Structures presenting kinematical inderterminacy are usually called mechanisms. This paper is entirely concerned with assemblies which reveal themselves to be mechanisms at a null value of the load. Among them a first distinction is made between infinitesimal and finite ones, the former being characterized by one or several directions of lower (but not zero) stiffness, whereas the latter show at least one finite admissible displacement for which none of the bars undergoes any elongation. Moreover, there exists the possibility to make a further distinction among the infinitesimal mechanisms, according to which is the order of the stiffness along the direction considered above. The way of evaluating this order is to perform a local analysis of the strain energy of the assembly, once the displacement field is parametrized in terms of a suitable parameter. By means of a finite element technique, this analysis can be easily performed through the numerical approach presented in this report.

How to recognize the order of infinitesimal mechanisms - a numerical approach

Bifurcation points require a special treatment in computational stability analysis of nonlinear structures. Tracing of the nonlinear equilibrium response, pinpointing of stability points and branch-switching are the foremost necessary procedures for bifurcation problems. The primary concern of this paper is to review the methodology, including all of these fundamental strategies for nonlinear bifurcation analysis. The underlying ideas in the background of the strategies are locally and globally convergent nonlinear solution methods. The global pinpointing procedures are recent contributions in the present paper, too. As the optional global paths to the target singular point, bypass, and homotopy path are introduced. The starting point, at which the global pinpointing will be initiated, may be an arbitrary equilibrium or nonequilibrium point. Regarding branch-switching, the homogeneous and particular solutions of singular stiffness equations are computed to predict the branching direction. Hilltop branching receives a special attention. Numerical examples including simple, multiple, and hilltop branching are presented.

Pinpointing bifurcation points and branch-switching

Shape formation and ultimate load behaviour of post-tensioned space trusses

This thesis is principally concerned with the shape formation of different forms of space trusses with non-zero Gaussian curvature by means of a post-tensioning technique. The space truss that is shaped and erected by a post-tensioning procedure, rather than by traditional techniques involving cranes and scaffolding, is called a post-tensioned and shaped space truss. The construction procedure for a post-tensioned and shaped space truss initially involves the assembly of a planar truss with a single-layer of chords, together with out-of-plane web members, at ground level. In proportion to the desired space shape, certain bottom chords are given gaps. The bottom chords comprise shorter tubes and strands that pass through the tubes and through the bottom joints. The initially too-short chords are used to create pre-defined gaps, while the tensioning strands are used to close the gaps. By means of post-tensioning, a planar layout can be deformed and erected to its desired space shape at the same time. In this thesis, the principles of the post-tensioned and shaped space trusses, together with the essential aspects that lead to shape formation and self-erection, are investigated theoretically and are verified by experimental models. In addition to theoretical work, seven domes and one hypar have been constructed by means of a post-tensioning method under laboratory conditions. Also, a pyramidal unit of a full-size practical space truss has been tested. After the shape formation tests, three post-tensioned and shaped domes and one hypar space truss have been loaded to failure in order to observe the ultimate load carrying capacity. The post-tensioned and shaped space trusses have satisfactory ultimate load capacity. The post-tensioning process may increase or reduce the ultimate load capacity of posttensioned and shaped space trusses, due to the resulted prestress forces in some critical members. Compared with the simplicity in construction and erection procedure, the post-tensioned and shaped space trusses still have evident advantages in economy, even if the ultimate load capacity is reduced. Furthermore, the ultimate load capacity of a posttensioned and shaped space truss can be improved by stiffening only a few critical members according to the test results.

Shape formation of space trusses

A simple theory using the stiffness concept is presented for linkage-motion analysis. The nonlinear incremental mechanism equations are established and their solution procedure is proposed. The kinematic system is first regarded as an instable elastic structure. The tangent stiffness equations are updated for current mechanism configurations. The obtained singular stiffness matrix is then used to predict incremental nodal displacements. The stress-associated deformation is designated as numerical error. The proposed stiffness method follows successively the mechanism motion in an incremental-iterative manner. Computed numerical examples show that the elastic deformation can easily be removed with a few iterations. The proposed stiffness approach to kinematics may be incorporated into existing finite-element computer program systems and can be applied to general three-dimensional mechanisms.

Yukio Usuda

Papers

Stiffness Formulations of Planar Kinematics