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Homogenization Methods in Materials and Structures
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Homogenization Methods in Materials and Structures

A special issue of Applied Sciences (ISSN 2076-3417). This special issue belongs to the section "Civil Engineering".

Deadline for manuscript submissions: closed (31 August 2020) | Viewed by 42287

Special Issue Editors

Prof. Dr. Angelo Luongo

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Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, Piazzale Ernesto Pontieri, Monteluco di Roio, 67100 L’Aquila, Italy
Interests: continuum and structural mechanics; linear and nonlinear dynamics; stability and bifurcation of dynamical systems; buckling and postbuckling of elastic structures; localization phenomena; aeroelasticity; perturbation methods; computational mechanics
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Prof. Dr. Francesco dell’Isola

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International Research Center on Mathematics and Mechanics of Complex Systems, University of L’Aquila, Via Giovanni Gronchi 18, 67100 L’Aquila, Italy
Interests: continuum mechanics; porous media; piezo-electro-mechanical structures; nonlinear elasticity; second gradient materials; metamaterials; mechanics of living tissue; smart materials; composite materials; experimental mechanics; numerical mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Giuseppe Piccardo

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Dipartimento di Ingegneria Civile, Chimica e Ambientale, Università di Genova, 16126 Genova, Italy
Interests: dynamics; aerodynamics; structural engineering; finite element analysis turbulence modeling; mechanical engineering; wind tunnel testing; coding; engineering, applied and computational mathematics; numerical analysis
Dr. Daniele Zulli

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Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, 67100 L’Aquila, Italy
Interests: stability and nonlinear oscillations of elastic structures; perturbation methods and reduced order models; aeroelasticity; homogeneous models of structures
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Prof. Dr. Francesco D’Annibale

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Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, Piazzale Ernesto Pontieri, Monteluco di Roio, 67100 L’Aquila, Italy
Interests: control of elastic systems via added piezoelectric devices; stability and nonlinear oscillations of elastic systems under conservative and non-conservative actions; perturbation methods; damage constitutive models; computational mechanics; homogenization of beam-like structures
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Dr. Manuel Ferretti

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International Research Center on Mathematics and Mechanics of Complex Systems, University of L’Aquila, Via Giovanni Gronchi 18, 67100 L’Aquila, Italy
Interests: stability and nonlinear oscillations of elastic systems under conservative and non-conservative loads; dynamics of strings and beams with traveling masses; perturbation methods; mechanics of generalized continua; mechanics of woven fibrous composite reinforcements; homogenization of beam-like structures

Special Issue Information

Dear Colleagues,

This Special Issue is dedicated to academic researchers who want to propose studies on the homogenization of complex materials and structures, covering the whole process, from the idealization and design, to the real application.

The subject has received great attention in the last few years, involving researchers whose expertise exploits different scientific areas, including continuum mechanics, structural mechanics, acoustics, materials design, and 3D printing techniques.

Some of the topics considered for this Special Issue include, but are not limited to, the following:

  • The formulation of homogenous models of micro-structured materials and periodic structures;
  • The static, dynamic, and stability behavior of homogenous models of lattice members;
  • Analytical and numerical methods in materials and structures design;
  • Wave propagation in periodic media;
  • The experimental validation of the homogenization.

Prof. Dr. Angelo Luongo
Prof. Dr. Francesco dell’Isola
Prof. Dr. Giuseppe Piccardo
Prof. Dr. Daniele Zulli
Dr. Francesco D’Annibale
Dr. Manuel Ferretti
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Applied Sciences is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Homogenization 
  • Structural design 
  • Material design 
  • Statics
  • Dynamics
  • Stability

Published Papers (12 papers)

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Research

19 pages, 698 KiB  
Article
Static Response of Double-Layered Pipes via a Perturbation Approach
by Daniele Zulli, Arnaldo Casalotti and Angelo Luongo
Appl. Sci. 2021, 11(2), 886; https://doi.org/10.3390/app11020886 - 19 Jan 2021
Cited by 4 | Viewed by 1710
Abstract
A double-layered pipe under the effect of static transverse loads is considered here. The mechanical model, taken from the literature and constituted by a nonlinear beam-like structure, is constituted by an underlying Timoshenko beam, enriched with further kinematic descriptors which account for local [...] Read more.
A double-layered pipe under the effect of static transverse loads is considered here. The mechanical model, taken from the literature and constituted by a nonlinear beam-like structure, is constituted by an underlying Timoshenko beam, enriched with further kinematic descriptors which account for local effects, namely, ovalization of the cross-section, warping and possible relative sliding of the layers under bending. The nonlinear equilibrium equations are addressed via a perturbation method, with the aim of obtaining a closed-form solution. The perturbation scheme, tailored for the specific load conditions, requires different scaling of the variables and proceeds up to the fourth order. For two load cases, namely, distributed and tip forces, the solution is compared to that obtained via a pure numeric approach and the finite element method. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
►▼ Show Figures

