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Stability design of steel plane frames by second-order elastic analysis
EngineeringStructures,Vol. 19, No.
ELSEVIER
PII: S0141-0296( 96)00134-4
8, pp. 617-627, 1997 © 1997 Elsevier Science Ltd All fights reserved. PfintedinGreat Britain 0141-0296/97 $17.00+ 0.00
Stability design of steel plane flames by second-order elastic analysis Hirotaka Oda Takigami Steel Construction, Co., Kiyokawa-cho 2-1, Nakagawa-ku, Nagoya, Japan
Tsutomu Usami Department of Civil Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan (Received July 1996; revised version accepted October 1996)
A new stability design method for steel plane frames using secondorder elastic analysis is proposed. Although many current design methods are based on the effective buckling length concept, the effective buckling length may not always be obtained accurately. This requires that a stability design method be developed without using the effective buckling length. In this method, the introduction of equivalent initial deflection, which accounts for the reduction in strength due to various initial imperfections, is of the utmost importance. Therefore, a new formula for the equivalent initial deflection is proposed, and a methodology using the curvature of the buckling mode is introduced to allovwfor a systematic application of this formula to arbitrary shaped irregular frames. The validity and efficiency of the method are demonstrated by some examples. © 1997 Elsevier Science Ltd. Keywords:stability design, buckling, steel frame, second-order elastic analysis, equivalent initial deflection
I.
Introduction
form cross-sections. An alternative method to calculate the effective buckling length is to use a matrix eigenvalue analysis of the framework 4.5. Compared with the aforementioned chart method, this method can determine the effective buckling length of all members in a frame having an arbitrary shape under arbitrary loads with the help of a computer. This method can also be used to determine the effective buckling length of each section in a frame with nonuniform cross-sections. Although this method is thought to be very effective, it has a problem in that the effective buckling length of a member with small axial force becomes extremely large 5. For example, this sort of situation occurs at the upper part of a cable-stayed bridge tower or near the centre of a girder in a cable-stayed bridge. In order to overcome this problem, a new design method has been proposed 6. In this method, the stress resultants with consideration of the effect of geometrical nonlinearity are directly computed from the second-order elastic analysis, and the frame stability design is made using only the cross-sectional strength equation. This method requires the introduction of an adequate initial deflection (the equivalent initial deflection (EID)) which is greater than the geometrical initial deflection specified in a code in order to account
This paper presents a stability design method for steel frames based on the second-order elastic analysis, which is expected to take the place of the effective buckling length method. In the current stability design method for steel frames in many countries, a strength equation for the cross-section and a stability equation for the member are usedL The stability equation is an interaction strength equation for a beamcolumn, and takes into account the effect of geometrical and material nonlinearities. Using the stability equation based on the effective buckling length concept changes the strength of the frame to the strength of beam-columns. However, the effective buckling length cannot be calculated accurately in cases such as a nonrectangular frame or a frame with nonuniform cross-sections which are often used in bridge structures 2. Figure 1 shows some steel towers of cable-stayed bridges in Japan 3. For these kinds of irregular frames, the effective buckling length cannot be determined accurately using a chart (e.g., AISC nomograph 2) because the chart is based on the assumption of a rectangular frame with uni-
617
Stability design of steel plane frames: H. Oda and T. Usami
618
|
a
I
b
c
Figure 1 Towers of cable stayed bridges: (a) Meiko-Nishi bridge; (b) Yokohama Bay bridge; (c) Iwaguro-shima bridge for the effects of reduction in strength due to various initial imperfections, such as initial deflection and residual stresses in a structure. In the current design procedure for a steel frame, the first-order elastic analysis is generally used. Contrary to the current design method, the present method needs secondorder elastic analysis. This analysis is based on the linearized beam-column theory with the simplest treatment of geometrical nonlinearity 7, but it gives accurate enough solutions for practical bridges and building frames in many cases. Moreover, because of using the second-order elastic analysis, the effective buckling length is not used and the design formula becomes very simple. Recent advances in computer hardware have made it possible to use the second-order analyses for practical design 8. One of the most sophisticated methods in the analyses is the plastic-zone analysis. However, the computational cost and time are too large for routine design work. Its applications to date are limited to research studies and for special design problems. The feasible method for practical design use on personal computers and workstations will be the second-order elastic analysis with pseudo (or enlarged) initial deflection (i.e., E I D ) 6'9 o r the notional-load plastic-hinge analysis 1°. These notional quantities have the same physical meanings, i.e., these are quantities that account for the influence of all initial imperfections and the effect of plastification in the members. Although the me~hod used to specify these equivalent values has been proposed for a strut and a rectangular frame 9,1°, the method for a nonrectangular frame with various boundary and loading oonditions has not yet been shown. In this paper, a rational £o~rmla for EID and a methodology for imposing the initial de~l~efftion shape on a frame are proposed. The validity of the method is demonstrated with some examples. Using the plastic-hinge analysis, the load-displacement behaviour, so-called hinge-by-hinge behaviour, can be traced. Second-order elastic analysis, however, cannot be used to trace this behaviour because the stress redistribution is not considered. In spite of this, the second-order elastic analysis is useful for practical design. This is because the
load related to the first plastic hinge is close to the ultimate load in an optimally designed framed structure. This fact will be explained in Section 5 with an example. 2.
Fundamental
concept
The physical meaning of the equivalent initial deflection (EID) concept is explained with an illustrative example. Figure 2 shows a relation between the axial force, P, acting at two ends of a pin-ended column with initial crookedness and the maximum bending moment, M, at the midspan of the column. The solid curve is supposed to represent the rigorous relation between P and M obtained from secondorder plastic-zone analysis of the column with initial imperfections (i.e., initial crookedness L/1000 and residual stresses). The results of the second-order elastic analysis represented by the dashed curves depend on the magnitude of the initial deflection, 6o- When an appropriate value (i.e., fo) is selected, the point of intersection of the dashed curve and the cross-sectional strength line (elastic limit in this case) can be matched to the ultimate strength by the P
Second-order
|
_. 6 < f0 elastic a n a l y s i s
1"0 t K ~ •
,," - ~ " " "
.....
6>f0
P
Second-order
0.5
M=p.6
0.5
1.0
M M"-y"
Figure 2 Fundamental concept of equivalent initial deflection
Stability design of steel plane frames: H. Oda and 7-. Usami second-order plastic-zone analysis. For this reason, all the effects of initial imperfections must be included in EID, and its magnitude will be greater than that used in the second-order plastic-zone analysis.
619 M0/My-09[=Eq (10)]"]
1.0
/"/;
- o2
t. ~oml,~
-0.4
f ECCS-b
-08
.J
Eq (14) [Proposedformula]
Numerical procedures
The second-order elastic analysis discussed here is a matrix analysis of a framed structure based on the linearized beamcolumn theory 7. The calculation procedure for the stability design method by the second-order elastic analysis is summarized in the following steps
0.6
p t~'-M°
,
/ // / ,///~¢//"
/¢/"
//;//////i
M°-'~ D
0.4
/ / /
,//,"
0.8
3.
~
"
,;7
0.8
1.0
0.2 (1) Specify a loading state (2) Give the magnitude and shape of EID (discussed in the next section) (3) Calculate N (= axial force) in each member by the firstorder elastic analysis (4) Evaluate stress resultants N and M (= bending moment) by solving the following equilibrium equations ([KE] + [Ka(N)]){d} = {f}
(1)
where [Ke] is a total elastic stiffness matrix, [Ka(N)] is a total geometric stiffness matrix, {d} is a nodal displacement vector, {f} is a load vector. (5) Check whether or not the forces N and M satisfy the following interaction equation N M ---+ = 1 Ny M,
(2)
with M,, = M e (for class 2 cross-sections 9) = My (for class 3 cross-sections 9)
(3)
where Ny = squash load, My = yield moment and Mp = fully plastic moment.
4.
