Stability of multi-storey unbraced steel frames subjected to variable loading

Journal of Constructional Steel Research 63 (2007) 1506–1514 www.elsevier.com/locate/jcsr

Stability of multi-storey unbraced steel frames subjected to variable loading L. Xu ∗ , X.H. Wang Department of Civil and Environmental Engineering, University of Waterloo, ON, Canada, N2L 3G1 Received 2 September 2006; accepted 31 January 2007

Abstract Proposed in this paper is an approach of evaluating the elastic buckling loads for multi-storey unbraced steel frames subjected to variable loading or non-proportional loading. In the case of variable loading, the conventional assumption of proportional loading is abandoned, and different load patterns may cause the frame to buckle at different levels of critical loads. In light of the use of the storey-based buckling concept to characterize the lateral sway buckling of unbraced framed structures, the problems of determining the lower and upper bounds among all of the frame buckling loads associated with different load patterns are presented as a pair of minimization and maximization problems subjected to elastic stability constraints. The problems take into account the semi-rigid behaviour of beam-to-column connections and the lateral stiffness reduction of columns due to the presence of an axial compressive load. The minimization and maximization problems are then solved by a linear programming method; thus, the lower and upper bounds of the frame buckling loads subjected to variable loading are obtained. Parametrical studies on the influence of the connection rigidity to the lower and upper bounds of critical loads and the comparisons to the conventional proportional loading are also presented in this paper. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Frame stability; Buckling load; Unbraced steel frame; Storey-based buckling; Variable loading; Semi-rigid connection; Linear programming

1. Introduction Stability is the fundamental safety criterion for steel structures during their construction period and operation life. Although research on the stability of structures can be traced back to 250 years ago when Euler published his famous Euler equation on the elastic stability of bars in 1744, adequate solutions are still not available for many types of structures while subjected to certain load conditions. The stability of structures has been exercising the minds of many eminent engineers and applied mathematicians for several decades. In general, three types of methods are available for stability analysis of framed structures subjected to proportional loading, i.e. the system buckling method, effective-length methods and the storey-based buckling method [1–6]. Among these methods, the system buckling method is considered to be impractical as it involves solving the minimum positive eigenvalue from either a highly nonlinear or transcendental equation [1]. In the effectivelength based methods, the alignment chart method [2] is the most widely used method in current design practice. However, ∗ Corresponding author. Tel.: +1 519 888 4567x36882; fax: +1 519 888 6197.

E-mail address: [email protected] (L. Xu). c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.01.010

the alignment chart method adopts certain assumptions, which may result in inaccuracy of the estimated column strengths in the case that the assumptions are violated. The storeybased buckling method [7–10], which is an alternative to the effective-length method in design practice that does not adopt the simplifications associated with the alignment chart method, is efficient and provides accurate results. The method is based on the fact that lateral sway instability of an unbraced frame is a storey phenomenon involving the interaction of the lateral sway resistance of each column in the same storey and the total gravity load in the columns in that storey. In this study, a storeybased buckling method developed by Xu and Liu [9,10] will be adopted and extended to facilitate the stability analysis of multistorey unbraced steel frames subjected to variable loading. Current design practice concerning stability analysis and design of framed structures are almost exclusively based on the assumption of proportional loading, where the obtained stability capacity of the structure corresponds to a specified load pattern that may not apply to any other load pattern. Therefore, structural engineers have to anticipate the possible load patterns caused by various types of loads that may be encountered during the lifespan of the building, and this is usually accomplished by specifying different load

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Nomenclature E Young’s modulus of steel EI c,i j /L c,i j flexural stiffness of the column under axial load i, k, l index of storey I, Ic,i j moment of inertia of member and column, respectively j index of column Ki j effective length factor L c,i j length of column m number of columns in one storey n number of storeys of the frame Pe,i j Euler buckling load of individual column Pu,i j upper bound of axial loading for individual column Pi j applied axial load of individual column rl,i j , ru,i j end-fixity factors of the lower and upper ends of the column, respectively Rl,i j , Ru,i j rotational restraining stiffness of connected beams at lower and upper joints, respectively Si j lateral stiffness Sk lateral stiffness of the kth storey Z objective function associated with maximization and minimization problems, respectively βi j modification factor of the lateral stiffness φi j applied load ratio combinations in accordance with existing design standards if available. However, the worst load patterns are not always guaranteed in the load combinations specified in the standards or by the engineers due to the unpredictable nature of various types of loads. The variabilities of loads in both magnitudes and locations need to be accounted for when assessing the stability of structures; otherwise, public safety may be jeopardized. The study of stability of multi-storey unbraced steel frames subjected to variable loading is of primary importance as the variable loading accounts for the variability of applied loads and represents more realistic conditions in the life span of the structures. As the nature of the semi-rigid behaviour of the beam-to-column connections is another primary contributing factor to structural stability, connection rigidity will be considered in the current study.

