steelwise
Working With Single-Angle Members
By Amanuel Gebremeskel, P.E.
The inherent eccentricities of this popular shape require
the engineer’s attention and understanding.
Angles have been used in construction almost as
long as structural steel has been around, and were commonly used as components of built-up shapes. For example,
Bethlehem Steel made I-shaped members and channels
using angles attached to plates. Other producers used them
to build similar cross sections and other more exotic shapes.
More recently, angles have been used as braces, tension
members, struts and lintels. Angles also have been used in
double-angle and single-angle connections.
In spite of their long history of usage, the design of members composed of angles—and single angles in particular—
has not become as familiar to the engineering profession as
the design of other, more common shapes. This article highlights the information available today to help in this regard.
The AISC Specification
AISC first published a single-angle specification in the
1980s. Since then more research and testing has helped
to develop the knowledge base upon which single-angle
design is covered in the 2005 AISC Specification (and the
soon-to-be-released 2010 AISC Specification).
The current approach to single-angle design offers two
alternatives:
1. A comprehensive design approach that can be used to
design any single angle for axial and/or flexural loads.
This approach is more general and involves more effort
in calculations that typically are based upon the principal axes.
2. A simplified design approach that can be used with
greater expediency for specific common cases. Although
limited in scope, it allows an easier design process.
Amanuel Gebremeskel, P.E., is a
senior engineer in the AISC Steel
Solutions Center and secretary of the
AISC Committee on Specifications’
Task Committee 5, Composite Design.
MODERN STEEL CONSTRUCTION october 2010
For axial compression in angles without slender elements,
comprehensive analysis and design of single angles can be carried out using the provisions of Section E3, whereas a simplified design approach is provided for special cases in Section E5.
Table 4-11 in the 13th Edition AISC Steel Construction Manual
applies to the design of single angles for concentric axial loads.
For flexure without slender elements, the comprehensive
approach is provided in Section F10.2, with subsections (iii)
and (iv), while the simplified approach is provided in Section
F10.2, with subsections (i) and (ii). Local buckling and slenderness are addressed in Sections E7 and F10.3 for compression and flexure, respectively.
Single angles also may be loaded in combined axial force
and flexural. These are designed according to Section H2, and
the design of single angles with typical end connection configurations that result in eccentric axial loads is addressed in
Table 4-12 in the 13th Edition AISC Manual. These can be
used as design aids for single angles with combined loading
due to end attachments to one leg alone as described in the
explanation of the table on page 4-7 of the Manual.
Principal Axes
The principal axes of any shape define two orthogonal axes
that correspond to the maximum and minimum moments of
inertia for that section. The axis around which one finds the
minimum moment of inertia is called the minor principal axis
while the axis about which one finds the maximum moment
of inertia is called the major principal axis. From a structural
analysis point of view, bending the section about the minor
principal axis corresponds with the minimum internal energy
of the member. This means the structure is completely stable
when bent about this axis and cannot experience lateral-torsional buckling.
Unlike singly and doubly symmetric wide-flanges and
channels, single angles have principal axes that do not coincide
with their geometric axes (see Figure 1). Therefore, the design
of single angles requires some consideration of both of these
sets of axes. While loading typically occurs about the geometric axes, the strength usually is controlled by response that is
influenced by properties that relate to the principal axes.
Part 1 of the AISC Manual contains properties of single
angles about both geometric axes (X and Y) and the minor principal axis (Z). Part 17 of the AISC Manual contains equations
that allow for the calculation of section properties about one
axis when the properties are known about the other.
1
2
R
R1
R1 R
R1
R1
1
11
2
2
2
ANGLE
ANGLE
ANGLE
ANGLE
Axis
of moments
momentsthrough
through
Axis of
Axis ofcenter
moments
through
gravity
Axis of
moments
through
center
ofofgravity
center of gravity
centerb of gravity
b
b a
t
a
t
a
t
b
Z
Z t
Z
Y a
Y
Y
Z
W
Y 90˚
W
c
d
W
90˚
c
d
90˚ θ
c
d
X
X
θ
W
θ
X
X
90°
c
X
X y
d
AISC-Sample (LRFD) y May
20:34
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θ
Xy W
X
W
x
t
y
x
t
Z
W
x
t
Y
Z
Z
Y
xY
F10.]
