2017 SEAOC CONVENTION PROCEEDINGS
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Design of Chevron Gusset Plates
Rafael Sabelli, Director of Seismic Design Walter P Moore
San Francisco, California Leigh Arber, Senior Engineer
American Institute of Steel Construction Chicago, Illinois
Abstract The “Chevron Effect” is a term used to describe local beam
forces in the gusset region of a chevron (also termed inverted-
V) braced frame. These local forces are typically missed by
beam analysis methods that neglect connection dimensions.
Recent publications have shown how to correctly analyze for
these forces (Fortney & Thornton, AISC Engineering Journal,
Vol. 52, 2015). This study adds design solutions for addressing
high shears in the connection region, including reinforcement,
proportioning, and innovative detailing.
Introduction
Chevron (also termed inverted-V) braced frames are
commonly used in steel structures. In these frames two braces
connect to the beam midpoint. Typically the braces are below
the beam, forming an inverted “V,” although they may be
above, forming a “V,” or both above and below, forming a
two-story “X” with the beam at the center. Figure 1 shows
these configurations.
Fig. 1. Chevron braced frame configurations.
These frames are typically designed using centerline models,
and the beam forces and brace forces are in equilibrium at the
center connection. In typical design, a substantial gusset plate
is provided at the center, and force transfer between braces and
beams is accomplished over the length of the gusset plate.
Figure 2 shows such a gusset plate.
Fig. 2. Typical chevron gusset design.
Recent work by Fortney and Thornton (Fortney and Thornton,
2015) highlights the importance of careful connection analysis
in order to determine the local stresses induced by the gusset
connection in a chevron braced frame. In particular, Fortney
and Thornton derive expression for the local moments and
shears that result from distribution of brace forces over the
gusset-plate length. These forces can result in the need to
supplement the beam web with a doubler plate. An example of
such a condition is shown in the AISC Seismic Design Manual
(AISC, 2012).
This study applies the same concepts investigated by Fortney
and Thornton, but with the aim of providing engineers with
design equations to enable the selection of beams that do not
require reinforcement.
Consistent with ductile design of braced frames, it is assumed
that braces apply loads to the beams and do not provide
support. These forces are typically equal to the capacity of the
braces in the design of ductile systems, but the design
equations derived here are equally applicable to chevron
frames designed for wind or other cases that do not involve
capacity design.
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Symbols, Nomenclature, and Conventions
This study employs the following symbols and terms:
Hbel = Horizontal component of brace forces (braces below).
Lg = Gusset length.
Lbeam = Beam length.
Mbel = Moment at beam-to-gusset-interface due to brace
forces (braces below).
N = Concentrated force at beam flange.
P1 = Left-hand (lower) brace. Tension is positive.
P2 = Right-hand (lower) brace. Compression is positive.
P3 = Left-hand (upper) brace. Compression is positive.
P4 = Right-hand (upper) brace. Tension is positive.
Ru = Required strength.
Vb = Maximum beam shear (within connection region) due
to brace forces.
Vba = Beam shear (outside connection region) due to brace
forces.
Vbel = Vertical component of brace forces (braces below).
Vef = Effective beam shear strength.
Vn = Nominal beam shear strength.
Vz = Beam shear from moment transfer (for concentrated-
stress approach).
a = Length of beam from support to gusset edge (equal
to half the difference between the beam length and
the gusset length).
d = Beam depth.
dg = Gusset depth.
eb = Eccentricity from beam flange to beam centerline,
equal to half the beam depth.
ez = Length of moment arm (for concentrated-stress
approach).
k = Distance from outer face of flange to web toe of
fillet.
tg = Gusset thickness.
tw = Beam web thickness.
x = Distance from gusset edge toward beam midpoint.
z = Length of concentrated stress region at ends of
gusset (for concentrated-stress approach).
φ = Resistance factor
θ = Brace angle from horizontal.
τ = Horizontal shear stress.
Figure 3 shows dimensions noted on beam and gusset-plate
diagrams.
