Jtam-A4.dvi
JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
51, 4, pp. 927-936, Warsaw 2013
STABILITY OF A POROUS-CELLULAR CYLINDRICAL SHELL
SUBJECTED TO COMBINED LOADS
Tomasz Belica
University of Zielona Gora, Institute of Computer Science and Production Management, Zielona Góra, Poland
e-mail: t.belica@ibem.uz.zgora.pl
Krzysztof Magnucki
Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland and
Institute of Rail Vehicles, Tabor, Poznań, Poland e-mail: krzysztof.magnucki@put.poznan.pl
The subject of the paper is a metal foam circular cylindrical shell subjected to combined
loads.Combinations of the external pressure and axial load are taken into account.The shell
is simply supported on all outer edges. The mechanical properties of the metal foam vary
continuously in the thickness direction. A non-linear hypothesis of deformation of a plane
cross section of the shell is formulated. The field of displacements of any cross section and
non-linear geometric relationships are assumed. The system of partial differential equations
for the shell is derived on the basis of the principle of stationarity of the total potential
energy. This system is approximately solved by the Bubnov-Galerkin method. The critical
loads for shells are numerically determined. Results of the calculation are shown in figures.
Key words: stability, cylindrical shell, porous-cellularmaterial
1. Introduction
Thinwalled circular cylindrical shells are often used inmany branches of industry. These struc-
tures are the building base of elementary structural parts of complex systems. The potential
applications include lightweight structuresmade of homogeneous, sandwich andmultilayer com-
posites. Themain assessment criterion of the practical application efficiency of these structures,
except for the economic aspect, is a relatively lowmass-strength ormass-rigidity ratio. However,
these constructions are sensitive to loss of stability, therefore, calculations of critical loads of the
shells are important elements of the analysis of the shells strength. Homogeneous and sandwich
cylindrical shells have been extensively investigated. The results of research have been presen-
ted in manymonographs, e.g. Volmir (1967), Doyle (2001). Magnucki andOstwald (2001) paid
particular attention to three-layered shells with two external facesmade of a steel and relatively
weak foam. Shen (1996) described the problem of the shell subjected to external pressure and
axial loads. Marcinowski (2003) presented numerical, geometrically non-linear static analysis of
sandwich plates and shells. In this paper, the concept of a degenerated finite element is used
to model the mechanical behaviour of sandwich constructions. Błachut (2010) presented the
numerical and experimental study buckling of axially compressed cylindrical shells with a non-
uniform axial length. It is assumed in this paper that the initial imperfection of the length had
a sinusoidal shape along the compressed edge. A review of selected problems of some aspects of
the strength, static stability, and optimisation of horizontal pressure vessels was presented by
Błachut andMagnucki (2008).
Current technologies allow creation of constructions of porous materials. Various methods
of technological processes of porousmaterials were presented by Banhart (2001). Investigations
and properties of these materials were presented by Bart-Smith et al. (2001), Ramamurty and
Paul (2004).
928 T. Belica, K.Magnucki
Magnucki and Stasiewicz (2004), Magnucka-Blandzi (2008), Magnucki et al. (2006a), carried
out analytical investigations of strength and stability of porous-cellular beams and plates. Ana-
lytical investigations of stability of the porous cylindrical shells were presented by Malinowski
andMagnucki (2005),Magnucki at al. (2006b) for static problems, Belica andMagnucki (2007),
Belica at al. (2011) for dynamic problems. These authors assumed the non-linear description of
the deformation cross-section of the wall of the shell. The transverse shear deformation effect
was taken into account. The shear effect does not occur on the external surfaces of the shell.
The proposed model of deformation of a plane cross-section is included in the Higher Order
Theory group, and it is the generalization of well-known theories. The higher order hypotheses
including shear deformation of beams and plates were presented by Wang et al. (2000). That
monograph illustrates how the shear deformation theories provide accurate solutions compared
to the classical theory.
2. Theoretical model
The subject of the paper is an isotropic porous-cellular cylindrical shell subject to a combined
load: axial force and external pressure (Fig. 1). Basic dimensions and relations are determined
in the cylindrical coordinate system. The shell is simply supported on all outer edges.
Fig. 1. Metal foam circular cylindrical shell
The shell ismadeof aporousmetalwithmechanical properties varying through the thickness
of the shell. The external surfaces (z = t/2 and z =−t/2) aremade of a homogeneousmaterial.
The mechanical properties are symmetrical and continuous with regard to the middle surface
of the shell (Fig. 2). Therefore, the defined porous material is a generalization of multi-layered
structures.