Equivalent initial deflection (EID)
4.1. Magnitude of equivalent initial deflection A pin-ended column subjected to a concentric compressive force and equal moments at both ends is considered as the simplest structure (see Figure 3), and a formula to calculate the magnitude of EID is proposed. The magnitude of EID depends on the strength interaction equation of the cross-section. In the proposed method, the foregoing linear interaction equation (2) is used. The initial deflection of the column is assumed as
"fix Yo =fo " sin ~ -
(4)
where Yo = EID curve, fo = EID at midspan, L = column length. When the column is analysed by conventional beam-column theory, the following closed form solution of maximum bending moment at midspan. Mmo~., can be derived ~,
0.2
0.4
0.6
1.2
1.4
Figure3 Magnitude of initial deflection Mmax | N Mo kL Mu - 1 - N/NE Ny 91 -t- Muu sec ~-
(5)
where M0 = moment actin a ~ o t h ends, NE = Euler's column buckling load, k = ~IN/EI, E = elastic modulus, I = moment of inertia, and "0 = dimensionless expression for EID given by A r / = f o ~ (for class 3 cross-section)
(6a)
A =fo ~ (for class 2 cross-section)
(6b)
in which A = cross-sectional area, W= elastic section modulus and Z = plastic section modulus. Substituting equation (5) into equation (2), one obtains the following strength equation for the pin-ended column by the proposed method. N N/Ny Mo kL Ny + 1 - N/NE rl + -~. sec ~- = 1
(7)
The magnitude of "O can be determined by equating the strength obtained from equation (7) to the exact strength of the pin-ended column. Here, the following beam-column interaction strength equation will be used as an accurate enough equation. N 1 Mo + - 1 17, 1 - N/NE 34,
(8)
where N, is the ultimate strength of a concentric compression column. When equation (7) is compared with equation (8), an important physical meaning of EID can be seen, i.e., it represents the reduction in the axial strength from the squash load, Ny, to the ultimate column strength, N,, because the third term in equation (7) is nearly equal to the second term in equation (8). Figure 3 shows the values of EID at midspan of the column which are calculated by changing the moment at both ends (denoted as Mo/Mu) from 0.0 to 0.8. The ultimate axial load, N,, is calculated from the column strength curve-b of Eurocode 39 for a box column in equation (8). In this figure, the basic value of initial member rotation, qbo, for
620
Stability design of steel plane frames: H. Oda and T. Usami
unbraced frames expressed in equation (9) and the equivalent initial bow imperfection for centrally loaded columns in equation(10), both of which have been proposed in Eurocode 3 9, a r e also plotted. 1
qb° = ½00
(9)
"r/= 0.34 (A - 0.2)
(10)
where the slenderness ratio parameter, A, is defined as q
~=~1 K . L / ~ , r
(11)
~/A • N -
(12)
N~I
in which K = effective buckling length factor, r = radius of gyration of the cross-section, o~,.= yield stress, and A = eigenvalue from buckling analysis of the whole frame. It is to be noted that the computed result for Mo = 0 is coincident with equation(10). Equation(9) is reduced to equation (13) for convenience in plotting. A
Figure 4 shows the strength interaction curve of a pinended beam column by using the proposed EID and the interaction equation (8) which matches the exact strength well. Moreover, the interaction curve by use of equation (9), i.e., the initial member rotation in Eurocode 3 is compared. In this case, the initial deflection mode of the shape of snapped line shown in Figure 4 is considered. It can be seen from the figure that both the proposed E1D and equation (9) give accurate enough strength. However, a detailed comparison shows that the proposed EID is slightly conservative when moments are zero but equation (9) is slightly unconservative when A is large. 4.2. Initital deflection mode The next important problem is to determine the equivalent initial deflection mode for a given structure. For example, in the girder of a cable-stayed bridge in Figure 5, ~4its first buckling mode obtained from the linear buckling analysis is quite different from the collapse mode obtained from the second-order plastic-zone analysis. This is because only axial force is taken into consideration in the buckling analysis, but the effect of bending moments predominates in the second-order plastic-zone analysis. To determine which buckling mode or collapse mode is appropriate as the initial deflection mode, a two-span continuous beam-column (see Figure 6) has been analysed by
n=/o w 1.0
|
t
|
w
!
t
Interaction strcn~ equation .....
W
400W
Proposed EID
400KW
Mo
~-0.6
As can be seen from the figure, equations (9) or (10) are very close to the computed results in the case of the centrally loaded column (Mo/M, = 0.0), but it are low compared with the computed results in the range where A is large and as the moments are increased. The reason is that the amplification of the bending moment due to the socalled P-delta effect is large in these cases, resulting in a considerable reduction in the column strength. From these results, the following equations are proposed to calculate the magnitude of EID.
"~X
"~
~"
'%
............. Initial
~ "%
.o.=
"%'-°
• ",%
fo memberrotation 1/200
N I'M°
;%\
?. . . .
Mo
N
. ".°,° '~ ~'.,°.
",.. ,,~,~,
A
=~.=a,.(A-0.2) = ~2"(A-/3)
( 0 . 2 % A ~ 1.0) (1.0 ~ A)
(14)
Values of the coefficients oq, c~2 and/3 are given in Table 1, in which groups of cross-sections other than the box section are also included 9,~2,~3.
oo 0.0
Figure 4
0.5 Mo/Mp
1.0
Pin-ended beam column
Table 1 Values of coefficients a~, e2 and /3 in e q u a t i o n (14) Eurocode 36
F u k u m o t o and Itoh ~
~1 ~x2 /3
JSHB 1°
Group 1
Group 2
G r oup 3
a0
a
b
c
d
0.539 1.337 0.678
0.072 1.165 0.951
0.260 1.281 0.838
0.515 1.475 0.721
0.125 1.194 0.916
0.236 1.265 0.851
0.404 1.338 0.767
0.582 1.529 0.696
0.884 1.784 0.604
g r o u p 1 = pipe, b o x and rolled-H (wall thickness -< 40 ram) cross-sections g r o u p 2 = w e l d e d - H (wall thickness ~- 40 mm), tee, channel and angle cross-sections g r o u p 3 = rolled and welded-H (wall thickness > 40 ram) cross-sections
Stability design of steel plane frames: H. Oda and T. Usami (a)
621
;
(b)
(e)
Buckling mode
Collapse mode
Figure 5 Buckling and collapse mode of cable-stayed bridge: (a) model 1 with weak tower; (b) model 2 with weak girder; (c) model 3 with weak cable p
~
Q
A,,I,
!_-
,
o.,,.
A2,12
,
o.4L .I
L
~Q
A,,I,
P
o.81. L
2t
,,fo v'
'f
1
Buckli-~ Mode
~-y
Collapse Mode due to bending
/
1.0 ~
e- - 4 Second-orderPlastic Analy~ ( Collapse Mode ) Second-order Plastic Analysis( Buctdi.~ Mode ) o- - ~ $econd-c~der Elastic Analysis( Collapse Mode ) o, o Second-order Elastic Analysis( Bucklin8 Mode )
.
0.5
I
I
[
I
v
0.5
1.0
w
'
Q
Qy
Figure 6 Comparison of initial deflection modes imposing two different critical deflection modes. This beam-column was designed to have two stepped cross-sections near the midsupport in order for the three cross-sections to yield at the same time when the effective buckling length method is employed. Figure 6 shows the ultimate
strengths from the second-order elastic analysis and the second-order plastic-zone analysis using both modes. The magnitude of initial deflection for these modes is L/lO00 in the latter analysis, and in the former analysis, it is determined by a method that will be shown in the following section. Furthermore, the yield moment, My, is used in equation (3) for the second-order elastic analysis. From Figure 6, it can be seen that using an initial deflection shape similar to the first buckling mode gives lower ultimate strengths in either of the two analysis methods, and almost the same strengths as in the case of the collapse mode with a large lateral force. Accordingly, it will be adequate to use the first buckling mode as the initial deflection mode in any loading condition. Moreover, the first buckling mode obtained through the linear buckling analysis is appropriate because this mode provides the most serious reduction in the column strength.
4.3.
Application to a general frame
In the present section, we will show how to apply EID expressed by equation (14) to frames with arbitrary shapes and boundary conditions. First, we will discuss simple examples of columns with various boundary conditions as in Figure 7. These columns have an identical effective buckling length and are considered to have equal strength in the current specification. For example, when the strength of column (b) by the second-order elastic analysis is equal to the strength of the pin-ended column (a), which has the same effective buckling length as column (b), the maximum equivalent initial deflection of column (b) should be twice that of column (a), namely, the relative deflection from the inflection point (i.e., midcolumn) to the maximum curvature points (i.e., column ends) of column (b) should be identical to that of column (a). The same situation can be observed in other cases. However, it is not easy to give the initial deflection between the two inflection points and to identify the maximum curvature point for a frame. Therefore, EID is determined by enlarging the buckling mode of the column
Stability design of steel plane frames: H. Oda and T. Usami
622
x : Maximum Curvature Point ' • : Inflection Point
I
L
",
~4
!
M ('0)
(O
(d)
Figure 7 Equivalent initial deflection of column under various b o u n d a r y conditions until its maximum curvature is identical to the maximum curvature of the pin-ended column with the same effective buckling length. This method gives the proper equivalent initial deflection without considering the inflection point because the effect of the boundary condition has already been taken into account in the buckling mode. For frames with arbitrary shapes and boundary conditions, the calculation procedure of EID on the basis of the equivalent curvature is outlined as follows: (1) Carry out the eigenvalue analysis of a geometrically perfect frame subjected to a set of design load, and find the first buckling mode. Determine the slenderness ratio parameter, A, for each member from the foregoing equation (12). (2) Find the member that has the smallest value of A. As explained later, this member has the lowest column strength. (3) In order to apply the EID of a pin-ended column obtained from equation (14), calculate the end rotation, 0o, and the curvature, Ko, at the midspan of the column from the following equations
o,,
=
(K.L
f,=X r
(15)
,. f , 1 o', -- iX. L) = E (16) (4) Calculate the curvature of the member considered in step (2) from its buckling mode, and find the maximum curvature point (denoted point M in Figure 8) of this member. Calculate the equivalent initial curvature KO)M, in which the effect of the boundary condition is considered, at the point M from the following equations (the derivations are shown later)
KO)M = KO " sin~M
( 17 )
~M= K 2L = c o t '
(18)
\KM/Kol where KM and 0M, respectively, represent the curvature and the slope at the point M, XM is the x-directional coordinate of the point M when the member of interest is converted to a member with an effective length KL (see Figure 8). (5) Determine the initial deflection by proportionally enlarging the buckling mode until the curvature at
Figure 8
Initial equivalent deflection of frame
point M is equal to the foregoing equivalent initial curvature given by equation (17). Now, we will explain the physical meaning of the smallest value of A in step (2). In this connection, we consider a ratio of axial force N acting on each member in a frame to its ultimate axial strength N,. The member with the largest ratio N/N,, implies that it is the most unstable in a frame. Multiplying the eigenvalue A which is obtained by the linear buckling analysis, the ratio N/N, gives the following relation. A •N
N.
