Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 3, pp. 783-794, Warsaw 2016 DOI: 10.15632/jtam-pl.54.3.783 TIME-INDEPENDENT STOCHASTIC DESIGN SENSITIVITY ANALYSIS OF STRUCTURAL SYSTEMS WITH SECOND-ORDER ACCURACY Hanna Weber West Pomeranian University of Technology in Szczecin, Faculty of Civil Engineering and Architecture, Poland e-mail: weber@zut.edu.pl In the paper, the static sensitivity of complex structureswith respect to randomdesignpara- meters is presented.Using the adjoint systemmethod, basedon themean-point second-order perturbationmethod, the first twoprobabilisticmoments of time-independent sensitivity are formulatedwithmeans andcross-covariancesof randomdesignparametersas the inputdata. It enables one to obtain the second-order accuracy of the solution. The presented formu- lations are illustrated by a number of numerical examples. The influence of finite element mesh density for the obtained results is discussed using the analysis of the spatial bar dome. Keyword: design sensitivity, statics, random parameter, finite element 1. Introduction Since the beginning ofmankind, humans have had aspirations to overcome their weaknesses and limitations. It is the element of our nature, which became the basis of development in all fields of life. In the civil engineering there are amazing objects that crossed existing barriers of height, span and slenderness. These types of structures can not be computed by analytical methods because of complexity of the system.Hence, the Finite ElementMethod (FEM) has become the basis of contemporary structure analysis (Bathe, 1982; Zienkiewicz and Taylor, 1991; Kincayd andCheney, 2002). It is an approximatemethodwhichallows one to obtain highlyprecise results with a properly dense FEMmesh. A significant element of correct design is to model an object to reflect reality as closely as possiblewhile taking into accountmany factors including connections,materials, external loads, etc. (see Leet et al., 2010). Engineers’ aim is to create programs for optimal design of new or strengthen the existing structures. It involves findings the best solutions bearing in mindmany different aspects such as the maximum load, allowable displacement, cost, time and possibility of execution, etc. (Niczyj, 2003; Choi and Kim, 2010). A very important case is to find an answer to the question of how the changing of design parameters affects the structural response. It is a subject of study of sensitivity analysis that can be found in (Haug et al., 1986; Mroz and Haftka, 1986; Hien andKleiber, 1989; Drewko and Hien, 2005; Ding et al., 2012; Mroz and Bojczuk, 2012; Choi andKim, 2013). This type of computations by using deterministic variables has been extensively discussed in the literature. However, it has been proved that in systems with many degrees of freedom (MDOF), even small uncertainties in structural parameters or external loads may have a significant impact on the work and load capacity of the system. The stochastic analysis (Adomian, 1983; Hisada andNakagiri, 1981; Spanos andGhanem, 1989; Ghanem and Spanos, 1991; Kleiber andHien, 1992; Li andChen, 2009) includes randomness in the described factors, and for complex systems it needs to be considered by numerical methods because analytical solutions are in many cases impossible. There are three main trends in the analysis with random variables: perturbation approach (Greene et al., 2011; Liu et al., 2013), Monte Carlo simulation (Fishman, 1995; Rubinstein 784 H.Weber and Kroese, 2008), and Neumann’s expansion (Liu et al., 1986). The goal of this paper is a numerical nonstatistical analysis of statics and time-independent sensitivity forMDOF systems by random parameters. Starting from the stochastic version of the equilibrium equation, using the mean-point second-order perturbation method, the equations for the first two probabilistic moments of the sensitivity gradients with respect to the design parameters are derived, where themeans and cross-covariances of randomvariables are treated as the input data. Thereby, not only the deterministic values of static response sensitivity are obtained but also the accuracy of the received results as expected values and cross-covariances. These formulations are illustrated by numerical examples. 