method.
KEYWORDS
Thin-walled steel structure, Ductility, Pushover analysis, Stub-column, Residual stress, Initial deflection,
Box section, Pipe section, Frame, Cyclic loading.
INTRODUCTION
Thin-walled steel columns and frames have been widely used as substructures in urban highway bridges,
suspension and cable-stayed bridge towers in Japan as well as some other countries. But the need for
evaluating the seismic performance, such as the ductility capacity, of such structures has come into focus
following the damage and collapse observed dr~ the 1995 Hyogoken-nanbu earthquake (Fukumoto
1997; Galambos 1998). Steel beam-column members employed in bridge structures are characterized by
the use of relatively thin plates, which makes these structures vulnerable to damages caused by the local
and overall interaction buckling. However, the task of accounting for such buckling can be formidable
for a practical use where the balance between reliability and simplicity is required.
63
64 T. Usami et al.
A simplified ductility evalUation method for steel columns and frames composed of box sections was
previously proposed by the authors (Usami et al. 1995). An inelastic pushover analysis is utilized in the

method and the structural ultimate state is assumed to be attained when the compressive flange strain of
the most critical part reaches its failure strain. However, the method employs an empirical equation
based on isolated, simply supported plate under compression (Usami et al. 1995) to calculate the failure
strain, and consequently leads to somewhat conservative predictions for structures composed of
moderately thin plates. This is for the reason that the interactive effects between adjacent component
plates at their junctions are neglected.
In this paper, aiming at proposing more refined empirical equations for failure strains, thin-walled steel
box stub-columns are studied under combined action of compression and bending. Extensive parametric
analyses are carried out to investigate the effects of some parameters on the behavior of stub-columns
with and without longitudinal stiffeners. An elasto-plastic large deformation FEM analysis is employed.
Based on the parametric analyses, empirical equations for the ductility of box stub-columns are
developed. Besides, the ductility equations for isolated plates in compression and short cylinders in
compression and bending proposed in the previous studies (Usami et al. 1995; Gao et al. 1998a) are also
presented. By using the equations based on stub-columns, the previous ductility evaluation procedure for
box-sectioned structures (Usami et al. 1995) is refined and meanwhile, is extended to both box and
pipe-sectioned structures. A one-story frame with stiffened box sections and several cantilever columns
with unstiffened box sections, stiffened box sections, and pipe sections are investigated as examples to
demonstrate the application of the procedure. Moreover, the computed results are compared with
previous results obtained through cyclic tests or numerical analyses (Usami 1996; Gao et al. 1998b;
Nishikawa et al. 1999). The comparison illustrates the validity of the proposed method.
DUCTILITY OF BOX STUB-COLUMNS
Numerical Analytical Model
Both the box stub-columns with and without longitudinal stiffeners are studied. The analytical models of
such stub-columns are shown in Fig. 1, which represent a part of a long column between the diaphragms.
Due to the symmetry of geometry and loading, only a half or a quarter of the stub-column is analyzed. A
simply supported boundary condition is assumed along the column end plate boundaries to simulate the
local buckling mode of a long column, which would deforms into several waves along the length. To
Y
Y
P P