=
N~:(A)
N,,(A)
(19)
Both elastic buckling strength, N,, and inelastic strength,
N,,, are expressed as function of X. and it is evident that the ratio of NE/N, is larger for smaller A. Accordingly, the member which has the smallest A obtained by the linear buckling analysis has the largest N/N, ratio; i.e., the column is weakest against axial force in the frame. Next, we will derive equations(17) and (18). In the equivalent pin-ended column with length KL, the equivalent initial deflection, slope and curvature are expressed as Yo = f , ' sin~
(20)
)'(; = 0o ' cos~
( 21 )
y~; = K,. sin~
(22)
= ~x/KL
(23)
with
and a prime indicates differentiation with respect to x. From the condition that the ratio of slope to curvature at the point M in a frame equals the ratio in the equivalent pin-ended column, the following equations are obtained 0M = y(; = 00 cot~: KM
Thus,
re YO
KO
(24)
Stability design of steel plane frames: H. Oda and T. Usami
r
b = 68cm
235MPa
"1 A = 544cm 2 I = 419605cm 4
~D
o
II
623
v
/f j -118MPa
r = 27.77cm
II
-I
Figure 9
Sectional properties and residual
stresses
( OM/O0]
= cot-'
(25)
\KM/Ko/
which result in equations (17) and (18). 5. A p p l i c a t i o n to s t e e l f r a m e s w i t h s t e p p e d cross-section
5.1. Cross-section properties As illustrative examples, we now consider cantilever columns (Figures 10-12) and portal frames (Figures 13 and 14). Unless otherwise stated, the members have a box section and the residual compressive stress is assumed to be half of the yield stress as shown in Figure 9. With this cross-section, the ultimate strength of a pin-ended column, which has initial out-of-straightness of L/1000, is almost equal to the'column strength curve of ECCS-b, so that the equivalent initial deflection corresponding to ECCS-b will be used in the following examples. The height of the cantilever c_olumn is determined such that the slenderness parameter, A, is equal to 0.9 when the column has the uniform cross-section and is subjected to an axial thrust at the top. In the following examples, the Nu --
PI
N y (~ 1.o
~
~
Second-order Plastic-zone Analysis
o
o
Second-order
tA
ElasticAnalysis
t- - - t
Effective Buckling Length Method
D-.--D
Initial m e m b e r
rotation =
~[
1/200
,
~,
..
P2 B
0.9
m
M
0.8
C ~ N @-0.7
F.quivale~t ]~-'~don atthe Cohmm Top
0.6
1.0
0.5
0.0
P1 Pl +P2
Figure 10 Cantilever c o l u m n subjected to axial thrusts at top and midheight
cross-section will be stepped or another axial thrust will act at midheight, thereby changing the slenderness parameter. The height of a portal frame with one bay and one storey is also determined such that X = 0.9 when the columns have the same cross-section and are subjected to the same magnitude of axial thrusts at both column tops. In the following examples, the magnitude of an axial thrust acting on one column will be changed, resulting in a different slenderness parameter from the other column. Under these conditions, we will examine whether or not the ultimate strength can be evaluated through the proposed method.
5.2. Cantilever columns 5.2.1. Cantilever column subjected to axial forces at top and midheight Figure 10 shows a cantilever column subjected to axial forces of PI at the top and P2 at the midheight. The ratio of P~ to P2 is allowed to change in this example. Since the cross-section of the column is uniform and since the axial force on the upper member is smaller than the lower member, the slenderness parameter, A, of the upper member is large_r than that of the lower member. If P~ becomes zero, A of the upper member becomes infinite. The magnitude of equivalent initial deflection in the proposed method is determined at the fixed end of the column. This is because the value of A of the lower member is smaller and the maximum curvature point in this member is at the fixed end. Accordingly, EID is determined by adopting the equivalent curvature at that point. The magnitude of EID at the column top obtained in this way is shown in Figure 10 for each loading case. On the other hand, the magnitude of initial deflection at the column top used in the second-order plastic-zone analysis is set equal to 1/500 of the column length. This is because the effective buckling length factor is 2.0 for the cantilever column, so that this initial deflection is equivalent to that of L/1000 for a pin-ended column. However, Clark et al. ~5 have proposed that the actual tolerance of erection should be used in the second-order plastic zone analysis. Figure 10 shows that the strengths calculated by the proposed method agree well with those by the second-order plastic-zone analysis. The maximum difference between these strengths is 7.2%. Also, the figure shows the strengths using the initial member rotation of 1/200. The accuracy of these results is sufficient but they are not uniform for loading cases. When only an axial force exists at the
Stability design of steel plane frames: H. Oda and T. Usami
624
(a)
(b) "N
N
~,,/
= =~ m - ~ . ~ .
N,,/
0.5
"
"s.~mm~m,~,.
0..~
1.0 2V~ e
0.5
M-;
0.~
l.O M |
M--;
(c) lq 1.0 i"
'
' ~ll~l~Elmic/il~
I'N.
° .o
O.l
1.0 M 0 M,
Figure 77 Cantilever column subjected to combined bending moment and axial forces at top and midspan: (a) P~:P2= 50:50; (b) PI:P2= 2 0 : 8 0 ; (c) PI:P2= O: 1 0 0 midheight (i.e., Pj = 0 ) , the use of an initial member rotation of 1/200 is more conservative than EID, although the initial deflections at column top are the same for both cases. This is because the initial deflection at midheight for EID similar to the buckling mode is smaller than the initial member rotation, whose mode is a straight line. In Figure 10, the strengths predicted by the effective buckling length method are also shown. These strengths are calculated by substituting the slenderness parameter, A, obtained by the buckling analysis to the column strength equation. The strength of the upper member is smaller than that of the lower member because the value of ~ is larger in the upper member. In spite of this, the strength of the column is controlled by the lower member subjected to larger axial force. This implies that the ratio of axial force to axial strength is important. The strengths by the effective buckling length method are also close to those by secondorder plastic-zone analysis. Figure 11 indicates some examples of the column subjected to combined axial forces and bending moment. In these examples, equation(2) on the basis of plastic moment, M/,, is used in the proposed method. The proposed
method results in good agreement with the second-order plastic-zone analysis.
5.2.2. Cantilever column with stepped cross-sections A cantilever column with stepped cross-sections is presented in Figure 12. The member @ has the cross-section shown in Figure 9, but the member ~ has a different cross-section which is varied in width. This variation is represented as the ratio of moment of inertia, I~/12, in which I~ is the moment of inertia of member 0) and 12 is that of member @. The external load is only an axial force acting at the free end. In this example, the member with smaller cross-section has a smaller value of ~, because the magnitudes of axial force acting on both members are identical. Accordingly, the point in question tbr settlement of EID comes to the point C shown in Figure 12 when It/12 >-- 1.0, and the point jumps to point B when If/I: < 1.0. The quantity of EID at the free end which is determined by adopting the equivalent curvature to the points in question is indicated in Figure 12. Even if the point in question jumps from C to B, there is no discontinuity in the magnitude of the initial deflection.
Stability design of steel plane frames: H. Oda and T. Usami Pu
Nu N~
~lC~
0.8
:
_
0.7
I ~:
I2
L/2
=_
.... = ~d.~&r
I P
I,
B
0.9
_
P
aP
Pl~lic-zone
o----o Second-order Elastic Analy~
A L/2
625
f----
A- - .A Effe~iw Buckling Length Method ~--.--o Initial Member Rotation = 1/200 =
0.8
I I
L I
I
r
~///////////////A
0.6 0.7 0.5
0
O
0
0
h
0.6
loo fo
0.4
.
c
e
Second-order Elastic Analysis
Equivalent
A,- - -t Effective Buckling Length Method o-.-.o Initial Member Rotati¢m ,. 1/200
l-
1
Eqmvalent
150 Deflection at the C~lunm Top
Initial Deflection at the Free End I
t
I
I
1.00
0.25
0.04
0.00
-1200
Figure 14 Portal f r a m e with different cross-sections
I
I
1.2
I
1.0
t
I
t
I
0.8
I
I
0.6
I]
I
0.4
__
Figure 12 Cantilever c o l u m n with stepped cross-sections
= ~ Second-m'd~ Plastic-zone Analysis o - - - o Second-order Elastic Analysis aP t- - * Effective Buckling Length Method r - - I " 1:3-.--O Initial Member Rotation= 1/200 [ I i"
p ..
" "
"
0.9
/
0.8
0.7
~
_._...---;~.---==~-~
11
/W-""
_
./,,"/
~
~
.
h Equivalent Initial ~ Deflection at the ~100 Column Top
/ _
0.6
~-h
1
.....---1.
150
J
I
t
I
I
1.00
0.25
0.04
0.00
-1200 (~
Figure 13 Portal f r a m e with u n i f o r m cross-section
As discussed previously, this is because the equivalent curvature is evaluated to take the boundary conditions into consideration. Figure 12 shows that the strengths obtained by the proposed method are in good agreement with the ultimate strengths by the second-order plastic-zone analysis, and the strengths using the initial member rotation of 1/200 also agree well with the 'exact' ultimate strengths. Although those by the effective buckling length method also agree well with the ultimate strengths, the margin between the ultimate strength and the strength by the effective buckling length method is not constant while the margin between the ultimate strength and the strength by the proposed method is almost constant.