Fig 3. Chevron gusset geometry.
Beam Forces For clarity, brace forces are separated into vertical and
horizontal components. Assuming two braces below with
forces P1 and P2, the horizontal component is:
( )1 2 cos= + θbelH P P
The vertical component is:
( )1 2 sin= − θbelV P P
These forces on the beam-to-gusset interface are statically
determined. In addition to these vertical and horizontal forces,
there is a moment (required for static equilibrium):
=bel bel bM H e
Figure 4 shows free-body diagrams of the gusset plate.
Fig. 4. Free body diagram of gusset plate
Uniform stress approach
Typically, the stresses at the beam-to-gusset interface are
assumed to be distributed uniformly using the full length for
the vertical and horizontal forces and a plastic-section-
modulus approach for the moment (Fortney and Thornton,
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2015). Following this approach, the shear within the
connection region is described by the following equation:
( ) ( ) ( ) ( )( )1 2 1 2 1 2
2
sin sin 4 cos
2
for 02
− θ − θ + θ= − +
≤ ≤
b
g g
g
P P P P P P eV x x x
L L
Lx
The maximum shear in the connection region occurs at the
beam midpoint and is equal to:
( ) ( )1 2
2
cos
=
+ θ=
belb
g
g
MV
L
P P d
L
This shear is not equal to the vertical component of either of
the brace forces (P1sinθ or P2sinθ); it may be greater or smaller
than those values, depending on the geometry of the
connection. The difference between the two is the shear
carried by the gusset, Vg, presented later.
Note that this beam shear, Vb, is due only to the horizontal
components. The unbalanced vertical component does cause
shear in the beam, but this shear becomes zero at the beam
midpoint. Figure 5 shows a shear diagram for brace-induced
shears in a typical pin-end beam. Fixed-end beams may have
a sway-induced shear at midspan. Also, in certain loading
conditions gravity loading may cause a non-zero shear at the
midpoint.
Fig. 5. Brace-induced shears in pin-end beam (uniform stress
approach).
The beam shear is the result of both the eccentricity (the beam
depth) and the gusset length. These can be adjusted (within
reason) to provide a beam that does not require web
strengthening. Following this approach, the minimum gusset
length is:
( )( )1 2
2
cos
≥φ
+ θ≥
φ
belg
n
n
ML
V
P P d
V
The longer the gusset plate, the greater the portion of shear that
remains in the gusset and the less that is transferred to the
beam. In this sense, the gusset plate can be used as external
shear reinforcement for the beam, although the degree of
reinforcement is limited by the connection geometry.
Note that selection of a shallower beam reduces the required
gusset length. For beams with small moments due to vertical
unbalanced forces it is often more economical to select a
shallow beam rather than a deeper beam that would either have
to be reinforced for shear or be heavier to preclude the need
for reinforcement.
The shear outside the connection region is due to the
unbalanced vertical components of the brace forces. The
moment Mbel due to the brace horizontal components produces
no shear outside of the connection region.
1
2=ba belV V
Beam moments are described by the following equation:
( ) ( )
( ) ( ) ( )
21 2
1 2 1 2 2
cos 2
sin sin
2 2
+ θ= − +
− θ − θ+ −
bg g
g
P P xM x e x
L L
P P P Px a x
L
A simplified equation can be used to provide a liberal estimate
of the maximum brace-induced moment:
4 8
bel beam belb
V L MM ≤ +
This equation combines two maxima that do not occur in the
same location, and neglects an offsetting term.
The second component of this moment (which is a local effect
of the connection geometry) is typically small, but may be the
governing moment in cases with no unbalanced vertical force
from the braces. Gravity moments are typically at a maximum
at the beam midspan and should be combined with these.
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Concentrated stress approach
As noted above, the beam forces are a result of the assumed
stress distribution on the gusset-to-beam interface. The beam
shear may be reduced by increasing the moment arm over
which the moment is divided. Instead of increasing the gusset
length, however, the concentrated stress approach maximizes
the moment arm within a given gusset length. In this approach,
the moment (due to the braces’ horizontal components) is
assumed to be transferred at the ends of the gusset over lengths
z. The unbalanced force (due to the braces’ vertical
components) is transferred in the remaining center portion of
the gusset, between the end regions. Figure shows such a
stress distribution.
