Mechanics of Classical Sandwich Structures 3031251059, 9783031251054

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Andreas Öchsner

Mechanics of Classical Sandwich Structures

Mechanics of Classical Sandwich Structures

Andreas Öchsner

Mechanics of Classical Sandwich Structures

Andreas Öchsner Faculty of Mechanical and Systems Engineering Esslingen University of Applied Sciences Esslingen, Baden-Württemberg, Germany

ISBN 978-3-031-25105-4 ISBN 978-3-031-25106-1 (eBook) https://doi.org/10.1007/978-3-031-25106-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Lightweight concepts can be understood as the application of classic engineering concepts and disciplines to reduce the structural weight. Here, in particular, basic knowledge of applied mechanics, materials science, manufacturing technology and design theory is used. In addition to classic applications in the astronautics and space industry or automotive engineering, these now also extend to quite different areas, such as sports equipment or medical prostheses. Due to the reduced weight, a reduction in fuel consumption and thus also a reduction in pollutants can be achieved in the transport sector. This results in economic as well as ecological advantages. In engineering practice, however, there are usually quite complex systems, also optimized using commercial program packages. However, to introduce the different lightweight construction concepts as part of a university engineering course, one can also first use simple structural elements that are presented in the context of applied mechanics. The simplest elements here are rods and beams, which, along with springs, are assigned to the one-dimensional structural elements. Based on these elements, questions about the selection of materials and the geometric design and optimization of load-bearing structures can be discussed quite clearly, see [Öchsner(2019), (2020)]. Thus, this textbook follows this idea and treats the mechanical behavior of one-dimensional sandwich structures, a typical concept in the context of lightweight design. Such structures are composed of different constituent (e.g., layers) in order to achieve overall properties, which are better than for a single component alone. This book covers the basic mechanical load cases, i.e., tension/compression, bending and shear. Based on this knowledge, different failure modes, i.e., plastic yielding, and global and local instabilities are investigated. In addition, an introduction to classic optimization problems, i.e., the formulation of an objective function (e.g., the weight of a structure) and corresponding restrictions, is included. However, the consideration here is limited to one- or two-dimensional design spaces, i.e., with a maximum of two design variables. For such simple cases,

v

vi

Preface

the minimum of the objective function can often be determined using analytical or graphical methods. Esslingen, Germany October 2022

Andreas Öchsner

References Öchsner, A.: Leichtbaukonzepte anhand einfacher Strukturelemente: Neuer didaktischer Ansatz mit zahlreichen Übungsaufgaben. Springer Vieweg, Wiesbaden (2019) Öchsner, A.: Stoff-und Formleichtbau: Leichter Einstieg mit eindimensionalen Strukturen. Springer Vieweg, Wiesbaden (2020)

Contents

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4

2 Basic Mechanical Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bending Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Tensile/Compressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Shear Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Technical Sandwich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Bending Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Tensile/Compressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Shear Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Bending Deformation of Sandwich Beams . . . . . . . . . . . . . . . 2.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Knowledge Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Calculation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 11 12 14 15 27 28 33 40 40 40 43

3 Limit Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Global Instability Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Shear Failure of the Connecting Layer . . . . . . . . . . . . . . . . . . 3.1.3 Local Wrinkling of the Compressive Face Sheet (Bending Load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Local Antisymmetric Wrinkling of both Face Sheets (Compressive Load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Local Symmetric Wrinkling of both Face Sheets (Compressive Load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Knowledge Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Calculation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 51 52 60 65 67 67 69 75 vii

viii

Contents

4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Optimal Dimensioning of Sandwich Beams . . . . . . . . . . . . . . . . . . . . 4.1.1 Tensile or Compressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Bending Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Knowledge Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Calculation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 77 83 90 90 90 93

5 Short Solutions to the Supplementary Problems . . . . . . . . . . . . . . . . . . . 95 5.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Mechanics and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Second-Order Moment of Area . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Derivation of the Shear Stress Distribution for the Beam . . . 6.1.3 Derivation of the Euler Buckling Force for Homogeneous and Isotropic Euler-Bernoulli Beams . . . . 6.1.4 Newton’s Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Numerical Integration of Functions with Variables . . . . . . . . 6.2 Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 121 124 128 128 130 139

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Symbols and Abbreviations

The following lists explain the most important symbols and abbreviations that are used in the course of this book.

Latin Symbols (Capital Letters) A As AG B1 B1 C E EA EIy F Fg Fcr G I K L M N Q Rp0.2 V

Area, cross-sectional area Shear area Shear stiffness Factor Factor Constant Young’s modulus Average tensile stiffness Average bending stiffness Force, Yield condition Dead weight Critical force (buckling force) Shear modulus Second moment of area Buckling coefficient Length Lightweight index, Moment Normal force Shear force Initial yield stress Volume

ix

x

Symbols and Abbreviations

Latin Symbols (Small Letters) a b c d f g h hc h h F h F,n h C h C,n ks kt ktinit m mn p q r s t u w x y z

Geometric dimension Geometric dimension Damper constant, Constant of integration Diameter Function Gravitational acceleration, Function Geometric dimension Average sandwich thickness Layer thickness (sandwich) Face sheet thickness (sandwich) Length specific face sheet thickness (sandwich) Core thickness (Sandwich) Length specific core thickness (sandwich) Spring constant, Geometric and material parameter, Shear yield stress, Shear correction factor Tensile yield stress Initial tensile yield stress Mass, Length specific moment Length specific mass Distributed load in x-direction Distributed load in z-direction Radius Geometric dimension Geometric dimension Displacement Geometric dimension Cartesian coordinate Cartesian coordinate Cartesian coordinate

Greek Symbols (Capital Letters)   Argument  =

2πh K λ



Greek Symbols (Small Letters) α γ

Parameter, Angle Shear strain

Symbols and Abbreviations

γaB ε εA εAt εp0.2t κ λ ν ρ σ σcr σeff σi τ τaB τp φ φF ϕ

Shear strain at failure Strain Strain at failure Total strain at failure (including elastic part) 0.2-% strain limit (corresponding to Rp0.2 ) Curvature Parameter, Wave length Poisson’s ratio Density Stress, normal stress Critical stress Equivalent stress (effective stress) Principal stress (i = 1, 2, 3) Shear stress Shear strength Shear yield stress Rotation angle, rotation Fiber volume fraction Rotation angle, rotation

Mathematical Symbols × Multiplication sign

Indices, Superscripted . . .C . . .c . . .E . . .el . . .F . . .k . . .pl . . .t

Core Compression Euler Elastic Face sheet Layer index Plastic Tension

Indices, Subscripted . . .o . . .b

Outer Bending

xi

xii

. . .F . . .i . . .C . . .m . . .max . . .ref . . .s . . .t . . .c

Symbols and Abbreviations

Face sheet Internal Core Average value Maximum value Reference Shear Torsion, Tension Center

Abbreviations 1D One-dimensional CFRP Carbon fiber-reinforced plastic UD Unidirectional

Chapter 1

Introduction and Motivation

Lightweight design plays a central role in transportation (e.g. in the aerospace industry or in automotive engineering), since a reduction in weight is directly reflected in a reduction in fuel costs. As a rough estimate of the influence of weight on the fuel consumption of aircrafts, a reduction of 1% in weight can result in fuel savings— depending on the engine type—from 0.75 to 1% (see Ohrn 2007). If one takes the total fuel consumption of the Lufthansa fleet in 2015 of 8947.766 tons as an example (see Lufthansa 2016), depending on the kerosene price (see IATA 2017), there is a savings potential of several million euros per year. Figure 1.1 shows a much simpler example of a steel plate. The figure on the left (a) is a plate made of solid material, which has a mass of around 7.7 kg for the dimensions given. If the plate is designed as a hollow sphere structure (see Öchsner and Augustin 2009) with the same external dimensions, the result is a significantly reduced mass of around 0.446 kg or a reduction of 94%. From this it can be concluded that not only the material itself, but also other factors, such as the shape or the mesostructure, can have an impact on the ‘lightweight potential’ of a structure. The German-language technical literature on the subject of lightweight construction is quite extensive and covers various subject areas. A summary of some textbooks is given in Table 1.1. There is also specialized literature, for example with a focus on the automotive industry, see Siebenpfeiffer (2014), Friedrich (2017), Kurek (2011). It should also be noted that lightweight design includes different disciplines, such as strength of materials (see Linke and Nast 2015; Altenbach 2018), materials science (see Weißbach 2012) and design (see Pahl and Beitz 1997). Thus, Table 1.1 can be expanded as desired with classical literature from the basic subjects of an engineering degree in the corresponding disciplines.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_1

1

2

1 Introduction and Motivation

Fig. 1.1 a Steel plate with external dimensions b = 11 cm, t = 30 cm and h = 3 cm. Mass: m ≈ 7.7 kg; b hollow sphere structure made of steel with the same dimensions. Mass: m ≈ 0.446 kg

Fig. 1.2 Examples of sandwich plates with metallic hollow sphere core: a Carbon-fiber reinforced face sheets; b aluminium face sheets. Adapted from Öchsner and Augustin (2009), reprinted with permission from Springer Nature publishers

A typical design principle in the context of lightweight structures is to compose different materials, many times in the form of layers, to so-called sandwich structures, see Fig. 1.2 for some typical examples. The idea is to combine the different advantages of the single constituents and to achieve overall properties, which are better than the ones of the single components. The technical literature on the subject of sandwich structures is summarized in Table 1.2. This textbook is purely focused on one-dimensional, i.e. rod- and beam-like sandwich structures and thus offers a new didactic approach to conveying the basic ideas of these structural members. The restriction to one-dimensional elements allows a relatively simple representation using equations that are easy for students to understand. Thus, there is a focus on basic design concepts and the application of the fundamentals of applied mechanics and not on complicated mathematical derivations or

1 Introduction and Motivation

3

Table 1.1 Selected German-language textbooks on the subject of lightweight design Year

Author/Editor

SC

References

1955

K. Bobek, A. Heiß, F. Schmidt Stahlleichtbau von Maschinen

Title

–

Bobek et al. (1955)

1960

H. Hertel

Leichtbau: Bauelemente, Bemessungen und Konstruktionen von Flugzeugen und anderen Leichtbauwerken

x

Hertel (1960)

1982

H.-J. Dreyer

Leichtbaustatik

–

Dreyer (1982)

1986

J. Wiedemann

Leichtbau Band 1: Elemente

x

Wiedemann (1986)

1989

J. Wiedemann

Leichtbau Band 2: Konstruktion

x

Wiedemann (1989)

1989

B. Klein

Leichtbau-Konstruktion: Berechnungsgrundlagen und Gestaltung

x

Klein (1989)

1989

B. Klein

Übungen zur Leichtbau-Konstruktion

x

Klein (1989)

1992

F.G. Rammerstorfer

Repetitorium Leichtbau

x

Rammerstorfer (1992)

1996

H. Kossira

Grundlagen des Leichtbaus: Einführung in die Theorie dünnwandiger stabförmiger Tragwerke

–

Kossira (1996)

2009

H.P. Degischer, S. Lüftl

Leichtbau: Prinzipien, Werkstoffauswahl und Fertigungsvarianten

x

Degischer and Lüftl (2009)

2011

F. Henning, E. Moeller

Handbuch Leichtbau: Methoden, Werkstoffe, Fertigung

x

Henning and Moeller (2011)

2014

B. Hill

Bionik—Leichtbau

–

Hill (2014)

2018

A. Öchsner

Leichtbaukonzepte: Eine Einführung anhand einfacher Strukturelemente für Studierende

x

Öchsner (2018)

2019

A. Öchsner

Leichtbaukonzepte anhand einfacher Strukturelemente: Neuer didaktischer Ansatz mit zahlreichen Übungsaufgaben

x

Öchsner (2019)

The year refers to the first edition. The fourth column (SC) indicates content on the mechanics of sandwich structures (x)

algorithms. Anyone who has mastered these basics can also familiarize themselves relatively easily with more complicated subject areas of composite structures, such as plane two-dimensional structures.

4

1 Introduction and Motivation

Table 1.2 Selected books/reports on sandwich structures Year

Author/Editor

Title

References

1948

C. Libove, S.B. Batdorf

A General Small-Deflection Theory for Flat Sandwich Plates

Libove and Batdorf (1948)

1950

N.J. Hoff

Bending and Buckling of Rectangular Sandwich Plates

Hoff (1950)

1966

F.J. Plantema

Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells

Plantema (1966)

1969

H.G. Allen

Analysis and Design of Structural Sandwich Panels

Allen (1969)

1974

K. Stamm, H. Witte

Sandwichkonstruktionen: Berechnung, Fertigung, Ausführung (in German)

Stamm and Witte (1974)

1975

L. Librescu

Elastostatics and Kinetics of Anistropic and Heterogeneous Shell-Type Structures

Librescu (1975)

1995

D. Zenkert

An Introduction to Sandwich Construction

Zenkert (1995)

1997

D. Zenkert

The Handbook of Sandwich Construction

Zenkert (1997)

1998

A. Vautrin

Mechanics of Sandwich Structures

Vautrin (1998)

1999

J.R. Vinson

The Behavior of Sandwich Structures of Isotropic and Composite Materials

Vinson (1999)

2001

J.M. Davies

Lightweight Sandwich Construction

Davies (2001)

2005

J.R. Vinson

Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction

Vinson (2005)

2009

I.M. Daniel, E.E. Gdoutos, Y.D.S. Rajapakse

Major Accomplishments in Composite Materials and Sandwich Structures

Daniel et al. (2009)

2021

T.J. Hause

Sandwich Structures: Theory and Responses

Hause (2021)

2022

W. Ma, R. Elkin

Sandwich Structural Composites: Theory and Practice

Ma and Elkin (2022)

2022

S. Krishnasamy, C. Muthukumar, S.M.K. Thiagamani, S.M. Rangappa, S. Siengchin

Sandwich Composites: Fabrication and Characterization

Krishnasamy et al. (2022)

The year refers to the first edition

References Allen, H.G.: Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969) Altenbach, H.: Holzmann/Meyer/Schumpich Technische Mechanik Festigkeitslehre. Springer Vieweg, Wiesbaden (2018) Bobek, K., Heiß, A., Schmidt, F.: Stahlleichtbau von Maschinen. Springer, Berlin (1955) Daniel, I.M., Gdoutos, E.E., Rajapakse, Y.D.S.: Major Accomplishments in Composite Materials and Sandwich Structures. Springer, Dordrecht (2009) Davies, J.M.: Lightweight Sandwich Construction. Blackwell Science Ltd, Oxford (2001) Degischer, H.P., Lüftl, S.: Leichtbau: Prinzipien, Werkstoffauswahl und Fertigungsvarianten. WILEY-VCH, Weinheim (2009) Dreyer, H.-J.: Leichtbaustatik. Teubner, Stuttgart (1982)

References

5

Friedrich, H.E.: Leichtbau in der Fahrzeugtechnik. Springer Vieweg, Wiesbaden (2017) Hause, T.J.: Sandwich Structures: Theory and Responses. Springer, Cham (2021) Henning, F., Moeller, E.: Handbuch Leichtbau: Methoden, Werkstoffe, Fertigung. Hanser, München (2011) Hertel, H.: Leichtbau: Bauelemente, Bemessungen und Konstruktionen von Flugzeugen und anderen Leichtbauwerken. Springer, Berlin (1960) Hill, B.: Bionik-Leichtbau. Knabe, Weimar (2014) Hoff, N.J.: Bending and Buckling of Rectangular Sandwich Plates. NACA TN 2225, Washington (1950) IATA: Jet Fuel Price Monitor. http://www.iata.org/publications/economics/fuel-monitor/Pages/ index.aspx. Zugegriffen: 15 April 2017 Krishnasamy, S., Muthukumar, C., Thiagamani, S.M.K., Rangappa, S.M., Siengchin, S.: Sandwich Composites: Fabrication and Characterization. CRC Press, Boca Raton (2022) Klein, B.: Leichtbau-Konstruktion: Berechnungsgrundlagen und Gestaltung. Friedr. Vieweg & Sohn, Braunschweig (1989) Klein, B.: Übungen zur Leichtbau-Konstruktion. Friedr. Vieweg & Sohn, Braunschweig (1989) Kossira, H.: Grundlagen des Leichtbaus: Einführung in die Theorie dünnwandiger stabförmiger Tragwerke. Springer, Berlin (1996) Kurek, R.: Karosserie-Leichtbau in der Automobilindustrie. Vogel, Würzburg (2011) Linke, M., Eckart Nast, E.: Festigkeitslehre für den Leichtbau: Ein Lehrbuch zur Technischen Mechanik. Springer Vieweg, Wiesbaden (2015) Libove, C., Batdorf, S.B.: A General Small-Deflection Theory for Flat Sandwich Plates. NACA TR 899, Washington (1948) Librescu, L.: Elastostatics and Kinetics of Anistropic and Heterogeneous Shell-Type Structures. Noordhoff International Publishing, Leyden (1975) Lufthansa Group: Balance Issue 2016. https://www.lufthansagroup.com/fileadmin/downloads/en/ LH-sustainability-report-2016.pdf Zugegriffen: 25 April 2017 Ma, W., Elkin, R.: Sandwich Structural Composites: Theory and Practice. CRC Press, Boca Raton (2022) Öchsner, A., Augustin, C. (eds.): Multifunctional Metallic Hollow Sphere Structures: Manufacturing, Properties and Application. Springer, Berlin (2009) Öchsner, A.: Leichtbaukonzepte: Eine Einführung anhand einfacher Strukturelemente für Studierende. Springer Vieweg, Wiesbaden (2018) Öchsner, A.: Leichtbaukonzepte anhand einfacher Strukturelemente: Neuer didaktischer Ansatz mit zahlreichen Übungsaufgaben. Springer Vieweg, Wiesbaden (2019) Ohrn, K.E.: Aircraft energy use. In: Capehart, B.L. (eds.) Encyclopedia of Energy Engineering and Technology, vol. 1, pp. 24–30. CRC Press, Boca Raton (2007) Pahl, G., Beitz, W.: Konstruktionslehre: Methoden und Anwendung. Springer, Berlin (1997) Plantema, F.J.: Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells. Wiley, New York (1966) Rammerstorfer, F.G.: Repetitorium Leichtbau. Oldenbourg, München (1992) Siebenpfeiffer, W.: Leichtbau-Technologien im Automobilbau. Springer Vieweg, Wiesbaden (2014) Stamm, K., Witte, H.: Sandwichkonstruktionen: Berechnung, Fertigung, Ausführung. Springer, Wien (1974) Vautrin, A.: Mechanics of Sandwich Structures: Proceedings of the EUROMECH 360 Colloquium held in Saint-Étienne, France, 13–15 May 1997. Springer Science+Business Media, Dordrecht (1998) Vinson, J.R.: The Behavior of Sandwich Structures of Isotropic and Composite Materials. CRC Press, Boca Raton (1999) Vinson, J.R.: Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction. Springer, Dordrecht (2005) Weißbach, W.: Werkstoffkunde: Strukturen, Eigenschaften, Prüfung. Springer Vieweg, Wiesbaden (2012)

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1 Introduction and Motivation

Wiedemann, J.: Leichtbau Band 1: Elemente. Springer, Berlin (1986) Wiedemann, J.: Leichtbau Band 2: Konstruktion. Springer, Berlin (1989) Zenkert, D.: An Introduction to Sandwich Construction. Engineering Materials Advisory Services, Clifton upon Teme (1995) Zenkert, D.: The Handbook of Sandwich Construction. Engineering Materials Advisory Services, Clifton upon Teme (1997)

Chapter 2

Basic Mechanical Load Cases

2.1 Introductory Remarks In the case of material and form lightweight design, an attempt is made to achieve an optimal lightweight potential through a suitable choice and combination of different materials and their distribution in the cross section. Typical representatives of this come from the class of composite materials in their most general definition, with sandwich elements being considered in more detail below, see (Allen 1969; Stamm and Witte 1974). A typical sandwich element is shown in Fig. 2.1. Here, two top layers, which often consist of the same material, are connected by a core and thus kept at a defined distance. It is also noted that the top layers and the core usually have different tasks, see Hertel (1960). In the following, three different load cases are examined in more detail, see Fig. 2.2. For simplification, the beam theory according to Euler-Bernoulli and the bar theory (see Öchsner 2020) are used as a basis. Thus, a generalized, i.e. a combination of beam and bar member, composite beam is considered.

