1.3-GHz
Superconducting
Radio Frequency Cavity Stressand Buckling Analysis
1. Cryomodule Assemblycollideraccelerator,

gun, polarized positron source, electron andpositron damping rings, beam transport from
damping rings to the main linacs, two 11 km
mainlinacsutilizing

1.3

GHz

9-Cellsuperconducting radio frequency cavities (SRF)

The International Linear Collider (ILC) is anelectron-positron

major sub-system will include photocathode DC

thatrequires approximately 7800 1.3 GHz Niobium

operating at an average gradient of 31.5 MV/m,two beam delivery systems each 2.2 km longandcryogenic

plants

to

cool

down

the

9-cell cavities to attain 250 GeV centre-of-mass

cryomodules. The 7800 cavities will be housed

ILC will be a 31 km long linear collider whoseroughly 12.65 m long [2].

energy and is extendable up to 1 TeV [1, 2]. The

in approximately 850 cryomodules, which are

Fig. 1 Example of cryomodule cross-section for FNAL and KEK (Top) [3]; Different types ofcryomodules, ILC and Eu-XFEL design [2] (below)

９-２－ 1

There are two types of cryomodules for ILC: alevel of safety which is greater than or at least

Type A module with nine 1.3 GHz nine-cellcavities and Type B with eight nine-cell cavities

equal to acceptable standards to the highpressure gas safety authorities for the cavity

and one superconducting quadrupole package

operation.

located at the centre of the module, an example
of a cryomodule is shown in Fig. 1 (below). The

2.1.1. 1.3 GHz SRF Cavity Assembly

cryomodule consists of cavity packages, input

There are currently two types of 1.3 GHz SRF

couplers, the gas return pipe (GRP), magnetic

shields, two sets of thermal shields, coolingcavity design in operation: Tesla and Tesla-likecavities. The difference between the shapes ofpipes and the vacuum vessel. The cooling pipes

the cavity is minuscule but the design of the

includes the 2 K two-phase superfluid helium

Titanium Jacket, end group and Tuner varies inmm), 5 K shield line and 80 K or 40 K shield

Jacket design proposed for ILC’s SRF cavities isline. The cold mass at 2 K is enclosed with two

shown in the Fig. 2 (below) and its cut section

pipeline, 2 K gas helium return line (Φ 300

thermal shields at 5 K and 40 K which aremany facilities. The current Tuner and Ti

view in Fig. 2 (top).

cooled with LHe (liquid helium) [2]. A crosssectional view of cryomodule is shown in Fig. 1a.

2.

Pressure Vessel Compliance for
Cryomodules

The cryomodule has components that can beconsidered as pressure vessels and needs to becleared by high-pressure gas safety authorities
for operation. Most of the components like the 2
K GHe return line, 2 K two-phase line, 5 K
thermal shield line and 80 K thermal shield line
are stainless steel (SUS316L) pipes which
doesn’t create any issues with clearing highpressure safety regulations. However, the SRF
cavities are constructed of Niobium and encasedin a Ti helium jacket, which is filled withsuperfluid helium (another phase of liquid
helium) at 2.0 K for its operational conditions,hence the assembly is considered as a pressure
vessel [3]. The materials such as Niobium,
Titanium and NbTi alloys are not approved forpressure vessel design due to the lack of code

Fig. 2 Illustration of a 1.3 GHz Nb SRF cavity
assembly (top) [3]; 1.3 GHz SRF cavity
assembly for ILC (below) [2]data

The largest accelerator operating the Tesla

in

high

pressure

codes,

forroom

temperature and cryogenic temperatures (at4.21 K or below). Hence, to operate the SRF
cavity assembly it is necessary to show certain

cavities is the European Free Electron Laser(EU-XFEL). It is a 2.1 km superconducting
Linac to accelerate electron to 17 GeV by using

９-２－ 2

808 1.3 GHz 9-Cell SRF cavities [4]. Design ofvaluable and practical analysis tool for thethe Eu-XFEL’s 1.3 GHz Tesla cavity assembly isshown in the Fig. 3.

solution of boundary, initial, and eigenvalue

problems in various engineering disciplines [5].
The FEA method requires following steps.
• Discretization of the domain into finite
number of sub-domains.

• Selection of interpolating functions.
• Development of the elemental matrix of the
sub-domain.

• Assembly of the element matrices for each
subdomain to obtain global matrix for theentire domain.

• Imposition of boundary conditions.
• Solution of equations.
• Additional data-analysis if necessary.
In matrix notation, the global system of
equations can be cast into:

Fig. 3 Tesla cavity assemble at Eu-XFEL DESY
(top) and its cut view (below) [4]

Where K is the system stiffness matrix, u is thevector of unknowns, and F is the force vector.
Depending on the nature of problem, K maybe

3. Structural Analysis
3.1. Structural Analysis using FEA [5]The

FEA isapowerfulbe dependent on u and F maybe time dependent,

computational

technique for approximate solutions to a varietyof“real-world”

problems.

