Advances in Mechanics of Materials for Environmental and Civil Engineering (Advanced Structured Materials, 197) [1st ed. 2023] 3031371003, 9783031371004

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Advanced Structured Materials

Francesco dell’Isola Emilio Barchiesi Francisco James León Trujillo   Editors

Advances in Mechanics of Materials for Environmental and Civil Engineering

Advanced Structured Materials Volume 197

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials are reaching their limits in many applications, and new developments are required to meet the increasing demands on engineering materials. The performance of materials can be improved by combining different materials to achieve better properties than with a single constituent, or by shaping the material or constituents into a specific structure. The interaction between material and structure can occur at different length scales, such as the micro, meso, or macro scale, and offers potential applications in very different fields. This book series addresses the fundamental relationships between materials and their structure on overall properties (e.g., mechanical, thermal, chemical, electrical, or magnetic properties, etc.). Experimental data and procedures are presented, as well as methods for modeling structures and materials using numerical and analytical approaches. In addition, the series shows how these materials engineering and design processes are implemented and how new technologies can be used to optimize materials and processes. Advanced Structured Materials is indexed in Google Scholar and Scopus.

Francesco dell’Isola · Emilio Barchiesi · Francisco James León Trujillo Editors

Advances in Mechanics of Materials for Environmental and Civil Engineering

Editors Francesco dell’Isola Department of Civil Engineering Construction-Architectural and Environmental Engineering Università degli Studi dell’Aquila L’Aquila, Italy

Emilio Barchiesi Dipartimento di Architettura, design e urbanistica Università degli Studi di Sassari Alghero, Italy

Francisco James León Trujillo Carrera de Ingeniería Civil and Instituto de Investigación Científica Universidad de Lima Lima, Peru

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-37100-4 ISBN 978-3-031-37101-1 (eBook) https://doi.org/10.1007/978-3-031-37101-1 © 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

Contents

1 Experimental Research on the Influence of Polypropylene Macrofiber Thickness in Fiber-Reinforced Concrete Mechanical Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexandre Almeida Del Savio, Darwin La Torre, Bruno Gamboa, and Jennifer Zuñiga 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Mixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Specimens and Experimental Study . . . . . . . . . . . . . . . . . . . . . 1.2.4 FRC’s Fresh State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 FRC’s Mechanical Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Artificial Intelligence Applied to the Control and Monitoring of Construction Site Personnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexandre Almeida Del Savio, Ana Luna Torres, Daniel Cárdenas-Salas, Mónica Alejandra Vergara Olivera, and Gianella Tania Urday Ibarra 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Construction Site and Data Generation Devices . . . . . . . . . . . 2.3.2 Neural Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Implementation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Distance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 3 3 4 4 6 8 12 16 19

20 20 22 22 24 24 24 27 28 28

v

vi

Contents

3 Finite Element Model for End-Plate Beam-to-Column Connections Under Bending and Axial Forces . . . . . . . . . . . . . . . . . . . . . Israel Díaz-Velazco and Alexandre Almeida Del Savio 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Geometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Contact Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Constraint and Load Conditions . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Results Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Use of Residues from the Metallurgical Industry in Construction . . . George Power 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Overview of Residues from the Metallurgical Industry with Use in Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Iron and Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Aluminium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Zinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Silicon and Ferrosilicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 On the Random Axially Functionally Graded Micropolar Timoshenko-Ehrenfest Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriele La Valle and Giovanni Falsone 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Timoshenko-Ehrenfest Beam Theory . . . . . . . . . . . . . . . . . . . 5.2.2 Additional Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Approximated Closed Form Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Simply Supported Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Direct and Inverse Uncertainty Quantification . . . . . . . . . . . . . . . . . . 5.4.1 The Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 33 33 34 34 38 40 42 43 48 49 50 50 53 53 54 54 55 57 57 58 59 59 61 62 65 65 67 68 70 71 72 74 76 76 77

Contents

5.5 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Roughly Classical Timoshenko-Ehrenfest Beam . . . . . . . . . . 5.5.2 Micropolar Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Influence of Soil-Pile-Structure Interaction on Seismic Response of Reinforced Concrete Buildings . . . . . . . . . . . . . . . . . . . . . . . Ricardo Madrid, David Zegarra, Pablo Perez, and Miguel Roncal 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Soil Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Hardening Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Hardening Soil Model with Small-Strain Stiffness (HSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Earthquake Design Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Soil Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Evaluation of Piles Response . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Storey Lateral Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Inter-Storey Drift Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Effects on Natural Period of Vibration . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

81 81 84 86 87 87 91 92 93 93 96 96 98 100 100 100 103 104 106 107 110

Chapter 1

Experimental Research on the Influence of Polypropylene Macrofiber Thickness in Fiber-Reinforced Concrete Mechanical Strengths Alexandre Almeida Del Savio, Darwin La Torre, Bruno Gamboa, and Jennifer Zuñiga

Abstract Herein, the effects of polypropylene fiber thickness on the compressive strength, split tensile strength, and flexural strength or modulus of rupture of polypropylene fiber reinforced concrete (PPFRC) were investigated. In consequence, knurled straight polypropylene fibers with three different thicknesses of 0.75, 0.90, and 1.05 mm and a constant length of 50 mm were used in conjunction with three fiber weight dosages of 4.00, 5.00, and 6.00 kg/m3 and four water-cement ratios of 0.40, 0.45, 0.50 and 0.55. In total, forty different concrete mixes were prepared with four control samples. The mechanical behavior of PPFRC as a function of polypropylene fiber thickness was determined in conjunction with its fresh-state properties. The results showed a strong indirect proportional correlation between fiber thickness and compressive strength of PPFRC for mixtures with water-cement ratios of 0.45 and 0.50. On the other hand, there is no statistically significant correlation between the split tensile strength and the modulus of rupture with fiber thickness. Keywords Polypropylene fiber · Macrofiber · Fiber-reinforced concrete · Concrete mechanical strength · Split tensile strength · Flexural strength · Modulus of rupture · Knurled straight fiber

A. Almeida Del Savio (B) · D. La Torre · B. Gamboa · J. Zuñiga Civil Engineering Department, Universidad de Lima, Lima 15023, Perú e-mail: [email protected]; [email protected] D. La Torre e-mail: [email protected] B. Gamboa e-mail: [email protected] J. Zuñiga e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_1

1

2

A. Almeida Del Savio et al.

1.1 Introduction Concrete combines cement, water, fine, and coarse aggregates [1]. This construction material is the most widely used due to its low cost, availability, and extensive applicability [1, 2]. In addition, concrete is a ceramic material characterized by its high compressive strength compared to its tensile strength [1, 3–5]. This aspect constitutes one of the essential concrete weaknesses [2–4, 6]. The issue is that the originated stresses by service loads or volumetric changes exceed their maximum stress, entailing superficial cracking. This makes concrete susceptible to deteriorating due to chemical agents carried by penetrating water through cracks, turning into low durability [2, 7, 8]. In the last decades, new types of concrete have been developed to meet the construction industry’s requests. For example, self-compacting concrete, highstrength, ultra-high-strength concrete, hybrid concrete, lightweight concrete, and fiber-reinforced concrete [4, 9, 10]. The fibers that reinforce the concrete can be of various types. There exist metallic fibers, such as steel fibers, and non-metallic fibers, such as glass, natural, carbon, and synthetic fibers [1, 4, 9]. In terms of size, they are divided into micro- and macro-fibers [11, 12]. Micro-fibers have a length smaller than 30 mm and a diameter smaller than 0.30 mm. On the other hand, macro-fibers have lengths ranging between 30 and 60 mm, and a diameter usually superior to or equal to 0.30 mm [1]. Due to the concrete’s fragile behavior, fibers enhance properties such as ductility, energy absorption capacity, cracking appearance limitation, and crack thickness [2, 4, 13]. Fiber-reinforced concrete (FRC) is commonly used in structural elements such as tunnels, slabs, beams, and columns since they reduce the number of cracks [3, 14]. Metallic and synthetic fibers are mainly used to support tensile stress in construction components more likely to crack [15]. The fiber usage also enhances concrete’s postcracking behavior [16]. Polypropylene fibers improve the modulus of rupture of the element [3]. According to [11] y [10], concrete compressive and tensile strength increase as the volume fraction increases. In addition, the authors conclude that the fiber aspect ratio, with a constant diameter, influences and improves both mechanical strengths. It is inferred then that the dosage and the fiber aspect ratio impact the mechanical and dynamical properties of the fiber-reinforced concrete [17]. As exposed, research exists about the influence of fiber slenderness modifying the fiber length [18]. However, there is no more deep understanding of the influence of the diameter variation. Thus, it is significant to conduct a study that specifies the material´s characteristics, such as a constant length and a variable thickness, with the workability and mechanical properties of fiber-reinforced concrete. Moreover, this is a novel study because of the lack of similar research, and it is relevant to any application field of fiber-reinforced concrete. This research aims to evaluate the influence of thickness (0.75, 0.90, and 1.05 mm) of polypropylene fibers, with a constant length of 50 mm, on the concrete’s fresh state properties (slump, air content, unitary weight, and fresh state temperature); and its mechanical properties (compressive strength, split tensile strength and

1 Experimental Research on the Influence of Polypropylene Macrofiber …

3

flexural strength or modulus of rupture). Next, the methodology includes materials, the mixture design, and the details of the specimens used for each test. The tests on the concrete in fresh and hardened states are presented. The results were analyzed using the Pearson hypothesis test with a significance level of 0.10 to determine the existence of a correlation between variables. Finally, the conclusions of the study are presented.

1.2 Materials and Methods 1.2.1 Materials Portland CEM V type cement with a specific gravity of 3.15 was selected for this study. Coarse sand with particles between 0 and 4.75 mm and crushed stone with particles between 4.75 and 19.05 mm, were used herein. The coarse aggregate reaches the HUSO N° 67 limits, and both aggregates meet the ASTM C 136 and NTP 400.012 standards. Table 1.1 and Fig. 1.1 expose the aggregates´ physical properties and the granulometric curves, respectively. Knurled straight polypropylene (PP) POLYSTARK PS 50 fibers with three different thicknesses of 0.75, 0.90, and 1.05 mm, a constant length of 50 mm, and three different fiber weights dosages of 4.00, 5.00, and 6.00 kg/m3 , were used herein. Table 1.1 Aggregates physical properties

Property

Fine aggregate

Coarse aggregate

Unit

Description

Coarse sand

Crushed stone

–

Top size

3/8

1

In

Nominal maximum size

#4

3/4

In

Fineness modulus

2.98

6.33

–

Sieve #200 content

4.78

–

%

Dry mass density

2.62

2.75

g/cm3

SSD density

2.64

2.77

g/cm3

Bulk mass density

2.67

2.82

g/cm3

Absorption

1.10

0.90

%

Loose unit weight

1719.67

1624.91

kg/m3

Compacted unit 1950.53 weight

1764.10

kg/m3

4

A. Almeida Del Savio et al.

Fig. 1.1 Particle size distribution curves of aggregates

The fiber relative density and the weight per fiber are 0.92 and 0.032 g, respectively. The number of fibers per kilogram is greater than 38500 fibers, approximately. The tensile strength of the PP fibers is 540 MPa for each type used. To maintain the workability of the FRC, a liquid superplasticizer (SP) MAPEFLUID N200 was added to the concrete mixtures.

1.2.2 Mixture Design A total of 40 different concrete mixtures were determined with the ACI 211.1 method as a reference to meet a specific compressive strength of 35 MPa. The fiber thickness (d), fiber weight dosage (W f ), and the water-cement ratio (w/c) vary between 0.75, 0.90, and 1.05 mm; 4.00, 5.00, and 6.00 kg/m3 ; and 0.40, 0.45, 0.50, and 0.55; respectively. A superplasticizer (SP) was added to each mixture as 1.00% of the cement weight content. A general slump of 100 mm was taken as a design parameter. Table 1.2 describes all the mixture proportions used hereunder.

1.2.3 Specimens and Experimental Study Cylindrical specimens of 100 × 200 mm were made to evaluate the compressive strength and split tensile strength at 28th days, according to ASTM C39 [19] and ASTM C496 [20]. According to the ASTM C39 standard, to calculate the compressive strength, a monotonic load must be applied to a cylinder, with the load parallel to its longitudinal axis, with a speed of 0.25 MPa/s until failure. The compression

1 Experimental Research on the Influence of Polypropylene Macrofiber …

5

Table 1.2 Mix proportions per thickness d = 0.75 mm, d = 0.90 mm, and d = 1.05 mm Mixture code*

W f (kg/ m3 )

Cement (kg/m3 )

Water (kg/ m3 )

w/c

Wf / C%

Sand (kg/m3 )

Stone (kg/ m3 )

SP (kg/ m3 )

CC_0.40

–

513.00

205.00

0.40

–

768.00

864.00

5.13

XFRC_4_ 0.40

4.00

513.00

205.00

0.40

0.78

768.00

864.00

5.13

XFRC_5_ 0.40

5.00

513.00

205.00

0.40

0.97

767.00

863.00

5.13

XFRC_6_ 0.40

6.00

513.00

205.00

0.40

1.17

765.00

861.00

5.13

CC_0.45

–

444.00

200.00

0.45

–

802.00

903.00

4.44

XFRC_4_ 0.45

4.00

444.00

200.00

0.45

0.90

802.00

903.00

4.44

XFRC_5_ 0.45

5.00

444.00

200.00

0.45

1.13

801.00

901.00

4.44

XFRC_6_ 0.45

6.00

444.00

200.00

0.45

1.35

800.00

900.00

4.44

CC_0.50

–

390.00

195.00

0.50

–

831.00

935.00

3.90

XFRC_4_ 0.50

4.00

390.00

195.00

0.50

1.03

831.00

935.00

3.90

XFRC_5_ 0.50

5.00

390.00

195.00

0.50

1.28

830.00

934.00

3.90

XFRC_6_ 0.50

6.00

390.00

195.00

0.50

1.54

828.00

932.00

3.90

CC_0.55

–

345.00

190.00

0.55

–

856.00

963.00

3.45

XFRC_4_ 0.55

4.00

345.00

190.00

0.55

1.16

856.00

963.00

3.45

XFRC_5_ 0.55

5.00

345.00

190.00

0.55

1.45

854.00

961.00

3.45

XFRC_6_ 0.55

6.00

345.00

190.00

0.55

1.74

853.00

960.00

3.45

* The

value “X” at the beginning of every mixture code is replaced with the value of its respective thickness “d” (0.75, 0.90, and 1.05 mm)

resistance ( f c ) is obtained by dividing the maximum applied load (Pmax ) by the cylinder’s cross-sectional area (A), as indicated in Eq. (1.1). fc =

Pmax A

(1.1)

According to ASTM C496, to calculate the indirect tensile strength by division test, a diametrical monotonic load must be applied to a concrete cylinder until it breaks at a rate of 1 MPa/min. To do this, the cylinder is placed horizontally and the load vertically. The test tensile strength ( f sp ) is calculated using Eq. (1.2), where l

6

A. Almeida Del Savio et al.

is the height of the cylinder, 200 mm, and d is the diameter, 100 mm. f sp =

2 × Pmax π ×l ×d

(1.2)

In addition, 150 × 150 × 500 mm beams were manufactured according to the ASTM C78 standard [21] to carry out the bending test of beams at 4 points. For this, the beam is placed on two supports spaced 450 mm apart, and the loads are applied vertically in the central thirds, separated 150 mm from each other, at a rate of 1 MPa/min until failure. The flexural tensile strength or modulus of rupture ( f r ) is calculated using Eq. (1.3), where b is the base of the beam, d is the depth, and L is the length. fr =

Pmax × L b × d2

(1.3)

Furthermore, the FRC’s fresh state properties were measured under ASTM C143 [22] for the slump test; ASTM C231 [23] for concrete’s air content; ASTM C1064 [24] for concrete’s fresh state temperature; and ASTM C138 [25] for concrete’s fresh state unit weight. Finally, the concrete mixture sequence was performed: cement, coarse aggregate, and fine aggregate were mixed. Subsequently, water and additive were poured. Finally, fibers were homogeneously added to the mixture.

1.2.4 FRC’s Fresh State Properties The results of slump test (S), air content (AC), unit weight (γ), and fresh state temperature (T) for 36 different concrete mixtures are shown in Table 1.3. As shown in the table, S results vary between 76.20 an 114.30 mm, denoting high values in W f of 4 and 5 kg/m3 . Although, in general, the slump values are acceptable for every concrete mixture, it is noticed that for W f = 6 kg/m3 , the slump values are more likely set below the design value of 100 cm. On the other hand, AC values range between 1.00 and 1.60%, whereas the FRCs with d = 1.05 mm fibers usually have the highest AC values. In addition, the lowest AC is just found on FRCs of d = 0.90 mm and a W f = 6 kg/m3 , where, in contrast, lower fiber dosages have higher air content. The results for γ tests vary between 2416.00 and 2441.00 kg/m3 . Even though a decrease in γ is expected while the W f increases, there is no evident relationship between the fiber dosage and the unit weight in plain sight. Finally, the T values range from 20 to 29 °C. Furthermore, it is noticed that mixtures with w/c of 0.40 and 0.55 have the highest T values, despite the fiber dosages W f in general.

1 Experimental Research on the Influence of Polypropylene Macrofiber …

7

Table 1.3 Fresh state properties of FRC AC (%)

γ (kg/m3 )

T (°C)

76.20

1.60

2436.00

28.50

0.90

76.20

1.20

2437.00

29.00

1.05

88.90

1.40

2430.00

20.40

5.00

0.75

88.90

1.40

2434.00

28.00

0.40

5.00

0.90

76.20

1.50

2437.00

28.00

0.40

5.00

1.05

101.60

1.50

2427.00

20.50

0.75FRC_6_0.40

0.40

6.00

0.75

82.55

1.50

2436.00

28.00

0.90FRC_6_0.40

0.40

6.00

0.90

76.20

1.20

2434.00

28.00

1.05FRC_6_0.40

0.40

6.00

1.05

82.55

1.50

2432.00

20.50

0.75FRC_4_0.45

0.45

4.00

0.75

114.30

1.50

2436.00

21.30

0.90FRC_4_0.45

0.45

4.00

0.90

76.20

1.50

2438.00

23.00

1.05FRC_4_0.45

0.45

4.00

1.05

114.30

1.50

2432.00

22.40

0.75FRC_5_0.45

0.45

5.00

0.75

76.20

1.50

2440.00

21.50

0.90FRC_5_0.45

0.45

5.00

0.90

101.60

1.50

2437.00

23.00

1.05FRC_5_0.45

0.45

5.00

1.05

101.60

1.40

2436.00

22.40

0.75FRC_6_0.45

0.45

6.00

0.75

76.20

1.50

2441.00

21.50

0.90FRC_6_0.45

0.45

6.00

0.90

76.20

1.60

2431.00

23.00

1.05FRC_6_0.45

0.45

6.00

1.05

114.30

1.30

2433.00

20.50

0.75FRC_4_0.50

0.50

4.00

0.75

95.25

1.50

2425.00

28.00

0.90FRC_4_0.50

0.50

4.00

0.90

101.60

1.50

2433.00

20.00

1.05FRC_4_0.50

0.50

4.00

1.05

95.25

1.60

2428.00

25.00

0.75FRC_5_0.50

0.50

5.00

0.75

76.20

1.50

2433.00

28.00

0.90FRC_5_0.50

0.50

5.00

0.90

101.60

1.50

2425.00

20.00

1.05FRC_5_0.50

0.50

5.00

1.05

76.20

1.60

2432.00

28.00

0.75FRC_6_0.50

0.50

6.00

0.75

82.55

1.20

2416.00

27.00

0.90FRC_6_0.50

0.50

6.00

0.90

101.60

1.50

2429.00

20.00

1.05FRC_6_0.50

0.50

6.00

1.05

82.55

1.50

2435.00

26.00

0.75FRC_4_0.55

0.55

4.00

0.75

82.55

1.40

2430.00

28.00

0.90FRC_4_0.55

0.55

4.00

0.90

76.20

1.00

2434.00

28.00

1.05FRC_4_0.55

0.55

4.00

1.05

76.20

1.20

2436.00

27.00

0.75FRC_5_0.55

0.55

5.00

0.75

76.20

1.40

2426.00

28.00

0.90FRC_5_0.55

0.55

5.00

0.90

101.60

1.00

2438.00

28.50

1.05FRC_5_0.55

0.55

5.00

1.05

88.90

1.40

2438.00

27.50

0.75FRC_6_0.55

0.55

6.00

0.75

95.25

1.20

2438.00

28.00

0.90FRC_6_0.55

0.55

6.00

0.90

82.55

1.00

2434.00

29.00

1.05FRC_6_0.55

0.55

6.00

1.05

76.20

1.30

2436.00

28.50

Mixture code

w/c

W f (kg/m3 )

d (mm)

0.75FRC_4_0.40

0.40

4.00

0.75

0.90FRC_4_0.40

0.40

4.00

1.05FRC_4_0.40

0.40

4.00

0.75FRC_5_0.40

0.40

0.90FRC_5_0.40 1.05FRC_5_0.40

S (mm)

8

A. Almeida Del Savio et al.

1.2.5 FRC’s Mechanical Strengths FRC’s mechanical results on the 28th day are shown in Table 1.4. The values given are averages of at least three rupture tests with their respective coefficient of variation (CV). A statistical analysis of all the exposed results was conducted to evaluate if the variables have no correlation and the level of it between them. A p-value test with a significance level of 0.10 and a null hypothesis that states that the variables do not correlate with them were performed regarding the former statistical evaluation. Furthermore, Pearson’s correlation parameter “r” and a null hypothesis that states that there is no correlation between variables were also performed for the latter statistical evaluation. If the absolute value of Pearson’s correlation parameter “r” is between 0 and 5%, then the correlation is non-existent. Between 5 and 20% it is very weak, between 20 and 40% it is weak, between 40 and 60% it is medium, between 60 and 80% it is considerable, between 80 and 95% it is strong, and between 95 and 100% is perfect.

1.2.5.1

Compressive Strength

The results of FRC’s compressive strength tests are shown in Table 1.4. The values range from 25.94 to 54.44 Mpa. The average f c and CV for each water-cement ratio are f c0.40 = 49.60 Mpa (CVavg = 3.60%), f c0.45 = 48.71 Mpa (CVavg = 3.29%), f c0.50 = 42.38 MPa (CVavg = 3.60%) and f c0.55 = 38.97 MPa (CVavg = 3.60%), for w/c = 0.40, 0.45, 0.50 and 0.55, respectively. The values eliminated in the statistical analysis were f c = 25.94 MPa and f c = 33.58 MPa regarding 0.75FRC_6_0.40 and 0.90FRC_6_0.55 mixtures. These eliminated results were identified with a p-value test with a significance level of 0.10 across the series. The relationship between FRC’s compressive strength and fiber thickness is shown in Fig. 1.2. It can be seen that the trends are more negatively determined for watercement ratios of 0.55, 0.50, and 0.45. Later, a linear regression was performed for the fiber weight dosages and water-cement ratios. Pearson’s correlation coefficient for every series is denoted in Table 1.5 with their particular level of correlation. In almost all the concrete mixtures with w/c = 0.45 and w/c = 0.50, there is at least a strong correlation between the fiber thickness and the compressive strength, except for samples with w/c = 0.45 and W f = 5.00 kg/m3 in which it cannot be stated that the variables have a significant correlation between them.

