Technical Note
Lateral–Torsional Buckling of Suspended I-Shape
Lifting Beams
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David Duerr, P.E., M.ASCE1
Abstract: The equations commonly used in practice for the design of beams are based on the assumption that the members are restrained
against lateral displacement and twist at the ends of the unbraced length. Beams suspended by wire rope or other flexible elements, such as
lifting beams, are not restrained as such. Thus, the buckling behavior of suspended beams can be expected to vary from that predicted by the
common design equations. This note examines the lateral–torsional buckling behavior of suspended I-shaped beams. A correction factor by
which existing beam-design equations can be modified to provide a practical means of accounting for the buckling behavior of suspended
beams is evaluated. The use of this correction factor for I-shaped beams has been tested by comparison to published experimental data.
DOI: 10.1061/(ASCE)SC.1943-5576.0000263. © 2015 American Society of Civil Engineers.
Author keywords: Allowable stress design; Beams; Buckling; Design; Lifting; Standards and codes.
Introduction
The equations most commonly used in practice for the design of
steel beams (e.g., AISC 2010) are based on the assumption that the
members are restrained against lateral displacement and twist at
the ends of the unbraced length. Beams that are suspended by wire
rope, chains, or other flexible elements, as is the case with beams
used for lifting, are not restrained as such. Thus, the buckling
behavior of suspended lifting beams can be expected to vary from
that predicted by the common design equations.
A standard applicable to the design of lifting and spreader
beams (ASME 2014) introduced in its 2011 edition a buckling
strength–correction factor, C LTB , to be used in the design of
I-shaped beams suspended from flexible elements, rather than
framed into a structure. The current edition refined the definition of
C LTB . Otherwise, the basic beam-design equations in ASME
(2014) are similar to those in AISC (1989) and AISC (2010). The
purpose of this note is to present the results of a finite-element
analysis (FEA) study of the lateral–torsional buckling of I-shaped
beams modeled to represent suspension supports. The validity of
the ASME (2014) correction factor is demonstrated by comparison
to the FEA results and to test data (Dux and Kitipornchai 1990).
Elastic Buckling of I-Shape Lifting Beams by FEA
Elastic-buckling analyses have been performed using BASP, an
FEA program developed at the University of Texas at Austin to
solve elastic buckling problems (the program’s name is an
acronym for buckling analysis of stiffened plates). The models
used for the buckling analyses of I-shaped beams are illustrated
in Fig. 1. Beam cross sections were developed with depths of
152 mm with flange widths of 76 and 152 mm; 305 mm with
1
President, 2DM Associates, Inc., 9235 Katy Freeway, Suite 350,
Houston, TX 77024-1526. E-mail: Duerr@2DM.us
Note. This manuscript was submitted on November 6, 2014; approved
on April 10, 2015; published online on June 10, 2015. Discussion period
open until November 10, 2015; separate discussions must be submitted for
individual papers. This technical note is part of the Practice Periodical
on Structural Design and Construction, © ASCE, ISSN 1084-0680/
06015001(4)/$25.00.
© ASCE
flange widths of 76, 102, 152, and 203 mm; and 610 mm with
flange widths of 152, 203, 254 and 305 mm. Flange and web
thicknesses were 9.5 mm for the beams of 152 mm depth and were
12.7 mm for all other shapes. Various lengths of each cross section
were modeled to provide a range of length/width ratios.
The model of the restrained beam [Fig. 1(a)] is consistent with
the condition normally found in building structures. The load is
applied downward at the top, whereas the ends are supported
vertically and are restrained laterally and torsionally. The model of
the suspended beam [Fig. 1(b)] is consistent with a lifting beam.
The beam is lifted from a midspan attachment at the top, and rigging to the payload is attached at the bottom at each end. The ends
of the beam as modeled are laterally restrained only at the bottom.
Fifty models were analyzed with both support and load configurations. Forty-two of the models used elastic properties for steel,
and the remaining eight models used elastic properties for aluminum. The difference in the buckling behavior of the suspended
beam relative to that of the restrained beam is quantified by the ratio
of the buckling loads calculated using BASP. These ratios are
plotted against an expression that relates cross-sectional properties
to the length (Fig. 2). At shorter spans, the correcting effect of the
applied load acting away from the centroid of the suspended beam
results in buckling loads that are greater than those computed for the
restrained beam. Apreduction
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the buckling load was observed at
values of (Lb / bf ) / EI x / GJ greater than about 1.6. The proportions
of these beams are in the elastic buckling range as defined in AISC
(2010) and ASME (2014). No difference in the ratio of the buckling
loads was observed between the steel and aluminum models.