It

reliesondecomposition of domain (solid, liquid or gas)

into a finite number of sub-domains for which
systematic approximate solution is constructedby applying the variational or weighted residual
methods. FEA reduces the problem into finite
number of unknowns by dividing the domain
into elements and by expressing the unknown
field

variable

in

terms

of

the

(3 − 1)

Ku = F,assumedapproximating functions within each element.
These functions are also called interpolating
functions are defined in terms of the values ofthe field variables at specific points, referred as
nodes which are located along the elementboundaries and connects the adjacent elements.
The ability to discretize the irregular domains
with finite elements makes the method ai.e., F = F(t) [5].
3.1.1. Nodes
A node specifies the coordinate location in
space where degrees of freedom and actions ofthephysicalproblemexist.

The

nodal

unknowns in the matrix system of equations
represents on or more of the primary field
variables. Nodal variables assigned to anelement are called degrees of freedom (DOF) of
the element. The common nodes provide thecontinuity for the nodal variables. DOF of anode is dictated by the physical nature of aproblem and the element type [5].
3.1.2. Elements
The geometry can be discretized considering
the complexity of shape and the physical natureof the problem. There are various methods of

９-２－ 3

discretization methodologies like line elementfor 1D problems, area elements like triangular,
rectangular and quadrilaterals for 2D problems
and tetrahedrals, prisms and hexahedrals forvolume elements. The elements are identified
by element numbers and is defined by a specificsequence of global node numbers, as shown in
Fig. 4 [5].

Fig. 5 Discretization of a domain: element andnode (Top); free-body diagram of a linear
spring element (bottom) [5]This is related to force acting on the spring
𝑓𝑓! = ku = k(𝑢𝑢! − 𝑢𝑢" ),The equilibrium of forces requires that

Which yields

Fig. 4 line, area and volume elements with node
numbers at element level [5]3.2.1. Linear Spring System
It is suitable for simpler problems but isfundamental step of a typical finite element
analysis. In this approach a linear spring
system is considered in 1-D domain [5]. As
shown in Fig. 5, a linear spring with stiffness 𝑘𝑘

has two nodes and each nodes is subjected to
axial

loads

of

𝑓𝑓!

displacement of 𝑢𝑢!

positive direction.

and 𝑓𝑓" resulting inand 𝑢𝑢" in their defined

Due to the defined nodal forces the resulting
displacement is,u = 𝑢𝑢! − 𝑢𝑢" ,𝑓𝑓" = −𝑓𝑓! ,(3 − 4)𝑓𝑓" = k(𝑢𝑢" − 𝑢𝑢! ),(3 − 5)

Combining equation 3-3 and 3-5 and rewriting
the equations in matrix form, we get
3

3.2. Direct Approach for FEA(3 − 3)𝑘𝑘−𝑘𝑘𝑓𝑓−𝑘𝑘 𝑢𝑢!
4 5𝑢𝑢 6 = 7 ! 8 𝑜𝑜𝑜𝑜 𝑘𝑘 ($) 𝑢𝑢($) = 𝑓𝑓 ($) . (3 − 6)
𝑓𝑓"
"
𝑘𝑘

Here, 𝑢𝑢($) is the vector of nodal unknowns
representing displacement, 𝑘𝑘 ($) is the element(stiffness) matrix and 𝑓𝑓 ($) is the element(force) vector. The stiffness matrix can be($)
represented in its indicial form as 𝑘𝑘&' ,($)

𝑘𝑘 ($) ~ 𝑘𝑘&' .

(3 − 7)

where i and j are the row and the column($)
numbers. The coefficients, 𝑘𝑘&' , may beinterpreted as the force required at node 𝑖𝑖 toproduce unit displacement in node 𝑗𝑗 while all
the other nodes are fixed [5].

(3 − 2)９-２－ 4

3.2.2. Assembly of the Global System ofEquations
To model an engineering problem with finite
elements requires the assembly of element
characteristic (stiffness) matrices and element
right-hand-side vectors, leading the global
system of equations, as equation 3-1 [5].

F = B 𝑓𝑓 ($) ,The global system matrix and the global righthand-side vector are constructed by considering

the systems of linear springs shown in Fig. 6.
by equation 3-6 and rewritten as [5]The global system matrix, K, can be obtained
from the “expanded” element coefficientmatrices, 𝑘𝑘

($)

($)

𝑘𝑘
G !!
($)
𝑘𝑘"!

, by summation in the form(

K = B 𝑘𝑘 ($) ,$)!

(3 − 10)$)!

For an element equations for a spring is given(3 − 8)

Ku = F,((3 − 9)

($)

𝑘𝑘!"

($)

($)

𝑢𝑢
𝑓𝑓
H I !($) J = I !($) J .
($)
𝑘𝑘"" 𝑢𝑢"𝑓𝑓"(3 − 11)

In which 𝑘𝑘!! = 𝑘𝑘"" = 𝑘𝑘 ($) and 𝑘𝑘"! = 𝑘𝑘!" =
($)

($)

($)

($)

−𝑘𝑘 ($) . The subscripts corresponds to Node 1 and2. In accordance with equation 3-8 and 3-9 theglobal system matrix is (4 × 4) and is shown
below [5]
(!)