5.3. Portal frames A one-bay and one-storey portal frame to be analysed is shown in Figure 13. The vertical loads acting at the top of both columns are different, and the ratio of the load on the left column to that on the right column is represented by a. The flame is analysed for four cases of loading where a varies from 1.0 to 0.0. When a becomes zero, the slenderness parameter, A, of the left column becomes infinite. An initial deflection of h/430 at the column top is introduced in the second-order plastic-zone analysis, considering that the effective buckling length factor K is 2.328 for a = 1.0. If the value of a changes, the factor K also changes (K becomes smaller in these cases). However, the change of initial deflection due to the load condition is inconsistent with reality, and the strength computed by the second-order plastic-zone analysis is insensitive to the initial deflection when the residual stresses are considered. Accordingly, initial deflections are identical for all cases of a. The point in question as to EID is at the top of the right column. The magnitude of EID at that point for each case is indicated in the figure. In Figure 13 the ultimate load is represented by the load acting on the right column. It can be seen from Figure 13 that the difference between the ultimate strength by the second-order plastic-zone analysis and the predicted strength becomes large with decrease in a. When a is 1.0, both columns collapse at the same time, resulting in the total collapse of the frame. Where a 4 1.0, however, the frame does not collapse even if the right column reaches the limit state because of stress redistribution. Therefore, the margin between the ultimate strength and the predicted strength which is estimated at the limit state of the right column becomes large. In order to examine the efficiency of the proposed method in practical design, the frame was redesigned by the proposed method to have a different cross-section in each column. The results are shown in Table 2. Next, the redesigned frame was analysed by the second-order plasticzone analysis. As a result of this, as shown in Figure 14, a constant margin between the ultimate strength and the design load could be obtained, indicating that the proposed method can predict the strength of a frame which is designed optimally. Unlike this, the predicted strength by
Stability design of steel plane frames: H. Oda and 7-. Usami
626 Table2
D e s i g n e d cross-section o f portal f r a m e in Figure 14 (x ( r a t i o o f loads on c o l u m n s )
Square-sectional Dimension (mm) ( W i d t h x thickness)
Left c o l u m n Beam Right c o l u m n
1.00
0.25
0.04
0.00
680 x 20 680 x 20 680 x 20
680 x 6.4 680 x 20 680 x 19.0
680 x 2.0 680 x 20 680 × 20.4
680 x 1.9 680 x 20 680 x 20.2
the effective buckling length method does not give a constant margin with the ultimate strength. Now we consider the mode of the equivalent initial deflection from another point of view. As discussed earlier, the reason why the first buckling mode through the linear buckling analysis is employed for EID is that this mode seriously influences the reduction in strength. However, the magnitude of all the imperfections except for geometrical deflection (i.e., residual stresses etc.) is not in proportion to the first buckling mode of a frame. Notwithstanding this assumption, the reduction in strength, which is considered in the present method as the secondary moment produced by multiplying the initial deflection by axial force, increases as the axial force increases, thereby explaining the actual reduction caused by the initial imperfection. Specifically, for the portal frame, the initial deflection reduces the strength of the column, but does not reduce the strength of the beam despite the large initial deflection. This is because the axial force on the beam is very small. Therefore, it seems valid to connect all the imperfections to the first buckling mode.
Uniform Ca~s Snction
"//A
A : Efl~ti-¢¢ Buckling Length M ~ ) d
B : Sccoad-ordcrElastic Analya~B SteppedCm. Seaic~ NI l
Ny 0.6
I)2
~
Pl,...%
.,.....
~/A
C : Effe~ive Buckling Length Me~hod ~ D : Seco~d-ordm"F.]a~c Analy~.$
(~)
¢~) o_ ~
'
Ultimate S~ng~ ~ by Second-otde¢ Plastic-zone Analysis
wj o_ Mo
M o P1 :P2=5:95 Me.0. 4 My
0.5
0
0
0
0
Design Load
a
N~: Squash Load in the Ca~ of A
0.4 I
l
IIlIlNII A
B
A
1.0
[
[: member (I)
0.8
A ' : Cross ~.lioml Area
O.
6. Comparison with effective buckling length method
As shown in Section 5, the proposed method can accurately predict the ultimate strength of a frame. The effective buckling length method can also give a satisfactory prediction of the ultimate strength, because the strength of a frame is determined on the member with the smallest slenderness parameter and the parameter can be calculated accurately by the linear buckling analysis. Here, the remaining problem is that the effective buckling length of a member where the axial force is very small becomes extremely large and the member cannot be designed reasonably. To solve this problem, the optimality of a frame designed by both the proposed method and the effective buckling method will be compared. Four cases of cantilever columns are shown in Figure 15. The column length is the same as the one in the previous examples of cantilever columns. The external loads are a moment, M0, at the right end, the magnitude of which is 0.4 My, and axial forces at both the right end and the midheight. The ratio of axial forces, P]/Pz, was chosen to be 5/95 so that it produced extremely different slenderness parameters between the left and right members. The magnitude of axial forces as the design load is set equal to the critical strength which is evaluated by the effective buckling length method in the case of a column with the uniform cross-section shown in Figure 9. In this load condition, the cantilever column was designed to have a uniform cross-section or two stepped cross-sections by both the proposed method and the effective buckling length method. Furthermore, the ultimate strengths of the designed columns were obtained by the second-order plastic-zone analysis. Figure 15 shows the
ia dae caLseof A
I
I
I
I
A
B
C
D
Figure 15 C o m p a r i s o n of s e c o n d - o r d e r elastic analysis and effective buckling length m e t h o d : (a) ultimate s t r e n g t h ; (b) c r o s s - s e c t i o n a l area
ultimate strength and cross-sectional area of the designed columns. These four columns have almost the same strengths, which are close to the design load, thereby indicating that both methods give a good prediction of the ultimate strength. The ultimate strengths of the columns are controlled by the left member subjected to large axial force, so that in these four cases the cross-sectional areas are almost equal to each other. In spite of this, when the columns consist of two stepped cross-sections, the cross-sectional area of the right member subjected to small axial force designed by the proposed method is smaller than that by the effective buckling method. From this, we can conclude that the proposed method provides a reasonable result, even for a frame with small axial force members.
7.
Conclusions
A stability design method for steel frames based on the second-order elastic analysis has been presented. In the present method, the introduction of equivalent initial deflections (EID) corresponding to all the imperfections is of utmost importance. A new methodology has been proposed for the determination of EID. The following conclusions have been drawn from this study.
Stability design of steel plane frames: H. Oda and 7". Usami An improved EID formula has been presented using an analysis of a pin-ended beam column. The formula can conform to the multiple column strength curves. In addition, the reliability of the formula is confirmed by comparison with existing formulae. The physical meaning of EID is related to the reduction in the axial strength due to initial imperfections. Thus, the mode of EID can be adequately set to the first buckling mode by the linear buckling analysis in spite of the distribution of initial bending moment. A systematic methodology for determining EID for arbitrary shaped frames has been proposed. Some examples have demonstrated that the proposed method can predict the ultimate strength of framed structures with good accuracy. In contrast to the proposed method, the accuracy of the effective buckling length method was found to be not consistent. The initial member rotation of 1/200 in Eurocode 3 gives fairly good predictions for the cantilever columns or the rectangular frames in this paper. However, a different magnitude of rotation is necessary for cross-sections other than the box section. Moreover, it seems to be difficult to determine the initial deflection mode for arbitrary frames. The proposed method can give a more reasonable and economical design for a member subjected to considerably smaller axial force in a frame than the effective buckling length method. In this paper, the examples are limited to simple problems in order to explain the new concept of stability design method. The application to practical and complicated structures, e.g., arch bridges or cable-stayed bridges will be given in a subsequent paper.
627
References 1 Beedle, L. S. (Ed.) Stability of metal structures, a world view (2nd edn), Structural Stability Research Council, USA, 1991 2 Galambos, T. V. (Ed.) 'Guide to stability design criteria for metal compression member (4th edn), Wiley, New York, 1988 3 Subcommittee for surveying progress of steel structures in committee on steel structures, 'Cable-stayed bridges', Japan Society of Civil Engineers, 1990 (in Japanese) 4 Wang, C. K. Computer methods in advanced structural analysis, Intext Educational Publishers, 1973 5 Nishino, F. and Hasegawa, A. 'A practical design for compression members and frames using eigenvalue analysis', Preliminary report, IABSE Colloquium on stability of metal structures, Paris, 1983, IABSE, Zurich, 1983, pp. 497-500 6 Hasegawa, A. and Nishino, F. 'A proposal of structural design method based on the linearized finite displacement theory', Proc. 44th annual conference of JSCE, 1989, Vol. !, pp. 108-109 (in Japanese) 7 Allen, H. G. and Bulson, P. S. Background to buckling, McGrawHill, New York, 1980 8 Chen, W. F. and Toma, S. Advanced analysis of steel frames, CRC Press, Boca Raton, FL, 1994, Chap. 1 9 Eurocode 3 Editorial Group, 'Eurocode No. 3 Design of steel structures', 1990 10 Liew, J. Y. R., White, D. W. and Chen, W. F. 'Notational-load plastic-hinge method for frame design', J. Struct. Engng, ASCE 1994, 120 (5), 1434-1454 11 Timoshenko, S. P. and Gere, G. M. Theory of elastic stability (2nd edn), McGraw-Hill, New York, 1961 12 Itoh, Y. 'Ultimate strength variations of structural steel members', Doctoral thesis Nagoya University, Japan, 1984 13 Japan Road Association, 'Specification for highway bridges (JSHB), part 2: steel bridges', 1987 (English edition) 14 Nakamura, S. and Nanaura, T. 'Case studies for ultimate strength of cable-stayed bridges', Bridge Foundation Engng 1989, 29 (10), 3540 (in Japanese) 15 Clarke, M. J. and Bridge, R. Q. 'The inclusion of imperfections in the design of beam-columns', Proc. 1992 Annual Technical Session, SSRC, Pittsburgh, PA, pp. 327-346
ELSEVIER
PII: S0141-0296( 96)00134-4
8, pp. 617-627, 1997 © 1997 Elsevier Science Ltd All fights reserved. PfintedinGreat Britain 0141-0296/97 $17.00+ 0.00
Stability design of steel plane flames by second-order elastic analysis Hirotaka Oda Takigami Steel Construction, Co., Kiyokawa-cho 2-1, Nakagawa-ku, Nagoya, Japan
Tsutomu Usami Department of Civil Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan (Received July 1996; revised version accepted October 1996)
A new stability design method for steel plane frames using secondorder elastic analysis is proposed. Although many current design methods are based on the effective buckling length concept, the effective buckling length may not always be obtained accurately. This requires that a stability design method be developed without using the effective buckling length. In this method, the introduction of equivalent initial deflection, which accounts for the reduction in strength due to various initial imperfections, is of the utmost importance. Therefore, a new formula for the equivalent initial deflection is proposed, and a methodology using the curvature of the buckling mode is introduced to allovwfor a systematic application of this formula to arbitrary shaped irregular frames. The validity and efficiency of the method are demonstrated by some examples. © 1997 Elsevier Science Ltd. Keywords:stability design, buckling, steel frame, second-order elastic analysis, equivalent initial deflection
I.
