Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

POST-BUCKLING BEHAVIOR OF RESTRAINED THIN-WALLED STEEL

BEAMS USING GENERALIZED BEAM THEORY (GBT)

Cilmar Basaglia, cbasaglia@civil.ist.utl.pt

Dinar Camotim, dcamotim@civil.ist.utl.pt

Nuno Silvestre, nunos@civil.ist.utl.pt Department of Civil Engineering and Architecture, ICIST/IST, TU Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract. A numerical investigation on the local, distortional and global post-buckling behavior of restrained thin-walled

steel beams is reported. All the results are obtained by means of geometrically non-linear analyses based on a recently

developed Generalized Beam Theory (GBT) formulation that incorporates the influence of non-standard support conditions.

The results presented, most of which are compared with values yielded by shell finite element analyses (for validation purposes),

include critical buckling loads and mode shapes, post-buckling equilibrium paths, deformed configuration representations and

nornal stress distibutions. Taking advantage of the GBT modal nature, these results are discussed in detail and it is possible to

unveil, explain and/or shed some new light on a number of interesting and scarcely known behavioral aspects.

Keywords: Thin-walled steel members, Post-buckling behavior, Non-linear Generalized Beam Theory, GBT-based beam finite

element, Localized displacement restraints

1. INTRODUCTION

In order to adequately assess the structural efficiency of thin-walled steel members, one must acquire in-depth knowledge

about their buckling and post-buckling behaviors, a task which requires evaluating bifurcation stresses and determining

the corresponding post-buckling equilibrium paths and strength values (accounting for the effect of unavoidable initial

imperfections). In addition, the above information also plays a crucial role in the development, validation and calibration

of methodologies (formulae and/or procedures) intended for an efficient design of thin-walled steel members. However,

since those members exhibit low torsion stiffness and high susceptibility to local, distortional and/global deformations,

assessing their structural response constitutes a rather complex task that involves the performance of either (i) costly and

carefully planned experimental tests (e.g., Chodraui et al. 2006) or (ii) sophisticated, computer-intensive and time-consuming

(including data input and result interpretation) shell finite element analyses (e.g., Dinis and Camotim 2006).

One very promising alternative to the above approaches is the use of a beam finite elements based on a GBT geometrically

non-linear formulation, which is valid in the moderate-to-large deformation range and makes it possible to perform

numerically efficient (and structurally clarifying) elastic post-buckling analyses of prismatic thin-walled members (e.g.,

Silvestre and Camotim 2003 or Basaglia et al. 2009) with open cross-sections – however, up to now, these analyses

could not handle members, namely beams, exhibiting non-standard support conditions.

The aim of this paper is to present and discuss a set of numerical results that illustrate the application and capabilities of the

aforementioned GBT-based beam finite elements to assess the elastic post-buckling behavior of beams with non-standard

support conditions (e.g., localized supports stemming from bracing systems) – for the sake of completion, the paper also provides

a very brief overview of the procedures associated with the determination of the non-linear finite element stiffness matrices,

which incorporate the support condition effects. The numerical results concern lipped channel beams under uniformly distributed

loads, exhibiting or not localized displacement restraints and containing critical-mode initial imperfections. For validation purposes,

most GBT-based results are compared with numerical values yielded by ANSYS (SAS, 2004) shell finite element analyses.

2. GBT MAIN CONCEPTS AND PROCEDURES

In a GBT formulation, the cross-section displacement field is expressed as a combination of deformation modes,

leading to a very convenient and most unique form of expressing the member equilibrium equations − this is achieved

by performing a cross-section analysis, which allows for a much better understanding of the member structural behavior.

Figures 1(a)-(b) show the dimensions, elastic constants (Young’s modulus and Poisson’s ratio) and GBT discretization of the

lipped channel cross-section dealt with in this work. On the basis of the above discretization, the GBT cross-section

analysis leads to (i) 19 conventional (4 global, 2 distortional and 13 local), (ii) 16 shear and (iii) 16 transverse extension

modes (Silvestre and Camotim 2003) − Fig. 2 shows the main features of those that are more relevant to the analyses

carried out next (i.e. with significant contributions to the beam buckling mode shapes and post-buckling deformed configurations):

8 conventional, 2 shear and 1 transverse extension deformation modes.