2. Formulation of the stochastic sensitivity for statics in the finite element context Theobjective of structural sensitivity analysis is to consider the impactofdesignvariable changes on the system response. In this paper, the design variable vector is denoted by b = {ba}, a = 1,2, . . . ,A. It can be cross-sectional areas of structural elements, thickness of a shell or plate, mass density, Young’s modulus etc., while displacements, stresses or natural frequencies in main nodes can be treated as structural response measures. The result of deterministic computation is only one variable at a given point, while the stochastic analysis determinesalso theaccuracyof theobtainedvalue.Duringthe secondprocess, the system is described by random variables. The time-independent random variable vector is assumed in the form h = {hr}, r = 1,2, . . . , r̂ and defined by means hr = E[hr] and cross- -covariances Cov(hr,hs), r,s = 1,2, . . . , r̂. From the definition, given by Kleiber and Hien (1992), the first probabilistic moment of the random variable is expressed by E[hr] = +∞∫ −∞ hrp(hr) dhr (2.1) and the second Cov(hr,hs)=E[(hr −hr)(hs−hs)] =αrαshrhsµ(hr,hs) (2.2) where µ(hr,hs)= +∞∫ −∞ +∞∫ −∞ hrhsp(hr,hs)dhrdhs (2.3) and αr = √ Var(hr) hr = σ(hr) hr (2.4) In Equations (2.1)-(2.4) p(hr), p(hr,hs), µ(hr,hs), Var(hr), σ(hr) andα denote the probability density function, joint probability density function, adopted function of correlation, variance, standard deviation and the coefficient of variation, respectively. In this paper from this moment forth, the summation notation is included. In the following equations, two repeated indices imply the sum, which results in greater transparency of presen- tation. Let us consider a linear elastic complex structure withN degrees of freedom. The static response of the system can be formulated then by the functional (Kleiber and Hien, 1992) φ=G[qα(ba,hr),ba] α=1,2, . . . ,N (2.5) Time-independent stochastic design sensitivity analysis... 785 where qα is determined as the time-independent vector of generalized coordinates. In the sto- chastic analysis of systemswithmulti degrees of freedom (MDOF), the stiffnessmatrixKαβ and the nodal load vector Qα are explicit functions of both the design parameter and the random variable vector. In the static case, the considered system is time-independent, therefore damping and mass effects are omitted during computations. Consequently, the equilibrium equation is written in the stochastic form as Kαβ(ba,hr)qβ(ba,hr)=Qα(ba,hr) α,β=1,2, . . . ,N (2.6) A solution to Eq. (2.6) with respect to qβ (Eq. (2.7)) proves that the nodal displacement vector is an implicit function of ba and hr qβ(ba,hr)=K −1 αβ (ba,hr)Qα(ba,hr) (2.7) To evaluate the probabilistic distribution of the static structural response with respect to the design parameters, we tend to obtain an absolute partial derivative of the functional φ with respect to the design variable, i.e. dφ/dba. Assuming that the stiffnessmatrix and the nodal load vector are twice continuously differentiable with respect to ba, the chain rule of differentiation gives dφ dba = ∂G ∂ba + ∂G ∂qα dqα dba α=1,2, . . . ,N a=1,2, . . . ,A (2.8) According to Eq. (2.5), G is an explicit function of ba and qα, therefore, the partial derivati- ves ∂G/∂ba and ∂G/∂qα are known. The goal of the procedure is to find the absolute partial derivative of the general coordinate vector with respect to the design variable, i.e. dqα/dba. To formulate the static sensitivity problem, the so-called adjoint system method is used (Kleiber and Hien, 1992; Choi and Kim, 2010). The adjoint equation system is adopted in the form Kαβλβ = ∂G ∂qα α,β=1,2, . . . ,N (2.9) Seeing that the stiffness matrix Kαβ and the vector ∂G/∂qα are functions of the design and random variable, the solution to Eq. (2.9) with respect to λβ written as λβ(ba,hr)=K −1 αβ (ba,hr) ∂G ∂qα (ba,hr) (2.10) demonstrates that the adjoint variable vector is an implicit function of ba and hr and it is expressed as λ = {λα(ba,hr)}, α = 1,2, . . . ,N. Differentiating Eq. (2.6) with respect to the design variable and then solving it to obtain the vector dqα/dba and putting it into Eq. (2.8) leads to dφ dba = ∂G ∂ba +K−1 αβ ∂G ∂qα (∂Qα ∂ba − ∂Kαβ ∂ba qβ ) (2.11) SubstitutingEq. (2.10) intoEq. (2.11)we receive the equation for design sensitivity of the system (Kleiber and Hien, 1992) dφ dba = ∂G ∂ba +λα (∂Qα ∂ba − ∂Kαβ ∂ba qβ ) (2.12) It should be noted that all terms in Eq. (2.12) are simultaneously functions of the design para- meter and the randomvariable vectors.Now, all elements fromEqs. (2.6) and (2.9) are expanded in power series around the means hr, in accordance with the following equation (·) = (·)(h)+ [∂(·) ∂hr δhr + 1 2 ∂2(·) ∂hr∂hs δhrδhs ] h=h r,s=1,2, . . . , r̂ (2.13) 786 H.Weber where δhr denotes the first variations hr about the means hr, and for any small parameter ǫ it can be written as (Hien, 2003) δhr = ǫ(hr −hr) (2.14) while δhrδhs is called the second mixed variation of hr and hs about their means hr and hs δhrδhs = ǫ 2(hr −hr)(hs−hs) (2.15) All the functions of the random variable are expanded in Taylor series up to the second order and substituted to Eqs. (2.6) and (2.9). After rearranging the equations and comparing terms with the same order of ǫ, we receive the primary and adjoint systems of equations (compare, Kleiber and Hien, 1992): — one pair of systems ofN equations of the zeroth-order Kαβqβ =Qα Kαβλβ = ∂G ∂qα (2.16) — r̂ pairs of systems ofN equations of the first-order Kαβ dqβ dhr = ∂Qα ∂hr − ∂Kαβ ∂hr qβ Kαβ dλβ dhr = ∂2G ∂hr∂qα − ∂Kαβ ∂hr λβ (2.17) — one pair of systems ofN equations of the second-order Kαβ d2qβ dhrdhs ∣∣∣∣ h=h Cov(hr,hs)= [ ∂2Qα ∂hr∂hs − ∂2Kαβ ∂hr∂hs qβ −2 ∂Kαβ ∂hr dqβ dhs ] h=h Cov(hr,hs) Kαβ d2λβ dhrdhs ∣∣∣∣ h=h Cov(hr,hs)= [ ∂3G ∂hr∂hs∂qα − ∂2Kαβ ∂hr∂hs λβ −2 ∂Kαβ ∂hr dλβ dhs ] h=h Cov(hr,hs) (2.18) From the definition (Kleiber and Hien, 1992), the first probabilistic moment for the sensitivity gradient is given by E [ dφ dba ] = +∞∫ −∞ +∞∫ −∞ . . . +∞∫ −∞︸ ︷︷ ︸ A−fold dφ dba pA(b1,b2, . . . ,bA) db1db2 · · ·dbA (2.19) while the second probabilistic moment is expressed analogically as Cov (dφ dba , dφ dbb ) =E [(dφ dba −E [dφ dba ])(dφ dbb −E [dφ dbb ])] (2.20) Themean value of Eq. (2.12) may be written as E [ dφ dba ] =E [∂G ∂ba ] +E [ λα ∂Qα ∂ba ] −E [ λα ∂Kαβ ∂ba qβ ] (2.21) Substituting Eq. (2.12) into Eq. (2.21) gives an expression for the second probabilistic moment of the design sensitivity gradient in the form Cov (dφ dba , dφ dbb ) =E [(∂G ∂ba +λα (∂Qα ∂ba − ∂Kαβ ∂ba qβ ) −E [dφ dba ]) · (∂G ∂bb +λγ (∂Qγ ∂bb − ∂Kγδ ∂bb qδ ) −E [dφ dbb ])] (2.22) Time-independent stochastic design sensitivity analysis... 