Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 3, pp. 785-796, Warsaw 2012

50th Anniversary of JTAM

FINITE ELEMENT ANALYSIS OF BUCKLING OF STRUCTURES
AT SPECIAL PREBUCKLING STATES

Herbert A. Mang, Xin Jia

Vienna University of Technology, Institute for Mechanics of Materials and Structures, Vienna, Austria

e-mail: herbert.mang@tuwien.ac.at

Gerhard Höfinger

BPM Bauprozess Management GmbH, STRABAG AG, Austria

The consistently linearized eigenproblem is used to derive mathematical conditions in the
frame of the Finite Element Method (FEM) for loss of static stability of elastic structures
at prebuckling states characterized by a constant percentage bending energy of the strain
energy in the prebuckling regime. Special cases of such prebuckling states are membrane
stress states and pure bending. Buckling from a membrane stress state as a special case
within sensitivity analysis of buckling at a constant non-zero percentage bending energy in
the prebuckling regime is one of two examples serving the purpose to verify the existence of
hitherto unknown subsidiary conditions of buckling in the context of the FEM.

Key words: special prebuckling stress states, consistently linearized eigenproblem, finite
element method

1. Introduction

Specialprebucklingstates aredefinedas stateswithaconstantpercentagebendingenergy
of the strain energy in the prebuckling regime. Special cases of the constant percentage
bending energy are:

• zero percentage bending energy (membrane stress state) and

• zero percentage membrane energy (pure bending).

The consistently linearized eigenproblem (Helnwein, 1997) will be used to derivemathemati-
cal conditions for loss of static stability of elastic structures at special prebuckling states in the
frame of the Finite ElementMethod (FEM). The difference between buckling from amembrane
stress state in the frame of sensitivity analysis restricted to such stress states and buckling from
amembrane stress state, representing a special case of loss of stability at states with a constant
percentage bending energy of the strain energy in the prebuckling regime, will be brought out
theoretically and verified numerically.

The paper is organized as follows: In Section 2, the consistently linearized eigenproblemwill
be used for derivation of mathematical relations for general prebuckling states. These relations
will then be specialized for the stability limit. In Section 3, the relations derived in Section 2will
be specialized for the initially mentioned special prebuckling states. In Section 4, results from
a numerical investigation will be presented. Section 5 contains the conclusions drawn from this
work.



786 H.A. Mang et al.

2. General prebuckling states

The consistently linearized eigenproblem forFEanalysis of a conservative systemwithNdegrees
of freedom is defined as (Helnwein, 1997)

[K̃T +(λ
∗−λ)K̃T,λ]v

∗ =0 (2.1)

where

K̃T(λ) := K̃T(q(λ)) (2.2)

is the tangent stiffness matrix and

K̃T,λ(λ) := K̃T,λ(q(λ),λ) (2.3)

indicates differentiation of K̃T with respect to the loadmultiplier λ along a direction parallel to
the primary path q(λ) (Schranz et al., 2006). In (2.1), λ∗−λ is the eigenvalue corresponding to
the eigenvector v∗. λ∗ and v∗ are functions of λ. Equation (2.1) represents a set of N implicit
equations defining N curves in the (λ∗ − λ)-space. Thus, it has got N solutions (λ∗j,v

∗

j),
j ∈{1,2, . . . ,N}.

Writing (2.1) for the first eigenpair gives

[K̃T +(λ
∗

1−λ)K̃T,λ]v
∗

1 =0 (2.4)

Hence, the following orthogonality relations must hold

v∗kK̃Tv
∗

1 =0 v
∗

kK̃T,λv
∗

1 =0 k∈{2,3, . . . ,N} (2.5)

Derivation of (2.4) with respect to λ gives

[λ∗1,λK̃T,λ+(λ
∗

1−λ)K̃T,λλ]v
∗

1 +[K̃T +(λ
∗

1−λ)K̃T,λ]v
∗

1,λ =0 (2.6)

Premultiplication of (2.6) by v∗1 and consideration of (2.4) yields

λ
∗

1,λ =−(λ
∗

1−λ)
v∗1K̃T,λλv

∗

1

v∗1K̃T,λv
∗

1

(2.7)

Normalization of the eigenvector such that

|v∗1|=1 (2.8)

results in

v∗1v
∗

1,λ =0 (2.9)

Since the eigenvectors v∗j, j ∈{1,2, . . . ,N}, are a basis of R
N, v∗1,λ can be expressed as

v∗1,λ =
N∑

j=1

c1jv
∗

j (2.10)

