Variability response functions for statically determinate beams with arbitrary nonlinear constitutive laws
VVariability response functions for statically determinatebeams with arbitrary nonlinear constitutive laws
Amir Kazemi a,b , Javad Payandehpeyman c, ∗ a School of Civil Engg., Iran University of Science and Technology, Tehran 16846, Iran. b The State Plan and Budget Organization, Tehran 11499, Iran. c Department of Robotics Engg., Hamedan University of Technology, Hamedan 65155, Iran.
Abstract
The variability response function (VRF) is generalized to statically determinateEuler Bernoulli beams with arbitrary stress-strain laws following Cauchy elas-tic behavior. The VRF is a Green’s function that maps the spectral densityfunction (SDF) of a statistically homogeneous random field describing the cor-relation structure of input uncertainty to the variance of a response quantity.The appeal of such Green’s function is that the variance can be determinedfor any correlation structure by a trivial computation of a convolution integral.The method introduced in this work derives VRFs in closed form for arbitrarynonlinear Cauchy-elastic constitutive laws and is demonstrated through threeexamples. It is shown why and how higher order spectra of the random fieldaffect the response variance for nonlinear constitutive laws. In the general sense,the VRF for a statically determinate beam is found to be a matrix kernel whoseinner product by a matrix of higher order SDFs and statistical moments is in-tegrated to give the response variance. The resulting VRF matrix is uniqueregardless of the random field’s marginal probability density function (PDF)and SDFs.
Keywords:
Uncertainty quantification, variability response functions,stochastic Greens functions, Cauchy elasticity, nonlinear constitutive law. ∗ Corresponding author
Email addresses: [email protected], [email protected] (Amir Kazemi), [email protected] (Javad Payandehpeyman)
Preprint submitted to Elsevier October 15, 2018 a r X i v : . [ phy s i c s . c l a ss - ph ] J un . Introduction The concept of the variability response function (VRF) was introduced in thelate 1980s [1] and has developed extensively since then. The VRF is a means tosystematically derive the spectral effects of uncertain system parameters mod-eled by homogeneous random fields on the response of structures. The VRF isindependent from the marginal probability distribution function (PDF) and thespectral density function (SDF) of the random fields. Using VRFs for a responsequantity, one performs the sensitivity of analysis of the system response easilyfor random fields with different SDFs.Exact VRFs of displacement response were derived in [2, 3] for staticallydeterminate beams with linear elastic material. In [4], VRFs were derived forstatically determinate beams with power constitutive laws. The concept ofthe VRF was adapted in [5] to measure the variability of upscaled materialproperties of stochastic volume elements, and to derive VRFs for the effectiveflexibility of statically determinate beams.For statically indeterminate structures, exact VRFs have not been derived,yet Taylor expansion techniques were used in [6, 7, 8, 9] for the displacementresponse of structures whose uncertainty is given by two-dimensional randomfields. Also, the fast Monte Carlo methodology proposed in [10] was developed in[11, 12] to estimate the VRF efficiently. The method was later applied to generallinear finite element systems, including dynamic problems [13, 14, 15, 16] andits ansatz (i.e. the independence of the VRF from the mariginal PDF and SDFof the stochastic field) was examined through the Generalized VRF methodol-ogy introduced in [17] addressing the static indeterminacy of structures. Themethodology was employed to estimate the VRF for effective flexibility of stat-ically indeterminate beams in [18], statically indeterminate beams with powerconstitutive laws in [4], and two-dimensional structures in [19].The unconditional existence of the VRF nevertheless has neither been provednor disproved formally under general material nonlinearity. The GeneralizedVRF methodology, when applied to nonlinear constitutive laws, requires know-2ng the specific higher order spectral functions affecting response variability.Identifying these higher order terms requires knowing the VRF solution of astatically determinate structure with the same constitutive law.The derivation presented in this work shows that VRFs can be calculatedfor statically determinate beams having constitutive laws of arbitrary functionalform. The VRFs obtained through this method are a generalization of theclassical VRF. By a polynomial interpolation of the beam’s curvature in termsof the nominal resisting bending moment, response variance can be expressedas the inner product of a VRF matrix by a matrix of higher order SDFs andstatistical moments of the random field describing the resisting bending momentuncertainty. The new formulation results in the same formulas for the VRFsof a linear and square root constitutive law, as well as the same coefficientsof higher-order spectral functions [4]. Moreover, in a numerical example, theresponse variance of a stochastic cantilever beam having a bilinear constitutivelaw is derived using this new approach. Trivial deviation of the results fromMonte Carlo (MC) simulations shows that whenever an accurate polynomialinterpolation is used to model the curvature in terms of the resisting bendingmoment, the variance can be calculated by the VRFs precisely.
