U.S. patent application number 14/370747 was filed with the patent office on 2015-01-01 for buckling restrained brace with lightweight construction.
The applicant listed for this patent is Oregon State Board of Higher Education acting by and through Portland State University. Invention is credited to Peter Dusicka.
Application Number | 20150000228 14/370747 |
Document ID | / |
Family ID | 48745451 |
Filed Date | 2015-01-01 |
United States Patent
Application |
20150000228 |
Kind Code |
A1 |
Dusicka; Peter |
January 1, 2015 |
BUCKLING RESTRAINED BRACE WITH LIGHTWEIGHT CONSTRUCTION
Abstract
A buckling restrained brace comprises a core member, core
restrainer member sections and a jacket member. The core member has
two opposite ends. The core restrainer member sections are
configured to be arranged around the core member. The jacket member
comprises fiber reinforced polymers configured to be wrapped around
the core restrainer member sections and core member to couple the
core restrainer member sections to the core member such that the
core restrainer member sections and jacket member cooperate to
provide greater resistance to buckling of the core member when the
brace is subjected to compression. In some implementations, the
brace has a weight less than about 50% of a weight of a
conventional buckling restrained brace of similar length and having
a steel core and mortar-filled tubular core restrainer member of
comparable cross-sectional areas, respectively.
Inventors: |
Dusicka; Peter; (Portland,
OR) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Oregon State Board of Higher Education acting by and through
Portland State University |
Portland |
OR |
US |
|
|
Family ID: |
48745451 |
Appl. No.: |
14/370747 |
Filed: |
January 4, 2013 |
PCT Filed: |
January 4, 2013 |
PCT NO: |
PCT/US2013/020364 |
371 Date: |
July 3, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61584066 |
Jan 6, 2012 |
|
|
|
Current U.S.
Class: |
52/835 |
Current CPC
Class: |
E04H 9/028 20130101;
E04G 23/0218 20130101; E04C 3/29 20130101; G01N 2203/0298 20130101;
E04H 9/0237 20200501; G01N 3/32 20130101; G01N 2203/0268 20130101;
G01N 3/00 20130101; G01N 2203/0075 20130101; E04C 3/00 20130101;
E04H 9/02 20130101; E04C 2003/026 20130101; E04C 3/32 20130101 |
Class at
Publication: |
52/835 |
International
Class: |
E04C 3/00 20060101
E04C003/00 |
Goverment Interests
ACKNOWLEDGMENT OF GOVERNMENT SUPPORT
[0002] This invention was made with government support under
DTRT06-G-0017 awarded by the Department of Transportation. The
government has certain rights in the invention.
Claims
1. A buckling restrained brace, comprising: a core member having
two opposite ends; core restrainer member sections configured to be
arranged around the core member; a jacket member comprising fiber
reinforced polymers configured to be wrapped around the core
restrainer member sections and core member to couple the core
restrainer member sections to the core member, wherein the core
restrainer member sections and jacket member cooperate to provide
greater resistance to buckling of the core member when the brace is
subjected to compression.
2. The buckling restrained brace of claim 1, wherein the brace has
a weight less than about 50% of a weight of a conventional buckling
restrained brace of similar length and having a steel core and
mortar-filled tubular core restrainer member of comparable
cross-sectional areas, respectively.
3. The buckling restrained brace of claim 1, wherein the core
member has a cross section defining at least one pair of opposed
spaces configured to receive a respective number of core restrainer
member sections.
4. The buckling restrained brace of claim 1, wherein the core
member has a T-shaped cross section defining two opposed spaces,
wherein each of the two spaces is configured to receive one of the
core restrainer sections.
5. The buckling restrained brace of claim 4, wherein the core
member is comprised of two angled sub-members defining a T-shape
when positioned adjacent each other.
6. The buckling restrained brace of claim 1, wherein the core
member has a cross section defining at least four separated spaces,
wherein each of the spaces is configured to receive one of the core
restrainer sections.
7. The buckling restrained brace of claim 1, wherein the core
member is comprised of four angled sub-members, and the angled
sub-members are arranged such that the vertices are adjacent each
other in a cross section of the core member.
8. The buckling restrained brace of claim 1, wherein the core
member is comprised of two T-shaped sub-members, and the T-shaped
sub-members are arranged opposite to each other in a cross section
of the core member.
9. The buckling restrained brace of claim 1, wherein the core
restrainer members are tubular and have a rectangular cross
section.
10. The buckling restrained brace of claim 1, wherein the core
restrainer members are tubular and have a circular cross
section.
11. The buckling restrained brace of claim 1, wherein the jacket
member is sized to extend over an intermediate portion of the core
member between the two opposite ends.
12. The buckling restrained brace of claim 1, wherein the core
member is formed of a ductile material.
13. The buckling restrained brace of claim 1, wherein the core
member is formed of an aluminum alloy.
14. The buckling restrained brace of claim 1, wherein the core
member comprises at least two core member sections, further
comprising at least one spacer member positioned between the core
member sections.
15. The buckling restrained brace of claim 14, wherein the at least
one spacer member is formed of at least one of a plastic material
and fiber reinforced polymers.
16. The buckling restrained brace of claim 1, wherein the jacket
member comprises at least one layer of fabric material applied at
different angles relative to the core member.
17. The buckling restrained brace of claim 1, wherein the jacket
member comprises at least one layer of fabric material applied at
an angle of about 30 degrees relative to an axis of the core
member.
18. The buckling restrained brace of claim 1, wherein the core
member is configured to dissipate seismic energy through
substantially reversible cyclic plastic strain.
19. The buckling restrained brace of claim 1, wherein the core
member, core restrainer member sections, and jacket member are
constructed of materials selected to reduce corrosion from exposure
to environmental conditions.
20. The buckling restrained brace of claim 1, wherein the core
member, core restrainer sections and jacket member are configured
to allow the core and the core restrainer sections to translate
relative to each other under predefined loading conditions imposed
on the brace.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] The present application claims the benefit of U.S.
Provisional Application No. 61/584,066, filed Jan. 6, 2012, which
is incorporated herein by reference.
BACKGROUND
[0003] A buckling-restrained brace (BRB) is a structural element
designed to withstand cyclic loading in the form of repeated
tensile and compressive forces such as from an earthquake or an
explosive blast. BRBs add reinforcement and energy dissipation to
steel frame buildings to protect them from large deformations by
yielding in tension and compression, while at the same time
resisting failure due to buckling.
[0004] Conventional BRBs have a steel core member and a surrounding
tubular member filled with mortar that is designed to resist
buckling of the core member when the core member is subjected to
compression loading. Although conventional BRBs are adequate in
some situations, it would be desirable to provide BRBs having the
same energy dissipating performance while having a lower overall
weight, which among other advantages makes handling and
installation easier.
SUMMARY
[0005] Described below are embodiments of a buckling restrained
brace having a light-weight construction.
[0006] In an exemplary embodiment, a buckling restrained brace
comprises a core member, core restrainer member sections, and a
jacket member. The core member has two opposite ends. The core
restrainer member sections are configured to be arranged around the
core member. The jacket member comprises fiber reinforced polymers
configured to be wrapped around the core restrainer member sections
and core member to couple the core restrainer member sections to
the core member. The core restrainer member sections and jacket
member cooperate to provide greater resistance to buckling of the
core member when the brace is subjected to compression.
[0007] The brace can have a weight less than 50% of a weight of a
conventional buckling restrained brace of similar length and having
a steel core and mortar-filled tubular core restrainer member of
comparable cross-sectional areas, respectively.
