CONTINUUM-BASED FINITE ELEMENTS FOR SHAKEDOWN ANALYSIS OF PIPELINES AND PRESSURE VESSELS Ricardo Rodrigues Martins Tese de Doutorado apresentada ao Programa de Pós-graduação em Engenharia Mecânica, COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Doutor em Engenharia Mecânica. Orientadores: Nestor Alberto Zouain Pereira Lavinia Maria Sanábio Alves Borges Rio de Janeiro Setembro de 2013
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CONTINUUM-BASED FINITE ELEMENTS FOR SHAKEDOWN ANALYSIS OF
PIPELINES AND PRESSURE VESSELS
Ricardo Rodrigues Martins
Tese de Doutorado apresentada ao Programa de
Pós-graduação em Engenharia Mecânica,
COPPE, da Universidade Federal do Rio de
Janeiro, como parte dos requisitos necessários à
obtenção do título de Doutor em Engenharia
Mecânica.
Orientadores: Nestor Alberto Zouain Pereira
Lavinia Maria Sanábio Alves Borges
Rio de Janeiro
Setembro de 2013
iii
Martins, Ricardo Rodrigues
Continuum-based finite elements for shakedown
analysis of pipelines and pressure vessels/ Ricardo
Rodrigues Martins. – Rio de Janeiro: UFRJ/COPPE,
2013.
XV, 138 p.: il.; 29,7 cm.
Orientadores: Nestor Alberto Zouain Pereira
Lavinia Maria Sanábio Alves Borges
Tese (doutorado) – UFRJ/ COPPE/ Programa de
Engenharia Mecânica, 2013.
Referências Bibliográficas: p. 127-138.
1. Direct methods. 2. Shakedown. 3. Finite Element
Method. I. Pereira, Nestor Alberto Zouain, et al. II.
Universidade Federal do Rio de Janeiro, COPPE,
Programa de Engenharia Mecânica. III. Título.
To God and my family
iv
Acknowledgments
I would like to express my sincere gratitude to all who contributed to this work. First
of all, I fully acknowledge the fundamental support given by Petroleo Brasileiro
S.A. (PETROBRAS) which believes in the potential of its collaborators, giving
us excellent opportunities for continuous learning. In particular, I would like to
thank Petrobras Research and Development Center (CENPES ), in the person of
the managers, Arthur Curty Saad and Luiz Augusto Petrus Levy, and all those
from PETROBRAS who trusted in this project from the very beginning and made
it possible the conduction of this research.
I am fully thankful to my advisor, Prof. Nestor Zouain, for his availability,
guidance, patience and friendship. This work can be regarded in great part as a
result of his valuable ideas. All the steps of this thesis were carried out under
his careful supervision, and I am quite sure that this work would not be the same
without his support.
In a similar way, I would like to express my gratitude to my co-advisor, Prof.
Lavinia Borges. She provided excellent contributions in different phases of this work.
Her enthusiasm helped me to be always motivated, especially during the hard times.
I am totally grateful to Prof. Eduardo de Souza Neto, who accepted to be my
supervisor during my stay as visiting researcher at Swansea University. His contri-
butions and insights were very important and improved the final results of the thesis.
Moreover, his friendship created an exceptional environment to the conduction of
part of this work in Wales.
I would like to express my gratitude to my colleagues from Pipelines and Risers
Department (CENPES/PDEP/TDUT), especially those who I have been working
The above equations are used in Fig. 5.4 to verify the deformed shape of the arch for
the finite element models with R/h = 700. Three distinct FE meshes with increasing
degree of discretization are considered. It can be seen a very good agreement between
analytical and numerical results.
5.6.2 Collapse of a beam in tension plus bending
As an example of limit analysis consider a beam with rectangular cross section (h
is the height and b is the width) under uniform external axial load N and external
bending moment M , as sketched in Fig. 5.5.
The axial stress σx is a scalar function of the transversal coordinate z and the
equilibrium equations are:
N = b
∫ h/2
−h/2σxdz, M = −b
∫ h/2
−h/2σxzdz (5.118)
The material of the beam is perfectly plastic with yield stress σY . Plastic ad-
missibility is determined by:
P : −σY ≤ σx(z) ≤ σY (5.119)
The collapse limits for pure traction and bending are, respectively, NY = bhσY
and MY = bh2σY /4.
Considering that the system of loads is defined by the following non-dimensional
load parameters n = N/NY and m = M/MY and recalling that the loading (nc,mc)
producing instantaneous plastic collapse satisfies n2c + mc = 1, hence, the collapse
49
Figure 5.3: Nodal displacements and rotation for the semi-circular arch consideringvarious slenderness ratios
50
Figure 5.4: Deformation of the circular arch: analytical solution and FE results
Figure 5.5: Beam under axial traction and bending moment
51
Figure 5.6: Collapse limits for a beam section under axial traction and bendingmoment: analytical solution and FE results. Left: discontinuous stress field betweenlayers. Right: continuous stress field between layers.
factor α for the load (n,m) is determined analytically by solving
(αn)2 + αm = 1 (5.120)
Figure 5.6 depicts a comparison between the analytical solution given by (5.120)
and the results obtained with the CB beam element. The structure is modeled
with just one element and different number of layers: 2, 4, 8 and 16. The number
of layers are increased to assess the effect of enriching interpolated stress fields on
element results. The graph on the left presents the results obtained assuming that
the stress field is discontinuous between adjacent layers whereas the graph on the
right depicts the results obtained when continuity of stresses is enforced. It can be
seen that, the higher the number of layers the closer numerical and analytical results
are, as expected. For elements with 8 or more layers numerical and analytical results
are quite close.
Considering a given number of layers, elements with no continuity of stress fields
between layers are in general more accurate. For the elements with 8 layers the
maximum difference to the analytical collapse loads is less than 1% for the element
with discontinuous stress fields and less than 3% for the element with continuous
stress fields (see Tab. 5.1). This better performance of elements with no continuity
of stress is also expected since axial stress distribution through the thickness is not
continuous in the collapse of a elastic-perfectly plastic beam subjected to bending
plus traction.
52
Table 5.1: Maximum differences between analytical and FE resultsNumber of Elem. discont. stress field Elem. cont. stress field
On the other hand, elements without continuity of stress field between adjacent
layers are computationally much more expensive than the respective version with
enforced continuity between layers. Indeed, when the number of layers increases
the number of stress nodes of an element with discontinuous stress fields between
layers tends to be twice the number of stress nodes of an respective element with
continuous stress fields between layers. The stress nodes are the points where plastic
admissibility is verified by the optimization algorithm for shakedown analysis. Then,
if the number of stress nodes increases the computational cost increases as well.
It is worth emphasizing that for this example the accuracy of results of elements
with similar number of stress nodes are comparable as shown in Tab. 5.1.
5.6.3 Shakedown analysis in tension plus bending
The structure considered in this example is exactly the same presented in the previ-
ous example except for the loads which are slightly modified to allow a shakedown
analysis. In this case, the axial load N remains fixed whereas the external bending
moment M can vary, i.e., −M ≤ M ≤ M . The solution for this problem is well
known [10, 69] and is provided here in the form of the Bree diagram shown in Figure
5.7. The blue curve labeled with C in Figure 5.7 indicates the collapse limit for the
loads already given in the previous example. The red curve represents the elastic
shakedown limits, and the green line is the pure elasticity boundary. Labels AP
and IC indicate respectively alternating plasticity and incremental collapse, which
are the failure mechanisms for loads beyond elastic shakedown limits. The objective
of this example is to compare the analytical limit for elastic shakedown with finite
element results obtained with the mixed CB beam element.
The numerical models for this example are identical to those employed for limit
analysis. However, in this case only three number of layers are considered: 2, 4 and
8. The finite element results are presented in Figure 5.8. The graph on the left in
Figure 5.8 depicts results for the element with discontinuous stresses between layers.
The graph on the right, in turn, shows the results for the element with continuous
stress field within the element. Note that all simulations present the same outcome.
Indeed, even the element with 2 layers and continuity of stress, which was poor
53
Figure 5.7: Bree diagram for a beam under constant axial traction and a variableuniform bending moment, ZOUAIN [10].
for the determination of accurate collapse limit, is capable of capturing the correct
limits for shakedown in this particular case.
54
Figure 5.8: Bree diagram for shakedown analysis of a beam under constant axialtraction and a variable uniform bending moment:analytical solution and FE re-sults. Left: discontinuous stress field between layers. Right: continuous stress fieldbetween layers.
55
Chapter 6
Continuum-based Axisymmetric
Shell Element
In this chapter a three-node shell element is devised for the solution of axisymmetric
shakedown problems.
This element is analogous to the beam element presented in the previous chapter.
Indeed, due to the mathematical similarity between the axisymmentric problem and
the plane stress (or plane strain) problem, the equations for this axisymmetric shell
element can be readily derived from the equations of the CB beam element by
considering that [see 61, Cap.5]:
1. In a body with symmetry of revolution in geometry and loads, the two com-
ponents of displacement in any plane section passing through its axis of sym-
metry define completely the state of strain and, consequently, the stresses on
the body;
2. The global reference frame of an axisymmetric body is given by the radial and
axial coordinates of the point. By using this reference frame instead of the
global reference frame employed for the CB beam element, one can realize that
the interpolation functions of the axisymmetric element and beam element are
identical;
3. All integrations of the axisymmetric element must consider the volume defined
by rotating the area of the element around its axis of revolution;
4. In axisymmetric bodies any displacement in radial direction induces strains
and stresses in circumferential direction. Therefore, a third component of
strain and stress in circumferential direction must be considered in the axisym-
metric shell element. This is the most remarkable difference between the
axisymmetric finite element and the CB beam element.
