Aalborg Universitet

Stochastic Finite Elements in Reliability-Based Structural Optimization

Sørensen, John Dalsgaard; Engelund, S.

Publication date:1995

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Citation for published version (APA):Sørensen, J. D., & Engelund, S. (1995). Stochastic Finite Elements in Reliability-Based Structural Optimization.Dept. of Building Technology and Structural Engineering. Structural Reliability Theory, No. 155, Vol.. R9543

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INSTITUTTET. FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALBORG UNIVERSITET • AUC • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 155

Presented at WCSM0-95, Goslar, Germany 1995o~~

J. D. S0RENSEN & S. ENGELUND STOCHASTIC FINITE ELEMENTS IN RELIABILITY-BASED STRUCTU­RAL OPTIMIZATION DECEMBER 1995 ISSN 1395-7953 R9543

The STRUCTURAL RELIABILITY THEORY papers are issued for early dissemina­tion of research results from the Structural Reliability Group at the Department of Building Technology and Structural Engineering, University of Aalborg. These papers are generally submitted to scientific meetings, conferences or journals and should there­fore not be widely distributed. Whenever possible reference should be given to the final publications (proceedings, journals, etc.) and not to the Structural Reliability Theory papers.

I Printed at Aalborg University I

INSTITUTTET FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALBORG UNIVERSITET • AUC • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 155

Presented at WCSM0-95, Goslar, Germany 1995

J. D. S0RENSEN & S. ENGELUND STOCHASTIC FINITE ELEMENTS IN RELIABILITY-BASED STRUCTU­RAL OPTIMIZATION DECEMBER 1995 ISSN 1395-7953 R9543

STOCH ASTIC F INITE ELEMENTS I N R E LIABILITY-BASED STRUCTUR A L OPTIM I ZATION

J .D. S~rensen and S. E ngelund

Department of Building Technology and Structural Engineering Aalborg University, Sohngaardsholmsvej 57

DK-9000 Aalborg, Denmark

ABSTRACT

Application of stochastic finite elements in structural optimization is considered. It is . shown how stochastic fields modelling e.g. the modulus of elasticity can be discretized in

stochastic variables and how a sensitivity analysis of the reliability of a structural system with respect to optimization variables can be performed. A computer implementation is described and an illustrative example is given.

Key Wo:r;ds: Reliability analysis; Stochastic finite elements; Structural optimization; Stochastic fields.

1. INTRODUCTION

The development of effective First Order Reliability Methods (FORM), see (Madsen et al., 1986) and optimization algorithms in combination with sensitivity analysis methods has implied that a large number of decison problems in structural engineering can now be solved with a reasonable amount of resources. Examples are optimal reliability-based design, inspection and maintenance planning with respect to cracks in steel structures and optimal reliability-based experiment planning. These decision problems can be sol­ved on the basis of classical decision theory using Bayesian statistics, see e.g. (S0rensen et al., 1995) and (Kroon et al., 1995).

Uncertain quantities are generally modelled by stochastic variables and the reliability is estimated using FORM. However, uncertainties related to structural material char­acteristics in general have to be modelled by stochastic fields. Using stochastic finite elements the stochastic fields can be discretized. In relation to optimization of struc­tural systems the discretized stochastic fields have until now been characterized in an approximate way by stochastic variables related to the midpoint or the average value of the stochastic field in a given stochastic finite element discretization, see (Brenner, 1991) and (Der Kiureghian et al., 1988). However, a newly developed method, see e.g. (Deodatis, 1991 ), makes it possible to formulate 'exact' stochastic stiffness matrices by the so-called weighted integral method.

The purpose of the present paper is to demonstrate how reliability-based optimization problems involving random spatial fluctuations of material properties can be solved

2

using FORM analysis and weighted integrals in stochastic finite elements.

The solution of this optimization problem can be divided into four main tasks, namely (stochastic) finite element analyses, sensitivity analyses, reliability analyses and appli­cation of an optimization algorithm. It is discussed how these four tasks can be linked effectively, especially the implementation of stochastic finite elements in such a system is considered. The suggested methology is demonstrated by an illustrative example.

2. RELIABILITY-BASED STRUCTURAL OPTIMIZATION

As described in (Enevoldsen et al., 1994) an optimization (design) problem in reliability­based structural optimization can be formulated by

mm C(b) (1)

,i = 1, ... ,M (2)

B;(b) ~ 0 ,i = 1, . .. ,m (3)

b~ ~ b; ~ b'( , i = 1, ... , N (4)

where b = (b1 , ... , bN) are the design/optimization variables. The optimization variab­les are assumed to be related to size and shape parameters. The objective function C is often chosen as the structural weight.

