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Design of chevron braced frames based on the ultimate lateral strength I:
Theory
Article · January 2006
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Design of Chevron Braced Frames Based on the Ultimate Lateral
Strength I: Theory
E.M. Marino & A. Ghersi
Department of Structural and Environmental Engineering, University of Catania, Catania, Italy
M. Nakashima
Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan
ABSTRACT: According to EuroCode 8 (EC8), in chevron braced frames, braces both in tension and compression are assumed to resist the seismic force. The shear strength of a pair of chevron braces is based on
their buckling strength. The behavior factor q, which allows for the trade-off between the strength and ductility, is set at 2.5 value irrespective of the brace slenderness. This paper, which is the first of two companion
papers, shows that the EC8 method for the calculation of brace strength supplies significantly different elastic
stiffness and actual strength for different values of brace slenderness. A new method to estimate the strength
of a chevron brace pair is proposed, in which the yield strength (for the brace in tension) and the postbuckling strength (for the brace in compression) are considered. The new method ensures an identical elastic
stiffness and a similar strength regardless of the brace slenderness. The advantage of the new method over the
EC8 method is demonstrated for the capacity of such method to control the maximum story drift.
1 INTRODUCTION
Chevron braced frames are very commonly used in
earthquake-prone countries in which building frames
have to endure large lateral forces. Braces, however,
buckle, and the resistance decreases in the postbuckling regime, and alternating elongation and
buckling further aggravate the brace resistance. To
reflect such characteristics of braced frames, EuroCode 8 (CEN, 2003) stipulates rather low values for
the behavior factor q (2.5) for braced frames, which
determines a large design seismic force. Braces both
in tension and compression are assumed to resist the
seismic force. The shear strength of a pair of chevron braces is based on their buckling strength. The
EC8 design method does not take into account that,
even after the buckling of the brace in compression,
the brace in tension is still elastic and can sustain
further force before attaining the yielding (collapse
of the story). This provides a reserve of lateral
strength with respect to the design value, which is
larger for slender braces (characterised by a yielding
strength much larger than the buckling strength). As
a consequence, the EC8 design method may be
overly conservative for slender braces. By contrast,
according to the Japanese seismic code (BCJ, 1997)
the design of braced frames is based on the ultimate
lateral strength. The design strength of a pair of
chevron braces is taken to equal the sum of the postbuckling strength in compression, and the yielding
strength in tension. Furthermore, the design seismic
force is obtained by the Ds factor (conceptually the
inverse of q), which depends on λ . The BCJ design
method appears more logical. However, the use of
the Ds factor depending on λ makes the design
rather complex, because λ is not known a priori. In
consideration of the issues stated above, this paper
examines earthquake response of chevron braced
frames designed by EC8, with brace slenderness as
the major variable. Significant differences in both
the elastic stiffness and actual strength are noted for
designed frames with different brace slenderness
values. A new method to estimate the strength of a
brace pair arranged in a chevron form is proposed.
This method is conceptually similar to that stipulated
in BCJ, but it specifies a unique q value, given independent of the slenderness. This makes the proposed
method much simpler than the BCJ method, without
the loss of accuracy. The effectiveness of the new
method over the EC8 method is demonstrated by the
comparison of the seismic responses of chevron
braced frames designed by the two methods.
2 BEHAVIOR OF CHEVRON BRACES
In major earthquake conditions, braces experience
multiple cycles of inelastic deformation, including
alternating yielding and buckling. Such behavior
may be reasonably schematized by the relationship
between the axial force of the brace N and the corresponding axial displacement δ shown in Figure 1.
N
E
3 DESIGN OF CHEVRON BRACES
F
According to EC8, the seismic design force Vd is
specified based on the elastic spectrum representative of reference ground motions (probability of 10%
exceedance in 50 years) and reduced by the behavior
factor q. The behavior factor equals 2.5 for high ductility chevron braced frames and is specified independent of the slenderness of the braces. EC8 also
stipulates that the shear strength of a pair of chevron
braces be based on their buckling strength Nb. If the
story shear is resisted by a pair of chevron braces,
the cross-sectional area of the braces is estimated by
the following equation:
D
O
δ
C
B
A
Figure 1. Hysteretic behavior of a brace subjected to cyclic axial loading.
