Journal of Constructional Steel Research Volume 10 Issue None 1988 (Doi 10.1016 - 0143-974x (88) 90034-x) Egor P. Popov Michael D. Engelhardt - Seismic Eccentrically Braced Frames PDF
Journal of Constructional Steel Research Volume 10 Issue None 1988 (Doi 10.1016 - 0143-974x (88) 90034-x) Egor P. Popov Michael D. Engelhardt - Seismic Eccentrically Braced Frames PDF
Journal of Constructional Steel Research Volume 10 Issue None 1988 (Doi 10.1016 - 0143-974x (88) 90034-x) Egor P. Popov Michael D. Engelhardt - Seismic Eccentrically Braced Frames PDF
E g o r P. P o p o v & M i c h a e l D . E n g e l h a r d t
Department of Civil Engineering, Davis Hall, Universityof California,
Berkeley, California, CA 94720, USA
ABSTRACT
INTRODUCTION
/ I
loads. The ultimate strength of the link can be accurately estimated. Thus,
by designing the brace to be stronger than the link, the designer can be
assured with a high degree of confidence that the brace will not buckle,
regardless of the severity of the earthquake load. The rapid deterioration of
buckled braces under cyclic loads is well documented. 6Thus, the preclusion
of brace buckling in EBFs permits stable hysteretic behavior under the most
severe cyclic loading conditions. Note that the links not only limit brace
forces, but also the load transmitted to the columns, permitting reliable
design for column stability, and offering some possible advantages for
difficult foundation design problems.
The ductility and energy dissipation capacity of EBFs may be better
understood by comparing the actual behavior of typical frames under cyclic
load. Figure 2 shows typical experimentally obtained lateral load versus
displacement plots for an MRF, CBF, and an EBF. The full and stable
hysteretic loops in Fig. 2a illustrate the MRFs ability to sustain large
deformations without strength loss and are indicative of the excellent energy
dissipation capacity of a moment-resisting frame. In contrast, the loops in
Fig. 2b are pinched and deteriorate as the number of loading cycles in-
creases, demonstrating the rather poor energy dissipation capacity of con-
centrically braced frames. This poor behavior is a result of the buckling of
braces and their ensuing rapid deterioration under cyclic load. Finally, Fig.
2c illustrates the hysteretic behavior of a well-designed EBF. Because brace
buckling is prevented and because the link can sustain large deformations
without strength loss, full and stable hysteretic loops similar to those of the
MRF are obtained.
An additional benefit of eccentric over concentric bracing is the greater
architectural freedom permitted with EBFs. The offset braces in EBFs
provide larger spaces for doors, windows, or other openings in the frame.
In its 1985 edition of the N E H R P Recommended Provisions for the
Development of Seismic Regulations for New Buildings, 1° the Federal
Emergency Management Agency provides tentative recommendations for
the design of EBFs. The Structural Engineers' Association of California
carried the process further and in 1986 completed a concensus document on
Recommended Lateral Force Requirements, 1~including specific provisions for
the design and detailing of seismic-resistant EBFs. The number of recently
constructed buildings incorporating eccentric bracing as well as the recogni-
tion of EBFs by the model earthquake design codes in the USA attests to the
rapid acceptance of this framing concept by the building construction
industry.
The primary purpose of this paper is to present design recommendations
for links and connections in seismic-resistant EBFs. Some basic concepts on
the behavior, analysis, and design of EBFs are briefly reviewed and signi-
324 E. P. Popov, M. D. Engelhardt
°/~'~"
~", ~ 201Hiram . .
,; ;./ ~ o ~ " U j ~
(a)
70
~ H
60r T ~
2,,~ 9: / L
Xd>~-L//--./ 9~,95'~
,o~~
70 L l __ i
-3 -2 I 0
(b)
150
I00
i
5O
~o
a."
-50
-100
I
I
-15C -2 -I
~,INCHES
(c)
Fig. 2. Typical experimental frame behavior under cyclic lateral load. (a) MRF (after
WakabayashiT); (b) CBF; 8 (c) EBF. 9
Seismic eccentrically bracedframes 325
C H A R A C T E R I S T I C S OF EBFs
ca 8 = 6
I*1
Z
±
M.
L 4
I.-
ca 6
W
h/L 1.0
.75
p L
il.
