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Design of beam-columns in steel frames in the United States
Thin-Walled Structures 13 (1991) i-83
Design of Beam-Columns in Steel Frames in the United States
W. F. C h e n Department of Structural Engineering, School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 25 January 1990; accepted 25 September 1990)
ABSTRACT This paper attempts to give designers an insight into the recent developments of the design criteria of steel beam-columns in the USA. The important features include (1) the ultimate strength of columns and beams that are special cases of beam-columns, (2) the secondary effects (P-6 and P-A) that should be considered in the ultimate strength design of beam-columns, and (3) the simplified computer-based second-order elastic and inelastic analysis methods that are suitable for adoption in the practical design for beamcolumns in rigid frames. Recommended and proposed ultimate strength design interaction equations in the USA for different categories of beamcolumns are presented. Importantfeatures of these equations are summarized. Several illustrative examples are given. Directions of further research are indicated.
1 THE CATEGORY OF BEAM-COLUMNS O n e of the most i m p o r t a n t structural elements in a steel f r a m e w o r k is the beam-column. A b e a m - c o l u m n is the type of structural m e m b e r w h i c h is subjected to the c o m b i n e d action of b e n d i n g m o m e n t s a n d axial thrust. D e p e n d i n g o n the m a n n e r the loads are applied a n d the resulting deformations, b e a m - c o l u m n s can be classified into one of the following categories: I
Thin-Walled Structures 0263-8231/91/$03-50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
V£ F. Chert
(a) In-plane beam-columns in which the bending moment is acting in only one principal axis direction in addition to the axial thrust. For this type of beam-column, only in-plane detbrmation will occur upon application of the loads. (b) Lateral torsional buckling of beam-columns in which the bending moment is again acting in only one principal axis direction in addition to the axial thrust. However, unlike in-plane beamcolumns, out-of-plane deformation will result in this type of beam-column due to twisting of the member as a result of lateral torsional buckling. (c) Biaxially loaded beam-columns in which the bending moments are acting in two principal axis directions in addition to the axial thrust. As a result, deformation of this type of beam-column is a three-dimensional space action. According to how the ultimate capacities of these beam-columns are established, the beam-columns in each of the above categories can be further categorized into short or long-beam-columns as follows: (a) In short beam-columns, the effect of member lateral displacement on the magnitude of moment in the member is not significant. The limit state of this type of beam-column is strength." thus, the ultimate capacity of a short beam-column is reached when full plastic yielding of the material of the entire cross-section has occurred. (b) In long beam-columns, the effect of member lateral displacement on the magnitude of moment in the member is not negligible. The limit state of this type of beam-column is stability; thus, the ultimate capacity of a long beam-column is its stability limit load. Long beam-columns can be further subdivided into two cases: (a) The non-sway case where the only secondary moment that needs to be considered is the P-6 moment. (b) Thesway case where both theP-6 and P-A secondary moments are important and need to be accounted for in the analysis and design procedures. The purpose of this paper is to summarize recent research on the subject of beam-columns for each of these categories and to present the new design philosophy based on a limit state design approach (Section 2) and design equations in the form of interaction equations that are currently in use, or proposed, in the USA. Before proceeding to the discussion of the beam-column problem, it is more appropriate to
Design of beam-columns in steel frames in USA
3
discuss first the column problem (Section 3) and the beam problem (Sections 4 and 5). This is because not only will they represent special cases of beam-columns (Sections 6 and 7), but they also provide end points to which the beam-coiumn interaction equation is referenced. Finally, the second-order elastic and inelastic analysis methods for the design of beam-columns in frames are presented (Sections 8-10).
2 DESIGN PHILOSOPHIES To implement the mathematical theory of stability into engineering practice, it is necessary to review the various design philosophies and safety concepts upon which current design practice is based. Details of this implementation on various specific subjects will be given in the sections that follow. The philosophy of steel building design in the USA has undergone a metamorphosis from the allowable stress design (ASD) approach through the plastic design (PD) approach to the present stage of the load and resistancefactor design (LRFD) approach. The LRFD approach is the product of a culminated effort by researchers and practitioners to incorporate the latest research results and experience on material and structural behavior, under the influence of a variety of loading types, into a consistent and rational design format. One fundamental feature that demarcates the LRFD approach to steel design from the ASD and PD approaches is that LRFD is a probability-based design methodology which utilizes statistical evidence and reliability theory to substantiate the logistic and rationale of the design. Design considerations are guided by the identification of a set of limit states. A design is considered acceptable if none of the limit states is violated. Since a probabilistic mathematical model was used in the development of the load and resistance factors, it is possible to give proper weight to the degree of importance for each load effect and resistance. The final result is that a more uniform margin of safety can be realized. 2.1 Allowable stress design
The purpose of ASD is to ensure that the stress computed under the action of the working (i.e. service) loads of a structure do not exceed some predesignated allowable values. The allowable stresses are usually expressed as a function of the yield stress or ultimate stress of the material. The general format for an allowable stress design is thus
4
W.F. Chen R?/
m
F.S~ > E
Qni
(1)
i=1
where Rn = nominal resistance of the structural member expressed in units of stress; Qn = nominal working (or service) stresses computed under working load conditions; F.S. -- factor of safety; i = type of load (i.e. dead load, live load and wind load, etc.); m = number of load types. The left-hand side of eqn (1) represents the allowable stress of the structural member or component under various loading conditions (e.g. tension, compression, bending and shear). The right-hand side of the equation represents the actual or computed combined stress produced by various load combinations (e.g. dead load, live load and wind load). Formulas for the allowable stresses for various types of structural members under various types of loadings are specified in the AISC Specifications. L2 A satisfactory design is when the stresses in the member, computed using a first-order analysis under working load conditions, do not exceed their allowable values. One should realize that, in ASD, the factor of safety is applied to the resistance term, and safety is evaluated in the service load range.
2.2 Plastic design The purpose of PD is to ensure that the nominal maximum plastic strength of the structual member or component exceeds that of the factored load combinations. It has the format: m
R n > ?" E
Q"~'
(2)
i=l
where R,, = nominal plastic strength of the member, and ~" = load factor. In steel design, the load factor is given by the AISC Specifications L2 as 1.7 if Q, consists 0nly of dead and live gravity loads, or as 1-3 if Q,, consists of dead and live gravity loads plus wind or earthquake loads. The use of a smaller load factor for the latter case is justified in that the simultaneous occurrence of all these load effects is not very likely. Note that in plastic design, safety is incorporated in the load term and is evaluated at the ultimate (plastic strength) limit state.
Design of beam-columns in steelframes in USA
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2.3 Load and resistance factor design The purpose of LRFD is to ensure that the nominal resistance of the structural member or component exceeds that of the load effects. Two implied safety factors are used, i.e. one applied to the loads and the other to the nominal resistance. Thus, the load and resistance factor design has the format: m
(PRn >/ Z
(3)
YiQni
i=1
where R, = nominal resistance of the structural member; = resistance factor (< 1.0) 7; = load factor (usually >1.0) corresponding to Q,i. In the 1986 LRFD Specification of AISC, 3 the resistance factors were developed mainly through calibration,4 whereas the load factors were developed on the basis of statistical analysis. 5.6. In particular, a firstorder probability theory was used. The load and resistance factors for various types ofloadings and various load combinations are summarized in Tables 1 and 2, respectively. A satisfactory design is one in which the TABLE 1 Load Factors a n d Load C o m b i n a t i o n s a 1.4D 1.2D+ 1.2 D + 1.2D+ 1.2D + 0.9D-
1.6 L + O.5 (L~ or S or R ) 1.6 (Lr or S or R) + (0.5 L or 0.8 I4") i.3W+0.5L+0.5(LrorSorR) 1.5E + (0.5L or 0 . 2 S ) 1.3 W o r 1.5E
where D = dead load L = live load Lr = roof live load W = wind load S = snow load E = earthquake load R = n o m i n a l load due to initial rainwater or ice exclusive of the p o n d i n g contribution aThe load factor o n L in the third, fourth a n d fifth load c o m b i n a t i o n s shown above shall equal 1-0 for garages, areas occupied as places of public assembly a n d all areas where the live load is greater t h a n 100 psf.
6
IV.. b: Chen TABLE 2 Resistance Factors Member type and limit state
Tension member, limit state: yielding Tension member, limit state: fracture Pin-connected member, limit state: tension Pin-connected member, limit state: shear Pin-connected member, limit state: bearing Columns, all limit stales Beams, all limit states High-strength bolts, limit state: tension High-strength bolts, limit state: shear A307 bolts others
0.90 0.75 0.75 0.75 1.00 0.85 0.90 0-75 0.60 0.65
probability of the structural m e m b e r exceeding a limit state (e.g. yielding, fracture a n d buckling) is minimal. Based on the first-order secondm o m e n t probabilistic analysis, 7 the safety of a structural m e m b e r is m e a s u r e d by a reliability or safety index, 4 defined as
//-
In(R,,/Q,,) +
(4)
where R = m e a n resistance: Q = m e a n load effect: ~rR V~ = coefficient of variation of resistance = -~-; R VQ = coefficient of variation of load effect -
crQ
~),
in which ~7 is the s t a n d a r d deviation. The physical interpretation of the reliabili~ index,//, is shown in Fig. 1. It is the multiplier of the s t a n d a r d deviation v"V~ + V~ between the m e a n of the In(R/Q) distribution a n d the ordinate. Note that both the resistance R a n d the load Q are treated as r a n d o m parameters in LRFD, and so ln(R/Q) does not have a single value, but follows a distribution, The s h a d e d area in the figure represents the probability in which In(R/Q) < !, i.e. the probability that the resistance will be smaller than the load effect, indicating that a limit state has been exceeded. T h e larger the value of//, the smaller the area of the s h a d e d area. and so it becomes more i m p r o b a b l e that a limit state may be exceeded: thus, the m a g n i t u d e o f / / reflects the safety of the member.
Design of beam-columns in steel frames in USA
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PROBABILITY DENSITY
~tv/t/ 2 -
2
v.+ vo
_1 -[
raisin
In(R/Q)
In(RIQ)
Fig. 1. Reliability index in the AISC L R F D design.
In the development of the present LRFD Specification 3 the following target values for fl were selected: (i) [3 = 3.0 for members and fl = 4.5 for connectors under dead plus live and/or snow loading; (ii) fl = 2.5 for members under dead plus live load acting in conjunction with wind loading; (iii) fl = 1.75 for members under dead plus live load acting in conjunction with earthquake loading. A higher value of fl for connectors ensures that the connections designed are stronger than their adjoining members, A lower value offl for members under the action of a combination of dead, live, wind or earthquake loading reflects the improbability that these loadings will act simultaneously. From the above discussion, it can be seen that in ASD, the safety of the structural member is evaluated on the basis of service load conditions, whereas in PD or LRFD, safety is evaluated on the basis of the ultimate or limit load conditions. In addition to strength, the designer must also consider the stiffness of the structure. One important criterion related to stiffness is that the structure or structural component must not deflect excessively under service load conditions. Thus, regardless of the design
8
W.F. Chen
method, one should always investigate such serviceability requirements as deflection and vibration at service load conditions.
3 COMPRESSION MEMBERS A column can be thought of as a special case of beam-column in which the primary bending moments are absent. The derivation of any column equation can be related to one of the two following approaches: (a) The eigenvalue approach, where the load at which lateral displacement of a perfectly straight or ideal column commences is found by an eigenvalue analysis. The load is referred to as the buckling load and is defined as the load that corresponds to a state of neutral equilibrium of the perfectly straight column or the lowest load at which a bifurcation of equilibrium takes place. (b) The stability approach, in which the load-deflection curve of a column is traced. The peak point of this load-deflection curve is referred to as the stability limit load, which is the maximum load that the column can carry. 3.1 Pin-ended columns When a perfectly straight elastic column is loaded by an axial thrust, P, lateral deflection will start as the load reaches the Euler load, P~: p~
P,
where P, = A F,. •
)ql = L r x~ rr-~E
in which A = cross-sectional area, F,. = yield stress, L = length of column, r = radius of gyration, and E = modulus of elasticity. The Euler load P~ marks the point of transition from stable to unstable equilibrium of the column. Once the Euler load is reached, the deflection of the original straight elastic column will increase without bound. In order to take into account the nonlinear stress-strain behavior of the material once the proportional limit is exceeded prior to buckling, the reduced modulus (double modulus) or the tangent modulus theories can be used. In the reduced modulus theory, complete strain reversal at the convex side of the column at buckling is assumed. As a result, the elastic
Design of beam-columns in steel frames in USA
9
modulus E governs the relation of stress to strain at the convex side of the column whereas the tangent modulus Et governs the stress-strain relationship at the concave side during buckling. The reduced modulus load is given by
Er
Pr = Pe -~-
(6)
where the reduced modulus Er is a function of E, Et and the shape of the cross-section. The reduced modulus load Pr marks the point of bifurcation of equilibrium for a perfectly straight inelastic column, In the tangent modulus theory, the axial load is assumed to increase as the member bends subsequent to reaching the critical load. As a result, there is no unloading of any fiber over the cross-section, and the tangent modulus Et will govern the stress-strain relationship of the entire crosssection. The corresponding tangent modulus load is
El
P, = Pc ~-
(7)
The tangent modulus load is the lowest load at which bufurcation of equilibrium can take place. However, due to imperfections that are present in columns, experimentally determined maximum loads of columns tend to agree more with the tangent modulus load. In view of this, the Structural Stability Research Council (SSRC) (formerly Column Research C o u n c i l - CRC) recommended that the tangent modulus load best represents the maximum load that an inelastic column can carry. 3.1.1 CRC curve The CRC curve forms the basis for the column design curves in the USA. Its development was based on the eigenvalue concept and is composed of two regions: (a) An inelastic region (0 ~
-
l -
0 . 2 5 ~.~,
(8a)
P,. (b) An elastic region (X0 > v~), which is the Euler load: P P,.
-
1 , ;t~
(8b)
Since the CRC curve was developed on the basis of the eigenvalue approach, it is only valid for perfectly straight columns.
I~. F. Chen
10
3.1.2 A I S C A S D curve
To account for initial crookedness or other geometric imperfections that are inevitably present in columns, the CRC curve was divided by a variable factor of safety of
v.s. = 5+
v2/
(9)
in the inelastic region and a constant factor of safety of 23/12 in the elastic region to form the AISC ASD curve. 3.1.3 A I S C PD curve
The AISC PD curve was established by multiplying the AISC ASD curve in the inelastic range by 1.7, where 1.7 is the load factor for gravity load for plastically designed structures, The elastic range is not defined for the AISC PD curve since, under the provision, the nondimensional slenderness parameterk0 for columns designed using plastic design procedures must be less than v 2. 3.1.4 A I S C L R F D curve
The LRFD curve s was established by a curve-fitting procedure, with the SSRC curve 2 to be discussed further in Section 3.1.5. It has the simple form: P _ j'expl-0.419A,0l P,. ],0.877 A,~72
A,,, < 1.5 X,, > 1.5
(10)
Table 3 summarizes the expressions for the aforementioned curves and Fig. 2 compares the CRC curve and the three AISC curves with the three SSRC multiple column curves described in what follows. Note that the AISC column curves are to be used in conjunction with their respective design format as given by eqns (1), (2) or (3). 3.1.5 S S R C multiple column curves
The SSRC multiple column curves were developed on the basis of the stability concept. Load-deflection curves for a number of columns with measured residual stresses and initial crookedness of 0.001L were traced by numerical techniques, using a computer model. The peak point of each of the load-deflection curves gives the maximum load that the column can carry. By tracing the load-deflection curve of each column at different slenderness ratios, the column curve for that particular column can be obtained. A total of 112 column curves were traced s for different types of columns, which were then categorized, and a set of three
Design of beam-columns in steel frames in USA
11
0 0
E
tO
~z
0
Q
m
It . _J
~
0
>
0 tO
-
tO
E
CO
o 0 !-
L)
B
> tO
L) L)
o
~5 ee"
ffJ O tO
®
CO •
oJ >
o
=
;
~
0
¢J
Q.
tO C4
<
6
c
0 Q
m
OD
,¢~
t'M
Q
r.i
W. F Chen
12
TABLE
3
Summary of Column Curves
Column curves
CRC curve
Cblumn equations
-- = 1-Pv 4 P
e,,
AISC ASD curve
P
--
-
Zo <. x , ~
1
Zo >
4
=
P
12
1
p,.
23
Ag
AISC LRFD curve
P
Zo -<< ~, 2
"~-0 )" \ ' 2
1.7 AISC PD curve
x- 5
l-
-
.a.0 <
P P~
exp (-0"419/lo)
P
0.877
P,.
;t~
.~o < 1.5
J. 0 >
1.5
multiple c o l u m n curves was developed. Each is representative of a specific category of columns. The expressions for these three multiple column curves were given elsewhere? For design purposes, it is more convenient to represent these curves with the R o n d a l - M a q u o i equation: ~0 P P-~I =
1 +r/+Zo 2~ .....
1 , 2A~ v ' ( l + 17 + A.~)2 -4,,10
(11)
where 1/ = a(Z0 - 0.15)
a: /
0. 103 (SSRC curve 1) 0.293 (SSRC curve 2) 0.622 (SSRC curve 3)
These curves, in their original form, and the R o n d a l - M a q u o i curves, are shown in Fig. 3. S S R C curves 1, 2 and 3 are closely c o m p a r a b l e to the
Design of beam-columns in steel frames in USA
13
0 o o
e~
t~ to b.. ,.z
I
~D Q to
Q ,.a
to
A
,.z
.g o
h
"3
, ~D to
a "3
uJ
i
0
_d
0 rr" o)
0 Z 0
"6 ?
I I
I
I
to
¢y
I
I
6
o
Q o
o
~_lw"
o
r4
14
IV.. ~: Chen
European multiple column curves a0, b and d, respectively, recommended by the European Convention of Constructional Steelworks (ECCS) and are given in Ref. 11. Although the SSRC multiple column curves are more realistic than the CRC curve in representing column strength, the lack of smooth transition from one curve to the other means that there will be a jump in column strength, which is physically not acceptable. Furthermore, although both residual stresses and initial crookedness were considered in their development, these parameters are not clearly identified in the column equations. Lastly, since the initial crookedness used in the development of these curves was assumed to be 0.001L, they may not be representative of the strength of columns with initial crookedness which is not 0.001L. To overcome these drawbacks, Lui and Chen ~2 have developed a column equation based on a physical model, which is applicable to columns with any value of initial crookedness. The equation has the form e
+ (l +
- v/l
+ (1 +
-
where E
Et rr
E
in which P =
magnitude of initial crookedness pin-ended length of the column '
f -- shape factor about the axis of bending; c = distance from neutral axis to extreme fiber: = plastification parameter, which is a function of the cross-sectional shape, axis of bending and the material of the column. The validity of the above approach has been demonstrated 13 by comparing the results obtained from eqn (12) with that from an elasticplastic analysis of imperfect columns, using a computer model developed. Good correlations were observed.
Design of beam-columns in steelframes in USA 3.2
End-restrained
15
columns
The pin-ended column represents an anchor point to which other columns with different end conditions are referenced. Once the strength of a pin-ended Column is known, the strength of the same column with other end conditions can be obtained by the use of the effective length
factor, K. The elastic effective length factor is defined as K =
Pv~P~r
(13)
where Pcr = critical load of the end-restrained column. For a perfectly straight elastic column, the critical load can be found by writing the differential equation conforming to the specific end conditions. The eigenvalue to the characteristic equation of the differential equation is the critical load for that column. The effective length factor can then be found from eqn (13). A perfectly straight column restrained elastically against rotation at ends A and B by rotational springs with spring constants RkA and RkB, and restrained elastically against translation at end B by a translational spring with spring constant Tk is shown in Fig. 4(a). The corresponding free body diagram of the column and the rotational springs are shown in Fig. 4(b). The critical load of this column can be found by first writing moment equilibrium equations of the three free bodies and then setting R~8%Fm %a -
~
: %a
MBA~ j P
mk
Maaf ~ - ~
Tk &
MAB
TkA
El constant
A
P
jL~ P
MAB Tk ~
~
TkA
RkABA
Ca)
(b)
Fig. 4. An elastically restrained column with flexible joints in frames.
16
W. F. Chen
the determinant of the coefficient matrix of the equilibrium equations equal to zero, i.e. SIA + RkA
S1B
- S 2 ( A + B)
SiB
S1A + Rk8
- S 2 ( A + B)
- S 2 ( A + B)
- S 2 ( A + B)
det
-2S3(A + B) -
P
= 0
(14)
+ Tk
where E1 S1
-
A =
E1 L 2'
$2-
L'
S3
-
E1 L3
k L ( s i n k L - k L coskL) 2 - 2 cos k L - k L sin k L '
B
=
k L ( k L - sin k L ) 2 - 2 cos k L - k L sin k L
in which
The smallest value of P that satisfies eqn (14) is the critical load for the column of Fig. 4(a). The effective length factor can then be obtained from eqn (12). Although the effective length factor discussed above was derived from elastic analysis, limited studies ~4-~ have shown that this will be conservative in most cases for initially crooked elastic-plastic columns. In real structures, a column usually exists as part of a frame. The effective length factor for columns as part of a frame has been derived by Julian and Lawrence ~7 and LeMessurier. t8 ~9 By assuming that (i) the girders are rigidly connected to the columns at all joints, (ii) all columns of a story buckle simultaneously and, at the onset of buckling, the rotations of the ends of the girders are equal and opposite for braced frames and equal in magnitude and direction for unbraced frames, and (iii) all members are prismatic and the behavior is elastic, the following equations for the effective length factor can be determined.~7 For braced frame (sway prevented): GAGB
4
~ \
/
+ \
)
+ 2 tan(n/2K) rr/K
-
1(15)
For unbraced frame (sway permitted): GAGB(n/K) 2 - 36
6(GA + GB)
n/K ..........
tan(n/K)
0
(16)
Design of beam-columns & steel frames in USA
17
where GA and Ga are the stiffness distribution factor for t h e A th and B th ends of the column, respectively. They are defined as
G = ~,(I/L)co~um, E( I/L )girjer
(17)
The summation in the numerator covers the stiffnesses of all columns that come together at a joint, and the summation in the denominator covers the stiffnesses of all the beams that come together at the same joint. Equations (15) and (16) form the bases for the alignment charts that are used extensively by designers in the USA. Some modifications can be made to the G factor to make the alignment charts more versatile:
Braced frames 1. If the far ends of the girders are fixed, divide G by 2.0. 2. If the far ends of the girders are hinged, divide G by 1.5. 3. If the girders are connected to the columns by semirigid connections with rotational stiffness Rk at both ends, replace ~Z(l/L)girde r by ~(I'/L)girder, where 1
I' =
2EI
I
(18)
Unbraced frames 1. If the far ends of the girders are hinged, multiply G by 2.0. 2. If the far ends of the girders are fixed, multiply G by 1.5. 3. If the girders are connected to the columns by semirigid connections with rotational stiffness Rk at both ends, replace ~.,(I/L)girde r by ~.,(I'/L)girder, where 1
I' = 1+
6EI
I
(19)
Figure 5 shows a plot o f e q n (18) and (19), which can be readily used to determine the modified girder moment of inertia I'. For cases in which the columns become inelastic, a modified value of G was recommended by Yura 2° to be used for the alignment charts. The modified G is
G =
Z (EtI/L )col,,m, ~Z(El/L)gird~r
(20)
W. F. Chen
18 10
0.8 0.6 ~ 0.4
~
peSway ~ prevent~ed rmqtted
0.2 o-
ors
~.'o
;5
2:o
EI/RkL
2'.~
3'-o
3'.5
Fig. 5. Chart for modified beam moment of inertia to include the influence of the flexibility of beam-to-column connections. An alternative a n d more accurate approach to determine the K factor for columns in sway frames was proposed by LeMessurier. ~9The expression is
K~ _ .2~i [~,e+ ~.(cLe)l
(21)
where the subscript i refers to the i th c o l u m n of the story, a n d Z P = sum of all vertical forces acting on the story: Pi = axial force on c o l u m n i whose effective length factorKi is required a n d is calculated based on first-order analysis; Iz = m o m e n t of inertia of c o l u m n i; 6(GA + GB) + 36 /3 = 2(GA + GB) + GAGB + 3'
CL = [/3K2 L .2
1] in which K is calculated from eqn (16) or the align-
ment charts. To account for the inelastic behavior o f the column, Ki is m o d i f i e d as
rr:rli [ E P + ~](CLP)]
K~-
e,
where
PV
L
~(~
J
(22)
Design of beam-columns in steel frames in USA
19
In the LeMessurier equation, fl is a factor to account for the endrestraint effect of the column, and CL is a factor to account for the P-6 effect in the column. Alternatively, a simplified form of the LeMessurier equation can be written by assuming that: (1) the total gravity load associated with the sway buckling of a story is equal to the sum of the buckling loads of all columns in a story that provide the story sideway resistance; and (2) the individual column buckling load is calculated, using the effective length factor obtained from the alignment charts. The simplified equation has the form:
rr2EIi K~ -
EP
L~P, EP~k
(23)
Figure 6 shows a comparison of the effective length factor K for column CD of a leaned-column frame evaluated using eqns (21) and (22), respectively. Good agreement is generally observed. The use of eqn (21) or (23) is recommended by the author over the alignment charts method since they are applicable for cases such as the leaned column problem and for cases in which the axial forces in the columns of a story differ significantly. If a computer program for buckling analysis is available, a still more accurate evaluation of the K factor can be obtained using eqn (13), where Pe is the r u l e r buckling load of column i, and Per is the axial force in column i at incipient buckling of the frame. It is interesting to note here that, during the course of the development 8
Ct.P
p
+B [~ l_ LclIc 1 I. 6 ~L~
L e M e s s u r i e r K eqn(21)
---- Mo,~i,ie,J K
Itc
. ~
I.'7 ~-~ .-'J~"
.~4°1
_.-'.~'-
I I
~
2.0 I I
I G = ([c/Lc) I ( [ b / 2 L b )
Fig. 6. Comparison of effective length factor K for a leaned column system.
20
I4<. F. Chen
of the 1986 LRFD interaction equations (eqns (88) and (89)) tbr the design of beam-columns to be discussed in Section 7. much discussion has been focussed on the need and validity of using the effective length factor K in these equations. Although attempts were made to tbrmulate general interaction equations without using K, it was found that this was almost impossible if the interaction equations were to be versatile enough for a wide range of slenderness ratios and load combinations. The final decision was to retain K in these interaction equations. The AISC specification describes a procedure by which the effective length of factor K is determined from two alignment charts 17 that corresponds to the two cases as given in eqns (15) and (16). This will be discussed later in Section 7. Once the K factor for an end-restrained column is found, the strength of that column can be evaluated from the column strength formulas presented above for a pin-ended column by replacing A.0 by A~ where A.~. = K~0
(24)
4 FLEXURAL MEMBERS When an ideal I beam is loaded in the plane of the web. in-plane deformation will occur at commencement of loading and increase as the applied loads increase. As the magnitude of the loads approaches a certain critical value, out-of-plane deformation and twisting of the beam will increase rapidly and the beam will be in a state of instability. The critical load is referred to as the lateral-torsional buckling load of the beam. It is obtained as the eigenvalue of the bifurcation problem of the beam. The theoretical critical moment for the elastic lateral-torsional buckling of a beam for various support and loading conditions is given 9as Mc,- = C, ~
C2g + C3j +
Ceg + C3j + ~
+ n2EC w (25)
where g = distance from shear center to the point of application of the transverse load taken positive when the load is below the shear center; j = S + (1/21,)fAY(X 2 +y2)dA. in which
Design of beam-columns in steel frames in USA
21
S = distance from centroid of the cross section to the shear center; J = torsional constant; C,, = warping constant; /v = moment of inertia about the y-axis (weak axis); C~, C2, C3 = coefficients which are functions of loading and support conditions. 9 Beams used in practice usually fail at loads below Mcr due to the presence of geometrical and material imperfections. For such beams, lateral deformation and twist commence at the start of loading, and increase as the magnitude of the load increases. The maximum load is obtained from the peak point of the load-deflection curve, which is referred to as the ultimate load of the beam. To obtain this ultimate load, a load-deflection approach must be used and a numerical analysis is inevitable. Analytical and experimental studies to determine the ultimate loads of beams are available in the literature.2,. 22 In general, the presence of residual stresses and geometric imperfections will decrease the ultimate strength of the beam with the effect of residual stresses causing early yielding. Depending on the types ofloadings and residual stress distributions, the ultimate strength will be affected differently. The effect of geometrical imperfections aggravates the deformation and so the beam will change into an unstable state at a lower load, being more pronounced for slender beams. The present AISC Specifications '-2 for PD adopted an empirical formula based on test data of the form: Mo,, =
1.07
(L/rv)~/F,'~-,] . M-60 J 1rip, < Mp,,
(26)
where = L = = = p.v
plastic moment capacity about the x-axis (strong axis) (in kips); unbraced length of beam (in); radius of gyration about y-axis (weak axis) (in); yield stress (ksi).
This represents the ultimate lateral torsional capacity of lateral unsupported beams. Equation (26) is plotted in Fig. 7. In the same figure, the numerical results obtained by Galambos 23 and Vinnakota 24 are also plotted. It can be seen that eqn (26) is not conservative for Vinnakota's results. However, in Vinnakota's study, the worst effect on the maximum strength of the beam was chosen. The initial crookedness at midspan was taken as
W. F. Chen
22
8 E 0 o
"I=
L .~
c
¢:
...4.
/ 0
D'~ C
/
....3
"F..
~J ¢~ "U
\/
C
0
c
.-
>
L
/
/ o
0
!
O_ 121 ~
I
E=
k
Oc
E
e,
I
!
I
.'!