2.2 Bending Load The average bending stiffness E I y for such a composite results from Klein (2009); Öchsner (2014) as follows: EIy =

3  k =1

Ek



1 b 12

3  2    k 3 = h + bh k z ck E k I yk ,

(2.1)

k =1

where z ck is the vertical distance (i.e. in the z-direction) of the center of gravity of the partial body k to the total center of gravity. The layer thickness h can also be calculated using h k = z k−1 − z k . In Eq. (2.1), the total second moment of area was calculated using the fraction with regard to the center of gravity of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_2

7

8

2 Basic Mechanical Load Cases

Fig. 2.1 Structure of a sandwich element: 1, 3: top layer (skin, face), 2: core layer Fig. 2.2 Investigated load cases: a Bending, b tensile or compressive loading, and (c) shear loading

partial body and the additional contribution from the parallel-axis theorem (‘Steiner’ fraction). Thus, from Eq. (2.1) for a homogeneous element of width b and height h, 1 E I y = E I y = E 12 bh 3 would result. In the case of a transverse strain constraint of the cover layers (e.g. by the core or a significant dimension in the y-direction, i.e. b ≈ L), instead of E the modified modulus of elasticity E → E/(1 − ν 2 ) can be used, see Stamm and Witte (1974).

2.2 Bending Load

9

Fig. 2.3 Course of the a Normal stress component σx and b strain component εx under bending load

The stress distribution in the sandwich can be described using the following modified approach (see Klein 2009; Öchsner 2014): σx,k (z) =

My E k EIy

× z.

(2.2)

According to Fig. 2.3a, there are jumps in stress at the transition from one material to the next in the case of different stiffnesses (E k ). In contrast to this, however, the strain is assumed without jumps, i.e. a linear progression through the origin of the coordinate system, see Fig. 2.3b. The derivation of Eq. (2.2) is briefly presented below. Assuming that each layer k has the same curvature κ, the stress in each layer can be given as follows: σx,k = −E k z

d2 u z (x) = +E k zκ. dx 2

(2.3)

The internal bending moment results from integration over the stress distribution using:  My =

 zσx dA =

A

 =

 zσx,1 dA1 +

A1



z 2 E 1 κdA1 + A1

 zσx,2 dA2 +

A2



z 2 E 2 κdA2 + A2

zσx,3 dA3 A3

z 2 E 3 κdA3 A3

(2.4)

(2.5)

10

2 Basic Mechanical Load Cases

Fig. 2.4 General configuration of a cantilever sandwich beam

⎛ = ⎝E 1 =





 z 2 dA1 + E 2

A1

E 1 I y1

+

z 2 dA2 + E 1 A2

E 2 I y2

+



E 3 I y3



⎞ z 2 dA3 ⎠ κ

(2.6)

A3

κ = E I y κ.

(2.7)

Using Eq. (2.3), the stress relationship (2.2) results from the last equation. In the following, different sandwich structures with different material and geometry combinations are compared in relation to the lightweight potential. This can be done by calculating the so-called lightweight index M, i.e. the ratio between the externally applied force and the dead weight of the structure, see Klein (2009), Öchsner (2020) for details. The basic configuration for numerous examples, i.e. cantilever beam with end shear force, is again as in Fig. 2.4. A homogeneous aluminum beam now serves as reference [see Fig. 2.5, configuration (1)]. By using a foam core and then reducing the thickness of the cover layer (with the same outer dimensions of the cross section), an increase in the lightweight potential can be achieved [see Fig. 2.5, configurations (2) and (3)]. A configuration in which the core layer has been completely removed is given as a limiting case [see Fig. 2.5, configuration (4)]. It should be noted here that the calculation of the lightweight index was carried out with a limit value as the maximum external load. An alternative design concept can be realized by using carbon fiber-reinforced plastic (CFRP) (characteristic values for CFRP according to Klein (2009):  = 1.50 × 10−6 kg/mm3 , E = 120,000 MPa, R t = 1700 MPa; unidirectional layer (UD) with fiber volume fraction of φ F = 0.55). For case (4) according to Fig. 2.5 this now results in MCFRP = 4077.

2.3 Tensile/Compressive Load

11

Fig. 2.5 Lightweight index for different configurations of sandwich elements (L = 100 mm, b = h = 10 mm) and load criterion for bending load case (F0 = −200 N ∧ σmax < Rp0.2 ). Charkg acteristic values of the Al foam according to Klein (2009):  = 0.4 × 10−6 mm 3 , E = 2500 MPa, t c Rp0.2 = 4 MPa, Rp0.2 = 6 MPa

2.3 Tensile/Compressive Load In the following, the load case of tension or compression (see Fig. 2.2b) is considered in more detail, although instabilities such as buckling or wrinkling are not yet taken into account here. Under the influence of an external axial force, the stress and strain distributions shown in Fig. 2.6 result. It is important to assume that all layers are perfectly connected and that the strain in each individual layer is equal to the total strain, see Fig. 2.6b. The internal normal force results from integration over the stress distribution using:  Nx =

 σx dA =

A

 σx,1 dA1 +

A1

 σx,2 dA2 +

A2

σx,3 dA3

(2.8)

A3

= E 1 εx h 1 b + E 2 εx h 2 b + E 3 εx h 3 b   = E 1 h 1 b + E 2 h 2 b + E 3 h 3 b εx = E Aεx .

(2.9) (2.10)

The average tensile stiffness is thus: EA =

3  k =1

E k h k b.

(2.11)

12

2 Basic Mechanical Load Cases

Fig. 2.6 Course of the a Normal stress component σx and b strain component εx under tensile load

Substituting in Eq. (2.10) the total strain using Hooke’s law for a layer and considering that the internal normal force is equal to the external force F0 , i.e. N x = F0 = E A

σk , Ek

(2.12)

the relationship for the stress in the k-th layer is obtained as: σx,k =

F0 E k EA

.

(2.13)

2.4 Shear Load In the following, the shear load case (see Fig. 2.2c) is considered in more detail. In general, the shear stress distribution can be calculated according to Eq. ( 6.6) by integration from the normal stress distribution. For the top layers (see layers 1 and 3 in Fig. 2.1) this results, as an example, for 2 2 layer 3 ( h2 ≤ z ≤ h2 + h 3 ) in: h 2 +h 3 2

τzx,3 (z) = z

With the stress gradient

dσx,3 (x)  dz + c3 . dx

(2.14)

2.4 Shear Load

13

dσx,3 (x) d = dx dx



M y (x)E 3 E Iy

E 3 z dM y (x) E 3 Q z (x) = z E I y dx E Iy

=

z

(2.15)

the shear stress distribution results in:

τzx,3 (z) =

E 3 Q z (x) E Iy

h 2 +h 3 2

z  dz  + c3 =

E 3 Q z (x)

z

2E I y

⎡

h 2 ⎣ + h 3 2

⎤

2

− z 2 ⎦ + c3 . (2.16)

The constant of integration c3 can be determined using the condition that no shear stresses occur at the free edge, i.e. τzx,3 (h 2 /2 + h 3 ) = 0, as c3 = 0. This finally gives the shear stress distribution in layer 3: ⎡ ⎤

2 E 3 Q z (x) h 2 ⎣ τzx,3 (z) = + h 3 − z 2 ⎦ , (2.17) 2 2E I y or in layer 1: τzx,1 (z) =

E 1 Q z (x) 2E I y

⎡

h 2 ⎣ + h 1 2

⎤

2

− z2⎦ .

For the core layer (see layer 2 in Fig. 2.1) we get (− h2 ≤ z ≤ 2

(2.18)

h 2 ): 2

h 2

2 τzx,2 (z) =

dσx,2 (x)  dz + c2 . dx

(2.19)

z

With the stress gradient dσx,2 (x) E 2 Q z (x) = z dx E Iy

(2.20)

the shear stress distribution results in: h 2 ⎡ ⎤

2  2 2 2 E 2 Q z (x) E Q (x) h z ⎣ τzx,2 (z) = z  dz  + c2 = − z 2 ⎦ + c2 . 2 E Iy 2E I y z

(2.21)

14

2 Basic Mechanical Load Cases

Fig. 2.7 Course of the a Shear stress component τx z and the b shear strain γx z under shear load

The integration constant c2 results from the transition condition for the stress τzx between layers 2 and 3, i.e. τzx,2 (z = h 2 /2) = τzx,3 (z = h 2 /2), to

c2 =

E 3 Q z (x) 2E I y

⎡

h 2 ⎣ + h 3 2



2 −

h 2 2

2 ⎤  E 3 Q z (x)  ⎦= (h 2 + h 3 )h 3 . 2E I y (2.22)

Thus, the shear stress distribution in the core layer (layer ‘2’) results in:

τzx,2 (z) =

Q z (x) 2E I y

⎡

⎛

⎣E 2 ⎝

h 2

2

2

⎞

⎤   − z 2 ⎠ + E 3 h 2 + h 3 h 3 ⎦ .

(2.23)

The distribution of the shear stress over the height of the sandwich element is shown in Fig. 2.7a. Different parabolas form the distribution here. The distribution of the shear strain shows, because γzx (z) = Gτzxzx with different shear moduli in layers 2 and 3 (= 1), discontinuities, i.e. steps at the layer transitions, see Fig. 2.7b.

2.5 Technical Sandwich For the further explanations, a more technical configuration of a sandwich is considered, see Fig. 2.8. The face sheets of the symmetrical structure are generally significantly thinner than the core layer. Furthermore, the core layer is usually softer

2.5 Technical Sandwich

15

Fig. 2.8 Technical sandwich element: symmetrical structure with thin face sheets (F) and soft core (C)

than the face sheets.1 In addition, homogeneous and isotropic materials that deform linearly and elastically are assumed for all layers. As tasks2 of the face sheets in a technical sandwich, the following functions can be cited: • They absorb practically the entire bending moment. • They absorb practically the entire axial tensile or compressive load. The core, on the other hand, has the following tasks: • It practically absorbs the entire lateral force (shear). • Fixing, supporting and stabilizing the face sheets in their mutual position (i.e. vertical spacing of the face sheets, avoidance of the face sheets sliding relative to each other, ensuring the evenness—as a stabilizing elastic bedding—of the face sheets to avoid buckling/wrinkling). In many cases, the tensile and shear-resistant assemblage between the face sheets and the core is created by gluing.3 However, self-forming adhesion after the foaming of the core, welding, nailing, screwing or dowelling can also be found in technical applications, see Stamm and Witte (1974).

2.5.1 Bending Load For this case, according to Eq. (2.1), the average bending stiffness for the configuration according to Fig. 2.8 can be set as follows: A composite element with h C < h F , G C < G F is called an anti-sandwich, see Aßmus (2017). The following subchapters will explain certain facts in more detail. 3 The thickness of the adhesive layer is usually neglected when considering the different layers. 1 2

16

2 Basic Mechanical Load Cases

 E I y = 2E

F

 1 1 F 3 F F 2 b(h ) + bh (z c ) + E C b(h C )3 12 12

E F b(h F )3 E F bh F (h C + h F )2 E C b(h C )3 + + 6 2 12 E F b(h F )3 E F bh F (h c )2 E C b(h C )3 = + + . 6 2 12          =

E I y,F

E I y,FSt

(2.24) (2.25) (2.26)

E I y,C

The average bending stiffness is therefore made up of three parts: Part E I y,F , which describes the bending stiffness with regard to the partial center of area of the face sheet, part E I y,FSt , which describes the Steiner part (i.e., the additional part from the parallel axis theorem) of the face sheets with regard to the overall center of area and the part E I y,C , which relates to the bending stiffness of the core of the overall center of area.4,5   C C This results in stress distribution in the core − h2 ≤ z ≤ h2 according to Eq. (2.2) σx,C (z) =

M y (x)E C

× z, (2.27) E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12   C C or in the upper face sheet h2 ≤ z ≤ h2 + h F or in the lower face sheet   C C − h2 − h F ≤ z ≤ − h2 : σx,F (z) =

M y (x)E F E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12

× z.

(2.28)

The graphical representation of these two curves is shown in Fig. 2.9a. For certain geometry or material pairings, the bending stiffness can be further simplified according to Eq. (2.26). First, let us examine for what ratio h c /h F is the bending stiffness of the face sheets (E I y,F ) neglectable compared to the Steiner fraction (E I y,FSt ). If a limit value of 1% is set, the result from Eq. (2.26) is:

4

In the case of the core, this is identical to the partial center of area of the core. The symbol “” stands for a part in relation to a center of area. The abbreviation “St” for a “Steiner fraction” (resulting from the parallel axis theorem).

5

2.5 Technical Sandwich

17

Fig. 2.9 Normal stress distribution due to bending load: a Exact, b soft core, c soft core and thin face sheets

E F b(h F )3 6 E F bh F (h c )2 2 F 2

⇔

≤ 0.01,

(2.29)

(h ) ≤ 0.01, 3(h c )2  hc 1 ⇒ ≥ = 5.77. F h 0.03

(2.30) (2.31)

If one also considers the relationship h c = h C + h F in the last equation, the following limit value condition results: h C ≥ 4.77. h F

(2.32)

Thus, for thin face sheets, i.e. h F h C , the part of the bending stiffness of the face sheets (E I y,F ) can be neglected compared to the Steiner fraction (E I y,FSt ): E Iy ≈

E F bh F (h c )2 E C b(h C )3 + . 2 12       E I y,FSt

(2.33)

E I y,C

A further simplification can be made by using hc ≈ 1. h C

(2.34)

hc h C + h F h F = = 1 + ≈ 1, h C h C h C 

(2.35)

According to

< 0.01

18

2 Basic Mechanical Load Cases

Table 2.1 Classification of sandwich structures with regard to the thickness ratio of core to face sheet, see Allen (1969) Description Thickness ratio Thick face sheets

h C < 4.77 h F

Thin face sheets

100 ≥

Very thin face sheets

h C > 100 h F

h C ≥ 4.77 h F

this further simplification is permissible if h F h C < 0.01 or > 100 C h h F

(2.36)

holds. Thus, the classification of sandwich structures shown in Table 2.1 can also be made in relation to the thickness ratio of core to face sheet.6 Next, it is examined for which condition the bending stiffness of the core (E I y,C ) can be neglected compared to the bending stiffness of the Steiner fraction (E I y,FSt ). If a limit value of 1% is also set here, the result from Eq. (2.26) is: E C b(h C )3 12 E F bh F (h c )2 2 F F 6E h (h c )2 E C (h C )3

≤ 0.01,

(2.37)

≥ 100.

(2.38)

Thus, for soft cores,7 i.e. E C E F , the bending stiffness of the core can be neglected compared to the Steiner fraction: E Iy ≈

E F b(h F )3 E F bh F (h c )2 + . 6 2       E I y,F

(2.39)

E I y,FSt

this, Eq. (2.2) approximately gives the stress distribution in the core  From C C − h2 ≤ z ≤ h2 to

6 7

In the following, however, no distinction is made between ‘thin’ and ‘very thin’ face sheets. hc h F Taking typical values for applications of h C = 0.02 . . . 0.1 and h C ≈ 1, the resulting stiffness

relationships are

EF EC

= 833 . . . 167, see Allen (1969).

2.5 Technical Sandwich

19

σx,C (z) ≈ =

M y (x)E C E Iy

×z

(2.40)

M y (x)E C

×z E b(h ) E F bh F (h c )2 + 6 2 EC M y (x) × z ≈ 0, (2.41) = F × E b(h F )3 bh F (h c )2  +

1 6 2  C  C or in the upper face sheet h2 ≤ z ≤ h2 + h F or in the lower face sheet   C C − h2 − h F ≤ z ≤ − h2 : F

F 3

σx,F (z) ≈ =

M y (x)E F E Iy

×z

(2.42)

M y (x) b(h F )3 bh F (h c )2 + 6 2

× z.

(2.43)

The graphical representation of these two curves is shown in Fig. 2.9b. Both simplifications, i.e. thin face sheets and soft core can also be combined, so that for h F h C and E C E F the total average stiffness is only determined by the Steiner fraction of the face sheets: E Iy ≈

E F bh F (h c )2 . 2   

(2.44)

E I y,FSt

this, Eq. (2.2) approximately gives the stress distribution in the core  From C C − h2 ≤ z ≤ h2 to σx,C (z) ≈ 0,  C or in the upper face sheet h2 ≤ z ≤   C C − h2 − h F ≤ z ≤ − h2 :

h C 2

+ h F

(2.45)  or in the lower face sheet

20

2 Basic Mechanical Load Cases

σx,F (z) ≈

=

M y (x)E F E Iy

×z =

M y (x)

×z

bh F (h c )2 2

2M y (x) F bh (h C + h F )2

×z =

(2.46) 2M y (x)

bh F (h C )2 (1 +

h F 2 ) h C

×z



1

2M y (x) × z. = bh F (h C )2

(2.47)

Since the change in the normal stress in the face sheet with the z coordinate occurs over a very small thickness h F , the course of the function can also be approximated by the function value in the middle of the face sheet: 2M y (x) h C + h F × bh F (h C )2 2 F  M y (x) h M y (x) = = × h C 1 + . F C 2 C bh (h ) h bh F h C 

σx,F (z) ≈ σx,F (z =

h C 2

+

h F ) 2

=

(2.48) (2.49)

1

The graphical representation of the curves according to Eqs. (2.45) and (2.49) shows Fig. 2.9c. The various expressions of the average bending stiffness of the considered cases are shown in Table 2.2 for comparison. At the end of this section, some universal design curves should be derived. For a technical sandwich according to Fig. 2.8 under bending load (assume a cantilever with end shear force) derive normalized design curves depending on the normalized C . The normalization should be done with the limit without core core thickness h h F (h C = 0). The following relationships are sought: • • • •

average bending stiffness, strength (max. force) according to stress criterion, mass, lightweight index.

The derived relationships should be compared with the special case for thin face sheets and a soft core. The limit case to be considered for the normalization is shown in Fig. 2.10. For this limit case, the average bending stiffness E I y ,ref is: E I y ,ref = E F

3 3  1  2 b 2h F = E F b h F . 12 3

(2.50)

2.5 Technical Sandwich

21

Table 2.2 Formulations of the average bending stiffness E I y = Formulation of E I y

3  k =1

E k I yk

Remark General case

E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12          EI

y,F

E I y,FSt

EI

y,C

With bending stiffness of the face sheets and the core as well as Steiner’s share of the face sheets

Thin face sheets E F bh F (h c )2 E C b(h C )3 + 2 12       E I y,FSt

EI

With bending stiffness of the core and Steiner’s share of the face sheets

y,C

Soft core E F b(h F )3 E F bh F (h c )2 + 6 2       EI

y,F

With bending stiffness and Steiner’s share of the face sheets

E I y,FSt

Thin face sheets and soft core E F bh F (h c )2 2   

Only Steiner’s share of the face sheets

E I y,FSt

Fig. 2.10 Limit case of a technical sandwich element without a core layer

Based on the requirement that the maximum stress must not exceed the 0.2% initial yield stress, the maximum force that can be endured can be determined: σx,max =

F0 L 2 b(h F )3 3

!

