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Computer Assisted Mechanics and Engineering Sciences, 16: 279–290, 2009.
Copyright c
2009 by Institute of Fundamental Technological Research, Polish Academy of Sciences
Eigenvalue analysis for high telecommunication towers
with lognormal stiffness by the response function method
and SFEM
Marcin Marek Kamiński
Department of Steel Structures
Faculty of Civil Engineering, Architecture and Environmental Engineering
Technical University of Łódź, Al. Politechniki 6, 90-924 Łódź, Poland
e-mail: Marcin.Kaminski@p.lodz.pl
Jacek Szafran
Department of Steel Structures
Faculty of Civil Engineering, Architecture and Environmental Engineering
Technical University of Łódź, Al. Politechniki 6, 90-924 Łódź, Poland
(Received in the final form February 28, 2010)
The main aim of this paper is to demonstrate the application of the generalized stochastic perturbation
technique to model the lognormal random variables in structural mechanics. This is done to study proba-
bilistic characteristics of the eigenvibrations for the high telecommunication towers with random stiffness,
which are modeled as the linear elastic 3D trusses. The generalized perturbation technique based on the
Taylor expansion is implemented using the Stochastic Finite Element Method in its Response Function
version. The main difficulty here, in a comparison to this technique previous applications, is a necessity
of both odd and even order terms inclusion in all the Taylor expansions. The hybrid numerical approach
combines the traditional FEM advantages with the symbolic computing and its visualization power and
it enables for a verification of probabilistic convergence of the entire computational procedure.
Keywords: stochastic dynamics, Stochastic Finite Element Method, response function method, stochastic
perturbation technique
1. INTRODUCTION
The eigenvalue problems with random parameters [1, 7–10] are still important in stochastic computa-
tional mechanics, which were extensively solved before using the Monte-Carlo simulation, Karhunen-
Loeve or Taylor expansions as well as some algebraic approximations. Although the stochastic per-
turbation technique based on the Taylor expansion was usually employed for the analysis of the
Gaussian input random variables and fields [3, 4, 6], its application area is general and include all
random variables types. The only condition here is that the probabilistic moments included in the
specific equations for both input and output characterization must exist and must be computable.
The lognormal probability distribution function (PDF), similarly to the Gaussian one, has the ana-
lytical equations describing recursively all probabilistic moments and, being popular in engineering
applications, is now tested with the Generalized Stochastic Finite Element Method (GSFEM). This
recursive property is very unusual, because most of the distributions has the moments generating
functions, so their usage may be proposed in the numerical context only. First, we would like to
test the lognormal distributions having the same first two moments as the Gaussian – it needs
an extra symbolic solution to the nonlinear equations for new PDF parameters. Then, we use the
Taylor series based formulas with all, odd and even, components to derive the structural response
280 M. M. Kamiński, J. Szafran
probabilistic moments and it is done for the first few perturbation orders to notice the probabilistic
convergence of the method. As we employ the generalized stochastic method, the input coefficient of
variations takes the very large values (until 0.25), which is unusual for the structural applications,
however it illustrates the validity of this technique within the whole range of random dispersions
known from computational mechanics.
The structural application of the interest here is the telecommunication tower consisting of the
steel bars with the constant-cross sectional area designed in the linear elastic range. The eigenvi-
brations study is necessary for this structure since the natural exposition on the stochastic wind
loading, some other non-uniform and unpredictable atmospheric influences like the ice increasing
the cross-sectional areas. The second, more deterministic reason, is a necessity of the reliability
analysis, which is based on the normative fact that the difference between the induced vibrations
frequency and eigenfrequency must be larger than 25% of this last quantity (determined in a ran-
dom context below). Further computational studies will enable for a consideration of the stochastic
forced vibrations.
