Geometric Nonlinear Static And Dynamic Analysis Method Of Beam Structures And Its Application In Tower Crane

The complicated beam structures, which is represented by large luffing jib tower crane's metal skeletal structure, gets more and more abroad attention on its accurate and efficient geometric nonlinear static and dynamic analysis. Sponsored by the National Key Technology Research and Development Program (Grant No. 2006BAJ12B03) and the application background of D6560/50t—the large tonnage luffing jib tower crane for the main construction of large buildings , several theories and technologies for geometric nonlinear static and dynamic analysis of beam structures, such as global stability analysis, the large displacement analysis, free vibration analysis and transient dynamic analysis, are discussed in this dissertation.From the governing differential equation of lateral deflection of the tapered Bernoulli-Euler beam, the slope-deflexion equations are derived and transformed into the finite element formulation. The exact stiffness matrix of the tapered beam are proposed whose inertia moment are quadratic and quartic, respectively. The proposed exact stiffness matrix will lead to the exact numerical solution by modelling each member by only one element in the buckling analysis. The exact static shape functions of the two tapered beam are presented, and can be used in developing the exact stiffness matrix through classic finite element method. The integral and differential formulation of exact stiffness matrix of the tapered beam are proposed expressed in the exact shape function, and the differential formulation is more concise and effective. The model reduction technique of transfer matrix in the stability analysis of beam strcutures is presented. The noncollinear branch chain substructure with elastic supports is modelled as a super element, and the force-displacement relation between both ends is developed by the transfer matrix method, so the order of the model is greatly reduced and guarantee the accuracy of the computation.A Bernoulli-Euler beam mechanism for static analysis of large displacement, large rotation but small strain planar tapered beam structures is proposed using the Updated Lagrangian formulation and the moving coordinate method. The nonlinear effect of the bending distortions on the axial action is considered to manifest itself as an axial change in length. The aforementioned stiffness matrix with second-order effects is amended, by developing the auxiliary stiffness of bowing effect through the slope-deflexion equations and the exact shape fuctions, respectively. The moving coordinate method is employed for obtaining the large displacement total equilibrium equations, and the hinged-hinged moving coordinate system is constructed at the last updated configuration. The multipe load steps Newton-Raphson scheme is adopted for the solution of the nonlinear equations, and the global stiffness is modified due to the variation of axial load and configuration in each iteration.Unlike the dynamic stiffness method(DSM) to develop the dynamic stiffness matrix for free vibration of the uniform beam, the exact solutions of the differential equation governing the lateral vibration of an axially loaded uniform beam are found, and then the dynamic exact shape function are obtained by eliminating the intergal constants through the displacement boudary conditions. The differential formulation of dynamic stiffness matrix of the beam are proposed expressed in the dynamic exact shape function, and the differential formulation can be used to obtain the static exact stiffness matrix if the the static exact shape function is introduced. The principle of virtual work is adopted to elaborate the validity of the generalized differential formulation. To follow the Timoshenko magnification factor of lateral deflection, the approximated formula to compute axially loaded influence factor of the natrual frequencies for lateral vibration of Bernoulli-Euler beam is proposed, and the Wittrick-Williams algorithm and the dynamic stiffness matrix are used to prove that the maximum relative error of the proposed approximate formula is smaller than 2%, when axial load is between the postive and negative half of the first order Euler critical load.Due to the dynamic stiffness matrix cannot be used in transient dynamic analysis of beam structures, the lateral and axial displacement field are derived from the dynamic exact shape function of free lateral and axial vibration of the uniform beam, the mass matrix and stiffness matrix are developed by finite element method. Each element of the mass matrix and stiffness matrix is the transcendental function of the natrual frequencies and axial load. The second order effect is considered in the stiffness matrix while the self rotate inertia of the section is considered in the mass matrix. The dynamic equilibrium equations are presented for the transient dynamic analysis of beam structures, meanwhile the stable and efficient solution scheme is proposed to solve the equations.The validity and efficiency of the proposed theories and technologies are shown by solving various numerical examples found in the literatures. Finally, based on the former theories and technologies, the geometric nonlinear static and dynamic analysis of the D6560/50t's shuttle-type tapered luffing jib and overall structrues are implemented. From the result of the stability analysis, as the lifting range increases, the global Euler critical load of the luffing jib is monotonously decreasing, while the extreme lifting moment of the jib is increasing, and reach the peak value at the lifting range 75.5123m. The buckling safety factor of the overall structures is increasing as the lifting range increases. From the result of the dynamic stiffness analysis of the overall structures, the first frequency is dive motion of the luffing jib induced by the swinging of the crane shaft. when the luffing angle inceases, that means lifting range decreases, the first frequency is monotonously decreasing, and the frequency error due to the second order effect is increasing. When the luffing angle exceeds 57degree, the frequency error exceeds the engineering allowable error 5%, so it should consider the impact on the frequencies due to the axial load. From the result of the large displacement analysis of the overall structures, if the luffing angle is less than 63degree, the large displacement analysis can be replaced by the second-order effect analysis, the maximum relative error is smaller than the engineering allowable error 5%, while the luffing angle exceeds 63degree, only the large displacement analysis lead to the more accurate result. So the proposed theories and technologies provide strong support for the research and development of the D6560/50t luffing jib tower crane.