The conventional modes, based on the assumption of null membrane shear strains and transverse extensions, constitute the core

of GBT and can still be subdivided into (i) global modes (cross-section in-plane rigid-body motions: axial extension, major/minor

axis bending and torsion), (ii) distortional modes and (iii) local modes − the last two involve cross-section in-plane deformation

(distortion and/or transverse wall bending). The shear modes concern the non-linear variation of the warping displacements along

the cross-section mid-line (the cross-section experiences no in-plane deformation). The transverse extension modes involve

only in-plane displacements and obviously account for the cross-section deformation due to wall transverse extensions.

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

20

20

2

75

2.3

(mm)

E = 205 GPa

v = 0.3 End node

Natural node

Intermediate node

(a) (b) (c)

Figure 1. Lipped channel (a) dimensions, (b) GBT discretization, and (c) lateral restraints

1 2 3 4 5 6 7 8

20 21 36

Global Distortional Local Shear Transverse

Extension

Figure 2. Main features of the most relevant deformation modes: in-plane shape of 8 conventional modes, warping

displacements of 2 shear modes and perspective of the in-plane shape of 1 transverse extension mode

Once the displacement fields defining each cross-section deformation mode are known, the determination of the member

post-buckling behavior involves the solution of a one-dimensional problem, consisting of the system of non-linear differential

equilibrium equations

+−+−−−+−−− xx,x,jx,kx,jx,khjkxxxx,kkkhkkkhxx,kkkhxxxx,kkkh )(C)(C)(B)(D)(C φφφφφφφφφφφφ2

1

hx,x,jx,ix,kx,jx,ix,kkijh q.t.o.h)(C =+−+ φφφφφφ2

1 , (1)

where (i) (.),x≡ d(.)/dx, (ii) the summation convention applies to subscript k, (iii) φk(x) are mode amplitude functions

defined along the member length, (iv) the bar identifies the mode amplitude functions of the initial geometrical imperfections

and (v) the tensor C, B and D components are cross-section modal mechanical properties associated with the resistance to

longitudinal extensions, transverse extensions and shear strains, respectively. While the second-order tensor components

(Ckh, Dkh, Bkh) characterize the cross-section linear behavior, the third (Ckjh, etc.), fourth (Ckijh, etc.) and higher-order (h.o.t.,

not shown in (1)) ones are associated with its geometrically non-linear behavior. Note that the global modes 1 (axial extension

− C11 is the axial stiffness), 2+3 (major and minor axis bending − C22 and C33 are the bending stiffness values) and 4 (torsion

− C44 and D44 are the warping and St. Venant torsion stiffness values) are characterized by Bkh=0, since they involve only

cross-section rigid-body motions. On the other hand, all the remaining deformation modes (k ≥ 5) exhibit (i) primary and

secondary warping displacements and/or (ii) cross-section in-plane deformation, thus leading to non-null Cik, Dik, Bik components

with no obvious mechanical interpretation − this feature is shared by all higher-order mechanical properties (even those

corresponding to the rigid-body deformation modes).

The solution of the non-linear system (1) can be obtained by means of a GBT-based beam finite element formulation analogous

to that developed and implemented by Silvestre and Camotim (2003) − it approximates the modal amplitude functions φk(x) by linear

combinations of (i) Lagrange cubic polynomial primitives (axial extension and shear modes) and (ii) Hermite cubic polynomials

(transverse extension and remaining conventional modes). In order to incorporate non-standard support conditions, namely

intermediate localized displacement restraints, into the analysis, one must impose appropriate constraint conditions that vary

from case to case (Basaglia et al. 2009). In this work, the discretized system of non-linear algebraic equations is solved by means of

an incremental-iterative technique based on Newton-Raphson’s method and adopting a load or displacement control strategy.

3. NUMERICAL RESULTS: LIPPED CHANNEL BEAMS

One analyses two lipped channel beams with the cross-section displayed in Fig. 1(a), length L=200cm and simply

supported ends (i.e., end sections locally/globally pinned and free to warp) – they are subjected to a uniformly distributed

load q applied at the shear centre axis and differ only in the fact that (i) one is laterally unrestrained and (ii) the other is laterally

restrained by two pairs of rigid intermediate supports, which are located in the web-flange corners (see Fig. 1(c) – localized

displacement restraints) of the L/3 and 2L/3 cross-sections. As for the initial geometrical imperfections, they were obtained

through preliminary buckling analyses and exhibit the critical buckling mode shape of each beam analyzed.