56
Figure 6.1: Continuum-based axisymmetric shell element: displacement nodes (mas-ter and slave) and stress nodes
Although there are in literature various works focusing on the limit analysis of
axisymmetric structures, references on shakedown analysis are rare and shell ele-
ments are seldom used. Some numerical examples of shakedown analysis of axisym-
metric structures can be found, for instance, in [41, 70–76], most of them employing
only solid elements. In this work the outcomes of the proposed axisymmetric shell
element [77] are compared to the results from solid finite element models and avai-
lable analytical solutions after the detailed presentation of the element formulation.
In what follows we organize the sections with the same divisions used in the
previous chapter. Some intermediate steps of calculation presented for the CB beam
element are omitted in this chapter to make the text more concise. However, it is
our intention to make the implementation of each finite element easier by giving
in its respective chapter the equations in their final form. For this reason and for
the sake of clarity of the exposition, some formulae previously introduced may be
repeated in this chapter.
6.1 Geometry, displacements and strains
The global (fixed) orthonormal reference frame (see Fig 6.1) is denoted R =
er, ez ≡ ex1 , ex2 with ez corresponding to the axis of symmetry. The circum-
ferential direction, perpendicular to the axisymmetric half-plane, is represented by
eϕ = er ∧ ez.
We remark that equations describing the geometry and the displacements of this
CB axisymmetric element are exactly the same expressions presented in Sect. 5.1.1–
5.1.3 to define the geometry and the displacements of the CB beam element with
57
coordinates x and y replaced by r and z respectively.
6.1.1 Master and slave nodes
In the one-dimensional element i there are three master nodes and three correspon-
ding directors, respectively denoted as:
xa = raer + zaez, a=1:3, (6.1)
pa = cos θaer + sin θaez, a=1:3. (6.2)
The respective underlying continuum element has six slave nodes:
xα = rαer + zαez, α=s1:s6. (6.3)
Using the notation introduced in (5.4) we can also relate master and slave nodes
by the formulas (5.5) and (5.6), i.e.,
xa− = xa − ha
2pa, a=1:3, (6.4)
xa+ = xa + ha
2pa, a=1:3, (6.5)
where ha is the shell thickness at node a.
A fiber through the master node a and parallel to the director pa (a pseudo
normal) moves rigidly. Thus, the velocities va, va− and va+ of master and slave
nodes are related by the angular velocity ωaeϕ of the director pa through the same
equations (5.7)-(5.10) used in the CB beam formulation
va− = va − ha
2ωaeϕ ∧ pa ≡ va + ωaeϕ ∧
(xa− − xa
), a=1:3, (6.6)
va+ = va + ha
2ωaeϕ ∧ pa ≡ va + ωaeϕ ∧
(xa+ − xa
), a=1:3. (6.7)
or in the symbolic notation (5.11)[va−
va+
]= T i,a
[va
ωa
]where T a,i :=
[1 2 eϕ ∧ (xa− − xa)
1 2 eϕ ∧ (xa+ − xa)
]. (6.8)
Relations between velocities of master and slave nodes can be also collected in
the intrinsic equation (5.12) which is given in global coordinates as
[vs,i]R
=[T i]R [
vi]R, (6.9)
58
with
[vs,i]R
=[vs1r vs1z . . . vs6r vs6z
]R,T(6.10)
=[v1−r v1−
z . . . v3+r v3+
z
]R,T, (6.11)
[vi]R
=[v1r v1
z ω1 . . . v3r v3
z ω3]R,T
, (6.12)
and
[T a,i
]R=
1 0 za − za−
0 1 ra− − ra
1 0 za − za+
0 1 ra+ − ra
R
. (6.13)
6.1.2 Kinematics of the underlying continuum element
The geometry of the CB axissymetric shell element is mapped with the same equa-
tions employed for the CB beam element. The parent element domain is defined by
Ω := (ξ, η) ∈ [−1, 1]× [−1, 1] and the geometry mapping in the i-th finite element
domain Bi is given by (5.22),i.e.,
Ω 3 ξ = (ξ, η) 7→ x(ξ)|Bi =∑
α=s1:s6
gα(ξ)xα, (6.14)
with the symbol gα(ξ) denoting the Lagrange interpolation functions g1 = (1/4)ξ(ξ−1)(1− η), g2 = (1/4)ξ(ξ − 1)(1 + η), g3 = (1/2)(1− ξ)(1 + ξ)(1− η), g4 = (1/2)(1−ξ)(1 + ξ)(1 + η), g5 = (1/4)ξ(ξ + 1)(1− η) and g6 = (1/4)ξ(ξ + 1)(1 + η).
The curvilinear coordinate system R(ξ) = ex, ey, used to enforce the hy-
pothesis of zero transverse normal stress, is obtained by defining the base vectors
ex tangent and ey normal to the lamina (see Figure 6.1) as
ex :=x,ξ‖x,ξ‖
=r,ξe
r + z,ξez(
r2,ξ + z2
,ξ
)1/2, (6.15)
ey := eϕ ∧ ex =−z,ξer + r,ξe
z(r2,ξ + z2
,ξ
)1/2, (6.16)
where subscript “, ξ”denotes the corresponding derivative.
Change of basis from R(ξ) to R is accomplished by using the general formulas
for vectors and second order tensors (5.32) with
R(ξ) :=
[er · ex er · ey
ez · ex ez · ey
]. (6.17)
59
The Jacobian of the geometry mapping (J(ξ) in (5.34)) is given by[r,ξ r,η
z,ξ z,η
]R=
∑α=s1:s12
[rαgα,ξ rαgα,η
zαgα,ξ zαgα,η
]R. (6.18)
and the derivatives of the interpolation functions with respect to spatial coordinates
are obtained by substituting the inverse of the Jacobian in the equation (5.39), i.e.[gα,r
gα,z
]R=
1
r,ξz,η − r,ηz,ξ
[z,η −z,ξ−r,η r,ξ
]R [gα,ξ
gα,η
](6.19)
for α=s1:s6.
6.1.3 The interpolation of displacements and velocities
The assumed displacement and velocity fields are interpolated using exactly the same
equations employed for the CB beam. The expressions presented in Sect. 5.1.3 are
not repeated here because they can be used for the CB axisymmetric shell element
without any modification.
6.1.4 Enforcing bending theory hypotheses
We assume that in the direction perpendicular to the local laminar axis both strain
and stress components are zero and keep all other non-zero strain components pos-
sible in an axisymmetric deformation. Accordingly, we use the notation for planar
tensors introduced in Sect. 5.1.4 to write
d = dxex ⊗ ex + dϕe
ϕ ⊗ eϕ + d(xy)
[1√2
(ex ⊗ ey + ey ⊗ ex
)](6.20)
and
σ = σxex ⊗ ex + σϕe
ϕ ⊗ eϕ + σ(xy)
[1√2
(ex ⊗ ey + ey ⊗ ex
)]. (6.21)
The relevant compatibility equations for the CB axisymmetric shell element are
dx = vx,x, (6.22)
dϕ =vrr, (6.23)
d(xy) = 1√2
(vx,y + vy,x) . (6.24)
Note that the component dϕ does not appear in the CB beam element formulation.
The velocity in (global) radial direction vr in (6.23) is computed from the velo-
60
city components in curvilinear coordinates applying (5.32) with Ri,j given in (6.17).
Accordingly
vr = R1,1vx + R1,2vy. (6.25)
6.1.5 Computing strain in a generic point
The computation of the infinitesimal strain rate at a generic (Gauss) point of the un-
derlying continuum and in terms of the interpolation parameters [vi]R is performed
with the formula (5.67)
[∇symx v(ξ)]R(ξ) = [∇sym
x Nv(ξ)]R(ξ)RT (ξ)[T i]R [
vi]R.
Note that in this equation the vector of strain rates ∇symx v is given in local la-
minar components whereas the vector of nodal velocities [vi]R is written in global
components.
The above equation is calculated by following the same procedure explained for
the CB beam element in subsection 5.1.5, i.e.,
1. Find the curvilinear coordinate system R(ξk) = ex, ey given by (6.15) and
(6.16) and the rotation matrix R(ξk) using (6.17);
2. Compute, by using
[vs,i]R(ξk)
= RT (ξk)[T i]R [
vi]R
all slave node velocities in the curvilinear system of the generic point;
3. For α=s1:s6 apply the change of coordinates for slave node positions
xα = RT (ξk) xα; (6.26)
4. For α=s1:s6 find (gα,x, gα,y);
To this end, compute the Jacobian in the curvilinear frame using (6.18) with
(xa, ya) (from (6.26)) instead of (ra, za). Then, use (5.39) written for curvili-
near coordinates, i.e. analogous to (6.19).
