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Optimal and robust design of docking blocks with uncertainty
Engineering Structures 26 (2004) 499–510 www.elsevier.com/locate/engstruct
Optimal and robust design of docking blocks with uncertainty Y.S. Cheng a,, F.T.K. Au b, L.G. Tham b, G.W. Zeng a a
Department of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China b Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Received 28 October 2002; received in revised form 24 November 2003; accepted 26 November 2003
Abstract The positioning and stiffness allocation of docking blocks are an important decision when docking a ship. In this paper, the issues of both optimal design and robust design of docking blocks with uncertainty are discussed, and the corresponding mathematical models are proposed. To describe the uncertainties of various parameters, the proposed method employs the convex model in which the indeterminacy about the uncertain variables is expressed in terms of some sets to which these uncertain variables belong. In the optimisation of docking blocks, the factors considered include the uncertain ship girder loads and uncertain equivalent stiffnesses of blocks. Numerical examples show that uncertainties do affect the optimal solution and lead to an increase in volume of blocks compared to that from deterministic optimisation. A method of robust design with respect to the uncertain equivalent stiffnesses of blocks is also proposed. The robustness of the objective function is achieved by minimising the maximum value of unsatisfactory degree functions of the uncertainty parameters while the feasibility robustness is ensured by an optimisation conducting the worst-case analysis at a lower level. Numerical simulations show that the optimal result from minimisation of the maximum value of unsatisfactory degree functions is superior to that from direct maximisation of the minimum value of uncertainty parameters without the use of unsatisfactory degree functions. # 2003 Elsevier Ltd. All rights reserved. Keywords: Convex model; Docking blocks; Optimal design; Robust design; Uncertainty
1. Introduction A dry dock normally consists of a basin dug into the shore of a body of water and provided with a watertight gate on the waterside, used for major repairs and overhaul of vessels. When a ship is to be docked, the dry dock is flooded, and the gate opened. After the vessel is brought in, positioned properly and guyed, the gate is closed and the dock is pumped dry, bringing the craft gradually to rest on supporting keel and bilge blocks anchored to the floor. The positioning and stiffness allocation of these docking blocks are an important decision when docking a ship because mispositioning or mis-allocation of docking blocks may give rise to unreasonably large block reactions and consequently serious damage to both the docked ship and blocks. Docking block failure may also cause the disruption of docking schedules and extension of the Corresponding author. Tel.: +86-27-875-43758x801; fax: +86-27875-42146. E-mail address: [email protected] (Y.S. Cheng).
0141-0296/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2003.11.007
ship downtime. Any failure may even lead to the loss of lives. Docking analysis has therefore attracted attention of various researchers. Jiang et al. [1] developed a reliable, efficient computer program for predicting block reactions in both graving and floating docking analyses. Cheng and Zeng [2] proposed a mathematical model for optimising the positioning and allocation of docking blocks ignoring potential uncertainties in the design of docking blocks. Two-level optimisation techniques were employed to solve for the optimal solution in their study. However, uncertainties in material, geometric properties, loads, etc. are unavoidable in the design of engineering structures. Various types of wood are used in the construction of docking blocks. The elastic modulus of various types of wood vary greatly depending on its age, moisture content, and the effects of previous loading on its proportional limit. All these factors lead to uncertain block stiffness and hence the uncertain equivalent block stiffness, which encompasses the stiffness of a docking block and that of its supporting structure. Ship girder loads are also difficult to
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determine accurately. The ship girder loads actually vary from their nominal values. Therefore, the ship girder loads and docking block stiffness may be treated as uncertain variables in the design of docking blocks. These uncertainties are traditionally dealt with by using probabilistic models, and the reliability of a structure is measured with the probability of no failure. However, probabilistic analysis requires sufficient data on probabilistic parameters with which the probability density functions of random variables can be determined. In many engineering problems, although one may not be able to estimate accurately the probability distributions of random variables due to various reasons such as insufficient information, one can at least evaluate their bounds. The convex models [3–5] have been developed to handle such uncertainties in design as attractive alternatives to traditional probability models. In the studies of such non-probabilistic modelling of uncertainties, the indeterminacy about the uncertain variables may be expressed in terms of some sets to which these uncertain variables belong. These convex models may include local energy-bound convex model, integral energy-bound convex model, envelope-bound convex model, Fourier-envelope convex model, responsespectrum-envelope convex model, etc. [6]. Recently, the optimisation of structures with uncertainty has been studied by using anti-optimisation approaches or convex models. Elishakoff et al. [7] proposed a design approach for structural optimisation with uncertain but bounded loads. The optimisation problem was formulated as a two-level optimisation. At the first level, the minimisation of the structural weight under stress and displacement constraints is carried out, while finding the maximum structural responses under bounded loads is conducted at the second level. These two optimisation processes are nested in their formulation. Later, Pantelides and Booth [8] built on the formulation proposed in Ref. [7] but approached the problem with a multidimensional ellipsoidal model rather than a box. They applied the method to the optimisation of a two-span continuous reinforced concrete beam and a steel 10-bar truss with uncertain loads. Solving a nested optimisation problem is time consuming. Lombardi [9], and Lombardi and Haftka [10] therefore proposed a method to alternate between the first level optimisation and the second level one. The computational burden is greatly alleviated because these two optimisation processes are executed separately. Recently, Ganzerli and Pantelides [11] developed a convex model superposition method to predict the response of structures with uncertain loads. In this way, the second optimisation process is avoided and the computational efficiency is considerably improved. As an important design methodology of products, robust design is becoming more and more popular. Robust design aims to achieve a state of robustness so
that the performance of a design is least sensitive to the variability of uncertain variables. The fundamental principle of the robust design is to improve the quality of a product by minimising the effect of the causes of variation without eliminating the causes, as pointed out by Phadke [12]. This can be achieved by selecting suitable design variables to make the product performance insensitive to the various causes of variation. Many methods such as those by Rao et al. [13], Belegundu and Zhang [14], Parkinson et al. [15], Emch and Parkinson [16], Chen et al. [17], Parkinson [18,19], Lee and Park [20], Du and Chen [21], and Sandgren and Cameron [22] have been proposed to achieve the goal. In the theory of robust reliability of structures recently proposed by Ben-Haim [23,24], the reliability of a structure can be measured by the size of the convex model that is consistent with no failure. In other words, the reliability of a structure can be quantified as the degree of immunity to uncertainty. A reliable design can perform satisfactorily in the presence of great uncertainty and is relatively immune to unexpected variations. Such a design is robust with respect to uncertainty and therefore such reliability is named as robust reliability. Robust reliability actually expresses the intuitive meaning of reliability: dependability in the face of uncertainty [25]. An optimal design based on the theory of robust reliability may be viewed as a kind of robust design. This paper first presents a mathematical model for minimisation of the total volume of docking blocks with uncertainty. The emphasis is placed on investigating the effects of uncertain ship girder loads and uncertain equivalent stiffnesses of docking blocks on the optimal solution. The paper then proposes a robust design method for the docking blocks with uncertainty based on improved theory of robust reliability in which the unsatisfactory degrees of the uncertainty parameters, instead of the uncertainty parameters themselves, are used to measure the robust reliability. The robustness of the objective function is guaranteed by minimisation of the maximum unsatisfactory degree of the uncertainty parameters while the feasibility robustness is handled through the worst-case analysis with an optimisation process.
2. Theory and formulation 2.1. Conventional optimisation of docking blocks Consider a typical docking arrangement in which a ship is docked on N docking blocks resting on a supporting structure. Each docking block consists of soft wooden blocks with a total height of H1 and hard wooden blocks with a total height of H2 as shown in Fig. 1. The symbols B and C denote the width and thickness
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Fig. 2. Mechanical model for analysing block reactions and ship girder displacements.
Fig. 1.
A docking block.
of the block, respectively, while BL denotes the width of flat keel. If the weight of the ship girder, ship structures and docking block locations are known, a typical optimisation of docking blocks is to choose block dimensions properly to minimise the total volume of docking blocks and satisfy various constraints simultaneously. The formulation of such optimisation ignoring the uncertainties may be written as follows:
minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1
s:t: Gj ðXÞ 0;
ð1Þ
j ¼ 1;2; . . . ;Ng
where X is a vector whose components are the design variables, i.e. H1i, H2i, Bi, Ci, (i¼ 1, 2, . . .,N); V is the total volume of blocks; Gj is the jth constraint function to be satisfied and Ng is the total number of constraints. The constraints may include various limits of stress, stability and displacement that the governing regulation sets. The details of various constraints can be found in Ref. [2] for example. In the study, the ship is treated as a beam with nonuniform cross section, which is in accordance with the normal practice of naval architects. Each of the docking blocks is represented by a spring. If side blocks exist, the side blocks and the keel block can be considered to be elastic springs in parallel and represented by a combined stiffness at the ship centreline. This combined stiffness and its support structure can be viewed as two springs in series. The final equivalent stiffness can then be readily obtained and it is called the equivalent stiffness of the docking block in the paper. The block reactions and the displacements of the ship girder can be obtained by analysing the model as shown in Fig. 2 using the finite element method. As the number of design variables and the number of constraints are large, a two-level solution strategy can be used to solve for the optimal solution efficiently and the details are as follows.