⎡𝑘𝑘!!
⎢ (%)
⎢𝑘𝑘$!
K=⎢
⎢
⎢ 0
⎢
⎣ 0

Fig. 6 System of linear springs (top) andcorresponding FEA (below) [5]The parameter 𝐸𝐸 denotes the number of

elements

andtheexpandedelement($)(&),𝑘𝑘$! + 𝑘𝑘$! 0

0

0 ⎤

⎥
0 ⎥
⎥ , (3 − 12)($)(&)
𝑘𝑘$$ + 𝑘𝑘$$(%) ⎥+ 𝑘𝑘!$ ⎥
)
(')
+𝑘𝑘!!
⎥
(%)
(')
𝑘𝑘$!
𝑘𝑘$$ ⎦($)(&),𝑘𝑘!$ + 𝑘𝑘!$ -and the contribution of each element is summed
as(!)

global system matrix but have the rows andcolumns of zeros corresponding to the nodes notfrom the expanded element coefficient vectors,(!)($)
𝑘𝑘 + 𝑘𝑘) $$ (&)!! ++𝑘𝑘!!

And the global right-hand-side vector is (4 × 1)

characteristic matrices are the same size as theassociated with element (e). Similarly, theglobal right hand side vector, 𝐹𝐹, can be obtained

(!)

𝑘𝑘!$

𝑓𝑓!
⎡
⎤
𝑓𝑓!
(!)
($)
(&) ⎥
⎢
𝑓𝑓 + 𝑓𝑓! + 𝑓𝑓!
𝑓𝑓⎥.
K = 8 $ : = ⎢ $($)
𝑓𝑓&
⎢𝑓𝑓$ + 𝑓𝑓$(&) + 𝑓𝑓!(')⎥
⎢
⎥
𝑓𝑓'
(')
𝑓𝑓$
⎦
⎣

(3 − 13)

And the vector of unknowns, u, becomes

f(e), by summation in the form given below, andthe size of the global right-hand-side vector isalso dictated by the highest number among the
global node numbers [5].

９-２－ 5

(!)

𝑢𝑢!
⎡
⎤
𝑢𝑢!
(!)
($)
(&)
⎢
𝑢𝑢$
𝑢𝑢$ = 𝑢𝑢! = 𝑢𝑢! ⎥⎥.
u = 8𝑢𝑢 : = ⎢ ($)&⎢𝑢𝑢$ = 𝑢𝑢$(&) = 𝑢𝑢!(')⎥
𝑢𝑢'
⎢
⎥
(')
𝑢𝑢$
⎦
⎣

(3 − 14)

For the specific values
and

($)
($)
𝑘𝑘"! = 𝑘𝑘!" = −𝑘𝑘 ($)

1 −1 00(𝑒𝑒) −1
3 −2 0K = 𝑘𝑘 8:.
0 −2 3 −1
0
0 −1 1eigenvalues√5, and 𝜆𝜆& = 3 − √5eigenvectors are

are
.

($)

, the global system

matrix becomes

The

($)

𝑘𝑘!! = 𝑘𝑘"" = 𝑘𝑘 ($)(3 − 15)

𝜆𝜆! = 0, 𝜆𝜆$ = 2, 𝜆𝜆& = 3 −The

If a boundary condition is imposed like 𝑢𝑢! =0, the nodal force 𝑓𝑓! still remains an unknown

and other nodal displacements are unknowns,

and the corresponding nodal forces have valuesof 𝑓𝑓! = 0, 𝑓𝑓" = 𝑓𝑓+ = 0, and 𝑓𝑓, = 𝐹𝐹 . Then the
global system of equations are written as
(𝑒𝑒)

𝑘𝑘

corresponding

1
1
1 ($)
−1
(!)
u = 8 :,u = 8 :,
1
−1
1
1
1
1
2
−
2
+
√5
√5
u(&) = 8: , u(') = 8:.
−2 + √5−2 − √5
1
1

0
1 −1 00𝑓𝑓!
−1 3 −2 0 𝑢𝑢$
08:8 : = 8 :,0 −2 3 −1 𝑢𝑢&
00
0 −1 1 𝑢𝑢'
𝐹𝐹Leading to following solutionsAnd(3 − 16)

Each of the eigenvectors represents a possible
solution mode and their contribution is illustrated(3 − 17)

𝑢𝑢" =

𝐹𝐹

(𝑒𝑒)

𝑘𝑘

−𝑘𝑘 ($) 𝑢𝑢" = 𝑓𝑓1 ,(3 − 18)

3 𝐹𝐹
5 𝐹𝐹, 𝑢𝑢, =
.
(
)
𝑒𝑒
2 𝑘𝑘
2 𝑘𝑘(𝑒𝑒)(3 − 19), 𝑢𝑢+ =

And the unknown nodal force 𝑓𝑓! is determinedas 𝑓𝑓! = −𝐹𝐹. The physically acceptable solution
mode is shown in Fig. 8 below.

in Fig. 7.