Fig. 1. An axially loaded semi-rigid column.

ru,i j =

1 1 + 3EI c,i j /Ru,i j L c,i j

(1b)

where rl,i j and ru,i j are the end-fixity factors for the lower and upper ends of the column, respectively. The end-fixity factors in Eq. (1) define the stiffness of each end connection relative to the attachment member. For flexible, i.e. pinned connections, the rotational stiffness of the connection is idealized as zero; thus, the value of the corresponding end-fixity factor is zero. For fully restrained or so-called rigid connections, the end-fixity factor is unity, because the connection rotational stiffness is taken to be infinite. A semi-rigid connection has an end-fixity factor between zero and unity. Upon the introduction of the end-fixity factors, the lateral stiffness of an axially loaded column in an unbraced frame can be expressed as [10] Si j = βi j (φi j , rl,i j , ru,i j )

12EI c,i j L 3c,i j

(2)

where βi j (φi j , rl,i j , ru,i j ) is the modification factor of the lateral stiffness and can be defined in terms of the end-fixity factors and expressed as [10] βi j (φi j , rl,i j , ru,i j ) =

φi3j

12 a1 φi j cos φi j + a2 sin φi j × 18rl,i j ru,i j − a3 cos φi j + a4 φi j sin φi j

(3)

where 2. Lateral stiffness of axially loaded semi-rigid column For an axially loaded column in the semi-rigid unbraced frame as shown in Fig. 1, let EI c,i j /L c,i j be the flexural stiffness of the column under axial load, and let Rl,i j and Ru,i j be the rotational restraining stiffness provided by the immediately connected beams at the lower and upper joints in the ith storey and jth column. The effect of beam-to-column end rotational restraints can be characterized by the end-fixity factors [11] and can be expressed as follows: rl,i j =

1 ; 1 + 3EI c,i j /Rl,i j L c,i j

(1a)

a1 = 3[rl,i j (1 − ru,i j ) + ru,i j (1 − rl,i j )]

(4a)

a2 = 9rl,i j ru,i j − (1 − rl,i j )(1 − ru,i j )φi2j

(4b)

a3 = 18rl,i j ru,i j + [3rl,i j (1 − ru,i j ) + 3ru,i j (1 − rl,i j )]φi2j (4c) a4 = −9rl,i j ru,i j + 3rl,i j (1 − ru,i j ) + 3ru,i j (1 − rl,i j ) + (1 − ru,i j )(1 − rl,i j )φi2j φi j is the applied load ratio and defined as s Pi j L 2c,i j p φi j = = π Pi j /Pe,i j EI c,i j

(4d)

(5)

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in which Pi j is the column axial load, and Pe,i j is the Euler buckling load for the column with pinned connections. The modification factor of the lateral stiffness βi j (φi j , rl,i j , ru,i j ) takes into account the effects of both axial force and column end rotational restraints. A zero value of βi j (φi j , rl,i j , ru,i j ) indicates the column has completely lost its lateral stiffness and that lateral buckling of the column is about to occur. A column with a negative value of βi j (φi j , rl,i j , ru,i j ) signifies the column relies upon the lateral restraint provided by other columns in the same storey in order to maintain the axial load. A column with a positive value of βi j (φi j , rl,i j , ru,i j ) indicates that the column is laterally stable and can provide lateral support to other columns to sustain the stability of the storey. 3. Storey-based stability equation The lateral stability of single-storey unbraced steel frames subjected to variable loading was first investigated by Xu, et al. [12]. Based on the concept of storey-based buckling, the problem of determining the critical elastic buckling loads of the frames subjected to variable loading is expressed as a pair of minimization and maximization problems with stability constraints. The study finds out that in the case of variable loading, the difference between the maximum and minimum elastic buckling loads associated with lateral instability of the single-storey unbraced frames can be as high as 20% in some cases. The beam-to-column connections were considered either as purely pinned or fully rigid in the study. Further investigation was carried out on single-storey unbraced semi-rigid steel frames [13], and it was discovered that the difference between the minimum and maximum buckling loads is insignificant for frames in which the lateral stiffnesses of columns are approximately the same, such as conventional rigid frames. However, the difference can still be substantial in some cases with lean-on columns, but it is not as significant as in the case where connections are simplified as ideally pinned or fully rigid. In the consideration of elastic buckling for multi-storey unbraced frames, the concept of storey-based buckling indicates that lateral sway instability of an unbraced frame is a storey phenomenon involving the interaction of lateral stiffnesses among columns in the same storey. It states that the columns with larger stiffnesses are able to provide lateral support for the weaker columns in the same storey to resist the lateral sway instability, while the columns with negative stiffness depend on such lateral support to maintain lateral stability. Therefore, the condition for storey-based buckling in a lateral sway mode is that the sum of the lateral stiffness of the storey reduces to zero. Based on Eq. (2), the stability equation of a single-storey semi-rigid frame buckling in a lateral sway mode is given by [10] Si =