Computational implementation is based here on the Response Function Method enabling for
a reconstruction of the function relating the particular eigenfrequencies with the parameter being
randomized. Numerical results of the several FEM models of this tower with the Young modulus
varying around its expected value are embedded into the computer algebra system to derive the
additional response functions. Then, the partial derivatives are automatically calculated and inserted
into the analytical equations for any desired probabilistic moment. Once we have all the formulas
given analytically, a direct inclusion of the fluctuating perturbation parameter εas well as the
input coefficient of variation αinto the computations and visualization remains straightforward.
Computational analysis obeys here determination of up to fourth probabilistic moments and some
corresponding coefficients as the functions of this coefficient as well as of the theory order. As one
may suppose, the convergence of the expected values is the same like for the Gaussian distributions
but the remaining higher probabilistic moments would need decisively longer expansions than before.
Further numerical and theoretical efforts will be directly connected with a determination of the
reliability indices for those types of structural problems since a necessity introduced by the EU
engineering codes.
2. THE STOCHASTIC PERTURBATION METHOD FOR THE LOGNORMAL
DISTRIBUTIONS
Let us introduce the random variable b≡b(ω)and its probability density function as p(b). Then,
the expected values and the m-th central probabilistic moment are defined as
E[b]≡b0=Z+∞
−∞
bp (b)db, (1)
µm(b) = Z+∞
−∞
(b−E[b])mp(b)db. (2)
The basic idea of the stochastic perturbation approach is to expand all the input variables and the
state functions via Taylor series about their spatial expectations using some small parameter ε > 0.
In case of random quantity e=e(b), the following expression is employed:
e=e0+
∞
X
n=1
1
n!εn∂ne
∂bn(∆b)n,(3)
where
ε∆b=ε b−b0(4)
Eigenvalue analysis for high telecommunication towers with lognormal stiffness . . . 281
is the first variation of babout b0. Symbol (.)0represents the function value (.)taken at the expec-
tation b0. Let us analyze further the expected values of any state function f(b)defined analogously
to the formula (3) by its expansion via Taylor series with a given small parameter εas follows:
E[f(b), b] = Z+∞
−∞
f(b)p(b)db =Z+∞
−∞ !f0+
∞
X
n=1
1
n!εnf(n)(∆b)n"p(b)db. (5)
Let us remind that this power expansion is valid only if the state function is analytic in εand
the series converge; therefore, any criteria of convergence should include the magnitude of the
perturbation parameter. Perturbation parameter is taken as equal to one in numerous practical
computations. Contrary to the previous analyses in this area, now the lognormal random variable
is considered with the probability density function given as follows:
f(x) = 1
xσ√2πexp −(ln x−µ)2
2σ2,(6)
where µand σare its parameters. As one can easily demonstrate, the first two and k-th central
probabilistic moments for bare equal now to
E[b] = exp µ+σ2
2,(7)
Var(b) = eσ2−1e2µ+σ2(8)
and, generally
µk(b) = exp kµ +k2σ2
2.(9)
From the numerical point of view, the expansion provided in the formula (5) is carried out for the
summation over the finite number of components. Now, let us focus on an analytical derivation of
the probabilistic moments for the structural response function. It is easy to prove that [3]
E[f(b)] = f0(b) + ε∆bf,b +1
2ε2f,bb(b)µ2(b) + 1
3!ε3f,bbb(b)µ3(b)
+1
4!ε4f,bbbb(b)µ4(b) + 1
5!ε5f,bbbbb(b)µ5(b) + 1
6!ε6f,bbbbbb(b)µ6(b) + . . . ,
(10)
where (.),b,(.),bb denote for instance the first and the second partial derivatives with respect to
bevaluated at b0, respectively. Thanks to such an extension of the random output, any desired
efficiency of the expected values as well as higher probabilistic moments can be achieved by an
appropriate choice of the distribution parameters. Similar considerations lead to the fourth order
expressions for a variance. The following holds
Var(f) = Z+∞
−∞ f0+ε∆bf,b +1
2ε2(∆b)2f,bb
+1
3!ε3(∆b)3f,bbb +1
4!ε4(∆b)4f,bbbb −E[f]2
p(f(b)) db
=ε2e(2µ+2σ2)f,b2+ε3e(3µ+9
2σ2)f,bf,bb +ε4e(4µ+8σ2)1
3f,bf,bbb +1
4(f,bb)2.
