Abstract
This paper focuses on the weight reduction optimization of a forearm of a bucketwheel stacker reclaimer considering uncertainties of structural parameters, material properties, loads, and surrogate model. However, the optimization problem is a highdimensional problem, with dozens of independent variables, which has negative effects on the optimization efficiency. Considering that millions of iterations are required for the reliabilitybased 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 reliabilitybased 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 reliabilitybased 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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Funding
This work was supported by the National Natural Science Foundation of China (51605071 and U1608256).
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Appendices
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 (onefactoratime) 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 μ_{i} and standard deviation σ_{i} of their EEs. Suppose that r groups of OAT experiments are designed. The Morris method is expressed as
where x = [x_{1},x_{2},...,x_{n}] is the normalized vector of an input, each component of which is set to [0, 1], y(x) is the output, d_{i}(x) is the EE of the ith variable, Δ is the change of the ith variable, μ_{i} and σ_{i} 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 μ_{i} indicates that the overall influence of the ith variable is large and a large value of σ_{i} indicates that the ith variable is significantly nonlinear or has a significant correlation with others. Therefore, a variable with a large value of μ_{i} or σ_{i} should be regarded as a main factor.
The results of the sensitivity analysis performed on the weight, the maximum stress, the maximum displacement, the firstorder natural frequency under noload situation, and the secondorder natural frequency under fullload situation are shown in Fig. 13, where the center points represent μ_{i}s, the oneside lengths of the error bars represent σ_{i}s, and the ranges of μ_{i} ± σ_{i} for each variable are shown. Herein, if μ_{i} + σ_{i} ≥ 500 kg for the weight, μ_{i} + σ_{i} ≥ 8 MPa for the maximum stress, μ_{i} + σ_{i} ≥ 0.005 m for the maximum displacement, and μ_{i} + σ_{i} ≥ 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, l_{d}, l_{u}, t_{13}, t_{14}, t_{15}, t_{16}, t_{23}, t_{24}, t_{25}, w_{13}, w_{14}, w_{15}, w_{16}, w_{23}, w_{24}, w_{25}, w_{26}, w_{33}, w_{35}, w.
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)
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 bucketwheel stacker reclaimer to work normally at the maximum wind pressure, the wind pressure p is 250 Pa. The wind load is calculated as
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 doubleside structure, the wind load on the leeward side decreases due to the windshield of the windward side. It is calculated as
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 1.7pw_{3} on the windward side and 1.7ηpw_{3} on the leeward side.

(2)
Loads on the bucket wheel
Loads on the bucket wheel are mainly composed of its own weight m_{0}, the weight of material in the bucket wheel m_{1} and the cutting force F, where m_{0} is known in advance.
The weight of the material m_{1} is calculated at the situation of 1/4 full load of all the buckets (Li 2013), which is written as
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)
where P is the power of the cutting force, P_{h} is the power consumed for lifting the material, η is the transmission efficiency, P_{a} is the rated power of the drive motor, F is circumferential cutting force, v_{c} is the circumferential speed on the tip of the bucket wheel, Q_{L} 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

(3)
Loads on the conveyer
Loads on the conveyer are composed of the weight of the conveyer system m_{2} known in advance, and the weight of material m_{3} calculated as (Li 2013)
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 v_{b} 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 reliabilitybased weight optimization can be conducted conveniently with the help of the files.
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. Reliabilitybased weight reduction optimization of forearm of bucketwheel stacker reclaimer considering multiple uncertainties. Struct Multidisc Optim (2020). https://doi.org/10.1007/s0015802002627y
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Keywords
 Forearm of bucketwheel stacker reclaimer
 Reliabilitybased weight reduction optimization
 Multiple uncertainties
 Kriging model