Advanced engineering materials and modeling

Preface xiii Part 1 Engineering of Materials, Characterizations, and Applications 1 Mechanical Behavior and Resistance of Structural Glass Beams Chiara Bedon and Jan Belis 1.3.2 Mechan

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Preface xiii

Part 1 Engineering of Materials, Characterizations,

and Applications

1 Mechanical Behavior and Resistance of Structural Glass Beams

Chiara Bedon and Jan Belis

1.3.2 Mechanical and Geometrical Influencing

Parameters in Structural Glass Beams 8

1.4 Theoretical Background for Structural Members in LTB 141.4.1 General LTB Method for Laterally

1.4.2 LTB Method for Laterally Unrestrained (LU)

1.4.2.1 Equivalent Thickness Methods for

Laminated Glass Beams 181.4.3 Laterally Restrained (LR) Beams in LTB 231.4.3.1 Extended Literature Review on

1.5 Finite-element Numerical Modeling 311.5.1 FE Solving Approach and Parametric Study 321.5.1.1 Linear Eigenvalue Buckling Analyses (lba) 321.5.1.2 Incremental Nonlinear Analyses (inl) 35

2 Room Temperature Mechanosynthesis of Nanocrystalline

S.K Pradhan and H Dutta

2.1.5 The Merits and Demerits of Planetary Ball Mill 522.1.6 Review of Works on Metal Carbides by

Contamination? 712.4.5 Variation of Particle Size 72

2.4.7 High-Resolution Transmission Electron

2.4.8 Comparison Study between Binary and Ternary

2.5 Conclusion 80Acknowledgment 80References 80

Chiara Bedon and Filipe Amarante dos Santos

3.4.2.2 Mechanical Characterization of Materials 983.5 Discussion of Parametric FE Results 1013.5.1 Roof Glass Panel (M1) 1013.5.1.1 Short-term Loads and Temperature

Variations 1023.5.1.2 First-cracking Configuration 1063.5.2 Point-supported Façade Panel (M2) 1093.5.2.1 Short-term Loads and Temperature

4.10 Properties of Bagasse Pulps 136

References 185

Part 2 Computational Modeling of Materials

6 Calculation on the Ground State Quantum Potentials for the

G.H.E Alshabeeb and A.K Arof

6.2 Ground State in D-Dimensional Configuration Space

for ZnSxSe1-x Zincblende Structure 1946.3 Ground States in the Case of Momentum Space 196

6.5 Conclusions 201Acknowledgment 201References 201

7 Application of First Principles Theory to the Design of

Y Song, J H Dai, and R Yang

7.2 Basic Concepts of First Principles 2047.3 Theoretical Models of Alloy Design 2077.3.1 The Hume-Rothery Theory 2077.3.2 Discrete Variational Methodandd-Orbital

Method 2127.3.2.1 Discrete Variational Method 212

7.3.2.2 d-Electrons Alloy Theory 214

9.3 Scattering from Rough Sea Surfaces 2489.3.1 Numerical Validation and Monte Carlo

Simulations 2529.4 Scattering from Obstacles with Rough Surfaces or

9.4.1 Numerical Validation and Monte Carlo

Simulations 2599.4.2 Combining Perturbation Theory and

Transformation Optics for Weakly Perturbed Surfaces 2609.5 Scattering from Randomly Positioned Array of Obstacles 2649.5.1 Separate Transformation Media 2659.5.1.1 Numerical Validation & Monte Carlo

Simulations 2679.5.2 A Single Transformation Medium 2699.5.2.1 Numerical Validation & Monte Carlo

Simulations 2719.5.3 Recurring Scaling and Translation

Transformations 2729.5.3.1 Numerical Validation & Monte Carlo

Simulations 2749.6 Propagation in a Waveguide with Rough or

9.6.1 Numerical Validation and Monte Carlo

Simulations 279

References 284

10 Superluminal Photons Tunneling through Brain Microtubules

Luigi Maxmilian Caligiuri and Takaaki Musha

10.2 QED Coherence in Water: A Brief Overview 29110.3 “Electronic” QED Coherence in Brain Microtubules 29710.4 Evanescent Field of Coherent Photons and

Their Superluminal Tunneling through MTs 30110.5 Coupling between Nearby MTs and their Superluminal Interaction through the Exchange of Virtual

10.7 Brain Microtubules as “Natural” Metamaterials and

the Amplification of Evanescent Tunneling Wave

Amplitude 31510.8 Quantum Computation by Means of Superluminal

Photons 321

References 326

11 Advanced Fundamental-solution-based Computational

Hui Wang and Qing-Hua Qin

11.3.2 Implementation of the HFS-FEM 342 11.3.4 Recovery of Rigid-body Motion 34511.4 Applications in Functionally Graded Materials 345 11.4.1 Basic Equations in Functionally Graded