0
0
6
0
<
I
0 0
Xdl,.~
xn~
C,
r.:
23
Design of beam-columns in steelframes in USA
O.O01L in both principal planes, and the effect of residual stress was included. Real sections have additional strength reserves over design values; for example, the actual yield strength of the material is usually higher than its nominal value and strain hardening may set in before failure. Based on a study reported by Yura et al., 25 the LRFD Specification3 adopted the following expressions for the ultimate moment capacity of beams: For c o m p a c t sections (sections in which flange and web local buckling are precluded) defined as br
65
(27)
2,-7 <
(28)
tw
where bf tf hc tw Fy
= = = = =
width of flange (inches); thickness of flange (inches); depth of web clear of fillets (inches); thickness of web (inches); yield stress of the material (ksi);
the limit stage may be lateral-torsional buckling. Defining (Fig. Lp = 300rv
8): (29)
rvXl
Zr - (F.~-Z]Wr) v/(i -I- V/(1 + X 2 ( F v - Fr) 2)
(30)
Lb = unbraced length of member, where X,rv Sx G A Fr
-= = = --
Sx
-
,
X2-
~
(31)
radius of gyration about the weak axis (in); section modulus about strong axis (in3); shear modulus (ksi); cross-sectional area (in2); compressive residual stress in flange, taken as 10 ksi for rolled shapes and 16.5 ksi for welded shapes;
W. F. Chen
24
,..A e-,
Z C
0"
L
u ql r~ I,IJ
C 0 -J
E
13-
\
-J
\
\
L .-J
\
\
u (3 r~
E L l::x, ,,.j
u
¢'3
g5"
nl, N ' a : ) u ~ l . s ! s a J
II~JnXal~.
eu~uJoN
Design of beam-columns in steel frames in USA
25
then, i f L b ~< Lp M~_,.
=
IfLr, <
(32)
Mpx L b
~< Lr
< Mpx
(33)
= j'1-75 + I'05(MI/M2) + 0"3(MI/M2) 2 < 2.3 [1 for unbraced cantilevers
(34)
where
Cb
in which M~ is the smaller and M2 is the larger end-moment in the unbraced segment of the member; Mj/M2 is positive when the member bends in reverse curvature, negative when the member bends in single curvature. Mr
= Sx(Fy -
fr)
(35)
IfLb > Lr
Mu.,
27
=
CbMocr
=
Cb~bb
~.GJ+
71"-
vECw] <-
where Moor = elastic lateral torsional buckling load of a simply supported I beam under pure equal end bending moments. The equations given above are applicable only to compact nonhybrid I sections in which yielding or lateral-torsional instability is the limit state. The applicable expressions for other sections, including symmetric box sections, solid rectangular bars, hybrid sections and tee and doubleangle sections, are given elsewhere. 3 Figures 9(a) and (b) show a comparison of the variation of the nondimensionalized ultimate moment resistance M,/Mp as a function of the slenderness ratio Lb/ry of a beam using the LRFD as well as the ASD and PD design approaches. The beam is a W18 × 76 section with a yield stress F~. = 36 ksi and a compressive residual stress in the flange Fr = 10 ksi. The Young's modulus E was taken as 29 000 ksi and the shear modulus G as 0-385E. Figure 9(a) corresponds to the case in which M~/M2 = - 1 (i.e. Cb -- 1.0), and Fig. 9(b) corresponds to the case in whichM~/M2 = 1 (i.e. Cb = 2"3). Note that the ultimate moment resistance of the beam is larger for doublecurvature bending than for single-curvature bending.
26
W. 1~: Chen
Lp/ry = 50 1.00 M u / Mp 0.80
PD 0.60
0.40
0.201 W 18 X 7 6 :
SINGLE CURVATURE BENDING Cb = 1 i 50
2=5
J 75
L 100
~ 125
i 150
{ a )
Ll~/ry= 50
J 175
200 =
Lb/ ry
L r l ry = 153
1.00 M u / Mp 0.80
0.6C
Lc/~ri= 5 3 . 6
"
Lu/ ry=
0.40
0.20
W 18 X 7 6 :
DOUBLE C b
=
CURVATURE BENDING
2.3
I
£.
I
I
I
50
100
150
200
250
(b)
,-.I
300
I
350 Lb/ ry
Fig. 9. Comparison of beam curves tbr ASD, PD and LRFD.
4oo"
Design of beam-columns in steelframes in USA
27
5 D E S I G N I N G FLEXURAL MEMBERS WITH LRFD, SSRC AND ECCS APPROACHES It should come as no surprise, at this time that designing for lateraltorsional buckling is a complex and challenging problem. This is because the resistance of the beam against lateral-torsional buckling is dependent on many factors, such as the geometry and end conditions of the beam, the amount and nature of bracing, the type and manner of loading, etc. Nevertheless, there are two undebatable facts. A beam with a low slenderness ratio should be able to develop its full plastic moment Mp, and a beam with a large slenderness ratio should behave in a fully elastic manner, so that the elastic critical moment Mcr represents the ultimate moment capacity of the beam. The main problem is that the behavior of a beam with intermediate slenderness ratios is neither fully plastic nor fully elastic, hence, neither the plastic m o m e n t Mp nor the elastic critical moment Mcr can be used to represent the ultimate moment capacity Mu of the beam. The ultimate moment capacity Mu of a beam with an intermediate slenderness ratio should fall between Mp and Mcr and its value should provide a smooth transition from Mp to Mcr. In most specifications, the choice of this transition moment is rather empirical. For example, with the ASD method, a parabola is used to represent this transition range, and with the LRFD method, a straight line (eqn (33)) is used for this transition range. In the discussions that follow, two other empirical methods that represent the beam resistance in the transition range will be presented.
5.1 S S R C approach
In the SSRC approach, it is assumed that the stability behavior of beams is analogous to the stability behavior of columns, hence, the lateraltorsional buckling resistance of a beam is represented by the SSRC multiple-column curves. Using the Rondal-Maquoi approach form (eqn (11)) with P/Py replaced by Mu/Mp and A,o replaced by Ab, where Ab = v/np/Mcr, one obtains M. Mp
(1 + r / + X~,) - v/[(1 + r / + Ab2)2 -- 4A~,] 2A~
where 17 is defined the same as for columns in eqn (11).
(37)
28
W. I~2 Chen
5.2 ECCS approach In the E C C S a p p r o a c h , the n o m i n a l m o m e n t c a p a c i t y for the t r a n s i t i o n r a n g e is r e p r e s e n t e d by
( ,)in
Mp -
1 + A2b"
(38)
w h e r e n is a coefficient t h a t varies f r o m 2.5 to 1-5, d e p e n d i n g o n the criteria set by the v a r i o u s n a t i o n a l c o d e - w r i t i n g b o d i e s in W e s t e r n E u r o p e a n d Ab is the b e a m ' s s l e n d e r n e s s p a r a m e t e r , d e f i n e d as An =
V/ Mp/Mcr •
5.3 Design example and comparisons C a l c u l a t e the n o m i n a l m o m e n t c a p a c i t y o f a W16 x 36 s e c t i o n b e n t a b o u t its s t r o n g axis u s i n g the following: 1. A I S C L R F D a p p r o a c h . 2. S S R C a p p r o a c h . 3. E C C S a p p r o a c h . T h e b e a m is s i m p l y s u p p o r t e d a n d s u b j e c t e d to a u n i f o r m m o m e n t . Use E = 29 000 ksi, G/E = 0-385, F~, = 36 ksi, Lb = 150 in, Fr = 10 ksi.
Solution." W16 × 36 s e c t i o n properties: A = 10.6 in 2, At- = 3 in 2, d = 15-86 in, bf = 6.985 in, rT = 1-79 in, S, = 56.5 in3,!,, = 24.5 in4, r, = 1'52 in, J = 0-545 in 4, Z , = 64.0 in 3, Cw = 1450 in 6. C~, = 1.0 ( c o n s t a n t m o m e n t )
Mp = Z,F,. = 2304 in kips Me,- = CbMocr "-Moor
Lb
EI,.GJ +
EI,.ECw]
= 2765 in kips
1. AISC LRFD equation Lp =
300r,. /--~- -
76 in
I V/ L,. = ?iir~,2X F,. [1 + x"[1
:
219in
29
Design of beam-columns in steel frames in USA
Since Lp < Lb < Lr, therefore, by using eqn (34) with Mr = Sx(Fv we get from eqn (32): M~ = Cb Mp - (Mp - M~) L-~-L--~p
Fr),
= 1544inkips
2. S S R C equation
A,b = Mv~M~~ _ ~//23042765 - 0.913 Using SSRC curve 2, that is, where a = 0-293 in eqn (11) for r/, we have, from eqn (37): Mu = (1 + 1/+A,~,)- ~/[(1 + q + A,~,)2 - 4A,~] Mp = 1534in kips
3. ECCS equation
A,b =
M~_
0.913
Using n = 2.5 in eqn (38), we have M,. =
1
1 -t--/~"J
Mp = 1893 in kips
It is seen that the L R F D and SSRC equations give a comparable result, while the ECCS equation significantly overestimates the flexural strength of beams.
6 S E C O N D A R Y P-DELTA E F F E C T S One of the major differences between a beam and a beam-column is that, in addition to primary deflections and moments, there are secondary deflections and moments present in a beam-column due to the axial force in the beam-column acting on the primary deflections. In general, two types of secondary effects can be identified: the P-6 effect a n d the p-A effect. These secondary effects cause the m e m b e r to deform more and thus increase the stresses in the member. As a result, they have a weakening or destabilizing effect on the structure.To ensure a safe design, these effects must be considered.
30
W. F. Chen
6.1 P-6 effect Consider a beam-column, as shown in Fig. 10, with joint translation prevented; the forces M~, M2, Q and q will produce primary moment MI and primary deflection y~. The axial force P will act on the primary deflection to produce an additional momentMn and deflectionyn. These additional effects are the result of the so-called P-6 effect. This effect will increase the member instability; hence, it is also referred to as the member instability effect. The total moment along the beam-column will then be M = Ml+Mn
(39)
and the total deflection is Y = YI+Yn
(40)
In order to ensure a safe design, it is necessary to sort the maximum moment Mmax in the member. This maximum moment can be found by solving the differential equation of the beam-column with the proper boundary conditions at the ends. However, for design purposes, it is more convenient to use a simplified approach to get Mmax. By assuming that (i) the secondary moment Mn is in the form of a half sine wave, and (ii) the maximum deflection Ymax (= 6 = 61 + 6n) occurs at midspan, we can write Trx
-EIy;~ = MI, = P6 sin ~ -
(41) Q
P Mj ~
I
MI~
M2
M I = f(M I ,M2,Q,q,x)
I
M2 p
MII=py Fig. I0. P-6 effect,
Design of beam-columns in steelframes in USA
3I
Integrating eqn (41) twice and enforcing boundary conditions gives
Yn - E1
sin ~-1rx
(42)
from which the secondary deflection at midspan is P 6n = Ynlx=L/2 = 6 ~
(43)
since 6 = 6~ + 6 .
(44)
Substitution of eqn (43) into eqn (44) and solving for 6 gives
6 = (l_l-p/p)
6,
(45)
Now, further assuming that the maximum primary moment occurs at or near midspan, one can write Mmax = MImax+ P~
(46)
Upon substitution ofeqn (44) into eqn (45) and rearranging, one obtains m~x
= [ I +- v/P/Pe] P/Pe
(47)
6lee Mlmax
(48)
where v/-
l
Defining
Cm = 1 + ~P/Pe
(49)
equation (47) becomes
Mmax = (1 £ ~ / P e ) M'max = ArM'max
(50)
The term in parenthesis in eqn (50) is referred to as the moment amplification factor, Av. For beam-columns with different end conditions, the Euler buckling load Pe in eqns (49) and (50) should be replaced with the corresponding buckling load Pek = zr2EI/(KL) 2 to reflect the effect of end restraint on the buckling strength.
W . F . Chen
32
Figure 11 shows the values o f ~ and Cm for several different load cases and end conditions. Note that, because of the assumption used in writing eqn (46), the definition for Ig in eqn (48) is only applicable to the two simply supported cases (Cases 1 and 4). For the other cases in which the maximum first-order moment occurs at the end(s) (Cases 2, 3 and 5) or occurs both at midspan and at the ends (Case 6), the exact values of the maximum moments, taking into consideration the effect of the axial force P, are first evaluated; the values for ~, for these cases are then obtained from calibration using eqn (47). A special case arises when there are no transverse loadings on the beam-column. If the beam-column is subjected to end moments only. the Cm value for design is redefined as Cm = 0 " 6 - 0"4(M1/M2) >~ 0"4
Case
(51)
~
Cm
1 l[IlllllllllllllIllNl _
I
= illIlllIllI[lllIlllll]~,~
I I
1.0
I
!
-0.4
1-0.4 P/Pek
-0.4
1-0.4 PIPek
-0.2
1-0.2 PIPe,
-0.3
1-0.3 P/Pek
-0.2
1-0.2 P/r'ek
6
I~io 1 1 V ~ h . ~ ~f .z a n d C... f a c l o r s for b e a m - c o l u m n s undcr transverse loading.
Design of beam-columns in steel frames in USA
33
where Mi is the smaller and M, is the larger end moment. MI/M2 is positive when the member bends in reverse curvature, and negative when the member bends in single curvature. Note that, from eqn (51), the Cm value can be as low as 0-4, which means the moment amplification factor can be less than unity (Fig. 12) according to the present AISC Specification.2 This is not physically meaningful, because if the amplification factor is less than 1-0, the maximum moment of the beam-column occurs at one of the ends rather than within the span, and we should take A v = 1"0 instead, rather than taking the value that is less than 1.0. An improved AF factor has been proposed recently by Duan et aL 26An evaluation of the Cm factor, as used in the AISC LRFD specification, has been reported by Zhou and Chen. 27
6.2 p-A effect When lateral forces 12H act on a frame, the frame will deflect laterally until an equilibrium position is reached (Fig. 13(a)). The corresponding lateral deflection may be calculated on the basis of the original configuration and is referred to here as the first-order deflection and is denoted by A~. Now, if in addition to ZH, vertical forces Y'.P act on the frame, these forces will interact with the lateral displacement A~ caused by E H to drift the frame further until a new equilibrium position is reached. The lateral deflection that corresponds to the new equilibrium position is denoted by A (Fig. 13(b)). The phenomenon by which the vertical forces ZP interact with the lateral displacement is called the P-A 6
i
//
I
o
0
0.2
0.4
0.6
0.8
I .0
P
Fig. l]. M o m e n t magnification factor AI:.
I4(. F. Chen
34
:rH
/ /
/ /
/
I Ih
!
r ///,
/1
illl
{a) Fiqst ~ o r d e r
__A__
Analysis
~P
YH
1i..
~P
---
-77
/ /
/
I
I
I /II
_
1
III
(b) S e c o n d - o r d e r A n a l y s i s
Fig. 13. P-A effect.
effect. The consequence of this effect is an increase in drift and an increase in overturning moment. Since these additional deflection and overturning moments have detrimental effects on the stiffness and stability of the frame, they should be considered in design. In order to determine accurately the final deflection A and the actual moment M taking into account the P-A effect, a second-order analysis based on the deformed geometry of the frame is necessary. Second-order analysis usually entails an iterative process. However, for design purposes, it is more convenient to use simplified approaches without resort to iteration to estimate A and M. Some of these approaches are given below.
6.2.1 Story magnifier m e t h o d 2~-3° By assuming that (i) each story can behave independently of other stories, and (ii) the additional moments in the columns caused by the PA effect are equivalent to that caused by a lateral force o f £ P A / h , where h = story height, the sway stiffness of the story can be defined as
Design of beam-columns in steelframes in USA
horizontal force SF = lateral displacement
Y~H
Y-,H + Y_,PA/h
A1
A
35
(52)
Solving eqn (52) for A gives [
1 ~PAI
A = 1
AI
(53)
Y_,Hh
Thus, it can be seen that the final deflection A can be estimated from the first-order deflection A~ by multiplying the latter by an amplification factor (the quantity in square brackets). Since the story sway moments (moments induced as a result of swaying of the story) are directly proportional to the lateral deflections of the story, one can write M =
( 1 - Y,P-AI/Y, 1 H h ) Mlsway = AFMlsway
(54)
where M = maximum end moment accounting for the P-A effect; M~sway = maximum first-order end moment as a result of swaying of the story; Ar = moment amplification factor. A special case arises when swaying of the frame is due to asymmetry of the applied vertical loads (Fig. 14(a)) or to asymmetry of the frame itself(Fig. 14(b)). If this is the case, a fictitious support can be introduced to prevent the frame from swaying. As a result, a holding force will be developed in the support. The horizontal force that balances this holding force will then be used in place of E H in eqns (53) and (54) to obtain the final deflection and moment (Fig. 15). If swaying of the frame is due to both applied horizontal forces Y,H and asymmetry of applied vertical loads and/or frame geometry, a similar approach can be used. The frame subjected to these forces is analyzed with a fictitious support to prevent it from swaying. The force that balances the horizontal reaction of the support is used in place of Y,H for the amplification factor in eqns (53) and (54). The story magnifier method works for frames with stiffbeams in every story so that a point of inflection occurs in every column in a story. 6.2.2 Multiple-column magnifier m e t h o d 2°
The multiple-column magnifier method, also known as the modified effective length method, is a direct extension of eqn (45), under the
36
W. F. Chen
l
I
lrp
_ ~_ A
/
(a)
/~ ~-.'~
~
-
t
/ /
(b) T7
Fig. 14. P-D effect arising from load/frame asymmetry.
l'"
11/5.-:,:::-0-1,;o° I/ &
AI
/
I
IIEP
_12"
M=/'l~h~l ) MIs"aY
~_R
===~
I co,oooP Lj Eo,,,oo.o,= I/ L ""'" k
Fig. 15. Procedures to determine the second-order deflection A and moment M for asymmetric applied load/frame. postulation that, w h e n instability is to occur in a story, all c o l u m n s in that story will b e c o m e unstable simultaneously. As a result, the term P/Pe in eqn (45) is replaced by Ir,P/lC,Pe where the s u m m a t i o n is carried through all c o l u m n s in a story. Using the same a r g u m e n t as in the story magnifier m e t h o d that the sway m o m e n t s are directly proportional to the lateral deflections of the story, the m a x i m u m e n d m o m e n t M a c c o u n t i n g for the P-A effect takes the form as follows:
Design of beam-columns in steelframes in USA
1 p
M = 1
Mlsway = AFnlsway
37
(55)
EPek
where Mlswayis the m a x i m u m first-order end moment as a result of swaying of the story, and AF = 1/(1 -- (Y,P/ZPek)) is the moment amplifi-
cation factor. If the P-D effect is not significant, the values of the moment amplification factor evaluated using eqns (54) and (55) are closely comparable to each other.
6.2.3 Negative brace method 31 In the negative braced method, a fictitious pin-ended diagonal bracing member with axial stiffness EA, given by EA -
-(£e/h)L cos20
(56)
where h = story height, L = length of bracing member, a n d 0 = angle of inclination of the bracing member, is inserted in every story. The structure with the negative braces is then analyzed by a first-order analysis to determine the moment. This m o m e n t corresponds to M in eqns (54) and (55). In order to account for the decrease in column stiffness due to the presence of axial force, a flexibility factor, y, ranging from 1-0 to 1.22, can be inserted into the term Y,P (i.e. E~P) in eqns (54) to (56) to give a better estimate of M. The lower bound for ~, (i.e. y = 1) corresponds to the case in which the columns remain more or less straight after deflection, and the upper bound fory (i.e. y = 1.22) corresponds to the case in which the columns are on the verge of buckling, with the deformed shape approaching that of a sine curve. The above approaches give reasonably accurate results if the moment amplification factor is under 1.5. If the P-A effect is significant, the approach suggested by LeMessurier t9 is recommended. In his approach, the amplification factor is defined as 1
AF = 1-
EP
(57)
EpL - Z C L P
where ZPL = EHh/Ax and CL is as defined in eqn (21). The present AISC Specification,2 based on the model shown in Fig. 16
38
W. F Chen
®
+
t,--
.,,..
~-I
g
o
<
fl
L
t._
8 H
-r
Design of beam-columns in steelframes in USA
39
and assuming a sine curve for the P-A moment, developed a moment amplification factor based on the procedure outlined in Section 6.1 as 1 -
0.18--
AF =
P (58a)
p
Although eqn (58) is suggested in the AISC Specification, the specification recommends the use of Av
-
0.85
(58b)
p
Pok for the design of a sway-permitted frame. Equations (58a) and (58b) are unconservative if the P-A effect is large.
7 DESIGN INTERACTION EQUATIONS FOR BEAM-COLUMNS A beam-column constitutes one of the major components of a structural frame. Because of its importance, analytical and experimental investigations of the beam-column problem have never diminished over the years. Various analytical techniques and experimental verifications of beamcolumns have been presented and summarized in the Chen and Atsuta's two-volume book, t t. 22 Chen and Cheong-Siat-Moy's review paper 32 and Chen and Lui's state-of-the-art papers 33.34 and books. 35.36 It should be noted that the exact prediction of beam-column behavior and strength is most complex in the inelastic range. The resulting differential equations of equilibrium must be written with respect to the deformed geometry of the member, which is not known in advance. The extent ofplastification of material in the member is also not known in advance. As a result, iterative procedures are needed in a numerical analysis. Even with simplifying assumptions, the differential equations are often intractable, and recourse to numerical techniques is usually inevitable. Although with the aid of computer technology, numerical analysis does not pose any significant problem to an analyst, it nevertheless is rather unappealing to a designer. For design purposes, the strength of a beam-column is usually represented by an interaction equation of the general form: ~'
M~,'
< 1.0
(59)
W. F. Chen
4(1
where P, Mx, M>, are the applied axial load and moments and Pu, M,~,, Mu,. are the ultimate axial load and moment resistances of the columns and beams, discussed in Sections 3 and 4, respectively. Depending on how the loads are applied, a beam-column problem can be related to one of the following three categories: (a) in-plane beam-columns; (b) lateral torsional buckling of beam-columns: (c) biaxially loaded beam-columns. The recommended and proposed interaction equations to describe the behavior and strength of these classes of beam-column are summarized as follows.
7.1 In-plane beam-columns In this type of beam-column, the moment is applied about the y-axis so that in-plane stability will occur. Out-of-plane deformation for this type of beam-column is not likely even without bracing in the y-direction, since both the applied axial force and bending moments will tend to destabilize the member in the x-z plane (Fig. 17).
7.1.1 SSRC approach ~'37 At braced points: p~
+ 0-84 M,p.II < 1.0, M < Mp,
(60)
P which is valid for 0-4 < PII "<< 1.0 and
M
= Mp,.
(61)
P which is valid for0 < p, < 0.4
Between braced points." P CraM Pu, + -M~;I;-{i-'-P/P~,Ii < 1.0 where P = applied axial force:
(62)
Design of beam-columns in steel frames in USA
~
41
My M
x
Z J
Mx
My Fig. 17. Biaxially loaded beam-column.
yield load; maximum applied (primary) moment; plastic moment about weak axis; ultimate axial load producing failure in the absence of bending moment, about the y-axis, evaluated using K from alignment chart (Section 3.2); moment reduction factor (Section 6.1); Cnl Mu|' --" bending moment producing failure in the absence of axial load, about the weak axis (for weak axis, this moment capacity is equal to Mr,,,); Euler buckling load about the weak axis considering K-factor. PCl,
M= Mr,y PU)'
7.1.2 Kanchanalai and Lu approach 38 In their study, theoretical solutions were obtained for sway and nonsway beam-columns bent about their weak axis. The resulting interaction curves were curve-fitted to obtain the interaction equation as follows: P M PL,~ + mB3 ~
< n
(63)
42
W. F. Chen
where Pu~ = ultimate axial load capacity in the absence of b e n d i n g moment, evaluated using K = 1; Muy = Mp.v = full plastic m o m e n t about the y-axis (weak axis); B3 =
1
EPAI 1 - 1"2 EHh
are constants such that the n o n l i n e a r ultimate resistance curve can be approximated by three linear interaction equations, as follows:
m and n
P B~M + 1-00 . - 7 7 -<< 1-0 Pu~
(64)
EPAI 1 which is valid for XHh > and P B3M Pu~ + 0.85 ~ < 1.0
(65)
P B3M Pu--~+ 1-75 ~ < 1-75
(66)
Y~PAI 1 which is valid for - EHh < 3 Equations (64) to (66) are valid only for the m i n o r axis b e n d i n g case, but K a n c h a n a l a i and Lu suggested that values for m a n d n can be obtained in a similar m a n n e r for the major axis b e n d i n g case. If the m o m e n t is applied about the x-axis a n d full lateral bracing is provided in the x-direction, the m e m b e r will then buckle in the y-z plane (Fig. 17). 7.1.3 AISC and SSRC approach 1.9.37 At braced points." P
PZ + 0.85
M
< 1.0, M < Mp,
P which is valid for 0-15 < ~o~, < 1.0 and
(67)
Design of beam-columns in steel frames in USA
M = Mpx
43 (68)
P which is valid for 0 < ~,. < 0.15
Between braced points: P
--
e~,.
+
CmM M,x(1 - e/eex)
<
1.0
(69)
where the subscript x refers to member strength about the x-axis, and Mux =Mpx.
7.1.4 Cheong-Siat-Moy and Downs approach 39 Based on a curve-fitting of their theoretical analysis of beam-columns bending about the major axis, the following interaction equation is proposed as P
M
Pu-~+ flAy ~
< 1.0
(70)
where
Pu i = ultimate axial load capacity of column in the absence of bending moment, evaluated using K = 1; MHX = Mp.,. = full plastic moment about the x-axis (strong axis); = 0.9 + 0"IAF; in which fl = ratio of second-order deflection to first-order deflection, and AF = moment amplification factor.
Av -~
1
1
Y'PAI
< 1"5
EHh Equation (70) is a nonlinear interaction equation since/3 is a function of the amplification factorAv. It is valid for combined gravity and lateral loads with no gravity moments present. It is unsafe for the pure gravity loading case. If gravity moments coexist with moments caused by lateral forces, eqn (70) must be modified as follows:
P Cmfl (AvM+Mg) Po-~+ l - - ~ P e ~'/~.,. < 1-0
(71)
where Mg = moment due to gravity loads; Cm is defined in eqn (51); however, the value Of Cm/(1 -- P/Pe) must be greater than or equal to 1.0 in eqn (71); fl,Av defined as in eqn (70).
44
W. F. Chen
Although eqn (70) was derived on the basis of major axis bending, it has been demonstrated 3° that it also applies for minor axis bending, provided that Muy is used instead of M~. An interaction formula that applies for both major and minor axis bendings and for relatively large P-A effects was proposed by LeMessurier. 19 P
M
p~ +Av ~
< 1.0
(72)
where ultimate axial load in the absence of bending moment, evaluated, using K as defined in eqn (21); AF = moment amplification factor defined in eqn (57).
Ptl
7.2 Lateral-torsional buckling of beam-columns If the beam-column in Fig. 17 is not laterally braced, both in-plane and out-of-plane deformations may occur due to lateral torsional instability. The interaction equations for this case are as follows.
7.2.1 AISC and SSRC approach 1.9.37 At braced points: as eqns (67) and (68). P M p~m+ 0-85 ~
< 1.0,
M < mpx
(73a)
P which is valid for 0-15 < ~ < 1.0 and (73b)
M=Mpx P
which is valid for 0 < ff~ < 0.15
Between braced points: P
--+ Puy
CraM -<< 1.0 Mu.,.(1 - P/Pex)
(74)
where M~_~ = [1.07 - (L/G) vIF~/3160]Mpx <,Mpx (see eqn (26) and Fig. 7).
Design of beam-columns in steelframes in USA
45
7.2.2 Vinnakota approach 24 Based on the numerical analysis of laterally unsupported I beamcolumns, Vinnakota attempted to verify the validity of eqn (74). He suggested that for beam-columns with no joint translation, the P-6 moment magnifcation term Cm/(1 -P/Peg) can be omitted provided that P,y is evaluated using SSRC curve 2 (Fig. 3) and M ~ is evaluated from his numerical results (Fig. 7). Before proceeding any further, it is of interest to scrutinize eqns (69), (73a), (73b) and (74). These equations are contained in the present AISC Specification 2 for the PD of steel structures. If the moment magnification factor fro/(1 P/Pe) is equal to unity, then the stability limit state (eqns (69) and (74)) will always control the design, as shown in Fig. 18. If the moment magnification factor is less than unity, then the maximum moment occurs at one of the ends rather than within the span. With this in mind, a slenderness ratio above which stability will always govern the limit state can be determined. Stability will always govern when -
Cm
1 -P/Pe
-
> 1"0
(75)
rearranging, one can write
(L)~ /TT2EA(Ip-Cm)
(76,
or, in terms of the slenderness parameter, ,~o >/ /~---Y ( 1 - Cm)
(77)
P/Py 1.0
~
n.(67)
ecln (69) or (74)N N with Crn/(yP/Pe) ~ equals unity ~.~.
1-0
M/Mp
Fig. 18. Comparison of A|SC resistance and stability interaction formulae.
W. F. Chen
46
Thus, eqn (76) or (79) gives the slenderness criterion for which stabilitywill always control. The stability interaction equation (eqn (69) or (74)) has been shown 3° to be unconservative for cases in which a column pinned at both ends exists in a frame (see Fig. 19). For such frames, eqn (69) or (74) will underestimate the P-A effect consequently, it is more advisable to use the approaches suggested by Cheong-Siat-Moy and Downs 39 (eqn (70) and LeMessurier 19 (eqn (72)).
7.3 Biaxiaily loaded beam-columns The loading for this type of beam-column is shown in Fig. 17. It represents the most general case of beam-columns, i.e. all other cases can be represented after elimination of one or the other applied moments.
7.3.I SSRC approach 9.37 The interaction equations are as follows.
At braced points." P
Mx
p~m+ 0.85 ~
My
+ 0.85 ~
< 1.0
(78)
Between braced points: P + CmxMx + Cm_vMv • 1.0 Pu Mux(1 - P/Pc.,-) M.~(1 - P/P~y)
(79)
where MUA"~Mll.V = ultimate bending moments of the beam about the strong
and weak axes, respectively; P~x, P~v = Euler buckling loads about the strong and weak axes, respectively, considering the K factor; Pu = ultimate load of an axially loaded member, considering the K factor.
7.3.2 Chen approach 2~ Based on the theoretical studies of nonsway biaxially loaded beamcolumns, nonlinear interaction equations are presented. Details of the development are given in Chapter 13, Vol. 2, of the book by Chen and Atsuta 2~ or originally by Tebedge and Chen. 4°
Design of beam-columns in steelframes in USA
P1
47
P~
T__H
PINNED COLUMb
///
)
~
Fig. 19. Frame with column pinned at both ends.
At braced points." < 1.0
(80)
where Mx = required flexural strength about the x-axis (strong axis), and My = required flexural strength about the y-axis (weak axis).
Mpcx = 1-18Mpx[l - (P/Pv)] < Mp.,.
(81a)
1.19Mp.v[1 - (p/py)2] < Mpy
(81b)
Mpcy =
in which MpxandMpy are the plastic moment capacities about the strong and weak axes, respectively. a = 1-6
(P/e,,) 21n(P/p,,)
(82)
for W-sections with width/depth ratio 0.5 < bf/d < 1.0.