Introduction
form cross-sections. An alternative method to calculate the effective buckling length is to use a matrix eigenvalue analysis of the framework 4.5. Compared with the aforementioned chart method, this method can determine the effective buckling length of all members in a frame having an arbitrary shape under arbitrary loads with the help of a computer. This method can also be used to determine the effective buckling length of each section in a frame with nonuniform cross-sections. Although this method is thought to be very effective, it has a problem in that the effective buckling length of a member with small axial force becomes extremely large 5. For example, this sort of situation occurs at the upper part of a cable-stayed bridge tower or near the centre of a girder in a cable-stayed bridge. In order to overcome this problem, a new design method has been proposed 6. In this method, the stress resultants with consideration of the effect of geometrical nonlinearity are directly computed from the second-order elastic analysis, and the frame stability design is made using only the cross-sectional strength equation. This method requires the introduction of an adequate initial deflection (the equivalent initial deflection (EID)) which is greater than the geometrical initial deflection specified in a code in order to account
This paper presents a stability design method for steel frames based on the second-order elastic analysis, which is expected to take the place of the effective buckling length method. In the current stability design method for steel frames in many countries, a strength equation for the cross-section and a stability equation for the member are usedL The stability equation is an interaction strength equation for a beamcolumn, and takes into account the effect of geometrical and material nonlinearities. Using the stability equation based on the effective buckling length concept changes the strength of the frame to the strength of beam-columns. However, the effective buckling length cannot be calculated accurately in cases such as a nonrectangular frame or a frame with nonuniform cross-sections which are often used in bridge structures 2. Figure 1 shows some steel towers of cable-stayed bridges in Japan 3. For these kinds of irregular frames, the effective buckling length cannot be determined accurately using a chart (e.g., AISC nomograph 2) because the chart is based on the assumption of a rectangular frame with uni-
617
Stability design of steel plane frames: H. Oda and T. Usami
618
|
a
I
b
c
Figure 1 Towers of cable stayed bridges: (a) Meiko-Nishi bridge; (b) Yokohama Bay bridge; (c) Iwaguro-shima bridge for the effects of reduction in strength due to various initial imperfections, such as initial deflection and residual stresses in a structure. In the current design procedure for a steel frame, the first-order elastic analysis is generally used. Contrary to the current design method, the present method needs secondorder elastic analysis. This analysis is based on the linearized beam-column theory with the simplest treatment of geometrical nonlinearity 7, but it gives accurate enough solutions for practical bridges and building frames in many cases. Moreover, because of using the second-order elastic analysis, the effective buckling length is not used and the design formula becomes very simple. Recent advances in computer hardware have made it possible to use the second-order analyses for practical design 8. One of the most sophisticated methods in the analyses is the plastic-zone analysis. However, the computational cost and time are too large for routine design work. Its applications to date are limited to research studies and for special design problems. The feasible method for practical design use on personal computers and workstations will be the second-order elastic analysis with pseudo (or enlarged) initial deflection (i.e., E I D ) 6'9 o r the notional-load plastic-hinge analysis 1°. These notional quantities have the same physical meanings, i.e., these are quantities that account for the influence of all initial imperfections and the effect of plastification in the members. Although the me~hod used to specify these equivalent values has been proposed for a strut and a rectangular frame 9,1°, the method for a nonrectangular frame with various boundary and loading oonditions has not yet been shown. In this paper, a rational £o~rmla for EID and a methodology for imposing the initial de~l~efftion shape on a frame are proposed. The validity of the method is demonstrated with some examples. Using the plastic-hinge analysis, the load-displacement behaviour, so-called hinge-by-hinge behaviour, can be traced. Second-order elastic analysis, however, cannot be used to trace this behaviour because the stress redistribution is not considered. In spite of this, the second-order elastic analysis is useful for practical design. This is because the
load related to the first plastic hinge is close to the ultimate load in an optimally designed framed structure. This fact will be explained in Section 5 with an example. 2.
Fundamental
concept
The physical meaning of the equivalent initial deflection (EID) concept is explained with an illustrative example. Figure 2 shows a relation between the axial force, P, acting at two ends of a pin-ended column with initial crookedness and the maximum bending moment, M, at the midspan of the column. The solid curve is supposed to represent the rigorous relation between P and M obtained from secondorder plastic-zone analysis of the column with initial imperfections (i.e., initial crookedness L/1000 and residual stresses). The results of the second-order elastic analysis represented by the dashed curves depend on the magnitude of the initial deflection, 6o- When an appropriate value (i.e., fo) is selected, the point of intersection of the dashed curve and the cross-sectional strength line (elastic limit in this case) can be matched to the ultimate strength by the P
Second-order
|
_. 6 < f0 elastic a n a l y s i s
1"0 t K ~ •
,," - ~ " " "
.....
6>f0
P
Second-order
0.5
M=p.6
0.5
1.0
M M"-y"
Figure 2 Fundamental concept of equivalent initial deflection
Stability design of steel plane frames: H. Oda and 7-. Usami second-order plastic-zone analysis. For this reason, all the effects of initial imperfections must be included in EID, and its magnitude will be greater than that used in the second-order plastic-zone analysis.
619 M0/My-09[=Eq (10)]"]
1.0
/"/;
- o2
t. ~oml,~
-0.4
f ECCS-b
-08
.J
Eq (14) [Proposedformula]
Numerical procedures
The second-order elastic analysis discussed here is a matrix analysis of a framed structure based on the linearized beamcolumn theory 7. The calculation procedure for the stability design method by the second-order elastic analysis is summarized in the following steps
0.6
p t~'-M°
,
/ // / ,///~¢//"
/¢/"
//;//////i
M°-'~ D
0.4
/ / /
,//,"
0.8
3.
~
"
,;7
0.8
1.0
0.2 (1) Specify a loading state (2) Give the magnitude and shape of EID (discussed in the next section) (3) Calculate N (= axial force) in each member by the firstorder elastic analysis (4) Evaluate stress resultants N and M (= bending moment) by solving the following equilibrium equations ([KE] + [Ka(N)]){d} = {f}
(1)
where [Ke] is a total elastic stiffness matrix, [Ka(N)] is a total geometric stiffness matrix, {d} is a nodal displacement vector, {f} is a load vector. (5) Check whether or not the forces N and M satisfy the following interaction equation N M ---+ = 1 Ny M,
(2)
with M,, = M e (for class 2 cross-sections 9) = My (for class 3 cross-sections 9)
(3)
where Ny = squash load, My = yield moment and Mp = fully plastic moment.
4.