787 All functions of the random variables fromEqs. (2.21) and (2.22) are expanded in Taylor series in accordance with Eq. (2.13) and are premultiplied. Excludingmembers in orders higher than the second andaveraging the other terms, leads to an expression of themeanvalues of the design sensitivity gradient, cf. Hien and Kleiber (1991), Kleiber and Hien (1992) E [ dφ dba ] = [∂G ∂ba +λαAαa + 1 2 ( ∂3G ∂hr∂hs∂ba + d2λα dhrdhs Aαa+2 dλα dhr Bαsa+λαCαrsa ) Cov(hr,hs) ] h=h (2.23) where, for clarity purposes, the following equations are used Aαa = ∂Qα ∂ba − ∂Kαβ ∂ba qβ Bαra = ∂2Qα ∂hr∂ba − ∂2Kαβ ∂hr∂ba qβ − ∂Kαβ ∂ba dqβ dhr Cαrsa = ∂3Qα ∂hr∂hs∂ba − ∂3Kαβ ∂hr∂hs∂ba qβ −2 ∂2Kαβ ∂hr∂ba dqβ dhs − ∂Kαβ ∂ba d2qβ dhrdhs (2.24) The cross-covariances at dφ/dba and dφ/dbb are Cov (dφ dba , dφ dbb ) = [( ∂2G ∂hr∂ba ∂2G ∂hs∂bb +(AβbBαas+AαaBβbs)λα dλβ dhr − (∂G ∂ba Cβrsb+ ∂2G ∂hr∂ba Bβrb+ ∂3G ∂hr∂hs∂ba Aβb ) λβ − (∂G ∂bb Cαrsa+ ∂2G ∂hr∂bb Bαra+ ∂3G ∂hr∂hs∂bb Aαa ) λα − ( 2 ∂G ∂ba Bαsb+ ∂2G ∂hs∂ba Aαb+2 ∂G ∂bb Bαra+ ∂2G ∂hs∂bb Aαa ) λα dhr − (∂G ∂bb Aαa+ ∂G ∂ba Aαb ) d2λα dhrdhs +λαλβBαarBβbs+AαaAβb dλα dhr dλβ dhs ) Cov(hr,hs) − 1 4 ( ∂3G ∂ht∂hu∂bb +2 dλβ dht Bβub+Aβb d2λβ dhtdhu +λβCβtub ) · ( ∂3G ∂hr∂hs∂ba + d2λα dhrdhs Aαa+2 dλα dhr Bαsa+λαCαrsa ) Cov(hr,hs)Cov(ht,hu) ] h=h (2.25) with r,s,t,u = 1,2, . . . , r̂. Interestingly, Eq. (2.25) is obtained by including terms up to the second order, not only to the first like inHien andKleiber (1991), Kleiber andHien (1992). The procedure of obtaining and averaging particular members of the first two probabilisticmoments of the static design sensitivity canbe found indetail inWeber (2014), while this paper is confined to present only the final version of these equations. The equation for covariances at dφ/dba and dφ/dbb inWeber (2014) is obtained by using the expressions forE[dφ/dba] andE[dφ/dbb] as thefinal products.ToderiveEq. (2.25), themembers E[dφ/dba] and E[dφ/dbb] are determined by Eq. (2.23) and substituted into Eq. (2.22). After ordering particular members, a more concise expression for the second probabilistic moment of the static design sensitivity than that presented byWeber (2014) is received. 3. Numerical examples – results and discussion Deterministic and stochastic computations are executed by the finite element code POLSAP (see, Hien and Kleiber, 1990), properly adapted for this type of analysis. At the beginning of 788 H.Weber numerical illustrations, let us consider a three-bar truss system. The analytical equations of nodal displacements and their sensitivities with respect to cross-sectional areas of the elements, were derived for the model by Choi and Kim (2010). The goal of this example is to obtain not only the deterministic displacements of nodes, but also their expected values and standard deviations by POLSAP and compare them with the results given by equations formulated by Choi and Kim (2010). Fig. 1. Three-bar truss system (a) presented in Choi andKim (2010), (b) adopted in numerical computation To simplify the numericalmodel, theChoi andKim system (2010) (see Fig. 1a) is rotated as shown in Fig. 1b, and nodal loads are reduced to vertical and horizontal forces Fx = 18.33kN and Fy = 68.30kN that correspond to forces F1 = F2 = 50kN. All results shown in Tables 1 and 2 are given for the adopted global coordinate system xy presented in Fig. 1b. Therefore, to compare the displacements and sensitivities, the values received by equations from literature have to be transformed by using trigonometric functions with an angle of 30 degrees. The input data of the material and elements are adopted for simplification as: Young’s modulusE =200GPa and cross-sectional areas of specific barsA1 =A2 =A3 =5cm 2. One bar is assumed as one truss element in the numerical model. The cross-sectional areas of particular elements are adopted as randomdesign variableswith themeansE[Ar] = 5 cm 2, and correlation function defined by the