5. Finally, compute the infinitesimal strain tensor
[∇sym
x v(ξk)]R(ξk)
=[∇sym
x Nv(ξk)]R(ξk) [
vs,i]R(ξk)
. (6.27)
61
Here (6.27) is written, discarding zero strain components, as
vx,xvrr
1√2
(vx,y + vy,x)
R(ξk)
=[G1 . . .G12
]R(ξk)
vs1x
vs1y...
vs6x
vs6y
R(ξk)
(6.28)
with auxiliary matrices Gα defined by
Gα(ξ) :=
gα,x 0
R1,1
rgα
R1,2
rgα
1√2gα,y
1√2gα,x
R
, α=s1:s6. (6.29)
6.2 Stresses
Layers are determined by a partition of the parent domain Ω =⋃`=1:L Ω` given in the
same way as in the CB beam element, i.e.: first, we select a sequence of coordinates
−1 = η0, η1, . . . , ηL = 1. Then, each parent layer is set as Ω` = (−1, 1)× (η`−1, η`).
Spatial layers are the image of these L parent layers through the geometry mapping
(6.14).
A local coordinate η ∈ (−1, 1), to be used in the restricted domain of one single
layer `, is defined as
η :=η − a`b`
, (6.30)
where a` := 12
(η` + η`−1) and b` := 12
(η` − η`−1).
Bending normal stress and shear components in each layer ` are interpolated
with bilinear functions. Each layer has four stress nodes located at corners and
σx(ξ)|Bi =∑j=1:4
tj(ξ)σj/`x , (6.31)
σϕ(ξ)|Bi =∑j=1:4
tj(ξ)σj/`ϕ , (6.32)
σ(xy)(ξ)|Bi =∑j=1:4
tj(ξ)σj/`(xy), (6.33)
where the interpolation parameters σj|`x , σ
j|`ϕ and σ
j|`(xy) are interpreted as the stress
components of the j-th stress node of layer ` in its own laminar directions. Note that
the component σj|`ϕ , not present in the CB beam element, is a remarkable difference
in the CB axisymmetric shell formulation.
62
The interpolation functions are the same employed in the CB beam, then
tj(ξ) = 14(1 + ξjξ)(1 + ηj η) ≡ 1
4(1 + ξjξ)
[1 +
ηj − a`b2`
(η − a`)], (6.34)
with (ξj, ηj) denoting the coordinates of node j. These are the usual bilinear func-
tions, just restricted to one layer, where an appropriate transversal coordinate was
defined in (6.30). This change of variable is also used to compute integrals by
numerical quadrature.
The above interpolation of stress is cast in matrix form as
[σ(ξ)]R = [Nσ(ξ)][σ`]
(6.35)
with (1 3 is the identity matrix 3× 3)
[Nσ(ξ)] = [t11 3 . . . t41 3] (6.36)
by defining in each layer `
[σ(ξ)]R =[σx σϕ σ(xy)
]T, (6.37)[
σ`]
=[σ1|` σ2|` σ3|` σ4|`]T , (6.38)
and in each stress node t of layer `
[σt|`]
=[σt|`x σt|`ϕ σ
t|`(xy)
]T. (6.39)
6.3 Yield function
The yield function f(σ) for the axisymmetric CB shell element is obtained from
the von Mises yield criterion (5.78) by considering the non-zero stress components.
Thus,
f(σ) = σ2x + σ2
ϕ − σxσϕ +3
2csσ
2(xy) − σ2
Y . (6.40)
The parameter cs is added to the model to consider or not the contribution of the
shear stresses to yielding. This coefficient which is set equal 1 by default can also
be set equal to 0 to disregard the shear stress component.
The gradient and the Hessian of the yield function are then
∇σf =
2σx − 1
2σϕ − 1
3csσ(xy)
and ∇σσf =
2 −1 0
−1 2 0
0 0 3cs
(6.41)
63
6.4 Discrete strain operator
For obtaining the discrete strain operator for the axisymmetric CB shell element
we firstly define in the discrete setting for each layer ` a work-conjugate strain rate
vector [d`]
:=[d1|` d2|` d3|` d4|`
]T, (6.42)
where [dt|`]
=[dt|`x dt|`ϕ d
t|`(xy)
]T(6.43)
and t denotes the stress nodes of layer `.
Then, we follow the steps presented for the CB beam element in Sect. 5.4 to
identify the layer discrete strain operator as
B`,i := B`,islave
[T i]R
(6.44)
where
B`,islave :=
2π
∫Ω`
[As1,`,i . . .As6,`,i
]RRT r(ξ) detJ dξ dη
(6.45)
with the auxiliary matrix A defined as
A`,i(ξ) := [Nσ(ξ)]T [∇symx Nv(ξ)]R(ξ) . (6.46)
Then, by considering (6.29) and (7.57), the matrix A is computed as
A`,i(ξ) =
t11 3
t21 3
t31 3
t41 3
[G1 G2 . . . G5 G6
]R. (6.47)
Alternatively,
A`,i(ξ) =[As1,`,i . . .As6,`,i
]R, (6.48)
with
Aα,`,i =
t11 3
t21 3
t31 3
t41 3
[Gα
]R, (6.49)
64
or, explicitly,
Aα,`,i =
t1gα,x 0
R1,1
rt1gα
R1,2
rt1gα
1√2t1gα,y
1√2t1gα,x
......
t4gα,x 0
R1,1
rt4gα
R1,2
rt4gα
1√2t4gα,y
1√2t4gα,x
R
. (6.50)
Finally, the element strain operator is obtained by assembling the contributions
of each layer:
Bi =∑`
B`,i. (6.51)
The assembly in (6.51) is easily performed by means of well-known procedures
employed in finite element analysis and depends on whether or not continuity of
stress fields between layers is enforced. When stress fields are discontinuous between
layers, the assembly is simply
Bi =
B1,islave...
BL,islave
[T i]R. (6.52)
6.5 Discrete elastic relation
In this section we compute the discrete form of the elastic operator for the axisymme-
tric CB shell element denoted by E. This matrix is only used in the data preparation
for the analysis, when the discrete approximation of the ideally elastic stress fields
σE, for each extreme loading, are computed. This elastic pre-analysis gives rise to
the practical definition of the prescribed domain of loading variations introduced in
(2.20) and (2.32).
Firstly, we define the elastic stress-strain operator E considering both plane stress
σz = 0 and axisymmetric constraints. Then,
[E]R
=E
1− ν2
1 ν 0
ν 1 0
0 0 1− ν2
, (6.53)
where E and ν are Young’s modulus and Poisson’s ratio.
Hence, by following the demonstration steps presented for the CB beam element
65
Figure 6.2: Cylindrical shell under ring load
in section 5.4, we conclude that the inverse of the discrete elastic operator for an
element i is computed by assembling the contributions of each layer ` as
(Ei)−1
=∑`
(E`,i)−1
, (6.54)
where(E`,i)−1
denotes the inverse of the elastic relation for the layer ` of the i-th
element defined as
(E`,i)−1
:= 2π
∫Ω`
[Nσ(ξ)]T[E−1
]R[Nσ(ξ)] r(ξ) detJ dξ dη (6.55)
The discrete elastic relation Ei, can be generally calculated in two steps: firstly,
its inverse (Ei)−1 is obtained by assembling the contributions (6.55) of each element
layer ` using standard assembly procedures. Secondly, the assembled matrix is
inverted. Note, however, that the computational burden to obtain the matrix Ei is
greatly reduced when there is no imposed continuity on stress fields between layers.
In this case, the matrices (E`,i)−1 of each layer are calculated and inverted and then
assembled directly in the matrix Ei.
6.6 Numerical Examples
6.6.1 Limit analysis of cylindrical shells under a ring load
In this example we consider cylindrical shells, with different lengths, under the action
of a ring load applied in their central cross-section. The model is geometrically
defined by the semi-length of the cylinder L, the mean radius R and the shell
thickness h, as shown in Figure 6.2. The material of the shell is assumed to be
elastic-ideally plastic with von Mises yield criterion and yield stress σY .
66
Analytical solutions for this example were obtained by various authors, e.g. DE-
MIR [78], SAWCZUK and HODGE [79], ZOUAIN [59] and CHAKRABARTY [80],
by assuming different simplified yield functions in generalized variables. In the fol-
lowing, we discuss the qualitative differences of these approaches with respect to the
present CB shell modeling.
It was generally identified in the technical literature that these ring-loaded cy-
linders can be classified in long, medium and short tubes according to the following
types of collapse mechanism, exhibited under plastic collapse conditions:
Long cylinders are those whose collapse mechanism presents a central circum-
ferential plastic hinge surrounded by two symmetric zones undergoing distributed
plastic strain rates and two rigid zones at both ends. Two additional symmetric
plastic hinges, separating the central plastic zone and the rigid endings, are found
in analytical solutions based on uniform or sandwich shell assumptions combined
with Mises or Tresca plastic conditions. However, these symmetric plastic hinges
are not present in our CB shell solutions; a fact confirmed in our additional solution
using solid rotationally symmetric finite elements.
Medium cylinders exhibit one central plastic hinge and distributed plastic defor-
mation all along their length. No rigid regions are observed.
For short cylinders there is no plastic hinge and plastic collapse happens with
dissipation distributed along the whole tube.