Figure 1

Figure 1
<p>The double-layered beam with an annular cross-section: (<b>a</b>) complete view; (<b>b</b>) details of the cross-section.</p> Full article ">Figure 2
<p>Initial configuration of the beam.</p> Full article ">Figure 3
<p>Physical meaning of the distortion variables: (<b>a</b>) assumed trial function for ovalization and amplitude <math display="inline"><semantics> <msub> <mi>a</mi> <mi>p</mi> </msub> </semantics></math>; (<b>b</b>) assumed trial function for warping and amplitude <math display="inline"><semantics> <msub> <mi>a</mi> <mi>w</mi> </msub> </semantics></math>; (<b>c</b>) assumed trial function for longitudinal sliding of the layers under opposite bending and amplitude <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Response in the case of the uniformly distributed vertical load: (<b>a</b>) longitudinal displacement <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>/</mo> <mi>l</mi> <mspace width="0.166667em"/> <mo>(</mo> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>)</mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>/</mo> <mi>l</mi> </mrow> </semantics></math>; (<b>c</b>) transversal displacement <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mspace width="0.166667em"/> <mo>(</mo> <mi>rad</mi> <mo>)</mo> </mrow> </semantics></math>; (<b>d</b>) amplitude of ovalization <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>p</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>; (<b>e</b>) amplitude of relative sliding <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>; (<b>f</b>) amplitude of warping <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>w</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>. Blue lines: perturbation solution; light blue lines: finite difference method; gray lines: FEM.</p> Full article ">Figure 5
<p>Distributed force: evolution of the tip displacements while increasing the amplitude of the load (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>p</mi> <mo>˜</mo> </mover> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>p</mi> <mi>v</mi> </msub> <mi>l</mi> </mrow> <mrow> <mi>π</mi> <mi>R</mi> <mo>(</mo> <msub> <mi>G</mi> <mi>e</mi> </msub> <mi>h</mi> <mi>e</mi> <mo>+</mo> <msub> <mi>G</mi> <mi>i</mi> </msub> <mi>h</mi> <mi>i</mi> <mo>)</mo> </mrow> </mfrac> </mrow> </semantics></math>). Black dashed line: load condition of <a href="#applsci-11-00886-f004" class="html-fig">Figure 4</a>; blue line: perturbation solution; light blue line: finite difference method; gray line: FEM.</p> Full article ">Figure 6
<p>Response in the case of the uniformly distributed vertical load: (<b>a</b>) longitudinal displacement <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>/</mo> <mi>l</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>/</mo> <mi>l</mi> </mrow> </semantics></math>; (<b>c</b>) transversal displacement <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mspace width="0.166667em"/> <mo>(</mo> <mi>rad</mi> <mo>)</mo> </mrow> </semantics></math>; (<b>d</b>) amplitude of ovalization <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>p</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>; (<b>e</b>) amplitude of relative sliding <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>; (<b>f</b>) amplitude of warping <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>w</mi> </msub> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>. Blue lines: perturbation solution; light blue lines: finite difference method; gray lines: FEM.</p> Full article ">Figure 7
<p>Tip force: evolution of the tip displacements while the increasing of the amplitude of the load (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>P</mi> <mo>˜</mo> </mover> <mo>:</mo> <mo>=</mo> <mfrac> <msub> <mi>P</mi> <mi>v</mi> </msub> <mrow> <mi>π</mi> <mi>R</mi> <mo>(</mo> <msub> <mi>G</mi> <mi>e</mi> </msub> <msub> <mi>h</mi> <mi>e</mi> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>i</mi> </msub> <msub> <mi>h</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mfrac> </mrow> </semantics></math>). Black dashed line: load condition of <a href="#applsci-11-00886-f006" class="html-fig">Figure 6</a>; blue line: perturbation solution; light blue line: finite difference method; gray line: FEM.</p> Full article ">Figure A1
<p>FE model: (<b>a</b>) sketch of the tubular structure; (<b>b</b>) representation of the adopted mesh.</p> Full article ">
18 pages, 929 KiB  
Article
Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
by Jarosław Latalski and Daniele Zulli
Appl. Sci. 2020, 10(21), 7802; https://doi.org/10.3390/app10217802 - 3 Nov 2020
Cited by 11 | Viewed by 4273
Abstract
The use of the Generalized Beam Theory (GBT) is extended to thin-walled beams with curvilinear cross-sections. After defining the kinematic features of the walls, where their curvature is consistently accounted for, the displacement of the points is assumed as linear combination of unknown [...] Read more.
The use of the Generalized Beam Theory (GBT) is extended to thin-walled beams with curvilinear cross-sections. After defining the kinematic features of the walls, where their curvature is consistently accounted for, the displacement of the points is assumed as linear combination of unknown amplitudes and pre-established trial functions. The latter, and specifically their in-plane components, are chosen as dynamic modes of a curved beam in the shape of the member cross-section. Moreover, the out-of-plane components come from the imposition of the Vlasov internal constraint of shear indeformable middle surface. For a case study of semi-annular cross-section, i.e., constant curvature, the modes are analytically evaluated and the procedure is implemented for two different load conditions. Outcomes are compared to those of a FEM model. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
►▼ Show Figures

Figure 1

Figure 1
<p>Thin-walled beam in initial configuration (<b>a</b>) and details of the mid-curve <math display="inline"><semantics> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> of the cross-section (<b>b</b>).</p> Full article ">Figure 2
<p>Local sketch of the singly curved thin shell.</p> Full article ">Figure 3
<p>Cross-section and load conditions in case-studies: (<b>a</b>) load condition 1; (<b>b</b>) load condition 2.</p> Full article ">Figure 4
<p>In-plane (ip) and out-of-plane (op) components of the trial functions, from I to IV (rigid modes). Blue dashed line: initial configuration; black solid line: trial functions.</p> Full article ">Figure 5
<p>In-plane (ip) and out-of-plane (op) components of the trial functions, from V to VIII (deformation modes). Blue dashed line: initial configuration; black solid line: trial functions.</p> Full article ">Figure 6
<p>Solution for the load condition 1; S: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; W: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math>; N: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mi>π</mi> <mi>R</mi> </mrow> </semantics></math>; (<b>a</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math>; (<b>b</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>8</mn> </mrow> </semantics></math>; (<b>c</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>x</mi> </msub> </semantics></math>; (<b>d</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>y</mi> </msub> </semantics></math>; (<b>e</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>z</mi> </msub> </semantics></math>; (<b>f</b>) stress component <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>x</mi> <mi>M</mi> </msubsup> </semantics></math>; (<b>g</b>) stress component <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>x</mi> </msub> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Solid lines–analytical solution, dotted lines–FE solution.</p> Full article ">Figure 6 Cont.
<p>Solution for the load condition 1; S: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; W: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math>; N: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mi>π</mi> <mi>R</mi> </mrow> </semantics></math>; (<b>a</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math>; (<b>b</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>8</mn> </mrow> </semantics></math>; (<b>c</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>x</mi> </msub> </semantics></math>; (<b>d</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>y</mi> </msub> </semantics></math>; (<b>e</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>z</mi> </msub> </semantics></math>; (<b>f</b>) stress component <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>x</mi> <mi>M</mi> </msubsup> </semantics></math>; (<b>g</b>) stress component <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>x</mi> </msub> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Solid lines–analytical solution, dotted lines–FE solution.</p> Full article ">Figure 7
<p>Solution for the load condition 2; S: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; W: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math>; N: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mi>π</mi> <mi>R</mi> </mrow> </semantics></math>; (<b>a</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> </mrow> </semantics></math>; (<b>b</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>x</mi> </msub> </semantics></math>; (<b>c</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>y</mi> </msub> </semantics></math>; (<b>d</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>z</mi> </msub> </semantics></math>; (<b>e</b>) stress component <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>x</mi> <mi>M</mi> </msubsup> </semantics></math>; (<b>f</b>) stress component <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>x</mi> </msub> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Solid lines–analytical solution, dotted lines–FE solution.</p> Full article ">Figure 7 Cont.
<p>Solution for the load condition 2; S: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; W: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math>; N: <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mi>π</mi> <mi>R</mi> </mrow> </semantics></math>; (<b>a</b>) amplitudes <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> </mrow> </semantics></math>; (<b>b</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>x</mi> </msub> </semantics></math>; (<b>c</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>y</mi> </msub> </semantics></math>; (<b>d</b>) displacement <math display="inline"><semantics> <msub> <mi>u</mi> <mi>z</mi> </msub> </semantics></math>; (<b>e</b>) stress component <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>x</mi> <mi>M</mi> </msubsup> </semantics></math>; (<b>f</b>) stress component <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>x</mi> </msub> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Solid lines–analytical solution, dotted lines–FE solution.</p> Full article ">Figure A1
<p>Meshed FE model of the thin-walled cantilever and nodal points to record displacements.</p> Full article ">
23 pages, 13725 KiB  
Article
A Transitional Connection Method for the Design of Functionally Graded Cellular Materials
by Shihao Liang, Liang Gao, Yongfeng Zheng and Hao Li
Appl. Sci. 2020, 10(21), 7449; https://doi.org/10.3390/app10217449 - 23 Oct 2020
Cited by 7 | Viewed by 1803
Abstract
In recent years, the functionally graded materials (FGM) with cellular structure have become a hot spot in the field of materials research. For the continuously varying cellular structure in the layer-wise FGM, the connection of gradient cellular structures has become the main problem. [...] Read more.
In recent years, the functionally graded materials (FGM) with cellular structure have become a hot spot in the field of materials research. For the continuously varying cellular structure in the layer-wise FGM, the connection of gradient cellular structures has become the main problem. Unfortunately, the effect of gradient connection method on the overall structural performance lacks attention, and the boundary mismatch has enormous implications. Using the homogenization theory and the level set method, this article presents an efficient topology optimization method to solve the connection issue. Firstly, a simple but efficient hybrid level set scheme is developed to generate a new level set surface that has the partial features of two candidate level sets. Then, when the new level set surface is formed by considering the level set functions of two gradient base cells, a special transitional cell can be constructed by finding the zero level set of this generated level set surface. Since the transitional cell has the geometric features of two gradient base cells, the shape of the transitional cell fits perfectly with its connected gradient cells on both sides. Thus, the design of FGM can have a smooth connectivity with C1 continuity without any complex numerical treatments during the optimization. A number of examples on both 2D and 3D are provided to demonstrate the characteristics of the proposed method. Finite element simulation has also been employed to calculate the mechanical properties of the designs. The simulation results show that the FGM devised by the proposed method exhibits better mechanical performances than conventional FGM with only C0 continuity. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
►▼ Show Figures

