Hybrid finite element formulation for geometrically nonlinear buckling analysis of truss with initial length imperfection

. This paper presents a novel hybrid FEM-based approach for nonlinear buckling analysis of truss with initial length imperfection. The contribution deals with establishing two types of truss finite element (perfection and imperfection element) considering large displacement based on displacement formulation and mixed formulation. Therefore, the hybrid global equation system is developed by assembling perfection and imperfection truss elements. The incremental-iterative algorithm based on the arc-length method is used to establish calculation programs for solving geometrically nonlinear buckling analysis of truss with initial length imperfection. Using a written calculation program, the numerical test is presented to investigate the equilibrium path for plan truss with initial member length imperfection.


Introduction
Many truss members have initial geometric imperfections as a result of manufacturing, transporting, and handling processes. This initial member imperfection significantly influences the buckling behaviour of the truss structure. In recent years, many research works addressed the influence of geometrical imperfection on the behaviour of truss structures [1][2][3][4]. For solving the buckling problem of truss structure, the finite element method is considered the most popular and efficient method. In geometrical linear finite element analysis, the length imperfection usually is calculated by adding equivalent loads to the nodal external force vector. However, in geometrical nonlinear analysis, it cannot be used. Generally, the solution of nonlinear buckling problem of truss based on displacement finite element formulation requires the implementation of length imperfection to the mater stiffness matrix. The operation of incorporating length imperfection considerably increases the difficulty in constructing and solving nonlinear incremental balanced equations of the system. For escaping difficulties of the mathematical treatment of imperfection, in [5] the author proposed an approach to formulate the nonlinear buckling problem of truss with imperfection based on mixed finite element formulation. The mixed model has significant advantage over displacement-based formulation model but increases the solving system dimension. Nowadays, the hybrid finite element approach is widely used to solve the nonlinear contact mechanic problem such as displacement-based finite elements are difficult to solve [6][7][8][9]. In this work, the author proposes a novel hybrid finite element approach for constructing the solving system of equation. The main idea is establishing two types of truss finite element considering large displacement based on displacement formulation and mixed formulation. The global equation system is developed by assembling two types of proposed truss elements. The solving algorithm of geometrically nonlinear buckling analysis of truss system is built by employing arc length method due to its efficiency to predict the proper response and follow the nonlinear equilibrium path through limit. Therefore, a new incremental-iterative algorithm for solving constructed system of equation and calculation program is established. The numerical results are presented to verify the efficiency of the proposed method.

Equilibrium equations for the truss elements considering large displacements
For hybrid finite element formulation, the research proposed to discretize the truss system into two types of the truss elements: first type element eI -perfection truss element; eIIimperfection truss element with initial length imperfection e  (shown in Fig.1).

Fig. 2. Truss elements eI and eII considering large displacements
The following is designated , , , X Y X Y : i th and j th nodal coordinates in global coordinate system before and after deformation; 0 L và L : distance between i th and j th node before and after deformation; , , , u u u u and 1 2 3 4 , , , P P P P : nodal displacements and forces in global coordinates; e P : resultant external force at the i th cross section after deformation; 5 e u P N   : resultant external force at the i th cross section after deformation; A : cross sectional area of truss element; E : elastic modulus of material; N : axial load of truss element. The length of the truss element after deformation is defined as The axial deformation of perfection truss element and imperfection truss element are obtained Work of internal axial force can be computed for each truss element as following For each truss element, the virtual external work can be defined as Combining equations (3) and (4), getting total work done by the applied forces and the inertial forces of a mechanical system   Based on the principle of virtual work, in equilibrium the virtual work of the forces applied to a system is zero, from equation (5) Expressing axial force through deformation and adding deformation from the equation (2) to equation (6), having the system (7)   e  e  II  II  II  e  k   T  T  e  e  I  II  e k u u u u u u u u u P Input incremental loading into the equation (7) and express in matrix format  I  I  I  I  I   e  e  e  e  II  II  II  II  II  e Where the tangent stiffness matrices are written  e  e  II  I  II  I  e  e  e  e  e  I  II  I  II  I  II  e  e  e  I

Hybrid equation of truss system
The hybrid global equation of truss system (9) can be established by assembling all perfection and imperfection truss elements which were established above "m" is a number of truss elements and "n" is number of unknowns; Using arc length technique [10][11] the incremental-iterative algorithm is established for solving nonlinear system.

Numerical investigations
Based on proposed incremental-iterative algorithm, the calculation program to solve the example is written using Matlab software. The system is composed of bars made of the same material and had the same geometrical properties (system is shown in Fig. 3), having length imperfection 5  The unknowns of truss system are designated as shown in Fig. 3 for solving the nonlinear equation based on hybrid formulation n, including ; P u P u   The calculating results are load-displacement and equilibrium path shown in Fig. 4.

Conclusion
The presented hybrid model does not require implementing the length imperfection to the mater stiffness matrix in geometrically nonlinear buckling analysis of truss system. This formulation provides an effective remedy to overcome the mathematical difficulty associated with the displacement-based formulation in establishing solving algorithm. Comparison to the mixed-based formulation, the current formulation has advantage of decreasing the dimension of the solving system.