2. The response of stochastic beams
Suppose that the section modulus and constitutive law of a transverselyloaded statically determinate Euler-Bernoulli beam vary randomly along thebeam’s length as 1 σ ( x, ε ) S ( x ) ≡ f ( x ) σ ( ε ) S (1)where S and σ ( ε ) denote the nominal section modulus and constitutive lawrespectively, and f ( x ) is a zero-mean, statistically homogeneous random fieldbounded as f ( x ) > − y = ρε (2)3here y is the vertical coordinate from the neutral axis and ρ is the curva-ture radius, the resulted maximum strain ( (cid:15) ) along the beam, as an uncertainstructural response (or output) quantity, is the random field satisfying | M ( x ) | = 3 S ( x ) (cid:15) ( x ) (cid:90) (cid:15) ( x )0 σ ( x, ε ) εdε (3)where | M ( x ) | is the absolute value of static moment at section x . Eqs. (1) and(3) give | M ( x ) | = 11 + f ( x ) (cid:32) S (cid:15) ( x ) (cid:90) (cid:15) ( x )0 σ ( ε ) εdε (cid:33) . (4)Let m (cid:15) ( (cid:15) ) ≡ S (cid:15) (cid:90) (cid:15) σ ( ε ) εdε. (5)where m (cid:15) ( (cid:15) ) is the nominal resisting bending moment corresponding to themaximum strain (cid:15) . The nominal resisting bending moment in terms of curvature( k = 2 (cid:15)/h with h as the section’s height) takes the following form: m ( k ) ≡ S k (cid:90) k σ ( ξh/ ξdξ. (6)Employing the definition of nominal resisting bending moment, one concludesfrom Eq. (4) that m ( k ( x )) = | M ( x ) | (1 + f ( x )) (7)The asymptotic behavior of Eq. (7) is in accordance with that of the initialdefinition in Eq. (1): As f ( x ) → + ∞ , it requires k ( x ) → + ∞ (the infiniteflexibility case); contrariwise, when f ( x ) → −
1, it makes k ( x ) → k ( x ) = 0 for M ( x ) = 0. The reader is cautioned that f ( x ) must posses an upper-bound so that the resisting bending moment acquiremeaningful realizations and the MC simulation becomes feasible. Therefore, tohave a well-posed problem, distributions like the lognormal should be appliedto f ( x ) carefully [20, p. 9].Calculation of k ( x ) is required to find the second derivative of the beam’sdeflection and thereof the deflection itself. This is realized by finding the inverseof m ( · ) using Eqs. (6-7): k ( x ) = m − ( | M ( x ) | (1 + f ( x ))) . (8)4f m ( · ) is one-to-one, it is invertible as well. Therefore, the next step is toinvestigate whether m ( · ) is increasing or not, that is to say: m (cid:48) ( k ) = 3 S (cid:32) − k (cid:90) k σ (cid:18) hξ (cid:19) ξdξ + 1 k σ (cid:18) hk (cid:19) k (cid:33) > (cid:90) (cid:15) σ ( ε ) εdε < σ ( (cid:15) ) (cid:15) . (10)This inequality holds for almost every constitutive law. As shown in the schematicstress-strain curve of Fig. (1), the left hand side of the inequality is the momentof the dotted area with respect to the stress axis, while the right hand side isthat of the total shaded area. Figure 1: Schematic stress-strain curve to show the invertibility of the resisting bendingmoment in terms of curvature.