Fig. 2. Scheme of a porous-cellular shell structure
The porousmetal is of continuous structure. For each layer of the ζ coordinate, thematerial
is isotropic, while its mechanical properties vary on the thickness. The moduli of elasticity are
defined in the following form
E(ζ)= E1[1−e0cos(πζ)] G(ζ)= G1[1−e0cos(πζ)] (2.1)
Stability of a porous-cellular cylindrical shell subjected to combined loads 929
where e0 = 1 − (E0/E1) = 1 − (G0/G1) is the dimensionless parameter of the porosity
(0¬ e0 < 1); E0, E1 and G0, G1 are Young’s and the shear moduli for ζ =0 and ζ =±1/2,
respectively; ζ = z/t is the dimensionless coordinate; t is thickness of the shell. The relationship
between themoduli of elasticity is defined in the following form Gj = Ej/[2(1+ν)], for j =0,1.
The physicalmodel of a non-linear hypothesis of deformation of the shell plane cross-section
is shown in Fig. 3. The cross-section, being initially a planar surface, becomes curved after
the deformation. It is assumed that the boundaries of the curved surface of the deflected shell
cross-section remain perpendicular to the outer surfaces of the shell. This geometric model is
analogous to the broken line hypothesis applied to three layered structures.
Fig. 3. Deformation of the plane cross section scheme
The field of displacement is assumed in the following form
u(x,ϕ,ζ) = u0(x,ϕ)− t
[
ζ
∂w
∂x
−
1
π
ψ(x,ϕ)sin(πζ)
]
v(x,ϕ,ζ) = v0(x,ϕ)− t
[
ζ
∂w
r∂ϕ
−
1
π
φ(x,ϕ)sin(πζ)
]
w(x,ϕ,ζ) = w(x,ϕ,0)= w(x,ϕ)
(2.2)
where u0, v0 are the tangential displacements along the x and ϕ coordinates; φ and ψ are
the dimensionless functions of displacements; w is the deflection of the shell (the transverse
displacement along the z coordinate). The geometric relationships – components of the strain
field – take the following forms
εx =
∂u
∂x
+
1
2
(∂w
∂x
)2
εϕ =
∂v
r∂ϕ
−
w
r
+
1
2
( ∂w
r∂ϕ
)2
γxϕ =
∂u
r∂ϕ
+
∂v
∂x
+
∂w
∂x
∂w
r∂ϕ
γxz =
∂u
∂z
+
∂w
∂x
γϕz =
∂v
∂z
+
∂w
r∂ϕ
(2.3)
The physical relationships, according to Hooke’s law, are
σx =
E(ζ)
1−ν2
(εx+νεϕ) σϕ =
E(ζ)
1−ν2
(εϕ+νεx)
τxϕ = G(ζ)γxϕ τxz = G(ζ)γxz τϕz = G(ζ)γϕz
(2.4)
930 T. Belica, K.Magnucki
3. Equations of stability
The system of five partial differential equations obtained from the principle of stationarity of
the total potential energy of the porous cylindrical shell under external pressure and intensity
of the axial force
δ(Uε−W)= 0 (3.1)
where Uε is the potential energy of the elastic strain
Uε =
t
2
2π∫
0
L∫
0
1/2∫
−1/2
(σxεx+σϕεϕ+ τxϕγxϕ+ τxzγxz + τϕzγϕz)r dζ dx dϕ (3.2)
W is the work of the load
W =
2π∫
0
L∫
0
(pw)r dx dϕ+
1
t
2π∫
0
1/2∫
−1/2
(Nxu)r dζ dϕ (3.3)
Introducing Eqs. (2.1)-(2.4) and Eqs. (3.2) and (3.3) into principle (3.1), a system of five equili-
brium equations: δu0 (3.4)1, δv0 (3.4)2, δψ (3.5)1, δϕ (3.5)2, δw (3.6) are obtained