Web I
(a) Unstiffened (b) Stiffened [ s.s.: simply Supported Edge
Figure 1" Analytical model of box stub-columns
Ductility Issues in Th&-Walled Steel Structures 65
Figure 2: Residual stresses
impose a rotation of the edge, the end sections are constrained as rigid planes by using the multi-point
constraint (MPC) boundary conditions, and the rotation displacement is applied at any node on the
sections. The bending moment is obtained as the reaction force of the node. The general FEM program
ABAQUS (1998) and a type of four-node doubly curved shell element (S4R) included in its package are
employed in the elasto-plastic large deformation analysis.
An idealized rectangular form of residual stress distribution in each unstiffened panel, stiffened panel,
and stiffener plate, is adopted due to the welding (see Fig. 2). The initial geometrical deflections are also
considered. For unstiffened stub-columns, the shape is assumed to be sinusoidal in both flange and web
plates (see Fig. 3(a)). The maximum values ofthe initial deflections in the flange and web are assumed to
be B/500 and D/500 (where B and D are the breadth and depth of the box section), respectively. The
directions of the initial deflections are assumed inward for flange plates while outward for web plates.
The assumed initial deflection shape in the flange plate of stiffened stub-columns (Fig. 3(b)) are given by
following equations:
where
8=~5a+8 L (1)
a ,000 sinI;
1( )
= sin ~Z y cos • Z
150 ~ ~ (3)
in which cY c denotes the global initial deflections; CYL represents the local initial deflections; a is the
length of the stiffened stub-columns; n is the number of the subpanels divided by the stiffeners; m is the
number of half-waves of the local initial deflections in the longitudinal direction, which is assumed as an
integer giving the lowest failure strain and will be further discussed below. The initial deflections in the
web plates are calculated by replacing B and z in Eqs. (2) and (3) by D and x, respectively, but assumed
in opposite direction (outward).

A kind of steel stress-strain relation including a strain hardening part, proposed by Usami et al. (1995), is
utilized in this study to define the material characteristics (see Fig. 4). Here, % and 6y denote the yield
stress and strain, respectively; E is the elastic modulus (i.e., Young's modulus); 6,, is the strain at the
onset of strain hardening; E~ is the initial strain hardening modulus; and E' is the strain hardening
modulus assumed as
E' = E,~ exp(-~ s - o%t ) (4)
6y
66
T. Usami et al.
Figure 3: Initial deflections
where 2j is a material coefficient. Mild steel SS400
(equivalent to ASTM A36) is utilized in the analysis of
stub-columns, for which Cry = 235 MPa, E = 206 GPa, t,'
= 0.3, e n = 10 e y, 2j = 0.06, and
En=E/40.
In this study, the ductility of the stub-column is evaluated
by using the failure strain,
6u/zy,
which is defined as a point
corresponding to 95% of the maximum strength after the
peak in the bending moment versus average compressive
strain curve (Usami and Ge 1998).
Parametric Study
Figure 4: Material model
The behavior of thin-walled box stub-columns subjected to compression and bending is considerably
affected by the magnitude of axial load,
P/Py (Py
is the squash load), and the flange width-thickness ratio,
RI, which is defined as
a~ B I12(1- v2) IO'y

Ry :
: t 4n2x 2 E (5)
in which
O'cr
is the elastic buckling stress; n is the number of subpanels (for unstiffened plate, n = 1). For
stub-columns with stiffeners, the stiffener's slenderness ratio, 2 s , is another key parameter, given by:
- 1 a 1 ~-~y
A" = x/-Q r, n 3r E (6)
1
Q 2-~f [13 - ~/13z - 4Rf ]
(7)
13 - 1.33Rf + 0.868 (8)
in which rs is the radius of gyration of a T-shape cross section consisting of one longitudinal stiffener and
the adjacent subpanel and Q is the local buckling strength of the subpanel plate
(Structural Stability
1997). An alternative parameter reflecting the characteristics of the stiffener plate is the stiffener's
relative flexural rigidity, y, which is interdependent on 2,. Thus, in the present study, only 2 s is
considered in the ductility equations.
Ductility Issues in Thin-Walled Steel Structures
TABLE 1 TABLE 2
Parameters of Thin-walled Parameters of Stiffened
Unstiffened Box Stub-columns Box Stub-columns
D/B at = a/B Rf D/B a = a/B
Rf ?"
/ y *
3/4, 1.0, 0.5, 0.7, 0.2, 0.4, 0.45, 0.5, 0.67, 1.00, 0.5, 0.7, 0.3, 0.35, 0.4, 0.45, 1.0,
4/3 1.0 0.55, 0.6, 0.8 1.33 1.0, 1.5 0.5, 0.55, 0.6, 0.7 3.0
67
~s
0.180 "~

