1569 ch32
1569 ch32
1569 ch32
Structural Bracing
32.1 Introduction
This chapter presents an overview of aspects related to the design of structural bracing used in beams,
columns, and frame structures and is intended for practicing civil and structural engineers. Many of the
design guidelines presented were incorporated into the 2002 Load and Resistance Factor Design Manual
published by the American Institute of Steel Construction (AISC). The intended focus is on simplicity
and ease of implementation over exact formulations. The basis for the design formulations along with
a classification system for bracing systems is first presented. Design formulations are presented with
illustrative numerical examples. Finally, common faulty bracing details are presented.
0-8493-1569-7/05/$0.00+$1.50
# 2005 by CRC Press 32-1
(a) (b)
Comp Cross
flange frames
Diaphragms
Brace
(c) (d)
Metal
deck
Girder A B
FIGURE 32.1 Types of bracing: (a) relative; (b) nodal; (c) continuous; and (d) lean-on.
systems, such as diagonal bracing or shear walls, prevent the relative lateral movement of adjacent stories
or of adjacent points along the length of a member. Relative systems can be readily identified if a cut at
any location along the length of a braced member passes through the brace member itself. Nodal systems
control the movement only where they attach to the braced member and do not directly interact with
adjacent brace points. Cross-frames or diaphragms between two adjacent beams are considered nodal
braces. Continuous systems provide uninterrupted support along the entire length of a member, leaving
no unbraced length. Shear walls and roof or floor deck are examples of continuous bracing systems.
Lean-on systems rely on adjacent structural members to provide support. Lean-on bracing links together
adjacent structural members such that buckling of one member requires all members in the system to
buckle with the same lateral displacement.
32.3 Background
Structural bracing used to increase the strength of members must possess both sufficient strength and
stiffness [2]. Simple bracing design rules such as designing a brace to resist 2% of the member
compressive force address only the strength criterion. The stiffness of the brace along with the out-
of-straightness of the member has a direct effect on the magnitude of the brace force [1]. Design
recommendations based on perfectly straight members should not be used directly in design since
extremely large brace forces and displacements may result [3].
∆o P P
∆o ∆
∆
L
Initial out-
of-straightness
A
P P
deflections and corresponding brace forces are kept small by using brace stiffnesses greater than the ideal
stiffness. The plots developed in Figure 32.3 were based on an assumed initial out-of-straightness equal to
0.002L. Larger out-of-straightness values linearly increase the magnitude of the brace forces.
∆sh = shortening
In the AISC load and resistance factor design (LRFD) specification, when the axial stress is less than one
third the yield stress, Fy, the column is classified as elastic (t ¼ 1.0). At greater stress levels, the stiffness
reduction factor is given by
Pu ðPu =Py Þ
t ¼ 7:38 log ð32:1Þ
Py 0:85
where Pu is the required column strength and Py is the column squash load.
systems where a story may contain n0 columns, each having a random out-of-plumbness, an average
value for Do can be used [4]
pffiffiffiffiffi
Do ¼ 0:002L n0 ð32:3Þ
EXAMPLE 32.1
12⬘
18⬘
X
Pu ¼ ð3 bentsÞð125 þ 300 þ 175Þ ¼ 1800 kip
Bracing shear force:
Pbr ¼ 0:004ð1800Þ ¼ 7:2 kip
Bracing shear stiffness:
2ð1800Þ
¼ 400 kip/ft
bbr ¼
0:75ð12Þ
Design recommendations assume brace shear force and stiffness are perpendicular to column. Therefore,
for an inclined threaded rod (A36 steel):
Strength:
7:2 kip
Pbr ¼ ¼ 8:64 kip ¼ 0:9ð36ÞAg ðAg Þreq d ¼ 0:27 in:2
cos y ’
Stiffness:
Ag E ðAg Þreq d ¼ 0:43 in:2 Controls
cos2 y ¼ 400 k/ft ’
21:6 ft Use 34 in. dia. rod Ag ¼ 0:44 in:2
Consider effects of shortening
300 k
Dos ¼ sway due to shortening ¼ Dsh tan y
300ð12 12Þ
Dsh ¼ ¼ 0:095 in:
ð15:6Þð29,000Þ
.6⬘ 12
12⬘ 21 Dos ¼ 0:095 ¼ 0:063 in:
18
Dos 0:063
¼ ¼ 0:00044
L ð12 12Þ
18⬘
0:002 þ 0:0004
Pbr ¼ 8:64 kip ¼ 10:5 kip
0:002
ðAg Þreq d ¼ 0:32 in:2 < 0:43 in:2 stiffness still controls
’
Pcr
1.0
0.8 L
Pcr 0.6
Pe
Limit
0.4 full bracing
2EI
0.2 Pe =
L2 3.41
0
0 1 2 3 4
L/Pe
For nodal column bracing, the ideal brace stiffness is a function of the number of intermediate braces
[1,2]. For a single brace at midheight bi ¼ 2P/L. For many closely spaced braces the ideal stiffness
approaches bi ¼ 4P/L. Twice the ideal stiffness for the most severe case was adopted by the AISC LRFD
for many braces.