2.1.1. The first level optimisation: optimal allocation of stiffnesses of docking blocks The loaded area, Fi, and total height, Hi, of docking blocks are chosen as design variables. In this step, the exact stiffness of the docking block is not available because the specifications of the block have not been determined yet. The block stiffness and the chosen design variables are approximately related as K i ¼ ni E 1
Fi ; Hi
i ¼ 1;2; . . . ;N
ð2Þ
where E1 is the elastic modulus of the soft wood block. The coefficient ni is called the stiffness coordinative factor of the ith docking block to be exactly determined at the second level optimisation. At this level, constraints may include the strength of docking blocks, longitudinal strength of typical cross sections of the ship girder, the strength of keels and transverse bulkheads, stability of docking stiffeners, displacements of the ship girder and size limits, etc. The total volume of the blocks is chosen to be the objective function and it can be written as V¼
N X
Vi ;
ð3Þ
Vi ¼ gi Fi Hi
ð4Þ
i¼1
where the coefficient gi is called the volume coordinative factor of the ith docking block. Similar to the coefficient ni, gi is to be exactly determined in the second level optimisation. In this way, the total number of design variables is greatly reduced and is only about half of the total number of design variables of the original optimisation. The optimal stiffness allocation of docking blocks can now be determined and are output to the second level optimisation as equality constraints. 2.1.2. The second level optimisation: specification of docking blocks The parameters B, C, H1 and H2 are now chosen as design variables. The inequality constraint of the strength of the docking block, the equality constraint of the block stiffness being equal to the optimal stiffness obtained in the first level optimisation, and the size limits are only included at this level. The objective function is again the volume of the docking block
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under consideration. The optimisation is carried out for each of the docking blocks. Once the details of the blocks are obtained, the coordinative factors can then be exactly calculated. The updated coordinative factors are then returned to the first level optimisation to update the old ones. The two optimisation procedures are alternately performed until convergence of the coordinative factors is achieved. 2.2. Optimisation of docking blocks with uncertainty Consider a structure subjected to M uncertain parameters Pi (i ¼ 1,2,. . . ,M) that may not be conveniently described by probabilistic models. The parameters Pi may be loads, material properties, geometric i dimensions and so on. Their nominal values are P (i ¼ 1,2,. . .,M). Suppose that the deviation of each uncertain parameter from its nominal value is bounded. The indeterminacy about the uncertain variables may be expressed in terms of some sets to which these uncertain variables belong. The set of all possible realisations of uncertain parameters with the unknownbut-bounded information can be defined as 1 P1 P a1 ; CP ðaÞ ¼ P1 2 P2 P a2 ; . . . ; PM PM aM ð5Þ P P 2 M a ¼ fa1 ;a2 ; . . . ;aM g
ð6Þ
P ¼ fP1 ;P2 ; . . . ;PM g
ð7Þ
The parameter ai defines the boundary of the uncertain parameter Pi and describes the degree of uncertain variability of Pi. The parameter ai is called the uncertainty parameter corresponding to uncertain variable Pi. It is obvious that the greater the value of ai, the greater the variation of Pi. In the study, the uncertainties in ship girder loads and the equivalent stiffnesses of docking blocks are considered separately. The mathematical model for optimal design of docking blocks with uncertainty can be formulated as follows: minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1
s:t: max Gj ðX;PÞ 0; P2CP ðaÞ
ð8Þ
j ¼ 1;2; . . . ;Ng
In Eq. (8), the maximisation of Gj(X,P) over CP(a) is a process of seeking the worst value of P for each constraint. Therefore, the above mathematical model is a nested optimisation. As finding a solution to a nested optimisation problem is time-consuming, a technique developed by Lombardi [9], and Lombardi and Haftka [10] is utilised to decompose the problem described by
Eq. (8) into two optimisation processes as below: minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1 ðjÞ
s:t: Gj ðX;P Þ 0; and maxGj ðX;PÞ P i Pi P s:t: ai ; Pi
ð9Þ
j ¼ 1;2; . . . ;Ng
i ¼ 1;2; . . . ;M
ð10Þ
where P(j) in Eq. (9) is an optimal solution to Eq. (10), which gives the worst case prediction of the jth constraint for the given X. Note that only X is varied in Eq. (9), and only P is varied in Eq. (10). After the determination of optimal values of P(j) by Eq. (10), updated values of X are obtained using Eq. (9). These updated values of X are then substituted back into Eq. (10) again to get the updated P(j). Such iterations are repeated until the difference between the values of P(j) in consecutive cycles is within the prescribed tolerance. In essence, it means that the worst constraints in Eq. (9) still satisfy the requirements. Therefore, the optimal design of docking blocks with uncertainty defined by Eq. (8) can be obtained by solving Eqs. (9) and (10) alternately until convergence is achieved. 2.3. Robust design of docking blocks with uncertainty As stated previously, the uncertainty parameter ai is used to measure the reliability of a structure according to the theory of robust reliability. If failure does not occur even for a large ai, then the structure is reliable since it is robust with respect to uncertainty. On the other hand, if failure may occur even for a small ai, then the structure is vulnerable to uncertainty and therefore unreliable. Such a rationale is valid if the variation of all uncertain variables can be represented by one uncertainty parameter a and the robust reliability of the structure can then be measured with the largest a value consistent with no failure. However, direct use of the uncertainty parameter as a measure of robust reliability may create problems if multiple uncertainty parameters are required to describe bounded-but-unknown uncertain variables as used in Eq. (5). For example, suppose that there are two kinds of uncertain loads P1 and P2. The first load, P1, only fluctuates slightly and is very likely to be within the range of a1 ¼ 0:05. However, the second load, P2, fluctuates drastically and is within the range of a2 ¼ 0:20. Two design schemes are therefore possible. One can confine the variation of P1 and P2 to be within their respective ranges of a1 ¼ 0:06 and a2 ¼ 0:21 without failure. It is obvious that the design is safe and the robust reliability is 0.06 based on the theory of robust reliability. In another design scheme, the variation of
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P1 and P2 can be confined within the range of a1 ¼ 0:08 and a2 ¼ 0:14, respectively, without failure. The design has the robust reliability of 0.08 accordingly, but obviously the design is not as reliable because the real variation of P2 may be beyond the range of a2 ¼ 0:14. The properties of various uncertain variables need to be considered further in depth to obtain more reasonable and equitable results. It is very desirable to build a benchmark so that the uncertain variables with different properties and disparate units can be treated on a par with one another. To build such a benchmark, the idea of constructing the class function in physical programming [26] is adopted in the paper to define the unsatisfactory degree function for each of the uncertainty parameters. To construct the unsatisfactory degree functions, one should first define six ranges for the uncertainty parameters ai (i¼ 1,2, . . .,M) and then describe their unsatisfactory degrees in order of decreasing preference as follows:
Another feature of the unsatisfactory degree function is to make the highly undesirable range significantly worse than the undesirable range, and the undesirable range significantly worse than the tolerable range and so on. This feature reflects the fact that the value of the unsatisfactory degree function within the highly desirable range is already small and further reduction of the value of the unsatisfactory degree function is unnecessary. On the other hand, the value of the unsatisfactory degree function within the highly undesirable range is large and therefore requires significantly minimisation of the unsatisfactory degree function. The unsatisfactory degree function is schematically shown in Fig. 3 and the mathematical expression of the function for the range k (k¼ 2, 3, 4, 5) can be represented as follows: Uki ¼N1 ðnki ÞUiðk1Þ þ N2 ðnki ÞUik þ N3 ðnki ;kki ÞU0iðk1Þ þ N4 ðnki ;kki ÞU0ik 1 4 1 3 n ðn 1Þ4 2n þ 2 2 2 1 4 1 1 N2 ðnÞ ¼ n þ ðn 1Þ4 þ 2n 2 2 2 1 4 3 1 3 4 N3 ðn;kÞ ¼ k n ðn 1Þ n þ 8 8 2 8 3 1 1 1 N4 ðn;kÞ ¼ k n4 ðn 1Þ4 n þ 8 8 2 8 a a i iðk1Þ nki ¼ aik aiðk1Þ N1 ðnÞ ¼
1. Highly desirable range (ai ai1 ): an acceptable range but further increase of the uncertainty parameter, although desirable, is of minimal additional value. 2. Desirable range (ai2 ai ai1 ): an acceptable range that is desirable. 3. Tolerable range (ai3 ai ai2 ): an acceptable and tolerable range. 4. Undesirable range (ai4 ai ai3 ): a range that is acceptable, but undesirable. 5. Highly undesirable range (ai5 ai ai4 ): a range that is still acceptable, but is highly undesirable. 6. Unacceptable range (ai ai5 ): the range of values that the uncertainty parameter may not take. The parameters ai1 through ai5 are physically meaningful values that may be provided by the designer to quantify the preference with respect to the ith uncertainty parameter ai. The unsatisfactory degree function Ui possesses two primary features. One is to have the same value at each of the range boundaries regardless of the particular uncertainty parameters in question. Hence, the change in the unsatisfactory degree function will always be the same as one moves from one boundary to another within a given range. Such a feature has a normalising effect and results in favourable numerical conditioning properties. In other words, a benchmark is built for various uncertain variables by mapping the uncertain variables to a dimensionless scale via the unsatisfactory degree functions. Obviously the lower the unsatisfactory degree, the higher the robust reliability and the more robust the design. The maximum or worst value of the unsatisfactory degree functions should therefore be minimised.
503
kki ¼ aik aiðk1Þ @Ui 0 Uik ¼ @ai ai ¼aik
ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ ð18Þ
For range 1, the unsatisfactory degree function is defined by an exponential function as follows: U1i ¼ Ui1 exp ðU0i1 =ai1 Þðai ai1 Þ for ai ai1 ð19Þ
Fig. 3. A schematic unsatisfactory degree function.
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Fig. 4. A schematic diagram of a docked ship.