Vd = 2 N b cos θ = 2 χ A b f y cos θ
The brace loaded in compression, after buckling
(point A), starts deflecting laterally and exhibits
strength and stiffness degradation (branch B-C). But,
When the axial force is reversed the brace rapidly
recovers its elastic stiffness (branch C-E) until it
yields (point E). Details can be found in many studies (for example Nakashima and Wakabayashi,
1992; Nonaka, 1973; Tada and Suito, 1998;
Tremblay, 2002). These studies demonstrated that
the cyclic behavior of steel braces is influenced by
their slenderness. A stocky brace can sustain a compressive axial force close to its plastic capacity Ny,
because it does not buckle. In a slender brace, buckling occurs at a small axial force but strength degradation after buckling is after all not large. A brace
with intermediate slenderness buckles and exhibits
relatively large strength reduction in the postbuckling range [Tremblay, 2002]. EC8 establishes a
maximum limit for the normalized brace slenderness
λ , given by the following formula:
where Nb is the buckling axial force of one brace, Ab is
the area of the brace, θ is the angle of inclination of the
brace with respect to the beam longitudinal axis, and χ
is the ratio between the buckling and plastic strengths.
This ratio is given as a function of the braces’ slenderness according to EuroCode 3 [CEN, 1993].
This paper proposes a different procedure to estimate the strength of chevron braces. In the proposed
procedure, the shear strength provided by a pair of
chevron braces is assumed to be equal to the sum of
the post-buckling strength in compression (Nu), and
the yielding strength in tension (Ny), and is given as:
λ=
λ
π
fy
E
(1)
where fy and E are the steel’s yield stress and the
Young’s modulus, respectively. The normalized
slenderness cannot be larger than 2.0 according to
EC8. This limit includes stocky, intermediate and
slender braces. But, in real design, braces having
very small slenderness are impractical. For instance,
even considering a frame having very small story
height and span length (equal to 3.0 m and 4.0 m, respectively) and therefore very short chevron braces
(just 3.6 m long) and the braces’ radius of gyration
about the weak axis set at the largest value of wideflange cross-sections available in Italy (80 mm for
340 x 310 x 21 x 39 mm), we find that the normalized slenderness of the braces is 0.49. Here, a steel
grade whose fy equals 235 MPa is considered. Instead, can be easily shown that λ = 2.0 can be obtained in practical applications. In reference to these
considerations, the range of λ between 0.4 and 2.0
is considered in this study.
Vu = ( N y + N u ) cos θ
(2)
(3)
Post-buckling strength has to be specified when applying Eq. (3). Previous studies (for example
Tremblay, 2002) indicate that for intermediate and
slender braces the post-buckling resistance is relatively constant for large deformations. EC8 stipulates that Nu equals 30% of the yielding strength Ny.
The AISC seismic standards (AISC, 2002) stipulate
that Nu equals 30% of the buckling axial force Nb.
Supposing that the story shear is resisted by a pair of
chevron braces, the required cross-section of the
brace is given by the following equations, depending
on the method of estimation of Nu:
Vd = ( N y + N u ) cos θ = 1.3 Ab f y cos θ
(4a)
Vd = ( N y + N u ) cos θ = (1 + 0.3 χ) Ab f y cos θ
(4b)
4 FEATURES OF CHEVRON BRACED
FRAMES: EC8 VS. PROPOSED METHOD
The EC8 and proposed methods are compared for
the shear force and inter-story drift relationship.
Various single-story chevron braced frames having
one span of 8.0 m and a story height of 3.3 m are
considered. The design spectrum stipulated in EC8
for soil type C and the behavior factor q of 2.5 were
(a)
V (kN)
EC8 method
λ = 2.0
λ = 1.6
1000
λ = 1.2
500
Design
Shear force
λ = 0.4
λ = 0.8
20
d30(mm)
0
0
(b)
10
It is assumed to be 30% of Nb according to AISC
standard.
