Optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method

Finite Elements in Analysis and Design 63 (2013) 69–79

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Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel

Optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method Marcin Kamin´ski n,1, Marta Solecka Chair of Steel Structures, Department of Structural Mechanics, Faculty of Civil Engineering, Architecture and Environmental Engineering, Technical University of Lodz, Lodz, Poland

a r t i c l e i n f o

abstract

Article history: Received 28 January 2012 Received in revised form 27 July 2012 Accepted 7 August 2012 Available online 6 October 2012

The main objective here is an engineering optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method. This procedure is based on triple reliability analysis consisting of the limit states for the admissible eigenfrequencies, horizontal displacements and the reduced stresses; it obeys an application of the stainless steel as well as the aluminum as the structural materials. Such a reliability-based optimization follows strictly engineering practice, demands of the Eurocodes, cost minimization of the structures and future installations of the new transmission equipment. The reliability analysis is based on the FORM approach, while computational realization of the generalized SFEM guarantees reliable determination of the first four probabilistic moments of the structural response for any random dispersion in the design parameters. Numerical implementation has hybrid character and is based on interoperability of the FEM design system ROBOT and the computer algebra system MAPLE, where all probabilistic functions are programmed. & 2012 Elsevier B.V. All rights reserved.

Keywords: Engineering optimization Stochastic Finite Element Method Response function method Stochastic perturbation technique Reliability analysis

1. Introduction Telecommunication towers undoubtedly belong to the class of the lightweight structures specifically exhibited to the stochastic influence of the wind blow and because of their reliability need to be evaluated with respect to the strength, to maximum deflections and rotations as well as to their eigenfrequencies [8]. An uncertainty in the structural response of the towers and masts in general follows the stochastic wind pressure and flattering, quasi-periodic and temporary ice covers increasing both mass and effective surfaces of the structural elements. Temperatures fluctuations leading to the significant thermal stresses not necessarily uniformly influencing the entire structure (concerning dominating southern exposure to the natural heating) play also important role [20]. Geometrical imperfections in the connections (especially in welds) and in the structural elements, considering material defects resulting from manufacturing stage, may be decisive for an overall uncertainty magnitude. Because aluminum has much more micro-compounds in its total volume than stainless steel, randomization of the basic material properties is even more justified here and random dispersion of these properties should be significantly larger.

n

Corresponding author. E-mail address: [email protected] (M. Kamin´ski). 1 ¨ Polymer Forschung Dresden e.V., Visiting Professor, Leibniz-Institut fur 01069 Dresden, Hohe Strasse 6, Germany. 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.08.002

Designing and manufacturing of truss-type structures (towers, masts and antennas) are still relatively new and very modern area for the engineers and scientists. We notice continuous development of the telecommunication equipment and particularly mobile phones, which may demand the brand new structural extensions in the nearest future. Therefore, an optimization of the supporting structures’ shape and the materials design is still challenging issue – an example is alternative usage of the aluminum and steel based supporting towers and other lightweight structures of the similar shape [11]. It needs to be emphasized that since such structures are really some 3D combinations of the beam and bar elements, classical mathematical methods for the shape or topology optimization (where optimal shape can be ‘‘cutted off’’ from some given volume subjected to the specified boundary conditions) have no application here [2]. An optimal shaping procedure is left rather to an engineering intuition, experiments and practice as well as some kind of the robust optimization [4], where some small corrections are provided a posteriori using the FEM reanalysis. Some optimization conditions follow also technological demands on the telecommunication equipment efficiency, safety and maintenance, and then cannot be simply expressed in the structural mechanics language. On the other hand, the engineering reliability analysis is still being developed [12,13], concerning at least demands of the Eurocodes and acquisition of various stochastic methods in engineering practice. A necessity of structural reliability verification was introduced by Eurocode 0 (not only for the bridges’ and skyscrapers’ designing and maintenance [17]); our analysis is

