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.