[0008] The core member can have a cross section defining at least
one pair of opposed spaces configured to receive a respective
number of core restrainer member sections. The core member can have
a T-shaped cross section defining two opposed spaces, wherein each
of the two spaces is configured to receive one of the core
restrainer sections. The core member can be comprised of two-angled
sub-members defining a T-shape when positioned adjacent each other.
The core member can have a cross-section defining at least four
separated spaces, wherein each of the spaces is configured to
receive one of the core restrainer sections. The core member can be
comprised of four angled sub-members, and the angled sub-members
can be arranged such that the vertices thereof are adjacent but
spaced apart from each other in a cross section of the core
member.
[0009] The core member can be comprised of two T-shaped sub-members
arranged opposite to each other. The core restrainer members can be
tubular and have a rectangular cross section or a circular cross
section, and are sometimes referred to herein as "tubes".
[0010] The jacket member can be sized to extend over an
intermediate portion of the core member between the two opposite
ends.
[0011] The core member can be formed of a ductile material, such as
an aluminum alloy. The core member can comprise at least two core
member sections and at least one spacer member positioned between
the core member sections. The spacer member can be formed of a
plastic material or fiber reinforced polymers.
[0012] The jacket member can comprise at least one layer of
material applied at different angles relative to the core member.
In some implementations, two or more layers are used. The jacket
member can comprise at least one layer of material applied at an
angle of about 30 degrees relative to an axis of the core member.
The core member can be configured to dissipate seismic energy
through substantially reversible cyclic plastic strain. The core
member, core restrainer member sections, and jacket member can be
constructed of materials selected to reduce corrosion from exposure
to environmental conditions.
[0013] The core member, core restrainer sections, and jacket member
are configured to allow the core and the core restrainer sections
to translate relative to each other under pre-defined loading
conditions imposed on the brace.
BRIEF DESCRIPTION OF DRAWINGS
[0014] FIG. 1(a) is a perspective view of an implementation of a
buckling restrained brace having a light weight construction with
an intermediate portion of the brace cut away to show the relative
positions of various components.
[0015] FIG. 1(b) is an enlarged perspective view of a portion of
the brace of FIG. 1(a).
[0016] FIGS. 1(c), 1(d), 1(e) and 1(f) are end elevation views of
representative braces having different core member and core
restrainer member configurations.
[0017] FIG. 1(g) is a perspective view similar to FIG. 1(a) of
another implementation of a buckling restrained brace.
[0018] FIG. 1(h) is an enlarged perspective view of a portion of
the brace of FIG. 1(g).
[0019] FIGS. 2(a) and 2(b) are side elevation views of a
representative brace identifying various dimensions used in
modeling.
[0020] FIG. 2(c) is an end view of the brace of FIGS. 2(a) and 2(b)
showing bolted connections to gusset plates.
[0021] FIG. 3(a) is a set of diagrams showing a single degree of
freedom mechanical model for modeling the brace.
[0022] FIG. 3(b) is a drawing showing another model of the
brace.
[0023] FIG. 4 is a scatter plot of required restrainer stiffness
vs. restrainer length providing a comparison between analytical and
numerical results.
[0024] FIG. 5(a) is a drawing showing dimensions for two test
coupons.
[0025] FIG. 5(b) is a perspective view showing a test apparatus for
subjecting a test coupon to a predetermined loading.
[0026] FIG. 6 is a graph of stress versus axial strain showing the
brace's response to predetermined loading.
[0027] FIG. 7 is a graph showing maximum cyclic stress versus a
number of reversals for the brace of FIG. 6.
[0028] FIG. 8(a) is a graph of normalized stress versus axial
strain based on testing of another representative coupon.
[0029] FIG. 8(b) is a side elevation view of the representative
coupon tested in FIG. 8(a).
[0030] FIG. 9(a) is a perspective view of a core member showing how
it is modeled using finite element analysis.
[0031] FIG. 9(b) is a graph of axial load versus axial displacement
for the model of FIG. 9(a).
[0032] FIG. 10(a) is a cross section of a core member at an
intermediate point showing various dimensions used in modeling
brace end moments.
[0033] FIG. 10(b) is a diagram illustrating end moments or
rotations that the brace of FIG. 10(a) may experience during severe
seismic loading.
[0034] FIGS. 11(a) through 11(f) show axial load versus
log-displacement relationships for six groups of simulations.
[0035] FIG. 11(g) is a graph of the required restrainer stiffness
versus the end moment ratio for two groups of braces.
[0036] FIG. 12(a) is a perspective view of a brace showing yielding
prior to buckling.
[0037] FIG. 12(b) is a perspective view of the brace of FIG. 12(a)
showing its deformed shape after buckling.
[0038] FIG. 12(c) is an enlarged view of a portion of the brace of
FIGS. 12(a) and 12(b) that has been subjected to buckling showing
that a plastic hinge is created in the area of junction between its
full section and its intermediate section.
[0039] FIGS. 13(a) and FIG. 13(b) are graphs of axial load versus
axial displacement for one prototype.
[0040] FIGS. 13(c) and FIG. 13(d) are graphs of axial load versus
axial strain at mid-length for the prototype of FIGS. 13(a) and
13(b).
[0041] FIG. 13(e) is a plot of normalized cumulative-displacement
versus restrainer stiffness for the prototype of FIGS. 13(a) and
13(b).
DETAILED DESCRIPTION
[0042] Small to medium size concrete or steel buildings constructed
according to deficient legacy codes constitute a large portion of
today's backlog of structures requiring seismic retrofit. A number
of retrofit solutions are available to address these deficient
structures. However, many solutions impose great difficulties for
material handling and installation of traditional lateral elements
such as shear walls or conventional steel braces due to limited
access for heavy lifting equipment such as cranes and forklifts
Therefore, an ultra-lightweight lateral bracing system is desired
that allows for easy manual transport, lifting, erection, and
connection of required components to the existing structure without
deconstruction of exterior walls for access. By minimizing
disruption to building occupants, the building may remain partially
viable during construction resulting in decreased cost and thus
increasing the feasibility of elective upgrades. Described herein
is a new BRB, having an aluminum core member for seismic force
dissipation, and fiber reinforced polymers (FRP) arranged to couple
tubular core restrainer sections to the core, thus accomplishing
the goals of decreased installation weight, increased system
compactness and efficient energy dissipation.
[0043] Attempts to refine metallic seismic dissipaters originally
proposed by Skinner et al. (1975) have recently strayed from the
traditional steel core and mortar-filled steel tube restrainer BRBs
developed throughout the 1980s and 1990s (Watanabe et al. 1988;
Wada et al. 1989; Watanabe and Nakamura 1992; Black et al. 2002;
Black et al. 2004). Many variations have been presented (Xie 2005),
but those termed "lightweight" and constructed of bolted or welded
all-steel components for both the core and restrainer are the most
numerous (Mazzolani et al. 2004; Tremblay et al. 2006; Usami et al.
2008; D'Aniello et al. 2008; D'Aniello et al. 2009; Chao and Chen
2009; Ju et al. 2009; Mazzolani et al. 2009). Competing concepts
have been characterized as beneficial due to decreased installation
cost, having replaceable cores, ability to use low-skilled labor
for installation, compact for installation confined spaces, and use
in existing building retrofits.
[0044] Aluminum as an industrial material has been around for more
than a century but its incorporation into the primary structural
elements of buildings has been relatively slow with uses limited to
secondary systems such as curtain walls and auxiliary structures
such as awnings, canopies or similar structures. However, attempts
to utilize its high ductility and absence of cyclic hardening in
seismic force dissipating systems have begun to appear. Shape
memory braces constructed with super-elastic aluminum alloys that
allow a structure to re-center after a seismic event with little
permanent deformation (Mazzolani et al. 2004), replaceable shear
links constructed of low yield point aluminum installed in
concentrically braced frames or special truss moment frames (Rai
and Wallace 2000) and replaceable aluminum plate shear panels (Rai
2002, Mazzolani et al. 2004, Brando et al. 2009) have all been
proposed and tested with moderate success.