1.2.5.2

Split-Tensile Strength

The results of FRC’s split-tensile strength tests are shown in Table 1.4. The values range from 2.94 to 4.14 Mpa. The average f sp and CV for each water-cement ratio are f sp0.40 = 3.64 Mpa (CVavg = 6.09%), f sp0.45 = 3.71 Mpa (CVavg = 3.52%), for

1 Experimental Research on the Influence of Polypropylene Macrofiber …

9

Table 1.4 FRC’s mechanical strength results Mixture code

f c (Mpa)

f sp (Mpa)

CV (%)

f r (Mpa)

CV (%)

CC_0.40

54.44

CV (%) 4.50

3.83

7.94

–

–

0.75FRC_4_0.40

51.54

0.86

3.17

9.28

–

–

0.90FRC_4_0.40

48.16

4.29

3.37

5.13

–

–

1.05FRC_4_0.40

51.39

1.07

3.97

1.36

–

–

0.75FRC_5_0.40

53.47

3.49

3.87

3.55

–

–

0.90FRC_5_0.40

52.26

1.78

3.64

3.48

–

–

1.05FRC_5_0.40

53.28

4.17

4.08

9.90

–

–

0.75FRC_6_0.40

25.94

13.19

–

–

–

–

0.90FRC_6_0.40

51.44

0.52

2.94

5.41

–

–

1.05FRC_6_0.40

54.10

2.11

3.93

8.73

–

–

CC_0.45

53.27

0.38

3.11

3.49

6.12

5.44

0.75FRC_4_0.45

50.76

3.75

4.14

3.46

5.99

*

0.90FRC_4_0.45

47.51

5.35

2.98

2.06

5.58

*

1.05FRC_4_0.45

45.88

2.13

3.83

1.36

5.89

3.15

0.75FRC_5_0.45

51.50

2.47

3.66

7.13

6.02

*

0.90FRC_5_0.45

47.05

2.56

4.04

3.75

5.56

*

1.05FRC_5_0.45

48.49

7.16

3.89

5.18

6.18

*

0.75FRC_6_0.45

49.37

4.77

3.96

4.77

5.89

*

0.90FRC_6_0.45

47.76

2.94

3.92

3.54

5.84

*

1.05FRC_6_0.45

45.49

1.42

3.53

0.43

6.03

*

CC_0.50

47.55

9.30

–

–

–

–

0.75FRC_4_0.50

41.48

2.43

–

–

–

–

0.90FRC_4_0.50

42.75

4.75

–

–

–

–

1.05FRC_4_0.50

40.29

3.82

–

–

–

–

0.75FRC_5_0.50

42.29

3.76

–

–

–

–

0.90FRC_5_0.50

41.56

3.93

–

–

–

–

1.05FRC_5_0.50

42.63

2.78

–

–

–

–

0.75FRC_6_0.50

41.38

7.00

–

–

–

–

0.90FRC_6_0.50

42.39

4.68

–

–

–

–

1.05FRC_6_0.50

41.50

0.72

–

–

–

–

CC_0.55

41.60

5.26

–

–

–

–

0.75FRC_4_0.55

41.86

4.21

–

–

–

–

0.90FRC_4_0.55

34.33

8.64

–

–

–

–

1.05FRC_4_0.55

39.20

5.28

–

–

–

–

0.75FRC_5_0.55

40.26

2.50

–

–

–

–

0.90FRC_5_0.55

37.27

3.63

–

–

–

– (continued)

10

A. Almeida Del Savio et al.

Table 1.4 (continued) Mixture code

f c (Mpa)

f sp (Mpa)

CV (%)

f r (Mpa)

CV (%)

1.05FRC_5_0.55

39.26

2.32

–

–

–

–

0.75FRC_6_0.55

41.86

0.61

–

–

–

–

0.90FRC_6_0.55

33.58

2.54

–

–

–

–

1.05FRC_6_0.55

40.42

2.41

–

–

–

–

* Only

CV (%)

one test was performed

Fig. 1.2 Normalized FRC’s compressive strength versus fiber thickness

w/c = 0.40 and 0.45, respectively. None of the values were eliminated through the p-value test with a significance level of 0.10 across the series. The relationship between FRC’s split-tensile strength and fiber thickness is shown in Fig. 1.3. It can be noticed that the trends have a positive characteristic (except for series w/c = 0.40, W f = 6 kg/m3 and w/c = 0.40, W f = 4 kg/m3 ) where the more determined trends appear with a w/c = 0.45. Subsequently, a linear regression was performed for the fiber weight dosages and water-cement ratios. Pearson’s correlation coefficient for every series is denoted in Table 1.6 with their particular level of correlation. PPFRC_5_0.45 studied mixture depicted a perfect correlation among the variables.

4.00

5.00

6.00

0.40

0.45

0.45

0.45

0.50

0.50

0.50

0.55

0.55

0.55

PPFRC_6_0.40

PPFRC_4_0.45

PPFRC_5_0.45

PPFRC_6_0.45

PPFRC_4_0.50

PPFRC_5_0.50

PPFRC_6_0.50

PPFRC_4_0.55

PPFRC_5_0.55

PPFRC_6_0.55

CV Control variable IV Independent variable DV Dependent variable

6.00

0.40

5.00

4.00

6.00

5.00

4.00

6.00

5.00

4.00

0.40

PPFRC_4_0.40

PPFRC_5_0.40

CV2 = W f

CV1 = w/c

Type

d

d

d

d

d

d

d

d

d

d

d

d

IV

fc

fc

fc

fc

fc

fc

fc

fc

fc

fc

fc

fc

DV

Table 1.5 Pearson’s hypothesis test for the correlation between f c and d

0.351

0.584

0.265

0.901

0.881

0.914

0.941

0.710

0.847

0.225

0.637

0.597

r2 Correlation level Considerable Considerable Medium Strong Strong Perfect Perfect Strong Strong Medium Considerable Medium

r −0.773 −0.798 −0.474 −0.920 −0.843 −0.970 −0.956 −0.939 −0.949 −0.515 −0.764 −0.592

0.597

0.236

0.485

0.051

0.061

0.004

0.030

0.157

0.080

0.686

0.202

0.227

p

No rejection

No rejection

No rejection

Rejected

Rejected

Rejected

Rejected

No rejection

Rejected

No rejection

No rejection

No rejection

Null hypothesis

1 Experimental Research on the Influence of Polypropylene Macrofiber … 11

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A. Almeida Del Savio et al.

Fig. 1.3 Normalized FRC’s split-tensile strength versus fiber thickness

1.2.5.3

Flexural Strength or Modulus of Rupture

The results of FRC’s flexural strength tests for w/c = 0.45 are shown in Table 1.4. The values range from 5.56 to 5.18 Mpa. The average f r for the only water-cement ratio is f r0.45 = 5.91 Mpa. The available coefficients of variation are 5.44% and 3.15% in respect of CC_0.45 and 1.05FRC_4_0.45 concrete mixtures. None of the values were eliminated through the p-value test with a significance level of 0.10. The relationship between FRC’s modulus of rupture and fiber thickness is shown in Fig. 1.4. The figure shows that the trends are negatively determined, especially for the series w/c = 0.45, W f = 4 kg/m3 . Afterward, a linear regression was performed for the fiber weight dosages and water-cement ratios. Pearson’s correlation coefficients for the single series are denoted in Table 1.7 with their particular level of correlation. In none of the concrete mixtures requested for the flexural tests, there is a significant correlation between variables; despite the level of correlation given by the “r” Pearson’s parameter.

1.3 Conclusions This research is centered on the experimental strength assessment of polypropylenefiber reinforced concrete (PPFRC) to determine the influence of the fiber diameters on the resistance to compression and indirect traction, split test, and bending of PPFRC

0.40

0.40

0.45

0.45

0.45

PPFRC_6_0.40

PPFRC_4_0.45

PPFRC_5_0.45

PPFRC_6_0.45

CV Control variable IV Independent variable DV Dependent variable

4.00

0.40

PPFRC_4_0.40

PPFRC_5_0.40

6.00

5.00

4.00

6.00

5.00

CV2 = W f

CV1 = w/c

Type

d

d

d

d

d

d

IV

f sp

f sp

f sp

f sp

f sp

f sp

DV

Table 1.6 Pearson’s hypothesis test for the correlation between f sp and d

0.525

0.908

0.161

0.082

0.052

0.036

r2

0.725

0.953

Considerable

Perfect

Medium

Weak

−0.286 0.401

Weak

Very weak

−0.190 0.228

Correlation level

r

0.276

0.047

0.599

0.815

0.772

0.811

p

No rejection

Rejected

No rejection

No rejection

No rejection

No rejection

Null hypothesis

1 Experimental Research on the Influence of Polypropylene Macrofiber … 13

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Fig. 1.4 Normalized FRC’s modulus of rupture versus fiber thickness

beams. 50 mm long polypropylene macrofibers and three diameter variations, 0.75, 0.90, and 1.05 mm, were used as independent variables. In addition, the weight dosage of the fibers was modified to 4, 5 and 6 kg/m3 , and the water-cement ratio was 0.40, 0.45, 0.50, and 0.55. From the research results, it can be concluded that: 1. There is a strong indirect proportional correlation between fiber thickness and compressive strength of PPFRC for mixtures with water-cement ratios of 0.45 and 0.50 and weight fiber dosages of 4, 5, and 6 kg/m3 . The highest compressive strength reductions, within the experimental study, in comparison with the concrete patterns CC_0.45 and CC_0.50, respectively, were about 14.62% (mixture 1.05FRC_6_0.45) and 15.28% (mixture 1.05FRC_4_0.50). 2. There is only one statistically significant correlation between the fiber thickness and PPFRC’s split tensile strength. For concrete mixtures with w/c = 0.45 and W f = 5 kg/m3 , both variables have a perfect positive correlation. In the experimental results, PPFRC tensile strength increased by 17%, 30%, and 25% for the diameters of 0.75, 0.90, and 1.05 mm, respectively. 3. Statistically significant correlations between the fiber thickness and PPFRC’s modulus of rupture were not found. It was noticed that for a w/c = 0.45 and the three weight dosages, the modulus of rupture decreases as the fiber thickness increases, yet there was no statistically significant correlation. Although for the three analyzed cases, there is at least a weak correlation between variables.

6.00

0.45

PPFRC_6_0.45

CV Control variable IV Independent variable DV Dependent variable

5.00

0.45

4.00

0.45

PPFRC_4_0.45

PPFRC_5_0.45

CV2 = W f

CV1 = w/c

Type

d

d

d

IV

fr

fr

fr

DV

Table 1.7 Pearson’s hypothesis test for the correlation between f r and d

0.397

0.080

0.451

r2 Correlation level Considerable Weak Considerable

r −0.672 −0.283 −0.630

0.370

0.717

0.328

p

No rejection

No rejection

No rejection

Null hypothesis

1 Experimental Research on the Influence of Polypropylene Macrofiber … 15

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References 1. Yin S, Tuladhar R, Shi F, Combe M, Collister T, Sivakugan N (2015) Use of macro plastic fibres in concrete: a review. Constr Build Mater 93:180–188. https://doi.org/10.1016/j.conbui ldmat.2015.05.105 2. Sadiqul Islam GM, Gupta S (2016) Das evaluating plastic shrinkage and permeability of polypropylene fiber reinforced concrete. Int J Sustain Built Environ 5:345–354. https://doi. org/10.1016/j.ijsbe.2016.05.007 3. Lee J, Cho B, Choi E, Kim Y (2016) Experimental study of the reinforcement effect of macrotype high strength polypropylene on the flexural capacity of concrete. Constr Build Mater 126:967–975. https://doi.org/10.1016/j.conbuildmat.2016.09.017 4. Blazy J, Blazy R (2021) Polypropylene fiber reinforced concrete and its application in creating architectural forms of public spaces. Case Stud Constr Mater 14:e00549. https://doi.org/10. 1016/j.cscm.2021.e00549 5. Mezher TM, Hiswa AAMR, Shubber MS (2020) Improvement of mechanical properties of concrete by using polyproplene fibers and admixure. Defect Diffus Forum 398 DDF:167–172. https://doi.org/10.4028/www.scientific.net/DDF.398.167 6. Fallah S, Nematzadeh M (2017) Mechanical properties and durability of high-strength concrete containing macro-polymeric and polypropylene fibers with nano-silica and silica fume. Constr Build Mater 132:170–187. https://doi.org/10.1016/j.conbuildmat.2016.11.100 7. He B, Zhu X, Ren Q, Jiang Z (2020) Effects of fibers on flexural strength of ultra-highperformance concrete subjected to cryogenic attack. Constr Build Mater 265:120323. https:// doi.org/10.1016/j.conbuildmat.2020.120323 8. Ren Z, Hu P, Zou Y (2011) Experimental study on flexural toughness of steel-polypropylene fiber reinforced concrete. Adv Mater Res 261–263:156–160. https://doi.org/10.4028/www.sci entific.net/AMR.261-263.156 9. Al-Mwanes AO, Aghayari R (2020) Study of the effect adding the polypropylene fibers and chemical additives on the behavior of ultra-high-performance concrete. IOP Conf Ser Mater Sci Eng 928. https://doi.org/10.1088/1757-899X/928/2/022069 10. Altalabani D, Bzeni DKH, Linsel S (2020) Mechanical properties and load deflection relationship of polypropylene fiber reinforced self-compacting lightweight concrete. Constr Build Mater 252:119084. https://doi.org/10.1016/j.conbuildmat.2020.119084 11. Saidani M, Saraireh D, Gerges M (2016) Behaviour of different types of fibre reinforced concrete without admixture. Eng Struct 113:328–334. https://doi.org/10.1016/j.engstruct.2016. 01.041 12. Li B, Chi Y, Xu L, Shi Y, Li C (2018) Experimental investigation on the flexural behavior of steel-polypropylene hybrid fiber reinforced concrete. Constr Build Mater 191:80–94. https:// doi.org/10.1016/j.conbuildmat.2018.09.202 13. Karimipour A, Ghalehnovi M, de Brito J, Attari M (2020) The effect of polypropylene fibres on the compressive strength, impact and heat resistance of self-compacting concrete. Structures 25:72–87. https://doi.org/10.1016/j.istruc.2020.02.022 14. Yao Z, Li X, Fu C, Xue W (2019) Mechanical properties of polypropylene macrofiberreinforced concrete. Adv Mater Sci Eng 15. Xu L, Li B, Ding X, Chi Y, Li C, Huang B, Shi Y (2018) Experimental investigation on damage behavior of polypropylene fiber reinforced concrete under compression. Int J Concr Struct Mater 12. https://doi.org/10.1186/s40069-018-0302-3 16. Sadrinejad I, Madandoust R, Ranjbar MM (2018) The mechanical and durability properties of concrete containing hybrid synthetic fibers. Constr Build Mater 178:72–82. https://doi.org/10. 1016/j.conbuildmat.2018.05.145 17. Del Savio AA, la Torre Esquivel D, Carrillo J, Chi Yep EC (2022) Determination of polypropylene fiber-reinforced concrete compressive strength and elasticity modulus via ultrasonic pulse tests. Appl Sci 12:10375. https://doi.org/10.3390/app122010375

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18. Del Savio AA, la Torre Esquivel D, Cedrón JP (2022) Experimental volume incidence study and the relationship of polypropylene macrofiber slenderness to the mechanical strengths of fiber-reinforced concretes. Appl Sci 12:1–25. https://doi.org/10.3390/app12189126 19. ASTM ASTM C39/C39M—20 (2020) Standard test method for compressive strength of cylindrical concrete specimens. ASTM Inte 8 20. ASTM ASTM C496/C496M—17 (2020) Standard test method for splitting tensile strength of cylindrical concrete specimens. ASTM Int 5 21. ASTM ASTM Standard C78/C78M—18 (2018) Standard test method for flexural strength of concrete (using simple beam with third-point loading). ASTM Spec Tech Publ C78-02:1–4 22. ASTM ASTM C143/C143M—20 (2020) Standard test method for slump of hydraulic-cement concrete. ASTM Int 4 23. ASTM ASTMC231/C231M—17a (2020) Standard test method for air content of freshly mixed concrete by the pressure method. ASTM Int 10 24. ASTM ASTM C1064/C1064M—17 (2020) Standard test method for temperature of freshly mixed hydraulic-cement concrete. ASTM Int 3 25. ASTM ASTM C138/C138M—17a (2020) Standard test method for density (unit weight), yield, and air content (gravimetric) of concrete. ASTM Int 6

Chapter 2

Artificial Intelligence Applied to the Control and Monitoring of Construction Site Personnel Alexandre Almeida Del Savio, Ana Luna Torres, Daniel Cárdenas-Salas, Mónica Alejandra Vergara Olivera, and Gianella Tania Urday Ibarra

Abstract Many countries are working towards gradually lifting restrictions generated by the COVID-19 virus as post-quarantine measures. The construction industry has had to adapt to new forms of work with economic and physical restrictions. For physical restrictions, the most worrying one is the risk of contagion, as many studies have indicated that social distancing is one of the most effective biosecurity measures. In this research, a training process was executed on a neural network to ensure an adequate social distance policy in a construction environment to identify people inside construction sites. More specific training was carried out to identify people performing activities in a position other than being completely upright, as is usually the case with construction workers. The “You Only Look Once” (YOLO) version 4 algorithm was used to train 2 classes of objects, an upright person and a crouched person. More than one thousand images of a construction site were used as a data set, achieving an accuracy of 77.98%. This research presents the results and recommendations to detect the people and calculate the distance between them. Based on the distance calculation, an alert report can be generated for the work areas for the health and safety team to take preventive actions.

A. Almeida Del Savio (B) · A. Luna Torres · D. Cárdenas-Salas · M. A. Vergara Olivera · G. T. Urday Ibarra Instituto de Investigación Científica (IDIC), Universidad de Lima, Lima 15023, Peru e-mail: [email protected] A. Luna Torres e-mail: [email protected] D. Cárdenas-Salas e-mail: [email protected] M. A. Vergara Olivera e-mail: [email protected] G. T. Urday Ibarra e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_2

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Keywords Artificial intelligence · Machine learning · Computer vision techniques · Neural network models · Construction monitoring · YOLO · Social distance · COVID-19

2.1 Introduction New trends in the construction industry are oriented toward the automation of control and monitoring processes [1]. The risk of contagion by COVID-19 in construction workers is amplified due to the requirement of a physical approach for various tasks [2]. To enable preventive and corrective decision-making in these situations, it is necessary to have direct control over the eventualities that may occur during construction execution [3]. During the COVID-19 pandemic, it is utterly important to monitor adequate social distancing between people in construction areas due to possible infections of COVID19 because of the necessary rapprochement between workers. This monitoring system should work autonomously, raise alarms when construction workers are too close, support complex environments or visually noisy surroundings, work in real or nearreal time, and not require specialized cameras or equipment. The main objective of this research is to use computer vision techniques, with artificial intelligence, based on images and videos captured by static cameras for controlling and monitoring construction site personnel. Computer vision techniques allow the identification of people and estimate the distance between site personnel, using artificial intelligence models fed with images and videos from a construction project. The proposed solution uses static cameras that record images and videos as training data in an artificial intelligence system based on neural networks [4]. The trained system can detect personnel on the construction site and use artificial intelligence to calculate the distance between them, reducing the risk of contagion among workers. As a case study, information was obtained from the University Well Being Center construction site located at the Universidad de Lima campus.

2.2 Previous Works A system that could monitor the proximity of two entities on a construction site was developed by Kim et al. [5] from a deep neural network model for object recognition using the model You Look Only Once v3 (YOLOv3). More than four thousand images were used to train the model with three classes already identified (worker, excavator, and front loader) and managed to obtain Mean Average Precision (mAP) values of up to 90.82% [5]. The work carried out by Roberts and Golparvar-Fard [6] validated a visual identification method of excavator and truck activities performing excavation and material

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loading activities from videos. Using a deep learning method based on convolutional neural networks (CNNs), they achieved a mAP of 97.43% for excavators and 75.29% for trucks. On the other hand, Del Savio et al. [7] developed a computer vision technique using YOLOv4 object detection neural network to identify eight object classes in 1000 drone images and 1046 static camera images [8] of a construction site, achieving an accuracy varying between 73.56% and 93.76%. A YOLO-based object for detecting and classifying six classes (four-color helmets, person, and vest) was evaluated by Wang et al. [9] using a database with 1330 images of construction sites with different angles and distances. Of the YOLObased detectors, the best result was YOLO v5x with a mAP of 86.55%, while YOLO v3 obtained the lowest mAP with 81.99%. For helmet detection and classification in different situations, including stained images, partially occluded, or with resolution variations, a YOLO v3 model was improved by Wu et al. [10]. This model used the Densenet network as a part of its structure, making it easier to train and with higher accuracy. The results showed a difference between the YOLO v3 model and the proposed one of 2.44%. The original algorithm obtained 95.15% of mAP while the YOLO-Dense backbone, 97.59%. With the same problem, Hu et al. [11] proposed a real-time detection application for wearing helmets in construction sites based on YOLOv3, meeting the real-time requirements by guaranteeing high detection precision at a faster speed. The data set was extracted from video monitoring data of construction sites in different periods, obtaining a total of 2205 images. The proposed application resulted in a mAP of 93.5% and the original YOLOv3 method with 91.3%. By training the YOLOv3 method, Xiao and Kang [12] automatically checked the status of construction machines by comparing them with the construction schedule through cameras. The dataset contains 5000 images from construction videos, in which 4 classes were manually tagged to be trained with the YOLOv3 algorithm. The result showed a mAP of 87%. The precision for each class was 71%, 93%, 91%, and 93% for truck, excavator, loader, and backhoe, respectively. Another application of an improved YOLOv3 in image augmentation was developed by Zeng et al. [13] for real-time multi-scale equipment detection in far-field surveillance video. 90% of the data set of 2581 images, with different environmental conditions, were used as the training set. This model was tested in a real hydraulicengineering megaproject, and the result presented an average precision of 82.8%, 4.8% over the original YOLOv3 model. To conclude, a S-YOLO-C was proposed by Ren et al. [14] as a people-counting method in real-time. This people detection model was trained with two data sets with 5011 and 11,540 images. The results showed an average precision of 72% for the proposed model and 65.4% for the original YOLO model.

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2.3 Methodology This section describes the methodology used in this research to generate and collect the data from the construction site and the neural network training.

2.3.1 Construction Site and Data Generation Devices A four-story building under construction, with a total area of 11,696 m2 , located at Universidad de Lima, Lima, Peru, was adopted as the case study. The building construction started in 2020 and finished in 2022. Initially, a drone was used to obtain photographs from various angles around the area. Then, four cameras were installed at strategic points around the perimeter of the construction site at heights ranging between 12 and 35 m, as shown in Fig. 2.1, all pointing towards the construction site. These cameras captured high-resolution videos stored in the university’s servers for later processing. An incremental approach is proposed in Fig. 2.2. As a first step, a large set of images (frames) were extracted from the videos, strategically located around the construction area (Fig. 2.1). These images were manually classified using LabelImg (Fig. 2.3) and later used to train the neural network (Sect. 3.2 and Fig. 2.4). The aim of the proposed approach (Fig. 2.2) is to obtain the best Mean Average Precision (mAP) among two pieces of training performed, after which the distance between workers is calculated (Sect. 4.1).

Fig. 2.1 Location of static cameras and approximate angle of vision

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Fig. 2.2 Methodology of the system

Fig. 2.3 LabelImg, a program that was used for the manual classification process Fig. 2.4 Components of training methodology

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Table 2.1 Training results Training Training images (70%) Validation images (30%) Total images mAP (IoU = 0.5) 1

850

364

1214

77.98%

2

850

364

1214

77.57%

2.3.2 Neural Network Training This stage should be performed once per project since its purpose is to train the neuronal network with objects (classes) that need to be identified at the construction site. For this research, the classes trained were: “person” and “leaning_person.” The best Mean Average Precision (mAP) obtained using YOLOv4 for unobstructed people was 77.98%, as shown in Table 2.1 for Training 1.