The values and slopes of the unsatisfactory degree function at range boundaries must be properly chosen to guarantee the above desirable properties. The evaluation algorithm proposed by Messac [26] is employed in the study. The robust design of docking blocks with uncertain equivalent stiffness is to establish one that obtains the desirable values of unsatisfactory degree functions for all uncertainty parameters and to satisfy simultaneously all constraints. The uncertainty parameters represent the variation of uncertain equivalent stiffness of docking blocks. The constraints to be satisfied may include the stress and stability of each structural component, nodal displacements, block volume and size limit of design variables, etc. The robustness of the objective function can be achieved by minimising the maximum or worst value of the unsatisfactory degree functions. It tends to move the uncertainty parameters to the highly desirable range and results in higher robust reliability with respect to uncertainty. As for the feasibility robustness, the conventional method for linear worst-case analysis is suitable for linear problems and cases of small variations in uncertain variables. However, according to Emch and Parkinson [16], even though higher order models are used in the worst-case analysis, violation of the constraints may still occur for the case in which the maximum variation does not occur at endpoints of the tolerances but rather at some values in between. Therefore, in the study, an optimisation procedure is employed to search for the values of the worst constraints that can still satisfy the constraint requirements to ensure the feasibility or constraint robustness. The formulation of the robust design can then be written as follows: min½maxðU1 ; U2 ; . . . ; UM Þ a
s:t: V =Vallowable 1:0 0
previously. In essence, Eq. (20) means that the maximum or worst value among the M values of unsatisfactory degree functions is first identified for each set of trial values, and this is subsequently minimised among all sets of trial values. It is also noted that the optimisation is doubly nested because in each step of the optimisation process as expressed in Eq. (20), an optimisation process to determine the minimum volume V is required, which itself is nested as shown in Eq. (21). Again the decomposition technique described previously can be employed here to obtain the optimal solution efficiently. 3. Numerical examples A ship is docked on a supporting structure with 33 blocks as shown in Fig. 4. The distributions of the ship girder loads and the second moment of sectional area of the ship girder are shown in Figs. 5 and 6, respectively. For simplicity, the variable linking technique [27] is used in the study and 33 blocks are grouped under eight categories as shown in Table 1. The allowable stress, buckling load, displacement of the ship girder, allowable stress of the blocks are the same as in Ref. [2]. Various size limits are described as follows. The minimum and maximum of the height for all soft wood blocks are 5.0 and 100.0 cm, respectively. The minimum and maximum of the total height for all blocks are 25.0 and 220.0 cm, respectively. The minimum and maximum limits of the block thickness are 25.0 and 70.0 cm, respectively, while those for the block breadth are 60.0 and 100.0 cm, respectively. The nominal stiffness of the block support structure is assumed to be
ð20Þ
and V ¼ minV ¼ X
N X Bi Ci ðH1i þ H2i Þ i¼1
s:t: max Gj ðX;PÞ 0; P2CP ðaÞ
ð21Þ
j ¼ 1;2; . . . ;Ng
where Ui is the unsatisfactory degree function corresponding to ith uncertainty parameter, Vallowable is the limited total volume of the docking blocks under consideration and the other symbols have been defined
Fig. 5. Distribution of the ship girder loads along the longitudinal direction.
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Table 2 Deterministic optimal design: worst values of constraints under prescribed uncertainty in ship girder loads Constraint No.a
Fig. 6. Distribution of the second moment of sectional area of the ship girder along the longitudinal direction.
1421 kN/cm. In the analyses, the constrained variable metric method [28,29] is employed to solve for the optimal solution. 3.1. Results of optimal design of docking blocks with uncertainty 3.1.1. Effect of uncertainty in ship girder loads The uncertainty in ship girder loads can be estimated by measurements of member sizes and quantities of fuel, foods and ballasts, etc., and comparing the statistics against the design nominal values. To study the effect of uncertain ship girder loads on the design, two cases are considered. In the first case, the loads at stations 0–2, 5–7 and 18–20 are assumed to have 5% uncertainties, while they have 10% uncertainties in the second case. The deterministic optimal design scheme taking no account of uncertainty is first worked out and evaluated. The worst values of constraints for the deterministic optimal design scheme are evaluated subject to the prescribed uncertainties in ship girder loads and some of them are listed in Table 2. It is observed that the safety margins of structural strength reduce as the uncertainty degrees increase, and this can be seen in constraints 9–11, 13 and 14. All stress constraints of docking blocks and some strength constraints of ship structures are violated when the uncertainties are considered as shown in constraints 1–8, 12 and 15, which leads to potential damage of the ship structures and docking blocks. It is therefore necessary to take into account various existing uncertainties in the real structures to improve the resistance to variability of uncertain variables. Table 1 Categories of the docking blocks Block type
1
2
3
Block No. Block location
1 2 3–6 Aft part
4
5
6
7
8
7–9 10–16 17–23 24–31 32–33 Central part Fore part
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The worst constraint value maxðr=rallow 1:0Þ 0% uncertainty
5% uncertainty
10% uncertainty
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.277 0.351 0.130 0.000 0.498 0.170 0.000
0.029 0.025 0.022 0.012 0.005 0.016 0.030 0.034 0.257 0.340 0.120 0.019 0.486 0.164 0.014
0.059 0.047 0.043 0.024 0.011 0.033 0.061 0.069 0.237 0.330 0.110 0.037 0.474 0.158 0.029
a Constraints 1–8 are stress constraints of docking blocks and the others are strength constraints of ship structures.
The optimisation problem is again solved taking into account various constraints and the uncertainty in ship girder loads. Table 3 shows the optimal dimensions of blocks considering the uncertain ship girder loads from Eqs. (9) and (10). In this case, no constraints are violated although uncertainties in ship girder loads exist. The corresponding results of deterministic design are also included in Table 3 for comparison. It is observed that the total heights of docking blocks at stern, i.e. blocks 1–6, increase as the degrees of uncertainty in ship girder loads increase, which leads to the increase in total volume of docking blocks. The total volumes of blocks are 2:785 106 and 2:984 106 cm3 , respectively, for the cases of 5% and 10% uncertainties in ship girder loads. Compared with the volume in the benchmark design scheme, they have increased by 8.1% and 15.8%, respectively. Such results may be explained by the fact that the maximum load-carrying capability is the same for each docking block for all cases under consideration. The maximum loaded area and allowable stress for each block are actually the same and they are independent of the degree of uncertainty. As the uncertainty degree increases, the differences between the critical nominal block reactions from deterministic design and the worst reactions taking into account uncertainty increase, as shown in Table 4. The nominal reactions have to be kept low to satisfy block strength constraints under the worst combination of ship girder loads. The way to decrease the critical nominal reactions is to reduce the stiffness of blocks, and it consequently increases the total height of blocks and hence the volume. The decrease in stiffness of docking blocks may result in comparatively flat distribution of block reac-
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Table 3 Optimal block dimensions obtained taking into account uncertainty in ship girder loads Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 75.5 0.0
60.0 70.0 41.6 0.0
60.0 57.0 38.4 0.0
60.0 51.0 16.2 8.8
60.0 43.9 6.1 18.9
60.0 34.1 7.1 17.9
60.0 28.5 22.4 2.6
60.0 56.0 15.3 9.7
5% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 85.3 0.0
60.0 70.0 48.1 0.0
60.0 57.3 43.7 0.0
60.0 51.2 19.6 5.4
60.0 46.6 5.5 19.5
60.0 32.6 7.2 17.8
60.0 34.1 8.0 17.0
60.0 51.1 5.9 19.1
10% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 95.2 0.0
60.0 70.0 54.6 0.0
60.0 57.7 49.2 0.0
60.0 51.5 23.5 1.5
60.0 48.0 6.2 18.8
60.0 34.1 7.2 17.8
60.0 35.1 8.0 17.0
60.0 52.7 5.8 19.2
Table 4 Critical nominal reactions and worst reactions taking into account uncertain ship girder loads Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
R0 (kN) Rm (kN)
857.5 857.5
857.5 857.5
698.3 698.3
624.4 624.4
537.7 537.7
417.9 417.9
349.1 349.1
686.0 686.0
5% uncertainty
R0 (kN) Rm (kN)
833.2 857.5
836.7 857.5
687.5 702.2
619.6 627.2
568.4 571.0
393.7 399.0
405.4 418.1
603.5 625.3
10% uncertainty
R0 (kN) Rm (kN)
810.3 857.5
818.0 857.5
678.2 707.3
615.5 631.4
582.1 587.0
407.0 417.6
404.6 430.1
601.8 645.6
Note: R0 and Rm denote critical nominal and worst block reactions, respectively.
Fig. 7.
Nominal block reaction distribution with different ship girder load uncertainties.
Y.S. Cheng et al. / Engineering Structures 26 (2004) 499–510 Table 5 Deterministic optimal design: worst values of constraints under prescribed uncertainty in equivalent stiffness of blocks Constraint No.a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The worst constraint value maxðr=rallow 1:0Þ 0% uncertainty
5% uncertainty
10% uncertainty
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.277 0.351 0.130 0.000 0.498 0.170 0.000
0.063 0.077 0.062 0.088 0.067 0.070 0.055 0.049 0.238 0.306 0.078 0.065 0.469 0.132 0.053
0.129 0.154 0.125 0.183 0.137 0.146 0.111 0.099 0.199 0.260 0.025 0.131 0.440 0.094 0.107
a Constraints 1–8 are stress constraints of docking blocks and the others are strength constraints of ship structures.