The constraints in (2) are related to the reliability of single elements of the structural systerri. (3;, i = 1, ... ,M are reliability indices for M failure modes. fJiin, i = 1, ... ,M are the corresponding lower limits on the reliabilities. The deterministic inequality constraints in (3) ensure that response characteristics such as displacements and stresses do not exceed codified critical values . The inequality constraints can also include general design requirements for the design variables. The constraints in ( 4) are simple bounds.

Alternatively, the optimization problem (1)- ( 4) can be formulated as a system reliability­based optimization problem if the constraint (2) is exchanged with a constraint on the system reliability index W estimated on the basis of the M failure modes in (2). Furt­her, a more general optimization problem can be formulated if the objective function in (1) is exchanged with the total expected costs in the design lifetime, see (S0rensen et al., 1994).

The reliability indices in (2) are assumed to be determined on the basis of limit state functions which can be written

g;(b,x(b), y(b,x(b))) = 0 i = 1, ... ,M (5)

3

where x are realizations of stochastic variables X. Realizations x where g :::; 0 corre­sponds to failure states, while realizations x where g > 0 corresponds to safe states. y are performances such as displacements and stresses.

The stochastic variables X can depend on the optimization variables b, for example if the expected value f.lX; of X; or the standard deviation CTX; of X; are related to an optimization variable by e.g. bj = f.LX;.

The deterministic constraints in (3) are assumed to be related to the limit state functions g; by

i = l, ... ,m. (6)

where xd are design values calculated from, e.g. xj = /j(f.lxi + kjCTXi ), where kj 1s a factor defining the characteristic value and /j is a partial safety factor.

3. STOCHASTIC FINITE ELEMENTS

If the properties of the structural elements are uncertain and depend on the spatial coordinates then they have to be modelled by stochastic fields. In the following it is shown how stochastic fields can be discretized in combination with structural finite elements,

v.e v,' j

e r V.e 3 Ei (j .. 6

v.e Le V.e 2 5

Figure 1. Two-dimensional beam element.

As an example a structural syst.em modelled by two-dimensional beam elements is con­sidered. The presentation in this section can easily be extended to truss and three­dimensional beam elements. Element e has the length U, the displacement vector is denoted ve = ( vf, ... , v~) and the corresponding nodal forces p = ( Qf, Mt, Nt, Qj, MJ, NJ), see figure 1.

Based on the flexibility method of structural analysis the local stiffness matrix kc for element e can be obtained from

(7)

where He is the compatibility matrix. The flexibility matrix Fe is related to the internal

4

forces Mie, MJ and NJ by

l rL' x-L• x-L• 1 d

Jo ---y;e---y;e J<E<(x) X

Fe rL· X x-L' 1 d = Jo £e ---y;e I•E•(x) X

0

rL' X x-L" 1 d Jo y;e----y;;- J•E•(x) X

rL' X X 1 d Jo V L< f<E•(x) X

0

(8)

where it is assumed that the moment of inertia Ie and the cross-sectional area Ae are constant in element e. Ee denotes the modulus of elasticity. It is assumed that Ee is uncertain and that it can be modelled by a stochastic field {Ee(x),x E Ln} where Ln denotes the total length of all beam elements in the structure. The stochastic field { Ee ( x)} is modelled by

1 1 Ee(x) = Eo (1 + !E(x )) (9)

where E0 is the expected value of {Ee(x)} and {/E(x)} is a homogeneous Gaussiap stochastic field with zero mean and covariance function

(10)

S !E ( w) is the spectral density function of { f E ( x)}.

If (9) is introduced in (8) the flexibility matrix can be written

(11)

where Xf, X2 and X3 are zero mean normally distributed stochastic variables defined by

and

L e

Xi= 1 xi-1 fE(x)dx i = 1,2,3

F~ =

L'

[

3Egfe Le

- 6E~Je

0

L' - 6E0I 0

Le

3E0 I~

0

L<E0! 0 0

0

0 0 j L'

E'A' 0

~]

(12)

r-t 0 _}.J F~ = 0 0

0 (13)

1 1

~] [= L<'f:I• L< 2 E~I~

F~ = 1 £<2E0Je L• 2 E0 I~

0 0

5

If two stochastic variables X? and Xj are related to the same finite element the cava-· riances become

(14)

If X? and Xj are related to finite elements in two different elements (14) can easily be generalized such that Cov[X?, Xj] can be estimated.