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focused on the towers initially designed according to the deterministic limit states. We are going to verify whether they have correct reliability indices or not and if they can be further optimized in this context. Rationally designed steel tower usually appears to automatically remain in the safe regime [8], while aluminum structures may need some additional stiffening of certain elements. Computational comparison provided here consists of a single steel and two aluminum structures being slightly below and above reliability limits. It is widely known, that the reliability measured with some indices must be computed not only for the load capacity and maximum deflections of some structures but also for their vibrations and fatigue under dynamic loads; it may demand more advanced stochastic computer methods [1,14,16,18,19]. These are the main reasons to compare aluminum and steel manufactured telecommunication structures in the presence of uncertainty of material properties and to consider their optimization. We study random fluctuations of the eigenfrequencies, stresses and deflections for the same towers to discuss their reliability issues assuming that Young modulus of both materials is a truncated Gaussian random variable with the given expected value; the coefficient of variation is taken as the extra input parameter in this analysis. These towers have identical height and spatial distribution of the structural components and, naturally, different cross-sections of the particular members’ groups. The generalized stochastic perturbation technique [6] is employed to achieve this goal because of the expected time savings (compared to the Monte-Carlo technique) and a capability to determine up to the fourth order probabilistic moments and coefficients [1,10,19]. Contrary to the previous models [6,8,10], now full tenth order stochastic expansions are used to recover all these moments, which significantly increases a final accuracy of the probabilistic results. This stochastic technique is not supported by the Direct Differentiation Method (DDM); we employ the Response Function Method (RFM) [7], similar to the Response Surface Method (RSM) [15] explored in the reliability analysis. The major and very important difference is that the RFM uses higher order polynomial response relating a single input random variable with the structural output. The RSM is based rather on the first or second order approximations of this output with respect to multiple random structural parameters. An application of the RSM is impossible in current context because the second order truncation for the response eliminates all higher order terms necessary for the reliable computations of the probabilistic structural behavior. Furthermore, RSM has some statistical aspects and issues [15], while the RFM has purely deterministic character and exhibits some errors typical for the mathematical approximation theory methods. A major novelty in this elaboration is an application of such a SFEM procedure for an optimization of large scale spatial structures based on the reliability analysis. We find the set of minimum cross-sections in the given towers with randomized Young modulus, resulting in the set of minimum allowable reliability indices for displacements, stresses and eigenfrequencies. Computational analysis shows that this optimization procedure is quite independent of the structural material and proceeds from the top to the bottom of the tower through the sections with different profiles. Computational procedure is provided using the Finite Element Method engineering system ROBOT (widely employed in civil engineering branch), where all eigenfrequencies, stresses and displacements are determined as the functions of Young modulus for all towers. Further computations of the response functions, their symbolic differentiation, probabilistic moments and the coefficients as well as their visualization are all provided with the use of computer algebra system MAPLE. This analysis is

planned to be extended towards a full verification of the static and dynamic reliability, buckling fragility as well as including fatigue, ageing and corrosion phenomena [12] into the overall structure’s mathematical and computational models. It could be especially attractive using a fully extended version of some FEM systems implemented with the RFM and the stochastic reliability verification together with additional probabilistic visualization.

2. Variational formulation Let us consider the following linear elasto-dynamic problem consisting of [3,9]

 equations of motion: € x A O, t A ½t 0 ,1Þ DT r þ f^ ¼ ru,

ð1Þ

 constitutive equations: r ¼ Ce, xA O, t A ½t0 ,1Þ

ð2Þ

 geometric equations: e ¼ Du, x A O, t A ½t 0 ,1Þ

ð3Þ

 displacement boundary conditions: ^ xA @Ou , t A ½t 0 ,1Þ u ¼ u,

ð4Þ

 stress boundary conditions: ^ x A @Os , t A ½t 0 ,1Þ Nr ¼ t,

ð5Þ

 initial conditions: 0 ^ 0 , u_ ¼ u^_ , t ¼ t 0 u¼u

ð6Þ

Let us consider the variation of uðx, tÞ in some time moment t ¼ t denoted by duðx, tÞ. Using the above equations one can show that Z Z ^ quÞ ^ T du dð@OÞ ¼ 0 € T du dO þ ðNrtÞ ð7Þ  ðDT r þ f O

@Os

Assuming further that uðx,t 1 Þ ¼ 0, uðx,t 2 Þ ¼ 0 and that their variations also equal 0, an integration by parts leads to  Z t2  Z Z Z T T dT rT de dO þ f^ dudO þ ð8Þ t^ dudð@OÞ dt ¼ 0 t1

X

X

@X

The kinetic energy of the region O is of course defined as Z 1 ru_ T u_ dO ð9Þ T¼ 2 Assuming that the mass forces f^ and the surface loadings t^ are independent of the displacement vector u, one may modify Eq. (8) to the form Z t2 d ðTJp Þdt ¼ 0 ð10Þ t1

where J p is potential energy stored in O Z Z T T Jp ¼ U f^ u dO t^ u dð@OÞ ¼ 0

ð11Þ

and where U is the elastic strain energy Z U ¼ 12 eT Ce dO:

ð12Þ

X

O

@Xr

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Eq. (10) represents Hamilton principle widely used in structural dynamics as the basis for the Finite Element Method implementation [3,5,9].

Let us consider a discretization of the displacement field uðx, tÞ using the following form [3,9]: ð13Þ

where q is a vector of the generalized coordinates of the discretized system. Contrary to the classical formulation of the perturbation-based Stochastic Finite Element Method, we introduce the additional index a ¼1,y,M to distinguish between various solutions of the elasto-dynamic problem necessary to build up the response function. They are provided around the mean value of the input random parameter. Then, the strain tensor can be expressed as

ea ðx, tÞ ¼ BðxÞqa ðtÞ

ð14Þ

The discretized version of Hamilton principle is obtained as Z t2 Z t2 1 aT a a 1 aT a a  d q M q  q K q t þ d ðQ aT qa Þdt ¼ 0 ð15Þ d 2 2 t1

systems for the RFM as follows [6,8]: Ka qa ¼ Q a

4.1. Stochastic Taylor expansion Let us introduce the random variable b  bðoÞ and its probability density function as p(b). The expected values and mth central probabilistic moments are defined as [6–8] Z þ1 0 E½b  b ¼ bpb ðxÞdx ð27Þ 1

mm ðbÞ ¼

þ1

ðbE½bÞm pb ðxÞdx

ð16Þ

so that all its partial derivatives with respect to random Young modulus equal 0. The vector Q represents the external loadings acting on this structure. Finally, the global stiffness matrix equals to Z ð17Þ Ka ¼ BT Ca B dO