tions as shown in Fig. 7. It is understood that increased volumes are utilised to improve the resistances of blocks and ship structures to variation of uncertain ship girder loads, which actually increases the reliability of the structures. 3.1.2. Effect of uncertainty in equivalent stiffness of docking blocks As timber is naturally occurring, its properties such as strength, stiffness, etc. are rather variable. Various testing methods including stress grading do provide engineers with better knowledge of the material properties, but uncertainties still exist. The uncertainty in equivalent stiffness of docking blocks can be considered in a similar manner. The deterministic optimal design
507
scheme is again taken as the benchmark for comparison. Two cases of uncertainty are considered. In the first case, the uncertainties of equivalent stiffnesses of all docking blocks are assumed to be 5%, while they are taken to be 10% in the second case. The worst values of constraints of the deterministic design are estimated when there are prescribed uncertainties in equivalent stiffness of blocks and some of them are shown in Table 5. It is again observed that the safety margins of structural strength reduce as the uncertainty degrees increase, as for example in constraints 9–11, 13 and 14. In addition, all stress constraints of docking blocks and some strength constraints of ship structures are violated when the uncertainties are considered as in constraints 1–8, 12 and 15. All these demonstrate that the design based on deterministic conditions does not satisfy the requirements if potential uncertainties are taken into account. Hence, adjustments to the optimal deterministic design have to be made to allow for uncertainty in equivalent block stiffness. Based on the mathematical model represented by Eqs. (9) and (10), the optimal dimensions of blocks with uncertain equivalent stiffness are obtained and listed in Table 6. In this case, there exists no violation of constraints for the optimal design scheme as the uncertainties have been considered. The block dimensions resulting from the deterministic design are also given in Table 6 for comparison. Corresponding to the cases of 5% and 10% uncertainties in equivalent block stiffness, the total volumes of blocks are now 3:200 106 and 4:330 106 cm3 , respectively, which have increased by 24.2% and 68.18%, respectively, compared to the volume of the benchmark design. The explanation for the results given in Section 3.1.1 is still valid herein. For the sake of completeness, the maximum nominal and the worst reactions for each type of docking blocks are
Table 6 Optimal block dimensions obtained taking into account uncertainty in equivalent stiffness of blocks Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 75.5 0.0
60.0 70.0 41.6 0.0
60.0 57.0 38.4 0.0
60.0 51.0 16.2 8.8
60.0 43.9 6.1 18.9
60.0 34.1 7.1 17.9
60.0 28.5 22.4 2.6
60.0 56.0 15.3 9.7
5% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 100.0 5.1
60.0 70.0 61.1 0.0
60.0 57.2 54.4 0.0
60.0 54.0 27.7 0.0
60.0 53.7 6.1 18.9
60.0 36.6 7.2 17.8
60.0 30.4 22.6 2.4
60.0 58.3 15.3 9.7
10% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 100.0 110.0
60.0 70.0 83.8 0.0
60.0 57.2 73.0 0.0
60.0 56.1 39.7 0.0
60.0 56.6 12.3 12.7
60.0 47.9 5.4 19.6
60.0 36.3 22.3 2.7
60.0 51.8 25.0 0.0
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Table 7 Critical nominal reactions and worst reactions taking into account uncertain equivalent stiffness of blocks Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
R0 (kN) Rm (kN)
857.5 857.5
857.5 857.5
698.3 698.3
624.4 624.4
537.7 537.7
417.9 417.9
349.1 349.1
686.0 686.0
5% uncertainty
R0 (kN) Rm (kN)
806.6 857.5
797.7 857.5
659.7 700.3
607.5 660.3
616.4 656.7
418.7 448.4
352.5 371.8
681.2 714.8
10% uncertainty
R0 (kN) Rm (kN)
759.4 857.5
744.4 857.5
625.6 703.1
583.6 686.3
608.2 692.9
508.6 586.7
404.5 445.7
568.5 633.7
Note: R0 and Rm denote critical nominal and worst block reactions, respectively.
listed in Table 7 and the distributions of nominal reactions for various cases considered are shown in Fig. 8, respectively. Comparing the results obtained from uncertain ship girder loads and those resulting from uncertain equivalent block stiffness, it is easily found that the effect due to uncertainty in block stiffness on the design is larger than that due to uncertainty in ship girder loads. This may be due to the fact that the variation of loads towards the lower bounds is usually helpful to structural safety, while increase in loads normally needs a more demanding design. However, the variation of block stiffness towards either the lower or upper bound may cause large reactions somewhere and possibly lead to structural problems. Therefore, although the same
Fig. 8.
degree of uncertainty is assumed, the actual variation range of equivalent block stiffness is larger than that of the ship girder loads making the design more stringent. 3.2. Results of robust design of docking blocks with uncertainty The numerical example in Section 3.1 is again employed for demonstration of the robust design. Only the uncertainty in equivalent stiffness of docking blocks is considered in the robust design. For simplicity, only three uncertainty parameters are assumed, namely a1, a2 and a3 for the equivalent stiffnesses of docking blocks at aft, central and fore parts, respectively. In particular, the uncertainty parameter a1 defines the
Nominal block reaction distribution with different block stiffness uncertainties.
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Table 8 Range limits of unsatisfactory degree functions Uncertainty parameter
Unacceptable
0.0 0.0 0.0
a1 a2 a3
Highly undesirable
Undesirable
Tolerable
Desirable
Highly desirable
ai5
ai4
ai3
ai2
ai1
0.025 0.0 0.025
0.05 0.025 0.05
0.075 0.05 0.075
0.10 0.075 0.10
0.20 0.10 0.20
+1 +1 +1
Table 9 Optimal block dimensions of the robust design under limited total volume of blocks Parameters
B (cm) C (cm) H1 (cm) H2 (cm)
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
60.0 70.0 100.0 104.3
60.0 70.0 82.8 0.0
60.0 57.3 71.5 0.0
60.0 56.7 36.7 0.0
60.0 57.0 9.6 15.4
60.0 43.5 5.0 20.0
60.0 39.8 12.9 12.1
60.0 47.4 19.8 5.2
variation of the equivalent stiffnesses of blocks 1–6, while the uncertainty parameters a2 and a3 define those of blocks 7–23 and blocks 24–33, respectively. The corresponding range limits are shown in Table 8. The limiting total volume of the docking blocks Vallowable is set to be 4:240 106 cm3 . Table 9 gives the optimal dimensions of blocks obtained from robust design. The maximum or worst unsatisfactory degree for the optimal design is 0.4297 and the uncertainty parameters a1, a2 and a3 are 0.103, 0.77 and 0.103, respectively. It is understood that all of them are within the desirable range. For comparison, robust design is again carried out by maximising the minimum of uncertainty parameters directly without the use of the unsatisfactory degree functions. In this case, the optimal values for all uncertainty parameters are 0.095. It is readily found that the uncertainty parameters a1 and a3 are in the tolerable range while the uncertainty parameter a2 is in the desirable range. In this case, the maximum or worst unsatisfactory degree for the optimal design is 0.5546, which is larger than the maximum or worst unsatisfactory degree for the optimal design resulting from the proposed method using the unsatisfactory degree functions. The numerical results demonstrate that the optimal result obtained from minimisation of the maximum value of unsatisfactory degree functions is superior to that resulting from direct maximisation of the minimum value of uncertainty parameters. All uncertainty parameters are in the desirable range and the corresponding maximum value of unsatisfactory degree functions is also smaller. This is because direct maximisation of the minimum of uncertainty parameters implies that the design is optimal with respect to the global robust reliability and hence all uncertainty parameters tend to be uniform.
This process somehow treats all uncertain variables equally and ignores different properties of individual uncertain variables. However, the characteristics of individual uncertain variables in the proposed method are taken into consideration by setting their own range limits and more reasonable results can be obtained. 4. Conclusions The paper discusses issues of optimal design and robust design of docking blocks with uncertainty and proposes the corresponding mathematical models. The convex model is employed to describe the uncertainties in ship girder loads and equivalent stiffness of docking blocks. Numerical examples show that the uncertainties do affect the optimal solution and lead to increase in volumes of docking blocks compared to the optimal solution obtained from deterministic optimisation. Increased volumes are required to protect the structure from variability of uncertain variables. The effect of uncertainty in block stiffness on the design is larger than that due to uncertainty in ship girder loads. The robust design of docking blocks with uncertainty can be achieved by minimisation of the maximum unsatisfactory degree of the uncertainty parameters. The numerical results show that the design scheme resulting from minimising the maximum unsatisfactory degree is superior to that obtained from directly maximising the minimum of uncertainty parameters without the use of unsatisfactory degree functions. Acknowledgements The work described in this paper has been partially supported by the Scientific Research Foundation for
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the Returned Overseas Chinese Scholars, State Education Ministry of China and the Committee on Research and Conference Grants, The University of Hong Kong, China.
References [1] Jiang I, Grubbs K, Haith R, Santomartino V. DRYDOCK: an interactive computer program for predicting dry dock block reactions. Transaction of the Society of Naval Architects and Marine Engineers 1987;95:29–44. [2] Cheng YS, Zeng GW. Optimum disposition of wooden blocks during ship docking. Shipbuilding of China 1995;1:18–27 [in Chinese]. [3] Ben-Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics. Amsterdam: Elsevier Science Publishers; 1990. [4] Ben-Haim Y. A non-probabilistic concept of reliability. Structural Safety 1994;14:227–45. [5] Elishakoff I. Essay on uncertainties in elastic and viscoelastic structures: from A.M. Freudernthal’s criticisms to modern convex modelling. Computers and Structures 1995;56(6):871–95. [6] Ben-Haim Y, Chen G, Soong SS. Maximum structural response using convex models. Journal of Engineering Mechanics 1996;122(4):325–33. [7] Elishakoff I, Haftka RT, Fang J. Structural design under bounded uncertainty-optimization with anti-optimization. Computers and Structures 1994;53(6):1401–5. [8] Pantelides CP, Booth BC. Computer-aided design of optimal structures with uncertainty. Computers and Structures 2000;74(3):293–307. [9] Lombardi M. Optimization of uncertain structures using nonprobabilistic models. Computers and Structures 1998;67(1-3): 99–103. [10] Lombardi M, Haftka RT. Anti-optimization technique for structural design under load uncertainties. Computer Methods in Applied Mechanics and Engineering 1998;157:19–31. [11] Ganzerli S, Pantelides CP. Optimum structural design via convex model superposition. Computers and Structures 2000;74(6):639–47. [12] Phadke MS. Quality engineering using robust design. Englewood Cliffs (NJ): PTR Prentice-Hall; 1989.
