Weak Form Quadrature Element Analysis On Stability And Dynamic Response Of Composite Stiffened Laminates

Composite laminates have become one of the indispensable materials in the civil,mechanical,marine engineering and aerospace industries.Especially in the aerospace industry,composite laminates are favored for their light weight and excellent mechanical properties.For the purpose of aircraft weight reduction,composite laminates and their reinforced form are usually applied in the aircraft’s components like girders,fuselage skin,wings and tails.During service,these components may be subjected to various static loads,temperature loads,and impact loads due to high-velocity air currents.Therefore,it is necessary to investigate the thermal mechanical buckling and dynamic response of the composite laminates.Due to the complex geometry,complicated structural modeling of stiffened shells and large computational scale,the nonlinear buckling and dynamic response of stiffened shells is still a difficult problem.Especially,there is still a lack of investigations focusing on thermal bifurcation buckling and thermal mechanical bifurcation buckling of stiffened laminates.In view of the above problems,the thermal mechanical bifurcation buckling and dynamic response parameters of the cylindrical composite laminated shell and its reinforced form are analyzed in this paper.The research can help to optimize the structural design and effectively reduce the structural weight while ensuring safety.The weak form quadrature element method(abbreviated as QEM)is a differential equation solving method based on the variational principle and high-order interpolation approximation.Numerical integral and differential quadrature method are adopted in the numerical approximation.The high-order approximation of the field variables can greatly simplify the processing of dealing with the complex solution domain.The structures of composite stiffened laminates are complex,which leads to large computational efforts in the analysis of nonlinear bifurcation buckling and dynamic response.In order to reduce the computational efforts,improve the calculation efficiency and avoid the common locking problems,a shell model with five degrees of freedom and a beam model with six degrees of freedom are adopted in the formulation of stiffened laminates,basing on QEM.The thermal mechanical buckling and dynamic response of cylindrical stiffened shells are performed in this paper.The specific content is as follows:(1)Based on QEM,the composite laminate shells and stiffened shells model are built.Combined with arc length method and branch-switching method,a kind of path identification algorithm is constructed to accurately and efficiently identify the bifurcation points and track the bifurcation paths.The algorithm solves the problem that the general finite element program cannot detect the bifurcation buckling point due to the degradation of the global stiffness matrix of the structure at the bifurcation point,and realize the identification of all bifurcation point and the accurate tracking of all bifurcation paths of the composite laminate shells and stiffened shells.(2)The total Lagrangian method combined with the arc length method are used to formulate nonlinear equilibrium equations of cylindrical laminated shells.The bifurcation buckling points are distinguished by monitoring the eigenvalues of the tangent stiffness matrix.Disturbances corresponding to the eigenvectors are applied to lead the structure to the buckling path.It has been shown that the laminate sequences at the middle position have more influence than those at top and bottom position on the number of bifurcation points.The number of bifurcation points increase when the curvature increase or thickness decrease.(3)The bifurcation buckling computational process for laminated shells is extended to stiffened shells.It is indicated that the more 90o laminates or less stiffeners can increase the number of bifurcation points.However,the laminate sequence and scheme of stiffeners have little effect on the number of bifurcation points.In addition,increasing the height to breadth radio,number of stiffeners and the number of 90o laminates of the shell can rise the maximum value of the point load.(4)The temperature is introduced and the thermal bifurcation buckling of laminated cylindrical shells and stiffened shells are given.It has been found that the laminate sequences at the middle position have less influence than those at top and bottom position on the number of bifurcation points.The thermal buckling curves of quasi-isotropic shells rise monotonically,but snap-back phenomenon exists in anisotropic shells.Reducing the curvature of the cylindrical shell,increasing the thickness of the cylindrical shell,the number of ribs,and the height to breadth ratio of the stiffener can effectively reduce the number of bifurcation buckling paths.Additionally,under the thermal-mechanical load,the maximum load on the main buckling path of the cylindrical shell with or without stiffeners will increases when thetemperature rises,but the temperature has no effect on the number of bifurcation buckling paths.(5)The Newton-Raphson iteration method and Newmark method are used to derive the nonlinear dynamic equilibrium equations of the shell subjected to the impact load on the edges.The influence of ply angle,shell curvature,shell thickness,peak and duration of different impact loads,and different boundary conditions on the nonlinear dynamic response of the shell are investigated.It is found that the top and bottom laminate sequence have a great influence on the shape of the nonlinear dynamic response curve at the center point,but the middle laminate sequence has little effect on this.Increasing the number of 90o laminate at the middle position,reducing the curvature of the shell,and increasing the thickness of the shell can effectively improve the impact resistance of the cylindrical shell.The higher the peak impact load and the shorter the duration,the greater the deflection of the cylindrical shell.(6)The computational process of dynamic response for laminated shells is extended to stiffened shells.The results show that increasing the thickness of the shell,90o laminate at the middle position,the number of stiffeners,the height to breadth ratio of the stiffeners section and the reducing the curvature of the cylindrical shell all contribute to the improve the impact resistance.A parametric analysis on the thermal mechanical buckling and dynamic response of cylindrical laminate stiffened shell is presented in this paper.It has been proved that QEM has unique advantages in the computation.Parametric analysis provides important theoretical guidance for the design and manufacturing of cylindrical stiffened or unstiffened laminates.