Lateral Torsional Buckling
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Transcript of Lateral Torsional Buckling
MIDAS IT Lateral Torsional Buckling
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Lateral torsional buckling
1. Types of buckling
When a slender member is subjected to an axial force, failure takes place due to bending or torsion
rather than direct compression of the material. Such type of failure is known as buckling, which is one
of the main causes for structural failure and thus needs to be taken into account in design. [1] The
load, which causes buckling in a member, is referred to as the critical load or the buckling load.
Theoretical equations are well known for relatively simple structure types.
Buckling caused by flexure as in Fig. 1.1(a) is referred to as the Euler buckling (Axial-flexural
buckling). Torsional buckling and translational buckling also exist, which are divided into lateral-
torsional buckling and axial-torsional buckling. Lateral torsional buckling exhibits deformation in a
lateral direction as in Fig. 1.1(b) due to a shear direction load. Axial-torsional buckling exhibits
torsional deformation as in Fig. 1.1(c) due to an axial load. While the Euler buckling considers only the
effects of flexural moments, buckling needs to be considered for the effects of shear, moment and
torsion together.
When a thin member is subjected to axial and shear forces and bending moments individually or in
combination, the three types of buckling may occur individually or in combination depending on the
geometric configuration and boundary conditions. Irrespective of the type of buckling, buckling in a
member takes place at the lowest critical load. So finding the first buckling mode and the
corresponding buckling load is the prime task in buckling analysis.
(a) axial-flexural buckling (Euler buckling) (b) lateral-torsional buckling
(c) axial-torsional buckling
Fig. 1.1 Types of buckling
MIDAS IT Lateral Torsional Buckling
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2. Axial torsional buckling
In this section, we will review the properties of a structure, which exhibits axial-flexural buckling (Euler
buckling) and axial-torsional buckling.
2.1 Overview of analytical models
Fig. 2.1(a) is a simply supported column of a thin rectangular section subjected to a concentric axial
force for which we find the buckling loads. The structure is represented by a beam element model Fig.
2.1(b) and a plate element model Fig. 2.1(c). The beam element model consists of 48 beam elements,
and the plate element model consist of elements divided into 48 segments horizontally and 6
segments vertically. We will review the results of both models against the theoretical solution.
Case 1: Beam element (total 48 elements: divided into 48 elements in the horizontal dir.)
Case 2: Plate element (total 288 elements: divided into 48 and 6 elements in the horizontal and
vertical directions respectively)
(a) Model shape (top View)
(b) Case 1: Beam element model
(c) Case 2: Plate element model
Fig. 2.1 Structural geometry and boundary conditions
MIDAS IT Lateral Torsional Buckling
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2.2 Properties of analytical models
Analysis Type
Axial-torsional buckling
Unit System
N, mm
Dimension
Length 240mm
Element
Beam element
Plate element (thick type without drilling dof)
Material
Young’s modulus of elasticity E = 71,240N/mm2
Poission’s ratio ν = 0.31
Section Property
Beam element: solid rectangular 0.6×30mm
Plate element: thickness 0.6mm, width 5mm & height 5mm
Boundary Condition
Left end is pinned and right end is roller
Load
P = 1.0 N
2.3 Analysis results
Fig. 2.2 shows the results up to 11 modes from MIDAS for both beam element and plate element
models. Fig. 2.3 shows the mode shapes. The first 10 modes exhibit Euler buckling and the 11th
mode exhibits Axial-torsional buckling.
(a) Beam element model (b) Plate element model
Fig. 2.2 Analysis result (Buckling load)
MIDAS IT Lateral Torsional Buckling
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Mode 1 (Beam element model) Mode 1 (Plate element model)
Mode 2 (Beam element model) Mode 2 (Plate element model)
Mode 3 (Beam element model) Mode 3 (Plate element model)
Mode 4 (Beam element model) Mode 4 (Plate element model)
Mode 5 (Beam element model) Mode 5 (Plate element model)
Mode 6 (Beam element model) Mode 6 (Plate element model)
MIDAS IT Lateral Torsional Buckling
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Mode 7 (Beam element model) Mode 7 (Plate element model)
Mode 8 (Beam element model) Mode 8 (Plate element model)
Mode 9 (Beam element model) Mode 9 (Plate element model)
Mode 10 (Beam element model) Mode 10 (Plate element model)
Mode 11 (Beam element model) Mode 11(Plate element model)
Fig. 2.3 Buckling modes
The fact that the beam model is of a uni-axial structure, axial-torsional buckling shape can not be
viewed. So for Mode 11, we will refer to the plate model for the buckling shape.
MIDAS IT Lateral Torsional Buckling
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2.4 Theoretical solution
For a simply supported column subjected to an axial force, the axial-flexural buckling load (Euler
buckling load) is found as follows (Gere [1]).