Substitution of (2.10) into (2.9) and consideration of

v∗kv
∗

1 =0 k∈{2,3, . . . ,N} (2.11)



Finite element analysis of buckling of structures... 787

gives

c11 =0 (2.12)

Premultiplication of (2.6) by v∗k and consideration of (2.5), (2.10), and

[K̃T +(λ
∗

k−λ)K̃T,λ]v
∗

k =0 (2.13)

yields

c1k =−
λ∗1−λ

λ∗1−λ
∗

k

v∗kK̃T,λλv
∗

1

v∗
k
K̃T,λv

∗

k

k∈{2,3, . . . ,N} (2.14)

Introducing the abbreviation

A= K̃T +(λ
∗

1−λ)K̃T,λ (2.15)

into (2.4), gives

Av∗1 =0 (2.16)

The first, second, and third derivative of (2.16) with respect to λ are obtained as

A,λv
∗

1 +Av
∗

1,λ =0

A,λλv
∗

1+2A,λv
∗

1,λ+Av
∗

1,λλ =0

A,λλλv
∗

1+3A,λλv
∗

1,λ+3A,λv
∗

1,λλ+Av
∗

1,λλλ =0

(2.17)

where

A,λ =λ
∗

1,λK̃T,λ+(λ
∗

1−λ)K̃T,λλ

A,λλ =λ
∗

1,λλK̃T,λ+(2λ
∗

1,λ−1)K̃T,λλ+(λ
∗

1−λ)K̃T,λλλ

A,λλλ =λ
∗

1,λλλK̃T,λ+3λ
∗

1,λλK̃T,λλ+(3λ
∗

1,λ−2)K̃T,λλλ+(λ
∗

1−λ)K̃T,λλλλ

(2.18)

The focus of the present work is on the influence of special prebuckling states on loss of
stability. Apart from the fact that buckling in the formof snap-through is impossible for some of
these states, it is irrelevant to thisworkwhether loss of stability occurs in the formof bifurcation
buckling or snap-through. For the latter mode of buckling

dλ(λS)= 0 (2.19)

where λ=λS refers to the stability limit.Hence, λwouldnotbeagoodchoice forparameterizing
the equilibrium path in the vicinity of the snap-through point. A detailed account of treating
snap-through by means of the consistently linearized eigenproblem is given in Steinboeck et al.
(2008). For the aforementioned reasons and for the sake of simplicity, it is assumed that loss of
stability occurs in the form of bifurcation buckling.

At λ=λS

K̃Tv1 =0 (2.20)

Hence, following from (2.13)

λ∗1(λS)=λS v
∗

1(λS)=v1 (2.21)



788 H.A. Mang et al.

Substitution of (2.21) into (2.7) and (2.14) gives

λ∗1,λ(λS)= 0 (2.22)

and

c1k(λ)= 0 k∈{2,3, . . . ,N} (2.23)

respectively. Inserting (2.12) and (2.23) into (2.10) yields

v∗1,λ(λS)=0 (2.24)

indicating a singular point on the vector curve v∗1(λ). Specialization of (2.15) and (2.18) for the
stability limit results in

A= K̃T

A,λ =0

A,λλ =λ
∗

1,λλK̃T,λ−K̃T,λλ

A,λλλ =λ
∗

1,λλλK̃T,λ+3λ
∗

1,λλK̃T,λλ−2K̃T,λλλ

(2.25)

Because of (2.24) and (2.25)2, (2.17)1 is trivially satisfied for λ=λS. Specialization of (2.17)2,3
for λ=λS gives

[λ∗1,λλK̃T,λ−K̃T,λλ]v1+K̃Tv
∗

1,λλ =0 (2.26)

and

[λ∗1,λλλK̃T,λ+3λ
∗

1,λλK̃T,λλ−2K̃T,λλλ]v1+K̃Tv
∗

1,λλλ =0 (2.27)

respectively. Elimination of K̃T,λλv1 in (2.27) with the help of (2.26), followed by premultipli-
cation of the result by v1 and consideration of (2.20), yields

λ
∗

1,λλλ =−3λ
∗2
1,λλ+2

v1K̃T,λλλv1

v1K̃T,λv1
(2.28)

For buckling at general prebuckling states, for mechanical reasons beyond the scope of this
work,

λ∗1,λλ(λS)< 0 λ
∗

1,λλλ(λS)< 0
v1K̃T,λλλv1

v1K̃T,λv1
> 0 (2.29)

Because of

v1K̃T,λv1 < 0 (2.30)