AISC LRFD brace requirements for nodal column bracing are
8Pu
bbr ¼ ð32:7Þ
fLb
Pbr ¼ 0:01Pu ð32:8Þ
where f ¼ 0.75, Pu is the required compressive strength of the column, and Lb is the required brace spacing.
For n equally spaced braces, the ideal stiffness can be approximated as
Pu
bi ¼ Ni ð32:9Þ
Lb
where Ni 4 2/n. Using the recommended stiffness equal to twice the ideal stiffness and applying the
resistance factor gives
2Pu
bbr ¼ Ni ð32:10Þ
fLb
Equation 32.10 is based on equally spaced braces and is unconservative for unequal spacings. The
required stiffness for unequal brace spacings can be obtained using Winter’s rigid bar model [6]. In this
model, the column is represented using rigid links with ficticious hinges at brace locations and the braces
are represented using simple springs. Under the applied load, displacements are imposed at brace
locations and equilibrium is enforced to obtain Ni. This technique is illustrated in Example 32.2. For
a single nodal brace at any location along the length of a column, with the longest segment defined as L
and the shorter segment as aL, Ni can be conservatively determined using
1
Ni ¼ 1 þ ð32:11Þ
a
EXAMPLE 32.2
P P
PD ¼ 0:6bDð0:4LÞ
A 0.6D
P
bi ¼ 4:16
0.4L L
D Ni ¼ 4:16
B
D
Conservative approximation
0.6L
1
Ni ¼ 1 þ ¼ 2:5
C 0:4=0:6
0.4D
P P
The brace stiffness requirements for nodal bracing are inversely proportional to the unbraced
length, Lb. Closer-spaced braces require more stiffness because the derivations are based upon
allowing the column to reach a load that corresponds to buckling of the most critical unbraced length
with a K-factor equal to 1.0. In many instances, there are more potential brace points than necessary
to support the member forces required. Using the actual unbraced length may result in excessively
conservative stiffness requirements. Therefore, the maximum unbraced length that enables the col-
umn to reach the required loading, Lq, can be used. For example, say the column shown in
Figure 32.5 is supported against weak-axis buckling at three locations giving an unbraced length of
L. If a single brace at midheight giving an unbraced length of 1.5L would be sufficient to carry the
load on the column, then the required stiffness for the three braces could be conservatively estimated
using the permissible unbraced length of 1.5L in Equation 32.10 in place of the actual unbraced
length of L (see Example 32.3).
EXAMPLE 32.3
2ð96Þ x x
Mbr ¼ ¼ 48 kip in:
4
qffiffiffiffiffiffiffi
2L Pe
Pcr ¼ Pe þ b ð32:12Þ
p
where Pe is the Euler buckling load, L is the column length, and b is the brace stiffness per unit
length.
The continuous brace formulation given in Equation 32.12 can also be applied for equally spaced
, using
discrete braces by determining an equivalent brace stiffness per unit length, b
¼bn
b ð32:13Þ
L
where n is the number of braces within the column length, L. This method is accurate for two or more
discrete braces and is illustrated in Example 32.4.