Figure 1

Figure 1
<p>C<sup>0</sup> connectivity in FGM.</p> Full article ">Figure 2
<p>Transitional cell (middle) to connect two CBS (left and right) with C<sup>1</sup> continuity.</p> Full article ">Figure 3
<p>Configuration of the considered FGM.</p> Full article ">Figure 4
<p>FGM with maximum bulk modulus cellular structure.</p> Full article ">Figure 5
<p>Two different connection method for the design of FGM: (<b>a</b>) conventional FGM with C<sup>0</sup> connection; (<b>b</b>) FGM with C<sup>1</sup> connection by using the transitional connection method.</p> Full article ">Figure 6
<p>Loading and boundary conditions for optimized FGMs in ANSYS simulation analysis.</p> Full article ">Figure 7
<p>The stress distribution of optimized FGMs (specified gradient in volume fraction) with two different connection methods: (<b>a</b>) conventional FGM with C<sup>0</sup> connection; (<b>b</b>) FGM with C<sup>1</sup> connection by using the transitional connection method.</p> Full article ">Figure 8
<p>Two different porous structures with the maximum bulk modulus: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 9
<p>Three different loading and boundary conditions: (<b>a</b>) cantilever beam; (<b>b</b>) half MBB beam; (<b>c</b>) Loading on top.</p> Full article ">Figure 10
<p>Structural deformation under the cantilever beam type condition: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 10 Cont.
<p>Structural deformation under the cantilever beam type condition: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 11
<p>Strain energy comparison for two designs under different loading and boundary conditions.</p> Full article ">Figure 12
<p>Two different porous structures with the maximum shear modulus: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 13
<p>Structural deformation under the cantilever beam type condition: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 13 Cont.
<p>Structural deformation under the cantilever beam type condition: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 14
<p>Strain energy comparison for two designs under different loading and boundary conditions.</p> Full article ">Figure 15
<p>Designs with two connection methods: (<b>a</b>) FGM with mismatched interfaces by directly connecting the two types of CBSs; (<b>b</b>) FGM with smooth interfaces by using the transitional connection method.</p> Full article ">Figure 16
<p>Two kinds of 3D FGM by using the proposed transitional connection method: (<b>a</b>) 3D FGM with maximum bulk modulus CBSs; (<b>b</b>) 3D FGM with maximum shear modulus CBSs.</p> Full article ">Figure 17
<p>Porous structure with single type 3D maximum shear modulus CBS.</p> Full article ">Figure 18
<p>Loading and boundary conditions for the 3D porous structure.</p> Full article ">Figure 19
<p>Structural deformation under the considered loading and boundary conditions: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 19 Cont.
<p>Structural deformation under the considered loading and boundary conditions: (<b>a</b>) FGM design; (<b>b</b>) design with single type CBS.</p> Full article ">Figure 20
<p>3D FGM with smooth interfaces by using the transitional connection method.</p> Full article ">Figure 21
<p>Different perspectives of the 3D transitional cellular structure: (<b>a</b>) perspective 1, top surface connects perfectly with the maximum shear modulus CBS; (<b>b</b>) perspective 2, top surface connects perfectly with the maximum bulk modulus CBS.</p> Full article ">Figure 22
<p>Sectional view of the connection between different cells.</p> Full article ">
12 pages, 1319 KiB  
Article
On Effective Bending Stiffness of a Laminate Nanoplate Considering Steigmann–Ogden Surface Elasticity
by Victor A. Eremeyev and Tomasz Wiczenbach
Appl. Sci. 2020, 10(21), 7402; https://doi.org/10.3390/app10217402 - 22 Oct 2020
Cited by 11 | Viewed by 2051
Abstract
As at the nanoscale the surface-to-volume ratio may be comparable with any characteristic length, while the material properties may essentially depend on surface/interface energy properties. In order to get effective material properties at the nanoscale, one can use various generalized models of continuum. [...] Read more.
As at the nanoscale the surface-to-volume ratio may be comparable with any characteristic length, while the material properties may essentially depend on surface/interface energy properties. In order to get effective material properties at the nanoscale, one can use various generalized models of continuum. In particular, within the framework of continuum mechanics, the surface elasticity is applied to the modelling of surface-related phenomena. In this paper, we derive an expression for the effective bending stiffness of a laminate plate, considering the Steigmann–Ogden surface elasticity. To this end, we consider plane bending deformations and utilize the through-the-thickness integration procedure. As a result, the calculated elastic bending stiffness depends on lamina thickness and on bulk and surface elastic moduli. The obtained expression could be useful for the description of the bending of multilayered thin films. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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Figure 1

