Reliability-Based Design Optimization for Crane Metallic Structure Using ACO and AFOSM Based on China Standards

Table 1

Mathematical model of the RBDO for OTCMS.

Design variable mean values of the optimization problem
Design variable mean value
Symbol
Random design variables and parameters of the optimization problem
Variable and parameter
Symbol
Objective function: minimizing the cross-sectional area of the girder
is taken equal to 20 mm which facilitates welding and is set equal to 15 so as to guarantee the local stability of flange overhanging and the rail installation.
Inequality constraints of the RBDO problem
Reliability constraints	Limit state functions		Description
			Maximum stress combined at dangerous point () (1-1 section).
			Normal stress at dangerous point () (1-1 section).
			Normal stress at dangerous point () (1-1 section).
			Maximum shear stress at the middle of main web (2-2 section).
			Horizontal shear stress in the flange (2-2 section).
			Fatigue strength at dangerous point () (1-1 section)
			Fatigue strength at dangerous point () (1-1 section)
			Vertical static deflection of the midspan
			Horizontal static displacement of the midspan
			Vertical natural vibration frequency of the midspan
			Horizontal natural vibration frequency of the midspan
Deterministic constraints	Nature of constraint		Description
	Stability constraint		Height-to-width ratio of the box girder
	Boundary constraint		Upper bounds of design variables
	Boundary constraint		Lower bounds of design variables
Material permissible normal stress and permissible shear stress /; represents material yield stress; and are the welded joint tensile fatigue permissible stresses for points (4) and (5), respectively; and are the vertical and horizontal permissible static stiffnesses, which are taken equal to and , respectively, and is the span of the girder; and are the vertical and horizontal permissible dynamic stiffnesses, which are taken equal to 2~4 Hz and 1.5~2 Hz, respectively. and are the lower and upper bounds of the design variable .

where
, , ,   
, , , ,    
, , , ,   
are the bending moments, are the shear forces, and are the torques at different locations. denote the inertial moduli, denote the strength moduli, and denotes the top flange static moment. represent net area at different sections, represents the height of rail, and represents the height of girder at the end-span. is elastic modulus, is gravity constant, is impact factor, is computing coefficient of the crane bridge, and is the net elongation of the steel wire rope. The other parameters are the same as before.