equation, cf. Kleiber and Hien (1992) µ(Ar,As)= exp (−|xr−xs| λ ) exp (−|yr−ys| λ ) (3.1) where xr, xs and yr, ys designates the x- and y-coordinates of the mid-points of the next two elements in the model, respectively. The symbol λ denotes the decay factor depending on the unit system used in numerical analysis. Its value is selected in order to receive no diagonal and non-zero covariancematrix of the randomvariables. For this example,λ=300and the coefficient of variationα=0.1 are adopted as the input data. In the secondmoment perturbationmethod, the random variables must fulfill the condition about small fluctuation and continuity at hr, therefore, the selection of α-coefficient is a very important part of numerical computations, cf. Weber (2014). Time-independent stochastic design sensitivity analysis... 789 The correlation function shows a dependence between the random variables. The bigger the distance from one element to another, the smaller impact between them is observed. For the members situated at a considerable distance from each other, the described dependence tends to zero. Using Eq. (3.1), we obtain results equal to 1 on the main diagonal, and the values in the range from 0 to 1 for other elements. Additionally, the bigger the difference between the coordinates, the smaller results of the coefficient in the matrix, which satisfies the assumptions for the correlation function. The deterministic and expected values of displacements obtained by POLSAPare similar to those given byChoi andKim (2010) (see Table 1). Themaximum difference between analytical anddeterministic results is about0.3%,while thatbetweendeterministic and stochastic is 0.85%. Table 1.Non-zero nodal static displacements for the three-bar truss system, [cm] Node number Analytical values POLSAP Coordinate Deterministic Expected displacements values 1 x 0.073205 0.073416 0.073905 3 x −0.084108 −0.084146 −0.084863 3 y 0.145680 0.145868 0.147066 The functional of the structural response defined by Eq. (2.5) is accepted in numerical analysis in the form φ= |qα| qall −1< 0 (3.2) where qα, qall are the actual and allowable displacements in a selected node. To compare the sensitivity results obtained from Choi and Kim (2010) and POLSAP, qall = 1.0cm is assumed for every node, otherwise the values given by the equations from literature should be divided by the accepted qall. In Table 2, displacement design sensitivities for selected nodes are presented. Themaximum difference between analytical anddeterministic values is 0.43%.Thedeterministic and stochastic results of static sensitivity vary in the range of 2.5-3.0%. The standard deviation is 19.4% of expected values. Table 2.Displacement design sensitivity - bar cross-sectional areas as randomdesign variables, [1/cm2] Node number Displa- Obtained values POLSAP cement based on Choi Deterministic Expected Standard direction and Kim (2010) values values deviations cross-sectional area of el. no.2 as design variable 1 x −1.46410E-2 −1.46831E-2 −1.50737E-2 2.92693E-3 3 y −4.64102E-3 −4.66079E-3 −4.77790E-3 9.27750E-4 cross-sectional area of el. no.3 as design variable 3 x −1.41421E-2 −1.41423E-2 −1.45617E-2 2.82752E-3 3 y −2.44949E-2 −2.45133E-2 −2.52209E-2 4.89726E-3 The second numerical example is a spatial dome presented in Fig. 2 (compare Weber and Hien, 2010;Weber, 2014).Thegeometrical dimensions included in theanalysis are: basediameter 1000cmandheight 500cm.Eachbar in the structure ismodelled as a onebeamelementwith the following characteristics: Young’s modulusE =200GPa, mass density 7.85KNs2/m4, Poisson’s ratio ν =0.3 and cross-section areaA=20cm2. Nodes 1, 3, 5, 7 and 9 are supported by pins. 790 H.Weber For the static and sensitivity analysis, only one