Fig. 8 Physically acceptable solution mode forthe system of linear springs [5]3.3. FEA with ANSYS® Simulation Software

In this section, we will investigate the theory
behind the structure analysis conducted using

Fig. 7 possible solution modes for the system of
linear springs [5]ANSYS® Mechanical APDL. This section is

In order to have unique solution, the global
system matrix is rendered non-singular byeliminating the zero eigenvalue. This isachieved by introducing boundary condition to
suppress the translational mode of the solution
corresponding to the zero eigenvalue [5].

basically an outline of the theory behind
structural analysis without material andgeometric nonlinearities. It was referenced fromANSYS® Mechanical APDL theory reference[6].
In this analysis, the applied load and theboundary conditions doesn’t change with time.
However, material and geometrical non-

９-２－ 6

linearities can be modeled, such as material
plasticity, creeping phenomenon etc. In terms of
SRF cavity, we would like to know the stress at
which the cavity structure will permanently
change (stress intensity > 0.2% yield strength),rendering its structure permanently deformed.
3.3.1. Stress-strain Relationship in ANSYS®
The stress and strain for linear materials are
related through the following equation and thestress vectors are shown in Fig. 9 [6]:𝛼𝛼.5$ = Secant coefficient of thermal expansion in
the 𝑥𝑥 direction∆𝑇𝑇 = 𝑇𝑇 − 𝑇𝑇6$7

𝑇𝑇 = current temperature at the point inquestion

𝑇𝑇6$7 = reference strain free temperature

The flexibility or compliance matrix is:
1
⎡
𝐸𝐸
⎢ #
⎢−𝜈𝜈$#⎢ 𝐸𝐸$⎢−𝜈𝜈%#⎢
𝐸𝐸[D]!" = ⎢ %
⎢ 0
⎢
⎢
⎢ 0
⎢
⎢ 0
⎣

−𝜈𝜈#$
𝐸𝐸#
1
𝐸𝐸$
−𝜈𝜈%$
𝐸𝐸%

−𝜈𝜈#%
𝐸𝐸#
−𝜈𝜈$%
𝐸𝐸$
1
𝐸𝐸%

0

0

0

0

0

0

0

0

0

0

1
𝐺𝐺#$

0

0

0

Where the typical terms are:

0
0

1
𝐺𝐺$%
0

0 ⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥ . (3 − 23)
0 ⎥
⎥
⎥
0 ⎥
⎥
1 ⎥
𝐺𝐺#% ⎦

𝐸𝐸. = Young’s modulus in the x direction𝜈𝜈./ = major Poisson’s ratio

Fig. 9 Stress vector definition [6]{σ} = 𝐷𝐷{𝜀𝜀 $- },where:(3 − 20){σ} = Stress vector = [𝜎𝜎. 𝜎𝜎/ 𝜎𝜎0 𝜎𝜎./ 𝜎𝜎/0 𝜎𝜎.0 ]1

𝜈𝜈/. = minor Poisson’s ratio

𝐺𝐺./ = Shear modulus in the xy planeThe

stress-strain matrix.
{𝜀𝜀 } = {𝜀𝜀} − {𝜀𝜀 } = elastic strain vector{𝜀𝜀 23 } = thermal strain vector

The equation 3-20 can be inverted to:{ε} ={𝜀𝜀 23 }4!

+ [𝐷𝐷] {𝜎𝜎},where:

consideredtobe𝜈𝜈/. 𝜈𝜈./
=
.
𝐸𝐸/
𝐸𝐸.

(3 − 24)𝜈𝜈0/ 𝜈𝜈/0
=
.
𝐸𝐸0
𝐸𝐸/

(3 − 26)The𝜈𝜈./ , 𝜈𝜈/0 , 𝜈𝜈.0 , 𝜈𝜈/. , 𝜈𝜈0/ 𝑎𝑎𝑎𝑎𝑎𝑎 𝜈𝜈0.

(3 − 25)

arenot

independent quantities because of above three(3 − 21)

For the 3-D case, the thermal strain vector is:{𝜀𝜀 23 } = ∆𝑇𝑇[𝛼𝛼.5$ 𝛼𝛼/5$ 𝛼𝛼05$ 0 0 0]1 ,is𝜈𝜈0. 𝜈𝜈.0
=
.
𝐸𝐸0
𝐸𝐸.

23

{𝜀𝜀} = total strain vector = [𝜀𝜀. 𝜀𝜀/ 𝜀𝜀0 𝜀𝜀./ 𝜀𝜀/0 𝜀𝜀.0 ]1

matrixsymmetric, so that:[D] = Elasticity or elastic stiffness matrix or
$-

[D]4!

(3 − 22)

relations and therefore at least one quantity isrequired from either side of the equation fororthotropic materials. In ANSYS®, it isassumed that 𝐸𝐸. is larger than 𝐸𝐸/ , 𝜈𝜈./ islarger than 𝜈𝜈/. . Hence, 𝜈𝜈./ is referred as

９-２－ 7

“major’ Poisson’s ratio and 𝜈𝜈/. as “minor”

Poisson’s ratio. For isotropic materials (𝐸𝐸. =𝐸𝐸/ = 𝐸𝐸0 𝑎𝑎𝑎𝑎𝑎𝑎 𝜈𝜈./ = 𝜈𝜈/0 = 𝜈𝜈.0 ), so it doesn’t makeany difference with the provided data [6].