ASME Correction Factor
The reduced buckling strength of a suspended beam can be
accounted for in design through the addition of a correction factor
to the standard design equations. ASME (2014) is an allowable
strength design standard in which the allowable major axis
bending stress for I-shaped beams in the elastic buckling range is
defined as the greater of the values given by Eqs. (1) and (2). The
correction factor C LTB is defined in Eq. (3)
Fbx = C LTB
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π 2 EC b
Fy
≤
N d (Lb / r T )2 N d
(1)
Pract. Period. Struct. Des. Constr.
Fbx = C LTB
C LTB =
0:66EC b
Fy
≤
N d (Lb d / Af ) N d
2:00(EI x / GJ)
+ 0:275 ≤ 1:00
(Lb / bf )2
(2)
(3)
The value of C LTB is taken as 1.00 for a beam braced against
twist or lateral displacement of the compression flange at the ends
of the unbraced length. The curve in Fig. 2 is a plot of Eq. (3).
The performance of the C LTB factor with respect to the
FEAs can be evaluated by means of the ratio Rc as defined by
Eq. (4)
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Rc =
Suspended beam BASP Pcr
C LTB (Restrained beam BASP Pcr )
(4)
The values of Rc computed for the FEAs have a mean of 1.03
and a coefficient of variation of 0.05 for those beams with C LTB
less than 1.00.
The consistency of the results given by the ASME (2014)
below-the-hook (BTH-1) equations [Eqs. (1)–(3)] is shown
graphically in Fig. 3. The upper portion of the graph compares
the buckling loads calculated using BASP with the buckling loads
calculated with the BTH-1 equations for beams with torsionally
restrained ends. The lower portion of the graph compares the
BASP and BTH-1 values for suspended beams.
Comparison of Results with Test Data
(a)
The performance of the C LTB factor can be checked against test
results of suspended I-shaped beams (Dux and Kitipornchai 1990).
(b)
Fig. 1. Models for beam-buckling analyses: (a) model of torsionally
restrained beam; (b) model of suspended beam
Fig. 3. Below-the-hook (BTH-1) buckling loads versus BASP
buckling loads
Fig. 2. Comparison of BASP buckling analysis results
© ASCE
Fig. 4. Lifting beam test configuration
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Test results for 22 suspended beams of the configuration shown
in Fig. 4 are reported, in which α > a in 8 tests, α < a in 10 tests, and
α = a in the remaining 4 tests. These beams have been
analyzed using the ASME (2014) equations with the design factor
N d equal to 1.00. The ratio of the test load at buckling to the
calculated load has a mean of 1.01 and a coefficient of variation
of 0.06.
An example solution using the ASME (2014) design method is
illustrated in Tables 1 and 2. Table 1 lists the dimensions, material
properties, and test load at buckling for one of the Dux and
Kitipornchai (1990) test specimens. Table 2 shows intermediate
and final calculation results for the check of this beam using
the ASME (2014) equations. In an actual design application, the
strength of the beam between the upper lift points must also be
checked. The addition of this check shows that the critical section
is at the upper lift points, as shown in Table 2. The additional
calculations are not shown here for brevity.
Other theoretical studies of beams with less than full torsional
restraints consider the behavior of suspended beams due to selfweight (Dux and Kitipornchai 1989; Essa and Kennedy 1993).
Essa and Kennedy (1993) also present a lifting beam analysis
solution in which the beam is supported vertically at each end
and loaded by a single point load midspan. Neither of these
Table 1. Example Test Beam Properties
Property
d
bf
Ix
J
E
G
L
α
a
a1
a2
θ
Unit weight
WExp
Value
75.2 mm (2.961 in.)
31.4 mm (1.236 in.)
329,809 mm4 (0:792 in:4 )
808 mm4 (0:002 in:4 )
64,825 kPa (9,402 ksi)
26,185 kPa (3,798 ksi)
5,000 mm (196.85 in.)
2,000 mm (78.74 in.)
1,000 mm (39.37 in.)
57.6 mm (2.268 in.)
46.6 mm (1.835 in.)
32.15°
8.53 N/m (0.58 lbs/ft)
81.4 N (18.30 lbs)
W
Beam self-weight
Upper cable tension,
Tuc
Horizontal
component of Tuc
α−a
Ma
Pa
f bx
Lb
CLTB
Fbx (from BTH-1
Eq. 3-17)
Interaction
ratio = f bx / Fbx
© ASCE
Value
81:4 / 2 = 40:7 N (9.15 lbs)
42.66 N (9.59 lbs)
116.57 N (26.21 lbs)
98.70 N (22.19 lbs)
1.0 m (39.37 in.)