The global stiffness matrix K is obtained from

Ne Ne

K = ~TeTkeTe = ~TeTHeFe-!HeTTe (15) e=l e=l

where Ne is the number of finite elements and Te is the geometric transformation matrix from the local coordinates to global coordinates.

If P denotes the global nodal load vector, the global displacements v are obtained from the global finite element equations Kv = P. If distributed loads are applied then P becomes dependent on the stochastic variables Xf, ... ,X~, see (Deodatis, 1991 ).

The performances Yi in (5) are assumed to be either a displacement or a stress component in a finite element. Therefore, in general

Yi = Yi(b,X(b),v(b,X(b))) (16)

4. SENSITIVITY ANALYSIS

In First Order Reliability Methods (FORM) the generally correlated and non-normally distributed stochastic variables X are transformed into standardized and normally di­stributed stochastic variables U = (U1 , Uz, ... , Un) : X = T(U, b), see e.g. (Madsen et al., 1986). In the u-space the reliability index f3 is defined by

(17)

s.t. g(b, x(b, u), Yi(b, X(b ), v(b, X(b )))) = 0 (18)

The solution point u* is denoted the /3-point. If the failure function is approximately linear the probability of failure Pt of the failure mode can be determined with good accuracy from Pt ~<I>( -/3) where <I> is the standard normal distribution function.

6

Gradients with respect to u

In iteration algorithms used to solve the optimization problem (17) - (18) the gradient \lug of the failure function g with respect t.o u1, ... , Un is needed. \lug is obtained from

(19)

where T signifies the transpose and lxu and lyx indicate Jacobian matrices. Vyg, \lxg and lxu can usually be estimated analytically or numerically without significant errors and with relatively low cost within reliability programmes. lyx can be obtained as described below.

Gradients with respect to b

The reliability-based optimization problems in ( 1) - ( 4) can usually be solved using first. order optimization algorithms, such as the NLPQL algorithm, see (Schittkowski, 1986). These methods require the gradient of objective function and of all constraints.

The gradients of the reliability indices 8; with respect to the design variables b are obtained from, see (Madsen et al., 1986)

where the gradients \l~g, \l~g, \l~g and the Jacobians Jyb, lyx and lyv can be estima­ted analytically or numerically without significant. errors and with relatively low cost. The elements in J xb are determined as ~ = aTi ~~I ,b), where ui is the ,8-point for the ith limit state function.

The Jacobians Jvb and lvx can be obtained by quasi-analytical sensitivity analysis. If P does not depend on b then

dv _ 1 dK db·= -K db· V

J J

(21)

For size design variables He, Xf, X~ and Xj will not be dependent on b. Therefore

(22)

h dFg dFj' dFz d dF:J bt . d f (13) If e h w ere db· , db·, db· an db· are o ame rom . e.g. Xj = x2 t en J J J J

(23)

7

5. ALGORITHMS

The above sensitivity analysis is implemented in a. computer programme which can be used to solve the design problem in ( 1) -( 4 ). The computer programme consists of the following modules: • OPTI: A general optimization algorithm where the optimization problem in (1 )-( 4) is programmed, e.g. NLPQL, (Schittkowski, 1986). • RELI: A general reliability programme (e.g. PRADSS (S!Ilrensen, 1991)) which calculates the reliability indices f3 and sensitivities 1-f . • GFUN: Limit state functions • STAT: Calculation

J

of cova.ria.nces of stocha.stic variables defined by ( 12). • FEM: A general linear finite element programme. • SFEM: modifies the local stiffness matrices in FEM according to realizations of X and b using (7) and (11 ). • SENS: evaluates the sensitivity coefficients in (20)-(23).

6. EXAMPLE

The structure shown in figure 2 is considered. All structural elements are assumed to have a rectangular cross-section. The elements are divided into four groups, see table 1. The N = 9 design variables have initial values, lower and upper bounds: bi = 0.5 m, b~ = 0.3 m, b'f = 0.6 m, i=1,3,5,7,8 and b; = 0.02 m, b~ = 0.01 m, b'f = 0.04 m, i=2,4,6,9.

gtoup elements width height thickness 1 1,4,7,10 bl bl b2 2 2,5 ,8,11 bJ bJ b4 3 3,6,9,12 bs bs b6 4 13-21 b7 bs bg

Table 1. Structural elements.

