Advanced Engineering Materials and Modeling

By Ashutosh Tiwari, N. Arul Murugan and Rajeev Ahuja

The engineering of materials with advanced features is driving the research towards the design of innovative materials with high performances. New materials often deliver the best solution for structural applications, precisely contributing towards the finest combination of mechanical properties and low weight. The mimicking of nature's principles lead to a new class of structural materials including biomimetic composites, natural hierarchical materials and smart materials. Meanwhile, computational modeling approaches are the valuable tools complementary to experimental techniques and provide significant information at the microscopic level and explain the properties of materials and their very existence. The modeling also provides useful insights to possible strategies to design and fabricate materials with novel and improved properties. The book brings together these two fascinating areas and offers a comprehensive view of cutting-edge research on materials interfaces and technologies the engineering materials. The topics covered in this book are divided into 2 parts: Engineering of Materials, Characterizations & Applications and Computational Modeling of Materials. The chapters include the following:

• Mechanical and resistance behavior of structural glass beams
• Nanocrystalline metal carbides - microstructure characterization
• SMA-reinforced laminated glass panel
• Sustainable sugarcane bagasse cellulose for papermaking
• Electrospun scaffolds for cardiac tissue engineering
• Bio-inspired composites
• Density functional theory for studying extended systems
• First principles based approaches for modeling materials
• Computer aided materials design
• Computational materials for stochastic electromagnets
• Computational methods for thermal analysis of heterogeneous materials
• Modelling of resistive bilayer structures
• Modeling tunneling of superluminal photons through Brain Microtubules
• Computer aided surgical workflow modeling
• Displaced multiwavelets and splitting algorithms

• Language: English
• Publisher: Wiley
• Release date: Aug 12, 2016
• ISBN: 9781119242543

Ashutosh Tiwari

Professor Ashutosh Tiwari is Director at Institute of Advanced Materials, Sweden; Secretary General, International Association of Advanced Materials; Chairman and Managing Director of VBRI Sverige AB and AAA Innotech Pvt. Ltd; Editor-in-Chief, Advanced Materials Letters and Docent in the Applied Physics with the specialization of Biosensors and Bioelectronics from Linköping University, Sweden. Prof. Tiwari has several national and international affiliations including in the United States of America, Europe, Japan, China and India. His research focus is on the design and advanced applications of cutting-edge advanced materials for new age devices. He has more than 200 peer-reviewed primary research publications in the field of materials science and nanotechnology and has edited or authored over 50 books.

Book preview

Preface

The engineering of materials with advanced features is driving the research towards the design of innovative high-performance materials. New materials 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, leading to a new class of structural materials which include biomimetic composites, natural hierarchical materials and smart materials. Meanwhile, computational modeling approaches are valuable tools which are complementary to experimental techniques and provide significant information 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 interest. 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 properties 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 validation 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 characterization of materials and some of their applications relevant to the paper 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 structure 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 chemistry, 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 engineering and modeling. We acknowledge the contributors and publisher for their prompt response in order that this book could be published in a timely manner.

Editors

Ashutosh Tiwari, PhD, DSc

N. Arul Murugan, PhD

Rajeev Ahuja, PhD

10 June 2016

Part 1

ENGINEERING OF MATERIALS, CHARACTERIZATIONS, AND APPLICATIONS

Chapter 1

Mechanical Behavior and Resistance of Structural Glass Beams in Lateral–Torsional Buckling (LTB) with Adhesive Joints

Chiara Bedon1* and Jan Belis2

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

2Ghent University, Department of Structural Engineering, Laboratory for Research on Structural Models – LMO, Ghent, Belgium

*Corresponding author: [e-mail]

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 construction 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. 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 numerical 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 geometrical 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

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 materials, 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 technological possibilities are available [1, 2, 10–15]. Compared to traditional construction materials, the major influencing parameter in the design of structural glass elements – in addition to their high architectural and aesthetic impact – is given by the well-known brittle behavior and limited tensile resistance of glass. The use of thermoplastic interlayers alternated 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 represent the two most used typologies of connections and can both be employed in glass façades or roofs, e.g. to provide the mechanical interaction between the glass beams and the supported glass roof panels. While in the first case the bolted connectors and their related effects can often be rationally described in the form of infinitely rigid intermediate restraints, the configuration of glass beams with continuous adhesive 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 appropriate 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 primarily application in building envelopes, roofs, canopies, etc. and solutions in which transparency is mandatory. Major structural glass assemblies – often of complex geometry – are obtained by appropriate conjunct use of glass elements with metal frameworks and substructures (Figure 1.1).

Graphic

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].

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

Graphic

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].

