Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 3, pp. 443-457, Warsaw 2003

ELASTO-PLASTIC LIMIT LOADS OF CYLINDER-CONE

CONFIGURATIONS

Dieter Dinkler

Oliver Knoke

Institut für Statik, Technische Universität Braunschweig

d.dinkler@tu-bs.de, o.knoke@tu-bs.de

The paper presents a procedure for computation of the limit loads of
imperfection-sensitive shells including plastic deformations.On the basis
of strainenergyof shells, theworstcase scenario for criticalperturbations
at fixed load levels is investigated. Limit loads of cylindrical and conical
shells under combined actions are compared to European design rules.
The concept is applied tomore general buckling cases of combined shell
structures.
Key words: stability, buckling, imperfection sensitivity, perturbation,
plasticity

1. Introduction

Imperfection-sensitivity is a widely discussed phenomenon of thin-walled
structures under shear or compression. Many investigations have been pre-
sented in this field pursuing the aim to improve the predictability of failure
loads that are of interest for design. Thus, publications are dealing very often
with safety design concepts for light-weight structures and their experimental
and numerical validation. In practice, the shell structures may be complex in
geometry, boundaries and loading ormay consist of different single shells. For
applicability of the design rules, the combined shell structures have to be re-
duced to uncoupled single substructures.Here the constructional requirements
for rigid boundaries arise, what requires expensive ring stiffeners in order to
decouple the buckling behaviour of the shell.

In contrast to classical concepts of the bifurcation theory, a perturbation
theory has been first developed byKröplin et al. (1985) and applied to nume-
rical investigations of imperfection-sensitive cylindrical shells. Improvements



444 D.Dinkler, O.Knoke

lead to a designmodel for structures under static loads presented byDuddeck
et al. (1990). Amore general model is presented inDinkler (1992) for structu-
res under time-dependent perturbations. The perturbation theory is used for
cylindrical and conical shells, but could be also applied tomore complex buc-
kling cases, if their behaviour is comparable to the known buckling cases. The
perturbation theory leads to a proper prediction of buckling loads of realistic
structures and fulfil the safety requirements of the design rules.

2. Model equations

A mixed formulation of the governing equations including elasto-plastic
material properties is applied to describe the deformation behaviour of thin
shells. Under the assumption of moderate rotations and elastic material be-
haviour, the mixed formulation of the Kirchhoff-Love shell theory including
geometrical nonlinearities is given by

Π =

∫

A

{
ααβñ

αβ +ϕβm
αβ
∣∣∣
α
−
1

2
ñαβDαβρλñ

ρλ−
1

2
mαβBαβρλm

ρλ
}
dA−

(2.1)

−

∫

A

{
uαp
α+u3p3

}
dA+

∫

su

mβϕβ ds−

∫

sσ

{
uαn

α+u3n3
}
ds

The effects of stresses normal to the shell surface and rotations around the
axes normal to shell’s surface are neglected. Dαβρλ and Bαβρλ are the mem-
brane and bendingflexibility, respectively. Symmetrical normal forces ñαβ are
defined by

ñαβ =nαβ + bαρm
ρβ (2.2)

and strains and slopes of the middle surface by

ααβ =
1

2
(uα
∣∣∣
β
+uβ

∣∣∣
α
−2bαβu

3+ϕαϕβ)

(2.3)

ϕα =u
3,α+b

ρ
αuρ

Primaryvariables are the stress resultants ñαβ,mαβ anddisplacements uα,
u3. States of equilibrium satisfy the condition Π → stationary and δΠ =0.
Yield conditions for stress resultants have been first discussed by Ilyushin

(1947, 1956). Amodel to consider the plastification through the cross-section



Elasto-plastic limit loads... 445

is given by Eggers andKröplin (1978). It is developed on the basis of the von
Mises yield condition,which couples the stress tensorwith theone-dimensional
yield stress. For the given shellmodel, yielding produced by normal forces and
moments is considered. The elastic limit state Y0 and plastic limit state Y1 of
the cross-section may be written in terms of the cross-section as