SINGLE
ANGLES
t
Fig. 1: Geometric Y(X andZ Y) axes and principal
A(6R − a )
+ 2a )
A(12R
21
2a )
2 22− a2 ) =A(12R
I 2(6=RA(6R
I1 = A
− a2 −
) a2A
R 12 +1248
a+
+ )a2 )
) (=12A(12R
I = A(6R 24
1
I1 =II11I2=
== I22 = � 24 = = �
48
24
48
48
� 6R24
2 − a2 �
� 12R 2 + a2
�
r 1 = r 2 = 6R 22 − a22 = 12R 122 1+ a22
+a 2
−2a = 12R 1248
6R
24
2
r1 = r2 =
a
r = r = 6 R − 24
= 12 R48
1 + a
24 =
48
r1 = r12 = 2
 √
√

24
48
2K
2=K 2K
tan =2θImportant
Other
Section Properties
tan
2θ
tan
2θ = − II2K
y − Ix
tan 2θ I=
y I yx− I x
2
2 + at
If the evaluation
of the moment
of inerI y − Ix
b
2 + ct y = d
2
A = t(b + c)x =b
b 22 +
+ ct
ct
d 22d++atat
b2(b
++ctc)yy=
d2(b
++atc)
A t=(bt(b
A =single
cangles
x ==
=
++
)c)x
tia of
about
the
principal
axes
A = t(b
+
c)x
= 2(b
y
=
2(b
c)
(b +
+ c)
cabout
)
2(+
b
+ cYY
)
K = Product of22(b
Inertia
XX
and
+
c)
2(b
+
c)
K = Product
of Inertia
about XX of
andthe
YY section
is important,
the
evaluation
abcdt
K
=
Product
of
Inertia
about
XX
and
YY
K = Product
of Inertia about X X and Y Y
= ±
abcdt
abcdt
4(b the
+ c) same axes is even more
±
moduli ==about
abcdt
+ c)
= ± ±14(b
4(b + c) 3
3 − a(y − t)3 )
=
+itbyis
I
4(
b
+ c)− y)
1 (t(d
useful.I Additionally,
important to recogx
1
3
= (t(d − y)33 + by 33 − a(y − t)33 )
+ by3 − a(y − t) )3
I xx 1= 3 (t(d − y)
3 3 angle
1
nizeIxthat
the
single
3
(t1(d(t(b
=I y =
− y−) x)+ by
−3 −
a ccan
(y(x −−thave
) 3)) as many
+
dx
t)
I y 3= 13(t(b − x)33 + dx 33 − c (x − t)33 )
as three
moduli
about
one
axis. For
(t(b
−
x)
+
dx
−
c
(x
−
t)
)
I y =section
1 33I x sin2 θ3+ I y cos
3 2 θ + K sin 2θ
2−
(tI x(bsinangles
−22 x
dx
ccorrespond
(xsin
− 2θ
t) 3 )
IY =I zI z==
θ )+ I+y cos
θ+
K
2
unequal-leg
two
to the
I z 3= I x sin θ2 + I y cos 2θ + K sin 2θ
θ + I y sin
I w = I x2 cos
22 θ − K sin 2θ
2+θwhile
=
I xlegs
cos
+
Icos
sinone
θ−
K
sin
2θ θto the heel.
I=w Ithe
toesIzof
relates
sin
θ
I
θ
+
K
sin2
y
2
2
x
Y
I = I x cos θ + I y sin θ − K sin 2θ
K iswnegative
2 when heel of2angle, ith respect to
Ixofcos
θ + heel
IinYunequal-leg
sin
θ − ith
K sin2
θpostitive
Iwis =negative
When
evaluating
single
Kcenter
when
ofor
angle,
respect
to angles
gravity,
1st
3rd quadrant,
K is negative
whenisheel
of angle,
ith respect
to
of
gravity,
is inheel
1st orof3rd
quadrant,
Kfor
iscenter
negative
when
angle,
withpostitive
respect
to
when
in
2nd
or 4th
center
of gravity,
is axial
inquadrant.
1st or and
3rd
quadrant,
postitive
combined
flexural
loading,
whenofingravity,
2nd or 4th quadrant.
center
in 1st or 3rd quadrant, positive
when in 2nd or 4thisquadrant.
this can
make
the
calculation quite lengthy.
when
in16.1-281
2nd
or 4th
quadrant.
Several articles published in AISC’s
(W and Z) axes of single angle.
The horizontal component of deflection being approximately 60 Engineering
percent of the Journal provide further insight
vertical deflection means that the lateral restraining force required
intoto achieve
working with single-angle mempurely verticalThe
deflection
must
be
60
percent
of
the
applied
load
value
(or
produce
bers:
“Evaluating
Single-Angle Compresimportance
of
evaluating
secBEAMS
AND
CHANNELS
BEAMS
ANDCHANNELS
CHANNELS
BEAMS
AND
a moment 60
percent
of the
applied
which
is very axes
significant.
