Experimental data on single angle compression members with single- and double-bolt end connection patterns is used to develop adjustment factors for effective buckling lengths. Connection length and clamping effect parameters are proposed and evaluated based on test data. Results from a total of 24 equal leg test angles with single- and double- bolts are used. Slenderness ratios considered ranged from 158 to 312. Angle sizes ranged from 1½” x 1½” x⅛” to 3½” x 3½” x ¼”. Estimated connection lengths were about 12.5% of the member length for single-bolt joints and24.7% for double-bolt joints. Computed effective length factors ranged from 0.753 to 0.875. Results seem to indicate that it is possible to define and calculate connection length of a lattice tower angle member. Suggestions for incorporating connection length issues in routine designs are made.
Structural steel angle members are the basic load-carrying elements in electrical transmission towers. These members are usually connected by gusset plates or directly bolted to other members through one leg. Tower joints involve single, double, triple or multiple bolted angles; all bolts are installed snug-
tight without any pretension. The buckling or compressive strength of these angles is a function of several parameters: bi-axial eccentricity of the loads (Figure 1), magnitude of restraint provided at the ends, slenderness ratios and pattern of failure either through flexural buckling or combined torsional-flexural buckling .
Theoretically, an angle strut is a restrained, eccentricallyandbi-
axially loaded, thin-walled beam-column whose buckling
strength is governed by end joint stiffness, and thereby, the
effective member length for buckling. However, the exact nature
of end restraint effects in bolted angle columns is too complex
to assess . Previous studies on bolted angles indicate that end
connection effects are noticeable even for a single-bolted joint
and that end restraints play an important role for members with
slenderness ratios over 120 . It was also shown that for slender
2- and 3- bolted angles, test buckling loads are consistently above
theoretical values .
ASCE Standard 10-15  gives the current design procedure
for tower angles. It defines effective slenderness in two categories: short columns ( L/r ≤120 ), controlled by eccentricity of loads;
and long columns (120 ≤ L/r ≤ 250), controlled by end restraint.
(See Notation for definition of parameters). Since the angle is a
thin-walled open section, various limits on width-to-thickness
ratios of angles are also defined, which in turn control the design
compressive stress of the angle column. (See Appendix A for
In general, laboratory tests of angles are focused on
determining the failure or buckling load; little effort is directed
towards assessing the actual connection length parameter. Part of
the difficulty is in defining what exactly constitutes the connection
or how far the clamping effect of bolting extends from the bolt(s).
One relatively recent study  proposed the concept of ‘equivalent
reference length’ or ‘characteristic length’ of a connection that
allows a comparison of the connection with the connected
member. Such knowledge of connection length can help provide a
basis for quantifying its translational and rotational stiffness and
eventually develop the stiffness matrix of the connection element
for inclusion into finite element analysis programs. This paper
attempts to address that issue.
The objective of this paper is to:
a. Propose a definition for an angle column connection
length including bolt clamping effect for 1- and 2-bolt angles,
b. Utilize test data on single and double bolted angles to
derive effective length factors for buckling,
c. Relate the effective length factors to connection lengths,
d. Propose adjustment to ASCE equations for slenderness
to give more accurate buckling capacities.
Only elastic buckling of equal-leg single angles with identical
end connections is considered in this paper. Residual stresses
and initial imperfections, although often important from buckling
perspectives, are not considered here.
The clamping effect of the bolts in a connection is known to
extend over a finite distance from the member end. Figure 2 shows
the idealization of connection and its length Lc as proposed in this
study and the assumed extent of the clamping zone. Each strut
of length L is taken to consist of identical end connections. It is
obvious that connection length and the clamping effect increases
with number of bolts, thus decreasing the effective member
buckling length. The effective beam-column length LE is taken as
center-to-center of connections. From Figure 2:.
The parameter Ke is an effective slenderness coefficient which
quantifies the influence of connection length Lc and the clamping
A minimum connection length LCM can be also defined just in
terms of the end distance ‘e’ and bolt spacing ‘s’.
Traditionally, angles with a single bolt are not considered to
provide any rotational restraint  and are treated as pin-ended
columns. In reality, members with more than one bolt in the end
connection scan be considered as those with intermediate level
restraint. The effective slenderness coefficient Ke in this case falls
in-between those of the pin-ended case and fixed case. Since there
are no guidelines to use for 3- bolt situations and beyond, the
associated coefficient should be determined empirically from test
data. Some previous studies  indicate that member behavior
approaches that of a fixed-ended column (k≈ 0.50) as the number
of bolts in the end connections is increased. It is generally seen
that this situation occurs when the joint is fully welded or when
the number of bolts approaches 5.