Y0 =
1

2

(
A11+3|A12|+

9

4
A22−S0

)
=0

(2.4)

Y1 =
1

2

(
A11+

1

2
A22+

√

(A12)2+
(1
2
A22
)2
−S1
)
=0

with

A11 =n
αβJαβρλn

ρλ

A12 =
4

t
nαβJαβρλm

ρλ (2.5)

A22 =
16

t2
mαβJαβρλm

ρλ

and with the one-dimensional reference forces S0 =S1 = fyt, where fy is the
yield stress, t is the shell thickness. The non-zero elements of the material
tensor Jαβρλ are for isotropic von Mises yielding

Jαααα =1.0 Jααββ =−
1

2
Jαβαβ =

3

4
α 6=β (2.6)

By increasing of the normal forces and moments, the yield surface changes
gradually from the elastic limit (2.4)1 to the plastic limit (2.4)2. Intermediate
states are approximated by the yield condition

Y =
bY0+ε

aY1
b+εa

=0 (2.7)

with the normalized equivalent strain

ε=
εpleq
εeleq

(2.8)

For ideal elasto-plastic material behaviour, the parameters a and b are
determined by comparison to a layer model; we obtain

a=
9

10
b=
1

15
(2.9)



446 D.Dinkler, O.Knoke

To consider the elasto-plastic material behaviour Eq. (2.1) is constrained
by the yield condition Y . In order to take into account the path-dependency
of plastic strains, the rate formulation of the potential is extended to

dΠtot = dΠ−

∫

A

(dY ·dλ) dA→ stationary (2.10)

with the total differential dY of the yield condition and the Lagrangian mul-
tiplier dλ. dΠtot is interpreted as the elasto-plastic potential. Integration over
the artificial time interval ∆t yields the incremental formulation

∆Πtot =∆Π−

∫

A

(∆Y ·∆λ) dA→ stationary (2.11)

which is the basis for numerical investigations. Variation of the incremental
potential leads to the extended governing equations of Prandtl-Reuss type and
to the consistency condition.

3. Perturbation theory and limit load criteria

Perturbations or imperfections of ideal configurations may be of different
kind with respect to loads, geometry and material. Fig.1a shows the load-
deformation behaviour of axially loaded cylindrical shells for different geome-
trical imperfections gp. Increasing the amplitude gp of geometrical imperfec-
tions leadsdecreasingof the loadmaximum P.Consideringaplaneata certain
load level P0 perpendicular to the load axis, the behaviour of a snap-through
becomes obvious, see Fig.1c. F denotes the fundamental state of equilibrium,
M the maximum of restoring forces, N the unstable state of equilibrium in
the neighbourhood of F. Stable states S belong to the deep post-buckling
region.
In case of perturbation loads orthogonal to the design load a completely

similar behaviour can be observed. In dependence on the axial load level,
the behaviour with respect to perturbation loads may be interpreted as a
snap-through as before, see Fig.1b. This means that perturbation loads and
imperfectionsmaybe interpreted by a similarmodel andmay lead to the same
process and deformation behaviour and could therefore be evaluated by the
samemechanical measure.
Consider a perfect shell under a certain fundamental load level P0 atwhich

certain states of equilibrium in the post-buckling region exist. The perturba-
tion of the fundamental state of equilibrium F may lead to a transition from



Elasto-plastic limit loads... 447

Fig. 1. Load-deformation behaviour of axially loaded cylindrical shells

the pre- to the post-buckling region, if the perturbation is large enough. This
means that a limit statemust exist between the pre- andpost-buckling regions
and it separates both regions. This limit state is the statically indifferent state
of equilibriumwith respect to the perturbations, M in Fig.1. In the following
it will be called a critical state of equilibrium and the strain energy that is
stored by the structure during the deformation from the fundamental state
of equilibrium F to the critical one M will be called the critical strain ener-
gy Πcr. In the cases when the strain energy Πp induced by perturbations is
larger than the critical strain energy, the structure will buckle; otherwise it
remains in the pre-buckling region