BEAMS
ANDvalue)
CHANNELS
sion Struts Using an Effective Slenderness
tion
properties
about
the
principal
Transverse
force
oblique
Transverse
force
oblique
through
center
of
gravity
Transverse
force
oblique
Transverse
force
Approach,”
for single
angles
iscenter
illustrated
inand
Figure
2. Leigh
through
gravity
Lateral-torsional
buckling
isthrough
limited
byof
Moblique
Lay, 1978;
and Lay,by Leroy A. Lutz (4th Quarter
e (Leigh
center
of
gravity
through
center
of
gravity
F
F about
2006),
“Towards the Simplified Design of
1984) in Equation
F10-4a,
whichangle
is based
on is bent
Consider
a Fsingle
that
F
F
F
Y
Ybraced
2 2 Columns,” by ChrisSingle-Angle
F axis and not
the geometric
F against
I
=
I x2 sin
I y cos
Y
Y
4
I
=
I
sin
θ2 2+ θIY+Beam
cos
θθ
3 3x
xY
2.33Eb txY
I 3 = I x sin
θ + I y cos222 2θ
2 2
xthan
x Y deformation
I xIJ.
sin
θ2++
I ysin
cos
Mcr =
× the ends. topher
and
lateral
other
θ
+
I
sin
I4 I=3I 4I=
cos
θEarls
I
θθ θ D. Christian Keelor
x Y 2 at
x cos
y
x
x=
Y
2
2
� 22 θ + I y sin 2 θ
�
θ (1 + 3 cos �)(K
θ
I 4 = I x cos
x l)
x
I
=
I
cos
θ
+
I
sin
θ � “Design of Single
y
x
x
y
�
4
θ
θ

(1st
Quarter
2007),
As
it tends to naturally
X
X�
f b = yM� y sin θx+x and
cos �
θ
θ
��the beam θis loaded,


y
x
I
I
X
f
=
M
sin
+
cos
θ
θ
f
=
M
sin
θ
+
cos
θ
x
y
θ XX
θ
b
X
fbb = M
sin θ +
cos Major
θ
t2
0.156(1
+ 3 cos2of
Angles
Bent
Principal Axis,”
I x About
IxM is
IY IImoment
deflect
direction
thel)2load.
Howythe
3�)(K
3 2 in the
X3
I x bending
where
due to force F.
y
+ sin �3X3
(C-F10-1)
3 sin � +
3
where
bendingJ.
moment
due
to force
F.
where
MM
isis
moment
due
to force
F.
3 it also tends to
3
3 deflect
b4 33 in the direction
by
Christopher
Earls.
All
are
available
at
where
M
isbending
bending moment
due
to force
F.
ever
3
3
3
3
X
X
www.aisc.org/epubs
as
free
downloads
to
of least
resistance,
which
corresponds
with
X
X
X
X
y
y
X y
X
y
(the general
for the
critical
of an Yequal-leg
angle)members
with
AISC
and may be purchased by
theexpression
minor
principal
axis. moment
y
Y
y
y
Y
Y
y stress is compressive
� = 45◦ or theThis
condition
where
the
angle
tip
(see Figure
Y that
Y in a total deflection
others.
results
4
4
Y
Y
4
4 the flexural
C-F10.3). occurs
Lateral-torsional
buckling can
limit
strength of the
4geometric
in 4the
direction
of also
both
4
4
cross section when the maximum angle tip stress is tensile from geometric axis
axes. For such cases it is difficult to evaluate Another Reference
flexure, especially with use of the flexural strength limits in Section F10.2. Using
In addition to the information available in
◦ first yield or the propensity of the member
� = 45 in Equation C-F10-1, the resulting expression is Equation F10-4b with
the
Specification and Manual, Whitney
to oflaterally
resolving
theOF STEELAISC
AMERICAN
INSTITUTE
CONSTRUCTION
a +1 instead
−1 as thebuckle
last term.without
A
IINSTITUTE
OF STEEL CONSTRUCTION
AMERICAN
MERICAN
NSTITUTE
OF STEEL
CONSTRUCTION
McNulty,
P.E., recently self-published a guide
load and response into
components
that
Stress at the tip of the angle leg parallel to the applied bending to
is of the design called the Single-Angle
single-angle
are parallel to the principal axes. Some- axis
same sign as the maximum stress at the tip of the other leg when the single angle
thing similar can be said of an axially loaded Design Manual. It is devoted to the specifics
is unrestrained. For an equal-leg angle this stress is about one-third of the maxof the design of angles and has chapters that
single
angle.
Its tendency
in Euler
imum stress.
It is only
necessary
to check to
the fail
nominal
bending strength based
get
flexural
buckling
will
be
about
the
axis
of
on the tip of the angle leg with the maximum stress when evaluatinginto
suchthe
an details of equal-leg and unequalleg combined
single angles in tension, shear, compresleast
which corresponds
with the
angle. Since
thisresistance
maximum moment
per Section F10.2(ii)
represents
andthese
flexure (including interaction). The
minor
principal
axis. F10-5 represents the design sion,
principal axis
moments
and Equation
limit for
interested reader can find this reference at
www.lulu.com/singleangle.
Conclusion
The design of single angles is more complicated than that of other more common
shapes. Nonetheless, the versatility of single
angles in construction has made them popular. Provisions and recommendations exist
in the AISC Specification, AISC Manual, and
other references to assist the engineer who
wants to design single angles. Fig. C-F10.2. Geometric axis bending of laterally unrestrained equal-leg angles.
Fig. 2: Deflection of single angle due to load
about
geometric
axis.Steel Buildings, March 9, 2005
Specification
for Structural
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
october 2010 MODERN STEEL CONSTRUCTION