2 2
2
zcr
n EIP
L
where,
n : Buckling mode (1, 2, … )
L : Length of the element
E = Young’s modulus of elasticity
zI = Moment of inertia about local z-axis
Substituting the material and section properties into the above equation, the buckling load is found as:
2
2
71,240 0.546.592
240crP N
For a simply supported column subjected to an axial force, the axial-torsional buckling load is found
as follows (Timoshenko and Gere [2]).
2(1 )
xx xxcr
y z y z
GI A I A EP
I I I I
E = Young’s modulus of elasticity
G = Shear modulus of elasticity
= Poisson’s ratio
yI = Moment of inertia about local y-axis
zI = Moment of inertia about local z-axis
xxI = Torsional moment of inertia
Substituting the material and section properties into the above equation, the buckling load is found as:
2.132784 18 71,240
2(1 ) 1,350 0.54 2(1 0.31)
772.920 N
xxcr
y z
I A EP
I I
MIDAS IT Lateral Torsional Buckling
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2.5 Comparison of results
Axial-flexural buckling (Euler buckling) occurs in the Modes 1-10, and Axial-torsional buckling occurs
in the Mode 11. Both beam and plate element models show the results close to the theoretical results.
Mode Buckling type Theoretical
Solution (N)
Beam model Plate model
Critical
load (N)
Error (%) Critical
load (N)
Error (%)
1 Euler buckling 6.592 6.592 0.000 6.606 0.212
2 Euler buckling 26.367 26.365 0.008 26.581 0.812
3 Euler buckling 59.325 59.316 0.015 60.331 1.696
4 Euler buckling 105.467 105.440 0.026 108.358 2.741
5 Euler buckling 164.791 164.728 0.038 171.145 3.856
6 Euler buckling 237.300 237.171 0.054 249.123 4.982
7 Euler buckling 322.991 322.758 0.072 342.693 6.100
8 Euler buckling 421.866 421.480 0.091 452.259 7.204
9 Euler buckling 533.924 533.326 0.112 578.261 8.304
10 Euler buckling 659.166 658.288 0.133 721.193 9.410
11 Axial-torsional 772.920 772.920 0.000 778.084 0.668
MIDAS IT Lateral Torsional Buckling
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3. Lateral-torsional buckling
In this section, we will review the lateral-torsional buckling through an example.
3.1 Overview of analytical models
Fig. 3.1 shows a cantilever beam of a thin rectangular section subjected to a concentric axial force
and a concentric shear force. We will find the buckling loads. The structure is represented by beam
and plate element models, which are divided into 10, 20 and 40 segments horizontally. We will review
the results of each model against the theoretical solution.
Case 1: Both beam and plate elements (divided into 10 elements in the horizontal dir.)
Case 2: Both beam and plate elements (divided into 20 elements in the horizontal dir.)
Case 3: Both beam and plate elements (divided into 40 elements in the horizontal dir.)
Fig. 3.1 Structural geometry and boundary conditions
3.2 Properties of analytical models
Analysis Type
Lateral torsional buckling
Unit System
lbf, in
Dimension
Length 20 in
Element
Beam element and plate element (thick type without drilling dof)
MIDAS IT Lateral Torsional Buckling
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Material
Young’s modulus of elasticity E = 1.0^8 lb/in2
Poisson’s ratio ν = 2/3
Section Property
Beam element : solid rectangular 0.05×1 in
Plate element : thickness 0.05in, width 1.0 in
Boundary Condition
Left end is fixed and right end is free
Load
P = 1.0 lbf
3.3 Analysis results
4 Buckling modes are found. Lateral-torsional buckling occurs in all the 4 modes. The analysis results
for the beam and plate element models are as follows.
MIDAS IT Lateral Torsional Buckling
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Case 1: Both beam and plate elements (10 elements)
Beam element model
1st mode Buckling load
Top view
Isometric view
Plate element model
1st mode Buckling load
Top view
Isometric view
MIDAS IT Lateral Torsional Buckling
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Case 2: Both beam and plate elements (20 elements)
Beam element model
1st mode Buckling load
Top view
Isometric view
Plate element model
1st mode Buckling load
Top view
Isometric view
MIDAS IT Lateral Torsional Buckling
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Case 3: Both beam and plate elements (40 elements)
Beam element model
1st mode Buckling load
Top view
Isometric view
Plate element model
1st mode Buckling load
Top view
Isometric view
MIDAS IT Lateral Torsional Buckling
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3.4 Theoretical solution
The buckling load for a cantilever beam of a thin rectangular section subjected to a set of concentric
axial and shear forces at the tip is found as follows (Timoshenko and Gere [2]).