Fig. 6. Stress distribution for concentrated stress approach.
Figure 7 shows a shear diagram corresponding to this stress
distribution.
Fig. 7. Brace-induced shears in pin-end beam with non-
uniform stress distribution at connection.
The moment arm ez is:
= −z ge L z
The shear from the moment transfer is thus:
= belz
z
MV
e
Note that in this case the maximum shear does not occur at the
beam midpoint. It is a combination of the shear due to the
unbalanced force and the shear due to delivery of the moment.
The maximum shear is given by the following equation:
12= +b bel zV V V
This shear may be set less than or equal to the design shear
strength of the beam in order to preclude the need for shear
reinforcement. For a given gusset length the maximum
moment transfer can be achieved by the highest concentration
of stress at the ends. At a maximum, stiffeners at the gusset
edges and within the throat of the beam may be used to create
a moment arm equal to the gusset length Lg. Short of that, the
concentrated stress may be limited by the web tensile strength
or the gusset strength. (Typically it is the former.)
Assuming the gusset length is optimized, the concentrated
stress will be maximized such that the full beam shear strength
is utilized. Considering that some of the beam shear strength
is utilized in resisting the unbalanced force, the remaining
beam shear strength that can be utilized for the moment
transfer is:
12= φ −ef n belV V V
Considering these limits the minimum length over which this
force can be transferred by the gusset is:
ef
y g
Vz
F t≥
φ
The minimum length over which this force can be transferred
by the beam in web local yielding (AISC Specification Section
J10.2) is:
5ef
y w
Vz k
F t≥ −
φ
For simplicity, the latter term may be neglected. Hereafter it is
assumed that the beam web is the limiting factor. This method
sets the beam required shear strength equal to its design
strength:
1
2= += φ
b ef bel
n
V V V
V
The corresponding moment arm is:
≥ belz
ef
Me
V
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The corresponding minimum gusset length is:
( ) ( ) 11 22
12
cos2
g z
efbel
ef y w
n bel
n bel y w
L e z
VM
V F t
dP P V V
V V F t
≥ +
= +φ
+ θ φ −≥ +
φ − φ
For gusset lengths greater than this value, the length z is:
2
2 4
g g bel
y w
L L Mz
F t= − −
φ
For the case with stiffeners used to transfer the vertical force
Vef, the second term becomes zero. Shorter gussets may be
used, but only if Vef is increased (e.g, the beam is reinforced).
Once again, note that using a shallower beam can be effective
in reducing the required gusset length.
Local beam limit states such as web crippling should be
evaluated. The concentrated force to be considered is:
12u n belR V V= φ −
or
=−bel
ug
MR
L z
The bearing length may be taken equal to z.
Beam Moment The approach to beam moment described earlier assumes that
the vertical stresses are as shown in Figure 6, and that the
horizontal stresses are uniform and equal to:
( )1 2 cos
g g
P P
L t
+ θτ =
Fortney and Thornton (2015) describe conditions in which the
beam moment determined using these assumptions (if not
considered in design) may necessitate reinforcement. In this
study the authors propose an alternative approach to beam
moment employing the lower-bound theorem to allow an
alternative (non-uniform) shear stress distribution and thereby
demonstrate the adequacy of the beam. In this approach is the
horizontal shear stress at each point x along the gusset length
is determined such that its effect on moment (due to
eccentricity from the beam centerline) negates the incremental
moment due to vertical shear:
( ) ( )
( )( )
b
b
e x dx V x dx
V xx
e
τ =
τ =
This horizontal stress distribution results in zero moment due
to connection forces. This stress should be considered in gusset
analysis and in the weld sizing. For the uniform-stress
approach (with non-uniform horizontal shear) the maximum
shear stress is:
( )( )1 2
max
cos
g g
P P d
L t
+ θτ =
This maximum stress occurs at the beam midpoint.