∂2u0
∂x2
+
1−ν
2
∂2u0
r2∂ϕ2
+
1+ν
2
( ∂w
r∂ϕ
∂2w
r∂x∂ϕ
+
∂2v0
r∂x∂ϕ
)
+
∂w
∂x
(∂2w
∂x2
+
1−ν
2
∂2w
r2∂ϕ2
−
ν
r
)
=0
∂2v0
r2∂ϕ2
+
1−ν
2
∂2v0
∂x2
+
1+ν
2
(∂w
∂x
∂2w
r∂x∂ϕ
+
∂2u0
r∂x∂ϕ
)
+
∂w
r∂ϕ
( ∂2w
r2∂ϕ2
+
1−ν
2
∂2w
∂x2
−
1
r
)
=0
(3.4)
and
C2
∂
∂x
(∇2w)−C3
(∂2ψ
∂x2
+
1−ν
2
∂2ψ
r2∂ϕ2
+
1+ν
2
∂2φ
r∂x∂ϕ
)
+C4
1−ν
2t2
ψ =0
C2
∂
r∂ϕ
(∇2w)−C3
( ∂2φ
r2∂ϕ2
+
1−ν
2
∂2φ
∂x2
+
1+ν
2
∂2ψ
r∂x∂ϕ
)
+C4
1−ν
2t2
φ =0
(3.5)
and
E1t
1−ν2
{
C0
{
−
[1
2
(∂w
∂x
)2
+
∂u0
∂x
](∂2w
∂x2
+ν
∂2w
r2∂ϕ2
+
ν
r
)
−
[1
2
( ∂w
r∂ϕ
)2
+
∂v0
r∂ϕ
−
w
r
](
ν
∂2w
∂x2
+
∂2w
r2∂ϕ2
+
1
r
)
+(1−ν)
∂2w
r∂x∂ϕ
(∂w
∂x
∂w
r∂ϕ
+
∂u0
r∂ϕ
+
∂v0
∂x
)}
+C1t
2(∇4w)
−C2t
2
[ ∂
∂x
(∇2ψ)+
∂
r∂ϕ
(∇2φ)
]}
= p
(3.6)
where the Laplace operator is defined as ∇2 = ∂2/∂x2 +∂2/(r2∂ϕ2), the bi-harmonic Laplace
operator has the following form ∇4 = ∂4/∂x4+2∂4/(r2∂x2∂ϕ2)+∂4/(r4∂ϕ4) and constants are
C0 =1−
2
π
e0 C1 =
1
12
−
π2−8
2π3
e0 C2 =
1
π2
(2
π
−
1
4
5e0
)
C3 =
1
π2
(1
2
−
2
3π
e0
)
C4 =
1
2
−
4
3π
e0
Stability of a porous-cellular cylindrical shell subjected to combined loads 931
The loads in the directions x (Nx) and ϕ (Nϕ) and the tangential load Sxϕ in the plane xϕ
are defined. Now, inserting a stress function F(x,ϕ) and a displacement function Φ(x,ϕ), we
obtain
ψ =
∂Φ
∂x
φ =
∂Φ
r∂ϕ
(3.7)
The normal and tangential loads expressed with stress functions may be written as
Nx =
∂2F
r2∂ϕ2
Nϕ =
∂2F
∂x2
Sxϕ =−
∂2F
r∂x∂ϕ
(3.8)
The systemof equations (3.4)-(3.6) is reduced to twodifferential equations thanks to introducing
functions (3.7) and (3.8). When we use the equation of strain continuity, a system of three
fundamental equations of stability is obtained
E1t
3
1−ν2
[C1∇
4(w)−C2∇
4(Φ)]−L(w,F)−
1
r
∂2F
∂x2
−p =0
1
C0E1t
∇
4(F)=−
1
2
L(w,w)−
1
r
∂2w
∂x2
C2∇
4(w)−C3∇
4(Φ)+C4
1−ν
2t2
∇
2(Φ)= 0
(3.9)
where the non-linear operators are defined as
L(w,F) =
∂2w
∂x2
∂2F
r2∂ϕ2
−2
∂2w
r∂x∂ϕ
∂2F
r∂x∂ϕ
+
∂2w
r2∂ϕ2
∂2F
∂x2
L(w,w) = 2
∂2w
∂x2
∂2w
r2∂ϕ2
−2
( ∂2w
r∂x∂ϕ
)2
The boundary conditions for x = 0 and x = L are formulated in the form
∂2F
r2∂ϕ2
∣∣∣∣
x=0;x=L
= N0x Mx
∣∣∣
x=0;x=L
=0 w
∣∣∣
x=0;x=L
=0 (3.10)
where
Mx = t
1/2∫
−1/2
ζσx dζ =−
E1t
3
1−ν2
[
C1
(∂2w
∂x2
+ν
∂2w
r2∂ϕ2
)
−C2
(∂ψ
∂x
+ν
∂φ
r∂ϕ
)]
The system of three equations (3.9) is approximately solved. The two unknown functions are
assumed in the following forms
w(x,ϕ) = w1 sinX cosY +2w2 sin
2X
Φ(x,ϕ) = w1αΦ1 sinX cosY +2w2αΦ2 sin
2X
(3.11)
where: X = mπx/L and Y = nϕ, n is the number of waves on the circuit, m is the number of
half-waves in the longitudinal direction of the shell, w1 and w2 are the amplitude parameters of
the deflection surface in formofwaves occurring along axial and circumferential directions of the
shell, respectively. These functions satisfy boundary conditions (3.10) for the simply supported
shell.