Figure 1
<p><span class="html-italic">N</span>th layered plate of thickness <span class="html-italic">h</span>. Here <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, and the <span class="html-italic">i</span>th layer has a thickness of <math display="inline"><semantics> <msub> <mi>h</mi> <mi>i</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Dimensionless bending stiffness <math display="inline"><semantics> <mrow> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mo>=</mo> <msub> <mi>D</mi> <mi>eff</mi> </msub> <mo>/</mo> <mi>D</mi> </mrow> </semantics></math> vs. thickness <span class="html-italic">h</span> (solid red curve). The dashed blue curve corresponds to the Gurtin–Murdoch model; the horizonal dashed black line describes the classic bending stiffness. Finally, the vertical dashed green line marks the characteristic length <span class="html-italic">l</span> defined as <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mo>(</mo> <msup> <mi>λ</mi> <mi>S</mi> </msup> <mo>+</mo> <mn>2</mn> <mi>μ</mi> <mo>)</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>D</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Dimensionless bending stiffness <math display="inline"><semantics> <mover> <mi>D</mi> <mo>¯</mo> </mover> </semantics></math> vs. number of layers <span class="html-italic">N</span>.</p> Full article ">
20 pages, 824 KiB  
Article
Static and Dynamic Responses of Micro-Structured Beams
by Francesco D’Annibale, Manuel Ferretti and Angelo Luongo
Appl. Sci. 2020, 10(19), 6836; https://doi.org/10.3390/app10196836 - 29 Sep 2020
Cited by 6 | Viewed by 1699
Abstract
In this study, we developed a one-dimensional Timoshenko beam model, embedded in a 3D space for static and dynamic analyses of beam-like structures. These are grid cylinders, that is, micro-structured bodies, made of a periodic and specifically designed three-dimensional assembly of beams. Derivation [...] Read more.
In this study, we developed a one-dimensional Timoshenko beam model, embedded in a 3D space for static and dynamic analyses of beam-like structures. These are grid cylinders, that is, micro-structured bodies, made of a periodic and specifically designed three-dimensional assembly of beams. Derivation is performed in the framework of the direct 1D approach, while the constitutive law is determined by a homogenization procedure based on an energy equivalence between a cell of the periodic model and a segment of the solid beam. Warping of the cross-section, caused by shear and torsion, is approximatively taken into account by the concept of a shear factor, namely, a corrective factor for the constitutive coefficients of the equivalent beam. The inertial properties of the Timoshenko model are analytically identified under the hypothesis, and the masses are lumped at the joints. Linear static and dynamic responses of some micro-structured beams, taken as case studies, are analyzed, and a comparison between the results given by the Timoshenko model and those obtained by Finite-Element analyses on 3D frames is made. In this framework, the effectiveness of the equivalent model and its limits of applicability are highlighted. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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Figure 1
<p>Study object: (<b>a</b>) grid beam; (<b>b</b>) equivalent beam model.</p> Full article ">Figure 2
<p>Single cell of the grid beam, and its deformation modes (bracings not shown): (<b>a</b>) 3D view of the <math display="inline"><semantics> <mi>EX</mi> </semantics></math> mode; (<b>b</b>) top view of the <math display="inline"><semantics> <mi>TO</mi> </semantics></math> mode; (<b>c</b>) lateral view of the <math display="inline"><semantics> <msub> <mi>SH</mi> <mi>ν</mi> </msub> </semantics></math> mode, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math>; (<b>d</b>,<b>e</b>) lateral view of the <math display="inline"><semantics> <msub> <mi>FL</mi> <mi>ν</mi> </msub> </semantics></math> mode, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Grid beam <span class="html-italic">x</span>-fibers layout: (<b>a</b>) case study I; (<b>b</b>) case study II.</p> Full article ">Figure 4
<p>Static response of the equivalent model vs. the discrete FE model, for case study I-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>) lateral and torsional displacements; (<b>b</b>) vertical displacements of the cross-section joints at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>h</mi> <mo>,</mo> <mspace width="0.166667em"/> <mo>ℓ</mo> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>ℓ</mo> </mrow> </semantics></math>, respectively. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 5
<p>Static response of the equivalent model vs. the discrete FE model, for case study I-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>: (<b>a</b>) lateral and torsional displacements; (<b>b</b>) vertical displacements of the cross-section joints at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>h</mi> <mo>,</mo> <mspace width="0.166667em"/> <mo>ℓ</mo> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>ℓ</mo> </mrow> </semantics></math>, respectively. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 6
<p>Deformed shape of the equivalent model vs. the discrete FE model, for case study I-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>) lateral view at <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>,<b>c</b>) top views at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>ℓ</mo> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>ℓ</mo> </mrow> </semantics></math>, respectively. Green dots: discrete FE solution. Continuous blue line: homogenized beam model. Amplification factor for displacements in the Figure is taken as equal to 10.</p> Full article ">Figure 7
<p>Static response of the equivalent model vs. discrete FE model, for case study I-b and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>) lateral displacement; (<b>b</b>) torsional displacement. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 8
<p>Static response of the equivalent model vs. discrete FE model, for case study II-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>) lateral displacement; (<b>b</b>) torsional displacement. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 9
<p>Modal shapes of the equivalent model vs. discrete FE model, for case study I-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>,<b>b</b>) First and second flexural mode; (<b>c</b>,<b>d</b>) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 10
<p>Modal shapes of the equivalent model vs. discrete FE model, for case study I-a and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>: (<b>a</b>,<b>b</b>) First and second flexural mode; (<b>c</b>,<b>d</b>) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">Figure 11
<p>Modal shapes of the equivalent model vs. discrete FE model, for case study I-b and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>: (<b>a</b>,<b>b</b>) First and second flexural mode; (<b>c</b>,<b>d</b>) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.</p> Full article ">
24 pages, 8804 KiB  
Article
Homogenization of Ancient Masonry Buildings: A Case Study
by Simona Di Nino and Daniele Zulli