[0045] FRP has successfully been used in structures since the 1970s
and has been commonly employed in applications bonded to concrete
or steel members requiring strengthening or repair (Zhao and Zhang
2007). More pertinent applications recently have been developed
that increase ductility of steel members. For instance, bonded
unidirectional sheets wrapped around special truss moment frame
chord members enhanced cyclic response of plastic hinge behavior
(Ekiz et al. 2004). FRP strips bonded to compression elements of
flexural members (Accord and Earls 2006), webs of WT compression
members (Harries et al. 2009), and HSS columns (Shaat and Fam 2006,
2007, 2009) have also been reported to delay local buckling of
elements subjected to compression. Although applications where the
FRP is not bonded to the substrate that it serves to reinforce are
rare, they are emerging as an effective method for precluding
compression buckling. Pilot tests of a single steel angle fit with
a pultruded FRP square tube and wrapped with GFRP (glass fiber
reinforced polymer) fabric was experimentally loaded in cyclic
push-pull testing and achieved an ultimate compressive strength of
35% of the tensile strength before global buckling (Dusicka and
Wiley 2008). Small-scale monotonic experiments and finite element
modeling of rectangular steel bars surrounded by PVC or mortar
blocks and wrapped with CFRP (carbon fiber reinforced polymer)
fabric have achieved compression loads up to P.sub.max/P.sub.y=1.53
(Ekiz and El-Tawil 2008). Experimental full-scale cyclic tests of
pinned and semi-fixed end steel angles similarly wrapped with
mortar blocks and CFRP fabric achieved up to a 270% increase in
energy dissipation over bare steel angles and compression loads up
to P.sub.max/P.sub.y=0.90 (El-Tawil and Ekiz 2009).
[0046] Described below are developments of high-performance BRBs,
and specifically, a new ultra-lightweight BRB, designed for a
typical model building using analytical models developed from
established buckling theory and experimental cyclic coupon testing
of a candidate 6061-T6511 aluminum alloy for development of a
calibrated constitutive model and finite element simulations.
Analytical models considered both a single degree of freedom (SDOF)
and an established Euler buckling model which provided an initial
required restrainer stiffness and strength for a given axial design
force and core length. Monotonic numerical simulations of a
prototype brace were performed to examine the effect of restrainer
stiffness with two different core reduced section lengths and three
degrees of applied end moment. Cyclic simulations were used to
assess if a predictable and reliable cumulative plastic ductility
and energy dissipation was possible at a considerable story drift
ratio. As part of the parametric investigation, end moment effect
due to frame drift was considered by using an upper bound approach
which considers plastic hinging of the unrestrained section of
core.
[0047] The new brace utilizes materials readily available in many
sizes and profiles to allow customization of the core-restrainer
configuration as shown in FIG. 1(a). Although not directly a part
of research, a review of past literature and practicality led to
the following considerations for development: (1) stock extruded
aluminum profiles for the core members should lower procurement and
fabrication costs; (2) bi-planar symmetry of the brace
cross-section should eliminate potential for global buckling in a
weak direction; (3) non-tapered core cross section dimensions
should allow a tight fit to the restrainer tubes without shimming;
(4) back to back core elements should be continuously supported by
high modulus FRP spacers to prevent core rippling; (5) sufficient
space should be provided at the tip of core elements to allow
Poisson expansion; (6) unrestrained sections of the core should be
sufficiently robust to prevent local or torsional buckling modes;
(7) the core should be fabricated without welding to prevent
material embrittlement and fatigue notching; (8) a reduced core
section should be used to direct plastic straining to the
mid-length of the brace away from the vulnerable unrestrained
areas; (9) axial independence between the core and restrainer
should be maintained using a frictionless interface of grease or
other lubricant between the FRP tubes and aluminum; and lastly (10)
GFRP should be used to prevent galvanic reaction as is present with
CFRP and aluminum.
Brace Geometry and Estimate of Strain Demands
[0048] Seismic forces, story drift, axial displacement, frame
geometry and end connections were established within the context of
a model building based on the SAC 3-story office building located
in Los Angeles, Calif. (FEMA 2000). The building consisted of 9.14
m [30 ft] square bays and measured 36.6 m by 54.9 m [120 ft by 180
ft] with a story height h.sub.i=3.96 m [13 ft]. Seismic design
criteria were taken from the current edition of the building code
as follows: S.sub.s=2.15 g, S.sub.ds=1.43 g, R=7 and C.sub.d=5.5
(ASCE 2005). Two adjacent BRB frames (BRBFs) in an inverted v-brace
configuration were centered on each of the four perimeter column
lines. An equivalent lateral force procedure with 5% minimum
eccentricity was used to determine the seismic base shear and
distribution to the individual stories and frames. A brace design
force of P.sub.u=1070 kN [241 k] at the first level was calculated
using the assumption of equal tension and compression stiffness of
the BRBs.
[0049] FIGS. 2(a)-(c) show a definition of the brace geometry with
a two-step core profile. An end to end core length L.sub.b=4.83 m
[190 in] was generated using assumed W21.times.111 beams and
W14.times.176 columns to remain consistent with previous literature
reports on testing of full-scale BRBFs (Fahnestock et al. 2007).
Selection of the brace reduced section length L.sub.c was
subsequently made by considering axial stiffness of the brace
required to limit the inelastic story drift to 2.5%, the maximum
allowed by code for a regular structure (ASCE 2005). Calculation of
the elastic story drift ratio D.sub.ie/h.sub.i for a non-prismatic
core neglecting the contribution of the much stiffer beams and
columns was previously cited by Tremblay et al. (2006) in Eq. (1)
where .gamma.=L.sub.c/L.sub.b and .eta.=A.sub.1/A.sub.3,
F.sub.y=core nominal specified yield strength, E.sub.c=core Young's
modulus and .theta.=brace angle with horizontal.
D ie h i = .phi. F y E c [ .gamma. + .eta. ( 1 - .gamma. ) sin
.theta. cos .theta. ] ( 1 ) ##EQU00001##
[0050] By rearranging Eq. (1) algebraically to solve for .gamma.,
Eq. (2) is given. D.sub.ie/h.sub.i=0.45% was calculated by dividing
the inelastic story drift of 2.5% by the deflection amplification
factor C.sub.d. Using the variables .theta.=42.degree.,
E.sub.c=69.6 GPa [10,100 ksi], .phi.=0.9, F.sub.y=241 MPa [35 ksi]
and .eta.=0.456, .gamma.=0.481 was calculated which represents a
2.31 m [91.4 in] long reduced section. The final reduced section
length was increased to 2.44 m [96 in] to give .gamma.=0.5 which is
termed the Group B brace. Another geometry was created for the
parametric study to examine higher expected axial strain and
stiffness with L.sub.c=1.47 m [58 in], or .gamma.=0.3, which is
termed the Group A brace.
.gamma. = ( 1 - .eta. ) - 1 [ ( D ie h i ) E c sin .theta. cos
.theta. .phi. F y - .eta. ] ( 2 ) ##EQU00002##
[0051] Table 1 shows all prototype brace dimensions.
