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-lateral-torsional buckling. Lateral 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.

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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

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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.

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Mode 1 (Beam element model) Mode 1 (Plate element model)

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Mode 11 (Beam element model) Mode 11(Plate element model)

Fig. 2.3 Buckling modes

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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 z cr

n

EI

P

L





where, n : Buckling mode (1, 2, … ) L : Length of the element

E = Young’s modulus of elasticity

z

I

= 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 0.54

6.592

240

cr

P









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 xx cr y z y z

GI A

I A

E

P

I









E = Young’s modulus of elasticity G = Shear modulus of elasticity



= Poisson’s ratio

y

I

= Moment of inertia about local y-axis

z

I

xx

I

= 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

xx cr y z

I A

E

P

I

















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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

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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

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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.

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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

Top view

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Beam element model

Top view

Isometric view

Plate element model

Top view

Isometric view

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Beam element model

Top view

Isometric view

Plate element model

Top view

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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 xx cr z xx

I I

P

EI GI

E

L







where,

L = Length of the cantilever beam E = Young’s modulus of elasticity G = Shear modulus of elasticity



= Poisson’s ratio

z

I

xx

I

Substituting the material and section properties into the above equation, we find:

5 5 8 2 2

4.013

4.013 (1.041667 10 ) (4.035417 10 )

10 2(1

)

20 2(1 2 / 3)

11.266 lbf

z xx cr

I I

P

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%)

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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

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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.

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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

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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

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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

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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

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(a) 1st Mode

(b) 2nd Mode

(c) 3rd Mode

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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

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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 xx cr z xx

I I

E

M

EI GI

L









where,

L = Length of the cantilever beam E = Young’s modulus of elasticity G = Shear modulus of elasticity

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z

I

xx

I

Substituting the material and section properties into the above equation, the critical buckling load is found as: 5 5 8

(1.041667 10 ) (4.035417 10 )

10 2(1

)

20 2(1 2 / 3)

176.396 lbf in

z xx cr

I I

M

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

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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.