•v1
4
> ~ 8 K ~ "°7~
.J
i*J
~" 2
0 i
°o'.o o'.z '
0.4 '
0.6 0'.8 ~.o 0.0 0.2 Ore4 J
0.6 ola 1.0
e/L e/L
(a) (b)
Fig. 3. V a r i a t i o n o f elastic lateral stiffness with e/L for two simple EBFs.12
326 E P. Popov, M. D. Engelhardt
I I I I
0.2 0.4 0.6 0.8 I0
elL
Fig. 4. Variation of first natural period with e/L for a five-sto~ EBF.
IO(~F P~ P ~,
BII . . I L ] ~ L ~I
01 I I I I I
0 0.2 0.4 0.6 0.8 1.0
e/L
Fig. 5. Variation of flame plastic capacity with e/L. 13
Forces in links
I
.m
M
V ................................ , , ~ : : I
(a) (b)
Fig. 6. Typical force distributions in beams and links of EBFs under lateral load.
e "1
I)M°
V V
Ve = M o + M b
e L e
] , ,
I, u .I
(a) (b) (c)
Fig. 8. Energy dissipation mechanisms.
___ ~-0
WO
o I I I I I
0 0.2 04 0.6 0.8 ~.0
e/L
E X P E R I M E N T A L RESULTS ON EBFs
Since about 1977, a great deal of experimental research has been conducted
on EBFs, verifying their excellent structural characteristics and providing a
n u m b e r of useful design rules. This section briefly reviews significant experi-
mental work on EBFs, and summarizes key results from each series of tests.
The vast majority of experimental work has been conducted on shear links
that meet the criteria of eqn (1). Extrapolation of results to longer links is
not r e c o m m e n d e d , as the behavior and failure mechanisms of long links
differ substantially from those of shear links.
The excellent overall behavior of EBFs with shear links was demonstrated in
a series of tests on one-third scale, three-story EBFs in 1977.2'3 In addition to
verifying the advantages of EBFs for seismic-resistant design, these tests
also demonstrated the extensive strain hardening that occurs in shear links.
As noted earlier, the braces in EBFs are designed to be stronger than the
links in order to preclude brace buckling. Based on this first series of tests, it
was r e c o m m e n d e d that braces be designed assuming the ultimate link shear
force reaches at least 1.5 Vp, where Vp is the shear capacity of the section as
defined previously. This recommendation has been adopted by both
S E A O C 11 and N E H R P . 1° As observed in later experiments, somewhat
larger link forces may be generated due to the presence of a composite
Seismic eccentrically braced frames" 331
Fig. 10. Set-up for testing links with equal end moments.12
concrete deck and overstrength of the web, i.e. the actual yield strength of
the web exceeding the nominal specified yield strength.
A second series of one-third scale, three-story EBF tests 9 was conducted
in which beam sections with thinner webs were used, more accurately
modeling the typical W sections used in building construction. The results of
one test are shown in Fig. 2(c). Due to the more realistic, thinner webs used
in these tests, inelastic web buckling (buckling after shear yielding had
occurred) was observed in the links, resulting in deterioration of link
performance.
In order to study the effects of inelastic web buckling in links and to better
understand link behavior, two series of tests were conducted on isolated
links. The experimental set-up used for both series of tests, illustrated in Fig.
10, subjects the link to constant shear force with equal end moments and no
axial force, similar to the link forces shown in Fig. 6(b). A total of twenty-
eight links was tested in these two series of experiments.
In the first series, ~2.15 fifteen full-size links were subjected to quasi-
statically applied cycles of increasing relative end displacement. An example
of link behavior from this first series is shown in Fig. 11. Both specimens
shown are W18 × 40 sections, 28 inches in length (e = 1-11 MJVp) of A36
steel. The unstiffened specimen illustrated in Fig. 1l(a) experienced severe
web buckling shortly after shear yield had occurred, causing deterioration of
load-carrying capacity. The pinched hysteretic loops indicate poor energy
dissipation and ductility. The specimen provided with three pair of stiffeners
(Fig. l l(b)) showed dramatic improvement in performance. The specimen
332 E. P. Popov, M. D. Engelhardt
200 200
I O0 IO0
g~
o
; ° !/
- I O0 -I00 y'//
Jj
-200 -200
I , I , I , I , I , I
-3.0 -2.0 -I.0 IO 2.0 30 -3.0 -2.0 -I.0 I0 2.0 3.0
DISPLACEMENT (IN) DISPLACEMENT (IN)
(a) (b)
Fig. 11. Hysteretic behavior of (a) unstiffened shear link; (b) stiffened shear link. 12
achieved large inelastic rotations and the hysteretic loops remained full for a
large number of severe loading cycles, indicating enormous energy
dissipation capacity. For shear links, the plastic rotation y can be closely
estimated by the relative end displacement of the link divided by the link
length. The elastic component of the relative end displacement for shear
links is very small and can be neglected when computing y. The stiffened
specimen achieved relative end displacements of +3 in, giving a plastic
rotation capacity -/of about _+0.10 radian. Other tests have confirmed that
plastic rotation capacities of y = -4--0.10 radian can be achieved by well-
stiffened shear links. Note also that the stiffened specimen achieved an
ultimate shear strength of approximately 210 kips. The nominal shear yield
capacity of an A36 W18 × 40 section is Vp = 112 kips. The actual shear
Seismic eccentrically braced frames 333
capacity of this specimen, based on coupon tests on the web (Fy = 39.5 ksi
for the web) was 122 kips. Thus, this specimen achieved an ultimate shear
strength of about 1.9 times the nominal Vp or 1.7 times the actual Vp.