TABLE-US-00001 TABLE 1 Brace prototype dimensions L.sub.b L.sub.c
L.sub.c2 L.sub.c3 L.sub.r L.sub.o L.sub.s L.sub.tr A.sub.1 A.sub.2
A.sub.3 m m cm cm m cm cm cm cm.sup.2 cm.sup.2 cm.sup.2 Group [in]
[in] [in] [in] [in] [in] [in] [in] [in.sup.2] [in.sup.2] [in.sup.2]
A 4.83 1.47 119 48.3 3.40 78.7 25.4 15.2 51.1 78.7 112 [190] [58]
[47] [19] [134] [31] [10] [6] [7.92] [12.2] [17.4] B 4.83 2.44 71.1
48.3 3.40 33.0 25.4 15.2 51.1 78.7 112 [190] [96] [28] [19] [134]
[13] [10] [6] [7.92] [12.2] [17.4]
[0052] Approximation of the average inelastic strain in the
two-step core at maximum story drift is required to determine
material strain demand. Using Eq. (3) and the previously defined
variables, .epsilon..sub.c=3.22% and 2.25% was calculated for the
Group A and B braces at 2.5% story drift, respectively. These
values fall within strain amplitudes reported for previous BRB
tests of 1% to 2% for longer core lengths and 3% to 5% for shorter
core lengths (Tremblay et al. 2006). Selection of L.sub.c should
target an appropriate inelastic strain suitable for use with
established cyclic properties of the core material as well as brace
stiffness required to meet a target design story drift. Typically,
connection details, intermediate section overlap length and axial
shortening requires .gamma..ltoreq.0.5 as a practical limit. In the
prototype, .gamma. was maximized by extension of the unwrapped
tubes a distance of L.sub.s to prevent local buckling of the
intermediate section while still allowing the full section to slide
through the restrainer.
c = C d L c [ D ie cos .theta. - ( n .phi. F y ( L b - L c ) E c )
] ( 3 ) ##EQU00003##
SDOF System Analytical Model
[0053] Transverse displacement of the slender core member during
buckling imparts flexural demand on the restrainer through
application of a force with an unknown distribution function w(x).
Effort to resist this displacement was conservatively modeled as a
simple span restrainer beam pinned at the end of length L.sub.r
assuming the full section of the core rigidly cantilevers from the
firmly bolted gusset plate as shown in FIG. 2(c). Force interaction
between the core and the restrainer was then established using a
SDOF mechanical model with axially inextensible truss members in
which an assumed plastic hinge exists at the mid-length as shown in
FIG. 3(a). Flexural stiffness of the elastic restrainer serves to
prevent transverse bifurcation of the core hence increasing the
critical buckling load P.sub.cr. The plastic hinge was justified by
first considering the internal core moment by combining elastic
column Eqs. (4) and (5) for a pinned-pinned column and solving for
the internal moment at mid-length M.sub.int.sup.p at a given
transverse displacement .DELTA..sub.t in Eq. (6). Tangent modulus
theory was used to account for core material non-linearity by
replacing E.sub.c with E.sub.ct.
M int = E c I c y '' ( 4 ) y ( x ) = .DELTA. t sin ( .pi. x L r ) (
5 ) M int p ( L r 2 ) = .pi. 2 .DELTA. t L r 2 E ct I c ( 6 )
##EQU00004##
[0054] The resisting moment M.sub.res provided by the bundled tube
restrainer was calculated from equilibrium on the column
half-length shown in FIG. 3a and Eq. (7) where E.sub.r and I.sub.r
are the Young's modulus and moment of inertia of the restrainer,
respectively. Comparison of M.sub.int.sup.p to M.sub.res showed
approximately two orders of magnitude difference at a common
transverse displacement .DELTA..sub.t. For this exercise, the
modulus of elasticity for the pultruded composite tubes was taken
as E.sub.r=19.3 GPa [2800 ksi] and I, was calculated from four 10.8
cm by 6.35 mm [4.25 in by 1/4 in] square tubes acting compositely.
Tangent modulus E.sub.ct of the core was taken as 1% of Young's
modulus to account for strain hardening. The sharp transition
between elastic and plastic behavior negates the need for an
incremental approach accounting for material non-linearity.
M res = 48 E r I r .DELTA. t L r 3 ( L r 2 ) 2 - .DELTA. t 2 ( 7 )
##EQU00005##
[0055] Effect of the restrainer was simulated by an elastic spring
with stiffness k.sub.s providing a force F exerted at the
mid-length of the core. The spring stiffness is taken from the
elastic deflection of a beam loaded at mid-span as
F/.DELTA..sub.t=48E.sub.rI.sub.r/L.sub.r.sup.3. This relationship
is substituted for k.sub.s in Eq. (8) and gives the critical
buckling load all in terms of known variables after using the
classical eigenvalue solution for the SDOF system. Eq. (9) solves
for the required restrainer stiffness E.sub.rI.sub.r/L.sub.r.sup.3
which can be used for intial restrainer sizing.
P cr = k s L r 4 = 12 E r I r L r L r 3 ( 8 ) E r I r L r 3 = P u
12 L r ( 9 ) ##EQU00006##
Euler Analytical Model
[0056] Previous research by Black et al. (2002) has used the Euler
column with a distributed force interaction w(x) between the core
and restrainer assuming Hooke's law, geometric perfection,
concentric load application and small displacements. Introduction
of a continuous support is shown diagrammatically in FIG. 3(b) as a
pinned-pinned column of length L.sub.r supported by an infinite
number of axially rigid connector bars connected to the restrainer
also spanning length L.sub.r. By beginning with force equilibrium
on an arbitrary length of column x and introducing sinusoidal
displacement functions originating from Eq. (5), Eq. (10) was
achieved. Eq. (11) arrives from solution of the differential
equation for the critical buckling load P.sub.cr where an
undetermined additional factor of safety is proposed by Black
(2002) to account for geometric imperfections and material
non-linearity of the core. The E.sub.cI.sub.c term for the core can
be omitted due to the much lower contribution as compared with
E.sub.rI.sub.r as previously explained and the effective length
factor K=1 for the pinned-pinned column. Removal of the
E.sub.cI.sub.c term likewise removes the need for incremental
analysis considering material non-linearity of the core. Eq. (12)
similarly solves for the required restrainer stiffness.
p 2 y ( x ) x 2 + E c I c 4 y ( x ) x 4 + E r I r 4 y ( x ) x 4 = 0
( 10 ) P cr = .pi. 2 ( KL r ) 2 ( E c I c + E r I r ) ( 11 ) E r I
r L r 3 = P u .pi. 2 L r ( 12 ) ##EQU00007##
[0057] The SDOF and Euler analytical design methods are plotted in
FIG. 4 as continuous functions illustrating required
E.sub.rI.sub.r/L.sub.r.sup.3 for lengths ranging from L.sub.r=2 m
[78.6 in] to 4.25 m [167 in] and loads P.sub.u=225 kN [50 k] to
1350 kN [300 k]. If no degree of conservatism is provided, the
present prototype brace requires E.sub.rI.sub.r/L.sub.r.sup.3=30.1
kN/m [0.172 k/in] and 36.6 kN/m [0.209 k/in] for the SDOF and Euler
methods, respectively. The prototype to be considered in the
numerical simulations, used a restrainer stiffness of 33.4 kN/m
[0.191 lain] provided by four 4.25 in.times.0.25 in bundled tubes.
The SDOF model resulted in required stiffness values equal to 82%
of the Euler model, indicating that it may be unconservative.
However, restrainer stiffness was selected to fall in between these
values.