A basic idea of the stochastic perturbation approach is to expand the state functions via Taylor series about their spatial expectations using some small parameter e 40. In case of random quantity e¼e(b), the following expression is employed: 1 X

e ¼ e0 þ

1 n n!

e

n¼1

@n e n n ðDbÞ @b

Since the 3D bar and beam elements are used in further computations, their first partial derivatives with respect to random input parameter differ from 0 by only. Considering the assumptions that ð18Þ

we finally obtain the dynamic equilibrium system Ma qa þ Ka qa ¼ Q a

ð19Þ

which represents equations of motion of the discretized system. We complete this equation usually with the component Ca qa getting Ma qa þ Ca qa þKa qa ¼ Q a

ð20Þ

and we decompose the damping matrix as Ca ¼ a0 Ma þ a1 Ka

ð21Þ

where the coefficients a0 and a1 are determined using the specific eigenfunctions for this problem. Then Ma qa þ a0 Ma qa þ a1 Ka qa þ Ka qa ¼ Q a

ð22Þ

where no summation over the doubled indices a is applied here. As it is known [4], the case of undamped free vibrations leads to the following algebraic system: Ma qa þ Ka qa ¼ 0

ð23Þ

and the solution qa ¼ Aa sinoa t leads to the relation 2

a a

M A oa sinoa t þK A sinoa t ¼ 0

ð30Þ 0

is the first variation of b about b , which is the expectation of b. Let us analyze further the expected values of any state function f(b) defined analogously to the formula (27) by its expansion via Taylor series as follows: Z þ1 Z þ1  X1   0 1 n ðnÞ E f ðbÞ ¼ f ðbÞpb ðxÞdx ¼ f þ e f Dbn pb ðxÞdx n ¼ 1 n! 1

ð31Þ From the numerical point of view, an expansion provided by the formula (31) is carried out over the finite number of components to assure satisfactory accuracy. Let us focus on analytical derivation of the probabilistic moments for the structural response. It is easy to prove that the general 10th order expansion results for Gaussian variables in the following formula:



  @2 f

0 E f ðbÞ ¼ f ðbÞ b ¼ b0 þ 12e2 m2 ðbÞ 2 b ¼ b0

@b @10 f

1 10 ð32Þ þ    þ 10!e m10 ðbÞ 10 b ¼ b0 @b Let us mention that we multiply here by the relevant order probabilistic moments of the input random variables to get the algebraic formulas in 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 the parameters). Finally, one may recover the kurtosis and the skewness from their well-known definitions as

kðf ðbÞÞ ¼

m4 ðf ðbÞÞ 3 s4 ðf ðbÞÞ

bðf ðbÞÞ ¼

m3 ðf ðbÞÞ s3 ðf ðbÞÞ

ð33Þ

The reliability index for the particular state functions equals ð24Þ

So that, for sinoa t a 0 and for Aa a 0 there holds Ma o2a þKa ¼ 0

ð29Þ

eDb ¼ eðbb0 Þ

1

O

dqðt1 Þ ¼ 0, dqðt2 Þ ¼ 0

ð28Þ

where

O

a a

Z

1

t1

The global mass matrix is defined as Z Ma ¼ ra ðxÞBT ðxÞBðxÞ dO

ð26Þ

4. Stochastic perturbation-based equations

3. Computational implementation

ua ðx, tÞ ¼ /ðxÞqa ðtÞ,

71

Rðf ðbÞÞ ¼

E½f^ f a  sðf^ f Þ

ð34Þ

a

ð25Þ

Obviously, Eq. (20) for the time independent generalized coordinates returns the well known linear statics equilibrium

where the pair ðf^ ; f a Þ denotes the admissible value of the given state function and its computed maximum value. Some civil engineering codes state for instance that this difference cannot be smaller than 25% of the structural eigenfrequency, so that

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72

Eq. (34) may serve for the straightforward estimation of the reliability for the structures subjected to the dynamic excitations. 4.2. Structural response determination via the least squares technique

2Db n1

ð35Þ

First we define the residuals r i ðub Þ between the trial points and the target function’s coefficients ^ r i ðub Þ ¼ uðiÞ b f ðb, ub Þ

ð36Þ

to determine the components of the vector u^ b . It precedes using minimization of the functional Sðub Þ ¼

m X

r 2i ðub Þ:

ð37Þ

i¼1

We minimize it to find the coefficients in f ðb, u^ b Þ @Sðub Þ ðjÞ

@u^ b

¼2

m X

r i ðub Þ

@r i ðub Þ

i¼1

ðjÞ

@u^ b

¼ 0; j ¼ 1,:::,n:

ð38Þ

Since the LHS derivatives combine the independent variable and the coefficients, so that these gradient equations do not have a closed solution and some initial values must be adopted for the coefficients. It may be resolved via some iterative approximation as ðjÞ ðjÞ ðjÞ ðjÞ u^ b ffi ðk þ 1Þ u^ b ¼ ðkÞ u^ b þ Du^ b

ð39Þ ðjÞ

where k denotes an iteration number and Du^ b stands for the socalled shift vector. The following Taylor linearization is applied at each iteration: 



f ðbi , u^ b Þ ffi f bi , ðkÞ u^ b þ

n @f ðb , ðkÞ u ^ bÞ X i ðjÞ

j¼1

@u^ b

ðjÞ

ðjÞ

ðu^ b ðkÞ u^ b Þ

ð40Þ

ðjÞ Since ð@r i =@u^ b Þ ¼ Jbij , there holds

r i ðub Þ ¼ DuðiÞ b

n X

Jbis DDs

s¼1 ðiÞ ðkÞ DuðiÞ b ¼ ub f ðbi , ub Þ ðkÞ ^ ðsÞ DDs ¼ u^ ðsÞ b  ub

s¼1

i¼1

The Least Squares Method (LSM) is provided to determine an analytical polynomial interrelation between the state function (displacements, stresses and, independently, eigenvibrations frequencies) and the given input random parameter b. These polynomial approximations are inserted into the Taylor expansions as well as the formulas (32) and (A.1)–(A.3) to determine probabilistic moments of the state functions provided that we have a single input random parameter. In case of multiple random variables we repeat this procedure for each input random variable separately and combine them through the cross-correlations. We consider a set of m data points ðbi ,uðiÞ b Þ for b ¼1,y,N, some nonlinear continuous response function ub ¼ f ðbÞ and a curve (approximating polynomial) ub ¼ f ðb, u^ b Þ depending on the paraðjÞ meters u^ b , j ¼1,y,n, where m Z n. The set of trial values is directly taken from the FEM experiments (Eqs. (23) or (26)) for 0 0 the set of input parameters b1 ,:::,bn that origin from equidistant partition of an interval ½bDb,b þ Db. Usually a ratio Db=b is in a range of the few percents, which follows a specific type of this variable in a given boundary problem and its physical limitations. A recursive formula for bi holds true when we provide a partition of ½bDb,b þ Db into n  1 subintervals bi ¼ bDb þ i

Substituting these expressions into the gradient equation (38), one obtains ! m n X X b 2 J bij DuðiÞ  J D D ¼0 ð42Þ s is b

ð41Þ

which effectively become a system of the linear equations: m X n X

n X

J bij J bis DDs ¼

i¼1s¼1

J bij DuðiÞ b ,

i¼1

j ¼ 1,:::,n

ð43Þ

Matrix notation is introduced to get finally ðJT JÞDD ¼ JT Du

ð44Þ

Further determination of the probabilistic moments and characteristics is based on determination of the partial derivatives of the response function with respect to the input random parameter [6–8]: @k f k

@b

¼

k Y

nk

ðniÞan b

i¼1

þ

k Y

ðniÞan1 b

nðk þ 1Þ

þ    þ ank

ð45Þ

i¼2

It should be underlined that further time savings and efficiency increase may be obtained by an application of the weighting procedure into the least squares technique. Weighting procedure may enlarge an influence of the results close to the expectation of the input random variable on the resulting probabilistic moments. Then, probabilistic convergence of the perturbation method is apparently faster and approximating polynomial order may be reduced.

5. Perturbation-based optimization procedure Let us consider a linear elastic spatial truss structure built up with n segments and two sets of discrete design variables Ali and Abi for i¼1,y,p. They simply denote the cross-section areas of the legs and braces belonging to the given segment indexed with i. These discrete variables are taken from the engineering tables, so that Ali  Alij and Abi  Abij , where j¼1,y,r indices their all available discrete values, respectively. Let us assume that an increasing value of an index j is equivalent to the larger cross-section. Let us introduce further a random variable of the problem – that is Young modulus of the structural material denoted by e given uniquely by its expectation E[e] and standard deviation s(e). The design objective is minimization of the structural crosssections Ali and Abi for i¼1,..,p with respect to the boundary conditions given by Eqs. (4) and (5) and the constraints imposed on the displacements, stresses and eigenfrequencies of the tower. The first two of them are simply defined as

bq ðAi Þb^ q Z0

ð46Þ

br ðAi Þb^ r Z 0

ð47Þ

where, according to Eurocode 0, we provide b^ q ¼ b^ r ¼ 3. The indices in Eqs. (46) and (47) are computed using the First Order Reliability Method (FORM) as the ratio of expectation to the standard deviation of the limit function, which is simply a difference between allowable and maximum values for displacements and stresses. Further, the reliability index for eigenfrequencies is calculated and we employ the following limit function for this purpose: g o ¼ of ow Z1:25ow ow ¼ 0:25ow

ð48Þ

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73

must be positive. A set of deterministically designed variables is a very good initial point in our procedure, because probabilistic case is a little bit more demanding. Of course, the maximum displacements are noticed at the top of the tower, whereas maximum stresses – in the lowest leg.

Fig. 1. Static scheme and photo of the telecommunication tower.

Table 1 Structural elements of the telecommunication tower. Segment no. Steel tower 1–2 3–4 5–6 7–8 9 10 Aluminum tower no. 1 1–2 3 4 5–6 7–8 9 10 Aluminum tower no. 2 1–2 3 4 5–6 7–8 9 10

Tower legs (mm)

Tower braces (mm)

Ø Ø Ø Ø Ø Ø

219.1  16.0 219.1  16.0 193.7  16.0 139.7  12.5 139.7  12.5 88.9  8.0

Ø Ø Ø Ø Ø Ø

159.0  12.5 133.0  10.0 101.6  11.0 88.9  10.0 76.1  8.0 76.1  8.0

Ø Ø Ø Ø Ø Ø Ø

323.9  17.5 323.9  17.5 193.7  17.5 193.7  17.5 139.7  12.5 139.7  12.5 108.0  8.0

Ø Ø Ø Ø Ø Ø Ø

219.1  12.5 168.3  17.5 168.3  17.5 139.7  11.0 114.3  12.5 101.6  8.0 101.6  8.0

Ø Ø Ø Ø Ø Ø Ø

457.0  17.5 323.9  17.5 193.7  17.5 193.7  17.5 139.7  12.5 139.7  12.5 108.0  8.0

Ø Ø Ø Ø Ø Ø Ø

219.1  12.5 168.3  17.5 168.3  17.5 139.7  11.0 114.3  12.5 101.6  8.0 101.6  8.0

Both omegas in this formula denote the forced and eigenvibration frequencies. So that, the third constraint here is

bx ðAi Þb^ x ¼

E½0:25ow  ^ b Z 0 sð0:25ow Þ x

ð49Þ

similar to the previous case b^ x ¼ 3. Reliability-based optimization process uses deterministically optimized tower structure and starts from the top segment, where the cross-sections of legs and braces Ali and Abi have minimum values for all i ¼1,yp; according to the engineering intuition there holds Ali þ 1 ZAli and Abiþ 1 Z Abi . Deterministic optimization deals with a structure subjected to the same boundary conditions, no input random variable, where the given constraints are replaced with the differences in-between allowable and maximum values of displacements, stresses and eigenvibration frequencies; they all

We start our optimization procedure from the top section, where the tower is fragile at most and then, we continue section

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by section to its bottom according to the given flowchart. We employ the stress condition each time first, because it is almost always fulfilled, then we control the displacements, because they are more sensitive to the cross-sectional design. Finally, we check the condition imposed on the eigenfrequency based reliability index. We increase the legs cross sectional area and then, the rebars, if necessary. It needs to be mentioned that each change of the cross-sectional area results in sequential determination of the response function and the relevant probabilistic moments. That is why we start from the least sensitive condition and progress towards the most sensitive – to minimize the number of iterations with the LSM procedure. It is especially important to separate elastostatics from the eigenvibration FEM solutions, because the response function determination needs quite different algorithms and, probably, even various partitions. As it is proven by some SFEM tests, final verification of the eigenmodes does not demand frequently any further changes in cross-sections.

6. Numerical illustrations Computational analysis has been provided on the example of the telecommunication tower with the height equal to 60.0 m discretized with 280 two-noded linear elastic beams and bars with 150 nodal points (see Fig. 1 – structural drawing, photo and static scheme). All geometrical data for the specific cross-sections are given in Table 1 (round steel and aluminum pipes with the given diameters and thicknesses). The data collected in this table show that the optimization of aluminum tower is carried out by the proportional increasing of all the subgroups cross-sections in the steel tower, preserving the given reliability conditions. All towers consist of 10 segments, each having 6.0 m height, and are fully restrained at the bottom legs. The lower part is built up with nine segments exhibiting 5% geometrical convergence (7.90 m distance of the legs at the very bottom), while the top segment has parallel legs (with the distance equal to 2.50 m). The legs, according to their aerodynamic loadings and relatively large length, are designed as the full round cross-sections joined inbetween the segments with bolts. The stiffening bars provided in the cross scheme are welded to the legs through the joining rectangular plates. The FEM analysis includes the dead load of the tower, its equipment together with the supporting structural elements, cables and exploitation installations as well as ice covers, static pressure from the wind blow and exploitation moving loads. We consider three design cases – (1) the steel tower designed fulfilling the traditional limit states and reliability

conditions, (2) the aluminum version of this tower fulfilling the traditional limit states and the reliability condition on eigenfrequencies as well as (3) the aluminum tower fulfilling both traditional limit states and all the reliability conditions also. By the aluminum version of the steel tower we further understand the tower having all connections and nodes in the same positions with slightly modified cross-sections but the external shape and cross-section type remains the same (steel versus aluminum round pipes and fully round cross-sections for the braces and the legs). As it is seen from a comparison of the detailed legs and braces’ cross-sections, traditional designing procedures are decisively less restrictive than these based on the reliability indices. Aluminum tower no. 2 having almost the same cross-sections at the top needs about 30% larger elements at the bottom (mainly according to the horizontal deflection index) and it drastically influences the overall designing procedure. It is especially important because the exploitation and maintenance of such structures usually assumes some extra loads resulting from the new antennas and additional equipment, so that certain capacity reserves are always important in this case. Basic input random variable of the problem is Young modulus with the expected values for the steel members taken as E½es  ¼ 210 GPa, where E½ea  ¼ 70 GPa is adopted for aluminum. This uncertainty follows the fact that this modulus is always determined experimentally with some statistical evidence, which shows that its standard deviation for aluminum is larger than for structural steels. Response functions of the eigenfrequencies as well as horizontal displacements are obtained in nine numerical FEM experiments consisting of E[es]¼241C363 GPa and E[ea]¼50C90 GPa with equidistant partitions of these intervals. The response function for the steel tower eigenfrequencies was found using traditional polynomial interpolation, while the additional functions for the displacements – thanks to the sixth order approximants in the LSM. Analysis of both aluminum towers was entirely provided with the use of parabolic response functions; generally it is advised to apply the lowest possible order of the LSM approximation verified by the consistency with the given FEM data set (visually or by some fitness test). The results of computational modeling for horizontal displacements probabilistic moments and associated reliability indices are given in Figs. 2–6, whereas analogous output data for the tower eigenfrequencies are given in Figs. 7–11. In case of displacements we notice fluctuations of expected values, coefficients of variation, kurtosis, skewnesses and reliability indices all as the functions of vertical coordinate (related to the corresponding nodes in the mesh) and, at the same time, as the function of the input coefficient of

Fig. 2. Expected values of the displacements for the steel and aluminum towers.

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75

Fig. 3. Coefficients of variation for the displacements for the steel and aluminum towers.

Fig. 4. Kurtosis of the displacements for the steel and aluminum towers.

Fig. 5. Skewness of the displacements for the steel and aluminum towers.

variation – by taking aðeÞ ¼ ½0:025,0:05,0:075:0:10,0:125,0:15. The eigenvibrations analysis is restricted to the first 10 eigenvalues only given as the continuous functions of the same parameter varying in the interval aðeÞ A ½0:0,0:10. Each time we compare the steel tower (left diagram) with traditional design of aluminum tower (graph in the middle) with the aluminum structure respecting all limit states

and reliability indices (right graph). Usually limit state for the stresses is less demanding than for the deflections of such a structure and so that we restricted a presentation to the second category by only. Comparison of the expected values presented as a function of the vertical coordinate and input coefficient of variation (see Fig. 2) shows that the larger distance from the terrain level, the

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Fig. 6. Reliability index in displacements analysis for the steel and aluminum towers.

Fig. 7. Expected values of the eigenvalues for the steel and aluminum towers.

Fig. 8. Coefficients of variation for the eigenvalues for the steel and aluminum towers.

larger expectation of the horizontal deflection of the specific node; as one may expect also this increase is nonlinear. The coefficient of variation of Young modulus has rather no influence on expectations computed for all the towers. The most apparent is a difference of the horizontal displacement for the aluminum

tower top – from about 80 mm (deterministic design only) up to 40 mm (full limit states plus reliability indices), while steel structure deformations are very close to the stiffer aluminum structure rather. The coefficients of variation given in Fig. 3 are almost independent of the tower type and have a linear constant

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77

Fig. 9. Kurtosis of the eigenvalues for the steel and aluminum towers.

Fig. 10. Skewness of the eigenvalues for the steel and aluminum towers.

Fig. 11. Reliability index in eigenvibrations analysis for the steel and aluminum towers.

distribution along the height with some small variations at the bottom of all structures. Since the computational model is linear elastic, then the larger input coefficient of variation, the larger output value of this coefficient. The ratio of both coefficients is almost equal to 1 (except the lowest point, not influencing the entire reliability), so

that this system neither amplifies nor damps an initial uncertainty. Kurtosis and skewness of the displacements examined shown in Figs. 4 and 5 show clearly that neither for the steel structure nor for aluminum towers the output probability density function is not Gaussian – the larger input random fluctuation, the smaller similarity to the bell-shaped distribution.

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´ ski, M. Solecka / Finite Elements in Analysis and Design 63 (2013) 69–79 M. Kamin

Definitely larger variations for both coefficients are noticed for the steel structure, however close to a(e) ¼0.05 all coefficients are very close to 0. Kurtosis of the steel structure displacement is almost two times larger than for the aluminum towers (the stiffer one has slightly larger values than the first aluminum structure), but the differences in-between the skewnesses for steel and aluminum towers are of about 20%. As before, we notice that the larger input coefficient of variation, the higher values of skewness coefficient and kurtosis. Their distributions along the tower height are more less constant in the interval h¼[20 m, 60 m] showing some fluctuations within two lowest segments. Of course, higher order statistics do not affect the reliability index itself, however give a clear information that first two moments give incomplete information about the random structural response of this structure for larger input random variations. Final graph in this series – Fig. 6 – shows the reliability indices variations, which quite naturally decrease for increasing value of the vertical coordinate of the node considered. It is noticed that the larger input coefficient of variation, the larger output reliability index (its absolute value for negative exceptions). It is clear that weaker aluminum tower has insufficient cross-sections of the legs and braces, because the reliability index at the same top is negative (admissible values starts from 3 up to 5, depending on the exploitation time). As it could be expected, the minimum value of this index is reached always at the top of a structure and then, it grows up almost parabolically to the maximum for the lowest node with free horizontal displacements. Of course, according to the buckling and stress limit conditions as well as their reliability demands, we cannot keep constant value of this index at the level given by Eurocode 0 by systematic decreasing of all cross-sections. It needs to be underlined that verification of this index for the top of tower does not allow for further decreasing of structural members used in the middle braces of a structure. Inappropriate usage of too small cross-section will immediately affect these pictures with the index smaller than admissible. As it is known from engineering practice, we need to control the eigenfrequencies conditions for high and lightweight structures and that is why we compare their expectations in Fig. 7. Quite clearly they are almost insensitive to the input coefficient of variation and also very similar to each other (steel structure exhibits usually some small values for the given number of the eigenmode). Coefficients of variation for all eigenfrequencies behave in exactly the same way (and all are the same) – linearly depending on the input coefficients of variation. It is apparent that they are exactly two times smaller than the input coefficient, so that the systems exhibit significant probabilistic damping. Kurtosis (Fig. 9) is also positive everywhere and starts from 0 (the same as coefficient of variation) but then grows nonlinearly in the convex mode up to the maximum for a(e)¼0.10. This maximum in case of the steel tower is the largest one, then – slightly smaller for the stiffer aluminum tower and definitely the smallest for the weaker aluminum structure. There is no clear interrelation in-between this kurtosis and the eigenmode number – first eigenfrequency for a weaker structure is the smallest for all a(e) and the largest everywhere for stronger aluminum tower, while somewhere in-between the others in case of the steel tower. The skewnesses given in Fig. 10 are also irregular in this context, nevertheless they all start from 0 and all have negative values with linear proportionality to the input coefficient of variation. Contrary to the random displacements the eigenfrequencies appear to have a distribution quite close to the Gaussian variables does not requiring extended statistical information and limited to the first two probabilistic moments only. Finally, we examine the reliability index defined on the

eigenvibrations as the function of a(e). It has exactly the same values for all eigenfrequencies and with relatively small error the same values computed for various towers. Of course, the larger input coefficient of variation, the smaller value of this index; this interrelation seems to be close to the inverse proportionality. Intersecting these curves with b ¼5 it is obtained that the largest admissible interval of the input coefficient of variation is even wider than from 0.0 up to 0.05 as in the optimization problem statement. Short comparison against Fig. 6 leads to the conclusion that the basis for reliability-based optimization of these towers is the reliability index defined on the admissible horizontal displacements.

7. Concluding remarks (1) The main conclusion driven by the performed computational analysis is the fact that decisive reliability criterion for the multi-parametric optimization of the towers structure is that connected with horizontal displacements. The resulting random fields of displacements are non-Gaussian quantities, so that neither first two probabilistic moments information nor reliability index formula given in Eurocode 0 are not sufficient for the very exact optimization of such structures. Most probably, looking for parametric multipliers of the stiffness matrices structures, the other design parameters randomized using truncated Gaussian distribution will also return the non-Gaussian structural response. A simple virtual transformation of the steel tower into its aluminum counterpart through an extension of all the crosssections using the ratio of both materials’ Young moduli leads to significant underestimation of the necessary and optimal crosssections here. The RFM-SFEM hybrid method return satisfactory response functions of even parabolic character, which significantly speeds up a probabilistic convergence of the stochastic perturbation technique itself. The reliability analysis is also straightforward procedure with the Stochastic Finite Element Method technique proposed, provided that the direct difference in-between induced frequency of vibrations and the eigenfrequency is declared in percents with respect to this last quantity. Otherwise, full stochastic forced vibrations analysis is necessary, which needs further extensive developments of the SFEM procedures. (2) There is no doubt that the computer algebra system plays a crucial role in this computational strategy – one may try to use this hybrid strategy with the response function method in addition to the other probability density functions, especially for the lognormal variables, where all central moments of any order have additional analytical forms. Otherwise, some further numerical techniques must be employed to recover these moments for the needs of specific input random variables configuration. The structural open research issue may be, for instance, the SFEM analysis of stochastic earthquake vibrations applied at the foundations of such towers, influencing significantly their stochastic reliability. Separate problems that need to be taken into account in further optimization strongly affecting reliability are somewhat similar to these typical for the turbine blades [19]. They obey thermal processing of both steel and aluminum, some modifications of the aluminum alloys to better fit into this design needs as well as the joining methods for both materials and their joints compliance under various external loads for both materials.

Acknowledgment The first author would like to acknowledge the Research Grant NN 519 386 636 from Polish National Science Center in the period 2009-2011.

´ ski, M. Solecka / Finite Elements in Analysis and Design 63 (2013) 69–79 M. Kamin

Appendix The additional perturbation-based variances as well as third and fourth central probabilistic moments are derived as ! @f @f 1 @2 f @2 f 2 @f @3 f Varðf ðbÞÞ ¼ e2 m2 ðbÞ þ e4 m4 ðbÞ þ @b @b 4 @b2 @b2 3! @b @b3 !  2 3 3 1 @ f @ f 1 @4 f @2 f 2 @5 f @f þ e6 m6 ðbÞ þ þ 3! @b3 @b3 4! @b4 @b2 5! @b5 @b

þ e8 m8 ðbÞ

1 @2 f @6 f 1 @f @7 f 1 @4 f @4 f 1 @3 f @5 f þ þ þ 720 @b2 @b6 2520 @b @b7 576 @b4 @b4 360 @b3 @b5 5

þ e10 m10 ðbÞ

4

1 @2 f @8 f 1 @f @6 f 1 @3 f @7 f 1 @f @6 f þ þ þ 40320 @b2 @b8 14400 @b5 @b6 15120 @b3 @b7 8640 @b4 @b6

!

The third order probabilistic moment has essentially longer expansion which is given as Z

þ1

1 Z þ1

¼ 1



  3 f ðbÞE f ðbÞ pb ðxÞdx

 2 2 !    3 @f 3 @f @ f 0 f þ e Dbþ :::E f ðbÞ pb ðxÞdx ffi e4 m4 ðbÞ @b 2 @b @b2

0 !3 1  2 4 1 @f @2 f @3 f 1 @f @ f 1 @2 f A þ þ þ e6 m6 ðbÞ@ 2 @b @b2 @b3 8 @b @b4 8 @b2 0 !2  2 6 1 @f @ f 1 @3 f @2 f þ þ e8 m8 ðbÞ@ 2 240 @b @b6 24 @b3 @b 1 @2 f 32 @b2

!2

þ e10 m10 ðbÞ þ e10 m10 ðbÞ

@4 f

!

1 @f @2 f @5 f 1 @f @3 f @4 f þ 5 4 2 40 @b 24 @b @b3 @b4 @b @b @b 0 1 !2 !2 !2 1 @4 f @2 f 1 @2 f @6 f 1 @3 f @4 f A 10 @ þ e m10 ðbÞ þ þ 2 6 4 384 @b4 960 @b2 288 @b3 @b @b @b þ

þ

1 @f @2 f @7 f 1 @f @3 f @6 f þ 1680 @b @b2 @b7 720 @b @b3 @b6

!

 2 8 ! 1 @f @4 f @5 f 1 @2 f @3 f @5 f 1 @f @ f þ þ 480 @b @b4 @b5 240 @b2 @b3 @b5 13440 @b @b8

ðA:2Þ Similarly, the fourth probabilistic moment is approximated as 0 1 4  3 3  2 2 !2 @f 2 @f @ f 3 @f @ f A þ e6 m6 ðbÞ@ þ 2 @b 3 @b @b3 2 @b @b 0 !4  2 2 4  3 5 1 @2 f 1 @f @ f @ f 1 @f @ f þ þ þ e8 m8 ðbÞ@ 2 2 4 16 @b 4 @b @b @b 30 @b @b5

m4 ðf ðbÞÞ ¼ e4 m4 ðbÞ



! !2  2 3 !2 1 @f @ f 1 @f @2 f @3 f þ 3 3 6 @b 2 @b @b2 @b @b  3 7  2 2 6 1 @f @ f 1 @f @ f @ f þ e10 m10 ðbÞ þ 1260 @b @b7 120 @b @b2 @b6 þ

!3 !  2 4 !2 1 @f @ f 1 @f @3 f þ þ 4 96 @b 54 @b @b3 @b 0 1 !3 !2 !2 !2 2 4 2 3 2 5 1 @ f @ f 1 @ f @ f 1 @f @ f @ f 10 A þ e m10 ðbÞ@ þ þ 5 4 3 48 @b2 24 @b2 40 @b @b2 @b @b @b þ e10 m10 ðbÞ

!  2 3 5 1 @f @ f @ f 1 @2 f @f @3 f @4 f þ 3 5 2 3 4 60 @b @b @b 12 @b @b @b @b

ðA:3Þ

!

ðA:1Þ

m3 ðf ðbÞÞ ¼

79

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