F × h F = Rp0.2 .

(2.51)

22

2 Basic Mechanical Load Cases

This results in the maximum bearable force: F0,ref =

2 F  2b h F Rp0.2 3L

.

(2.52)

The mass results from density and volume: m ref = F V = F L A = 2F Lbh F .

(2.53)

Finally, the lightweight index of the reference configuration is: Mref =

F Rp0.2 F0,ref = . m ref × g 3F g L 2F h

(2.54)

The average bending stiffness of the sandwich with three layers results from Eq. (2.26) as follows:

EIy =

E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12          E I y,F

E I y,FSt

(2.55)

E I y,C

 3  2  3  h F 2 F 3h F h C + h F E C h C = E b + + 3 4 4 8E F ⎡

2

3 ⎤ 2 F  F 3 1 3 h C 1 EC h C ⎦, = E b h ⎣ + +1 + F F F 3 4 4 h 8 E h   

(2.56)

E I y ,ref

or in the normalized representation (see Fig. 2.11): ⎡

2

3 ⎤ EIy 1 3 h C 1 EC h C ⎦. =⎣ + +1 + 4 4 h F 8 EF h F E I y ,ref

(2.57)

For thin face sheets and soft cores, this relationship can be significantly simplified: EIy E I y ,ref

2 3 h C ≈ +1 . 4 h F

(2.58)

The mass of the sandwich is composed of the contribution of the face sheets (see Eq. (2.53)) and the core:

2.5 Technical Sandwich

23

Fig. 2.11 Normalized average bending stiffness for sandwich elements

Fig. 2.12 Normalized mass for sandwich elements



1 C h C m =  A L +  A L = 2 Lbh 1 + ,    2 F h F F

F

C

C

F

F

(2.59)

m ref

or in the normalized representation (see Fig. 2.12):

m 1 C h C =1+ . m ref 2 F h F

(2.60)

24

2 Basic Mechanical Load Cases

The maximum force that can be endured under a stress criterion8 results from Eq. (2.2) to:

σx,F,max =

M y,max E F E Iy

× z max =

F0 L E F E Iy

×

h C + h F 2

!

F = Rp0.2 .

(2.61)

This results in the maximum force:

F0 =

F Rp0.2

L ED

×

1 h C 2

+ h F

× E Iy

2 F  F 3 E b h h F LE 3 + h ⎡ 2

2

3 ⎤ 1 3 h C 1 EC h C ⎦ ×⎣ + +1 + 4 4 h F 8 EF h F 2 F  2b h F Rp0.2 h F = × h C 3L + h F    =

F Rp0.2

× D

1

C

(2.62)

×

(2.63)

2

F0,ref

⎡

2

3 ⎤ 1 3 h C 1 EC h C ⎦, ×⎣ + +1 + 4 4 h F 8 EF h F

(2.64)

or in the normalized representation (see Fig. 2.13): ⎡

2

3 ⎤ F0 1 1 3 h C 1 EC h C ⎦.  C⎣ + = +1 + F0,ref 4 4 h F 8 EF h F 1 + 21 h F h

(2.65)

For thin face sheets and soft cores, this relationship can be significantly simplified:

2 F0 1 3 h C  C × ≈ +1 . F0,ref 4 h F 1 + 21 h h F

(2.66)

Finally, the lightweight index of the sandwich based on Eqs. (2.65) and (2.60) results in:

8

Here, for the sake of simplicity, it is assumed that the failure occurs in the face sheet. Core failure should be excluded at this point.

2.5 Technical Sandwich

25

Fig. 2.13 Normalized maximum force for sandwich elements

2 F  2b h F Rp0.2 M=

=

×

h F h 2

C

+ h F

C C 1  h g2F Lbh F 1 + 2 F h F ⎡

2

3 ⎤ 1 3 h C 1 EC h C ⎦ ×⎣ + +1 + 4 4 h F 8 EF h F 3L

F0 = mg

F Rp0.2 L2 3F g h F

  

×

1+

1 2



1 

h C h F

(2.67)



Mref

⎡

2

3 ⎤ 1 3 h C 1 EC h C ⎣ + ⎦ +1 + 4 4 h F 8 EF h F

× , 1 C h C 1+ 2 F h F or in the normalized representation (see Fig. 2.14): ⎡

2

⎤ C C C 3 1 3 h 1 E h ⎣ + ⎦ +1 + F F 4 4 h 8 E h F M



= . Mref 1 h C 1 C h C 1+ × 1+ 2 h F 2 F h F

(2.68)

(2.69)

26

2 Basic Mechanical Load Cases

Fig. 2.14 Normalized lightweight index for sandwich elements under bending load: a Density ratio C = 51 and b density ratio F C F

=

1 100

For thin face sheets and soft cores, this relationship can be significantly simplified:

2 3 h C +1 4 h F M



. = Mref 1 h C 1 C h C 1+ × 1+ 2 h F 2 F h F

(2.70)

2.5 Technical Sandwich

27

2.5.2 Tensile/Compressive Load For this case, according to Eq. (2.11), the mean tensile stiffness for the configuration according to Fig. 2.8 can be set as follows:

EA =

3 

E k h k b = E F h F b + E C h C b + E F h F b

k =1 C = 2E F h F b + E C h  b . E AF

(2.71)

E AC

The mean tensile stiffness is therefore made up of two parts: The part E AF , which describes the tensile stiffness of the face sheets, and the part E AC , which describes the tensile stiffness of the core.   C C This results in the stress distribution in the core − h2 ≤ z ≤ h2 according to Eq. (2.13) F0 E C , (2.72) 2E F h F b + E C h C b  C  C or in the upper face sheet h2 ≤ z ≤ h2 + h F or in the lower face sheet   C C − h2 − h F ≤ z ≤ − h2 : σx,C =

σx,F =

F0 E F . 2E F h F b + E C h C b

(2.73)

The graphical representation of these two courses is shown in Fig. 2.15a. For soft cores, i.e. E C E F , the tensile stiffness of the core can be neglected compared to the face sheets:   EC E A ≈ E F 2h F b + F h C b = 2E F h F b. E 

(2.74)

1

This results in the normal stress distribution (see Fig. 2.15b) in the core

F0 E C EC F0 = F σx,C ≈ ≈ 0, 2E F h F b  E 2h F b

1

(2.75)

28

2 Basic Mechanical Load Cases

Fig. 2.15 Normal stress distribution under tensile load: a Exact and b soft core

 C or in the upper face sheet h2 ≤ z ≤   C C − h2 − h F ≤ z ≤ − h2 : σx,F ≈

h C 2

+ h F

 or in the lower face sheet

F0 E F F0 = . F F 2E h b 2h F b

(2.76)

The approximation formulas do not change if additional thin layers are taken into account.

2.5.3 Shear Load The exact shear stress distributions result from Eqs. (2.17) to (2.23) for the face sheet ⎤

2 h C ⎣ τzx,F (z) = + h F − z 2 ⎦ 2 2E I y ⎡ ⎤

2 C h E F Q z (x) ⎣ + h F − z 2 ⎦ 2

, = E F b(h F )3 E F bh F (h c )2 E C b(h C )3 2 + + 6 2 12 E F Q z (x)

or for the core layer

⎡

(2.77)

(2.78)

2.5 Technical Sandwich

29

Fig. 2.16 Shear stress distribution under shear load: a Exact, b soft core, c soft core and thin face sheets

⎛ ⎞ ⎤

C 2   h ⎣E C ⎝ τzx,C (z) = − z 2 ⎠ + E F h C + h F h F ⎦ 2 2E I y ⎡ ⎛ ⎞ ⎤

C 2   h Q z (x) ⎣ E C ⎝ − z 2 ⎠ + E F h C + h F h F ⎦ 2

= . E F b(h F )3 E F bh F (h c )2 E C b(h C )3 2 + + 6 2 12 Q z (x)

⎡

(2.79)

(2.80)

These two parabolic curves are shown in Fig. 2.16a. For soft cores, i.e. E C E F , the average bending stiffness can be approximated using Eq. (2.39) and according to the procedure in Sect. 2.4, the shear stress distribution in the face sheet results from integration over the normal stress gradient: h C +h F 2

h C +h F 2 F

dσx,F (x)  E Q z (x)   z dz + c3 dz + c3 = dx E Iy z z ⎡ ⎤

2 E F Q z (x) h C ⎣ = + h F − z 2 ⎦ + c3 . (2.81) 2 2E I y

τzx,F (z) =

The constant of integration c3 can also be determined here as c3 = 0 using the condition that no shear stresses occur at the free edge. This finally gives the shear stress distribution in the face sheet:

30

2 Basic Mechanical Load Cases

⎤

2 h C ⎣ τzx,F (z) = + h F − z 2 ⎦ 2 2E I y ⎡ ⎤

2 Q z (x) h C ⎣ = b(h F )3 + h F − z 2 ⎦ . 2 + b(h F )(h c )2 E F Q z (x)

⎡

(2.82)

3

At the transition point between the core and the face sheet, the following value results from Eq. (2.82): τzx,F



h C 2



=

Q z (x) b(h ) 3

F 3

+ b(h F )(h c )2

2

⎤ C 2   h C h 2 ⎣ ⎦ + h C h F + h F − 2 2   Q z (x) h C + h F Q z (x)h c    = =  F2 (h F )2 (h ) 2 2 b + (h ) b + (h ) c c 3 3 ⎡

=

Q z (x) 2E I y

E F h F h c .

(2.83)

(2.84)

Accordingly, the shear stress for the core is: h C

2 τzx,C (z) = z



dσx,C (x) dz +c2 . dx  

(2.85)

= 0, see Eq. (2.41)

The integration constant c2 results from the transition condition for the stress τzx between the core and the face sheet, i.e. identical stresses τzx,C (z = h C /2) = τzx,F (z = h C /2): c2 =

Q z (x) 2E I y



E F h F h c = b

Q z (x)h c (h F )2 3

+ (h c )2

,

(2.86)

and thus also the constant shear stress in the core: τzx,C (z) =

Q z (x) 2E I y



E F h F h c =

These two courses are shown in Fig. 2.16b.

b

Q z (x)h c (h F )2 3

+ (h c )2

.

(2.87)

2.5 Technical Sandwich

31

For soft cores, i.e. E C E F , and thin face sheets, i.e. h F h C , the shear stress distribution in the face sheet results from integration over the normal stress gradient: h C +h F 2

τzx,F (z) =

dσx,F (x)  dz + c3 = dx

h C +h F 2

z

z h C +h F 2

Q z (x) = bh F h C

Q z (x) dz  + c3 bh F h C

  Q z (x) h C F dz + c3 = + h − z + c3 . bh F h C 2

z



(2.88) Here, too, the constant of integration c3 can be determined as c3 = 0 using the condition that no shear stresses occur at the free edge. This finally gives the shear stress distribution in the face sheet for soft cores and thin face sheets: 

 Q z (x) h C F τzx,F (z) = + h − z . bh F h C 2 At the transition point between core and face sheet, Eq. (2.89) results in the following value: 

  C Q z (x) h C h C Q z (x) F h τzx,F 2 = + h − . (2.89) = bh F h C 2 2 bh C Alternatively, the last relation can also be expressed as follows: τzx,F



h C 2



=

Q z (x) Q z (x) Q z (x) Q z (x) =  ≈ = . (2.90) C + h F ) h F bh C b(h bh c C bh 1 + h C

Accordingly, the shear stress for the core is: h C

2 τzx,C (z) = z



dσx,C (x) Q z (x) Q z (x) dz +c2 = = . C dx bh bh c  

(2.91)

= 0, see Eq. (2.45)

These two courses are shown in Fig. 2.16c. Finally, all stress courses are shown comparatively in Tables 2.3, 2.4 and 2.5.

+

M y (x) bh F h C

6

b(h F )3

bh F (h c )2 2

M y (x) ×z

E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12

M y (x)E F

Face sheets σx,F (z) ×z

M y (x)E C

0

0

E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12

Core σx,C (z)

Approximations with soft core (E C E F ) and soft core with thin cover layers (E C E F , h F h C )

E C E F , h F h C

EC EF

Exact

Approach ×z

Graphic

Table 2.3 Comparison of the normal stress distributions due to bending stress in the sandwich according to the Euler-Bernoulli beam theory

32 2 Basic Mechanical Load Cases

2.5 Technical Sandwich

33

Table 2.4 Comparison of the normal stress distributions due to tensile stress in the sandwich according to the bar theory Approach Face sheets σx,F (z) Core σx,C (z) Graphic Exact F0 E F F 2E h b +

EC EF

F

E C h C b

F0 2h F b

F0 E F F 2E h b +

C

E C h C b

0

Soft core approximations (E C E F )

2.5.4 Bending Deformation of Sandwich Beams A simple theory for determining the deflection of technical sandwich beams is presented in the following. Here it is assumed that the total deflection u z (x) is composed of a bending component u z, b (x) and a shear component u z, s (x) (see Fig. 2.17): u z (x) = u z, b (x) + u z, s (x).

(2.92)

In this simple theory for sandwich beams with thin face sheets and a soft core— also known as the method of partial deflections in the literature (Klein 1989)—the bending deformation is calculated using the classical differential equations (Öchsner 2014):

d2 d2 u z, b (x) E Iy = qz (x), dx 2 dx 2

d d2 u z, b (x) E Iy = −Q z (x), dx dx 2 E Iy

d2 u z, b (x) = −M y (x), dx 2

where the mean bending stiffness according to Eq. (2.44) is to be used:

(2.93) (2.94) (2.95)

6

+

Q z (x)

E F b(h F )3

bh F h C

Q z (x)



h C

2

+ h F +

2

12

E C b(h C )3

− z2 ⎦

⎤

+ h F − z



⎡ ⎤

2 h C ⎣ + h F − z2 ⎦ 2 2

2

E F bh F (h c )2

2

h C

b(h F )3 + b(h F )(h c ) 3

2



E F Q z (x) ⎣

Face sheets τzx,F (z) ⎡

Q z (x)h c

6

bh C

≈

bh c

Q z (x)

+



E F b(h F )3

2

2

+ 12

E C b(h C )3

⎤   − z 2 ⎠ + E F h C + h F h F ⎦

⎞

E F bh F (h c )2

2 h C

(h F )2 + (h c )2 3

Q z (x)

b



2



Q z (x) ⎣ E C ⎝

Core τzx,C (z) ⎡ ⎛

Approximations with soft core (E C E F ) and soft core with thin cover layers (E C E F , h F h C )

E C E F , h F h C

EC EF

Exact

Approach

Graphic

Table 2.5 Comparison of the shear stress distributions due to shear force loading in the sandwich according to the Euler-Bernoulli beam theory

34 2 Basic Mechanical Load Cases

2.5 Technical Sandwich

35

Fig. 2.17 Bending of sandwich beams: a Undeformed, b bending deformation, c shear deformation, d total deformation

E Iy ≈

E F bh F (h c )2 . 2   

(2.96)

E I y,FSt

To determine the shear deformation, the shear stress9 in the core for sandwich beams with thin face sheets and soft core is considered according to Eq. (2.91) τzx,C (x) =

Q z (x) Q z (x) = , bh C bh c

(2.97)

or the shear strain (angular distortion) is considered using Hooke’s law (τ = Gγ ) : γzx,C (x) =

Q z (x) Q z (x) = . C Gbh Gbh c

(2.98)

For further derivation of the shear differential equation, consider a deformed sandwich element under pure shear deformation, see Fig. 2.18.

9

This relationship can also be seen as the basic equation of equilibrium, i.e. between the inner reactions (here: τx y ) and the outer loads (here: Q z , as a reaction to the outer shear forces).

36

2 Basic Mechanical Load Cases

Fig. 2.18 Technical sandwich beam (the thickness of the cover layers is overdrawn): a Undeformed initial state, b pure shear deformation (see Fig. 2.17c)

From the right-angled triangle 1 2 3 , the following geometric relationship results: 1 2 2 3

=

1 2 = |tan(γ )| ≈ |γ |. h K

(2.99)

Note that the shear strain γ is negative (γ < 0) in the way drawn in Fig. 2.18b. If one now looks at the right-angled triangle 123 (see Fig. 2.18b and the details in Fig. 2.19), a further geometric relationship results: 12 23

=

du z, s 12 = |tan(α)| ≈ |α| ≈ . hc dx

(2.100)

Finally, the kinematic relationship results from the geometric identity 1 2 = 12: du z, s h C = × γ. dx hc

(2.101)

2.5 Technical Sandwich

37

Fig. 2.19 Derivation of the shear differential equation

If one also considers the relation for the shear strain according to Eq. (2.98), i.e. the combination of the equilibrium with the constitutive law, the shear differential equation results in: du z, s Q z (x) Q z (x) = = . (2.102) h2 dx Gbh c Gb cC h

By means of A=

bh 2c h C

≈ bh c

(2.103)

du z, s Q z (x) = , dx AG

(2.104)

the following results after simplification

where AG represents the shear stiffness of the sandwich with thin face sheets and a soft core. The different differential equations for calculating the deflection are shown in Table 2.6 for comparison. In the following, the deflection for a sandwich beam under 3-point bending with point load is calculated using the partial deflection method, see Fig. 2.20. For this configuration, the internal shear force and bending moment distributions in the range 0 ≤ x ≤ L/2 result in (see also Fig. 5.6): Q z (x) = −

F0 F0 x und M y (x) = − . 2 x

(2.105)

The bending and shear deformations result in general—assuming thin face sheets and a soft core—by integrating the differential equations according to Table 2.6 to:

38

2 Basic Mechanical Load Cases

Table 2.6 Various differential equations for calculating the deflection for sandwich beams with thin face sheets and a soft core using the partial deflection method Differential equation

Stiffness Bending deformation



d2 d2 u z, b (x) = qz (x) E Iy 2 dx dx 2

d d2 u z, b (x) E Iy = −Q z (x) dx dx 2 E Iy

E Iy ≈

E F bh F (h c )2 2

d2 u z, b (x) = −M y (x) dx 2 Shear deformation

du z, s Q z (x) = dx AG C

AG C =

bh 2c GC h C

≈ bh c G C

Fig. 2.20 Sandwich beam under 3-point bending with point load: a Boundary conditions and external load; b beam cross-section

u z, b (x) =

1 E Iy

u z, s (x) = −



F0 x 3 + c1 x + c2 , 12

F0 x + c3 . 2G C A

(2.106) (2.107)

Using the boundary conditions u z, b (0) = u z, s (0) = 0 and du z, bdx(L/2) = 0 the three 2 0L constants of integration result to c1 = − F16 , c2 = 0 and c3 = 0. Thus, the total deflection of the sandwich beam along the x axis is:

2.5 Technical Sandwich

39

Fig. 2.21 Ratio of partial deflections as a function of core layer shear modulus: sandwich beam under 3-point bending with point load

u z (x) = u z, b (x) + u z, s (x) ⎛ ⎞ 3 F0 L 3 x x F0 L x ⎠− = × ⎝4 −3 ×2 , C L L 4 AG L 48E I y or the maximum value of the deflection at x = L2 : L F0 L 3 F0 L uz − . =− 2 48E I y 4 AG C The ratio of the partial deflections at x =

L 2

(2.108) (2.109)

(2.110)

is:

3

u z, b = u z, s

F0 L 48E I y F0 L 4 AG C

=

1 GC L2 × F C F. 6 E h h

(2.111)

Considering a concrete example of a sandwich beam with thin face sheets and a soft core, i.e., L = 2000 mm, h C = 150 mm, h F = 5 mm, E F = 74,000 MPa and G C = 11 MPa, the resulting ratio is 0.132 = 13.2%. The evaluation of Eq. (2.111) as a function of the shear modulus of the core layer is shown in Fig. 2.21. It can be seen that for small values of the shear modulus of the core layer there are definitely clear contributions to the shear deformation. This is the case, although the beam is more than ten times longer (L = 2000 mm) than high (10 × 160 = 1600 mm) and thus at a homogeneous beam, the shear component is usually negligible in a first approximation.