Mang and Höfinger (2012)

v1K̃T,λλλv1 < 0 (2.31)



Finite element analysis of buckling of structures... 789

3. Special prebuckling states

Asmentioned at the beginning, special prebuckling states are defined as states with a constant
percentage bending energy of the strain energy in the prebuckling regime. For such prebuckling
states, (2.17)3 disintegrates into (Mang, 2011)

A,λλv
∗

1,λ =0 ∧ A,λλλv
∗

1 +3A,λv
∗

1,λλ+Av
∗

1,λλλ =0 (3.1)

At the stability limit, (3.1)1 is trivially satisfied and (3.1)2 degenerates to (2.27), as is the case
with (2.17)3 for buckling at general prebuckling states. However, instead of (2.29), for buckling
at special prebuckling states

λ
∗

1,λλλ ­ 0 =⇒
v1K̃T,λλλv1

v1K̃T,λv1
­ 0 (3.2)

In contrast to (2.31), the numerator in (3.2)2 may become zero. For a constant non-zero per-
centage buckling energy of the strain energy in the prebuckling regime

λ
∗

1,λλ(λS)< 0 λ
∗

1,λλλ(λS)= 0
v1K̃T,λλλv1

v1K̃T,λv1
=
3

2
λ
∗2
1,λλ(λS) (3.3)

3.1. Membrane stress state

A membrane stress state represents a special case of a state with a constant percentage
bending energy of the strain energy, namely, one with zero percentage bending energy. For such
a case, (2.17)2 disintegrates into (Mang, 2011)

A,λλv
∗

1 =0 ∧ 2A,λv
∗

1,λ+Av
∗

1,λλ =0 (3.4)

Derivation of (3.4)1 with respect to λ gives

A,λλλv
∗

1+A,λλv
∗

1,λ =0 (3.5)

Substitution of (3.1)1 into (3.5) yields

A,λλλv
∗

1 =0 (3.6)

Substitution of (3.6) into (3.1)2 results in

3A,λv
∗

1,λλ+Av
∗

1,λλλ =0 (3.7)

At the stability limit, taking (2.24) and (2.25)2, into account

A,λλv1 =0 A,λλλv1 =0

Av∗1,λλ =0 Av
∗

1,λλλ =0
(3.8)

Making use of (2.25)3,4, and (2.25)1, gives

[λ∗1,λλK̃T,λ−K̃T,λλ]v1 =0

[λ∗1,λλλK̃T,λ+3λ
∗

1,λλK̃T,λλ−2K̃T,λλλ]v1 =0
(3.9)

and



790 H.A. Mang et al.

v∗1,λλ =0 v
∗

1,λλλ =0 (3.10)

respectively, noting that v∗1,λλ and v
∗

1,λλλ are not eigenvectors of A(λS)= K̃T .
Premultiplication of (3.9)1 by v

∗

k and consideration of (2.5)2 yields

v∗kK̃T,λλv1 =0 k∈{2,3, . . . ,N} (3.11)

Premultiplication of (3.9)2 by v
∗

k and consideration of (2.5)2 and (3.11) results in

v∗kK̃T,λλλv1 =0 k∈{2,3, . . . ,N} (3.12)

In contrast to (2.5), the orthogonality relations (3.11) and (3.12) are restricted to the stability
limit.
Buckling from amembrane stress state obeys (3.2) and (3.9)-(3.12). Specialization of (3.9)2

for λ∗
1,λλλ =0, which is a special case of (3.2)1, gives

[3λ∗1,λλK̃T,λλ−2K̃T,λλλ]v1 =0 (3.13)

Elimination of K̃T,λλv1 in (3.13) with the help of (3.9)1 yields

[3λ∗21,λλK̃T,λ−2K̃T,λλ]v1 =0 (3.14)

An eigenvector of a squarematrix cannot correspond to two distinct eigenvalues (Wylie, 1975).
Hence, the eigenvalue of (3.14) is obtained as

λ
∗

1,λλ =0 (3.15)

Substitution of (3.15) into (3.9)1 and (3.13) results in the following remarkable subsidiary buck-
ling conditions (Höfinger, 2010)

K̃T,λλv1 =0 ∧ K̃T,λλλv1 =0 (3.16)

Themechanicalmeaning of this special case is buckling fromamembrane stress state as a special
case in the frame of sensitivity analysis of buckling at a constant non-zero percentage bending
energy of the strain energy in the prebuckling regime. Satisfaction of (3.3)2 by (3.15) and (3.16)2
proves this interpretation.