vertical force is taken into account Fz =1000kN. For clarity purposes, the dead weight of the structure is omitted during compu- tations. As it is well known, this type of bar structures loaded only in nodes is usuallymodelled as a truss system. However, the goal of this presentation is to show the influence of complexi- ty of the finite element mesh on static sensitivity results. Therefore, in this system, the rigid connections between bars are assumed. A comparison of the results obtained for the truss and beammodel of this structure can be found inWeber (2014). Fig. 2. Spatial dome (a) front, (b) bird eye – view In static and sensitivity computations, the cross-sectional areas of elements are adopted as random design variables with the mean values A = 20cm2. The correlation function and functional response of the system are assumed as in Eqs. (3.1) and (3.2). The decay factor λ=200 and the coefficient of variation α=0.1 are used in numerical computations. The largest vertical deflection occurs at the top of the dome, which is predictable because of the point of force application. The deterministic and expected values of the nodal displacements in different directions vary by about 1%, which is acceptable. The considered system is symmetric in terms of geometry, supported conditions and external load.Numerical computations give the same results in correspondingnodes,which confirms that the model input to the program is correct. However, for clarity purposes, Table 3 presents the values of displacements only for selected nodes. Table 3. Selected nodal static displacements for the spatial dome, [cm] Node Coordinate Deterministic Expected Difference number displacements values [%] 31 z −2.049777 −2.070121 0.99 29 x −0.260314 −0.262972 1.02 30 x −0.162236 −0.163888 1.02 20 y −0.144629 −0.145915 0.89 19 y −0.100229 −0.101091 0.86 Both deterministic and stochastic analysis of static sensitivity study the vulnerability of various node displacements with respect to cross-sectional areas of different elements. The value Time-independent stochastic design sensitivity analysis... 791 of qall is chosen separately for every node andmovement direction, according to Eq. (3.2). The most significant results of static displacement sensitivity for selected nodes, with the cross- sectional areas of elements as the random design variables, are given in Table 4. The differences between deterministic and expected values are about 2.5-3%, while the standard deviations are equal to 20% of expected values. In both examples, all computations have been conducted in centimeters and kilonewtons as basic units. Table 4.Displacement design sensitivity – bar cross-sectional areas as randomdesign variables, [1/cm2] Node number Displ. direct. qall [cm] Design Deterministic values Expected values Standard deviationsel. number 31 z 2.5 72 −1.94449E-3 −2.00421E-3 3.88446E-4 10 z 1.0 29 −2.18042E-3 −2.23223E-3 3.78821E-4 29 x 1.0 71 9.89193E-3 1.01891E-2 1.97174E-3 30 x 1.0 72 6.14086E-3 6.32574E-3 1.22490E-3 20 y 1.0 32 −3.17786E-3 −3.25960E-3 5.7216E-4 19 y 1.0 40 −3.59847E-3 −3.69325E-3 6.59437E-4 Considering various examples of the static displacement sensitivity allows one to spot a pattern. In most instances, a point displacement is the most sensitive with respect to, e.g., the cross-sectional areas of elements in the immediate vicinity of the considered point. However, in some cases there are certain derogations of the pattern when a displacement of a node turns out to be themost sensitive with respect to the cross-section of an element not lying directly by the examined node. Then sensitivity analysis allows one to find the key element of the considered displacement. This computation changes our view