The curvature or k ( x ) in Eq. (8) can be approximated by the polynomialinterpolation of m − ( · ). According to Weierstrass approximation theorem, anycontinuous function like m − ( · ) defined over a closed interval is uniformly ap-proximated by a polynomial as accurately as desired. Without loss of generality,this closed interval in mathematical texts is supposed as [0 ,
1] or [ − ,
1] to whicharbitrary intervals are easily mapped [21, p. 509]. The reader can choose amongvarious polynomial interpolation forms to model curvature in terms of resistingbending moment as a polynomial function.5n this paper, the monomial form of polynomial interpolation is employed toapproximate m − ( · ), where the polynomial coefficients are calculated throughan explicit formulation constructed from the Vandermonde matrix [22, p. 2].The monomial form of polynomial interpolation finds the unique polynomial of N th -degree crossing N + 1 points such that the curvature is expressed as k ( x ) = N (cid:88) i =0 λ i ( | M ( x ) | (1 + f ( x ))) i (11)to fit k = [ k , k , ..., k N +1 ] T and m = [ m ( k ) , m ( k ) , ..., m ( k N +1 )] T . The coeffi-cients λ i are calculated by solving a linear system of equations as follows: λ = V − k (12)with V defined as the square Vandermonde matrix: V = m ( k ) m ( k ) · · · m N ( k )1 m ( k ) m ( k ) · · · m N ( k )... ... ... . . . ...1 m ( k N +1 ) m ( k N +1 ) · · · m N ( k N +1 ) . (13)It is noteworthy that the interpolation form is called monomial because the basesfor the interpolating N th -degree polynomial are selected as 1 , m ( k ) , m ( k ) , ...,m N ( k ) which are monomials.Note that a major concern for the convergence of polynomial interpolationis Runge’s phenomenon which is the oscillation at the edges of the fitting in-terval including equispaced interpolation points. To minimize the effect of thisphenomenon in polynomial interpolation, Chebyshev nodes should be used asfitting data [23, Ch. 13]. For a fitting interval of [0 , k u ], the abscissas of suchnodes are determined by m ( k n ) = 12 m ( k u ) + 12 m ( k u ) cos (cid:18) n − N + 2 π (cid:19) . (14)The ordinates of the Chebyshev nodes are hence: k n = m − (cid:18) m ( k u ) + 12 m ( k u ) cos (cid:18) n − N + 2 π (cid:19)(cid:19) (15)6hich can be estimated by an interpolation within the pairs of ( k, m ( k )) usingEqs. (6) and (14). Noteworthy is the fact that, according to Eq. (8), a validinterpolation requires k u ≥ m − (max( | M ( x ) | (1 + f ( x ))) . (16)As the kinematic relationship in Eq. (2) states, the beam’s signed curvatureis given by u (cid:48)(cid:48) ( x ) = k ( x )sgn( M ( x )) (17)which means that the signed curvature is positive under positive static moment.Importing Eq. (11) in Eq. (17) yields u (cid:48)(cid:48) ( x ) = N (cid:88) i =0 λ i ( | M ( x ) | (1 + f ( x ))) i sgn( M ( x )) (18)which is solved as u ( x ) = (cid:90) x N (cid:88) i =0 λ i ( | M ( s ) | (1 + f ( s ))) i sgn( M ( s )) G ( x, s ) ds (19)where G ( x, s ) is the Green’s function for the differential equation in Eq. (18)along with imposed boundary conditions on u ( x ).
3. The VRFs
The response variance, i.e.