[13] Rao RR, Sundararaju K, Balakrishna C, Prakash BG. Multiobjective insensitive design of structures. Computers and Structures 1992;45(2):349–59. [14] Belegundu AD, Zhang SH. Robustness of design through minimum sensitivity. Journal of Mechanical Design 1992;114(2):213–7. [15] Parkinson A, Sorensen C, Pourhassan N. A general approach for robust optimal design. Journal of Mechanical Design 1993;115(1):74–80. [16] Emch G, Parkinson A. Robust optimal design for worst-case tolerances. Journal of Mechanical Design 1994;116(4):1019–25. [17] Chen W, Allen JK, Tsui KL, Mistree F. A procedure for robust design: minimizing variations caused by noise factors and control factors. Journal of Mechanical Design 1996;118(4):479–85. [18] Parkinson DB. Robust design by variability optimisation. Quality and Reliability Engineering International 1997;13(2):97–102. [19] Parkinson DB. The application of a robust design method to telerancing. Journal of Mechanical Design 2000;122(2):149–54. [20] Lee KH, Park GJ. Robust optimisation considering tolerances of design variables. Computers and Structures 2001;79(1):77–86. [21] Du XP, Chen W. Efficient uncertainty analysis methods for multidisciplinary robust design. AIAA Journal 2002;40(3):545–52. [22] Sandgren E, Cameron TM. Robust design optimization of structures through consideration of variation. Computers and Structures 2002;80(20-21):1605–13. [23] Ben-Haim Y. Robust reliability of structures. Advances in Applied Mechanics 1997;33:1–41. [24] Ben-Haim Y. Reliability of vibrating structures with uncertainty inputs. The Shock and Vibration Digest 1998;30(2):106–13. [25] Ben-Haim Y, Fryba L, Yoshikawa N. Robust reliability of dynamically loaded beam on an uncertain foundation. In: Kumar B, Topping BHV, editors. Computing developments in civil and structural engineering. Edinburgh: Civil-Comp Press; 1999, p. 71–5. [26] Messac A. Physical programming: effective optimization for computational design. AIAA Journal 1996;34(1):149–58. [27] Rao SS. Engineering optimization, theory and practice, 3rd ed. New York: John Wiley & Sons; 1996. [28] Yu J, Zhou J. Optimization programming OPB-1: theory and applications. Beijing (PR China): Mechanical Industry Press; 1989 [in Chinese]. [29] Powell MJD. Algorithms for nonlinear constraints that use Lagrangian functions. Mathematical Programming 1978;14: 224–48.
Optimal and robust design of docking blocks with uncertainty Y.S. Cheng a,, F.T.K. Au b, L.G. Tham b, G.W. Zeng a a
Department of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China b Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Received 28 October 2002; received in revised form 24 November 2003; accepted 26 November 2003
Abstract The positioning and stiffness allocation of docking blocks are an important decision when docking a ship. In this paper, the issues of both optimal design and robust design of docking blocks with uncertainty are discussed, and the corresponding mathematical models are proposed. To describe the uncertainties of various parameters, the proposed method employs the convex model in which the indeterminacy about the uncertain variables is expressed in terms of some sets to which these uncertain variables belong. In the optimisation of docking blocks, the factors considered include the uncertain ship girder loads and uncertain equivalent stiffnesses of blocks. Numerical examples show that uncertainties do affect the optimal solution and lead to an increase in volume of blocks compared to that from deterministic optimisation. A method of robust design with respect to the uncertain equivalent stiffnesses of blocks is also proposed. The robustness of the objective function is achieved by minimising the maximum value of unsatisfactory degree functions of the uncertainty parameters while the feasibility robustness is ensured by an optimisation conducting the worst-case analysis at a lower level. Numerical simulations show that the optimal result from minimisation of the maximum value of unsatisfactory degree functions is superior to that from direct maximisation of the minimum value of uncertainty parameters without the use of unsatisfactory degree functions. # 2003 Elsevier Ltd. All rights reserved. Keywords: Convex model; Docking blocks; Optimal design; Robust design; Uncertainty
1. Introduction A dry dock normally consists of a basin dug into the shore of a body of water and provided with a watertight gate on the waterside, used for major repairs and overhaul of vessels. When a ship is to be docked, the dry dock is flooded, and the gate opened. After the vessel is brought in, positioned properly and guyed, the gate is closed and the dock is pumped dry, bringing the craft gradually to rest on supporting keel and bilge blocks anchored to the floor. The positioning and stiffness allocation of these docking blocks are an important decision when docking a ship because mispositioning or mis-allocation of docking blocks may give rise to unreasonably large block reactions and consequently serious damage to both the docked ship and blocks. Docking block failure may also cause the disruption of docking schedules and extension of the Corresponding author. Tel.: +86-27-875-43758x801; fax: +86-27875-42146. E-mail address: [email protected] (Y.S. Cheng).
0141-0296/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2003.11.007
ship downtime. Any failure may even lead to the loss of lives. Docking analysis has therefore attracted attention of various researchers. Jiang et al. [1] developed a reliable, efficient computer program for predicting block reactions in both graving and floating docking analyses. Cheng and Zeng [2] proposed a mathematical model for optimising the positioning and allocation of docking blocks ignoring potential uncertainties in the design of docking blocks. Two-level optimisation techniques were employed to solve for the optimal solution in their study. However, uncertainties in material, geometric properties, loads, etc. are unavoidable in the design of engineering structures. Various types of wood are used in the construction of docking blocks. The elastic modulus of various types of wood vary greatly depending on its age, moisture content, and the effects of previous loading on its proportional limit. All these factors lead to uncertain block stiffness and hence the uncertain equivalent block stiffness, which encompasses the stiffness of a docking block and that of its supporting structure. Ship girder loads are also difficult to
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determine accurately. The ship girder loads actually vary from their nominal values. Therefore, the ship girder loads and docking block stiffness may be treated as uncertain variables in the design of docking blocks. These uncertainties are traditionally dealt with by using probabilistic models, and the reliability of a structure is measured with the probability of no failure. However, probabilistic analysis requires sufficient data on probabilistic parameters with which the probability density functions of random variables can be determined. In many engineering problems, although one may not be able to estimate accurately the probability distributions of random variables due to various reasons such as insufficient information, one can at least evaluate their bounds. The convex models [3–5] have been developed to handle such uncertainties in design as attractive alternatives to traditional probability models. In the studies of such non-probabilistic modelling of uncertainties, the indeterminacy about the uncertain variables may be expressed in terms of some sets to which these uncertain variables belong. These convex models may include local energy-bound convex model, integral energy-bound convex model, envelope-bound convex model, Fourier-envelope convex model, responsespectrum-envelope convex model, etc. [6]. Recently, the optimisation of structures with uncertainty has been studied by using anti-optimisation approaches or convex models. Elishakoff et al. [7] proposed a design approach for structural optimisation with uncertain but bounded loads. The optimisation problem was formulated as a two-level optimisation. At the first level, the minimisation of the structural weight under stress and displacement constraints is carried out, while finding the maximum structural responses under bounded loads is conducted at the second level. These two optimisation processes are nested in their formulation. Later, Pantelides and Booth [8] built on the formulation proposed in Ref. [7] but approached the problem with a multidimensional ellipsoidal model rather than a box. They applied the method to the optimisation of a two-span continuous reinforced concrete beam and a steel 10-bar truss with uncertain loads. Solving a nested optimisation problem is time consuming. Lombardi [9], and Lombardi and Haftka [10] therefore proposed a method to alternate between the first level optimisation and the second level one. The computational burden is greatly alleviated because these two optimisation processes are executed separately. Recently, Ganzerli and Pantelides [11] developed a convex model superposition method to predict the response of structures with uncertain loads. In this way, the second optimisation process is avoided and the computational efficiency is considerably improved. As an important design methodology of products, robust design is becoming more and more popular. Robust design aims to achieve a state of robustness so
that the performance of a design is least sensitive to the variability of uncertain variables. The fundamental principle of the robust design is to improve the quality of a product by minimising the effect of the causes of variation without eliminating the causes, as pointed out by Phadke [12]. This can be achieved by selecting suitable design variables to make the product performance insensitive to the various causes of variation. Many methods such as those by Rao et al. [13], Belegundu and Zhang [14], Parkinson et al. [15], Emch and Parkinson [16], Chen et al. [17], Parkinson [18,19], Lee and Park [20], Du and Chen [21], and Sandgren and Cameron [22] have been proposed to achieve the goal. In the theory of robust reliability of structures recently proposed by Ben-Haim [23,24], the reliability of a structure can be measured by the size of the convex model that is consistent with no failure. In other words, the reliability of a structure can be quantified as the degree of immunity to uncertainty. A reliable design can perform satisfactorily in the presence of great uncertainty and is relatively immune to unexpected variations. Such a design is robust with respect to uncertainty and therefore such reliability is named as robust reliability. Robust reliability actually expresses the intuitive meaning of reliability: dependability in the face of uncertainty [25]. An optimal design based on the theory of robust reliability may be viewed as a kind of robust design. This paper first presents a mathematical model for minimisation of the total volume of docking blocks with uncertainty. The emphasis is placed on investigating the effects of uncertain ship girder loads and uncertain equivalent stiffnesses of docking blocks on the optimal solution. The paper then proposes a robust design method for the docking blocks with uncertainty based on improved theory of robust reliability in which the unsatisfactory degrees of the uncertainty parameters, instead of the uncertainty parameters themselves, are used to measure the robust reliability. The robustness of the objective function is guaranteed by minimisation of the maximum unsatisfactory degree of the uncertainty parameters while the feasibility robustness is handled through the worst-case analysis with an optimisation process.
2. Theory and formulation 2.1. Conventional optimisation of docking blocks Consider a typical docking arrangement in which a ship is docked on N docking blocks resting on a supporting structure. Each docking block consists of soft wooden blocks with a total height of H1 and hard wooden blocks with a total height of H2 as shown in Fig. 1. The symbols B and C denote the width and thickness
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Fig. 2. Mechanical model for analysing block reactions and ship girder displacements.
Fig. 1.