Appl. Sci. 2020, 10(19), 6687; https://doi.org/10.3390/app10196687 - 24 Sep 2020
Cited by 5 | Viewed by 1761
Abstract
With the aim of evaluating local and global dynamic mechanisms of a vast and historical masonry building, a homogeneous structural model is proposed here. It is realized with the assembly of othotropic plates and Timoshenko and pure shear beams as well. The identification [...] Read more.
With the aim of evaluating local and global dynamic mechanisms of a vast and historical masonry building, a homogeneous structural model is proposed here. It is realized with the assembly of othotropic plates and Timoshenko and pure shear beams as well. The identification of the constitutive parameters is carried out after realizing refined finite element models of building portions, and imposing energy or displacement equivalence with the corresponding homogeneous versions, depending on the complexity of the involved schemes. The outcomes are compared with those provided by experimental investigations, and help to give insight and interpretation on the dynamic behavior of the building. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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<p>(<b>a</b>) plan of the first level: (red) middle line and highlighted (gray filled area) longitudinal arm; (<b>b</b>) sections. All the units are in meters.</p> Full article ">Figure 2
<p>Long longitudinal arm details: (<b>a</b>) plan of the first floor; (<b>b</b>) perspectives of the two external facades. All the units are in meters.</p> Full article ">Figure 3
<p>Fine finite element model: (<b>a</b>) overview; (<b>b</b>) details of slabs and transverse walls.</p> Full article ">Figure 4
<p>Homogenization scheme.</p> Full article ">Figure 5
<p>Homogenization scheme in the plane <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> of: (<b>a</b>) the LW model; (<b>b</b>) a couple of localized transverse walls.</p> Full article ">Figure 6
<p>Scheme of the equivalent plates of the homogenized model.</p> Full article ">Figure 7
<p>(<b>a</b>) cell geometry; (<b>b</b>) modified I domain under extensional strains: subtraction of the regions at almost zero-strain.</p> Full article ">Figure 8
<p><math display="inline"><semantics> <msub> <mi>S</mi> <mi>x</mi> </msub> </semantics></math> identification. Deformed configuration of the fine model in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 9
<p>(<b>a</b>) scheme of the shear beam equivalent to the <span class="html-italic">i</span>-plate; (<b>b</b>) comparison between analytical (pink line) and numerical (black line) deformations in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane for the plate 5; quantities in meters.</p> Full article ">Figure 10
<p><math display="inline"><semantics> <msub> <mi>S</mi> <mi>y</mi> </msub> </semantics></math> identification. Deformed configuration of the fine model in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 11
<p>(<b>a</b>) scheme of the equivalent shear beam system with axis <span class="html-italic">y</span>; (<b>b</b>) comparison between analytical (pink line) and numerical (black line) deformations in the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; quantities in meters.</p> Full article ">Figure 12
<p>Overview of the finite element global homogenized model.</p> Full article ">Figure 13
<p>(<b>a</b>) building portion affected by the dynamic tests of the campaign 1—second floor. First vibrating mode: natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.54</mn> </mrow> </semantics></math> and (<b>b</b>) modal shape in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 14
<p>(<b>a</b>) building portion affected by the dynamic tests of the campaign 2—first, second, third, fourth floor. First vibrating mode: natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.54</mn> </mrow> </semantics></math> and modal shapes in the (<b>b</b>) 3D space, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane, and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 15
<p>First natural mode of the LW fine model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0.54</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 16
<p>First natural mode of the LTW fine model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.38</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 17
<p>First natural mode of the SLTW fine model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.42</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 18
<p>First natural mode of the LW homogenized model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0.46</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 19
<p>First natural mode of the LTW homogenized model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.23</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 20
<p>First natural mode of the SLTW homogenized model. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.44</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 21
<p>Natural modes of the global LTW homogenized model. Modal shape in the (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 22
<p>Natural modes of the global SLTW homogenized model. Modal shape in the (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure A1
<p>First natural mode of the LW fine model without slabs. Natural frequency <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0.24</mn> </mrow> </semantics></math> Hz. Modal shape in the: (<b>a</b>) 3D space; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">
16 pages, 663 KiB  
Article
Buckling of Planar Micro-Structured Beams
by Manuel Ferretti and Francesco D’Annibale
Appl. Sci. 2020, 10(18), 6506; https://doi.org/10.3390/app10186506 - 18 Sep 2020
Cited by 2 | Viewed by 1737
Abstract
In this paper, a Timoshenko beam model is formulated for buckling analysis of periodic micro-structured beams, uniformly compressed. These are planar grid beams, whose micro-structure consists of a square lattice of equal fibers, modeled as Timoshenko micro-beams. The equivalent beam model is derived [...] Read more.
In this paper, a Timoshenko beam model is formulated for buckling analysis of periodic micro-structured beams, uniformly compressed. These are planar grid beams, whose micro-structure consists of a square lattice of equal fibers, modeled as Timoshenko micro-beams. The equivalent beam model is derived in the framework of a direct one-dimensional approach and its constitutive law, including the effect of prestress of the longitudinal fibers, is deduced through a homogenization approach. Accordingly, micro–macro constitutive relations are obtained through an energy equivalence between a cell of the periodic model and a segment of the equivalent beam. The model also accounts for warping of the micro-structure, via the introduction of elastic and geometric corrective factors of the constitutive coefficients. A survey of the buckling behavior of sample grid beams is presented to validate the effectiveness and limits of the equivalent model. To this purpose, results supplied by the exact analyses of the equivalent beam are compared with those given by finite element models of bi-dimensional frames. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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Figure 1