Materials 345 11.4.2 MFS for Functionally Graded Materials 346 11.4.3 HFS-FEM for Functionally Graded Materials 34911.5 Applications in Composite Materials 353 11.5.1 Basic Equations of Composite Materials 353 11.5.2 MFS for Composite Materials 356 11.5.2.1 MFS for the Matrix Domain 356 11.5.2.2 MFS for the Fiber Domain 356 11.5.2.3 Complete Linear Equation System 357 11.5.3 HFS-FEM for Composite Materials 358 11.5.3.1 Special Fundamental Solutions 358 11.5.3.2 Special n-Sided Fiber/Matrix

Elements 359

Acknowledgments 362

References 362

12 Understanding the SET/RESET Characteristics of Forming

P Bousoulas and D Tsoukalas

12.2 Experimental Methodology 372

12.3.1 Resistive-Switching Performance 376 12.3.2 Resistive-Switching Model 379

13 Advanced Materials and Three-dimensional Computer-aided

Luis Miguel Gonzalez-Perez, Borja Gonzalez-Perez-Somarriba Gabriel Centeno, Carpóforo Vallellano and

Juan Jose Egea-Guerrero

14.1 An Algorithm with Splitting of Wavelet

Transformation of Splines of the First Degree 439

14.1.2 Examples of Wavelet Decomposition

14.1.3 “Orthonormal” Wavelets 446 14.1.4 An Example of Function of Harten 45014.2 An Algorithm for Constructing Orthogonal to

Polynomials Multiwavelet Bases 452 14.2.1 Creation of System of Basic Multiwavelets

of Any Odd Degree on a Closed Interval 452 14.2.2 Creation of the Block of Filters 455 14.2.3 Example of Orthogonal to Polynomials

14.2.4 The Discussion of Approximation on a Closed

Interval 45914.3 The Tridiagonal Block Matrix Algorithm 460 14.3.1 Inverse of the Block of Filters 460 14.3.2 Example of the Hermite Quintic Spline

Function Supported on [−1, 1] 461 14.3.3 Example of the Hermite Septimus Spline

Function Supported on [−1, 1] 463 14.3.4 Numerical Example of Approximation

14.3.5 Numerical Example with Two Ruptures

of the First Kind and a Corner 46714.4 Problem of Optimization of Wavelet Transformation

of Hermite Splines of Any Odd Degree 471 14.4.1 An Algorithm with Splitting for Wavelet

Transformation of Hermite Splines of

Modeling of Routes and Surfaces of Highways 488

References 490

Index 495

The engineering of materials with advanced features is driving the research towards the design of innovative high-performance materials New mate-rials often deliver the best solutions for structural applications, precisely contributing to the finest combination of mechanical properties and low weight Furthermore, these materials mimic the principles of nature, lead-ing to a new class of structural materials which include biomimetic com-posites, natural hierarchical materials and smart materials Meanwhile, computational modeling approaches are valuable tools which are com-plementary to experimental techniques and provide significant informa-tion at the microscopic level and explain the properties of materials and their existence itself The modeling further provides useful insight to propose possible strategies to design and fabricate materials with novel and improved properties Depending upon the pragmatic computational models of choice, approaches vary for the prediction of the structure- and element-based approaches to fabricate materials with properties of inter-est This book brings together the engineering materials and modeling approaches generally used in structural materials science.

Research topics on materials engineering, characterization, applications and their computational modeling are covered in this book In general, computational modeling approaches are routinely used as cost-effective and complementary tools to get information about the materials at the microscopic level and to explain their electronic and magnetic proper-ties and the way they respond to external parameters like temperature and pressure In addition, modeling provides useful insight into the construct

of design principles and strategies to fabricate materials with novel and improved properties The use of modeling together with experimental vali-dation opens up the possibility for designing extremely useful materials that are relevant for various industries and healthcare sectors This book has been designed in such a way as to cover aspects of both the use of experimental and computational approaches for materials engineering and fabrication Chapters 1 through 6 are devoted to experimental character-ization of materials and some of their applications relevant to the paper

xv

industry and healthcare sectors Chapters 7 through 13 are devoted to computational materials modeling and their fabrication using atomistic- and finite-element-based approaches Specifically discussed in Chapters 7 and 8 are first-principles-based modeling approaches to predict the struc-ture and electronic properties of extended systems The remaining chapters contribute with theoretical approaches to understanding hybrid materials and stochastic electromagnets and to modeling complex processes like tunneling of superluminal photons