A docking block.
of the block, respectively, while BL denotes the width of flat keel. If the weight of the ship girder, ship structures and docking block locations are known, a typical optimisation of docking blocks is to choose block dimensions properly to minimise the total volume of docking blocks and satisfy various constraints simultaneously. The formulation of such optimisation ignoring the uncertainties may be written as follows:
minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1
s:t: Gj ðXÞ 0;
ð1Þ
j ¼ 1;2; . . . ;Ng
where X is a vector whose components are the design variables, i.e. H1i, H2i, Bi, Ci, (i¼ 1, 2, . . .,N); V is the total volume of blocks; Gj is the jth constraint function to be satisfied and Ng is the total number of constraints. The constraints may include various limits of stress, stability and displacement that the governing regulation sets. The details of various constraints can be found in Ref. [2] for example. In the study, the ship is treated as a beam with nonuniform cross section, which is in accordance with the normal practice of naval architects. Each of the docking blocks is represented by a spring. If side blocks exist, the side blocks and the keel block can be considered to be elastic springs in parallel and represented by a combined stiffness at the ship centreline. This combined stiffness and its support structure can be viewed as two springs in series. The final equivalent stiffness can then be readily obtained and it is called the equivalent stiffness of the docking block in the paper. The block reactions and the displacements of the ship girder can be obtained by analysing the model as shown in Fig. 2 using the finite element method. As the number of design variables and the number of constraints are large, a two-level solution strategy can be used to solve for the optimal solution efficiently and the details are as follows.
2.1.1. The first level optimisation: optimal allocation of stiffnesses of docking blocks The loaded area, Fi, and total height, Hi, of docking blocks are chosen as design variables. In this step, the exact stiffness of the docking block is not available because the specifications of the block have not been determined yet. The block stiffness and the chosen design variables are approximately related as K i ¼ ni E 1
Fi ; Hi
i ¼ 1;2; . . . ;N
ð2Þ
where E1 is the elastic modulus of the soft wood block. The coefficient ni is called the stiffness coordinative factor of the ith docking block to be exactly determined at the second level optimisation. At this level, constraints may include the strength of docking blocks, longitudinal strength of typical cross sections of the ship girder, the strength of keels and transverse bulkheads, stability of docking stiffeners, displacements of the ship girder and size limits, etc. The total volume of the blocks is chosen to be the objective function and it can be written as V¼
N X
Vi ;
ð3Þ
Vi ¼ gi Fi Hi
ð4Þ
i¼1
where the coefficient gi is called the volume coordinative factor of the ith docking block. Similar to the coefficient ni, gi is to be exactly determined in the second level optimisation. In this way, the total number of design variables is greatly reduced and is only about half of the total number of design variables of the original optimisation. The optimal stiffness allocation of docking blocks can now be determined and are output to the second level optimisation as equality constraints. 2.1.2. The second level optimisation: specification of docking blocks The parameters B, C, H1 and H2 are now chosen as design variables. The inequality constraint of the strength of the docking block, the equality constraint of the block stiffness being equal to the optimal stiffness obtained in the first level optimisation, and the size limits are only included at this level. The objective function is again the volume of the docking block
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under consideration. The optimisation is carried out for each of the docking blocks. Once the details of the blocks are obtained, the coordinative factors can then be exactly calculated. The updated coordinative factors are then returned to the first level optimisation to update the old ones. The two optimisation procedures are alternately performed until convergence of the coordinative factors is achieved. 2.2. Optimisation of docking blocks with uncertainty Consider a structure subjected to M uncertain parameters Pi (i ¼ 1,2,. . . ,M) that may not be conveniently described by probabilistic models. The parameters Pi may be loads, material properties, geometric i dimensions and so on. Their nominal values are P (i ¼ 1,2,. . .,M). Suppose that the deviation of each uncertain parameter from its nominal value is bounded. The indeterminacy about the uncertain variables may be expressed in terms of some sets to which these uncertain variables belong. The set of all possible realisations of uncertain parameters with the unknownbut-bounded information can be defined as 1 P1 P a1 ; CP ðaÞ ¼ P1 2 P2 P a2 ; . . . ; PM PM aM ð5Þ P P 2 M a ¼ fa1 ;a2 ; . . . ;aM g
ð6Þ
P ¼ fP1 ;P2 ; . . . ;PM g
ð7Þ
The parameter ai defines the boundary of the uncertain parameter Pi and describes the degree of uncertain variability of Pi. The parameter ai is called the uncertainty parameter corresponding to uncertain variable Pi. It is obvious that the greater the value of ai, the greater the variation of Pi. In the study, the uncertainties in ship girder loads and the equivalent stiffnesses of docking blocks are considered separately. The mathematical model for optimal design of docking blocks with uncertainty can be formulated as follows: minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1
s:t: max Gj ðX;PÞ 0; P2CP ðaÞ
ð8Þ
j ¼ 1;2; . . . ;Ng
In Eq. (8), the maximisation of Gj(X,P) over CP(a) is a process of seeking the worst value of P for each constraint. Therefore, the above mathematical model is a nested optimisation. As finding a solution to a nested optimisation problem is time-consuming, a technique developed by Lombardi [9], and Lombardi and Haftka [10] is utilised to decompose the problem described by
Eq. (8) into two optimisation processes as below: minV ¼ X
N X
Bi Ci ðH1i þ H2i Þ
i¼1 ðjÞ
s:t: Gj ðX;P Þ 0; and maxGj ðX;PÞ P i Pi P s:t: ai ; Pi
ð9Þ
j ¼ 1;2; . . . ;Ng
i ¼ 1;2; . . . ;M
ð10Þ
where P(j) in Eq. (9) is an optimal solution to Eq. (10), which gives the worst case prediction of the jth constraint for the given X. Note that only X is varied in Eq. (9), and only P is varied in Eq. (10). After the determination of optimal values of P(j) by Eq. (10), updated values of X are obtained using Eq. (9). These updated values of X are then substituted back into Eq. (10) again to get the updated P(j). Such iterations are repeated until the difference between the values of P(j) in consecutive cycles is within the prescribed tolerance. In essence, it means that the worst constraints in Eq. (9) still satisfy the requirements. Therefore, the optimal design of docking blocks with uncertainty defined by Eq. (8) can be obtained by solving Eqs. (9) and (10) alternately until convergence is achieved. 2.3. Robust design of docking blocks with uncertainty As stated previously, the uncertainty parameter ai is used to measure the reliability of a structure according to the theory of robust reliability. If failure does not occur even for a large ai, then the structure is reliable since it is robust with respect to uncertainty. On the other hand, if failure may occur even for a small ai, then the structure is vulnerable to uncertainty and therefore unreliable. Such a rationale is valid if the variation of all uncertain variables can be represented by one uncertainty parameter a and the robust reliability of the structure can then be measured with the largest a value consistent with no failure. However, direct use of the uncertainty parameter as a measure of robust reliability may create problems if multiple uncertainty parameters are required to describe bounded-but-unknown uncertain variables as used in Eq. (5). For example, suppose that there are two kinds of uncertain loads P1 and P2. The first load, P1, only fluctuates slightly and is very likely to be within the range of a1 ¼ 0:05. However, the second load, P2, fluctuates drastically and is within the range of a2 ¼ 0:20. Two design schemes are therefore possible. One can confine the variation of P1 and P2 to be within their respective ranges of a1 ¼ 0:06 and a2 ¼ 0:21 without failure. It is obvious that the design is safe and the robust reliability is 0.06 based on the theory of robust reliability. In another design scheme, the variation of
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P1 and P2 can be confined within the range of a1 ¼ 0:08 and a2 ¼ 0:14, respectively, without failure. The design has the robust reliability of 0.08 accordingly, but obviously the design is not as reliable because the real variation of P2 may be beyond the range of a2 ¼ 0:14. The properties of various uncertain variables need to be considered further in depth to obtain more reasonable and equitable results. It is very desirable to build a benchmark so that the uncertain variables with different properties and disparate units can be treated on a par with one another. To build such a benchmark, the idea of constructing the class function in physical programming [26] is adopted in the paper to define the unsatisfactory degree function for each of the uncertainty parameters. To construct the unsatisfactory degree functions, one should first define six ranges for the uncertainty parameters ai (i¼ 1,2, . . .,M) and then describe their unsatisfactory degrees in order of decreasing preference as follows:
Another feature of the unsatisfactory degree function is to make the highly undesirable range significantly worse than the undesirable range, and the undesirable range significantly worse than the tolerable range and so on. This feature reflects the fact that the value of the unsatisfactory degree function within the highly desirable range is already small and further reduction of the value of the unsatisfactory degree function is unnecessary. On the other hand, the value of the unsatisfactory degree function within the highly undesirable range is large and therefore requires significantly minimisation of the unsatisfactory degree function. The unsatisfactory degree function is schematically shown in Fig. 3 and the mathematical expression of the function for the range k (k¼ 2, 3, 4, 5) can be represented as follows: Uki ¼N1 ðnki ÞUiðk1Þ þ N2 ðnki ÞUik þ N3 ðnki ;kki ÞU0iðk1Þ þ N4 ðnki ;kki ÞU0ik 1 4 1 3 n ðn 1Þ4 2n þ 2 2 2 1 4 1 1 N2 ðnÞ ¼ n þ ðn 1Þ4 þ 2n 2 2 2 1 4 3 1 3 4 N3 ðn;kÞ ¼ k n ðn 1Þ n þ 8 8 2 8 3 1 1 1 N4 ðn;kÞ ¼ k n4 ðn 1Þ4 n þ 8 8 2 8 a a i iðk1Þ nki ¼ aik aiðk1Þ N1 ðnÞ ¼
1. Highly desirable range (ai ai1 ): an acceptable range but further increase of the uncertainty parameter, although desirable, is of minimal additional value. 2. Desirable range (ai2 ai ai1 ): an acceptable range that is desirable. 3. Tolerable range (ai3 ai ai2 ): an acceptable and tolerable range. 4. Undesirable range (ai4 ai ai3 ): a range that is acceptable, but undesirable. 5. Highly undesirable range (ai5 ai ai4 ): a range that is still acceptable, but is highly undesirable. 6. Unacceptable range (ai ai5 ): the range of values that the uncertainty parameter may not take. The parameters ai1 through ai5 are physically meaningful values that may be provided by the designer to quantify the preference with respect to the ith uncertainty parameter ai. The unsatisfactory degree function Ui possesses two primary features. One is to have the same value at each of the range boundaries regardless of the particular uncertainty parameters in question. Hence, the change in the unsatisfactory degree function will always be the same as one moves from one boundary to another within a given range. Such a feature has a normalising effect and results in favourable numerical conditioning properties. In other words, a benchmark is built for various uncertain variables by mapping the uncertain variables to a dimensionless scale via the unsatisfactory degree functions. Obviously the lower the unsatisfactory degree, the higher the robust reliability and the more robust the design. The maximum or worst value of the unsatisfactory degree functions should therefore be minimised.