m X j=1

Si j =

m X j=1

βi j (φi j , rl,i j , ru,i j )

12EI c,i j L 3c,i j

= 0.

(6)

Fig. 2. (m − 1)-bay n-storey unbraced frame.

For the multi-storey frame shown in Fig. 2, once the lateral stiffness of any one storey vanishes, the frame becomes laterally unstable. Therefore, the lateral instability of a multi-storey frame can be defined as the case where there is at least one storey of the frame, say storey k, having its lateral stiffness vanished; that is, Sk becomes zero. Based on Eq. (6), the lateral stability equation for multi-storey unbraced frames is given by [10] ! n m n Y X Y 12EI c,i j = 0. (7) Si = βi j (φi j , rl,i j , ru,i j ) L 3c,i i=1 i=1 j=1 This equation implies that if any one of the stories fails to maintain its lateral stability, the storey-based buckling of an unbraced multi-storey frame will occur. It is impractical to evaluate column critical buckling load in multi-storey frames directly from Eq. (3) due to the transcendental relationship of βi j (φi j , rl,i j , ru,i j ) and φi j . Applying the first-order Taylor series expansion, Eq. (3) for a column j in the ith storey of a multi-storey frame is simplified as follows [9,10] βi j (φi j , rl,i j , ru,i j ) = β0,i j (rl,i j , ru,i j ) − β1,i j (rl,i j , ru,i j )φi2j (8) where β0,i j , β1,i j are given by β0,i j (rl,i j , ru,i j ) =

rl,i j + ru,i j + rl,i j ru,i j 4 − rl,i j ru,i j

(9a)

β1,i j (rl,i j , ru,i j ) =

2 ) − (34 − r 2 2 8(5 + ru,i u,i j )ru,i j rl,i j + (8 + ru,i j + 3ru,i j )rl,i j j

30(4 − rl,i j ru,i j )2

.

(9b) Substituting Eq. (9) into Eq. (6), the lateral stiffness of column i j becomes ! EI c,i j Pi j Si j = 12 β0,i j − β1,i j (10) L c,i j L 3c,i j where L c,i j and Pi j are the length and applied axial load of column j in the ith storey respectively. Consequently the stability equation for the multi-storey unbraced frame of Eq. (7)

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as follows: Maximize: Z =

n X m X

(12a)

Pi j

i=1 j=1

Subject to: Sk = 12

m X EI k j

L 3k j

j=1

β0,k j

n β1,k j X − Pi j L k j i=k

! ≥0 (12b)

Fig. 3. Decomposed single storey model.

0≤

i=1

Si =

n Y i=1

m X j=1

12

EI c,i j L 3c,i

β0,i j −

Pi j ≤ Pu,i j =

i=1

becomes the following equation [10] n Y

n X

π 2 EI

ij 2 K i j L i2j

(k = 1, 2 . . . n; i = 1, 2 . . . n; j = 1, 2 . . . m) Pi j β1,i j L c,i j

(12c)

!! = 0.