2.4 Implementation and Results 2.4.1 Distance Calculation For the initial implementation, videos were recorded from static cameras showing workers interacting at close distances and in wider ranges. Images like those shown in Fig. 2.5 were obtained from the recorded videos to be analyzed. Then, the YOLOv4 model was used to obtain the two coordinates (x, y) in pixels of every detected object, being x1 and y1 the coordinates of the first object, and x2,y2

Fig. 2.5 Original image

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Fig. 2.6 Image with 12 people detected

the coordinates of the second object. The Euclidean distance between an object and every other detected object can be calculated using the following formula [15]: d=

√ (x2 − x1)2 + (y2 − y1)2

However, to identify when a person is too close to another, a custom-selected threshold distance of 175 pixels was used. In Fig. 2.6, 12 people were detected (P0– P11), with 7 were at risk (P0, P1, P2, P4, P6, P9, and P11) due to the social distance rule violation. Each detection is drawn with a bounding box around the object, using green for objects that do not violate the distance threshold and red otherwise. Figure 2.7 depicts a section of the image where the system detected three workers at an apparent close distance from each other, violating the social distance limits (Table 2.2). A problem found with this approach is that the system detects two people that are not at a real close distance. For example, at ground level, as shown in Fig. 2.8, the calculated Euclidean distance “d” is relatively small. Even though it is evident to the human brain that workers P4 and P5 are not close to one another, their Euclidean distance would be below the threshold, and thus the system would wrongly emit an alert. Another issue detected is that, if this approach were to be used in videos, false alerts could be triggered if people are in close contact during short periods (in the range of seconds), like when, for example, they are walking by each other.

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Fig. 2.7 Image with 12 people detected and a zoom-in of three close objects violating the social distance limits Table 2.2 Distance calculation results

ObjectId

Confidence %

Top location (x, y)

Distance to object (pixels)

P0

100%

1284, 838

P4: 133 P6: 91

P4

99%

1405, 781

P0: 133

P6

91%

1211, 783

P0: 91

Fig. 2.8 Euclidean distance “d” between objects when the camera is at ground level

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2.4.2 Proposed Solution A solution to the Euclidean distance calculation for an object location in 3D would need two perpendicular cameras, roughly at the same vertical level, pointing both at the same area. This setting requires exact matching between objects identified in two images, increasing the calculation complexity. It is also important to point out that due to construction site physical restrictions might not be possible to install two perpendicular cameras. Another way to calculate the distance between objects is to use two cameras separated at a certain distance. This setting also requires exact matching between objects identified in two images from two installed cameras at approximately the same location [16, 17]. In order to overcome the two-camera requirement, a simplified solution is proposed based on a single camera and a small number of markers placed at the site at known distances from the camera (Fig. 2.9). In this way, smaller bounding box areas indicate a greater distance from the camera. The distance between two points, including depth (3D, Fig. 2.10), could be calculated considering the relative position between bounding boxes. Also, construction personnel is generally crouched or bent over, so the bounding box area for this special case needs to be included in the calculation.

Fig. 2.9 The proposed solution, with the on-site physical markers

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Fig. 2.10 3D proposed distance calculation

2.5 Conclusions A YOLOv4 neural network was trained to identify people in two postures (standing and bent over), reaching a 77.98% precision for the detection process of unobstructed people. The Euclidean distances between detected objects were calculated based on one camera, a simplified solution for 3D distance calculation between detected objects, based on the bounding box area and a few physical markers whose locations are known at the construction site. The proposed solution for 3D distance calculation can also be extended to calculate distances between people and equipment, people and hazard areas, among other configurations. The next step of this research will be focused on training the neural network to identify the physical markers placed on-site and calibrate the mathematical formula for distance calculation based on proximity to markers and bounding box area. Additionally, the object detection precision and speed can be improved to allow near real-time videos to detect objects. Finally, the proposed neural network technique could be combined with genetic algorithm search-based optimization techniques [18] to find the most suitable position for the camera and the markers on-site.

References 1. Yang J, Park MW, Vela P, Golparvar-Fard M (2015) Construction performance monitoring via still images, time-lapse photos, and video streams: now, tomorrow, and the future. Adv Eng Inform 29:1–14. https://doi.org/10.1016/j.aei.2015.01.011 2. Pasco RF, Fox SJ, Johnston SC, Pignone M, Meyers LA (2020) Estimated association of construction work with risks of COVID-19 infection and hospitalization in texas. AMA Netw Open 3(10):2020. https://doi.org/10.1001/jamanetworko-pen.2020.26373

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3. Kim C, Son H, Kim C (2012) Automated construction progress measurement using a 4D building information model and 3D data. Autom Constr 31:75–82. https://doi.org/10.1016/j. autcon.2012.11.041 4. Xu B, Chen Z (2018) Multi-level fusion based 3d object detection from monocular images. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2345–2353 5. Kim D, Liu M, Lee S, Kamat V (2019) Remote proximity monitoring between mobile construction resources using camera-mounted UAVs. Autom Constr 99:168–182 6. Roberts D, Golparvar-Fard M (2019) End-to-end vision-based detection, tracking and activity analysis of earthmoving equipment filmed at ground level. Autom Constr 105:102811 7. Del Savio AA, Luna A, Cárdenas-Salas D, Vergara M, Urday G (2021) The use of artificial intelligence to identify objects in a construction site. In: International conference on artificial intelligence and energy system (ICAIES), in virtual mode, Jaipur, India, pp 1–8. https://doi. org/10.26439/ulima.prep.14933 8. Del Savio AA, Luna A, Cárdenas-Salas D, Vergara M, Urday G (2022) Dataset of manually classified images obtained from a construction site. Data Brief 42:2022. https://doi.org/10. 1016/j.dib.2022.108042 9. Wang Z, Wu Y, Yang L, Thirunavukarasu A, Evison C, Zhao Y (2021) Fast personal protective equipment detection for real construction sites using deep learning approaches. Sensors 21(10):3478. https://doi.org/10.3390/s21103478 10. Wu F, Jin G, Gao M, Zhiwei HE, Yang Y (2019) Helmet detection based on improved YOLO V3 deep model. In: 2019 IEEE 16th International conference on networking, sensing and control (ICNSC), pp 363–368. https://doi.org/10.1109/ICNSC.2019.8743246 11. Hu J, Gao X, Wu H, Gao S (2019) Detection of workers without the helmets in videos based on YOLO V3. In: 2019 12th International congress on image and signal processing, biomedical engineering and informatics (CISP-BMEI), pp 1–4 12. Xiao B, Kang SJ (2019) Deep learning detection for real-time construction machine checking. In: Proceedings of the 36th international symposium on automation and robotics in construction (ISARC) 13. Zeng T, Wang J, Cui B, Wang X, Wang D, Zhang Y (2021) The equipment detection and localization of large-scale construction jobsite by far-field construction surveillance video based on improving YOLOv3 and grey wolf optimizer improving extreme learning machine. Constr Build Mater 291:123268. https://doi.org/10.1016/j.conbuildmat.2021.12326 14. Ren P, Wang L, Fang W, Song S, Djahel S (2020) A novel squeeze YOLO-based real-time people counting approach. Int J Bio-Inspired Comput 16(2):94–101. https://doi.org/10.1504/ ijbic.2020.109674 15. Ahamad H, Zaini N, Latip MFA (2020) Person detection for social distancing and safety violation alert based on segmented ROI. In: 10th IEEE International conference on control system, computing and engineering (ICCSCE), pp 113–118. https://doi.org/10.1109/ICCSCE 50387.2020.9204934 16. Mussabayev RR, Kalimoldayev MN, Amirgaliyev YN, Tairova AT. 17. Mussabayev R (2018) Calculation of 3D coordinates of a point on the basis of a stereoscopic system. Open Eng 8(1):109–177. https://doi.org/10.1515/eng-2018-0016 18. Del Savio AA, De Andrade SAL, Vellasco PCGS, Martha LF (2005) Genetic algorithm optimization of semi-rigid steel structures. In: Proceedings of the eighth international conference on the application of artificial intelligence to civil, structural and environmental engineering. Civil-Comp Proc 82(24). https://doi.org/10.4203/ccp.82.24

Chapter 3

Finite Element Model for End-Plate Beam-to-Column Connections Under Bending and Axial Forces Israel Díaz-Velazco and Alexandre Almeida Del Savio

Abstract The assessment of steel beam-to-column connections is a fundamental piece in the design process of steel structures according to the guidelines established by standards such as Eurocode 3—Part 1.8 and ANSI/AISC 358-16. In addition, the finite element analysis is an alternative path to determine the behavior of steel beam-to-column connections in contrast to the analytical methods. Furthermore, the axial force in a semi-rigid steel structure connection is usually negligible compared to a bending force. However, there are some scenarios where the influence of axial forces cannot be ignored due to their elevated value. Therefore, a finite element model for end-plate beam-to-column connections is proposed considering the actions of bending and axial forces. The numerical simulation results were validated with experimental data, and an approximate representation of the physical phenomena was obtained. Keywords Bending and axial forces · Finite element method · Semi-rigid connections · Steel structures · Eurocode 3 · ANSI/AISC 358.16 · Steel connections · End-plate connections

3.1 Introduction Historically, steel structure buildings have been built for decades, mainly in economically and technologically developed countries [1]. Nevertheless, they had suffered significant failures, causing human and economic losses as a result of earthquakes, for instance, Northridge (1994) and Kobe (1995) [2]. This fact marked an important point in structural engineering because the structure’s performance was not fully considered. Moreover, it was the main reason for revising design procedures and standards, especially in beam-column connections [3]. I. Díaz-Velazco (B) · A. Almeida Del Savio Universidad de Lima, Avenida Javier Prado Este 4600—Santiago de Surco, Lima 15023, Perú e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_3

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In the case of Northridge (1994), several steel moment-frame buildings were found to have experienced brittle fractures of beam-to-column connections. Generally, steel moment-frame buildings damaged by the Northridge earthquake fulfill the basic intent of the building codes. They experienced little structural damage but did not collapse [4]. However, in the case of Kobe (1995), approximately 1067 older steel buildings collapsed or were damaged beyond repair, and 90 modern steel buildings were rated as having collapsed. Damage to members in modern buildings included plastification, excessive distortion, and local buckling near the column ends, and many fractures were observed in beam-to-column connections [2]. After that, based on the failures during the Northridge earthquake, the Federal Emergency Management Agency (FEMA) worked together with universities and professionals to create a team called as SAC Joint Venture to figure out the damage caused during the earthquake and learn how to reduce it in future events [5]. As a result of the research, the recommendations given by the SAC Joint Venture were taken by the American Institute for Steel Construction (AISC), which indicates testing real-scale connections and then proving their ductility [4]. In addition, the results were taken and included in the standard ANSI/AISC 358-16 [3], in which the design and detailed procedure is described for some kinds of semi-rigid connections. According to FEMA 350 [4], a finite-element model will only provide information on forces and deformations at places in the structure where a modeling element is inserted. Therefore, when nonlinear deformations are expected in a structure, the designer must anticipate the location of the plastic hinges and insert nonlinear finite elements at these locations if the model captures the inelastic behavior. Moreover, according to ANSI/AISC 358-16 [3], prequalification of the moment connection is based on two major research and testing programs. Both programs combined largescale tests with extensive finite element studies showing the finite element analysis as an alternative path to determine the behavior of steel beam-to-column connections. Typically, the axial force in a semi-rigid steel structure connection is negligible compared to the bending force [6]. Nevertheless, beam-to-column connections can be subjected to simultaneous bending moments and axial forces [7, 8]. This way, although the axial force transferred from the beam is usually low, it may attain values that significantly reduce the joint flexural capacity [9]. These conditions may be found in Vierendeel girder systems, regular sway frames under significant horizontal loading (seismic or extreme wind), irregular frames (especially with incomplete stories) under gravity/horizontal loading, and pitched-roof frames [10]. Therefore, there are some scenarios where the influence of axial compression or tension shows significantly elevated values that cannot be ignored. To deal with this issue, a finite element model for end-plate beam-to-column connections is proposed considering the actions of bending and axial forces.

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3.2 Methodology The first step in the methodology is developing a geometric steel joint model. After, a static structural analysis is conducted because the research only explores static simulations, moment versus rotation curves, and, more specifically, solid bodies that interact under axial load and bending moment. In this way, the Static Structural module of ANSYS was chosen to determine displacements, stress, deformations, and forces in structures or components caused by loads that do not induce significant inertia and damping effects [11]. Subsequently, it sets the mechanical properties in the elastic and plastic zone, the finite element meshing, the contact conditions, constraints, and the load’s application, taking into account the balance system, solution procedure, and results visualization, to represent the phenomenon in the steel connection under bending moment and axial forces. In the simulation of bolted connections, the contact between surfaces is considered because it defines the precision of the results in the finite element model (FEM). Thus, contact conditions must be modeled to obtain consistent results with the real behavior of the semi-rigid connection. In the end, the results of the proposed FEM model were validated with experimental tests carried out by Simoes et al. [12], who obtained the moment versus rotation curves from experimental tests in elastic and plastic zones for semi-rigid connections. Next, the geometric model, the mechanical properties, the meshing, the contact conditions, the constraints and load conditions, the solution procedure, and the visualization results are detailed.

3.2.1 Geometric Model The methodology to generate a finite element model for end-plate beam-to-column connection begins with developing a geometric model of the steel connection. The geometric model of the steel semi-rigid connection was developed in SpaceClaim [13], as shown in Fig. 3.1. SpaceClaim is a 3D modeling software from ANSYS [11] that enables creating, editing, or modifying geometry without the complexity associated with traditional CAD systems. In addition, SpaceClaim allows downloading bolts, washers, and nuts from manufacturers on the web.

3.2.2 Mechanical Properties The material in the elastic zone was defined to behave as an isotropic linear elastic material with Young’s modulus. In the plastic zone, steel responses with an isotropic bi-linear hardening model with a tangent modulus equal to 0.5 percent of its elastic modulus were adopted [14, 15]. Furthermore, the material mechanical properties

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a. Isometric view.

b. Front view.

Fig. 3.1 Geometric model in SpaceClaim

in the beam, column, and plate, as Young’s modulus, yield strength, and ultimate tensile strength, were obtained from experimental tests conducted by Simoes et al. [12] shown in Table 3.1. Finally, the properties for each component of the steel connection are established in ANSYS, obtaining an isotropic bi-linear hardening model as shown in Fig. 3.2 for the case of the end-plate connection. A tangent modulus equal to 1000 MPa for all components in the steel connection was also adopted. The end-plate joint mechanical properties are indicated in Table 3.2.

3.2.3 Meshing The next step is to set the finite element meshing. The generation of the finite element (FE) mesh can be controlled by two parameters related to the maximum length of a FE: the length of the elements in the region near the connection (5–10 mm) and the length of the elements in the region far from the connection (20–35 mm) [16]. These parameters were used to obtain the FE mesh displayed in Fig. 3.3. The hexahedron element is used in the beam, while the tetrahedron element is employed in column, plate, bolt, washer, and nut. The element order is quadratic because of the shape function of grade 2. Furthermore, this can be used to interpolate the displacement fields [17].

3.2.4 Contact Condition The contact conditions influence the FEM model convergence and precision. Generally, when two separate surfaces touch each other to become tangent, it’s assumed that both surfaces are in contact [18]. The solution to a contact problem involves first identifying which points on a boundary interact and second the insertion of

3 Finite Element Model for End-Plate Beam-to-Column Connections … Table 3.1 Steel mechanical properties [12]

35

Beam IPE240 Specimen

f y (MPa)

f u (MPa)

Young’s modulus (MPa)

Web_1

366.45

460.36

201483

Web_2

358.93

454.70

202836

Web_3

371.86

449.32

211839

Web_4

380.25

455.99

201544

Web_5

375.79

459.49

211308

Web_6

379.12

461.98

210128

Web_7

342.72

453.40

190443

Web_8

332.32

438.76

200127

Web average

363.4

454.3

203713

Standard deviaton

17.64

7.49

7214

Flange_1

365.83

444.52

215739

Flauge_2

331.62

448.30

213809

Flauge_3

340.75

448.77

212497

Flange_4

346.42

450.50

216924

Flange_5

355.40

458.90

221813

Flange_6

349.22

455.88

213589

Flauge_7

312.13

443.81

214147

Flange_8

319.73

435.20

213257

Flange average

340.14

448.23

215222

Standard deviaton

18.08

7.38

3017

Specimen

f y (MPa)

f u (MPa)

Young’s modulus (MPa)

Web_1

392.63

491.82

205667

Web_2

399.38

495.29

204567

Web_7

340.16

454.39

218456

Web_8

355.92

467.69

199055

Web average

372.02

477.29

206936

Standard deviaton

28.56

19.59

8206

Flange_1

344.92

410.06

232937

Flange_2

350.09

472.93

210434

Flauge_7

337.94

450.53

222665

Flange_8

338.84

461.63

217132

Flange average

342.95

448.79

220792

Standard deviaton

5.68

27.39

9516

Column HEB240

End-plate (continued)

36

I. Díaz-Velazco and A. Almeida Del Savio

Table 3.1 (continued)

Specimen

f y (MPa)

f u (MPa)

Young’s modulus (MPa)

Ep1

365.39

504.45

198936

Ep2

374.75

514.44

(not available)

Ep3

380.91

497.81

199648

Ep4

356.71

497.08

202161

Web average

369.44

503.45

200248

Standard deviation

10.62

8.05

1694.36

600 500

Stres (MPa)

400 300 200 100 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Strain

Fig. 3.2 Bilinear isotropic hardening for the end-plate connection Table 3.2 End-plate joint mechanical properties

Properties

Value

Unit

Density

7850

kg/m3

Isotropic elasticity Derive from

Young’s Modulus

Young’s modulus

2.0025E + 05

Poisson’s ratio

0.3

Bulk modulus

1.6687E + 11

Pa

Shear modulus

7.7018E + 10

Pa

Yield strength

369.44

MPa

Tangent modulus

1000

MPa

Tensile ultimate strength

503.45

MPa

MPa

Bilinear isotropic hardening

3 Finite Element Model for End-Plate Beam-to-Column Connections …

(a) Meshing - Isometric view.

37

(b) Meshing - Front view.

Fig. 3.3 Finite element mesh

appropriate conditions to prevent penetration [19]. In this way, they can transmit compressive forces in a normal direction and friction forces in a tangential direction and often do not transmit normal tensile forces [18]. Contact problems are inherently nonlinear, where an abrupt change in stiffness can occur when bodies come in and out of contact [20]. Many approaches may be used to insert the constraint for any nodal pair where the gap between surfaces in contact should be negative or zero, for instance, pure penalty, augmented Lagrange, and Lagrange multiplier [19]. The augmented Lagrange approach is shown in Eq. (3.1), in which Fnor mal is the finite contact force, knor mal is the contact stiffness, x penetration is the penetration, and λ is the extra term lambda. Fnor mal = knor mal · x penetration + λ

(3.1)

The augmented Lagrange approach was employed instead of pure penalty because this formulation is less sensitive to the contact stiffness coefficient. However, it may require additional iterations in some problems if the deformed mesh is too distorted [11]. According to ANSYS, due to the extra term λ, the augmented Lagrange method is less dependent on the contact stiffness [18]. Furthermore, it is important to define two contact surfaces called contact and target in the model for every contact region. After that, the conditions in the interaction of surfaces are established, such as the type of contact, the formulation used to solve the phenomenon (Augmented Lagrange), the detection method, and the normal stiffness factor value. Moreover, it is necessary to carry out an adequate interface treatment because if an initial gap is present and a force is applied, initial contact may not be established. One body can fly away relative to another. The convergence is unattainable when the body’s movement occurs at the analysis’s beginning; this motion cannot happen in a static structural analysis. Interface treatment is a mathematical adjustment where nodes and elements are not altered. In this manner, the position of the contact surface is interpreted as being offset by a specified amount [21]. Furthermore, adjusting to

38

I. Díaz-Velazco and A. Almeida Del Savio Face A B C D E F G H I J

Contact and target bodies Nut - Washer 1 Washer 1 - Column Column - Plate Plate - Washer 2 Washer 2 - Bolt

Contact type Frictional Frictional Frictional Frictional Frictional

Fig. 3.4 Lateral contact between connection elements: frictional

touch is chosen as an interface treatment because any initial gap is closed, and any initial penetration is ignored, creating an initial stress-free state [11]. Next, it is detailed the configurations adopted for elements in contact with the semi-rigid connection, such as plates, beam, column, bolts, washers, and nuts. The frictional contact type was used for the lateral contact region between nut-washer 1, washer 1-column, column-plate, plate-washer 2, and washer 2-bolt, as displayed in Fig. 3.4. Frictional was also used for the cylindrical contact region between the boltplate, bolt-column, bolt-washer 1, and bolt-washer 2, as shown in Fig. 3.5. Bonded contact was used for the cylindrical contact region between the nut and the bolt, as shown in Fig. 3.6. The lateral contact region among the beam-plate represents welded connection, as shown in Fig. 3.7. The contact conditions used in Ansys for frictional contact type are presented in Table 3.3. In the case of bonded contact types, they were established as programcontrolled according to the recommendations given by the Ansys manual [11].

3.2.5 Constraint and Load Conditions The next steps are the node constraints of the finite elements generated and the application of loads considering a system in equilibrium. According to Díaz et al. [16], using two different loads and support conditions is possible. In the first type, as shown in Fig. 3.8, the supports were located on the top face of the beams, and

3 Finite Element Model for End-Plate Beam-to-Column Connections … Face A B C D E F G H

Contact and target bodies Bolt - Plate Bolt - Column Bolt - Washer 1 Bolt - Washer 2

39

Contact type Frictional Frictional Frictional Frictional

Fig. 3.5 Cylindrical contact between connection elements: frictional Fig. 3.6 Cylindrical contact between nut-bolt: bonded

Contact and target bodies Nut - Bolt

Contact type Bonded

the load was applied to the bottom plate of the column and modeled as an upward pressure limit condition. In the second type, as shown in Fig. 3.9, the column was fixed and supported at both ends, and the load was applied to the beam’s free end. It was adopted as the second type configuration in this study due to being largely employed by researchers in the following studies “A parametric study of FE of bolted connections from beam to column RWS with cellular beams” [22] and “Analysis of the seismic performance of the site-connections of beam bolted to a column in modularized prefabricated steel structures” [23]. For this configuration, the constraint is fixed supports located at faces A and B (Fig. 10a). In addition, a line pressure was used over line C (Fig. 10a) because the vertical load was applied through a hydraulic actuator in the experimental test developed by Simoes et al. [12], which was located

40

I. Díaz-Velazco and A. Almeida Del Savio

Fig. 3.7 Lateral contact between beam-plate: bonded

Table 3.3 Contact conditions—frictional

Contact and target bodies Beam - Plate

Contact type Bonded

Definition Type

Frictional

Friction coefficient

0.6

Behavior

Symmetric

Advanced Formulation

Augmented Lagrange

Detection method

On Gaus point

Normal stiffness

Factor

Normal stiffness factor

0.01

Update stiffness

Each iteration

Geometric modification Interface treatment

Adjust to touch

a meter away from the face of the column flange. After that, the compressive axial force application of the hydraulic jack in the testing procedure [12] was represented by a compressive force over face D in the simulation software. Finally, the analytical model in 2D for constraints and load conditions employed in this research is displayed in Fig. 10b.