503
kki ¼ aik aiðk1Þ @Ui 0 Uik ¼ @ai ai ¼aik
ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ ð18Þ
For range 1, the unsatisfactory degree function is defined by an exponential function as follows: U1i ¼ Ui1 exp ðU0i1 =ai1 Þðai ai1 Þ for ai ai1 ð19Þ
Fig. 3. A schematic unsatisfactory degree function.
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Fig. 4. A schematic diagram of a docked ship.
The values and slopes of the unsatisfactory degree function at range boundaries must be properly chosen to guarantee the above desirable properties. The evaluation algorithm proposed by Messac [26] is employed in the study. The robust design of docking blocks with uncertain equivalent stiffness is to establish one that obtains the desirable values of unsatisfactory degree functions for all uncertainty parameters and to satisfy simultaneously all constraints. The uncertainty parameters represent the variation of uncertain equivalent stiffness of docking blocks. The constraints to be satisfied may include the stress and stability of each structural component, nodal displacements, block volume and size limit of design variables, etc. The robustness of the objective function can be achieved by minimising the maximum or worst value of the unsatisfactory degree functions. It tends to move the uncertainty parameters to the highly desirable range and results in higher robust reliability with respect to uncertainty. As for the feasibility robustness, the conventional method for linear worst-case analysis is suitable for linear problems and cases of small variations in uncertain variables. However, according to Emch and Parkinson [16], even though higher order models are used in the worst-case analysis, violation of the constraints may still occur for the case in which the maximum variation does not occur at endpoints of the tolerances but rather at some values in between. Therefore, in the study, an optimisation procedure is employed to search for the values of the worst constraints that can still satisfy the constraint requirements to ensure the feasibility or constraint robustness. The formulation of the robust design can then be written as follows: min½maxðU1 ; U2 ; . . . ; UM Þ a
s:t: V =Vallowable 1:0 0
previously. In essence, Eq. (20) means that the maximum or worst value among the M values of unsatisfactory degree functions is first identified for each set of trial values, and this is subsequently minimised among all sets of trial values. It is also noted that the optimisation is doubly nested because in each step of the optimisation process as expressed in Eq. (20), an optimisation process to determine the minimum volume V is required, which itself is nested as shown in Eq. (21). Again the decomposition technique described previously can be employed here to obtain the optimal solution efficiently. 3. Numerical examples A ship is docked on a supporting structure with 33 blocks as shown in Fig. 4. The distributions of the ship girder loads and the second moment of sectional area of the ship girder are shown in Figs. 5 and 6, respectively. For simplicity, the variable linking technique [27] is used in the study and 33 blocks are grouped under eight categories as shown in Table 1. The allowable stress, buckling load, displacement of the ship girder, allowable stress of the blocks are the same as in Ref. [2]. Various size limits are described as follows. The minimum and maximum of the height for all soft wood blocks are 5.0 and 100.0 cm, respectively. The minimum and maximum of the total height for all blocks are 25.0 and 220.0 cm, respectively. The minimum and maximum limits of the block thickness are 25.0 and 70.0 cm, respectively, while those for the block breadth are 60.0 and 100.0 cm, respectively. The nominal stiffness of the block support structure is assumed to be
ð20Þ
and V ¼ minV ¼ X
N X Bi Ci ðH1i þ H2i Þ i¼1
s:t: max Gj ðX;PÞ 0; P2CP ðaÞ
ð21Þ
j ¼ 1;2; . . . ;Ng
where Ui is the unsatisfactory degree function corresponding to ith uncertainty parameter, Vallowable is the limited total volume of the docking blocks under consideration and the other symbols have been defined
Fig. 5. Distribution of the ship girder loads along the longitudinal direction.
Y.S. Cheng et al. / Engineering Structures 26 (2004) 499–510
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Table 2 Deterministic optimal design: worst values of constraints under prescribed uncertainty in ship girder loads Constraint No.a
Fig. 6. Distribution of the second moment of sectional area of the ship girder along the longitudinal direction.
1421 kN/cm. In the analyses, the constrained variable metric method [28,29] is employed to solve for the optimal solution. 3.1. Results of optimal design of docking blocks with uncertainty 3.1.1. Effect of uncertainty in ship girder loads The uncertainty in ship girder loads can be estimated by measurements of member sizes and quantities of fuel, foods and ballasts, etc., and comparing the statistics against the design nominal values. To study the effect of uncertain ship girder loads on the design, two cases are considered. In the first case, the loads at stations 0–2, 5–7 and 18–20 are assumed to have 5% uncertainties, while they have 10% uncertainties in the second case. The deterministic optimal design scheme taking no account of uncertainty is first worked out and evaluated. The worst values of constraints for the deterministic optimal design scheme are evaluated subject to the prescribed uncertainties in ship girder loads and some of them are listed in Table 2. It is observed that the safety margins of structural strength reduce as the uncertainty degrees increase, and this can be seen in constraints 9–11, 13 and 14. All stress constraints of docking blocks and some strength constraints of ship structures are violated when the uncertainties are considered as shown in constraints 1–8, 12 and 15, which leads to potential damage of the ship structures and docking blocks. It is therefore necessary to take into account various existing uncertainties in the real structures to improve the resistance to variability of uncertain variables. Table 1 Categories of the docking blocks Block type
1
2
3
Block No. Block location
1 2 3–6 Aft part
4
5
6
7
8
7–9 10–16 17–23 24–31 32–33 Central part Fore part
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The worst constraint value maxðr=rallow 1:0Þ 0% uncertainty
5% uncertainty
10% uncertainty
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.277 0.351 0.130 0.000 0.498 0.170 0.000
0.029 0.025 0.022 0.012 0.005 0.016 0.030 0.034 0.257 0.340 0.120 0.019 0.486 0.164 0.014
0.059 0.047 0.043 0.024 0.011 0.033 0.061 0.069 0.237 0.330 0.110 0.037 0.474 0.158 0.029
a Constraints 1–8 are stress constraints of docking blocks and the others are strength constraints of ship structures.
The optimisation problem is again solved taking into account various constraints and the uncertainty in ship girder loads. Table 3 shows the optimal dimensions of blocks considering the uncertain ship girder loads from Eqs. (9) and (10). In this case, no constraints are violated although uncertainties in ship girder loads exist. The corresponding results of deterministic design are also included in Table 3 for comparison. It is observed that the total heights of docking blocks at stern, i.e. blocks 1–6, increase as the degrees of uncertainty in ship girder loads increase, which leads to the increase in total volume of docking blocks. The total volumes of blocks are 2:785 106 and 2:984 106 cm3 , respectively, for the cases of 5% and 10% uncertainties in ship girder loads. Compared with the volume in the benchmark design scheme, they have increased by 8.1% and 15.8%, respectively. Such results may be explained by the fact that the maximum load-carrying capability is the same for each docking block for all cases under consideration. The maximum loaded area and allowable stress for each block are actually the same and they are independent of the degree of uncertainty. As the uncertainty degree increases, the differences between the critical nominal block reactions from deterministic design and the worst reactions taking into account uncertainty increase, as shown in Table 4. The nominal reactions have to be kept low to satisfy block strength constraints under the worst combination of ship girder loads. The way to decrease the critical nominal reactions is to reduce the stiffness of blocks, and it consequently increases the total height of blocks and hence the volume. The decrease in stiffness of docking blocks may result in comparatively flat distribution of block reac-
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Table 3 Optimal block dimensions obtained taking into account uncertainty in ship girder loads Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 75.5 0.0
60.0 70.0 41.6 0.0
60.0 57.0 38.4 0.0
60.0 51.0 16.2 8.8
60.0 43.9 6.1 18.9
60.0 34.1 7.1 17.9
60.0 28.5 22.4 2.6
60.0 56.0 15.3 9.7
5% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 85.3 0.0
60.0 70.0 48.1 0.0
60.0 57.3 43.7 0.0
60.0 51.2 19.6 5.4
60.0 46.6 5.5 19.5
60.0 32.6 7.2 17.8
60.0 34.1 8.0 17.0
60.0 51.1 5.9 19.1
10% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 95.2 0.0
60.0 70.0 54.6 0.0
60.0 57.7 49.2 0.0
60.0 51.5 23.5 1.5
60.0 48.0 6.2 18.8
60.0 34.1 7.2 17.8
60.0 35.1 8.0 17.0
60.0 52.7 5.8 19.2
Table 4 Critical nominal reactions and worst reactions taking into account uncertain ship girder loads Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
R0 (kN) Rm (kN)
857.5 857.5
857.5 857.5
698.3 698.3
624.4 624.4
537.7 537.7
417.9 417.9
349.1 349.1
686.0 686.0
5% uncertainty
R0 (kN) Rm (kN)
833.2 857.5
836.7 857.5
687.5 702.2
619.6 627.2
568.4 571.0
393.7 399.0
405.4 418.1
603.5 625.3
10% uncertainty
R0 (kN) Rm (kN)
810.3 857.5
818.0 857.5
678.2 707.3
615.5 631.4
582.1 587.0
407.0 417.6
404.6 430.1
601.8 645.6
Note: R0 and Rm denote critical nominal and worst block reactions, respectively.