(11)

4. Decomposition of multi-storey frames Eq. (11) defines the stability condition of multi-storey unbraced frames. It is difficult for practitioners to facilitate stability analysis with direct use of Eq. (11) because of the highorder of nonlinearity. From previous research, Liu and Xu [10] proposed a strategy of decomposing a multi-storey frame into a series of single storey frames to evaluate the column effective length factor. In the case of single storey frames, the beam-tocolumn rotational restraints are directly applied to the upper ends of connected columns. For a multi-storey frame, floor beams provide rotational restraints for both the lower and upper columns at a joint. Fig. 3 illustrates a deformed profile of the single storey model decomposed from a typical storey of the multi-storey frame shown in Fig. 2. 5. Maximum and minimum frame-buckling loads In the investigation of stability of multi-storey unbraced frames subjected to variable loading, each individual applied load on the frame can vary independently; therefore, the load patterns are not predefined. As different load patterns may result in different magnitudes of frame buckling loads, the most critical or so-called lower bound of the buckling loads, corresponding to the worst load patterns, is the one associated with the minimum magnitude of the total applied load for the frame. The upper bound of the frame buckling loads, corresponding to the most favorable load pattern, is the one with the maximum magnitude of total applied load for the frame. The lower and upper bounds of the frame buckling loads together with their corresponding load patterns present a clear characterization of the buckling capacity of unbraced frames subjected to variable loading. After decomposing the multi-storey frames into a series of single storey frames, the problems of investigating the stability of multi-storey unbraced frames subjected to variable loading can be formulated as two problems, seeking of both the upper and lower bounds of frame buckling loads. The problem of seeking the upper bound of frame buckling loads can be stated

where n is the number of stories in the frame, and m is the number of columns in one storey. Pi j is the applied load associated with column i j and is the variable of the maximization problem. Z is the objective function corresponding to the maximum elastic buckling loads of the frame, and it is the sum of variables Pi j . Pu,i j is the non-sway buckling load of an individual column. Eq. (12b) represents the storey-based stability condition for the kth storey of the frame, in which the column stiffness modification factor βi j (φi j , rl,i j , ru,i j ) is defined in Eq. (8). In the case that the lateral stiffness of storey Sk is greater than zero, the storey is laterally stable; otherwise, the storey becomes laterally unstable if Sk = 0. Eq. (12c) is a side constraint for each of the applied column loads which is to be less than an associated upper bound load, Pu,i j . The problem of seeking the lower bound of the buckling loads for a multi-storey unbraced frame subjected to variable loading can be stated as, ( ) n X m X Minimum Z = min Z l = Pi j l = 1, 2, 3 . . . n (13) i=1 j=1 where n is the number of stories in the frame, and m is the number of columns in one storey. Z l (l = 1, 2, 3 . . . n) is obtained from the minimization problem as follows, Minimize: Z l =

n X m X

(14a)

Pi j

i=1 j=1

Subject to: Sl = 12

m X EI l j j=1

L l3j

β0,k j

n β1,l j X Pi j − L l j i=l

! =0 (14b)

Sk = 12 0≤

n X i=1

m X

EI k j

j=1

L 3k j

β0,k j −

Pi j ≤ Pu,i j =

β1,k j Lkj

n X

! Pi j

≥0

(14c)

i=k

π 2 EI i j K i2j L i2j

(k = 1, 2 . . . n; k 6= l; i = 1, 2 . . . n; j = 1, 2 . . . m).

(14d)

In both the maximum and minimum frame-buckling problems, the upper bound of applied column load, π 2 EI c,i j /K i2j L 2c,i j , is imposed to ensure that the magnitude of the applied

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Fig. 5. Two-bay two-storey steel frame — Case 2.

Fig. 4. Two-bay two-storey steel frame — Case 1.

load will not exceed the buckling load associated with nonsway buckling of the individual column, in which the column effective length factor associated with non-sway buckling can obtained from [13] K i2j =

[π 2 + (6 − π 2 )ru,i j ] × [π 2 + (6 − π 2 )rl,i j ] [π 2 + (12 − π 2 )ru,i j ] × [π 2 + (12 − π 2 )rl,i j ]

(15)

where rl,i j and ru,i j are the end-fixity factors provided by the beams connected at the lower and upper ends of the column, respectively. It is noticed that both the formulation and procedure of seeking the minimum buckling load are different from those of the maximum buckling loads. First, an equality constraint, Eq. (14b), is imposed in the minimization problem to ensure that the minimum value of the loads obtained from Eq. (14) will result in lateral instability at least in one storey, say storey l in this case. Second, the minimization problem shown in Eq. (14) needs to be solved n times with l = 1, 2, 3, . . . , n, and the minimum buckling load obtained from Eq. (13) is the minimum of the minimum buckling loads associated with the instability of each storey. It is also noticed that Eqs. (12) and (14) are linear programming problems. Thus, the minimum or maximum frame-buckling loads subjected to variable loading can be solved with the use of a linear programming algorithm such as the simplex method.