(11)
282 M. M. Kamiński, J. Szafran
The third order probabilistic moments are derived including the lowest orders only as
µ3(f(b), b) = Z+∞
−∞
(f(b)−E[f(b)])3p(b)db
=Z+∞
−∞ f0+εf,b∆b+1
2ε2f,bb(∆b)2+1
3!ε3f,bbb(∆b)3+. . . −E[f(b)]3
p(b)db
=ε3µ3(b)f,b3+3
2ε4µ4(b)f,b2f,bb +3
8ε5µ5(b)f,b f,bb2+f,b2f,bbb.
(12)
Finally, the fourth probabilistic moment is approximated with the first few perturbation terms as
µ4(f(b), b) = Z+∞
−∞
(f(b)−E[f(b)])4p(b)db
=ε4µ4(b)f,b4+ 2ε5µ5(b)f,b3f,bb +3
2ε6µ6(b)f,b2f,bb2.
(13)
Let us mention that it is necessary to multiply in each of these equations by the relevant order
probabilistic moments of the input random variables to get the algebraic form convenient for any
symbolic computations. Therefore, this method in its generalized form is convenient for all the
random distributions, where the above mentioned moments may be analytically derived (or at least
computed for a specific combination of those distributions parameters).
3. VARIATIONAL FORMULATION
Let us consider the following set of partial differential equations adequate to the linear elastodynamic
problem consisting of [2, 5]
•the equations of motion
DTσ+ˆ
f=ρ¨u,x∈Ω, τ ∈[t0,∞),(14)
•the constitutive equations
σ=Cε,x∈Ω, τ ∈[t0,∞),(15)
•the geometric equations
ε=Du,x∈Ω, τ ∈[t0,∞),(16)
•the displacement boundary conditions
u=ˆu,x∈∂Ωu, τ ∈[t0,∞),(17)
•the stress boundary conditions
Nσ =ˆ
t,x∈∂Ωσ, τ ∈[t0,∞),(18)
•the initial conditions
u=ˆu0,˙u=ˆ
˙u0, τ ∈[t0,∞).(19)
Eigenvalue analysis for high telecommunication towers with lognormal stiffness . . . 283
It is assumed that all the state functions appearing in this system are sufficiently smooth functions
of the independent variables xand τ. Let us consider the variation of u(x, τ )in some time moment
τ=t, denoted by δu(x, τ ). Using the above equations one can show that [5]
−Z
Ω
(DTσ+ˆ
f−ρ¨u)TδudΩ + Z
∂Ωσ
(Nσ −ˆ
t)Tδu d(∂Ω) = 0.(20)
Assuming further that the displacement function u(x, t)has known values at the initial moment
u(x, t1) = 0 and at the end of the process u(x, t2) = 0, the variations of this function also equal
zero at those time moments
δu(x, t1) = 0, δu(x, t2) = 0.(21)
Integrating by parts with respect to the variables xand τwe can obtain that
Zt2
t1
δT −Z
Ω
σTδεdΩ + Z
Ω
ˆ
fTδudΩ + Z
∂Ω
ˆ
tTδud(∂Ω)
dτ = 0,(22)
where the kinetic energy of the region Ωis defined as [5]
T=1
2Zρ˙uT˙udΩ.(23)
We also notice that
δε=Dδu,x∈Ω, τ ∈[t0,∞).(24)
Next, we introduce the assumption that the mass forces ˆ
fand the surface loadings ˆ
tare independent
from the displacement vector u, which means that the external loadings do not follow the changes
in the domain initial configuration. Therefore, Eq. (22) can be modified to the following statement:
δZt2
t1
(T−Jp)dτ = 0,(25)
where Jpmeans the potential energy stored in the entire domain Ω
Jp=U−Z
Ω
ˆ
fTudΩ−Z
∂Ωσ
ˆ
tTud(∂Ω) = 0,(26)
whereas the variation is determined with respect to the displacement function and Uis the elastic
strain energy given by the formula
U=1
2Z
Ω
εTCε dΩ.(27)
It is well known that Eq. (26) represents the Hamilton principle widely used in structural dynamics
in conjunction with the Finite Element Method approach.