Corrugated metal deck is a common type of continuous lateral bracing and acts like a shear
diaphragm with the properties of a relative brace. The stiffness and strength properties of the metal
Pcr
20
Pe = EI
2
EQ. 12.5
L2
L
15 n =3
= k/ in. per in.
P EQ. 12.4
10
Pcr n =2
5
n =1
0
0 200 400 600 800
L2/Pe
∆ Metal deck
F 2Pu
F br = =
∆ Lb
F = 0.004Pu
F/b
G⬘=
Lb ∆/Lb
∆/L
2Pu
G⬘req’d =
b
deck are generally defined in a per unit width basis (e.g., shear stiffness ¼ G 0 kip/rad per ft width).
The bracing requirements for the shear diaphragm can be determined from the relative brace
requirements presented in Section 32.5.2, as shown in Figure 32.7. Properties for corrugated deck can
be obtained from the Steel Deck Institute Diaphragm Design Manual [7]. The required shear dia-
phragm stiffness per unit width is
0 2Pu
Greq’d ¼ fb ð32:14Þ
Dividing the perpendicular brace force requirement by the diaphragm width gives the required shear
strength per unit width, Su
0:004Pu
Su ¼ ð32:15Þ
b
It should be noted that the brace force requirements given in Equation 32.13 are in addition to other
load demands placed on the diaphragm.
EXAMPLE 32.4
EXAMPLE 32.5
50 k 450 k P
Sway capacity — Using P concept
8 ft
Pu 450 1
¼ ¼ 0:735 > \ inelastic
Fy Ag ð36Þð17:0Þ 3
0:735
t ¼ 7:38ð0:735Þlog ¼ 0:342
W10 × 33
W12 × 58
0:85
8 ft
p2 ð29,000Þð107Þ
fPn ¼ 0:85ð0:342Þð0:877Þ ¼ 94 k
ð288Þ2
Column B (W10 33)
8 ft
Sway Pu 50 1
mode ¼ ¼ 0:143< \ elastic t ¼ 1:0
Fy Ag ð36Þð9:71Þ 3
p2 ð29,000Þð171Þ
fPn ¼ 0:85ð1:0Þð0:877Þ ¼ 440 k
B A ð288Þ2
P
Using P concept
X
X
fPn ¼ 94 þ 440 ¼ 534 k > Pu ¼ 50 þ 450 ¼ 500 k OK
where xbr, ybr are the coordinates of axis of restraint with respect to column centroid, d is the column
depth, and Pey is the Euler load based on a column length between points of zero twist.
To compensate for the assumption in the derivation of Equations 32.16 and 32.17 that the brace
is infinitely stiff, the maximum factored column load should be limited to 0.90PT [3].
(a) (b)
Brace point (axis of restraint)
x x
ybr
d
FIGURE 32.9 Buckling of a column about a restrained axis: (a) lateral brace at flange and (b) buckled shape.
(a) (b)
Moment connection
Stiffener at least
half depth
FIGURE 32.10 Typical torsional brace details: (a) control twist with struts and (b) moment connection with stiffener.
If column loads greater than PT are required, torsional bracing must be provided. Two typical bracing
schemes are shown in Figure 32.10. For continuous girts with moment connections, twisting restraint is
provided. However, partial depth stiffeners should be used to control web distortion. The design
requirements for torsional bracing are based on the nodal requirements presented in Section 32.5.3
and are obtained by introducing equal and opposite brace forces on each flange. The magnitude of these
forces is based on the assumption that each flange carries one-half of the total column load. The resulting
brace moment, MT ¼ 0.5Pbrd. Using the angle of twist y ¼ D/d as shown in Figure 32.9b, the stiffness
requirement bT ¼ MT/y ¼ 0.5Pbrd2/D reduces to
bT ¼ 0:5bbr d 2 ð32:18Þ
where bbr is the nodal brace stiffness requirement from Section 32.5.3.