Figure 1
<p>Study object: (<b>a</b>) grid beam; (<b>b</b>) equivalent beam model in the reference and current configuration.</p> Full article ">Figure 2
<p>Single cell of the grid beam, and its deformation modes: (<b>a</b>) extensional (<math display="inline"><semantics> <mi>EX</mi> </semantics></math>) mode; (<b>b</b>) shear (<math display="inline"><semantics> <mi>SH</mi> </semantics></math>) mode; (<b>c</b>) flexural (<math display="inline"><semantics> <mi>FL</mi> </semantics></math>) mode.</p> Full article ">Figure 3
<p>Non-dimensional critical load <math display="inline"><semantics> <mi>μ</mi> </semantics></math> vs. the number of cells <span class="html-italic">n</span>, for the case study I and for different boundary conditions: (<b>a</b>) Hinge–roller (H-R); (<b>b</b>) clamp–free (C-F); (<b>c</b>) clamp–roller (C-R); (<b>d</b>) clamp–slider (C-S). Green curve (A): Timoshenko model; blue curve (B): Finite-element (FE) solution; yellow (C) and red (D) curves: Timoshenko and Euler–Bernoulli models without constitutive geometric effect.</p> Full article ">Figure 4
<p>Critical mode in the case study I, when <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>, and for different boundary conditions: (<b>a</b>) H-R; (<b>b</b>) C-F; (<b>c</b>) C-R; (<b>d</b>) C-S. Continuous green curve: Timoshenko model; blue dots: FE solution.</p> Full article ">Figure 5
<p>Non-dimensional critical load <math display="inline"><semantics> <mi>μ</mi> </semantics></math> vs. the number of cells <span class="html-italic">n</span>, for the case study II and for different boundary conditions: (<b>a</b>) H-R; (<b>b</b>) C-F; (<b>c</b>) C-R; (<b>d</b>) C-S. Green curve (A): Timoshenko model; blue curve (B): FE solution; yellow (C) and red (D) curves: Timoshenko and Euler–Bernoulli models without constitutive geometric effect.</p> Full article ">Figure 6
<p>Critical mode in the case study II, when <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, and for different boundary conditions: (<b>a</b>) H-R; (<b>b</b>) C-F; (<b>c</b>) C-R; (<b>d</b>) C-S. Continuous green curve: Timoshenko model; blue dots: FE solution.</p> Full article ">
20 pages, 25911 KiB  
Article
Impacts of Compaction Load and Procedure on Stress-Deformation Behaviors of a Soil Geosynthetic Composite (SGC) Mass—A Case Study
by Meenwah Gui, Truc Phan and Thang Pham
Appl. Sci. 2020, 10(18), 6339; https://doi.org/10.3390/app10186339 - 11 Sep 2020
Cited by 2 | Viewed by 3078
Abstract
Fill compaction in the construction of Geosynthetic Reinforced Soil (GRS) mass is typically performed by operating a vibratory or roller compactor, which in turns imposed a compaction load in direction perpendicular to the wall face. The compaction process resulted in the development of [...] Read more.
Fill compaction in the construction of Geosynthetic Reinforced Soil (GRS) mass is typically performed by operating a vibratory or roller compactor, which in turns imposed a compaction load in direction perpendicular to the wall face. The compaction process resulted in the development of the so-called compaction-induced stress (CIS), which may subsequently increase the stiffness and strength of the fill material. Compaction process is normally simulated using one of the following compaction procedures—(i) a uniformly distributed load acting on the top surface of each soil lift, (ii) a uniformly distributed load acting on the top and bottom surface of each soil lift, and (iii) a moving strip load with different width. Uncertainties such as compaction procedures, compaction and surcharge loads led to the disparity in studying the mechanism of GRS mass. This paper aimed to study the impact of compaction load, compaction procedure, surcharge load and CIS on the stress-deformation behavior of GRS mass via the simulation of a 2 m high Soil Geosynthetic Composite (SGC) mass and a 6 m high GRS mass. The results were examined in terms of reinforcement strains, wall lateral displacements, and net CIS. Results from the analysis show the important impacts of compaction conditions on the stress-deformation behavior of SGC mass and the CIS. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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<p>(<b>a</b>) Schematic diagram of a moving compaction load. (<b>b</b>) Stress paths of a soil element at depth <span class="html-italic">z</span> as the compaction load moves away from I-I.</p> Full article ">Figure 2
<p>(<b>a</b>) Schematic diagram and configuration of Soil-Geosynthetic Composite (SGC) mass. (<b>b</b>) Simulated compaction procedures adopted for SGC mass in FE analysis.</p> Full article ">Figure 3
<p>Effect of compaction procedures (Types-I, II, and III) on reinforcement strains of SGC mass, under <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>600</mn> </mrow> </semantics></math> kPa and <span class="html-italic">q</span> of 10, 30, 44, 60 and 70 kPa—for reinforcements at depths: (<b>a</b>) 0.4 m; and (<b>b</b>) 1.2 m, from the specimen’s top surface.</p> Full article ">Figure 4
<p>Effect of the widths of strip load (<span class="html-italic">w</span> = 0.175, 0.3, 0.7 m) on reinforcement strains of SGC mass, under <span class="html-italic">p</span> of 600 kPa, compaction procedure Type-III and <span class="html-italic">q</span> of 10, 30, 44, 60 and 70 kPa—for reinforcements at depths: (<b>a</b>) 0.4 m; and (<b>b</b>) 1.2 m, from the top surface.</p> Full article ">Figure 5
<p>Effect of surcharge loads (<span class="html-italic">p</span> = 200, 600, 800 kPa) on reinforcement strains of SGC mass, under compaction procedures Type-I, II and III and <span class="html-italic">q</span> of 44 and 70 kPa—for reinforcements at depths: (<b>a</b>) 0.4 m; and (<b>b</b>) 1.2 m, from the top surface.</p> Full article ">Figure 6
<p>Effect of compaction load <span class="html-italic">q</span> and compaction procedures on open face lateral displacement of SGC mass: for case of <span class="html-italic">p</span> = 1000 kPa, <span class="html-italic">q</span> = 20, 44 and 70 kPa: (<b>a</b>) Type-I; (<b>b</b>) Type-II; (<b>c</b>) Type-III (<span class="html-italic">w</span> = 0.7 m); (<b>d</b>) Type-III (<span class="html-italic">w</span> = 0.35 m); (<b>e</b>) Type-III (<span class="html-italic">w</span> = 0.175 m).</p> Full article ">Figure 7
<p>Effect of surcharge load <span class="html-italic">p</span> and compaction procedures on open face lateral displacement of SGC mass, for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>44</mn> </mrow> </semantics></math> kPa, <span class="html-italic">p</span> = 400, 1000, 2000 and 2500 kPa: (<b>a</b>) Type-I; (<b>b</b>) Type-II; (<b>c</b>) Type-III (<span class="html-italic">w</span> = 0.7 m); (<b>d</b>) Type-III (<span class="html-italic">w</span> = 0.35 m); (<b>e</b>) Type-III (<span class="html-italic">w</span> = 0.175 m).</p> Full article ">Figure 8
<p>Effect of compaction procedures on lateral displacement of SGC mass for case of <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> kPa and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>44</mn> </mrow> </semantics></math> kPa.</p> Full article ">Figure 9
<p>(<b>a</b>) Schematic diagram and configuration of the back-to-back Geo-synthetic Reinforced Soil (GRS) mass. (<b>b</b>) Heavy compaction procedures for GRS wall mass adopted in FE analysis.</p> Full article ">Figure 10
<p>Distribution of lateral stresses along the centre line of the hypothetical GRS mass during construction stage, for cases of <span class="html-italic">q</span> = 44 and 70 kPa, and compaction procedures: (<b>a</b>) Type-I; (<b>b</b>) Type-II; (<b>c</b>) Type-III (<span class="html-italic">w</span> = 0.7 m).</p> Full article ">Figure 11
<p>Comparison of CIS between analytical and numerical results for <span class="html-italic">q</span> = 44 and 70 kPa: (<b>a</b>) Type-I; (<b>b</b>) Type-II; (<b>c</b>) Type-III (<span class="html-italic">w</span> = 0.7 m).</p> Full article ">Figure 12
<p>Incremental net compaction-induced stress corresponded to GRS mass construction using compaction procedure Type-II.</p> Full article ">
23 pages, 8098 KiB  
Article
An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
by Annalisa Greco, Ilaria Fiore, Giuseppe Occhipinti, Salvatore Caddemi, Daniele Spina and Ivo Caliò
Appl. Sci. 2020, 10(9), 3212; https://doi.org/10.3390/app10093212 - 5 May 2020
Cited by 18 | Viewed by 2706
Abstract
