Abstract This study is devoted to strain-based formulation for a curved beam. Arches with parabolic geometry, which have a variety of applications, belong to this structural type. Dependency of the curvature radius to the arch length creates some complexities in the solution process. To analyze these complex structures, a two-node beam with six degrees of freedom is suggested by utilizing closed-form solution and the stiffness-based finite element method. Considering the effect of shear deformation, and incorporating equilibrium conditions into the finite element model, lead to the exact strains. Displacements and explicit stiffness matrix are found based on these exact strains. To validate the efficiency of the author's formulation, seven numerical tests are performed. The outcomes demonstrate that by employing only a single element, the locking-free answers can be found.

Utilizing the straingradient notation process and the free formulation, an efficient way of constructing membrane elements will be proposed. This strategy can be utilized for linear and geometric non-linear problems. In the suggested formulation, the optimization constraints of insensitivity to distortion, rotational invariance and not having parasitic shear error are employed. In addition, the equilibrium equations will be established based on some constraints among the strain states. The authors' technique can easily separate the rigid body motions, and those belong to deformational motions. In this article, a novel triangular element, named SST10, is formulated. This element will be used in several plane problems having irregular mesh and complicated geometry with linear and geometrically nonlinear behavior. The numerical outcomes clearly demonstrate the efficiency of the new formulation.

In this paper, two efficient elements for analyzing thin plate bending are proposed. They are a triangular element (THS) and a quadrilateral element (QHS), which have 9 and 12 degrees of freedom, respectively. Formulations of these elements are based on hybrid variational principle and analytical homogeneous solution of thin plate equation. Independent fields in hybrid functional are internal stress and boundary displacement field. The internal stress field has been calculated using analytical homogeneous solution and boundary field is related to the nodal degree of freedoms by the boundary interpolation functions. To calculate these functions, the edges of element are assumed to behave like an Euler-Bernoulli beam. The high accuracy and efficiency of the proposed elements are demonstrated in the severe tests.

This study is devoted to the analysis of the Reissner-Mindlin plate bending. In this paper, the hybrid-Trefftz strategy will be utilized. Two novel and efficient elements are formulated in details. They are a Triangular element (THT) and a quadrilateral element (QHT), which have 9 and 12 degrees of freedom, respectively. In this approach, two independent displacement fields are defined; one within the element and the other on the edges of the element. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. Boundary field is related to the nodal degree of freedoms by the boundary interpolation functions. To calculate these functions, the edges of the element are assumed to behave like a Timoshenko beam. The high accuracy and efficiency of the proposed elements and absence of the shear locking in these formulations are all proven, using various numerical tests.