0.751
Nevertheless, to propose ductility equations for comprehensive applications, the influence of box
cross-sectional shape (say, square or rectangle) and the aspect ratio
(a =a/B)
has to be surveyed. And for
stiffened stub-columns, the critical local initial deflection mode along the length direction giving the
lowest ductility should be first determined. The thickness of the plates is assumed as 20ram and the
considered axial force, P, ranges from
O.OPy
to
0.SPy.
Other pertinent parameters are given in Tables 1
and 2, where 9" * represents the optimum value of 9" obtained from elastic buckling theory
("DIN 4114"
1953).
Through parametric analyses, following conclusions are drawn for unstiffened stub-columns: (1) The
effects of the cross-sectional shapes and column aspect ratios on the stub-column ductility are
insignificant and the present empirical equation is based on the models with square sections and aspect
ratios equal to 0.7; (2) Referring to the computed
e,/ey
versus
R z
and
P/Py
relations presented in Fig. 5, it
is observed that the failure strain decreases as the increase of either
R z
or
P/Py;
(3) Considering the

effects of axial loads, an equation of failure strain,
eu/ey,
versus flange width-thickness ratio,
R z,
is fitted
as follows:
~;, 0.108(1- P / Py
)1.09
~.~
: (Rf
-0.2) 3"2~ + 3.58(1- P
/
py)O.839 < 20.0 (9)
The applicable range of this equation is
R/=
0.2 0.8,
D/B
= 0.75 1.33, and
P/Py
= 0.0 0.5. It should
be noted that when the failure strain, 6u/ey (which is the average strain in the compressive flange),
exceeds 20.0, the local maximum strain would be very large (say, 5% or larger) and the numerical
analysis results would become unreliable. Thus, the upper bound of 6,,/6y is limited as 20.0 at present
time although the consequent prediction will be on the safe side for some cases.
As for the stiffened stub-columns, the observations from the parametric analysis can be concluded as: (1)
The critical local initial deflection mode along the length of stiffened stub-columns varies with different
aspect ratios and the corresponding number of half-waves (m) is found as 2, 3, 4 and 5 for aspect ratios of
0.5, 0.7, 1.0 and 1.5, respectively; (2) The buckling mode of stub-columns has almost same shapes as the
assumed initial deflection mode; (3) The influences of box cross-sectional shape and the aspect ratio on
the ductility of stub-columns are not obvious and for simplification, they can be neglected in the design

formulas of failure strain; (4) The effects of flange width-thickness ratio and stiffener's slenderness
~,
~0.1s
ratio should be considered together and a combined parameter *-/,~s is introduced. Inversely
proportional relations of the failure strains to this combined parameter and the axial load are found (see

018
Fig. 6). On this basis, an equation of
eu/~y
versus
Rye. s " ,
considering the effect of axial load, are fitted
as follows:
~'u _ 0"8(1- P / Py )0"94
w
/,• ~-
0.18
~'Y ~"'~f"s
- 0"168) lzzs
+ 2.78(1 - P / P, )0.68
~_~
20.0
(10)
Here,
R/ranges
from 0.3 to 0.7, ~, is in a scope from 0.18 to 0.75, and
P/Py
is between 0.0 and 0.5. And
68 T. Usami et al.
Figure 5 Failure strains of unstiffened