The book is written for readers from diverse backgrounds across try, physics, materials science and engineering, medical science, pharmacy, environmental technology, biotechnology, and biomedical engineering It offers a comprehensive view of cutting-edge research on materials engi-neering and modeling We acknowledge the contributors and publisher for their prompt response in order that this book could be published in a timely manner

chemis-Editors Ashutosh Tiwari, PhD, DSc

N Arul Murugan, PhD Rajeev Ahuja, PhD

10 June 2016

Part 1

ENGINEERING OF MATERIALS, CHARACTERIZATIONS, AND

APPLICATIONS

Ashutosh Tiwari, N Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (1–48) © 2016 Scrivener Publishing LLC

Ashutosh Tiwari, N Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (3–48) © 2016 Scrivener Publishing LLC

Mechanical Behavior and Resistance

of Structural Glass Beams in Lateral–Torsional Buckling (LTB) with Adhesive Joints

Chiara Bedon 1 * and Jan Belis 2

1 University of Trieste, Department of Engineering and Architecture, Trieste, Italy

2 Ghent University, Department of Structural Engineering, Laboratory

for Research on Structural Models – LMO, Ghent, Belgium

Abstract

Glass is largely used in practice as an innovative structural material in the form

of beams or plate elements able to carry loads Compared to traditional struction materials, the major influencing parameter in the design of structural glass elements – in addition to their high architectural and aesthetic impacts – is given by the well-known brittle behavior and limited tensile resistance of glass

con-In this chapter, careful attention is paid to the lateral–torsional buckling (LTB) response of glass beams laterally restrained by continuous adhesive joints, as in the case of glass façades or roofs Closed-form solutions and finite-element numer-ical approaches are recalled for the estimation of their Euler’s critical buckling moment under various loading conditions Nonlinear buckling analyses are then critically discussed by taking into account a multitude of mechanical and geo-metrical aspects Design recommendations for laterally restrained glass beams in LTB are finally presented

Keywords: Lateral–torsional buckling (LTB), glass beams, analytical models,

finite-element modeling, structural adhesive joints, composite sections,

incremental buckling analysis, imperfections, buckling design methods,

buckling curve

*Corresponding author: bedon@dicar.units.it

1.1 Introduction

Glass is largely used in practice as an innovative structural material, e.g in the form of beams or plate elements able to carry loads Often, structural glass components are used in structures in combination with other mate-rials, such as timber [1–6] or composites [1, 7–9] However, especially in façades, roofs, and building envelopes, the use of glass panels combined with steel frames, aluminum bracing systems, or cable nets represents one

of the major configurations, for which a wide set of case studies and nological possibilities are available [1, 2, 10–15] Compared to traditional construction materials, the major influencing parameter in the design

tech-of structural glass elements – in addition to their high architectural and aesthetic impact – is given by the well-known brittle behavior and lim-ited tensile resistance of glass The use of thermoplastic interlayers alter-nated to two (or more) glass sheets in the form of laminated glass (LG) elements – despite the high sensitivity of the bonding foils to the effects

of temperature and load-duration – represents the typical solution for buildings, automotive applications, etc due to the intrinsic ductility and post-breakage resistance

In those cases, the typical configurations for structural glass assemblies are often derived – and properly modified, to account for the brittle behavior of glass – from practice of traditional construction materials (e.g steel structures and sandwich structures) The connections used in such LG assemblies are traditionally properly designed and well-calibrated mechanical connections (e.g steel fasteners and bolted joints) able to offer a certain structural interaction among multiple glass components However, due to continuous scientific (material) improvements, technological innovations and architectural demands, recent design trends are often oriented towards the minimization of mechanical joints and toward the development of frameless glazing systems, in which glass to glass interaction is provided by chemical connections such as sealant joints or adhesives only This is the case for beams, such as glass elements used in practice as stiffeners for façade or roof panels, where the coupling between them is often provided by continuous adhesive joints From a structural point of view, the effect of such joints can be compared to a partially rigid shear connection, and consequently its mechanical effectiveness should be properly taken into account

Bolted point fixings or continuous adhesive joints currently resent the two most used typologies of connections and can both be employed in glass façades or roofs, e.g to provide the mechanical inter-action between the glass beams and the supported glass roof panels

rep-While in the first case the bolted connectors and their related effects can often be rationally described in the form of infinitely rigid intermedi-ate restraints, the configuration of glass beams with continuous adhe-sive joints requires appropriate studies and related analytical methods Adhesive joints are in fact characterized by moderate shear stiffness, and consequently they act as a continuous, flexible joint between the beams and the connected panels Adhesives of common use in practice are also characterized by moderate shear/tensile resistance; hence, an appropri-ate design approach should be taken into account for them, regardless of possible LTB phenomena.

