Projects per year

### Abstract

In an eﬀort to design according to the true behaviour of structures focus is on increasing the accuracy of the individual aspects of the structural models. The present thesis addresses cyclic plasticity models and is organised in ﬁve parts; the ﬁrst four parts focus on diﬀerent aspects of computational cyclic plasticity models. The ﬁfth part focuses on the eﬀects of using improved cyclic plasticity models. The sum of the parts will support the ability to design according to the true behaviour of a structure.

The ﬁrst part addresses the development of a cyclic plasticity model. A cyclic plasticity model with parameter evolution is presented based on three potentials; a speciﬁc energy deﬁning the constitutive relations, a yield function deﬁning the size and shape of the elastic domain in the form of a yield surface, and a plastic ﬂow potential deﬁning the evolution of the plastic strains. The cyclic plasticity model exhibits kinematic hardening and the translation of the center of the yield surface is limited by a surface similar to the yield surface deﬁning an ultimate capacity. The parameter evolution enables modelling of eﬀects as cyclic hardening/softening.

The second part focuses on developing a generic ﬁrst-order yield surface format usable for e.g. anisotropic materials and plastic hinges in beam members and joints. The format is deﬁned as a sum of square roots of quadratic terms that individually would represent ellipsoids and the surface is thereby convex. The format will be homogeneous for most yield surfaces of interest resulting in improved algorithmic properties. It is shown to be possible to locally reduce the curvature of the yield surface while still having a single-equation format.

The third part describes how a frame element can include the four most important eﬀects in analysis of tubular structures with cyclic plasticity: an elastic initially imperfect member, elastic local joint ﬂexibility and plastic mechanisms at the member ends and in the joints. The frame element is based on an equilibrium format, splitting element displacements into a set of deformations and a set of rigid body motions. The deformations and thereby the ﬂexibilities are additive. The element has an explicit stiﬀness matrix that only requires inversion of a matrix of maximum size 4 × 4. A standard full format element including rigid body motions is obtained by use of the equilibrium conditions.

In the fourth part a robust return algorithm is developed. The return algorithm is based on satisfying the generalized strain evolution equations in the ﬁnal state in combination with ensuring the ﬁnal stress state is located on the yield surface. The robustness is increased by making a second order approximation of the generalized stress increment leading to a two-step return algorithm. First a mid-step is made to obtain information and subsequently a full step is made with the information obtained at the mid-step.

In the ﬁnal part the eﬀects of cyclic plasticity are discussed including the eﬀects of elasto-plastic buckling and plastic deformation for complex structures. The permanent change of the geometry reduces characteristic stiﬀness and capacities of the structure. The previously developed models have been used to update a recognized computer code making it more robust and increasing the ability to represent the true behaviour of frame structures.

The ﬁrst part addresses the development of a cyclic plasticity model. A cyclic plasticity model with parameter evolution is presented based on three potentials; a speciﬁc energy deﬁning the constitutive relations, a yield function deﬁning the size and shape of the elastic domain in the form of a yield surface, and a plastic ﬂow potential deﬁning the evolution of the plastic strains. The cyclic plasticity model exhibits kinematic hardening and the translation of the center of the yield surface is limited by a surface similar to the yield surface deﬁning an ultimate capacity. The parameter evolution enables modelling of eﬀects as cyclic hardening/softening.

The second part focuses on developing a generic ﬁrst-order yield surface format usable for e.g. anisotropic materials and plastic hinges in beam members and joints. The format is deﬁned as a sum of square roots of quadratic terms that individually would represent ellipsoids and the surface is thereby convex. The format will be homogeneous for most yield surfaces of interest resulting in improved algorithmic properties. It is shown to be possible to locally reduce the curvature of the yield surface while still having a single-equation format.

The third part describes how a frame element can include the four most important eﬀects in analysis of tubular structures with cyclic plasticity: an elastic initially imperfect member, elastic local joint ﬂexibility and plastic mechanisms at the member ends and in the joints. The frame element is based on an equilibrium format, splitting element displacements into a set of deformations and a set of rigid body motions. The deformations and thereby the ﬂexibilities are additive. The element has an explicit stiﬀness matrix that only requires inversion of a matrix of maximum size 4 × 4. A standard full format element including rigid body motions is obtained by use of the equilibrium conditions.

In the fourth part a robust return algorithm is developed. The return algorithm is based on satisfying the generalized strain evolution equations in the ﬁnal state in combination with ensuring the ﬁnal stress state is located on the yield surface. The robustness is increased by making a second order approximation of the generalized stress increment leading to a two-step return algorithm. First a mid-step is made to obtain information and subsequently a full step is made with the information obtained at the mid-step.

In the ﬁnal part the eﬀects of cyclic plasticity are discussed including the eﬀects of elasto-plastic buckling and plastic deformation for complex structures. The permanent change of the geometry reduces characteristic stiﬀness and capacities of the structure. The previously developed models have been used to update a recognized computer code making it more robust and increasing the ability to represent the true behaviour of frame structures.

Original language | English |
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Place of Publication | Kgs. Lyngby |
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Publisher | Technical University of Denmark |

Number of pages | 128 |

ISBN (Electronic) | 978-87-7475-516-6 |

Publication status | Published - 2018 |

Series | DCAMM Special Report |
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Number | S238 |

ISSN | 0903-1685 |

## Projects

- 1 Finished

## Cyclic Yielding of Tubular Structures

Tidemann, L., Krenk, S., Tychsen, J., Wægter, J., Høgsberg, J. B., D'Aniello, M., Ristinmaa, M. & Tidemann, L.

15/12/2014 → 07/05/2018

Project: PhD

## Cite this

Tidemann, L. (2018).

*Cyclic Yielding of Tubular Structures*. Technical University of Denmark. DCAMM Special Report, No. S238