Equivalent initial deflection (EID)
4.1. Magnitude of equivalent initial deflection A pin-ended column subjected to a concentric compressive force and equal moments at both ends is considered as the simplest structure (see Figure 3), and a formula to calculate the magnitude of EID is proposed. The magnitude of EID depends on the strength interaction equation of the cross-section. In the proposed method, the foregoing linear interaction equation (2) is used. The initial deflection of the column is assumed as
"fix Yo =fo " sin ~ -
(4)
where Yo = EID curve, fo = EID at midspan, L = column length. When the column is analysed by conventional beam-column theory, the following closed form solution of maximum bending moment at midspan. Mmo~., can be derived ~,
0.2
0.4
0.6
1.2
1.4
Figure3 Magnitude of initial deflection Mmax | N Mo kL Mu - 1 - N/NE Ny 91 -t- Muu sec ~-
(5)
where M0 = moment actin a ~ o t h ends, NE = Euler's column buckling load, k = ~IN/EI, E = elastic modulus, I = moment of inertia, and "0 = dimensionless expression for EID given by A r / = f o ~ (for class 3 cross-section)
(6a)
A =fo ~ (for class 2 cross-section)
(6b)
in which A = cross-sectional area, W= elastic section modulus and Z = plastic section modulus. Substituting equation (5) into equation (2), one obtains the following strength equation for the pin-ended column by the proposed method. N N/Ny Mo kL Ny + 1 - N/NE rl + -~. sec ~- = 1
(7)
The magnitude of "O can be determined by equating the strength obtained from equation (7) to the exact strength of the pin-ended column. Here, the following beam-column interaction strength equation will be used as an accurate enough equation. N 1 Mo + - 1 17, 1 - N/NE 34,
(8)
where N, is the ultimate strength of a concentric compression column. When equation (7) is compared with equation (8), an important physical meaning of EID can be seen, i.e., it represents the reduction in the axial strength from the squash load, Ny, to the ultimate column strength, N,, because the third term in equation (7) is nearly equal to the second term in equation (8). Figure 3 shows the values of EID at midspan of the column which are calculated by changing the moment at both ends (denoted as Mo/Mu) from 0.0 to 0.8. The ultimate axial load, N,, is calculated from the column strength curve-b of Eurocode 39 for a box column in equation (8). In this figure, the basic value of initial member rotation, qbo, for
620
Stability design of steel plane frames: H. Oda and T. Usami
unbraced frames expressed in equation (9) and the equivalent initial bow imperfection for centrally loaded columns in equation(10), both of which have been proposed in Eurocode 3 9, a r e also plotted. 1
qb° = ½00
(9)
"r/= 0.34 (A - 0.2)
(10)
where the slenderness ratio parameter, A, is defined as q
~=~1 K . L / ~ , r
(11)
~/A • N -
(12)
N~I
in which K = effective buckling length factor, r = radius of gyration of the cross-section, o~,.= yield stress, and A = eigenvalue from buckling analysis of the whole frame. It is to be noted that the computed result for Mo = 0 is coincident with equation(10). Equation(9) is reduced to equation (13) for convenience in plotting. A
Figure 4 shows the strength interaction curve of a pinended beam column by using the proposed EID and the interaction equation (8) which matches the exact strength well. Moreover, the interaction curve by use of equation (9), i.e., the initial member rotation in Eurocode 3 is compared. In this case, the initial deflection mode of the shape of snapped line shown in Figure 4 is considered. It can be seen from the figure that both the proposed E1D and equation (9) give accurate enough strength. However, a detailed comparison shows that the proposed EID is slightly conservative when moments are zero but equation (9) is slightly unconservative when A is large. 4.2. Initital deflection mode The next important problem is to determine the equivalent initial deflection mode for a given structure. For example, in the girder of a cable-stayed bridge in Figure 5, ~4its first buckling mode obtained from the linear buckling analysis is quite different from the collapse mode obtained from the second-order plastic-zone analysis. This is because only axial force is taken into consideration in the buckling analysis, but the effect of bending moments predominates in the second-order plastic-zone analysis. To determine which buckling mode or collapse mode is appropriate as the initial deflection mode, a two-span continuous beam-column (see Figure 6) has been analysed by
n=/o w 1.0
|
t
|
w
!
t
Interaction strcn~ equation .....
W
400W
Proposed EID
400KW
Mo
~-0.6
As can be seen from the figure, equations (9) or (10) are very close to the computed results in the case of the centrally loaded column (Mo/M, = 0.0), but it are low compared with the computed results in the range where A is large and as the moments are increased. The reason is that the amplification of the bending moment due to the socalled P-delta effect is large in these cases, resulting in a considerable reduction in the column strength. From these results, the following equations are proposed to calculate the magnitude of EID.
"~X
"~
~"
'%
............. Initial
~ "%
.o.=
"%'-°
• ",%
fo memberrotation 1/200
N I'M°
;%\
?. . . .
Mo
N
. ".°,° '~ ~'.,°.
",.. ,,~,~,
A
=~.=a,.(A-0.2) = ~2"(A-/3)
( 0 . 2 % A ~ 1.0) (1.0 ~ A)
(14)
Values of the coefficients oq, c~2 and/3 are given in Table 1, in which groups of cross-sections other than the box section are also included 9,~2,~3.
oo 0.0
Figure 4
0.5 Mo/Mp
1.0
Pin-ended beam column
Table 1 Values of coefficients a~, e2 and /3 in e q u a t i o n (14) Eurocode 36
F u k u m o t o and Itoh ~
~1 ~x2 /3
JSHB 1°
Group 1
Group 2
G r oup 3
a0
a
b
c
d
0.539 1.337 0.678
0.072 1.165 0.951
0.260 1.281 0.838
0.515 1.475 0.721
0.125 1.194 0.916
0.236 1.265 0.851
0.404 1.338 0.767
0.582 1.529 0.696
0.884 1.784 0.604
g r o u p 1 = pipe, b o x and rolled-H (wall thickness -< 40 ram) cross-sections g r o u p 2 = w e l d e d - H (wall thickness ~- 40 mm), tee, channel and angle cross-sections g r o u p 3 = rolled and welded-H (wall thickness > 40 ram) cross-sections
Stability design of steel plane frames: H. Oda and T. Usami (a)
621
;
(b)
(e)
Buckling mode
Collapse mode
Figure 5 Buckling and collapse mode of cable-stayed bridge: (a) model 1 with weak tower; (b) model 2 with weak girder; (c) model 3 with weak cable p
~
Q
A,,I,
!_-
,
o.,,.
A2,12
,
o.4L .I
L
~Q
A,,I,
P
o.81. L
2t
,,fo v'
'f
1
Buckli-~ Mode
~-y
Collapse Mode due to bending
/
1.0 ~
e- - 4 Second-orderPlastic Analy~ ( Collapse Mode ) Second-order Plastic Analysis( Buctdi.~ Mode ) o- - ~ $econd-c~der Elastic Analysis( Collapse Mode ) o, o Second-order Elastic Analysis( Bucklin8 Mode )
.
0.5
I
I
[
I
v
0.5
1.0
w
'
Q
Qy
Figure 6 Comparison of initial deflection modes imposing two different critical deflection modes. This beam-column was designed to have two stepped cross-sections near the midsupport in order for the three cross-sections to yield at the same time when the effective buckling length method is employed. Figure 6 shows the ultimate
strengths from the second-order elastic analysis and the second-order plastic-zone analysis using both modes. The magnitude of initial deflection for these modes is L/lO00 in the latter analysis, and in the former analysis, it is determined by a method that will be shown in the following section. Furthermore, the yield moment, My, is used in equation (3) for the second-order elastic analysis. From Figure 6, it can be seen that using an initial deflection shape similar to the first buckling mode gives lower ultimate strengths in either of the two analysis methods, and almost the same strengths as in the case of the collapse mode with a large lateral force. Accordingly, it will be adequate to use the first buckling mode as the initial deflection mode in any loading condition. Moreover, the first buckling mode obtained through the linear buckling analysis is appropriate because this mode provides the most serious reduction in the column strength.
4.3.
Application to a general frame
In the present section, we will show how to apply EID expressed by equation (14) to frames with arbitrary shapes and boundary conditions. First, we will discuss simple examples of columns with various boundary conditions as in Figure 7. These columns have an identical effective buckling length and are considered to have equal strength in the current specification. For example, when the strength of column (b) by the second-order elastic analysis is equal to the strength of the pin-ended column (a), which has the same effective buckling length as column (b), the maximum equivalent initial deflection of column (b) should be twice that of column (a), namely, the relative deflection from the inflection point (i.e., midcolumn) to the maximum curvature points (i.e., column ends) of column (b) should be identical to that of column (a). The same situation can be observed in other cases. However, it is not easy to give the initial deflection between the two inflection points and to identify the maximum curvature point for a frame. Therefore, EID is determined by enlarging the buckling mode of the column
Stability design of steel plane frames: H. Oda and T. Usami
622
x : Maximum Curvature Point ' • : Inflection Point
I
L
",
~4
!
M ('0)
(O
(d)
Figure 7 Equivalent initial deflection of column under various b o u n d a r y conditions until its maximum curvature is identical to the maximum curvature of the pin-ended column with the same effective buckling length. This method gives the proper equivalent initial deflection without considering the inflection point because the effect of the boundary condition has already been taken into account in the buckling mode. For frames with arbitrary shapes and boundary conditions, the calculation procedure of EID on the basis of the equivalent curvature is outlined as follows: (1) Carry out the eigenvalue analysis of a geometrically perfect frame subjected to a set of design load, and find the first buckling mode. Determine the slenderness ratio parameter, A, for each member from the foregoing equation (12). (2) Find the member that has the smallest value of A. As explained later, this member has the lowest column strength. (3) In order to apply the EID of a pin-ended column obtained from equation (14), calculate the end rotation, 0o, and the curvature, Ko, at the midspan of the column from the following equations
o,,
=
(K.L
f,=X r
(15)
,. f , 1 o', -- iX. L) = E (16) (4) Calculate the curvature of the member considered in step (2) from its buckling mode, and find the maximum curvature point (denoted point M in Figure 8) of this member. Calculate the equivalent initial curvature KO)M, in which the effect of the boundary condition is considered, at the point M from the following equations (the derivations are shown later)
KO)M = KO " sin~M
( 17 )
~M= K 2L = c o t '
(18)
\KM/Kol where KM and 0M, respectively, represent the curvature and the slope at the point M, XM is the x-directional coordinate of the point M when the member of interest is converted to a member with an effective length KL (see Figure 8). (5) Determine the initial deflection by proportionally enlarging the buckling mode until the curvature at
Figure 8
Initial equivalent deflection of frame
point M is equal to the foregoing equivalent initial curvature given by equation (17). Now, we will explain the physical meaning of the smallest value of A in step (2). In this connection, we consider a ratio of axial force N acting on each member in a frame to its ultimate axial strength N,. The member with the largest ratio N/N,, implies that it is the most unstable in a frame. Multiplying the eigenvalue A which is obtained by the linear buckling analysis, the ratio N/N, gives the following relation. A •N
N.