3.2.6 Solution Procedure The methods available for solving systems of equations can be divided into two types: direct and iterative solvers. Direct methods will produce the exact solution without round-off and other errors. The direct methods depend on the computer’s work with a finite word length. The errors from round-off and truncation may lead to poor or even useless results [24].

3 Finite Element Model for End-Plate Beam-to-Column Connections …

41

Fig. 3.8 First load and support condition [16]

Fig. 3.9 Second load and support condition [16]

This method operates on fully assembled system equations and demands far larger storage space. It works well for relatively small equation systems [17]. The iterative methods start with an initial approximation and apply a suitably chosen algorithm leading to successively better approximations. The main advantages are the simplicity and uniformity of the operations, which make them suitable for digital computers, and their insensitivity to the growth of round-off errors [24].

42

I. Díaz-Velazco and A. Almeida Del Savio Geometry Face A Face B Line C Face D

a.

Type Fixed Support Fixed Support Line Pressure (Vertical Load) Force (Axial)

FEM model in Ansys.

b.

Analytical model in 2D.

Fig. 3.10 Constraint and load conditions

This solver avoids full assembly of the system matrices to save significantly on storage. It works well for relatively larger systems [17]. In summary, the iterative solver was adopted because pre-conditioning plays an important role in accelerating the convergence process [17]. An adequate interface treatment reduces computational time. In addition, when the process converges, we can expect to get a good approximate solution [24].

3.2.7 Results Visualization After solving the system equation, the results can be visualized so that it’s easy to interpolate, analyze and present from a vast volume of digital data. In that regard, most processors allow the user to show 3D objects conveniently and colorfully on screen. In addition, tools are available for the user to produce iso-surfaces or vectors of variables [17]. This way, the present research results are displayed in the post-process. The moment versus rotation curve is obtained from the results of the FEM model. The rotation is calculated using Eq. (3.2), in which δ is the vertical displacement

3 Finite Element Model for End-Plate Beam-to-Column Connections …

43

Fig. 3.11 Rotation and vertical displacement

of the beam, L is equal to 1 m (distance between vertical force, F, and face of the column flange), θ is the rotation of the beam. Furthermore, the moment is estimated from Eq. (3.3), in which M is the bending moment of the connection and F is the vertical force applied a meter away from the face of the column flange. Rotation and vertical displacement are shown in Fig. 3.11. δ=θ·L

(3.2)

M =F·L

(3.3)

3.3 Results The status and pressure of the contact region between plate-column are indicated in Fig. 3.12. At the first point for axial load −345 kN and bending moment 10 kN-m (Fig. 12a), the contact region’s status between column-plate is sticking in most of the area, and pressure occurs at the same zone. At the second point for axial load, −345 kN and bending moment 20 kN-m (Fig. 12b), the contact region’s status between the column-plate continues sticking, and the pressure maintains most of the area. At the third point for axial load −345 kN and bending moment 30 kN-m, the contact region’s status between the column-plate start to separate at the top as the color yellow indicates in Fig. 12c, and the pressure begins to move towards below. At the fourth point, for axial load −345 kN and bending moment 40 kN-m (Fig. 12d), the separation between column-plate increases at the top side, and most of the sticking area moves towards the bottom side of the plate. At the fifth to eleventh points for the bending moment from 50 to 110 kN-m, the sticking area is approximately 25% of the contact region, concentrated on the bottom sides, as shown in Fig. 12e–g. The pressure’s behavior in this range of bending moments is similar and centered at the lower part, but its value is enhanced.

44

I. Díaz-Velazco and A. Almeida Del Savio

a. Fa = -345 kN-m; M = 10 kN-m

b.

Fa = -345 kN-m; M = 20 kN-m

Fig. 3.12 Status and pressure of contact region between plate-column for compressive forces

3 Finite Element Model for End-Plate Beam-to-Column Connections …

c. Fa = -345 kN-m; M = 30 kN-m

d. Fa = -345 kN-m; M = 40 kN-m

e. Fa = -345 kN-m; M = 50 kN-m

f. Fa = -345 kN-m; M = 72.4 kN-m

Fig. 3.12 (continued)

45

46

I. Díaz-Velazco and A. Almeida Del Savio

Fig. 3.12 (continued)

(g) Fa = -345 kN-m; M = 110 kN-m

The directional deformation in axis Y at the seventh point for axial load -345 kN and bending moment 72.4 kN-m can be seen in Fig. 3.13. Additionally, the equivalent stress for the column and plate at a seventh point is presented in Fig. 3.14.

Fig. 3.13 Directional deformation in axis Y: Fa = −345 kN-m; M = 72.4 kN-m

3 Finite Element Model for End-Plate Beam-to-Column Connections …

a.

Column stress - Isometric view.

c.

Plate stress - Profile view.

b.

d.

47

Column stress - Front view.

Plate stress - Front view.

Fig. 3.14 Equivalent stress: Fa = −345 kN-m; M = 72.4 kN-m

The comparison between the moment-versus-rotation curve from the experimental procedures and the FEM model is presented in Fig. 3.15. Subsequently, each component’s failure of the semi-rigid connection is analyzed at every point with their equivalent stresses. At the first point for axial load, −345 kN, and bending moment 10 kN-m, all components such as plate, beam, column, bolts, washers, and nuts are below yield strength. All components continue below yield strength at the second to third points for axial load −345 kN and bending moments of 20 and 30 kN-m, respectively. At the fourth point for axial load -345 kN and bending moment of 40 kN-m, the column stress is over yield strength and under ultimate tensile strength. At the fifth point for axial load −345 kN and bending moment of 50 kN-m, column, beam, and washer stress are over yield and below ultimate tensile strength, respectively. At the sixth to eighth points for axial load −345 kN and bending moments of 60, 70 and 80 kN-m, respectively, washer stress is over ultimate tensile strength.

48

I. Díaz-Velazco and A. Almeida Del Savio

Fig. 3.15 Moment versus rotation curves—FEM and experimental models

Likewise, column, beam, and plate stress are over-yield and below ultimate tensile strength. At the ninth and tenth points for axial load −345 kN and bending moments of 88.1 and 100 kN-m, respectively, washer stress is greater than ultimate tensile strength. Moreover, column, beam, plate, and bolt stress are higher than yield strength and lower than ultimate tensile strength. The eleventh point for axial load, −345 kN, and bending moment 110 kN-m, washer, bolt, and column stress is over ultimate tensile strength. In addition, the beam and plate continue over yield strength and below ultimate tensile strength. It’s important to mention that the experimental curves have some uncertainties due to strain measurement for bolts in tension, where holes were done in the bolt head to place strain gauges to determine the axial deformation [12].

3.4 Discussion Figure 3.12 shows the pressure and the contact state between plate-column for different bending moment levels, keeping the compression force constant. The pressure moves towards the bottom side of the plate. The sticking state is concentrated in the same area. The upper part of the plate is separated from the column as the bending moment value increases in the semi-rigid connection. The steel connection model can represent the division between the plate column at the top generated by the bending moments. Simultaneously, compression force and bending moments produce pressure at the lower part of the steel connection. The joint rotation is evaluated from the vertical displacement, as appreciated in Fig. 3.11. For example, rotation at point 7 (Fig. 3.15) is calculated in mrad from the maximum directional deformation in the Y-axis, as shown in Fig. 3.13. It is worth

3 Finite Element Model for End-Plate Beam-to-Column Connections …

49

mentioning that this procedure was used to calculate the rotation for the studied bending moments, which is shown in Fig. 3.15. At the seventh point, the column’s biggest equivalent stress develops below the flange’s lower holes, as seen in Fig. 14a. In addition, elevated stress distribution in the column’s web can be observed in Fig. 14b. To reduce the stress values in the column’s web, it would be desirable to place a plate parallel to this area, as recommended by international standards for steel connections. Likewise, stress reduction in the column flange can be achieved by positioning a perpendicular stiffener to the web and flange below the lower holes, which international standards advise. Therefore, the finite element model can represent the critical points of equivalent stress, where the standards suggest using stiffeners or plates to enhance the connection’s performance. The highest equivalent stress of the plate, at the seventh point, occurs at the height of the upper holes, as shown in Fig. 14c, d. The stress distribution is compatible with the bending moment increases. The plate will begin to deform around a stress fiber, whose position is located in the higher holes. Finally, from the first to the seventh points (Fig. 3.15), the moment versus rotation curve developed from the FEM model presents higher rotation values than the experimental one. On the other hand, the curve has lower rotation values than experimental results from the eighth to the eleventh points for the same bending moment values.

3.5 Conclusion A step-by-step methodology was presented to build a finite element model for semirigid connections, including contact elements modeling. This methodology was used to propose a finite element model for end-plate beam-to-column connections, considering the actions of bending moments and axial compressive forces. The results obtained from the analysis were compared with experimental data developed by Simoes et al. [12], showing a rotation difference of 43–55% from the first to the third point, 73–116% from the fourth to the seventh points, and 26–66% from to the eighth to the ninth points. The proposed FEM model can identify critical points of equivalent stress, where it is normally suggested to employ stiffeners or plates to improve semi-rigid joint performance. The end-plate for the beam-to-column connection has the highest stress distribution at the height of the upper holes. Furthermore, the plate’s bottom applies more pressure over the column as the bending moment increases. Therefore, the model can be used to comprehend the plate’s behavior for different bending moment values and identify critical stress areas.

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I. Díaz-Velazco and A. Almeida Del Savio

3.6 Future research Further studies should be conducted considering the welded connections between the plate-beam to reduce the stress distribution due to the sudden geometry change in the beam’s area near the connection. Additionally, the proposed FEM model can be tested to: (1) Simulate the prequalified connections from ANSI-AISC-358-16 standards considering the influence of meshing, contact conditions, constraints, and load conditions; (2) Simulate steel connections under bending and axial force and consider the local buckling in the nearby region to the semi-rigid connection; (3) Perform a study similar to the one presented in this paper but considering the actions of bending moments and tension forces; (4) Compare forces at the bolts between the FEM model and experimental data to better understand the semi-rigid steel connection behavior.

References 1. Crisafulli FJ (2018) Diseño simorresistente de construcciones de acero. Asociación Latinoamericana del Acero 2. FEMA (2000) FEMA 355E: state of the art report on past performance of steel moment-frame buildings in earthquakes. SAC Joint Venture, CA, USA 3. ANSI/AISC (2016) ANSI/AISC 358-16: Prequalified connections for special and intermediate steel moment frames for seismic applications. AISC, Chicago, USA 4. FEMA (2000) FEMA 350: recommended seismic design criteria for the new steel momentframe buildings. SAC Joint Venture, CA, USA 5. Panillo G, Chacón M, Riera H (2018) Desarrollo y programación de conexiones sismorresistentes tipo BFP y RBS conforme a la normative ANSI/AISC 358–16. Revista Gaceta Tércnica 19(2):51–68 6. Del Savio AA, Nethercot DA, Vellasco PCGS, De Lima LRO, Andrade SAL, Martha LF (2010) An assessment of beam-to-column endplate and baseplate joints including: the axial-moment interaction. Adv Steel Constr 6(1):548–566. https://doi.org/10.18057/IJASC.2010.6.1.2 7. Del Savio AA, Nethercot DA, Vellasco PCGS, Andrade SAL, Martha LF (2009) Generalised component-based model for beam-to-column connections including axial versus moment interaction. J Constr Steel Res 65(8–9):1876–1895. https://doi.org/10.1016/j.jcsr.2009.02.011 8. Del Savio AA, Nethercot DA, Vellasco PCGS, Andrade SAL, Martha LF (2007) A component method model for semi-rigid end-plate beam-to-column joints including the axial versus bending moment interaction. In: Proceedings of the 5th international conference on advances in steel structures, ICASS 2007, vol 3, pp 481–486 9. Del Savio AA, De Andrade SAL, Martha LF, Vellasco PCG, De Lima LRO (2006) Semi-rigid portal frame finite element modelling including the axial versus bending moment interaction in the structural joints. Proc Int Colloq Stab Duct Steel Struct SDSS 2006:389–396 10. Del Savio AA, Nethercot DA, Vellasco PCGS, Andrade SAL, Martha LF (2007) Developments in semi-rigid joint moment versus rotation curves to incorporate the axial versus moment interaction. In: Proceedings of the 3rd international conference on steel and composite structures, ICSCS07—Steel and composite structures, pp 381–387

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11. ANSYS (2013) ANSYS mechanical user guide. ANSYS, Pensilvania, USA 12. Simões da Silva L, De Lima LRO, Vellasco PCGS, De Andrade SA (2004) Behaviour of flush endplate beam-to-column joints under bending and axial force. Steel Compos Struct 4(2):77–94 13. ANSYS (2021) Discovery space claim. ANSYS, Pensilvania, USA 14. Movaghatti S, Rahai A (2010) Numerical analysis of steel HSS beam-column retrofitted with CFRP. In: The 2nd international conference on composites: characterization, fabrication and application 15. Tsavdaridis KD, D’Mello C (2012) Optimisation of novel elliptically-based web opening of perforated steel beams. J Constr Steel Res 16. Díaz C, Victoria M, Querin OM et al (2018) FE model of three-dimensional steel beam-tocolumn bolted extended end-plate joint. Int J Steel Struct 18(3):843–867 17. Liu GR, Quek SS (2014) The finite element method: a practical course, 2nd edn. ButterworthHeinemann, Oxford, UK 18. ANSYS (2013) ANSYS mechanical introduction to structural nonlinearities. Lecture 3: Introduction to contact. ANSYS, Pensilvania, USA 19. Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 2: Solid mechanics. Butterworth-Heinemann, Oxford, UK 20. ANSYS (2013) ANSYS mechanical introduction to structural nonlinearities. Lecture 1: Overwiev. ANSYS, Pensilvania, USA 21. ANSYS (2013) ANSYS mechanical advanced connections. Lecture 2: Interface Treatments. ANSYS, Pensilvania, USA 22. Tsavdaridis KD, Papadopoulos T (2016) A FE parametric study of RWS beam-to-column bolted connections with cellular beams. J Constr Steel Res 116:92–113 23. Liu X, Cui X, Yang Z, Zhan X (2017) Analysis of the seismic performance of site-bolted beam to column connections in modularized prefabricated steel structures. Adv Mater Sci Eng 1–19 24. Rao SS (2018) The finite element method in engineering, 6th edn. Butterworth-Heinemann, Oxford, UK

Chapter 4

Use of Residues from the Metallurgical Industry in Construction George Power

Abstract The metallurgical industry produces a large amount of residues and wastes that can be successfully applied in building and construction. Key sectors, like iron and steel, aluminium, copper, zinc, lead, silicon and ferrosilicon, were analyzed, the residues identified, and potential applications described. The most important residue is metallurgical slag, which finds applications in cement making, admixture to mortar and concrete mixes, aggregate in concrete and asphalt and others. Bulky residues from aluminium and zinc hydrometallurgical processes, which take up a lot of land space to dispose, are more difficult to use, but several applications were also found. Keywords Metallurgical industry · Construction · Residues · Slag · Waste

4.1 Introduction The metallurgical industry comprises the beneficiation of ores, smelting and refining of metals, and generates about 8% of greenhouse gas (GHG) emissions [21]. With 1’878 Mt/a (million metric tons per year) crude steel production [35], iron and steel dwarf the other, nonferrous metals, for instance, aluminium, 65 Mt/a, copper, 25 Mt/a, zinc, 12 Mt/a, etc. [30]. Therefore, it is not surprising that the largest part (91%) of the Greenhouse Gas (GHG) emissions in the metallurgical industry comes from iron and steel production. Apart from the emissions, metallurgical processes also generate large amounts of residues or by-products in form of slag, ash, dust, and sludge, which are reused in the industry in some cases, but mostly dumped or landfilled. The main objective of this paper is to describe these residues, their production volumes, and how they can be applied in the building and construction industry, and so, incorporating them in the circular economy and fulfilling several G. Power (B) Faculty of Engineering, Department of Civil Engineering, Universidad de Lima, Lima, Peru e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_4

53

54

G. Power

goals of the 2030 Agenda for Sustainable Development [33]. A review of recent papers and books on this topic was carried out and the findings summarized.

4.2 Overview of Residues from the Metallurgical Industry with Use in Construction 4.2.1 Iron and Steel The production of crude steel follows one of the two conventional routes: (1) smelting of iron ore with coke and fluxes in the blast furnace (BF) to produce pig iron, followed by conversion in the basic oxygen furnace (BOF); or (2), direct reduction of iron ore in a rotary kiln or other type of furnace and melting of direct reduced iron (DRI) in an electric arc furnace (EAF). In both cases, large amounts of steel scrap (up to 60%) are melted to recover materials, energy, and for temperature control. Crude steel from BOF or EAF is refined in a ladle furnace and fed to the continuous casting machine to produce slabs, blooms, and billets, which are then transformed into the commercial forms of sheet, profile, rail, reinforcement bar (rebar), wire, etc. Smelting of iron ore in the blast furnace produces large amounts of slag, equivalent to ca. 400 Mt/a [34, 35] for the current steel production. The term slag refers to a mixture of silicates and different metal oxides that floats over the liquid metal bath and carries away most impurities. An overview of slag production in the iron and steel industry is given in Fig. 4.1. This residue, in form of ground granulated blast furnace slag (GGBFS) has long been known as a partial replacement of clinker in Portland Cement (PC) production, also known as supplementary cementitious material (SCM). It is considered a raw material for composite cements in standards like ASTM C595/C595M and EN 197–1, so, strictly speaking, it is not a residue but an intermediate material in cement making. GGBFS is obtained by rapidly chilling the molten slag in water to produce a material with a glassy and porous structure that has hydraulic properties and in Blast furnace slag (BFS)

Iron ore

Air cooled blast furnace slag

Blast furnace (BF) Ground granulated blast furnace slag (GGBFS)

Other materials (coke, limestone) Pig iron

BOF slag

Steel scrap Basic oxygen furnace (BOF) Other materials Steel scrap

Refined steel Crude steel

Electric arc furnace (EAF) Other materials

Other materials Ladle furnace Ladle slag

EAF slag

Fig. 4.1 Overview of slag production in the iron and steel industry

4 Use of Residues from the Metallurgical Industry in Construction

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cement and concrete mixes even increases flexural and compressive strength [3]. The rapid cooling already causes the larger slag pieces to break in small grains, which are then ground to sand grain size. Later, GGBFS is added to the cement ball mill, together with clinker and gypsum, to obtain a compound cement. Also, finely ground slag may be added to a cement blend, or a combination of grinding with clinker and blending with cement is used. According to the ASTM C595/C595M-21 standard, the slag constituent must be less than 25% of the mass of the slag-modified PC [1], on the other hand, the EN 197–1:2011 standard specifies different qualities of blast furnace cement (CEM III/A, B and C) with up to 95% of GGBFS [31]. The production of 1 ton of clinker generates about one ton of CO2 emissions, so, in theory, 480 Mt of blast furnace slag could reduce CO2 emissions by the same amount. Of course, this is not considering the energy consumption and emissions in the production of steel, and in the transport and processing of the slag. Blast furnace slag that is slowly cooled on air has a different, more crystalline structure. It does not exhibit the same hydraulic properties as the slag that has been quenched in water. This type of slag, as well as steelworks slag (from BOF or EAF), also known as dump ferrous slag (DFS), is allowed to cool and solidify in a pit, and sometimes sprayed with water to generate cracks and facilitate digging. Slag from BOF or EAF has a higher metal content and density than BF slag, has good bonding properties with cement and GGBFS, good absorption and is better suited as aggregate for concrete or asphalt mixes. For each ton of electric furnace steel, about 110 kg of slag are generated [34], this ratio is similar for the BOF [18]. Considering that 73.2% of crude teel production comes from the BOF and the reminder from EAF (the amount of steel produced by other processes like Open Hearth is negligible), the amount of DFS generated can be estimated in 206 Mt/a. Aceros Arequipa, the largest steelworks in Peru, reports processing of EAF slag by crushing and screening to a product called ecogravilla, or eco small gravel. About 27 thousand metric tons of this eco-gravel were recovered in 2019 and used mainly in prefabricated concrete eco-blocks for construction. Some eco-gravel was also used in road construction to nearby farms and communities. During slag processing, the very fine fraction is recovered and currently used for concrete and mortar mixes as partial replacement of cement [5]. Another use of slag from steelworks is in form of fibers as insulation material [3]. Like glass, slag can be converted to fibers by dropping it in molten state through nozzles with pressurized air. The compacted fibers, termed as slag wool, find applications as thermal and acoustic insulation materials [19].

4.2.2 Aluminium Among nonferrous metals, aluminium1 is the most widely used. In 2020, about 65 Mt were produced in 2020. The main aluminium ore is bauxite, a mixture of hydrated 1

Aluminum in American English spelling.

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oxides of aluminum (30% to 56% Al2 O3 content), with lower amounts of iron, titanium, and silica. In the Bayer process, bauxite is leached with caustic soda and refined to almost pure alumina (Al2 O3 ). A simplified process of alumina production is shown in Fig. 4.2. Alumina is then electrolytically reduced to pure aluminium metal in molten phase with fluorite as a fluxing agent (Hall-Héroult process); this is typically carried out in a separate plant. Besides the residues and emissions of the aluminium smelter plant, the refining of bauxite to alumina generates a bulky residue, also known as red mud, inasmuch as 2.5 to 3 tons per ton of refined aluminium metal (1 to 1.2 tons per ton of alumina), although figures as high as 4–5 tons of red mud per ton of metal are also mentioned. This bauxite residue is a red sludge that contains iron (which causes the red color), titanium and silicon oxides, together with undissolved aluminium from bauxite and residual sodium hydroxide and water. The actual amount of bauxite residue generated depends on the type of ore, process conditions and final composition of the residue. World estimates vary from 70 to 205 Mt/a [2, 14, 17, 23, 26]. There are different possible uses of bauxite residue in construction, for instance in cement production (clinkers, composite cements, and alkali-activated cements), as raw material and additive in iron and steel production, as a pigment in concrete and brick, as road base, as embankment and backfill material, also capping of landfill sites and for soil improvement [6, 23]. The use of red mud in porous asphalt, as an alternative filler to limestone powder was studied by Zhang et al. [37]. In India, an important aluminium producer, applications of red mud in brick manufacturing, especially interlocking blocks, and important material of construction in that country [9]. In Australia, another important producer of aluminium, use of coarse bauxite residue in roadway construction was studied [15]. Despite all the research work and Fig. 4.2 Simplified block flow diagram of alumina production

Bauxite

NaOH makeup

Digestion (pressure leaching) Water

Sodium hydroxide

Settling and washing

Bauxite residue (red mud)

Precipitation Alumina hydrate Calcination

Alumina

Water

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test applications in industry, total usage of bauxite residue is estimated between 3 and 4 Mt/a, which is small compared to the total amount generated.

4.2.3 Copper Global production of copper in 2020 was about 25 million metric tons [10, 32]. Primary copper is produced by either pyrometallurgical route (smelting, converting, and refining), for sulfide ores, or hydrometallurgical route (leaching, solvent extraction, and electrowinning), for low grade oxide ores. Another source of copper production is scrap recycling. It is estimated that 17% of copper production is secondary (recycled from scrap), and 16% from the hydrometallurgical route [22]. That leaves some 67% of copper production by the pyrometallurgical route, and only this process generates slag that can be used in construction. Some 2.2 tons of slag are generated for each ton of primary copper [12], that equates to roughly 37 million metric tons of copper slag which can have many uses in construction: as admixture to concrete and mortar, as supplementary cementitious material in cement making, in road pavement, and geotechnical applications [8, 12, 20].