Fig. 6. Two-bay two-storey steel frame — Cases 3 and 5.

6. Numerical example A two-bay and two-storey unbraced rigid steel frame shown in Fig. 4 is presented to demonstrate the validity and efficiency of the foregoing proposed method for stability analysis of multistorey unbraced steel frames subjected to variable loading. The frame is a benchmark example for stability analysis of rigid frames and has been investigated by Lui [14] and subsequently by Liu and Xu [10] to validate different analytical methods. In the current study, the stability analysis of multi-storey unbraced frames subjected to variable loading accounts for the semirigid behaviour of beam-to-column connections (Figs. 4–7). The end-fixity factors associated with beam-to-column and column base connections are described in Table 1. The Young’s modulus of steel is E = 2 × 105 MPa where the reference

Fig. 7. Two-bay two-storey steel frame — Case 4.

moment of inertia for beams and columns is I = 8.3246 × 107 mm4 . The dimensions of frames and the moment of inertia of each member are shown in Figs. 4–7, and following the procedures described in the previous section, the critical buckling loads associated with the two-bay two-storey unbraced steel frames subjected to variable loading can be obtained from solving the maximization and minimization problems stated in Eqs. (12)– (14) with the simplex method. For the five cases described in Table 1, the values of the coefficients including the end-

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L. Xu, X.H. Wang / Journal of Constructional Steel Research 63 (2007) 1506–1514 Table 1 2-bay 2-storey steel frames with different connections Case

Column base connections

1 2 3 4 5

Rigid: r Rigid: r Rigid: r Rigid: r Rigid: r

=1 =1 =1 =1 =1

Beam-to-column connections Interior column

Exterior column

Rigid: r = 1 Rigid: r = 1 Semi-rigid: r = 0.8 Pinned: r = 0 Semi-rigid: r = 0.2

Rigid: r = 1 Semi-rigid: r = 0.8 Semi-rigid: r = 0.8 Rigid: r = 1 Semi-rigid: r = 0.2

Table 2 Results of the unbraced steel frames shown in Fig. 4 – Case 1 Col. i j

rl,i j

11 12 13

1 1 1

21 22 23

0.6350 0.8338 0.8257

αi j β0,i j

ru,i j 0.6350 0.8338 0.8257

0.7838 0.8945 0.9195 P Critical storey buckling loads Pi j =

K i2j

Pu,i j (kN)

S1 = 0, S2 ≥ 0 Max. (kN)

S1 ≥ 0, S2 = 0 Max. (kN) S1 = 0

Min. (kN)

Min. (kN)

3.103 2.637 1.971

0.330 0.285 0.287

183 700.0 144 700.0 108 500.0

54 840.43 0.0 0.0

0.0 107 395.8 0.0

33 152.76 0.0 0.0

0.0 0.0 0.0

1.286 1.787 0.773

0.391 0.310 0.306

106 200.0 133 900.0 57 750.0

54 840.43 0.0 0.0

0.0 0.0 0.0

76 528.09 0.0 0.0

0.0 73 228.3 8.3

109 680.9

Difference of max. & min. storey-buckling loads

107 395.8

109 680.9

2.1%

73 236.6 49.8%

Difference of max. & min. frame-buckling loads

49.8%

Proportional loading

108 200.0

Difference of proportional loading & min. frame-buckling loads

47.7%

Table 3 Results of the unbraced steel frames shown in Fig. 5 – Case 2 Col. i j

rl,i j

11 12 13

1 1 1

21 22 23

0.5680 0.8150 0.7810

ru,i j 0.5680 0.8150 0.7810

0.7310 0.8810 0.8950 P Critical frame-buckling loads Pi j

αi j β0,i j

K i2j

Pu,i j (kN)

S1 = 0, S2 ≥ 0 Max. (kN) S2 = 0

Min. (kN) S2 = 0

S1 ≥ 0, S2 = 0 Max. (kN) S1 = 0

Min. (kN)