1.3 Glass Beams in LTB

1.3.1 Susceptibility of Glass Structural Elements to Buckling Phenomena

The exposure of structural components in general to significant compression, shear, bending, or a combination of them is the first cause of buckling 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 susceptibility 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.

Graphic

Figure 1.3 Buckling phenomena in columns, beams, and plates.

The presence of rather unconventional materials, in particular, represents 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, geometrical features, initial imperfections, etc., should be properly taken into account to prevent possible LTB failure mechanisms.

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 characterized by specific mechanical and geometrical aspects that should be properly 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).

Table 1.1 Soda lime silica glass properties [25].

Graphic

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).

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 typically 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 connection 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 Gint strictly depends on several conditions (e.g. time loading, temperature (Figure 1.6)).

Graphic

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.

Graphic

Figure 1.6 Mechanical constitutive law of common interlayers for structural glass applications. Data provided for PVB films [26, 27] in the form of shear modulus G as a function of load duration, for different temperatures.

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.

Common options in structural glass applications are in fact represented by metal clamp fixings, drilled fixings, and auxiliary metal connectors (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 requirements must be satisfied, to avoid possible local failure mechanisms in glass, etc.

Graphic

Figure 1.7 Examples of mechanical joints in structural glass applications. Pictures derived from (a) [30], (b) [31], (c), and (d) [32].

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 adhesive joints at the interface between the beams themselves and the supported 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 in the direction of the applied external loads. In this hypothesis, under the action of loads applied on the glass panels (e.g. distributed pressures due to live loads on the roof), the adhesive joint behaves as continuous, 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 parameters, 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 various adhesives of common use for glass-to-glass, glass-to-steel, and glass-to-timber connections.

Graphic

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.

Table 1.2 Mechanical properties of common adhesive types for structural glass applications.

n.a.: not available.

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).

Graphic

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 ladh = 100 mm) with different square section size (wadh = 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 ky = 0.184 N/mm² the average elastic stiffness 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 εu ≈ (du – wadh)/wadh equal to ≈416% and ≈406% for series A and B, respectively, and du denoting the maximum displacement at failure. The obtained average ultimate elongation εu,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 σu,avg, this parameter was also derived from experimental measurements as the average ratio between the failure load Fu of each specimens and the corresponding resisting cross-sectional area Aadh, hence resulting in σu,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 imperfections, 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 several 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.

For clarity of presentation, let us consider the monolithic beam depicted in Figure 1.10, with the given nominal dimensions and coordinate 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 member with regard to buckling and instability has been first expressed in the form of a buckling reduction factor χLT, being a normalized factor strictly related to the member’s susceptibility to buckling phenomena and to the so-called slenderness parameter . While is typically expressed as the inverse square root of Euler’s critical moment , e.g. in the form:

(1.1)

equationGraphic

Figure 1.10 LTB of a laterally unrestrained monolithic beam under constant bending moment My. (a) Elevation, (b) axonometry, and (c) transversal cross section.

with

Mpl denoting the plastic moment,

Wz = ht³/6 the elastic resistant modulus, and

σyk the yielding stress of steel.

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

(1.2)

equation

where

(1.3)

equation

(1.4)

equation

and the imperfection factor αimp depends on the cross-sectional properties, the steel grade and the buckling case (e.g. weak or strong axis) under consideration.

In accordance with this formulation, the LTB verification of a given member can be considered satisfied when the maximum applied design moment My,Ed does not exceed the buckling design resistance Mb,Rd:

(1.5)

equation

with γM1 the partial safety factor for steel.

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

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:

(1.6)

equation

where

kz is the effective length factors for lateral bending,

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

zG 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),

Iz is the second moment of area of the section, about the minor axis,

It is the torsion constant,

Iw is the warping constant.

Table 1.3 Loading/restraint coefficients for the calculation of Euler’s critical moment of simply supported, fork-end restrained beams [41].

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 example, proposed for LU glass beams in LTB [42–44], by assuming in (Eq. 1.6) the characteristic tensile resistance of glass fyk ≡ σRk and γM = 1.4 as a partial safety factor. The advantage of this approach is given to its general formulation, but namely represents the extension of earlier consistent studies on the topic (e.g. [45, 46]).

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

(1.7)

equation

where E ≡ Eg and G ≡ Gg in Eq. (1.7) represent Young’s and shear moduli of glass, respectively, while Iz = ht³/12 signifies the moment of inertia about the minor z-axis and It ≈ ht³/3 (for h/t > 6) is the torsional moment of inertia. Extended comparison proposed in [45] and [46] highlighted the good correlation 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. In the latter case, viscoelastic FE calculations were assessed toward equivalent 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.

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 analytical 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).

Graphic

Figure 1.11 Calibration of design buckling curves for LU beams in LTB, by assuming different geometrical imperfection amplitudes.

