Assistant Professor

Structural Design & Mechanics

AI Lab for Design, Analysis, and Optimization

Faculty of Architecture & the Built Environment

Delft University of Technology

# Beam-column elements, discretized with elastoplastic fibers at the section level to account for material nonlinearities and equipped with corrotational transformation at the element level to account for geometrical nonlinearities, constitute a powerful tool in the analysis of structural engineering systems. Based on the interpolated fields included in the description of their underlying variational principle, various formulations exist, giving rise to displacement-based, force-based, mixed, and hybrid elements.

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# The major goal of all these formulations is to provide elements that accurately capture nonlinear responses without excessively encumbering computational complexity. As a result of the Newton schemes which are predominant in their numerical solutions, the major driver of complexity is the inversion of global or elemental matrices (cubic complexity with the degrees of freedom, under standard inversion methods). The dimensions of these matrices depend on the degrees of freedom of the model and, hence, on the number of unknown variables to be determined (this is greatly associated with the mesh discretization density).

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# Based on this remark, there are at least two restrictions we can lift: we can develop efficient numerical schemes beyond Netwon-based methods to reduce the need for costly matrix inversions, or develop formulations that reduce the number of unknown variables involved in inversions without of course compromising accuracy, or, at best, combine both the previous.

# In this research, the analysis problem is formulated as a nonlinear programming problem where the saddle point of a discretized variational principle is sought. This renders the problem solvable by the abundance of optimization schemes existing in the literature, also beyond Newton schemes. To reduce the number of unknowns, exact beam kinematics following the Reissner-Simo theory are utilized, which allow us to avoid the inaccuracies introduced by corrotational transformations and reduce the discretization density.

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# The performance of the element based on the above principles can be seen below in various benchmark applications. Comparisons with the OpenSees corrotational flexibility elements and Abaqus quadrilateral elements are provided to assess the accuracy of the results. Results include elastic and inelastic cases, with or without shear deformable sections.

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# For Lees' frame (shown above), the effect of shear deformations is studied below, underscoring that shear flexibility can significantly complicate the final equilibrium path in highly nonlinear displacement and rotation regions. It is notable that for the case of deep structural members (last case) the snap-through pattern of slender cases is at parts reversed.

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# The deformed shape response and respective equilibrium path of a two-story frame is also shown below. Validation is again based on the OpenSees flexibility-based elements with corrotational transformation and Abaqus quadrilateral elements.

# It is overall shown that the developed formulation captures highly-nonlinear responses with only one element per member, thus outperforming its counterparts, especially in highly nonlinear regions. In terms of the total number of unknown variables, the figures below show a comparison with the corrotational flexibility-based elements.

# It is observed that the number of total unknowns is considerably reduced. Per element, the matrices to be inverted become slightly bigger (from 3x3 to 8x8). It should be noted that, in both formulations, element matrices have fixed or bounded dimensions, and elemental inversions are conducted independently (linear complexity with the number of elements). What is not bounded is the number of elements itself, which can indefinitely increase for an indefinitely large structural system with a highly accurate mesh. Thus, the critical factor is ultimately the global stiffness matrix of the structure, the dimensions of which are drastically reduced by the proposed formulation (in the case of Lee's frame for example, for the same level of accuracy, the global stiffness matrix dimensions are reduced from 32x32 to 8x8).

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# References:

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Lyritsakis, C.M., Andriotis, C.P., and Papakonstantinou, K.G., “Geometrically exact hybrid beam element based on nonlinear programming”, International Journal of Numerical Methods in Engineering (under review), 2020. [Link]

Lyritsakis, C.M., Andriotis C.P., and Papakonstantinou, K.G., “Nonlinear programming approach to a shear-deformable hybrid beam element for large displacement inelastic analysis”, 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN), Crete, Greece, 2019. [Link]

Andriotis, C.P., Papakonstantinou, K.G., and Koumousis, V.K., “Nonlinear programming hybrid beam-column element formulation for large displacement elastic and inelastic analysis”, Journal of Engineering Mechanics, 144 (10), 04018096, 2018. [Link]

Andriotis C.P., I. Gkimousis, and V.K. Koumousis, “Modeling reinforced concrete structures using smooth plasticity and damage models”, Journal of Structural Engineering, vol. 142, no. 2, p. 04015105, 2015. [Link]

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Faculty of Architecture & the Built Environment

Delft University of Technology

Julianalaan 134, 2628 BL, Delft

email: c.andriotis [at] tudelft [dot] nl

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