=
N~:(A)
N,,(A)
(19)
Both elastic buckling strength, N,, and inelastic strength,
N,,, are expressed as function of X. and it is evident that the ratio of NE/N, is larger for smaller A. Accordingly, the member which has the smallest A obtained by the linear buckling analysis has the largest N/N, ratio; i.e., the column is weakest against axial force in the frame. Next, we will derive equations(17) and (18). In the equivalent pin-ended column with length KL, the equivalent initial deflection, slope and curvature are expressed as Yo = f , ' sin~
(20)
)'(; = 0o ' cos~
( 21 )
y~; = K,. sin~
(22)
= ~x/KL
(23)
with
and a prime indicates differentiation with respect to x. From the condition that the ratio of slope to curvature at the point M in a frame equals the ratio in the equivalent pin-ended column, the following equations are obtained 0M = y(; = 00 cot~: KM
Thus,
re YO
KO
(24)
Stability design of steel plane frames: H. Oda and T. Usami
r
b = 68cm
235MPa
"1 A = 544cm 2 I = 419605cm 4
~D
o
II
623
v
/f j -118MPa
r = 27.77cm
II
-I
Figure 9
Sectional properties and residual
stresses
( OM/O0]
= cot-'
(25)
\KM/Ko/
which result in equations (17) and (18). 5. A p p l i c a t i o n to s t e e l f r a m e s w i t h s t e p p e d cross-section
5.1. Cross-section properties As illustrative examples, we now consider cantilever columns (Figures 10-12) and portal frames (Figures 13 and 14). Unless otherwise stated, the members have a box section and the residual compressive stress is assumed to be half of the yield stress as shown in Figure 9. With this cross-section, the ultimate strength of a pin-ended column, which has initial out-of-straightness of L/1000, is almost equal to the'column strength curve of ECCS-b, so that the equivalent initial deflection corresponding to ECCS-b will be used in the following examples. The height of the cantilever c_olumn is determined such that the slenderness parameter, A, is equal to 0.9 when the column has the uniform cross-section and is subjected to an axial thrust at the top. In the following examples, the Nu --
PI
N y (~ 1.o
~
~
Second-order Plastic-zone Analysis
o
o
Second-order
tA
ElasticAnalysis
t- - - t
Effective Buckling Length Method
D-.--D
Initial m e m b e r
rotation =
~[
1/200
,
~,
..
P2 B
0.9
m
M
0.8
C ~ N @-0.7
F.quivale~t ]~-'~don atthe Cohmm Top
0.6
1.0
0.5
0.0
P1 Pl +P2
Figure 10 Cantilever c o l u m n subjected to axial thrusts at top and midheight
cross-section will be stepped or another axial thrust will act at midheight, thereby changing the slenderness parameter. The height of a portal frame with one bay and one storey is also determined such that X = 0.9 when the columns have the same cross-section and are subjected to the same magnitude of axial thrusts at both column tops. In the following examples, the magnitude of an axial thrust acting on one column will be changed, resulting in a different slenderness parameter from the other column. Under these conditions, we will examine whether or not the ultimate strength can be evaluated through the proposed method.
5.2. Cantilever columns 5.2.1. Cantilever column subjected to axial forces at top and midheight Figure 10 shows a cantilever column subjected to axial forces of PI at the top and P2 at the midheight. The ratio of P~ to P2 is allowed to change in this example. Since the cross-section of the column is uniform and since the axial force on the upper member is smaller than the lower member, the slenderness parameter, A, of the upper member is large_r than that of the lower member. If P~ becomes zero, A of the upper member becomes infinite. The magnitude of equivalent initial deflection in the proposed method is determined at the fixed end of the column. This is because the value of A of the lower member is smaller and the maximum curvature point in this member is at the fixed end. Accordingly, EID is determined by adopting the equivalent curvature at that point. The magnitude of EID at the column top obtained in this way is shown in Figure 10 for each loading case. On the other hand, the magnitude of initial deflection at the column top used in the second-order plastic-zone analysis is set equal to 1/500 of the column length. This is because the effective buckling length factor is 2.0 for the cantilever column, so that this initial deflection is equivalent to that of L/1000 for a pin-ended column. However, Clark et al. ~5 have proposed that the actual tolerance of erection should be used in the second-order plastic zone analysis. Figure 10 shows that the strengths calculated by the proposed method agree well with those by the second-order plastic-zone analysis. The maximum difference between these strengths is 7.2%. Also, the figure shows the strengths using the initial member rotation of 1/200. The accuracy of these results is sufficient but they are not uniform for loading cases. When only an axial force exists at the
Stability design of steel plane frames: H. Oda and T. Usami
624
(a)
(b) "N
N
~,,/
= =~ m - ~ . ~ .
N,,/
0.5
"
"s.~mm~m,~,.
0..~
1.0 2V~ e
0.5
M-;
0.~
l.O M |
M--;
(c) lq 1.0 i"
'
' ~ll~l~Elmic/il~
I'N.
° .o
O.l
1.0 M 0 M,
Figure 77 Cantilever column subjected to combined bending moment and axial forces at top and midspan: (a) P~:P2= 50:50; (b) PI:P2= 2 0 : 8 0 ; (c) PI:P2= O: 1 0 0 midheight (i.e., Pj = 0 ) , the use of an initial member rotation of 1/200 is more conservative than EID, although the initial deflections at column top are the same for both cases. This is because the initial deflection at midheight for EID similar to the buckling mode is smaller than the initial member rotation, whose mode is a straight line. In Figure 10, the strengths predicted by the effective buckling length method are also shown. These strengths are calculated by substituting the slenderness parameter, A, obtained by the buckling analysis to the column strength equation. The strength of the upper member is smaller than that of the lower member because the value of ~ is larger in the upper member. In spite of this, the strength of the column is controlled by the lower member subjected to larger axial force. This implies that the ratio of axial force to axial strength is important. The strengths by the effective buckling length method are also close to those by secondorder plastic-zone analysis. Figure 11 indicates some examples of the column subjected to combined axial forces and bending moment. In these examples, equation(2) on the basis of plastic moment, M/,, is used in the proposed method. The proposed
method results in good agreement with the second-order plastic-zone analysis.
5.2.2. Cantilever column with stepped cross-sections A cantilever column with stepped cross-sections is presented in Figure 12. The member @ has the cross-section shown in Figure 9, but the member ~ has a different cross-section which is varied in width. This variation is represented as the ratio of moment of inertia, I~/12, in which I~ is the moment of inertia of member 0) and 12 is that of member @. The external load is only an axial force acting at the free end. In this example, the member with smaller cross-section has a smaller value of ~, because the magnitudes of axial force acting on both members are identical. Accordingly, the point in question tbr settlement of EID comes to the point C shown in Figure 12 when It/12 >-- 1.0, and the point jumps to point B when If/I: < 1.0. The quantity of EID at the free end which is determined by adopting the equivalent curvature to the points in question is indicated in Figure 12. Even if the point in question jumps from C to B, there is no discontinuity in the magnitude of the initial deflection.
Stability design of steel plane frames: H. Oda and T. Usami Pu
Nu N~
~lC~
0.8
:
_
0.7
I ~:
I2
L/2
=_
.... = ~d.~&r
I P
I,
B
0.9
_
P
aP
Pl~lic-zone
o----o Second-order Elastic Analy~
A L/2
625
f----
A- - .A Effe~iw Buckling Length Method ~--.--o Initial Member Rotation = 1/200 =
0.8
I I
L I
I
r
~///////////////A
0.6 0.7 0.5
0
O
0
0
h
0.6
loo fo
0.4
.
c
e
Second-order Elastic Analysis
Equivalent
A,- - -t Effective Buckling Length Method o-.-.o Initial Member Rotati¢m ,. 1/200
l-
1
Eqmvalent
150 Deflection at the C~lunm Top
Initial Deflection at the Free End I
t
I
I
1.00
0.25
0.04
0.00
-1200
Figure 14 Portal f r a m e with different cross-sections
I
I
1.2
I
1.0
t
I
t
I
0.8
I
I
0.6
I]
I
0.4
__
Figure 12 Cantilever c o l u m n with stepped cross-sections
= ~ Second-m'd~ Plastic-zone Analysis o - - - o Second-order Elastic Analysis aP t- - * Effective Buckling Length Method r - - I " 1:3-.--O Initial Member Rotation= 1/200 [ I i"
p ..
" "
"
0.9
/
0.8
0.7
~
_._...---;~.---==~-~
11
/W-""
_
./,,"/
~
~
.
h Equivalent Initial ~ Deflection at the ~100 Column Top
/ _
0.6
~-h
1
.....---1.
150
J
I
t
I
I
1.00
0.25
0.04
0.00
-1200 (~
Figure 13 Portal f r a m e with u n i f o r m cross-section
As discussed previously, this is because the equivalent curvature is evaluated to take the boundary conditions into consideration. Figure 12 shows that the strengths obtained by the proposed method are in good agreement with the ultimate strengths by the second-order plastic-zone analysis, and the strengths using the initial member rotation of 1/200 also agree well with the 'exact' ultimate strengths. Although those by the effective buckling length method also agree well with the ultimate strengths, the margin between the ultimate strength and the strength by the effective buckling length method is not constant while the margin between the ultimate strength and the strength by the proposed method is almost constant.