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 appropriate estimation of both the equivalent bending stiffness and torsional stiffness 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 analytical 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 formulations are recalled for LG beams. Analytical models are proposed for symmetric 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 sandwich structural elements to glass sections. The mentioned analytical approach, in particular, is based on the concepts of equivalent bending stiffness EIz,eff and equivalent torsional stiffness GIt, where EIz,eff is calculated depending on the specific loading condition (constant bending moment My, distributed load q, concentrated load F at mid-span), while GIt depends on the geometrical/mechanical properties of the cross section only.

The expression proposed for EIz,eff is given as a function of the slenderness of the beam (t1, h, L0), the elastic stiffness of glass (E), the thickness of the interlayer (tint), and its mechanical properties (Gint). Based on [45], according to the laminated cross section of Figure 1.4, case (b), and considering the beam subjected to a constant bending moment My, as proposed in Figure 1.10, EIz,eff is in fact defined as follows:

(1.8)

equation

with

(1.9)

equation

(1.10)

equation

(1.11)

equation

(1.12)

equation

(1.13)

equation

(1.14)

equation

Based on the same approach, the equivalent torsional stiffness GIt was also derived from the classical theory of sandwich elements (Figure 1.11 [45]). For a symmetric two-layer LG beam, in particular, GIt can be calculated as follows:

(1.15)

equation

where the expression for It,1 and It,comp are listed in Table 1.4.

Table 1.4 The Stamm–Witte equivalent parameters for the calculation of the torsional stiffness term in layered cross sections [45].

Specifically, Eq. (1.15) takes into account the effective torsional contribution It,comp due to the adopted interlayer. Stamm and Witte originally derived the expressions, partly collected in Table 1.4 [45], for the estimation of this torsional stiffness term, typically occurring in a faced soft core within a flat sandwich panel subjected to a torsional moment MT. Their model basically applies to layered elements in which the cross section 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.

Graphic

Figure 1.12 Qualitative torsional behavior of a LG beam in accordance with the analytical model recalled in [45].

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 teq, e.g. on the concept of an equivalent, monolithic flexural stiffness EIz,eff = hteq³/12 with

(1.16)

equation

and

(1.17)

equation

the shear transfer coefficient representative of the shear transfer contribution of the adopted interlayer, where Is,WB in Eq. (1.17) is equal to Is/h (Eq. 1.9). Due to the shear transfer coefficient Γb, the effective stiffness of the interlayer can be rationally taken into account within a range conventionally comprised between an abs layered limit (e.g. Gint → 0) and full monolithic limit (e.g. Gint → ∞).

Analytical calculations highlighted that based on Eq. (1.17) the flexural stiffness EIz,eff = f(teq) 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 GIt, 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 corresponding to a null shear stiffness of the interlayers (Gint → 0), e.g. by introducing a parameter f ≥ 1 so that

(1.18)

equation

is the equivalent torsional stiffness of the laminated member, where

(1.19)

equation

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 strengthening 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 applications and traditional steel–concrete or timber–concrete composite sections, where almost fully rigid connections at the beam-to-roof interfaces (namely consisting in steel stud connectors) generally ensure the occurrence 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 possible 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 unrestrained Euler’s critical buckling moment. The same authors implemented also a further analytical approach for the LTB assessment of I beams continuously restrained along a flange by accounting for the buckling resistance 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 analytical model. Bruins [58] numerically 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 My) and laterally restrained by single, elastic, discrete connectors, highlighting through parametric FE numerical studies and earlier experiments that partial elastic 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 closed-form solutions of practical use. Often, however, these analytical models are difficult to obtain.

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 requiring the use of sophisticated FE numerical models and computationally expensive simulations.

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 ky, as well as by the typically high slenderness ratios of glass beams and fins.

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 originally 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. ky = ∞).

Graphic

Figure 1.13 Reference transversal cross section for the analytical model of laterally restrained beam presented in [54].

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 My can in fact be described by [54]

(1.23a)

(1.23b)

equation

where

ky represents the translational (shear) rigidity of the continuous elastic restraint, per unit of length, along the y-axis;

kθ is the rotational rigidity of the continuous elastic restraint, per unit of length, about the x-axis;

My is the applied bending moment;

qz represents a possible transversal distributed load;

zM is the distance between the continuous lateral restraint and the middle x-axis;

zq is the distance between the point of introduction of qz load and the x-axis;

v represents the vertical deflection of the beam, in the z-direction;

θx is the rotation of the cross-section about the longitudinal x-axis of the beam,

while Iz, It, Iw are defined according to Eq. (1.6).

For the full mathematical and matrix formulation of the reference analytical model, the reader is referred to [54].

