Lateral Torsional Buckling

MIDAS IT

Lateral Torsional Buckling

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 lateraltorsional 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

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MIDAS IT

Lateral Torsional Buckling

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

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MIDAS IT

Lateral Torsional Buckling

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)

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Lateral Torsional Buckling

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)

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Lateral Torsional Buckling

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.

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MIDAS IT

Lateral Torsional Buckling

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]).

Pcr 

n 2 2 EI z L2

where, n : Buckling mode (1, 2, … ) L : Length of the element E = Young’s modulus of elasticity

I z = Moment of inertia about local z-axis Substituting the material and section properties into the above equation, the buckling load is found as:

Pcr 

 2  71, 240  0.54 2402

 6.592 N

For a simply supported column subjected to an axial force, the axial-torsional buckling load is found as follows (Timoshenko and Gere [2]).

Pcr 

GI xx A I A E  xx  I y  I z I y  I z 2(1   )

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

 = Poisson’s ratio I y = Moment of inertia about local y-axis

I z = Moment of inertia about local z-axis I xx = Torsional moment of inertia Substituting the material and section properties into the above equation, the buckling load is found as:

Pcr 

I xx A E 2.132784 18 71, 240    I y  I z 2(1   ) 1,350  0.54 2(1  0.31)

 772.920 N

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Lateral Torsional Buckling

2.5 Comparison of results Mode

Buckling type

Theoretical Solution

(N)

Beam model Critical

Error (%)

load (N)

Plate model Critical

Error (%)

load (N)

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

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.

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MIDAS IT

Lateral Torsional Buckling

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)

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Lateral Torsional Buckling

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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Lateral Torsional Buckling

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

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MIDAS IT

Lateral Torsional Buckling

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

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MIDAS IT

Lateral Torsional Buckling

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

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MIDAS IT

Lateral Torsional Buckling

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]).

Pcr 

I z I xx 4.013 4.013 EI z GI xx  2 E 2 L L 2(1   )

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

 = Poisson’s ratio I z = Moment of inertia about local z-axis I xx = Torsional moment of inertia Substituting the material and section properties into the above equation, we find:

Pcr 

I z I xx 4.013 4.013 (1.041667 105 )  (4.035417 105 ) 8 E   10  L2 2(1   ) 202 2(1  2 / 3)

 11.266 lbf

3.5 Comparison of results Unit : lbf Case

Critical load for 1st buckling Theoretical solution

1 2 3

11.266

MIDAS Beam element (error)

Plate element (error)

11.293

(0.24%)

11.815

(4.87%)

11.272

(0.05%)

11.808

(4.81%)

11.267

(0.01%)

11.809

(4.82%)

Both beam and plate element models show the results close to the theoretical results.

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MIDAS IT

Lateral Torsional Buckling

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

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Lateral Torsional Buckling

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 th

the 12 mode, buckling occurs at the arch part. This shows the importance of lateral buckling in such a structure. st

Fig. 4.3 shows the 1 buckling mode when lateral buckling is considered.

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Lateral Torsional Buckling

st

Fig. 4.3 1 mode when lateral buckling is considered Fig. 4.4 shows the similarity in buckling loads and shapes between the 12

th

mode of the case

st

considering lateral buckling and the 1 mode of the case without considering lateral buckling.

st

(a) 1 mode without considering lateral buckling

th

(b) 12 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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Lateral Torsional Buckling

buckling

Fig. 4.5 shows the similarity in buckling loads and shapes between the 13

th

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

th

(b) 13 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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Lateral Torsional Buckling

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

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Lateral Torsional Buckling

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.

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Lateral Torsional Buckling

st

(a) 1 Mode

nd

(b) 2 Mode

rd

(c) 3 Mode Fig. 5.3 Buckling modes considering lateral buckling

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Lateral Torsional Buckling

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 lateraltorsional 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

(b) Plate element model

(c) Plate element model

(Point moment)

(Quasitangential moment)

(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

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Lateral Torsional Buckling

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

(c) Plate element model

(Quasitangential moment)

(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]).

M cr 

 L

EI z GI xx 

E L

I z I xx 2(1  )

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

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Lateral Torsional Buckling

I z = Moment of inertia about local z-axis I xx = Torsional moment of inertia Substituting the material and section properties into the above equation, the critical buckling load is found as:

M cr 

 L

E

I z I xx  (1.041667 105 )  (4.035417 105 )  108  2(1   ) 20 2(1  2 / 3)

 176.396 lbf  in

6.5 Comparison of results Unit: lbf·in Mode

Theoretical

Beam element

solution

model

Quasitangential

(Point moment)

(Point moment)

moment

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

1

Plate element model Semitangential moment

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 lateraltorsional 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

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Lateral Torsional Buckling

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.

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