100
Brace
Beam A
L
68
L
Beam B
L
68
Moment diagrams
Mid-depth
Top flange
Bottom flange
Ineffective
Plan of buckled shape of Beam B midspan brace
Compression
flange
Compression
flange
FIGURE 32.12 Lateral buckling of cantilever and simple beams: (a) max moment at max twist and (b) zero
moment at max twist.
a beam. In most configurations, this places the brace the furthest distance from the center of twist of the
buckled cross-section as shown in Figure 32.12a. The exception to this rule is for cantilever beams
(Figure 32.12b) where the position of the buckled cross-section reveals that a brace located at the tension
flange is most beneficial.
When a beam is bent in reverse curvature, the compression and tension flanges switch at the point of
inflection. Beams such as these, in which both the top and bottom flanges encounter compression along the
span, have more severe bracing requirements than beams where compression resides only in one flange [3].
In these situations, lateral bracing is required on both flanges to prevent twist of the cross-section.
The vertical position of transverse loads on beams has significant influence on the effectiveness of
lateral bracing. Loads applied higher on the cross-section, such as at the top flange, have more pro-
nounced destabilizing effects while loads applied lower on the cross-section tend to provide added
stability when compared to centroidal loading. The effect of top-flange loading is even greater when
lateral bracing is located near the centroid of the section. In these situations, the center of twist shifts to
a position closer to mid-depth and the centroidal brace becomes almost totally ineffective as shown in
Figure 32.13. Therefore, centroidal lateral beam braces are not recommended due to the effects of both
cross-section distortion and load position.
The load position effect described above is based on the assumption that the load remains vertical and
passes through the original point of contact on the member as it buckles. For many structural systems,
the load transferred to beams is applied through secondary members or a floor slab. When loading is
through a slab, for example, a restoring torque is created by a tipping effect during buckling as illustrated
in Figure 32.14. This tipping effect has been shown to significantly increase the lateral buckling capacity
even if the slab is only resting (not positively attached) on the top flange [12]. Unfortunately, the benefits
of tipping are severly limited by distortion of the cross-section and are difficult to quantify. As a result,
the beneficial effects of tipping are generally neglected.
The AISC LRFD lateral beam brace requirements were based on the following design recommenda-
tions developed by Yura [3]:
The brace stiffness requirement for both relative and nodal beam bracings is
2Ni ðCb Pf ÞCt Cd
bbr ¼ ð32:19Þ
fLb
The brace strength requirement for relative bracing is
Mu Ct Cd
Pbr ¼ 0:004 ð32:20Þ
hoh
and for nodal bracing it is
Mu Ct Cd
Pbr ¼ 0:01 ð32:21Þ
hoh
where
Ni ¼ 1.0 for relative bracing
¼ (4 2/n) for nodal bracing
n ¼ number of intermediate braces
Pf ¼ beam compressive flange force
¼ p2 EIyc =Lb2
Iyc ¼ out-of-plane moment of inertia of the compression flange
Ct ¼ top-flange loading factor
¼ 1.0 for centroidal loading
¼ 1 þ (1.2/n) for top-flange loading
Cd ¼ 1 þ (MS/ML)2 for reverse-curvature bending
¼ 1.0 for single-curvature bending
MS ¼ smallest moment causing compression in each flange
ML ¼ largest moment causing compression in each flange
Cb ¼ nonuniform moment modification factor
12:5Mmax
¼
2:5Mmax þ 3MA þ 4MB þ 3MC
Mmax ¼ absolute value of maximum moment in unbraced segment
MA ¼ absolute value of moment at quarter point of unbraced segment
MB ¼ absolute value of moment at midspan of unbraced segment
MC ¼ absolute value of moment at three-quarter point of unbraced segment
The brace force requirements were developed assuming an initial lateral displacement of the
compression flange equal to 0.002Lb and vary in direct proportion to the actual out-of-straightness.