Although this particular specimen experienced a rather unusually high
degree of strain hardening, it does illustrate that the code-specified ultimate
strength of a shear link (1.5 times the nominal V~) is not overly conservative
and can be exceeded.
In the first series of link tests, two longer links were also tested. Both were
W12 × 22 sections, 36 inches in length (e = 2.15 MflV~). The unstiffened
specimen showed poor behavior due to web and flange buckling combined
with lateral torsional buckling. The other W12 × 22 specimen was provided
with a pair of stiffeners placed 6 inches from each end of the link in an
attempt to delay flange buckling. These stiffeners were very effective in
controlling flange buckling and lateral torsional buckling and significantly
improved the behavior of the link, which achieved a plastic rotation capacity
of about ~/= ---0.055 radian. However, overall ductility and energy dissi-
pation were still inferior to the stiffened shear links. Further, the stiffened
long link ultimately failed in a brittle manner by tearing of the flanges in the
heat-affected zone at the weld. This points to a potential danger in long
links. The very high bending strains required to achieve significant plastic
rotations in long links may induce brittle failure at flange-to-column welds.
However, this single test on a long link is insufficient to draw any firm
conclusions.
In the second series of isolated link tests,16'17 thirteen additional full-size
links were tested using the set-up in Fig. 10. All specimens were 36 inches
long and were either W18 × 40 or W18 × 60 sections of A36 steel. All
specimens in this series were shear links meeting the criteria of eqn (1). The
objectives of these tests were to determine the effects of loading history,
stiffener detail and spacing, and end connection details.
One specimen was designed to investigate the effects of providing shear
links with one-sided stiffeners. The hysteretic loops for this specimen,
shown in Fig. 12, were remarkably similar to those of an earlier specimen
which was identical except that it employed pairs of stiffeners placed on both
sides of the web. Additional tests also confirmed the adequacy of one-sided
stiffeners. At present, SEAOC 11 and N E H R P 1° permit one-sided inter-
mediate stiffeners on links whose overall depth is less than 24 in. For links of
deeper sections, stiffeners are required on both sides of the web. The use of
one-sided stiffeners reduces fabrication costs.
In some E B F configurations, one end of the link is connected to a column,
as in Figs l(a) and 1(c). Therefore, several link end connection details were
employed in order to gain insight into the link-to-column connection. The
end connection details included:
334 E. P. Popov, M. D. Engelhardt
I00 -L \
;2!!
DISPLACEMENT (IN)
200
I00
~ o
I
m
-IO0
-2OO
-4 -2 0 2
DISPLACEMENT (IN)
I-
(a)
+N
w
65".
T ~JACK
57
JACK 5 15½"
~-~'-e 5~
(b)
Fig. 14. Set-up for testing links with initially unequal end moments. (a) Schematic; (b)
actual. 13
338 E. P. Popov, M. D. Engelhardt
5o
-50 I
/, - 5 0 ~
8(IN) 8(IN)
(a) (b)
Fig. 15. Shear links with unequal end momentswith (a) no axial force; (b) axial force.13
capacities have clearly degraded. Further, it appears that the longer
the link, the more severe is the deterioration. EBF framing arrange-
ments should therefore always be chosen to minimize axial force in
the links. If axial force in the link is unavoidable, link length should be
reduced. Recommendations for this situation are given in Ref. 14.
(4) Interaction between bending moment and shear force can be
neglected when predicting the yield limit state of a link. That is, even
in the presence of very high shear, the fully plastic m o m e n t can be
taken as Mp, rather than a reduced value based on the flanges only.