40

2 Basic Mechanical Load Cases

2.6 Supplementary Problems 2.6.1 Knowledge Questions • Describe a sandwich element. • Give typical properties in terms of geometry and materials for a technical sandwich element. • Describe the function of the face sheets and the core in a technical sandwich. • Name typical joining techniques between core and face sheets. • Name typical types of loads that can occur in a sandwich element. • Sketch the typical course of normal stress and normal strain versus the height for a symmetrical sandwich with three layers (core ’softer’ than face sheets) under bending load. • Sketch the typical course of normal stress and normal strain versus the height for a symmetrical sandwich with three layers (core ’softer’ than face sheets) under tensile loading. • Sketch the typical behavior of the shear stress and shear strain versus the height for a symmetrical sandwich with three layers (core ’softer’ than face sheets) under shear loading. • Sketch the course of the normal stress in a technical sandwich under bending load for: (a) exact consideration, (b) soft core and (c) soft core with thin face sheets. • What is the thickness ratio of the core to the face sheet in terms of ’thick face sheets’? • What is the thickness ratio of the core to the face sheet in terms of ’thin face sheets’? • What is the ratio of the thickness of the core to the face sheet in terms of ’very thin face sheets’? • Sketch the course of the normal stress in a technical sandwich under tensile loading for: (a) exact consideration and (b) soft core. • Sketch the course of the shear stress in a technical sandwich under shear force loading for: (a) exact consideration, (b) soft core and (c) soft core with thin face sheets.

2.6.2 Calculation Problems 2.6.1 Average Bending Stiffness of a Sandwich Determine the general average bending stiffness E I y for the sandwich shown in Fig. 2.22, which consists of three layers and a cross-section as an I-profile. Afterwards, simplify the result for the special case bC = bF /3 and h F = h C /4.

2.6 Supplementary Problems

41

Fig. 2.22 Sandwich of three layers and cross section as I-profile

2.6.2 Calculation of the Shear Stress Distribution for a Circular Cross-Section A beam with a circular cross-section (radius R) is given. Calculate the shear stress distribution τzx over the cross-section under the influence of a shear force Q z (x). Consider the distribution in the center of the circular cross-section, i.e. for y = 0. 2.6.3 Limit Case of the Normalized Stiffness of a Sandwich Simplify the expression for the normalized stiffness of a sandwich according to Eq. (2.57) for the limiting case E F = E C = E and h F = h C = 13 . This is a homogeneous body of total height ‘1’. Clarify how the result to understand (why is the ratio not equal to ‘1’?). 2.6.4 Deflection of a Sandwich Beam with Distributed Load Using the Partial Deflection Method For the sandwich beam shown in Fig. 2.23, the bending line u z (x) and the maximum value of the deflection must be determined. It should be assumed that it is a beam with thin face sheets and a soft core. Furthermore, calculate the general ratio of the partial deflections and also provide the numerical value for L = 2000 mm, h C = 150 mm, h F = 5 mm, E F = 74,000 MPa and G C = 11 MPa. 2.6.5 Deflection of a Sandwich Beam (Cantilever) with Distributed Load Using the Partial Deflection Method For the sandwich beam shown in Fig. 2.24, the bending line u z (x) and the maximum value of the deflection must be determined. It should be assumed that it is a beam with thin face sheets and a soft core. Furthermore, calculate the general ratio of the partial deflections and also provide the numerical value for L = 2000 mm, h C = 150 mm, h F = 5 mm, E F = 74,000 MPa and G C = 11 MPa.

42

2 Basic Mechanical Load Cases

Fig. 2.23 Sandwich beam with constant distributed load: a Boundary conditions and external load; b beam cross-section

Fig. 2.24 Sandwich beam with constant distributed load: a Boundary condition and external load; b beam cross-section

Fig. 2.25 Sandwich beam with shear force: a Boundary condition and external load; b beam cross-section

References

43

2.6.6 Deflection of a Sandwich Beam (Cantilever) with Shear Force Using the Partial Deflection Method For the sandwich beam shown in Fig. 2.25, the bending line u z (x) and the maximum deflection value must be determined. It should be assumed that this is a beam with thin face sheets and a soft core. Furthermore, calculate the general ratio of the partial deflections and also provide the numerical value for L = 2000 mm, h C = 150 mm, h F = 5 mm, E F = 74,000 MPa and G C = 11 MPa.

References Allen, H.G.: Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969) Aßmus, M., Bergmann, S., Eisenträger, J., Naumenko, K., Altenbach, H.: Consideration of nonuniform and non-orthogonal mechanical loads for structural analysis of photovoltaic composite structures. In: Altenbach, H., Goldstein, R., Murashkin, E. (eds.) Mechanics for Materials and Technologies, pp 73–122. Springer, Cham (2017) Hertel, H.: Leichtbau: Bauelemente, Bemessungen und Konstruktionen von Flugzeugen und anderen Leichtbauwerken. Springer, Berlin (1960) Klein, B.: Leichtbau-Konstruktion: Berechnungsgrundlagen und Gestaltung. Friedr. Vieweg & Sohn, Braunschweig (1989) Klein, B.: Leichtbau-Konstruktion: Berechnungsgrundlagen und Gestaltung. Vieweg+Teubner, Wiesbaden (2009) Öchsner, A.: Elasto-Plasticity of Frame Structure Elements: Modeling and Simulation of Rods and Beams. Springer, Berlin (2014) Öchsner, A.: Computational Statics and Dynamics: An Introduction Based on the Finite Element Method. Springer, Singapore (2020) Öchsner, A.: Stoff- und Formleichtbau: Leichter Einstieg mit eindimensionalen Strukturen. Springer Vieweg, Wiesbaden (2020) Stamm, K., Witte, H.: Sandwichkonstruktionen. Springer, Wien (1974)

Chapter 3

Limit Load

3.1 Failure Modes The various failure modes for technical sandwich beams are shown in Table 3.1. Here, between failure mechanisms of the face sheets (yielding, instability and local deformation excess) and the core (yielding and instability) can be distinguished. The failure of the connecting layer between core and face sheet and the global instability failure must be considered as further failure mechanisms. The various forms of instability are shown in Table 3.2 with a distinction between global and local forms of failure.

3.1.1 Global Instability Failure To derive the buckling formula1 for a sandwich beam with thin face sheets and a soft core, consider a configuration hinged on both sides under a compressive force F, see Fig. 3.1a. The entire deformation is composed of the pure bending component and the shear component, see Fig. 3.1b. The equilibrium is now established for the first time on the deformed member,2 see Fig. 3.1b. The equilibrium of moments at the cutting site x yields (see Fig. 3.2a):  

  M y = 0 ⇔ −F × u z, b + u z, s + M y (x) = 0,

(3.1)

1

For a better understanding, it is recommended here that the reader first understands the buckling force derivation according to Euler for homogeneous and isotropic Euler-Bernoulli beams, see Appendix 6.1.3. 2 For the derivation of the classical Euler-Bernoulli differential equations, the equilibrium is established on the non-deformed member, see Öchsner (2014). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_3

45

46

3 Limit Load

Table 3.1 Failure modes in technical sandwich beams, cf. Stamm and Witte (1974) Failure mode Load case (section) Face sheets • Failure by reaching the yield stress or strength under tension or compression • Exceeding the instability limit under compressive loading, i.e. local wrinkling (local instability) of the compressively loaded face sheet • Exceeding the instability limit under compressive loading, i.e. local symmetrical or asymmetrical wrinkling (local instability) of both face sheets • Strong local deformations of the face sheets due to local force application Core • Failure by reaching the shear yield stress or shear strength

Bending (2.5.1, 2.5.2) Bending (3.1.3) Compression (3.1.5, 3.1.4) –

Bending, shear force (2.5.3) • Exceeding the instability limit under shear loading, e.g. buckling of the Bending, shear force webs of a honeycomb core (–) Other causes • Failure of the joining (e.g. adhesive layer) between the face sheet and Bending (3.1.2) core Compression (3.1.1) • Global instability failure (overall instability), i.e. buckling of the sandwich beam

  ⇒ M y (x) = +F × u z, b + u z, s = +Fu z .

(3.2)

Thus, the following formulation for the bending deformation can be specified using the bending differential equation according to Eq. (2.95): E Iy

  d2 u z, b = −M y (x) = −F × u z, b + u z, s , dx 2

(3.3)

or after a one-time differentiation with respect to the x coordinate: dM y (x) d3 u z, b = −F × E Iy =− 3 dx dx



 du z, b du z, s + . dx dx

(3.4)

The following geometric relationship for small angles can be derived from the right-angled force triangle according to Fig. 3.2b:   Qz du z, b du z, s du z, b du z, s du z = sin + ≈ + = . F dx dx dx dx dx

(3.5)

3.1 Failure Modes

47

Table 3.2 Instabilities in technical sandwich beams: (a) buckling, (b) symmetrical wrinkling of the face sheets, (c) asymmetrical wrinkling of the face sheets, (d) wrinkling of the compressively loaded face sheet, cf. Stamm and Witte (1974) Schematic representation Load case (section) (a)

Global instability Compression (3.1.1)

(b)

Local instability Compression (3.1.5)

(c)

Compression (3.1.4)

(d)

Bending (3.1.3)

The last relation can be transformed immediately using the shear differential equation (2.104) after the first-order derivative of the shear deformation: F du z, s  = dx AG C 1 −

F AG C

×

du z, b , dx

(3.6)

where the area according to Eq. (2.103) is given as A = bh 2c /h C ≈ bh c . If one now inserts Eq. (3.6) into the differential equation according to (3.4), the result is: E Iy

F du z, b d3 u z, b  −F× = −F × C dx 3 dx AG 1 −

or transformed to:

F AG C

×

du z, b , dx

(3.7)

48

3 Limit Load

Fig. 3.1 Sandwich beam hinged on both sides under compressive load: a Initial configuration and b deformation

Fig. 3.2 Sandwich beam hinged on both sides under compressive load: a Cutting free at location x; b force triangle (without consideration of the current signs)

F d3 u z, b +  dx 3 E Iy 1 −   λ2

or as:

F AG C



du z, b = 0, dx

d3 u z, b du z, b = 0. + λ2 3 dx dx

(3.8)

(3.9)

The general solution for such a differential equation results in: u z (x) = c1 × sin(λx) + c2 × cos(λx) + c3 .

(3.10)

3.1 Failure Modes

49

The first- and second-order derivatives of this general function result in: d1 u z, b = c1 α cos(λx) − c2 λ sin(λx), dx 1 d2 u z, b = −c1 λ2 sin(λx) − c2 λ2 cos(λx). dx 2

(3.11) (3.12)

If one inserts this second-order derivative into the differential equation according   2 to (3.3), i.e. M y (x) = F u z, b + u z, s = −E I y d dxu z,2 b , the total deflection is obtained as:  E Iy  2 c1 λ sin(λx) − c2 λ2 cos(λx) , u z, b + u z, s = − (3.13) F or with λ2 =

 F F E I y 1− AG C

finally to:

u z, b (x) + u z, s (x) = u z (x) =

c1 sin(λx) + c2 cos(λx) 1−

F AG C

.

(3.14)

Using the boundary conditions u z (0) = 0 and u z (L) = 0, the unknown constants c1 and c2 can be approached. The first boundary condition, i.e. u z (0) = 0, results in: 0 = c1 × sin(0) + c2 × cos(0) ⇔ c2 = 0.

(3.15)

From the second boundary condition, i.e. u z (L) = 0, the result is: 0 = c1 × sin(λ × L).

(3.16)

If the product is to be zero, one of the two factors, i.e. c1 or sin(λ × L) must be zero. c1 = 0 is a trivial solution (with c1 = c2 = 0 according to Eq. (3.14): u z = 0, thus no deformation). Therefore sin(λ × L) = 0 has to be looked at more closely. The condition sin(λ × L) = 0 means that λ × L = k × π with k = 0, 1, 2, . . . (see Fig. 6.6). The condition λ × L = 0 would mean F = 0 (see definition of λ according to Eq. (3.8)). This results in the reasonable condition: λ × L = π ⇔ λ2 =

π2 . L2

(3.17)

If one inserts the last result into the definition equation of λ according to Eq. (3.8), the result is:

50

3 Limit Load

F π2 =  L2 E Iy 1 −

F AG C

,

(3.18)

or after a short transformation: Fcr = 1

π 2 E Iy L2 π2 E I + L 2 ×AGy C

=

FcrE 1+

FcrE AG C

,

(3.19)

where FcrE = π 2 E I y /L 2 is the Euler buckling force for homogeneous beams, see Eq. (6.23). Furthermore, it should be noted that the area according to Eq. (2.103) is given as A = bh 2c /h C ≈ bh c . For other support conditions, the length L in the equation for the buckling force according to Euler can be replaced by the so-called buckling length L cr (see Table 6.2). Thus, a total of four support cases can be calculated. Equation (3.19) is also often represented in the following form (see Allen 1969): 1 1 1 = E+ , (3.20) Fcr Fcr AG C where the following special cases can be distinguished: • G → ∞ ⇒ Fcr = FcrE . • G finite ⇒ Fcr < FcrE . • G small ⇒ Fcr → AG C . An alternative way to derive the buckling formula is shown in Plantema (1966). For this purpose, the bending and shear differential equations (see (2.95) or (2.104)) are rearranged for the second-order derivatives: M y (x) d2 u z b =− , 2 dx E Iy

(3.21)

1 dQ z (x) d2 u z, s = . 2 dx AG C dx

(3.22)

Both formulations can be composed additively to the total curvature3 : d2 u z, b d2 u z, s M y (x) 1 dQ z (x) d2 u z, . = + =− + 2 2 2 dx dx dx AG C dx E Iy

(3.23)

If one uses in the last expression of the differential equation the bending moment according to Eq. (3.2), i.e. M y (x) = +Fu z (x), and the shear force according to z (x) Eq. (3.5), i.e. Q z (x) = F dudx , the following differential equation results after transformation: 3

u z (x) To be precise, the following applies to the curvature: κ(x) = − d dx 2 . 2

3.1 Failure Modes

51

F d2 u z (x) +  2 dx E Iy 1 −  

F AG C

λ2

 u z (x) = 0 ⇔

d2 u z + λ2 u z (x) = 0. dx 2

(3.24)

The solution is as shown in the Appendix 6.1.3 and one finally gets the result according to Eq. (3.19).

3.1.2 Shear Failure of the Connecting Layer To assess whether the connecting layer (e.g. an adhesive layer) fails, the shear stress at the point ±h C /2 must be evaluated. According to the exact theory [see Eq. (2.80)], the shear stress is:  τzx

h C 2

 

= 2

   Q z (x) E F h C + h F h F E F b(h F )3 E F bh F (h c )2 E C b(h C )3 + + 6 2 12

.

(3.25)

For soft cores, i.e. E C  E F , the result is (see Eq. (2.84)):  τzx

h C 2



 Q z (x)h c Q z (x) h C + h F =  F2  F2 = b (h3 ) + (h c )2 b (h3 ) + (h c )2 =

Q z (x) 2E I y

E F h F h c .

(3.26)

For soft cores, i.e. E C  E F , and thin face sheets, i.e. h F  h C , the shear stress results in [see Eq. (2.89)]:  τzx

h C 2

 =

Q z (x) Q z (x) = . bh C bh c

(3.27)

Typical material properties of adhesive layers are summarized in Table 3.3.

52

3 Limit Load

Table 3.3 Mechanical properties of some adhesives: G: shear modulus; τp : shear yield limit; τaB : shear strength; γaB : shear strain at failure Adhesive Manufacturer G in MPa τp in MPa τaB in MPa γaB in % Araldite AV138 Hysol EA 9394 Hysol EA 9321 Supreme 10HT Araldite AV 119 Hysol EA 9359.3 Araldite 2015 Sikaflex 256 Redux 326 DP-8005

Huntsman

1559

Epoxide 25.0

30.2

5.50

Loctite

1140

25.0

40.4

8.36

Loctite

1030

20.0

33.0

6.35

Master Bond

1460

37.1

37.1

16.1

Huntsman

1260

47.0

47.0

50.7

Loctite

660.0

35.3

35.3

63.0

Huntsman

560.0

20.0

40.3

8.26

330

37.9

3.70

8.40

180

14.0 Polyurethane Sika 1.351 8.26 Bismaleimide Hexcel comp. 1615 37.9 Modified acrylate 3M 178.6 5.3

Based on da Silva et al. (2018)

3.1.3 Local Wrinkling of the Compressive Face Sheet (Bending Load) Local wrinkling of the face sheets in the bending load case can occur in the face sheet loaded in compression.4 The mechanical mechanical replacement model model shown in Fig. 3.3 was proposed in the literature as a modeling approach, see Allen (1969). Here it is assumed that the tensile face sheet remains completely planar since it is modeled as a rigid layer. Furthermore, the core is modeled as a homogeneous and isotropic continuum.5 The critical wrinkling stress is generally found to be  1  2 σcr = B1 E F 3 E C 3 ,

4

(3.28)

In the bending load case, one face sheet is loaded in tension and the other in compression. In other modeling approaches, the core layer was assumed to be the elastic spring bedding of the face sheet. However, such approaches neglect the shear stiffness in the x z plane and are therefore inadequate. 5

3.1 Failure Modes

53

Fig. 3.3 Simplified model for wrinkling of the compression-loaded top layer (bending load case). Adapted from Allen (1969)

where the factor B1 depends on the geometry and material parameters. Using the relationship between the elastic constants for isotropic materials, i.e. GC =

EC , 2(1 + ν C )

(3.29)

this results in the alternative relationship for the wrinkling stress:

1 1 1   σcr = B1 2(1 + ν C ) 3 × E F E C G C 3 = B1 × E F E C G C 3 .  

(3.30)

B1

The following conservative approximation for the factor B1 is proposed by Hoff and Mautner (1945), Allen (1966): σcr ≈

1  F C C 1 × E E G 3. 2

(3.31)

For sandwich  structures with thin face sheets and a soft core, under the condition 1/3

h E k = h < 0.25, the following values for B1 were determined in Gough C EC et al. (1939), Allen (1969): F

F

B1 (ν C = 0.25) = 0.575,

(3.32)

B1 (ν = 0.50) = 0.543,

(3.33)

C

which results in the following values for B1 :

54

3 Limit Load

Fig. 3.4 Position of the local minimum of the function B1 = B1 (): influence of the parameter  F 1/3 h F E k = h C EC

1 B1 (ν C = 0.25) = 0.575 2(1 + 41 ) 3 = 0.7804,

1 B1 (ν C = 0.50) = 0.543 2(1 + 21 ) 3 = 0.7831.

(3.34) (3.35)

For other material and geometry combinations, the following calculation must be  F 1/3 h F E , i.e. carried out: For specific values of ν C and k = h C EC f (Θ) k2Θ 2 + 12  k k2Θ 2 1 2 (3 − ν K ) sinh(Θ) cosh(Θ) + (1 + ν K )Θ = + × , 12 k Θ (1 + ν K )(3 − ν K )2 sinh2 (Θ) − (1 + ν K )3 Θ 2

B1 =

(3.36) (3.37)

C

whereby the argument as Θ = 2πh is given and λ represents the wavelength of the λ wrinkle wave of the compressive face sheet (see Figs. 3.4 and 3.5). One can see from Fig. 3.4 that an enlargement of the k value with constant ν C leads to a shift of the minimum to smaller  values. From Fig. 3.5 it can be seen that the influence of the Poisson’s ratio is not so dominant. The local minimum of the function B1 = B1 (Θ) results from the condition ∂ B1 () k 2 Θ 1 ∂ f (Θ) ! = + × = 0, ∂ 6 k ∂

(3.38)

where the Newton’s method can be used for the numerical determination of the root.6 If one repeats the procedure of determining the minimum for a value range of the geometry and material factor k with a constant Poisson’s ratio ν C , determination 6

A Python3 program is provided in the Appendix 6.2 to automatically evaluate Eq. (3.38) for given value ranges of k and ν C .