Test data on twenty four (24) single, equal-leg angles is
selected from published literature [7-9]. The test angles chosen
for this study ranged from 1½” x 1½” x ⅛” to 3½” x 3½” x
¼”.Slenderness ratios ranged from 150 to 312 (elastic buckling).
Yield strength of steel varied from 36.1ksi (249MPa) to 46.7ksi
(322MPa). Bolts used were ⅝” (15.9mm) in diameter with all bolt
holes sized 11/16” (17.5mm). All angles were tested in a manner
that simulates the actual joint situation in a lattice tower (i.e.)
unrestrained rotation in space. The testing setup also ensured
that load is applied at an eccentricity as in a real tower. For details
of the testing machine, instrumentation, loading process etc. the
reader is referred to the paper cited under Reference .
Table 1 & 2 shows the results of calculations for effective
slenderness coefficient Ke and connection length LC for one-bolt
and two-bolt angles, respectively. The minimum connection length
LCM is also calculated and shownin terms of the end distance ‘e’ and
bolt spacing ‘s’ of the test specimens. Figure 3 shows the variation
of Ke with the number of bolts in the end connection. Assuming
fixity to correspond with 5 bolts and Ke = 0.50, the Ke values for
3- and 4-bolt connections are approximately extrapolated to be
0.680 and 0.610, respectively. These two data points are shown
with a triangle symbol on the plot.
As seen in Table 1 & 2, and in Figure 3, the effective slenderness
coefficients derived from test results are consistently less than
1.00. As anticipated, the value of Ke decreased as the number
of bolts increased. The value obtained for the single-bolt case
is 0.875 instead of the usually assumed k =1.0 ; which means,
contrary to assumptions that such a joint does not provide any
restraint, there is a certain clamping effect associated with the
bolt. For the two-bolt case, the Ke value is 0.753 which indicates
a larger restraint than that of a single-bolt.
Connection lengths LC also proportionately increased as the
number of bolts increased. From Tables 1 & 2, it can be seen that
the connection lengths derived from test results are consistently
non-zero and clearly indicate a clamping effect. The average value
obtained for the single-bolt case is 0.125; which means, 12.5% of
member length L and indicating a finite clamping effect associated
with the bolt. For the two-bolt case, the average LC value is 0.247
or 24.7% of member length L, at each end. This indicates a
larger clamping effect than a single-bolt. This observation can be
extended to connections with 3 and 4 bolts as well.
In comparison with the minimum connection length LCM
defined in terms of end distance and bolt spacing, the actual
connection lengths were between 4 to 6 times that of the value
of LCM .
To verify if the computed effective slenderness coefficients
Ke can be used to obtain more accurate design capacities, the
parameter is applied to two test angles selected from the set used
in this study. The capacity of the two angles (one single-bolted and
one double-bolted) is computed using the ASCE 10-15 procedure
and then adjusted using the effective slenderness parameter Ke.
Appendix B shows the calculations. In both cases, the angles were
first assumed as having no restraint at ends (Equation 3a) and
then the slenderness is modified with Ke . Results show that the
adjusted design capacities are very close to the test loads.
In the preceding sections, a definition for the connection
length of an angle beam-column in a lattice tower is proposed
where the clamping effect of the bolts is quantified. Test data on
single-bolt and double-bolt angles is utilized to develop simple
expressions for effective length factors. Connection lengths were
determined from the calculated effective slenderness coefficient.
The validity of making a simple modification to ASCE design
equations using ke is examined.
Although modest, this study showed that it is possible to
define and determine connection effects in angle columns in lattice
transmission towers using carefully measured test data as a basis.
Only a limited number of angle sections and slenderness ratios
are studied in this paper. The results reported in this study are by
no means exhaustive and further studies are warranted before the
concepts discussed here can be generalized. Future investigations
may include a larger database of test results, encompassing
moreangle sizes, unequal leg angles, slenderness levels, and
effects of bolt size and connection geometries. An effort in any of
those directions will be a worthwhile undertaking whose goal is
to prescribe a more rational basis to the issue of quantifying end
restraint effects in transmission towers and thereby more robust