5.3. Portal frames A one-bay and one-storey portal frame to be analysed is shown in Figure 13. The vertical loads acting at the top of both columns are different, and the ratio of the load on the left column to that on the right column is represented by a. The flame is analysed for four cases of loading where a varies from 1.0 to 0.0. When a becomes zero, the slenderness parameter, A, of the left column becomes infinite. An initial deflection of h/430 at the column top is introduced in the second-order plastic-zone analysis, considering that the effective buckling length factor K is 2.328 for a = 1.0. If the value of a changes, the factor K also changes (K becomes smaller in these cases). However, the change of initial deflection due to the load condition is inconsistent with reality, and the strength computed by the second-order plastic-zone analysis is insensitive to the initial deflection when the residual stresses are considered. Accordingly, initial deflections are identical for all cases of a. The point in question as to EID is at the top of the right column. The magnitude of EID at that point for each case is indicated in the figure. In Figure 13 the ultimate load is represented by the load acting on the right column. It can be seen from Figure 13 that the difference between the ultimate strength by the second-order plastic-zone analysis and the predicted strength becomes large with decrease in a. When a is 1.0, both columns collapse at the same time, resulting in the total collapse of the frame. Where a 4 1.0, however, the frame does not collapse even if the right column reaches the limit state because of stress redistribution. Therefore, the margin between the ultimate strength and the predicted strength which is estimated at the limit state of the right column becomes large. In order to examine the efficiency of the proposed method in practical design, the frame was redesigned by the proposed method to have a different cross-section in each column. The results are shown in Table 2. Next, the redesigned frame was analysed by the second-order plasticzone analysis. As a result of this, as shown in Figure 14, a constant margin between the ultimate strength and the design load could be obtained, indicating that the proposed method can predict the strength of a frame which is designed optimally. Unlike this, the predicted strength by
Stability design of steel plane frames: H. Oda and 7-. Usami
626 Table2
D e s i g n e d cross-section o f portal f r a m e in Figure 14 (x ( r a t i o o f loads on c o l u m n s )
Square-sectional Dimension (mm) ( W i d t h x thickness)
Left c o l u m n Beam Right c o l u m n
1.00
0.25
0.04
0.00
680 x 20 680 x 20 680 x 20
680 x 6.4 680 x 20 680 x 19.0
680 x 2.0 680 x 20 680 × 20.4
680 x 1.9 680 x 20 680 x 20.2
the effective buckling length method does not give a constant margin with the ultimate strength. Now we consider the mode of the equivalent initial deflection from another point of view. As discussed earlier, the reason why the first buckling mode through the linear buckling analysis is employed for EID is that this mode seriously influences the reduction in strength. However, the magnitude of all the imperfections except for geometrical deflection (i.e., residual stresses etc.) is not in proportion to the first buckling mode of a frame. Notwithstanding this assumption, the reduction in strength, which is considered in the present method as the secondary moment produced by multiplying the initial deflection by axial force, increases as the axial force increases, thereby explaining the actual reduction caused by the initial imperfection. Specifically, for the portal frame, the initial deflection reduces the strength of the column, but does not reduce the strength of the beam despite the large initial deflection. This is because the axial force on the beam is very small. Therefore, it seems valid to connect all the imperfections to the first buckling mode.
Uniform Ca~s Snction
"//A
A : Efl~ti-¢¢ Buckling Length M ~ ) d
B : Sccoad-ordcrElastic Analya~B SteppedCm. Seaic~ NI l
Ny 0.6
I)2
~
Pl,...%
.,.....
~/A
C : Effe~ive Buckling Length Me~hod ~ D : Seco~d-ordm"F.]a~c Analy~.$
(~)
¢~) o_ ~
'
Ultimate S~ng~ ~ by Second-otde¢ Plastic-zone Analysis
wj o_ Mo
M o P1 :P2=5:95 Me.0. 4 My
0.5
0
0
0
0
Design Load
a
N~: Squash Load in the Ca~ of A
0.4 I
l
IIlIlNII A
B
A
1.0
[
[: member (I)
0.8
A ' : Cross ~.lioml Area
O.
6. Comparison with effective buckling length method
As shown in Section 5, the proposed method can accurately predict the ultimate strength of a frame. The effective buckling length method can also give a satisfactory prediction of the ultimate strength, because the strength of a frame is determined on the member with the smallest slenderness parameter and the parameter can be calculated accurately by the linear buckling analysis. Here, the remaining problem is that the effective buckling length of a member where the axial force is very small becomes extremely large and the member cannot be designed reasonably. To solve this problem, the optimality of a frame designed by both the proposed method and the effective buckling method will be compared. Four cases of cantilever columns are shown in Figure 15. The column length is the same as the one in the previous examples of cantilever columns. The external loads are a moment, M0, at the right end, the magnitude of which is 0.4 My, and axial forces at both the right end and the midheight. The ratio of axial forces, P]/Pz, was chosen to be 5/95 so that it produced extremely different slenderness parameters between the left and right members. The magnitude of axial forces as the design load is set equal to the critical strength which is evaluated by the effective buckling length method in the case of a column with the uniform cross-section shown in Figure 9. In this load condition, the cantilever column was designed to have a uniform cross-section or two stepped cross-sections by both the proposed method and the effective buckling length method. Furthermore, the ultimate strengths of the designed columns were obtained by the second-order plastic-zone analysis. Figure 15 shows the
ia dae caLseof A
I
I
I
I
A
B
C
D
Figure 15 C o m p a r i s o n of s e c o n d - o r d e r elastic analysis and effective buckling length m e t h o d : (a) ultimate s t r e n g t h ; (b) c r o s s - s e c t i o n a l area
ultimate strength and cross-sectional area of the designed columns. These four columns have almost the same strengths, which are close to the design load, thereby indicating that both methods give a good prediction of the ultimate strength. The ultimate strengths of the columns are controlled by the left member subjected to large axial force, so that in these four cases the cross-sectional areas are almost equal to each other. In spite of this, when the columns consist of two stepped cross-sections, the cross-sectional area of the right member subjected to small axial force designed by the proposed method is smaller than that by the effective buckling method. From this, we can conclude that the proposed method provides a reasonable result, even for a frame with small axial force members.
7.
Conclusions
A stability design method for steel frames based on the second-order elastic analysis has been presented. In the present method, the introduction of equivalent initial deflections (EID) corresponding to all the imperfections is of utmost importance. A new methodology has been proposed for the determination of EID. The following conclusions have been drawn from this study.
Stability design of steel plane frames: H. Oda and 7". Usami An improved EID formula has been presented using an analysis of a pin-ended beam column. The formula can conform to the multiple column strength curves. In addition, the reliability of the formula is confirmed by comparison with existing formulae. The physical meaning of EID is related to the reduction in the axial strength due to initial imperfections. Thus, the mode of EID can be adequately set to the first buckling mode by the linear buckling analysis in spite of the distribution of initial bending moment. A systematic methodology for determining EID for arbitrary shaped frames has been proposed. Some examples have demonstrated that the proposed method can predict the ultimate strength of framed structures with good accuracy. In contrast to the proposed method, the accuracy of the effective buckling length method was found to be not consistent. The initial member rotation of 1/200 in Eurocode 3 gives fairly good predictions for the cantilever columns or the rectangular frames in this paper. However, a different magnitude of rotation is necessary for cross-sections other than the box section. Moreover, it seems to be difficult to determine the initial deflection mode for arbitrary frames. The proposed method can give a more reasonable and economical design for a member subjected to considerably smaller axial force in a frame than the effective buckling length method. In this paper, the examples are limited to simple problems in order to explain the new concept of stability design method. The application to practical and complicated structures, e.g., arch bridges or cable-stayed bridges will be given in a subsequent paper.
627
References 1 Beedle, L. S. (Ed.) Stability of metal structures, a world view (2nd edn), Structural Stability Research Council, USA, 1991 2 Galambos, T. V. (Ed.) 'Guide to stability design criteria for metal compression member (4th edn), Wiley, New York, 1988 3 Subcommittee for surveying progress of steel structures in committee on steel structures, 'Cable-stayed bridges', Japan Society of Civil Engineers, 1990 (in Japanese) 4 Wang, C. K. Computer methods in advanced structural analysis, Intext Educational Publishers, 1973 5 Nishino, F. and Hasegawa, A. 'A practical design for compression members and frames using eigenvalue analysis', Preliminary report, IABSE Colloquium on stability of metal structures, Paris, 1983, IABSE, Zurich, 1983, pp. 497-500 6 Hasegawa, A. and Nishino, F. 'A proposal of structural design method based on the linearized finite displacement theory', Proc. 44th annual conference of JSCE, 1989, Vol. !, pp. 108-109 (in Japanese) 7 Allen, H. G. and Bulson, P. S. Background to buckling, McGrawHill, New York, 1980 8 Chen, W. F. and Toma, S. Advanced analysis of steel frames, CRC Press, Boca Raton, FL, 1994, Chap. 1 9 Eurocode 3 Editorial Group, 'Eurocode No. 3 Design of steel structures', 1990 10 Liew, J. Y. R., White, D. W. and Chen, W. F. 'Notational-load plastic-hinge method for frame design', J. Struct. Engng, ASCE 1994, 120 (5), 1434-1454 11 Timoshenko, S. P. and Gere, G. M. Theory of elastic stability (2nd edn), McGraw-Hill, New York, 1961 12 Itoh, Y. 'Ultimate strength variations of structural steel members', Doctoral thesis Nagoya University, Japan, 1984 13 Japan Road Association, 'Specification for highway bridges (JSHB), part 2: steel bridges', 1987 (English edition) 14 Nakamura, S. and Nanaura, T. 'Case studies for ultimate strength of cable-stayed bridges', Bridge Foundation Engng 1989, 29 (10), 3540 (in Japanese) 15 Clarke, M. J. and Bridge, R. Q. 'The inclusion of imperfections in the design of beam-columns', Proc. 1992 Annual Technical Session, SSRC, Pittsburgh, PA, pp. 327-346