ARTICLE | doi:10.20944/preprints202107.0672.v1
Subject: Engineering, Automotive Engineering Keywords: generalized nonlinear constitutive law; finite element analysis; nonlinear materials; composite structures
Online: 29 July 2021 (16:11:28 CEST)
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. Preferably 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 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.
ARTICLE | doi:10.20944/preprints202106.0178.v1
Subject: Keywords: generalized nonlinear constitutive law; composite beam; Timoshenko beam; Bernoulli beam
Online: 7 June 2021 (12:48:26 CEST)
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.
ARTICLE | doi:10.20944/preprints202106.0165.v1
Subject: Engineering, Automotive Engineering Keywords: generalized nonlinear constitutive law; steel frame; Timoshenko; Bernoulli
Online: 7 June 2021 (10:51:17 CEST)
The article presents a modified finite element (FE) based algorithm for nonlinear analysis of 2D beam structures, which takes into account the influence of the shear forces. The method proposed enables using complex materials with nonlinearities without the need of implementing advanced constitutive models in FE routines. It can be directly integrated with commonly available FE software for linear analysis of beam structures, so its functionality can be easily extended also with material nonlinearities. The presented approach adopts the generalized constitutive law algorithm to iteratively modify the stiffness of beam element. To address an influence of a shear stiffness, a Timoshenko beam element was used. The methodology was implemented and its performance was verified on several numerical examples. For validation, the displacements and the ultimate loads were compared with the values from a commercial FE software. The results shown a good correlation between the reference model and the method proposed.
ARTICLE | doi:10.20944/preprints202106.0153.v1
Subject: Engineering, Automotive Engineering Keywords: generalized nonlinear constitutive law; trapezoidal sheet plates
Online: 7 June 2021 (08:33:56 CEST)
In the paper, a modified nonlinear finite element method for analysis of trapezoidal plates geometrically reduced to shallow-shell Reissner-Mindlin formulation is presented. Due to the method proposed the complex plate cross-section and nonlinear materials may be modelled and no implementation of advanced constitutive law via user subroutines is needed. The generalized nonlinear constitutive law is used to update the stiffness of the plate element. The method enables modeling of complicated cross-sections, such as steel trapezoidal sheets, metal facing sandwich panels or reinforced concrete. Additionally, for those geometrically complex sections an advanced nonlinear material may be adopted. To verify the proposed method, a selected trapezoidal sheets were modeled in a commercial software as full 3D shell structures. By comparing displacements and forces, it was shown that both models behave almost identically, however, the simplified model has about 300-400 times less degrees of freedom, thus it is much more efficient.
ARTICLE | doi:10.20944/preprints202105.0388.v1
Subject: Engineering, Automotive Engineering Keywords: corrugated cardboard; edge crush test; orthotropic elasticity; digital image correlation
Online: 17 May 2021 (13:27:32 CEST)
The article focuses on derivation of simplified predictive models for identification of overall compressive stiffness and strength of corrugated cardboards. As a representative example an unsymmetrical 5-ply sample (with E and B flute) was used in this study. In order to exclude unreliable displacement measurement in the standard edge crush test, the virtual strain gauges are used. Video extensometry is employed here to collect measurements from the outer surfaces of the sample on both sides. Additional data allows for obtaining real force-displacement curves, which are used in the validation procedure. To emulate the experimental results, beside simple analytical model, also a 3D numerical model fully reflecting the geometry of the corrugated board, based on finite elements method was also built. In both cases a good agreement between the experimental results and the analytical and numerical calculations was observed. This proves that the proposed analytical model can be successfully used to determine the overall stiffness and compressive strength of the corrugated board, provided that the geometry and properties of all the layers of the board are known. The simple model presented in this work enables quick and reliable design and prototyping of new assemblies without the need to manufacture them.
ARTICLE | doi:10.20944/preprints202109.0019.v1
Subject: Engineering, Mechanical Engineering Keywords: corrugated cardboard; edge crush test; orthotropic elasticity; digital image correlation, tensile stiffness, compressive stiffness, sandwich panel
Online: 1 September 2021 (14:20:05 CEST)
The standard edge crush test (ECT) allows to determine the crushing strength of the corrugated cardboard. Unfortunately, this test cannot be used to estimate the compressive stiffness, which is an equally important parameter. It is because, any attempt to determine this parameter using current lab equipment quickly ends in a fiasco. The biggest obstacle is obtaining a reliable measurement of displacements and strains in the corrugated cardboard sample. In this paper, we present a method that not only allows to reliably identify the stiffness in the loaded direction of orthotropy in the corrugated board sample, but also the full orthotropic material stiffness matrix. The proposed method uses two samples: (a) traditional, cut crosswise to the wave direction of the corrugated core, and (b) cut at an angle of 45 degrees. Additionally, in both cases, an optical system with digital image correlation (DIC) is used to measure the displacements and strains on the outer surface of samples. The use of a non-contact measuring system allows to avoid using the measurement of displacements from the crosshead, which is burdened with a large error. Apart from the new experimental configuration, the article also proposes a simple algorithm to quickly characterize all sought stiffness parameters. The obtained results are finally compared with the results obtained in the homogenization procedure of the cross-section of the corrugated board. The results were consistent in both cases.