4.2.4 Zinc World zinc mine and refined zinc production in 2020 was 12 Mt and 13.7 Mt, respectively [11, 13]. About 80% to 85% of refined zinc is produced by the RLE (roastingleaching-electrowinning) process (Myrim et al. 2004) [16]. Although this type of process does not generate slag, it produces a bulky residue known as jarosite residue (JR), a basic hydrous sulfate of iron and ammonium, which is mostly being disposed in impermeabilized ponds. A simplified block flow diagram of the RLE process can be seen in Fig. 4.3. In China, about 1 Mt/a of residues are produced [16], which is equivalent to 23% of zinc mine production, so the world JR production can be estimated in 2.76 Mt/a. Numerous studies have been carried out on the recovery of valuable metals from JR, or converting it to an inert material, as landfill capacities become exhausted [16, 34]. Despite the high iron content (30% to 50%), JR cannot be used as source of iron in steelmaking, because it contains about 5% of zinc, lead and other impurities. Myrim et al. [19] analyze potential applications of JR in construction. In principle, jarosite residue can be melted with some additional minerals as fluxes and converted into a glassy material that can be used as aggregate in concrete or asphalt mixes, or as pavement subbase. However, not only the metal contents are lost in this case, but some leaching of heavy metals may occur when exposed to longer contact with water. The solution proposed by the authors in that paper is a mixture of JR with ferrous slag from steelworks and some alkaline aluminium-surface cleaning waste to neutralize the residue.

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Zinc sulfide concentrate Air

Roasting

Zinc dust Calcine

Neutral leaching

Recycle liquor

Purification

Electrolysis

Zinc

Cu, Cd, etc. Calcine, ammonia

Precipitation

Spent electrolyte

Hot strong leaching Jarosite residue

Fig. 4.3 Simplified block flow diagram of RLE zinc process

Another residue from zinc production is slag from smelting of zinc ores in retorts or the combined zinc and lead blast furnace, known as Imperial Smelting Furnace, or ISF [34]. Typical production of slag is 0.9 ton per ton of zinc produced [27]. The slag is a granulated glassy material with a grain size between coarse and fine sand, but with higher density, due to its metal content. ISF slag is quite stable, but some leaching of metals may occur with prolonged exposure to water. In a road construction test performed at Avonmouth, UK, sand in the concrete mixture was replaced with 50% and 75% (by volume) ferrosilicate slag from ISF, achieving compressive strengths around 40 and 50 MPa after 7 and 28 days, respectively. The tests also showed that no significant leaching of zinc and arsenic was taking place with the ISF slag concrete [36].

4.2.5 Lead World refined lead production in 2020 was 11.7 Mt. Different metallurgical processes are applied for smelting lead concentrates, among them the blast furnace, Imperial Smelting, and various direct smelting methods (Kaldo, Kivcet, TSL, QSL, etc.); all processes have in common the generation of slag with similar characteristics to the slags of other nonferrous metals, so it can be used as an aggregate in concrete and asphalt mixes. The amount of slag generated varies from 0.4 to 2.5 tons per ton of metal produced, lead content in slag ranges from 3 to 6% [25, 28].

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4.2.6 Silicon and Ferrosilicon Silicon, for the semiconductor, metallurgical and chemical industries, and ferrosilicon, used for deoxidizing and alloying steel, are both produced by fusing silica (silicon dioxide from quartz sand) with coke, and some iron in the case of ferrosilicon, in electric-arc furnaces. A by-product known as silica fumes, microsilica, or nanosilica, is also generated. Silica fumes are composed of very fine (150 nm diameter) almost spherical dust particles of amorphous SiO2 (>85%) and other elements (Fe2 O3 , CaO, Al2 O3 ), and are recovered in filters attached to the furnaces [4, 26]. About 35% of dust, related to the end product, is produced [3]. With an annual production of 8 million metric tons of silicon content combined—silicon and ferrosilicon—[32], one can estimate a production of 2.8 Mt of silica fumes. Silica fumes are added in dosages of 8% to 12% to concrete and polymer concrete mixtures. The extreme fineness of the particles that can fill better the voids between the components of the concrete mix and results in very high compressive strengths (up to 180 MPa) in concrete and polymer concrete [3]. Silica fumes are also added to Portland cement, in dosages of 6–10%, for instance in CEM II/A-D, EN 197–1:2011 standard.

4.3 Results and Discussion The different metallurgical residues and byproducts mentioned in Sect. 4.2 offer a great opportunity of incorporating them in a circular economy and fulfilling various Sustainable Development Goals (SDGs) as described in the 2030 Agenda for Sustainable Development [33]. Among these goals are SDG 7, Affordable and clean energy; SDG 9, Industry, innovation, and infrastructure; SDG 11, Sustainable cities and communities; SDG 12, Responsible consumption and production; and, last but not least, SDGs related to Climate action, Life below water, and Life on land, respectively. Table 4.1 summarizes the findings on various residues from the metallurgical industry. Of course, not all wastes have been covered here, like nickel or tin slag, but the most important were considered. From the table it can be seen that the amount of residues, especially from the steel and aluminium industries, is staggering, up to 800 Mt/a, equivalent to 19% of global cement production. Alone GGBFS would worth between 30 and 46 billion US$, according to prices from 2018 [7]. The most important difficulty in using these residues, however, remains in their limited availability near the application site. Shipping costs have increased in the last years, not only because of the pandemic, but also fluctuations in oil production, shipping capacities, conflicts, etc. A small cement producer in Peru, for instance, imports GGBFS from Japan for a compound cement type MS (moderate sulfate resistance). With 25% substitution of clinker by GGBFS and considering shipping, only a 21% GHG emissions is reached, whereas a 100% substitution would give 84% GHG emissions reduction [29].

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Table 4.1 Summary of metallurgical residues and byproducts with use in construction-related industries Industry sector

Production (Mt/a)

Residue or by-product Type

Estimated amount (Mt/a)

Average Applications composition (%)

Iron and steel

1, 877.5

Ground granulated blast furnace slag (GGBFS), water cooled

400

31–50 CaO; 27–45 SiO2 ; 7–24 Al2 O3 ; 1–18 MgO; 0.3–2 FeO/ Fe2 O3 ; 0.1–2.3 MnO; 0.3–0.6 S; < 0.1 P2 O5

Supplementary cementitious material (SCM) in compound cements, mortar, and concrete mixes

Dump ferrous 151 slag (DFS) from BOF, air cooled

35–45 CaO; 12–17 SiO2 ; 1–3.4 Al2 O3 ; 3–15 MgO; 10–25 FeO/ Fe2 O3 ; 5–15 MnO; 0–0.3 SO3 ; 0.2–4 P2 O5

Admixture to mortar and concrete, aggregate in concrete and asphalt, eco-blocks for building construction, road construction

Dump ferrous 55 slag (DFS) from EAF, air cooled

40–60 CaO; 10–30 SiO2 ; 2–9 Al2 O3 ; 3–8 MgO; 10–30 FeO/Fe2 O3 ; 2–5 MnO; 0.1–0.6 SO3 ; 0–1.2 P2 O5

Aluminium

65

Bauxite 162–195 residue (BR), a.k.a. red mud

25–37 Al2 O3 ; 30–48 Fe2 O3 ; 2.3–17 SiO2 ; 2.8–6.1 TiO2 ; 2.4–9.7 Na2 O; 1.2–3.7 CaO

Cement production, iron and steel production, brick manufacturing, porous asphalt, roadway construction

Copper

25

Slag from reverberatory furnace, flash smelting and others

0.2–2.1 Cu; 29–51 Fe (total); 5.1–9.5 Fe3 O4 ; 24–38.9 SiO2 ; 2.9–15.6 Al2 O3 ; 2–5.9 CaO; 0.1–3.5 MgO; 0.4–1.7 MnO; 0.3–1 S

Admixture to concrete and mortar, supplementary cementitious material, road pavement, geotechnical applications

37

(continued)

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Table 4.1 (continued) Industry sector

Production (Mt/a)

Residue or by-product Type

Estimated amount (Mt/a)

Average Applications composition (%)

Zinc

12

Jarosite residue (JR)

2.76

5 SiO2; 0.5 MgO; 2 CaO; 1.7 Al2 O3 ; 0.3 MnO; 5 Zn; 20–30 FeO; 30 SO3 ; 6.8 Na2 O/ K2 O/NH3

Ferrosilicate slag from blast furnace, retort or ISF

2.2

18.5 CaO; 19.5 Road construction SiO2 ; 41.5 FeO; 9.5 ZnO; 1 PbO; 8 Al2 O3 ; 0.5 MgO; 1.5 other

4.7–29

20–28.7 Fe (total); 21.4–35 SiO2 ; 3.6–10 Al2 O3 ; 16–23.1 CaO; 5.4–10 MgO; 1.4–5 MnO; 0.4–2 S; 2–6 PbO, 2–12 ZnO; 2.2 C

Aggregate in concrete or asphalt mixes

87.3–90.4 SiO2 ; 0.5–1.2 Al2 O3 ; 0.6–4.8 Fe2 O3 ; 0.4–1 CaO; 1.5–4.5 MgO; 0.6–1.5 K2 O; 0.4–1.3 Na2 O; 0.2–1.1 SO3

Admixture to concrete, polymer concrete, concrete for hydraulic applications, compound cements

Lead

11.7

Ferrosilicate slag from blast furnace, ISF or direct smelting

Silicon and ferrosilicon

8

Silica fumes, 2.8 a.k.a. microsilica or nanosilica

Aggregate in concrete or asphalt mixes, pavement subbase (stabilized with ferrous slag and neutralized with alkaline Al-surface cleaning waste

4.4 Conclusions Following conclusions can be derived from the previous sections in this paper: • The most abundant and useful residue in the metallurgical industry is slag, which is used in different ways in construction, mainly in cement making, as admixture to mortar and concrete mixes and as aggregate (replacement of gravel), so that it can be considered a by-product, rather than a waste from the metallurgical industry.

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• Silica fumes is another valuable by-product from silicon and ferrosilicon production and find direct application in compound cements and as a filler in concrete mixes. • Bauxite and Jarosite residues, from the aluminium and zinc industry, respectively, are more difficult to implement, as they vary more in composition, and a mixture with other materials is required. Applications are still limited and under current investigation. • The use of any these residues will depend ultimately on their amount available and proximity to the intended application point.

References 1. ASTM - American Society for Testing and Materials (2021) ASTM C595/C595M-21. Standard specification for blended hydraulic cements. https://www.astm.org/c0595_c0595m-21.html 2. Abdel-Raheem M, Gómez Santana LM, Piñeiro Cordava MA, Olazaran Martínez B (2017) Uses of red mud as a construction material. In: AEI 2017: resilience of the integrated building - proceedings of the architectural engineering national conference 2017 388–399. https://doi. org/10.1061/9780784480502.032 3. Barbuta M, Bucur RD, Cimpeanu SM, Paraschiv G, Bucur D (2015) Wastes in building materials industry. In: Pilipavicius V (ed) Agroecology, pp 81–99. IntechOpen. https://doi.org/10. 5772/59933 4. Black L (2016) Low clinker cement as a sustainable construction material. In: Khatib JM (ed) Sustainability of construction materials, pp 415–457. https://doi.org/10.1016/B978-0-08-100 370-1.00017-2 5. CAASA - Corporación Aceros Arequipa S.A (2019) Reporte de sostenibilidad 2019. https:// investors.acerosarequipa.com/storage/reporte-sostenibilidad/July2020/RS_ESP_2019.pdf 6. Carlson C (n.d.) Sustainability in industry: employing red mud in construction materials. Feeco International. https://feeco.com/sustainability-in-industry-employing-red-mud-in-construct ion-materials/#:~:text=Using%2020%2D40%25%20red%20mud,reducing%20building% 20and%20foundation%20costs 7. Curry KC (2018) USGS Minerals Yearbook. Slag—Iron and Steel [Advance release]. United states geological service. https://pubs.usgs.gov/myb/vol1/2018/myb1-2018-iron-steel-slag.pdf 8. Dhir OBE RK, De Brito J, Mangabhai R, Lye CQ (2017) Sustainable construction materials: copper slag. Elsevier Woodhead Publishing 9. Garg A, Yadav H (2015) Study of red mud as an alternative building material for interlocking block manufacturing in construction industry. Int J Mater Sci Eng 3(4):295–300. https://doi. org/10.17706/ijmse.2015.3.4.295-300 10. Garside M (2021) Refinery production of copper worldwide 2000–2020. https://www.statista. com/statistics/254917/total-global-copper-production-since-2006/ 11. Garside M (2022) Zinc production in major countries 2010–2021. https://www.statista.com/ statistics/264634/zinc-production-by-country/ 12. Gorai B, Jana RK, Premchand (2003) Characteristics and utilisation of copper slag—a review. Resour Conserv Recycl 39:299–313. https://doi.org/10.1016/S0921-3449(02)00171-4 13. Government of Canada (2022) Zinc facts. https://www.nrcan.gc.ca/our-natural-resources/min erals-mining/minerals-metals-facts/zinc-facts/20534 14. Jain S (2014) Red mud as construction material using bioremediation [Master Thesis, National Institute of Technology, Rourkela, India]. https://core.ac.uk/download/pdf/53190437.pdf 15. Jitsangiam P, Nikraz H (2013) Coarse bauxite residue for roadway construction materials. Int J Pavement Eng 14(3):265–273. https://doi.org/10.1080/10298436.2012.705843

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16. Li J, Ma H (2018) Zinc and lead recovery from jarosite residues produced in zinc hydrometallurgy by vacuum reduction and distillation. Green Process Synth 7:552–557. https://doi.org/ 10.1515/gps-2017-0079 17. Lima MSS, Thives LP, Haritonovs V, Bajars K (2017) Red mud application in construction industry: review of benefits and possibilities. In: IOP conferences of series: materials science and engineering, vol 251. https://doi.org/10.1088/1757-899X/251/1/012033 18. Madhavan N, Brooks GA, Rhamdhani MA, Rout BK, Schrama FNH, Overbosch A (2020) General mass balance for oxygen steelmaking. Ironmak Steelmaking Process, Prod Appl 48(1):40–54. https://doi.org/10.1080/03019233.2020.1731252 19. Mymrin VA, Ponte HA, Impinnisi PR (2005) Potential application of acid jarosite wastes as the main component of construction materials. Constr Build Mater 19:141–146 20. Nazer A, Pavez O, Toledo I (2013) Effect of type cement on the mechanical strength of copper slag mortars. Revista Escola de Minas, Ouro Preto 66(1):85–90. https://www.redalyc.org/art iculo.oa?id=56425762011 21. Our World in Data (2020) Emissions by sector. https://ourworldindata.org/emissions-by-sector 22. Pietrzyk S, Tora B (2018) Trends in global copper mining – a review. IOP Conf Ser: Mater Sci Eng 427:012002. https://doi.org/10.1088/1757-899X/427/1/012002 23. Raahauge BE, Evans K, Bach M, Thomas D, Ter Weer P-H (2019) Alumina. In: Dunne RC, Komar Kawatra S, Young CA (eds) SME mineral processing & extractive metallurgy handbook. Society of mining, metallurgy and exploration (SME), pp 1511–1536 24. Siddique R, Kunal, Mehta A (2020) Utilization of industrial by-products and natural ashes in mortar and concrete development of sustainable construction materials. Silica fumes. In: Harries KA, Sharma B (eds) Nonconventional and vernacular construction materials. characterisation, properties and applications, 2nd edn. Woodhead Publishing-Elsevier, pp 259–263 25. Siegmund A (2019) Lead and bismuth. In: Dunne RC, Komar Kawatra S, Young CA (eds) SME mineral processing & extractive metallurgy handbook. Society of mining, metallurgy and exploration (SME), pp 1807–1838 26. Silveira NCG, Martins MLF, Bezerra ACS, Araújo FGS (2021) Red mud from the aluminium industry: production, characteristics, and alternative applications in construction materials—a review. Sustainability 13:12741. https://doi.org/10.3390/su132212741 27. Sinclair R (2005) The extractive metallurgy of zinc. The Australasian Institute of Mining and Metallurgy 28. Sinclair R (2009) The extractive metallurgy of lead. The Australasian Institute of Mining and Metallurgy 29. Sotomayor A, Power G (2019) Tecnologías Limpias y Medio Ambiente en el Sector Industrial Peruano. Fondo Editorial de la Universidad de Lima, Casos Prácticos 30. Statista Research Department (2021) Lead metal refined production volume worldwide 2006– 2020. https://www.statista.com/statistics/264872/world-production-of-lead-metal/ 31. UNE - Normalización Española (2021) UNE-EN 197–1:2011. Cement - Part 1: composition, specifications and conformity criteria for common cements. https://tienda.aenor.com/normaune-en-197-1-2011-n0048623 32. United States Geological Service—USGS (2021) Mineral commodity summaries. https://pubs. usgs.gov/periodicals/mcs2021/mcs2021.pdf 33. United Nations—UN (2015) Transforming our world: the 2030 agenda for sustainable development. Department of economic and social. Affairs sustainable development. https://sdgs.un. org/publications/transforming-our-world-2030-agenda-sustainable-development-17981 34. Wang GC (2016) The utilization of slag in civil infrastructure construction. Elsevier Woodhead Publishing. 35. World Steel Association (2021) World steel in figures. https://worldsteel.org/wp-content/upl oads/2021-World-Steel-in-Figures.pdf 36. Wrap (2012) Aggregates case study. Ferrosilicate slag from zinc production as aggregates bound in cement. https://www.aggregain.org.uk 37. Zhang H, Li H, Zhang Y, Wang D, Harvey J, Wang H (2018) Performance enhancement of porous asphalt pavement using red mud as alternative filler. Constr Build Mater 160:707–713. https://doi.org/10.1016/j.conbuildmat.2017.11.105

Chapter 5

On the Random Axially Functionally Graded Micropolar Timoshenko-Ehrenfest Beams Gabriele La Valle and Giovanni Falsone

Abstract This work aims to propose a simplified method for the random analysis of axially functionally graded micropolar Timoshenko-Ehrenfest beams (ATBs). First, new approximated closed form solutions in terms of displacements are derived. Secondly, the probability density functions (pdfs) of the displacements are obtained starting from the pdfs of the constitutive parameters, and vice-versa. The probability transformation method (PTM) and the Monte Carlo (MC) simulation are both applied. The choice of studying random axially functionally graded micropolar beams lies into many reasons: the growing importance of the functionally graded (FG) materials within the scientific panorama; the generic nature of the mathematical model, which proves useful to study beams where the assumption of constant and deterministic stiffnesses along the axis does not hold anymore; the recent application of micropolar continua in the analyses of nano- and micro-electromechanical systems (NEMS and MEMS).

5.1 Introduction Micropolar elasticity adds to the classical one a kinematic field related to the microrotation of each particle of the continuum. The second half of the XIX century is characterized by a renaissance of the micropolar models, summarized by the Cosserat brothers in 1909 [16]. New theoretical results can be found in the works by Eremeyev et al. [21], Altenbach and Eremeyev [5]. The main advantages are the capability of forecasting length scale effects and of describing with greater accuracy the mechanical behaviors of composite materials. To this regard, discrete and continuum micropolar models were developed by Misra and Poorsolhjouy [42, 43], Giorgio et al. G. La Valle (B) · G. Falsone University of Messina, C.da Di Dio, 98158 Sant’Agata, Messina ME, Italy e-mail: [email protected] G. Falsone e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_5

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[28, 29], Turco et al. [56], Steigmann [55], Shirani and Steigmann [50]. Other sources of potential concern the design of new metamaterials [4, 8, 13–15, 27, 53]. Although 3D models are the most accurate ones for describing mechanical structures, since from the fundamental works by Euler and Bernoulli, simplified 1D models attracted the attention of scientists. The so called “beam theory” is widely applied in many fields of engineering - civil, electrical, and mechanical engineering. Most works on the analysis of beams deal with the Euler-Bernoulli beam theory (EBT) [17, 46], which is not adequate for describing the mechanical behaviors of nonslender beams, i.e., beams characterized by a relatively large thickness-to-length ratio. To overcome these limitations, shear deformation theories were developed. Some of them were brought together by Wang et al. [58]. The first in order of time, and the simplest one, is the Timoshenko-Ehrenfest beam theory (TBT), which can consider shear and bending deformations. About the development of the cited theory, we recall the paper by Elishakoff [20]. It has been studying in applied mathematics [18], structural mechanics [12], computational mechanics [11, 24], and alternatives were proposed and analyzed by [9, 26, 30, 31, 57]. Recently, the development of functionally graded beam models, characterized by continuously varying mechanical properties [33, 47], became crucial in the field of mechanics [23]. Functionally graded micropolar beams, i.e., functionally graded beam models developed in the field of micropolar elasticity, can be useful for describing micro- and nano-scale devices, where scale effects are commonly observed [38, 48]. They can also play a significant role for describing non-standard mechanical systems which show a chiral behavior. In this regard, we can refer to the works by De Angelo et al. [6], Misra et al. [44] and Nejadsadeghi et al. [45]. In the last years, more complex beam models have been developed. Among these, Faghidian [2] and Faghidian et al. [3] proposed the use of the non-local modified gradient theory and higher-order unified gradient elasticity, where the non-local integral elasticity and the modified strain gradient theory have been merged. Non-local elasticity can capture the peculiar characteristics at micro/nanoscale. On the other hand, higher order theories can consider new boundary conditions on the edges and corners. All the listed models involve constitutive parameters closely related to the micro/meso-structure of the beams, that, consequently, should be described as random fields. Considering the above, the present paper aims to discuss the response and the stochastic identification of random axially functionally graded micropolar Timoshenko-Ehrenfest beams (ATBs). In the authors’ opinion, the ATB is the simplest conceivable model able to describe scale effects. The work is organized in five sections. First, the Euler-Lagrange equations are derived in the field of the isotropic axially functionally graded micropolar continuum via the Principle of Virtual Work [7, 54]. Secondly, by approximating the axial, bending and shear stiffnesses as piecewise functions, new approximated closed form solutions are proposed for the transversal and horizontal displacements. The generalized functions are applied (the use of generalized functions in the solution of the beam bending problems was addressed by [10, 22, 39]). Finally, the direct and inverse stochastic problems are

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defined, and their solutions by means of the Monte Carlo (MC) method and the probability transformation method (PTM) are proposed. Several numerical applications are performed.

