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Surface of limit state in nonlinear material and its relation with plasticity condition

Tadeusz Michał Wegner
Archive of Mechanical Engineering | 2000 | vol. 47 | No 3 | 205-223
Keywords limit analysis nonlinear materials strength of materials strength hypothesis
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Abstract

In this work, the author implements concepts and methods of analysis of nonlinear elasticity theory in a simplified description of elastic-plastic properties of materials. Taking the principle of conservation of energy as the theoretical basis, the author formulates a criterion that makes it possible to examine the stability of internal equilibrium in deformed material whose nonlinear properties are defined by strain energy density function. The formulae allowing for assessment of complex states of strain in the aspect of material strength were derived on the assumption of small deformation. These formulae can replace mathematical relationships traditionally known as strength hypotheses. The example included in the paper presents the method of determining, in the space of strain state components, the areas where permanent deformation or destruction of material is possible because of strain state stability. Characteristic parameters used in the example are obtained in a static tensile test on specimen 01· constructional carbon steel of ordinary grade. The results of the analysis, based on the formulated strength hypothesis on stability of strain state, are compared with those resulting from the Huber's hypothesis on energy of non-dilatational strain.
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Authors and Affiliations

Tadeusz Michał Wegner

Application of the generalized nonlinear constitutive law in numerical analysis of hollow-core slabs

Natalia Staszak Tomasz Garbowski Barbara Ksit
Archives of Civil Engineering | 2022 | vol. 68 | No 2 | 125-145 | DOI: 10.24425/ace.2022.140633
Keywords generalized nonlinear constitutive law finite element analysis nonlinear materials composite structures
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Abstract

The non-linear analysis of hollow-core concrete slabs requires the use of advanced numerical techniques, proper constitutive models both for concrete and steel as well as particular computational skills. If prestressing, cracking, crack opening, material softening, etc. are also to be taken into account, then the computational task can far exceed the capabilities of an ordinary engineer. In order for the calculations to be carried out in a traditional design office, simplified calculation methods are needed. They should be based on the linear finite element (FE) method with a simple approach that takes into account material nonlinearities. In this paper the simplified analysis of hollow-core slabs based on the generalized nonlinear constitutive law is presented. In the proposed method a simple decomposition of the traditional iterative linear finite element analysis and the non-linear algebraic analysis of the plate cross-section is used. Through independent analysis of the plate cross-section in different deformation states, a degraded plate stiffness can be obtained, which allows for iterative update of displacements and rotations in the nodes of the FE model. Which in turn allows to update the deformation state and then correct translations and rotations in the nodes again. The results obtained from the full detailed 3D nonlinear FEM model and from the proposed approach are compared for different slab cross-sections. The obtained results from both models are consistent.
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Authors and Affiliations

Natalia Staszak
1
ORCID: ORCID
Tomasz Garbowski
1
ORCID: ORCID
Barbara Ksit
2
ORCID: ORCID
  1. Poznan University of Life Sciences, Department of Biosystems Engineering, Wojska Polskiego 50, 60-627 Poznań
  2. Poznan University of Technology, Institute of Building Engineering, Piotrowo 5, 60-965 Poznan, Poland

Application of the generalized nonlinear constitutive law in 2D shear flexible beam structures

Damian Mrówczyński Tomasz Gajewski Tomasz Garbowski
Archives of Civil Engineering | 2021 | vol. 67 | No 3 | 157-176 | DOI: 10.24425/ace.2021.138049
Keywords generalized nonlinear constitutive law finite element analysis nonlinear materials composite structures Timoshenko beam element
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Abstract

The paper presents a modified finite element method for nonlinear analysis of 2D beam structures. To take into account the influence of the shear flexibility, a Timoshenko beam element was adopted. The algorithm proposed enables using complex material laws without the need of implementing advanced constitutive models in finite element routines. The method is easy to implement in commonly available CAE software for linear analysis of beam structures. It allows to extend the functionality of these programs with material nonlinearities. By using the structure deformations, computed from the nodal displacements, and the presented here generalized nonlinear constitutive law, it is possible to iteratively reduce the bending, tensile and shear stiffnesses of the structures. By applying a beam model with a multi layered cross-section and generalized stresses and strains to obtain a representative constitutive law, it is easy to model not only the complex multi-material cross-sections, but also the advanced nonlinear constitutive laws (e.g. material softening in tension). The proposed method was implemented in the MATLAB environment, its performance was shown on the several numerical examples. The cross-sections such us a steel I-beam and a steel I-beam with a concrete encasement for different slenderness ratios were considered here. To verify the accuracy of the computations, all results are compared with the ones received from a commercial CAE software. The comparison reveals a good correlation between the reference model and the method proposed.
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Bibliography


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Authors and Affiliations

Damian Mrówczyński
1
ORCID: ORCID
Tomasz Gajewski
2
ORCID: ORCID
Tomasz Garbowski
3
ORCID: ORCID
  1. Research and Development Division, FEMAT Sp. z o.o., Romana Maya 1, 61-371, Poznan, Poland
  2. Poznan University of Technology, Institute of Structural Analysis, Piotrowo 5, 60-965 Poznan, Poland
  3. Poznan University of Life Sciences, Department of Biosystems Engineering, Wojska Polskiego 50, 60-627 Poznan, Poland

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