Dynamic analyses and seismic assessments of multi-storey buildings at the urban level require large-scale simulations and computational procedures based on simplified but accurate numerical models. For this aim, the present paper proposes an equivalent non-uniform beam-like model, suitable for the dynamic analysis of [...] Read more.
Dynamic analyses and seismic assessments of multi-storey buildings at the urban level require large-scale simulations and computational procedures based on simplified but accurate numerical models. For this aim, the present paper proposes an equivalent non-uniform beam-like model, suitable for the dynamic analysis of buildings with an asymmetric plan and non-uniform vertical distribution of mass and stiffness. The equations of motion of this beam-like model, which presents only shear and torsional deformability, were derived through the application of Hamilton’s principle. The linear dynamic behaviour was evaluated by discretizing the continuous non-uniform model according to a Rayleigh–Ritz approach based on a suitable number of modal shapes of the uniform shear-torsional beam. In spite of its simplicity, the model is able to reproduce the dynamic behaviour of low- and mid-rise buildings with a significant reduction of the computational burden with respect to that required by more general models. The efficacy of the proposed approach was tested, by means of comparisons with linear Finite Element Model (FEM) simulations, on three multi-storey buildings characterized by different irregularities. The satisfactory agreement, in terms of natural frequencies, modes of vibration and seismic response, proves the capability of the proposed approach to reproduce the dynamic response of complex spatial multi-storey frames. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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<p>Aerial view of (<b>a</b>) residential blocks and (<b>b</b>) their beam-like idealization.</p> Full article ">Figure 2
<p>Conceptual representation, starting from (<b>a</b>) the 3D structure; (<b>b</b>) the generic kth floor (or inter-storey) up to the (<b>c</b>) sub-beam element; and the (<b>d</b>) the proposed beam-like model.</p> Full article ">Figure 3
<p>Scheme of (<b>a</b>) the eigen-vector matrix and (<b>b</b>) the j-th eigen-vector.</p> Full article ">Figure 4
<p>Plan archetypes: (<b>a</b>) Type 1 low- or mid-rise and (<b>b</b>) Plan B mid- or high-rise buildings.</p> Full article ">Figure 5
<p>Accelerograms of (<b>a</b>) Santa Venerina (2018) and (<b>b</b>) L’Aquila (2009).</p> Full article ">Figure 6
<p>Four storey building benchmark (Type 1): (<b>a</b>) column cross section plan, and (<b>b</b>) the FEM model.</p> Full article ">Figure 7
<p>Four-storey benchmark. Modal shapes comparison between the (<b>a</b>) FEM and (<b>b</b>) beam-like models.</p> Full article ">Figure 8
<p>Four-storey benchmark. Santa Venerina earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 9
<p>Four-storey benchmark. L’Aquila earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 10
<p>Four-storey benchmark. Maxima floor displacements in the (<b>a</b>) Santa Venerina and (<b>b</b>) L’Aquila earthquakes.</p> Full article ">Figure 11
<p>Eight storey building benchmark (Type 2): (<b>a</b>) the column cross section plan, and (<b>b</b>) the FEM model.</p> Full article ">Figure 12
<p>Eight -storey benchmark. Modal shapes comparison between the (<b>a</b>) FEM and (<b>b</b>) beam-like models.</p> Full article ">Figure 13
<p>Eight-storey benchmark. Santa Venerina earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 14
<p>Eight-storey benchmark. L’Aquila earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 15
<p>Eight-storey benchmark. Maxima floor displacements in the (<b>a</b>) Santa Venerina and (<b>b</b>) L’Aquila earthquakes.</p> Full article ">Figure 16
<p>Six storey building benchmark (Model 2): (<b>a</b>) the column cross section plan and (<b>b</b>) the FEM model.</p> Full article ">Figure 17
<p>Six-storey benchmark. Modal shapes comparison between the (<b>a</b>) FEM and (<b>b</b>) beam-like models.</p> Full article ">Figure 18
<p>Six-storey benchmark. Santa Venerina earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 19
<p>Six-storey benchmark. L’Aquila earthquake. Displacements time history along the (<b>a</b>) x and (<b>b</b>) y direction.</p> Full article ">Figure 20
<p>Six-storey benchmark. Maxima floor displacements in (<b>a</b>) Santa Venerina and (<b>b</b>) L’Aquila earthquakes.</p> Full article ">Figure 21
<p>Six-storey benchmark. Maxima floor displacements in the L’Aquila earthquake for a different number of shape functions adopted in the Rayleigh–Ritz discretization.</p> Full article ">
20 pages, 6942 KiB  
Article
Experimental Study on the Flexural Behavior of over Reinforced Concrete Beams Bolted with Compression Steel Plate: Part I
by Shatha Alasadi, Payam Shafigh and Zainah Ibrahim
Appl. Sci. 2020, 10(3), 822; https://doi.org/10.3390/app10030822 - 23 Jan 2020
Cited by 12 | Viewed by 4731
Abstract
The purpose of this paper is to investigate the flexural behavior of over-reinforced concrete beam enhancement by bolted-compression steel plate (BCSP) with normal reinforced concrete beams under laboratory experimental condition. Three beams developed with steel plates were tested until they failed in compression [...] Read more.
The purpose of this paper is to investigate the flexural behavior of over-reinforced concrete beam enhancement by bolted-compression steel plate (BCSP) with normal reinforced concrete beams under laboratory experimental condition. Three beams developed with steel plates were tested until they failed in compression compared with one beam without a steel plate. The thicknesses of the steel plates used were 6 mm, 10 mm, and 15 mm. The beams were simply supported and loaded monotonically with two-point loads. Load-deflection behaviors of the beams were observed, analyzed, and evaluated in terms of spall-off concrete loading, peak loading, displacement at mid-span, flexural stiffness (service and post-peak), and energy dissipation. The outcome of the experiment shows that the use of a steel plate can improve the failure modes of the beams and also increases the peak load and flexural stiffness. The steel development beams dissipated much higher energies with an increase in plate thicknesses than the conventional beam. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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Figure 1
<p>Detailing of all the over-reinforced beam specimens (<b>a</b>) control beam (CB) and (<b>b</b>) bolts compression steel plates the (BCSP-6), BCSP-10, and BCSP-15 beam (all dimensions in mm).</p> Full article ">Figure 2
<p>Details of the reinforcement for the over-reinforced specimens (all dimensions in mm).</p> Full article ">Figure 3
<p>Mold with reinforcement and location of steel-plate.</p> Full article ">Figure 4
<p>Casting the over-reinforced beam and after one day of the cast and location of bolts.</p> Full article ">Figure 5
<p>View of (<b>a</b>) curing for all specimens and (<b>b</b>) all the over-reinforced beam specimens.</p> Full article ">Figure 6
<p>(<b>a</b>) Schematic of the test-setup for over-reinforced beam specimens and (<b>b</b>) experimental for the BCSP beams test-setup.</p> Full article ">Figure 7
<p>Modes of failure and crack patterns of all the over-reinforced specimens.</p> Full article ">Figure 7 Cont.
<p>Modes of failure and crack patterns of all the over-reinforced specimens.</p> Full article ">Figure 8
<p>Modes of failure and crack patterns of all the CB, BCSP-6, BCSP-10, and BCSP-15 beam specimens at 80% peak load.</p> Full article ">Figure 9
<p>Load–strain relationships for all the over-reinforced beam specimens’ strains up until failure.</p> Full article ">Figure 10
<p>Location varies of load–displacement curves for all the over-reinforced beam specimens.</p> Full article ">Figure 11
<p>Maximum width of flexural crack for all the over-reinforced beam specimens.</p> Full article ">Figure 12
<p>Typical load-displacement curves for evaluating the energy absorbed (EA) for all the over-reinforced beam specimens.</p> Full article ">Figure 13
<p>Energy absorption (EA) capacity for all the over-reinforced beam specimens.</p> Full article ">Figure 14
<p>Flexural stiffness (K<sub>0.3Ppeak</sub>, K<sub>spall</sub>, K<sub>yield</sub>, and K<sub>Ppeak</sub>) of all the over-reinforced beam specimens.</p> Full article ">Figure 15
<p>Modes of failure and crack patterns of over-reinforced beams with helical reinforcements (beam 5) [<a href="#B7-applsci-10-00822" class="html-bibr">7</a>].</p> Full article ">Figure 16
