Reliability-based weight reduction optimization of forearm of bucket-wheel stacker reclaimer considering multiple uncertainties


This paper focuses on the weight reduction optimization of a forearm of a bucket-wheel stacker reclaimer considering uncertainties of structural parameters, material properties, loads, and surrogate model. However, the optimization problem is a high-dimensional problem, with dozens of independent variables, which has negative effects on the optimization efficiency. Considering that millions of iterations are required for the reliability-based optimization, the finite element model can cause overwhelming computational cost. In addition, due to its complex structure and working conditions, multiple uncertainties exist in practical applications and affect the reliability of a design, especially the uncertainty of the surrogate model. To address these challenges, the sensitivity analysis is performed to improve the optimization efficiency by selecting main factors. The Kriging model with high accuracy is constructed to reduce the computational cost. In order to improve the optimization efficiency further, the deterministic optimization is performed firstly, and the optimal design is used as the initial point of the reliability-based optimization algorithm. For estimating the reliability, the multiple uncertainty models are constructed. Finally, according to the design requirements and taking the multiple uncertainties into account, the reliability-based optimization is proposed and carried out. The result proves that the weight is reduced greatly and the reliability is kept at a high level.

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  1. Alshafiee M, AlAlaween WH, Markl D et al (2019) A predictive integrated framework based on the radial basis function for the modelling of the flow of pharmaceutical powders. Int J Pharm 568:118542

    Article  Google Scholar 

  2. Cheng K, Lu ZZ, Zhou YC, Shi Y, Wei YH (2017) Global sensitivity analysis using support vector regression. Appl Math Model 49:587–598

    MathSciNet  Article  Google Scholar 

  3. Choi SK, Grandhi RV, Canfield RA (2007) Reliability-based structural design. Springer, London

    Google Scholar 

  4. Han D, Zheng JR, Zhou JY (2016) A multi-objective optimization method using Kriging model and parallel point adding strategy. Mech Sci Technol Aerosp Eng 35(11):1715–1720. [in Chinese]

  5. He HW, Yi L, Peng JK (2017) Combinatorial optimization algorithm of MIGA and NLPQL for a plug-in hybrid electric bus parameters optimization. Energy Procedia 105:2460–2465

    Article  Google Scholar 

  6. Hu QG, Liu YP (2014) Mechanical reliability design and application. Publishing House of Electronics Industry, Beijing [in Chinese]

    Google Scholar 

  7. Huang ZL, Jiang C, Zhou YS, Zheng J, Long XY (2017) Reliability-based design optimization for problems with interval distribution parameters. Struct Multidiscip Optim 55(2):513–528

    MathSciNet  Article  Google Scholar 

  8. Li PF (2013) Finite element analysis and optimal design for pitch structures of bucket wheel stacker-Reclaimer. Harbin Engineering University, Dissertation [in Chinese]

    Google Scholar 

  9. Li X, Gong CL, Gu LX, Jing Z, Fang H (2019) A reliability-based optimization method using sequential surrogate model and Monte Carlo simulation. Struct Multidiscip Optim 59:439–460

    MathSciNet  Article  Google Scholar 

  10. Ma JG (2018) Research on the reliability analysis method in optimal design considering uncertain factors. Shenyang University of Technology, Dissertation [in Chinese]

    Google Scholar 

  11. Niu ZJ, Sun ZJ, Zhu H, Zhang J (2017) Response surface model and genetic algorithm-based multi-objective optimization of Stator structures of hollow-type traveling wave ultrasonic motors. Proc Inst Mech Eng C J Mech Eng Sci 231(12):2187–2199

    Article  Google Scholar 

  12. Richter GM, Acutis M, Trevisiol P, Latiri K, Confalonieri R (2010) Sensitivity analysis for a complex crop model applied to durum wheat in the Mediterranean. Eur J Agron 32(2):127–136

    Article  Google Scholar 

  13. Saltelli A, Tarantola S, Campolongo F, Ratto M (2004) Sensitivity analysis in practice: a guide to assessing scientific models. John Wiley & Sons, Chichester

    Google Scholar 

  14. Shang P, Hu YZ, He LH, Guan YM (2013) The modal analysis of the Main steel structure of bucket wheel stacker reclaimer. Adv Mater Res 690-693:3121–3124

    Article  Google Scholar 

  15. Sun W, Peng X, Dou J, Wang LT (2020) Surrogate-based weight reduction optimization of forearm of bucket-wheel stacker Reclaimer. Struct Multidiscip Optim 61:1287–1301

    Article  Google Scholar 

  16. Touhami HB, Lardy R, Barra V, Bellocchi G (2013) Screening parameters in the pasture simulation model using the Morris method. Ecol Model 266:42–57

    Article  Google Scholar 

  17. Wang D (2012) The vibration characteristics of the Stacker & Reclaimer Cantilever Beam. Tsinghua University, Dissertation [in Chinese]

    Google Scholar 

  18. Wang FK, Mamo T (2018) A hybrid model based on support vector regression and differential evolution for remaining useful lifetime prediction of lithium-ion batteries. J Power Sources 401:49–54