In the analytical solution given by [78] the cylinder was idealized as a uniform
shell made of elastic-ideally plastic material with Tresca yield criterion. The mecha-
nisms of collapse were identified in [78], depending on the non-dimensional length
of the cylinder
ξL =L√Rh
. (6.56)
Accordingly, for uniform Tresca shells, ring-loaded cylinders can be classified as
follows:
ξL ≥ 2.336↔ long cylinder, (6.57)
0.725 < ξL < 2.336↔ medium cylinder, (6.58)
ξL ≤ 0.725↔ short cylinder. (6.59)
The above categories are used here for the definition of the geometries of the solid
models employed to validate the proposed CB shell element.
Earlier analytical solutions for the particular case of long cylinders are found in
[79]. These solutions were obtained by the combination of two different shell models
– the uniform shell and the sandwich shell – with two different yield criteria – the
von Mises and the Tresca criterion. Lately, ZOUAIN [59] provided the analytical
67
solution for cylinders of different lengths considering a sandwich shell model and
the Tresca yield criterion. More recently, this problem of cylinder under ring load
was revisited in [80] where analytical solutions were given for sandwich and uniform
shells considering different yield criteria. A recent application of this example in the
verification of a numerical method for limit analysis is found in [81].
We remark that all analytical solutions aforementioned were obtained under the
assumption of Kirchhoff-Love shell theory where fibers initially normal to the mid-
plane of the shell are regarded to remain plane and normal after deformation. This
hypothesis, generally used for thin shells, differs from the kinematic assumptions of
the the proposed CB axisymmetric shell element (see Sect. 6.1.4).
In the present example, finite element approximations are obtained for cylin-
ders with radius to thickness ratio R/h = 30. The ring load is applied as a non-
dimensional reference load f0, same as that used in [78], i.e.,
f0 =
√M0
RN0
=1
2
√h
R, (6.60)
where
N0 = hσY and M0 =h2
4σY (6.61)
are, respectively, the limit loads corresponding to axial force and pure bending.
Due to symmetry, only a half of the cylinder is modeled with n-layer CB axisym-
metric shell elements without continuity of stresses between layers. Mesh density as
well as the number of layers are determined by a mesh sensitivity study comprising
two steps:
(i) a coarser mesh made of 4-layer elements is refined by adding more elements
until convergence;
(ii) the number of layers of the finer mesh defined in the previous step is gradually
increased until a new convergence.
The accuracy of CB shell results is verified by solid models built with 6-node
mixed triangular axisymmetric elements with their limit loads calculated using the
same algorithm employed with the CB shell models [10, 22]. In this verification,
cylinders with four different lengths are considered: a short (ξL = 0.511), a transition
short-medium (ξL = 0.741), a medium (ξL = 1.716) and a long (ξL = 3.536) cylinder.
The interpolation of the triangular solid element is made in a similar fashion
than the CB shell element, i.e. displacements are assumed quadratic and continuous
whereas stresses are linear and discontinuous. This mixed interpolation scheme has
been successfully used in shakedown analyses as well as other applications [22, 82].
68
Figure 6.3: Finite element meshes
The mesh density of solid models is also determined by a mesh refinement proce-
dure to guarantee convergence. As a result, regions of the model presenting higher
plastic strain rates are meshed with a higher density of elements. In Fig. 6.3
is illustratively given the coarser and the finer meshes used for the long cylinder
(ξL = 3.536) model. Figure 6.3a shows the meshes for the CB shell and Fig. 6.3b
depicts the meshes employed in solid models.
To illustrate the convergence pattern obtained in both mesh refinement proce-
dures, it is shown in Fig. 6.4 two graphs corresponding to the long cylinder case.
Figure 6.4a depicts the limit loads computed in step (i) of the mesh refinement pro-
cedure for the CB shell, whereas Fig. 6.4b shows the results obtained in step (ii). In
both graphs the results of the respective solid models with different levels of mesh
refinement are also presented for comparison.
Next, we show in Fig. 6.5 the different collapse mechanisms computed with the
CB shell element for the long, medium and short cylinders.
Figure 6.6 depicts an example of plastic strain rate plot, which corresponds to
instantaneous collapse of the short cylinder (ξL = 0.511); there we can see the plastic
strain rates distributed along the length of the cylinder, as expected.
The load multipliers (µ = F/f0), calculated with both shell and solid models,
are provided in Table 6.1. Note that both results are very close, with maximum
difference of 1%. However, the difference in number of degrees of freedom between
shell and solid models makes clear the advantage of using the CB shell element in
69
Figure 6.4: Example of mesh refinement results: long cylinder (ξL = 3.536)
Figure 6.5: Collapse mechanisms of cylinders under ring load
70
Figure 6.6: Plastic strain rates for a short cylinder (ξL = 0.511) under collapse dueto a ring load
this case.
Table 6.1: Load multipliers (µ) obtained with CB shell models and solid axisymme-tric models.
CB shell models Solid models
ξL stress stress
d.o.f. nodes µ d.o.f. nodes µ
0.511 49 192 1.000 1,581 1,104 0.999
0.741 91 360 1.337 8,613 6,240 1.341
1.716 127 504 1.684 7,477 5,376 1.701
3.536 127 672 1.899 15,093 11,004 1.918
The collapse load computed analytically by SAWCZUK and HODGE [79] for
a long cylinder, considering a uniform shell and the von Mises yield criterion, was
µ = 1.949. This result is 2.63% higher than the collapse load calculated in the
present work with the CB shell. However, the analytical solution was computed in
[79] neglecting the effect of shear stresses. Thus, that result can be considered as
less precise than the collapse loads obtained here, with both shell and solid models.
Moreover, by setting to zero the shear coefficient cs in the yield function of the CB
elements, (5.81), we obtain a result comparable to the solution from [79]. In fact,
with cs = 0 the collapse load predicted by the CB shell model is µ = 1.938 which is
only 0.56% less than the analytical result from [79].
Finally, we plot in Fig. 6.7 the collapse load as a function of the cylinder length.
This graph is constructed employing the CB shell element with 8 layers. The figure
also shows, for comparison purpose, the results of the four solid models and also
the analytical solutions given in [78] and [59], the latter calculated considering the
Tresca’s yield criterion and respectively uniform or sandwich shells.
71
Figure 6.7: Collapse load versus non-dimensional cylinder’s length
6.6.2 Limit and shakedown analysis of pressure vessels with
ellipsoidal and torispherical heads
In this example we employ the CB shell element to calculate collapse and shakedown
limits for pressure vessels with ellipsoidal and torispherical heads under internal
pressure. The geometry and material properties are taken from [72, 76]. In both
references the material was assumed as elastic-perfectly plastic with von Mises yield
criterion and axisymmetric solid elements were used with different techniques to
calculate the limit loads.
The pressure vessels have the same thickness in the cylindrical part and in the
head. Their geometry, schematically given in Fig. 6.8, is defined by non-dimensional
parameters. For pressure vessels with ellipsoidal heads these parameters are a/b and
a/h. For pressure vessels with torispherical heads the parameter r/a is also used.
The radius R of the spherical part of the torispherical head can be calculated
with the recursive equation
R = b+√
(r + c−R) (r − c−R) (6.62)
where the parameter c := a−r denotes the distance from the the axis of the cylinder
to center of the torus. Note that R > r + c.
The transition sphere-torus occurs at the critical radius given by
rcrit = c
[1 +
(R
r− 1
)−1]. (6.63)
72
Figure 6.8: Geometrical parameters of pressure vessels
The geometries considered in [76] were:
• for ellipsoidal head
a/b = 2, 2.5, 3, 4
a/h = 10, 25, 100, 300
• for torispherical head
a/b= 1.0, 1.5625, 2.0, 2.5, 3.33333
r/a= 0.12, 0.20, 0.30
a/h= 10, 100/7, 25, 50
and the the finite element models were carried out in [76] with the ABAQUS code
neglecting large displacement effects. The material properties employed were the
Young modulus E = 70 GPa, the Poisson’s ratio ν = 0.3 and the yield stress
σY = 20 MPa.
The estimate of the collapse pressure, sometimes cumbersome when calculated
via incremental analyses, was made in [76] by monitoring the pressure-deflection
73
curve. The collapse pressure was defined in [76] as the applied pressure from which
a very small increase in pressure produced a large increase in displacement.
The computation of shakedown loads, even more difficult by incremental analy-
sis, was made in [76] by using an iterative procedure. The methodology used by
[76] consisted of varying the load between zero and an arbitrary pressure p and mo-
nitoring in each cycle the displacement of the top of the head, the plastic strain,
and the residual stress field. If the displacement was repeated at the next cycle and
the plastic strain did not change during a cycle, and the residual stress field at zero
pressure did not violate the yield condition, then the pressure vessel was regarded as
shaking down and the pressure p was increased, otherwise, the pressure was reduced.
The procedure was repeated until the difference between pressures to shake down
and not to shake down showed less than 1.0% according to that authors.
The influence of the size of knuckle radius r in failure mechanism of a torispherical
head was investigated in [72]. The parameter R/a was used instead of parameter
a/b. Two cases were examined: r/a = 0.1 and r/a = 0.4, both with R/a = 1.5 and
a/h = 25.
Instead of using incremental analyses, it was employed in [72] a direct approach
to calculate the limit loads. The method used in [72] consisted of finding the collapse
and shakedown loads making use of Melan’s static theorem. The formulation was
posed as a conic quadratic optimization problem. The pressure vessel heads were
discretized in [72] with 6-node triangular axisymmetric solid elements using uniform
meshes.