Πp >Πcr → buckling (3.1)



448 D.Dinkler, O.Knoke

Thus, aperturbation theorymaybeestablished on thebasis of the assump-
tion of a potential, first for the elastic material behaviour. The total potential
of the considered structures is split up to the strain energy and the external
potential with respect to external actions

Πtot =Πstr+Πext (3.2)

The external potential is described by the design actions p0 and the per-
turbations pp

Πext(u,p0,pp)=−

∫
u(p0+pp) dA (3.3)

Theprinciple of a stationary value of the potential leads to the equilibrium
conditions, if its first variation with respect to the displacement field vanishes

δΠtot = δu
(∂Πstr
∂u
+
∂Πext
∂u

)
=0 (3.4)

Equation (3.4) may be used for computations of states of equilibrium with
respect to p0 or a perturbation load pp, that belongs to the critical state
(p0+pp,u0+up).
If the sensitivity of the ideal structure against perturbations is taken into

account, theworst case of perturbations has to be investigated. Theworst case
is defined by theminimum of perturbation energy thatmay initiate buckling.
This means that in addition to Eq. (3.4), the minimality condition for the
potential with respect to perturbation loads must vanish

δΠtot =upδpp =0 (3.5)

as presented in Dinkler and Schäfer (1997). By means of Eq. (3.3) and (3.4),
δpp may be replaced by the displacement field up, that is conjugated to the
perturbation load. From

pp =
∂Πstr
∂up

it follows that δpp = δup
∂

∂up

(∂Πstr
∂up

)
(3.6)

Considering Eq. (3.6) in Eq.(3.5) yields the condition of the critical state
of equilibrium for the worst perturbation

δup
(∂2Πstr
∂u2p

)
up =0 (3.7)

Physical meaning of this condition may be interpreted similarly to the snap-
through behaviour: a state up has to be investigated, whereby the tangential
stiffness at the state up in the direction of up vanishes.



Elasto-plastic limit loads... 449

The interpretation of Eq. (3.7) as a nonlinear eigenvalue problem is obvio-
us, since the right-hand side vanishes and the coefficient of up depends on up
itself. If the critical state is given by

ucr =uF +λup (3.8)

uF holds the stable fundamental state of equilibrium and up – the normalized
perturbation field, scaled by λ to the distance between uF and the critical
state. Considering Eq. (3.8) and Eq. (3.7), the nonlinear eigenvalue problem
for the worst perturbation leads to

δup
[∂2Πstr
∂u2p

(uF)+λ
∂2ΠNLstr
∂u2p

(up)
]
up =0 (3.9)

The first coefficient of Eq. (3.9) describes the tangential stiffness of the funda-
mental state of equilibrium, the secondpart the change of stiffness due to λup.
If the critical deformation field λup is known, the critical strain energy Πcr
depending on λup may be computed analogously to ∆Πstr.

Fig. 2. Limit states for elasto-plastic material behaviour

In case of plastic material behaviour, the limit states depend on plastifi-
cation. If thematerial starts yielding between the fundamental state of equili-
brium and the critical state, the nonlinear eigenvalue problem, Eq. (3.9), has
to be solved incrementally from the fundamental state to the critical state,
with iterative correction of the plastic parts of the solution, see Fig.2. The



450 D.Dinkler, O.Knoke

modified nonlinear eigenvalue problemmay be written as

δui+1p

[∂2Πstr
∂u2p

(ui,ε
pl
i )+λi+1

∂2ΠNLstr
∂u2p

(ui+1p )
]
ui+1p =0 (3.10)

with index i for the increment. Here

ui =uF +
∑
αλiu

i
p 0<α¬ 1.0 (3.11)

is applied. α is a proper multiplier to scale the size of the increment. Eq.
(3.10) has to be solved several times until the critical state is reached. Each

intermediate state ui, ε
pl
i has to satisfy all the governing equations and the

yield condition, what is automatically achieved by the algorithm. In case of
elasto-plastic material behaviour, the critical strain energy Πcr is the sum for
each increment.