40:7(1:0) + 8:53(1:5)2 / 2 = 50:298 N ⋅ m
(445.18 lbs-in.)
0:00 N ⋅ m (0.00 lbs-in.)
50,298(75:2 / 2)/ 329,809 = 5:734 MPa (832 psi)
2α = 4,000 mm (157.48 in.)
0.40
5.532 MPa (802 psi)
1.036
Comments on Practical Considerations
The study reported here, along with the experimental and theoretical studies reported in the referenced papers, clearly indicates
that the buckling strength of suspended I-shaped beams of certain
proportions is less than that of beams with laterally and torsionally
restrained supports. Because the common steel design equations
(e.g., AISC 1989 and earlier editions of ASME 2014) have been
used for the design of lifting beams for many years, the following
question may be asked: Why have there not been many buckling
failures of lifting beams?
The reduction in buckling strength of an I-shaped beam only
occurs in relatively long and slender beams. Practical aspects of
lifting beam design most likely have restricted beams in practice to
proportions that are stout enough that this loss of buckling strength
is not realized. Although the typical proportions and details of
suspended lifting beams may preclude buckling failures due to
reduced strength, it is still appropriate for designers to have a practical tool for addressing this behavior.
Conclusions
The study presented in this note examines the lateral–torsional
buckling behavior of suspended I-shaped beams, such as those
used as lifting beams. The buckling strength of suspended beams
has been analyzed through the performance of a study of beams of
various proportions using a suitable FEA program. A correction
factor by which existing beam-design equations can be modified to
provide a practical means of accounting for the buckling behavior
of suspended beams is evaluated. The use of this factor has been
tested by comparison to experimental data.
Notation
Table 2. Example Beam Design Solution by ASME (2014)
Quantity
sources reports any experimental test results. Analyses made
using the methods described in these papers do not correlate well
with the BTH-1 approach, but a comparison shows that the
BTH-1 method is conservative across a wide range of beam
proportions and support configurations. Comparing the Essa and
Kennedy (1993) lifting beam analysis method with the BTH-1
method shows somewhat less scatter than the self-weight
analysis.
The following symbols are used in this paper:
Af = area of the compression flange;
a = horizontal distance from beam midspan to upper
lift point;
a1 = vertical distance from beam shear center to upper
cable attachment;
a2 = vertical distance from beam shear center to lower
load attachment;
bf = width of the compression flange;
C b = lateral–torsional buckling modification factor
dependent upon moment gradient;
C LTB = ASME BTH-1 lateral–torsional buckling strength
coefficient;
d = beam depth;
E = modulus of elasticity;
Fy = specified minimum yield stress;
G = shear modulus of elasticity;
I x = major axis moment of inertia;
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J = torsional constant;
L = overall beam length;
Lb = the greater of the maximum distance between
supports or the distance between the two points of
applied load that are farthest apart;
N d = ASME BTH-1 nominal design factor;
Pcr = critical buckling load;
r T = radius of gyration of a section comprising the compression flange plus one-third of the compression web
area, taken about an axis in the plane of the web;
Tuc = upper cable tension;
W = point load applied to beam;
WExp = experimental buckling load;
α = horizontal distance from beam midspan to point
load; and
θ = angle of upper cable to horizontal.
© ASCE
References
AISC. (1989). Specification for structural steel buildings—Allowable
stress design and plastic design, 9th Ed., Chicago.
AISC. (2010). Specification for structural steel buildings, 14th Ed.,
Chicago.
ASME. (2014). “Design of below-the-hook lifting devices.” BTH-1-2014,
New York.
BASP [Computer software]. Austin, TX, University of Texas at Austin.
Dux, P. F., and Kitipornchai, S. (1989). “Stability of I-beams under
self-weight lifting.” Steel Constr., 23(2), 2–11.
Dux, P. F., and Kitipornchai, S. (1990). “Buckling of suspended Ibeams.” J. Struct. Eng., 10.1061/(ASCE)0733-9445(1990)116:7(1877),
1877–1891.
Essa, H. S., and Kennedy, D. J. L. (1993). “Distortional buckling of
steel beams.” Structural Engineering Rep. No. 185, Dept. of Civil
Engineering, Univ. of Alberta, Edmonton, AB, Canada.
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