The shakedown analyses of [72] demonstrated that alternate plasticity was the
critical failure mode for the pressure vessel with lower knuckle radius while the other
pressure vessel was more prone to fail due to incremental or instantaneous collapse.
This conclusion corroborates the fact that the knuckles with small radii are elastic
stress concentrators that can lead to alternate plasticity failure of the pressure vessel.
No material properties were listed in [72]. Nevertheless, for this example in
particular, the relevant material property is the Poisson’s ratio, assumed here as
ν = 0.3. The exact Young modulus is not important in this case because the
hypothetically elastic (unbounded) stress field only depends on the Poisson’s ratio,
the boundary conditions and the applied mechanical loads. The exact value of yield
stress is also irrelevant because the limit pressure is linearly dependent on the yield
stress and the applied internal pressure p is normalized with respect to the reference
value
p0 = σYh
a. (6.64)
In this work, the finite element models are carried out with the CB shell elements
with 8 layers and stress fields discontinuous between adjacent layers. Mesh density
74
as well as number of layers are defined using a mesh refinement procedure equivalent
to that described in the previous example (see Sect. 6.6.1).
Limit loads calculated with the CB shell models are listed in Table 6.2 for pres-
sure vessels with elliptical heads and in Table 6.3 for pressure vessels with torisphe-
rical heads.
Table 6.2: Limit loads calculated with CB shell element for pressure vessels withellipsoidal heads. pE = elastic limit; pLA = collapse load; pSD = shakedown load;Mch = failure mechanism; AP = alternate plasticity; IC = incremental collapse.
100 presented the failure with plastic strain rates distributed in the intersection of
the head and the cylindrical part.
The mechanisms of instantaneous collapse found for the set of pressure vessels
with torispherical head are similar to those described for the pressure vessels with
ellipsoidal head. Depending on the geometry, the weakest part can be the head or
the cylinder. In this case, the radius of the knuckle has an important effect on the
head strength. To illustrate, we provide in Fig. 6.11 the collapse mechanisms of the
models investigated in [72]. It can be seen that for the pressure vessel with r/a = 0.1
the collapse occurs in the head whereas for the pressure vessel with r/a = 0.4 the
cylinder is the weakest part.
The difference between the CB shell results and the results from literature [72, 76]
is presented in Table 6.4 for the pressure vessels with ellipsoidal head and in Table
76
Figure 6.9: Collapse load versus thickness ratio for pressure vessels with ellipsoidalheads
6.5 for the pressure vessels with torispherical head. This difference is computed as
Difference(%) =
(CB shell
reference− 1
)× 100 (6.65)
Note that only the collapse loads calculated in [72], which are independent from
elastic properties, are comparable to the CB shell results. For this reason, differences
with [72] for elastic limits and shakedown loads are intentionally omitted in Table
6.5.
Considering the set of models investigated, the following conclusions can be
drawn from the results:
• Regarding the elastic limit (pE)
- The elastic limit decreases as the ratio a/b of the pressure vessels increases;
- The discrepancy with results from literature increases as the wall thickness
increases. The CB shell results are slightly higher (2.2-5.7%) than the results
from [76] for pressure vessels with radius to thickness ratio a/h = 10. The
results are in very good agreement for structures with a/h > 10.
Assuming that all elastic results from solid models [76] are accurate, the rea-
sons for the higher differences observed for models with higher thicknesses can
be:
(a) the limitations of the shell model. When the wall of the pressure vessel
becomes excessively thick, the actual elastic stress state in the knuckle of the
vessel head, where the curvature is higher, is no longer well reproduced by
the shell hypothesis of plane stress state and/or the kinematic assumption of
fibers passing through master nodes remaining straight.
(b) the difference in the way that pressure load is applied. In solid models,
77
Figure 6.10: Examples of collapse mechanisms for pressure vessels with ellipsoidalheads under internal pressure.
78
Figure 6.11: Examples of collapse mechanisms for pressure vessels with torisphericalheads under internal pressure.
Table 6.4: Difference between limit loads calculated with CB shell and results fromYEOM and ROBINSON [76] for pressure vessels with ellipsoidal heads. pE = elasticlimit; pLA = collapse load; pSD = shakedown load
the internal pressure is obviously applied in the inner surface of the pressure
vessel. For shell elements the pressure is applied in the shell surface which
coincides with the mean surface of the pressure vessel. For thin-walled pressure
vessels this difference is insignificant. However, when the wall is thick the
total load applied in regions of high curvature in the shell model is higher
79
Table 6.5: Difference between limit loads calculated with CB shell and results fromYEOM and ROBINSON [76] and MAKRODIMOPOULOS [72] for pressure vesselswith torispherical heads. pE = elastic limit; pLA = collapse load; pSD = shakedownload
than the total load applied in the respective regions of the solid model. The
effect of this difference can be demonstrated by re-running the CB shell model
corresponding to the torispherical vessel with a/b = 2.5, r/a = 0.12 and
a/h = 10. In this new model we applied a correction factor to the pressure
load to make it compatible to the total load applied in the solid model. As a
result the difference in the elastic limit decreased from 5.7% to 4%.
• Regarding the collapse loads (pLA)
- The instantaneous collapse limit also decreases as the ratio a/b of the pressure
vessel increases;
80
- The CB shell results are very close to the results from references [72, 76].
The maximum discrepancy is only 2.8% in absolute value.
• Regarding the shakedown limit (pSD)
- For the models presenting 2pE < pLA the mechanism of failure is alternate
plasticity and the shakedown load is pSD = 2pE. For the remaining models the
mechanism of failure is incremental collapse and the limit load is pSD ≈ pLA;
- The comparison of shakedown loads shows the larger discrepancies (up to
8.5%) between the CB shell results and the results from [76]. For models
with low a/h, such differences can be partially explained by the same reasons
presented for the elastic limits. Nevertheless, differences can also be originated
by drawbacks faced in [76] when the shakedown limits were calculated by
means of incremental analyses.
Indeed, taking the case of models which mechanism of failure is alternate
plasticity, the shakedown load must be theoretically equal to twice the elastic
limit since the load domain has only two vertices, one of them being zero [84].
Therefore, considering that the computation of the elastic limit is much easier
than calculating the shakedown limit, one can expect that elastic limits from
[76] are more accurate than their shakedown loads. Consequently, differences
observed in shakedown loads should be smaller, i.e. equal to differences obser-
ved for elastic limits. The differences in shakedown loads should be, at least,
closer to the differences in elastic limits assuming, in this case, that elastic
limits from [76] are given in a nodal basis and that some smoothing technique
was used to compute them.
Note that, for some models of ellipsoidal vessels which incremental collapse
is the failure mechanism, the difference between the shakedown limit and the
collapse load calculated in [76] is slightly higher than that observed with the
respective CB shell models. This larger difference may be due to difficulty
encountered in [76] in computing the shakedown load by means of incremental
analysis when the applied load approaches the collapse limit and the stiffness
matrix becomes near singular.
Finally we build an additional model of the torispherical vessel with a/b = 2.5,
r/a = 0.12 and a/h = 10 which presented the maximum differences listed in Ta-
ble 6.5 using the 6-node triangular mixed finite element described in subsection
6.6.1. The objective is to compare the elastic stresses of this solid model with the
corresponding CB shell model.
After a mesh refinement procedure, we obtain the following limit loads from the
solid model: pE = 0.337, pLA = 1.128 and pSD = 0.674. The shakedown mechanism
81
Figure 6.12: Equivalent elastic stress plots from solid (left) and shell (right) modelsfor the pressure vessel with torispherical head. Geometrical parameters of the vessel:a/b = 2.5, r/a = 0.12 and a/h = 10.
of this vessel is alternate plasticity, as correctly predicted by the respective CB shell
model, and the shakedown limit is twice the elastic limit, as expected.
Figure 6.12 shows the plots of von Mises elastic stresses in the knuckle of the
vessel for both solid (left) and shell (right) models. This is the region of higher
stresses in this vessel. The shell model is not taking into account the previously
discussed difference in the way that pressure load is applied.
The maximum von Mises stress computed by the solid model is 7.1% above the
respective stress in shell model. This difference should be 4.6% whether correction in
the applied pressure were considered. Despite this difference, we note from Fig. 6.12
that the shell model can reproduce quite well the stress state in the knuckle with
some slight discontinuities observed when the curvature of the wall changes.
In this case, the difference in the value computed by the solid and the shell model
for the load amplification factor preventing shakedown is acceptable for engineering
purposes.
Note that, as a general rule, care must be taken when using structural elements
whether alternate plasticity is identified as the critical mechanism due to the local
nature of this phenomenon. When there are regions of concentrated stresses in thick-
walled shells, the confirmation of the load amplification factor computed by the shell
model is accomplished by using solid elements associated to a mesh convergence
study. To this end, sub-modeling techniques may be helpful.
82
Chapter 7
Continuum-based
Three-Dimensional Shell Element
In this chapter a six-node triangular shell element is presented for the solution of
3-D shakedown problems. The element is formulated employing the same strategy
used in the development of the beam element and axisymmetric shell elements pre-
sented in the two previous chapters. Hence, the element is built combining the
continuum-based approach [12, 13, 67] and a two-field mixed formulation. In this
case displacements and velocities are quadratic along the element mid-surface and
linear through its thickness. In each layer, stresses are interpolated linearly with
respect to mid-surface parameters and also linearly across the layer thickness. Con-
tinuity of stresses through layers may or may not be enforced, giving two alternative
versions of the finite element.