This result contradicts predictions from simple plastic theory (see for
example Ref. 18), but is confirmed by all available link tests.
Neglecting M - V interaction permits simplifications in analysis and
design of links.
tL.--, L r II
(a)
To
7'-21/2" ~ L?11=
" -I"'o 21/2 " .ILTI:
'- " 5'-21/21' i
A C
(b)
16'-0" TEST I _[
-I
(c)
t . 14'-0" TEST2
providing lateral restraint for the link, and obtaining experimental data on
the effective width of composite floor beams under cyclic load.
A schematic of the experimental set-up is shown in Fig. 16. A total of eight
links was tested in this program. Four tests simulated interior links, such as
in Fig. 1(b); the other four simulated links next to columns. In each case, one
bare steel and three composite links were tested. All links were W12 × 19
sections, 19 inches in length (e = 1-2 MJVp). The composite beams were
constructed by casting a lightweight concrete slab on a ribbed metal deck,
with the ribs oriented perpendicular to the test beam. Shear studs welded to
the steel beams were used to develop composite action.
Some principal conclusions from this test series are as follows:
(1) The overall hysteretic behavior of composite shear links is very
similar to that of bare steel shear links. The composite links yield in
shear and dissipate energy primarily through web yielding, as do the
bare steel shear links. A photo of a deformed composite link shows
essentially the same type of behavior as a bare steel link (Fig. 17).
Composite shear links achieve the same plastic rotations as bare links.
(2) Damage to the concrete floor deck due to link deformation is
localized in the link region. Damage occurs in the form of cracking
and spalling of the slab above the link, slippage of the shear studs, and
US-Japan tests
WlSx40 I 1
/ /.,,f"'- BUCKLEDREGION
OF GUSSET
Link length
where a is the stiffener spacing, d is the beam depth, and tw is the web
thickness. For values of y between _+0.06 and _+0.03, it is appropriate to
interpolate.
After the designer estimates the required plastic rotation at a link, eqns
(3) and (4) provide the stiffener spacing needed to achieve it. For shear
links, stiffeners should be equally spaced along the length of the link. It must
be emphasized that the stiffener spacing criteria given above have been
developed for and are applicable to shear links that satisfy eqn (1).
Definitive recommendations for locating stiffeners in longer links are not yet
available.
EBF DESIGN
Some basic concepts and key aspects of EBF design are reviewed in this
section. Proper design of links and connections in EBFs requires an under-
standing of the overall EBF design philosophy. The material in this section is
applicable to EBFs with shear links that meet the criteria of eqn (1), and may
not necessarily be appropriate for EBFs with longer links.
Preliminary considerations
An early decision in the EBF design process is the choice of bracing
arrangement. EBF types favored by the authors are shown in Fig. 1. The
arrangement of Fig. l(a) is useful for narrow bays. This EBF type should
preferably be used in symmetrically opposing pairs to maintain overall
symmetry in the bracing system. The EBF arrangement of Fig. l(b) has the
advantage of symmetry, and since the links are not adjacent to the columns,
link-to-column connections are avoided. The EBF type in Fig. l(c) also has
the advantage of symmetry, and, as previously noted, reduced rotation
demand at the links. As an additional guideline in choosing a bracing
arrangement, it is generally best to avoid brace-to-beam angles less than
about 40 degrees. As the brace-to-beam angle becomes smaller, very large
axial forces are generated in the beam segment adjoining the link, leading to
potential strength and stability problems in this member. Also, as noted
earlier, bracing arrangements that transfer large axial forces through the
links should be avoided.
344 E. P. Popov, M. D. Engelhardt
The designer must also choose a link length at the preliminary stages.
Again, it is emphasized that based on the present state of research on link
behavior, the use of shear links meeting the criteria of eqn (1) is recom-
mended. Lengths on the order of 1.0 to 1.3 Mp/Vpappear to be particularly
effective. A useful guide at the early stages of design is to choose a link
length on the order of 1.5 to 2.0 times the nominal beam depth. For
example, if the designer anticipates using W18 sections, a preliminary link
length of about 27-36 inches would provide a reasonable starting point. Of
course, after member sizes are chosen, link length should be checked against
eqn (1). In general, it is possible to use longer links while still satisfying eqn
(1) by choosing heavier beam sections.