2.8640 2.5840 1.8780

0.346 0.289 0.297

175 300.0 142 700.0 104 900.0

31 398.2 0.0 0.0

0.0 32 391.8 0.0

31 398.2 0.0 0.0

0.0 0.0 0.0

1.1240 1.7290 0.7484

0.427 0.318 0.323

97 460.0 130 600.0 54 780.0

72 761.4 0.0 0.0

0.0 69 597.8 0.0

72 761.4 0.0 0.0

0.0 14 817.8 54 780.0

101 989.6

104 159.6

=

Difference of max. & min. storey-buckling loads Difference of max. & min. frame-buckling loads Proportional loading Difference of proportional loading & min. frame-buckling loads

fixity factors, the effective length factors and the buckling loads associated with the maximum and minimum frame-buckling loads, together with their relative differences, are presented in Tables 2–6. Also presented in the tables are the coefficients associated with the column initial lateral stiffness modification factors, αi j β0,i j , where αi j (i = 1, 2 and j = 1, 2, 3) is the coefficient of normalization correlated with the reference moment of inertia of each column, and the upper limits of an individual column. The values of proportional loading together with their relative difference with the minimum frame-buckling loads are also given in these tables.

104 159.6

2.1%

69 597.8 49.6%

49.6% 103 200.0 48.3%

For Case 1, in which both the column base and beam-tocolumn connections are rigidly connected, it is observed from Table 2 that the maximum frame-buckling load, 109 680.9 kN, is achieved when lateral instability takes place in the first or the second stories of the frame. The minimum storeybuckling loads associated with lateral instability of the first and second stories are 107 395.8 kN and 73 236.6 kN, respectively. Therefore, the relative difference between the maximum and minimum frame-buckling loads is 49.8%, which is significant. It is also observed from Table 2 that the load patterns associated with the maximum and minimum frame-buckling loads are

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Table 4 Results of the unbraced steel frames shown in Fig. 6 – Case 3 Col. i j

rl,i j

11 12 13

1 1 1

21 22 23

0.5403 0.7722 0.7613

ru,i j 0.5403 0.7722 0.7613

0.7084 0.8504 0.8843 P Critical frame-buckling loads Pi j

αi j β0,i j

K i2j

Pu,i j (kN)

S1 = 0, S2 ≥ 0 Max. (kN) S2 = 0

Min. (kN) S2 = 0

S1 ≥ 0, S2 = 0 Max. (kN) S1 = 0

Min. (kN)

2.7670 2.4650 1.8360

0.353 0.299 0.301

171 900.0 138 100.0 103 300.0

31 397.4 0.0 0.0

0.0 17 726.9 16 191.7

31 397.4 0.0 0.0

0.0 0.0 0.0

1.0590 1.6000 0.6960

0.442 0.337 0.331

93 930.0 123 300.0 53 460.0

69 092.0 0.0 0.0

0.0 16 191.7 49 321.3

69 092.0 0.0 0.0

0.0 12 007.5 53 460.0

100 489.4

99 431.6

100 489.4

65 467.5

=

Difference of max. & min. storey-buckling loads

1.1%

Difference of max. & min. frame-buckling loads

53.5% 53.5%

Proportional loading

99 890.0

Difference of proportional loading & min. frame-buckling loads

52.6%

Table 5 Results of the unbraced steel frames shown in Fig. 7 – Case 4 Col. i j

rl,i j

11 12 13

1 1 1

21 22 23

0.4200 0.0 0.6580

ru,i j 0.4200 0.0 0.6580

αi j β0,i j 2.364 0.782 1.682

0.5920 0.790 0.0 0.0 0.8200 0.601 P Critical frame-buckling loads Pi j = Difference of max. & min. storey-buckling loads

K i2j

Pu,i j (kN)

S1 = 0, S2 ≥ 0 Max. (kN) S2 = 0

Min. (kN) S2 = 0

S1 ≥ 0, S2 = 0 Max. (kN) S1 = 0

Min. (kN)