5.2 Euler-Lagrange Equations The present section aims to derive the Euler-Lagrange equations for beams in the field of the linear micropolar theory. We consider a beam of length L, and of rectangular cross section w × h, where w and h stand for the width and the height, respectively. About the external reference system, the z-coordinate is chosen along the length, the y-coordinate along the thickness and the x-coordinate along the width. The applied external loads are assumed to be functions of z, so that the kinematic fields of the displacements and rotations can be approximatively expressed as depending on z and y. The Euler-Lagrange equations are formulated by using the Timoshenko beam theory and the Principle of Virtual Work (PVW): 0 = δΦ − δW

(5.1)

where δΦ is the internal deformation energy, and δW is the virtual work done by external loads consisting in forces and couples per unit volume and area. According to the micropolar theory, the internal deformation energy can be written as: { ( δΦ = V

) ∂φ ∂φ δεi j + δκi j d V ∂εi j ∂κi j

(5.2)

where summation on repeated indices is implied; here, the symbol φ denotes the specific deformation energy, which, in the field of the linearized model, results a function of the strain tensor ε and of the wryness tensor κ defined as follows: εi j =

∂u i + ei jk ϕk ∂x j

κi j =

∂ϕi ∂x j

i, j = 1, 2, 3

(5.3)

where ei jk is the Levi-Civita symbol; u i (x1 , x2 , x3 ) is the ith component of the displacement field u, ϕi (x1 , x2 , x3 ) the ith component of the rotation field ϕ; x1 = x, x2 = y, x3 = z. For an isotropic linear micropolar material, the specific deformation energy can be expressed as (see the general form in [34, 35, 41]): φ (ε, κ) =

1 λΓ σ (λ E , μ E , S ε) : S ε + ξ R || A ε||2 + T r 2 [κ] + 2 2 ( ( ) ) μΓ + ξΓ μΓ − ξΓ || S κ||2 + || A κ||2 + 2 2

(5.4)

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G. La Valle and G. Falsone

where σ (λ E , μ E , S ε) is the relationship between σ and S ε given by the Hook’s law; σ is the stress tensor whose components are given by σi j = ∂φ/∂εi j ; the symbol “:” stands for the scalar product in the space of second-order tensors; λ E , μ E are the classical Lamé parameters; ξ R is a constitutive parameter related to the micro-macro relative rotation; λΓ , μΓ and ξΓ are three material constants related to the wryness tensor. The symbols T r [·] and ||·|| stand for the trace and the Euclidean norm of the assigned tensors. Moreover, the symbols S ε and A ε represent the symmetric and the skew symmetric part of ε: Sε

=

ε + εT 2

Aε

=

ε − εT 2

(5.5)

In the same way, S κ and A κ are the symmetric and skew symmetric part of κ, respectively: κ + κT κ − κT (5.6) Sκ = Aκ = 2 2 It is necessary and sufficient condition for the nonlinearized deformation energy density φ to be nonnegative define that: μ E ≥ 0 3λ E + 2μ E ≥ 0 μΓ − ξΓ ≥ 0

ξR ≥ 0

μΓ + ξΓ ≥ 0

3λΓ + (μΓ + ξΓ ) ≥ 0

(5.7)

For an axially functionally graded continuum, each constitutive parameter varies along the axis: λ E = λ E (z), μ E = μ E (z), ξ R = ξ R (z), λΓ = λΓ (z), μΓ = μΓ (z), ξΓ = ξΓ (z). This kind of model can be applied not only to structural elements made by axially functionally graded materials, but also every time some uncertainties and/or defects along the axis appear. Since λ E and μ E are the classical Lamé parameters, they are connected to the Young modulus, Y , the shear deformation modulus, G, and the Poisson coefficient, ν, by the following relationships: λE =

Yν Y μE = G = (1 + ν)(1 − 2ν) 2(1 + ν)

(5.8)

In this work, the material parameters will be modelized as random fields variable along the beam axis.

5.2.1 Timoshenko-Ehrenfest Beam Theory The local displacements u i (x, y, z) as implied by the Timoshenko-Ehrenfest beam hypotheses are given by:

5 On the Random Axially Functionally Graded Micropolar …

u 3 (x, y, z) = u(z) + yϕ(z) u 1 (x, y, z) = 0

u 2 (x, y, z) = v(z)

ϕ1 (x, y, z) = ϕ(z)

ϕ2 (x, y, z) = ϕ3 (x, y, z) = 0

69

(5.9) (5.10) (5.11)

where u and v denote the axial and transversal displacements generalized to the cross section; ϕ the microrotation generalized to the cross section. In the monodimensional approximation processing, the local kinematic unknown fields, u i and ϕi , are written in terms of the generalized ones, u, v and ϕ. By replacing Eqs. (5.9, 5.10 and 5.11) into Eq. (5.4), we get: [ ] [ ]2 Y (z) du (z) 2 G (z) dv (z) + ϕ (z) + + φ (ε, κ) = 2 dz 2 dz ][ ] ] [ [ C (z) Y (z) dϕ (z) 2 du (z) dϕ (z) + y2 + Y (z) y + 2 2 dz dz dz

(5.12)

where we assume ξ R ≈ 0 to preserve the classical shear behavior at the macro-scale (see Sect. 5.2.2; the exposed results can be generalized also for ξ R /= 0). The field C (z), named by the authors “Cosserat modulus”, defines and engineering parameter related to the wryness deformation, and it is given by: μΓ (z) = C (z)

(5.13)

Then, by replacing Eq. (5.12) into Eqs. (5.1 and 5.2), we arrive at the equilibrium equations: ( ) d du (z) Y (z) A + f (z) = 0 (5.14) dz dz [ )] ( d dv (z) G (z) A + ϕ (z) + q (z) = 0 dz dz [ ] ) ( d dϕ (z) dv (z) − G (z) A + ϕ (z) = 0 (C (z) A + Y (z) I ) dz dz dz

(5.15)

(5.16)

with the boundary conditions: [

du (z) δu Y (z) A dz

[

( G (z) A

]L =0

(5.17)

0

) ]L dv (z) + ϕ (z) δv = 0 dz 0

(5.18)

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G. La Valle and G. Falsone

[

dϕ (z) δϕ (C (z) A + Y (z) I ) dz

]L =0

(5.19)

0

where f (z) and q (z) are the axial and transversal external loads, respectively. The derived formula are the simplest conceivable ones to consider scale effects. For instance, we could think to replace the Young modulus Y (z) with an equivalent Young modulus Ye (z) = Y (z) η (z), where η (z) allows to achieve more accurate mechanical or thermal deformations; to replace the shear stiffness G (z) A with an improved one G (z) A/χ (z), in which χ is known as shear correction factor. Different shear correction factors have been proposed in the literature. To this regard, we can refer to the work by Faghidian [1], where a unified approach to set forth a unified formulation of Cowper’s formula for shear coefficients is presented. An overview on the three most popular shear correction factors used in the Timoshenko beam theory can also be found in [40]. More generally, an axial stiffness a (z), a bending stiffness b (z), and a shear stiffness t (z) can be defined. The Timoshenko-Ehrenfest beam equations become: ( ) d du (z) a (z) + f (z) = 0 (5.20) dz dz [ )] ( d dv (z) t (z) + ϕ (z) + q (z) = 0 dz dz

(5.21)

[ ] ) ( d dϕ (z) dv (z) b (z) − t (z) + ϕ (z) = 0 dz dz dz

(5.22)

with the boundary conditions:

[

]L [ du (z) δu = 0 a (z) dz 0

(5.23)

) ]L dv (z) t (z) + ϕ (z) δv = 0 dz 0

(5.24)

(

[

dϕ (z) b (z) δϕ dz

]L =0

(5.25)

0

5.2.2 Additional Remarks The proposed equations show that, once known the displacements, it is possible to derive at most a (z), t (z), b (z), and vice-versa. In the next sections, it is proven that if a (z), t (z), b (z) are assumed to be random fields, we can derive the stochastic

5 On the Random Axially Functionally Graded Micropolar …

71

information of the displacements once known the stochastic information of a (z), t (z), b (z), respectively. The vice-versa still holds. Another important aspect is related to the relationship between the moment of inertia and the area of the cross section. It is useful to notice that, if micro-beams are analyzed, the moment of inertia, I , is negligible with respect to the area, A: b (z) = C (z) A + Y (z) I ≈ C (z) A I ≪ A

(5.26)

Then, if G (z), Y (z) and C (z) are random fields, and the geometric quantities are deterministic, we need just to stochastically characterize a (z), t (z), b (z) to stochastically characterize Y (z), G (z) and C (z). In the same way, for beams of the typical dimension of civil engineering, we have: b (z) = C (z) A + Y (z) I ≈ Y (z) I I ≫ A

(5.27)

In this case, the stochastic description of t (z) and b (z) leads to the stochastic description of G (z) and Y (z). Both Eqs. (5.26 and 5.27) refer to Eqs. (5.14, 5.15 and 5.16). The previous statements are numerically verified in Sect. 5.5. In Sect. 5.2.1 we fixed ξ R ≈ 0 to preserve a classical behavior at the macro-scale. With ξ R /= 0, we would have found a new additional term related to the shear depending on ξ R A non-negligible, in general, with respect to G A.

5.3 Approximated Closed Form Solutions The present section aims to find approximated closed form relationships between the displacements in a point of the axis z, and the axial, bending and shear stiffnesses. To this purpose, they are all approximated up by means of the Heaviside function U: a (z) ≈ a0 +

n−1 Σ

(ai − ai−1 ) U (z − z i )

(5.28)

(bi − bi−1 ) U (z − z i )

(5.29)

(ti − ti−1 ) U (z − z i )

(5.30)

i=1

b (z) ≈ b0 +

n−1 Σ i=1

t (z) ≈ t0 +

n−1 Σ i=1

where n is the number of discretization intervals. The properties of the generalized functions are used. The results are obtained for simply supported and cantilever beams. The same procedure is easily generalizable for other conditions of constraints.

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G. La Valle and G. Falsone

5.3.1 Simply Supported Beams If Eq. (5.21) is integrated one time with respect to z, we get: (

) dv (z) + ϕ (z) = −q (1) (z) + k1 t (z) dz

(5.31)

Hereafter, given a generic function f (z), the symbol f (i) (z) stands for a function whose ith derivative is equal to f (z). By adding Eq. (5.31) to Eq. (5.22), it is derived: [ ] d dϕ (z) b (z) = −q (1) (z) + k1 dz dz

(5.32)

Equation (5.32) can be integrated to obtain: dϕ (z) −q (2) (z) + k1 z + k2 = dz b(z)

(5.33)

If we replace Eq. (5.29) into Eq. (5.33), the latter can be written as: dϕ (z) −q (2) (z 0 ) + k1 z 0 + k2 ≈ + dz b0 (5.34) ) n−1 ( Σ −q (2) (z i ) + k1 z i + k2 −q (2) (z i−1 ) + k1 z i−1 + k2 − + U (z − z i ) bi bi−1 i=1 Since for a simply supported beam, two of the four boundary conditions are: | dϕ (z) || =0 dz |z=0

| dϕ (z) || =0 dz |z=L

we get also: k2 = 0

k1 =

q (2) (L) L

(5.35)

(5.36)

To shorten the equations, it is useful to introduce the function: qˆ (z) = −q (2) (z) +

q (2) (L) z L

(5.37)

Position (5.37) is needed for convenience and not for theoretical reasons. Considering the above, Eqs. (5.36), (5.37) and (5.33) lead to:

5 On the Random Axially Functionally Graded Micropolar …

) n−1 ( dϕ (z) qˆ (2) (z 0 ) Σ qˆ (2) (z i ) qˆ (2) (z i−1 ) ≈ + − U (z − z i ) dz b0 bi bi−1 i=1

73

(5.38)

which can be integrated to derive: ) n−1 ( (2) Σ qˆ (2) (z 0 ) qˆ (z i ) qˆ (2) (z i−1 ) ϕ (z) ≈ k3 + z+ − R (z − z i ) b0 bi bi−1 i=1

(5.39)

where R (z) is the generalized 1th order Ramp function, defined as follows: { R (z − z i ) =

0 z − zi

z ≤ zi z > zi

(5.40)

By replacing Eq. (5.39) into Eq. (5.31), we arrive to: ) n−1 ( dv (z) qˆ (1) (z 0 ) Σ qˆ (1) (z i ) qˆ (1) (z i−1 ) ≈ −k3 + + − U (z − z i ) + dz t0 ti ti−1 i=1 ) n−1 ( (2) Σ qˆ (2) (z 0 ) qˆ (z i ) qˆ (2) (z i−1 ) z− − R (z − z i ) − b0 bi bi−1 i=1

(5.41)

Equation (5.41) can be integrated and an analytical expression for the transversal displacement is achieved: ) n−1 ( (1) Σ qˆ (1) (z 0 ) qˆ (z i ) qˆ (1) (z i−1 ) z+ − R (z − z i ) + t0 ti ti−1 i=1 ) n−1 ( qˆ (2) (z 0 ) z 2 Σ qˆ (2) (z i ) qˆ (2) (z i−1 ) − − Q (z − z i ) − 2 b0 bi bi−1 i=1 (5.42) where Q denotes the 2th order ramp function: v (z) ≈ −k3 z + k4 +

{ Q (z − z i ) =

0 (z−z i )2 2

z ≤ zi z > zi

(5.43)

If the other two boundary conditions, valid for a simply supported beam, are imposed: v (0) = 0

v (L) = 0

the integration constant k3 assumes the following values:

(5.44)

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G. La Valle and G. Falsone

[ ] ) n−1 ( (1) Σ 1 qˆ (1) (z 0 ) qˆ (z i ) qˆ (1) (z i−1 ) k3 = L+ − R (L − z i ) + L t0 ti ti−1 i=1 [ ] ) n−1 ( 1 qˆ (2) (z 0 ) L 2 Σ qˆ (2) (z i ) qˆ (2) (z i−1 ) − − − Q (L − z i ) + L 2 b0 bi bi−1 i=1

(5.45)

while the integration constant k4 becomes equal to zero, k4 = 0. Let us observe that, the derived solution (5.42) can be specified for different b (z) and t (z) (see Sect. 5.2). In the same way, it is possible to solve the differential equation governing the axial behavior. By integrating Eq. (5.20) two times with respect to z, we get: ) f (1) (z i ) f (1) (z i−1 ) − R (z − z i ) ai ai−1 (5.46) The constants k1 and k2 are derived by imposing: Σ f (1) (z 0 ) z− u (z) ≈ −k1 z + k2 − a0 i=1 n−1

u (0) = 0

(

u (L) = 0

(5.47)

We obtain: k2 = 0

[ ] ) n−1 ( (1) Σ 1 f (1) (z 0 ) f (z i ) f (1) (z i−1 ) − L− − k1 = R (L − z i ) L a0 ai ai−1 i=1

(5.48)

(5.49)

5.3.2 Cantilever Beam For a cantilever beam, the boundary conditions change. They are detailed below: v (0) = 0 | dϕ (z) || =0 dz |z=L

ϕ (0) = 0 | | dv (z) + ϕ (z)|| =0 dz z=L

(5.50)

(5.51)

The same steps performed into Sect. 5.3.1 can be repeated. To avoid making the paper redundant, the authors directly give the derived solutions. The only technical difficulty is to observe that, from Eq. (5.22), the following implication holds:

5 On the Random Axially Functionally Graded Micropolar …

| | dv (z) + ϕ (z)|| =0 ⇒ dz z=L

75

| d 2 ϕ (z) || =0 dz 2 |z=L

since: [( ]| ) n−1 Σ d dϕ (z) || b0 + (bi − bi−1 ) U (z − z i ) | | dz dz i=1

= bn−1 z=L

(5.52)

| d 2 ϕ (z) || (5.53) dz 2 |z=L

Σn−1 Let us note that the derivative of b (z) = b0 + i=1 (bi − bi−1 ) U (z − z i ) is equal to zero in z = L. It is not needed to introduce the delta di Dirac, δ (z − z i ), since we are looking for the derivative in a point of continuity for b (z). In particular, the derivative of b (z) is equal to zero for every point z ∈ [0, L] such that z /= z i , for i = 1, . . . , n − 1. An approximated closed form solution can be obtained for the transversal displacement: ) n−1 ( (1) Σ qˆ (1) (z 0 ) qˆ (z i ) qˆ (1) (z i−1 ) z+ − R (z − z i ) + t0 ti ti−1 i=1 ) n−1 ( qˆ (2) (z 0 ) z 2 Σ qˆ (2) (z i ) qˆ (2) (z i−1 ) − − Q (z − z i ) − 2 b0 bi bi−1 i=1

v (z) ≈

(5.54)

where, in this case, qˆ (2) (z) is defined as: qˆ (2) (z) = −q (2) (z) + q (1) (L) z + q (2) (L) − q (1) (L) L

(5.55)

For simply supported and cantilever beams, once fixed a coordinate z, the relationship between the constitutive parameters and the displacement in z: (

1 1 1 1 1 1 , ,..., , , ,..., t0 t1 tn−1 b0 b1 bn−1

) → v (z)

(5.56)

is linear. This statement holds, in general, for isostatic beams. By defining an augmented vector containing not only the inverse of the constitutive parameters but also appropriate constants of integrations, a linear application of the kind given by Eq. (5.56) can also be built for static indeterminate beams. Hence, without loss of generality, we focus just on simply supported and cantilever beams for studying the stochastic identification of a Timoshenko-Ehrenfest beam model.

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G. La Valle and G. Falsone

5.4 Direct and Inverse Uncertainty Quantification All mechanical systems are affected by random aspects, which could be related to geometric, material defects and non-deterministic loads. In this section, the mechanical properties are modelized as uncertain parameters. In accordance with the common notation, random objects are denoted by means of uppercase letters. The quantities a (z), b(z) and t (z) are assumed to be random fields, denoted by A(z), B(z) and T (z). Consequently, the constants ai , bi and ti (see Eqs. (5.28, 5.29 and 5.30)) are replaced with random variables, denoted by Ai , Bi and Ti . In the same way, the kinematic fields u(z) and v(z) are described as random fields, U (z) and V (z). We focus just on the random horizontal and vertical displacements since they are easier to be empirically measured than the cross-section rotations, and, consequently, more useful for the identification of random constitutive parameters.

5.4.1 The Direct Problem The direct problem consists in finding the probability density functions (pdfs) of the random variables Ai , Bi and Ti , for i = 0 . . . n − 1, once known the pdfs related to the horizontal and vertical displacements. Looking at Eqs. (5.42, 5.46 and 5.54), it is trivial to state that, a vector transformation between the inverse of the constitutive parameters in each interval of the beam and the vertical displacements can be obtained. In other words, a vector transformation V = h( D) can be identified, which can link with each other the random vectors D and V defined as follows: ( D=

1 1 1 1 1 1 , ,..., , , ,..., T0 T1 Tn−1 B0 B1 Bn−1

)T

V = (V (ξ0 ) , . . . , V (ξn−1 ) , V (η0 ) , . . . , V (ηn−1 ))T

(5.57)

(5.58)

where ξi and ηi are coordinates along the axis included between z i and z i+1 : z i < ξi < z i+1 , z i < ηi < z i+1 , for i = 0 . . . n − 1. A lot of methods already existing in the literature can be applied to derive the pdf of the random vector V once known the pdf of the random vector D. Among these, we recall the probability transformation method (PTM) for linear maps developed by Falsone and Settineri [25], the PTM for nonlinear transformation maps between input and output variables given by La Valle et al. [37], the interval analysis [49], the Monte Carlo (MC) simulation method [19, 36].

5 On the Random Axially Functionally Graded Micropolar …

77

5.4.2 The Inverse Problem In Sect. 5.4.1, the possibility to introduce an application h between V and D, V = h ( D), is discussed. If h is an invertible function, an inverse map g = h−1 can be defined which can arrange the vector V to D, D = g (V ). In this case, the pdf of D can be derived once known the pdf of V via the PTM and the MC simulation. If g does not exist, we can still search an approximated function, gˆ , that approximatively behaves like an inverse function of h, i.e., D ≈ gˆ (V ). Also in this case, the pdfs of the entrances of D can be derived once known the pdf of V via the PTM and the MC simulation. If neither of the two listed approaches is applicable, for every V , a useful approximated, or exact, numerical solution D of the equation h( D) − V = 0 can often still be evaluated. Generating n random samples of V , n approximated or exact numerical outcomes of D can be obtained by numerically solving the equation h( D) − V = 0. Hence, the pdf of D can be derived via the MC simulation. When one of these three scenarios occurs, a simple identification of random constitutive parameters can be performed (See Soize [51, 52] for more advanced technique).

5.4.2.1

Simply Supported Beam

Let us focus on the simply supported beam model. In Sect. 5.4.1, h is defined as a function able to arrange the vector D, containing the inverse of the constitutive parameters (see Eq. (5.57)), to the vector V , containing the transversal displacements in some points of the axis (see Eq. (5.58)). In general, to perform direct and inverse random analyses, it is preferable to work with matrices. Since, for a simply supported beam, the function h is a linear map (see Eqs. (5.42 and 5.45)), the matrix H, for which V = h( D) = H D, can be associated to h. For the authors convenience, H is written as a block matrix: ( ) H1 H2 H= (5.59) H3 H4 whose submatrices elements are: ( )[ ( ) ( )] (H1 )i j = qˆ (1) z j−1 R ξi−1 − z j−1 − R ξi−1 − z j + [ ] ) ) ( ( ( ) R L − z R L − z j−1 j ξi−1 + ξi−1 +qˆ (1) z j−1 − L L ( )[ ( ) ( )] (H2 )i j = qˆ (2) z j−1 −Q ξi−1 − z j−1 + Q ξi−1 − z j + [ ( ] ) ) ( ( ) Q L − z j−1 Q L − zj (2) z j−1 ξi−1 − ξi−1 qˆ L L

(5.60)

(5.61)

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G. La Valle and G. Falsone

( )[ ( ) ( )] (H3 )i j = qˆ (1) z j−1 R ηi−1 − z j−1 − R ηi−1 − z j + [ ] ) ) ( ( ( ) R L − z j−1 R L − zj (1) z j−1 − ηi−1 + ηi−1 +qˆ L L ( )[ ( ) ( )] (H4 )i j = qˆ (2) z j−1 −Q ηi−1 − z j−1 + Q ηi−1 − z j [ ( ] ( ) ) ( ) Q L − z j−1 Q L − zj (2) qˆ z j−1 ηi−1 − ηi−1 L L

(5.62)

(5.63)

for i = 1 . . . n, where ξi−1 and ηi−1 are different coordinates along the axis included between z i−1 and z i and ξi−1 /= ηi−1 (see also Sect. 5.4.1). Equations (5.60, 5.61, 5.62 and 5.63) are obtained after simple but cumbersome mathematical steps that are omitted. Finally, we are allowed to write the linear system: V = H D = h ( D)

(5.64)

The approximated closed form solutions for the displacements proposed in Eqs. (5.42, 5.46 and 5.54), are characterized by a discretization of loads and constitutive parameters in n intervals. The reader could be interested in more compact and practical expressions, where the stochastic fields related to the constitutive parameters are meshed in m pieces such that m ≤ n. For instance, the reader may be looking for the stochastic response and/or identification of a beam for which a coarse discretization of the constitutive stochastic fields is acceptable. That is the case of beams characterized by mechanical properties which do not vary much along the axis or it could also be a necessity led by experimental measures of displacements in just few points of the beam (few strain gauges available). Hereby, a simplified version of the system (5.64) is derived able to solve the previous issues. Let us assume the random variables Bi and Ti equal into groups of l = 2n/m. In summary, the number n defines the mesh of the quantities related to the external loads; m the mesh of the constitutive stochastic fields. Under these hypotheses, the elements of D are also equal into group of l. The first terms of each group of the ˜ containing m l equal objects of D can be collected to create the reduced vector D, different entrances of D. The correspondent elements of V generate a reduced vector V˜ , whose dimension is also m. In detail, by involving an index notation, we fix: D˜ j = Dl j−l+1

V˜ j = Vl j−l+1

(5.65)

for j = 1, . . . , m. Considering the above, the matrix H is simplified by summing the columns which refer to the same random variables inside the vector D. The matrix ˜ is obtained whose generic element is given by: H

5 On the Random Axially Functionally Graded Micropolar …

H˜ i j =

l−1 Σ

Hli−l+1,l j−l+1+k

79

(5.66)

k=0

for j = 1 . . . m. The stochastic linear problem (5.64) is reduced as below: ˜ D ˜ V˜ = H

(5.67)

˜ is a m × m matrix. Both H and H ˜ are not always invertible. The singularity where H ˜ has a clear physical significance. There are different combinations of of H and H constitutive parameters Yi , G i and Ci able to give the same displacements. Never˜ can be found able to theless, via an optimized raw reduction algorithm, a matrix G approximatively transform the direct problem to the inverse one: ˜ V˜ ˜ ≈G D