    Article  Google Scholar 

  19. Wang J, Sun ZL, Yang Q, Li R (2017) Two accuracy measures of the Kriging model for structural reliability analysis. Reliab Eng Syst Saf 167:494–505

    Article  Google Scholar 

  20. Wang LT, Sun W, Long YY, Yang X (2018) Reliability-based performance optimization of tunnel boring machine considering geological uncertainties. IEEE Access 6:19086–19098

  21. Wu S, Guo RQ, Li Z, Zhou Y (2014) Structural analysis and optimization Design of Bucket-Wheel Body of bucket-wheel stacker/Reclaimer. Machinery Des Manuf 11:5–8 [in Chinese]

  22. Wu ZP, Wang DH, Okolo NP, Hu F, Zhang WH (2016) Global sensitivity analysis using a Gaussian radial basis function metamodel. Reliab Eng Syst Saf 154:171–179

    Article  Google Scholar 

  23. Wu XH, Fang YD, Zhan ZF, Liu X, Guo G (2017) A corrected surrogate model based multidisciplinary design optimization method under uncertainty. SAE Int J Commer Veh 10(1):106–112

    Article  Google Scholar 

  24. Xing L, Song XG, Scott K, Pickert V, Cao WP (2013) Multi-variable optimisation of PEMFC cathodes based on surrogate Modelling. Int J Hydrog Energy 38(33):14295–14313

    Article  Google Scholar 

  25. Yang MY (2013) Bucket wheel stacker reclaimer forearm rack structure of static and dynamic analysis. Kunming University of Science and Technology, Dissertation [in Chinese]

    Google Scholar 

  26. Yang GB, Li Y, Chen DF (2015) Analysis of reclaiming process of bucket wheel stacker-reclaimer based on DEM simulation. In: International Conference on Human Centered Computing. Springer International Publishing, In, pp 69–79

    Google Scholar 

  27. Zhang XQ (2018) Reliability analysis and optimal design methods for mechanical systems under Aleatory and epistemic uncertainties. University of Electronic Science and Technology of China, Dissertation [in Chinese]

    Google Scholar 

  28. Zhang ZH, Xu L, Flores P, Lankarani HM (2014) A Kriging model for dynamics of mechanical systems with revolute joint clearances. J Comput Nonlinear Dyn 9:031013

    Article  Google Scholar 

  29. Zhang S, Sun YJ, Cheng Y, Huang P, Oladokun MO, Lin Z (2018) Response-surface-model-based system sizing for nearly/net zero energy buildings under uncertainty. Appl Energy 228:1020–1031

    Article  Google Scholar 

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This work was supported by the National Natural Science Foundation of China (51605071 and U1608256).

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Correspondence to Lintao Wang.

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Appendix 1: Morris method

Since the number of model evaluations required for a sensitivity analysis is positively correlated with the number of variables, and the Morris method requires fewer evaluations than other methods (Saltelli et al. 2004), the Morris method is used for the sensitivity analysis in this paper. The Morris method calculates elementary effects (EE) for each variable based on several groups of OAT (one-factor-a-time) experiments (Richter et al. 2010; Touhami et al. 2013), and analyzes the overall influences and nonlinearities of the variables on the output by calculating the mean of the absolute values μi and standard deviation σi of their EEs. Suppose that r groups of OAT experiments are designed. The Morris method is expressed as

$$ {d}_i(x)=\frac{y\left({x}_1,\cdots, {x}_{i-1},{x}_i+\varDelta, {x}_{i+1},{x}_n\right)-y\left(\mathbf{x}\right)}{\varDelta } $$
$$ {\mu}_i=\frac{1}{r}\sum \limits_{j=1}^r\left|{d}_i^{(j)}\left(\mathbf{x}\right)\right| $$
$$ {\sigma}_i=\sqrt{\frac{1}{r}\sum \limits_{j=1}^r{\left({d}_i^{(j)}(x)-\frac{1}{r}\sum \limits_{k=1}^r{d}_i^{(k)}\left(\mathbf{x}\right)\right)}^2} $$

where x = [x1,x2,...,xn] is the normalized vector of an input, each component of which is set to [0, 1], y(x) is the output, di(x) is the EE of the ith variable, Δ is the change of the ith variable, μi and σi represent the overall influence and the nonlinearity of the ith variable on the output, respectively, and they are two indicators for evaluating the sensitivity in the Morris method. A large value of μi indicates that the overall influence of the ith variable is large and a large value of σi indicates that the ith variable is significantly nonlinear or has a significant correlation with others. Therefore, a variable with a large value of μi or σi should be regarded as a main factor.