0.383 0.50 0.319

158 400.0 82 570.0 97 700.0

20 200.9 0.0 20 192.2

0.0 34 575.9 0.0

20 200.9 0.0 20 192.2

0.0 0.0 0.0

0.521 1.0 0.367

79 700.0 41 570.0 48 150.0

14 431.9 0.0 13 898.4

0.0 30 024.1 0.0

14 431.9 0.0 13 898.4

0.0 0.0 27 688.9

68 723.4

64 600.0 6.4%

68 723.4

Difference of max. & min. frame-buckling loads

27 688.9 148%

148%

Proportional loading

66 530.0

Difference of proportional loading & min. frame-buckling loads

140.3%

Table 6 Results of the unbraced steel frames shown in Fig. 6 – Case 5 Col. i j

rl,i j

11 12 13

1 1 1

21 22 23

0.1840 0.3840 0.3590

ru,i j 0.1840 0.3840 0.3590

0.2870 0.4850 0.5590 P Critical frame-buckling loads Pi j

αi j β0,i j

K i2j

Pu,i j (kN)

S1 = 0, S2 ≥ 0 Max. (kN) S2 = 0

Min. (kN) S2 = 0

S1 ≥ 0, S2 = 0 Max. (kN) S1 = 0

Min. (kN)

1.648 1.531 1.114

0.446 0.392 0.399

136 000.0 105 300.0 78 070.0

0.0 16 941.8 16 941.8

32 322.9 0.0 0.0

0.0 16 941.8 16 941.8

0.0 0.0 0.0

0.312 0.651 0.295

0.746 0.575 0.556

55 730.0 72 300.0 31 840.0

0.0 13 252.9 13 252.9

26 821.4 0.0 0.0

0.0 13 252.9 13 252.9

0.0 13 252.9 13 252.9

60 389.5

59 144.3

60 389.5

26 505.8

=

Difference of max. & min. storey-buckling loads

2.1%

127%

Difference of max. & min. frame-buckling loads

127%

Proportional loading

60 200

Difference of proportional loading & min. frame-buckling loads

127%

different. The load pattern corresponding to the maximum frame-buckling loads tends to place the loads only on exterior columns 11 and 21. In contrast to that, the load pattern associated with the minimum frame-buckling load applies the loads only on the interior columns which are laterally stiffer

than the exterior ones. Also found in Case 1 is that there are two different loading patterns associated with the maximum framebuckling load of 109 680.9 kN, which causes lateral instability of the first or the second storey, and the first and second stories are not becoming laterally instable simultaneously.

L. Xu, X.H. Wang / Journal of Constructional Steel Research 63 (2007) 1506–1514

In Case 2, the column base and the interior beam-tocolumn connections are rigid. The exterior column is semirigidly connected with the corresponding end-fixity factor being 0.8. The presence of semi-rigid connections yields a flexible frame which is evidenced by decreasing the magnitudes of coefficients αi j β0,i j compared to that of Case 1. Consequently, the maximum frame-buckling load of Case 2 reduces to 104 159.6 kN, and the corresponding minimum frame-buckling load decreases to 69 597.8 kN, which yields the relative difference between the maximum and minimum frame-buckling loads to be 49.6%, which is significant. The load patterns associated with the maximum frame-buckling load including load locations and magnitudes are identical for the first and second stories. There are two different load patterns associated with minimum frame-buckling loads. It is also observed that the first and second storey become unstable simultaneously when they are subjected to maximum framebuckling loads. For Case 3, the beam-to-column connections for both the interior and exterior columns are semi-rigidly connected with the corresponding end-fixity factor being 0.8. Compared with Cases 1 and 2, the Case 3 frame is more flexible; thus, the magnitudes of the maximum and minimum framebuckling loads are reduced to 100 489.4 kN and 65 467.5 kN, respectively, which leads to a difference of 53.8% between the buckling loads. Like Case 2, it is found that lateral instability occurs simultaneously for both first and second stories when they are subjected to the maximum frame-buckling load. It is also seen that the load patterns are identical when the frames are subjected to maximum frame-buckling loads. Table 5 presents the results of Case 4, in which the column base are rigid connections. The beam-to-column connections for the exterior columns are rigid, and for the interior columns, they are pinned. The maximum and minimum frame-buckling loads are 68 723.4 kN and 27 688.9 kN, respectively. The load patterns corresponding to the maximum frame-bucking loads applied to the exterior columns which are characterized by the rigid beam-to-column connections. The difference between the maximum and minimum frame-buckling loads is 148%, which is quite significant. It is observed that the load patterns associated with the maximum frame-buckling load are the same for both the first and second stories. It is also found that lateral instability occurs simultaneously for both first and second stories when they are subjected to the maximum framebuckling load. In Case 5, the column base connection is rigid, while the beam-to-column connections for both the interior and exterior columns are quite flexible with the corresponding end-fixity factor being 0.2. The maximum and minimum frame-buckling loads are 60 389.5 kN and 26 505.8 kN as shown in Table 6, which yields a considerable difference of 127%. The load patterns corresponding to the maximum and minimum framebuckling loads tend to apply the loads both on the interior and exterior columns. Similar to Cases 2–4, the load patterns associated with the maximum frame-buckling loads is the same and also the lateral instability occurs simultaneously for

