A general hybrid force-based method for structural analysis

The form of the energy function (i.e. Total, Hellinger-Reissner, Hu-Washizu or Complementary energy functions) has a significant influence on FEM performance. Motivated by the ability of the force-based method to satisfy the equilibrium equation and ability of the displacement-based method to satisfy compatibility equation, this thesis proposes a mathematical framework, namely the ‘Hybrid Force-Based Method’ which employs two physical concepts; the Total and Complementary Potential Energy functions. Satisfaction of both the Total and Complementary Potential Energy function is critical to the success of the Hybrid Force-Based Method. The Hybrid Force-Based Method is constructed using these two independent energy functions in order to perform inelastic structural analyses. The method has been proposed, implemented and evaluated across the entire structure, element, section and material domains first considering each domain separately and then in combination. The equilibrium and compatibility equations are satisfied simultaneously by discretisation of these two equations, and accuracy is controlled by specifying the upper and lower bounds of the results. Outcomes following evaluation of the proposed method can be classified into the following three categories: (i) structure-level performance (see Chapter 2), (ii) material-level performance (see Chapter 3), and (iii) element level performance (see Chapter 4). The proposed Hybrid Force-Based Method is constructed by deriving the governing equations directly from the Total and Complementary Potential Energy functions, leading to two distinct variants of the hybrid approach (i) the so-called ‘augmented Hybrid Force-Based Method’, and (ii) the so-called ‘unaugmented Hybrid Force-Based Method’. A number of numerical posterior process tests were devised and used to demonstrate the performance of these two variations of the hybrid method (see Sections 2.9.4.1 and 2.9.5.1) to demonstrate those methods ability in convergence in contrast to the Large Increment Method. Due to the occurrence of numerical instabilities experienced when using various established solution algorithms in solving the fundamental equations at the material level, within implicit approach (such as the Standard Implicit Method, the Cutting Plane Method, and the Closest Point Projection Method). A new form of the constitutive equation solver is proposed in Sections 3.9, referred to as the General Implicit Method (GIM). It is shown that the GIM can be implemented both in the strain and stress domains, and is therefore appropriate for use in both the displacement- and the force-based solution family of methods. The GIM is then evaluated by comparing its predictions to those of other common solution algorithms for inelastic analysis. Performance evaluation involves the use of a new error indicator that guarantees the uniqueness and accuracy of a solution in both the stress and the strain domains. Three iso-error maps serve to emphasis the accuracy, reliability, and computational performance of the General Implicit Method as a solution method compared to those are evaluated for the defined Stress Increment Ratio. The fundamental equations at the element level are followed, based on structured fibre discretisation. The decomposition of the various degrees of freedom into deformational and rigid-body motion serve as a mechanism by which independent equilibrium equations can be determined for each element. The subsequent equation is able to involve axial force, torsion, and both in and out of plane moments while a general form of shear strain distribution is also involved. The original form of the solution at the cross section of the elements leads to novel governing equations that are based on the characteristics of the hybrid force-based approach. The numerical evaluation in Section 4.11.7.1 demonstrates the performance of the proposed method. The newly defined error indicators demonstrate the accuracy and computational performance of the method and the uniqueness of the solution in satisfying both the equilibrium and compatibility equations for Euler-Bernoulli, Timoshenko, and Reddy strain distributions across the element section. Further to the structured fibre, distributed, semi-distributed and concentrated inelastic approach elements, as a simplified form of the element are implemented and evaluated. Although performance of those original formulations is evaluated independently in in comparison with the conventional approaches, compatibility of those as an important issue is followed as well. The numerical evaluation demonstrates higher accuracy and reliability by following the proposed method, further to the higher computational performance respect to the conjugate approaches