3.4. Eigenvalue Buckling

3.4.1. Euler’s Buckling Load
The criteria for failure for a structure until
now has been considered as the stress generated
in a body due to its own weight or the externalload should not exceed the yield stress of its
construction

material.

However,whenastructure has a dimension disproportionallylarger than the others like a long slender beam,

it is susceptible to failure under compressive
force. The compressive force can be thestructure’s own weight or some kind of external

Fig. 10 Buckled column in equilibrium (a);determining bending moment as a function of x(b) [7]The bending moment for the column is 𝑀𝑀 =−𝐹𝐹𝐹𝐹, where 𝜈𝜈 the column’s deflection at is 𝑥𝑥.

kind of failure is quite sudden and should be

Substituting this expression in a relationshipbetween the beam’s deflection and the bending
moment 𝜈𝜈" = 𝑀𝑀/𝐸𝐸𝐸𝐸 we get

avoided for structure’s integrity [7].

Whereforce. The sheer size difference will cause thelateral deflection to be larger than the axial
displacement called as buckling failure. This𝜈𝜈" + 𝜆𝜆" 𝜈𝜈 = 0,

3.4.2. Euler Buckling Load for a pinnedpinned column
An example of buckling load for a pinnedpinned slender column subjected to axial forcesat the ends is shown in this section. To
determine the expression for buckling load, it is𝜆𝜆" =force ‘F’ necessary to hold it in equilibrium. It is
accomplished by proceeding to determine the
distribution of the column’s deflection in termsof 𝐹𝐹. In Fig. 10, a free-body diagram by passing

a plane through the column at an arbitrary
position 𝑥𝑥 [7].

(3 − 28)

Where EI is the flexural rigidity, E is theYoung’s modulus and I is the moment of inertia.
The general solution of second order differential
equation 3-27 isassumed that the column has already buckled

and seek to determine the value of compressive

𝐹𝐹
,
𝐸𝐸𝐸𝐸

(3 − 27)𝜈𝜈 = 𝐵𝐵 sin 𝜆𝜆𝜆𝜆 + 𝐶𝐶 cos 𝑥𝑥 .

(3 − 29)

Where B and C are constants. From theboundary condition that deflection is 0 at 𝑥𝑥 =0, 𝜈𝜈|.)8 𝑖𝑖𝑖𝑖 0, we get C = 0. The deflection is also
0 at 𝑥𝑥 = 𝐿𝐿.

𝜈𝜈|.)9 = 0,

Bsin 𝜆𝜆𝜆𝜆 = 0.

(3 − 30)(3 − 31)

If B = 0, then 𝜈𝜈 = 0 which does satisfy equation

3-27 but we are seeking a buckled solution,
hence if 𝐵𝐵 ≠ 0 then

９-２－ 8(3 − 32)

sin 𝜆𝜆𝜆𝜆 = 0.

The parameter 𝜆𝜆 depends on the compressive

Where K is the column effective length factor
and its values are given in Fig. 12.

force F and equation 3-32 has infinite numberof roots for 𝜆𝜆. It is satisfied when𝜆𝜆 =Where n is any integer.

𝑛𝑛𝑛𝑛
.
𝐿𝐿

(3 − 33)

Substituting equation 3-33 into 3-28 and 329, we obtain axial load as in equation 3-34 anddeflection as in equation 3-35, with various
modes of deflections shown in Fig. 11 [7]:𝑛𝑛" 𝜋𝜋 " 𝐸𝐸𝐸𝐸.
𝐿𝐿"
𝑛𝑛𝑛𝑛𝑛𝑛𝜈𝜈 = Bsin.
𝐿𝐿(3 − 34)𝐹𝐹 =(3 − 35)

Fig. 12 Mode of failure of a slender column withdifferent boundary conditions
3.4.4. Eigenvalue Buckling with ANSYS®
The search for the load that can cause
structural instability and bifurcation is known
as buckling load. The finite element equilibrium
equations for this type of analysis involve thesolution of homogeneous algebraic equations
whose lowest eigenvalue corresponds to thebuckling load, and the eigenvector represents
the primary buckling mode. Eigen value
buckling is used to calculate the theoretical
buckling load of a linear elastic structure. Since,
it assumes the structure exhibits linearly
elastic behavior, the predicted buckling loads
are overestimated (unconservative). This isFig. 11 Deflection distributions for increasing
values of n [7]3.4.3. Euler’s Buckling Loadboundary conditions

for

other

Some of the other end conditions that can beapplied to a slender column is shown in Fig. 12

available for valid structural degrees of freedom
only and the structure fails suddenly with ahorizontal force deflection curve. The structure
is considered to have constant stiffness effects[6].

and their Euler’s buckling load is given by𝐹𝐹 =𝑛𝑛" 𝜋𝜋 " 𝐸𝐸𝐸𝐸.
𝐾𝐾𝐿𝐿"

(3 − 36)

９-２－ 9

3.5. Non-linear Structural Analysis

This is a good tool to confirm the buckling loadofastructure,

when

there

is

a

small

deformation in the structure due to some

unknown external factors or when the structureis already deformed due to known or unknown
factors.