3 The third alternative (AM-3) differs from AM-1
in the design shear distribution. The force distribution stipulated in UBC97 (ICBO, 1997) is
adopted, i.e. the total seismic force is distributed
according to the following formulas:
Proposed method, N u = 0.3 N y
V (kN)
1000
λ = 0.4
λ = 0.8
Fi =
j =1
(c)
10
d30(mm)
20
Proposed method, N u = 0.3 N b
V (kN)
1000
λ = 0.4
500
λ = 1.2
λ = 1.6
λ = 0.8
Design
Shear force
λ = 2.0
0
0
10
20
d30(mm)
Figure 2. Effects of brace slenderness on monotonic inelastic behavior of chevron braces: (a) EC8 method, (b) proposed method with Nu = 0.3 Ny, (c) proposed method with
Nu = 0.3 Nb.
adopted. The design shear force was 0.35 in terms of
the base shear coefficient for all analyzed frames.
Braces having various slenderness values ( λ from
0.4 to 2.0, and λ from 37 to 186) were considered.
For each slenderness, the corresponding crosssection area of the brace Ab was evaluated by means
of Eqs. (2) and (4a) or (4b). Here we assume that the
surrounding beams and columns would remain rigid.
All frames were pushed to large inelastic deformations and the corresponding monotonic story
shear - displacement relationships are represented in
Figure 2. Figures 2a, 2b and 2c refer to the frames
designed by EC8 and the proposed methods with Nu
= 0.3 Ny and Nu = 0.3 Nb, respectively.
5 DYNAMIC BEHAVIOR OF CHEVRON
BRACED FRAMES
The EC8 method is compared with the proposed
method for estimating the strength of chevron braces.
Three alternatives were considered for the proposed
method according to the post-buckling strength and
the design shear distribution along the height.
1 In the first alternative (AM-1) the cross-section of
the braces is determined based on the linear distribution of the design shear. The post-buckling
strength of the compressive brace Nu is assumed
to equal 30% of Ny in accordance with EC8.
2 The second alternative (AM-2) differs from AM1 only in the estimation of post-buckling strength.
(5b)
where N is the number of stories, Ft is the portion of
the design base shear Vd applied at the top of the
building in addition to FN, Fi is the design seismic
force applied to level i, wi is the weight evaluated for
the seismic design situation at i-th floor, hi is the
height above the base to level i, and T is the fundamental period of the system.
5.1 Structural systems
The first example was a steel frame having a plan
view and the arrangement of chevron braces shown
in Figure 3. Steel grade with fy = 235 MPa was used
for all members. All beam-to-column connections
were taken to be pinned, and seismic actions were
sustained only by the chevron braces located on the
perimeters. The slenderness of the braces was taken
as the variable, and various braces with λ from 0.4
to 2.0 were adopted. Note that the slenderness was
assumed constant for each case.
Story mass was estimated by taking into account
the dead and live load of 5.0 kN/m2. The total seismic force was evaluated by means of the design
spectrum proposed by EC8 for soil type C, characterized by a peak ground acceleration ag equal to 0.35 g.
Two values of behavior factor q were considered: 2.5
and 4.5. q = 2.5 corresponds to what EC8 stipulates.
q = 4.5 was adopted to examine how the response
would be aggravated for a larger q (meaning a smaller
strength). For the braces, virtual cross-sections were
selected so that the braces would possess exactly the
8.0 m
8.0 m
8.0 m
3.3 m
0
hj
Ft = 0.07 T Vd
λ = 2.0
0
j
24.0 m
λ = 1.6
(5a)
N
∑w
λ = 1.2
500
Design
Shear force
(Vd − Ft ) wi hi
24.0 m
8.0 m
Figure 3. Plan layout of analyzed frame and arrangement of
chevron braces
Table 1. Fundamental periods (in sec) of analyzed frames.
ence is no more than 9% and 17% for q to equal 2.5
and 4.5, respectively.