In many applications, EBFs are combined with MRFs in the same
structure. For example, perimeter moment frames combined with EBFs in
the building core have been used in several tall buildings. The redundancy
provided by MRF-EBF combined systems is recognized by both SEAOC tl
and NEHRW ° in the form of reduced lateral forces. In many instances,
moment-resisting connections are used at all beam-column joints within the
EBF itself, not only at the link-to-column connections. Though moment
connections at the non-link end of beams are not essential in an EBF, they
provide additional redundancy and safety.
Strength and ductility are the two key design requirements for any seismic-
resistant structure. In a well-designed EBF, the strength and ductility of the
frame are directly related to the strength and ductility of the links. As a
result of this relationship, the basic design philosophy for EBFs can be
summarized as follows:
(1) Size the links to provide the required level of frame strength; detail
the links to provide the required level of ductility.
(2) Design and detail the other frame members to be stronger than the
links so that the strength and ductility of the links, and therefore the
frame, can be fully developed.
With this approach, the links are designed for code or other specified
earthquake forces. All other frame members, however, are not designed for
code level forces, but rather for the forces generated by the fully yielded and
strain-hardened links. These represent the maximum forces that can occur
in these members regardless of earthquake magnitude. If this design philo-
sophy is followed, the maximum strength and ductility of the EBF will be
achieved by assuring that yielding in the frame is restricted to the links. This
approach is often referred to as capacity design, because the other frame
Seismic eccentrically braced frames 345
members are designed for the capacity of the links. This is analogous to the
usual approach to MRF design wherein yielding is restricted to the beam
ends by assuring that the columns are stronger than the beams.
Sizing members in EBFs to achieve this design philosophy is best
accomplished through the use of plastic design procedures. A straight-
forward plastic design technique based on a generalized portal method of
analysis has been developed for EBFs by Kasai. 13 Some highlights are
described below.
The first members to be sized in an EBF are the links. For most EBF
configurations, there is a remarkably simple relationship between the link
shear force and the lateral forces on the frame. A highly simplified free body
diagram of a portion of a K-braced EBF is illustrated in Fig. 20. In this
figure, Vcum is the accumulated story shear from the top of the structure down
to the level under consideration, and Vti,k is the resulting shear force in the
link. The forces not shown on the free-body diagram either tend to cancel
each other out or are small enough to be neglected. By summing moments
about point A, one obtains the following relationship:
h
V,,,~ = -= ~ . , . (5)
L
// \ I
/ x I
/ / \ I
/ \ I
/ \ I
I"~ ~,I VLINK X I
. . . . -co~, ~ \ t
A,
limiting y to ___0.06 radian. After the required rotation has been computed,
link stiffener spacing can be computed from eqns (3) and (4).
This section provides recommended details for links and selected con-
nections in seismic-resistant EBFs. Developing the ductility needed to ride
out a major earthquake is largely dependent on the proper detailing of the
critical frame elements and connections. The details illustrated in this
section are believed to be both safe and practical, based on the present state
of research on EBFs. 25 Many of the recommendations contained in this
section, particularly regarding links and link-to-column connections, are
consistent with recently adopted provisions of SEAOC" and NEHRP. 10
Link details
Key elements in developing the full strength and rotation capacity of shear
links are proper stiffening and lateral bracing. Typical stiffening of shear
links is illustrated in Figs 21 and 22. Two-sided, full-depth stiffeners must be
provided at the link end (at the locations marked 'Lateral brace on this line'
in Figs 21 and 22). Intermediate stiffeners, equally spaced according to eqns
(3) and (4), may be single sided for beam depths less than 24 inches, but are
required on both sides of the web in deeper beams. If the web of the link is
L A T E R A L BRACE
ON THIS LINE , cab~
/
,/
LATERALBRACE
ONTHISLINE
e _1
I" -I
1__ /~
" ~ T U B E ~
Connection details
strength of the link. SEAOC '~ requires full-penetration flange welds and a
welded web connection capable of developing the shear capacity of the link.
Either fillet welding the web to a shear tab, as shown, or providing a full
penetration weld between the web and the column flange is acceptable. The
welding sequence should be chosen to minimize locked-in stresses due to
restraint. As previously noted, for the severe service intended for links,
bolted web connections show inadequate ductility due to bolt slippage and
should not be used. The authors also recommend avoiding connections of
links to column webs. The reliability of connections to column webs has not
been firmly established experimentally, and design recommendations
cannot be provided.
Some recommended brace-to-link connection details are illustrated in
Figs 2l and 22. Many other satisfactory details can be devised. SEAOC ~
recommends designing brace connections for the compression strength of
the brace. It must also be recognized that in certain cases significant bending
m o m e n t s will be developed at the brace end, which must be considered in
the brace connection design. Large brace end moments are most likely to be
encountered when longer shear links, approaching the upper length limit of
eqn (1), are used in combination with a very shallow brace-to-beam angle.