3.1 Failure Modes

55

Fig. 3.5 Position of the local minimum of the function B1 = B1 (): influence of the Poisson’s ratio ν C of the core for (a) k = 1 und (b) k = 4

diagrams, as in Figs. 3.6, 3.7 and 3.8 are generated. These diagrams allow the critical factor B1 to be read off easily without the need for numerical iteration to determine the minimum. The disadvantage, however, is that each of these diagrams is only valid for a specific Poisson’s ratio. From the diagrams in Figs. 3.6, 3.7 and 3.8 it can be seen that the B1 value approaches a constant value for k ≤ 0.25. These values are summarized in Fig. 3.9 as a function of the Poisson’s ratio. One can see a slightly falling trend of the B1 value. Furthermore, Figs. 3.6, 3.7 and 3.8 show that the B1 value and the normalized wavelength are monotonically increasing functions.

56

3 Limit Load

Fig. 3.6 Determination diagram of the factor B1 and the wavelength λ for the case ν C = 41 . Based on Allen (1969)

3.1 Failure Modes

Fig. 3.7 Determination diagram of the factor B1 and the wavelength λ for the case ν K = 0.3

57

58

3 Limit Load

Fig. 3.8 Determination diagram of the factor B1 and the wavelength λ for the case ν C = 21 . Based on Allen (1969)

3.1 Failure Modes

Fig. 3.9 Determination diagram of the factor B1 for the case k ≤ 0.25

59

60

3 Limit Load

3.1.4 Local Antisymmetric Wrinkling of both Face Sheets (Compressive Load) Local antisymmetric wrinkling can occur in compression-loaded sandwich beams and is characterized by the face sheets deforming antisymmetrically to the center line in the transverse direction, see Fig. 3.10. In the case of thin face sheets and a soft core, the critical wrinkling stress σcr can be approximated according to the basic equation from Sect. 3.1.3, i.e.  1  2 σcr = B1 E F 3 E C 3 .

(3.39)

However, another function f (Θ) must be used to calculate the factor B1 : f (Θ) k2Θ 2 + 12 k  k2Θ 2 1 2 cosh(Θ) − 1 = + × . 12 k Θ (1 + ν K )(3 − ν K ) sinh(Θ) + (1 + ν K )2 Θ

B1 =

(3.40) (3.41)

Here, too, the determination of the local minimum of the function B1 () allows the factor B1 for Eq. (3.39) to be determined. When applying Newton’s method for small values of k, it should be noted that the gradient of the function B1 () has local extreme values close to  = 0, see Fig. 3.11 b. It is therefore useful for convergence if the start value of the Newton’s iteration (i = 0 ) is to the right-hand side of the root. Determination diagrams for the factor B1 as a function of the geometry and material factor k are shown in Figs. 3.12, 3.13 and 3.14. These diagrams allow the critical factor B1 to be read off easily without the need for numerical iteration to determine

Fig. 3.10 Simplified representation of the antisymmetric wrinkling of the face sheets (compression load case). Based on Allen (1969)

3.1 Failure Modes

61

Fig. 3.11 Determination of the root of the derivative of the function B1 = B1 (): a Large range of values with root at  = 75.85 and b local extrema for  → 0

the minimum. The disadvantage here, however, is that each of these diagrams is only valid for one specific Poisson’s ratio ν C .7 Furthermore, one can see from Figs. 3.12, 3.13 and 3.14 that the curves for B1 and for the normalized wavelength hλ C only go up to a certain abscissa value k, which is determined by the following relationship: 

7

1 − νC 8(1 + ν C )

1 3

.

(3.42)

A Python3 program is provided in the Appendix 6.2 to automatically evaluate Eq. (3.38) for given value ranges of k and ν C .

62

3 Limit Load

Fig. 3.12 Determination diagram of the factor B1 and the wavelength λ for the case ν C = antisymmetric wrinkling. Based on Allen (1969)

1 4

in

3.1 Failure Modes

63

Fig. 3.13 Determination diagram of the factor B1 and the wavelength λ for the case ν C = 0.3 in antisymmetric wrinkling

64

3 Limit Load

Fig. 3.14 Determination diagram of the factor B1 and the wavelength λ for the case ν C = 0.5 in antisymmetric wrinkling. Based on Allen (1969)

3.1 Failure Modes

65

From the diagrams in Figs. 3.12, 3.13 and 3.14 it can be seen that the B1 value approaches a constant value for k ≤ 0.20. These values are summarized in Fig. 3.15 as a function of the Poisson’s ratio. One can see a slightly falling trend of the B1 value. Furthermore, Figs. 3.12, 3.13 and 3.14 show that the B1 value is a monotonically decreasing function and the normalized wavelength is a monotonically increasing function.

3.1.5 Local Symmetric Wrinkling of both Face Sheets (Compressive Load) Local symmetric wrinkling can occur in compression-loaded sandwich beams and is characterized by the face sheets deforming symmetrically to the centerline in the transverse direction, see Fig. 3.16. In the case of thin face sheets and a soft core, the critical wrinkling stress σcr can also be approximated according to the basic equation from Sect. 3.1.3, i.e.  1  2 σcr = B1 E F 3 E C 3 .

(3.43)

However, another function f (Θ) should be used to calculate the factor B1 : k2Θ 2 f (Θ) + 12 k  2 2 1 2 cosh(Θ) + 1 k Θ + × = . 12 k Θ 3 sinh(Θ) − Θ

B1 =

(3.44) (3.45)

Here, too, the determination of the local minimum of the function B1 () allows the factor B1 for Eq. (3.43) to be determined. As in the other cases, the smaller value of B1 occurs with larger Poisson’s ratios ν C . Therefore, Fig. 3.17 provides a determination diagram for the factor B1 as a function of the geometry and material factor k. Here, too, small values of k, i.e. k ≤ 0.25, result in a constant value of 0.630 for ν C = 0.0. Furthermore, Fig. 3.17 shows that the B1 value and the normalized wavelength are monotonically increasing functions. The following conclusions can be drawn from the diagrams for antisymmetric (see Figs. 3.12, 3.13 and 3.14) and symmetric wrinkling (see Fig. 3.17), see Allen 1969: • For very small values of k, i.e. k < 0.2, there are similar values for B1 . Thus, antisymmetric and symmetric wrinkling occur with more or less equal probability. 1  3 1−ν C • In the range 0.2 < k < 8(1+ν antisymmetric wrinkling occurs at lower C) stresses since B1 () is a monotonically decreasing function for antisymmetric wrinkling, but monotonically increasing for symmetric wrinkling.

66

3 Limit Load

Fig. 3.15 Determination diagram of the factor B1 for the case k ≤ 0.20 in antisymmetric wrinkling

3.2 Supplementary Problems

67

Fig. 3.16 Simplified representation of the symmetric wrinkling of the face sheets (compression load case). Based on Allen (1969)

1  3 1−ν C • For k > 8(1+ν no antisymmetric wrinkling occurs and only symmetric wrinC) kling needs to be considered. 1  3 1−ν C • For k = 8(1+ν the B1 value of antisymmetric wrinkling is the most conserC) vative of all three wrinkling cases. • For sufficiently long beams, global instability failure (see Sect. 3.1.1) occurs before local buckling. At the end of this subchapter, the various formulations for calculating the factor B1 are summarized and compared in Table 3.4.

3.2 Supplementary Problems 3.2.1 Knowledge Questions • Describe typical failure modes of the face sheets of a sandwich beam. • Describe typical failure modes of the core of a sandwich beam. • Which failure modes can occur in a sandwich apart from the failure of the face sheets and the core? • Under which types of loading can local wrinkling occur in a sandwich? • Name the different types of instability that can occur in a sandwich beam. • Name typical core materials for sandwich beams. • What is the magnitude of the shear strength of classic adhesives? • Which conservative approximation formula can be used to estimate the local wrinkling of the pressure face sheet (bending load case)? • Give the simple formula for the buckling force of a sandwich considering Euler’s buckling force.

68

3 Limit Load

Fig. 3.17 Determination diagram of the factor B1 and the wavelength λ for the case ν C = 0.0 in symmetric wrinkling. Based on Allen (1969)

3.2 Supplementary Problems

69

Table 3.4 Determination of the factor B1 =

k2 Θ2 12

+

f (Θ) k

for local wrinkling

Case (section)

Function f (Θ)

Bending (3.1.3)

(3 − ν K ) sinh(Θ) cosh(Θ) + (1 + ν K )Θ 2 × Θ (1 + ν K )(3 − ν K )2 sinh2 (Θ) − (1 + ν K )3 Θ 2

Compression, asymmetric (3.1.4)

2 cosh(Θ) − 1 × K Θ (1 + ν )(3 − ν K ) sinh(Θ) + (1 + ν K )2 Θ

Compression , symmetric (3.1.5)

2 cosh(Θ) + 1 × Θ 3 sinh(Θ) − Θ

(ν K = 0)

Based on Allen (1969)

Fig. 3.18 Sandwich beam with constant distributed load: a Boundary conditions and external load; b beam cross-section

• Sketch the bending moment distribution of a sandwich beam under symmetric 3-point and 4-point bending.

3.2.2 Calculation Problems 3.2.1 Failure Analysis of a Sandwich Beam with Distributed Load The corresponding proof of strengths must be carried out for the sandwich beam shown in Fig. 3.18. The sandwich consists of a hard foam core material bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the distributed load q0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 150 mm, h F = 5 mm. • Material properties of the hard foam core: E C = 30 MPa, ν C = 0.364, C C C = 0.5 MPa, Rm = 0.90 MPa, σdB = 0.38 MPa. τaB

70

3 Limit Load

Fig. 3.19 Sandwich beam (cantilever) with constant distributed load: a Boundary conditions and external load; b beam cross-section

F • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 = 364 MPa. • Material property of the adhesive: τaB = 37.1 MPa. N . • External load: q0 = 2.5 mm

3.2.2 Failure Analysis of a Sandwich Beam (Cantilever) with Distributed Load The corresponding proofs of strength must be carried out for the sandwich beam shown in Fig. 3.19. The sandwich consists of a hard foam core material bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the distributed load q0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 150 mm, h F = 5 mm. • Material properties of the hard foam core: E C = 30 MPa, C C C C ν = 0.364, τaB = 0.5 MPa, Rm = 0.90 MPa, σdB = 0.38 MPa. F = 364 MPa. • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 • Material property of the adhesive: τaB = 37.1 MPa. N . • External load: q0 = 2.5 mm 3.2.3 Failure Analysis of a Sandwich Beam Under 4-Point Bending The corresponding proof of strength must be carried out for the sandwich beam shown in Fig. 3.20. The sandwich is constructed from a synthetic core material bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the force F0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 50 mm, h F = 50 mm. • Material properties of the synthetic core: E C = 25,000 MPa, ν C = 0.4, C C C τaB = 40 MPa, Rm = 80 MPa, σdB = 60 MPa.

3.2 Supplementary Problems

71

Fig. 3.20 Sandwich beam under 4-point bending: a Boundary conditions and external loads; b beam cross-section

Fig. 3.21 Sandwich beam under 3-point bending: a Boundary conditions and external load; b beam cross-section F • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 = 364 MPa. • Material property of the adhesive: τaB = 37.1 MPa. • External load: F0 = 2500 N.

3.2.4 Failure Analysis of a Sandwich Beam Under 3-Point Bending with Point Load The corresponding proofs of strength must be carried out for the sandwich beam shown in Fig. 3.21. The sandwich consists of a plastic core bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the force F0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 100 mm, h F = 25 mm. • Material properties of the plastic core: E C = 1500 MPa, ν C = 0.4, C C C τaB = 40 MPa, Rm = 80 MPa, σdB = 60 MPa.

72

3 Limit Load

Fig. 3.22 Sandwich beam under 3-point bending with distributed load: a Boundary conditions and external load; b beam cross-section

F • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 = 364 MPa. • Material property of the adhesive: τaB = 37.1 MPa. • External load: F0 = 5000 N.

3.2.5 Failure Analysis of a Sandwich Beam Under 3-Point Bending with Distributed Load The corresponding proofs of strength must be carried out for the sandwich beam shown in Fig. 3.22. The sandwich consists of a plastic core bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the distributed load q0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 100 mm, h F = 25 mm. • Material properties of the plastic core: E C = 1500 MPa, ν C = 0.4, C C C = 40 MPa, Rm = 80 MPa, σdB = 60 MPa. τaB F = 364 MPa. • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 • Material property of the adhesive: τaB = 37.1 MPa. • External load: q0 = 10 N/mm. 3.2.6 Failure Analysis of a Sandwich Beam Under 5-Point Bending with Point Loads The corresponding proofs of strength must be carried out for the sandwich beam shown in Fig. 3.23. The sandwich consists of a plastic core bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the forces F0 is already multiplied by a sufficient safety factor.

3.2 Supplementary Problems

73

Fig. 3.23 Sandwich beam under 5-point bending with point loads: a Boundary conditions and external loads; b beam cross-section

Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 100 mm, h F = 25 mm. • Material properties of the plastic core: E C = 1500 MPa, ν C = 0.4, C C C = 40 MPa, Rm = 80 MPa, σdB = 60 MPa. τaB F • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 = 364 MPa. • Material property of the adhesive: τaB = 37.1 MPa. • External load: F0 = 2500 N. 3.2.7 Failure Analysis of a Sandwich Beam Under 5-Point Bending with Distributed Load The corresponding proofs of strength must be carried out for the sandwich beam shown in Fig. 3.24. The sandwich consists of a plastic core bonded to two aluminum face sheets. The thickness of the adhesive layer can be neglected in the calculation. Furthermore, it should be assumed that the value of the distributed load q0 has already been multiplied by a sufficient safety factor. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 100 mm, h F = 25 mm. • Material properties of the plastic core: E C = 1500 MPa, ν C = 0.4, C C C = 40 MPa, Rm = 80 MPa, σdB = 60 MPa. τaB F = 364 MPa. • Material properties of the face sheets: E F = 74,000 MPa, Rp0.2 • Material property of the adhesive: τaB = 37.1 MPa. • External load: q0 = 5 N/mm.

74

3 Limit Load

Fig. 3.24 Sandwich beam under 5-point bending with distributed loads: a Boundary conditions and external loads; b beam cross-section Fig. 3.25 Global instability failure of a sandwich beam clamped on both sides under compressive loading

3.2.8 Global Instability Failure of a Sandwich Beam Clamped on Both Sides Under Compressive Loading Derive the buckling force for the sandwich beam shown in Fig. 3.25 assuming thin face sheets and a soft core. Compare the result with the classic solution according to Euler for the 4th support condition. 3.2.9 Instability Failure of a Sandwich Beam Hinged on Both Ends Under Compressive Loading For the sandwich beam shown in Fig. 3.26, calculate the critical stresses for global and local instability failure assuming thin face sheets and a soft core. Furthermore, sketch the course of the global buckling stress as a function of the beam length L. Given are: • Geometric dimensions: L = 2000 mm, b = 200 mm, h C = 100 mm, h F = 5 mm. • Material properties of the isotropic core: E C = 200 MPa, ν C = 0.4. • Material properties of the face sheets: E F = 74,000 MPa.

References

75

Fig. 3.26 a Sandwich beam hinged on both sides under compression; b beam cross-section

References Allen, H.G.: Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969) Allen, H.G.: Optimum design of sandwich struts and beams. In: Plastics in Building Structures, Proceedings of a Conference Held in London, 14–16 June 1965. Pergamon Press, Oxford (1966) da Silva, L.F.M., Öchsner, A., Adams, R.: Handbook of Adhesion Technology. Springer, Cham (2018) Gough, G.S., Elam, C.F., Tipper, G.H., De Bruyne, N.A.: The stabilisation of a thin sheet by a continuous supporting medium. Aeronaut. J. 44(349), 12–43 (1940) Hoff, N.J., Mautner, S.E.: The buckling of sandwich-type panels. J. Aeronaut. Sci. 12(3), 285–297 (1945) Öchsner, A.: Elasto-Plasticity of Frame Structure Elements: Modeling and Simulation of Rods and Beams (2014) Plantema, F.J.: Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells. Wiley, New York (1966) Stamm, K., Witte, H.: Sandwichkonstruktionen. Springer, Wien (1974)

Chapter 4

Optimization

4.1 Optimal Dimensioning of Sandwich Beams 4.1.1 Tensile or Compressive Load The following derivations for the optimization of a sandwich beam under tensile or compressive loading are limited to sandwich beams with thin face sheets and a soft core. A detailed description can be found in Allen (1966). The general configuration with the geometric dimensions used can be seen in Fig. 4.1. In the case of pure tensile loading for a technical sandwich beam, there acts only tensile stress in both face sheets (see Fig. 2.15). This means that the acting force F must follow the following condition: F ≤ 2bh F Rp0.2 .

(4.1)

The case of compressive loading is much more complex, since global buckling (see Sect. 3.1.1), local wrinkling (see Sects. 3.1.4 and 3.1.5) and yield failure (see Sect. 2.5.2) can occur. According to Eq. (3.19), global buckling under compression occurs at the following critical force: FcrE , (4.2) F= FE 1 + AGcrC where FcrE =

π 2 E Iy L2

with E I y ≈

E F bh F (h c )2 2

is the Euler buckling force for homogebh 2

neous beams and the area is given according to Eq. (2.103) as A = hcC ≈ bh c . For the further derivations, this approximation is further simplified to A ≈ bh C1 (see Fig. 4.2).