The main assumptions of the second order differential system given by Eqs. (1.23a) and (1.23b) are that:

the geometrical and mechanical properties of the beam are kept constant along the buckling length L0,

the resisting cross section remains flat and undistorted after bending of the beam,

the beam is fork supported at the ends, while the continuous lateral joints introduce translational (ky) and rotational (kθ) restraints only.

Based on the definitions given above, the critical buckling moment for the simply supported, fork-end restrained beam of Figure 1.10a subjected to a constant, positive bending moment My is given by:

(1.24)

equation

where nR ≥ 1 is an integer able to minimize the critical load given by Eq. (1.24).

The main effect of this assumptions is that for a rectangular cross section composed of glass (e.g. Figure 1.10b), Iω = 0. A second aspect is strictly related to the torsional contribution of adhesive joints, that at first instance could be taken equal to kθ = 0. As a result, the LR critical buckling moment can be calculated as follows:

(1.25)

equation

In accordance with Eq. (1.25), for a given t × h × L0 monolithic glass beam under a positive, constant bending moment My, it can be also concluded that:

in presence of a flexible adhesive joint (0 < ky < ∞)

(1.26)

equation

while as far as the adhesive joint is weak (ky = 0), the critical buckling moment is given by

(1.27)

equation

where in both the cases is expressed by Eq. (1.7).

The amplification factor RM = f (ky, b, t, L0, zM, nR) > 1 of Eq. (1.26) is representative of the effects deriving from a multitude of aspects, e.g. the joint shear rigidity ky, the beam aspect ratio, its elastic bending and torsional stiffnesses, as well as the position of the applied restraints (zM) or the number nR of half-sine waves able to minimize, based on Eq. (1.25), the expected critical buckling moment .

When zM = h/2 (Figure 1.3b) and h/t > 6, for example, Eqs. (1.25) and (1.26) leads to

(1.28)

equation

and in the specific case of glass (E = 70 GPa and G ≈ 0.41E [25]) to

(1.29)

equation

1.4.3.3 LR Glass Beams Under Positive Bending Moment My

Eqs. (1.25) and (1.29) – although not fully exhaustive for the description of the expected LTB behavior – can provide a first assessment of the effects due to continuous, adhesive joints acting as flexible lateral restraints along the top edge of a given glass beam in LTB, both in terms of expected magnifying factors RM as well as corresponding critical buckling shapes.

Some qualitative analytical calculations derived from Eqs. (1.25) and (1.29) are in fact collected in Figure 1.14 to investigate the sensitivity of the LTB theoretical resistance of glass beams with several properties, when modifying some of the input parameters. In doing so, various geometrical (beam thickness t, height h, buckling length L0) and mechanical (joint shear stiffness ky) parameters were properly modified within a sufficiently wide range of practical interest for structural glass applications (1000 mm ≤ L0 ≤ 6000 mm, 6 mm ≤ t ≤ 25 mm, 100 mm ≤ b ≤ 350 mm, with 0 < ky < ∞ the shear joint stiffness). Several RM,i magnifying coefficients given by Eq. (1.29) were first calculated – for each beam/adhesive joint stiffness – as a function of a specific number of critical half-sine waves nR, with 1 ≤ nR ≤ 20. The actual critical coefficient RM was then detected as the minimum of the so-calculated RM,i values.

Graphic

Figure 1.14 Analytical estimation of the non-dimensional RM coefficient (Eq. 1.29) for monolithic glass beams under positive bending moment. Effects of the number of half-sine waves nR on the expected strengthening contribution of continuous lateral restraints.

In general, the actual critical RM coefficient representative of the lowest critical buckling moment should in fact be properly estimated, by considering not only the assigned mechanical/geometrical properties of a given LR beam, but also the geometrical configuration (e.g. number of half-sine waves nR) leading to LTB collapse.

In this context, the major difference between the LTB response and theoretical resistance of a LU or LR glass beam is strictly related to nR. While for LU beams the lowest buckling load is associated to a single half-sine shape (e.g. nR = 1), this is not the case of LR beams.

Some practical examples are proposed in Figure 1.14 for a selected set of geometrical configurations. In Figure 1.14a, in particular, the lower envelope of the RM,i. values calculated as described above is represented by the black thick line.

As shown, it is clear that for a given beam geometry, as far as the adhesive shear stiffness ky increases, the corresponding fundamental buckling shape modifies (nR > 1). Consequently, calculations derived from Eq. (1.25) with a reference single sine-shaped configuration (e.g. nR = 1) would strongly underestimate the actual LTB resistance and overall response of the examined LR beams.

Once assessed the correlation between half-sine waves and adhesive joints, further analytical calculations were carried out by changing the beams geometrical properties (e.g. in the form of slenderness ratios and torsional stiffness). In doing so, the nominal dimensions only (e.g. thickness,