Fig. 7.
Nominal block reaction distribution with different ship girder load uncertainties.
Y.S. Cheng et al. / Engineering Structures 26 (2004) 499–510 Table 5 Deterministic optimal design: worst values of constraints under prescribed uncertainty in equivalent stiffness of blocks Constraint No.a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The worst constraint value maxðr=rallow 1:0Þ 0% uncertainty
5% uncertainty
10% uncertainty
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.277 0.351 0.130 0.000 0.498 0.170 0.000
0.063 0.077 0.062 0.088 0.067 0.070 0.055 0.049 0.238 0.306 0.078 0.065 0.469 0.132 0.053
0.129 0.154 0.125 0.183 0.137 0.146 0.111 0.099 0.199 0.260 0.025 0.131 0.440 0.094 0.107
a Constraints 1–8 are stress constraints of docking blocks and the others are strength constraints of ship structures.
tions as shown in Fig. 7. It is understood that increased volumes are utilised to improve the resistances of blocks and ship structures to variation of uncertain ship girder loads, which actually increases the reliability of the structures. 3.1.2. Effect of uncertainty in equivalent stiffness of docking blocks As timber is naturally occurring, its properties such as strength, stiffness, etc. are rather variable. Various testing methods including stress grading do provide engineers with better knowledge of the material properties, but uncertainties still exist. The uncertainty in equivalent stiffness of docking blocks can be considered in a similar manner. The deterministic optimal design
507
scheme is again taken as the benchmark for comparison. Two cases of uncertainty are considered. In the first case, the uncertainties of equivalent stiffnesses of all docking blocks are assumed to be 5%, while they are taken to be 10% in the second case. The worst values of constraints of the deterministic design are estimated when there are prescribed uncertainties in equivalent stiffness of blocks and some of them are shown in Table 5. It is again observed that the safety margins of structural strength reduce as the uncertainty degrees increase, as for example in constraints 9–11, 13 and 14. In addition, all stress constraints of docking blocks and some strength constraints of ship structures are violated when the uncertainties are considered as in constraints 1–8, 12 and 15. All these demonstrate that the design based on deterministic conditions does not satisfy the requirements if potential uncertainties are taken into account. Hence, adjustments to the optimal deterministic design have to be made to allow for uncertainty in equivalent block stiffness. Based on the mathematical model represented by Eqs. (9) and (10), the optimal dimensions of blocks with uncertain equivalent stiffness are obtained and listed in Table 6. In this case, there exists no violation of constraints for the optimal design scheme as the uncertainties have been considered. The block dimensions resulting from the deterministic design are also given in Table 6 for comparison. Corresponding to the cases of 5% and 10% uncertainties in equivalent block stiffness, the total volumes of blocks are now 3:200 106 and 4:330 106 cm3 , respectively, which have increased by 24.2% and 68.18%, respectively, compared to the volume of the benchmark design. The explanation for the results given in Section 3.1.1 is still valid herein. For the sake of completeness, the maximum nominal and the worst reactions for each type of docking blocks are
Table 6 Optimal block dimensions obtained taking into account uncertainty in equivalent stiffness of blocks Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 75.5 0.0
60.0 70.0 41.6 0.0
60.0 57.0 38.4 0.0
60.0 51.0 16.2 8.8
60.0 43.9 6.1 18.9
60.0 34.1 7.1 17.9
60.0 28.5 22.4 2.6
60.0 56.0 15.3 9.7
5% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 100.0 5.1
60.0 70.0 61.1 0.0
60.0 57.2 54.4 0.0
60.0 54.0 27.7 0.0
60.0 53.7 6.1 18.9
60.0 36.6 7.2 17.8
60.0 30.4 22.6 2.4
60.0 58.3 15.3 9.7
10% uncertainty
B (cm) C (cm) H1 (cm) H2 (cm)
60.0 70.0 100.0 110.0
60.0 70.0 83.8 0.0
60.0 57.2 73.0 0.0
60.0 56.1 39.7 0.0
60.0 56.6 12.3 12.7
60.0 47.9 5.4 19.6
60.0 36.3 22.3 2.7
60.0 51.8 25.0 0.0
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Table 7 Critical nominal reactions and worst reactions taking into account uncertain equivalent stiffness of blocks Case
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
0% uncertainty
R0 (kN) Rm (kN)
857.5 857.5
857.5 857.5
698.3 698.3
624.4 624.4
537.7 537.7
417.9 417.9
349.1 349.1
686.0 686.0
5% uncertainty
R0 (kN) Rm (kN)
806.6 857.5
797.7 857.5
659.7 700.3
607.5 660.3
616.4 656.7
418.7 448.4
352.5 371.8
681.2 714.8
10% uncertainty
R0 (kN) Rm (kN)
759.4 857.5
744.4 857.5
625.6 703.1
583.6 686.3
608.2 692.9
508.6 586.7
404.5 445.7
568.5 633.7
Note: R0 and Rm denote critical nominal and worst block reactions, respectively.
listed in Table 7 and the distributions of nominal reactions for various cases considered are shown in Fig. 8, respectively. Comparing the results obtained from uncertain ship girder loads and those resulting from uncertain equivalent block stiffness, it is easily found that the effect due to uncertainty in block stiffness on the design is larger than that due to uncertainty in ship girder loads. This may be due to the fact that the variation of loads towards the lower bounds is usually helpful to structural safety, while increase in loads normally needs a more demanding design. However, the variation of block stiffness towards either the lower or upper bound may cause large reactions somewhere and possibly lead to structural problems. Therefore, although the same
Fig. 8.
degree of uncertainty is assumed, the actual variation range of equivalent block stiffness is larger than that of the ship girder loads making the design more stringent. 3.2. Results of robust design of docking blocks with uncertainty The numerical example in Section 3.1 is again employed for demonstration of the robust design. Only the uncertainty in equivalent stiffness of docking blocks is considered in the robust design. For simplicity, only three uncertainty parameters are assumed, namely a1, a2 and a3 for the equivalent stiffnesses of docking blocks at aft, central and fore parts, respectively. In particular, the uncertainty parameter a1 defines the
Nominal block reaction distribution with different block stiffness uncertainties.
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Table 8 Range limits of unsatisfactory degree functions Uncertainty parameter
Unacceptable
0.0 0.0 0.0
a1 a2 a3
Highly undesirable
Undesirable
Tolerable
Desirable
Highly desirable
ai5
ai4
ai3
ai2
ai1
0.025 0.0 0.025
0.05 0.025 0.05
0.075 0.05 0.075
0.10 0.075 0.10
0.20 0.10 0.20
+1 +1 +1
Table 9 Optimal block dimensions of the robust design under limited total volume of blocks Parameters
B (cm) C (cm) H1 (cm) H2 (cm)
Block No. 1
2
3–6
7–9
10–16
17–23
24–31
32–33
60.0 70.0 100.0 104.3
60.0 70.0 82.8 0.0
60.0 57.3 71.5 0.0
60.0 56.7 36.7 0.0
60.0 57.0 9.6 15.4
60.0 43.5 5.0 20.0
60.0 39.8 12.9 12.1
60.0 47.4 19.8 5.2
variation of the equivalent stiffnesses of blocks 1–6, while the uncertainty parameters a2 and a3 define those of blocks 7–23 and blocks 24–33, respectively. The corresponding range limits are shown in Table 8. The limiting total volume of the docking blocks Vallowable is set to be 4:240 106 cm3 . Table 9 gives the optimal dimensions of blocks obtained from robust design. The maximum or worst unsatisfactory degree for the optimal design is 0.4297 and the uncertainty parameters a1, a2 and a3 are 0.103, 0.77 and 0.103, respectively. It is understood that all of them are within the desirable range. For comparison, robust design is again carried out by maximising the minimum of uncertainty parameters directly without the use of the unsatisfactory degree functions. In this case, the optimal values for all uncertainty parameters are 0.095. It is readily found that the uncertainty parameters a1 and a3 are in the tolerable range while the uncertainty parameter a2 is in the desirable range. In this case, the maximum or worst unsatisfactory degree for the optimal design is 0.5546, which is larger than the maximum or worst unsatisfactory degree for the optimal design resulting from the proposed method using the unsatisfactory degree functions. The numerical results demonstrate that the optimal result obtained from minimisation of the maximum value of unsatisfactory degree functions is superior to that resulting from direct maximisation of the minimum value of uncertainty parameters. All uncertainty parameters are in the desirable range and the corresponding maximum value of unsatisfactory degree functions is also smaller. This is because direct maximisation of the minimum of uncertainty parameters implies that the design is optimal with respect to the global robust reliability and hence all uncertainty parameters tend to be uniform.
This process somehow treats all uncertain variables equally and ignores different properties of individual uncertain variables. However, the characteristics of individual uncertain variables in the proposed method are taken into consideration by setting their own range limits and more reasonable results can be obtained. 4. Conclusions The paper discusses issues of optimal design and robust design of docking blocks with uncertainty and proposes the corresponding mathematical models. The convex model is employed to describe the uncertainties in ship girder loads and equivalent stiffness of docking blocks. Numerical examples show that the uncertainties do affect the optimal solution and lead to increase in volumes of docking blocks compared to the optimal solution obtained from deterministic optimisation. Increased volumes are required to protect the structure from variability of uncertain variables. The effect of uncertainty in block stiffness on the design is larger than that due to uncertainty in ship girder loads. The robust design of docking blocks with uncertainty can be achieved by minimisation of the maximum unsatisfactory degree of the uncertainty parameters. The numerical results show that the design scheme resulting from minimising the maximum unsatisfactory degree is superior to that obtained from directly maximising the minimum of uncertainty parameters without the use of unsatisfactory degree functions. Acknowledgements The work described in this paper has been partially supported by the Scientific Research Foundation for
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the Returned Overseas Chinese Scholars, State Education Ministry of China and the Committee on Research and Conference Grants, The University of Hong Kong, China.
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