Research On Stability And Nonlinear Large Displacements Of Non-uniform Beam-bar System In Crane

Hoisting machinery with complex truss structure is the typical heavy lifting machinery, which has slender structure and can bearing complex load form. The analysis of the stability and geometric nonlinear is important to ensure working safely. In order to use materials rationally and reduce the weight of structural, crane metal structures are usually fabricated as non-uniform cross-section lattice components or their composite structure. The special boundary conditions and structure types induce greater difficulties in the calculation of the theoretical analysis and design, which are the focus points of many industries and scholars. Therefore, by the supporting of the National Key Technology Research and Development Program (Grant No.2011BAJ02B01-02), this paper researches on stability, large displacement geometric nonlinear and torsional behavior with the background and application object of the tapered lattice crane’s beam-bar structures.Based on the vertical and horizontal bending theory, the lateral displacement and stability of tapered cantilever are analyzed, whose inertia moment varies quadratic in axial. Using differential equation method, the deflection differential equation of non-uniform Bernoulli-Euler beam with axial force effects is established. The deflection exact expression and instability characteristic equation of the non-uniform cantilever are deduced under complex loads. Approximate formula of tapered cantilever member axial force influence coefficients is presented according to the Timoshenko factor and the exact factor. The results show that, while the taper range in commonly engineering, and the ratio of the axial force and Euler critical force is less than0.6, then the error of approximate expression is less than2%. Therefore, the tip deflection of the quadratic inertia moment variation cantilever can be approximated using a unified amplification factor formula multiplied by the maximum deflection caused by transverse load only. Meanwhile, based on the above research, axial force impact factor, impact factor of elastic support and buckling characteristic equation are expressed with taking into account the second-order effects and the effects on lateral displacement and stability of tapered cantilever by elastic constraints.Based on the deflection differential equations of the non-uniform beam, considering the shear deformation effect and the second order effects, the angle displacement equation of tapered Timoshenko beam, whose moment of inertia changes quadratic, is deduced and written as finite element formulation. Precise element tangent stiffness matrix of tapered beam with secondary changes in moment of inertia, taking into account the effect of shear deformation, is obtained. The precise beam elements can realize the transformation between the tapered beam and the uniform beam, taking into account of shear deformation or not, and with good performance degradation. The result of classical example shows that the accurate numerical solution can be obtained by simply performing as one element in analysis of stability and second-order effects, using the precision tapered beam element; For slender beams, shear deformation effects can be ignored, but when the beam is relatively short, the height is small, and the nonlinear effects has to be considered, the shear deformation effect must be included for achieving satisfactory accuracy; Shear deformation increases the lateral displacement of the structure and diminished the buckling capacity of beams.In order to study the torsional stiffness problem of space non-uniform beam-bar structure, space truss are decomposed into flat truss, and according to the equivalent force distribution principle of space truss in torsion, the complex spatial truss torsional problem is switched into simple single non-uniform cross-section truss’ bending problem in this paper, and then, the torsional stiffness expressions of the spatial tapered structure are deduced. First, taking the single-chip tapered truss structure as research object, the each member’s force and lateral displacement expressions of the tapered truss with different arrangement in the form of webs are given. Then, based on the flexibility coefficient of monolithic truss structure, considering comprehensively the effects of lateral stiffness and torsional stiffness caused by the chords, webs of space truss structure, the lateral displacement and torsional stiffness formulas are deduced for rectangular space truss structure, and the distribution coefficient formula of torque equivalent couple is also given. Final, the effect on lateral stiffness of tapered space beam caused by webs is addressed. When the member slenderness is relatively large, the effect of the lateral stiffness caused by webs can be ignored, and the non-uniform cross-section space frame structure can be equivalent to a solid-web tapered beam with moment of inertia quadratic variation. Result of examples show that the application of this method to calculate the torsional stiffness of space truss structure is absolutely correct and effective.Based on the virtual displacement principle and the updated Lagrangian (U.L.) format, a calculation method of Timoshenko beam element to analyze the geometrically nonlinear large displacement is proposed. By using angular displacement independent interpolation method, element interpolation function of tapered beam is presented by considering the shear deformation effect. Element geometrical nonlinear incremental virtual work equation of tapered plane beam with moment of inertia secondary variation is established by taking into account the axial force, shear, bending effect and its coupling terms simultaneously. Large displacement element tangent stiffness matrix of tapered beam is obtained. The presented method strictly takes account into the effect caused by shear deformation and tapered cross-section factors. It can be well degenerated into a uniform section beam element while the taper coefficient is1. Therefore, the proposed analysis method is suitable for large displacement geometric nonlinear structural analysis of tapered beam, uniform beam and other composite forms. Finally, large displacement geometric nonlinear analysis and calculation procedures of tapered beam-bar system considering the shear deformation are programed, and the numerical results show that, the proposed geometric nonlinear large displacement analysis modeling method and programe are correct.