1

This means that the entire shear stress acts only in the core.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_4

77

78

4 Optimization

Fig. 4.1 Sandwich beam under compression: a general configuration and b beam cross-section

Fig. 4.2 Approximation of the shear area: a A ≈ bh c and b A ≈ bh C

The failure of the face sheets occurs at the following force: F = 2bh F σcr ,

(4.3)

where for the critical stress σcr the lower value of the 0.2% initial yield stress (Rp0.2 ) or the wrinkling stress is to be used. To simplify the following derivations, the wrinkling 1/3  . stress according to Eq. (3.31) is approximated using σcr ≈ 21 × E F E C G C To optimize a sandwich structure, the weight is usually reduced to a minimum. The total mass is composed proportionally of the face sheets and the core: m = V

(4.4)

=  V +  V =  h bL +  2h bL   = b C h C L + F 2h F L , C

C

F

F

C

C

F

F

(4.5) (4.6)

or as length-related mass: mn =

  m = b C h C + F 2h F . L

(4.7)

4.1 Optimal Dimensioning of Sandwich Beams

79

If one replaces the volume-related mass in Eq. (4.7), i.e. the density, through the volume-related costs of the core and face sheets, m n can be interpreted as the length-related costs of the sandwich beam. If one uses the length-specific normalizations for the face sheet, i.e. h F,n = h F /L, and the core, i.e. h C,n = h C /L, Eqs. (4.2), (4.3) and (4.7) can be formulated as follows: FcrE 1+ h F (h C )2 × L 2 ×L F C 2 + h Lh 2 π2 EF

FcrE AG C

bL ×

1 GC  C,n 2

≥F

(4.8)

≥F

(4.9)

  F,n  h F,n h F C,n , = 2 g1 h , h ≥ h F,n h C,n bL + π2 EF GC

(4.10)

  F , g2 h F,n , h C,n = 2h F,n σcr ≥ bL

(4.11)

or Eq. (4.3)

or Eq. (4.7)    m n  C C,n m f h F,n , h C,n = = =  h + 2F h F,n . (4.12) bL b   Equation (4.12), i.e. f h F,n , h C,n , can be regarded as an objective function   which is to be minimized under the constraints g1 h F,n , h C,n and g2 h F,n , h C,n . A graphical representation of the objective function f in Fig. 4.3 shows that it is an inclined plane (O ABC) through the origin. The constraints gi according to Eqs. (4.10) and (4.11) can first be illustrated in a h C,n − h F,n coordinate system. To do this, both equations are solved for h F,n : h F,n ≥

F × π 22E F  bL F h C,n h C,n − bL ×

h F,n ≥

F . 2bLσcr

1 GC

,

(4.13) (4.14)

Figure 4.4 shows the course of the two limit curves g1 and g2 in the h C,n − h F,n plane. The gray area is the common permissible area. If one transfers the limit curves from Fig. 4.4 to the three-dimensional representation of Fig. 4.3 and projects both curves onto the inclined plane f , Fig. 4.5 results. The points lying along the curve D  E  F  on the inclined plane can be expressed as follows: From the first constraint in the formulation of Eq. (4.13) follows

80

4 Optimization

A

  Fig. 4.3 Schematic representation of the objective function f h F,n , h C,n . Adapted from Allen (1966) Fig. 4.4 Normalized face thickness as a function of the normalized core thickness based on the functions g1 and g2 according to Eqs. (4.10) and (4.11). Only areas above the two constraints are allowed

h F,n ≥

2 π2 EF h C,n h C,n × bL F



−

1 GC

.

(4.15)

Inserting this relationship into the objective function f according to Eq. (4.12) results in

4.1 Optimal Dimensioning of Sandwich Beams

81

  Fig. 4.5 Schematic representation of the objective function f h F,n , h C,n and the constraints  F,n    g1 h , h C,n and g2 h F,n , h C,n . Adapted from Allen (1966)

  f h C,n = 2F

2

 π E h C,n h C,n × 2

F

bL F

−

1 GC

 + C h C,n

(4.16)

4F

=

 π E h C,n h C,n × 2

F

bL F

−

1 GC

 + C h C,n .

(4.17)

  The schematic course of the function f h C,n is shown in Fig. 4.6. The aim of the optimization is now to determine the minimum at the point G( f G , h C,n G ) in order to minimize the weight or the costs. This minimum can be obtained using the condition2   ∂ f h C,n ! = 0. (4.18) ∂h C,n However, it should be noted that the minimum G must be located within the permissible range, see Fig. 4.5. To assess the admissibility of point G, two cases must be distinguished. In the C,n case of h C,n G ≤ h E , i.e. the point G lies on the monotonously falling curve   section D E , the optimum point has been found, since both constraints g1 and g2 are C,n fulfilled. For the second case with h C,n G > h E , the point lies on the monotonically   decreasing curve section F E . However, only the constraint g1 is fulfilled here. The 2

In the Appendix 6.2, a Python3 program is provided to automatically carry out the evaluation of Eq. (4.18).

82

4 Optimization

Fig. 4.6 Normalized objective function as a function of the normalized core thickness

next point with minimum function value of f is E, i.e. the intersection of the two constraints. Therefore, in this case, point E is the optimal point with minimum weight. These two facts are shown again in the h C,n − h F,n coordinate system in Fig. 4.7. In order to be able to distinguish between cases with regard to the point E, its coordinates in a general representation are helpful. The point of intersection E of the constraints g1 and g2 in the h C,n − h F,n coordinate system results from equating Eqs. (4.13) and (4.14), i.e. F × π 22E F F  bL = C,n C,n F 1 2bLσ cr h E h E − bL × G C  2  2 F F 4σcr C,n ⇔ h E − = + 2 F C C 2bLG 2bLG π E   2 4σcr F F C,n

⇒ h E = + 2 F + , 2bLG C π E 2bLG C

(4.19)

(4.20)

(4.21)

or as complete coordinates of point E: ⎛    ⎜

C,n F,n E h E , h E = ⎝

F 2bLG C

⎞

2 +

4σcr F F ⎟ + , ⎠. π2 EF 2bLG C 2bLσcr

(4.22)

4.1 Optimal Dimensioning of Sandwich Beams

83

Fig. 4.7 Determination of the optimum point in the h C,n − h F,n coordinate system: a point G for C,n h C,n G ≤ h E , b point E C,n for h G > h C,n E

4.1.2 Bending Load The following derivations for optimizing a sandwich beam under bending loading are again limited to sandwich beams with thin face sheets and a soft core. A detailed description can be found in Allen (1966). The general configuration with the geometric dimensions used can be taken from Fig. 4.8. It should be noted here that the external loads (e.g. single forces or distributed loads) were not drawn in since they are case-specific. In the case of a bending load, local wrinkling (see Sect. 3.1.3) can occur in the pressure-loaded face sheet and/or plastic yield failure can occur in the tensile or compressive region (see Sect. 2.5.1).

84

4 Optimization

Fig. 4.8 Sandwich beam for optimization under bending loads: a general configuration and b beam cross-section

To ensure that the face sheets do not fail, the following relationship (see Eq. 2.49) must be fulfilled: M y,max < σcr , (4.23) σx,F ≈ bh F h C where for the critical stress σcr the lower value of the 0.2% initial yield stress (Rp0.2 ) or the wrinkling stress is to be used. To simplify the following derivations, the wrinkling 1/3  . stress is approximated according to Eq. (3.31) using σcr ≈ 21 × E F E C G C The following relationship (see Eq. 2.91) must be fulfilled so that no shear failure of the core or the connecting layer between core and cover layer occurs τzx,C ≈

Q z,max < τp , bh C

(4.24)

where τp represents the shear yield stress of the core or the shear yield stress of the interlayer (see Table 3.3 for adhesives). A maximum deflection is often specified as a boundary condition. According to the partial deflection method, two differential equations (see Table 2.6) have to be solved and the expression for the deflection depends on the boundary and loading conditions. For the special case of a sandwich beam under 3-point bending with a concentrated load in the middle (see Fig. 2.20), the maximum deflection results from Eq. (2.110). If the limit value is specified as a fraction of the beam length as r1 L, the following additional condition results3 : F L3 48E I y

+

FL < r1 L , 4 AG C

(4.25)

or with E I y ≈ E F bh F (h c )2 /2 and A = bh 2c /h C ≈ bh c or with the additional simplifications h c ≈ h C , i.e. E I y ≈ E F bh F (h C )2 /2 and A ≈ bh C : 3

The force F was assumed to be in the positive z direction in order to avoid the minus sign in Eq. (2.110).

4.1 Optimal Dimensioning of Sandwich Beams

85

FL 2F L 3 + < r1 L . 48E F bh F (h C )2 4bh C G C

(4.26)

In order to optimize a sandwich structure, the weight is usually reduced to a minimum here as well. The total mass is composed proportionally of the face sheets and the core, see Eq. (4.4):   m = V = b C h C L + F 2h F L ,

(4.27)

or as length-related mass: mn =

  m = b C h C + F 2h F . L

(4.28)

If one substitutes the volume-related mass in Eq. (4.28), i.e. the density, through the volume-related costs of the core and face sheets, m n can be interpreted as the length-related costs of the sandwich beam. If one uses the length-specific normalizations h F,n = h F /L and h C,n = h C /L, Eqs. (4.23), (4.24), (4.26) and (4.28) can be formulated as follows: • Constraints gi : g1 (h C,n , h F,n ) =

M y,max C,n h h F,n bL 2

< σcr ,

Q z,max < τp , h C,n bL 2F F g3 (h C,n , h F,n ) = + < r1 . 48E F bLh F,n (h C,n )2 4bLG C h C,n

g2 (h C,n , h F,n ) =

(4.29) (4.30) (4.31)

• Objective function f :   mn m = = C h C,n + 2F h F,n . f h F,n , h C,n = bL b

(4.32)

The constraints gi according to Eqs. (4.29)–(4.31) can be illustrated again in a h C,n − h F,n coordinate system. To do this, the three equations are solved for h F,n : g1 :

h F,n g1 >

M y,max , 2 bL σcr h C,n

(4.33)

g2 :

h C,n g2 >

Q z,max , bLτp

(4.34)

g3 :

h F,n g3 >

2F 48E F bL(h C,n )2 . r1 − 4bLGFC h C,n

(4.35)

86

4 Optimization

Fig. 4.9 Normalized face thickness as a function of the normalized core thickness based on the functions g1 , g2 and g3 according to Eqs. (4.33)–(4.35). Only the areas above the two constraints g1 and g3 and to the right of g2 are allowed

Figure 4.9 shows the three limit curves g1 , g2 and g3 in the h C,n − h F,n plane. The gray area is the common permissible area. The poles of the limit curve g3 are h C,n = 0 and F h C,n = . (4.36) 4bLG Cr1 It should be noted here that for g3 the case of a sandwich beam under 3-point bending with a concentrated load in the middle was assumed4 and thus |M y,max | = F L/4 and |Q z,max | = F/2 results. The intersection points of the constraint curves (see Fig. 4.9) can be determined as follows: The point E results from the intersection of the limit curves g1 and g2 : E=



h C,n E

   F,n  h E =



  Q x,max  M y,max τp .  bLτp  Q z,max Lσcr

(4.37)

The point A results from the intersection of the limit curves g1 and g3 :     F,n h A = h C,n  A A    2 M y,max r1 F M y,max  2Lσcr  = +  F C M y,max r1 48E 4bLG  FbL 2 σ 2Lσcr + cr 48E F

4

⎞ M y,max 4bLG C

⎠ .

(4.38)

At this point it is pointed out again that Eqs. (4.26) and (4.35) are subject to this assumption and for other cases, i.e. supports and loads, have to be adjusted.

4.1 Optimal Dimensioning of Sandwich Beams

87

The point D results from the intersection of the limit curves g2 and g3 :     F,n h D = h C,n =  D D



⎞  2bL Fτp2 Q x,max   ⎠ .  Fτp bLτp  48E F Q 2 z,max r 1 − 4G C Q z,max

(4.39)

The minimum C of the objective function f along the constraints g3 for u z,max results from inserting h F,n g3 according to Eq. (4.35) into the objective function (4.32), i.e.   f h C,n = C h C,n + 2F h F,n = C h C,n + 2F

2F 48E F bL(h C,n )2 , r1 − 4bLGFC h C,n

(4.40)

and subsequent differentiation according to the variable h C,n : ∂ f (h C,n ) ! = 0 ⇒ h CC,n . ∂h C,n

(4.41)

The root can be determined numerically, for example using Newton’s method. The coordinates of the point sought are then obtained using Eq. (4.35):  C,n  C = h C  h CF,n . The minimum B of the objective function f along the constraints g1 for yielding/wrinkling results from inserting h F,n g1 according to Eq. (4.33) into the objective function (4.32), i.e.   f h C,n = C h C,n + 2F h F,n = C h C,n + 2F

M y,max , 2 bL σcr h C,n

(4.42)

and subsequent differentiation according to the variable h C,n : ∂ f (h C,n ) M y,max ! = C − 2F  2 = 0. C,n 2 C,n ∂h bL σcr h

(4.43)

The last equation can be written in terms of the quantity sought, i.e. h C,n , and the point we are looking for finally results in: ⎛    F M y,max  F,n ⎝ h = B = h C,n 2 × ×  B B C  bL 2 σcr

 ⎞   1 C M y,max  ⎠ . (4.44)  2 × F × bL 2 σ cr 

88 Fig. 4.10 Optimization ranges for sandwich under bending load: a case 1–3, b case 4–7, c case 8–9

4 Optimization

4.1 Optimal Dimensioning of Sandwich Beams

89

Depending on the position of these points relative to each other5 different cases can be distinguished, see Fig. 4.10. For cases 1–3 according to Fig. 4.10a, the second pole of the limit curve g3 (see Eq. 4.36) lies to the right of the limit curve g2 (see Eq. 4.34): F Q z,max > . (4.45) C 4bLG r1 bLτp Here, the following cases can be distinguished: Case 1: h CC,n < h C,n A applies, i.e. the minimum exists on the limit curve g3 for the deflection. C,n Case 2: h C,n B > h A applies, i.e. the minimum exists on the limit curve g1 for yielding/wrinkling. C,n C,n Case 3: h CC,n ≮ h C,n A and h B ≯ h A applies, i.e. no minimum exists on the limit curves. The optimum point is represented by A (simultaneous yielding/wrinkling and deflection limit). For cases 4–7 according to Fig. 4.10b, the second pole of the limit curve g3 (see Eq. 4.36) is to the left of the limit curve g2 (see Eq. 4.34) and furthermore both straight lines lie to the left of point A (see Eq. 4.38): Q z,max F < < h C,n A . C 4bLG r1 bLτp

(4.46)

Here, the following cases can be distinguished: C,n C,n Case 4: h C,n D < h C < h A applies, i.e. the minimum exists on the deflection limit curve g3 . C,n Case 5: h C,n B > h A applies, i.e. the minimum exists on the yielding/wrinkling limit curve g1 .

If neither case 4 nor case 5 is applicable, either A or D represents the optimal point: Case 6: Point A represents the optimum (simultaneous yielding/wrinkling and deflection limit). Case 7: Point D represents the optimum (simultaneous shear failure and deflection limit). For cases 8–9 according to Fig. 4.10c, the limit curve g2 (see Eq. 4.34) lies to the right of point A (see Eq. 4.38): Q z,max > h C,n A . bLτp

5

(4.47)

A Python3 program is provided in the Appendix 6.2, to automatically evaluate the points A–E.

90

4 Optimization

(a)

(b)

Fig. 4.11 Optimization of a sandwich beam under compressive loading: a general configuration and b beam cross-section

Here, the following cases can be distinguished: C,n Case 8: h C,n B > h E applies, i.e. the minimum exists on the yielding/wrinkling limit curve g1 . C,n Case 9: h C,n B < h E applies, i.e. no minimum exists on the limit curves. The optimum point is represented by E (simultaneous yielding/wrinkling and shear failure).

It should be noted here that cases 1 and 2 or 4 and 5 could also occur simultaneously.

4.2 Supplementary Problems 4.2.1 Knowledge Questions • Which different failure modes have to be considered when optimizing a technical sandwich beam under (a) tensile and (b) compressive loading? • Which different failure modes have to be considered when optimizing a technical sandwich beam under bending load?

4.2.2 Calculation Problems 4.2.1 Optimization of a Sandwich Beam Under Compressive Loading For the sandwich beam shown in Fig. 4.11, optimize the core and face sheet thicknesses assuming thin face sheets and a soft core. Given are: • Geometric dimensions: L = 2540 mm, b = 305 mm. • Material properties of the core: E C = 6.8948 MPa, G C = 3.4474 MPa, C = 240 kg/m3 .

4.2 Supplementary Problems

91

Fig. 4.12 Optimization of a sandwich beam under bending load due to a single force: a general configuration and b beam cross-section

• Material properties of the face sheets: E F = 68,948 MPa, F = 2691 kg/m3 , F = 247 MPa. Rp0.2 • External load: case (a): F = 2670 N, case (b) 10 × F = 26,700 N. 4.2.2 Optimization of a Sandwich Beam Under Bending Load Due to a Single Force For the sandwich beam shown in Fig. 4.12, optimize the core and face sheet thicknesses assuming thin face sheets and a soft core. Given are: • Geometric dimensions: L = 2540 mm, b = 305 mm. • Material properties of the core: E C = 6.8948 MPa, G C = 3.4474 MPa, C = 240 kg/m3 , τpC = E C /50. • Material properties of the face sheets: E F = 68,948 MPa, F = 2691 kg/m3 , F = 247 MPa. Rp0.2 • External load: F0 = 2667 N. Furthermore, r1 L with r1 = 0.003 can be assumed for the maximum deflection. 4.2.3 Optimization of a Sandwich Beam Under Bending Load Due to a Distributed Load For the sandwich beam shown in Fig. 4.13, optimize the core and face sheet thicknesses assuming thin face sheets and a soft core. Given are: • Geometric dimensions: L = 2540 mm, b = 305 mm. • Material properties of the core: E C = 6.8948 MPa, G C = 3.4474 MPa, C = 240 kg/m3 , τpC = E C /50. • Material properties of the face sheets: E F = 68,948 MPa, F = 2691 kg/m3 , F = 247 MPa. Rp0.2 • External load: q0 = 1.05 N/mm. Furthermore, r1 L with r1 = 0.003 can be assumed for the maximum deflection.

92

4 Optimization

Fig. 4.13 Optimization of a sandwich beam under bending load due to a distributed load: a general configuration and b beam cross-section

Fig. 4.14 Optimization of a homogeneous beam under bending load due to a single force: a general configuration and b beam cross-section

4.2.4 Optimization of a Homogeneous Beam Under Bending Load Due to a Single Force For the homogeneous beam shown in Fig. 4.14, optimize the dimensions b and h of the cross-section for minimum weight. The maximum normal and shear stress, the maximum deflection, and the maximum height-to-width ratio (h ≤ 20b) in order to avoid instabilities, are to be taken into account as constraints. Given are: • Geometric dimensions: L = 2540 mm. • Material properties of the beam: E = 68,948 MPa,  = 2691 kg/m3 , Rp0.2 = 247 MPa, τp = Rp0.2 /2. • External load: F0 = 2667 N. Furthermore, r1 L with r1 = 0.03 or r1 = 0.003 can be assumed for the maximum deflection.

Reference

93

Reference Allen, H, G.: Optimum Design of Sandwich Struts and Beams. In: Plastics in Building Structures, Proceedings of a Conference Held in London, 14–16 June 1965. Pergamon Press, Oxford (1966)

Chapter 5

Short Solutions to the Supplementary Problems

5.1 Chapter 2 2.6.1 Average Bending Stiffness of a Sandwich The generalization of Eq. (2.1) for different layer widths bk results in: EI y =

3 

Ek



1 k b 12



hk

3

 2  . + bk hk zck

(5.1)

k =1

Application to our three-layer problem gives: EI y =

E F bF (hF )3 E F bF hF (hC + hF )2 E C bC (hC )3 + + . 6 2 12

For the special case bC = EI y =

bF 3

and hF =

hC 4

(5.2)

the simplified formula results in:

 bF (hF )3  × 114E F + 16E C . 9

(5.3)

2.6.2 Calculation of the Shear Stress Distribution for a Circular Cross-section The starting point is again an infinitesimal beam element as in Fig. 6.2. However, now the cross-section is as in Fig. 5.1. The forces equilibrium in the x direction results in:



dσx (x) dx dA + τxz 2y dx = 0 . σx (x) dA − σx (x) + dx

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_5

(5.4)

95

96

5 Short Solutions to the Supplementary Problems

Fig. 5.1 Circular cross-section for deriving the shear stress distribution τxz

Let us consider now Eq. (6.7) from the Appendix 6.1.2: z dMy (x) Qz (x) × z dσx (x) =+ = . dx Iy dx Iy Consequently: τxz =

Qz (x) 2yIy

(5.5)

z dA .