In

this

case,

large

deflection

methodology or arclength method can beapplied to model the sudden large displacement

Fig. 13 Types of buckling problems [6]A static solution is needed to establish thein a structure due to buckling.

stiffening of the structure under the applied
load (stress stiffening). There are several

4. 1.3 GHz SRF Cavity Assembly FEA

buckling modes (theoretically, infinitely many!)

on a 3-cell 1.3-GHz Niobium SRF cavity with its
Titanium jacket, its structure is as shown inin a structure. The first buckling mode is the
one requiring the smallest load. A linear
structure with a force buckling curve like Fig
13a is well modelled by a linear buckling

analysis, whereas for curve like Fig 13b, a large
deflection analysis is appropriate (with large
deflection ON). The buckling problem isformulated as eigenvalue problem [6]:where:([𝐾𝐾] + 𝜆𝜆& [𝑆𝑆]){𝜓𝜓& } = 0,

(3 − 37)[𝐾𝐾] = Stiffness matrix

The stress and buckling analysis is performed

Fig. 14. The cavity shape and Titanium jacket issimilar to Tesla-like that was designed at KEK.
This simulation is just an example and should
not be considered as the finalized results for the
stress and buckling analysis for Tesla-likecavity.
4.1. Static Structure Analysis

For the high-pressure gas safety authority
the SRF cavity with its Titanium vessel shouldbe able to withstand maximum allowable
working pressure (MAWP) of 0.2 MPa or 2 bar[𝑆𝑆] = Stress matrix

absolute pressure. The cavity normal operating

𝜆𝜆& = ith eigenvalue (used to multiply loads
generated in [𝑆𝑆]pressure is 0.003 MPa or 0.03 bar absolute,
which is roughly the saturated vapor pressure

The eigenvectors are normalized so that thelargest component is 1.0. Thus, the stresses

design and the materials used for this assembly
are not standardized the design must be proven𝜓𝜓& = ith eigenvector of displacementsof 2.0 K superfluid helium. Moreover, since this(when output) may only be interpreted as arobust. For this design, the simulation isrelative distribution of stresses. By default, theBlock Lanczos and Subspace Iteration methods

divided in three case studies:find buckling modes in the range of negative• Case A: In this case the assembly is in room
temperature condition (40 °C) with helium gas

infinity to positive infinity [6].

between the SRF cavity and Ti jacket. The
MAWP between the SRF cavity and Titanium
Jacket is 0.2 MPa and the tuner displacement is
0.65 mm. The location of tuner displacement

９-２－ 10can be seen in Fig. 14. The displacement iscomponent is described in detail in next

considered equally on either jackets, hence,
0.325 mm on one jacket and -0.325 mm on thesections.

other tank. There is vacuum outside Ti jacket
and inside SRF cavity.
• Case B: In this case, the assembly is cooled

down to operational conditions by fillingsuperfluid helium between the SRF cavity andits Ti helium jacket, i.e, it is cooled down from
40 °C to -271.4 °C (1.8 K). The MAWP between
the SRF cavity and Ti Jacket is 0.2 MPa and thetuner displacement is 0.65 mm. There isvacuum outside Ti jacket and inside of the SRF
cavity.
• Case C: In this case, the assembly is cooled

down to operational conditions by fillingsuperfluid helium between the SRF cavity andits Ti helium jacket, i.e, it is cooled down from

40 °C to -271.4 °C (1.8 K). The MAWP between
the SRF cavity and Titanium helium Jacket is
0.2 MPa and the tuner displacement ismaximized to 3 mm. There is vacuum outside Ti
jacket and inside the SRF cavity.
Thelayout

of

the

cases

in

ANSYS®

Workbench is shown in Fig. 15 and eachFig. 15 3-cell 1.3 GHz Tesla-like cavity with its
Titanium jacket
4.1.1. Engineering Data
In this section, the necessary material
properties are inserted to provide necessary
information regarding the elastic and plastic

behavior of the materials. Examples of some of
the included properties are Young’s modulus,
Poisson’s ratio, Shear modulus, Coefficient ofthermal expansion etc. It is possible to vary
properties with respect to temperatures tooeither in tabular or in equation form.

For this simulation, we will only consider theelastic properties of Nb and Ti. The properties
that were entered in the engineering data areshown in Fig. 16 and the properties are also
listed in table 1:

Fig. 14 3-cell 1.3 GHz Tesla-like cavity with its Titanium jacket

９-２－ 11

９-２－ 12

subsequent simulations in the quantities suchas stress, strain, deformation etc.
• Geometry: All the solids or surface geometries

are listed here. The materials are assigned toindividual solids, as shown in Fig. 18. The

cavities are assigned Niobium and the jackets
are assigned Titanium.
Fig. 20 Discretization of the geometry bymeshing
4.1.4. Setup
In this section, the external factors andFig. 18 Material assignment in ANSYS® Model• Symmetry: This is an optional setting. As theboundary conditions are provided to the model.
The analysis settings and boundary conditions
are shown in Fig. 21:

cavity and tank structure can be considered axisymmetrical, only a half 3D model of thestructure was modeled, and the other half wasconsidered as a symmetry. In this case, themodel is symmetrical in z-axis and the faces in
red are with which the cavity model issymmetrical to, as seen in Fig. 19.