Var [ u ( x )] = E [ u ( x )] − E [ u ( x )] , can be writtenas Var [ u ( x )] = N (cid:88) i =0 N (cid:88) j =0 (cid:90) x (cid:90) x λ i λ j ×| M ( s ) | i | M ( s ) | j sgn( M ( s ))sgn( M ( s )) × G ( x, s ) G ( x, s ) (cid:0) R ∗ ij ( τ ) − µ ∗ i µ ∗ j (cid:1) ds ds (20)with E [(1 + f ( s )) i ] = µ ∗ i (the i th moment), E [(1 + f ( s )) i (1 + f ( s )) j ] = R ∗ ij ( τ )(the ij th autocorrelation function), and τ = s − s . It is worth noting theexplicit dependence of the response variance on the higher order correlations of7 ( x ) for arbitrarily nonlinear constitutive law. By using Wiener-Khinchin theo-rem, which states that R ij ( τ ) = (cid:82) + ∞−∞ S ij ( κ ) exp( i κτ ) dκ , Eq. (20) is expressedas Var [ u ( x )] = (cid:90) + ∞−∞ VRF ( x, κ ) : ( S ( κ ) − δ ( κ ) M ) dκ (21)where : denotes the Frobenius inner product, S and M are the matrices ofSDFs and statistical moments of the random field 1 + f ( x ) with the followingcomponents: S ij ( κ ) ≡ S ∗ ij ( κ ) = i (cid:88) p =0 j (cid:88) q =0 (cid:18) ip (cid:19)(cid:18) jq (cid:19) S pq ( κ ) , (22) M ij ≡ µ ∗ i µ ∗ j = i (cid:88) p =0 j (cid:88) q =0 (cid:18) ip (cid:19)(cid:18) jq (cid:19) µ p µ q , (23)where asterisks denotes that the parameter belongs to 1 + f ( x ), rather than f ( x ) (for which no asterisk is used). The matrix VRF is given by the followingvector multiplication:
VRF ( x, κ ) = V † ( x, κ ) V ( x, κ ) (24)where V † is the conjugate transpose of V and V i ( x, κ ) ≡ (cid:90) x λ i | M ( s ) | i sgn( M ( s )) G ( x, s ) exp( i κs ) ds. (25)Note that R ∗ ij ( τ ) − µ ∗ i µ ∗ j is zero when i = 0 and/or j = 0. Therefore, the sumsin Eq. (20) could start from i = 1 and j = 1. Besides, a correct interpolationof curvature with respect to nominal resisting bending moment requires λ = 0as a result of m ( k = 0) = 0.
4. Parametric Examples
The derivations in sections 2 and 3 are based on the definition of randomfield for the reciprocal of section modulus by stress, i.e. Eq. (1), rather than forthe elastic flexibility. Therefore, it is critical to examine whether this assump-tion is robust and leads to the same VRFs for linear and a class of non-linearconstitutive laws as shown in [2, 3, 4] respectively:8 .1. Linear constitutive law
Let the nominal constitutive law be σ ( ε ) = Eε . As a result of stochasticmaterial and cross-section (i.e. Eq. (1)), resisting bending moment is a randomfield along the beam as obtained in Eq. (7) with the nominal value given as m ( k ) = 3 Sk (cid:90) k E (cid:18) hξ (cid:19) ξdξ = EIk (26)where I = bh /
12 is the moment of inertia of the cross section. Consideran interpolation of m − ( · ) by a polynomial of second degree using a set ofthree points (0 , k , m ( k )), and ( k , m ( k )). Note that two points sufficesinasmuch as m − ( · ) is linear, yet three points are selected to show that addingpoints in the interpolation does not alter the results for the linear constitutivelaw. The Vandermonde matrix according to Eq. (13) becomes: V = m ( k ) m ( k )1 m ( k ) m ( k ) = αk α k αk α k (27)where α = EI . Introducing the inverse of V into Eq. (12) yields: λ λ λ = 1 α ( k k − k k ) α ( k k − k k ) 0 0 α ( − k + k ) α k − α k − α ( − k + k ) − αk αk k k (28)where the polynomial coefficients are solved as λ λ λ = /α . (29)Using λ in Eqs. (24) and (25) gives VRF ( x, κ ) = (cid:18) EI (cid:19) (cid:90) x (cid:90) x M ( s ) M ( s ) G ( x, s ) G ( x, s ) exp( i κτ ) ds ds (30)and S ( κ ) − δ ( κ ) M = S ∗ ( κ ) − δ ( κ ) µ ∗ µ ∗ = S ( κ ) (31)9qs. (45) and (46) are exactly the widely-known VRF and SDF for a linearconstitutive law [2, 3]. Note that it can be shown that λ i = 0 for all i (cid:54) = 1 whensolving Eqs. (27) and (28) by assuming a higher degree polynomial and solvingfor vector λ . Let the constitutive law be σ ( ε ) = E √ ε . Resisting bending moment is arandom field along the beam as obtained in Eq. (7) with the nominal valuegiven as m ( k ) = 3 Sk (cid:90) k E (cid:18) hξ (cid:19) . ξdξ = 12 EI √ h √ k. (32)As a starting point, assume a forth-degree polynomial for interpolating m − ( · ).The nodes for interpolation have [0 k k k k ] as ordinates and [ m ( k i )] asabscissas. Note that employing parametric nodes and thus an arbitrary fittinginterval obviates the need to control Eq. (16), because one may assume k ≥ k u without loss of generality. The Vandermonde matrix becomes V = βk . β k β k . β k βk . β k β k . β k βk . β k β k . β k βk . β k β k . β k (33)where β = (12 EI ) / (5 √ h ). Introducing Eq. (48) into Eq. (12) gives λ λ λ λ λ = /β . (34)Using λ in Eqs. (21-25) yields VRF ( x, κ ) = (cid:90) x (cid:90) x (cid:18) b h E (cid:19) | M ( s ) | | M ( s ) | sgn( M ( s ))sgn( M ( s )) G ( x, s ) × G ( x, s ) exp( i κτ ) ds ds (35)and S ( κ ) − δ ( κ ) M = S ∗ ( κ ) − δ ( κ ) µ ∗ µ ∗ = 4 S ( κ ) + 4 S ( κ ) + S ( κ ) − δ ( κ ) σ f . (36)which agree with Eqs. (30) and (31) in Ref. [4]. Mathematical induction canshow that λ i = 0 for all i (cid:54) = 2 when assuming a higher degree polynomial andsolving for vector λ .
5. Numerical Example
To show the method’s efficiency for the estimation of VRFs for arbitraryCauchy elastic materials, a bilinear constitutive law is examined for the staticallydeterminate beam shown in Fig. (2) with M = 3500, q ( x ) = 50, L = 16, b = 1, h = √
12, and G ( x, s ) = x − s . The nominal constitutive law is σ ( ε ) = E ε ε ≤ . . E ( ε + 0 . ε > .
002 (37)with E = 7 × . The resisting bending moment is considered as a statisti-cally homogeneous random field as derived in Eq. (7) whose nominal value isobtained by introducing Eq. (37) into Eq. (6). The monomial form of polyno-mial interpolation is employed to model m − ( · ) as suggested in section 2. Theanalytically derived VRF is verified by comparing the predicted variance usingthe VRFs, i.e. Eq. (21), for the vertical displacement at x = 16 with thatcomputed by brute-force MC simulation for three different random field modelsof f ( x ) as discussed below. 11 .2. The associated random field The MC simulation employs translation from an underlying U-Beta randomfield to a target one (an associated field) with a target marginal cumulativedistribution function (CDF) P f [24, 25, 26]. The underlying random field variessinusoidally with random phase angles θ uniformly distributed on [0 , π ] asfollows g ( x ) = √ σ g cos ( κ δ x + θ ) (38)where √ σ g is the amplitude and κ δ is a certain wave number determining thespectral content of the field. The underlying U-Beta random field has SDF givenas S = σ g / δ ( κ + κ δ ) + δ ( κ − κ δ )], and the values used in this example are κ δ = π/
2, and σ g = 1 / √ f ( x ) = P − f ◦ P g ( g ( x )) = A ( g ( x )) (39) Figure 2: Cantilever analysed in the numerical example from Ref. [4] P g denotes the CDF of the underlying field given as P g ( g ( x )) = 1 − π arccos (cid:32) g ( x ) √ σ g (cid:33) . (40)In this example, the three associated fields considered have uniform (UN),truncated Gaussian (TG), and Lognormal (LN) marginal distributions. Therandom field f ( x ) is realized by mapping g ( x ) as follows: the mapping for UNis given as f ( x ) = ( a u − a l ) P g ( g ( x )) + a l , (41)the mapping for TG is f ( x ) = a l s Φ − ( P g ( g ( x ))) + m < a l s Φ − ( P g ( g ( x ))) + m a l ≤ s Φ − ( P g ( g ( x ))) + m ≤ a u a u a u < s Φ − ( P g ( g ( x ))) + m , (42)and the mapping for LN is given as f ( x ) = exp (cid:0) s