Coupon Testing
[0058] Alloy 6061-T6511 is a relatively inexpensive heat-treated
structural aluminum that is available in many extruded profiles
that are conformable to square or round FRP tubes. This alloy has
proven to be reliable when limited to the elastic range, but
reversed cyclic behavior has not been reported for
.DELTA..epsilon..sub.r/2.gtoreq.4% and required investigation to
determine its cyclic behavior such as is reported for plate steels
(Dusicka et al. 2007). FIGS. 5(a) and 5(b) define the monotonic
tension and cyclic push-pull coupons machined from 0.875 inch round
bar and test setup which used a MTS load frame with a +/-445 kN
[+/-110 k] capacity. The apparatus was manually controlled using a
LVDT at a constant strain rate d.epsilon./dt for both the monotonic
and cyclic tests. Cyclic tests were performed with a triangular
waveform load history and began with a tensile excursion.
Individual cyclic tests subjected the coupon to constant total
strain amplitudes .DELTA..epsilon..sub.r/2=2, 3 and 4% at a cyclic
strain ratio R.sub..epsilon.=-1. Table 2 summarizes experimental
results for each of the specimens.
TABLE-US-00002 TABLE 2 6061-T6511 coupon test results Coupon ID
Material Property T1 C1 C2 C3 C4 f.sub.0.2, MPa [ksi] 296 [42.9]
297 [43.0] 283 [41.1] 289 [41.9] 297 [43.0] f.sub.u, MPa [ksi] 317
[46.0] 321 [46.5] -- -- -- .epsilon..sub.y, % 0.38 0.33 0.481 0.427
0.464 .epsilon..sub.u, % 21.9 22.3 -- -- -- A.sub.f, mm.sup.2
[in.sup.2] 53.6 [0.0831] 53.1 [0.0823] -- -- -- .sigma..sub.true,
MPa [ksi] 414 [60.1] 378 [54.8] -- -- -- .mu. 0.816 0.805 0.852
0.835 0.814 .DELTA..epsilon..sub.t/2, % -- -- 2.0 3.0 4.0
d.epsilon./dt, .times. 10.sup.3 s.sup.-1 0.05 0.10 0.10 0.10 0.10
2N.sub.f -- -- 48 36 22 Notes: A.sub.f = measured cross sectional
area at failure surface .sigma..sub.true = true fracture stress
(P.sub.u/A.sub.f) .mu. = ratio of core nominal specified yield
strength/experimental yield strength (F.sub.y/f.sub.0.2) 2N.sub.f =
number of reversals to fracture
[0059] Since experimental yield strength exceeded nominal specified
yield strength by approximately 25%, a normalization factor
.mu.=F.sub.y/f.sub.0.2 was introduced. This value was used in
creation of the material constitutive model in order to remain
consistent with the material yield strength assumptions made during
analytical modeling.
[0060] Monotonic results indicated strain hardening equal to 0.89%
of Young's modulus until 5% elongation followed by 0.43% softening
until tensile fracture (FIG. 6). Cyclic loops have a bi-asymptotic
shape and are comprised of a linear elastic, smooth non-linear
elastic-plastic transition and an approximately linear plastic
region (FIG. 7). Transitions between the elastic and plastic
regions were less abrupt for the cyclic tests as compared to the
monotonic primarily due to the Bauschinger effect. Anomalies at the
zero stress level can be attributed to looseness in the double nuts
holding the specimens in the fixture. Closeness of the loops
indicated that cyclic softening occurred at a very low rate,
especially for the 2% test. This is better illustrated in FIG. 7 as
a plot of maximum cyclic stress versus number of reversals where
cyclic softening is negligible after an initial cycle of hardening.
Cyclic softening increased minimally as the strain amplitude was
increased from 2% to 4%. Consequently, isotropic cyclic hardening
was relatively slow compared to kinematic hardening. This absence
of isotropic hardening has been witnessed in similar tests on
6060-T6 aluminum tested to strain amplitudes of 0.4%, 0.8%, and
1.2% (Hopperstad et al. 1995).
[0061] Cyclic coupon tests on 6061-T651 alloy have been reported
that achieved 2N.sub.f=142 at .DELTA..epsilon..sub.i/2=2.5% and
R.sub..epsilon.=-1 (Brodrick and Spiering 1972). This is up to 3
times greater than was achieved in the present tests at similar
strain amplitudes. Slight bending in the 4% strain coupon was
witnessed and is manifested in the hysteresis plots by a bend in
the linear portion of the loop beginning at zero stress. Uniaxial
stress may have been similarly compromised in the 2% and 3% coupons
leading to premature fatigue failure due to non-uniform cross
section strain distribution. Therefore, use of an hourglass shaped
coupon without a prismatic center section is recommended for strain
ranges greater than 2% to control buckling (ASTM 2004). Use of the
current data for definition of a cyclic material model is not
anticipated to significantly affect results since the linear
portion of the curve is not used in its definition. However,
suitability of the candidate alloy for use in high cyclic strain
applications requires further experimental study.
Numerical Simulations
Cyclic Material Constitutive Model
[0062] Representative prediction of cyclic material behavior uses a
calibrated general nonlinear combined kinematic-isotropic
constituitive model. This model has proven to be capable of
simulating the Bauschinger effect, cyclic hardening with plastic
shakedown and relaxation of the mean stress (Simulia 2010).
Experimental data from coupon C3 normalized by .mu..sub.ave=0.810
was selected which is in between the approximate inelastic core
strains of 2.25% and 3.22%. The calibration procedure used a 3D
finite element model of the test coupon comprised of C3D4
tetrahedral continuum elements as depicted in FIG. 8(b).
Convergence of the fine mesh was studied by varying the number of
degrees of freedom and the element polynomial. Coupon simulations
were set to run in displacement control for two full cycles to
verify calibration. Superposition of three backstresses effectively
captured the shape of the experimental hysteresis plots in the
Bauschinger region by accounting for strain ratcheting effects.
Superposition of the experimental and numerical results for the 2%,
3% and 4% strain amplitudes are illustrated in FIG. 8(a) with
reasonable correlation. Isotropic hardening was not used due to its
negligible influence on cyclic behavior as shown by the stable
maximum cyclic stress plots.
Finite Element Model Configuration
[0063] Numerical models were created using commercially available
finite element analysis and post-processor software (Simulia 2010)
and configured as shown in FIG. 9(a). Core angles were modeled as
four separate 3D planar extrusions meshed with fully integrated,
general purpose 4-node shell (S4) elements capable of modeling
large membrane strains and the restrainer was modeled as a single
1D beam meshed with Timoshenko (B31) elements. Beam element section
properties were assigned using equivalent square tubes
representative of the x-x and y-y flexural stiffness of the bundled
restrainer tubes acting compositely. Core and restrainer nodes were
connected with slide-plane connectors to the angle tips and slot
connectors to the angle vertex to decouple axial interaction and
allow Poisson deformation. The spacing between slotted connectors
was kept at a constant 25.4 mm [1 in] leaving one unsupported node
in between the connector nodes. However, in the present study no
local buckling imperfections were assigned to promote rippling
between the connectors. Boundary conditions were assigned to
reference points (RP) positioned at the end of the gusset plate a
distance L.sub.c3 from the end of the brace. Load eccentricity for
end moments was introduced by offsetting the RP from the centroid a
distance e.sub.l in the positive y-direction to simulate gusset
plate rotation from frame drift and single curvature of the brace.
The applied end moment was directly proportional to axial load.
