Assumptions of linear buckling analysis:

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FEM analysis of buckling

CMCE Lecture 5/2, Civil Engineering, II cycle, specialty BEC

Jerzy Pamin

Institute for Computational Civil Engineering

Civil Engineering Department, Cracow University of Technology e-mail: JPamin@L5.pk.edu.pl

With thanks to:

M. Radwańska, M. Słoński, A. Wosatko ANSYS, Inc. http://www.ansys.com ROBOT http://www.autodesk.com

Comp.Meth.Civ.Eng., II cycle

Buckling phenomenon [1,2,6]

Assumptions of linear buckling analysis:

I one-parameter loading, varying proportionally to load parameter λ P = λP^∗

I loading is conservative, i.e. does not change direction during structure deformation

I structure (bar, panel, shell) is ideal, with no geometrical, material or load imperfection which would disturb ideal pre-buckling state

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Buckling phenomenon cont’d

Buckling occurs when increasing load reaches critical value P_cr = λ_crP^∗, where P^∗ denotes so-called configurational load for which λ = 1.

Characteristic feature of buckling as one of loss of stability phenomena is the significant change of deformation mode of structural system which experiences compressive stresses as a whole or in some part.

Source: E. Ramm, Buckling of Shells, Springer-Verlag, Berlin 1982

Examples of buckling phenomenon

Static criterion of the buckling as one of loss of stability types consists in examination of close pre- and post-buckling states. The phenomenon is presented for

I simply supported bar,

I deep cantilever beam,

I unidirectionally compressed panel, simply supported along circumference,

I cyllindrical shell under normal pressure, clamped along lower edge.

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Buckling of a bar

Before buckling

The bar:

I has straight axis,

I is only compressed (is not bent).

After buckling

The bar:

I has curved axis,

I exhibits compression and bending.

Buckling of deep cantilever beam

Before buckling

I The beam is bent in plane by vertical force applied at the free end

X

Y

Beam displacements in pre-buckling state

After buckling

I lateral buckling (warp, twist) occurs due to coupled bending and torsional deformation

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Buckling of deep cantilever beam cont’d

Z X

Buckling modes (forms)

Buckling of panel compressed in one-direction

Before buckling

Ideal membrane state:

I Panel with ideal medium plane,

I Constant compressive loading along one direction in the medium plane.

After buckling

Bending occurs:

I non-zero displacements perpendicular to medium plane,

I non-zero curvatures and bending moments.

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Buckling of compressed panel (ANSYS, [3])

First and second buckling mode

Third and fourth buckling mode

Buckling of cylindrical shell under external radial pressure

Before buckling

In the shell:

I axisymmetric conditions,

I in large part of the long shell pure membrane state,

I bending in vicinity of clamped edge (flexure) state.

After buckling:

Significant disturbance of axisymmetry:

I waves along circumference,

I number of half-waves is different for subsequent critical multipliers of the applied loading.

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Buckling of cylindrical shell cont’d (ANSYS, [3])

Subsequent buckling modes

General buckling analysis [1,2,6]

Energetic buckling criterion

Energetic buckling criterion consists in the analysis of an increment of potential energy Π during transition from pre- to post-buckling state.

One considers two adjacent equilibrium states:

I pre-buckling state I for which

δΠ^{(I )} = 0

I post-buckling state II for which

δΠ^{(II )} = δΠ^{(I )}+ δ∆Π = 0

I energetic criterion of critical state δ∆Π = 0.

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Algorithm of FEM buckling analysis

Matrix equation describing the loss of stability viz. buckling [K₀ + λK_σ(s^∗)]v = 0

or

{K₀+ λ[K_σ(s^∗) + K_u1(g^∗)]}v = 0

where:

I linear stiffness matrix of the system K₀

I initial stress matrix K_σ(s^∗) and initial displacement matrix K_u1(g^∗)

I critical loading multiplier to be determined λ_cr

I respective post-buckling form represented by eigenvector v = ∆d

Pre-buckling statics

Stage I of algorithm:

1. Compute the global stiffnesss matrix K₀

2. Compute nodal forces representing initial loading configuration P^∗, i.e. for loading multiplier λ = 1 (one-parametr loading assumed P = λP^∗)

3. Take boundary conditions into account

4. Solve equation set K₀· d^∗ = P^∗, to obtain nodal displacements in pre-buckling state: d^∗ = K⁻¹₀ · P^∗

5. From displacements of the system d^∗ extract element dofs d^e∗ and compute in each element:

I displacement gradients g^e∗ and

I generalized stresses s^e∗.