Direct welding of a W-section brace to the beam, as in Fig. 21, is particularly
effective in cases where large end moments may develop in the brace. It also
avoids the rather large gusset and splice plates sometimes needed for a
W-section bracing connection, though care is required in the fabrication and
erection process to avoid fit-up problems. Figure 22 shows recommended
connections for rectangular and square tube braces. These details are in-
tended to avoid the gusset buckling type of failure observed in the Tsukuba
test. Bending moments in the beam produce large compressive stresses
along the edge of the gusset nearest the link. Stiffening of this edge, as
shown in Fig. 22, is therefore recommended. The connection can also be
made more compact by cutting the brace end parallel to the beam and
locating it as close to the beam as practical.
Nominally, the brace centerline should intersect the beam centerline at
the end of the link, as shown on the left side of Fig. 22. However, analytical
studies have shown that it is acceptable for the brace and beam centerlines to
intersect somewhat inside the link, as shown in Fig. 21 and on the right side
of Fig. 22. This will, in some cases, permit a more compact brace con-
nection. The centerlines should not, however, intersect outside of the link.
Some suggested details for the nominally concentric connection at the
non-link end of the brace are illustrated in Figs 23 and 24. The case of the
brace framing into a moment-resisting beam-column connection is shown in
Fig. 23. An alternative approach to this particular connection would be to
provide a welded rather than a bolted beam-web connection. The advant-
Seismic eccentrically braced frames 351
Fig. 23. Typical detail for brace at moment beam-column connection .25
ONEPCSHEAR
TAB
"'~-~~ ~ ~ ONESIDE
/~/~/m.// L//v \ATSHEAR
$- ~'" ~ S ATT~B OF BM
A (a)
It
W"IP'~ A i
l~XONEPCS
. HEARTAB
TI~B OF BM.
(b)
Fig. 24. Typical detail for brace at non-moment beam-column connection. 25 (a) Beam-to-
column flange connection. (b) Section A-A.
shear tab may not by itself be adequate for this purpose. For the detail
shown in Fig. 24, additional restraint against twisting is provided by the
extended plates at the top and bottom of the beam. A more compact brace
connection can sometimes be achieved by offsetting the work point from the
column centerline to, for example, the column face, as shown in Fig. 24. The
same concept can be used to advantage for the type of connection in Fig. 23.
The additional m o m e n t produced by an offset in the work point should be
included in the column design.
CONCLUSIONS
ACKNOWLEDGMENTS
REFERENCES
1. Spurr, H. W., Wind Bracing. McGraw-Hill Book Co., New York, 1930.
2. Roeder, C. W. & Popov, E. P., Eccentrically braced frames for earthquakes. J.
Struct. Div., ASCE, 104, No. 3 (March 1978) 391-412.
3. Roeder, C. W. & Popov, E. P., Inelastic behavior of eccentrically braced steel
frames under cyclic loadings. Report No. UCB/EERC-77/18, Earthquake
Engineering Research Center, University of California, Berkeley, USA, 1977.
4. Libby, J. R., Eccentrically braced frame construction--A case history. Engin-
eering Journal, AISC, 4th qtr, 1981.
5. Merovich, A. T., Nicoletti, J. P. & Hartle, E., Eccentric bracing in tall
buildings. J. Struct. Div., ASCE, 108, No. 9 (September 1982).
6. Black, R. G., Wenger, W. A. B. & Popov, E. P., Inelastic buckling of steel
struts under cyclic load reversals. Report No. UCB/EERC-80/40, Earthquake
Engineering Research Center, University of California, Berkeley, USA, 1980.
7. Wakabayashi, M., et al., Inelastic behavior of full-scale steel frames with and
without bracings. Bulletin of the Disaster Prevention Research Institute, Kyoto
University, Kyoto, Japan, 24, Part 1 (March 1974) 1-23.
8. Maison, B. F. & Popov, E. P., Cyclic response prediction for braced steel
frames. J. Struct. Div., ASCE, 106, No. 7 (July 1980) 1401-16.
9. Manheim, D. N., On the design of eccentrically braced frames. DEng thesis,
Department of Civil Engineering, University of California, Berkeley, USA,
February 1982.
10. N E H R P (National Earthquake Hazards Reduction Program) Recommended
354 E. P. Popov, M. D. Engelhardt