This chapter, in this context, aims to present an extended review of glass beams in LTB, including a discussion of the main influencing parameters, mechanical properties, geometrical aspects, available analytical methods, and finite-element (FE) approaches A detailed discussion of the LTB mechanical response of glass beams, laterally unrestrained or restrained by means of continuous adhesive joints, will then be proposed

1.2 Overview on Structural Glass Applications in Buildings

Structural glass applications are mainly associated, in current practice, to aesthetic, architectural or thermal, and acoustic requirements Glass is,

in fact, synonymous of transparency and lightness, hence finds ily application in building envelopes, roofs, canopies, etc and solutions in which transparency is mandatory Major structural glass assemblies – often

primar-of complex geometry – are obtained by appropriate conjunct use primar-of glass elements with metal frameworks and substructures (Figure 1.1)

Structural configurations combining glass elements with timber ponents (Figure 1.2) also represent a solution of large interest for design-ers and engineers, especially in those applications aiming to strong energy efficiency [24]

compres-Figure 1.1 Example of structural glass applications in buildings, in conjunction with metal

frameworks and substructures Pictures taken from (a) [16], (b) [17], (c) [18], and (d) [19].

failure mechanisms (Figure 1.3) As far as these structural elements are slender and/or affected by several influencing parameters, such as initial geometrical imperfections, eccentricities, and residual stresses, the suscep-tibility to buckling phenomena increases and represents an important issue

to be properly predicted and prevented This is the case of both isotropic and orthotropic plates, beams, columns, but also laminates and composites

in general

The presence of rather unconventional materials, in particular, resents one of the major influencing parameters to be properly assessed, especially in the presence of materials whose mechanical properties can be affected by time/temperature-dependent degradation In structural glass beams, a multitude of effects strictly related to mechanical properties, geo-metrical features, initial imperfections, etc., should be properly taken into account to prevent possible LTB failure mechanisms

rep-Figure 1.2 Example of structural glass applications in buildings, in conjunction with timber

components and assemblies Pictures taken from (a) [20], (b) [21], (c) [22], and (d) [23].

L0

L0

1.3.2 Mechanical and Geometrical Influencing Parameters

in Structural Glass Beams

Structural glass beams find primary applications in façades and roofs in the form of stiffeners There, both mechanical and adhesive joints can be used to provide a certain structural interaction between the glass beams and the supported panels (see Sections 1.3.3 and 1.3.4)

Compared to beams composed of traditional construction materials, such as steel, the out-of-plane bending response of glass fins is character-ized by specific mechanical and geometrical aspects that should be prop-erly taken into account when assessing their expected structural response.First, glass is a material characterized by a relatively small modulus

of elasticity E compared to steel, see Table 1.1 and Figure 1.4), and by a

typical brittle elastic tensile behavior with limited characteristic strength (Figure 1.4b)

Figure 1.4 Mechanical properties of monolithic glass for structural applications

(a) Qualitative comparison of glass tensile behavior vs traditional construction materials, such as steel and concrete; (b) tensile constitutive law of glass, depending on the adopted pre-stressing technique (FT = fully tempered, HS = heat-strengthened, AN = annealed).

Concrete

HS FT

Steel

Table 1.1 Soda lime silica glass properties [25].

Symbol Unit Soda lime silica glass

Although thermal or chemical pre-stressing processes can increase the reference characteristic tensile strength of annealed glass (AN) by a factor

of about two (for heat-strengthened glass, HT) or even three (in the case

of fully tempered glass, FT), the occurrence of both local or global failure mechanisms due to the tensile peaks should be properly prevented

Careful consideration should be given to glass especially in the vicinity

of supports and point fixings, as well as to LG cross sections (Figure 1.5), representing the majority of structural glass applications but being typi-cally characterized by the presence of two (or more) glass layers and one (or more) intermediate foils able to act in the form of a flexible shear con-nection only between them Common interlayers are, in fact, composed of PVB [26, 27], SG [28], or Ethylene vinyl acetate (EVA) [29] components,

e.g by thermoplastic films whose shear stiffness G int strictly depends on several conditions (e.g time loading, temperature (Figure 1.6))

Also in the case of cross sections composed of multiple glass layers (e.g Figure 1.5), glass beams are moreover characterized by relatively high

slenderness ratios, e.g large h/t ratios with long spans L.