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Buckling analysis

Stage II of algorithm:

1. Generate:

- initial stress (geometrical) matrices for each element K^e_σ(s^∗e) and the whole structure K_σ(s^∗)

- optionally initial displacement matrix K_u1(g^∗)

2. Formulate non-standard (generalized) eigenproblem representing linearized buckling problem: [K₀ + λ(K_σ+ K_u1)]v = 0

or initial buckling problem: [K₀ + λK_σ]v = 0

3. Solve the eigenproblem to determine the pairs (λ₁, v₁), . . ., (λ_N, v_N) where:

I N – number of dofs

I λ_i – eigenvalue - critical loading multiplier

I v_i = ∆d_i – eigenvector - post-buckling deformation mode

Buckling of ideal panel/plate [4] – input data

I dimensions: L_x = L_y = 1.16 m, h = 0.012 m

I material data: E = 2.05 · 10⁸ kN/m², ν = 0.3

I configurational loading along perimeter which represents in-plane bending: |px ,max ,min^∗ | = 1.0 kN/m

I two options of boundary conditions along circumference:

a) simply supported (hinged, right) b) fully supported (clamped, left)

FEM discretization, loading and options of boundary conditionss

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Panel buckling

Assumptions:

I ideally flat medium plane,

I loading acts exactly in the medium plane,

I the one-parameter loading is governed by λ factor.

Buckling analysis of ideal panel under pure in-plane bending

Loading which causes pure in-plane bending prior to buckling Computations:

I numerical (FEM packages ANKA and ROBOT) – approximate solution

I analytical – exact solution

Panel buckling for in-plane bending

Computation of critical load:

Loading and deformation in pre-buckling state

Analytical solution for:

I simply supported panel p_cr^ib,analit = ^25.6·π²^·D^m

L²_x = 6077 kN/m

I clamped panel pcr^ib,analit = ^39.0·π²^·D^m

L²_x = 9259 kN/m Numerical solution (ANKA, mesh 8 × 8 ES) for:

I simply supported panel pcr^ib,MES = 6028 kN/m

I clamped panel p_cr^ib,MES = 11304 kN/m

Numerical solution (ROBOT, mesh 12 × 12 ES) for:

I simply supported panel pcr^be,MES = 6241 kN/m

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In-plane bending in pre-buckling state

Distribution of membrane force n_x

for the simply supported (left) and clamped (right) panel

In-plane bending, buckling modes

First two buckling modes for simply supported panel (ROBOT)

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In-plane bending, buckling modes

First two buckling modes for clamped panel (ROBOT)

Buckling of I-beam – input data

I dimensions: L_x = L_y = 1.16 m, h_s = 0.012 m, h_p = 0.018 m

I material data: E = 2.05 · 10⁸ kN/m², ν = 0.3

I configurational loading along beam sections:

|p^∗_{x ,min,max}| = 1.0 kN/m

I two options of buckling analysis (ROBOT):

option 1: local buckling of the web

option 2: buckling of beam segment (web+flanges)

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Option 1: web buckling

Local web buckling:

I for isolated web panel, in reality connected to flanges and ribs, different boundary conditions can be imposed along the connection lines

I in limiting cases one can assume:

a) hinged support along whole circumference b) clamped support along whole circumference

I hence the actual situation is inbetween

I former computations can be used to consider the web buckling

Option 2: I-beam buckling

Buckling analysis for the beam:

I discrete model in ROBOT for I-beam composed of web (12 × 12 elements) and two flanges (4 × 12), loaded by bending in the plane of the web

I numerical results (ROBOT):

I pcr^be,MES = 9068 kN/m

I comparison of critical forces computed with FEM (ROBOT):

I for isolated web:

- simply supported (ss) - clamped (cl)

I whole beam segment (be) p^ss,MES < p^be,MES < p^{cl ,MES}

6241 kN/m < 9068 kN/m < 11666 kN/m

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Bending of I-beam in pre-buckling state

Membrane force distribution n_x in pre-buckling state

Buckling modes for I-beam

Two buckling modes for I-beam segment subjected to bending (ROBOT)

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Buckling analysis of cylindrical shell using ABAQUS package [5]

References

[1] M. Radwańska. Ustroje powierzchniowe, podstawy teoretyczne oraz rozwiązania analityczne i numeryczne. Wydawnictwo PK, Kraków, 2009.

[2] Z. Waszczyszyn, C. Cichoń, M. Radwańska. Stability of Structures by Finite Elements Methods. Elsevier, 1994.

[3] M. Bera. Analiza utraty stateczności wybranych tarcz i powłok sprężystych metodą elementów skończonych. Praca dyplomowa, Politechnika Krakowska, Kraków, 2006.

[4] M. Radwańska, E. Pabisek. Zastosowanie systemu metody elementów skończonych ANKA do analizy statyki i wyboczenia ustrojów powierzchniowych. Pomoc dydaktyczna PK, Kraków 1996.

[5] M. Chojnacki. Projekt zbiornika stalowego i nieliniowa analiza wyboczenia powłoki z imperfekcjami. Praca dyplomowa, Politechnika Krakowska, Kraków, 2014.

[6] M. Radwańska, A. Stankiewicz, A. Wosatko, J. Pamin. Plate and Shell Structures. Selected Analytical and Finite Element Solutions. John Wiley & Sons, 2017.