(5.68)

for every l = 2n/m ≥ 2. ˜ just for l ≥ 2, this result is general enough. Although we succeed in finding G In fact, for any fixed m, it is possible to choose a value of n equal to 2n/l, and any level of accuracy can be achieved. High values of n are always advisable for accurate results. The same steps as before can be repeated for solving the axial stochastic problem. Briefly, by considering the approximated closed form solution (5.46), let us introduce a matrix K , which can discretize the axial solution of the equilibrium equation (5.20) and whose generic element, K i j , is given by: ( )[ ( ) ( )] K i j = − f (1) z j−1 R ξi−1 − z j−1 − R ξi−1 − z j + [ ] ) ) ( ( ( ) R L − z R L − z j−1 j ξi−1 + ξi−1 − f (1) z j−1 − L L

(5.69)

and the vectors S and U: ( S=

1 1 1 , ,..., A0 A1 An−1

)T

U = (U (ξ0 ) , . . . , U (ξn−1 ))T

(5.70)

(5.71)

˜ whose Finally, let us introduce the reduced matrix K˜ , and the vectors S˜ and U, components are equal to: K˜ i j =

l−1 Σ k=0

K li−l+1,l j−l+1+k

(5.72)

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G. La Valle and G. Falsone

S˜ j = Sl j−l+1

U˜ j = Ul j−l+1

(5.73)

for j = 1 . . . m. In this case, l = n/m. The reduced direct stochastic axial problem can be written as: (5.74) U˜ = K˜ S˜

5.4.2.2

Cantilever Beam

Analogous techniques can be used for solving the direct and inverse stochastic problems for a cantilever beam. Also in this case, the application h, which arranges the vector D to the vector V , is linear (see Eq. 5.54). We are led to the introduction of the associated block matrix H, whose generic entrances are equal to: ( )[ ( ) ( )] (H1 )i j = qˆ (1) z j−1 R ξi−1 − z j−1 − R ξi−1 − z j

(5.75)

( )[ ( ) ( )] (H2 )i j = qˆ (2) z j−1 −Q ξi−1 − z j−1 + Q ξi−1 − z j

(5.76)

( )[ ( ) ( )] (H1 )i j = qˆ (1) z j−1 R ηi−1 − z j−1 − R ηi−1 − z j

(5.77)

( )[ ( ) ( )] (H2 )i j = qˆ (2) z j−1 −Q ηi−1 − z j−1 + Q ηi−1 − z j

(5.78)

where ξi−1 and ηi−1 are different coordinates along the axis included between z i−1 and z i : z i−1 < ξi−1 < z i , z i−1 < ηi−1 < z i , ξi−1 /= ηi−1 , for i = 1 . . . n (see Sect. 5.4.1) and qˆ (2) is given by: qˆ (2) (z) = −q (2) (z) + q (1) (L) z + q (2) (L) − q (1) (L) L

(5.79)

We can conclude that the vectors D and V are linked with each other by means of ˜ the vectors D ˜ and V˜ can be defined as into the matrix H and, again, the matrix H, Eqs. (5.65 and 5.66) to consider a different discretization for loads and parameters. ˜ such that Unlike for the simply supported beam, it is not easy to find the matrix G, ˜ V˜ , without significant numerical errors. Nevertheless, for any vectors V and ˜D ≈ G ˜ by numerically solving the equations V˜ , we can find the corresponding D and D ˜ ˜ ˜ H D − V = 0 and H D − V = 0 (the same notation of Sect. 5.4.2.1 is maintained). Efficient numerical methods need to be used. The largest implication of not knowing ˜ is that the PTM cannot be applied for the stochastic identification of the system. G On the contrary, the MC method is useful for this purpose.

5 On the Random Axially Functionally Graded Micropolar …

81

Table 5.1 Dimensionless mean values of the random variables Yi for i = 0 . . . 4 μY0 /μYr μY1 /μYr μY2 /μYr μY3 /μYr μY4 /μYr 0.9

0.8

0.85

0.9

0.95

5.5 Numerical Applications Hereafter, the stochastic identification of axially functionally graded micropolar Timoshenko beams of different dimensions is performed. First, a simply supported beam of the usual size of civil engineering is studied. Secondly, a micro-beam is analyzed. The pdfs of Y (z), C(z), G(z) and ν(z) (assumed to be random fields stepwise defined and meshed in five intervals) are derived in different points of the beam axis via the MC method (3 · 105 samples) and the PTM. Finally, a cantilever beam is studied via the MC method (3 · 105 samples). The axial, a(z), bending, b(z), and shear stiffnesses, t (z), are fixed equal to a(z) = Y (z)A, b(z) = Y (z)I + C(z)A, t (z) = G(z) A/χ , where the geometric quantities are assumed to be deterministic. We choose the correction factor χ given by the energetic approach and Jourawsky formula (see [40]). Since it depends just on geometric quantities, it is considered deterministic.

5.5.1 Roughly Classical Timoshenko-Ehrenfest Beam In this subsection, a simply supported axially graded Timoshenko-Ehrenfest beam model is analyzed. The dimensions of the beam are fixed equal to: w = 3 · 10−1 m

h = 6 · 10−1 m

L = 5h

(5.80)

where w denotes the width and h the height of the cross-section, L the length of the axis. Since we are dealing with rectangular beams, χ = 1.2. The transversal load is fixed equal to 10 KN/m, q(z) = 10 KN/m. About the mechanical properties, we consider a discretization of the constitutive stochastic fields in five intervals, Y (z) = Σn−1 Σn−1 and G (z) = G 0 + i=1 Y0 + i=1 (Yi − Yi−1 ) U (z − z i ), (G i − G i−1 ) U (z − z i ). The following partition of the domain [0, L] is chosen: 0 = z 0 < z 1 = Δz < z 2 = 2Δz < · · · < z 5 = 5Δz = L. In each interval, the moduli Yi and G i , for i = 0 . . . 4, are assumed to be Gaussian random variables. Their mean values μYi and μG i , and correlation factors CvYi and CvG i are listed in Tables 5.1, 5.2, 5.3 and 5.4, where μYr = 200000 MPa and μG r = 76923 MPa. In Fig. 5.1, we show the deterministic deflection obtained in the) field of the ( Σn−1 μYi − μYi−1 and G (z) = Timoshenko-Ehrenfest theory with Y (z) = μY0 + i=1 ) Σn−1 ( μG 0 + i=1 μG i − μG i−1 . The related curve is denoted by ATBT. Moreover, the deflections derived in the field of the Timoshenko-Ehrenfest and Euler-Bernoulli

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Table 5.2 Dimensionless mean values of the random variables G i for i = 0 . . . 4 μG 0 /μG r μG 1 /μG r μG 2 /μG r μG 3 /μG r μG 4 /μG r 0.9

0.8

0.85

0.9

0.95

Table 5.3 Coefficients of variation of the random variables Yi for i = 0 . . . 4 CvY0 CvY1 CvY2 CvY3 CvY4 2.5%

1%

2%

1.5%

2.5%

Table 5.4 Coefficients of variation of the random variables G i for i = 0 . . . 4 CvG 0 CvG 1 CvG 2 CvG 3 CvG 4 2.5%

1%

Fig. 5.1 Deflection v (z) of the mean ATBT, mean homogeneous TBT, mean homogeneous EBT

2%

1.5%

2.5%

v(z)[mm] 12 10 v(z) ATBT

8

v(z) TBT

6

v(z) EBT

4 2 500

1000

1500

2000

2500

3000

z[mm]

Σn−1 theories are shown under the following positions: Y (z) = i=0 μYi /n and G (z) = Σn−1 μ /n . The related curves are denoted by TBT and EBT, respectively. For i=0 G i what concern the stochastic identification, a numerical experiment is performed. In detail, an “experimental” measurement is simulated by following the steps listed below. First, the Young moduli, Yi , and shear moduli, G i , are assumed to be independent gaussian random variables, whose properties can be found in Tables 5.1, 5.2, 5.3 and 5.4. Secondly, the direct problem is solved and the pdfs of the displacements are derived by means of the transformation shown in Eq. (5.64), V = H D, and the PTM in the frequency formulation (see [25]). Now, starting from the pdfs of the displacements, the pdfs of Bi and Ti can be derived in each of the five intervals of ˜ V˜ . The matrix G ˜ ˜ ≈G the beam via the transformation proposed in Eq. (5.68), D is obtained via the powerful row reduction algorithms implemented by the software Mathematica 12. The procedure is summarized below: pYi , pG j → pV˜1 ...V˜10 → p D˜1 ... D˜ 10 → p

1 I D˜ i+5

,p

χ A D˜ j+1

,

≈ pYi , pG j

(5.81)

5 On the Random Axially Functionally Graded Micropolar … Fig. 5.2 Probability density function pY3 of the Young modulus in the 4th interval

83

pY3 (y3 ) 0.00015

0.00010 pY3 (y3 ) via MC 0.00005

pY3 (y3 ) input pY3 (y3 ) via PTM 170 000 175 000 180 000 185 000 190 000

Fig. 5.3 Probability density function pG 3 of the shear modulus in the 4th interval

y3 [MPa]

pG3 (g3 ) 0.0004 0.0003 0.0002

pG3 (g3 ) via MC pG3 (g3 ) input

0.0001

pG3 (g3 ) via PTM 66 000

68 000

70 000

Table 5.5 Mean values of the random variables Ni , for i = 0 . . . 4 μ N1 μ N2 μ N3 μ N0 0.3

0.3

0.3

0.3

72 000

74 000

g3 [MPa]

μ N4 0.3

for i, j = 0 . . . 4. The output pdfs of Bi /I = Yi + Ci A/I and χ Ti /A = G i are approximatively the same of the starting gaussian ones related to Yi and G i (see Figs. 5.2 and 5.3). The MC method and the PTM in the frequency formulation are both applied. To derive the pdfs, p Ni , of the Poisson coefficients in each interval, the classical relationship between Yi , G i and Ni is considered (see Eq. (5.8)). For every i = 0 . . . 4, the random variable X = G i is defined, and the PTM in the classical form is applied to the vector random variables (Ni , G i )T and (G i , Yi )T . Once obtained the joint pdf p Ni G i , the pdf p Ni is derived from the saturation with respect to G i . About the application of the PTM in stochastic problems where the input and output variables have different dimensions, see [32]. Although the random variables Ni do not follow a gaussian distribution (see Fig. 5.4), their mean values and coefficients of variation are reported in Tables 5.5 and 5.6. It is noteworthy, the coefficients of variation of Ni are significantly bigger than the ones fixed for the random variables Yi and G i .

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Fig. 5.4 Probability density function p N3 of the Poisson coefficient in the 4th interval

pN3 ( 3 ) 15

10

pN3 ( 3 ) via MC

5

pN3 ( 3 ) via PTM

0.20

0.25

0.30

0.35

0.40

ν3

Table 5.6 Coefficients of variation of the random variables Ni , for i = 0 . . . 4 Cv N1 Cv N2 Cv N3 Cv N4 Cv N0 15.3%

6.1%

12.3%

9.2%

15.3%

Table 5.7 Dimensionless mean values of the random variables Ci and for i = 0 . . . 4 μC0 /μCr μC1 /μCr μC2 /μCr μC3 /μCr μC4 /μCr 0.9

0.8

0.85

0.9

0.95

Table 5.8 Coefficients of variation of the random variables Ci for i = 0 . . . 4 CvC1 CvC2 CvC3 CvC4 CvC0 2.5%

1%

2%

1.5%

2.5%

5.5.2 Micropolar Timoshenko Beam Now, a simply supported axially functionally graded Timoshenko-Ehrenfest beam model, with a clear micropolar behavior, is studied. The geometrical properties are fixed equal to: (5.82) w = 3 · 10−6 m h = 6 · 10−6 m L = 5h The considered 1D element is much smaller than the ones usually involved in civil engineering. Together with the random variables Yi and G i defined in Tables 5.1, 5.2, 5.3 and 5.4, we introduce the random variables CiΣ , able to describe the Cosserat modun−1 lus in the ith interval of the beam, C (z) = C0 + i=1 (Ci − Ci−1 ) U (z − z i ). Since they are assumed to follow a Gaussian distribution, specifying their mean values and coefficients of variation is sufficient to fully characterize these random constitutive parameters. In this regard, see Tables 5.7 and 5.8, where μCr = 100000 MPa. The applied procedure consists of the following steps:

5 On the Random Axially Functionally Graded Micropolar … Fig. 5.5 Deflection v(z) Of the mean ATBT and homogeneous TBT

85

v(z)[mm] 0.00010 0.00008 0.00006 v(z) ATBT

0.00004

v(z) TBT

0.00002 0.005

Fig. 5.6 Probability density function pC3 of the shear modulus in the 4th interval

0.010

0.015

0.020

0.025

z[mm] 0.030

pC3 (c3 ) 0.00015

0.00010 pC3 (c3 ) via MC 0.00005

pC3 (c3 ) input pC3 (c3 ) via PTM 80 000

pCi , pG j → pV˜1 ...V˜10 → p D˜1 ... D˜ 10 → p

85 000

1 A D˜ i+5

90 000

,p

χ A D˜ j+1

,

95 000 100 000

≈ pCi , pG j

c3 [MPa]

(5.83)

for i, j = 0 . . . 4. The main difference between Eq. (5.83) and Eq. (5.81) is that the pdfs p1/( A D˜ i+5 ) are approximable up to pCi . This result is highlighted in Fig. 5.6, and it is due to the small dimensions of the studied structural element. Thanks to the geometrical properties (see Eq. (5.82)), the moment of inertia I is negligible with respect to the area of the section, A (see Sect. 5.2.2). Then, the output pdfs of Bi /A = Yi I / A + Ci are approximable up to the pdfs of Ci . On the other hand, the Young modulus can be identified by means of a similar procedure applied to ˜ It consists of the the axial equilibrium described by transformation (5.74), U˜ = K˜ S. following steps: (5.84) pYi → pU˜1 ...U˜ 5 → p S˜1 ... S˜5 → p ˜1 ≈ pYi A Si+1

for i = 0 . . . 4. The derived pdfs of Yi , G i and Ni are the same of the ones reported in Figs. 5.2, 5.3 and 5.4. In Fig. 5.5, it is shown the deflection of the( Timoshenko-Ehrenfest beam ) Σn−1 μYi − μYi−1 , G (z) = μG 0 + model obtained by fixing Y (z) = μY0 + i=1 ) ) Σn−1 ( Σn−1 ( i=1 μG i − μG i−1 , and C (z) = μC0 + i=1 μCi − μCi−1 . The related curve Σn−1 is denoted by ATBT. Moreover, the deflection obtained by Y (z) = i=0 μYi /n, Σn−1 Σn−1 G (z) = i=0 μG i /n and C (z) = i=0 μCi /n is shown. The latter is denoted by

86 Fig. 5.7 Deflection v(z) of the mean ATBT, mean homogeneous TBT, mean homogeneous EBT

G. La Valle and G. Falsone v(z)[mm] 30 25 20

v(z) ATBT v(z) TBT

15

v(z) EBT

10 5 500

1000

1500

2000

2500

3000

z[mm]

TBT. The deflection related to the Euler-Bernoulli model is omitted since it is significantly different from the ones derived via the Timoshenko-Ehrenfest ones (Fig. 5.6).

5.5.3 Cantilever Beam Similar numerical applications are performed for a cantilever beam under the same geometrical and loads hypotheses. About the constitutive parameters, the same assumptions as before are maintained for the first four intervals. The fifth interval is supposed to be equal to the fourth, Y4 = Y3 , G 4 = G 3 , C4 = C3 . In general, if the constitutive fields are meshed into m intervals, the constraints Ym−1 = Ym−2 , G m−1 = G m−2 , Cm−1 = Cm−2 , must be added to guarantee the reliability of the following procedure. The pdfs are obtained for both, the roughly classical Timoshenko-Ehrenfest beam and the purely micropolar model. The input and output pdfs of Yi , G i and Ni , for i = 0 . . . 4, are the same of those shown in Figs. 5.2, 5.3, 5.4 and 5.5. However, they are derived via the MC method (the PTM is not applied anymore). First, Yi , G i and Ci for i = 0 . . . 4 are assumed to be gaussian random variables (see Tables from 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7 and 5.8). Secondly, samples of V˜l , ˜ defined in l = 1 . . . 10, and U˜ k , k = 1 . . . 5, are generated through the matrix H ˜ ˜ Sect. 5.4.2.2. The corresponding outcomes D j and Sk are evaluated by numerically ˜ D ˜ − V˜ = 0 and K˜ S˜ − U˜ = 0 via the powerful tools of solving the equations H the software Mathematica 12, which computes a numerical Gröbner basis using an efficient monomial ordering, then uses eigensystem methods to extract numerical roots. The output pdfs of Yi , G i and Ci are derived for beams of usual size in civil engineering and micro-beams. In Figs. 5.7 and 5.8, we compare the responses obtained in the field of the Timoshenko-Ehrenfest and( Euler-Bernoulli beam models where the Young ) Σn−1 μYi − μYi−1 , the shear modulus G (z) = μG 0 + modulus Y (z) = μY0 + i=1 ) ) Σn−1 ( Σn−1 ( i=1 μG i − μG i−1 and the Cosserat modulus C (z) = μC0 + i=1 μCi − μCi−1 ;

5 On the Random Axially Functionally Graded Micropolar … Fig. 5.8 Deflection v(z) of the mean ATBT and mean homogeneous TBT

87

v(z)[mm] 0.0004 0.0003 0.0002

v(z) ATBT v(z) TBT

0.0001 0.005

0.010

0.015

0.020

0.025

z[mm] 0.030

Σn−1 Σn−1 Σn−1 Y (z) = i=0 μYi /n, G (z) = i=0 μG i /n, C (z) = i=0 μCi /n. The different curves are denoted by ATBT, TBT and EBT, respectively. As for the simply supported micro-beam, the curve EBT derived for the cantilever micro-beam is omitted.

5.6 Conclusion The stochastic identification of a micropolar axial functionally graded TimoshenkoEhrenfest beam is discussed in the present work. After deriving approximated closed form solutions for transversal and axial displacements, some “experimental” measurements are simulated. Then, the probability density functions (pdfs) of the involved constitutive parameters in each point of the axis are derived for simply supported and cantilever beams. About the main results, it is proven that one 3D constitutive micropolar parameter can be evaluated by stochastically identifying a micropolar Timoshenko micro-beam; the direct and inverse problem of a simply supported random axially functionally graded micropolar Timoshenko beam can be analytically solved. Big coefficients of variation of the Poisson coefficients with respect to the other material moduli have been obtained. The used techniques are applicable for studying beams characterized by random axial defects, bleeding phenomena and step beams for which the new approximated closed form solutions related to the displacements are extremely useful. In the authors’ knowledge, the proposed approaches and solutions are the simplest ones findable in the literature. Consequently, they can find applications in several technical contexts.

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Chapter 6

Influence of Soil-Pile-Structure Interaction on Seismic Response of Reinforced Concrete Buildings Ricardo Madrid, David Zegarra, Pablo Perez, and Miguel Roncal

Abstract It is well known that Peru is located in one of the most active seismic zones in the world and seismic design is a primary concern. Building constructions has increased in Peru during the last few years, especially on the north coast where loose soil deposits predominate. Soil conditions have increased the need of pile foundations. However, the soil-pile-structure interaction is usually not considered in design of superstructures. The soil-foundation system beneath the superstructure influences the seismic performance of buildings. The soil-pile-structure interaction is influenced by the highly non-linear behaviour of the soil, where the interface pilesoil seems to play an important role in design. This study focusses on the influence of soil-pile-structure interaction on seismic response of buildings. A dynamic finite element model of a typical building founded on sandy soil is used to investigate the ground response and earthquake behaviour of a reinforced concrete building based on frames and plates. The results are presented in terms of the most important design parameters such as the lateral displacement of the piles, shear stress, bending moments, structural inter-storey drifts and storey lateral displacements. Effect on ground response in term of spectral acceleration and changes in natural period are also discussed.

R. Madrid (B) · D. Zegarra · P. Perez Universidad de Lima, Lima, Peru e-mail: [email protected] D. Zegarra e-mail: [email protected] P. Perez e-mail: [email protected] M. Roncal National University of Engineering, Lima, Peru e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. dell’Isola et al. (eds.), Advances in Mechanics of Materials for Environmental and Civil Engineering, Advanced Structured Materials 197, https://doi.org/10.1007/978-3-031-37101-1_6

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6.1 Introduction The soil-structure interaction (SSI) is a well-studied topic in geotechnical community. It is recognised that SSI may change the frequency of ground motion effecting the seismic performance of buildings. Mylonakis and Gazetas [1] explained that the SSI may increase the natural period of the structure and increase the ductility demand, depending on the characteristics of the motion and the structure. Also, Mylonakis et al. [2] shown that the collapse of Hanshin Expressway during the 1997 Kobe earthquake was due to SSI effects that modified the frequency of ground motion and the fundamental period of the structure. Failure of piles supported building were also observed by Tokimatsu et al. [3] during the 2011 Tohoku earthquake. Although several research has shown recently that the effect of SSI is essential for designing of ductile reinforced concrete building frames, bridges or other structures resting on soft and very soft soil [4, 5], the Peruvian code of seismic design (E-030) ignore the SSI effects on building response. Different approaches are used to study numerically the SSI. The direct approach is more time consuming, but permitted the modelling of soil behaviour, foundation and structure as a whole [6–10] while in the sub-structure approach, foundation impedance functions of sub-structure systems are estimated first and assigned for the seismic analysis of super-structures [11–14]. The last one is most expanded in practice engineering. Also, analytical solutions based on plasticity theorems have been proposed. Sessa et al. [15] present an interesting description of several methods to evaluate capacity surfaces of reinforced concrete based on plasticity approach. Limit analysis was shown to produce accurate results capable of reproducing the nonlinear response and safety checks of reinforced concrete cross sections. Marmo et al. [16] developed an effective approach to deal with the problem of soil-structure interaction of a homogeneous isotropic half-space. The method was later extended by Marmo et al. [17] to cover the solutions to the elastic equilibrium problem of transversely isotropic media. The nonlinearity of soft soils and piles increase the complexity of seismic analysis of soil-structure interaction making their effect not usually taken into account. Soilpile-structure response have been described by different researchers mainly in terms of storey displacements, inter-storey drifts and fundamental period of the structures. Their effect on piles response is less explored. Hokmabadi et al. [8] and Hokmabadi and Fatahi [18] observed that the storey displacements and inter-storey drifts were amplified in the case of pile-supported structures. Also, the main factors affecting the structures response was identified as the group effect of piles [19, 20] and the number of piles in a group, spacing of piles and diameter of pile [21, 22]. Very few research works are available on the effects of soil-pile structure interaction with advance constitutive models to assess the structural response of buildings. To account this problem into consideration, two constitutive models: the Hardening Soil model (HS model) and the more advanced Hardening Soil model with consideration of small strain stiffness (HSS model) is used in this study. A dynamic finite element

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model which includes elastic structural elements to describe the response of superstructure and piles was chosen to use the direct approach. Plaxis 3D software was used to carried out the analyses. Three different records of strong ground motion were adjusted to match the design spectra according to the Peruvian code. Results are presented in terms of the most significant design parameters such as pile lateral displacement, structural inter-storey drifts and storey lateral displacements, shear stress and bending moment of piles. In addition, the influence of piles in the natural frequency of the structure and soil is discussed.