4. Application to cylindrical and conical shells

Toobtain aproper buckling indicator for comparison of different shells and
fundamental load conditions, the critical strain energy may be normalized by
the bending stiffness, see Dinkler and Spohr (1998)

B=
Et3

12(1−ν2)
(4.1)

since buckling is directly connected to bending. The buckling indicator may
be defined as

πcr =
Πcr
B

(4.2)

Figure 3 presents numerical investigations for cylindrical and conical shells
with various cone angles under axial compression in case of elastic material
behaviour.Conical shells are representedat the slenderness r/tof the reference
cylinder, where buckling occurs.

Normalized buckling loads for some values of the buckling indicator πcr
are shown in comparison with experiments and the German design rule
DAStRi 013. For the value of the buckling indicator πcr of about πcr =0.025,
numerical buckling loads for cylindrical and conical shells coincide extremely
well with lower bounds of the experiments. However, the value of the buckling
indicator, that fits the experiments, depends considerably on thediscretisation



Elasto-plastic limit loads... 451

Fig. 3. Buckling loads of axially loaded shells in comparison to design rules
DASt Ri 013

of the numerical computation, since local buckles need very finemeshes to be
simulated. For short compact shells, the influence of boundary conditions on
the buckling mode increases, what is considered directly by the buckling in-
dicator πcr. This may be the reason for the discrepancy with the empirical
design values at this range of slenderness r/t.

Figure 4 presents the application of the concept to elasto-plastic buckling
of cylindrical shells in comparison to the German design rule DIN 18800 and
the European recommendations ECCS. The numerical results may be inter-
preted depending on the slenderness of shells. In case of small slenderness
λSx < 0.4 rotational elasto-plastic buckling leads to very high limit loads in
comparison to the yield stress, since local buckling does not occur. Slender
shells exhibit a purely elastic buckling behaviour as indicated in Fig.3. For
moderate slenderness 0.4<λSx < 1.25, amixedmode buckling occurs, where
the procedure of Section 3.3 has to be applied. Nonetheless, the limit loads
for shells with slenderness λSx < 0.5 are below the minimum load capacity
of the post-buckling region, where procedure 3.3 fails. In these cases the limit
load is computed in the direction up of the lowest load level, where procedure
3.3 converges. A good coincidence of numerical results with the design rules is
obvious.



452 D.Dinkler, O.Knoke

Fig. 4. Buckling loads of axially loaded shells in comparison to design rules

5. Application to combined shells

Combinations of single shells leading to combined shell structures yield a
wide range of different shell types with different load-carrying and buckling
behaviour. As the first example, unstiffened combinations of cylindrical and
conical shells under axial compression are investigated. Fig.5 shows the critical
buckling shapes in case of an elastic material.

Fig. 5. Critical buckling shapes (a) concave cylinder-cone configuration, (b) convex
configuration, (c) single conical shell, α=20◦,R′max =150,R

′

min =96, l= r,
tcyl = tcone

Without stiffeners at the cylinder-cone connections, the deformed confi-
guration in the prebuckling region is dominated by large radial deformation



Elasto-plastic limit loads... 453

at these connections. Concave connections show large inward-looking radial
deformation. The resulting bending moments and circumferential forces are
similar to single, simply supported shells. Convex connections, with outward-
looking radial deformations, lead to contrary bending moments and circum-
ferential forces. Nevertheless, the buckling shape for concave and convex con-
nection is comparable to the single conical shell. It is a local buckle near the
shell connection limited to the conical shell. Overall buckling does not occur.

Figure 6 shows the buckling loads of cylinder-cone configurations in com-
parison to theGerman design ruleDAStRi 013. The shells are represented at
the slenderness r/t of the reference cylinder, where buckling occurs. The coin-
cidence between the calculated buckling loads and the design rule is obvious,
especially concave connections are comparable to single cylindrical or conical
shells. The numerical buckling loads of convex connections show a slightly dif-
ferent behaviour, founded in the differences of deformation and stresses of the
prebuckling region.