Φ − ( P g ( g ( x ))) + m (cid:1) + a l , (43)where a l , a u , m and s are defined in Table (1). Simulations for g ( x ) are obtainedthrough the simulation of random variable θ as given in Eq. (38). Table 1: Parameters of the PDFs for f ( x ) PDF a l a u m s σ f UN -0.80 0.80 n/a n/a 0.46TG -0.90 0.90 0.00 1.00 0.67LN -0.40 n/a -1.03 0.47 0.20The SDFs of the associated fields are obtained as follows. Due to the shiftinvariance of the U-beta random field (i.e. g ( x + 2 π ) = g ( x )) and the one-to-onemapping of the associated field, the autocorrelation function of f ( x ) is given by R ij ( τ ) = 12 π (cid:90) π A i ( √ σ g cos ( θ )) A j ( √ σ g cos ( κ δ τ + θ )) dθ (44)13hich is an even function representable by the following Fourier series: R ij ( τ ) = a ( i, j )2 + ∞ (cid:88) η =1 a n ( i, j ) cos ( ηκ δ τ ) (45)with a η ( i, j ) = 12 π (cid:90) π (cid:90) π cos ( ηξ ) A i ( √ σ g cos ( θ )) A j ( √ σ g cos ( ξ + θ )) dθdξ. (46)Corresponding higher order SDFs, obtained by taking the Fourier transform ofEq. (45), are expressed as S ij ( κ ) = a ( i, j )2 δ ( κ ) + 12 ∞ (cid:88) η =1 a η ( i, j ) ( δ ( κ + ηκ δ ) + δ ( κ − ηκ δ )) (47)where δ ( · ) is the Dirac’s delta function. The statistical moments are also givenby µ i = 12 π (cid:90) π A i ( √ σ g cos ( θ )) dθ. (48) As shown in Fig. (3), m − ( · ) (the dotted blue line) is fitted by the monomialform of polynomial interpolation (the solid red line) with different degrees usingEqs. (11-16). The data of m − ( · ) is a set of ordered pairs obtained by inter-changing the first and second elements of the pairs ( k, m ( k )) generated by Eq.(6) within the curvature domain [0 , . f ( x ) having the PDFof UN and TG. Yet for f ( x ) with the PDF of LN, one should assure that uppertails do not affect the variance significantly. Fig. (4) shows that increasingthe truncation value of the LN-based f ( x ) more than one hardly changes theresponse variance in the MC simulation. Therefore, the mentioned curvaturedomain produces an accurate response for LN truncated as f ( x ) ≤ VRF and S ( κ ) − δ ( κ ) M using a fourthdegree polynomial interpolation of curvature-resisting bending moment. Theresponse variance using the interpolation with different degrees of polynomialare shown in Fig. (6) in comparison with that of the MC simulations. Fig.14 Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Resisting bending moment, m. C u r v a t u r e , k . m −1 (.) from Eq. (6)Polynomial Interp. Figure 3: The monomial form of polynomial interpolation of m − ( · ) is performed for thebilinear constitutive law in Eq. (37) using Eqs. (11-16). The green points illustrate theChebyshev nodes defined by Eqs. (14) and (15), and a set of N + 1 nodes means a polynomialof N th degree. −1 −3 max(f(x)) V a r( u ( x )) Figure 4: The variance of the tip vertical displacement using 10,000 MC simulations fordifferent truncation values of f ( x ) having the PDF of LN, indicating the insignificance of thePDF tail’s effect on the variance for truncation values larger than one. (6-a) shows convergence for the interpolation-based approach as the polynomialdegree increases. The converged variances derived analytically are very closeto the variances determined through MC simulation, as illustrated in Fig. (6).While the MC simulation starts converging after about 1000 simulations, themethod presented in this paper converges well after a polynomial of fifth to 10thdegree. The relative error of analytical results with respect to the variance using10,000 MC simulations are shown in Fig. (6-c) for different polynomial degrees.Note that the responses generated in the MC simulation unlike Eq. (19) donot involve the polynomial interpolation of curvature-resisting bending momentand are calculated directly by u ( x ) = (cid:90) x k ( s )sgn( M ( s )) G ( x, s ) ds (49)where curvatures are given by Eqs. (6-7) using linear interpolation within thepairs ( k, m ( k )). Such approach guarantees that the MC simulation, as theonly verification benchmark, is not subject to the approximations of polynomialinterpolation. 