stub-columns
Figure 6 Failure strains of stiffened
stub-columns
this equation is applicable to stiffened box stub-columns with a from 0.5 to 1.5 and B/D from 0.67 to
1.33. Moreover, it should be noted that this equation is fitted to give slightly smaller prediction of failure
strains for the cases with smaller -" = 0.~s
K:% . This is for the reason that the numerical results of present
study are based on monotonically loading conditions and when applied to long columns with small
018
R/~ " , they are found to yield larger ductility predictions compared with the cyclic experimental and
numerical results as presented later.
DUCTILITY OF ISOLATED PLATES UNDER UNIAXIAL COMPRESSION
For comparison, this paper also presents the failure strain equations based on isolated plates under
uniaxial compression (Usami et al. 1995; Usami and Ge 1998). They are defined as follows:
Unstiffened olates: ~" 0.07
= + 1.85 _< 20.0 (11)
" % (R: -0.2y "~
Stiffened nlates: e, 0.145
: + 1.19 < 20.0 (12)
6, (x-, - 0.2)
TM
Equation (11) is plotted in Fig. 5 and some computed results of stiffened plates (Usami and Ge 1998) are
~ ~'- o 18
also plotted in Fig. 6, in the form of E / E y versus K/~s " 9 It is observed that the failure strains of
stub-columns subjected to compression and bending are larger than those of isolated plates under pure
compression. When the axial load is so large as to approach the pure compression state two procedures
will give similar predictions.
DUCTILITY OF SHORT CYLINDERS
The ductility of thin-walled steel short cylinders in compression and bending has been also investigated
in a previous study (Gao et al. 1998a). Analytical models similar to those used for box stub-columns,

which have been presented above, were employed for the cylinders. Main parameters controlling their
behaviors are found to be the magnitude of the axial force and the radius-thickness ratio parameter, Rt,
which is in the form of
Rt = 6y = ~/3(1 - v 2) oy d (13)
o~, E 2t
Here d and t denote the diameter and thickness of the cylinder, respectively. And an empirical equation is
proposed as follows:
Ductility Issues in Th&-Walled Steel Structures
~,,, O.120+4P/Py)
6 f = (R t -
0.03) 1"4s
(1 + P / P,)5 + 3.6(1 - P / P,) <__ 20.0
69
(14)
DUCTILITY EVALUATION PROCEDURE FOR THIN-WALLED STEEL STRUCTURES
By using the empirical equations of failure strains
given above, a ductility evaluation procedure is
proposed for practical structures composed of thin-
walled steel beam-column members. It is applicable
for the design of a new structure or evaluation of a
existing structure, which are in the form of
cantilever-typed columns or framing structures. The
procedure involves the following steps:
1. Based on the general layout and loading condition
of the structure, establish the analytical model as Figure 7: Pushover analysis model
shown in Fig. 7, by using beam elements, which
facilitate the FEM modeling procedure but do not
account for local buckling. Neither the residual stresses nor initial deflections are take into
consideration. The material model defined in Fig. 4 is also utilized for the pushover analysis.
2. Carry out a planar pushover analysis. This procedure involves applying the constant vertical loads

and incrementally increased lateral loads to represent the relative inertia forces which are generated
at locations of sustained mass. An elastoplastic large displacement analysis is employed to account
for the second-order effects.
3. The pushover analysis is terminated once the failure criterion is attained and this state is taken as the
ultimate state of the structure, based on which the ductility capacity, 8u/By, of the structure can be
determined.
Like the previous study (Usami et al. 1995), the failure of a structure composed of the thin-walled steel
box members is assumed when the average strain over an effective failure length in the compressive
flange (or in the maximum compressive meridional fiber for pipe sections) reaches its failure stain (Eqs.
(9), (10) and (13)). The effective failure length, I,, of a box-sectioned member is assumed as the smaller
one between 0.7 times of the flange width and the distance between two adjacent diaphragms (Usami et
al. 1995). For pipe-sectioned structures, based on the observations in the previous studies (Gao et al.
1999a and 1999b), an empirical equation is proposed here to define the effective failure length:
1 -1)d (15)
I t = 1.2(Rt ~.0s
The critical parts could be more than one place in a framing structure and all of them should be checked
(see Fig. 7(b)). In a thin-walled steel structure, however, the excessive deformation tends to intensify in
a local part and consequently the redistribution of the plastic stress becomes unexpected. Thus, once the
failure criterion at any one of the critical parts is satisfied, the ultimate state of such a structure is though
to be reached.
NUMERICAL EXAMPLES
To demonstrate its implementation, the proposed ductility evaluation procedure is applied to some
cantilever-typed columns with box or pipe sections and a one-story rigid frame composed of box section