The proposed CB shell element differs from the family of mixed two-field trian-
gular shell elements of Bathe and co-workers [85–87] where displacement and strain
fields are interpolated and combined in a particular way with the main objective
of avoiding locking, usually present in pure displacement formulations. This shell
element also differs from the mixed (strain-displacement) triangular shell element
originally proposed by Argyris and co-workers [88–91] and further modified by Cor-
radi and Panzeri [62, 63] for application in sequential limit analysis.
In what follows we provide details for implementation of this CB shell element.
Sections are organized in the same way as in the two previous chapters. Note
that some intermediate steps of calculation presented for the CB beam element are
omitted in this chapter to make the text more concise. On the other hand, our
intention is to make the implementation of each finite element easier by giving in its
respective chapters equations in their final form. For this reason and for the sake
of clarity of the exposition, some formulae introduced in previous chapters may be
repeated here.
After the presentation of the element formulation we show the examples of ap-
Thus, in view of (5.39), the derivatives of the interpolation functions with respect
to spatial coordinates are gα,x
gα,y
gα,z
=1
detJCJ
gα,ξ
gα,η
gα,ζ
(7.31)
for α=s1:s12.
7.1.3 The interpolation of displacements and velocities
The assumed displacement and velocity fields are then
u(ξ)|Bi =∑
α=s1:s12
gα(ξ)uα v(ξ)|Bi =∑
α=s1:s12
gα(ξ)vα. (7.32)
The equations (7.32) above can also be recast in the compact intrinsic form of (5.43)
u(ξ)|Bi = Nv(ξ)us,i v(ξ)|Bi = Nv(ξ)vs,i,
where, in this case, the vector vs,i, collecting velocities of all slave nodes, is defined
in (7.14) and
us,i =[us1 us2 . . . us11 us12
]i,T(7.33)
≡[u1− u1+ . . . u6− u6+
]i,T, (7.34)
Nv(ξ) :=[g11 3 g21 3 . . . g111 3 g121 3
]. (7.35)
The velocity field written in curvilinear components, i.e. [v(ξ)]R, is given in
terms of the parameters of interpolation written in global components, i.e. [vi]R, by
(5.52)
[v(ξ)]R(ξ) = [Nv(ξ)]RT (ξ)[T i]R [
vi]R.
where the matrices [Nv(ξ)], R(ξ) and[T i]R
of the three-dimensional CB shell
element are analogous to the respective matrices of the elements presented in the
88
previous chapters, i.e., T i is given in (7.13), the linear map Nv which is intrinsic is
[Nv(ξ)] =
g1 0 0 g12 0 0
0 g1 0 . . . 0 g12 0
0 0 g1 0 0 g12
, (7.36)
and the “rotation” matrix is defined as
R(ξ) := diag(R,R,R,R,R,R,R,R,R,R,R,R), (7.37)
with R(ξ) given in (7.28).
7.1.4 Enforcing bending theory hypotheses
We employ the notation for planar tensors introduced in Sect. 5.1.4. Then, assuming
that in the direction perpendicular to the local laminar surface both strain and stress
components are zero we have
d = dxex ⊗ ex + dye
y ⊗ ey + d(xy)
[1√2
(ex ⊗ ey + ey ⊗ ex
)]+
d(yz)
[1√2
(ey ⊗ ez + ez ⊗ ey
)]+ d(xz)
[1√2
(ex ⊗ ez + ez ⊗ ex
)](7.38)
and
σ = σxex ⊗ ex + σye
y ⊗ ey + σ(xy)
[1√2
(ex ⊗ ey + ey ⊗ ex
)]+
σ(yz)
[1√2
(ey ⊗ ez + ez ⊗ ey
)]+ σ(zx)
[1√2
(ex ⊗ ez + ez ⊗ ex
)]. (7.39)
The relevant compatibility equations for the 3-D CB shell element are then
dx = vx,x, (7.40)
dy = vy,y, (7.41)
d(xy) = 1√2
(vx,y + vy,x) , (7.42)
d(yz) = 1√2
(vy,z + vz,y) , (7.43)
d(xz) = 1√2
(vx,z + vz,x) . (7.44)
7.1.5 Computing strain at a generic point
The computation of the infinitesimal strain rate at a generic (Gauss) point of the un-
derlying continuum and in terms of the interpolation parameters [vi]R is performed
89
by using the formula (5.67)
[∇symx v(ξ)]R(ξ) = [∇sym
x Nv(ξ)]R(ξ)RT (ξ)[T i]R [
vi]R.
Note that in this equation the vector of strain rates ∇symx v is given in local la-
minar components whereas the vector of nodal velocities [vi]R is written in global
components.
The above equation is calculated by following the same procedure explained for
the CB beam element in Sect. 5.1.5, i.e.,
1. Find the curvilinear coordinate system R(ξk) = ex, ey, ez given by (7.26)
and (7.25) and the rotation matrix R(ξk) using (7.28);
2. Compute, by using
[vs,i]R(ξk)
= RT (ξk)[T i]R [
vi]R
all slave node velocities in the curvilinear system of the generic point;
3. For α=s1:s12 apply the change of coordinates for slave node positions
xα = RT (ξk) xα; (7.45)
4. For α=s1:s12 find (gα,x, gα,y, gα,z);
To this end, compute the Jacobian in the curvilinear frame using (7.29) with
(xa, ya, za) (from (7.45)) instead of (xa, ya, za). Then, use ∇xgα = J−T∇ξgαwritten for curvilinear coordinates, i.e. analogous to (7.31).
5. Finally, compute the infinitesimal strain tensor
[∇sym
x v(ξk)]R(ξk)
=[∇sym
x Nv(ξk)]R(ξk) [
vs,i]R(ξk)
. (7.46)
Here (7.46) is written, discarding the transversal strain, as
vx,x
vy,y1√2
(vx,y + vy,x)1√2
(vy,z + vz,y)1√2
(vx,z + vz,x)
R(ξk)
=[G1 . . .G12
]R(ξk)
vs1x
vs1y
vs1z...
vs12x
vs12y
vs12z
R(ξk)
(7.47)
90
with auxiliary matrices Gα defined by
Gα(ξ) :=
gα,x 0 0
0 gα,x 0
1√2gα,y
1√2gα,x 0
0 1√2gα,z
1√2gα,y
1√2gα,z 0 1√
2gα,x
R
, α=s1:s12. (7.48)
7.2 Stresses
Stresses are interpolated by employing a similar strategy to that used to interpolate
stresses in the beam element (see Sect. 5.2) and in the axisymmetric shell element
(see Sect. 6.2).
The parent domain Ω =⋃`=1:L Ω` is divided to map each element layer. To this
end a sequence of coordinates −1 = ζ0 < ζ1 < · · · < ζL = 1 is selected. Then, each
parent layer is defined as Ω` := (ξ, η, ζ) ∈ (0, 1) × (0, 1) × (ζ`−1, ζ`), ξ + η ≤ 1.The image of these L parent layers through the geometry mapping (7.23) defines
the spatial layers.
The local coordinate ζ ∈ (−1, 1), to be used in the restricted domain of one
single layer `, is defined as
ζ :=ζ − a`b`
, (7.49)
where a` := 12
(ζ` + ζ`−1
)and b` := 1
2
(ζ` − ζ`−1
).
Stress components in each layer ` are interpolated by linear functions in ξ, η and
ζ. Then, each layer has six stress nodes located at corners and
σx(ξ)|Bi =∑j=1:6
tj(ξ)σj|`x , (7.50)
σy(ξ)|Bi =∑j=1:6
tj(ξ)σj|`y , (7.51)
σ(xy)(ξ)|Bi =∑j=1:6
tj(ξ)σj|`(xy), (7.52)
σ(yz)(ξ)|Bi =∑j=1:6
tj(ξ)σj|`(yz), (7.53)
σ(zx)(ξ)|Bi =∑j=1:6
tj(ξ)σj|`(zx), (7.54)
where the interpolation parameters σj|`x , σ
j|`y , σ
j|`(xy), σ
j|`(yz) and σ
j|`(zx) are interpreted as
the stress components of the j-th stress node of layer ` in its own laminar directions.
91
The interpolation functions are
tj(ξ) = 12(ξjξ + ηjη + ψjψ)(1 + ζj ζ) (7.55)
or
tj(ξ) = 12(ξjξ + ηjη + ψjψ)
[1 +
ζj − a`b2`
(ζ − a`)], (7.56)
with (ξj, ηj, ψj) denoting the triangular coordinates of node j. These are the usual
linear functions, just restricted to one layer, where an appropriate transversal co-
ordinate was defined in (7.49). This change of variable is also used to compute
integrals by numerical quadrature.
