Design and implementation of consistent 3D frame analysis program

zenodo.org

A consistent 3D frame analysis program allowing for non-prismatic beams with non-symmetrical cross-sections is developed in MATLAB. For non-symmetrical cross-sections the elastic and shear center are generally distinct and no obvious choice of beam axis is available. A user defined reference axis is therefore introduced in relation to which the cross-section centers are described. Beams meet consistently at frame corners by meeting at their reference axes, hereby forming a three dimensional reference axis for the frame. Although an infinity of reference axes can be chosen for any given frame, it is found that it is important to choose one, so that beams get lengths that corresponds well with reality. The element stiffness matrix implemented incorporates both shear flexibility from Timoshenko beam theory and warping flexibility from Vlasov beam theory. It is a combination of two matrices formulated using the principle of complementary energy by assuming section force fields. The torsion problem is formulated using section force fields from the differential equation for inhomogeneous torsion. The result is only consistent for prismatic beams, but seems not to be completely wrong for non-prismatic beams. The extension and bending problems are formulated using section force fields that allow for static equilibrium in elements. This combined with cross-section properties defined by MATLAB function handles gives a consistent result where one element is enough to model any beam. Uniform distributed loads are introduced consistently via energy equivalent nodal forces derived here in this report. Supports can be offsetted from nodes to where the user desires. This is done by transforming the equation system to offsetted coordinates, where it can be solved with the constrains from the offsetted supports. The transformation is achieved via a translation matrix valid for cross-sections. Offsets of supports should therefore stay in the vicinity of nodes, as this matrix cannot be used to introduce effects long into beams. Displacements along elements are found in an inconsistent manner, by using displacement fields for a prismatic beam without shear flexibility and without correcting for distributed loads. However, they still allow for smooth plots to be made of the deformed structure. Section forces along elements are found in a consistent manner, by using the section force fields from the formulation of the stiffness matrix and correcting for distributed loads. Note: The 7x7 translation matrix J on pages 22-23, which translates section forces between the elastic/shear centers and the reference axis, is based on a 6x6 matrix from lecture slides "Lecture 3: Finite elements by complementary energy" from the course Thin-Walled Beam Structures, Department of Mechanical Engineering, Technical University of Denmark, Fall 2011.