[0064] Consideration of thin and thick shell formulations on
elastic buckling behavior was made by examining load vs. axial
displacement behavior of braces loaded monotonically to an enforced
axial displacement of 50.8 mm [2 in] with adequately and
inadequately restrained cores. For these simulations, a nominal
material yield strength F.sub.y was used along with a nominal 1%
post-yield hardening. The difference between tension and
compression yield load, buckling load and post-buckling path are
shown to be negligible in FIG. 9(b), indicating that transverse
shear flexibility is not important for global buckling modes with
thickness to characteristic length ratios less than 1/15. Model
convergence was also studied by examining element strains at both
the elastic and plastic regions for varying number of degrees of
freedom. To stimulate global buckling, geometric imperfections were
introduced into the mesh from the first four buckling modes.
Maximum global out-of-straightness of L.sub.b/1000 was assigned for
modes 1 & 2 and L.sub.b/4000 for modes 3 & 4.
Effect of Brace End Moments
[0065] BRBF in-plane drift may introduce end moments or rotations
into the brace during severe seismic loading as shown in FIG. 10(b)
for an inverted v-brace configuration. This effect causes
additional flexural demand on the restrainer above those caused by
ideal column buckling models. An upper bound end moment that
utilizes the available plastic moment of the core's intermediate
section M.sub.p' was used to quantify this effect which can be
determined by performing a rigorous non-linear push-over analyses
of the BRB/BRBF assembly which is beyond the scope of this
research. The available plastic moment of the axially loaded member
is reduced below the value of F.sub.yZ as is shown diagrammatically
in FIG. 10(a) when all four core angles act compositely by shear
transfer occurring at the bolted connections. Using this
interaction of axial load and moment with an axial load equal to
the core nominal yield load P.sub.yc=A.sub.1F.sub.y, the following
values were calculated: d.sub.1=8.43 cm [3.32 in], d.sub.2=15.7 mm
[6.16 in], Z=215 cm.sup.3 [13.1 in.sup.3] and M.sub.p'=5186 kN-cm
[459 k-in]. To express this moment as a ratio of the upper bound,
the variable .PSI.=M.sub.app/M.sub.p' was introduced where
M.sub.app=maximum applied end moment and .PSI.<1. Additionally,
the relationship .PSI.M.sub.p' can be converted to a
load-eccentricity relationship for use in numerical simulations as
shown in Eq. (13) where the core nominal yield load is used
neglecting the contribution of post-yield hardening and e.sub.1 is
calculated for a desired end moment effect.
e 1 = .psi. M p ' P yc ( 13 ) ##EQU00008##
Monotonic Simulations
[0066] Parametric numerical simulations of monotonically loaded
prototypes were studied in displacement control for comparison with
the proposed analytical models. Variables used were L.sub.c,
E.sub.rI.sub.r/L.sub.r.sup.3 and .PSI.M.sub.p' as shown in Table
3.
TABLE-US-00003 TABLE 3 Numerical simulation parameters and results
General Parameters Cyclic Results Simulation Simulation Mono/
L.sub.c E.sub.rI.sub.r/L.sub.r.sup.3 Monotonic Results
.SIGMA.P.DELTA..sub.a .SIGMA.P.DELTA..sub.a Group ID Cyclic m [in]
kN/m [k/in] .psi. P.sub.e/P.sub.yc P.sub.max/P.sub.yc
.DELTA..sub.a/.DELTA..sub.t kN/m .SIGMA.P.sub.o.DELTA..sub.o 1A
58-R1-M0 M 1.47 [58] 31.5 [0.180] 0 0.859 0.730 0.341 .gamma. = 0.3
58-R2-M0 M 1.47 [58] 42.2 [0.241] 0 1.15 0.934 0.362 58-R3-M0 M
1.47 [58] 74.6 [0.426] 0 2.04 1.43 2.34* 2A 58-R1-M1 M 1.47 [58]
42.2 [0.241] 0.25 1.15 0.908 0.370 .gamma. = 0.3 58-R2-M1 M 1.47
[58] 55.1 [0.315] 0.25 1.50 1.13 0.398 58-R3-M1 M 1.47 [58] 83.6
[0.477] 0.25 2.28 1.42 1.78* 3A 58-R1-M2 M 1.47 [58] 55.1 [0.315]
0.5 1.50 0.883 0.379 .gamma. = 0.3 58-R2-M2 M 1.47 [58] 70.4
[0.402] 0.5 1.92 1.09 0.407 58-R3-M2 M 1.47 [58] 93.2 [0.532] 0.5
2.54 1.39 1.66* 1B 96-R1-M0 M/C 2.44 [96] 31.5 [0.180] 0 0.859
0.720 0.340 3990 0.529 .gamma. = 0.5 96-R2-M0 M/C 2.44 [96] 42.2
[0.241] 0 1.15 0.923 0.362 5100 0.677 96-R3-M0 M/C 2.44 [96] 66.4
[0.379] 0 1.81 1.29 3.59* 6740 0.895 2B 96-R1-M1 M/C 2.44 [96] 42.2
[0.241] 0.25 1.15 0.898 0.371 4884 0.649 .gamma. = 0.5 96-R2-M1 M/C
2.44 [96] 55.1 [0.315] 0.25 1.50 1.12 0.401 6019 0.799 96-R3-M1 M/C
2.44 [96] 74,6 [0.426] 0.25 2.04 1.29 2.01* 6653 0.883 3B 96-R1-M2
M/C 2.44 [96] 42.2 [0.241] 0.5 1.15 0.873 0.379 4700 0.624 .gamma.
= 0.5 96-R2-M2 M/C 2.44 [96] 55.1 [0.315] 0.5 1.50 1.08 0.409 5760
0.765 96-R3-M2 M/C 2.44 [96] 83.6 [0.477] 0.5 2.28 1.28 1.82* 6630
0.880 Notes: P.sub.e = restrainer buckling load = .pi..sup.2
E.sub.rI.sub.r/L.sub.r.sup.2 P.sub.max = maximum compressive axial
load .SIGMA.P.DELTA..sub.a = cumulative load-displacement
.SIGMA.P.sub.o.DELTA..sub.o = area of ideal trapezoidal hysteresis
*indicates successful BRB simulation
[0067] Target axial displacement .DELTA..sub.bm relating to 2.5%
story drift as calculated from the model building was multiplied by
two as specified by the cyclic loading protocol for "Qualifying
Cyclic Tests of Buckling-Restrained Braces" (AISC 2005).
[0068] FIGS. 11(a)-(f) show axial load vs. log-displacement
relationships for the six groups of simulations. Each dual plot
illustrates the ability of the trial to meet the target axial
displacement before reaching the failure criteria. The failure
criteria were defined as buckling or reaching a limiting transverse
displacement at the mid-length of the brace. Maximum transverse
displacement .DELTA..sub.t.sup.max=8.79 cm [3.47 in] is denoted as
a dashed line and was calculated by considering f.sub.b=206 MPa [30
ksi] and c=11.8 cm [4.63 in] as measured from the baseline four
10.8 cm by 6.35 mm [4.25 in by 0.25 in] tube configuration. The
relationship given in Eq. (14) was derived from M.sub.r=FL.sub.r/4,
f.sub.b=M.sub.r/S.sub.r, S.sub.r=I.sub.r/c and
.DELTA..sub.t=FL.sub.r.sup.3/48E.sub.rI.sub.r where M.sub.r and
S.sub.r are the flexural moment when flexurally loaded by a point
load at mid-span and elastic section modulus of the restrainer,
respectively.
.DELTA. t max = f b L r 2 12 cE r ( 14 ) ##EQU00009##
[0069] Failure points of inadequately restrained braces are denoted
by white markers on the plots while end of simulation points for
adequately restrained braces are denoted by black markers.