Fig. 21 Boundary conditions and analysis
settings• Analysis settings: In this section, we canbasically set the number of steps, switchinglarge deflection ON or OFF for non-linear
structural analysis and other controls, as shownin Fig. 22.

Fig. 19 Symmetry region for the half 3D model(in red)• Meshing: In this section, the uploaded CAD

geometryisdiscretizedusinghexahedral

meshes. Mesh dependency studies have to be
carried out and the maximum possible size ofthe meshes were determined. In this case themaximum mesh size was selected to be 3 mm.
The geometry was discretized in to 87,440
number of elements and 470,000 nodes, as
shown in Fig. 20.

Fig. 22 Analysis settings for the simulation• Pressure: The inner faces of the Ti jacket andthe outer faces of the SRF cavities are selected
to provide pressure condition normal to them.

９-２－ 13

For all cases, pressure was set to 0.2 MPa, as

rotation is allowed along any direction, as

seen in Fig. 23.

shown in Fig. 25.

Fig. 23 Pressure boundary condition (in red)
• Displacement: This is provided to the crosssectional inner surface of the Tank 1 and Tank

2, respectively, as seen in Fig 24, to simulate thetuner movement, which stretches the cavity
structure. There is a virtual spring with a
stiffness coefficient of 206 N/mm connecting
these surfaces, which replaces the Ti bellow(added in connection setting). Displacement
condition can used to fix or provide movement

to a line or surface in any direction. In this case,the cavity was stretched in ±x-axis and thecavity movement in other directions were kept
free to deform or move.

Fig. 25 Remote displacement boundary
condition
4.1.5. Solution and Results
In this section, the results of the converged
solution are shown. It is possible to view andplot many of the quantities such as Von-misses
stress, Von-misses strain, Stress intensity,
Deformation (total or directional), strain energy,
linear stress etc. Moreover, with a probing tool

it is possible to determine all of the above listed
quantities at a node, element, line or a plane.
Usually, the results are visually represented ina gradient color format but that can be modified
according to designers’ preference. It is possible
to form charts, graphs and many othergraphical and visual form of the simulation
results.

The stress intensity on the cavity assembly
for Cases A, B and C are shown in Fig. 26 andsummarized in Table 2 and 3. The maximum
stress intensity in all cases occurred at the
stiffener ring weld section. It is mainly due toFig. 24 Displacement boundary condition• Remote Displacement: This was not used toprovide any displacement but to fix the end
flanges in Y-direction. As the cavity structure isin a string in a cryomodule and is connected toanothercavitystructurewithabellow

assembly, as seen in Fig. 1, it cannot be fixed in
all three directions of the space. Hence, only aline contact is fixed in Y-direction and nothe stretching effect on the cavity from tuner
displacement. For the high-pressure gas safety
regulations, the mechanical properties of thematerials used to manufacture the pressure
vessel should satisfy the following conditions: >1.5 times the maximum stress intensity for
0.2% Y.S and > 4 times the maximum stressintensity for T.S, on individual components.

９-２－ 14

Table 2 Maximum stress intensity on
various components of cavity assembly
Case

Table 3 Maximum stress intensity on
various sections of SRF cavity

Nb Half

Ti Tank

Nb-Ti weld

Cells [MPa]

[MPa]

[MPa]

A

70

5

12

B

76

5

23

C

300

18

65

Stiffener

Iris

EquatorRing Weld

Weld

Region weld

[MPa]

[MPa]

[MPa]

A

147

22

12

B

161

24

14

C

668

111

67

Case

Case A

Case B

Case C

Fig. 26 Stress intensity on the cavity assembly for various cases

９-２－ 15

4.2. ANSYS® Eigenvalue Bucklingits

In this FEA, we would determine the pressure
at which the cavity structure will buckle. As
theorized before this analysis only takes theelastic region of the materials in consideration.
For this analysis, Case B is taken intomeshing

andconnections

are

also

transferred to eigenvalue buckling system.
4.2.4. Setup
In this case, the same boundary conditions

were transferred to the eigenvalue bucklingconsideration, as the tuner movement is thesystem.

would be easier to buckle in Case B rather thanconsidered as a variable on each nodes of outer

least out of all cases and the cavity structure

condition

However,isthesuppressedpressureasitboundarywouldbeCase C, as larger tuner movement makes thesurface of SRF cavity and inner surface of Ti

case structure of Eigenvalue buckling inpressure of the SRF cavity structure. The

cavity structure more difficult to buckle. The
ANSYS® is shown in Fig. 27.

helium jacket, to determine the buckling
various boundary conditions are shown in Fig
28.