Fig. 6. Buckling loads of axially loaded cylinder-cone configurations

Elastic buckling of unstiffened cylinder-cone configurations is comparable
to the buckling cases of single shells and the calculated limit loads are such
high as the limit loads of stiffened structures. Expensive ring stiffeners are not
necessary.

Figure 7a shows the load-deformation behaviour of an unstiffened cylinder-
cone-cylinder configuration under axial compression for elastic and elasto-
plastic material, experimental results from Swadlo (2001). In case of elasto-
plastic material, the deformation behaviour is completely different from the



454 D.Dinkler, O.Knoke

Fig. 7. Comparison of experiment, elastic and inelastic buckling behaviour,
deformed configuration and critical buckling shape

case of elastic material. The ultimate load is much smaller than the bifurca-
tion load and also the lower buckling load. There is no kind of snap-back in
the load deformation path, rather a slow reduction of the ultimate load by
increasing deformations.

The behaviour is caused by large radial deformations at the shell connec-
tions, which lead to high circumferential tension nϕ, see Fig.8a. In connection
with the axial compression forces nα, material starts to yield at a very low
load level compared to the stiffened structure, see Fig.8c.

The failure of the entire structure is not a failure in terms of stability
but rather a material failure at the shell connections. Coupled material and
stability failure could only occur in case of very thin shells or material with
high yield stress. To obtain the limit as high as those for single shells, ring
stiffeners are necessary to reduce the circumferential tension stresses, what
may help us to avoid the material failure at the shell connections.

6. Conclusions

The paper presents a procedure of evaluating imperfection sensitivity of
shells and a buckling indicator for the design of shells. In contrast to other
methods and failure criteria, it is not intended to get the lower bounds of the



Elasto-plastic limit loads... 455

Fig. 8. Membrane forces and interaction

loaddeformationbehaviour,buta comparisonof thequantities of perturbation
with experiments. So far, the authors know from literature the perturbation
energyconcept is theonly concept, that is able todescribequality andquantity
of the imperfection sensitivity of elasto-plastic buckling shells in very good
agreement with experiments by means of a single scalar value π.

The concept is validated for buckling cases of cylindrical and conical
shells. The coincidence between numerical results and the German design ru-
le DASt Ri 013 is extremely good. In case of elasto-plastic behaviour, there
is a good agreement between the numerical investigations and the European
recommendations ECCS,what shows remarkable additional advantages of the
German design rule DIN 18800.



456 D.Dinkler, O.Knoke

In case of an elasticmaterial, the buckling behaviour of unstiffened combi-
ned shells such as cylinder-cone configurations is comparable to the validated
buckling cases of single shells. Limit loads of a unstiffened structures are such
high as the limit loads of comparable stiffened structures. Expensive ring stif-
feners are not necessary. Investigations concerning to inelastic buckling show
a completely different failure modes as compared to single shells. Here, ring
stiffeners at the shell connections are essential to increase the limit load of the
entire structure and can not be neglected.

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Elasto-plastic limit loads... 457

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Sprężysto-plastyczne obciążenia graniczne w układach

walcowo-stożkowych

Streszczenie

W pracy przedstawiono metodę wyznaczania obciążeń granicznych dla powłok
wrażliwych na niedoskonałości, z uwzględnieniem odkształceń plastycznych. Opiera-
jąc się na energii odkształcenia powłok rozważonoprzypadek najgorszego scenariusza
krytycznych perturbacji przy ustalonych poziomach obciążeń.Wartości obciążeń gra-
nicznych dla powłok cylindrycznych i stożkowych porównano z przyjętymi wEuropie
metodami projektowania. Koncepcja została zastosowana do analizy bardziej ogól-
nych przypadków utraty stateczności konstrukcji powłokowych.

Manuscript received January 24, 2003; accepted for print March 5, 2003