Finally, the above interpolation of stress is cast a matrix form correspondent to
(5.73) as
[σ(ξ)]R = [Nσ(ξ)][σ`], (7.57)
with (1 5 is the identity matrix 5× 5)
[Nσ(ξ)] = [t11 5 . . . t61 5] (7.58)
by defining in each layer `
[σ(ξ)]R =[σx σy σ(xy) σ(yz) σ(zx)
]T, (7.59)[
σ`]
=[σ1|` σ2|` σ3|` σ4|` σ5|` σ6|`]T (7.60)
and in each stress node t of layer `
[σt|`]
=[σt|`x σ
t|`y σ
t|`(xy) σ
t|`(yz) σ
t|`(zx)
]T. (7.61)
7.3 Yield function
The yield function f(σ) for the 3-D CB shell element is obtained from the von Mises
yield criterion (5.78) by considering the plane stress condition (σz = 0) as
f(σ) = σ2x + σ2
y − σxσy −3
2
(σ2
(xy) + csσ2(yz) + csσ
2(zx)
)− σ2
Y , (7.62)
The parameter cs denotes a coefficient that can be set equal to 0 or 1 to consider
or not the contribution of the shear stresses to yielding. Therefore, if cs = 0 a
Kirchhoff-Love-type shell theory is simulated. Otherwise, cs = 1 and shear stresses
are considered as in Mindlin-Reissner-type shell elements.
92
The gradient and the Hessian of the yield function are then
∇σf =
2σx − σy2σy − σx
3σ(xy)
3csσ(yz)
3csσ(zx)
and ∇σσf =
2 −1 0 0 0
−1 2 0 0 0
0 0 3 0 0
0 0 0 3cs 0
0 0 0 0 3cs
(7.63)
7.4 Discrete strain operator
For obtaining the discrete strain operator for the three-dimensional CB shell element
we firstly define in the discrete setting for each layer ` a work-conjugate strain rate
vector [d`]
:=[d1|` d2|` d3|` d4|` d5|` d6|`
]T, (7.64)
where [dt|`]
=[dt|`x d
t|`y d
t|`(xy) d
t|`(yz) d
t|`(xz)
]T(7.65)
and t denotes the stress nodes of layer `.
Then, we follow the steps presented for the CB beam element in Sect. 5.4 to
identify the layer discrete strain operator as
B`,i := B`,islave
[T i]R. (7.66)
where
B`,islave :=
∫Ω`
[As1,`,i . . .As12,`,i
]RRT detJ dξ dη dζ
(7.67)
with the auxiliary matrix A defined as
A`,i(ξ) := [Nσ(ξ)]T [∇symx Nv(ξ)]R(ξ) . (7.68)
Then, by considering (7.48) and (7.57), the matrix A is computed as
A`,i(ξ) =
t11 5
t21 5
t31 5
t41 5
t51 5
t61 5
[G1 G2 . . . G11 G12
]R. (7.69)
Alternatively,
A`,i(ξ) =[As1,`,i . . .As12,`,i
]R, (7.70)
93
with
Aα,`,i =
t11 5
t21 5
t31 5
t41 5
t51 5
t61 5
[Gα
]R, (7.71)
or, explicitly,
Aα,`,i =
t1gα,x 0 0
0 t1gα,x 0
1√2t1gα,y
1√2t1gα,x 0
0 1√2t1gα,z
1√2t1gα,y
1√2t1gα,z 0 1√
2t1gα,x
......
...
t6gα,x 0 0
0 t6gα,x 0
1√2t6gα,y
1√2t6gα,x 0
0 1√2t6gα,z
1√2t6gα,y
1√2t6gα,z 0 1√
2t6gα,x
R
. (7.72)
Finally, the element strain operator is obtained by assembling the contributions
of each layer:
Bi =∑`
B`,i. (7.73)
The assembly in (7.73) is easily performed by means of well-known procedures
employed in finite element analysis and depends on whether or not continuity of
stress fields between layers is enforced. When stress fields are discontinuous between
layers, the assembly is simply
Bi =
B1,islave...
BL,islave
[T i]R. (7.74)
7.5 Discrete elastic relation
In this section we compute the discrete form of the elastic operator for the 3-D
CB shell element denoted by E. This matrix is only used in the data preparation
for the analysis, when the discrete approximation of the ideally elastic stress fields
94
σE, for each extreme loading, are computed. This elastic pre-analysis gives rise to
the practical definition of the prescribed domain of loading variations introduced in
(2.20) and (2.32).
Firstly, we define the elastic stress-strain operator E considering the plane stress
constraint σz = 0. Then,
[E]R
=E
1− ν2
1 ν 0 0 0
ν 1 0 0 0
0 0 1− ν 0 0
0 0 0 1− ν 0
0 0 0 0 1− ν
, (7.75)
where E and ν are Young’s modulus and Poisson’s ratio.
Hence, by following the demonstration steps presented for the CB beam element
in section 5.4, we conclude that the inverse of the discrete elastic operator for an
element i is computed by assembling the contributions of each layer ` as
(Ei)−1
=∑`
(E`,i)−1
, (7.76)
where(E`,i)−1
denotes the inverse of the elastic relation for the layer ` of the i-th
element defined as
(E`,i)−1
:=
∫Ω`
[Nσ(ξ)]T[E−1
]R[Nσ(ξ)] detJ dξ dη dζ. (7.77)
7.6 Eliminating singularities in rotational stiff-
ness
Since our shell element has six degrees-of-freedom per node we must deal with the
fact that, when the element is flat, its stiffness matrix has singularities associated
with rotations/angular velocities about the directors [12]. This problem is addressed
with a procedure analogous to that employed by BENSON et al. [13]. In this case, a
small rotational stiffness is added to the element stiffness matrix on a node-by-node
basis,
Kaω ←Ka
ω + skpa ⊗ pa, (7.78)
where Kaω is the 3×3 sub-matrix associated with master node a, k is the maximum
value in the sub-matrix diagonal which is multiplied by the small non-dimensional
constant s, and pa is the director of master node a. BENSON et al. [13] suggest a
choice of s within the range [10−6, 10−4]. Here the value s = 10−6 has proven to be
95
an adequate choice in all numerical examples considered.
7.7 Numerical Examples
7.7.1 Straight pipe under combined loading
In the following the 3-D CB shell is used to calculate collapse loads and the shake-
down limits for straight segments of pipes subjected to the combination of bending
moment M and internal pressure p or axial force F .
Limit analysis of a thin-walled pipe
In this first example the objective is to obtain the collapse load diagram for a thin-
walled pipe with closed ends under the combined action of bending moment M and
internal pressure p. The pipe is geometrically defined by its mean radius r and wall
thickness t. The material is elastic-perfectly plastic with yield stress σY .
The analytical solution for this problem considering the von Mises yielding cri-
terion is [93] (M
MC
)2
+
(p
pC
)2
= 1, (7.79)
with the collapse bending moment MC and the collapse pressure pC calculated under
thin-shell hypotheses, i.e.
MC = 4r2tσY , pC =2√3
t
rσY . (7.80)
The finite element approximation to this problem is obtained for a pipe with
σY = 200 MPa and r/t = 25. The length of the model, L = 20r, was determined by
a convergence study to avoid boundary effects.
Due to the geometry of the problem, the pipe is represented by a quarter sym-
metry model. Kinematic boundary conditions are properly imposed to guarantee
the correct structural behavior. Rigid body motion in the direction parallel to the
symmetry planes is avoided by restraining an arbitrary node lying on the symmetry
plane transversal to the pipe.
Bending moment is applied as a linear pressure distribution at the free end cross
section. Internal pressure, p, is applied assuming closed ends condition with an
equivalent axial force equal to pπr2 applied as a constant pressure distribution at
the free end cross section.
The pipe is modeled with 4-layer elements without continuity of stresses between
layers. The mesh, depicted in Fig. 7.2, has 504 elements and 6118 degrees of freedom.
96
Figure 7.2: Finite element mesh for thin-walled pipe (r/t = 25)
Figure 7.3: Collapse load curve of thin-walled pipe (r/t = 25) under internal pressureand bending
The results obtained with the CB shell are compared to the analytical solution
in Fig. 7.3. This comparison demonstrates that the CB shell is able to calculate
limit loads in very close agreement with thin shell theory. For the five points shown
in Fig. 7.3 the maximum relative error is 0.022%. Nevertheless, thin pipes subject
to bending can buckle under smaller loads, mainly due to ovalization of the cross
section.
Limit analysis and shakedown of a thick-walled pipe
In this subsection we analyze a thick walled pipe (r/t = 5) to avoid the large
displacement effects mentioned in the thin-walled pipe example. In this case, the
pipe is geometrically defined by its internal and external radii, ro and ri, respectively.
Firstly we calculate the exact collapse load diagram for a thick-walled pipe under
the combined action of a bending moment M and an axial load F .
Figure 7.4 shows the stress field σx(y) in the pipe cross section when the limit
load is attained considering the hypothesis of elastic-perfectly plastic material. In
97
Figure 7.4: Stress field in the cross section of a thick-walled pipe under maximumcombination of axial tension and bending
this figure we also define the variable y corresponding to the distance between the
axis z and the line separating the part of the cross section in tension from the part
in compression. According to this definition, if y > 0 then F is positive (tension),
and if y < 0 then F is negative (compression).
The collapse axial load is computed as F (y) = 2σY A with A denoting the area
of the section between y = 0 and y = y.
Then, if | y |≤ ri
F (y) = 2σY r2o
[arcsin
(y
ro
)+y√r2o − y2
r2o
]
− 2σY r2i
[arcsin
(y
ri
)+y√r2i − y2
r2i
], (7.81)
otherwise
F (y) = 2σY r2o
[arcsin
(y
ro
)+y√r2o − y2
r2o
]− σY πr2
i . (7.82)
Likewise, for the bending moment, assuming y > 0, we have M (y) = 2σY Q with
Q denoting the first moment of area of the part of the section between y = y and
y = ro.