1.3.3 Mechanical Joints

Glass elements can be used in constructions in different ways, including

a variety of metal point fixings and connectors able to provide a certain restraint and interaction between multiple structural components

Figure 1.5 Typical cross sections of common use in structural glass applications (edge

chamfers are neglected) (a) Monolithic cross section or (b), (c), and (d) laminated cross sections.

t int

Common options in structural glass applications are in fact sented by metal clamp fixings, drilled fixings, and auxiliary metal connec-tors (Figure 1.7) Characterized by strong aesthetic impact, mechanical point supports generally provide a fully rigid restraint to the joined glass panels, hence allowing to minimize the presence of metal frameworks and substructures On the other hand, specific design rules and require-ments must be satisfied, to avoid possible local failure mechanisms in glass, etc.

repre-1.3.4 Adhesive Joints

An important aspect that should be properly taken into account in the design of structural glass assemblies in general, but in particular for the LTB calculation of glass beams, is represented by the presence of adhe-sive joints at the interface between the beams themselves and the sup-ported elements Some examples are provided in Figure 1.8a and b, while Figure 1.8c and d presents a schematic view of a typical adhesive joint, with the corresponding analytical model As shown in Figure 1.8c, the typical joint consists in fact of small, continuous layers of adhesive and special setting blocks, which act as spacers during application and curing

of the adhesive In addition, they provide an appropriate joint stiffness

Figure 1.6 Mechanical constitutive law of common interlayers for structural glass

function of load duration, for different temperatures.

Figure 1.7 Examples of mechanical joints in structural glass applications Pictures

derived from (a) [30], (b) [31], (c), and (d) [32].

(b) (a)

(d) (c)

in the direction of the applied external loads In this hypothesis, under the action of loads applied on the glass panels (e.g distributed pres-sures due to live loads on the roof), the adhesive joint behaves as con-

tinuous, infinitely rigid link in the z-direction, while the same joint acts

as a flexible shear connection toward possible out-of-plane phenomena

(e.g y-direction) In general, when designing an adhesive connection for

structural glass applications, several aspects should be properly taken into account The strength of a given adhesive joint – compared to mechanical fasteners – is in fact strictly related to a multitude of influencing param-eters, such as the joint geometrical properties (e.g shape, thickness), its mechanical properties (e.g the type of adhesive), the duration of loading (e.g due to possible degradation of the reference mechanical properties), and further environmental parameters including temperature, moisture,

UV light, time curing, adhesion, etc For analytical calculations or refined

FE studies and analyses related to the LTB response of laterally restrained glass beams, as shown in Sections 1.4 and 1.5, the actual mechanical

properties of common adhesive joints represent a key input parameter

In Table 1.2, some nominal mechanical properties are proposed for ous adhesives of common use for glass-to-glass, glass-to-steel, and glass-to-timber connections

vari-Alternatively, the simplest way to determine the constitutive law of a given adhesive type for structural applications in glass beams and fins takes the form of a pure shear test

Shear tests, for example, were performed at Ghent University [39] on a series of adhesive specimens composed of Dow Corning 895 (DC 895) [37], a one-component sealant largely used in practice for glass structures (Figure 1.9)

Figure 1.8 Typical examples of structural glass applications with adhesive joints Pictures

taken from (a) [33] and (b) [34] (c and d) Schematic view of a typical adhesive joint for application in a glass roof (c) Overview and (d) transversal cross section of the reference analytical model.

W adh

h adh

Table 1.2 Mechanical properties of common adhesive types for structural glass

Young’s modulus (MPa)

n.a.: not available.

Figure 1.9 Progressive shear failure of a structural silicone specimen [39].

Displacement-controlled shear tests were carried out at 23 °C, with

a constant speed deformation of 5 mm/min as recommended by ETAG

002 [40] Small sealant samples (total length l adh = 100 mm) with different

square section size (w adh = 6 and 15 mm, in accordance with [37]) were tested, in accordance with Figure 1.8 An almost linear elastic behavior

was noticed, up to failure, with k y = 0.184 N/mm2 the average elastic ness per unit of length All the shear tests generally highlighted in fact an almost stable behavior for the specimens, attaining large displacements before failure, with an ultimate elongation ≈ (d – w )/w equal

stiff-to ≈416% and ≈406% for series A and B, respectively, and du denoting the maximum displacement at failure The obtained average ultimate elonga-tion u,avg resulted well comparable to nominal values of structural sealants available in commerce (e.g Table 1.2) In terms of ultimate shear/tensile stress u,avg, this parameter was also derived from experimental measure-

ments as the average ratio between the failure load F u of each specimens

and the corresponding resisting cross-sectional area A adh, hence resulting

in u,avg = 0.94 N/mm2 Again, the so-calculated strength was found to be

in rather good agreement with the nominal ultimate tensile resistance

of common structural sealants and adhesives for glass applications (e.g Table 1.2)

1.4 Theoretical Background for Structural

Members in LTB

1.4.1 General LTB Method for Laterally Unrestrained (LU)

Members

In practice, the possible buckling failure of structural members composed

of traditional construction materials (e.g steel, timber or concrete beams, columns, panels) is usually checked by means of standardized design methods In them, normalized buckling curves properly calibrated are used

to express the effective design resistance of a given structural element, as a function of Euler’s theoretical buckling load and a multitude of influencing parameters, strictly related to material properties, geometrical imperfec-tions, residual stresses, defects, eccentricities, etc For safe design purposes, Euler’s buckling load is in fact conventionally reduced from the ideal value,

by means of well-calibrated imperfection factors and buckling reduction coefficients