ARTICLE | doi:10.20944/preprints202107.0678.v1
Subject: Engineering, Automotive Engineering Keywords: corrugated board; box strength estimation; packaging flaps; crease line shifting
Online: 30 July 2021 (09:08:44 CEST)
In the modern world, all manufacturers strive for the optimal design of their products. This general trend is recently also observed in the corrugated board packaging industry. Colorful prints on displays, perforations in shelf-ready-packaging and various types of ventilation holes in trays, although extremely important for ergonomic or functional reasons, weaken the strength of the box. To meet the requirements of customers and recipients, packaging manufacturers outdo each other in new ideas for the construction of their products. Often the aesthetic qualities of the product become more important than the attention to maintaining the standards of the load capacity of the packaging (which, apart from their attention-grabbing functions, are also intended to protect transported products). The particular flaps design (both top and bottom) and their influence on the strength of the box is investigated in this study. The updated analytical-numerical approach is used here to predict the strength of the packaging with various flap’s offsets. Experimental results indicated a significant decrease in the static load-bearing capacity of packaging in the case of shifted flap creases. The simulation model proposed in our previous work has been modified and updated to take into account also this effect. The results obtained by the model presented in the paper are in satisfactory agreement with the experimental data.
ARTICLE | doi:10.20944/preprints202106.0009.v1
Subject: Engineering, Mechanical Engineering Keywords: corrugated cardboard; numerical homogenization; strain energy equivalence; perforation; creasing; flexural stiffness; torsional stiffness
Online: 1 June 2021 (09:40:56 CEST)
The corrugated board packaging industry is increasingly using advanced numerical tools to design and estimate the load capacity of its products. That is why numerical analyzes are becoming a common standard in this branch of manufacturing. Such trend causes either the use of advanced computational models that take into account the full 3D geometry of the flat and wavy layers of corrugated board, or the use of homogenization techniques to simplify the numerical model. The article presents theoretical considerations that extend the numerical homogenization technique already presented in our previous work. The proposed here homogenization procedure also takes into account the creasing and / or perforation of corrugated board, i.e. processes that undoubtedly weaken the stiffness and strength of the corrugated board locally. However, it is not always easy to estimate how exactly these processes affect the bending or torsional stiffness. What is known for sure is that the degradation of stiffness depends, among other things, on the type of cut, its shape, the depth of creasing, as well as their position or direction in relation to the corrugation direction. The method proposed here can be successfully applied to model smeared degradation in a finite element or to define degraded interface stiffnesses on a crease line or a perforation line.
ARTICLE | doi:10.20944/preprints202105.0681.v1
Subject: Materials Science, Biomaterials Keywords: numerical homogenization; prefabricated floor slab; concrete; composite structure; strain energy equivalence
Online: 27 May 2021 (14:34:45 CEST)
The need for quick and easy deflection calculations of various prefabricated slabs causes that simplified procedures and numerical tools are used more and more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Using the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on RVE boundary and generalized strains.
ARTICLE | doi:10.20944/preprints202105.0403.v1
Subject: Engineering, Mechanical Engineering Keywords: corrugated cardboard; converting; numerical homogenization; strain energy equivalence; finite element method; shell structures; transverse shear
Online: 18 May 2021 (10:01:06 CEST)
Corrugated cardboard is an ecological material, mainly because, in addition to virgin cellulose fibers also the fibers recovered during recycling process are used in its production. However, the use of recycled fibers causes slight deterioration of the mechanical properties of the corrugated board. In addition, converting processes such as printing, die-cutting, lamination, etc. cause micro-damage in the corrugated cardboard layers. In this work, the focus is precisely on the crushing of corrugated cardboard. A series of laboratory experiments were conducted, in which the different types of single-walled corrugated cardboards were pressed in a fully controlled manner to check the impact of the crush on the basic material parameters. The amount of crushing (with a precision of 10 micrometers) was controlled by a precise FEMat device, for crushing the corrugated board in the range from 10 to 70 % of its original thickness. In this study, the influence of crushing on bending, twisting and shear stiffness as well as a residual thickness and edge crush resistance of corrugated board was investigated. Then, a procedure based on a numerical homogenization, taking into account a partial delamination in the corrugated layers to determine the degraded material stiffness was proposed. Finally, using the empirical-numerical method, a simplified calculation model of corrugated cardboard was derived, which satisfactorily reflects the experimental results.
ARTICLE | doi:10.20944/preprints202105.0667.v1
Subject: Materials Science, Biomaterials Keywords: corrugated cardboard; converting; crushing; numerical homogenization; strain energy equivalence; finite element method; shell structures; transverse shear
Online: 27 May 2021 (13:13:08 CEST)
As long as the non-contact digital printing is not a common standard in the corrugated packaging industry, corrugated board crushing is a real issue that affects the load capacity of the boxes. Crushing mainly occurs during the converting of corrugated board (e.g. analog flexographic printing or laminating) and is a process that cannot be avoided. However, as show in this study, it can be controlled. In this work, extended laboratory tests were carried out on the crushing of double-walled corrugated board. The influence of fully controlled crushing (with a precision: ±10 μm) in the range from 10 to 70 % on different laboratory measurements was checked. Most of the typical mechanical tests were performed e.g. edge crush test, four-point bending test, shear stiffness test, torsional stiffness test, etc. on reference and crushed specimens. The residual thickness reduction of the crushed samples was also controlled. All empirical observations and performed measurements were the basis for building an analytical model of crushed corrugated board. The proven and verified model was then used to study the crushing effect of the selected corrugated board on the efficiency of simple packages with various dimensions.