Fig. 27 Case structure of Eigenvalue bucklingsimulation
4.2.1. Engineering Data
In this case, the engineering data is the sameasinstaticstructure

analysis,astheengineering data is linked, the elastic
mechanical properties are transferred to thegeometry for Eigenvalue buckling system.
Fig. 28 Boundary condition for structure
analysis for buckling simulation

4.2.2. Geometry
The geometry remains the same as thegeometry considered for the static structure
analysis. Hence, the same geometry is

4.2.5. Analysis Settings for the Solver

forwarded to this system by linking them.

equations for Eigenvalue buckling analysis. The
pre-stress load pattern is set to No, since we

4.2.3. Model
As the same geometry is transferred from thestatic structure analysis system, the model with

Mechanical APDL solves the governing

would like to provide pressure as the variable todetermine the load multiplier to the initialpressure condition of 0.2 MPa. In this case, wewould like to determine 2 modes of failure,

９-２－ 16

hence maximum number of modes to find are

4.2.6. Results and discussions

set to 2, as seen in Fig. 29.

The buckling load multiplier for the cavity
structure was determined to be 385.2 for the 1st
Eigen mode and 386.6 for the 2nd Eigen mode,
as seen in Fig. 30. This provides the bucklingpressure of 77.1 MPa (385.2 × 0.2 MPa) for both
Eigen modes. As the mode of failure were at thesections of similar thickness and similar innature, the load multipliers were equal to eachother. The buckled behavior of the structures
can be seen in Fig. 31.

For a 9-cell cavity, the 1st mode of Eigenvalue
buckling occurs at 48 MPa with similarboundary conditions. The mode of failure for a
9-cell cavity structure is similar to the one
shown for 3-cell cavity structure.

Fig. 29 Analysis setting (top) and nodal
pressure boundary condition (below)

Fig. 30 Load multiplier for the buckling modes

1st Eigen mode

2nd Eigen mode
Fig. 31 Load multiplier for the buckling modes

９-２－ 17

4.3.2. Boundary Conditions

4.3. Non-linear Structural Analysis

This analysis is conducted to verify the

The boundary conditions remained the same

results obtained from the Eigen value bucklingas the ones previously used for structure

analysis, and to study the effect of deformationsin the cavity structure on its buckling pressure.

analysis of Case B, except the pressure
boundary condition, which was incrementally

In this study, the buckled structure from Case B

increased from 0 – 80 MPa in 8 steps towas considered for a non-linear structural

analysis with large deflection set to ON todetermine the buckling pressure, as shown in
Fig. 33. At the buckling pressure, the solution

determine the buckling pressure.

will not converge due to large deformation of thestructure, as seen on Fig. 34.

4.3.1. Case structure
The case structure for this simulation is acontinuation of the Eigen value buckling

analysis, where any of the determined bucklingmode can be transferred to a static structure
analysis. Moreover, it is possible to control thelevel of buckling in the structure that would betransferred with scale factor, which can bevaried from 0 to > 1, where 0 being the original
structure and 1 being the buckled structure thatwould be transferred to the static structure
analysis. Here, the 1st Eigen mode wastransferred to four static structure analysis
studies with the scale factors being varied from

Fig. 33 Incremental pressure boundary
condition

0.005, 0.01, 0.1 and 1, as shown in Fig. 32.

Fig. 34 Buckled cavity structure
4.3.3. Results and Discussions
The buckling pressure was determined byplacing a total deformation probe on a point on
the cavity structure where the maximum
buckling occurs. The location of the deformation
probe is shown in Fig. 35. The total deformation
Fig. 32 Case structure for non-linear structural
analysis

obtained from the deformation probe wasplotted against the pressure condition for

９-２－ 18

various scaling factors and is shown in Fig. 36.

results are different for 9-cell cavity). Moreover,The buckling pressure was 74 MPa for astructure with scale factor of 0.005 andthe buckling pressure and its Eigen modes wereapproximately 54 MPa for the scale factor of 1.
From the plot it can theorized, that for elastic
material properties, the cavity will not buckle
upto 54 MPa pressure. The buckling pressure
usually should be > 4 times the MAWP, and in
this case the cavity structure has sufficientstrength to qualify for that criterion.

Fig. 35 Deformation probe location on the
structure after buckling for scale factor of
0.005 at 73.5 MPa pressure

determined.

REFERENCES[1] A. Yamamoto et al., “Ingot Nb based SRF
technology for the International Linear Collider”,
AIP Conference Proceedings 1687, 030005 (2015);
doi:10.1063/1.4935326[2] The International Linear Collider: A Global Project,
arXiv:1903.01629 [hep-ex][3] AIP Conference Proceedings 1434, 1575 (2012);
https://doi.org/10.1063/1.4707088[4] B. Aune et al., “Superconducting TESLA Cavities”,
Phys. Rev.ST Accel. Beams 3, 092001 (2000)[5] E Madenci and I Guven., The finite element method
and applications in engineering using ANSYS®,
second edition, Springer 2015.
[6] P. Kohnke, ANSYS® Mechanical APDL theoryreference, ANSYS® Inc.
[7] A. Bedford and K.M Leichti, Mechanics of materials,
second edition, Springer.

Fig. 36 Pressure v/s total displacement for the
deformation probe on the Nb SRF cavity

5. Summary
In this lecture, the basics of finite element
analysis for static structure and buckling
failure was detailed. An example of 3-cell 1.3
GHz Tesla-like cavity was considered todetermine the stress generated on the cavity forthe high-pressure gas safety conditions (the９-２－ 19