The structural stability represents an essential phase of design for eral structural typologies and for steel structures in particular so that the first analytical approaches for a standardized buckling verification have been implemented for steel members first

sev-For clarity of presentation, let us consider the monolithic beam depicted in Figure 1.10, with the given nominal dimensions and coor-dinate system The beam is simply supported by fork bearings on its

span L0 along the x-axis, while (zy) denotes the plane of its general cross

section In EN-1993-1-1: 2005 [41], the capacity of a structural ber with regard to buckling and instability has been first expressed in the form of a buckling reduction factor , being a normalized factor

mem-strictly related to the member’s susceptibility to buckling phenomena and to the so-called slenderness parameter λLT While λLT is typically expressed as the inverse square root of Euler’s critical moment M cr( )E , e.g

in the form:

pl cr E

cr E

M M

W M

with

M pl denoting the plastic moment,

W z = ht3/6 the elastic resistant modulus, and

yk the yielding stress of steel

The buckling reduction factor LT representative of the capacity of the member with regard to stability is generally calculated as:

Figure 1.10 LTB of a laterally unrestrained monolithic beam under constant bending

proper-In accordance with this formulation, the LTB verification of a given member can be considered satisfied when the maximum applied design

moment M y,Ed does not exceed the buckling design resistance M b,Rd:

with M1 the partial safety factor for steel

The major advantage of Eqs (1.1–1.5) is represented by the general ity of the method, once the non-dimensional slenderness ratio λLT of a given cross-section (e.g the geometrical and mechanical properties) are known

valid-At the same time, the loading and boundary condition is implicitly taken into account in the form of correction factors able to modify the general expression for Euler’s critical moment value:

k L

k k

I I

z w w z

2

0 2 2

π

2 2

where

k z is the effective length factors for lateral bending,

k w is the effective length factor for warping (=1, unless special provisions are provided),

z G is the distance between the point of application of the load and the middle axis,

C1 and C2 are coefficients depending on the loading and end restraint condition (Table 1.3),

I z is the second moment of area of the section, about the minor axis,

I t is the torsion constant,

I is the warping constant

1.4.2 LTB Method for Laterally Unrestrained (LU) Glass Beams

In the past years, several experimental, analytical, and numerical studies have been dedicated to the assessment of the LTB response in LU glass beams

A buckling design approach strictly related to Eq (1.5) was, for ple, proposed for LU glass beams in LTB [42–44], by assuming in (Eq 1.6)

exam-the characteristic tensile resistance of glass f yk≡ Rk and M = 1.4 as a partial

safety factor The advantage of this approach is given to its general tion, but namely represents the extension of earlier consistent studies on the topic (e.g [45, 46])

formula-In the hypotheses of a rectangular cross section for a fork-end restrained glass beam, Eq (1.6) leads in fact to

cr E

0

, (1.7)

where E ≡ E g and G ≡ G g in Eq (1.7) represent Young’s and shear moduli of

glass, respectively, while I z = ht3/12 signifies the moment of inertia about the

minor z-axis and I t ≈ ht3/3 (for h/t > 6) is the torsional moment of inertia

Extended comparison proposed in [45] and [46] highlighted the good relation between analytical critical load predictions derived from Eq (1.7) and detailed FE models, both for monolithic and laminated cross sections belonging to beams in LTB with various geometrical and mechanical aspects

cor-In the latter case, viscoelastic FE calculations were assessed toward lent thickness approaches (e.g Section 1.4.2.1) applied to LG beams under well-defined load-time and temperature conditions The main advantage deriving from the application of equivalent thickness-based methods to LG elements is given by the assumption of fully monolithic glass sections with equivalent bending and torsional stiffnesses, hence resulting in simplified but rational and practical design methods, especially for buckling purposes

equiva-Table 1.3 Loading/restraint coefficients for the calculation of Euler’s critical

moment of simply supported, fork-end restrained beams [41]

Free rotation about

the weak axis

In [43, 44], based on classical Euler’s buckling moment definitions (e.g Eq 1.7) and the standardized method proposed by the Eurocode 3 for steel structures (Eq 1.5), calibration of the imperfection factors defining

χLT was then carried out on the base of LTB experimental data available in literature for monolithic and LG beams, as well as extended FE and ana-lytical calculations Figure 1.11b presents the result of this calibration, for the so-called “Eurocode-based” design buckling curves for glass beams in LTB, compared to previous studies (Figure 1.11a)

1.4.2.1 Equivalent Thickness Methods for Laminated Glass Beams

When applying equivalent thickness methods to LG sections in LTB, two main aspects should be properly taken into account, namely the appropri-ate estimation of both the equivalent bending stiffness and torsional stiff-ness required in Eq (1.7)

Several formulations are available in literature for this purpose In [48], for example, extended assessment of some of these existing ana-lytical models based on the equivalent thickness concept and primarily intended for the calculation of the critical LTB moment in three-layered sandwich beams was discussed and further extended for the LTB analysis

of LU LG beams, after an appropriate validation toward FE viscoelastic and experimental data In the following sections, some of these formula-tions are recalled for LG beams Analytical models are proposed for sym-metric cross sections composed of two glass layers only (e.g Figure 1.5, case b)

1.4.2.1.1 Method I

For the analysis of the LTB behavior of LG members, Luible [45] first applied the analytical formulations originally developed for sand-wich structural elements to glass sections The mentioned analytical approach, in particular, is based on the concepts of equivalent bending

stiffness EI z,eff and equivalent torsional stiffness GI t , where EI z,eff is culated depending on the specific loading condition (constant bending

cal-moment M y , distributed load q, concentrated load F at mid-span), while

GI t depends on the geometrical/mechanical properties of the cross tion only

sec-The expression proposed for EI z,eff is given as a function of the

slender-ness of the beam (t1, h, L0), the elastic stiffness of glass (E), the thickness

of the interlayer (t int ), and its mechanical properties (G int) Based on [45],

Figure 1.11 Calibration of design buckling curves for LU beams in LTB, by assuming

different geometrical imperfection amplitudes.

FE, M cost [45]

(θ 0= L0/270 h rad)

FE, q dist [45]

(θ 0= L0/270 h rad) TEST, F conc [45]

according to the laminated cross section of Figure 1.4, case (b), and

consid-ering the beam subjected to a constant bending moment M y, as proposed

in Figure 1.10, EI z,eff is in fact defined as follows:

s

int int ( 2 1)2

where the expression for I t,1 and I t,comp are listed in Table 1.4

Specifically, Eq (1.15) takes into account the effective torsional

con-tribution I t,comp due to the adopted interlayer Stamm and Witte originally derived the expressions, partly collected in Table 1.4 [45], for the estima-tion of this torsional stiffness term, typically occurring in a faced “soft”

core within a flat sandwich panel subjected to a torsional moment M

Figure 1.12 Qualitative torsional behavior of a LG beam in accordance with the analytical

model recalled in [45].

Shear stresses due to the interlayer

Shear stresses due to torsion

of the glass layers

Table 1.4 The Stamm–Witte equivalent parameters for the calculation of the

torsional stiffness term in layered cross sections [45]

1 33/

,tanh( / )/

2

λλ

It,LT

4

21

21

Their model basically applies to layered elements in which the cross

sec-tion is uniform along the total length L0 Largely used for the analysis of sandwich elements and recalled in several handbooks [49–51], Eq (1.15) has been applied successfully to LG elements

1.4.2.1.2 Method II

An alternative analytical model for the lateral–torsional buckling (LTB) verification of LG beams has been assessed in [48] In that case, the

theoretical model was based on the Wölfel–Bennison expression for the

equivalent thickness t eq, e.g on the concept of an equivalent, monolithic flexural stiffness EI z eff, =ht eq3 /

the shear transfer coefficient representative of the shear transfer

contri-bution of the adopted interlayer, where I s,WB in Eq (1.17) is equal to I s /h

(Eq 1.9) Due to the shear transfer coefficient b, the effective stiffness of the interlayer can be rationally taken into account within a range conven-

tionally comprised between an “abs” layered limit (e.g G int → 0) and “full”

monolithic limit (e.g G int → ∞)

Analytical calculations highlighted that based on Eq (1.17) the

flex-ural stiffness EI z,eff = f(t eq) exactly coincides, for the boundary and loading conditions considered in this contribution, with calculations provided

by exact analytical models (e.g derived for example from Newmark’s theory of beams with partially rigid interaction [42]) To be used for LTB purposes, the “Method II” further requires the calculation of the

torsional stiffness term GI t, that also in this case is calculated based on

Eq (1.15)

1.4.2.1.3 Other Available Formulations

The so-called “Method I” and “Method II” represent two analytical approaches

of large use for structural glass applications Other formulations – with almost the same effects – are anyway available in the literature

Based on [52], for example, the torsional stiffness of laminated cross

sections is calculated as a multiple of the “abs” torsional stiffness sponding to a null shear stiffness of the interlayers (G int → 0), e.g by intro-

corre-ducing a parameter f ≥ 1 so that

is the equivalent torsional stiffness of the laminated member, where

1 2 1

2

int int

int int

int int

(1.19)

in the case of a symmetric laminated cross section as given in Figure 1.5,

case (b), while specific expressions are provided for “f ” as far as the cross

section is unsymmetrical or not, and composed of two or three glass foils, respectively (e.g Figure 1.5, cases a and c)

1.4.3 Laterally Restrained (LR) Beams in LTB

Differing from the reference LU theoretical configuration depicted in Figure 1.10a, the typical glass beam supporting a roof or façade panel is usually connected in practice to the adjacent construction elements by means of continuous structural silicone sealant joints acting as linear shear flexible connections between the LU glass beam and the supported plates (see also Figure 1.8)

In these hypotheses, it is expected that the lateral restraint provided by the sealant joints could improve the LTB resistance of the LU reference beams (Figure 1.10) However, at the same time, the actual strengthen-ing and stiffening contribution deriving from sealant joints on the LTB response of LU glass beams must be first properly assessed This latter aspect represents in fact a crucial difference between structural glass appli-cations and traditional steel–concrete or timber–concrete composite sec-tions, where almost fully rigid connections at the beam-to-roof interfaces (namely consisting in steel stud connectors) generally ensure the occur-rence of possible LTB phenomena

1.4.3.1 Extended Literature Review on LR Beams

LTB of structural beams with lateral restraints has been widely investigated and assessed in the past years However, careful consideration has been primarily focused on the LTB response of steel members whose behavior is not directly comparable to structural glass beams and fins

In [53], research studies have been in fact dedicated to the typical LTB response of doubly symmetric steel I beams, with careful attention for pos-sible distortional buckling phenomena in the steel webs Khelil and Larue proposed in [54–56] a simple analytical model for the assessment of the critical buckling moment in steel I sections with LR tensioned flanges,

highlighting that the presence of rigid continuous lateral restraints in steel I beams under LTB can have a weak influence, compared to their unre-strained Euler’s critical buckling moment The same authors implemented also a further analytical approach for the LTB assessment of I beams con-tinuously restrained along a flange by accounting for the buckling resis-tance of an equivalent, isolated “T” profile The latter approach, due to its basic assumptions, typically resulted in conservative analytical predictions for the LTB resistance of rigidly LR steel I beams Conversely, the main advantage of this method was given by the availability of the Appendix values of practical use for designers.

The LTB behavior of thin-walled cold-formed steel channel members partially restrained by steel sheeting has been assessed, under various

boundary conditions, by Chu et al [57], based on an energy-based

analyti-cal model Bruins [58] numerianalyti-cally investigated the LTB response of steel

I section profiles under various loading conditions (e.g distributed load q, mid-span concentrated force F, constant bending moment M y) and later-ally restrained by single, elastic, discrete connectors, highlighting through parametric FE numerical studies and earlier experiments that partial elas-tic restraints can have significant influence on the overall LTB response.The effects deriving from initial geometrical curvatures with different shape were also emphasized by means of FE simulations, while simple equations were proposed as strength design method for “rigid” discrete lateral restraints Further assessment of the main structural effects deriving from discrete rigid supports on the buckling behavior of steel beams and braced columns can be found also in [59–62]

1.4.3.2 Closed-form Formulation for LR Beams in LTB

As far as a certain lateral restraint is provided to a given structural member

in LTB, the most efficient tool for design purposes is represented by form solutions of practical use Often, however, these analytical models are difficult to obtain

closed-As also highlighted in [54–56], when continuous elastic lateral restraints are introduced to prevent LTB of beams, the analytical description of the corresponding structural phenomenon becomes rather complex, and closed-form, practical expressions, and simplified analytical models can

be derived only for simple loading/boundary conditions, thus ing the use of sophisticated FE numerical models and computationally expensive simulations

requir-A further difficulty in elastic buckling calculations is given, when

applying the analytical model of Larue et al [54] for fully rigid laterally

restrained steel members to glass beams with continuous sealant joints, by

the intrinsic mechanical properties of the sealant joints themselves, namely

characterized by relatively small shear stiffnesses k y, as well as by the cally high slenderness ratios of glass beams and fins

typi-In this work, based on Figure 1.13, the LTB behavior of LR monolithic glass beams is first investigated by means of the analytical approach origi-nally proposed in [54] for the prediction of the elastic LTB moment of doubly symmetrical steel I beams with fully rigid and continuous lateral

restraints (e.g k y = ∞).

With reference to the loading condition depicted in Figure 1.10a and

to the schematic cross-sections provided in Figure 1.13, the elastic LTB behavior of the LR beam subjected to a constant, positive bending moment

M y can in fact be described by [54]

k y represents the translational (shear) rigidity of the continuous elastic

restraint, per unit of length, along the y-axis;

k is the rotational rigidity of the continuous elastic restraint, per unit of

length, about the x-axis;

M is the applied bending moment;

Figure 1.13 Reference transversal cross section for the analytical model of laterally

restrained beam presented in [54].

z M

k y k

Deformed configuration