2 2
4.013 4.013
2(1 )
z xxcr z xx
I IP EI GI E
L L
where,
L = Length of the cantilever beam
E = Young’s modulus of elasticity
G = Shear modulus of elasticity
= Poisson’s ratio
zI = Moment of inertia about local z-axis
xxI = Torsional moment of inertia
Substituting the material and section properties into the above equation, we find:
5 58
2 2
4.013 4.013 (1.041667 10 ) (4.035417 10 )10
2(1 ) 20 2(1 2 / 3)
11.266 lbf
z xxcr
I IP E
L
3.5 Comparison of results
Unit : lbf
Case Critical load for 1st buckling
Theoretical
solution
MIDAS
Beam element (error) Plate element (error)
1
11.266
11.293 (0.24%) 11.815 (4.87%)
2 11.272 (0.05%) 11.808 (4.81%)
3 11.267 (0.01%) 11.809 (4.82%)
Both beam and plate element models show the results close to the theoretical results.
MIDAS IT Lateral Torsional Buckling
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4. An arch example
In this section we will examine the effects of lateral buckling in an arch bridge. Buckling loads and
shapes are examined for the cases considering lateral buckling and without considering lateral
buckling. Consideration of lateral buckling is meant to consider shear and bending deformations. This
example is examined by assuming that the bridge deck provides no lateral restraint to the girders.
4.1 Overview of analytical model
Fig. 4.1 shows an arch bridge, which is simply supported at each end. It is subjected to dead load,
pedestrian load and vehicular load. The girders are thin and long, which are prone to lateral buckling.
(a) Dead load
(b) Pedestrian load
(c) Vehicular load
Fig. 4.1 Analytical model and loads
MIDAS IT Lateral Torsional Buckling
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4.2 Analysis results
Fig. 4.2 shows the results corresponding to the cases considering lateral buckling and without
considering lateral buckling. As expected, the buckling loads for the case considering lateral buckling
are less than those of the case without considering it.
(a) Lateral buckling not considered (b) Lateral buckling considered
Fig. 4.2 Comparison of buckling loads for the cases considering lateral buckling and
without considering lateral buckling
When lateral buckling is not considered, buckling occurs only at the arch part. However, when lateral
buckling is considered, the buckling modes from 1 to 11 take place at the bridge deck girders. Only at
the 12th mode, buckling occurs at the arch part. This shows the importance of lateral buckling in such
a structure.
Fig. 4.3 shows the 1st buckling mode when lateral buckling is considered.
MIDAS IT Lateral Torsional Buckling
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Fig. 4.3 1st mode when lateral buckling is considered
Fig. 4.4 shows the similarity in buckling loads and shapes between the 12th mode of the case
considering lateral buckling and the 1st mode of the case without considering lateral buckling.
(a) 1st mode without considering lateral buckling
(b) 12th mode considering lateral buckling
Fig. 4.4 Comparison of buckling modes between the cases of
considering lateral buckling and without considering lateral
MIDAS IT Lateral Torsional Buckling
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buckling
Fig. 4.5 shows the similarity in buckling loads and shapes between the 13th mode of the case
considering lateral buckling and the 2nd mode of the case without considering lateral buckling.
(a) 2nd mode without considering lateral buckling
(b) 13th mode considering lateral buckling
Fig. 4.5 Comparison of buckling modes between the cases of
considering lateral buckling and without considering lateral buckling
MIDAS IT Lateral Torsional Buckling
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5. An example of a silo ceiling frame
In this section we will examine lateral buckling of an industrial structure frame.
5.1 Overview of analytical model
Fig. 5.1 shows a frame, which is simply supported at the ends of the girders. Concentrated loads of
0.2tonf exert at each node in the gravity direction. At the intersection, 0.4tonf is applied. The girders
are thin and long, which are subjected to only vertical loads without the presence of axial forces.
(a) Boundary conditions
(b) Loading
Fig. 5.1 Boundary conditions and loading
MIDAS IT Lateral Torsional Buckling
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5.2 Analysis results
We now seek buckling loads for this example in which no axial forces exist. Without considering
lateral buckling (shear and bending deformations), buckling loads can not be obtained. MIDAS finds
buckling loads considering axial direction as well as shear and bending deformations.
Fig. 5.2 shows the buckling loads obtained from MIDAS.
Fig. 5.2 Buckling loads considering lateral buckling
The table below compares the results of MIDAS and MSC Nastran, which are almost identical.
Unit : tonf
Mode MIDAS MSC Nastran Difference
1 8.326 8.326 0.000
2 9.648 9.648 0.000
3 10.132 10.131 0.001
Fig. 5.3 shows the buckling modes 1 to 3 for this example considering lateral buckling.