(5.6)

√ Taking into account dA = 2ydz and y = R2 − z 2 , the shear stress distribution results in:  Qz (x)  2 R − (z  )2 . (5.7) τxz = 3Iy The maximum shear stress results for z  = 0: τxz,max =

Qz (x)R2 4Qz (x) 4Qz (x) . = = 3Iy 3π R2 3A

2.6.3 Limit Case of the Normalized Stiffness of a Sandwich

1 1 3 EI y 27 + 4+ = = . 4 4 8 8 EIy ,ref Sandwich of height ‘1’: EI y =  3 1 EIy ,ref = 12 b 23 = 2b . Ratio: 81

1 b b13 = 12 ; 12 EI y 81 = 2×12 EIy ,ref

(5.8)

(5.9)

reference configuration of height ‘ 23 ’: =

27 . 8

5.1 Chapter 2

97

2.6.4 Deflection of a Sandwich Beam with Distributed Load Using the Partial Deflection Method • General solution:  q0 x4 c1 x3 c2 x2 + + + c3 x + c4 , uz, b (x) = − 24 6 2 EIy

 1 q0 x2 q0 Lx uz, c (x) = − + c5 . AG C 2 2 1



(5.10) (5.11)

• Particular solution: Using the boundary conditions uz, b (0) = uz, s (0) = uz, b (L) = 0 and My (0) = My (L) = 0 the constants of integration result in c2 = c4 = c5 = 0, c1 = q20 L and 3

0L c3 = − q24 . This gives the particular solution:

⎛  ⎛ 

3 ⎞

⎞ 4 2 x x x x x q0 L2 ⎝ ⎝− ⎠− ⎠. uz (x) = − −2 + + C L L L 2AG L L 24EIy q0 L4

(5.12) • Maximum value of deflection: uz

L 2

=−

5q0 L4 384EIy

−

q0 L2 . 8AG C

• Ratio of the partial deflections:   L2 G C 5 uz, b L2 L = × F C F = 0.165 = 16.5%. 24 E h h uz, s 2

(5.13)

(5.14)

2.6.5 Deflection of a Sandwich Beam (Cantilever) with Distributed Load Using the Partial Deflection Method • General solution:  q0 x4 c1 x3 c2 x2 + + + c3 x + c4 , uz, b (x) = − 24 6 2 EIy 

1 q0 x2 − q0 Lx + c5 . uz, c (x) = AG C 2 1



(5.15) (5.16)

98

5 Short Solutions to the Supplementary Problems

• Particular solution: Using the boundary conditions uz, b (0) = uz, s (0) = 0, duz,dxb (0) = 0, My (L) = 0 and Qz (L) = 0 the constants of integration result in c3 = c4 = c5 = 0, c1 = q0 L and c2 = − 21 q0 L2 . This gives the particular solution: ⎛

⎛

4

3

2 ⎞

2 ⎞ 1 x 1 x x 1 x q0 L2 1 x ⎝ ⎠− ⎝− ⎠. uz (x) = − − + + C 3 L 2 L AG 2 L L 2EIy 12 L q0 L4

(5.17) • Maximum value of deflection: uz (L) = −

q0 L4 8EIy

−

q0 L2 . 2AG C

(5.18)

• Ratio of the partial deflections: L2 G C 1 uz, b (L) 1 L2 GAC = × = × D C F = 0.397 = 39.7%. uz, s (L) 4 2 E h h EIy

(5.19)

2.6.6 Deflection of a Sandwich Beam (Cantilever) with Shear Force Using the Partial Deflection Method • Internal reactions: The internal reactions for the problem according to Fig. 2.25 are shown in Fig. 5.2. • General solution:



  1 Lx2 x3 − uz, b (x) = −F0 (5.20) + c1 x + c2 , 2 6 EIy uz, c (x) = −

F0 x + c3 . AG C

(5.21)

• Particular solution: Using the boundary conditions uz, b (0) = uz, s (0) = 0 and duz,dxb (0) = 0 the constants of integration result in c1 = c2 = c3 = 0. This gives the particular solution: ⎛ 

3 ⎞

 2 x x F0 L x ⎝3 ⎠− uz (x) = − − . L L AG C L 6EIy F0 L3

(5.22)

• Maximum value of deflection: uz (L) = −

F0 L3 3EIy

−

F0 L . AG C

(5.23)

5.2 Chapter 3

99

Fig. 5.2 Internal reactions for a sandwich beam according to Fig. 2.25, i.e. cantilever with end shear force F0 at x = L in negative z-direction: a shear force and b bending moment diagram

• Ratio of the partial deflections: L2 G C uz, b (L) 2 = × F C F = 0.529 = 52.9%. uz, s (L) 3 E h h

(5.24)

5.2 Chapter 3 3.2.1 Failure Analysis of a Sandwich Beam with Distributed Load The internal reactions and the corresponding maximum values can be taken from Fig. 5.3.

100

5 Short Solutions to the Supplementary Problems

Fig. 5.3 Internal reactions for a sandwich beam with distributed load (configuration: hinged on both sides with constant distributed load q0 in negative z-direction): a shear force and b bending moment diagram

Checking whether one of the simplifying theories is applicable: 6E F hF (hc )2 = 526.77 > 100, E C (hC )3 hC = 30.0 > 4.77. hF

(5.25) (5.26)

Thus, the theory for soft cores and thin face sheets can be applied. • Maximum normal stress in the face sheets: σx,F = 8.33 MPa < RFp0.2 .

(5.27)

• Maximum shear stress in the core: C . τzx,C = 0.081 MPa < τaB

• Shear stress in the adhesive layer:

(5.28)

5.2 Chapter 3

101

τzx = 0.081 MPa < τaB .

(5.29)

• Local wrinkling of the compression face sheet: With B1 = 0.567 it follows for the critical stress: σcr = 229.904 MPa > σx,D .

(5.30)

Thus, there is no failure due to local wrinkling of the face sheet. 3.2.2 Failure Analysis of a Sandwich Beam (Cantilever) with Distributed Load The internal reactions and the corresponding maximum values can be seen in Fig. 5.4. Checking whether one of the simplifying theories is applicable: 6E F hF (hc )2 = 526.77 > 100, E C (hC )3 hC = 30.0 > 4.77. hF Fig. 5.4 Internal reactions for a sandwich beam with distributed load (configuration: cantilever beam with constant distributed load q0 in negative z-direction): a shear force and b bending moment diagram

(5.31) (5.32)

102

5 Short Solutions to the Supplementary Problems

Thus, the theory for soft cores and thin face sheets can be applied. • Maximum normal stress in the face sheets: σx,F = 33.22 MPa < RFp0.2 .

(5.33)

• Maximum shear stress in the core: C . τzx,C = 0.163 MPa < τaB

(5.34)

• Shear stress in the adhesive layer: τzx = 0.163 MPa < τaB .

(5.35)

• Local wrinkling of the compression face sheet: With B1 = 0.567 it follows for the critical stress: σcr = 229.904 MPa > σx,D .

(5.36)

Thus, there is no failure due to local wrinkling of the face sheet. 3.2.3 Failure Analysis of a Sandwich Beam Under 4-Point Bending The internal reactions and the corresponding maximum values can be seen in Fig. 5.5. Checking whether one of the simplifying theories is applicable: 6E F hF (hc )2 = 71.04 < 100, E C (hC )3 hC = 1.0 < 4.77. hF

(5.37) (5.38)

Thus, the exact theory must be applied. • Maximum normal stress in the face sheets: σx,F = 1.709 MPa < RFp0.2 .

(5.39)

5.2 Chapter 3

103

Fig. 5.5 Internal reactions for sandwich under 4-point bending: a shear force and b bending moment diagram

• Maximum normal stress in the core:  C ). σx,C = 0.192 MPa < RCm (< σdB

(5.40)

• Maximum shear stress in the core: C τzx,C = 0.119 MPa < τaB .

(5.41)

• Shear stress in the adhesive layer: τzx = 0.114 MPa < τaB .

(5.42)

• Local wrinkling of the compression face sheet: With B1 = 0.768 it follows for the critical stress: σcr = 27,555.434 MPa > σx,F . Thus, there is no failure due to local wrinkling of the face sheet.

(5.43)

104

5 Short Solutions to the Supplementary Problems

Fig. 5.6 Internal reactions for a sandwich under 3-point bending with point load F0 in negative z-direction: a shear force and b bending moment diagram

3.2.4 Failure Analysis of a Sandwich Beam Under 3-Point Bending with Point Load The internal reactions and the corresponding maximum values can be seen in Fig. 5.6. Checking whether one of the simplifying theories is applicable: 6E F hF (hc )2 = 115.625 > 100, E C (hC )3 hC = 4.0 < 4.77. hF

(5.44) (5.45)

Thus, the simplifying theory for soft cores can be applied. • Maximum normal stress in the face sheets: σx,F = 4.737 MPa < RFp0.2 .

(5.46)

5.2 Chapter 3

105

Fig. 5.7 Internal reactions for a sandwich beam under 3-point bending with negative distributed load q0 in the range 3L/8 ≤ x ≤ 5L/8: a shear force and b bending moment diagram

• Maximum shear stress in the core: C . τzx,C = 0.0987 MPa < τaB

(5.47)

• Shear stress in the adhesive layer: τzx = 0.0987 MPa < τaB .

(5.48)

• Local wrinkling of the compression face sheet: With B1 = 0.648 it follows for the critical stress: σcr = 3564.461 MPa > σx,F . Thus, there is no failure due to local wrinkling of the face sheet.

(5.49)

106

5 Short Solutions to the Supplementary Problems

3.2.5 Failure Analysis of a Sandwich Beam Under 3-Point Bending with Distributed Load The internal reactions and the corresponding maximum values can be seen in Fig. 5.7. Checking whether one of the simplifying theories is applicable: 6E F hF (hc )2 = 115.625 > 100, E C (hC )3 hC = 4.0 < 4.77. hF

(5.50) (5.51)

Thus, the simplifying theory for soft cores can be applied. • Maximum normal stress in the face sheets: σx,F = 4.145 MPa < RFp0.2 .

(5.52)

• Maximum shear stress in the core: C . τzx,C = 0.0987 MPa < τaB

(5.53)

• Shear stress in the adhesive layer: τzx = 0.0987 MPa < τaB .

(5.54)

• Local wrinkling of the compression face sheet: With B1 = 0.648 it follows for the critical stress: σcr = 3564.461 MPa > σx,F .

(5.55)

Thus, there is no failure due to local wrinkling of the face sheet. 3.2.6 Failure Analysis of a Sandwich Beam Under 5-Point Bending with Point Loads The free body diagram of the configuration and a system under consideration of symmetry is shown in Fig. 5.8. The model according to Fig. 5.8b can be used to determine the internal reactions. The corresponding curves are shown in Fig. 5.9. Checking whether one of the simplifying theories is applicable:

5.2 Chapter 3

107

Fig. 5.8 Sandwich beam under 5-point bending with concentrated loads: a free body diagram; b free body diagram considering symmetry

6E F hF (hc )2 = 115.625 > 100, E C (hC )3 hC = 4.0 < 4.77. hF

(5.56) (5.57)

Thus, the simplifying theory for soft cores can be applied. • Maximum normal stress in the face sheets: σx,F = 1.579 MPa < RFp0.2 . • Maximum shear stress in the core:

(5.58)

108

5 Short Solutions to the Supplementary Problems

Fig. 5.9 Internal reactions for a sandwich beam under 5-point bending with concentrated loads: a shear force and b bending moment diagram

C τzx,C = 0.0658 MPa < τaB .

(5.59)

• Shear stress in the adhesive layer: τzx = 0.0658 MPa < τaB .

(5.60)

• Local wrinkling of the compression face sheet: With B1 = 0.648 it follows for the critical stress: σcr = 3564.461 MPa > σx,F .

(5.61)

Thus, there is no failure due to local wrinkling of the face sheet. 3.2.7 Failure Analysis of a Sandwich Beam Under 5-Point Bending with Distributed Load The internal reactions are shown in Fig. 5.10. Checking whether one of the simplifying theories is applicable:

5.2 Chapter 3

109

Fig. 5.10 Internal reactions for a sandwich beam under 5-point bending with distributed load: a shear force and b bending moment diagram

6E F hF (hc )2 = 115.625 > 100, E C (hC )3 hC = 4.0 < 4.77. hF

(5.62) (5.63)

Thus, the simplifying theory for soft cores can be applied. • Maximum normal stress in the face sheets: σx,F = 0.177 MPa < RFp0.2 .

(5.64)

• Maximum shear stress in the core: C . τzx,C = 0.0658 MPa < τaB

• Shear stress in the adhesive layer:

(5.65)

110

5 Short Solutions to the Supplementary Problems

Fig. 5.11 Sandwich beam clamped on both sides under compressive load: a cutting free at location x; b force triangle (without considering the current sign)

τzx = 0.0658 MPa < τaB .

(5.66)

• Local wrinkling of the compression face sheet: With B1 = 0.648 it follows for the critical stress: σcr = 3564.461 MPa > σx,F .

(5.67)

Thus, there is no failure due to local wrinkling of the face sheet. 3.2.8 Global Instability Failure of a Sandwich Beam Clamped on Both Sides Under Compressive Loading The configuration for determining the internal reactions is shown in Fig. 5.11. This results in the following internal reactions: My (x) = Fuz (x) − MyR , Qz (x) = F

(5.68)

duz (x) . dx

(5.69)

The describing differential equation results in: MR d2 uz (x) F 2 2 y with λ2 = + λ u (x) = λ  z dx2 F EIy 1 −

1 AG C

.

(5.70) MR

From the general solution,1 i.e. uz (x) = c1 cos(λx) + c2 sin(λx) + Fy , results z (0) with the corresponding boundary conditions, i.e. uz (0) = 0 and dudx = 0, the inteMR

gration constants to c1 = − Fy and c2 = 0. Finally, the additional boundary condition uz (L) = 0 results in the condition for determining the buckling force: cos(λx) = 1. This results in the buckling force for the boundary conditions under consideration: 1

This can be determined with a computer algebra system (e.g. Maxima).

5.2 Chapter 3

111

Fcr =

1+

4π 2 EIy L2 1 4π 2 EIy C AG × L2

=

FcrE 1+

FKE AG C

.

(5.71)

3.2.9 Instability Failure of a Sandwich Beam Hinged on Both Ends Under Compressive Loading • Verification of conditions for soft core and thin face sheets 6E F hF (hc )2 = 122.38 ≥ 100, E C (hC )3 hC = 20.0 ≥ 4.77. hF

(5.72) (5.73)

• Global buckling Fcr =

σcr =

1+

π 2 EIy L2 1 π 2 EIy AG C × 2 L

b(hC

=

FcrE 1+

= 614,081.55 N,

FcrE AG C

Fcr = 27.91 MPa. + 2hF )

(5.74)

(5.75)

• Local wrinkling, antisymmetric k = 0.359,  = 2.362,

(5.76) (5.77)

B1 = 0.492, σcr = 706.236 MPa.

(5.78) (5.79)

Local wrinkling, symmetric k = 0.359,

(5.80)

 = 4.754, B1 = 0.651,

(5.81) (5.82)

σcr = 935.132 MPa.

(5.83)

• Course of the critical buckling stress (Fig. 5.12) Buckling stress limit for L → 0: lim σcr =

L→0

1 × A

1 L2 1 L2

×

π 2 EIy L2

+

π 2 EIy AG C

→ G C.

(5.84)

112

5 Short Solutions to the Supplementary Problems

Fig. 5.12 Course of the critical buckling stress as a function of the beam length

5.3 Chapter 4 4.2.1 Optimization of a Sandwich Beam Under Compressive Loading • Conversion of densities to consistent units Since the stiffnesses are given in MPa = N/mm2 , the densities must be converted to the consistent unit N/mm3 . The result is the following: F = 2691 kg/m3 = 2691 × 9.81 × 10−9 N/mm3 ,  = 240 kg/m = 240 × 9.81 × 10 C

3

−9

N/mm . 3

(5.85) (5.86)

Case (a): F = 2670 N: • Calculation of point E     F,n = 0.0191215; 2.923250 × 10−5 . E hC,n E ; hE

(5.87)

• Calculation of point G     F,n = 0.00820895; 1.711632 × 10−4 . ; h G hC,n G G • Optimal design C,n Since hC,n G ≤ hE holds, point G results as the optimal geometry with:

(5.88)

5.3 Chapter 4

113

Fig. 5.13 Normalized objective function depending on the normalized core thickness with the position of the minima for different loads

hCG = hC,n G × L = 20.85 mm, hFG =

hF,n G

× L = 0.43 mm.

(5.89) (5.90)

Under certain circumstances, however, one still has to respect the minimum thicknesses of sheet metal. Case (b): 10 × F = 26,700 N: • Calculation of point E     F,n = 0.0242730; 2.923250 × 10−4 . ; h E hC,n E E

(5.91)

• Calculation of point G     F,n = 0.0225247; 3.589804 × 10−4 . G hC,n G ; hG

(5.92)

• Optimal design C,n Since hC,n G ≤ hE holds, point G results as the optimal geometry with: hCG = hC,n G × L = 57.21 mm, hFG

=

hF,n G

× L = 0.91 mm.

(5.93) (5.94)

Under certain circumstances, however, one still has to respect the minimum thicknesses of sheet metal. The position of the two minima is shown in Fig. 5.13. it is possible to conclude that the position of the minimum shifts further to the right with increasing external load.

114

5 Short Solutions to the Supplementary Problems

4.2.2 Optimization of a Sandwich Beam Under Bending Load Due to a Single Force • Second pole of the function g3 in the hC,n -hF,n coordinate system hC,n =

F = 0.0832179. 4bLG C r1

(5.95)

• Intersection points in the hC,n -hF,n coordinate system Point E (g1 –g2 )     F,n h = E = hC,n  E E



  F0  τp  2bLτp  2σcr

= (0.0124827| 0.00116961) .

(5.96) (5.97)

Point A (g1 –g3 )     D,n A = hC,n = A  hA



4 Lr1



F0 2Lσcr + D 48E 16bG K  = (0.130717| 1.116902 × 10−4 .

   F0    4bLσcr hK,n

(5.98) (5.99)

• Minima of the objective function f along the constraints Point B (minimum of f along g1 ) ∂f (hC,n ) ! = 0 ⇒ hC,n B = ∂hC,n



F0 1 F × C× . 2  bLσcr

 ⎛ ⎞   F   F0  1 C F0 1  C,n  F,n  ⎠ × × B = hB  hB = ⎝ × × 2 C bLσcr  8 F bLσcr  = (0.0180942| 8.0687762 × 10−4 .

(5.100)

(5.101) (5.102)

Point C (minimum of f along g3 ) ∂f (hC,n ) ! = 0 ⇒ hC,n (Newton’s iteration). C ∂hC,n     F,n C = hC,n C  hC = (0.0967536| 5.295226 × 10

 −4

(5.103)

(5.104) .

(5.105)

5.3 Chapter 4

115

Fig. 5.14 Normalized face thickness as a function of normalized core thickness based on functions g1 , g2 and g3 for a bending beam with center load

• Optimum point It is case 1 and therefore point C is the optimal point. Optimal dimensions: hC = 245.754 mm, hF = 1.345 mm. With these geometric dimensions, the condition for thin face sheets and a soft core is also met: 6E F hF (hc )2 = 328.4 ≥ 100, E C (hC )3 hC = 182.7 ≥ 4.77. hF

(5.106) (5.107)

• Graphical representation in the hC,n -hF,n coordinate system (see Fig. 5.14) 4.2.3 Optimization of a Sandwich Beam Under Bending Load Due to a Distributed Load • Objective function and constraints   f hF,n , hC,n = C hC,n + 2F hF,n . q0 < σcr , 8hC,n hF,n b q0 g2 (hC,n , hF,n ) = < τp , 2hC,n b 10q0 q0 + < r1 . g3 (hC,n , hF,n ) = F F,n C,n 2 C 384E bh (h ) 8bG hC,n g1 (hC,n , hF,n ) =

(5.108)

(5.109) (5.110) (5.111)

116

5 Short Solutions to the Supplementary Problems

Representation of the constraints in the hC,n -hF,n coordinate system: q0 , 8bσcr hC,n q0 > , 2bτp

g1 :

hF,n g1 >

(5.112)

g2 :

hC,n g2

(5.113)

g3 :

hF,n g3 >

10q0 384E F b(hC,n )2 . 0 r1 − 8bG Cqh C,n

(5.114)

• Intersection points in the hC,n -hF,n coordinate system Point E (g1 –g2 )   q0  τp  2bτp  4σcr  = (0.0124827| 5.848035 × 10−4 .