Substitution of the equations (3.11) for Eq. (3.9)2 provides the stress function
F = C0E1t[w
2
1(αf1cos2X −αf2cos2Y )+w1w2(αf3 sin3X cosY −αf4 sinX cosY )
−w2rαf5cos2X +w1rαf6 sinX cosY ]−
1
2
(N0xr
2ϕ2+N0ϕx
2)
(3.12)
932 T. Belica, K.Magnucki
where
αf1 =
k21
32
αf2 =
1
32k21
αf3 =
2k21
(9+k21)
2
αf4 =
2k21
(1+k21)
2
αf5 =
(k1
2n
)2
αf6 =
αf4
2n2
k1 =
nL
mπr
Equations (3.9)1 and (3.9)3 are solved with the use of the orthogonalization Bubnov-Galerkin
(B-G) method
2π∫
0
L∫
0
ℜ(x,ϕ)sinX cosY r dx dϕ =0
2π∫
0
L∫
0
ℜ(x,ϕ)sin2Xr dx dϕ =0 (3.13)
where ℜ(x,ϕ) stands for the left side of Eqs. (3.9)1 and (3.9)3.
The following parameters are obtained with equation (3.9)3
αΦ1 =
C2
C3+k2C4
αΦ2 =
C2
C3+k3C4
k2 =
4k3
1+k21
k3 =
1−ν
8
( L
mπt
)2
(3.14)
Substitution of expressions (3.11), (3.12) and (3.14) forEq. (3.9)1, and using of theB-Gmethod,
gives a system of equations
1
E1t
(Nx+Nϕk
2
1)−αw1−αw3w̃
2
1 +αw5w̃2−αw4w̃
2
2 =0
8
Nx
E1t
w̃2−8αw2w̃2−αw4w̃
2
1w̃2+αw6w̃
2
1 =0
(3.15)
From equation (3.15)2, the parameter is designated w̃2 and substituted to equation (3.15)1.
Finally, a non-linear algebraic equation enabling the analysis of stability loss of the investigated
shell is given
N0 =
E1t
k0+k
2
1(1−k0)
[αw1+(αw3−αw5αw7)w̃
2
1 +αw4α
2
w7w̃
4
1] (3.16)
where w̃1 = w1/t and w̃2 = w2/t are dimensionless parameters of the deflection; N0 [N/mm] is
the external load; k0 is the dimensionless parameter of the load, 0¬ k0 ¬ 1; Nϕ [N/mm] is the
intensity of the circumferential load, Nϕ =(1−k0)N0 = pr; Nx [N/mm] is the intensity of the
axial load, Nx = k0N0; p [MPa] is the external pressure; and the dimensionless parameters are
αw1 = C0αf6+
t2
1−ν2
(mπ
L
)2
(1+k21)
2(C1−C2αΦ1)
αw2 =
4
1−ν2
(
mπt
L
)2
(C1−C2αΦ2)+C0αf5
αw3 =2C0
(nt
r
)2
(αf1+αf2) αw4 =2C0
(nt
r
)2
(αf3+αf4)
αw5 =
1
2
C0
t
r
(k21 +4αf4) αw6 = C0
t
r
(8αf1+αf4)
αw7 =
αw6
8αw2+αw4w̃
2
1 −8
N0k0
E1t
The static upper critical load of the shell has been obtained after the removal of geometric
non-linear relationships (components of the strain) in Eqs. (3.16)
N0,cr =min
m,n
{ E1
k0+k
2
1(1−k0)
[
C0αf6+
t2
1−ν2
(mπ
L
)2
(1+k21)
2(C1−C2αΦ1)
]}
(3.17)
Stability of a porous-cellular cylindrical shell subjected to combined loads 933
4. Numerical calculations
Equation (3.16) has been solved by a numerical-iterative method. It has been assumed that the
maximummistake equaled 0.01%. In the first iteration, the worth of parameter αw7 is assumed
without the element 8N0k0/(E1t).
The condition of allowable stress is assumed σeq,max ¬ σall, where σall – allowable stress for
the shell material, σeq,max –maximum equivalent stress
σeq,max =
N0,cr
C0t
√
1−3k0(1−k0) (4.1)
The detailed numerical analysis of the porous shell is carried out for the following data:
E1 =7.06 ·10
4MPa, ν =0.33, σall =90MPa, L/r =3, r/t =200 (600).
The shells have been investigated for different coefficients of porosity for different loads:
external pressure, axial load and combinations of these loads. The results of these calculations
are presented in Table 1 and in the Figs. 4-7. In Table 1, the upper and lower critical loads of
the shells are presented. Defining the lower critical loads, N0,low assumed the lowest value N0
for the shell for different numbers m and n.
Table 1.The upper and lower critical loads of the studied shells
e0 N0,cr/t [MPa] m n N0,low/t [MPa] m n N0,low/N0,cr
External pressure Nϕ = N0; (r/t =200)
0 8.02 1 6 5.80 1 5 0.72
0.5 6.33 1 6 4.57 1 5 0.72
0.9 4.98 1 6 3.59 1 5 0.72
Combined loads Nx =0.9N0; Nϕ =0.1N0; (r/t =200)
0 62.96 1 6 37.23 1 5 0.59
0.5 49.72 1 6 26.45 1 4 0.53
0.9 37.00 1 5 19.16 1 4 0.52
Axial load Nx = N0; (r/t =600)
0 71.96 4 13 13.74 5 10 0.19
0.5 53.8 3 11 10.43 5 10 0.19
0.9 38.66 4 12 7.68 5 9 0.20
In Figs. 4-6, the equilibrium paths of the cylindrical shells for the following geometric data
is presented: L/r = 3 and r/t = 200 for the shell subjected to the external pressure (Fig. 4)
and combined loads (Fig. 5); L/r = 3 and r/t = 600 for the shell subjected to the axial load
(Fig. 6).
Fig. 4. Equilibrium paths of the cylindrical shells subjected to external pressure
934 T. Belica, K.Magnucki
Fig. 5. Equilibrium paths of the cylindrical shells subjected to combined loads
Fig. 6. Equilibrium paths of the cylindrical shells subjected to the axial load
The calculation results are presented for different values of the parameters m, n. The influ-
ence of the load parameter k0 on the relationship between the upper and lower critical loads is
shown in Fig. 7.
Fig. 7. Upper and lower critical loads depending on the parameter k0
5. Conclusions
The problem of the stability of isotropic circular cylindrical shells made of a porous-cellular
material, subjected to combined loads is presented in the paper. On the basis of the non-
-linear description of the deformation cross-section of thewall of the shell, relations between the
parameters of deflection and load considering the shear effect have been obtained. The condition
Stability of a porous-cellular cylindrical shell subjected to combined loads 935
of free external surfaces of the shell is satisfied simultaneously. The introduced equations have
the universal character in the range of:
• the change of physical properties of thematerial (parameter e0); for e0 =0, the structure
is homogeneous, and for 0 < e0 < 1, it is a porous-cellular one;
• the combination of the external load (parameter k0); if k0 = 0, then only the external
pressure works, if k0 = 1, then there is an axial load, and if 0 < k0 < 1, then there is a
combined external load.
The solution of static stability problemof porous cylindrical shellswere presentedbyMalinowski
andMagnucki (2005),Magnucki at al. (2006b). In thoseworks, only geometrically linear relations
between the displacement anddeformationwere assumed.On the basis of these foundations, one
can calculate theupper critical loads.Amoredifficult problem is the non-linear stability analysis
of the shells, presented in this work. This analysis allows the study of dependence between the
external loads and deflection of the shell as well as the designation of the lower critical loads.
This is particularly important since, in most cases, the real value of the load causing buckling
of the shell is in the range between the upper and lower theoretical value of the critical load.
This problem confirmed the results of experimental and theoretical study conducted by many
researchers due to small imperfections that every real shell (deviation from the ideal geometric
shape of shell, non-homogenous of shell material, irregular load distribution, etc.) has. It is
worth noting that larger imperfections of the shell cause reduction of the real value of the load
at which the shell is buckled (even below the lower critical load). The exact consideration of
various factors in theoretical studies of shells is not possible (only approximate solutions exist).
The influence of imperfections on the buckling of porous-cellular cylindrical shells will be the
subject of the further research.
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Manuscript received January 17, 2013; accepted for print March 26, 2013