MIDAS IT Lateral Torsional Buckling
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(a) 1st Mode
(b) 2nd
Mode
(c) 3rd
Mode
Fig. 5.3 Buckling modes considering lateral buckling
MIDAS IT Lateral Torsional Buckling
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6. Cautionary notes
When a moment is applied to a structure consisted of thin plates, buckling analysis results in different
solutions depending on how the load is applied. This section explains the characteristics of the lateral-
torsional buckling algorithm adopted in MIDAS.
6.1 Overview of analytical model
Fig. 6.1 shows different models representing a cantilever beam subjected to a tip moment. The first
model is a beam element model. The next two models are plate element models with two different
ways of applying the acting moment. Buckling analysis results will be compared among different
models. A concentrated moment is applied to the beam element model. Quasitangential moment and
Semitangential moment are applied to the plate element models.
(a) Beam element model
(Point moment)
(b) Plate element model
(Quasitangential moment)
(c) Plate element model
(Semitangential moment)
Fig. 6.1 Representation of the external bending moment
6.2 Properties of analytical models
Analysis Type
Lateral-torsional buckling
Unit System
lbf, in
Dimension
Length 20 in
Element
Beam element
Plate element (thick type without drilling dof)
Material
Young’s modulus of elasticity E = 108 lb/in2
Poission’s ratio ν = 2/3
Section Property
Beam element : solid rectangular 0.05×1 in
MIDAS IT Lateral Torsional Buckling
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Plate element (single coupling force): thickness 0.05 in, width 1.0 in, height 1.0 in
Plate element (double coupling force): thickness 0.05 in, width 0.5 in, height 0.5 in
Boundary Condition
Left end is fixed and right end is free
Load
M = 1.0 lbf∙in
P = 1.0 lbf (Quasitangential moment, Moment arm: 1 in)
P = 0.5 lbf (Semitangential moment, Moment arm: 1 in)
6.3 Analysis results
Fig. 6.2 shows the results of 10 buckling modes for the three models.
(a) Beam element model (b) Plate element model
(Quasitangential moment)
(c) Plate element model
(Semitangential moment)
Fig. 6.2 Critical load for the external bending moment
6.4 Theoretical solution
For a cantilever beam of a thin rectangular section subjected to a concentrated moment, the buckling
load is found as: (Timoshenko and Gere [2]).
2(1 )
z xxcr z xx
I IEM EI GI
L L
where,
L = Length of the cantilever beam
E = Young’s modulus of elasticity
G = Shear modulus of elasticity
= Poisson’s ratio
MIDAS IT Lateral Torsional Buckling
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zI = Moment of inertia about local z-axis
xxI = Torsional moment of inertia
Substituting the material and section properties into the above equation, the critical buckling load is
found as:
5 58 (1.041667 10 ) (4.035417 10 )
102(1 ) 20 2(1 2 / 3)
176.396 lbf in
z xxcr
I IM E
L
6.5 Comparison of results
Unit: lbf·in
Mode Theoretical
solution
(Point moment)
Beam element
model
(Point moment)
Plate element model
Quasitangential
moment
Semitangential moment
1 176.396 176.576 90.334 187.623
2 176.576 272.256 188.094
3 534.047 457.972 567.759
4 534.047 650.121 569.799
5 904.542 851.507 962.798
6 904.542 1065.150 967.869
When buckling loads due to moment loads are sought, and if torsional displacement occurs at the
point of moment load application, it is cautioned that the results differ depending on the use of nodal
moments or coupling forces. There are largely two algorithms for reflecting the effects of lateral-
torsional buckling. One approach is to consider nodal rotation as small rotation, and the other is to
consider it as large rotation (Saleeb et al. [3]). MIDAS uses the large rotation approach. The large
rotation approach consistently reflects torsion and bending at the points of reentrant corners, which is
implemented in high quality commercial software. The user must use caution when using the large
rotation approach in that a coupling force representing a nodal moment is based on Fig. 6.1(c)
Semitangential moment rather than Fig. 6.1(b) Quasitangential moment.
The models 6.1(a) and 6.1(c) produce similar results. The difference comes from the points of load
MIDAS IT Lateral Torsional Buckling
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application and elements. However, the model 6.1(b) produces drastically different results.
6. Reference
1. James M. Gere, Mechanics of Materials, 5th Edition, 2001, Thomson
2. Timoshenko, S.P., and Gere, J.M., (1961). Theory of Elastic Stability, McGraw-Hill, New York.
3. Saleeb, A.F, Chang T.Y.P, Gendy A.S., (1992). “Effective modeling of spatial buckling of beam
assemblages, accounting for warping constraints and rotation-dependency of moments,” Int. J. Num.
Meth. Eng., Vol. 33, 469–502.