Fig. 1. Two-hinged arches (solid line: thrust line arch, dashed line: modified configuration)

Anexample for sucha sensitivity analysis, in the frameof theFEM, is aparameterized family
of two-hinged arches, subjected to a uniformly distributed load p (Mang and Höfinger, 2012).
The design parameter ∆κ refers to the deviation of the geometric form of the axis of the arch
from a quadratic parabola for which ∆κ = 0, representing a thrust-line arch (Fig. 1). Hence,



Finite element analysis of buckling of structures... 791

for ∆κ= 0, buckling occurs from a membrane stress state. Numerical results from sensitivity
analysis of the mentioned family of arches will be presented in Section 4.
The general case of (3.2)1 is characterized by λ

∗

1,λλλ > 0. It refers to sensitivity analysis
restricted to buckling frommembrane stress states.
An example for such a sensitivity analysis is a vonMises trusswith an elastic spring attached

to the load point (Fig. 2). The stiffness of the spring is given as κk where k is a constant and
κ is the variable design parameter. P is the reference load. Numerical results from sensitivity
analysis of the von Mises truss will be presented in Section 4.

Fig. 2.Von Mises truss with an elastic vertical spring attached to the load point)

InMang and Höfinger (2012) it is shown that

v1K̃T,λλq,λλ =0 (3.17)

is a necessary and sufficient condition for buckling from a membrane stress state. The general
case

K̃T,λλ 6=0 q,λλ 6=0 (3.18)

represents a nonlinear stability problemwith nonlinear prebuckling paths. The two special cases

K̃T,λλ 6=0 q,λλ =0

K̃T,λλ =0 q,λλ 6=0
(3.19)

show that linear stability problems and linear prebuckling paths need not bemutually conditio-
nal. The third special case is obtained as

K̃T,λλ =0 q,λλ =0 (3.20)

For the special case of a linear stability problem

K̃T =K0+λKσ (3.21)

where K0 is the constant small-displacement stiffness matrix and Kσ is the constant initial
stress matrix evaluated with the help of the stresses obtained from the first step of the analysis
(Zienkiewicz and Taylor, 1989). Substitution of (3.21) and of

K̃T,λ =Kσ (3.22)

into (2.4) gives

[K0+λ
∗

1Kσ]v
∗

1 =0 (3.23)

Since K0 and Kσ are constant matrices (Fig. 3)

λ∗1 = const ∧ v
∗

1 =v1 = const (3.24)



792 H.A. Mang et al.

Fig. 3. λ∗
1
−λ diagram for a linear stability problem

3.2. Pure bending

Purebendingrepresents the second special case of a statewith a constantpercentage bending
energy of the strain energy, namely, onewith zero percentagemembrane energy. For such a case,
(2.17)1 disintegrates intoMang (2011)

A,λv
∗

1 =0 ∧ Av
∗

1,λ =0 (3.25)

Since v∗1,λ is not an eigenvector of A

v∗1,λ(λ)=0 ∀λ (3.26)

Thus,

v∗1(λ)=v1 = const (3.27)

Substitution of (2.18)1 and (3.27) into (3.25)1 gives

[λ∗1,λK̃T,λ+(λ
∗

1−λ)K̃T,λλ]v1 =0 (3.28)

Premultiplication of (3.28) by

v∗k(λ)=vk = const (3.29)

and consideration of (2.5)2 yields

vkK̃T,λλv1 =0 ∀λ k∈{2,3, . . . ,N} (3.30)

In contrast to (3.11), (3.30) is not restricted to the stability limit. In Aminbaghai and Mang
(2012) it is shown that

K̃T =K0+λKσ+KL (3.31)

where KL(q(λ)) denotes the large-displacement stiffnessmatrix (Zienkiewicz andTaylor, 1989).
For λ=0

KL =0 (3.32)

Specialization of (2.4) for λ=0, considering (3.31) and (3.32), gives

K0+λ
∗

1(Kσ+KL,λ)v1 =0 (3.33)



Finite element analysis of buckling of structures... 793

where

(Kσ+KL,λ)v1 =0 (3.34)

(Aminbaghai andMang, 2012), which requires

λ
∗

1 =∞ (3.35)

(Fig. 4).
At the stability limit

λ
∗

1,λλλ(λS)= 0 λ
∗

1,λλ(λS)> 0 (3.36)

as follows from (3.3)2 and Fig. 4. Hence, the curvature of the curve λ
∗

1(λ) becomes aminimum
at S.

