Keywords: three-dimensional frames, large displacements, large rotations, non-linear material behaviour.

This paper describes an algorithm for the structural analysis of frames subject to large displacements and rotations . The deformations are assumed to be small. The description of the deformed geometry is fully Lagrangian. The displacement method is used. The equilibrium configurations are iteratively computed by using a classical Newton-Raphson algorithm. The robustness of this approach is enhanced by introducing a limitation in the displacement and rotation increments in the Newton-Raphson iterative steps. This allows for very large loading steps without convergence problems, which is illustrated by means of examples with linear elastic behaviour.

In the first part of the paper the two-dimensional case is presented, first in a general way, defining the kinematic quantities which characterize the motion and deformation of a bar element. Local and global reference coordinates are defined, as well as the tools used for the necessary conversions. After presenting an example with linear elastic material which illustrates the main features of the algorithm, the case of non-linear material behaviour is presented. To this end, a finite element is developed and the functions used to interpolate the longitudinal and transverse displacements within the bar element are defined. Numerical techniques are described to compute the fixity forces and the stiffness matrix, for a given deformation state, emphasizing the way as numerical derivatives required for the iterative computation of the axial deformation and of the material stiffness matrix of the bar are obtained. Finally, the case of a bar made of reinforced concrete with a rectangular cross-section is analysed.

The second part of the paper deals with the extension of the techniques developed for the two-dimensional case to three-dimensional frames. The main difference resides on the treatment of rotations, which are scalars in the two-dimensional case, but have three components in the three-dimensional case. Furthermore, while in the case of infinitesimal rotations they may be described by a vector in the three-dimensional space and the order of application of each component does not influence the result, in the case of finite rotations they can no longer be treated as vectors and the order of application of the components does influence the final result. To this end, a set of three successive finite rotations is used in order to define the rotated position and deformation of each node of the frame. The approaches used to compute the fixation forces and the stiffness matrix of the bar element, in local and global coordinates are subsequently described, as well as the way the local reference systems are initially defined and updated during the iterative process. While in the two-dimensional case the structural stiffness matrix coincides with the Jacobian matrix of the equations describing the equilibrium conditions, this is not the case in presence of three-dimensional large rotations. Thus, the stiffness matrix must be converted to the Jacobian matrix, in order to preserve the high convergent rate of the Newton-Raphson algorithm, in the case of moderate finite rotations, or even to achieve convergence, in the case of large rotations. An example is analysed illustrating the main features of the algorithm, especially its robustness. Finally, some considerations about the extension to non-linear material behaviour of the three-dimensional version of the algorithm are presented.

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