of the importance of individual structural members in the system and thereby should have a significant role in modern design. To show the influence of finite element mesh complexity on the results of stochastic static sensitivity, the second model of the spatial dome is created, where one bar is divided into four beam elements, cf. Fig. 3. It yields a system consisting of 320 members. Thereby, we can determine not only which element but also which part of it generates the displacement at a specific point themost sensitive with respect to the cross-section area. Thismay be useful while examining themodel in terms of the predicted path of failure or structure strengthening. Fig. 3. Selected element number The comparison of results received for the first and secondnumericalmodels are summarized in Table 5. It is clear that the sensitivity values obtained for the 320 element scheme is about 792 H.Weber four times less than for the 80 element one. It seems to be natural, because we examine the sensitivity with respect to the cross-sectional areas of the elements four times shorter than at the beginning. Careful analysis presented by Weber (2014) gives the following dependence: the higher the number of finite elements in themesh, the smaller impact of changes in cross-section areas of elements on the nodal displacements. Table 5.Displacement design sensitivity - bar cross-sectional areas as randomdesign variables, [1/cm2] Node number Displ. direct. qall [cm] Design Deterministic values Expected values Standard deviationsel. number 31 z 2.5 72a −5.18116E-4 −5.34095E-4 1.04516E-4 72b −4.71989E-4 −4.86661E-4 9.51452E-5 72c −4.62112E-4 −4.76374E-4 9.25389E-5 72d −4.88431E-4 −5.03158E-4 9.66361E-5 29 x 1.0 71a 2.41918E-3 2.49298E-3 4.87353E-4 71b 2.50572E-3 2.58152E-3 5.02619E-4 71c 2.51620E-3 2.59250E-3 5.05291E-4 71d 2.45059E-3 2.52538E-3 4.94002E-4 4. Concluding remarks In the stochastic static sensitivity analysis by using the second order perturbationmethod, the samemember, for example the cross-section area of an element, is both the design and random variable. It is significant in terms of the cost of numerical computations. It allows us to obtain complex results in the form of deterministic and expected values, and standard deviations. On the basis of the presented formulation, using the first two perturbation moments of random variables as the input data, we obtain the means and cross-covariances of the static design sensitivity with the second order accuracy. We receive results with the same precision as those in theMonte Carlo simulation, but by considering an r̂ order system of equations no r̂3. It is known that even small uncertainties in design parameters may have a large influence on results of displacements and internal forces, therefore this type of analysis seems to be very important in the design. Sensitivity analysis sometimes can provide a completely new insight into the work of a structure and the meaning of particular members in the considered system. It allows one to find themost sensitive point that determines the stability of the entire system. In this paper, only design sensitivitywith randomparameters for the static case is presented. It seems thatdynamic computations need tobean integral part of analysis of complex structures. Only by using both static and dynamic stochastic design sensitivity, we can determine the optimal solution for a system while taking into account all relevant aspects. Therefore, the dynamic sensitivity by deterministic and randomparameters will be the subject of furtherwork. Acknowledgments I would like to express my gratitude to promoter of my doctoral dissertation Professor Tran Duong Hien, for providing a computer programme for this research. References 1. Adomian G., 1983, Stochastic Systems, Academic Press 2. Bathe K.-J., 1982,Finite Element Procedures in Engineering Analysis, Prentice-Hall Time-independent stochastic design sensitivity analysis... 793 3. Choi K.K., Kim N.-H., 2010, Structural Sensitivity Analysis and Optimization, Springer 4. Ding J., Pan Z., Chen L., 2012, Parameter identification ofmultibody systems based on second order sensitivity analysis, International Journal of Non-Linear Mechanics, 47, 1105-1110 5. Drewko J., Hien T.D., 2005, First- and second-order sensitivities of beams with respect to cross-sectional cracks,Archive of Applied Mechanics, 74, 309-324 6. Fishman G.S., 1995,Monte Carlo: Concept, Algoritms and Aplications, Springer 7. Ghanem R.G., Spanos P.D., 1991, Stochastic Finite Elements: A Spectral Approach, Springer 8. Greene M.S., Liu Y., Chen W., Liu W.K., 2011, Computational uncertainty analysis in mul- tiresolution materials via stochastic constitutive theory, Computer Methods in Applied Mechanics and Engineering, 200, 309-325 9. Haug E.J., Choi K.K., Komkov V., 1986, Design Sensitivity Analysis of Structural Systems, Academic Press 10. Hien T.D., 2003,Numerical Analysis of Stochastic Systems,Wyd. PS 11. HienT.D.,KleiberM., 1989,Computational aspects in structural design sensitivity analysis for statics and dynamics,Computers and Structures, 33, 939-950 12. Hien T.D., Kleiber M., 1990, POLSAP — A Finite Element Code for Deterministic and Sto- chastic Analyses of Large 3D Structures, IPPTPAN 13. Hien T.D., Kleiber M., 1991, Stochastic structural design sensitivity of static response, Com- puters and Structures, 38, 5/6, 659-667 14. Hisada T., Nakagiri S., 1981, Stochastic finite elementmethod for structural safety and relabi- lity,Proceedings of 3rd International Conference on Structural Safety and Reliability, 395-402 15. Kincayd D., Cheney W., 2002, Numerical Analysis. Mathematic of Scientific Computing, 3rd ed.,Wadsworth Group 16. Kleiber M., Hien T.D., 1992,The Stochastic Finite Element Method, Willey 17. Leet K.M., Uang Ch.-M., GilbertA.M., 2010,Fundamentals of Structural Analysis, 4th ed., Mc Graw-Hill 18. Li J., Chen J., 2009, Stochastic Dynamics of Structures,Wiley 19. Liu W.K., Belytschko T., Mani A., 1986, Randomfield finite elements, International Journal for Numerical Methods in Engineering, 23, 1831-1845 20. Liu Y., Greene M.S., Chen W., Dikin A.D., Liu W.K., 2013, Computational microstructu- re characterization and reconstruction for stochastic multiscale material design, Computer-Aided Design, 65-76 21. Mroz Z., Bojczuk D., 2012, Shape and topology sensitivity analysis and its application to structural design,Archive of Applied Mechanics, 82, 1541-1555 22. Mroz Z., HaftkaR.T., 1986, First- and second-order sensitivity analysis of linear and nonlinear systems,AIAA Journal, 24, 1187-1192 23. Niczyj J., 2003,Multi-Criteria Optimization of Reliability and Estimation of the Technical State of Bar Structures Using Fuzzy Sets Theory (in Polish),Wyd. PS 24. RubinsteinR.Y.,KroeseD. P., 2008,Simulation and theMonte CarloMethod, 2nd ed.,Willey 25. SpanosP.D.,GhanemR.G., 1989,Stochasticfinite elementexpansion for randommedia,Journal of Engineering Mechanics, 115, 5, 1035-1053 26. Weber H., Hien T.D., 2010, Elimination of beat effects in structures by added lumped mass, Pomiary Automatyka Kontrola, 56, 617-619 794 H.Weber 27. Weber H., 2014, Numerical analysis of static and dynamic sensitivity of complex structu- ral systems with random parameters, Ph.D Thesis, West Pomeranian University of Technology Szczecin 28. Zienkiewicz O.C., Taylor R.M., 1991,The Finite Element Method, McGraw-Hill Manuscript received May 18, 2015; accepted for print November 24, 2015