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

3.1. Preliminary buckling analyses

Figures 3(a1)–(b2) provide the critical buckling mode shapes of the two beams analyzed. While the left hand side figures are

three-dimensional views of the ANSYS buckling modes, obtained by considering a discretization into a fine mesh of shell

elements (SHELL181), their right hand side counterparts display the associated GBT modal amplitude functions φk(x),

corresponding to a dicretization into only 8 beam elements. The observation of the GBT and ANSYS buckling results prompts

the following remarks:

(i) The critical loads yielded by the GBT and ANSYS analyses practically coincide – one has (i1) qcr.GBT=0.608 kN/cm

and qcr.ANSYS=0.610 kN/cm, for the unrestrained beam, and (i2) qcr.GBT=0.671 kN/cm and qcr.ANSYS=0.664 kN/cm, for

the restrained beam – maximum difference equal to 1.05%.

(ii) While the unrestrained beam critical buckling mode combines participations from global (3-4), distortional (5-6) and local

(7-8) modes, only the local and distortional ones have perceptible contributions to the restrained beam buckling mode.

(iii) The distortional modes 5-6, which play a dominant role in both cases, are not qualitatively affected by the restraints:

3 half-wave contributions spanning the whole beam length and “opposing” each other − therefore, only the upper

flange-lip assembly experiences visible motions. However, note that while the unrestrained beam exhibits outer half-waves

with larger amplitudes, the restrained beam dominant half-wave is the central one – this can be readily confirmed

by comparing the upper (compressed) flange-lip assembly motions in Figs. 3(a1) and (b1).

(iv) Since the unrestrained beam critical buckling mode is predominantly distortional, the efficiency of the lateral restraints

considered (preventing the web-flange corner horizontal displacements – see Fig. 1(c) ) is rather limited − this explains why the

critical buckling load increase is only about 10%. Indeed, the lateral restraints basically brace the beam against the global

deformation modes (modes 3-4 in this case).

(a1)

(a2)

-1.0

0.0

1.0

0 40 80 120 160 200

5

6 7

8 (×5) 4 (×5) 3

(b1)

(b2)

-1.0

0.0

1.0

0 40 80 120 160 200

5

7 (×5)

6

8 (×5)

L (cm)

Figure 3. ANSYS and GBT-based buckling mode shape of the (a1)–(a2) unrestrained and (b1)–(b2) restrained beams 3.2. Post-buckling analyses

In the post-buckling analyses performed in this work, the two beams contain critical-mode initial imperfections with

different amplitudes: (i) w0=0.4cm for the unrestrained beam (w0 is the mid-span the web-flange corner lateral displacement)

and (ii) v0=–0.02cm for the restrained beam (v0 is the mid-span flange-lip corner vertical displacements − see Fig. 4(a).

Figures 4 and 5 display the post-buckling results concerning the unrestrained and restrained beams. Figures 4(a) and 5(a)

show the equilibrium paths (q vs. w and/or v) yielded by the (i) shell (ANSYS) and (ii) beam (GBT) finite element analyses

– the solid and dashed curves correspond to mid-span flange-lip corner vertical displacements (v) and web-flange corner lateral

displacements (w). As for the modal participation diagrams depicted in Figs. 4(b) and 5(b), they provide information about the

evolution of the “relative participations” of the various GBT deformation modes on the beam mid-span cross-section deformed

configuration along the post-buckling equilibrium path. Finally, Figs 4(c) and 5(c) depict the evolution of the mid-span

cross-section deformed configurations yielded by the ANSYS analyses. After observing these post-buckling results, one

is led to the following conclusions:

(i) Virtually “exact” beam post-buckling behaviors are obtained by means of GBT analyses including only deformation modes

1–8+20+21+36 (unrestrained beam) and 1+2+5–8+20+21+36 (restrained beam) – for v<5.0cm, the differences between the

GBT and ANSYS equilibrium paths never reach 3.5%. Moreover, note that the GBT analyses involve only a small fraction of

the number of d.o.f. required by the ANSYS ones: 200 (unrestrained beam) or 168 (restrained beam) against over 12500.