284 M. M. Kamiński, J. Szafran
4. COMPUTATIONAL IMPLEMENTATION
4.1. The Response Function Method in elastodynamics
Let us consider a discretization of the displacement field u(x, τ )using the following forms [2, 5]:
uα
3×1(x, τ )∼
=ϕ3×N(e)(x)qα
N(e)×1(τ),uα
3×1(x, τ )∼
=Φ3×N(x)rα
N×1(τ),(28)
where qis a vector of the generalized coordinates for the considered finite element, ris a vector
for the generalized coordinates of the entire discretized system, N(e)is the total number of the
e-th finite element degrees of freedom, Nis the total number of degrees of freedom in the structure
model. The generalized coordinates vector for the entire structure model is composed from the finite
element degrees of freedom and the transformation matrix as
rα
N×1=aN×N(e)qα
N(e)×1,(29)
ϕand Φare the corresponding shape function matrices (local and global). Contrary to the classi-
cal formulations of both FEM and the perturbation-based Stochastic Finite Element Method we
introduce here the additional index α= 1, . . . , M to distinguish between various solutions of the
elastodynamic problem necessary to build up the response function (around the mean value of the
input random parameter). The strain tensor can be expressed as
ε
ε
εα
6×1(x, τ ) = B6×N(e)(x)qα
N(e)×1(τ) = ˜
B6×N(x)rα
N×1(τ).(30)
Let us denote by Ethe total number of finite elements in the model. Then, the Hamilton principle
is obtained as
δZt2
t1!1
2
E
X
e=1
(qα)Tmα
N(e)×N(e)qα−1
2
E
X
e=1
(qα)Tkα
N(e)×N(e)qα+
E
X
e=1 Qα
N(e)T
qα"dτ = 0,(31)
so that
δZt2
t11
2(˙rα)TMα˙rα−1
2(rα)TKαrα+ (Rα)Trαdτ = 0.(32)
The element and global mass matrices are defined as
mα
N(e)×N(e)=Z
Ωe
ρα(x)BN(e)×6(x)T
B6×N(e)(x)dΩ(33)
and
Mα
N×N=Z
Ω
ρα(x)e
BN×6(x)Te
B6×N(x)dΩ.(34)
The stiffness matrices at the element and at the global scales are defined as follows
kα
N(e)×N(e)=Z
Ω(e)BN(e)×6T
Cα
6×6B6×N(e)dΩ(35)
and
Kα
N×N=Z
Ω(e)e
BN×6T
Cα
6×6e
B6×NdΩ.(36)
Eigenvalue analysis for high telecommunication towers with lognormal stiffness . . . 285
Hence, equation (32) can be rewritten with those substitutions as
(˙rα)TMαδr−Zt2
t1(¨rα)TMα+ (rα)TKα−(Rα)Tδrdτ = 0.(37)
Considering the assumptions
δr(t1) = 0, δr(t2) = 0,(38)
we finally obtain the dynamic equilibrium system
Mα¨rα+Kαrα=Rα,(39)
which represents the equations of motion of the discretized system. We complete this equation with
the component Cα
N×Nrα
N×1getting
Mα¨rα+Cα˙rα+Kαrα=Rα.(40)
Then we decompose the damping matrix as [6]
Cα=α0Mα+α1Kα,(41)
where the coefficients α0and α1are determined using the specific eigenfunctions for this problem,
so that
Mα¨rα+α0Mα˙rα+α1Kα˙rα+Kαrα=Rα,(42)
where no summation over the doubled indices αis applied here. As it is known, the case of undamped
and free vibrations leads to the system
Mα¨rα+Kαrα=0(43)
and the solution rα=Aαsin ωαtleads to the relation
−MαAαω2
αsin ωαt+KαAαsin ωαt=0,(44)
so that for sin ωαt6= 0 and Aα6=0there holds
−Mαω2
α+Kα=0.(45)
When the index αis postponed, then the stochastic problem is solved in a straightforward manner
analogously to the previous methods and the methodology follows the successive solutions of the
increasing order equations proposed in Section 3.