Inadequately restrained braces generally exhibited a failure
progression in compression as shown in FIGS. 12(a) to 12(c) and
described as follows: 1) uniform axial stress and yielding at the
reduced section with transverse bending; 2) increasing transverse
displacement and bending stress at the ends of the restrainer; 3)
plastic local buckling of the core angle legs leading to hinging;
and 4) overall global buckling.
[0070] Numerical results are given in Table 3 for each simulation.
Restrainer stiffness used in the third simulation for each group
was determined by an iterative process of increasing
E.sub.rI.sub.r/L.sub.r.sup.3 to achieve stable BRB performance with
P.sub.max/P.sub.yc>1 and .DELTA..sub.a/.DELTA..sub.t>1.28
which represents target axial displacement over
.DELTA..sub.t.sup.max. Application of end eccentricity had a
degradation effect on the .DELTA..sub.a/.DELTA..sub.t ratio, but
successful simulations were able to remain in relatively straight
axial alignment. FIG. 11(g) shows the linear effect of application
of end eccentricity on required E.sub.rI.sub.r/L.sub.r.sup.3. Slope
of the lines remained constant between the Group A and B braces
demonstrating that reduced section length has insignificant effect.
Furthermore, end eccentricity may be accounted for by superimposing
from 34.4 to 37.2 kN/m of additional stiffness per unit of .PSI.
which effectively doubles required E.sub.rI.sub.r/L.sub.r.sup.3 for
this prototype. FIG. 4 illustrates scatter plot comparison between
analytical and numerical results. Numerical simulations resulted in
approximately two times greater required
E.sub.rI.sub.r/L.sub.r.sup.3 than analytical for the examined brace
length indicating that a degree of conservatism of two or greater
may be required to account for material non-linearity, load
eccentricity, reasonable transverse displacement as well as
possible local buckling effects near the end of the restrainer.
Although, it is recognized that further study is required to
determine the degree of conservatism required for other brace
lengths since only one length was considered in this research.
Accounting for this larger required stiffness, four 5 in by 5/16 in
or 51/2 in by 5/16 bundled tubes would be required for .PSI.=0 and
.PSI.=0.5, respectively. This is within reasonable practical limits
for brace compactness and promotes the notion that the described
brace is a viable concept.
Cyclic Simulations
[0071] The objectives included to assess energy dissipation
potential and assert numerical model stability and repeatability
when subjected to cyclic axial force and rotational demand when
subjected to a minimum cumulative inelastic axial deformation of
200 times the yield deformation (AISC 2005). Numerical formulation
used the calibrated cyclic constitutive model and did not include
simulation of material fatigue failure.
[0072] Table 3 shows test parameters for the Group B brace.
Representative hysteresis plots for Group 1B prototypes for load
vs. axial displacement and load vs. transverse displacement are
given in FIGS. 13(a)-13(d). Inadequately restrained braces
exhibited large transverse displacement along with pinched
hysteresis loops on the compression excursions while adequately
restrained braces exhibited full symmetrical loops with minimal
transverse displacement. Group 1B hysteresis plots of load vs.
axial strain at the mid-length of the reduced section demonstrate
tension side strain ratcheting for the inadequately restrained
brace (96-R1-M0) and nearly symmetrical loops for the adequately
restrained brace (96-R3-M0). A strain shift of approximately 2.5%
is witnessed toward the tension side due to incomplete strain
reversal during compression excursions due to transverse
displacement. Average achieved material strain over a gage length
of 2.54 cm [1 in] at the mid-length of the reduced section was
numerically measured at the 1.0.DELTA..sub.bm cycle as +2.89% to
-1.72% and +3.20% to -2.52% for the R1 and R3 braces, respectively.
This correlates reasonably well with .epsilon..sub.t=+/-2.25% as
calculated from Eq. (3) with the caveat that positive tension
strains are approximately 25% greater than compression strains in
an adequately restrained brace due to the strain ratchetting.
[0073] Cumulative energy dissipation .SIGMA.P.DELTA..sub.a and
load-strain .SIGMA.P.epsilon..sub.t were determined by numerically
integrating the area under the curve for each of the nine
simulations. Table 3 shows these results along with those
normalized by an ideal trapezoidal hysteresis plot
.SIGMA.P.sub.o.DELTA..sub.o=7530 kN-m [66,700 k-in]. Adequately
restrained braces achieved nearly 90% of the energy dissipation of
the idealized hysteresis. Inadequately restrained braces ranged
from 53% to 83% showing a marked improvement. End eccentricity also
had a significant effect on cumulative displacement and strain
demand as witnessed in FIG. 13(e) where a steeper slope is present
at higher values of .PSI. up to the plateau of
.SIGMA.P.DELTA..sub.a=6600 kN/m [58,500 k-in].
.SIGMA.P.epsilon..sub.t plots exhibit a lower slope reaching a
plateau of 2900 kN-m/m [652 k-in/in].
FRP Wrap Design
[0074] Bundled FRP tubes should work compositely to achieve
greatest strength and stiffness. Maximum expected shear flow
q.sub.max between the tubes was approximated by utilizing the same
Euler column buckling model and force equilibrium method. Since
failure of the bundled tube assembly should be controlled by the
flexural moment of the tubes and not shear failure of the wrap,
shear flow was determined using the previously related
.DELTA..sub.t.sup.max. Previously defined values of c, f.sub.b and
L.sub.r were used with the tube arrangement to calculate
q.sub.max=8.60 kN/cm [4.91 k/in] at the end of the restrainer. A
multi-layer wet layup GFRP uniaxial fabric can resist this shear
flow. Proprietary wrap systems are common and typically exhibit
ultimate tensile strengths of 582 MPa [84.4 ksi] in the primary
fiber direction and have an effective laminate thickness of 1.27 mm
[0.05 in]. A truss-like mechanism was conceived using two layers of
wrap along the entire length of the restrainer oriented at
+/-30.degree. from the longitudinal axis to resist shear flow in
tension through the primary fibers. The allowable shear strength of
the wrap was calculated as 9.85 kN/cm [5.63 k/in] using a degree of
conservatism of 1.5 to account for additional extreme fiber
longitudinal stress imparted by bending of the restrainer assembly.
Although, additional stress in the wrap from bending is expected to
be minimal since the modular ratio of the fabric and tubes is
unity.
Weight Reduction
[0075] The described prototype brace was calculated to weigh 200 kg
[440 lb] or 27% and 41% the weight of a traditional mortar-filled
tube and all-steel BRB of similar length, core area and restrainer
dimensions. Thus, the described prototype brace weighs less than
50% of a comparable conventional brace. For the mortar-filled tube,
a single square tube of comparable size (8 in.times.1/4 in) was
used. Since the nominal yield strength of common steel and 6061-T6
aluminum are almost identical, similar core sizes were considered
fair comparison. Nominal unit weight for mild steel, aluminum and
concrete mortar were taken as 7860 kg/m.sup.3 [490 lbs/ft.sup.3],
2650 kg/m.sup.3 [165 lbs/ft.sup.3] and 2410 kg/m.sup.3 [150
lbs/ft.sup.3], respectively. This comparison serves to highlight
the considerable weight savings that can be realized with the
described brace.