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both first and second stories when they are subjected to the maximum frame-buckling load. For reasons of comparison, the frame-buckling strengths associated with storey-based buckling subjected to proportional loading for the frames in the foregoing cases are also presented in Tables 2–6. It is found that the differences between the proportional and the minimum loading are 47.7%, 48.3%, 52.6%, 140.3% and 127%, respectively. A concern may be raised from such significant differences in this particular example and other studies [12,13,15] as to the appropriateness of using the conventional proportional loading approach to evaluate frame-buckling strength for unbraced steel frames such as the ones investigated in this example. 7. Conclusions A method of evaluating the critical frame-buckling loads for multi-storey unbraced steel frames subjected to variable loading is proposed in this study. The validity of the proposed method for the stability analysis of multi-storey unbraced steel frames subjected to variable loading are demonstrated by two-bay twostorey unbraced steel frames. In this study, the effects of the connection behaviour on the critical frame-buckling loads in a variable case, as well as cases with different rigidities of beamto-column connections are considered. From this study, it is concluded that: The stability analysis established for a single storey unbraced frame subjected to variable loading is extended to the analysis of multi-storey unbraced frames. In contrast to current frame stability analysis involving only proportional loading, the approach proposed in this study permits individual applied loads on the frame to vary independently, therefore, the load patterns are not predefined. The analysis method presented in this study can be characterized by the column lateral stiffness modification factor βi j which provides a quantitative measurement of the stiffness interactions among the columns in a storey to resist lateral instability. The concept of storey-based buckling is employed in this study to formulate the problem of determining the critical frame-buckling loads to be a pair of constrained minimization and maximization problems. The variables and objective functions of the minimization and maximization problems are the applied column loads and the summation of applied column load variables. The stability constraints are imposed to ensure that lateral instability occurs in at least one storey of the frame. For each variable, an upper limit is imposed to ensure that the magnitude of the applied load will not exceed the buckling load associated with the non-sway buckling of the individual column. The maximum and minimum frame-buckling loads and their associated load patterns can be obtained by solving the maximization and minimization problems, respectively, with a linear programming method. They represent the lower and upper bounds of the frame buckling loads of the structures, which characterize the stability capacity of the frame under extreme loading conditions. The frame-buckling loads associated with proportional loading are always between the

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L. Xu, X.H. Wang / Journal of Constructional Steel Research 63 (2007) 1506–1514

maximum and minimum loads subjected to variable loading. The study also finds that there might be several different load patterns associated with the maximum and minimum framebuckling loads. It is also observed from the presented example that the corresponding maximum frame-buckling load is always associated with the lateral instability of both the first and second storey simultaneously, which indicates that further increase in any one of the applied loads is impossible as each storey has already reached the limit state of lateral instability. This study reveals that the differences between the maximum and the minimum frame-buckling loads could be substantial for multi-storey unbraced steel frames. The significant differences demonstrated in the presented example resulted from the sizeable differences between the lateral stiffnesses among columns in the same storey, which are primarily due to the differences among the moments of inertia of column sections. Therefore, for the frames for which there is not substantial variation in column size and connection rigidity, the differences between the maximum and the minimum framebuckling loads are not as significant as demonstrated in the presented example. However, to ensure that the minimum frame-buckling strength of the frame is being accounted for in the design, the stability analysis of the frames subjected to variable loading proposed herein is recommended for the frames in either of the following cases: 1. There is a considerable variation in lateral stiffness among columns in the same storey of any storey of the frame; 2. There is a considerable variation in connection stiffness among beam-to-column connections in the same storey of any storey of the frame or column base connection; and 3. There is an expected substantial volatility in applied loads. Finally, the proposed stability analysis of multi-storey unbraced frames subjected to variable loading takes into consideration the volatility of live loads during the life span of structures and buckling characteristics of the frame under any possible load patterns. As the minimum and maximum framebuckling loads and the associated load patterns clearly define the stability capacities of frames under extreme load cases, the proposed method provides crucial information concerning the

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