<p>Modes of failure and crack patterns of over-reinforced beams with stirrups (3B) and fiber + stirrups (4B) [<a href="#B9-applsci-10-00822" class="html-bibr">9</a>].</p> Full article ">Figure 17
<p>Modes of failure and crack patterns of over-reinforced beams with a side-steel plate (<b>a</b>) SBSP and (<b>b</b>) SBWP [<a href="#B12-applsci-10-00822" class="html-bibr">12</a>].</p> Full article ">
14 pages, 1364 KiB  
Article
Free Damping Vibration of Piezoelectric Cantilever Beams: A Biparametric Perturbation Solution and Its Experimental Verification
by Zhi-Xin Yang, Xiao-Ting He, Dan-Dan Peng and Jun-Yi Sun
Appl. Sci. 2020, 10(1), 215; https://doi.org/10.3390/app10010215 - 26 Dec 2019
Cited by 7 | Viewed by 6525
Abstract
As an intelligent material, piezoelectric materials have been widely used in many intelligent fields, especially in the analysis and design of sensors and actuators; however, the vibration problems of the corresponding structures made of the piezoelectric materials are often difficult to solve analytically, [...] Read more.
As an intelligent material, piezoelectric materials have been widely used in many intelligent fields, especially in the analysis and design of sensors and actuators; however, the vibration problems of the corresponding structures made of the piezoelectric materials are often difficult to solve analytically, because of their force–electric coupling characteristics. In this paper, the biparametric perturbation method was used to solve the free damping vibration problem of piezoelectric cantilever beams, and the perturbation solution of the problem solved here was given. A numerical example was given to discuss the influence of the piezoelectric properties on the vibration of piezoelectric cantilever beams. In addition, related vibration experiments of the piezoelectric cantilever beams were carried out, and the experimental results were in good agreement with the theoretical results. The results indicated that the biparametric perturbation solution obtained in this study is effective, and it may serve as a theoretical reference for the design of sensors and actuators made of piezoelectric materials. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Scheme of the mechanical model of the piezoelectric cantilever beam.</p> Full article ">Figure 2
<p>Scheme of the experimental specimens and piezoelectric cantilever beam: (<b>a</b>) PZT-5 piezoelectric ceramic specimens; (<b>b</b>) the piezoelectric cantilever beam.</p> Full article ">Figure 3
<p>Scheme of the measuring instruments and experimental device: (<b>a</b>) the non-contact laser displacement sensor; (<b>b</b>) the integral measuring device.</p> Full article ">Figure 4
<p>The time–displacement curves under the initial displacement of 0.475 mm.</p> Full article ">Figure 5
<p>The time–displacement curves under the initial displacement of 0.750 mm.</p> Full article ">Figure 6
<p>The time–displacement curves under the initial displacement of 1.342 mm.</p> Full article ">
21 pages, 33223 KiB  
Article
An Experimental and Numerical Study on the Flexural Performance of Over-Reinforced Concrete Beam Strengthening with Bolted-Compression Steel Plates: Part II
by Shatha Alasadi, Zainah Ibrahim, Payam Shafigh, Ahad Javanmardi and Karim Nouri
Appl. Sci. 2020, 10(1), 94; https://doi.org/10.3390/app10010094 - 20 Dec 2019
Cited by 7 | Viewed by 9089
Abstract
This study presents an experimental investigation and finite element modelling (FEM) of the behavior of over-reinforced simply-supported beams developed under compression with a bolt-compression steel plate (BCSP) system. This study aims to avoid brittle failure in the compression zone by improving the strength, [...] Read more.
This study presents an experimental investigation and finite element modelling (FEM) of the behavior of over-reinforced simply-supported beams developed under compression with a bolt-compression steel plate (BCSP) system. This study aims to avoid brittle failure in the compression zone by improving the strength, strain, and energy absorption (EA) of the over-reinforced beam. The experimental program consists of a control beam (CB) and three BCSP beams. With a fixed steel plate length of 1100 mm, the thicknesses of the steel plates vary at the top section. The adopted plate thicknesses were 6 mm, 10 mm, and 15 mm, denoted as BCSP-6, BCSP-10, and BCSP-15, respectively. The bolt arrangement was used to implement the bonding behavior between the concrete and the steel plate when casting. These plates were tested under flexural-static loading (four-point bending). The load-deflection and EA of the beams were determined experimentally. It was observed that the load capacity of the BCSP beams was improved by an increase in plate thickness. The increase in load capacity ranged from 73.7% to 149% of the load capacity of the control beam. The EA was improved up to about 247.5% in comparison with the control beam. There was also an improvement in the crack patterns and failure modes. It was concluded that the developed system has a great effect on the parameters studied. Moreover, the prediction of the concrete failure characteristics by the FE models, using the ABAQUS software package, was comparable with the values determined via the experimental procedures. Hence, the FE models were proven to accurately predict the concrete failure characteristics. Full article
(This article belongs to the Special Issue Homogenization Methods in Materials and Structures)
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<p>Dimension details of all the specimens: (<b>a</b>) The CB and (<b>b</b>) the BCSP-6, BCSP-10, &amp; BCSP-15 beams (all dimensions are in mm). BCSP: Bolt-compression steel plate. CB: Control beam.</p> Full article ">Figure 1 Cont.
<p>Dimension details of all the specimens: (<b>a</b>) The CB and (<b>b</b>) the BCSP-6, BCSP-10, &amp; BCSP-15 beams (all dimensions are in mm). BCSP: Bolt-compression steel plate. CB: Control beam.</p> Full article ">Figure 2
<p>Details of the reinforcement of all the over-reinforced specimens. Over-RC: Over-reinforced concrete.</p> Full article ">Figure 3
<p>Reinforcement and location of the steel plate.</p> Full article ">Figure 4
<p>Over-reinforced concrete beam experiment test setup for all of the tested specimens. LVDT: Linear variable displacement transducer.</p> Full article ">Figure 5
<p>Modes of failure and crack patterns of all of the specimens: (<b>a</b>) CB; (<b>b</b>) BCSP-6; (<b>c</b>) BCSP-10 and (<b>d</b>) BCSP-15.</p> Full article ">Figure 6
<p>Load-strain relationships for all the beam specimens’ applied strains until failure.</p> Full article ">Figure 6 Cont.
<p>Load-strain relationships for all the beam specimens’ applied strains until failure.</p> Full article ">Figure 7
<p>Load-displacement plots for all of the over-reinforced concrete beam specimens.</p> Full article ">Figure 8
<p>The response of the concrete to uniaxial loading under (<b>a</b>) tension and (<b>b</b>) compression.</p> Full article ">Figure 9
<p>Uniaxial stress-strain and damage-strain relationships.</p> Full article ">Figure 10
<p>Mesh discretization for all the over-reinforced specimens.</p> Full article ">Figure 11
<p>Comparison between the mesh sizes for all FE models.</p> Full article ">Figure 12
<p>Comparison of the load-displacement curves derived from the experimental and numerical studies.</p> Full article ">Figure 13
<p>The von Mises stresses for all the beam specimens.</p> Full article ">Figure 14
<p>Comparison of the energy absorption between the experiment and FEM.</p> Full article ">Figure 15
<p>Comparison between (<b>a</b>) the experimental tests with the FE analysis of the (<b>b</b>) compressive and (<b>c</b>) tensile damage cracks in the CB at the point of failure.</p> Full article ">Figure 16
<p>Comparison between (<b>a</b>) the experimental tests with the FE modelling of the (<b>b</b>) compressive and (<b>c</b>) tensile damage cracks in the BCSP-6 beam at the point of failure.</p> Full article ">Figure 17
<p>Comparison between (<b>a</b>) the experimental tests with the FE modelling of the (<b>b</b>) compressive and (<b>c</b>) tensile damage cracks in the BCSP-10 beam at the point of failure.</p> Full article ">Figure 18
<p>Comparison between (<b>a</b>) the experimental tests with the FE modelling of the (<b>b</b>) compressive and (<b>c</b>) tensile damage cracks in the BCSP-15 beam at the point of failure.</p> Full article ">
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