(ii) As expected, the restrained beam exhibits a considerable post-critical strength reserve − this strength reserve is very

small in the unrestrained beam. This difference stems from the natures of the two beam critical buckling modes and

corresponding post-buckling behaviors: the lateral restraints have the net effect of “replacing” the global modes appearing

in the unrestrained beam by the local modes characterizing the restrained beam.

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 1.0 2.0 3.0 4.0 5.0

GBT v

GBT w

● ○ ANSYS

Displacement (cm)

qcr

q (kN/cm)

w

v

(a)

0

25

50

75

100

0.00 0.10 0.20 0.30 0.40 0.50

q (kN/cm)

(%)

2

3

4 5

6

7

8 > 20

(b)

0.15 0.30 0.35 0.40 0.45 0.50kN/cm

(c)

Figure 4. Unrestrained beam (a) post-buckling equilibrium paths, (b) GBT mid-span cross-section modal participation

diagram and (c) evolution of the ANSYS mid-span cross-section deformed configuration (magnified by a factor 2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 1.0 2.0 3.0 4.0 5.0

v

GBT v

○ ANSYS

Displacement (cm)

qcr

q (kN/cm)

(a)

0

25

50

75

100

0.00 0.15 0.30 0.45 0.60 0.75 0.90

q (kN/cm)

(%)

2

5

6 6

7

7 8 8 > 20

5

(b)

0.40 0.50 0.60 0.70 0.80 0.90kN/cm

(c)

Figure 5. Restrained beam (a) post-buckling equilibrium paths, (b) GBT mid-span cross-section modal participation

diagram and (c) evolution of the ANSYS mid-span cross-section deformed configuration (magnified by a factor 2)

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

(iii) The unrestrained beam geometrically non-linear behavior is governed by the global (2-4) and distortional (5+6)

deformation modes, associated with marginal (global) and fairly small (distortional) post-buckling strength reserves –

the latter prevail in the early loading stages (highly influenced by the initial imperfection shape) and also become gradually

more dominant as post-buckling progresses. On the other hand, major axis flexure (mode 2) governs the early loading stages

of the restrained beam behavior − then, it is progressively “replaced” by the distortional (5+6) and local (7) modes.

(iv) The shear and transverse extension deformation modes, which only emerge in the advanced post-buckling stages,

have minute contributions to the deformed configurations of both beams. Nevertheless, they are a bit more relevant

in the restrained beam.

(v) The observation of the restrained beam modal participation diagram shows that modes 5 (for 0.107 kN/cm) and 6-8

(for 0.278 kN/cm) have null contributions to the mid-span cross-section deformed configuration. This fact, which does not

occur in the unrestrained beam, stems from the predominance of major axis flexure (mode 2), which “forces” modes

5-8 to invert their amplitude signs. In other words, forces the upper flange-lip assembly to move downwards, while the

initial imperfections correspond to an upward motion. The same does not happen in the unrestrained beam: the upper

flange-lip assembly always moves upwards (even if such motion decreases in the intermediate loading stages).

Figs. 6 and 7 show the unrestrained and restrained beam deformed configurations and longitudinal normal stresses distributions

corresponding to the load level q=0.5kN/cm and yielded by GBT analyses. Note that the deformed configurations are 3D

representations of results obtained by means of GBT-based beam finite elements − they were obtained on the basis of the

displacement field, expressed as a linear combination of the contributions of the various deformation modes considered, by

using an adaptation of the GMEC post-processing code (Paccola 2004) that was specifically developed to perform this task.

59.57

45.52

31.48

17.43

3.39

-10.67

-24.70

-38.75

-52.79

-66.84

Stress

(kN/cm2)

com

pre

ssio

n

ten

sio

n

s

Figure 6. GBT 3D deformed configuration (magnifying factor of 4) and normal stress of the unrestrained beam (q=0.5 kN/cm)

48.38

37.79

27.19

16.59

5.99

-4.61

-15.22

-25.82

-36.42

-47.02

Stress

(kN/cm2)

com

pre

ssio

n

ten

sio

n

Figure 7. GBT 3D deformed configuration (magnifying factor of 4) and normal stress of the restrained beam (q=0.5 kN/cm)

Proceedings of PACAM XI 11th Pan-American Congress of Applied Mechanics Copyright © 2009 by ABCM January 04-08, 2010, Foz do Iguaçu, PR, Brazil