Fig. 4. λ∗
1
−λ diagram for the buckling from a pure bending stress state (lateral torsional buckling)

For all other cases of the buckling at prebuckling states characterized by a constant non-zero
percentage buckling energy of the total strain

λ∗1,λλλ(λS)= 0 λ
∗

1,λλ(λS)< 0 (3.37)

Hence, lateral torsional buckling is not a special case of these cases.

4. Numerical investigation

4.1. Sensitivity analysis of two-hinged arches subjected to a uniformly distributed load
(Fig. 1)

The span of the arches l is chosen as 6m, the rise of the thrust-line arch h as 2.4m, and the
side length of the constant square cross-section as 0.07m. The geometric form of the axis of the
arch is given as (Mang and Höfinger, 2012)

x∈ [0, l] y=
4h

l2
x(l−x)+∆κsin

(l−x
l
π
)

(4.1)

Themodulus of elasticity is assumed as 2.1 ·1011N/m2. FEAP (Taylor, 2001) was used for
sensitivity analysis of bifurcation buckling of the arches bymeans of beam elements. The system
was discretized, using 100 beam elements available in the FEAP version 7.5. This discretization
was sufficient to obtain numerically stable results for the load-displacement relations. For the
chosen configuration of arches, bifurcation buckling with an antisymmetric buckling mode is



794 H.A. Mang et al.

relevant (Mang and Höfinger, 2012). Figure 5 shows the Euclidean norms ‖K̃T,λλv1‖2 and

‖K̃T,λλλv1‖2 as functions of thedesignparameter ∆κ. Theywere computed, employing a scheme

for numerical differentiation of higher order of theglobal tangent stiffnessmatrix K̃T(λ) byusing
function values at five interpolation nodes. As soon as the discretization was fine enough to get
reliable data for the load-displacement relations, no significant dependency of the calculated
values of the norms on the number of elements was observed. For the special case of a thrust-
line arch

‖K̃T,λλv1‖2 =0 ‖K̃T,λλλv1‖2 =0 (4.2)

which confirms (3.16).

Fig. 5. Sensitivity analysis of bifurcation buckling of a family of two-hinged arches: (a) ‖K̃T,λλv1‖2 and

(b) ‖K̃T,λλλv1‖2 as functions of ∆κ representing the deviation from a thrust-line arch (Mang and
Höfinger, 2012)

4.2. Sensitivity analysis of a von Mises truss with an elastic spring attached to the load
point (Fig. 2)

The length of the two bars in the undeformed configuration L is chosen as 100cm, the
corresponding rise h as 30.9cm, the side length of the square cross-section as 17cm, the elastic
modulus as 2.8 · 1011kN/cm2, and the vertical reference load P as 1N. The value of k in
the expression for the spring constant κk, where κ ∈ R is a scaling parameter, was taken
as 1N/cm. To avoid a multiple bifurcation point, only one half of the truss is analyzed. A
detailed analytical treatment of similar structures can be found in Schranz et al. (2006) and
Steinboeck et al. (2008). The truss was designed such that for κ = 0 the bifurcation point
is relatively close to the snap-through point (Höfinger, 2010). With increasing spring stiffness,
the distance of the snap-through point from the bifurcation point is increasing. The structure
was discretized by means of 30 FEAP beam elements for finite displacements. Figure 6 serves
the purpose of verification of (3.17) for the general case of a nonlinear stability problem with
nonlinear prebuckling paths, characterized by

‖K̃T,λλq,λλ‖2 6=0 (4.3)

(Fig. 6a). However, apart fromnumerical noise for relatively small values of κ, the bilinear form
v1K̃T,λλq,λλ vanishes (Fig. 6b), which proves (3.17).

5. Conclusions

• The characteristic feature of special prebuckling states, defined as states with a constant
percentage bending energy of the strain energy in the prebuckling regime, is disintegra-
tion of the third derivative of the mathematical formulation of the consistently linearized
eigenproblem with respect to the load multiplier λ (see (3.1)).



Finite element analysis of buckling of structures... 795

Fig. 6. Sensitivity analysis of bifurcation buckling of a von Mises truss with an elastic spring attached
to the load point: (a) ‖K̃T,λλq,λλ‖2 and (b) v1K̃T,λλq,λλ as functions of the scaling parameter κ of the

spring stiffness (Höfinger, 2010)

• The characteristic feature of buckling from a membrane stress state, representing the
special state of zero percentage bending energy of the total strain energy, is disintegration
of the second derivative of the mathematical formulation of the mentioned eigenproblem
with respect to λ, in addition to disintegration of the third derivative (see (3.4) and (3.1)).

• For buckling fromamembrane stress state, obtained as a special case in the frame of sensi-
tivity analysis of buckling froma state of constant percentage bending energy of the strain
energy, the buckling mode is also the eigenvector of the second and the third derivative
of the tangent stiffness matrix with respect to λ (see (3.16)). This remarkable result was
verified numerically by means of sensitivity analysis of two-hinged arches subjected to a
uniformly distributed load, containing a thrust-line arch as a special case.

• Thedifference between such a sensitivity analysis and one that is restricted to the buckling
frommembrane stress states is reflected by λ∗1,λλλ =0 (see (3.3)) and λ

∗

1,λλλ > 0 (see the
general case of (3.2)).

• Apreviously derived necessary and sufficient condition for the buckling from amembrane
stress state (see (3.17)) was verified numerically by means of sensitivity analysis of a
von Mises truss with an elastic spring attached to the load point and the spring stiffness
serving as a variable design parameter.

• The characteristic feature of lateral torsional buckling, representing the state of zero per-
centagemembrane energy of the total strain energy, is disintegration of the first derivative
of the mathematical formulation of the consistently linearized eigenproblem with respect
to λ (see (3.25)). For this special case, for λ = 0, λ∗1 = ∞ (see (3.35)). At the stability
limit, λ∗

1,λλλ =0 and λ
∗

1,λλ > 0 (see (3.37)), indicating aminimumof the curvature of the
curve λ∗1(λ).

References

1. Aminbaghai M., Mang H.A., 2012, Characteristics of the solution of the consistently linearized
eigenproblem for lateral torsional buckling,Bulletin of the Georgian National Academy of Sciences,
in print

2. Helnwein P., 1997, Zur initialen Abschätzbarkeit von Stabilitätsgrenzen auf nicht linearen Last-
VerschiebungspfadenelastischerStrukturenmittels derMethodederFinitenElemente [Onab initio
estimates of stability limits on nonlinear load-displacement paths of elastic structures bymeans of
the finite element method], Ph. D. Thesis, Vienna University of Technology, Vienna, Austria



796 H.A. Mang et al.

3. Höfinger G., 2010, Sensitivity analysis of the initial postbuckling behavior of elastic structures,
Ph.D. Thesis, Vienna University of Technology, Vienna, Austria

4. Mang H.A., 2011, Categorization of buckling bymeans of spherical geometry, Research proposal
submitted to the Austrian Science Fund

5. Mang H.A., Höfinger G., 2012, Bifurcation buckling from amembrane stress state, Internatio-
nal Journal for Numerical Methods in Engineering, in print

6. Mang H.A., Höfinger G., JIA X., 2010, On the predictability of zero-stiffness postbuckling,
ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik, 90, 10/11, 837-846

7. Schranz C., Krenn B., Mang H.A., 2006, Conversion from imperfection-sensitive into
imperfection-insensitive elastic structures, II: numerical investigation, Computer Methods in Ap-
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8. Steinboeck A., Jia X., Rubin H.,MangH.A., 2008,Remarkable postbuckling paths analyzed
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in Engineering, 76, 156-182

9. Taylor R.L., 2001, FEAP – a Finite Element Analysis Program, Version 7.4, User Manual,
Department of Civil and Environmental Engineering, University of California at Berkeley

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Analiza wyboczenia konstrukcji w specjalnych przypadkach wyboczenia wstępnego za
pomocą metody elementów skończonych

Streszczenie

W pracy przedyskutowano warunki matematyczne w ramach metody elementów skończonych dla
niesprzecznie zlinearyzowanego zagadnienia własnego struktur sprężystychw celu określenia granicy sta-
tycznej stateczności tych struktur, gdy te poddane zostająwyboczeniuwstępnemu scharakteryzowanemu
stałym udziałem energii zginania w stosunku do całkowitej energii odkształcenia. Szczególnym przy-
padkiem wyboczenia wstępnego jest stan naprężeń powłokowych (brak zginania) oraz czyste zginanie.
Wyboczenie przy wstępnych naprężeniach membranowych, jako specjalny przykład analizy wrażliwości
wyboczenia na obecność niezerowej energii odkształceń giętych, jest jednym z dwóch przypadków zbada-
nych dla weryfikacji istnienia nieznanych, uzupełniających warunków wyboczenia w kontekście metody
elementów skończonych.

Manuscript received February 3, 2012; accepted for print March 26, 2012