q = 2.5
λ
EC8
AM-1
AM-2
AM-3
0.4
0.8
1.2
1.6
2.0
0.669
0.632
0.508
0.390
0.314
0.514
0.514
0.514
0.514
0.514
0.513
0.510
0.492
0.475
0.467
0.508
0.508
0.508
0.508
0.508
5.2 Dynamic analyses
q = 4.5
λ
EC8
AM-1
AM-2
AM-3
0.4
0.8
1.2
1.6
2.0
1.200
1.137
0.774
0.523
0.421
0.791
0.791
0.791
0.791
0.791
0.789
0.779
0.725
0.677
0.653
0.767
0.767
0.767
0.767
0.767
required design strength. Such selection automatically satisfies EC8 condition that the ratio of the
provided to the demanded strength of the braces be
almost uniform along the height. Large stiffness and
strength was assigned for all columns and beams.
For the chevron frame thus designed, its serviceability limit state is checked according to the provisions stipulated in EC8. The inter-story drift angle ∆r
demanded by the moderate earthquake condition is
given as:
∆ r = ν q ∆e
(6)
where ν is the reduction factor to account for the
magnitude of the moderate earthquake. The factor is
set at 0.5 for ordinary buildings. ∆e is the elastic inter-story drift angle due to the design seismic force
(the force reduced by q). The inter-story drift should
not exceed a specific limit. According to EC8, the
limit is 1% for buildings that adopt ductile attachment details for non-structural elements. For the
frames designed by the EC8 method, the largest value
(0.52%) was obtained for the frame with stocky braces
( λ = 0.4) and with q = 4.5, and the smallest (0.07%)
was obtained for the frame with slender braces
( λ = 2.0) and with q = 2.5. The frames designed by
the proposed method show the same inter-story drift
angles regardless of the brace slenderness. It was
0.19% for q = 2.5 and to 0.34% for q = 4.5.
The fundamental periods (T) of the designed
frames having different brace slenderness are summarized in Table II. For both values of q, the fundamental period T of the frames designed by the
EC8 method is strongly influenced by the slenderness. It is larger for frames with stocky braces. For q
= 2.5, T varies from 0.669 sec to 0.314 sec for λ
from 0.4 to 2.0. By contrast, T is practically independent of λ when the proposed method is applied.
For the frames designed by the proposed methods T
is about 0.5 sec for q to equal 2.5 and about 0.7 sec
for q to equal 4.5. For those designed by the AM-2
method, T is slightly dependent on λ , but the differ-
Nonlinear dynamic analyses were carried out by
means of a frame analysis program code named
CLAP (Ogawa and Tada, 1994). This program
adopts member-by-member modeling with plastic
hinges assigned at member ends. Yield criteria take
into account the interaction between axial force and
bending moment. A Rayleigh viscous damping was
used and set at 3% for the first two modes of vibration. Strain hardening after reaching the member
strength was set equal to 3% of the elastic stiffness.
The geometrical nonlinearity, i.e., the P-∆ effect,
was considered. The gravity load considered in the
analyses included the vertical load carried by both
the braced frames and gravity columns.
In the numerical model, only braces were allowed
to yield and buckle. Large strength and stiffness
were assigned for columns and beams. Such treatment was practical because the columns and beams
were pinned. In the analysis, the brace was modeled
by two beam elements. Each element can resist an
axial force, shear force, and bending moment, and a
plastic hinge is formed at each end after yielding. An
initial lateral deflection at the mid-length equal to
0.1% of the brace length was imposed. The lumped
mass assigned at mid-length was equal to half the
brace weight. Details about the brace modeling are
found in (Tada and Suito, 1998). The numerical
model can reproduce the brace hysteretic behavior
shown in Figure 1.
A suite of ground motions (Somerville et al.,
1997) adopted in the FEMA/SAC project in the
United States was used in this study. The suite, comprising 20 records, represents seismic events having
a probability of exceedance of 10 percent in 50 years
in the Los Angeles area. Figure 5 shows the average
spectrum of the twenty ground motions, together
with the EC8 spectrum used in design. Except for
the short-period range (not greater than 0.5 sec), the
average 5% damped spectrum is similar to the
design spectrum. Note that the used suite of accelerograms satisfied all the requirements stipulated in
EC8 for the use of recorded accelerograms and the
Se
2
(m/s )
15.0
SAC Earthquakes
10% / 50 years
Average spectrum
10.0
5.0
EC8 spectrum
Soil C, damp.