4.2. The Direct Differentiation Method in the eigenvalue problem
Let us consider a deterministic eigenproblem in its matrix description for its further stochastic
expansion
K−˜ω2Mϕ=0.(46)
Its zeroth order version looks like
K0− ˜ω02M0ϕ0=0.(47)
286 M. M. Kamiński, J. Szafran
After some algebraic transformation one can get the first order equation in the following form:
K,b −2˜ω0˜ω,bM0−(˜ω0)2M,bϕ0=− K0−(˜ω0)2M0ϕ,b.(48)
The next differentiation of Eq. (47) with respect to the input random variable breturns
K,bbϕ0+ 2K,bϕ,b +K0ϕ,bb −2˜ω0(˜ω,b)2M0ϕ0−2˜ω0˜ω,bbM0ϕ0
−4˜ω0˜ω,b(M,b φ0+M0ϕ,b)−( ˜ω0)2(M,bbϕ0+ 2M,b ϕ,b+M0ϕ,bb) = 0.(49)
It is quite clear here that the generalized version of the stochastic perturbation technique based on
the n-th order Taylor series expansion may lead to the very complex equation corresponding to the
highest order closure of the entire system. One demonstrates [4] that Eq. (46) may be rewritten as
n
X
k=0 n
kK(n−k)ϕk=
n
X
k=0 n
kk−1
X
l=0 k−1
l(2˜ω)(k−(l+1)) ˜ω(l+1)
n−k
X
m=0 n−k
mM(m)ϕ(n−k−m).(50)
The solution to this equation makes it possible to determine up to the n-th order eigenvalues together
with the corresponding eigenvectors.
5. NUMERICAL ILLUSTRATION
Computational analysis has been provided on the example of the steel telecommunication tower
with the height equal to 52.0 meters presented schematically in Fig. 1.
Fig. 1. Static scheme of the telecommunication tower
Eigenvalue analysis for high telecommunication towers with lognormal stiffness . . . 287
The entire structure has been discretized using the two-noded 183 linear space structure finite
elements (3D truss elements) joined in 66 nodal points and fully supported at the ground level. All
the structural members have been manufactured with the stainless steel with Young modulus equal
to E= 205 GPa treated here as the input lognormal random variable; analogous numerical studies
with the Gaussian random variables were provided in [4]. The expected value E[E]and the coeffi-
cient of variation α(E)of this variable were the input parameter and they were used to determine
this distribution input parameters. Therefore, the following nonlinear equations system was solved
symbolically using the computer algebra system MAPLE 13, to determine those coefficients
(exp µ+σ2
2=E[E]
exp σ2−1exp 2µ+σ2=α2(E)E2[E](51)
Since full analytical parametric solution to this system with respect to α(E)was impossible, the
solution variability with respect to the specific values of this parameter was computed and presented
in Table 1 below. The combinations of µand σwere further used in the SFEM computations to
verify the probabilistic convergence of the moments in this method with the lognormal input.