Notation
TABLE-US-00004 [0076] A.sub.1 = core reduced cross sectional area
A.sub.2 = core intermediate cross sectional area A.sub.3 = core
full cross sectional area c = distance from NA to restrainer
extreme fiber C.sub.d = deflection amplification factor D.sub.ie =
elastic story drift E.sub.c = core Young's modulus E.sub.ct = core
tangent modulus E.sub.r = restrainer Young's modulus f.sub.0.2 =
experimental 0.2% offset yield strength f.sub.b = restrainer
ultimate bending stress F = transverse restrainer force F.sub.y =
core nominal specified yield strength h.sub.i = story height, level
"i" I.sub.c = core moment of inertia I.sub.r = restrainer moment of
inertia k.sub.s = equivalent restrainer spring stiffness K =
effective length factor L.sub.b = brace end to end length L.sub.c =
core reduced section length L.sub.r = restrainer length M.sub.app =
maximum applied end moment M.sub.int.sup.p = core internal moment
M.sub.p' = available plastic moment of intermediate section M.sub.r
= restrainer moment at mid-length M.sub.res = restrainer resisting
moment P = brace applied axial load P.sub.cr = critical buckling
load P.sub.u = design axial force P.sub.yc = core nominal yield
load q.sub.max = maximum wrap shear flow R.sub..epsilon. = cyclic
strain ratio .gamma. = L.sub.c/L.sub.b .DELTA..sub.a = core axial
displacement .DELTA..sub.bm = brace disp. at design story drift
.DELTA..sub.t = restrainer transverse displacement at mid-length
.DELTA..sub.t.sup.max = max. transverse disp. of restrainer
.epsilon..sub.c = core inelastic strain .epsilon..sub.t = total
experimental strain .eta. = A.sub.1/A.sub.3 .theta. = brace angle
with horizontal .mu. = F.sub.y/f.sub.0.2 .PSI. =
M.sub.app/M.sub.p'
EXEMPLARY EMBODIMENTS
[0077] Referring to FIGS. 1(a) and 1(b), an exemplary embodiment of
a buckling restrained brace 10, sometime referred to as a "full
cruciform" type, is shown. The brace has an elongate core member 12
with opposite ends 14, 16. At least one spacer member 18 is
positioned on the core member 12. In the illustrated
implementation, there is a first spacer member 18 oriented along
one plane of the core member 12 and a second space member oriented
perpendicular to the first spacer member. There are one or more
core restrainer member sections 20 arranged adjacent the spacer
member and around the core member 12. The core restrainer member
sections 20 are coupled together with the spacer member 18 and the
core member 12 by a jacket 19 comprising fiber reinforced polymer
fabric that is configured to be wrapped around the assembled core
member and core restrainer member sections with the spacer member
sandwiched therebetween.
[0078] In exemplary embodiments, the core member is made of
aluminum, although other materials with suitable ductility could be
used. In the illustrated embodiments, the ends 14, 16 of the core
member 12 are exposed.
[0079] In the embodiment of FIGS. 1(a) and 1(b), there are four
core restrainer member sections 20, but any suitable number of
sections may be used. The core restrainer member sections 20 can
have a rectangular (or square) cross-section as shown in FIGS.
1(a), 1(b), 1(c) and 1(e), a circular cross section as shown for
the core restrainer members 24 in FIGS. 1(d) and 1(f), or any other
suitable cross-section. As illustrated, the core restrainer
sections may have a hollow tubular configuration over at least a
portion of their length.
[0080] The core member may be comprised of a single member or
several sub-members. In the illustrated implementation, the core
member 12 is comprised of comprised of four angles 22a, 22b, 22c,
and 22d arranged such that adjacent side surfaces are in contact
with each other and the vertices are adjacent each other and
oriented toward the center as shown. As shown in FIGS. 1(e) and
1(f), the core member 12 can be comprised of two tee members 26a,
26b arranged opposite each other, i.e., with the respective
uninterrupted side surfaces facing each other. In some embodiments,
the multiple sub-members are separated from each other, e.g., by
the interposed spacer member(s), over at least an intermediate
portion of the length of the brace 10. Also, exemplary
configurations define at least one pair of separated spaces (such
as two pair or four spaces as shown in FIGS. 1(c)-1(f)) for
receiving the core restrainer member sections.
[0081] In some embodiments, a debonding material such as PTFE can
be applied between adjacent surfaces of the core restrainer member
sections 20 and the core member 12 to ensure that there is no
coupling or bonding between the adjacent surfaces. In some
embodiments, no such debonding material is used.
[0082] Referring to FIGS. 1(g) and 1(h), another exemplary
embodiment of a buckling restrained brace 210 is shown. The brace
210 is similar to the brace 10 of FIGS. 1(a) and 1(b), except the
brace 210 has an elongate core member 212 formed in a T-shape
(shown inverted in the figures), and there are two core restrainer
member sections 220 received in the spaces defined on either side
of the core member 212. As in the case of the brace 10, a jacket
219 of fiber reinforced polymer fabric is wrapped around the core
restrainer member sections 220 and the core member 212.
[0083] In specific implementations, the core restrainer member
sections 20, 200 are made of fiber reinforced polymers. The core
member 12, 212 is made of a suitable material, such as, e.g., an
aluminum alloy.
[0084] In the illustrated embodiment, the brace 210 does not
include any spacer member, but one or more spacer members can be
provided if desired or if required in certain circumstances.
[0085] This analytical and numerical study focusing on global
buckling demonstrated the ability to develop a new
ultra-lightweight buckling-restrained brace for potential
application in existing building seismic retrofit situations
similar to a representative 3-story office model building.
Calculated required restrainer stiffness from a newly developed
SDOF model and previously established Euler buckling model was
compared with monotonic and cyclic numerical simulations of
prototypes with varying restrainer stiffness, reduced section
length and applied end moment. The following presents a summary of
results:
[0086] (1) A common structural aluminum coupon was tested
cyclically to develop a hysteresis for use in creation of a
constitutive model for cyclic numerical modeling. Excellent
correlation was illustrated with a general nonlinear combined
hardening model using finite element simulations. Test results
indicated that low monotonic strain hardening and negligible cyclic
hardening make 6061-T6511 a potentially suitable candidate for
seismic applications.
[0087] (2) An important contribution was made to modeling BRB
behavior using numerical finite element simulations to verify
existing analytical based design methods. Monotonic numerical
results indicated that a degree of conservatism of two or greater
was required for the considered brace length when using proposed
analytical methods to account for geometric and material
non-linearity, local buckling at the unrestrained core and limiting
transverse bending stress on the restrainer. SDOF and Euler
buckling models achieved similar results with only 13% difference.
Further research was recommended to examine the effect different
brace lengths have on the degree of conservatism required to
achieve BRB performance.
[0088] (3) BRB applied end moment was quantified using an upper
bound approach in lieu of performing specific frame analyses in
order to account for story drift two times greater than the maximum
2.5% given in typical building codes. The restrainer demand from
applied end moment was determined by monotonic simulations to be
one of linear superposition with conventional buckling demand.
Additional required stiffness of 34.4 to 37.2 kN/m per unit of
.PSI. was required for the brace length examined. This relationship
was shown to hold for restrainer length ratios of .gamma.=0.3 and
0.5 indicating that there may be potential for using the method as
a quick and easy design aid to account for end moment effects on
BRB performance.
[0089] (4) Cyclic simulations indicated that reliable BRB
performance was achieved with approximately 90% efficiency as
compared to an ideal trapezoidal hysteresis if adequate restrainer
stiffness was provided. Lower values of 53% to 83% were typical
with inadequate restraint with most of the cumulative ductility
occurring from yielding in the tension excursions. Material strains
achieved in the simulations correlated reasonably well with those
estimated by simple analytical methods and were approximately +3.2%
to -2.52% for the design story drift. Material strains were shown
to be asymmetrical due to strain ratcheting caused by transverse
bending.
[0090] In view of the many possible embodiments to which the
disclosed principles may be applied, it should be recognized that
the illustrated embodiments are only preferred examples and should
not be taken as limiting in scope. Rather, the scope of protection
is defined by the following claims.
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