As for the longitudinal normal stresses, they may be expressed as

xx,kk )s(Eu)s,x( φσ = , (2)

where (i) s is a coordinate defined along the cross-section mid-line (see Fig. 6) and (ii) uk(s) is the longitudinal displacement

profile associated with deformation mode k – once again, the summation convention applies to subscript k. After observing

the results displayed in Figs. 6 and 7, the following conclusions may be drawn:

(i) There is an excellent correlation between the mid-span cross-section deformed configurations (most deformed and

acted by the highest bending moment) provided by the ANSYS (see Figs. 4(c) and 5(c)) and GBT analyses.

(ii) In both beams the local and distortional modes occur mainly in the beam mid-span region. On the other hand, the participation

of the global modes is felt along the whole unrestrained beam length.

(iii) In the unrestrained beam, (iii1) high compressive stresses develop at the central half-length around the top web-flange edge and

(iii2) almost equally high tensile stresses appear at the bottom lip free edge close to the mid-span cross-section. In the

restrained beam, similarly high compressive and tensile stresses occur at the central third-length and involve the whole

width of the top and bottom flanges, respectively.

4. CONCLUSION

Initially, a very brief overview of the most relevant concepts and procedures involved in the derivation and numerical (beam

finite element) implementation of a geometrically non-linear GBT formulation was presented − this formulation makes it possible to

analyze the elastic post-buckling behavior of open-section thin-walled members, including the effect of localized restraints.

Then, in order to illustrate the application and capabilities of this novel computational tool, a set of numerical results were

presented and discussed − they concerned unrestrained and restrained simply supported lipped channel beams subjected to

uniformly distributed loads and containing critical-mode initial geometrical imperfections. Most GBT-based results (post-

buckling equilibrium paths, deformed configurations, modal participation diagrams and normal stress distributions) were compared

with numerical values yielded by ANSYS shell finite element analyses – a virtually perfect agreement was found in all cases.

Taking full advantage of the GBT modal nature, it was possible (i) to discuss the results in great structural detail and also (ii) to

unveil and/or shed some new light on a number of interesting and scarcely known behavioral aspects, namely the relative

importance of the local, distortional and global deformations as post-buckling progresses.

5. ACKNOWLEDGEMENTS

The first author gratefully acknowledges the financial support provided by the CAPES Foundation, Ministry of Education of

Brazil, through scholarship nº BEX 3932/06-0.

6. REFERENCES Basaglia C., Camotim D. and Silvestre N., 2009. “Non-linear GBT formulation for open-section thin-walled members with

arbitrary support conditions”, Proceedings of The Twelfth International Conference on Civil, Structural and Environmental

Engineering Computing (CC 2009 – Funchal, 1-4/9). (in press − full paper in CD-ROM Proceedings)

Chodraui G.M.B, Munaiar Neto J., Malite M. and Schafer B.W., 2006. “On the stability of cold-formed steel members under axial

compression”, Proceedings of International Colloquium on Stability and Ductility of Steel Structures (SDSS 2006 − Lisboa,

6-8/9), D. Camotim, N. Silvestre, P.B. Dinis (eds.), IST Press, Lisboa, pp. 381-388.

Dinis P.B. and Camotim D., 2006. “On the use of shell finite element analysis to assess the local buckling and post-buckling

behaviour of cold-formed steel thin-walled members”, Book of Abstracts of III European Conference on Computational

Mechanics: Solids, Structures and Coupled Problems in Engineering (III ECCM − Lisboa, 5-9/6), C.A.M. Soares et al.

(eds.), Springer, pp. 689. (full paper in CD-ROM Proceedings).

Paccola R.R., 2004. Physically Non-Linear Analysis of Anisotropic Laminated Plates and Shells Interacting or not with a Three-

Dimensional Viscoelastic Continuum by means of a Combined BEM/FEM Approach, Ph.D. thesis, University of Sao Paulo at

São Carlos, Brazil. (Portuguese)

Silvestre N. and Camotim D., 2003. “Non-linear generalised beam theory for cold-formed steel members”, International

Journal of Structural Stability and Dynamics, Vol. 3, No. 4, pp. 461-490.

Swanson Analysis Systems Inc (SAS), 2004. ANSYS Reference Manual (Version 8.1).

7. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.