Thus, if | y |≤ ri then
M (y) = 43σY
[(r2o − y2
)3/2 −(r2i − y2
)3/2], (7.83)
otherwise
M (y) = 43σY(r2o − y2
)3/2. (7.84)
The parametric equations (7.81),(7.82), (7.83) and (7.84) are used to build the
exact collapse curve varying the parameter y between zero and ro. This exact
solution is used to verify the results obtained with the CB shell for a pipe with
98
Figure 7.5: Collapse load curve of thick-walled pipe (r/t = 5) under axial force andbending
ro = 11 mm and ri = 9 mm.
The material properties used in this example are: E = 200 GPa, ν = 0.3,
σY = 200 MPa. The length of the model is the same employed in the previous
example as well as the boundary conditions, the element options and mesh. Bending
moment and axial force are respectively applied at the free end cross section as linear
and constant pressure distributions.
The comparison between exact and finite element results is depicted in Figure
7.5. In this figure axial force and bending moment are normalized by the respective
collapse loads, FC and MC , calculated for thick-walled pipes:
FC = σY π(r2o − r2
i
), MC = 4
3σY(r3o − r3
i
). (7.85)
As can be seen, the CB shell also performs well for this problem. Figure 7.6 shows
the plot of plastic strain rates for this example corresponding to point A in the limit
load diagram (see Fig. 7.5). Figure 7.6 looks like a plot from a full 3-D analysis
because the results are plotted for the underlying continuum elements of the CB
shell.
Finally, we give in Fig. 7.8 the collapse load diagram and the shakedown limit
for the same pipe considering the combined action of internal pressure and bending.
The load domains for both limit and shakedown analyses are shown in Fig. 7.7. Note
in Fig. 7.7 (b) that the load domain for shakedown is the combination of a fixed
(constant) internal pressure and a fully-reversed bending moment.
In the diagrams of Fig. 7.8 the applied bending load is normalized by the limit
99
Figure 7.6: Plastic strain rates for a thick-walled pipe (r/t = 5) under collapse dueto the loads (F/FC ,M/MC) = (0.4924, 0.7162)
Figure 7.7: Load domains for (a) the limit analysis and (b) the shakedown analysisof a thick-walled pipe (r/t = 5) under internal pressure and bending
100
Figure 7.8: Collapse load curve and shakedown limit for a thick-walled pipe (r/t = 5)under internal pressure and bending
load MC given in (7.85). Internal pressure is also normalized by the respective
analytical limit, pC , calculated for a thick-walled straight pipe [50]:
pC =2√3σY ln
(rori
). (7.86)
In the shakedown diagram depicted in Fig. 7.8, cyclic bending loads above seg-
ment AB will produce alternate plasticity whereas cyclic bending loads above seg-
ment BC will cause failure due to incremental collapse.
The load domain in point A consists of the load M varying cyclically between
two extremes of equal magnitude and opposite sign. Therefore, for this point, the
alternate plasticity limit is equal to the elastic limit [84]. The moment ME corres-
ponding to the elastic limit for a thick-walled pipe is
ME =σY π (r4
o − r4i )
4ro. (7.87)
Thus, the exact value of the maximum bending moment preventing alternate plas-
ticity is M exact = 0.71874MC . The limit load obtained with the CB shell is
MFE = 0.71053MC which is 1.2% below the exact solution.
101
Figure 7.9: Pipe bend geometry and loads
7.7.2 Shakedown analysis of pipe bend subjected to internal
pressure and bending moment
Next, we calculate the collapse load diagram and the shakedown limit for a pipe
bend with attached straight pipe sections subjected to the combination of internal
pressure p and in-plane bending moment M (see Fig. 7.9). For shakedown analysis
the applied load consists of constant internal pressure and cyclic opening bending
(0 ≤M ≤Mmax).
This type of structure has been studied by many researchers. A list of references
can be found in the following recent works on this subject: ABDALLA et al. [49],
CHEN et al. [50], TRAN et al. [51]. Most of the previous works on pipe bends are
restricted to limit analysis and just a few focus on shakedown.
The bend is defined by its mean radius r, bend radius of curvature R, and
thickness t. This geometry can be classified by the curvature factor, cf , defined as
cf :=R/r
r/t=Rt
r2. (7.88)
Accordingly, bends with cf ≤ 0.5 are considered highly curved.
The geometrical data and material properties of this example are: r = 10 mm,
t = 2 mm, R = 20 mm, L = 160 mm, E = 200 GPa, ν = 0.3, σY = 200 MPa.
Thus, the present example consists of a thick-walled (r/t = 5) and highly curved
102
Figure 7.10: Finite element mesh using CB shell elements
(cf = 0.4) pipe bend. These properties are taken from an example given in [50].
The finite element model is built, for a quarter of the pipe bend geometry, em-
ploying 6-layer elements without continuity of stresses between layers. Internal pres-
sure assuming closed end condition and bending moment is applied as explained in
the previous example.
The mesh, shown in Fig. 7.10, has 720 elements and 9005 degrees of freedom.
This mesh density is similar to the mesh used in [50], however in the cited work three-
dimensional solid elements were employed with a different technique to perform the
shakedown analysis.
The results obtained with the CB shell are presented in Fig. 7.11. This figure
also shows the collapse load diagram and the shakedown limit found by CHEN et al.
[50]. There are also 5 points from elastic-plastic incremental analyses performed
using Rik´s method in ABAQUS code which were used by these authors to verify
their collapse load results. We remark that all results are normalized by the collapse
limits MC and pC given in (7.85) and (7.86).
It can be seen in Fig. 7.11 that collapse load results obtained with the CB shell
are very close to the results found in [50] and verified with ABAQUS (also in [50]).
The segment BC of the shakedown limit diagram obtained with the CB shell is also
in very good agreement with the results provided in [50]. Cyclic loads above segment
BC cause failure due to incremental collapse. On the other hand, the limit indicated
by segment AB is considerably different from that found in [50]. Cyclic loads above
segment AB produce alternate plasticity.
The determination of the limit against alternate plasticity is much simpler than
the problem of elastic shakedown formulated in Sect. 2.3. Indeed, the formulation
103
Figure 7.11: Collapse load curve and shakedown limit for a thick-walled (r/t = 5)and highly curved (h = 0.4) pipe bend under internal pressure and bending. CL =collapse load; SD = shakedown limit
for alternate plasticity is obtained neglecting the equilibrium constraints so that
yield condition can be checked independently at each point of the body [10, 84, 94].
The accurate determination of the limit preventing alternate plasticity exclusivelly
rely on the accurate computing of elastic stresses.
The load domain represented by point A in the shakedown diagram corresponds
to the bending moment M varying between zero and the maximum load. For this
particular type of load domain the alternate plasticity limit is simply twice the
elastic limit [84]. Then, if the exact elastic solution of this problem were known the
exact limit against alternate plasticity would be easily calculated as the division of
twice the yield stress by the maximum von Mises elastic stress.
When the exact solution is not available, better estimates of the alternate plas-
ticity limit are achieved by improving the finite element elastic results. Then, in
order to check the elastic results produced by the CB shell element, a new model is
built using ANSYS. This model employs quadratic solid elements to represent the
pipe bend and part of the attached straight pipe. The remaining part of the pipe,
far from the region of interest, is modeled with quadratic shell elements. Shell-solid
connection is made using ANSYS multipoint constraint (MPC) feature. The solid
element chosen from ANSYS element library is SOLID186 with displacement for-
mulation. Both full and reduced integration schemes are tested and compared. The
shell element utilized is SHELL 281.
104
Figure 7.12: Shell-solid model for the pipe bend. Detail of the coarser mesh employedin bend region
A mesh solution convergence study was conducted for the part modeled with
solid elements. A coarser mesh was built with four elements through the thickness,
ten around the radius of the bend and twenty around the circumference of the pipe
(see Fig. 7.12). The number of elements in each direction of pipe bend was doubled
and tripled and the maximum von Mises elastic stress for each mesh was computed
to calculate the respective estimates for the load multiplier µ preventing alternate
plasticity. The results of the convergence study are summarized in Table 7.1 which
also provides the result obtained with the CB shell for comparison. It can be seen
that the estimate for the alternate plasticity limit calculated with the CB shell
element is only 2.2% less than the limit computed with the most refined mesh of
ANSYS (mesh 3) considering the full integration scheme or 1.8% less than the limit
calculated with the ANSYS model (mesh 3) for the reduced integration technique.
Figures 7.13 and 7.14 show, respectively, the von Mises stress plot corresponding to
the ANSYS model (full integration) and the CB shell model.
7.7.3 Limit analysis of a cylinder-cylinder intersection
In this example we calculate the collapse load diagram for a cylinder-cylinder inter-
section. The structure is subjected to internal pressure p and a bending moment
M acting on the nozzle. The shell is made of a material with yield stress σY = 234
MPa and is geometrically defined by its internal diameter D = 285 mm and wall
thickness s = 15 mm. The material of the nozzle has yield stress σY = 343 MPa and
its internal diameter and wall thickness are, respectively, d = 20 mm and t = 7.5