5%
0.0
0.0
0.5
1.0
1.5
2.0
2.5
T (sec)
Figure 4. Comparison between EC8 spectrum and average
spectrum of adopted ground motions.
fundamental periods of the analyzed frames are not
smaller than 0.5 sec in the majority of cases (Tab. 1).
values of the data.
Chevron braced frames are prone to develop a
story collapse mechanism (Tremblay, 2001; Tremblay and Robet, 2001; Tremblay, 2003), and the P-∆
instability amplifies the drift concentrations in particular stories. The results clearly indicate this tendency, showing that inter-story drifts concentrate at
the first story. It is also notable that large inter-story
drifts are present at the top story when linear distribution of the seismic force was adopted (AM-1).
The results of AM-3 obtained by means of the
UBC97 force distribution show considerable reduc-
5.3 Results
The maximum inter-story drift was used as an index
to assess the seismic performance of the analyzed
frames. Figure 5 shows the distribution of maximum
inter-story drift angles. Here, the value in each story
is the median of the twenty maximum inter-story
drift angles obtained for the twenty ground motions.
The median refers to the exponent of the natural log
(a)
EC8, q = 2.5
N
(b)
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
4
0%
1%
2%
3%
0
∆
AM-1, q = 2.5
N
0%
(d)
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
1%
2%
3%
∆
AM-1, q = 4.5
N
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
2
2
0
0%
1%
2%
3%
N
0
∆
AM-2, q = 2.5
0%
(f)
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
1%
2%
3%
∆
AM-2, q = 4.5
N
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
2
2
0
0%
(g)
close to 5%
out of scale
2
0
(e)
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
2
(c)
EC8, q = 4.5
N
1%
2%
3%
AM-3, q = 2.5
N
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
6
4
2
0
∆
0%
(h)
N
6
4
1%
2%
3%
∆
AM-3, q = 4.5
λ = 0.4
λ = 0.8
λ = 1.2
λ = 1.6
λ = 2.0
2
0
0
0%
1%
2%
3%
∆
∆
Figure 5. Median of maximum inter-story drift angles: (a) EC8, q = 2.5, (b) EC8, q = 4.5, (c) AM-1, q = 2.5, (d) AM-1,
q = 4.5, (e) AM-2, q = 2.5, (f) AM-2, q = 4.5, (g) AM-3, q = 2.5, (h) AM-3, q = 4.5.
0%
1%
2%
3%
tion in the inter-story drift at the top story. The
UBC97 force distribution considers higher mode effects, and a larger design shear is assigned to higher
stories compared to the linear distribution. Frames
designed with q = 4.5 exhibit larger drifts than those
with q = 2.5, especially at the first story.
In accordance with the results obtained in
(Tremblay, 2003), the maximum inter-story drifts of
the frames designed according to EC8 are strongly
influenced by λ (Fig. 5a and Fig. 5b). Frames with
slender braces ( λ = 1.6 and 2.0) and designed by
q = 2.5 sustain drifts that are smaller and relatively
uniformly distributed along the height (Fig. 5a),
seemingly because of their larger strength and stiffness. On the contrary, the maximum drifts concentrate significantly at the first story for intermediate
and stocky braces ( λ = 0.4, 0.8 and 1.2), in which
both the stiffness and strength are relatively similar.
As shown in Figure 5a and Figure. 5b, the maximum
inter-story drift angles are about 0.3% and 1.8% for
λ = 2.0 and 0.8 when q = 2.5, and about 0.8% and
5% when q = 4.5.