The results of the sensitivity analysis performed on the weight, the maximum stress, the maximum displacement, the first-order natural frequency under no-load situation, and the second-order natural frequency under full-load situation are shown in Fig. 13, where the center points represent μis, the one-side lengths of the error bars represent σis, and the ranges of μi ± σi for each variable are shown. Herein, if μi + σi ≥ 500 kg for the weight, μi + σi ≥ 8 MPa for the maximum stress, μi + σi ≥ 0.005 m for the maximum displacement, and μi + σi ≥ 0.01 Hz for the natural frequencies, the ith variable is considered as a variable with great influence on the outputs. It is found that all the variables have great influences on the weight while only some of them on the other outputs. Since the goal is to minimize the weight which is positively correlated with all the variables, main factors can be selected based on the latter four outputs and the remaining variables are set to their lower bounds. Finally, the selected main factors are h, ld, lu, t13, t14, t15, t16, t23, t24, t25, w13, w14, w15, w16, w23, w24, w25, w26, w33, w35, w.

Fig. 13

Sensitivity analysis results. (a) Sensitivity analysis on weight. (b) Sensitivity analysis on the maximum stress. (c) Sensitivity analysis on the maximum displacement. (d) Sensitivity analysis on the first-order natural frequency. (e) Sensitivity analysis on second-order natural frequency

Appendix 2: Load calculation

Loads on the forearm can be divided into three parts: wind load, loads on the bucket wheel and loads on the conveyer.

  1. (1)

    Wind load

According to the design rules of cranes, the maximum wind pressure during the work is set to 250 Pa. In order to allow the bucket-wheel stacker reclaimer to work normally at the maximum wind pressure, the wind pressure p is 250 Pa. The wind load is calculated as

$$ {F}_{\mathrm{w}}= CpA{\sin}^2\theta $$

where C is the wind load coefficient, set as 1.7, A is the area of the windward surface, and θ is the angle between the wind direction and the normal direction of the surface. In order to maximize the effect of the wind load, θ is set to 90°.

For a double-side structure, the wind load on the leeward side decreases due to the windshield of the windward side. It is calculated as

$$ {F}_{\mathrm{w}}^{\prime }=\eta {F}_{\mathrm{w}} $$

where η is the windshield coefficient, determined by the interval ratio and the full ratio of the windward side.

As the wind load is applied as a line pressure, and according to (36), where l is the length of the beam and f is the line pressure, the value of the line pressure is set as 1.7pw3 on the windward side and 1.7ηpw3 on the leeward side.

$$ {F}_{\mathrm{w}}=1.7 pl{w}_3= fl $$
$$ {F}_{\mathrm{w}}^{\prime }=1.7\eta pl{w}_3={f}^{\prime }l $$
  1. (2)

    Loads on the bucket wheel

Loads on the bucket wheel are mainly composed of its own weight m0, the weight of material in the bucket wheel m1 and the cutting force F, where m0 is known in advance.

The weight of the material m1 is calculated at the situation of 1/4 full load of all the buckets (Li 2013), which is written as

$$ {m}_1=\frac{1}{4} V\rho z $$

where V is the volume of a bucket, ρ is the density of the material, and z is the number of the buckets.

The cutting force is calculated according to the power of the drive motor, which is written as (Li 2013)

$$ P+{P}_{\mathrm{h}}=\eta {P}_{\mathrm{a}} $$
$$ P=\frac{F{v}_{\mathrm{c}}}{1000} $$
$$ {P}_{\mathrm{h}}={Q}_{\mathrm{L}}\rho gh $$

where P is the power of the cutting force, Ph is the power consumed for lifting the material, η is the transmission efficiency, Pa is the rated power of the drive motor, F is circumferential cutting force, vc is the circumferential speed on the tip of the bucket wheel, QL is the theoretical productivity, and h is the lifting height of the material, approximately equal to the radius of the bucket wheel.

There is also a pressure along with the radius of the bucket wheel from the material on the bucket. According to the empirical formula (Li 2013; Yang 2013), it is written as

$$ {F}^{\prime }=0.3F $$
  1. (3)

    Loads on the conveyer

Loads on the conveyer are composed of the weight of the conveyer system m2 known in advance, and the weight of material m3 calculated as (Li 2013)

$$ {m}_3=\frac{fQL}{3600{v}_{\mathrm{b}}} $$

where f is the dynamic load factor, Q is the rated production capacity (the unit is kg/h), L is the length of the forearm, and vb is the speed of the conveyer.

Appendix 3: Replication of results

In order to facilitate the replication of results in this paper, the MATLAB code files for calculating the POFs of a design are provided as supplementary material. Table 11 shows brief descriptions of all the files. The reliability-based weight optimization can be conducted conveniently with the help of the files.

Table 11 MATLAB code files in the supplementary material

There are 4 files in total, where the “ModelInfo.mat” contains all the messages used to construct the Kriging model. “POF_calculation.m” is the main program which is used to calculate the POFs of all the constraints under a design. “rlh.m” is used to generate a random Latin hypercubic sampling. “ReturnX.m” is used to convert each design variable value to a number between [0, 1].

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Sun, W., Peng, X., Wang, L. et al. Reliability-based weight reduction optimization of forearm of bucket-wheel stacker reclaimer considering multiple uncertainties. Struct Multidisc Optim (2020).

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  • Forearm of bucket-wheel stacker reclaimer
  • Reliability-based weight reduction optimization
  • Multiple uncertainties
  • Kriging model