Between braced points: (
\ M-~x]
t~ + , CmvMv)l ~
< l-0
(83)
where Crux and Cmy = equivalent moment factors, defined in eqn (51). /3 =
0.4 + P/Pv + bf/d > 1.0 for bf/d > 0.3 1-0 forbf/d < 0.3
(84)
48
W. F. Chen
Me and My are evaluated on the basis of a first-order analysis, and Mu~x = Mu~ [1 - (P/Pu)] [1 - (e/eex)l
(85)
(P/&)! 11 - (P/Pey)l
(86)
Muc.v = Mu.v[1 in which
M~.~ = [1.07 - (L/rv)~ff~J3160]Mpx << Mpx
(87)
where bf is the flange width, and d is the member depth. The validity of eqns (80) and (83) has been checked by a number of researchers and is summarized in Chapter 13, Vol. 2, of the book by Chen and Atsuta) ~ An experimental verification of these two equations was given by Anslijn and Massonnet. 4j Eighty-one beam-columns were tested and the experimental results were compared with the results predicted by the interaction equations. Excellent correlation was reported. 42 The most recent analytical and experimental verification of these two equations was given by Cai et aL 43 Further discussion of the above interaction equations can be found in the review papers by Chen and Cheong-Siat-Moy 32 and Chen and Lui, -~3and the recent book by Chen and Lui. 36
7.3.3 LRFD approach 3 The AISC L R F D Specification gives the following bilinear interaction formulas for biaxially loaded beam-columns:
jO
which is valid for ~ > 0.2 O~ru and 20cP,,
+
M,-
+
.,.. < 1.0,
(])bMux 4)hMuy
(89)
P which is valid f o r - - < 0.2 0cPu where subscripts x and y denote the strong and weak axes, respectively, and M = BIMNT + B2MLT
(90)
Design of beam-columns in steel frames in USA
BI
Cme
_
> 1"0
49
(91)
1 ----
eek
1
B2 -
~PA!
1
(92)
EHh
or alternatively
1 B2 -
1 Cm
Mr~rr =
ML+ = Pek =
EP = A~ = Y,H = h = Pu = Mu = ~t, = q~c =
EP
(93)
ZPek
defined in eqn (51); required first-order m o m e n t in member assuming there is no lateral translation in the frame (Fig. 20(b)); required first-order m o m e n t in member as a result of lateral translation of the frame only (Fig. 20(c)); Euler load, n2EI/(KL) 2, evaluated using K < 1.0 for the B~ factor application in eqn (91), and using K/> 1 for the B2 factor application in eqn (93) in the plane of bending of the frame corresponding to braced and unbraced frames, respectively; axial load on all columns in a story; first-order translation deflection of the story under consideration; sum of all story horizontal forces producing A~; story height; ultimate axial strength evaluated using the LRFD column curve with K factor presented in Section 3; ultimate moment resistance using the LRFD beam formulas presented in Section 4; resistance factor for flexural = 0-90; resistance factor for compression = 0.85.
With some modifications, Chen's equations are also included in the 1986 LRFD Specification as alternative interaction equations for nonsway beam-columns. Note that eqns (88) and (89) are bilinear interaction equations (the terms P/Pu, Mx/M~,.,Mv/M,y are combined linearly) and are applicable to both nonsway and sway cases, whereas eqns (80) and (83) are nonlinear interaction equations, which can be used to assess the strength of beam-columns more accurately; however, they are applicable only to nonsway beam-columns.
50
W. F. Chen
nO w
b-. Z "1"
v! ~3 QT.
.ira
o
~3 3~
it
W rr -am
Z e¢ 0
-g
Design of beam-columns in steelframes in USA
51
In order to use eqns (87) and (89), the flame must be analyzed twice, first as a nonsway case, followed by a sway case. The first-order moment obtained from the nonsway case is magnified byB~ to account for the P-6 effect, and the first-order moment obtained from the sway case is magnified by B2 to account for the P-A effect. Thus, both secondary effects are taken into consideration. The two magnified moments are then added together to produce the design moment in eqn (90). This approach is rather conservative since the two magnified moments may not coincide in an actual situation; nevertheless, this is a safe approach. In the development of the LRFD interaction equations (eqns (88) and (89), the following design guidelines were established: 44 1. The equations should be applicable to a wide range of problems such as strong- and weak-axis bending, sway and nonsway frames, laterally loaded columns, imperfect columns with various slenderness ratios and degrees of end restraint, and leaned column systems (Fig. 19). They should account for both inelastic behavior and second-order effects. 2. The equations should clearly distinguish between the load effects and the resistances so that second-order elastic analysis can be easily accommodated. 3. The equations should be based on the load effects obtained from second-order elastic analysis since, at the present time, secondorder inelastic analysis is not readily accessible for design office use.
4. The equations should not be more than 5% unconservative when compared to second-order plastic-zone solutions. 5. The equations should not need to consider strength and stability separately as in previous specifications, since, in general, all columns of finite length fail by some combination of inelastic bending and stability effects. 6. The same ultimate strength should be predicted for similar members. 7. Adjustment of effective length based on column inelasticity should be allowed for. With the above guidelines in place, the interaction equations expressed by eqns (88) and (89) were developed by curve-fitting to a set of numerical second-order inelastic flame analyses data generated by Kanchanalai. 45 Generally speaking, the interaction equations give a good representation for all strong-axis cases for 0 < KL/rx < 100. It is rather conservative for weak axis cases for 0 < K L / r v < 40, and it is moderatively conservative for both strong and weak axis cases for KL/ r > 120.46-49
W. F Chen
52
Note that, unlike the ASD and PD interaction equations in which both the yielding and stability interaction equations are needed in the design process, only one interaction equation is needed if the LRFD approach is used. The applicable equation is determined by the term P/OcPu. If P/q~cPu/> 0-2, eqn (88) is applicable, and if P/OcPu < 0.2, eqn (89) is applicable. Another feature of the LRFD approach that is different from the ASD and PD approaches is that the P-8 and P-A moment magnification effects are located independently, as is evident from eqn (89). Recall that, in the ASD or PD approach, if the member is subjected to sway, the factor Cm is taken as 0.85; therefore, the moment magnification factor is 0"85/(l'P/Pek) and this is applied to the total first-order moment of the member regardless of whether it is caused by gravity load (M~q.) or lateral load (MET). In addition to eqns (88) and (89), the LRFD Specification2 also recommends a set of nonlinear interaction equations in its Appendix, which are valid for nonsway members with end moments Mx and M~,. These equations are given by eqns (80) and (83), with the following definitions for Mpcand M~c. Mpcx = 1.2Mpx[1 -
(P/Pv)] < Mpx
(94)
Mpcy = 1.2Mpy[l - (p/p,,)2] < Mpy
(95)
Mucx = Mu~ [l - (P/O ~Pu)][ 1 - (P/P¢x)]
(96)
Mucy = Mu.v11 - (P/epcPu)111 - (P/Pev)]
(97)
The interaction equations (80) and (82) have a jump from a long to a short member, i.e. for a very short member, the stability interaction equation (eqn (83)) does not always reduce to the strength equation, eqn (80). For short-beam columns in which the weak-axis bending and axial force are dominant, and for double curvature bending, the LRFD bilinear interaction equations, eqns (88) and (89), may result in an overly conservative design. To this end, the following nonlinear interaction equation is proposed: 5°
(Mx~ 2 (My~ ~ M*~] + k M*~] < 1.0
(98)
where a = 1.2+
-0.03
~
>~ 1.0
(99)
Design of beam-columns in steelframes in USA
M * = Mu~ 1 -
~
53 (100)
1-
(101)
1.3+ 0.002 (-~x)
(102)
Mu*y = Muy
and ( = 3.0+0.035
~
~
1
) 1.0
(103)
in which B I is calculated from eqn (91), but with Cm evaluated, using the equation: Cm = 1 + 0"25(P/Pe) -- 0"6(P/Pe) ~/3 (M~/M2 + 1)
(104)
and B2 is calculated from eqn (92) or (93). In eqns (103) and (104), M t / M 2 is the ratio of the smaller to larger end moments. Its value is taken as positive if the member bends in reverse curvature, and negative if the member bends in single curvature. The ability o f e q n (98) to represent the actual inelastic behavior of a beam-column under biaxial bending and axial compression has been firmly established by Duan and Chen. s° Generally speaking, eqn (98) gives a better fit of the numerical data than do the AISC LRFD nonlinear interaction equations.
8 ANALYSIS AND DESIGN OF BEAM-COLUMNS IN FRAMES Discussion of the behavior and strength of beam-columns in this paper has so far focussed primarily on isolated members. In reality, most structural members exist as parts of a framework and their behavior and strength are, therefore, influenced by the behavior of other members of the frame. To illustrate some aspects of this interaction between the beams and columns in a frame, it is instructive to consider the following simple example. Shown in Fig. 21 is a simple braced frame consisting of a beam and a column. The beam is loaded by a uniformly distributed load w. After the full value ofw is reached, the column is then loaded by a monotonically increasing concentric load of P until failure. The behavior of the beam and column will now be studied as P increases from zero to its ultimate value. In performing the analysis, the following assumptions are made:
W.F. Chen
54
(applied after W has reached its
full value) W
Xb
El
El
Xc __
A
L Fig. 21. T w o - m e m b e r frame.
1. The axial force in the beam is negligible. 2. The axial force in the column is represented by P. Figure 22 shows the maximum beam moment, (Mmax)b~am, the maximum column moment, (Mmax)column, and the beam end moment, MRc, equal to --MBA (the column-end moment) nondimensionalized by the quantity wL2/8 as a function of the applied column force P nondimensionalized by the Euler load. The maximum beam moment is obtained as the larger value of the beam end moment M~c and the maximum in-span beam moment (MR)max. Similarly, the maximum column moment is obtained as the larger value of the column-end moment MBA and the maximum in-span column moment (Mc)max. The sign convention used in the figure is that a positive beam moment will cause tension in the bottom fiber of the beam, and a positive column moment will cause tension in the outside fiber of the column. Also shown in Fig. 22 is the maximum moment in the column as predicted by the AISC approach (eqn (50)), i.e. MAR = 0. (Mmax)colL,,,,,, =
=
Crll P
0"6 ]
l_i_ P
MBA =
MRA
0"6 - 0"4(MAB/MRA ) P
MBA
(io5)
Design of beam-columns in steel frames in USA
55
o.
c
E
E
\\ ~J
E
\ ¢5
.
i
i
I
I
t~
I
E O~ E 6 ¢1) q..
to
/
i
,,
[
= ..:
!
/
/
o.
~, T o
ei ¢q "7 o i
°! I I I I
o
56
W. F. Chen
The lower dashed line was obtained by using an effective length factor K = 1 in calculating Pe for the column, while the upper dashed line was obtained by using K = 0.839 (from alignment chart discussed in Section 3.2) in calculating P~. Observations regarding the behavior of the two-member frame can be made from the figure, and are as follows: 1. As the applied column force P increases, the magnitude of the maximum column moment and the maximum beam moment both increase. Nevertheless, the locations of these maximum moments vary as a function of P. The location of the maximum in-span column moment is given by rr 2k
xc-
L /
V
(106) P
1 -p~,
The location of the maximum in-span beam moment is given by Xb
L -
2
MBc _ L wL
L
2+8 -x
rt2P/Pe [ 3 _ 3rr x/p~_~ cot(rr p x / ~ ) + (rr2V,p ~
)]
(107)
In writing the above equation, the expression for MBc with kL = rrv/P/P ~was used. It should be noted that the above expressions for xc and Xb are valid only if the calculated values fall within the range 0 to L. If the calculated values fall outside this range, the location of the maximum moment is at the end rather than within the span of the member. Figure 23 shows a plot of the variation ofxc and Xb, nondimensionalized by the length of the member L as a function of P/P~. For the column, the location of the maximum moment shifts from the upper end to the middle of the member as P increases from 0 to P~. For the beam, the location of the maximum moment shifts from Xb = 0-562L at P = 0 to Xb = 0"5L at P = P~. . The change in values in (mmax)bcam, (mmax)column and M~c implies that the moment in the structure is being redistributed as the applied column force P increases. This change in moment distribution is revealed in Fig. 24 in which the bending moment diagrams for the frame at various values of P/P~ are shown. Note that there is a reversal in moment at the joint as P/P~ exceeds unity. In other words, when P/P~ < 1 (Fig. 25(a)), the beam is inducing
Design of beam-columns in steel frames in USA
57
Column 1.2 Beam
--~
1.0
0.8
P/Pe
0.6 Column
Beam
0.4
0.2
i 0.2
I 0.4
Xb L
0.5 or
0.6
0.8
1.0
Xc L
Fig. 23. Variation of the location of maximum beam and column moments with
P/Pe.
m o m e n t to the c o l u m n (Fig. 24(b)); however, as P/Pe > 1 (Fig. 25(b)), the b e a m is restraining the c o l u m n against buckling. At P/Pe = 1 (Fig. 24(c)), the b e a m end m o m e n t is zero, indicating that the b e a m is neither inducing m o m e n t to the c o l u m n nor restraining it from buckling. It is also worth noting that as P/Pe (Fig. 24(d), (Mmax)beam/(wL2/8) will exceed unity, which implies that if the designer is to rely on the beam to restrain the column, i.e. to design the c o l u m n with a K factor less than unity, the beam must be designed to carry a m a x i m u m in-span m o m e n t that exceeds wL~-/8 (i.e. the m a x i m u m m o m e n t of a simply supported beam). This is in sharp contrast to the c o m m o n notion that c o l u m n end m o m e n t s do not c h a n g e in a braced frame because the P-6 m o m e n t s are zero at the ends. This clearly shows that this is not true. Consequently.
58
W. F. Chen
.E 0
hE
E
h ~
"~
E
~I~
Design of beam-columns in steelframes in USA P
P
~
59
MBA
MBA
i P
PPe Fig. ZS. Beam and column interaction through the end moment MBA (a)
=
--MBc.
second-order effects will change beam moments as in unbraced frames. For example, when P/Pe = 1-1, the beam m o m e n t is 1-12 x (wL2/8), which will require a larger beam cross-section. 3. The AISC formula for the maximum strength of a column (eqn (50)) gives a good correlation with the exact result, if an effective length K = 0.839 is used in computing the critical load in the magnification term (the upper dashed line in Fig. 22). If an effective length factor K = 1 is used, then the formula will underestimate the column moment (the lower dashed line in Fig. 22). From observations 2 and 3, it can be concluded that, for braced frames, it is advisable to use an effective length factorK = 1 in the first term of the interaction equation (eqn (69)), but not in the second term where one should use an effective length factor K < 1.
9 SECOND-ORDER ELASTIC ANALYSIS FOR RIGID FRAME DESIGN The need for a sec~ond-0rder elastic analysis of steel frames is increasing in view of the recent advent of the AISC/LRFD Specification. As a simplified second-order elastic analysis, the specification uses two modifiers, B~ and B2 factors, 3 which are applied directly to the results of a first-order analysis. 22There are several other methods ~8-2°"29-31available for second-order analysis, based on approximately the same philosophy as the B~ and B2 factor analysis in the LRFD Specification. However,
60
W. F. Chen
these methods do not always give satisfactory results, even for design purposes, because they are rather approximate in nature and their application is mostly limited to rectangular frames with rigid connections and small displacements. Furthermore, it is cumbersome and tedious to calculate the modifiers applied to the result of a first-order analysis. In view of the present availability of workstations and future development of mini- and micro-computers, it is more convenient and realistic to use a computer-based method as long as the method can be easily implemented. 5~.52 Several computer-based second-order elastic analysis methods, simple enough for use in practical design, are presented as follows. At present, workstations do not have enough capacity to utilize ready-made programs based on finite element techniques, Hence, the method presented here uses the analytical solutions to reduce the number of degrees of freedom in a discrete analysis. These analytical solutions are derived from two kinds of governing differential equations, which are approximated with the assumptions of relatively small displacements, considering that extremely large displacements seldom have an important effect on the design of frames. The difference between the two equations is whether or not they consider bowing deformation in the calculation of axial displacement. When bowing deformation is ignored, the stiffness equation is basically the same as the one which uses stability functions. ~3,54 However, different from the customary stiffness equation, the equation presented here is expressed by a power series without truncation in order to eliminate the numerical problem under small axial force (Chen and Lui, 1987). Both of the derived stiffness equations are nonlinear algebraic equations, hence, iteration has to be used to obtain solutions. Herein, the secant stiffness method is employed as an iterative procedure, because the programming for this method is simpler compared with that using the tangent stiffness. Indeed, iterative procedures are essential to the solution of nonlinear problems, but it is desirable to reduce the number of iterations to an absolute minimum because iteration is costly. From this viewpoint, the accuracy of the solutions obtained after one cycle of iteration was also investigated with regard to the applicability to practical design.
9.1 Derivations of governing differential equations The governing differential equations for a second-order analysis are formulated through the principle of virtual work, introducing the usual beam assumptions. This is mainly because the equilibrium equations,
Design of beam-columns in steel frames in USA
61
consistent with the compatibility equations, can be easily derived by purely mathematical manipulations. 52 As shown in Fig. 26, consider a plane member 1,2 subjected to a uniformly distributed force py acting perpendicular to the member axis before deformation. A Cartesian coordinate system (x, y) as well as displacement components (u. v) are defined as the initial configuration of the member. By using the displacement components defined above, the nonlinear Green strain tensor eij for plane problems is given by
egg - Ox + -2 ~ Ox ) + ~ Ov
1 [(Ou)2
eyy -- Oy +
(108a)
(Ov~2]
2kk- ) + k y)
(108b)
J
1 Ov +Ou Ou Ou ---OY exy = -~ ~ Oy + O---x~ + Ox
(108c)
If the derivatives of axial displacement are negligibly small compared with those of deflection, the following condition holds:
Ou Ou dv Ov << Ox' Oy Ox'Oy
(109)
_1
1
IllIlIiIl U01 m~
e--~ "
M1
i V01
2
X
02==
I
~v
V02
~ - J ' M2 S Fig. 26. Coordinate system for a plane member.
W.F. Chen
62
Considering the condition of eqn (109), eqns (108) can be simplified as follows:
e,, - Ox + -2 \ Ox } IOV
1 (Or) 2
¢y,'- ay + ] t ay)
l ( Ov Ou Ov Or) exy - 2 0 x x + O y y + O--x-O-yy
(110a)
(1 lOb)
(110c)
Herein, the usual beam assumptions are introduced, i.e. the assumptions of no change of cross-sectional shape and the Bernoulli-Euler hypothesis where the transverse plane is assumed plane and normal to the beam axis throughout deformation. These assumptions are mathematically expressed in terms of the strain tensor as
eyy = O, exy = 0
(111)
Using the displacement components (u0, v0) on the centroidal axis, the displacement field which satisfies eqn (111) can be given by u = u0-y
dv0 dx
v = v0
(ll2a) (l12b)
It should be noted that u0 and v0 are both functions ofx. Substituting eqn (112) into eqn (110), the nonzero component of the strain tensor is expressed by the displacement components on the centroidal axis:
e~x -
d.o dx
d2v0 l(dv0)2
Y-d~+ 2\dxJ
(113)
This strain-displacement relationship is what is customarily used in a second-order finite element analysis for plane frames. 37 55.s6 In order to obtain the exact analytical solution corresponding to the customary finite element solution, the governing differential equations are derived through the principle of virtual work. Using the strain tensor component exx of eqn (112) and the corresponding stress component t~xxin addition to the components of external forces and displacements, as shown in Fig. 26, the equation of virtual work for the plane member is given by
Design of beam-columns in steelframes in USA I
6II=
63
l
fo fAtrxx6exx dA d x - fopy 6Vo d x - [nx(Ni 8Uoi-[- Si¢~l~oi + M,6v'oi)]~ = 0
(114) I
where fAdA indicates the integration over the cross-sectional area, f, dx over the length of the member, and nx has the values of - 1 and 1, respectively, at nodes 1 and 2. The virtual strain 6egg can be calculated by taking the variation ofeqn (113):
6exx = 6 u 6 - y 6 v ~ ' + V'o6V'o
(115)
in which the notation (-)' is introduced for simplicity to express the differentiation with respect to x. Substituting eqn (115) into eqn (114) and integrating by parts leads to 81-I = [(N - nxNi)SUoi + (Nv'o + M' - nxSi)SVoi , 2_ {( Nv; + - (M + nxMi)6Voi]l fo [N'6uo + M')'
+ py}6vo]dx = 0
(116)
where
N=
;AaxxdA,
M=
fAaxxydA
(117)
These stress resultants correspond to axial force and bending moment, respectively. Equilibrium equations and the associated boundary conditions are obtained from the necessary and sufficient conditions for eqn (116) to hold for any virtual displacements. The terms in the bracket under the integal sign yield the equilibrium equations: N' = 0
(Nv'o + M')' + py
(l18a) =
0
(l18b)
and the integrated terms give the associated boundary conditions at nodes 1 and 2:
Uo = Uoi or N = nxNi
(l19a)
v0 = v0~ or Nv'o + M' = nxSi
(l19b)
v~ = ai or M = -nxMi,
(i = 1, 2)
(ll9c)
where Si and a; are vertical forces and end rotations of the member.
64
W. F. Chen
The stress resultant-displacement relations are obtained by substituting eqn (112) into eqn (117). I f t h e x axis is selected such that it coincides with the centroidal axis of the member, these relations are simplified as
{
1
}
N = EA u ~ + ~ ( v ~ ) 2
(120a)
M = - E I v~'
(120b)
9.2 Analytical solutions of the governing equations Equations (118) to (120) are the governing equations to analyze the member. From eqn (118a), the axial force N is constant because there are no distributed forces in the axial direction. Therefore, eqn (118b) can be solved with respect to v0 independent of eqn (l18a). By using the mechanical and the geometrical boundary conditions at node 1, the solution is expressed by N~ > 0 Vo _
l
(121a)
Vol + a l
1
,gl
~-/sin yx + ( ~
[ (yx)~ 1
+ py 2(y1)4 + ~
(yx - sin y x )
,Q1
(~[j2 (1 - cos yx)
]
(cos yx - 1)
(121b)
N~ < 0
(121c)
1)0
VOI
l
l
+ ~ - s i n h yx + ( - ~ ( s i n h y x - y x )
( ~ 2 (cosh yx - 1)
[ (Yx)2 + (~/)4-(1 - cosh yx)]
+ p v L 20,l)4
(121d)
where Y = v / { N , I / E L ~,
-
S,l 2 El" ~
-
mJ pd3 E l ' py - E1
(122a-d)
Substituting eqns (121) into eqns (120) and making use of the boundary conditions at node 1, u0 can also be obtained as follows: Nt u0 = u 0 1 + ~ x - ~
1 fx ,/0
v~2dx
(123)
Design of beam-columns in steel frames in USA
65
X
where the expressions for the integral I, v62 dx can be obtained in close forms corresponding to NI > 0 and N~ < 0, respectively. These expressions are given elsewhere. 52
9.3 Stiffness equation expressed by power series From the analytical solutions obtained in Section 9.2, a stiffness equation can be derived. If the stiffness equation is expressed by such trigonometric functions as shown in Section 9.2, it differs according to whether the axial force N~ is positive or negative. 5L52 Furthermore, if the axial force approaches zero, the coefficients of the stiffness equations become indefinite, which causes numerical instability in a structural analysis. Therefore, three kinds of different stiffness equation have to be used, according to the value of the axial force, in order to avoid numerical difficulty. Such a procedure is very cumbersome and, hence, power series is introduced here. This expression is convenient for computer-aided analysis because the expressions by the series are free from numerical instability when the axial force approaches zero. Consider the following power series expansions for the trigonometric functions: co
sinM ]. sinhMJ = yl+YlE
1 (-+ Nt)" ( 2 n + 1)!
( 124a )
n=l
cosrl ~ = I + S "
1 ( + RI)" /_..., (2n)!
cosh ylJ
(124b)
n=l
lfll = Nml2/EI = rr2Ni/Pe,
Pe = rt2El/12
(124c, d)
where E1 = sectional rigidity and I = length of the member. The nondimensional stiffness equation can be obtained from eqns (121) and (122) in the form: r
AI2/l S, I
0
0
1201 60: 403
s,I
sym
1 (,~ -(AI'-/21) l I vll-dx
-AI2/I
0
0
0
- 1201
60_~
-p,./2
0 Al'-/l
-60! 0
204 0
-(fi,./12){3/(203 + 04)}
1201
-602 403
L ;J
+(Al'-/21) ~
v~2dx
-p.,/2 -(p,/12){3/(2~3 + 04)}
(125)
66
W. I~ Chen
where ai = ui/l,
0; = vi/1
1l £V,o2dx ,
(126a,b)
a~f, + S ~ . [ 2 + l Q ~ f ~ + c t , ' , ~ , f 4 + a , A ,
IL[.s+g,~/IL[6
+ P.~f7 + P v a , f 8 + pvg,f9 + PflQ~fl0
(127)
The functions ~pj andf. (i = l to 10) in eqns (125) and (127) are expressed in terms of the axial force N~ in the form of a power series and are given elsewhere.SL 52 It should be noted that in eqn (125) the bowing effect is treated as a quantity equivalent to the distributed load in the direction of the x axis. Functions ~p,.a n d f . are mathematically expressed by an infinite series expansion. However, it is impossible to consider all the terms of an infinite series in a numerical analysis. Hence, the number of terms necessary for a numerical convergence has been examined by Goto and Chen.SL 52 Following the usual procedure in stiffness matrix analysis, structural stiffness equations can be obtained by assembling the element stiffness matrix given by eqn (125). The structural stiffness equations, thus obtained, are nonlinear and an iterative procedure is, therefore, required to solve these equations. Such a procedure is also given by Goto and Chen.SL 52
9.4 Stiffness equation neglecting bowing effect The solution obtained by using eqn (125) can be accepted as exact for the frames with moderate rotational displacements. However, the equations can be simplified by ignoring the bowing effect, since this effect will generally have little influence on the results. /
By neglecting the terms( v;2dx, eqn (125) is reduced to .10
0 AI 2
NI Si
?
-AI~ ]
0
12@1 6@3
0
-12@1
4@~
0
-6@3
2@4
Al'?
o
o
0
g,
0
0
fa,
6@3
I Ol
-~,: -p,,
0 -2
sym
12~pl
-60, 403
31
~; 4~;) i'
a, -P,
(128)
Design of beam-columns in steel frames in USA
67
This stiffness equation is basically the same as the one which is often referred to in analysis by stability functions.53.54However, as stated in the previous section, eqn (128) is expressed by the exact series expansion and, hence, this equation can be used whatever the quantity of the axial force may be. This is different from the customary analysis using stability functions.
9.5 Numerical examples of rigidly connected elastic frames Three kinds of typical rectangular frames are analyzed to demonstrate the accuracy of the workstations and microcomputer-based methods with the derived element stiffness equations given in the preceding section, when they are applied to structural analysis. In addition, the results are compared with those obtained by the method using B1 and B2 factors in LRFD. 3Numerical studies on the effects of joint flexibility on the behavior and strength of steel frames are given elsewhere.57"58
9.5.1 Multistory rectangular frames with rigid connections The rectangular frames analyzed are shown in Fig. 27. along with their design load. The maximum column moments for each story of the frame are calculated under the loads up to five times as much as the design load. The ratios of the moments by second-order analysis to those by first-order analysis are summarized for three kinds of frames, respectively, in Tables 4 to 6, classified according to the method of analysis. The methods examined are of five kinds. The first is the most exact method with eqn (128). The second and third use the same element stiffness equation of eqn (128) ignoring the bowing effects. The difference is that the third method is simplified in its iterative procedure, that is, the TABLE 4 Maximum Column Moment in the One-story Frame (node 4) (Fig. 27(a))
Applied load Design load
1.0 2.0 3.0 4.0 5.0
'Exact~ with eqn (125)
1-010 1,020 1.032 1.043 1-056
"Simplified', with eqn (128)
1.011 1,023 1-036 1.049 1.063
With eqn (128) a)qer one cycle of iteration 1.011 1.023 1.036 1.049 1.063
LRFD (BI and Be factor method) with eqn (92) with eqn (93) 1.008 1,014 1,023 1.029 1,038
1.008 1-016 1.024 1.033 1.043
Moments are nondimensionalized by corresponding moments obtained by first-order analysis.
68
IV.. F. Chen
0.137 kips/in 7.28 kips, ®
w12x30
1
® o o3 >¢
031
eql 12.976 I kips
0.18848 kips/in ~)
rT
354.33 In
W14x30
- -
I
I ~
2.88 kips
~)
0.1~88 klp;/In
5.456 kips
0.155 kips/In
t i t i l t~ W16x31
5.76 kips (~
w
W21x44
,x.
;2 I_
288 in
_j
L
I
300 in
®1!
t
Fig. 27. Three typical rectangular frames.
procedure with one cycle of iteration. Fourth and fifth are the methods with B~ and B2 factors recommended by the AISC/LRFD Specification,3 respectively, using eqns (92) and (93) for the calculation of the B2 factor. All the proposed methods except for the one with the simplified iterative procedure are continued until the absolute ratio of an unbalanced force to the corresponding nodal force becomes less than 1 0 - 4 . As is clear from the formulation of the element stiffness equation, the results obtained from eqn (125) can be accepted as the most accurate ones in the tables. Throughout Tables 6 to 8, it can be seen that the results from eqn (128) are comparable with those from eqn (125) whether the method with eqn (128) uses the simplified iterative procedures or not. On the other hand, the accuracy of the method with B1 and B2 factors varies from case to case, and the results for one- and two-story frames are not so accurate when compared with the method using eqn (128). Moreover, the results oftheB~ andB2 factor method differ according to which of the two equations is used for the calculation of the B2 factor.
Design of beam-columns in steel frames in USA
69
TABLE 5 Maximum Column Moments in the Two-story Frame (Fig. 27(b))
Applied load Design load
'Exact', with eqn (125)
"Simplified', with eqn (128)
With eqn (128) aider one cycle of iteration
LRFD (BI and B2factor method) with eqn (92) with eqn (93)
First story (node 4) 1.0 2-0 3.0 4.0 5.0
1.006 1.012 1.019 1.027 1.036
1.007 1.014 1.022 1.031 1.040
1.007 1.014 1.022 1.031 1.040
1-010 1.021 1.032 1.044 1.057
1-012 1.025 1.040 1.055 1.071
1.005 1.010 1.015 1.020 1.025
1.005 1.010 1.015 1.020 0.026
1.002 1.005 1.008 1-010 1-013
1.002 1-004 1.006 1.009 1.011
Second story (node 6) 1.0 2.0 3.0 4.0 5.0
1.005 1.009 1.014 1.019 1-025
TABLE 6 Maximum Column Moments in the Three-story Frame (Fig. 27(c))
Applied load Design load
'Exact', with eqn (125)
'Simplified', with eqn (128)
With eqn (128) after one cycle of iteration
LRFD (Bi and B2factor method) with eqn (92) with eqn (93)
First story (node 4) 1.0 2.0 3.0 4.0 5.0
1.021 1.043 1.068 1.095 1.126
1.021 1.045 1.070 1.099 1. 130
1.021 1.044 1.070 1.098 !. 130
1.022 1.045 1.070 1-097 1. 127
1-016 1.034 1.053 1.072 1.093
1.022 1-046 1.073 1.103 1-137
1.021 1.044 1.070 1.099 1. 132
1.021 1.044 1.070 1.099 1. 132
1.023 1.049 1.074 1.104 i. 138
1.025 1.054 1.085 1.120 1. 159
1.005 1.010 1-016 1.022 1.028
1-006 1.013 1.021 1.028 1-037
1.006 1.013 1-020 1.028 1-037
1.004 1-008 1.012 1.016 1.021
1.005 1.010 1.016 1.023 1.030
Secondary story (node 6) 1.0 2.0 3.0 4.0 5.0
Third story (node 8) i.0 2.0 3-0 4.0 5-0
70
W. F. Chen
those for the derivations of eqn (128). Furthermore, the calculation of these magnifying factors is based on a first-order analysis, and no correction is made on the factors by iteration. Hence, it will be possible to interpret that this B~ and B2 factor method uses one cycle of iteration. Judging from these facts, B~ and B 2 f a c t o r methods could not be more accurate than the method with eqn (128), using one cycle of iteration.
9.5.2 General remarks Several methods of second-order elastic analysis have been developed here, primarily for the analysis using workstations and microcomputers. Their accuracy is examined with regard to the applicability to design, where the accuracy of the B ~and B2 factor analysis is recommended by the LRFD specification is also examined. The methods presented here use analytical solutions to reduce the number of degrees of freedom in a discrete analysis. Moreover, the stiffness equations derived from these analytical solutions are expressed in series expansion without truncation to avoid numerical difficulty. Therefore, these methods always give accurate solutions without increasing the number of degrees of freedom. Comparing their accuracy with the B~ and B2 factor method, the present methods give more accurate solutions because they use fewer assumptions in their formulations. Moreover, the methods developed have no such restrictions as theBj andB2 factor method has on the shape of structures. Regarding the applicability to design analysis, the method using one cycle of iteration is simplest and yet accurate enough for practical purposes. This method can also be easily implemented into the customary computer programs for the usual elastic first-order analysis without increasing its capacity. Taking all these factors into account, the method with one cycle of iteration is considered to be a better practical method for an elastic second-order frame analysis. It is, therefore, recommended for general use.
10 PRACTICAL SECOND-ORDER INELASTIC ANALYSIS FOR RIGID FRAME DESIGN In recent years much effort has been devoted to the development and validation of simplified second-order inelastic analysis methods for practical use. As pointed out in Section 9, current practice in the USA is to perform elastic analysis (first-order elastic analysis with moment magnification factors B~ and B2 or direct second-order elastic analysis) to -k,~:~ ,~, . . . . . . ;,.,~,j . . . . . t ~ fc~r d e ~ i ~ n Material nonlinearity is
Design of beam-columns in steel frames in USA
71
accounted for implicitly in the design interaction equations. It is well known that, as the frame deforms into the inelastic range, the distribution of forces a n d moments will differ from the case when the frame remains fully elastic. Thus, if inelasticity is not accounted for in the analysis, the forces and moments calculated will not be truly representative of the actual response of the frame. The incompatibility that exists between the analysis and design assumptions will severely undermine the philosophy of a limit states approach to design. Recognizing the eventual need for incorporating advanced analysis methods in the provisions, researchers and practitioners have collaborated in a joint effort to establish guidelines which would set the standard for the use of such methods of analysis in a limit state setting. In this regard, twotask groups have recently been set up to address the problem: The AISC Task Committee 117 -- Inelastic Analysis and Design, and the SSRC Task Group 29 m Second-Order Inelastic Analysis for Frame Design. The main objectives for the task groups are: 1. To provide a dictionary and a uniform definition of terms that pertain to nonlinear analysis and nonlinear behavior. 2. To categorize and detail behavioural effects such as quantifying the magnitude and distribution of residual stresses, member out-ofstraightness and member out-of-plumbness. 3. To identify benchmark problems for use in calibration, verification and comparison of second-order inelastic analysis procedures. 4. To identify situations in which advanced analysis will be appropriate and should be performed. 5. To assess the accuracy and feasibility of various approaches for inelastic analyses. 6. To implement and codify innovative analysis and design approaches which incorporate all essential elements of nonlinear effects. Table 7 summarizes various behavioral effects which should be addressed in a valid limit states design. 59It should be mentioned that the incorporation of all these factors in design will undoubtedly further complicate the already cumbersome design process. An important task for the task groups is, therefore, to identify circumstances and situations in which simplifying assumptions with regard to structural behavior can be applied without incurring noticeable errors. In what follows, three recently proposed simplified second-order inelastic analysis methods which take into account only the in-plane geometrical and material nonlinear effects will be described. The ability of each method to model frame response will also be demonstrated. The other factors which are located in Table 9 will not be discussed here. A discussion of some of these factors can be found elsewhere (Chen and Lui. 1991)) 6
72
W F. Chen TABLE 7
Behavioral Effects of Steel Frames ~9 1. Physical characteristics (a) Initial imperfections -- camber, sweep and twist of members, and out-of-plane erection of frame (b) Residual stresses present in members prior to loading as a result of manufacturing, fabrication and erection processes (c) Member shape (e.g. unsymmetrical cross-section and tapered profile) (d) Connectin type -- pinned, semi-rigid and rigid (e) External and internal restraints (e,g. bracing and support conditions) (f) Miscellaneous initial strains (e.g. environmental thermal loading) (g) Construction sequence 2. Response phenomena -- geometrical effects (a) Influence of axial force on member bending stiffness (often called the P-6 effect) (b) Effect of relative horizontal joint displacements (often called the P-A effect) (c) Changes in member chord length resulting from axial strain and bowing (the latter often called curvature shortening) (d) Shearing deformation in members (e) Local buckling and other local distortions 3. Response phenomena -- material effects (a) Nonlinear stress-strain relationship, including strain hardening and elastic unloading at plastic hinges or in inelastic zones (b) Spread of inelastic zones in members (c) Inelastic interaction of axial force, biaxial bending, shear and torsion 4. Response phenomena -- coupled geometric and material effects (a) Finite joint size (b) Panel zone deformations (c) Connection deformations (d) Contributions of slabs, infilling, and secondary systems, to strength and stiffness 5. Loading effects (a) Nonproportional loading (b) Variable repeated loading (c) Dynamic effects 6. Uncertain~' (a) Variability in load effect -- temporal and spatial (b) Variability in resistance
10.1 Inelastic cross-section method The inelastic element model used by Lui 6° is shown in Fig. 28. In this model, yielding is assumed to concentrate in specific regions in a member. For a column, these regions are assumed to be located at the ends o f the member. For a beam, these regions are assumed to be located
Design of beam-columns in steel frames in USA
2,o
r
73
!
Inelastic
Elastic
Fig. 28. Inelastic element model.
at the ends and at a point within the span where the moment is expected to be the largest. Although spread of yield along the member length is ignored, plastification over the cross-section is accounted for by the use of an effective flexural rigidity (EI)c~ in the inelastic regions. The calculation of this effective E1 is shown schematically in Fig. 29. As can be seen, the variation of E1 is assumed to be linear from the cross-section first yield envelope to the cross-section ultimate strength (limit) envelope. The equation for the ultimate strength envelope was adopted from Duan and Chen. 5° To account for the presence of geometrical nonlinearity, a modified form of the stability function is used in formulating the element stiffness matrix. The validity of the approach is demonstrated in Fig. 30 in which the load-deflection behavior of a three-story one-bay frame, tested by Yarimci, 6~ is plotted and compared with results obtained using the inelastic cross-section method. Reasonably good agreement is observed.
P W8 x 31 1.0
0.8
0.6
0.4
S SS
0,2
,,
lo
\
\
s S "e
I 0.0
0.2
'Y l''" 0.4
I 0.6
I~ X 0.8
X
~ M,
1.0
Fig. 29. Schematic representation of (El)effective.
Mpx
W. F. Chen
74
2.0
1.6
,j
. ~ , ~ <
1.64 1.63
P1 =23 kips P2 =20 kips
"/
1.2
P2 P, P, P2
v
_o
0.8
Inelastic
cross-section ID
1 II__H =q ~ I 11_..H
method
.J
.....
0.4
Experimntal
iq ' 1 5 ' =1 Beams: W10x25 Columns: MSx 18.9
0.0
I
I
1
0.5
1.0
1.5
I
2.0
I
I
2.5
3.0
3.5
Lateral deflection of first story (inch)
Fig. 30. C o m p a r i s o n of the i n e l a s t i c cross-section m e t h o d with e x p e r i m e n t a l results. (1 ft = 0.305 m; 1 in = 2 . 5 4 c m ; 1 kip = 4 . 3 4 N )
10.2 Modified plastic hinge method
The modified plastic hinge method 62accounts for cross-sectional plastification by the use of a plastification factor p in the member stiffness formulation. Spread of yield along member length is ignored. The plastification factor p is defined as 0 < Pk < Mk-M~c < 1 M p c -- M,.c
(129)
where Mk is the moment at the k th end of the member, Myc is the yield moment reduced for the presence of axial force and residual stresses, and Mpc is the plastic moment reduced for the presence of axial force. They are obtained, respectively, from the equations: M~,~
0.9Mp ( -
f
P
1
Mp~ = M p [ 1 - ( ~ )
O.~p v
'3]
)
(130) (131)
In the above equation, Mp is the plastic moment capacity of the crosssection,./is the shape factor of the cross-section, P is the axial force in the member, and P,, is the yield load of the cross-section.
75
Design of beam-columns in steelframes in USA
'51.
Deflected position
J r2
ds
i,.._
dI
rl K~. %
Fig. 31. Six degree-of-freedom plane beam-column element. For the element shown in Fig. 31, the King et al. 62 element stiffness relationships has the form:
{r,} AE -~-
r2
r3
0
0
(K;, + 2K~ + K~) + P L2 L
(K~ + X],) L
0
0
(K~ + K~) L
K~,
0
-(K;~ + K;i ) L
K~j
-AE L
0
0
AE -L
0
0
0
-(K~i+2K~+K'ii) L'-
-(K~,+K~) L
0
(K~,+2K~+K~) LY"
0
(K;, + K~) L
K;,
0
-(K,~ + K~) L,
=
r4
r5 r~
-AE L
0
0
-(K~ + 2K~ + K~) L2
0
P L
(K~ + K~) L
-(K,~+K~i ) L
::] d4 d5 d~
K~j
(132)
where Ki'i =
(
K~i - Kij ~
Pj
)
(1 - p;)
(133a)
K'ii = Kii(l - Pi) (1 - P i)
(133b)
K]i = Kii(l - Pi) (1 - pj)
(133c)
K~/ =
(133d)
K~i - K(i ~
p~ (1 - P.i)
in which Pi a n d p j are plastification factors evaluated at the i th andjth ends of the member, respectively, and
76
W. F. Chen
4El
2PL
K, = K~ = ~ -
+ ~-
2EI
PL
L
30
K u = Kj; -
44P2L 3
+ 25 O00EI
(134)
26p2L 3 25 O00EI
(135)
Note that, when pg = pj = 0, the member is fully elastic and eqn (132) represents the second-order elastic stiffness relationship of the member. The ability of the modified plastic hinge method to model inelastic frame response is demonstrated in Fig. 32. In this figure, the numerical results obtained using King's method are compared with the numerical results obtained by Alvarez and Birnstie163 using the plastic zone method. Good agreement is observed.
10.3 Beam-column strength The beam-column strength method 64 for inelastic frame analysis treats each column in a frame as an individual element. The beam-column interaction equation for different slenderness ratios is taken as the ultimate strength (limit) surface for each element. When the load combination of the element reaches this ultimate strength surface, the element is considered to have failed. Since the failure criterion is based on the ultimate strength of the members that comprise the frame, the
50
:27.591
o P ~ _ ~ Q
= 75 k
Q
40
/
_° "o o
i ir
= 179 k
30 J- -
_J
Q q = 3 . 6 kilt
-
~
-
~
Q = 272 k
20
~
10
Plasticzonemethod
/
• 0.'4
lOW49~..
t_
10'__~
I-
M o d i f i e d plastic hinge m e t h o d A
0.12
f
0.16
O.J8
1.10
1.2
I
1.4
1
1.6
Lateral d e f l e c t i o n at top left joint (inch)
Fig. 32. Comparison of the modified plastic hinge method with the plastic zone method.
Design of beam-columns in steelframes in USA
77
proposed method takes into account the effect of spread of yield along member length on the inelastic stability response of a frame. For uniaxial bending about the strong axis, the beam-column interaction equation used for the beam-column strength method has the form: +
= 1
(136)
where P is the axial force, Mx is the moment, Mpx is the plastic moment capacity, and Pu is the axial load capacity of the member given by Pu
= -~(1 - 0"25~'c2)Py f°rA'c < v/2 (py/A2c for Ac > v/2
(137a, b)
in which A,c is given by eqn (24) and Py is the yield load. fl = 1.3 + 0.002(KL/rx)
(138)
f o rMI ~ <
1 r/ = 1 +0.006
-40
,,,
/> 1 f o r ~
0.5 (139)
> 0.5
Equation (136) is referred to by Chen et al. 64 as the elastic K-factor surface. The ultimate strength surface is obtained from the elastic K-factor surface by settingK = 1. In addition, an initial yield surface is defined as 70% of the elastic K-factor surface. The stiffness of the member is assumed to reduce linearly from this initial yield surface to the ultimate strength surface once the load combination of the member reaches the initial yield surface. Thus, if we define p as the distance between the initial yield surface and the ultimate strength surface, the member stiffness matrix can be expressed by eqn (132) with Pi,Pi replaced byp. In the beam-column strength method, the K factor is assumed to change from its elastic value to unity when the member fails. When the member stiffness reduces as the load combination increases, the K factor of the member is updated according to the following rules. Ifgelasti c
> 1, then K'
=
Kelastic:
(I) Kr = K ' -
1
(2) K.ew = 1 + Kr(1 - 19) (3) K' = Knew
(140)
78
If K e l a s t i
IV.. F. Chen c
< 1, then K'
Kelastic:
=
(1) Kr = 1 - K ' (2) Knew
=
1 --
gr(1 - p)
(140)
(3) K' = Knew where Kr is the difference between the current K factor and 1, K,ew is the updated K factor in each load step due to the reduction of the beamcolumn stiffness, and K' is the new K factor to be used for the next load step. The validity of the above approach for inelastic frame analysis is demonstrated in Fig. 33 in which the load-deflection response of a sixstory two-bay frame is shown. The dotted line represents the results .
.
.
.
.
Plastic
Zone Method
Beam-Column E = 20,500
1.2
SLrength Method
kN/cm 2
Fy = 2 3 . 5 0 k N / c m 2 I m p e r f e c t i o n :7/Jo = 1 / 4 5 0
_•
Load R a t i o
1.0 • 31.TkNIm !
H2•
10 2 ] k N
l|
l l i l l i i l !
=
ti
~ J ] I| E
0.8 HI=20I-~ kN Z =
I I llli;ti
i J l l l l i l i IPE
o
300
m
H,
0.6
~'
=
l i l l t l l i ' l
o
o
o
0.4
., H~
.
|l;li;;;i
z[
o
o
11111,1;i
l0
15
[ l i l i t l i ; IPE 360
] i l i l J l l ;
E = 205 k t . ; / m m I
60m
5
IPE 330
m
o
0.2
i i l l ] l l t ; IPE 300
! IPE (.00
b ~
f;0rn
20
25
Lateral DefiectAon at Right Top Joint (era) Fig. 33. Load-deflection curves o f Vogel six-story frame.
Design of beam-columns in steelframes in USA
79
obtained using the plastic zone method, 65'66 while the solid line was obtained using Chen's method. 64 Excellent correlation is observed.
11 SUMMARY A beam-column is a structural member which is subjected to both axial forces and bending moments. A column and a beam can be regarded as special cases of a beam-column. For short beam-columns the limit state will probably be that of strength. For slender beam-columns in unbraced frames, two types of instability effects can be identified: (a) Member stability orP-6 effect in which the axial force acting on the deformation of the member relative to its chord produces secondary destabilizing moments. (b) Frame stability orP-A effect in which the gravity load acting on the lateral displacement of the frame produces secondary destabilizing overturning moments. Both these effects weaken the strength and stiffness of the structure and so they must be accounted for in design. For design purposes, these secondary effects can be accounted for accurately by using a computer-based second-order elastic analysis method, or approximately by using the moment magnification factors (Bt and B2). The first-order moments evaluated from first-order analysis are multiplied by these magnification factors to approximate the total moments that will be used for design. The design of beam-columns is facilitated by the use of interaction equations. These interaction equations provide a smooth transition between ultimate strengths of columns and beams (Pu and Mu) to give safe combinations of the applied loads (P and M). Interaction equations may be linear or nonlinear. Linear interaction equations have the advantage of simplicity, but they are not as accurate as their nonlinear counterparts. Interaction equations, either recommended or proposed in the USA, are summarized and discussed in this paper. The introduction of the limit states philosophy has revolutionized the concept of steel design. When a structure is designed according to its limit of usefulness, a more accurate assessment of the response of the structure is desirable. Although the use of a first-order elastic analysis is still common practice in the design profession, there has been a trend towards the use of more elaborate analysis methods for design. Since almost all structures behave in some nonlinear fashion prior to reaching their limit of usefulness, a valid limit state design should, therefore,
80
Iv.. F. Chen
include all factors that affect the response of the structure to loadings. A summary of such factors has been given in Table 7. The use of a general analysis method which is capable of incorporating all these factors and at the same time simple enough for routine office use seems to be unrealistic at present. However, steps are moving toward that direction. In particular, the AISC L R F D Specification 3has provided provisions for a direct second-order elastic analysis for design, and the Australian Limit State Steel Design Specification (AS4100) 67 and the Eurocode 68 have both attempted to provide provisions for a second-order inelastic analysis for design. With the advancements made in computer hardware and software, computer-aided analysis and design will definitely play an important role in structural design. The adoption and proliferation of the limit states approach to design have provided engineers with a vehicle to explore innovative analysis and design procedures. In this paper, the current AISC L R F D interaction equations for beamcolumn design have been reviewed. The importance of an accurate assessment of the effective length factor K has been addressed. New interaction equations capable of representing the inelastic behavior of beam-column have also been presented. To conclude, three simplified second-order inelastic analysis methods suitable for practical use have been described. REFERENCES 1. AISC, Specification for the Design. Fabrication and Erection of Structural Steel .for Buildings. American Institute of Steel Construction, Chicago, Nov. i 978. 2. AISC, Allowable Stress Design Specification .for Structural Steel Buildings. American Institute of Steel Construction Chicago, 1989. 3. AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, Oct. 1986, ! st edn, 313 pp. 4. Ravindara, M. K. & Galambos, T. V., Load and resistance factor design for steel. J. Struct. Div., ASCE, 104(ST9)(1978) 1337-54. 5. Ellingwood, B., MacGregor, J. G., Galambos, T. V. & Cornell, C. A., Probability-based load criteria, load factors and load combinations. J. Struct. Div.. ASCE, 108(ST5) (1982) 978-97. 6. ANSI, Building Code Requirements for minimum Design Loads in Buildings and other Structures, ANSI, A58.1, American National Standards Institute, New York (1982). 7. Ang, A. H. S. & Corneil, C. A., Reliability bases of structural safety and design. J. Struct. Div.. ASCE. 100(STg) (1974) 1755-69. 8. Bjorhovde, R., Deterministic and Probabilistic Approaches to the Strength of Steel Columns, PhD Dissertation, Department of Civil Engineering, Lehigh University, Bethlehem, PA, 1972.
Design of beam-columns in steelframes in USA
81
9. Johnston, B. G., (ed.), SSRC Guide to Stability Design Criteria for Metal Structures. 3rd edn, John Wiley, New York, 1976, 616 pp. 10. Rondal, J. & Maquoi, R., Single equation for SSRC column strength curves. Technical Notes, J. Struct. Div., ASCE, 105(STI) (1979) 247-50. 11. Chen, W. F. & Atsuta, T., In-plane behavior and design. Theory of BeamColumns. Vol. 1, McGraw-Hill, New York, 1976, 512 pp. 12. Lui, E. M. & Chen, W. F., Generalized column e q u a t i o n - A physical approach. In Advances in Tall Buildings. ed. L. S. Beedle. Council on Tall Buildings and Urban Habitat, Van Nostrand Reinhold, New York, 1986, pp 323-52. 13. Lui, E. M. & Chen, W. F., Simplified approach to the analysis and design of columns with imperfections. Engineering J.. AISC, 21(2) (1984)99-177. 14. Sugimoto, H. & Chen, W. F., Small end restraint effects on strength of Hcolumns. J. Struct. Div.. ASCE, 108(ST3) (1982) 661-81. 15. Lui, E. M. & Chen, W. F., End restraint and column design using LRFD. Engineering J., AISC, 20(1) (1983) 29-39. 16. Chapius, J. & Galambos, T. V., Restrained crooked aluminium columns. J. Struct. Div.. ASCE. 108(ST3) (1982) 511-24. 17. Julian, O. G. & Lawrence, L. S., Notes on J and L Nomographs for Determination of Effective Lengths. Unpublished Report, Jackson and Moreland Engineers, Boston, MA, 1959. 18. LeMessurier, W. J., A practical method of second order analysis, Part I. Engineering J..AISC. 13(4) (1976) 89-96. 19. LeMessurier, W. J., A practical method of second order analysis, Part II. Engineering J.,AISC, 14 (1977) 49-67. 20. Yura, J. A., The effective length of columns in unbraced frames, Engineering J.. AISC. 8(2) (1971) 37-42. 21. Chen, W. F. & Atsuta, T., Space behavior and design. Theory of BeamColumns. Vol. 2, McGraw-Hill, New York, 1977, 732 pp. 22. Chen, W. F. & Lui, E. M., Effects of joint flexibility on the behavior of steel frames. Computers and Structures. 26(5) (1987) 719-32. 23. Galambos, T. V., Inelastic lateral buckling of beams. J. Struct. Div., ASCE. 89(ST5) (1963) 217-42. 24. Vinnakota, S., Verification of the SSRC Interaction Formula for Lateral Buckling of Beam-Columns. SSRC-TG3 Report, March 1982. 25. Yura, J. A., Galambos, T. V. & Ravindra, M. K., The bending resistance of steel beams. J. Struct. Div.. ASCE 104 (STg)(1978) 1355-69. 26. Duan, L., Sohal, I. S. & Chen, W. F., on beam-column moment amplification factors, Engineering J.. 26(4) (1989) 130-5. 27. Zhou, S. P. & Chen, W. F., C,, factor in LRFD.J. Struct. Eng.. ASCE. 113(ST8) ! 738-54. 28. Rosenblueth, E., Slenderness effects in buildings. J. Struct, Div.. ASCE. 91(ST1) (1965) 229-52. 29. Stevens, L. K., Elastic stability of practical multistory frames. Proc. Inst. Civil Engrs. Vol. 36, London, UK, 1967. 30. Cheong-Siat-Moy, F., Consideration of secondary effects in frame design. J. Struct. Div.. ASCE. 103(STI0)(1977)2005-19. 31. Nixon, D., Beaulieu, D. & Adams, P. F., Simplified second-order frame analysis. Can, J. Civil Engng, 2(4) (1975). 32. Chen, W. F. & Cheong-Siat-Moy, F.. Limit states design of steel beam-
82
W. F. Chen
columns -- A state-of-the-art review. J. Solid Mech. Arch., 5( I ) (1980) 29-73. 33. Chen, W. F. & Lui, E. M., Stability design criteria for steel members and frames in the United States. J. Construct. Steel Res., 5 31-71. 34. Chen, W. F. & Lui, E. M., Analysis and design of steel frames in the USA -90's and beyond, keynote lecture. Proc. Int. Conf. on Steel and Aluminium Structures, Singapore, May 1991. 35. Chen, W. F. & Lui, E. M., Structural Stability: Theory and Implementation. Elsevier, New York, March 1987, 490 pp. 36. Chen, W. F. & Lui, E. M., Stability Design of Steel Frames, Blackwell Scienti tic, Oxford, UK, 1991, 373 pp. 37. Galambos, T. V. (ed.), SSRC Guide to Stability Design Criteria .for Metal Structures, 4th edn, Wiley-lnterscience, New York, 1987. 38. Kanchanalai, T. & Lu, L. W., Analysis and design of frame columns under minor axis bending. Engineering J., AISC, 16(2) (1979) 29-41. 39. Cheong-Siat-Moy, F., & Downs, T., New interaction equation for steel column design. J. ofStruct. Div,, ASCE, 106, (ST5), (1980) 1047-62. 40. Tebedge, N. & Chen, W. F., Design criteria for H-columns under biaxial bending. J. Struct. Div.. ASCE, 100(ST3) (1974) 579-98. 41. Anslijn, R. & Massonnet, C. E., New Tests on Steel I Beam-Columns in Mild-Steel Subjected to Thrust and Biaxial Bending. Technishe Hochschule, Darmstadt, Germany, 1978, pp. 103-123. 42. Anslijn, R., Tests on Steel I Beam-Columns Subjected to Thrust and Biaxial Bending. Construction Metailique, MT 157, Aug. 1983. 43. Cai, C. S., Liu, X. L. &Chen, W. F., Further verifications of the beam-column equations. J. Struct. Engng. ASCE, 116(2) (1991) 501-13. 44. Yura, J. A., Elements for Teaching Load and Resistance Factor Design: Combined Bending and Axial Load. Lecture Notes, University of Texas at Austin, 1988, pp. 55-71. 45. Kanchanalai, T., The Design and Behavior of Beam-Columns in Unbraced Steel Frames. AISI Project No. 189, Report No. 2, Civil Engineering/ Structures Research Laboratory, University of Texas at Austin, 1977, 300 pp. 46. Chen, W. F., Zhou, S. P. & Duan, L., Second-order inelastic analysis of braced portal frames. J. Singapore Soc Structural Engrs, I(1), December (1990) 5-15. 47. Zhou, S. P., Duan, L. & Chen, W. F., Comparison of design equations for steel beam-columns. Struct. Eng. Rev. 2(!) (1990) 45-53. 48. Zhou, S. P., Duan, L. & C h e n , W. F., The P-A effect on portal steel frames. Proc. Int. Conf. on Steel and Aluminium Structures, Singapore, May 1991. 49. Liew, R. J. Y., White, D. W. & Chen, W. F., Beam-column design in steel frameworks. In Constructional Steel Design: An International Guide. eds. R. Bjorhovde, & A. Colson. Elsevier, London, 1991. 50. Duan, L. & Chen. W. F., Design interaction equation for steel beamcolumns. J. Struct. Enging.. ASCE. 115(5) (1989) 1225-43. 51. Goto, Y., & Chen, W. F., On the computer-based design analysis for the flexibility jointed frames. J. Construct. Steel Res., 8 (1987) 203-31. 52. Goto, Y., & C h e n , W. F., Second-order elastic analysis for frame design. J. Struct. Enging.. ASCE. 113(7) (1987) 1501-19. 53. Livesley, R. K. & Chandler, D. B., Stability Functions for Structural Frameworks. Manchester University Press, Manchester, UK, 1956.
Design of beam-columns in steelframes in USA
83
54. Home, M. R. and Merchant, W., The Stability of Frames. Pergamon Press, Oxford, UK, 1965. 55. Connor, J. J., Logcher, R. D. & Chan, S. C., Nonlinear analysis of elastic frames structures. J. Struct. Div., ASCE, 94(ST6) (1968) 1525-47. 56. Mallet, R. H. & Marcal, P. V., Finite element analysis of nonlinear structures. J. Struct. Div., ASCE, 94(ST9) (1968) 2081-310. 57. Chen, W. F. & Zhou, S. P., Inelastic analysis of steel braced frames with flexible joints. J. Solids and Structures, 23(5) (1987) 631-49. 58. Lui, E. M. & Chen, W. F., Steel frame analysis with flexible joints. In Joint Flexibility in Steel Frames, ed. W. F. Chen. Elsevier Applied Science, London, 1987, pp. 161-202. 59. McGuire, W., Structural analysis in load and resistance factor design. Introductory remarks at a Seminar on Frame Analysis in LRFD, dISC National Steel Conference. 1990. 60. Lui, E. M., Nonlinear stability analysis of frameworks- from theory to practice. 1990 Structures Congress, Baltimore, MD, 1990. 61. Yarimci, E., Incremental Inelastic Analysis of Framed Structures and Some Experimental Verifications, PhD Thesis, Department of Civil Engineering, Lehigh University, Bethlehem, PA,~1966. 62. King, W. S., White, D. W. & Chen, W. F., A Modified Plastic Hinge Method for Second-Order Inelastic Analysis of Steel Rigid Frames. Structural Engineering Report No. STR-90-17, School of Civil Engineering, Purdue University, West Lafayette, IN, 1990, 34 pp. 63. Alvarez, R. J. & Birnstiel, C., Inelastic analysis of multistory multibay frames. J. Struct. Div., ASCE, 95(11) (1969) 2477-503. 64. Chen, W. F., King, W. S. & White, D. W., A Beam-Column Strength Method for Second-Order Inelastic Analysis of Steel Frames. Structural Engineering Report No. CE-STR-90-18, School of Civil Engineering, Purdue University, West Lafayette, IN, 1990, 29 pp. 65. Vogel, U., Calibrating frames. Stahlbau. 10 (1985) 1-7. 66. Toma, S. & Chen, W. F., Calibration Frame in Europe. StructUral Engineering Report No. CE-STR-91-1, School of Civil Engineering, Purdue University, West Lafayette, IN, 1991. 67. Standards Australia, Australian Standard for Design of Steel Structures, AS 4100, Committee B D / I - Steel Structures, Working Group BD/1/-/7Methods of Structural Analysis, 1990. 68. Eurocode No. 3,Design of Steel Structures: Part I -- General Rules and Rulesfor Buildings. Vol. 1, Edited Draft Issue 3, 1990.
Design of Beam-Columns in Steel Frames in the United States
W. F. C h e n Department of Structural Engineering, School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 25 January 1990; accepted 25 September 1990)
ABSTRACT This paper attempts to give designers an insight into the recent developments of the design criteria of steel beam-columns in the USA. The important features include (1) the ultimate strength of columns and beams that are special cases of beam-columns, (2) the secondary effects (P-6 and P-A) that should be considered in the ultimate strength design of beam-columns, and (3) the simplified computer-based second-order elastic and inelastic analysis methods that are suitable for adoption in the practical design for beamcolumns in rigid frames. Recommended and proposed ultimate strength design interaction equations in the USA for different categories of beamcolumns are presented. Importantfeatures of these equations are summarized. Several illustrative examples are given. Directions of further research are indicated.
1 THE CATEGORY OF BEAM-COLUMNS O n e of the most i m p o r t a n t structural elements in a steel f r a m e w o r k is the beam-column. A b e a m - c o l u m n is the type of structural m e m b e r w h i c h is subjected to the c o m b i n e d action of b e n d i n g m o m e n t s a n d axial thrust. D e p e n d i n g o n the m a n n e r the loads are applied a n d the resulting deformations, b e a m - c o l u m n s can be classified into one of the following categories: I
Thin-Walled Structures 0263-8231/91/$03-50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
V£ F. Chert
(a) In-plane beam-columns in which the bending moment is acting in only one principal axis direction in addition to the axial thrust. For this type of beam-column, only in-plane detbrmation will occur upon application of the loads. (b) Lateral torsional buckling of beam-columns in which the bending moment is again acting in only one principal axis direction in addition to the axial thrust. However, unlike in-plane beamcolumns, out-of-plane deformation will result in this type of beam-column due to twisting of the member as a result of lateral torsional buckling. (c) Biaxially loaded beam-columns in which the bending moments are acting in two principal axis directions in addition to the axial thrust. As a result, deformation of this type of beam-column is a three-dimensional space action. According to how the ultimate capacities of these beam-columns are established, the beam-columns in each of the above categories can be further categorized into short or long-beam-columns as follows: (a) In short beam-columns, the effect of member lateral displacement on the magnitude of moment in the member is not significant. The limit state of this type of beam-column is strength." thus, the ultimate capacity of a short beam-column is reached when full plastic yielding of the material of the entire cross-section has occurred. (b) In long beam-columns, the effect of member lateral displacement on the magnitude of moment in the member is not negligible. The limit state of this type of beam-column is stability; thus, the ultimate capacity of a long beam-column is its stability limit load. Long beam-columns can be further subdivided into two cases: (a) The non-sway case where the only secondary moment that needs to be considered is the P-6 moment. (b) Thesway case where both theP-6 and P-A secondary moments are important and need to be accounted for in the analysis and design procedures. The purpose of this paper is to summarize recent research on the subject of beam-columns for each of these categories and to present the new design philosophy based on a limit state design approach (Section 2) and design equations in the form of interaction equations that are currently in use, or proposed, in the USA. Before proceeding to the discussion of the beam-column problem, it is more appropriate to
Design of beam-columns in steel frames in USA
3
discuss first the column problem (Section 3) and the beam problem (Sections 4 and 5). This is because not only will they represent special cases of beam-columns (Sections 6 and 7), but they also provide end points to which the beam-coiumn interaction equation is referenced. Finally, the second-order elastic and inelastic analysis methods for the design of beam-columns in frames are presented (Sections 8-10).
2 DESIGN PHILOSOPHIES To implement the mathematical theory of stability into engineering practice, it is necessary to review the various design philosophies and safety concepts upon which current design practice is based. Details of this implementation on various specific subjects will be given in the sections that follow. The philosophy of steel building design in the USA has undergone a metamorphosis from the allowable stress design (ASD) approach through the plastic design (PD) approach to the present stage of the load and resistancefactor design (LRFD) approach. The LRFD approach is the product of a culminated effort by researchers and practitioners to incorporate the latest research results and experience on material and structural behavior, under the influence of a variety of loading types, into a consistent and rational design format. One fundamental feature that demarcates the LRFD approach to steel design from the ASD and PD approaches is that LRFD is a probability-based design methodology which utilizes statistical evidence and reliability theory to substantiate the logistic and rationale of the design. Design considerations are guided by the identification of a set of limit states. A design is considered acceptable if none of the limit states is violated. Since a probabilistic mathematical model was used in the development of the load and resistance factors, it is possible to give proper weight to the degree of importance for each load effect and resistance. The final result is that a more uniform margin of safety can be realized. 2.1 Allowable stress design
The purpose of ASD is to ensure that the stress computed under the action of the working (i.e. service) loads of a structure do not exceed some predesignated allowable values. The allowable stresses are usually expressed as a function of the yield stress or ultimate stress of the material. The general format for an allowable stress design is thus
4
W.F. Chen R?/
m
F.S~ > E
Qni
(1)
i=1
where Rn = nominal resistance of the structural member expressed in units of stress; Qn = nominal working (or service) stresses computed under working load conditions; F.S. -- factor of safety; i = type of load (i.e. dead load, live load and wind load, etc.); m = number of load types. The left-hand side of eqn (1) represents the allowable stress of the structural member or component under various loading conditions (e.g. tension, compression, bending and shear). The right-hand side of the equation represents the actual or computed combined stress produced by various load combinations (e.g. dead load, live load and wind load). Formulas for the allowable stresses for various types of structural members under various types of loadings are specified in the AISC Specifications. L2 A satisfactory design is when the stresses in the member, computed using a first-order analysis under working load conditions, do not exceed their allowable values. One should realize that, in ASD, the factor of safety is applied to the resistance term, and safety is evaluated in the service load range.
2.2 Plastic design The purpose of PD is to ensure that the nominal maximum plastic strength of the structual member or component exceeds that of the factored load combinations. It has the format: m
R n > ?" E
Q"~'
(2)
i=l
where R,, = nominal plastic strength of the member, and ~" = load factor. In steel design, the load factor is given by the AISC Specifications L2 as 1.7 if Q, consists 0nly of dead and live gravity loads, or as 1-3 if Q,, consists of dead and live gravity loads plus wind or earthquake loads. The use of a smaller load factor for the latter case is justified in that the simultaneous occurrence of all these load effects is not very likely. Note that in plastic design, safety is incorporated in the load term and is evaluated at the ultimate (plastic strength) limit state.
Design of beam-columns in steelframes in USA
5
2.3 Load and resistance factor design The purpose of LRFD is to ensure that the nominal resistance of the structural member or component exceeds that of the load effects. Two implied safety factors are used, i.e. one applied to the loads and the other to the nominal resistance. Thus, the load and resistance factor design has the format: m
(PRn >/ Z
(3)
YiQni
i=1
where R, = nominal resistance of the structural member; = resistance factor (< 1.0) 7; = load factor (usually >1.0) corresponding to Q,i. In the 1986 LRFD Specification of AISC, 3 the resistance factors were developed mainly through calibration,4 whereas the load factors were developed on the basis of statistical analysis. 5.6. In particular, a firstorder probability theory was used. The load and resistance factors for various types ofloadings and various load combinations are summarized in Tables 1 and 2, respectively. A satisfactory design is one in which the TABLE 1 Load Factors a n d Load C o m b i n a t i o n s a 1.4D 1.2D+ 1.2 D + 1.2D+ 1.2D + 0.9D-
1.6 L + O.5 (L~ or S or R ) 1.6 (Lr or S or R) + (0.5 L or 0.8 I4") i.3W+0.5L+0.5(LrorSorR) 1.5E + (0.5L or 0 . 2 S ) 1.3 W o r 1.5E
where D = dead load L = live load Lr = roof live load W = wind load S = snow load E = earthquake load R = n o m i n a l load due to initial rainwater or ice exclusive of the p o n d i n g contribution aThe load factor o n L in the third, fourth a n d fifth load c o m b i n a t i o n s shown above shall equal 1-0 for garages, areas occupied as places of public assembly a n d all areas where the live load is greater t h a n 100 psf.
6
IV.. b: Chen TABLE 2 Resistance Factors Member type and limit state
Tension member, limit state: yielding Tension member, limit state: fracture Pin-connected member, limit state: tension Pin-connected member, limit state: shear Pin-connected member, limit state: bearing Columns, all limit stales Beams, all limit states High-strength bolts, limit state: tension High-strength bolts, limit state: shear A307 bolts others
0.90 0.75 0.75 0.75 1.00 0.85 0.90 0-75 0.60 0.65
probability of the structural m e m b e r exceeding a limit state (e.g. yielding, fracture a n d buckling) is minimal. Based on the first-order secondm o m e n t probabilistic analysis, 7 the safety of a structural m e m b e r is m e a s u r e d by a reliability or safety index, 4 defined as
//-
In(R,,/Q,,) +
(4)
where R = m e a n resistance: Q = m e a n load effect: ~rR V~ = coefficient of variation of resistance = -~-; R VQ = coefficient of variation of load effect -
crQ
~),
in which ~7 is the s t a n d a r d deviation. The physical interpretation of the reliabili~ index,//, is shown in Fig. 1. It is the multiplier of the s t a n d a r d deviation v"V~ + V~ between the m e a n of the In(R/Q) distribution a n d the ordinate. Note that both the resistance R a n d the load Q are treated as r a n d o m parameters in LRFD, and so ln(R/Q) does not have a single value, but follows a distribution, The s h a d e d area in the figure represents the probability in which In(R/Q) < !, i.e. the probability that the resistance will be smaller than the load effect, indicating that a limit state has been exceeded. T h e larger the value of//, the smaller the area of the s h a d e d area. and so it becomes more i m p r o b a b l e that a limit state may be exceeded: thus, the m a g n i t u d e o f / / reflects the safety of the member.
Design of beam-columns in steel frames in USA
7
PROBABILITY DENSITY
~tv/t/ 2 -
2
v.+ vo
_1 -[
raisin
In(R/Q)
In(RIQ)
Fig. 1. Reliability index in the AISC L R F D design.
In the development of the present LRFD Specification 3 the following target values for fl were selected: (i) [3 = 3.0 for members and fl = 4.5 for connectors under dead plus live and/or snow loading; (ii) fl = 2.5 for members under dead plus live load acting in conjunction with wind loading; (iii) fl = 1.75 for members under dead plus live load acting in conjunction with earthquake loading. A higher value of fl for connectors ensures that the connections designed are stronger than their adjoining members, A lower value offl for members under the action of a combination of dead, live, wind or earthquake loading reflects the improbability that these loadings will act simultaneously. From the above discussion, it can be seen that in ASD, the safety of the structural member is evaluated on the basis of service load conditions, whereas in PD or LRFD, safety is evaluated on the basis of the ultimate or limit load conditions. In addition to strength, the designer must also consider the stiffness of the structure. One important criterion related to stiffness is that the structure or structural component must not deflect excessively under service load conditions. Thus, regardless of the design
8
W.F. Chen
method, one should always investigate such serviceability requirements as deflection and vibration at service load conditions.
3 COMPRESSION MEMBERS A column can be thought of as a special case of beam-column in which the primary bending moments are absent. The derivation of any column equation can be related to one of the two following approaches: (a) The eigenvalue approach, where the load at which lateral displacement of a perfectly straight or ideal column commences is found by an eigenvalue analysis. The load is referred to as the buckling load and is defined as the load that corresponds to a state of neutral equilibrium of the perfectly straight column or the lowest load at which a bifurcation of equilibrium takes place. (b) The stability approach, in which the load-deflection curve of a column is traced. The peak point of this load-deflection curve is referred to as the stability limit load, which is the maximum load that the column can carry. 3.1 Pin-ended columns When a perfectly straight elastic column is loaded by an axial thrust, P, lateral deflection will start as the load reaches the Euler load, P~: p~
P,
where P, = A F,. •
)ql = L r x~ rr-~E
in which A = cross-sectional area, F,. = yield stress, L = length of column, r = radius of gyration, and E = modulus of elasticity. The Euler load P~ marks the point of transition from stable to unstable equilibrium of the column. Once the Euler load is reached, the deflection of the original straight elastic column will increase without bound. In order to take into account the nonlinear stress-strain behavior of the material once the proportional limit is exceeded prior to buckling, the reduced modulus (double modulus) or the tangent modulus theories can be used. In the reduced modulus theory, complete strain reversal at the convex side of the column at buckling is assumed. As a result, the elastic
Design of beam-columns in steel frames in USA
9
modulus E governs the relation of stress to strain at the convex side of the column whereas the tangent modulus Et governs the stress-strain relationship at the concave side during buckling. The reduced modulus load is given by
Er
Pr = Pe -~-
(6)
where the reduced modulus Er is a function of E, Et and the shape of the cross-section. The reduced modulus load Pr marks the point of bifurcation of equilibrium for a perfectly straight inelastic column, In the tangent modulus theory, the axial load is assumed to increase as the member bends subsequent to reaching the critical load. As a result, there is no unloading of any fiber over the cross-section, and the tangent modulus Et will govern the stress-strain relationship of the entire crosssection. The corresponding tangent modulus load is
El
P, = Pc ~-
(7)
The tangent modulus load is the lowest load at which bufurcation of equilibrium can take place. However, due to imperfections that are present in columns, experimentally determined maximum loads of columns tend to agree more with the tangent modulus load. In view of this, the Structural Stability Research Council (SSRC) (formerly Column Research C o u n c i l - CRC) recommended that the tangent modulus load best represents the maximum load that an inelastic column can carry. 3.1.1 CRC curve The CRC curve forms the basis for the column design curves in the USA. Its development was based on the eigenvalue concept and is composed of two regions: (a) An inelastic region (0 ~
-
l -
0 . 2 5 ~.~,
(8a)
P,. (b) An elastic region (X0 > v~), which is the Euler load: P P,.
-
1 , ;t~
(8b)
Since the CRC curve was developed on the basis of the eigenvalue approach, it is only valid for perfectly straight columns.
I~. F. Chen
10
3.1.2 A I S C A S D curve
To account for initial crookedness or other geometric imperfections that are inevitably present in columns, the CRC curve was divided by a variable factor of safety of
v.s. = 5+
v2/
(9)
in the inelastic region and a constant factor of safety of 23/12 in the elastic region to form the AISC ASD curve. 3.1.3 A I S C PD curve
The AISC PD curve was established by multiplying the AISC ASD curve in the inelastic range by 1.7, where 1.7 is the load factor for gravity load for plastically designed structures, The elastic range is not defined for the AISC PD curve since, under the provision, the nondimensional slenderness parameterk0 for columns designed using plastic design procedures must be less than v 2. 3.1.4 A I S C L R F D curve
The LRFD curve s was established by a curve-fitting procedure, with the SSRC curve 2 to be discussed further in Section 3.1.5. It has the simple form: P _ j'expl-0.419A,0l P,. ],0.877 A,~72
A,,, < 1.5 X,, > 1.5
(10)
Table 3 summarizes the expressions for the aforementioned curves and Fig. 2 compares the CRC curve and the three AISC curves with the three SSRC multiple column curves described in what follows. Note that the AISC column curves are to be used in conjunction with their respective design format as given by eqns (1), (2) or (3). 3.1.5 S S R C multiple column curves
The SSRC multiple column curves were developed on the basis of the stability concept. Load-deflection curves for a number of columns with measured residual stresses and initial crookedness of 0.001L were traced by numerical techniques, using a computer model. The peak point of each of the load-deflection curves gives the maximum load that the column can carry. By tracing the load-deflection curve of each column at different slenderness ratios, the column curve for that particular column can be obtained. A total of 112 column curves were traced s for different types of columns, which were then categorized, and a set of three
Design of beam-columns in steel frames in USA
11
0 0
E
tO
~z
0
Q
m
It . _J
~
0
>
0 tO
-
tO
E
CO
o 0 !-
L)
B
> tO
L) L)
o
~5 ee"
ffJ O tO
®
CO •
oJ >
o
=
;
~
0
¢J
Q.
tO C4
<
6
c
0 Q
m
OD
,¢~
t'M
Q
r.i
W. F Chen
12
TABLE
3
Summary of Column Curves
Column curves
CRC curve
Cblumn equations
-- = 1-Pv 4 P
e,,
AISC ASD curve
P
--
-
Zo <. x , ~
1
Zo >
4
=
P
12
1
p,.
23
Ag
AISC LRFD curve
P
Zo -<< ~, 2
"~-0 )" \ ' 2
1.7 AISC PD curve
x- 5
l-
-
.a.0 <
P P~
exp (-0"419/lo)
P
0.877
P,.
;t~
.~o < 1.5
J. 0 >
1.5
multiple c o l u m n curves was developed. Each is representative of a specific category of columns. The expressions for these three multiple column curves were given elsewhere? For design purposes, it is more convenient to represent these curves with the R o n d a l - M a q u o i equation: ~0 P P-~I =
1 +r/+Zo 2~ .....
1 , 2A~ v ' ( l + 17 + A.~)2 -4,,10
(11)
where 1/ = a(Z0 - 0.15)
a: /
0. 103 (SSRC curve 1) 0.293 (SSRC curve 2) 0.622 (SSRC curve 3)
These curves, in their original form, and the R o n d a l - M a q u o i curves, are shown in Fig. 3. S S R C curves 1, 2 and 3 are closely c o m p a r a b l e to the
Design of beam-columns in steel frames in USA
13
0 o o
e~
t~ to b.. ,.z
I
~D Q to
Q ,.a
to
A
,.z
.g o
h
"3
, ~D to
a "3
uJ
i
0
_d
0 rr" o)
0 Z 0
"6 ?
I I
I
I
to
¢y
I
I
6
o
Q o
o
~_lw"
o
r4
14
IV.. ~: Chen
European multiple column curves a0, b and d, respectively, recommended by the European Convention of Constructional Steelworks (ECCS) and are given in Ref. 11. Although the SSRC multiple column curves are more realistic than the CRC curve in representing column strength, the lack of smooth transition from one curve to the other means that there will be a jump in column strength, which is physically not acceptable. Furthermore, although both residual stresses and initial crookedness were considered in their development, these parameters are not clearly identified in the column equations. Lastly, since the initial crookedness used in the development of these curves was assumed to be 0.001L, they may not be representative of the strength of columns with initial crookedness which is not 0.001L. To overcome these drawbacks, Lui and Chen ~2 have developed a column equation based on a physical model, which is applicable to columns with any value of initial crookedness. The equation has the form e
+ (l +
- v/l
+ (1 +
-
where E
Et rr
E
in which P =
magnitude of initial crookedness pin-ended length of the column '
f -- shape factor about the axis of bending; c = distance from neutral axis to extreme fiber: = plastification parameter, which is a function of the cross-sectional shape, axis of bending and the material of the column. The validity of the above approach has been demonstrated 13 by comparing the results obtained from eqn (12) with that from an elasticplastic analysis of imperfect columns, using a computer model developed. Good correlations were observed.
Design of beam-columns in steelframes in USA 3.2
End-restrained
15
columns
The pin-ended column represents an anchor point to which other columns with different end conditions are referenced. Once the strength of a pin-ended Column is known, the strength of the same column with other end conditions can be obtained by the use of the effective length
factor, K. The elastic effective length factor is defined as K =
Pv~P~r
(13)
where Pcr = critical load of the end-restrained column. For a perfectly straight elastic column, the critical load can be found by writing the differential equation conforming to the specific end conditions. The eigenvalue to the characteristic equation of the differential equation is the critical load for that column. The effective length factor can then be found from eqn (13). A perfectly straight column restrained elastically against rotation at ends A and B by rotational springs with spring constants RkA and RkB, and restrained elastically against translation at end B by a translational spring with spring constant Tk is shown in Fig. 4(a). The corresponding free body diagram of the column and the rotational springs are shown in Fig. 4(b). The critical load of this column can be found by first writing moment equilibrium equations of the three free bodies and then setting R~8%Fm %a -
~
: %a
MBA~ j P
mk
Maaf ~ - ~
Tk &
MAB
TkA
El constant
A
P
jL~ P
MAB Tk ~
~
TkA
RkABA
Ca)
(b)
Fig. 4. An elastically restrained column with flexible joints in frames.
16
W. F. Chen
the determinant of the coefficient matrix of the equilibrium equations equal to zero, i.e. SIA + RkA
S1B
- S 2 ( A + B)
SiB
S1A + Rk8
- S 2 ( A + B)
- S 2 ( A + B)
- S 2 ( A + B)
det
-2S3(A + B) -
P
= 0
(14)
+ Tk
where E1 S1
-
A =
E1 L 2'
$2-
L'
S3
-
E1 L3
k L ( s i n k L - k L coskL) 2 - 2 cos k L - k L sin k L '
B
=
k L ( k L - sin k L ) 2 - 2 cos k L - k L sin k L
in which
The smallest value of P that satisfies eqn (14) is the critical load for the column of Fig. 4(a). The effective length factor can then be obtained from eqn (12). Although the effective length factor discussed above was derived from elastic analysis, limited studies ~4-~ have shown that this will be conservative in most cases for initially crooked elastic-plastic columns. In real structures, a column usually exists as part of a frame. The effective length factor for columns as part of a frame has been derived by Julian and Lawrence ~7 and LeMessurier. t8 ~9 By assuming that (i) the girders are rigidly connected to the columns at all joints, (ii) all columns of a story buckle simultaneously and, at the onset of buckling, the rotations of the ends of the girders are equal and opposite for braced frames and equal in magnitude and direction for unbraced frames, and (iii) all members are prismatic and the behavior is elastic, the following equations for the effective length factor can be determined.~7 For braced frame (sway prevented): GAGB
4
~ \
/
+ \
)
+ 2 tan(n/2K) rr/K
-
1(15)
For unbraced frame (sway permitted): GAGB(n/K) 2 - 36
6(GA + GB)
n/K ..........
tan(n/K)
0
(16)
Design of beam-columns & steel frames in USA
17
where GA and Ga are the stiffness distribution factor for t h e A th and B th ends of the column, respectively. They are defined as
G = ~,(I/L)co~um, E( I/L )girjer
(17)
The summation in the numerator covers the stiffnesses of all columns that come together at a joint, and the summation in the denominator covers the stiffnesses of all the beams that come together at the same joint. Equations (15) and (16) form the bases for the alignment charts that are used extensively by designers in the USA. Some modifications can be made to the G factor to make the alignment charts more versatile:
Braced frames 1. If the far ends of the girders are fixed, divide G by 2.0. 2. If the far ends of the girders are hinged, divide G by 1.5. 3. If the girders are connected to the columns by semirigid connections with rotational stiffness Rk at both ends, replace ~Z(l/L)girde r by ~(I'/L)girder, where 1
I' =
2EI
I
(18)
Unbraced frames 1. If the far ends of the girders are hinged, multiply G by 2.0. 2. If the far ends of the girders are fixed, multiply G by 1.5. 3. If the girders are connected to the columns by semirigid connections with rotational stiffness Rk at both ends, replace ~.,(I/L)girde r by ~.,(I'/L)girder, where 1
I' = 1+
6EI
I
(19)
Figure 5 shows a plot o f e q n (18) and (19), which can be readily used to determine the modified girder moment of inertia I'. For cases in which the columns become inelastic, a modified value of G was recommended by Yura 2° to be used for the alignment charts. The modified G is
G =
Z (EtI/L )col,,m, ~Z(El/L)gird~r
(20)
W. F. Chen
18 10
0.8 0.6 ~ 0.4
~
peSway ~ prevent~ed rmqtted
0.2 o-
ors
~.'o
;5
2:o
EI/RkL
2'.~
3'-o
3'.5
Fig. 5. Chart for modified beam moment of inertia to include the influence of the flexibility of beam-to-column connections. An alternative a n d more accurate approach to determine the K factor for columns in sway frames was proposed by LeMessurier. ~9The expression is
K~ _ .2~i [~,e+ ~.(cLe)l
(21)
where the subscript i refers to the i th c o l u m n of the story, a n d Z P = sum of all vertical forces acting on the story: Pi = axial force on c o l u m n i whose effective length factorKi is required a n d is calculated based on first-order analysis; Iz = m o m e n t of inertia of c o l u m n i; 6(GA + GB) + 36 /3 = 2(GA + GB) + GAGB + 3'
CL = [/3K2 L .2
1] in which K is calculated from eqn (16) or the align-
ment charts. To account for the inelastic behavior o f the column, Ki is m o d i f i e d as
rr:rli [ E P + ~](CLP)]
K~-
e,
where
PV
L
~(~
J
(22)
Design of beam-columns in steel frames in USA
19
In the LeMessurier equation, fl is a factor to account for the endrestraint effect of the column, and CL is a factor to account for the P-6 effect in the column. Alternatively, a simplified form of the LeMessurier equation can be written by assuming that: (1) the total gravity load associated with the sway buckling of a story is equal to the sum of the buckling loads of all columns in a story that provide the story sideway resistance; and (2) the individual column buckling load is calculated, using the effective length factor obtained from the alignment charts. The simplified equation has the form:
rr2EIi K~ -
EP
L~P, EP~k
(23)
Figure 6 shows a comparison of the effective length factor K for column CD of a leaned-column frame evaluated using eqns (21) and (22), respectively. Good agreement is generally observed. The use of eqn (21) or (23) is recommended by the author over the alignment charts method since they are applicable for cases such as the leaned column problem and for cases in which the axial forces in the columns of a story differ significantly. If a computer program for buckling analysis is available, a still more accurate evaluation of the K factor can be obtained using eqn (13), where Pe is the r u l e r buckling load of column i, and Per is the axial force in column i at incipient buckling of the frame. It is interesting to note here that, during the course of the development 8
Ct.P
p
+B [~ l_ LclIc 1 I. 6 ~L~
L e M e s s u r i e r K eqn(21)
---- Mo,~i,ie,J K
Itc
. ~
I.'7 ~-~ .-'J~"
.~4°1
_.-'.~'-
I I
~
2.0 I I
I G = ([c/Lc) I ( [ b / 2 L b )
Fig. 6. Comparison of effective length factor K for a leaned column system.
20
I4<. F. Chen
of the 1986 LRFD interaction equations (eqns (88) and (89)) tbr the design of beam-columns to be discussed in Section 7. much discussion has been focussed on the need and validity of using the effective length factor K in these equations. Although attempts were made to tbrmulate general interaction equations without using K, it was found that this was almost impossible if the interaction equations were to be versatile enough for a wide range of slenderness ratios and load combinations. The final decision was to retain K in these interaction equations. The AISC specification describes a procedure by which the effective length of factor K is determined from two alignment charts 17 that corresponds to the two cases as given in eqns (15) and (16). This will be discussed later in Section 7. Once the K factor for an end-restrained column is found, the strength of that column can be evaluated from the column strength formulas presented above for a pin-ended column by replacing A.0 by A~ where A.~. = K~0
(24)
4 FLEXURAL MEMBERS When an ideal I beam is loaded in the plane of the web. in-plane deformation will occur at commencement of loading and increase as the applied loads increase. As the magnitude of the loads approaches a certain critical value, out-of-plane deformation and twisting of the beam will increase rapidly and the beam will be in a state of instability. The critical load is referred to as the lateral-torsional buckling load of the beam. It is obtained as the eigenvalue of the bifurcation problem of the beam. The theoretical critical moment for the elastic lateral-torsional buckling of a beam for various support and loading conditions is given 9as Mc,- = C, ~
C2g + C3j +
Ceg + C3j + ~
+ n2EC w (25)
where g = distance from shear center to the point of application of the transverse load taken positive when the load is below the shear center; j = S + (1/21,)fAY(X 2 +y2)dA. in which
Design of beam-columns in steel frames in USA
21
S = distance from centroid of the cross section to the shear center; J = torsional constant; C,, = warping constant; /v = moment of inertia about the y-axis (weak axis); C~, C2, C3 = coefficients which are functions of loading and support conditions. 9 Beams used in practice usually fail at loads below Mcr due to the presence of geometrical and material imperfections. For such beams, lateral deformation and twist commence at the start of loading, and increase as the magnitude of the load increases. The maximum load is obtained from the peak point of the load-deflection curve, which is referred to as the ultimate load of the beam. To obtain this ultimate load, a load-deflection approach must be used and a numerical analysis is inevitable. Analytical and experimental studies to determine the ultimate loads of beams are available in the literature.2,. 22 In general, the presence of residual stresses and geometric imperfections will decrease the ultimate strength of the beam with the effect of residual stresses causing early yielding. Depending on the types ofloadings and residual stress distributions, the ultimate strength will be affected differently. The effect of geometrical imperfections aggravates the deformation and so the beam will change into an unstable state at a lower load, being more pronounced for slender beams. The present AISC Specifications '-2 for PD adopted an empirical formula based on test data of the form: Mo,, =
1.07
(L/rv)~/F,'~-,] . M-60 J 1rip, < Mp,,
(26)
where = L = = = p.v
plastic moment capacity about the x-axis (strong axis) (in kips); unbraced length of beam (in); radius of gyration about y-axis (weak axis) (in); yield stress (ksi).
This represents the ultimate lateral torsional capacity of lateral unsupported beams. Equation (26) is plotted in Fig. 7. In the same figure, the numerical results obtained by Galambos 23 and Vinnakota 24 are also plotted. It can be seen that eqn (26) is not conservative for Vinnakota's results. However, in Vinnakota's study, the worst effect on the maximum strength of the beam was chosen. The initial crookedness at midspan was taken as
W. F. Chen
22
8 E 0 o
"I=
L .~
c
¢:
...4.
/ 0
D'~ C
/
....3
"F..
~J ¢~ "U
\/
C
0
c
.-
>
L
/
/ o
0
!
O_ 121 ~
I
E=
k
Oc
E
e,
I
!
I
.'!
0
0
6
0
<
I
0 0
Xdl,.~
xn~
C,
r.:
23
Design of beam-columns in steelframes in USA
O.O01L in both principal planes, and the effect of residual stress was included. Real sections have additional strength reserves over design values; for example, the actual yield strength of the material is usually higher than its nominal value and strain hardening may set in before failure. Based on a study reported by Yura et al., 25 the LRFD Specification3 adopted the following expressions for the ultimate moment capacity of beams: For c o m p a c t sections (sections in which flange and web local buckling are precluded) defined as br
65
(27)
2,-7 <
(28)
tw
where bf tf hc tw Fy
= = = = =
width of flange (inches); thickness of flange (inches); depth of web clear of fillets (inches); thickness of web (inches); yield stress of the material (ksi);
the limit stage may be lateral-torsional buckling. Defining (Fig. Lp = 300rv
8): (29)
rvXl
Zr - (F.~-Z]Wr) v/(i -I- V/(1 + X 2 ( F v - Fr) 2)
(30)
Lb = unbraced length of member, where X,rv Sx G A Fr
-= = = --
Sx
-
,
X2-
~
(31)
radius of gyration about the weak axis (in); section modulus about strong axis (in3); shear modulus (ksi); cross-sectional area (in2); compressive residual stress in flange, taken as 10 ksi for rolled shapes and 16.5 ksi for welded shapes;
W. F. Chen
24
,..A e-,
Z C
0"
L
u ql r~ I,IJ
C 0 -J
E
13-
\
-J
\
\
L .-J
\
\
u (3 r~
E L l::x, ,,.j
u
¢'3
g5"
nl, N ' a : ) u ~ l . s ! s a J
II~JnXal~.
eu~uJoN
Design of beam-columns in steel frames in USA
25
then, i f L b ~< Lp M~_,.
=
IfLr, <
(32)
Mpx L b
~< Lr
< Mpx
(33)
= j'1-75 + I'05(MI/M2) + 0"3(MI/M2) 2 < 2.3 [1 for unbraced cantilevers
(34)
where
Cb
in which M~ is the smaller and M2 is the larger end-moment in the unbraced segment of the member; Mj/M2 is positive when the member bends in reverse curvature, negative when the member bends in single curvature. Mr
= Sx(Fy -
fr)
(35)
IfLb > Lr
Mu.,
27
=
CbMocr
=
Cb~bb
~.GJ+
71"-
vECw] <-
where Moor = elastic lateral torsional buckling load of a simply supported I beam under pure equal end bending moments. The equations given above are applicable only to compact nonhybrid I sections in which yielding or lateral-torsional instability is the limit state. The applicable expressions for other sections, including symmetric box sections, solid rectangular bars, hybrid sections and tee and doubleangle sections, are given elsewhere. 3 Figures 9(a) and (b) show a comparison of the variation of the nondimensionalized ultimate moment resistance M,/Mp as a function of the slenderness ratio Lb/ry of a beam using the LRFD as well as the ASD and PD design approaches. The beam is a W18 × 76 section with a yield stress F~. = 36 ksi and a compressive residual stress in the flange Fr = 10 ksi. The Young's modulus E was taken as 29 000 ksi and the shear modulus G as 0-385E. Figure 9(a) corresponds to the case in which M~/M2 = - 1 (i.e. Cb -- 1.0), and Fig. 9(b) corresponds to the case in whichM~/M2 = 1 (i.e. Cb = 2"3). Note that the ultimate moment resistance of the beam is larger for doublecurvature bending than for single-curvature bending.
26
W. 1~: Chen
Lp/ry = 50 1.00 M u / Mp 0.80
PD 0.60
0.40
0.201 W 18 X 7 6 :
SINGLE CURVATURE BENDING Cb = 1 i 50
2=5
J 75
L 100
~ 125
i 150
{ a )
Ll~/ry= 50
J 175
200 =
Lb/ ry
L r l ry = 153
1.00 M u / Mp 0.80
0.6C
Lc/~ri= 5 3 . 6
"
Lu/ ry=
0.40
0.20
W 18 X 7 6 :
DOUBLE C b
=
CURVATURE BENDING
2.3
I
£.
I
I
I
50
100
150
200
250
(b)
,-.I
300
I
350 Lb/ ry
Fig. 9. Comparison of beam curves tbr ASD, PD and LRFD.
4oo"
Design of beam-columns in steelframes in USA
27
5 D E S I G N I N G FLEXURAL MEMBERS WITH LRFD, SSRC AND ECCS APPROACHES It should come as no surprise, at this time that designing for lateraltorsional buckling is a complex and challenging problem. This is because the resistance of the beam against lateral-torsional buckling is dependent on many factors, such as the geometry and end conditions of the beam, the amount and nature of bracing, the type and manner of loading, etc. Nevertheless, there are two undebatable facts. A beam with a low slenderness ratio should be able to develop its full plastic moment Mp, and a beam with a large slenderness ratio should behave in a fully elastic manner, so that the elastic critical moment Mcr represents the ultimate moment capacity of the beam. The main problem is that the behavior of a beam with intermediate slenderness ratios is neither fully plastic nor fully elastic, hence, neither the plastic m o m e n t Mp nor the elastic critical moment Mcr can be used to represent the ultimate moment capacity Mu of the beam. The ultimate moment capacity Mu of a beam with an intermediate slenderness ratio should fall between Mp and Mcr and its value should provide a smooth transition from Mp to Mcr. In most specifications, the choice of this transition moment is rather empirical. For example, with the ASD method, a parabola is used to represent this transition range, and with the LRFD method, a straight line (eqn (33)) is used for this transition range. In the discussions that follow, two other empirical methods that represent the beam resistance in the transition range will be presented.
5.1 S S R C approach
In the SSRC approach, it is assumed that the stability behavior of beams is analogous to the stability behavior of columns, hence, the lateraltorsional buckling resistance of a beam is represented by the SSRC multiple-column curves. Using the Rondal-Maquoi approach form (eqn (11)) with P/Py replaced by Mu/Mp and A,o replaced by Ab, where Ab = v/np/Mcr, one obtains M. Mp
(1 + r / + X~,) - v/[(1 + r / + Ab2)2 -- 4A~,] 2A~
where 17 is defined the same as for columns in eqn (11).
(37)
28
W. I~2 Chen
5.2 ECCS approach In the E C C S a p p r o a c h , the n o m i n a l m o m e n t c a p a c i t y for the t r a n s i t i o n r a n g e is r e p r e s e n t e d by
( ,)in
Mp -
1 + A2b"
(38)
w h e r e n is a coefficient t h a t varies f r o m 2.5 to 1-5, d e p e n d i n g o n the criteria set by the v a r i o u s n a t i o n a l c o d e - w r i t i n g b o d i e s in W e s t e r n E u r o p e a n d Ab is the b e a m ' s s l e n d e r n e s s p a r a m e t e r , d e f i n e d as An =
V/ Mp/Mcr •
5.3 Design example and comparisons C a l c u l a t e the n o m i n a l m o m e n t c a p a c i t y o f a W16 x 36 s e c t i o n b e n t a b o u t its s t r o n g axis u s i n g the following: 1. A I S C L R F D a p p r o a c h . 2. S S R C a p p r o a c h . 3. E C C S a p p r o a c h . T h e b e a m is s i m p l y s u p p o r t e d a n d s u b j e c t e d to a u n i f o r m m o m e n t . Use E = 29 000 ksi, G/E = 0-385, F~, = 36 ksi, Lb = 150 in, Fr = 10 ksi.
Solution." W16 × 36 s e c t i o n properties: A = 10.6 in 2, At- = 3 in 2, d = 15-86 in, bf = 6.985 in, rT = 1-79 in, S, = 56.5 in3,!,, = 24.5 in4, r, = 1'52 in, J = 0-545 in 4, Z , = 64.0 in 3, Cw = 1450 in 6. C~, = 1.0 ( c o n s t a n t m o m e n t )
Mp = Z,F,. = 2304 in kips Me,- = CbMocr "-Moor
Lb
EI,.GJ +
EI,.ECw]
= 2765 in kips
1. AISC LRFD equation Lp =
300r,. /--~- -
76 in
I V/ L,. = ?iir~,2X F,. [1 + x"[1
:
219in
29
Design of beam-columns in steel frames in USA
Since Lp < Lb < Lr, therefore, by using eqn (34) with Mr = Sx(Fv we get from eqn (32): M~ = Cb Mp - (Mp - M~) L-~-L--~p
Fr),
= 1544inkips
2. S S R C equation
A,b = Mv~M~~ _ ~//23042765 - 0.913 Using SSRC curve 2, that is, where a = 0-293 in eqn (11) for r/, we have, from eqn (37): Mu = (1 + 1/+A,~,)- ~/[(1 + q + A,~,)2 - 4A,~] Mp = 1534in kips
3. ECCS equation
A,b =
M~_
0.913
Using n = 2.5 in eqn (38), we have M,. =
1
1 -t--/~"J
Mp = 1893 in kips
It is seen that the L R F D and SSRC equations give a comparable result, while the ECCS equation significantly overestimates the flexural strength of beams.
6 S E C O N D A R Y P-DELTA E F F E C T S One of the major differences between a beam and a beam-column is that, in addition to primary deflections and moments, there are secondary deflections and moments present in a beam-column due to the axial force in the beam-column acting on the primary deflections. In general, two types of secondary effects can be identified: the P-6 effect a n d the p-A effect. These secondary effects cause the m e m b e r to deform more and thus increase the stresses in the member. As a result, they have a weakening or destabilizing effect on the structure.To ensure a safe design, these effects must be considered.
30
W. F. Chen
6.1 P-6 effect Consider a beam-column, as shown in Fig. 10, with joint translation prevented; the forces M~, M2, Q and q will produce primary moment MI and primary deflection y~. The axial force P will act on the primary deflection to produce an additional momentMn and deflectionyn. These additional effects are the result of the so-called P-6 effect. This effect will increase the member instability; hence, it is also referred to as the member instability effect. The total moment along the beam-column will then be M = Ml+Mn
(39)
and the total deflection is Y = YI+Yn
(40)
In order to ensure a safe design, it is necessary to sort the maximum moment Mmax in the member. This maximum moment can be found by solving the differential equation of the beam-column with the proper boundary conditions at the ends. However, for design purposes, it is more convenient to use a simplified approach to get Mmax. By assuming that (i) the secondary moment Mn is in the form of a half sine wave, and (ii) the maximum deflection Ymax (= 6 = 61 + 6n) occurs at midspan, we can write Trx
-EIy;~ = MI, = P6 sin ~ -
(41) Q
P Mj ~
I
MI~
M2
M I = f(M I ,M2,Q,q,x)
I
M2 p
MII=py Fig. I0. P-6 effect,
Design of beam-columns in steelframes in USA
3I
Integrating eqn (41) twice and enforcing boundary conditions gives
Yn - E1
sin ~-1rx
(42)
from which the secondary deflection at midspan is P 6n = Ynlx=L/2 = 6 ~
(43)
since 6 = 6~ + 6 .
(44)
Substitution of eqn (43) into eqn (44) and solving for 6 gives
6 = (l_l-p/p)
6,
(45)
Now, further assuming that the maximum primary moment occurs at or near midspan, one can write Mmax = MImax+ P~
(46)
Upon substitution ofeqn (44) into eqn (45) and rearranging, one obtains m~x
= [ I +- v/P/Pe] P/Pe
(47)
6lee Mlmax
(48)
where v/-
l
Defining
Cm = 1 + ~P/Pe
(49)
equation (47) becomes
Mmax = (1 £ ~ / P e ) M'max = ArM'max
(50)
The term in parenthesis in eqn (50) is referred to as the moment amplification factor, Av. For beam-columns with different end conditions, the Euler buckling load Pe in eqns (49) and (50) should be replaced with the corresponding buckling load Pek = zr2EI/(KL) 2 to reflect the effect of end restraint on the buckling strength.
W . F . Chen
32
Figure 11 shows the values o f ~ and Cm for several different load cases and end conditions. Note that, because of the assumption used in writing eqn (46), the definition for Ig in eqn (48) is only applicable to the two simply supported cases (Cases 1 and 4). For the other cases in which the maximum first-order moment occurs at the end(s) (Cases 2, 3 and 5) or occurs both at midspan and at the ends (Case 6), the exact values of the maximum moments, taking into consideration the effect of the axial force P, are first evaluated; the values for ~, for these cases are then obtained from calibration using eqn (47). A special case arises when there are no transverse loadings on the beam-column. If the beam-column is subjected to end moments only. the Cm value for design is redefined as Cm = 0 " 6 - 0"4(M1/M2) >~ 0"4
Case
(51)
~
Cm
1 l[IlllllllllllllIllNl _
I
= illIlllIllI[lllIlllll]~,~
I I
1.0
I
!
-0.4
1-0.4 P/Pek
-0.4
1-0.4 PIPek
-0.2
1-0.2 PIPe,
-0.3
1-0.3 P/Pek
-0.2
1-0.2 P/r'ek
6
I~io 1 1 V ~ h . ~ ~f .z a n d C... f a c l o r s for b e a m - c o l u m n s undcr transverse loading.
Design of beam-columns in steel frames in USA
33
where Mi is the smaller and M, is the larger end moment. MI/M2 is positive when the member bends in reverse curvature, and negative when the member bends in single curvature. Note that, from eqn (51), the Cm value can be as low as 0-4, which means the moment amplification factor can be less than unity (Fig. 12) according to the present AISC Specification.2 This is not physically meaningful, because if the amplification factor is less than 1-0, the maximum moment of the beam-column occurs at one of the ends rather than within the span, and we should take A v = 1"0 instead, rather than taking the value that is less than 1.0. An improved AF factor has been proposed recently by Duan et aL 26An evaluation of the Cm factor, as used in the AISC LRFD specification, has been reported by Zhou and Chen. 27
6.2 p-A effect When lateral forces 12H act on a frame, the frame will deflect laterally until an equilibrium position is reached (Fig. 13(a)). The corresponding lateral deflection may be calculated on the basis of the original configuration and is referred to here as the first-order deflection and is denoted by A~. Now, if in addition to ZH, vertical forces Y'.P act on the frame, these forces will interact with the lateral displacement A~ caused by E H to drift the frame further until a new equilibrium position is reached. The lateral deflection that corresponds to the new equilibrium position is denoted by A (Fig. 13(b)). The phenomenon by which the vertical forces ZP interact with the lateral displacement is called the P-A 6
i
//
I
o
0
0.2
0.4
0.6
0.8
I .0
P
Fig. l]. M o m e n t magnification factor AI:.
I4(. F. Chen
34
:rH
/ /
/ /
/
I Ih
!
r ///,
/1
illl
{a) Fiqst ~ o r d e r
__A__
Analysis
~P
YH
1i..
~P
---
-77
/ /
/
I
I
I /II
_
1
III
(b) S e c o n d - o r d e r A n a l y s i s
Fig. 13. P-A effect.
effect. The consequence of this effect is an increase in drift and an increase in overturning moment. Since these additional deflection and overturning moments have detrimental effects on the stiffness and stability of the frame, they should be considered in design. In order to determine accurately the final deflection A and the actual moment M taking into account the P-A effect, a second-order analysis based on the deformed geometry of the frame is necessary. Second-order analysis usually entails an iterative process. However, for design purposes, it is more convenient to use simplified approaches without resort to iteration to estimate A and M. Some of these approaches are given below.
6.2.1 Story magnifier m e t h o d 2~-3° By assuming that (i) each story can behave independently of other stories, and (ii) the additional moments in the columns caused by the PA effect are equivalent to that caused by a lateral force o f £ P A / h , where h = story height, the sway stiffness of the story can be defined as
Design of beam-columns in steelframes in USA
horizontal force SF = lateral displacement
Y~H
Y-,H + Y_,PA/h
A1
A
35
(52)
Solving eqn (52) for A gives [
1 ~PAI
A = 1
AI
(53)
Y_,Hh
Thus, it can be seen that the final deflection A can be estimated from the first-order deflection A~ by multiplying the latter by an amplification factor (the quantity in square brackets). Since the story sway moments (moments induced as a result of swaying of the story) are directly proportional to the lateral deflections of the story, one can write M =
( 1 - Y,P-AI/Y, 1 H h ) Mlsway = AFMlsway
(54)
where M = maximum end moment accounting for the P-A effect; M~sway = maximum first-order end moment as a result of swaying of the story; Ar = moment amplification factor. A special case arises when swaying of the frame is due to asymmetry of the applied vertical loads (Fig. 14(a)) or to asymmetry of the frame itself(Fig. 14(b)). If this is the case, a fictitious support can be introduced to prevent the frame from swaying. As a result, a holding force will be developed in the support. The horizontal force that balances this holding force will then be used in place of E H in eqns (53) and (54) to obtain the final deflection and moment (Fig. 15). If swaying of the frame is due to both applied horizontal forces Y,H and asymmetry of applied vertical loads and/or frame geometry, a similar approach can be used. The frame subjected to these forces is analyzed with a fictitious support to prevent it from swaying. The force that balances the horizontal reaction of the support is used in place of Y,H for the amplification factor in eqns (53) and (54). The story magnifier method works for frames with stiffbeams in every story so that a point of inflection occurs in every column in a story. 6.2.2 Multiple-column magnifier m e t h o d 2°
The multiple-column magnifier method, also known as the modified effective length method, is a direct extension of eqn (45), under the
36
W. F. Chen
l
I
lrp
_ ~_ A
/
(a)
/~ ~-.'~
~
-
t
/ /
(b) T7
Fig. 14. P-D effect arising from load/frame asymmetry.
l'"
11/5.-:,:::-0-1,;o° I/ &
AI
/
I
IIEP
_12"
M=/'l~h~l ) MIs"aY
~_R
===~
I co,oooP Lj Eo,,,oo.o,= I/ L ""'" k
Fig. 15. Procedures to determine the second-order deflection A and moment M for asymmetric applied load/frame. postulation that, w h e n instability is to occur in a story, all c o l u m n s in that story will b e c o m e unstable simultaneously. As a result, the term P/Pe in eqn (45) is replaced by Ir,P/lC,Pe where the s u m m a t i o n is carried through all c o l u m n s in a story. Using the same a r g u m e n t as in the story magnifier m e t h o d that the sway m o m e n t s are directly proportional to the lateral deflections of the story, the m a x i m u m e n d m o m e n t M a c c o u n t i n g for the P-A effect takes the form as follows:
Design of beam-columns in steelframes in USA
1 p
M = 1
Mlsway = AFnlsway
37
(55)
EPek
where Mlswayis the m a x i m u m first-order end moment as a result of swaying of the story, and AF = 1/(1 -- (Y,P/ZPek)) is the moment amplifi-
cation factor. If the P-D effect is not significant, the values of the moment amplification factor evaluated using eqns (54) and (55) are closely comparable to each other.
6.2.3 Negative brace method 31 In the negative braced method, a fictitious pin-ended diagonal bracing member with axial stiffness EA, given by EA -
-(£e/h)L cos20
(56)
where h = story height, L = length of bracing member, a n d 0 = angle of inclination of the bracing member, is inserted in every story. The structure with the negative braces is then analyzed by a first-order analysis to determine the moment. This m o m e n t corresponds to M in eqns (54) and (55). In order to account for the decrease in column stiffness due to the presence of axial force, a flexibility factor, y, ranging from 1-0 to 1.22, can be inserted into the term Y,P (i.e. E~P) in eqns (54) to (56) to give a better estimate of M. The lower bound for ~, (i.e. y = 1) corresponds to the case in which the columns remain more or less straight after deflection, and the upper bound fory (i.e. y = 1.22) corresponds to the case in which the columns are on the verge of buckling, with the deformed shape approaching that of a sine curve. The above approaches give reasonably accurate results if the moment amplification factor is under 1.5. If the P-A effect is significant, the approach suggested by LeMessurier t9 is recommended. In his approach, the amplification factor is defined as 1
AF = 1-
EP
(57)
EpL - Z C L P
where ZPL = EHh/Ax and CL is as defined in eqn (21). The present AISC Specification,2 based on the model shown in Fig. 16
38
W. F Chen
®
+
t,--
.,,..
~-I
g
o
<
fl
L
t._
8 H
-r
Design of beam-columns in steelframes in USA
39
and assuming a sine curve for the P-A moment, developed a moment amplification factor based on the procedure outlined in Section 6.1 as 1 -
0.18--
AF =
P (58a)
p
Although eqn (58) is suggested in the AISC Specification, the specification recommends the use of Av
-
0.85
(58b)
p
Pok for the design of a sway-permitted frame. Equations (58a) and (58b) are unconservative if the P-A effect is large.
7 DESIGN INTERACTION EQUATIONS FOR BEAM-COLUMNS A beam-column constitutes one of the major components of a structural frame. Because of its importance, analytical and experimental investigations of the beam-column problem have never diminished over the years. Various analytical techniques and experimental verifications of beamcolumns have been presented and summarized in the Chen and Atsuta's two-volume book, t t. 22 Chen and Cheong-Siat-Moy's review paper 32 and Chen and Lui's state-of-the-art papers 33.34 and books. 35.36 It should be noted that the exact prediction of beam-column behavior and strength is most complex in the inelastic range. The resulting differential equations of equilibrium must be written with respect to the deformed geometry of the member, which is not known in advance. The extent ofplastification of material in the member is also not known in advance. As a result, iterative procedures are needed in a numerical analysis. Even with simplifying assumptions, the differential equations are often intractable, and recourse to numerical techniques is usually inevitable. Although with the aid of computer technology, numerical analysis does not pose any significant problem to an analyst, it nevertheless is rather unappealing to a designer. For design purposes, the strength of a beam-column is usually represented by an interaction equation of the general form: ~'
M~,'
< 1.0
(59)
W. F. Chen
4(1
where P, Mx, M>, are the applied axial load and moments and Pu, M,~,, Mu,. are the ultimate axial load and moment resistances of the columns and beams, discussed in Sections 3 and 4, respectively. Depending on how the loads are applied, a beam-column problem can be related to one of the following three categories: (a) in-plane beam-columns; (b) lateral torsional buckling of beam-columns: (c) biaxially loaded beam-columns. The recommended and proposed interaction equations to describe the behavior and strength of these classes of beam-column are summarized as follows.
7.1 In-plane beam-columns In this type of beam-column, the moment is applied about the y-axis so that in-plane stability will occur. Out-of-plane deformation for this type of beam-column is not likely even without bracing in the y-direction, since both the applied axial force and bending moments will tend to destabilize the member in the x-z plane (Fig. 17).
7.1.1 SSRC approach ~'37 At braced points: p~
+ 0-84 M,p.II < 1.0, M < Mp,
(60)
P which is valid for 0-4 < PII "<< 1.0 and
M
= Mp,.
(61)
P which is valid for0 < p, < 0.4
Between braced points." P CraM Pu, + -M~;I;-{i-'-P/P~,Ii < 1.0 where P = applied axial force:
(62)
Design of beam-columns in steel frames in USA
~
41
My M
x
Z J
Mx
My Fig. 17. Biaxially loaded beam-column.
yield load; maximum applied (primary) moment; plastic moment about weak axis; ultimate axial load producing failure in the absence of bending moment, about the y-axis, evaluated using K from alignment chart (Section 3.2); moment reduction factor (Section 6.1); Cnl Mu|' --" bending moment producing failure in the absence of axial load, about the weak axis (for weak axis, this moment capacity is equal to Mr,,,); Euler buckling load about the weak axis considering K-factor. PCl,
M= Mr,y PU)'
7.1.2 Kanchanalai and Lu approach 38 In their study, theoretical solutions were obtained for sway and nonsway beam-columns bent about their weak axis. The resulting interaction curves were curve-fitted to obtain the interaction equation as follows: P M PL,~ + mB3 ~
< n
(63)
42
W. F. Chen
where Pu~ = ultimate axial load capacity in the absence of b e n d i n g moment, evaluated using K = 1; Muy = Mp.v = full plastic m o m e n t about the y-axis (weak axis); B3 =
1
EPAI 1 - 1"2 EHh
are constants such that the n o n l i n e a r ultimate resistance curve can be approximated by three linear interaction equations, as follows:
m and n
P B~M + 1-00 . - 7 7 -<< 1-0 Pu~
(64)
EPAI 1 which is valid for XHh > and P B3M Pu~ + 0.85 ~ < 1.0
(65)
P B3M Pu--~+ 1-75 ~ < 1-75
(66)
Y~PAI 1 which is valid for - EHh < 3 Equations (64) to (66) are valid only for the m i n o r axis b e n d i n g case, but K a n c h a n a l a i and Lu suggested that values for m a n d n can be obtained in a similar m a n n e r for the major axis b e n d i n g case. If the m o m e n t is applied about the x-axis a n d full lateral bracing is provided in the x-direction, the m e m b e r will then buckle in the y-z plane (Fig. 17). 7.1.3 AISC and SSRC approach 1.9.37 At braced points." P
PZ + 0.85
M
< 1.0, M < Mp,
P which is valid for 0-15 < ~o~, < 1.0 and
(67)
Design of beam-columns in steel frames in USA
M = Mpx
43 (68)
P which is valid for 0 < ~,. < 0.15
Between braced points: P
--
e~,.
+
CmM M,x(1 - e/eex)
<
1.0
(69)
where the subscript x refers to member strength about the x-axis, and Mux =Mpx.
7.1.4 Cheong-Siat-Moy and Downs approach 39 Based on a curve-fitting of their theoretical analysis of beam-columns bending about the major axis, the following interaction equation is proposed as P
M
Pu-~+ flAy ~
< 1.0
(70)
where
Pu i = ultimate axial load capacity of column in the absence of bending moment, evaluated using K = 1; MHX = Mp.,. = full plastic moment about the x-axis (strong axis); = 0.9 + 0"IAF; in which fl = ratio of second-order deflection to first-order deflection, and AF = moment amplification factor.
Av -~
1
1
Y'PAI
< 1"5
EHh Equation (70) is a nonlinear interaction equation since/3 is a function of the amplification factorAv. It is valid for combined gravity and lateral loads with no gravity moments present. It is unsafe for the pure gravity loading case. If gravity moments coexist with moments caused by lateral forces, eqn (70) must be modified as follows:
P Cmfl (AvM+Mg) Po-~+ l - - ~ P e ~'/~.,. < 1-0
(71)
where Mg = moment due to gravity loads; Cm is defined in eqn (51); however, the value Of Cm/(1 -- P/Pe) must be greater than or equal to 1.0 in eqn (71); fl,Av defined as in eqn (70).
44
W. F. Chen
Although eqn (70) was derived on the basis of major axis bending, it has been demonstrated 3° that it also applies for minor axis bending, provided that Muy is used instead of M~. An interaction formula that applies for both major and minor axis bendings and for relatively large P-A effects was proposed by LeMessurier. 19 P
M
p~ +Av ~
< 1.0
(72)
where ultimate axial load in the absence of bending moment, evaluated, using K as defined in eqn (21); AF = moment amplification factor defined in eqn (57).
Ptl
7.2 Lateral-torsional buckling of beam-columns If the beam-column in Fig. 17 is not laterally braced, both in-plane and out-of-plane deformations may occur due to lateral torsional instability. The interaction equations for this case are as follows.
7.2.1 AISC and SSRC approach 1.9.37 At braced points: as eqns (67) and (68). P M p~m+ 0-85 ~
< 1.0,
M < mpx
(73a)
P which is valid for 0-15 < ~ < 1.0 and (73b)
M=Mpx P
which is valid for 0 < ff~ < 0.15
Between braced points: P
--+ Puy
CraM -<< 1.0 Mu.,.(1 - P/Pex)
(74)
where M~_~ = [1.07 - (L/G) vIF~/3160]Mpx <,Mpx (see eqn (26) and Fig. 7).
Design of beam-columns in steelframes in USA
45
7.2.2 Vinnakota approach 24 Based on the numerical analysis of laterally unsupported I beamcolumns, Vinnakota attempted to verify the validity of eqn (74). He suggested that for beam-columns with no joint translation, the P-6 moment magnifcation term Cm/(1 -P/Peg) can be omitted provided that P,y is evaluated using SSRC curve 2 (Fig. 3) and M ~ is evaluated from his numerical results (Fig. 7). Before proceeding any further, it is of interest to scrutinize eqns (69), (73a), (73b) and (74). These equations are contained in the present AISC Specification 2 for the PD of steel structures. If the moment magnification factor fro/(1 P/Pe) is equal to unity, then the stability limit state (eqns (69) and (74)) will always control the design, as shown in Fig. 18. If the moment magnification factor is less than unity, then the maximum moment occurs at one of the ends rather than within the span. With this in mind, a slenderness ratio above which stability will always govern the limit state can be determined. Stability will always govern when -
Cm
1 -P/Pe
-
> 1"0
(75)
rearranging, one can write
(L)~ /TT2EA(Ip-Cm)
(76,
or, in terms of the slenderness parameter, ,~o >/ /~---Y ( 1 - Cm)
(77)
P/Py 1.0
~
n.(67)
ecln (69) or (74)N N with Crn/(yP/Pe) ~ equals unity ~.~.
1-0
M/Mp
Fig. 18. Comparison of A|SC resistance and stability interaction formulae.
W. F. Chen
46
Thus, eqn (76) or (79) gives the slenderness criterion for which stabilitywill always control. The stability interaction equation (eqn (69) or (74)) has been shown 3° to be unconservative for cases in which a column pinned at both ends exists in a frame (see Fig. 19). For such frames, eqn (69) or (74) will underestimate the P-A effect consequently, it is more advisable to use the approaches suggested by Cheong-Siat-Moy and Downs 39 (eqn (70) and LeMessurier 19 (eqn (72)).
7.3 Biaxiaily loaded beam-columns The loading for this type of beam-column is shown in Fig. 17. It represents the most general case of beam-columns, i.e. all other cases can be represented after elimination of one or the other applied moments.
7.3.I SSRC approach 9.37 The interaction equations are as follows.
At braced points." P
Mx
p~m+ 0.85 ~
My
+ 0.85 ~
< 1.0
(78)
Between braced points: P + CmxMx + Cm_vMv • 1.0 Pu Mux(1 - P/Pc.,-) M.~(1 - P/P~y)
(79)
where MUA"~Mll.V = ultimate bending moments of the beam about the strong
and weak axes, respectively; P~x, P~v = Euler buckling loads about the strong and weak axes, respectively, considering the K factor; Pu = ultimate load of an axially loaded member, considering the K factor.
7.3.2 Chen approach 2~ Based on the theoretical studies of nonsway biaxially loaded beamcolumns, nonlinear interaction equations are presented. Details of the development are given in Chapter 13, Vol. 2, of the book by Chen and Atsuta 2~ or originally by Tebedge and Chen. 4°
Design of beam-columns in steelframes in USA
P1
47
P~
T__H
PINNED COLUMb
///
)
~
Fig. 19. Frame with column pinned at both ends.
At braced points." < 1.0
(80)
where Mx = required flexural strength about the x-axis (strong axis), and My = required flexural strength about the y-axis (weak axis).
Mpcx = 1-18Mpx[l - (P/Pv)] < Mp.,.
(81a)
1.19Mp.v[1 - (p/py)2] < Mpy
(81b)
Mpcy =
in which MpxandMpy are the plastic moment capacities about the strong and weak axes, respectively. a = 1-6
(P/e,,) 21n(P/p,,)
(82)
for W-sections with width/depth ratio 0.5 < bf/d < 1.0.
Between braced points: (
\ M-~x]
t~ + , CmvMv)l ~
< l-0
(83)
where Crux and Cmy = equivalent moment factors, defined in eqn (51). /3 =
0.4 + P/Pv + bf/d > 1.0 for bf/d > 0.3 1-0 forbf/d < 0.3
(84)
48
W. F. Chen
Me and My are evaluated on the basis of a first-order analysis, and Mu~x = Mu~ [1 - (P/Pu)] [1 - (e/eex)l
(85)
(P/&)! 11 - (P/Pey)l
(86)
Muc.v = Mu.v[1 in which
M~.~ = [1.07 - (L/rv)~ff~J3160]Mpx << Mpx
(87)
where bf is the flange width, and d is the member depth. The validity of eqns (80) and (83) has been checked by a number of researchers and is summarized in Chapter 13, Vol. 2, of the book by Chen and Atsuta) ~ An experimental verification of these two equations was given by Anslijn and Massonnet. 4j Eighty-one beam-columns were tested and the experimental results were compared with the results predicted by the interaction equations. Excellent correlation was reported. 42 The most recent analytical and experimental verification of these two equations was given by Cai et aL 43 Further discussion of the above interaction equations can be found in the review papers by Chen and Cheong-Siat-Moy 32 and Chen and Lui, -~3and the recent book by Chen and Lui. 36
7.3.3 LRFD approach 3 The AISC L R F D Specification gives the following bilinear interaction formulas for biaxially loaded beam-columns:
jO
which is valid for ~ > 0.2 O~ru and 20cP,,
+
M,-
+
.,.. < 1.0,
(])bMux 4)hMuy
(89)
P which is valid f o r - - < 0.2 0cPu where subscripts x and y denote the strong and weak axes, respectively, and M = BIMNT + B2MLT
(90)
Design of beam-columns in steel frames in USA
BI
Cme
_
> 1"0
49
(91)
1 ----
eek
1
B2 -
~PA!
1
(92)
EHh
or alternatively
1 B2 -
1 Cm
Mr~rr =
ML+ = Pek =
EP = A~ = Y,H = h = Pu = Mu = ~t, = q~c =
EP
(93)
ZPek
defined in eqn (51); required first-order m o m e n t in member assuming there is no lateral translation in the frame (Fig. 20(b)); required first-order m o m e n t in member as a result of lateral translation of the frame only (Fig. 20(c)); Euler load, n2EI/(KL) 2, evaluated using K < 1.0 for the B~ factor application in eqn (91), and using K/> 1 for the B2 factor application in eqn (93) in the plane of bending of the frame corresponding to braced and unbraced frames, respectively; axial load on all columns in a story; first-order translation deflection of the story under consideration; sum of all story horizontal forces producing A~; story height; ultimate axial strength evaluated using the LRFD column curve with K factor presented in Section 3; ultimate moment resistance using the LRFD beam formulas presented in Section 4; resistance factor for flexural = 0-90; resistance factor for compression = 0.85.
With some modifications, Chen's equations are also included in the 1986 LRFD Specification as alternative interaction equations for nonsway beam-columns. Note that eqns (88) and (89) are bilinear interaction equations (the terms P/Pu, Mx/M~,.,Mv/M,y are combined linearly) and are applicable to both nonsway and sway cases, whereas eqns (80) and (83) are nonlinear interaction equations, which can be used to assess the strength of beam-columns more accurately; however, they are applicable only to nonsway beam-columns.
50
W. F. Chen
nO w
b-. Z "1"
v! ~3 QT.
.ira
o
~3 3~
it
W rr -am
Z e¢ 0
-g
Design of beam-columns in steelframes in USA
51
In order to use eqns (87) and (89), the flame must be analyzed twice, first as a nonsway case, followed by a sway case. The first-order moment obtained from the nonsway case is magnified byB~ to account for the P-6 effect, and the first-order moment obtained from the sway case is magnified by B2 to account for the P-A effect. Thus, both secondary effects are taken into consideration. The two magnified moments are then added together to produce the design moment in eqn (90). This approach is rather conservative since the two magnified moments may not coincide in an actual situation; nevertheless, this is a safe approach. In the development of the LRFD interaction equations (eqns (88) and (89), the following design guidelines were established: 44 1. The equations should be applicable to a wide range of problems such as strong- and weak-axis bending, sway and nonsway frames, laterally loaded columns, imperfect columns with various slenderness ratios and degrees of end restraint, and leaned column systems (Fig. 19). They should account for both inelastic behavior and second-order effects. 2. The equations should clearly distinguish between the load effects and the resistances so that second-order elastic analysis can be easily accommodated. 3. The equations should be based on the load effects obtained from second-order elastic analysis since, at the present time, secondorder inelastic analysis is not readily accessible for design office use.
4. The equations should not be more than 5% unconservative when compared to second-order plastic-zone solutions. 5. The equations should not need to consider strength and stability separately as in previous specifications, since, in general, all columns of finite length fail by some combination of inelastic bending and stability effects. 6. The same ultimate strength should be predicted for similar members. 7. Adjustment of effective length based on column inelasticity should be allowed for. With the above guidelines in place, the interaction equations expressed by eqns (88) and (89) were developed by curve-fitting to a set of numerical second-order inelastic flame analyses data generated by Kanchanalai. 45 Generally speaking, the interaction equations give a good representation for all strong-axis cases for 0 < KL/rx < 100. It is rather conservative for weak axis cases for 0 < K L / r v < 40, and it is moderatively conservative for both strong and weak axis cases for KL/ r > 120.46-49
W. F Chen
52
Note that, unlike the ASD and PD interaction equations in which both the yielding and stability interaction equations are needed in the design process, only one interaction equation is needed if the LRFD approach is used. The applicable equation is determined by the term P/OcPu. If P/q~cPu/> 0-2, eqn (88) is applicable, and if P/OcPu < 0.2, eqn (89) is applicable. Another feature of the LRFD approach that is different from the ASD and PD approaches is that the P-8 and P-A moment magnification effects are located independently, as is evident from eqn (89). Recall that, in the ASD or PD approach, if the member is subjected to sway, the factor Cm is taken as 0.85; therefore, the moment magnification factor is 0"85/(l'P/Pek) and this is applied to the total first-order moment of the member regardless of whether it is caused by gravity load (M~q.) or lateral load (MET). In addition to eqns (88) and (89), the LRFD Specification2 also recommends a set of nonlinear interaction equations in its Appendix, which are valid for nonsway members with end moments Mx and M~,. These equations are given by eqns (80) and (83), with the following definitions for Mpcand M~c. Mpcx = 1.2Mpx[1 -
(P/Pv)] < Mpx
(94)
Mpcy = 1.2Mpy[l - (p/p,,)2] < Mpy
(95)
Mucx = Mu~ [l - (P/O ~Pu)][ 1 - (P/P¢x)]
(96)
Mucy = Mu.v11 - (P/epcPu)111 - (P/Pev)]
(97)
The interaction equations (80) and (82) have a jump from a long to a short member, i.e. for a very short member, the stability interaction equation (eqn (83)) does not always reduce to the strength equation, eqn (80). For short-beam columns in which the weak-axis bending and axial force are dominant, and for double curvature bending, the LRFD bilinear interaction equations, eqns (88) and (89), may result in an overly conservative design. To this end, the following nonlinear interaction equation is proposed: 5°
(Mx~ 2 (My~ ~ M*~] + k M*~] < 1.0
(98)
where a = 1.2+
-0.03
~
>~ 1.0
(99)
Design of beam-columns in steelframes in USA
M * = Mu~ 1 -
~
53 (100)
1-
(101)
1.3+ 0.002 (-~x)
(102)
Mu*y = Muy
and ( = 3.0+0.035
~
~
1
) 1.0
(103)
in which B I is calculated from eqn (91), but with Cm evaluated, using the equation: Cm = 1 + 0"25(P/Pe) -- 0"6(P/Pe) ~/3 (M~/M2 + 1)
(104)
and B2 is calculated from eqn (92) or (93). In eqns (103) and (104), M t / M 2 is the ratio of the smaller to larger end moments. Its value is taken as positive if the member bends in reverse curvature, and negative if the member bends in single curvature. The ability o f e q n (98) to represent the actual inelastic behavior of a beam-column under biaxial bending and axial compression has been firmly established by Duan and Chen. s° Generally speaking, eqn (98) gives a better fit of the numerical data than do the AISC LRFD nonlinear interaction equations.
8 ANALYSIS AND DESIGN OF BEAM-COLUMNS IN FRAMES Discussion of the behavior and strength of beam-columns in this paper has so far focussed primarily on isolated members. In reality, most structural members exist as parts of a framework and their behavior and strength are, therefore, influenced by the behavior of other members of the frame. To illustrate some aspects of this interaction between the beams and columns in a frame, it is instructive to consider the following simple example. Shown in Fig. 21 is a simple braced frame consisting of a beam and a column. The beam is loaded by a uniformly distributed load w. After the full value ofw is reached, the column is then loaded by a monotonically increasing concentric load of P until failure. The behavior of the beam and column will now be studied as P increases from zero to its ultimate value. In performing the analysis, the following assumptions are made:
W.F. Chen
54
(applied after W has reached its
full value) W
Xb
El
El
Xc __
A
L Fig. 21. T w o - m e m b e r frame.
1. The axial force in the beam is negligible. 2. The axial force in the column is represented by P. Figure 22 shows the maximum beam moment, (Mmax)b~am, the maximum column moment, (Mmax)column, and the beam end moment, MRc, equal to --MBA (the column-end moment) nondimensionalized by the quantity wL2/8 as a function of the applied column force P nondimensionalized by the Euler load. The maximum beam moment is obtained as the larger value of the beam end moment M~c and the maximum in-span beam moment (MR)max. Similarly, the maximum column moment is obtained as the larger value of the column-end moment MBA and the maximum in-span column moment (Mc)max. The sign convention used in the figure is that a positive beam moment will cause tension in the bottom fiber of the beam, and a positive column moment will cause tension in the outside fiber of the column. Also shown in Fig. 22 is the maximum moment in the column as predicted by the AISC approach (eqn (50)), i.e. MAR = 0. (Mmax)colL,,,,,, =
=
Crll P
0"6 ]
l_i_ P
MBA =
MRA
0"6 - 0"4(MAB/MRA ) P
MBA
(io5)
Design of beam-columns in steel frames in USA
55
o.
c
E
E
\\ ~J
E
\ ¢5
.
i
i
I
I
t~
I
E O~ E 6 ¢1) q..
to
/
i
,,
[
= ..:
!
/
/
o.
~, T o
ei ¢q "7 o i
°! I I I I
o
56
W. F. Chen
The lower dashed line was obtained by using an effective length factor K = 1 in calculating Pe for the column, while the upper dashed line was obtained by using K = 0.839 (from alignment chart discussed in Section 3.2) in calculating P~. Observations regarding the behavior of the two-member frame can be made from the figure, and are as follows: 1. As the applied column force P increases, the magnitude of the maximum column moment and the maximum beam moment both increase. Nevertheless, the locations of these maximum moments vary as a function of P. The location of the maximum in-span column moment is given by rr 2k
xc-
L /
V
(106) P
1 -p~,
The location of the maximum in-span beam moment is given by Xb
L -
2
MBc _ L wL
L
2+8 -x
rt2P/Pe [ 3 _ 3rr x/p~_~ cot(rr p x / ~ ) + (rr2V,p ~
)]
(107)
In writing the above equation, the expression for MBc with kL = rrv/P/P ~was used. It should be noted that the above expressions for xc and Xb are valid only if the calculated values fall within the range 0 to L. If the calculated values fall outside this range, the location of the maximum moment is at the end rather than within the span of the member. Figure 23 shows a plot of the variation ofxc and Xb, nondimensionalized by the length of the member L as a function of P/P~. For the column, the location of the maximum moment shifts from the upper end to the middle of the member as P increases from 0 to P~. For the beam, the location of the maximum moment shifts from Xb = 0-562L at P = 0 to Xb = 0"5L at P = P~. . The change in values in (mmax)bcam, (mmax)column and M~c implies that the moment in the structure is being redistributed as the applied column force P increases. This change in moment distribution is revealed in Fig. 24 in which the bending moment diagrams for the frame at various values of P/P~ are shown. Note that there is a reversal in moment at the joint as P/P~ exceeds unity. In other words, when P/P~ < 1 (Fig. 25(a)), the beam is inducing
Design of beam-columns in steel frames in USA
57
Column 1.2 Beam
--~
1.0
0.8
P/Pe
0.6 Column
Beam
0.4
0.2
i 0.2
I 0.4
Xb L
0.5 or
0.6
0.8
1.0
Xc L
Fig. 23. Variation of the location of maximum beam and column moments with
P/Pe.
m o m e n t to the c o l u m n (Fig. 24(b)); however, as P/Pe > 1 (Fig. 25(b)), the b e a m is restraining the c o l u m n against buckling. At P/Pe = 1 (Fig. 24(c)), the b e a m end m o m e n t is zero, indicating that the b e a m is neither inducing m o m e n t to the c o l u m n nor restraining it from buckling. It is also worth noting that as P/Pe (Fig. 24(d), (Mmax)beam/(wL2/8) will exceed unity, which implies that if the designer is to rely on the beam to restrain the column, i.e. to design the c o l u m n with a K factor less than unity, the beam must be designed to carry a m a x i m u m in-span m o m e n t that exceeds wL~-/8 (i.e. the m a x i m u m m o m e n t of a simply supported beam). This is in sharp contrast to the c o m m o n notion that c o l u m n end m o m e n t s do not c h a n g e in a braced frame because the P-6 m o m e n t s are zero at the ends. This clearly shows that this is not true. Consequently.
58
W. F. Chen
.E 0
hE
E
h ~
"~
E
~I~
Design of beam-columns in steelframes in USA P
P
~
59
MBA
MBA
i P
P
=
--MBc.
second-order effects will change beam moments as in unbraced frames. For example, when P/Pe = 1-1, the beam m o m e n t is 1-12 x (wL2/8), which will require a larger beam cross-section. 3. The AISC formula for the maximum strength of a column (eqn (50)) gives a good correlation with the exact result, if an effective length K = 0.839 is used in computing the critical load in the magnification term (the upper dashed line in Fig. 22). If an effective length factor K = 1 is used, then the formula will underestimate the column moment (the lower dashed line in Fig. 22). From observations 2 and 3, it can be concluded that, for braced frames, it is advisable to use an effective length factorK = 1 in the first term of the interaction equation (eqn (69)), but not in the second term where one should use an effective length factor K < 1.
9 SECOND-ORDER ELASTIC ANALYSIS FOR RIGID FRAME DESIGN The need for a sec~ond-0rder elastic analysis of steel frames is increasing in view of the recent advent of the AISC/LRFD Specification. As a simplified second-order elastic analysis, the specification uses two modifiers, B~ and B2 factors, 3 which are applied directly to the results of a first-order analysis. 22There are several other methods ~8-2°"29-31available for second-order analysis, based on approximately the same philosophy as the B~ and B2 factor analysis in the LRFD Specification. However,
60
W. F. Chen
these methods do not always give satisfactory results, even for design purposes, because they are rather approximate in nature and their application is mostly limited to rectangular frames with rigid connections and small displacements. Furthermore, it is cumbersome and tedious to calculate the modifiers applied to the result of a first-order analysis. In view of the present availability of workstations and future development of mini- and micro-computers, it is more convenient and realistic to use a computer-based method as long as the method can be easily implemented. 5~.52 Several computer-based second-order elastic analysis methods, simple enough for use in practical design, are presented as follows. At present, workstations do not have enough capacity to utilize ready-made programs based on finite element techniques, Hence, the method presented here uses the analytical solutions to reduce the number of degrees of freedom in a discrete analysis. These analytical solutions are derived from two kinds of governing differential equations, which are approximated with the assumptions of relatively small displacements, considering that extremely large displacements seldom have an important effect on the design of frames. The difference between the two equations is whether or not they consider bowing deformation in the calculation of axial displacement. When bowing deformation is ignored, the stiffness equation is basically the same as the one which uses stability functions. ~3,54 However, different from the customary stiffness equation, the equation presented here is expressed by a power series without truncation in order to eliminate the numerical problem under small axial force (Chen and Lui, 1987). Both of the derived stiffness equations are nonlinear algebraic equations, hence, iteration has to be used to obtain solutions. Herein, the secant stiffness method is employed as an iterative procedure, because the programming for this method is simpler compared with that using the tangent stiffness. Indeed, iterative procedures are essential to the solution of nonlinear problems, but it is desirable to reduce the number of iterations to an absolute minimum because iteration is costly. From this viewpoint, the accuracy of the solutions obtained after one cycle of iteration was also investigated with regard to the applicability to practical design.
9.1 Derivations of governing differential equations The governing differential equations for a second-order analysis are formulated through the principle of virtual work, introducing the usual beam assumptions. This is mainly because the equilibrium equations,
Design of beam-columns in steel frames in USA
61
consistent with the compatibility equations, can be easily derived by purely mathematical manipulations. 52 As shown in Fig. 26, consider a plane member 1,2 subjected to a uniformly distributed force py acting perpendicular to the member axis before deformation. A Cartesian coordinate system (x, y) as well as displacement components (u. v) are defined as the initial configuration of the member. By using the displacement components defined above, the nonlinear Green strain tensor eij for plane problems is given by
egg - Ox + -2 ~ Ox ) + ~ Ov
1 [(Ou)2
eyy -- Oy +
(108a)
(Ov~2]
2kk- ) + k y)
(108b)
J
1 Ov +Ou Ou Ou ---OY exy = -~ ~ Oy + O---x~ + Ox
(108c)
If the derivatives of axial displacement are negligibly small compared with those of deflection, the following condition holds:
Ou Ou dv Ov << Ox' Oy Ox'Oy
(109)
_1
1
IllIlIiIl U01 m~
e--~ "
M1
i V01
2
X
02==
I
~v
V02
~ - J ' M2 S Fig. 26. Coordinate system for a plane member.
W.F. Chen
62
Considering the condition of eqn (109), eqns (108) can be simplified as follows:
e,, - Ox + -2 \ Ox } IOV
1 (Or) 2
¢y,'- ay + ] t ay)
l ( Ov Ou Ov Or) exy - 2 0 x x + O y y + O--x-O-yy
(110a)
(1 lOb)
(110c)
Herein, the usual beam assumptions are introduced, i.e. the assumptions of no change of cross-sectional shape and the Bernoulli-Euler hypothesis where the transverse plane is assumed plane and normal to the beam axis throughout deformation. These assumptions are mathematically expressed in terms of the strain tensor as
eyy = O, exy = 0
(111)
Using the displacement components (u0, v0) on the centroidal axis, the displacement field which satisfies eqn (111) can be given by u = u0-y
dv0 dx
v = v0
(ll2a) (l12b)
It should be noted that u0 and v0 are both functions ofx. Substituting eqn (112) into eqn (110), the nonzero component of the strain tensor is expressed by the displacement components on the centroidal axis:
e~x -
d.o dx
d2v0 l(dv0)2
Y-d~+ 2\dxJ
(113)
This strain-displacement relationship is what is customarily used in a second-order finite element analysis for plane frames. 37 55.s6 In order to obtain the exact analytical solution corresponding to the customary finite element solution, the governing differential equations are derived through the principle of virtual work. Using the strain tensor component exx of eqn (112) and the corresponding stress component t~xxin addition to the components of external forces and displacements, as shown in Fig. 26, the equation of virtual work for the plane member is given by
Design of beam-columns in steelframes in USA I
6II=
63
l
fo fAtrxx6exx dA d x - fopy 6Vo d x - [nx(Ni 8Uoi-[- Si¢~l~oi + M,6v'oi)]~ = 0
(114) I
where fAdA indicates the integration over the cross-sectional area, f, dx over the length of the member, and nx has the values of - 1 and 1, respectively, at nodes 1 and 2. The virtual strain 6egg can be calculated by taking the variation ofeqn (113):
6exx = 6 u 6 - y 6 v ~ ' + V'o6V'o
(115)
in which the notation (-)' is introduced for simplicity to express the differentiation with respect to x. Substituting eqn (115) into eqn (114) and integrating by parts leads to 81-I = [(N - nxNi)SUoi + (Nv'o + M' - nxSi)SVoi , 2_ {( Nv; + - (M + nxMi)6Voi]l fo [N'6uo + M')'
+ py}6vo]dx = 0
(116)
where
N=
;AaxxdA,
M=
fAaxxydA
(117)
These stress resultants correspond to axial force and bending moment, respectively. Equilibrium equations and the associated boundary conditions are obtained from the necessary and sufficient conditions for eqn (116) to hold for any virtual displacements. The terms in the bracket under the integal sign yield the equilibrium equations: N' = 0
(Nv'o + M')' + py
(l18a) =
0
(l18b)
and the integrated terms give the associated boundary conditions at nodes 1 and 2:
Uo = Uoi or N = nxNi
(l19a)
v0 = v0~ or Nv'o + M' = nxSi
(l19b)
v~ = ai or M = -nxMi,
(i = 1, 2)
(ll9c)
where Si and a; are vertical forces and end rotations of the member.
64
W. F. Chen
The stress resultant-displacement relations are obtained by substituting eqn (112) into eqn (117). I f t h e x axis is selected such that it coincides with the centroidal axis of the member, these relations are simplified as
{
1
}
N = EA u ~ + ~ ( v ~ ) 2
(120a)
M = - E I v~'
(120b)
9.2 Analytical solutions of the governing equations Equations (118) to (120) are the governing equations to analyze the member. From eqn (118a), the axial force N is constant because there are no distributed forces in the axial direction. Therefore, eqn (118b) can be solved with respect to v0 independent of eqn (l18a). By using the mechanical and the geometrical boundary conditions at node 1, the solution is expressed by N~ > 0 Vo _
l
(121a)
Vol + a l
1
,gl
~-/sin yx + ( ~
[ (yx)~ 1
+ py 2(y1)4 + ~
(yx - sin y x )
,Q1
(~[j2 (1 - cos yx)
]
(cos yx - 1)
(121b)
N~ < 0
(121c)
1)0
VOI
l
l
+ ~ - s i n h yx + ( - ~ ( s i n h y x - y x )
( ~ 2 (cosh yx - 1)
[ (Yx)2 + (~/)4-(1 - cosh yx)]
+ p v L 20,l)4
(121d)
where Y = v / { N , I / E L ~,
-
S,l 2 El" ~
-
mJ pd3 E l ' py - E1
(122a-d)
Substituting eqns (121) into eqns (120) and making use of the boundary conditions at node 1, u0 can also be obtained as follows: Nt u0 = u 0 1 + ~ x - ~
1 fx ,/0
v~2dx
(123)
Design of beam-columns in steel frames in USA
65
X
where the expressions for the integral I, v62 dx can be obtained in close forms corresponding to NI > 0 and N~ < 0, respectively. These expressions are given elsewhere. 52
9.3 Stiffness equation expressed by power series From the analytical solutions obtained in Section 9.2, a stiffness equation can be derived. If the stiffness equation is expressed by such trigonometric functions as shown in Section 9.2, it differs according to whether the axial force N~ is positive or negative. 5L52 Furthermore, if the axial force approaches zero, the coefficients of the stiffness equations become indefinite, which causes numerical instability in a structural analysis. Therefore, three kinds of different stiffness equation have to be used, according to the value of the axial force, in order to avoid numerical difficulty. Such a procedure is very cumbersome and, hence, power series is introduced here. This expression is convenient for computer-aided analysis because the expressions by the series are free from numerical instability when the axial force approaches zero. Consider the following power series expansions for the trigonometric functions: co
sinM ]. sinhMJ = yl+YlE
1 (-+ Nt)" ( 2 n + 1)!
( 124a )
n=l
cosrl ~ = I + S "
1 ( + RI)" /_..., (2n)!
cosh ylJ
(124b)
n=l
lfll = Nml2/EI = rr2Ni/Pe,
Pe = rt2El/12
(124c, d)
where E1 = sectional rigidity and I = length of the member. The nondimensional stiffness equation can be obtained from eqns (121) and (122) in the form: r
AI2/l S, I
0
0
1201 60: 403
s,I
sym
1 (,~ -(AI'-/21) l I vll-dx
-AI2/I
0
0
0
- 1201
60_~
-p,./2
0 Al'-/l
-60! 0
204 0
-(fi,./12){3/(203 + 04)}
1201
-602 403
L ;J
+(Al'-/21) ~
v~2dx
-p.,/2 -(p,/12){3/(2~3 + 04)}
(125)
66
W. I~ Chen
where ai = ui/l,
0; = vi/1
1l £V,o2dx ,
(126a,b)
a~f, + S ~ . [ 2 + l Q ~ f ~ + c t , ' , ~ , f 4 + a , A ,
IL[.s+g,~/IL[6
+ P.~f7 + P v a , f 8 + pvg,f9 + PflQ~fl0
(127)
The functions ~pj andf. (i = l to 10) in eqns (125) and (127) are expressed in terms of the axial force N~ in the form of a power series and are given elsewhere.SL 52 It should be noted that in eqn (125) the bowing effect is treated as a quantity equivalent to the distributed load in the direction of the x axis. Functions ~p,.a n d f . are mathematically expressed by an infinite series expansion. However, it is impossible to consider all the terms of an infinite series in a numerical analysis. Hence, the number of terms necessary for a numerical convergence has been examined by Goto and Chen.SL 52 Following the usual procedure in stiffness matrix analysis, structural stiffness equations can be obtained by assembling the element stiffness matrix given by eqn (125). The structural stiffness equations, thus obtained, are nonlinear and an iterative procedure is, therefore, required to solve these equations. Such a procedure is also given by Goto and Chen.SL 52
9.4 Stiffness equation neglecting bowing effect The solution obtained by using eqn (125) can be accepted as exact for the frames with moderate rotational displacements. However, the equations can be simplified by ignoring the bowing effect, since this effect will generally have little influence on the results. /
By neglecting the terms( v;2dx, eqn (125) is reduced to .10
0 AI 2
NI Si
?
-AI~ ]
0
12@1 6@3
0
-12@1
4@~
0
-6@3
2@4
Al'?
o
o
0
g,
0
0
fa,
6@3
I Ol
-~,: -p,,
0 -2
sym
12~pl
-60, 403
31
~; 4~;) i'
a, -P,
(128)
Design of beam-columns in steel frames in USA
67
This stiffness equation is basically the same as the one which is often referred to in analysis by stability functions.53.54However, as stated in the previous section, eqn (128) is expressed by the exact series expansion and, hence, this equation can be used whatever the quantity of the axial force may be. This is different from the customary analysis using stability functions.
9.5 Numerical examples of rigidly connected elastic frames Three kinds of typical rectangular frames are analyzed to demonstrate the accuracy of the workstations and microcomputer-based methods with the derived element stiffness equations given in the preceding section, when they are applied to structural analysis. In addition, the results are compared with those obtained by the method using B1 and B2 factors in LRFD. 3Numerical studies on the effects of joint flexibility on the behavior and strength of steel frames are given elsewhere.57"58
9.5.1 Multistory rectangular frames with rigid connections The rectangular frames analyzed are shown in Fig. 27. along with their design load. The maximum column moments for each story of the frame are calculated under the loads up to five times as much as the design load. The ratios of the moments by second-order analysis to those by first-order analysis are summarized for three kinds of frames, respectively, in Tables 4 to 6, classified according to the method of analysis. The methods examined are of five kinds. The first is the most exact method with eqn (128). The second and third use the same element stiffness equation of eqn (128) ignoring the bowing effects. The difference is that the third method is simplified in its iterative procedure, that is, the TABLE 4 Maximum Column Moment in the One-story Frame (node 4) (Fig. 27(a))
Applied load Design load
1.0 2.0 3.0 4.0 5.0
'Exact~ with eqn (125)
1-010 1,020 1.032 1.043 1-056
"Simplified', with eqn (128)
1.011 1,023 1-036 1.049 1.063
With eqn (128) a)qer one cycle of iteration 1.011 1.023 1.036 1.049 1.063
LRFD (BI and Be factor method) with eqn (92) with eqn (93) 1.008 1,014 1,023 1.029 1,038
1.008 1-016 1.024 1.033 1.043
Moments are nondimensionalized by corresponding moments obtained by first-order analysis.
68
IV.. F. Chen
0.137 kips/in 7.28 kips, ®
w12x30
1
® o o3 >¢
031
eql 12.976 I kips
0.18848 kips/in ~)
rT
354.33 In
W14x30
- -
I
I ~
2.88 kips
~)
0.1~88 klp;/In
5.456 kips
0.155 kips/In
t i t i l t~ W16x31
5.76 kips (~
w
W21x44
,x.
;2 I_
288 in
_j
L
I
300 in
®1!
t
Fig. 27. Three typical rectangular frames.
procedure with one cycle of iteration. Fourth and fifth are the methods with B~ and B2 factors recommended by the AISC/LRFD Specification,3 respectively, using eqns (92) and (93) for the calculation of the B2 factor. All the proposed methods except for the one with the simplified iterative procedure are continued until the absolute ratio of an unbalanced force to the corresponding nodal force becomes less than 1 0 - 4 . As is clear from the formulation of the element stiffness equation, the results obtained from eqn (125) can be accepted as the most accurate ones in the tables. Throughout Tables 6 to 8, it can be seen that the results from eqn (128) are comparable with those from eqn (125) whether the method with eqn (128) uses the simplified iterative procedures or not. On the other hand, the accuracy of the method with B1 and B2 factors varies from case to case, and the results for one- and two-story frames are not so accurate when compared with the method using eqn (128). Moreover, the results oftheB~ andB2 factor method differ according to which of the two equations is used for the calculation of the B2 factor.
Design of beam-columns in steel frames in USA
69
TABLE 5 Maximum Column Moments in the Two-story Frame (Fig. 27(b))
Applied load Design load
'Exact', with eqn (125)
"Simplified', with eqn (128)
With eqn (128) aider one cycle of iteration
LRFD (BI and B2factor method) with eqn (92) with eqn (93)
First story (node 4) 1.0 2-0 3.0 4.0 5.0
1.006 1.012 1.019 1.027 1.036
1.007 1.014 1.022 1.031 1.040
1.007 1.014 1.022 1.031 1.040
1-010 1.021 1.032 1.044 1.057
1-012 1.025 1.040 1.055 1.071
1.005 1.010 1.015 1.020 1.025
1.005 1.010 1.015 1.020 0.026
1.002 1.005 1.008 1-010 1-013
1.002 1-004 1.006 1.009 1.011
Second story (node 6) 1.0 2.0 3.0 4.0 5.0
1.005 1.009 1.014 1.019 1-025
TABLE 6 Maximum Column Moments in the Three-story Frame (Fig. 27(c))
Applied load Design load
'Exact', with eqn (125)
'Simplified', with eqn (128)
With eqn (128) after one cycle of iteration
LRFD (Bi and B2factor method) with eqn (92) with eqn (93)
First story (node 4) 1.0 2.0 3.0 4.0 5.0
1.021 1.043 1.068 1.095 1.126
1.021 1.045 1.070 1.099 1. 130
1.021 1.044 1.070 1.098 !. 130
1.022 1.045 1.070 1-097 1. 127
1-016 1.034 1.053 1.072 1.093
1.022 1-046 1.073 1.103 1-137
1.021 1.044 1.070 1.099 1. 132
1.021 1.044 1.070 1.099 1. 132
1.023 1.049 1.074 1.104 i. 138
1.025 1.054 1.085 1.120 1. 159
1.005 1.010 1-016 1.022 1.028
1-006 1.013 1.021 1.028 1-037
1.006 1.013 1-020 1.028 1-037
1.004 1-008 1.012 1.016 1.021
1.005 1.010 1.016 1.023 1.030
Secondary story (node 6) 1.0 2.0 3.0 4.0 5.0
Third story (node 8) i.0 2.0 3-0 4.0 5-0
70
W. F. Chen
those for the derivations of eqn (128). Furthermore, the calculation of these magnifying factors is based on a first-order analysis, and no correction is made on the factors by iteration. Hence, it will be possible to interpret that this B~ and B2 factor method uses one cycle of iteration. Judging from these facts, B~ and B 2 f a c t o r methods could not be more accurate than the method with eqn (128), using one cycle of iteration.
9.5.2 General remarks Several methods of second-order elastic analysis have been developed here, primarily for the analysis using workstations and microcomputers. Their accuracy is examined with regard to the applicability to design, where the accuracy of the B ~and B2 factor analysis is recommended by the LRFD specification is also examined. The methods presented here use analytical solutions to reduce the number of degrees of freedom in a discrete analysis. Moreover, the stiffness equations derived from these analytical solutions are expressed in series expansion without truncation to avoid numerical difficulty. Therefore, these methods always give accurate solutions without increasing the number of degrees of freedom. Comparing their accuracy with the B~ and B2 factor method, the present methods give more accurate solutions because they use fewer assumptions in their formulations. Moreover, the methods developed have no such restrictions as theBj andB2 factor method has on the shape of structures. Regarding the applicability to design analysis, the method using one cycle of iteration is simplest and yet accurate enough for practical purposes. This method can also be easily implemented into the customary computer programs for the usual elastic first-order analysis without increasing its capacity. Taking all these factors into account, the method with one cycle of iteration is considered to be a better practical method for an elastic second-order frame analysis. It is, therefore, recommended for general use.
10 PRACTICAL SECOND-ORDER INELASTIC ANALYSIS FOR RIGID FRAME DESIGN In recent years much effort has been devoted to the development and validation of simplified second-order inelastic analysis methods for practical use. As pointed out in Section 9, current practice in the USA is to perform elastic analysis (first-order elastic analysis with moment magnification factors B~ and B2 or direct second-order elastic analysis) to -k,~:~ ,~, . . . . . . ;,.,~,j . . . . . t ~ fc~r d e ~ i ~ n Material nonlinearity is
Design of beam-columns in steel frames in USA
71
accounted for implicitly in the design interaction equations. It is well known that, as the frame deforms into the inelastic range, the distribution of forces a n d moments will differ from the case when the frame remains fully elastic. Thus, if inelasticity is not accounted for in the analysis, the forces and moments calculated will not be truly representative of the actual response of the frame. The incompatibility that exists between the analysis and design assumptions will severely undermine the philosophy of a limit states approach to design. Recognizing the eventual need for incorporating advanced analysis methods in the provisions, researchers and practitioners have collaborated in a joint effort to establish guidelines which would set the standard for the use of such methods of analysis in a limit state setting. In this regard, twotask groups have recently been set up to address the problem: The AISC Task Committee 117 -- Inelastic Analysis and Design, and the SSRC Task Group 29 m Second-Order Inelastic Analysis for Frame Design. The main objectives for the task groups are: 1. To provide a dictionary and a uniform definition of terms that pertain to nonlinear analysis and nonlinear behavior. 2. To categorize and detail behavioural effects such as quantifying the magnitude and distribution of residual stresses, member out-ofstraightness and member out-of-plumbness. 3. To identify benchmark problems for use in calibration, verification and comparison of second-order inelastic analysis procedures. 4. To identify situations in which advanced analysis will be appropriate and should be performed. 5. To assess the accuracy and feasibility of various approaches for inelastic analyses. 6. To implement and codify innovative analysis and design approaches which incorporate all essential elements of nonlinear effects. Table 7 summarizes various behavioral effects which should be addressed in a valid limit states design. 59It should be mentioned that the incorporation of all these factors in design will undoubtedly further complicate the already cumbersome design process. An important task for the task groups is, therefore, to identify circumstances and situations in which simplifying assumptions with regard to structural behavior can be applied without incurring noticeable errors. In what follows, three recently proposed simplified second-order inelastic analysis methods which take into account only the in-plane geometrical and material nonlinear effects will be described. The ability of each method to model frame response will also be demonstrated. The other factors which are located in Table 9 will not be discussed here. A discussion of some of these factors can be found elsewhere (Chen and Lui. 1991)) 6
72
W F. Chen TABLE 7
Behavioral Effects of Steel Frames ~9 1. Physical characteristics (a) Initial imperfections -- camber, sweep and twist of members, and out-of-plane erection of frame (b) Residual stresses present in members prior to loading as a result of manufacturing, fabrication and erection processes (c) Member shape (e.g. unsymmetrical cross-section and tapered profile) (d) Connectin type -- pinned, semi-rigid and rigid (e) External and internal restraints (e,g. bracing and support conditions) (f) Miscellaneous initial strains (e.g. environmental thermal loading) (g) Construction sequence 2. Response phenomena -- geometrical effects (a) Influence of axial force on member bending stiffness (often called the P-6 effect) (b) Effect of relative horizontal joint displacements (often called the P-A effect) (c) Changes in member chord length resulting from axial strain and bowing (the latter often called curvature shortening) (d) Shearing deformation in members (e) Local buckling and other local distortions 3. Response phenomena -- material effects (a) Nonlinear stress-strain relationship, including strain hardening and elastic unloading at plastic hinges or in inelastic zones (b) Spread of inelastic zones in members (c) Inelastic interaction of axial force, biaxial bending, shear and torsion 4. Response phenomena -- coupled geometric and material effects (a) Finite joint size (b) Panel zone deformations (c) Connection deformations (d) Contributions of slabs, infilling, and secondary systems, to strength and stiffness 5. Loading effects (a) Nonproportional loading (b) Variable repeated loading (c) Dynamic effects 6. Uncertain~' (a) Variability in load effect -- temporal and spatial (b) Variability in resistance
10.1 Inelastic cross-section method The inelastic element model used by Lui 6° is shown in Fig. 28. In this model, yielding is assumed to concentrate in specific regions in a member. For a column, these regions are assumed to be located at the ends o f the member. For a beam, these regions are assumed to be located
Design of beam-columns in steel frames in USA
2,o
r
73
!
Inelastic
Elastic
Fig. 28. Inelastic element model.
at the ends and at a point within the span where the moment is expected to be the largest. Although spread of yield along the member length is ignored, plastification over the cross-section is accounted for by the use of an effective flexural rigidity (EI)c~ in the inelastic regions. The calculation of this effective E1 is shown schematically in Fig. 29. As can be seen, the variation of E1 is assumed to be linear from the cross-section first yield envelope to the cross-section ultimate strength (limit) envelope. The equation for the ultimate strength envelope was adopted from Duan and Chen. 5° To account for the presence of geometrical nonlinearity, a modified form of the stability function is used in formulating the element stiffness matrix. The validity of the approach is demonstrated in Fig. 30 in which the load-deflection behavior of a three-story one-bay frame, tested by Yarimci, 6~ is plotted and compared with results obtained using the inelastic cross-section method. Reasonably good agreement is observed.
P W8 x 31 1.0
0.8
0.6
0.4
S SS
0,2
,,
lo
\
\
s S "e
I 0.0
0.2
'Y l''" 0.4
I 0.6
I~ X 0.8
X
~ M,
1.0
Fig. 29. Schematic representation of (El)effective.
Mpx
W. F. Chen
74
2.0
1.6
,j
. ~ , ~ <
1.64 1.63
P1 =23 kips P2 =20 kips
"/
1.2
P2 P, P, P2
v
_o
0.8
Inelastic
cross-section ID
1 II__H =q ~ I 11_..H
method
.J
.....
0.4
Experimntal
iq ' 1 5 ' =1 Beams: W10x25 Columns: MSx 18.9
0.0
I
I
1
0.5
1.0
1.5
I
2.0
I
I
2.5
3.0
3.5
Lateral deflection of first story (inch)
Fig. 30. C o m p a r i s o n of the i n e l a s t i c cross-section m e t h o d with e x p e r i m e n t a l results. (1 ft = 0.305 m; 1 in = 2 . 5 4 c m ; 1 kip = 4 . 3 4 N )
10.2 Modified plastic hinge method
The modified plastic hinge method 62accounts for cross-sectional plastification by the use of a plastification factor p in the member stiffness formulation. Spread of yield along member length is ignored. The plastification factor p is defined as 0 < Pk < Mk-M~c < 1 M p c -- M,.c
(129)
where Mk is the moment at the k th end of the member, Myc is the yield moment reduced for the presence of axial force and residual stresses, and Mpc is the plastic moment reduced for the presence of axial force. They are obtained, respectively, from the equations: M~,~
0.9Mp ( -
f
P
1
Mp~ = M p [ 1 - ( ~ )
O.~p v
'3]
)
(130) (131)
In the above equation, Mp is the plastic moment capacity of the crosssection,./is the shape factor of the cross-section, P is the axial force in the member, and P,, is the yield load of the cross-section.
75
Design of beam-columns in steelframes in USA
'51.
Deflected position
J r2
ds
i,.._
dI
rl K~. %
Fig. 31. Six degree-of-freedom plane beam-column element. For the element shown in Fig. 31, the King et al. 62 element stiffness relationships has the form:
{r,} AE -~-
r2
r3
0
0
(K;, + 2K~ + K~) + P L2 L
(K~ + X],) L
0
0
(K~ + K~) L
K~,
0
-(K;~ + K;i ) L
K~j
-AE L
0
0
AE -L
0
0
0
-(K~i+2K~+K'ii) L'-
-(K~,+K~) L
0
(K~,+2K~+K~) LY"
0
(K;, + K~) L
K;,
0
-(K,~ + K~) L,
=
r4
r5 r~
-AE L
0
0
-(K~ + 2K~ + K~) L2
0
P L
(K~ + K~) L
-(K,~+K~i ) L
::] d4 d5 d~
K~j
(132)
where Ki'i =
(
K~i - Kij ~
Pj
)
(1 - p;)
(133a)
K'ii = Kii(l - Pi) (1 - P i)
(133b)
K]i = Kii(l - Pi) (1 - pj)
(133c)
K~/ =
(133d)
K~i - K(i ~
p~ (1 - P.i)
in which Pi a n d p j are plastification factors evaluated at the i th andjth ends of the member, respectively, and
76
W. F. Chen
4El
2PL
K, = K~ = ~ -
+ ~-
2EI
PL
L
30
K u = Kj; -
44P2L 3
+ 25 O00EI
(134)
26p2L 3 25 O00EI
(135)
Note that, when pg = pj = 0, the member is fully elastic and eqn (132) represents the second-order elastic stiffness relationship of the member. The ability of the modified plastic hinge method to model inelastic frame response is demonstrated in Fig. 32. In this figure, the numerical results obtained using King's method are compared with the numerical results obtained by Alvarez and Birnstie163 using the plastic zone method. Good agreement is observed.
10.3 Beam-column strength The beam-column strength method 64 for inelastic frame analysis treats each column in a frame as an individual element. The beam-column interaction equation for different slenderness ratios is taken as the ultimate strength (limit) surface for each element. When the load combination of the element reaches this ultimate strength surface, the element is considered to have failed. Since the failure criterion is based on the ultimate strength of the members that comprise the frame, the
50
:27.591
o P ~ _ ~ Q
= 75 k
Q
40
/
_° "o o
i ir
= 179 k
30 J- -
_J
Q q = 3 . 6 kilt
-
~
-
~
Q = 272 k
20
~
10
Plasticzonemethod
/
• 0.'4
lOW49~..
t_
10'__~
I-
M o d i f i e d plastic hinge m e t h o d A
0.12
f
0.16
O.J8
1.10
1.2
I
1.4
1
1.6
Lateral d e f l e c t i o n at top left joint (inch)
Fig. 32. Comparison of the modified plastic hinge method with the plastic zone method.
Design of beam-columns in steelframes in USA
77
proposed method takes into account the effect of spread of yield along member length on the inelastic stability response of a frame. For uniaxial bending about the strong axis, the beam-column interaction equation used for the beam-column strength method has the form: +
= 1
(136)
where P is the axial force, Mx is the moment, Mpx is the plastic moment capacity, and Pu is the axial load capacity of the member given by Pu
= -~(1 - 0"25~'c2)Py f°rA'c < v/2 (py/A2c for Ac > v/2
(137a, b)
in which A,c is given by eqn (24) and Py is the yield load. fl = 1.3 + 0.002(KL/rx)
(138)
f o rMI ~ <
1 r/ = 1 +0.006
-40
,,,
/> 1 f o r ~
0.5 (139)
> 0.5
Equation (136) is referred to by Chen et al. 64 as the elastic K-factor surface. The ultimate strength surface is obtained from the elastic K-factor surface by settingK = 1. In addition, an initial yield surface is defined as 70% of the elastic K-factor surface. The stiffness of the member is assumed to reduce linearly from this initial yield surface to the ultimate strength surface once the load combination of the member reaches the initial yield surface. Thus, if we define p as the distance between the initial yield surface and the ultimate strength surface, the member stiffness matrix can be expressed by eqn (132) with Pi,Pi replaced byp. In the beam-column strength method, the K factor is assumed to change from its elastic value to unity when the member fails. When the member stiffness reduces as the load combination increases, the K factor of the member is updated according to the following rules. Ifgelasti c
> 1, then K'
=
Kelastic:
(I) Kr = K ' -
1
(2) K.ew = 1 + Kr(1 - 19) (3) K' = Knew
(140)
78
If K e l a s t i
IV.. F. Chen c
< 1, then K'
Kelastic:
=
(1) Kr = 1 - K ' (2) Knew
=
1 --
gr(1 - p)
(140)
(3) K' = Knew where Kr is the difference between the current K factor and 1, K,ew is the updated K factor in each load step due to the reduction of the beamcolumn stiffness, and K' is the new K factor to be used for the next load step. The validity of the above approach for inelastic frame analysis is demonstrated in Fig. 33 in which the load-deflection response of a sixstory two-bay frame is shown. The dotted line represents the results .
.
.
.
.
Plastic
Zone Method
Beam-Column E = 20,500
1.2
SLrength Method
kN/cm 2
Fy = 2 3 . 5 0 k N / c m 2 I m p e r f e c t i o n :7/Jo = 1 / 4 5 0
_•
Load R a t i o
1.0 • 31.TkNIm !
H2•
10 2 ] k N
l|
l l i l l i i l !
=
ti
~ J ] I| E
0.8 HI=20I-~ kN Z =
I I llli;ti
i J l l l l i l i IPE
o
300
m
H,
0.6
~'
=
l i l l t l l i ' l
o
o
o
0.4
., H~
.
|l;li;;;i
z[
o
o
11111,1;i
l0
15
[ l i l i t l i ; IPE 360
] i l i l J l l ;
E = 205 k t . ; / m m I
60m
5
IPE 330
m
o
0.2
i i l l ] l l t ; IPE 300
! IPE (.00
b ~
f;0rn
20
25
Lateral DefiectAon at Right Top Joint (era) Fig. 33. Load-deflection curves o f Vogel six-story frame.
Design of beam-columns in steelframes in USA
79
obtained using the plastic zone method, 65'66 while the solid line was obtained using Chen's method. 64 Excellent correlation is observed.
11 SUMMARY A beam-column is a structural member which is subjected to both axial forces and bending moments. A column and a beam can be regarded as special cases of a beam-column. For short beam-columns the limit state will probably be that of strength. For slender beam-columns in unbraced frames, two types of instability effects can be identified: (a) Member stability orP-6 effect in which the axial force acting on the deformation of the member relative to its chord produces secondary destabilizing moments. (b) Frame stability orP-A effect in which the gravity load acting on the lateral displacement of the frame produces secondary destabilizing overturning moments. Both these effects weaken the strength and stiffness of the structure and so they must be accounted for in design. For design purposes, these secondary effects can be accounted for accurately by using a computer-based second-order elastic analysis method, or approximately by using the moment magnification factors (Bt and B2). The first-order moments evaluated from first-order analysis are multiplied by these magnification factors to approximate the total moments that will be used for design. The design of beam-columns is facilitated by the use of interaction equations. These interaction equations provide a smooth transition between ultimate strengths of columns and beams (Pu and Mu) to give safe combinations of the applied loads (P and M). Interaction equations may be linear or nonlinear. Linear interaction equations have the advantage of simplicity, but they are not as accurate as their nonlinear counterparts. Interaction equations, either recommended or proposed in the USA, are summarized and discussed in this paper. The introduction of the limit states philosophy has revolutionized the concept of steel design. When a structure is designed according to its limit of usefulness, a more accurate assessment of the response of the structure is desirable. Although the use of a first-order elastic analysis is still common practice in the design profession, there has been a trend towards the use of more elaborate analysis methods for design. Since almost all structures behave in some nonlinear fashion prior to reaching their limit of usefulness, a valid limit state design should, therefore,
80
Iv.. F. Chen
include all factors that affect the response of the structure to loadings. A summary of such factors has been given in Table 7. The use of a general analysis method which is capable of incorporating all these factors and at the same time simple enough for routine office use seems to be unrealistic at present. However, steps are moving toward that direction. In particular, the AISC L R F D Specification 3has provided provisions for a direct second-order elastic analysis for design, and the Australian Limit State Steel Design Specification (AS4100) 67 and the Eurocode 68 have both attempted to provide provisions for a second-order inelastic analysis for design. With the advancements made in computer hardware and software, computer-aided analysis and design will definitely play an important role in structural design. The adoption and proliferation of the limit states approach to design have provided engineers with a vehicle to explore innovative analysis and design procedures. In this paper, the current AISC L R F D interaction equations for beamcolumn design have been reviewed. The importance of an accurate assessment of the effective length factor K has been addressed. New interaction equations capable of representing the inelastic behavior of beam-column have also been presented. To conclude, three simplified second-order inelastic analysis methods suitable for practical use have been described. REFERENCES 1. AISC, Specification for the Design. Fabrication and Erection of Structural Steel .for Buildings. American Institute of Steel Construction, Chicago, Nov. i 978. 2. AISC, Allowable Stress Design Specification .for Structural Steel Buildings. American Institute of Steel Construction Chicago, 1989. 3. AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, Oct. 1986, ! st edn, 313 pp. 4. Ravindara, M. K. & Galambos, T. V., Load and resistance factor design for steel. J. Struct. Div., ASCE, 104(ST9)(1978) 1337-54. 5. Ellingwood, B., MacGregor, J. G., Galambos, T. V. & Cornell, C. A., Probability-based load criteria, load factors and load combinations. J. Struct. Div.. ASCE, 108(ST5) (1982) 978-97. 6. ANSI, Building Code Requirements for minimum Design Loads in Buildings and other Structures, ANSI, A58.1, American National Standards Institute, New York (1982). 7. Ang, A. H. S. & Corneil, C. A., Reliability bases of structural safety and design. J. Struct. Div.. ASCE. 100(STg) (1974) 1755-69. 8. Bjorhovde, R., Deterministic and Probabilistic Approaches to the Strength of Steel Columns, PhD Dissertation, Department of Civil Engineering, Lehigh University, Bethlehem, PA, 1972.
Design of beam-columns in steelframes in USA
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9. Johnston, B. G., (ed.), SSRC Guide to Stability Design Criteria for Metal Structures. 3rd edn, John Wiley, New York, 1976, 616 pp. 10. Rondal, J. & Maquoi, R., Single equation for SSRC column strength curves. Technical Notes, J. Struct. Div., ASCE, 105(STI) (1979) 247-50. 11. Chen, W. F. & Atsuta, T., In-plane behavior and design. Theory of BeamColumns. Vol. 1, McGraw-Hill, New York, 1976, 512 pp. 12. Lui, E. M. & Chen, W. F., Generalized column e q u a t i o n - A physical approach. In Advances in Tall Buildings. ed. L. S. Beedle. Council on Tall Buildings and Urban Habitat, Van Nostrand Reinhold, New York, 1986, pp 323-52. 13. Lui, E. M. & Chen, W. F., Simplified approach to the analysis and design of columns with imperfections. Engineering J.. AISC, 21(2) (1984)99-177. 14. Sugimoto, H. & Chen, W. F., Small end restraint effects on strength of Hcolumns. J. Struct. Div.. ASCE, 108(ST3) (1982) 661-81. 15. Lui, E. M. & Chen, W. F., End restraint and column design using LRFD. Engineering J., AISC, 20(1) (1983) 29-39. 16. Chapius, J. & Galambos, T. V., Restrained crooked aluminium columns. J. Struct. Div.. ASCE. 108(ST3) (1982) 511-24. 17. Julian, O. G. & Lawrence, L. S., Notes on J and L Nomographs for Determination of Effective Lengths. Unpublished Report, Jackson and Moreland Engineers, Boston, MA, 1959. 18. LeMessurier, W. J., A practical method of second order analysis, Part I. Engineering J..AISC. 13(4) (1976) 89-96. 19. LeMessurier, W. J., A practical method of second order analysis, Part II. Engineering J.,AISC, 14 (1977) 49-67. 20. Yura, J. A., The effective length of columns in unbraced frames, Engineering J.. AISC. 8(2) (1971) 37-42. 21. Chen, W. F. & Atsuta, T., Space behavior and design. Theory of BeamColumns. Vol. 2, McGraw-Hill, New York, 1977, 732 pp. 22. Chen, W. F. & Lui, E. M., Effects of joint flexibility on the behavior of steel frames. Computers and Structures. 26(5) (1987) 719-32. 23. Galambos, T. V., Inelastic lateral buckling of beams. J. Struct. Div., ASCE. 89(ST5) (1963) 217-42. 24. Vinnakota, S., Verification of the SSRC Interaction Formula for Lateral Buckling of Beam-Columns. SSRC-TG3 Report, March 1982. 25. Yura, J. A., Galambos, T. V. & Ravindra, M. K., The bending resistance of steel beams. J. Struct. Div.. ASCE 104 (STg)(1978) 1355-69. 26. Duan, L., Sohal, I. S. & Chen, W. F., on beam-column moment amplification factors, Engineering J.. 26(4) (1989) 130-5. 27. Zhou, S. P. & Chen, W. F., C,, factor in LRFD.J. Struct. Eng.. ASCE. 113(ST8) ! 738-54. 28. Rosenblueth, E., Slenderness effects in buildings. J. Struct, Div.. ASCE. 91(ST1) (1965) 229-52. 29. Stevens, L. K., Elastic stability of practical multistory frames. Proc. Inst. Civil Engrs. Vol. 36, London, UK, 1967. 30. Cheong-Siat-Moy, F., Consideration of secondary effects in frame design. J. Struct. Div.. ASCE. 103(STI0)(1977)2005-19. 31. Nixon, D., Beaulieu, D. & Adams, P. F., Simplified second-order frame analysis. Can, J. Civil Engng, 2(4) (1975). 32. Chen, W. F. & Cheong-Siat-Moy, F.. Limit states design of steel beam-
82
W. F. Chen
columns -- A state-of-the-art review. J. Solid Mech. Arch., 5( I ) (1980) 29-73. 33. Chen, W. F. & Lui, E. M., Stability design criteria for steel members and frames in the United States. J. Construct. Steel Res., 5 31-71. 34. Chen, W. F. & Lui, E. M., Analysis and design of steel frames in the USA -90's and beyond, keynote lecture. Proc. Int. Conf. on Steel and Aluminium Structures, Singapore, May 1991. 35. Chen, W. F. & Lui, E. M., Structural Stability: Theory and Implementation. Elsevier, New York, March 1987, 490 pp. 36. Chen, W. F. & Lui, E. M., Stability Design of Steel Frames, Blackwell Scienti tic, Oxford, UK, 1991, 373 pp. 37. Galambos, T. V. (ed.), SSRC Guide to Stability Design Criteria .for Metal Structures, 4th edn, Wiley-lnterscience, New York, 1987. 38. Kanchanalai, T. & Lu, L. W., Analysis and design of frame columns under minor axis bending. Engineering J., AISC, 16(2) (1979) 29-41. 39. Cheong-Siat-Moy, F., & Downs, T., New interaction equation for steel column design. J. ofStruct. Div,, ASCE, 106, (ST5), (1980) 1047-62. 40. Tebedge, N. & Chen, W. F., Design criteria for H-columns under biaxial bending. J. Struct. Div.. ASCE, 100(ST3) (1974) 579-98. 41. Anslijn, R. & Massonnet, C. E., New Tests on Steel I Beam-Columns in Mild-Steel Subjected to Thrust and Biaxial Bending. Technishe Hochschule, Darmstadt, Germany, 1978, pp. 103-123. 42. Anslijn, R., Tests on Steel I Beam-Columns Subjected to Thrust and Biaxial Bending. Construction Metailique, MT 157, Aug. 1983. 43. Cai, C. S., Liu, X. L. &Chen, W. F., Further verifications of the beam-column equations. J. Struct. Engng. ASCE, 116(2) (1991) 501-13. 44. Yura, J. A., Elements for Teaching Load and Resistance Factor Design: Combined Bending and Axial Load. Lecture Notes, University of Texas at Austin, 1988, pp. 55-71. 45. Kanchanalai, T., The Design and Behavior of Beam-Columns in Unbraced Steel Frames. AISI Project No. 189, Report No. 2, Civil Engineering/ Structures Research Laboratory, University of Texas at Austin, 1977, 300 pp. 46. Chen, W. F., Zhou, S. P. & Duan, L., Second-order inelastic analysis of braced portal frames. J. Singapore Soc Structural Engrs, I(1), December (1990) 5-15. 47. Zhou, S. P., Duan, L. & Chen, W. F., Comparison of design equations for steel beam-columns. Struct. Eng. Rev. 2(!) (1990) 45-53. 48. Zhou, S. P., Duan, L. & C h e n , W. F., The P-A effect on portal steel frames. Proc. Int. Conf. on Steel and Aluminium Structures, Singapore, May 1991. 49. Liew, R. J. Y., White, D. W. & Chen, W. F., Beam-column design in steel frameworks. In Constructional Steel Design: An International Guide. eds. R. Bjorhovde, & A. Colson. Elsevier, London, 1991. 50. Duan, L. & Chen. W. F., Design interaction equation for steel beamcolumns. J. Struct. Enging.. ASCE. 115(5) (1989) 1225-43. 51. Goto, Y., & Chen, W. F., On the computer-based design analysis for the flexibility jointed frames. J. Construct. Steel Res., 8 (1987) 203-31. 52. Goto, Y., & C h e n , W. F., Second-order elastic analysis for frame design. J. Struct. Enging.. ASCE. 113(7) (1987) 1501-19. 53. Livesley, R. K. & Chandler, D. B., Stability Functions for Structural Frameworks. Manchester University Press, Manchester, UK, 1956.
Design of beam-columns in steelframes in USA
83
54. Home, M. R. and Merchant, W., The Stability of Frames. Pergamon Press, Oxford, UK, 1965. 55. Connor, J. J., Logcher, R. D. & Chan, S. C., Nonlinear analysis of elastic frames structures. J. Struct. Div., ASCE, 94(ST6) (1968) 1525-47. 56. Mallet, R. H. & Marcal, P. V., Finite element analysis of nonlinear structures. J. Struct. Div., ASCE, 94(ST9) (1968) 2081-310. 57. Chen, W. F. & Zhou, S. P., Inelastic analysis of steel braced frames with flexible joints. J. Solids and Structures, 23(5) (1987) 631-49. 58. Lui, E. M. & Chen, W. F., Steel frame analysis with flexible joints. In Joint Flexibility in Steel Frames, ed. W. F. Chen. Elsevier Applied Science, London, 1987, pp. 161-202. 59. McGuire, W., Structural analysis in load and resistance factor design. Introductory remarks at a Seminar on Frame Analysis in LRFD, dISC National Steel Conference. 1990. 60. Lui, E. M., Nonlinear stability analysis of frameworks- from theory to practice. 1990 Structures Congress, Baltimore, MD, 1990. 61. Yarimci, E., Incremental Inelastic Analysis of Framed Structures and Some Experimental Verifications, PhD Thesis, Department of Civil Engineering, Lehigh University, Bethlehem, PA,~1966. 62. King, W. S., White, D. W. & Chen, W. F., A Modified Plastic Hinge Method for Second-Order Inelastic Analysis of Steel Rigid Frames. Structural Engineering Report No. STR-90-17, School of Civil Engineering, Purdue University, West Lafayette, IN, 1990, 34 pp. 63. Alvarez, R. J. & Birnstiel, C., Inelastic analysis of multistory multibay frames. J. Struct. Div., ASCE, 95(11) (1969) 2477-503. 64. Chen, W. F., King, W. S. & White, D. W., A Beam-Column Strength Method for Second-Order Inelastic Analysis of Steel Frames. Structural Engineering Report No. CE-STR-90-18, School of Civil Engineering, Purdue University, West Lafayette, IN, 1990, 29 pp. 65. Vogel, U., Calibrating frames. Stahlbau. 10 (1985) 1-7. 66. Toma, S. & Chen, W. F., Calibration Frame in Europe. StructUral Engineering Report No. CE-STR-91-1, School of Civil Engineering, Purdue University, West Lafayette, IN, 1991. 67. Standards Australia, Australian Standard for Design of Steel Structures, AS 4100, Committee B D / I - Steel Structures, Working Group BD/1/-/7Methods of Structural Analysis, 1990. 68. Eurocode No. 3,Design of Steel Structures: Part I -- General Rules and Rulesfor Buildings. Vol. 1, Edited Draft Issue 3, 1990.