Table 1. The parameters for the input lognormal distribution
α σ µ
0 0 26.04627582
0,025 0.02499609497 26.04596342
0,050 0.04996879205 26.04502738
0,075 0.07489485110 26.04347120
0,100 0.09975134473 26.04130065
0,125 0.1245158079 26.03852373
0,150 0.1491663795 26.03515052
0,175 0.1736819337 26.03119311
0,200 0.1980421997 26.02666546
0,225 0.2222278668 26.02158321
0,250 0.2462206761 26.01596351
0,275 0.2700034952 26.00982488
0,300 0.2935603781 26.00318697
Numerical results based on the SFEM application obey in turn the following moments: the
expected values for first (Fig. 2), third (Fig. 3) and ninth (Fig. 4) eigenvibrations frequencies and
further – standard deviations (Fig. 5), third (Fig. 6) and fourth (Fig. 7) eigenvibrations frequencies.
They are all given as the functions of the input coefficient of variation; first two moments are
additionally also shown as the functions of the perturbation order – from first to fourth (standard
deviations) or to ninth (expectations). Let us mention that the input coefficient of variation for
structural steels is usually smaller or equal to 0.1, however this range has been extended more
than twice to check the entire numerical method. The particular results show that the first two
moments for lower order computations are almost independent from this input coefficient and after
some critical order (like the fourth one for the expected values) we notice the relationship: the
higher analysis order – the larger dependence on the input coefficient of variation. A comparison
of the first three diagrams leads to the conclusion that the expectations of the eigenfrequencies
are not always directly proportional to the coefficient α– the third eigenvalue shows an inverse
proportionality, for instance. Let us note further that for all eigenfrequencies being analyzed the
probabilistic convergence is sufficient since there are no visual differences between the models based
on eighth and ninth order perturbations. It should be clearly underlined that the differences between
the neighbouring orders approximations are relatively larger than for the SFEM studies with the
Gaussian random variables in the same random dispersion range. Figure 5 documents clearly that the
288 M. M. Kamiński, J. Szafran
Fig. 2. The expected values for first eigenvibrations frequency
Fig. 3. The expected values for third eigenvibrations frequency
Fig. 4. The expected values for ninth eigenvibrations frequency
Eigenvalue analysis for high telecommunication towers with lognormal stiffness . . . 289
Fig. 5. The standard deviations for first eigenvibrations frequency
Fig. 6. Third probabilistic moments for first eigenvibrations frequency
Fig. 7. Fourth probabilistic moments for first eigenvibrations frequency
290 M. M. Kamiński, J. Szafran
fourth order analysis is decisively not sufficient for the precise estimation of the standard deviations
– the reason is in full expansion with the odd and even order perturbation terms. Higher moments
show similarity to the results with Gaussian input (Figs. 6 and 7) since the first few terms only may
be decisive for the overall values of those moments.
6. CONCLUDING REMARKS
The application of the stochastic perturbation technique based on Taylor expansion of the general
order together with the additional computational implementation for lognormal variables have been
shown above. Numerical analysis was provided using the Response Function Method, a traditional
FEM engineering package as well as symbolic computational routines programmed in MAPLE 13.
The steel telecommunication tower with the Young modulus randomized according to the lognormal
distribution was studied computationally in the context of the probabilistic moments of its eigenval-
ues. As it is documented here, the Taylor expansions are significantly longer and much more complex
than for the Gaussian distributions studied before [3, 4, 6]. Probabilistic convergence of first two
probabilistic moments together with the analysis order also seems to be much slower. A comparison
with the results obtained for the Gaussian distributions needs an extra solution for the nonlinear
equations system, so that the hybrid symbolic-traditional FEM computational technique is very
reasonable. This model has been introduced to study further the influence of the ice covers on the
overall behavior of the towers and masts leading in cold regions to frequent engineering failures.
The methodology may be also straightforwardly used in the stochastic reliability analyses for such
structures, where the limit function may be provided as a difference between the eigenvibrations
and those induced into this structure by the wind, for instance.
ACKNOWLEDGMENT
The first author would like to acknowledge the Research Grant NN 519 386 636 from the Polish
Ministry of Science and Higher Education, whereas the second author participated in European
Project “Fellowship supporting innovative scientific research of the PhD students”.
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