When focusing on the first story that sustains the
largest maximum drift angles in most cases, variation with respect to λ is significantly smaller with
the proposed method (AM-1, AM-2 and AM-3) than
with the EC8 method. In particular, the variation is
very small if the case with the stockiest brace ( λ =
0.4) is not included. Note that use of such a stocky
brace is rare as discussed earlier in Section 2. When
compared for the three alternatives (AM-1 to AM3), the top story’s inter-story drift is larger for slender braces (Fig. 5c to Fig. 5h). The AM-1 and AM-2
method lead to similar scatter in the top story’s interstory drift, indicating that either of the two equations
for the estimation of the post-buckling strength
[Equations (4a) and (4b)] is equally applicable. In
the AM-3 method, in which higher vibration modes
are taken into account, the scatter of maximum interstory drifts at the top story [for q = 2.5 (Fig. 5c)] is
reduced by 30% with respect to that of the frames
designed by the AM-1 method (Fig. 5g). In conclusion, the proposed method ensures significantly
more uniform maximum inter-story drifts, regardless
of the brace slenderness, than the EC8 method.
6 CONCLUSIONS
This paper examined the seismic performance of
steel braced frames configured in a chevron pattern.
The seismic behavior of chevron braced frames designed by EC8 method was carefully analyzed, and
the following findings were obtained.
1 The elastic stiffness and actual strength of chevron
braced frames differ significantly according to the
brace slenderness. Both quantities are much larger for frames with slender braces, and this has to
do with the method of strength estimation enforced
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in EC8, in which the strength of a brace pair is estimated based on the brace buckling strength.
2 A new method of strength estimation was proposed, in which the strength of a chevron brace
pair is estimated as the sum of the yield strength
of the brace in tension and the post-buckling
strength of the brace in compression. The proposed
method promises that the designed chevron brace
frames will have identical elastic stiffness, similar
actual strength, and therefore similar seismic performance regardless of the brace slenderness.
3 Two alternatives for the estimation of brace postbuckling strength, one based on the buckling
strength and the other based on the yield strength,
were explored. Both alternatives supply similar
performance. The one based on the yield strength
is found to be easier to handle in design.
REFERENCES
AISC 2002. Seismic provisions for structural steel buildings.
America Institute of Steel Construction, Chicago.
BCJ 1997. Structural provisions for building structures – 1997
Edition. Building Center of Japan, Tokyo. (in Japanese)
CEN 1993. EuroCode 3: Design of steel structures – Part 1-1:
General rules and rules for buildings, ENV 1993-1-1.
European Committee for Standardization, Bruxelles.
CEN 2003. Draft n. 6 of EuroCode 8: Design provisions for
earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. European Committee for Standardization, Bruxelles.
ICBO 1997. Uniform Building Code. International Conference
of Building Officials, Whittier.
Nakashima, M. and Wakabayashi, M. 1992. Analysis and design of steel braced frames in building structures. In Stability and ductility of steel structures under cyclic loading. Edited by Fukumoto, Y. and Lee, G.C. CRC Press, Boca
Raton, Fla.; 309–321.
Nonaka, T. 1973. An Elastic-plastic Analysis of a Bar under
Repeated Axial Loading. Int. Journal of Solid Structures; 9:
569–580.
Ogawa, K and Tada, M. 1994. Computer program for static and
dynamic analysis of steel frames considering the deformation of joint panel. In: Proceedings of the Seventeenth Symposium on Computer Technology of Information, Systems
and Applications, AIJ,. (in Japanese)
Somerville, P. et al. 1997. Development of ground motion time
histories for phase 2 of the FEMA/Sac steel project. SAC
Background Document. Report No. SAC/BD-99-03, SAC
Joint Venture, 555 University Ave., Sacramento.
Tada, M. and Suito, A. 1998. Static and dynamic post-buckling
behavior of truss structures. Engineering Structures; 20(46): 384–389.
Tremblay, R. 2001. Seismic Behavior and Design of Concentrically Braced Frames. Engineering Journal, AISC; 38(3):
148–166.
Tremblay, R. Robert, N. 2001. Seismic performances of lowand medium-rise chevron braced steel frames. Canadian
Journal of Civil Engineering; 28; 699–714.
Tremblay, R. 2002. Inelastic seismic response of steel bracing
members. Journal of Constructional Steel Research; 58:
665–701.
Tremblay, R. 2003. Achieving a Stable Inelastic Seismic Response for Multi-Story Concentrically Braced Steel Frames.
Engineering Journal, AISC; 40(2); 111–129.