The load P is assumed to be Weibull distributed with expected value = 0.25 MN and standard deviation = 0.0375 MN. The modulus of elasticity is modelled as a stochastic field with expected value 2.1 · 105 MPa and spectral density S!E(w) = tO'}b3w 2 exp( -blwl), where O'J = 0.1 and b = 1 m. The corresponding zero mean normally distributed stochastic variables X 1 - X 63 are defined by (12). The covariances are obtained from (14). It is assumed that the stochastic field {!E(x)} is independent between different structural elements.

A Serviceability Limit State (SLS) function is defined by

91 = Vmax - V4,hor (24)

where Vmax = 0.05 m is the maximum allowable displacement and v4,hor is the horisontal displacement of node 4. The minimum acceptable reliability index is f3iin = 3.0.

Ultimate Limit State (ULS) functions are defined by

g;=O'F-IO'il i=2,43 (25)

where O'F is the yield stress which is assumed to be Log-Normal distributed with expe­cted value = 360 MPa and standard deviation = 36 MPa. O'j is the maximum stress in the considered element. The minimum acceptable reliability index is f3inin = 4. 7.

8

The objective function is the volume of the structure. The optimization problem is sol­ved using the modules described above. The result is b1 = 0.600 m, b2 = 0.0298 m, b3 = 0.600 m, b4 = 0.0229 m, b5 = 0.600 m, b6 = 0.0210 m, h = 0.600 m, bs = 0.600 m, bg = 0.0251 m with volume 4.66 m 3 . The active constraints are ULS in elements 4, 5, 6 and 13.

p 8 12 16 4

@ @ @ 3 ® ® @

7 11 15 3

@ @ @ 2 ® ® @ --

6 10 14 2

@ 8 @

i) 0 0 @ ( E 0

"<:!" I

1 5 9 13 ~ 777 77 77;

4 .0m 4 .0m 4.0m -!'- -----~!'------ -,/'----------,!'

Figure 2.

7. CONCLUSIONS

It is shown how stochastic fields modelling the modulus of elasticity can be discretizcd in stochastic variables and how the reliability of a structural system can be estima­ted. Reliability-based optimization problems are formulated and sensitivity analyses are performed. A computer implementation is described where standard programmes for optimization, reliability analysis and structural analysis are coupled with special routines for stochastic finite elements and sensitivity analysis . An illustrative example IS given.

8. ACKNOWLEDGEMENTS

Part of this paper is supported by the Danish Technical Research Council within the research programme on 'Safety and Reliability'.

9. REFERENCES Brenner, C.E. (1991). Stochastic Finite Element Methods. Report 35-91, Institute of

Engineering Mechanics, University of Innsbruck. Deodatis, G. (1991). Weighted Integral Method. I: Stochastic Stiffness Matrix. ASCE,

9

Journal Engineering Mechanics, Vol. 117, No. 8, pp. 1851-1877. Der Kiureghian, A. & J.-B. Ke (1988). The Stochastic Finite Element Method in

Structural Reliability. Probabilistic Engineering Mechanics, Vol. 3, No. 2. Enevoldsen, I. & J.D. S0rensen (1994). Reliability-Based Optimization in Structural

Engineering. Structural Safety, Vol. 15, pp. 169-196. Kroon, LB., M.H. Faber & J.D. S0rensen (1995). Reliability-Based Experiment Plan­

ning with Linear Regression Models. Proc. OMAE-95, Copenhagen, Denmark. Madsen, H.O., S. Krenk & N.C. Lind (1986). Methods of Structural Safety. Prentice­

Hall. Schittkowski, K. (1986). NLPQL: A FORTRAN Subroutine Solving Constrained Non­

Linear Programming Problems. Annals of Operations Research. S0rensen, J.D. (1991). PRADSS: Program for Reliability Analysis and Design of Struc­

tural Systems. Aalborg University. S0rensen, J.D., P. Thoft-Christensen, A. Siemaszko, J.M.B. Cardoso & J.L.T. Santos,

(1994). Interactive Reliability-Based Optimal Design. Proc. 6th IFIP WG7.5, Assisi, Italy, Chapman & Hall.

S0rensen, J.D., M.H. Faber & LB. Kroon (1995) . Optimal Reliability-Based Planning of Experiments for POD Curves. Proc. ISOPE-95, The Hague, The Netherlands.

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J

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