16 V R F ( x , κ ) (1,1) 0 2−505 (1,2) 0 2012 (1,3) 0 2−1012 (1,4)0 2−505 V R F ( x , κ ) (2,1) 0 20510 (2,2) 0 2−4−202 (2,3) 0 2−2024 (2,4)0 2012 V R F ( x , κ ) (3,1) 0 2−4−202 (3,2) 0 20123 (3,3) 0 2−2024 (3,4)0 2−1012 κ V R F ( x , κ ) (4,1) 0 2−2024 κ (4,2) 0 2−2024 κ (4,3) 0 20510 κ (4,4) Figure 5: The VRF components at x = 16 using a fourth degree polynomial interpolationare plotted as blue lines. The terms of the components of S ( κ ) − δ ( κ ) M are represented for κ δ = π/ nκ δ , a n ( i, j )) and belong to 1 + f ( x ) with f ( x ) having thePDF of UN. The horizontal axes of the plots are the wave number ( κ ), and the plot titles( i, j ) indicate the component of the functions. The response variance is the sum of the bluecurves’ ordinates multiplied by a n ( i, j ) at nκ δ . −3 −2 −1 Polynomial degree V a r( u ( x )) (a) UNTGLN10 −3 −2 −1 Number of MC simulations V a r( u ( x )) (b) UNTGLN1 2 3 4 5 6 7 8 9 10−60−40−200204060
Polynomial degree T he v a r i an c e e rr o r ( % ) (c) UNTGLN
Figure 6: The tip displacement variance for the bilinear constitutive law using (a) the mono-mial form of polynomial interpolation and (b) MC simulation, and (c) the relative error ofanalytical method with respect to the results of 10,000 MC simulations. Note that calculationsare based on κ δ = π/ . Conclusion This paper generalizes the concept of the VRF to the response of stochasticstatically determinate Euler-Bernoulli beams having arbitrary functional formsof the constitutive law (i.e. Cauchy elastic materials). The new formulation issuch that once the inverse of the nominal resisting bending moment with respectto the beam’s curvature is interpolated by a polynomial function, the varianceis determined by the inner product of a VRF matrix with a matrix containingthe SDFs and statistical moments of the random field describing the resistingbending moment uncertainty. The interpolation-based approach certifies theclosed-form VRFs already obtained for root constitutive laws and is tested toestimate the response variance of a cantilever having a bi-linear constitutive lawby means of the VRF matrix. The accuracy of the VRFs is verified by the minordiscrepancies among the predicted response variance values from the VRF withthat obtained by MC simulation.Another significance of this work is that the derivations presented in thispaper open the possibility to compute VRFs for statically indeterminate struc-tures having arbitrary Cauchy elastic constitutive laws using the GeneralizedVariability Response Function (GVRF) method. For statically indeterminatestructures, the integrand in the expression for the response variance (e.g. Eq.(21)) cannot be separated into the product of a deterministic function (i.e. theVRF) and properties of the stochastic field (i.e. the SDF and higher orderstatistics). The GVRF method is a numerical technique to compute approx-imate VRFs and have been demonstrated on various statically indeterminate,linear structures [11, 12, 13, 14, 15, 16]. For nonlinear constitutive laws, GVRFscan only be approximated if the specific higher order statistical moments andcorrelation functions that affect response variance for statically determinatestructures, along with their relative contributions, are known [4].19 eferenceseferences