The term 2NiCt can be conservatively approximated as 10 for any number of nodal braces and 4 for
any number of relative braces. The term CbPf can also be conservatively approximated as Mu/hoh. Using
the worst-case top-flange loading (Ct ¼ 2.0) and the previous assumptions yields the AISC LRFD brace
requirements for lateral beam bracing (Example 32.6):
AISC LRFD brace requirements for relative lateral beam bracing:
4Mu Cd
bbr ¼ ð32:22Þ
fLb hoh
Mu Cd
Pbr ¼ 0:008 ð32:23Þ
hoh
AISC LRFD brace requirements for nodal lateral beam bracing:
10Mu Cd
bbr ¼ ð32:24Þ
fLb hoh
Mu Cd
Pbr ¼ 0:02 ð32:25Þ
hoh
where
f ¼ 0.75
Mu ¼ required flexural strength
ho ¼ distance between flange centroids
Cd ¼ 1 þ (MS/ML)2 for reverse-curvature bending
¼ 1.0 for single-curvature bending
Lb ¼ distance between braces
EXAMPLE 32.6
Section view
5 × 18 ft = 90 ft 1 × 10
Plan view
Stiffness
1
2 × 48
4:0ð1200 12Þð1:0Þ 1 × 16
bbr ¼ ð2:5 girdersÞ ¼ 18:1 k=in:
0:75ð18 12Þð49Þ ho = 49 in.
2
AE 2 Abr ð29,000Þ 1
cos y ¼ p ffiffi
ffi pffiffiffi ¼ 18:1
L br 9 12 5 5
Strength
ð1200 12Þð1:0Þ
Pbr ¼ 0:008 ð2:5 girdersÞ ¼ 5:88 k
49
pffiffiffi
5:88 5 1
Abr ¼ ¼ 0:41 in:2 Use L2 2 ðAg ¼ 0:944 in.2 Þ
ð0:9 36Þ 4
0:024Mu L
Mbr ¼ ð32:27Þ
nCb Lb
where
3:3E 1:5hoh tw3 ts bs3
bsec ¼ þ ð32:28Þ
hoh 12 12
2:4LMu2
bT ¼ ð32:29Þ
fnEIy Cb2
where
f ¼ 0.75
L ¼ span length
n ¼ number of nodal braced points within span
E ¼ modulus of elasticity
Iy ¼ out-of-plane moment of inertia of beam
Cb ¼ nonuniform moment modification factor
tw ¼ thickness of beam web
Torsional brace
Web
4tw
Ib
Mbr
6EIb 2EIb
T = T =
S S
S
P P
/S 2ES2hb2r
Tension PL
+2 T =
system hbr 2Ld3 S3
+
P P Ad Ah
–P
2Phbr 2Phbr
S S
S
P P
/S
Compression PL
system +2 AdES2 hb2r
hbr T =
–2
PL L3d
P /S P
2Phbr 2Phbr
S S
S
P P
2ES2hb2r
T =
8L3d S3
–2
/S
K-brace hbr
+
PL
PL
Ad Ah
+2
/S
P P
+P –P
2Phbr 2Phbr
S S
∆
Ad = Area of diagonal members
∆ + ∆b
Ah = Area of horizontal members T = M =
Ld = Length of diagonal members S
∆b
V V
b Joist
a b
Compression B
flange
Sect B–B
P
P
Siding
a
LG
Girt LG LC
LG
In-plane Out-of-plane
top of the column if there are braces at point b and consideration is given to the compression in the
flange when evaluating its stiffness. In general, a brace, such as a bottom chord extension from the joist,
should be used at point a. Beam web stiffeners at the column location will also be effective unless bottom
flange lateral buckling is critical.
Another common faulty bracing detail is shown in Figure 32.20. The girts that frame into the column
flange prevent weak-axis translation at the braced flange. Since the girts are discontinuous, they will not
prevent twist of the cross-section and will force the column to buckle about a restrained axis (see also
Figure 32.9b). For this column, there are three possible buckling modes: strong-axis flexural buckling
(Lb ¼ KLC), weak-axis flexural buckling (Lb ¼ KLG), and torsional buckling about a restrained axis
(Lb ¼ KLC, ybr ¼ a assuming no twist at column ends).
Nomenclature
A cross-sectional area of primary MS smallest moment causing compression
member in each flange along beam length
Abr cross-sectional area of brace member MT torsional brace moment
Ad area of diagonal member in Mu required bending strength
cross-frame Ni brace stiffness coefficient for nodal
Ah area of horizontal member in braces
cross-frame Pbr brace force
Ag gross cross-sectional area Pcr member buckling load
Cb bending coefficient dependent on Pe Euler column buckling load
moment gradient Pey Euler buckling load based on distance
Cd reverse-curvature bending factor between points of zero twist
Ct top-flange loading factor Pf beam compressive flange force
E modulus of elasticity PT torsional buckling load
ET tangent modulus of elasticity Pu required compressive strength of
Fy specified minimum yield stress column
G shear modulus of elasticity Py column squash load
G0 diaphragm shear stiffness per unit S cross-frame or diaphragm length
width Sx strong-axis section modulus
I moment of inertia Su required diaphragm shear strength
Ieff effective moment of inertia for singly per unit width
symmetric beam sections V shear force
Ireq’d required moment of inertia b orthogonal distance between point of
Ix, Iy moment of inertia about strong and restraint and weak axis of member
weak axes, respectively bs stiffener width for one-sided stiffeners
Iyc, Iyt out-of-plane moment of inertia of d member depth
compression and tension flanges, fb bending stress
respectively hbr deight of cross frame
J torsion constant for section torsional brace
K column effective length factor hoh distance between flange centroids
L member length n rumber of braces within span
Lb required brace spacing or laterally n0 rumber of columns in a story
unbraced length; length between rx, ry radius of gyration about strong and
points that are either braced against weak axes, respectively
lateral displacement of the compres- tw thickness of beam web
sion flange or braced against twist of ts thickness of web stiffener
the cross-section xbr, ybr coordinates of axis of restraint with
Lq maximum unbraced length for respect to column centroid
required forces yc, yt coordinates with respect to centroid of
MA absolute value of moment at quarter expreme compression and tension
point of unbraced beam segment fibers, respectively
MB absolute value of moment at midspan b continuous brace stiffness per unit
of unbraced beam segment length
MC absolute value of moment at bact stiffness provided by brace member
three-quarter point of unbraced bbr required lateral brace stiffness
beam segment bconn stiffness of brace connection
ML largest moment causing compression bsec web distortional stiffness including
in each flange along beam length any transverse web stiffeners if present
Mmax absolute value of max moment in bsys stiffness of brace system
unbraced beam segment bT nodal torsional brace stiffness
bTb required nodal torsional brace stiffness DT total column sway deflection
including web distortion f resistance factor
D translational displacement y complementary angle between
Do column initial out-of-straightness diagonal brace and axial member or
Dos column out-of-straightness due to twist of member cross-section
shortening t inelastic stiffness reduction
Dsh shortening or compression element factor
References
[1] Winter, G. (1958), ‘‘Lateral Bracing of Columns and Beams,’’ Trans. ASCE, Vol. 125, Part 1,
pp. 809–825.
[2] Winter, G. (1960), ‘‘Lateral Bracing of Columns and Beams,’’ Proc. ASCE, Vol. 84 (ST2), pp. 1561-
1–1561-22.
[3] Yura, J.A., Bracing, in Stability Design Criteria for Metal Structures, 5th Edition, Galambos, T.V.
Ed.; John Wiley & Sons, Inc., New York, 1998; Chapter 12.
[4] Chen, S. and Tong, G. (1994), ‘‘Design for Stability: Correct Use of Braces,’’ Steel Struct.,
J. Singapore Struct. Steel Soc., Vol. 5, No. 1, pp. 15–23.
[5] Timoshenko, S. and Gere, J. (1961), Theory of Elastic Stability, McGraw-Hill Book Company,
New York.
[6] Yura, J.A. (1994), ‘‘Winters Bracing Model Revisited,’’ 50th Anniversary Proc., Struc. Stability
Research Council, pp. 375–382.
[7] Luttrell, L.D. (1987), Diaphragm Design Manual, 2nd Edition, Steel Deck Institute, Fox River
Grove, IL.