    F,n h = E = hC,n  E E



(5.115) (5.116)

Point A (g1 –g3 )

 

q0 1 80bσcr q0  +  r1 b 384E F 8G C  8bσcr hC,n  = (0.100983| 7.228848 × 10−5 .

    F,n = A = hC,n A  hA

(5.117) (5.118)

• Minima of the objective function f along the constraints Point B (minimum of f along g1 ) ∂f (hC,n ) ! = 0 ⇒ hC,n B = ∂hC,n



2F q0 . 8bσcr C

 ⎛    F   q0 2 q 0 C,n  F,n  ⎝ B = hB  hB = 8bσcr C  8bσcr hC,n B  −4 . = (0.0127946| 5.705486 × 10

(5.119)



(5.120) (5.121)

Point C (minimum of f along g3 ) ∂f (hC,n ) ! = 0 ⇒ hC,n (Newton’s iteration). C ∂hC,n

(5.122)

5.3 Chapter 4

117

Fig. 5.15 Normalized face thickness as a function of normalized core thickness based on functions g1 , g2 and g3 for a bending beam with constant distributed load

    F,n C = hC,n C  hC = (0.0563599| 5.213426 × 10

 −4

(5.123) .

(5.124)

• Optimum point C,n C,n but hC,n ⇒ not a valid point, hC,n B > hE B < hA

hC,n D




3F0 , 4L2 τp hn

(5.141)

g3 : g4 :

F0 , 4Er1 L2 (hn )3 hn bng4 ≥ . 20 bng3 >

(5.142) (5.143)

Figure 5.16 shows the four constraints g1 , g2 , g3 and g4 in the hn − bn plane.

5.3 Chapter 4

119

Fig. 5.16 Determination of the optimum point in the hn − bn coordinate system: a point B as the intersection of g1 and g4 (r1 = 0.03). b Point C as the intersection of g3 and g4 (r1 = 0.003)

• Intersections Point A (g1 –g3 )    A = hnA  bnA =



  Rp0.2  54F0 E 2 r12 .  6Er1  L2 (Rp0.2 )3

(5.144)

Point B (g1 –g4 )  ⎞ ⎛ 1  1



 3 3    ⎜ 30F0 30F0  1 ⎟ B = hnB  bnB = ⎝ 2  ⎠. 2  L Rp0.2 20 L Rp0.2 

(5.145)

120

5 Short Solutions to the Supplementary Problems

Point C (g3 –g4 )  ⎞ ⎛ 1 

1

 1 5F 4 4  n  n  ⎜ 5F0 0  ⎟ C = hC  bC = ⎝  ⎠. Er1 L2  2 Er1 L2 

(5.146)

Note: The objective function f has no local minimum along g1 and g3 . The larger hn , the smaller f (strictly decreasing functions). According to Fig. 5.16, r1 = 0.03 results in point B and r1 = 0.003 in point C as the optimal point.

References Javanbakht, Z., Öchsner, A.: Advanced Finite Element Simulation with MSC Marc: Application of User Subroutines. Springer, Cham (2017) Merkel, M., Öchsner, A.: Eindimensionale Finite Elemente: Ein Einstieg in die Methode. Springer Vieweg, Berlin (2014) Öchsner, A., Merkel, M.: One-Dimensional Finite Elements: An Introduction to the FE Method. Springer, Berlin (2013) Öchsner, A.: Elasto-Plasticity of Frame Structure Elements: Modeling and Simulation of Rods and Beams. Springer-Verlag, Berlin (2014) Öchsner, A.: Computational Statics and Dynamics: An Introduction Based on the Finite Element Method. Springer, Singapore (2016) Öchsner, A.: A Project-Based Introduction to Computational Statics. Springer, Cham (2018)

Chapter 6

Appendix

6.1 Mechanics and Mathematics 6.1.1 Second-Order Moment of Area The second-order moments of area are generally defined as follows:  Iy =

z 2 dA,

(6.1)

y 2 dA.

(6.2)

A

 Iz = A

The formulas given in Table 6.1 can be used for simple geometric cross-sections.

6.1.2 Derivation of the Shear Stress Distribution for the Beam As a starting point for deriving the shear stress distribution for a beam, for example, a cantilever with shear force loading F0 can be considered, see Fig. 6.1. Furthermore, in the following we only consider a rectangular cross-section with side dimensions b × h. An infinitesimal beam element of this configuration is shown in Fig. 6.2. The internal reactions shown are drawn according to the common sign definition, see Öchsner (2018). Since there is no distributed load, i.e. qz = 0, the vertical force equilibrium gives Q z (x) ≈ Q z (x + dx). The next step is to replace the inner reactions, i.e. the shear force and the bending moment, by the corresponding normal and shear stresses. To do this, a small element © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner, Mechanics of Classical Sandwich Structures, https://doi.org/10.1007/978-3-031-25106-1_6

121

122

6 Appendix

Table 6.1 Second-order moments of area about the y- and z-axis Cross-section

Iy

Iz

π D4 64

=

π R4

π D4

4

64

π ba 3

πab3

4

4

a4

a4

12

12

bh 3

hb3

12

12

=

π R4 4

Fig. 6.1 General configuration of a cantilever with shear force

of height h/2 − z  is cut out in the vertical direction from the infinitesimal (in the horizontal direction) beam element, see Fig. 6.3. The horizontal forces equilibrium gives for this beam element: − σx (x) b(z)dz + σx (x + dx) b(z)dz − τzx (z)b(z)dx = 0,       dA

dA

(6.3)

6.1 Mechanics and Mathematics

123

Fig. 6.2 Infinitesimal beam element dx in the x-z plane with internal reactions Fig. 6.3 Infinitesimal beam element of dimensions dx × (h/2 − z  ). The overall configuration is shown in Fig. 6.1

or simplified after to a Taylor series expansion of the stress at (x + dx): dσx (x) dxdA − τzx (z)b(z)dx = 0, dx

(6.4)

dσx (x) dA − τzx (z)b(z) = 0. dx

(6.5)

or

Rearranged for τzx (z) = gives with

dσx (x) dA, b(z)dx

(6.6)

124

6 Appendix

d dσx (x) = dx dx



M y (x) ×z Iy

 =

Q z (x) z dM y (x) = × ×z Iy dx Iy

(6.7)

the shear stress distribution τzx (z) =

1 b(z)



Q z (x) Q z (x) × z  dA = Iy I y b(z)



z  dA =

Q z (x)H y (z) , I y b(z)

(6.8)

where H y (z) is the first-order axial moment of area for the part of the cross-section according to Fig. 6.3. Assuming a rectangular cross-section with dimensions b × h, this moment results in:   h/2 b h2   2 −z H y (z) = z dA = b z dz = 2 4 z ⎡  2 ⎤ 2 bh z ⎦. ⎣1 − = 8 h/2 



(6.9)

(6.10)

This gives finally the shear stress distribution for a beam with a rectangular crosssection (b × h) and the condition τzx = τx z to: ⎡  2 ⎤ Q z (x)h 2 z ⎣1 − ⎦ τx z (z) = 8I y h/2 ⎡  2 ⎤ 3Q z (x) z ⎦. ⎣1 − = 2A h/2

(6.11)

(6.12)

The maximum of the shear stress distribution results for z = 0: τx z,max =

Q z (x)h 2 3Q z (x) 3Q z (x) = . = 8I y 2bh 2A

(6.13)

6.1.3 Derivation of the Euler Buckling Force for Homogeneous and Isotropic Euler-Bernoulli Beams To derive the buckling formula according to Euler, consider a beam hinged on both sides under a compressive force F, see Fig. 6.4.

6.1 Mechanics and Mathematics

125

Fig. 6.4 Beam hinged on both sides under a compressive load: a initial configuration and b deformation Fig. 6.5 Cutting free at location x

The equilibrium is now established for the first time on the deformed component,1 see Fig. 6.4b. The equilibrium of moments at the location of the positive face (general position x) yields (see Fig. 6.5): 

M y = 0 ⇔ +F × u z − M y (x) = 0. ⇒ M y (x) = +F × u z .

(6.14)

(6.15)

The deformation of an Euler-Bernoulli beam is generally described by a differential equation. For beams with constant bending stiffness E I y , three classical formulations, based on the distributed load, based on the shear force distribution, or 1

For the derivation of the classical differential equations of Euler-Bernoulli beams under small deformations, the equilibrium is commonly established on the non-deformed component, see Öchsner (2020).

126

6 Appendix

based on the bending moment distribution, can be given, see Öchsner (2020). Using the formulation with the bending moment, we get in our case: d2 u z (x) = −F × u z (x), dx 2

(6.16)

d2 u z (x) + F × u z (x) = 0. dx 2

(6.17)

E Iy or transformed: E Iy

Using the abbreviation λ2 =

F E Iy

the following representation results:

d2 u z (x) + λ2 u z (x) = 0. dx 2

(6.18)

The general solution for such a differential equation is: u z (x) = c1 × cos(λx) + c2 × sin(λx).

(6.19)

Using the boundary conditions u z (0) = 0 and u z (L) = 0, the unknown constants c1 and c2 can be approached. From the first boundary condition, i.e. u z (0) = 0, one gets: 0 = c1 × cos(0) + c2 × sin(0) ⇔ c1 = 0.

(6.20)

From the second boundary condition, i.e. u z (L) = 0, one gets: 0 = c2 × sin(λ · L).

(6.21)

If the product is to be zero, one of the two factors, i.e. c2 or sin(λ × L), must be zero. c2 = 0 is a trivial solution (with c1 = c2 = 0 according to Eq. (6.19): u z = 0, thus no deformation). Therefore, sin(λ × L) = 0 must be looked at more closely. The condition sin(λ × L) = 0 means that λ × L = k × π with k = 0, 1, 2, · · · (see Fig. 6.6). The condition λ × L = 0 would mean F = 0 (see definition of λ). This results in the reasonable condition: λ × L = π ⇔ λ2 = And finally: Fcr = This results in the buckling shape:

π 2 E Imin . L2

π2 . L2

(6.22)

(6.23)

6.1 Mechanics and Mathematics

127

Fig. 6.6 Representation of the trigonometric function sin(x)

Table 6.2 Characterization of the classical cases according to Euler Case Boundary condition Buckling length 1 2 3 4

Free–fixed Hinged–hinged Hinged–fixed Fixed–fixed

L cr L cr L cr L cr

= 2L =L ≈ 0.7L = 21 L



 π·x u z (x) = c2 × sin , L

(6.24)

where the constant c2 remains undetermined. It should also be noted that the smaller value of I y and Iz is to be taken for Imin . Equation (6.23) can also be generalized according to Euler for other support cases by introducing the so-called buckling length L cr . This results in the generalized buckling force according to Euler: Fcr =

π 2 E Imin . L 2cr

(6.25)

The different formulations of the Euler buckling force are summarized in Table 6.2. The buckling stress σcr results from the buckling force using: σcr =

π 2 E Imin Fcr = . A AL 2cr

(6.26)

If one defines the so-called slenderness ratio λ by means of the geometric quantities, i.e.  L 2cr A A LK = L cr ⇔ λ2 = , (6.27) λ=  Imin Imin Imin A

128

6 Appendix

Fig. 6.7 Schematic representation of Newton’s iteration for the numerical determination of the root of a function

the buckling stress results in: σcr =

π2E . λ2

(6.28)

6.1.4 Newton’s Iteration Newton’s method is used for the numerical determination of the roots of functions, see Mitchell (1980). To derive the iteration scheme, a function f at a point x0 is expanded into a first-order Taylor series: f (x) ≈ f (x0 ) + f  (x0 ) × (x − x0 ) + · · · .

(6.29)

Assuming the root at the location x, i.e. f (x) = 0, the following iteration rule results (see also Fig. 6.7): f (xn ) . (6.30) xn+1 = xn −  f (xn ) The iteration is performed until the difference between two consecutive abscissa values is less than a given tolerance: tol = xn+1 − xn . It should also be noted that the convergence depends on the start value of the iteration. In order to select a meaningful start value, it may be necessary for the function to be represented graphically first.

6.1.5 Numerical Integration of Functions with Variables When calculating some particular problems, it is possible that difficult integrations have to be carried out. These integrals may contain general variables – such as in the integration in the limits from x = 0 to x = L – and these can complicate the calculation of the integral. By using computer algebra systems (e.g. maxima), some

6.1 Mechanics and Mathematics

129

Fig. 6.8 Approximation of the integration of a function (with variable), see Eq. (6.31)

of these integrals can be analytically evaluated. However, no solution is found for other integrals with variables and one has to assign concrete numerical values to the variables in order to be able to carry out a numerical integration. Alternatively, a simple numerical scheme should be presented here in order to be able to approximate integrals with variables numerically. For example, consider the following integral: ⎛    1 ⎞ 23 2 L x x ⎝− ⎠ dx. + (6.31) L L 0    y(x)

This integral can no longer be calculated analytically with the computer algebra system Maxima. Let us first consider the integrand, i.e. the function y(x), which is to be integrated, see Fig. 6.8a. It is a continuous function and all function values are greater than zero. The integral is represented by the area below the function graph (see the gray area in Fig. 6.8a).

130

6 Appendix

For the approximate calculation of the integral, the area under the function graph y(x) is approximated by a series of n rectangles of constant width x (see Fig. 6.8b), i.e. x =

L . n

(6.32)

The abscissa values of the center points of each rectangle result for 1 ≤ i ≤ n in: xi = (i − 1) × x +

x . 2

(6.33)

Thus, the area Ai of a single rectangle based on the ordinate value y(xi ) = yi and the interval width is: Ai = x × y(xi ) = x × yi

(6.34)

Finally, the entire approximation of the area, and thus the integral, results from the summation of the partial areas: L y(x)dx ≈ 0







n

i =1

Ai =

n

x × y(xi ).

(6.35)

i =1

A

The integration scheme according to Eqs. (6.32)–(6.35) can easily be implemented in a computer algebra system (see Fig. 6.9) or realized using a classical programming language (see Chap. 6.2). The results of the numerical approximation of the integral according to Eq. (6.31) are summarized in Table 6.3. Since no analytical solution to this problem is available, the difference between two approximations was evaluated to check the convergence of the calculation. For the given parameters one recognizes a quite good convergence. Finally, the integration scheme can also be tested on integrals for which there is an analytical solution, see Tables 6.4 and 6.5. Here, too, one can see that the integrals are approximated with sufficient accuracy from a certain interval width (the exact solution results for the constant function).

6.2 Computer Programs Factors for Local Instability Calculation The following Python program B1.py uses Newton’s method to calculate the factor B1 for Eqs. (3.28), (3.39) and (3.43) to be able to evaluate the critical crease stress σcr . Furthermore, the normalized wavelength hλ C is calculated. The program requires a

6.2 Computer Programs

131

Table 6.3 Approximation of the integration of a function (with variable), see Eq. (6.31) n A |A| = |Ai+1 − Ai | 1 2 4 8 16 32 64 128 256 512 1024 2056

0.125L 0.08118988160479111L 0.07481934508844082L 0.07382736735303455L 0.0736645275266041L 0.07363687899496056L 0.07363209336908189L 0.07363125646066039L 0.07363110932005427L 0.07363108338032945L 0.07363107880109886L 0.07363107799047096L

– 0.04381011839520889L 0.006370536516350292L 9.919777354062687 × 10−4 L 1.628398264304498 × 10−4 L 2.764853164353986 × 10−5 L 4.785625878675481 × 10−6 L 8.369084214948641 × 10−7 L 1.471406061159808 × 10−7 L 2.593972482645146 × 10−8 L 4.579230591938988 × 10−9 L 8.10627898140126 × 10−10 L

Table 6.4 Investigation of the convergence of the integration scheme according to Fig. 6.8 based on the function y(x) = 0 n A 1 2056 Exakt

5.0L 5.0L 5L

Table 6.5 Analysis of the convergence of the integration scheme according to Fig. 6.8 based on     1 1/2 2 the function y(x) = − Lx + Lx n 1 2 4 8 16 32 64 128 256 512 1024 2056 4112 Exakt

A 0.5L 0.4330127018922193L 0.4074209160795005L 0.3979911525764883L 0.3945858664122887L 0.3933689759908151L 0.3929364253352797L 0.3927830840016931L 0.3927287966897307L 0.3927095903102595L 0.392702797544798 L 0.3927003878749765L 0.3926995435169603L 5 π8L = 0.3926990816987241L

132

6 Appendix

Fig. 6.9 Numerical approximation of the integration of a function (with variable) using the computer algebra system Maxima, Makvandi (2020)

Python 3 installation,2 which provides the additional libraries sympy and numpy. Using klist, the start and end value or the associated subdivision into steps for the material and geometry parameter k can be specified, see line 8. A list of the Poisson’s numbers of the core ν C is specified in vlist, see line 9. By commenting out two of the three equations in the lines 56–58, one can distinguish between the cases in the Sects. 3.1.3, 3.1.4 and 3.1.4. The basics of the Python programming language can be found in Zhang (2015); Padmanabhan (2016); Linge and Langtangen (2016). B1.py 1 2 3

import sympy as sp import numpy as np sp.init_printing()

4 5

###PARAMETERS###

2

Thus, the program is called using python3 B1.py.

6.2 Computer Programs

133

6 7 8 9 10

precision=0.000001 klist=np.linspace(0.05,4.00,100) vlist=[0.50] x0=100

11 12 13

###EQUATIONS### v,k,x=sp.symbols(’v k x’)

14 15

expr=(k**2*x**2)/12 + (1/k)*((2/x) * (((3-v)*sp.sinh(x) * sp.cosh(x)+(1+v)*x)/((1+v)*(3-v)**2 *(sp.sinh(x))**2 -((1+v)**3)*x**2)))

16 17

expr2=(k**2*x**2)/12 + (1/k)*((2/x) * ((sp.cosh(x)-1)/((1+v)*(3-v)*sp.sinh(x) + ((1+v)**2)*x)))

18 19

expr3=(k**2*x**2)/12 + (1/k)*((2/x) * ((sp.cosh(x)+1)/(3*sp.sinh(x)-x)))

20 21

###FUNCTION###

22 23 24 25

26 27 28 29 30 31 32 33

def sympy_newton(expr,klist,vlist,precision,x0,label): """Takes a sympy expression, here with parameters v, k, and variable x, and lists of parameters k and v, and for a given x0 start value and precision calculates the minimum or maximum of function closest to x0, for each k and v. The function returns the k, v, wavelength, and y-value of the function at the minimum output is saved in a textfile, the name of which is specified with the variable label.""" v,k,x=sp.symbols(’v k x’) first_der=sp.diff(expr, x) second_der=sp.diff(first_der, x) function=sp.lambdify((x,v,k),expr,("numpy", "math", "mpmath", "sympy")) function_first_der=sp.lambdify((x,v,k),first_der,("numpy", "math", "mpmath", "sympy")) function_second_der=sp.lambdify((x,v,k),second_der,("numpy", "math", "mpmath", "sympy")) wavelength=float((2*sp.pi)/x0) with open(label+".txt", ’w’) as outfile:

34 35 36 37 38

print("{: