Limit and Shakedown Analysis under Uncertainty

Transcription

1 Limit and Shakedown Analysis under Uncertainty Manfred Staat 1 ABSTRACT Structural reliability analysis is based on the concept of a limit state function separating failure from safe states of a structure. Upper and lower bound theorems of limit and shakedown analysis are used for a direct definition of the limit state function for failure by plastic collapse or by inadaptation. Shakedown describes an asymptotic and therefore time invariant structural behavior under time variant loading. The limit state function and its gradient are obtained from a mathematical optimization problem. The method is implemented into a general purpose FEM code. Combined with FORM/SORM robust and precise analyses can be performed for structures with high reliability. This approach is particularly effective because the sensitivities which are needed by FORM/SORM are derived from the solution of the deterministic problem. Key words: Shakedown, Direct plasticity, FEM, Mathematical programming, Structural reliability, FORM/SORM 1. INTRODUCTION σ low cycle fatigue σ ε pure elastic behaviour ε σ elastic shakedown ratchetting ε 1 σ collapse ε σ mech. σ 0 Figure 1: The interaction diagram of the pressurized thin walled tube under thermal loading is called Bree-Diagram [1], [7] Designing and assessing of structures the engineer has to make decisions under uncertainty of the actual load carrying capacity of the structure. Uncertainty may originate from limited information about idealizations in the structural model and from random fluctuations of significant physical properties. The mechanical deterministic model and stochastic model depend on the definition of limit states. For instance, if the limit state of the structure is defined with respect to plastic collapse, then Young s modulus, hardening modulus and residual stress need not be modeled as random variables, because they all do not influence the limit load. Conversely, elastic buckling is governed by Young s modulus, residual stress, and geometry imperfections. Structural reliability analysis deals with all these uncertainties in a rational way. Safety assessment of structures requires deterministic mechanical models and analysis procedures that are capable of modeling limit states accurately. Additionally a full stochastic model of the random variables and numerical methods are also necessary for a reliability assessment. The probabilistic procedures depend on the underlying deterministic approach. Damage accumulation in low cycle fatigue (LCF) or plastic strain accumulation in ratcheting are evolution problems. In principle the possible structural responses, which are presented as icons in the Bree-Diagram, see Figure 1, may be reproduced in a detailed incremental plastic analysis. However, this assumes that the details of the load history (including any residual stress) and of the constitutive equations are known. Often it is economically not justified or even impossible to obtain all these necessary information. Therefore, it is an advantage of limit and shakedown analysis that it is a direct plasticity method which only considers the limiting state. The upper bound method calculates the least Khoa học & Ứng dụng 45

2 failure load and the lower bound method gives the largest safe load. Both solutions converge to the same exact load bearing capacity of the discretized structure (which may not be precisely the limit for the continuum problem). In more detail limit and shakedown analysis are optimization problems with an objective function to be minimized or maximized under some constraints. We present the solution of the deterministic shakedown problem in a form which has been developed in [2], [3], [4]. This kind of primal dual shakedown analysis has been extended to bounded linearly hardening plastic material in [5], [6] and to shell problems in [7], [8]. Alternative numerical methods are Sequential Quadratic Programming (SQP) with basis reduction for the lower bound problem [9], [10], interior point methods [36], Second Order Cone Programming (SOCP) [11], [12] or the Linear Matching Method (LMM) [13]. Shakedown analysis is an exact method of classical plasticity. However, geometrical nonlinearities [14], bounds on inelastic deformations, and temperature dependent Young s modulus pose some open problems [15]. The reliability of most structures under time variant loads such as cyclic loading will decrease or in other words the failure probability will increase. One could model the loading as a stochastic process and calculate the probability of this process to outcross the safe load domain. Shakedown analysis renders the reliability problem timeinvariant. This not only simplifies reliability analysis greatly but also cuts costs of the analysis because only the convex hull of the load domain and only little key information on material behavior are needed. Pioneering work on probabilistic limit and shakedown analysis has been collected in [16]. Follow-up work was firstly restricted to stochastic limit analysis of frames and linear programming [17], [18], [19], [20]. We have used shakedown analysis to calculate the so-called limit state function and reliability methods to compute failure probabilities by calling the limit state function in an inner loop. First Order Methods (FORM) and Second Order Methods (SORM) are wellestablished analytical methods which approximate the limit state function at the so-called design point (point most likely failure) and calculate exact failure probabilities for this approximation. FORM is also used to define partial safety factors in the Eurocodes for a reliability management with standard procedures using safety factors [37]. For several reasons they appear best suited for shakedown analysis: FORM is exact for linear limit state functions and for shakedown analysis the limit state function is linear or only weakly nonlinear, FORM/SORM solve optimization problems like shakedown analysis does and are very effective for high reliability and few uncertain variables. The costly sensitivities of a nonlinear FEM problem need not be provided because all needed sensitivities are already available during the solution of the shakedown problem. Therefore there is little argument to use modern stochastic simulation methods instead of FORM/SORM here except maybe of the potential occurrence of multiple design points. Our probabilistic approach was first established for the lower bound problem in [1], [9], [21] and then for the upper bound problem for shells [7], [22], [23]. Other contributions are [24], [25]. Stochastic programming is an alternative approach to limit and shakedown analysis under uncertainty. Under uncertainty the shakedown problem can be stated with random objective function or with random constraints. In the second approach, called chance constraint programming, a probability is set with which the constraint has to be satisfied. This has been considered for limit analysis of frames in [26] and for any structure in [1]. For large FEM problems the solution of the problem seems to be only feasible if all stochastic variables are normally distributed and a deterministic equivalent problem can be stated. The limit load corresponding to a defined probability is obtained. Engineering structures are no typical mass products and thus the stochastic approach to uncertainty may be questionable. One interesting alternative are fuzzy models which are similar to human reasoning but may be even more questionable for engineering structures. However, fuzzy programming has been proposed for limit analysis in [27], [28]. 2. SHAKEDOWN FORMULATION 2.1. Upper Bound Problem Inelastic structures under variable repeated or cyclic loading may work in four different regimes, which are presented in the Bree-diagram (figure 1) together with the evolution of the structural response: elastic, shakedown (adaptation), inadaptation (non-shakedown), and limit state (plastic collapse). Since for the elastic regime there are no plastic effects at all, whereas for the adaptation regime the plastic effects are restricted to the initial loading cycles and then they are followed by asymptotically elastic behavior, both regimes are considered as safe working ones and they constitute a foundation for the structural design. We do not consider elastic failure such as buckling or high cycle fatigue. The inadaptation phenomena such as low cycle fatigue and or ratchetting should be avoided since they lead to a rapid structural failure. At the limit load the structure loses instantaneously its load bearing capacity. Limit and shakedown analyses deal directly with the calculation of load capacity or the maximum load intensities that the structure is able to support. Consider a convex polyhedral load domain and a special loading path consisting of all load vertices ( ) of. At each load vertex the kinematical condition may not be satisfied, however the accumulated generalized strains over a load cycle in the time interval ( ) must be kinematically compatible. Let the fictitious elastic stress be denotewd by. In upper bound shakedown analysis we calculate the exterior approximation of the shakedown load domain α, i.e. we determine the minimum load factor for failure by non-shakedown. It converges to the same load factor as the interior approximation of the lower bound analysis which seeks the maximum load factor for survival by shakedown:. 46 Khoa học & Ứng dụng

3 According to Koiter s theorem, shakedown cannot occur if a kinematically admissible velocity field can be found so that the exterior work is larger than the internal plastic dissipation [38]. In a weaker form the shakedown conditions are considered over a cycle and the shakedown limit is the smaller one of the low cycle fatigue limit, and the ratcheting limit may be found by the following minimization: Then numerical form of eq. (1) is (3) (4) where is the weight at Gaussian point, (the superscript p is omitted for simplicity) is the strain vector corresponding to load vertex, at point where the total plastic energy dissipation over a cycle in a perfectly plastic structure is (1) is the fictitious elastic stress vector corresponding to load vertex k, at point i, u is the nodal displacement vector and is deformation matrix. D and are square matrices, (5) (2) The upper bound of the shakedown multiplier is the solution of constraint nonlinear programming problem (1). For the numerical solution we regularize the nondifferentiable dissipation function with a very small value, where, such that. Constraint (1b) is the definition of plastic strain accumulation. The plastic strain rate may not necessarily be compatible, but must be compatible. This is expressed by constraints (1d) and (1e). Constraint (1c) is the incompressibility condition and (1g) is the normalized work of the external actions Problem Discretization The whole structure is discretized into finite elements with Gaussian points, where is number of Gaussian points in each element. If the load domain is convex, it is sufficient to check if shakedown will happen at all vertices of. So the load domain can be discretized into a finite number of load combinations,, and is the total number of vertices of. By these discretizations, the shakedown analysis is reduced to checking shakedown conditions at all Gaussian points and all load vertices, instead of checking for whole structure and all load histories in the domain. For the later probabilistic analysis we write the yield stress as product of a reference value and a stochastic stress variable, For the sake of simplicity, we define some new plastic strain, fictitious elastic stress, deformation matrix, regularization parameter respectively as Then eq. (4) becomes Dealing with the nonlinear constrained optimization problem above, let us write the penalty function for the compatibility condition (8b) and the incompressibility condition (8c) (6) (7) (8) Khoa học & Ứng dụng 47

4 where is a penalty parameter such that. For the sake of simplicity, let be constant. Following (8) the modified kinematic formulation (8) becomes The corresponding Lagrange function of (9) is (9) (10) (11) The so-called Karush-Kuhn-Tucker (KKT) conditions and the strict complementary slackness must hold at the optimal point in the form equilibrium condition and is a time invariant residual stress. The structural shakedown takes place due to development of permanent residual stresses which, imposed on the actual stresses, shift them towards purely elastic behaviour. Residual stresses are a result of kinematically inadmissible plastic strains introduced to the structure by overloads. They clear out effects of all preceding smaller loads. They also avoid any plastic effects in the future provided that the loads are smaller than the initial overload. Therefore, in limit and shakedown analyses the knowledge of the exact load history is not necessary. Only the load (limits) count and the envelopes should be taken into consideration which makes the reliability problem time-invariant. By duality (14) both optimization problems yield the exact solution because the maximum safe load and the minimum overload are the same. 3. STRUCTURAL RELIABILITY ANALYSIS 3.1. Limit State Function (12) By applying Newton s method to solve the system (12) we obtain the Newton directions and, which assure that suitable steps along them will lead to a decrease of the objective function in (8). If the relative improvement between two steps is smaller than a given constant, the algorithm stops and the shakedown limit factor is obtained. Details of the iterative algorithm can be found in [7]. If = 1 the problem reduces to limit analysis and denotes the limit load factor. Otherwise an upper bound of the shakedown factor is obtained Lower Bound Problem There is the dual lower bound problem which can be derived from Melan s lower bound theorem [39], [40]. The discretized problem is The behavior of a structure is influenced by various typically uncertain parameters (loading type, loading magnitude, dimensions, or material data,...). All parameters are described by random variables which are collected in the n-dimensional random vector of basic variables. We will restrict us to those basic variables for which the joint density exists and the joint distribution function is given by (15) The deterministic safety margin R - S is based on the comparison of a structural resistance (threshold) R and loading S. With R, S functions of the structure fails for any realization with non positive limit state function, i.e. (16) Here the fictitious elastic stress (13) is calculated from the Different definitions of limit state functions for various failure modes are suggested in Table 1. The limit state function defines the limit state hypersurface which separates the failure region from the safe region. Figure 2 shows the densities of two random variables R, S which are generally unknown or difficult to establish. The failure probability is the probability that is non-positive. This means 48 Khoa học & Ứng dụng

5 (FORM/SORM) Figure 2: Basic R - S problem in, presentation on one axis, (17) In general it is not possible to calculate analytically from the failure integral, because of the complex structure of the failure region V. Additionally, it is not necessary that the limit state function is given explicitly but only in algorithmic form. A FE-analysis of structures with one initial data set gives only one value of the limit state function. For the probabilistic shakedown problem the loads,, and the yield limit are considered as random variables. Let us restrict ourselves to the case of homogeneous material, where the yield limit is the same at every point of the structure. Then we always can write where is a constant reference value and Y is a random variable First- and Second-Order Reliability Methods For the general cases, there are analytical methods like FORM/SORM and several simulation methods to compute the failure probability. Direct Monte Carlo Simulation (MCS) becomes increasingly expensive with the increase of the structural reliability. Acceptable failure probabilities might be in the range of to. They are even much lower in nuclear reactor technology. For a validation that the failure probability is less than an accepted limit, the sample size required for direct MCS must be at least leading to a minimum sample size in the range of to. Such a large number exceeds particularly for complex FE models, available resources by far. The numerical effort can be reduced considerably by Response Surface Methods (RSM) and by Importance Sampling, Directional Sampling [41], Latin Hypercube Sampling or a combination of these variance reduction methods. An obvious strategy is to concentrate the simulation around the design point so that about 50% of the simulations fall into the failure region. However, the most effective analysis is based on First- and Second-Order Reliability Methods (FORM/SORM) if gradient information is available. First and Second Order Reliability Methods (FORM/SORM) are analytical probability integration methods. Therefore, the defined problem has to fulfill the necessary analytical requirements (e.g. FORM/SORM apply to problems, where the set of basic variables is continuous). The numerical effort depends on the number of stochastic variables but not on (contrary to MCS). The failure probability is computed in three steps. Transformation of the vector of basic random variables into a vector of uncorrelated basic random variables, Linear or quadratic approximations region in the U-space, of the failure Table 1: Different limit state functions in structural reliability Khoa học & Ứng dụng 49

6 Computation of the failure probability due to the approximations. FORM/SORM is an exact solution of the approximated stochastic problem. Transformation The basic variables are transformed into standard normally distributed variables ( : normal distribution with zero means,, unit variance,, and independent). Such a transformation is always possible for continuous random variables. If the variables are mutually independent, with distribution functions, each variable can be transformed separately by the Gaussian normal distribution into (definition of in eq. (20) below). For dependent random variables other isoprobabilistic transformations (generalized Nataf [29] or Rosenblatt [30]) can be used. The function is the corresponding limit state function in U-space. The dimension of the U-space depends on the dependencies of the random variables and is not necessarily equal to the dimension of the X-space. However, the transformation to U-space is exact and not an approximation [31]. Approximation In FORM, with is approximated by its Taylor expansion at the socalled design point (so that ) Figure 3: Safe and failure regions with their linear and quadratic approximations in U-space. For a linear limit state function FORM gives the exact failure probability. If the limit state function is not linear in U-space a quadratic approximation of the failure region V gives closer predictions of. These second order methods (SORM) may be either based on a nonlinear optimization algorithm or on correction of a FORM analysis. FORM/SORM give the exact solution of an approximate problem. A quadratic approximation generated in SORM [7] and of the failure region V is The failure region V is linearly approximated by (18) (19) (21) where H ist the scaled Hessian matrix of the limit state function and The vector is proportional to the sensitivities The failure event is equivalent to the event, such that an approximation of the failure probability is (20) because the random variable is normally distributed. The failure probability depends only on, such that it is called safety (or reliability) index. If it is possible to derive analytically from the input data, the probability is calculated directly from the cumulative distribution function of the standard normal distribution (Gaussian distribution function). (22) with are n - 1 principle curvatures at the design point. The calculation of normally needs the second derivatives of the limit state function Calculation of the Design Point In order to apply FORM/SORM, the design point must be identified. This leads to a nonlinear, inequality constrained optimization problem as follows 50 Khoa học & Ứng dụng

7 (23) Applying SQP for solving the optimization problem (22), the quadratic programming sub-problem (25) at iteration k simplifies with and : where a coefficient 1/2 is added for technical reasons. Many algorithms have been suggested to deal with this problem. In and [1], [9], [19], [21] good results have been obtained with Rackwitz s simple gradient search algorithm, which is based on a linearization of the limit state function at each step. However, this algorithm is only guaranteed to converge towards a locally most likely failure point in each sequence of points on the failure surface if the safe region is quasi-convex or concave. The KKT conditions for this problem are (27) (28) A more general algorithm is the Sequential Quadratic Programming (SQP), [32], [33]. The SQP method, also known as successive or recursive quadratic programming, employs Newton s method (or quasi-newton methods) to solve the KKT conditions for the original problem directly. As a result, the accompanying subproblem turns out to be the minimization of a quadratic approximation to the Lagrangian function optimized over a linear approximation of the constraints. Consider the inequality constrained nonlinear optimization problem (24) where x is an n-dimensional parameter vector containing the design variables and j the set of active constraints. A basic SQP algorithm optimizes the Lagrangian function over a linear approximation of the constraints, where is a matrix of Lagrange multpliers. In each iterate an appropriate search direction is defined as the solution to: minimize linearized contraints. under the Then a quadratic programming subproblem of the form (25) is obtained and must be solved in each iteration. Let be the optimal solution, the corresponding multiplier of this subproblem, then the new iterate is obtained by (26) Applying Newton s method for solving these equations leads to the system with the solutions (29) (30) as the Lagrange multiplier and search direction for the next iteration. If then together with the Lagrange multiplier yields the optimal solution for the problem (23), i.e. the design point is actually found. The calculation of necessary derivatives is considered as the sensitivity analysis and will be discussed in the next section. There is only the potential occurrence of multiple design points which need extra consideration in FORM/SORM [7], [18], [19]. 4. SENSITIVITY OF THE LIMIT STATE FUNCTION The reliability analysis described above can be carried out now with the help of a probabilistic limit and shakedown analysis. From the results of the finite element analysis, the necessary derivatives of the limit state function based shakedown analysis can be determined analytically. This represents a considerable reduction of computing time comparing with the other methods, e.g. the difference approximation, and makes such an efficient and cost-saving calculation of the reliability of the structure. Contrary to the numerical calculation, the analytical calculation is faster and more exact. The necessary data for the calculation of the derivatives are available after the execution of the deterministic shakedown analysis since they are based on the upper bound load factor. The derivatives must be calculated at each iteration in the physical x space and then transformed into the standard Gaussian u space by where is a suitable step length parameter. Khoa học & Ứng dụng 51

8 4.1. Definition of the Limit State Function (31) where is the stress vector at Gauss point i and load vertex k due to the load case. The derivatives of the limit load factor versus yield stress variable Y can be determined in the same way and have the form As mentioned above, the limit state function contains the parameters of structural resistance and loading. If we defined the limit load factor as follows (32) where, are limit load and actual load of the structure. For the sake of simplicity, the limit state function can be normalized with the actual load and then becomes (33) The load factor can be calculated by the nonlinear program (8). It can be seen that the limit state function is the function of yield stress variable and load variables. The actual load is defined in n components by using the concept of a constant reference load as follows 4.3. Second Derivatives of the Limit State Function (38) In the SORM algorithm, as discussed above, the Hessian matrix which summarizes second partial derivatives of the limit state function is needed. They can be obtained from a direct analytical derivation of the first derivatives. By taking derivatives of (38) versus, Y and referring to (37) one has, (39) (34) where is the realization of the basic load variable ( ). The corresponding actual stress field can also be described in the same way (35) From (6) and (35), the corresponding normalized stress fields are obtained 4.2. First Derivatives of the Limit State Function (36) In order to get the sensitivities of the limit state function, one requires that i.e. the partial derivatives of the limit load factor must be available. Let be the solutions of the optimization problem (8). At this optimal point, the first derivative of the limit load factor versus load variable can be calculated as follows [7] (40) For we refer to Tran et al. It is obvious from (40) that for the case of homogeneous material the shakedown load factor is actually a linear function of the yield stress, i.e. the optimization variables, u are independent of Y. Moreover, the Lagrange multiplier is a linear function of Y. A flow chart of the algorithm is presented in Figure 4 which also shows the derivatives which are needed because of the transformation in the u space. The sensitivity analysis and algorithm for the lower bound method are presented in [1], [9] Special Case of Probabilistic Shakedown Analysis Consider a special case of probabilistic shakedown analysis, where is a stochastic one-parameter load, i.e. with a reference load we have and where is the stress vector due to referent load. In this case, the nonlinear optimization (10) becomes (41) (37) where NV = 2 for shakedown analysis and NV = 1 for limit analysis. If we now define a new strain vector 52 Khoa học & Ứng dụng

9 (46) These above derivatives can also be easily obtained by using the same procedure as applied for the material variable Y. By this way, it is easy to prove that,, and then we find again the above expressions finally. 5. NUMERICAL APPLICATIONS The probabilistic limit and shakedown algorithm described above has been programmed and implemented in the finite element package Code_Aster in [7]. The 4-noded quadrangular isoparametric flat shell element, which is based on Kirchhoff s hypothesis, was applied. Higher order shell elements were not available in former versions of Code_Aster, [34]. In all numerical examples, the structures are made of elastic-perfectly plastic material. For each test case, some existing analytical and numerical solutions found in literature are briefly represented and compared. The finite element discretisations were realized by the personal pre- and post-processor GiD, [35] Square Plate with a Central Hole Figure 4: Flowchart of the probabilistic limit and shakedown analysis, [7] and new displacement vector, then (41) can be rewritten as follows A square plate with central circular hole as shown in Figure 5 is considered. This is a standard test for deterministic shakedown analysis. Under uncertainty it was also studied analytically and numerically in [1], [9], [21] using lower bound limit and shakedown analysis, FORM with Rackwitz s algorithm and volume finite elements. The two following cases are examined. (42) with (43) Since the constraint is now no longer dependent on the load variable X, it follows that (44) (45) Khoa học & Ứng dụng 53

10 ( 1 RL) µ r µ s 0.8µ r µ s ( ) RL σ σ 0.64σ 2 2 r + σs β = = + (51) r s Figure 5: FE-mesh and geometrical dimensions of square plate, [7] Limit load analysis In case of a square plate of length L with a hole of diameter 2R (see Figure 5) and R/L = 0.2 subjected to the uniaxial tension the exact limit load for plane stress is given by with the yield stress. Thus the limit load depends linearly of the realization of the yield stress basic variable X. The load S is a homogeneous uniaxial tension on one side of the plate. The magnitude of the tension is the second basic variable Y. The limit load of every realization of Y is The limit state function is defined by (47) (48) The normally distributed random variables X and Y with means and variances respectively, yield with and the transformed limit state function In Table 2 the probability of a plastic collapse from the numerical limit analyses are compared with the analytic values resulting from the exact solution for the case that both variables are normally distributed with standard deviations and. The different accuracies of the deterministic limit analyses seem to cause the small differences between the FORM solutions. The numerical error involves both deterministic shakedown analysis and reliability analysis and is very small and acceptable. for because the density of the normal distribution is symmetric. For both normal and log-normal distributions SORM solutions are almost the same of FORM due to weak linearity of the limit state function. The weak nonlinearity may occur in the case of multi-parameter loads or by extreme value distribution of the random variables. But to our experience the improvement by SORM is rarely needed. Shakedown load analysis For this case, the tension varies within the range and only the amplitudes but not the uncertain complete load history influences the solution. Consider the case where the maximal magnitude is a random variable and the minimal magnitude is held zero. From the deterministic analysis we got the shakedown load. The numerical probability of failure for normally distributed variables are presented in Table 2, compared with the analytical solutions, which are calculated from (52) It is worth to note that the shakedown probabilities of failure are considerably larger than those of limit analysis. Thus, the loading conditions should be considered carefully when assessing the load carrying capacity of the structure Pipe-Junction under Internal Pressure With realizations U it may be written (49) of the new random variable (50) A pipe-junction is considered under internal pressure which can be constant or vary within the range. The FE mesh and the geometrical data are introduced in Figure 6. Both yield limit and pressure are considered as random variables. Numerical deterministic analyses lead to a collapse pressure and a shakedown limit. If both material resistance and load (stress) random variables are supposed to be normally distributed with means, and standard deviations, respectively, then the reliability index may be given [8] such that the safety index (with R/L = 0.2) is 54 Khoa học & Ứng dụng

11 [7] [7] [1], [9] [7] [7] Table 2: Numerical results and comparison for normal distributions ( ) d=15 t=3.44 s=3.44 (53) Numerical failure probabilities for limit analysis (probabilities of plastic collapse) for normal distributions are presented in Table 3 compared with the corresponding analytical solutions. Both random variables have standard deviations. It is shown that the numerical results of FORM and SORM are very close to the exact ones. Besides normal and log-normal distributions, the Weibull distribution is frequently used for modeling loads and resistance. Table 3 also shows the numerical results of FORM/SORM for the case that both random variables have a Weibull distribution. The Weibull distributions lead to much larger failure probabilities. No analytical results are known. The limit state function is weakly nonlinear after the transformation into the u space. In this case SORM gives slightly improved results. Table 4 shows the failure probabilities for shakedown analysis which are considerably larger than those of limit analysis. Comparing the two distributions similar observations as for limit analysis are made. Figure 6: FE-mesh and geometrical dimensions of pipe-junction, [7] 6. CONCLUSIONS A probabilistic model of finite element limit and shakedown analysis has been presented, in which the loading and strength of the material are considered as random variables. The procedure involves a deterministic limit and shakedown analysis for each probabilistic iteration, which is based on the kinematical approach. Different kinds of distribution of basic variables can be considered for calculation of the failure probability of the structure with FORM/SORM. A nonlinear optimization was implemented, which is based on the Sequential Quadratic Programming for finding the design point. Nonlinear sensitivity analyses are also performed for computing the Jacobian and the Hessian of the limit state function. D=39 Khoa học & Ứng dụng 55

12 Table 3: Numerical results for limit analysis, [7] Table 4: Numerical results for shakedown analysis, [7] The advantage of the method is that sensitivity analyses are obtained directly from a mathematical optimization with no extra computational cost. Two examples were tested against literature with analytical method and with a numerical method using volume elements. The proposed method appears to be capable of identifying good estimates of the failure probability, even in the case of very small probabilities. The limit state function is linear with the limit and shakedown analysis. Then the use of FORM is sufficient and SORM is only necessary if the linearity is lost after the transformation into the U-space. 1 Faculty of Medical Engineering and Technomathematics, Aachen University of Applied Sciences Heinrich-Mußmann-Str. 1, Jülich, Germany m.staat@fh-aachen.de, 56 Khoa học & Ứng dụng

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