[8] Yura, J.A. (1971), ‘‘The Effective Length of Columns in Unbraced Frames,’’ Eng. J. Am. Inst. Steel
Const., Vol. 8, No. 2, pp. 37–42.
[9] Mutton, B.R. and Trahair, N.S. (1973), ‘‘Stiffness Requirements for Lateral Bracing,’’ ASCE
J. Struct. Div., Vol. 99, No. ST10, pp. 2167–2182.
[10] Tong, G. and Chen, S. (1988), ‘‘Buckling of Laterally and Torsionally Braced Beams,’’ J. Const. Steel
Res., Vol. 11, pp. 41–55.
[11] Flint, A.R. (1951), ‘‘The Stability of Beams Loaded Through Secondary Members,’’ Civil Eng.
Public Works Rev., Vol. 46, No. 537–8 (see also pp. 259–260).
[12] Yura, J.A. (1993), ‘‘Fundamentals of Beam Bracing,’’ Proc. Struc. Stability Research Council Annual
Technical Session, ‘‘Is Your Structure Suitably Braced?’’ Milwaukee, April, 20 pp.
Further Reading
Akay, H.U., Johnson, C.P. and Will, K.M. (1977), ‘‘Lateral and Local Buckling of Beams and Frames,’’
ASCE J. Struct. Div., Vol. 103, No. ST9, pp. 1821–1832.
Ales, J.M. and Yura, J.A. (1993), ‘‘Bracing Design for Inelastic Structures,’’ Proc., SSRC Conf. ‘‘Is Your
Structure Suitably Braced?,’’ Milwaukee, April, pp. 29–37.
American Institute of Steel Construction (1995), Manual of Steel Construction: Load & Resistance Factor
Design, Vol. 1, 2nd Edition, Chicago.
American Society of Civil Engineers (1971), ‘‘Commentary on Plastic Design in Steel,’’ ASCE Manual
No. 41, 2nd Edition, New York.
Essa, H.S. and Kennedy, D.J.L. (1995), ‘‘Design of Steel Beams in Cantilever-Suspended-Span
Construction,’’ ASCE J. Struct. Div., Vol. 121, No. 11, pp. 1667–1673.
Gil, H. (1966), ‘‘Bracing Requirements for Inelastic Steel Members,’’ PhD dissertation, The University of
Texas at Austin, May, 156 pp.
Helwig, T.A., Yura, J.A., and Frank, K.H. (1993), ‘‘Bracing Forces in Diaphragms and Cross Frames,’’
Proc., SSRC Conf., ‘‘Is Your Structure Suitably Braced?’’ Milwaukee, April, pp. 129–140.
Horne, M.R. and Ajmani, J.L. (1971), ‘‘Design of Columns Restrained by Side Rails,’’ Struct. Eng., Vol. 49,
No. 8, pp. 329–345.
Horne, M.R. and Ajmani, J.L. (1972), ‘‘Failure of Columns Laterally Supported on One Flange,’’ Struct.
Eng., Vol. 50, No. 9, pp. 355–366.
Lutz, A.L. and Fisher, J. (1985), ‘‘A Unified Approach for Stability Bracing Requirements,’’ Eng. J., Am.
Inst. Steel Constr., Vol. 22, No. 4, pp. 163–167.
Medland, I.C. and Segedin, C.M. (1979), ‘‘Brace Forces in Interbraced Column Structures,’’ ASCE
J. Struct. Div., Vol. 105, No. ST7, pp. 1543–1556.
Milner, H.R. and Rao, S.N. (1978), ‘‘Strength and Stiffness of Moment Resisting Beam-Purlin
Connections,’’ Civil Eng. Trans., Inst. of Engrg, Australia, CE 20(1), pp. 37–42.
Nakamura, T. (1988), ‘‘Strength and Deformability of H-Shaped Steel Beams and Lateral Bracing
Requirements,’’ J. Const. Steel Res., Vol. 9, 217–228.
Plaut, R.H. (1993), ‘‘Requirements for Lateral Bracing of Columns with Two Spans,’’ ASCE J. Struct. Div.,
Vol. 119, No. 10, pp. 2913–2931.
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