For the concentrated-stress approach the maximum horizontal
shear stress (corresponding to the shear-stress distribution that
results in zero moment) is:
max2 n
g
Vdt
φτ =
This stress is quite high and thus negating the moment is not
generally a suitable approach if the concentrated-stress
approach is used to limit beam shear. In such cases the
horizontal stress may be assumed to be transferred over the
length ez. The shear stress would thus be:
( )1 2 cos
z g
P P
e t
+ θτ =
In this case brace-induced moment in the beam would be:
2 4
ef bel beamb
V z V LM = +
The first term is the connection-induced moment and is at a
maximum at a point along the gusset a distance z from its edge.
The beam should be evaluated considering this moment in
combination with the axial force. Note that the accumulation
of axial force in the beam is a consequence of this horizontal
stress and is thus at a maximum at the location of maximum
moment.
Weld sizing
Under either the uniform-stress approach or the concentrated-
stress approach the weld adequacy should be evaluated using
AISC 360 methods, such as the instantaneous center of
rotation, which represents both weld strength and the limits on
weld ductility (assuming the weld connects rigid elements).
Forces across the gusset-to-beam interface are Hbel, Vbel, and
Mbel.
However, for designs employing the concentrated-stress
approach stresses may redistribute along the weld due to beam
inelasticity. Conformance with the design methods described
above indicates adequacy of the system under those
conditions. For the concentrated-stress approach the weld in
the zones z should be evaluated for the vertical force Vef.
Welds in the center region (Lg– 2z), the vertical force is Vbel and
the horizontal force is Hbel.
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Combinations of forces
The beam forces derived are for braces below with opposite
forces (one brace in tension and the other in compression).
These forces may be combined with gravity-induced forces in
the beam, and with shear due to flexural restraint for fixed-end
beams. While the diagrams show the left brace in tension and
the right brace in compression, forces corresponding to the
opposite case are easily determined by using negative values
for the brace forces.
For the two-story-X configuration brace induced shears and
moments will be additive for the typical case in which the
story shears are in the same direction. Gusset plates may be of
different lengths above and below, but for simplicity they may
be set to be equal. The modified equation for minimum gusset-
plate length to preclude the need for reinforcement is:
( )( ) 11 2 3 42
12
cos2
+ + + θ φ −≥ +
φ − φn bel
gn bel y
dP P P P V VL
V V F t
The vertical unbalance force includes the effects of all braces.
Gusset forces
Statics require that certain forces be transferred across the
midpoint of the gusset. These forces are:
( )1
1 22
12
cos= − θ=
g
g g g
H P P
M d H
For the uniform-stress method:
( )11 22 sin= + θ −g bV P P V
For the concentrated-stress method:
( )11 22 sin= + θ −g efV P P V
Conclusions
This study provides design equations that can be used in the
selection of beams in chevron braced frames that will have
sufficient shear strength without the need for web
reinforcement. The design method allows engineers to use the
gusset plate as external reinforcement for the beam web. These
equations can be used to assess the effects of beam depth and
gusset length on the beam shear demand in order to optimize
beam selection.
Dr. Paul Richards of Brigham Young University is currently
investigating the chevron effect through inelastic finite
element analysis. His work will study the distribution of shear
between the beam and the gusset, the impact of gusset yielding
on the connection region, and the behavior of existing frames
which were designed without consideration of the chevron
effect. The research project will conclude in 2018.
References
AISC (2016). Specification for Structural Steel Buildings,
ANSI/AISC 360-16, American Institute of Steel
Construction, Chicago, IL, July 7.
AISC (2012), Seismic Design Manual, 2nd ed., American
Institute of Steel Construction, Chicago, IL.
Fortney, Patrick J. and William A. Thornton. (2015), “The
Chevron Effect – Not an Isolated Problem.” AISC
Engineering Journal, 2015, Qtr 2.