6.2 Soil Constitutive Models Two well-known constitutive models were proved in this investigation. They are all based on the framework of elasto-plasticity and both are implemented in the Plaxis code. A brief description based on the Plaxis Manual is presented below. For more details the reader is referred to Benz et al. [23].

6.2.1 Hardening Soil Model The hardening soil model is an advanced model which introduce a yield surface that is not fixed in principal stress space, as it can expand due to plastic straining. This model uses in its formulation two types of hardening: shear hardening that respond to irreversible strains due to primary deviatoric loading and the compression hardening to model plastic strain due to primary compression in oedometer loading or isotropic loading. This feature makes the model able to simulate the behaviour of several soil types, from soft soils to stiff soils. In fact, the model can be considered as an extension of the well-known based on elasticity hyperbolic model [24]. When consider stress path of standard drained triaxial tests the model gives the hyperbolic stress strain curve. The Hardening Soil model, supersedes the hyperbolic model by three features: It is based on the theory of plasticity, includes soil dilatancy and introduces a yield cap. In triaxial p-q space the shear hardening function is defined as: f =

q 2 Ei 1 −

q qa

−

2q −γp E ur

(6.1)

where: Ei =

2E 50 2 − Rf

(6.2)

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Fig. 6.1 Yield loci for various values of γp and m = 0.5 [25]

( ref

E 50 = E 50

( E ur = q

ref E ur

c cosφ − σ3, sinφ c cosφ − pr e f sinφ c cosφ − σ3, sinφ c cosφ − pr e f sinφ

ref

)m (6.3) )m (6.4) ref

The failure ratio is R f = qaf ; E 50 is the reference stiffness modulus and E ur is the reference unloading–reloading modulus corresponding to the reference pressure pref . It is observed that the shape of the yield loci depends on the exponent m. Figure 6.1 shows the shape of yield loci for increasing values of γ p and m = 0.5. The HS model involves a relationship between rates of volumetric plastic strain p ε˙ v and shear plastic strain γ˙ p , as follow: ε˙ vp = sin(ψm )γ˙ p

(6.5)

The mobilised dilatancy angle ψm follows the stress-dilatancy theory [26, 27] for larger values of the mobilised friction angle φm , whereas for small mobilised friction angles and for negative values of ψm , is taken zero. sinψm =

sinφm − sinφcv 1 − sinφm sinφcv

(6.6)

where φcv is the critical state friction angle and the mobilised friction angle φm is defined as: sinφm =

σ1,

σ1, − σ3, + σ3, − 2ccotφ

(6.7)

A second yield surface is introduced to close the elastic region and simulate compressive behaviour. The cap yield surface takes the form of an ellipse with its centre point in the origin. The magnitude of the yield cap is determined by the isotropic preconsolidation stress pp and its aspect ratio is controlled by Mpp (see Fig. 6.2). The ellipse is used both as a yield surface and as a plastic potential (associated plasticity) and is defined as follow:

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Fig. 6.2 Yield surfaces in the p-q plane [29]

fc =

q2 2 + p , − p 2p M2

(6.8)

where M is an auxiliary parameter related to K0 . The hardening law relating p˙ p to pc volumetric cap strain ε˙ v is: ε˙ vpc ref

in which K s

=

Ks Kc ref Ks

[(

p p + c cot φ pr e f + c cot φ

)−m ]

p˙ p

(6.9)

is the reference bulk modulus in unloading/reloading: ref

K sr e f =

E ur 3(1 − 2νur )

(6.10)

and Ks/Kc is the ratio of bulk moduli in isotropic swelling and primary isotropic compression, which can be approximate as: ref

K s /K c ≈ ref

E ur

ref

E oed

ref

K 0nc (1 + 2K 0nc )(1 − 2νur )

(6.11)

In this way, K 0nc , E ur and E edo determine the magnitude of M and Ks /Kc respecref tively. E 50 controls the magnitude of the plastic strains that are associated with the ref shear yield surface and E edo is used to control the magnitude of plastic strains from the yield cap. When using the model in dynamic calculations, the stiffness parameters Eur need to be selected such that the model correctly predicts wave velocities in the soil. The Hardening Soil model will generate plastic strains when mobilizing the soil’s material strength (shear hardening) or increasing the soil’s preconsolidation stress (compaction hardening). However, for stress cycles within the current hardening contours the model will only generate elastic strains and no (hysteretic) damping, nor accumulation of strains or pore pressure. It should be necessary to define Rayleigh

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damping to simulate soil’s damping characteristics. The standard code damping was assumed in this soil model.

6.2.2 Hardening Soil Model with Small-Strain Stiffness (HSS) The HSS model is an enhanced version of the Hardening Soil model (HS) implemented in Plaxis [25], which include in its formulation, the very small-strain soil stiffness and its non-linear dependency on strain amplitude. This feature makes it better for simulating the soil response to an earthquake. Following the formulation of the HS model, the model distinguishes between two types of hardening: shear hardening used to simulate irreversible deformations due to a deviatoric load and compression hardening to simulate irreversible plastic deformations due to isotropic compression. The model incorporates the Mohr–Coulomb failure criterion to define soil resistance, but unlike the perfect elasto-plastic Mohr–Coulomb model, the yield surface of the HSS model can expand in the principal stress space due to plastic deformations. Additionally, the HSS model allows to simulate the non-linear behaviour of the soil in small strain range. For which, it incorporates a non-linear relation between degradation of stiffness and increase in deformations following the hyperbolic law suggested by Dos Santos and Correia [28]. The model introduces two additional parameters γ0.7 and G0 . The parameter γ0.7 is the shear deformation for which the stiffness modulus has degraded to 70% of its initial value in small strain, G0 . The stress dependency of G0 is taken into account with the power law, as follow: ( G0 =

ref G0

ccosφ − σ3, sinφ ccosφ − pr e f sinφ

)m (6.12)

More details about the HSS model can be found in Benz [29].

6.3 Earthquake Design Definition Acceleration time histories were constructed by spectrally matching of the spectral accelerations in the time-domain to the Peruvian Seismic Code E.030 design response spectrum [30]. This design response spectrum was built considering acceleration zone (Z) factor of 0.45 which represents de peak ground acceleration of the coastal region of Peru; soil (S) amplification factor of 1.05, and 5% of damping ratio. This design response spectrum represents an event with 10% probability of exceedance in 50 years or 475-year of return period.

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Time-domain spectral matching adds wavelets in the time domain to improve the spectral deficiencies of the seed motion. The wavelets introduce less energy into the acceleration time history and preserve the non-stationary characteristics of the initial time history [31]. Each wavelet is applied to the time history so that the time of the maximum spectral response in the adjusted time history occurs at the same time as the maximum spectral response in the unadjusted time history. The fundamental assumption is that the peak response does not change as a result of adding the wavelet adjustment [31]. Typically, several iterations are required to obtain reasonable convergence between the spectrally-matched acceleration time history and the design response spectrum. A final baseline correction is often necessary to remove any permanent offset on the displacement time history introduced during the spectral matching procedure. The spectral matching in this study was accomplished using the computer program SeismoMatch 2020 which is an application capable of adjusting earthquake accelerograms to match a specific target response spectrum, using the wavelets algorithm proposed by Abrahamson [32] and Hancock et al. [31]. In general, there are often insufficient recordings in the empirical earthquake strong motion database to satisfy all the conditions often required in the state of art for the engineering practice. Bommer and Acevedo [33] and Abrahamson [34] suggests that in selecting acceleration time-histories, the key earthquake parameters that will affect the nonstationary character of the waveform are magnitude, distance, and directivity (for sites located close to large faults). For earthquake magnitude, the recordings should be within 0.5 magnitude units of the selected design earthquake (e.g., mean earthquake magnitude from deaggregation results). For distance selection, the proximity to the seismic source is very important. If multiple recordings are used, then they should be selected so that they have a range of non-stationary characteristics from various ground motions induced by multiple earthquakes. Abrahamson [34] also argues that finding an acceleration record with the same type of faulting (e.g. normal, reverse, strike-slip) as the design earthquake is not very important. The subsurface soil condition is also less relevant for spectral matching because the spectral matching process will correct for the differences in frequency content between different soil conditions [35]. The acceleration time histories were selected from the following earthquakes: • 2001, (8.4 Mw) Arequipa earthquake • 2007, (7.9 Mw) Pisco-Ica earthquake • 2010, (8.8 Mw) Maule-Chile. Time histories records were obtained from University of Chile and CISMID (Peruvian-Japanese Center for Seismic Research and Disaster Mitigation). Figure 6.3 shows the acceleration time-history and the spectral matching of the seismic used in the numerical simulations.

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Fig. 6.3 Acceleration time-history and spectral matching of seismic used in the numerical analyses

6.4 Numerical Model In this study, a 10-story reinforced concrete building, typical of those currently being built on the north coast of Peru, was considered. The width of the building is 12 m in North direction and 22.5 m in South direction (‘X’ and ‘Y’ direction in the model, respectively), and the height of the building is 27.7 m. The seismic response of the soil-pile-structure interaction was studied using geotechnical finite element code PLAXIS 3D. This program was chosen because its ability to simulate nonlinear behaviour of soils and because it includes structural elements to model the building components (plates, beams, columns) and piles. Plates elements are shell elements used to model thin two-dimensional structural elements with a flexural rigidity. According to Mindlin’s plate theory (Bathe,), deflection in plate element is due to shearing as well as bending. Building slabs and plates were modelled using ‘plate element’. The thickness of the floors [36] lab was considered to be 250 mm while raft foundation was 600 mm thickness. Beams and columns were modelled using 3D ‘beam element’, which is a special structural element with a significant bending stiffness and axial stiffness. Piles were modelled with the ‘embedded beam element’. Elastic material properties were adopted for modelling the building

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structure, including raft and pile foundations. The value of modulus of elasticity √ of concrete was estimated using the relation suggested by ACI 318–11: EC = 4700 f’c (MPa), where f’c is the compressive strength of concrete. The dead weight and super-imposed loads (5 kN/m2 ) were adopted according to the E-020 Peruvian code. The live load was assigned as surface pressure on each floor slabs. The soil medium was considered as semi-infinite. The horizontal distance of the soil lateral boundaries was 600 m in Y direction and 360 m in X direction. As suggested by Rayhani and El Naggar [37], the model boundary must be at least 5 times more than the width of the structure, which is far outweighed in this case. The bedrock depth of 30 m was considered in the numerical analysis (see Fig. 4a). Ten nodes tetrahedral elements were used for discretised soil clusters. Stage construction process was used to model the sequence of construction of the building. Twelve phases were needed to define the model. After the initial stress computation has finished, the plastic computation phases corresponding to the activation of structural elements of each storey is started. Once the building construction is completed, the static load representing dead and live loads was activated. Finally, full dynamic computation is carried out using the three acceleration seismic records described previously. The fully fixed and horizontally fixed boundary condition are considered at base and sides, respectively, in static analyses. Special viscous boundaries were assigned during dynamic analyses. For reducing the analysis running time and also for convergence of results, fine meshing was chosen in this study. The piles load capacity was calculated using the static analysis suggested by AASHTO [38]. 15 piles of 20 m long and 0.80 m in diameter were designed to support the building, the adopted spacing of 3.2 m was enough to avoid any group effect. The individual capacity of the piles was computed at 2730 kN. The structure modelled and piled raft foundation is shown in Fig. 6.4.

Fig. 6.4 3D Finite element model for soil-pile-structure interaction analysis. a View of the central part of the model, b piled—raft model, c plan view of foundation

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Table 6.1 Material parameters of HS model Model

φ [°]

c [kN/m2 ]

ψ [°]

m

E50 [kN/ m2 ]

Eoed [kN/ m2 ]

Eur [kN/m2 ]

k [m/d]

Plane strain

25

1

0

0.5

1 × 104

1 × 104

3 × 104

8.64 × 10–4

0.8

6×

6×

1.8 ×

8.64 × 10–4

30

1

0

104

104

105

6.5 Soil Parameters Foundation soils in the study area, are mainly formed by medium compactness to dense sands stratifications and medium consistency clays. Soil parameters were stablished from SPT tests, Multichannel Analysis of Surface waves (MASW) and Microtremor Array Measurement (MAM) tests. The stratigraphic model was defined with 2 layers: the first layer about 10 m, characterized by an SPT N value 30 blows/ft) and Vs that varies between 300 and 450 m/s up to 30 m depth. The groundwater level is about −4 m to −5 m. The model parameters adopted to simulate this soil conditions are indicated in Table 6.1 (HS model) and Table 6.2 (HSS model). Figure 6.5 shows the Versus wave profile measured with the MASW and MAM tests compared to the wave profile computed by the HSS model. The relationship of the reduction factor of shear modulus and damping ratio with the cyclic shear strain adopted in the HSS model is shown in Fig. 6.6. Rollins et al. [39] proposed Gs /G0 curves are also shown in this figure.

6.6 Results and Discussion The results of this study revealed important insight of the influence of piles on the seismic interaction of soil-pile-structure that can contribute to enhanced geotechnical and structural design of buildings. Pile response is first examinate based on lateral displacement, shear forces and bending moment. Then the storey lateral displacement and inter-storey drift values are presented. Finally, the influence of piles over the spectral acceleration of the ground and frequency vibration of building is discussed.

6.6.1 Evaluation of Piles Response The computed values of maximum pile horizontal displacements for the three seismic records analysed are shown in Fig. 6.7. Data reported correspond to the central pile, which is a good representative of the entire pile foundation. Both models (HS and HSS model) were used to compute the displacement. The maximum lateral displacements

1

1

25

30

Plane strain

c [kN/m2 ]

φ [°]

Model

0

0

ψ [°]

Table 6.2 Material parameters of HSS model

0.8

0.5

m

1.8 ×

6×

6× 105

3 × 104

104

1 × 104

104

1 × 104

Eur [kN/m2 ]

Eoed [kN/m2 ]

E50 [kN/m2 ]

2.2 ×

105

1 × 105

G0 [kN/m2 ]

1×

10–4

1 × 10–4

γ0.7

8.64 × 10–4

8.64 × 10–4

k [m/d]

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Fig. 6.5 Shear wave velocity profile with depth, measured and computed with the HSS model

1.20E+00

Fig. 6.6 Relationship between a Gs/G0 and cyclic shear strains, b damping ratio and cyclic shear strain

1.00E+00 G0/Gur=4

Gs/G0

8.00E-01

G0/Gur=5.6

6.00E-01

Rolling et al (1998)

4.00E-01 2.00E-01 0.00E+00 0.000001 0.00001

0.0001

0.001

0.01

Cyclic shear strain

0.1

1

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Fig. 6.7 Maximum pile lateral displacement computed from several earthquake

were observed at time 75 s for the Arequipa earthquake (42 mm and 90 mm), 25 s for the Pisco earthquake (98 mm and 52 mm) and 40 s for the Chile earthquake (25 mm and 40 mm). Although there is no specific allowable lateral displacement in the Peruvian Code, it is accepted that 1 inch is an acceptable value for buildings. The tolerance is largely exceeded in all computations; however, the maximum lateral displacement is obtained when the HSS model is used, which is expected due to a larger deformation during the seismic event produces a degradation of soil stiffness. Visuvasam and Chandrasekaran [6] identified that the lateral displacements of the piles are greater as the number of storeys increases, due to greater flexibility of the superstructure. In addition, decreased soil stiffness and decreased pile spacing result in larger pile displacements. Although the shear stresses and bending moments of the piles seem not to be affected by the earthquake used in the analysis, the bending moment are about 25% lower when the small strain stiffness is considered in the constitutive model (Figs. 6.8 and 6.9). This means that the selection of the constitutive model has a great impact on the final design of the piles.

6.6.2 Storey Lateral Displacement Computation of maximum lateral displacements of the building stories during the three selected earthquakes were carried out with the two constitutive models described above. That means, foundation soil behaviour is characterized as a non

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Fig. 6.8 Shear stresses in the piles. a Arequipa earthquake. b Pisco earthquake. c Chile Earthquake

Fig. 6.9 Bending moments in the piles. a Arequipa earthquake. b Pisco earthquake. c Chile Earthquake

linear material while the structure behaviour is linear. The results considering piles and raft foundation is shown in Fig. 6.10 and without piles in Fig. 6.11. According to the results, the use of the HS model generated more lateral displacement in the structure compared to the results obtained with the HSS model, in all the earthquakes tested, which is to be expected due to the absence of energy dissipation (damping of the material) of this model. In general, the pile foundation tended to reduce the lateral displacement by almost 2 to 3 times when compared to shallow foundation.

6.6.3 Inter-Storey Drift Ratio The inter-storey drift ratio (IDR) is defined as the relation of the relative lateral displacement and the corresponding storey height: IDR = ( Xi + 1 − Xi)/( Hi + 1 − Hi). Where Xi + 1 = storey lateral displacement at i + 1th storey; Xi = storey lateral displacement at ith storey; Hi + 1 = height at i + 1th storey level and Hi = height at ith storey level. Peruvian code E-030 stablishes a permissible value of 0.7%

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Fig. 6.10 FEM predictions of the maximum lateral displacement of the 3D model with piles. a X direction, b Y direction

Fig. 6.11 FEM predictions of the maximum lateral displacement of the 3D model without piles. a X direction, b Y direction

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Fig. 6.12 FEM predictions of the inter-storey drift of the model with piles. a X direction, b Y direction

for concrete structures. The results (Figs. 6.12 and 6.13) show that the IDR values computed with the HSS model are sligthly lower than the corresponding values computed with the HS model in almost all the cases studied, but the difference is more notable in the case of shallow foundations (Fig. 6.13). Also note that for the Pisco earthquake neither the pile foundation nor the shallow foundation meet the permissible value of 0.7%, whatever the model used. It is clear from the results that pile foundations show a better structural response and with some further refinement in the structural design, the allowable IDR could be met.

6.6.4 Effects on Natural Period of Vibration The comparison between the pseudo spectral acceleration computed at the base of the model (input earthquake) and at the ground surface (base of building) is shown in Fig. 6.14. Two models were computed, one corresponding to a foundation with piles and the other without piles and the soils response in both models was examinated. It is observed in all the models that the natural period of soil increases from approximately 0.5–0.8 s when piles are included in the model. Also notice that, ground vibration is amplified between periods of 0.6–1.5 s and an attenuation of the soil response is observed for period lower than 0.6 s. It is seems that the pile stiffness acts as a

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Fig. 6.13 FEM predictions of the inter-storey drift of the model without piles. a X direction, (b) Y direction

filter, absorbing the high frequency components of the motion and amplifying the low frequency component of ground vibrations. Figure 6.15 shows the seismic response of the structure (data correspond to the center of gravity of the upper slab). The model without piles shows a fundamental vibration of about 0.9 Hz, while the model with piles the frequency of vibration increases to 1.5 Hz. The higher frequency obtained in the model with piles is explained by the increase in the stiffness of the soil-pile assembly, mainly produced by the high stiffness of concrete pile, and where the main factors that impact on the stiffness of the assembly are the distance between piles and pile diameter. Taking into account that the natural period of vibration of 30-m-high concrete buidings is around 0.9 s (T = CH3/4 , C = 0.075 for concrete frame buidings), care must be taken with this type of soil stiffening, to avoid the always inappropriate resonance phenomenon between the ground and the building.

6.7 Conclusions A 3D dynamic finite element model of a typical building and pile foundation was used to investigate the ground response and earthquake behaviour of a reinforced concrete building based on frames and plates. The soil behaviour was simulated with two advance nonlinear constitutive models, one of them including the small strain

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Fig. 6.14 Computation of Pseudo spectral acceleration (PSA) at the base and ground surface using different earthquakes (the left side corresponds to models without piles, the right side to models with piles)

stiffness behaviour. The building structure was simulated with special models based on the Mindlin’s theory. The acceleration time histories were defined after an iterative procedure of matching the spectral accelerations in the time domain of the Peruvian Seismic Code E-030. Design response spectrum was built considering parameters appropriated for the north coast of Peru: zone (Z) factor of 0.45, soil (S) factor of 1.05 and 5% of damping ratio. The design response spectrum represents an event with 10% probability of exceedance in 50 years or a 475 years of return period. Three seismic records were selected for analysis. The lateral displacement of the pile seems to be affected by the interaction with the building, which produces a greater lateral displacement. Of course, the taller the building, the greater the impact on the piles. Increasing the pile number and diameter can improve the foundation performance. Lateral storey displacement and drift ratio were also calculated. An almost linear displacement distribution was observed along the building height with an inflection point around the 4th floor, which is greatly reduced by the inclusion of piles in

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Fig. 6.15 Computation of the Fourier Amplitude of the top of building using different earthquakes (the left side corresponds to models without piles, the right side to models with piles)

the foundation. In addition, the drift between floors is significantly reduced in the foundation with piles. Contrary to what is expected, the soil-pile-structure interaction may increase the natural period of the ground. It is seems that the pile stiffness behaves as a filter, attenuating the high frequency components of the motion and amplifying the low frequency component of ground vibrations. The seismic response of the building is also affected by the soil-pile-structure interaction. The increase in the stiffness of the soil-pile system seems to produce a higher frequency of vibration of the building. Therefore, ignoring the soil-pilestructure interaction in the analysis and design of buildings may lead to erroneous results and it is recommended that the entire soil–pile–structure foundation system should be evaluated through a robust procedure. Finally, further research shall include modelling of various pile distributions and lengths and their influence on reinforced concrete structures considering global torsion effects, as described and modelled by [40]. Also, the computational strategies described by Sessa et al. [41, 42] could be useful to analyses the structure response.

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References 1. Mylonakis G, Gazetas G (2000) Seismic soil–structure interaction: beneficial or detrimental? J Earthquake Eng 4(3):277–301 2. Mylonakis G, Gazetas G, Nikolaou S, Michaelides O (2000) The role of soil on the collapse of collapse of 18 piers of the Hanshin expressway in the Kobe earthquake. In: Proceedings of the 12th world conference on earthquake engineering. Auckland, New Zealand, 1074, pp 1–7 3. Tokimatsu K, Tamura S, Suzuki H, Katsumata K (2012) Building damage associated with geotechnical problems in the 2011 Tohoku Pacific earthquake. Soils Found 52(5):956–974 4. Massumi A, Tabatabaiefar HR (2007) Effects of soil–structure interaction on seismic behaviour of ductile reinforced concrete frames. World housing congress Malaysia, pp 1–8 5. Tabatabaiefar HR, Massumi A (2010) A simplified method to determine seismic responses of reinforced concrete moment resisting building frames under influence of soil–structure interaction. Soil Dyn Earthq Eng 30:1259–1267 6. Visuvasam J, Chandrasekaran SS (2019) Effect of soil-pile-structure interaction on seismic behaviour of RC building frames. Innov Infrastruct Solut 4:45. https://doi.org/10.1007/s41 062-019-0233-0 7. Bagheri M, Ebadi JM, Samali B (2018) Effect of seismic soil–pile–structure interaction on mid- and high-rise steel buildings resting on a group of pile foundations. Int J Geomech 18(9):04018103 8. Hokmabadi AS, Fatahi B, Tabatabaiefar SHR, Samali B (2012) Effects of soil–pile–structure interaction on seismic response of moment resisting buildings on soft soil. In: Proceedings of 3rd international conference on new developments in soil mechanics and geotechnical engineering. New East University, Nicosia, North Cyprus 9. Ghandil M, Behnamfar F (2017) Ductility demands of MRF structures on soft soils considering soil–structure interaction. Soil Dyn Earthq Eng 92:203–214 10. Nguyen QV, Fatahi B, Hokmabadi AS (2017) Influence of size and load bearing mechanism of piles on seismic performance of buildings considering soil–pile–structure interaction. Int J Geomech 04017007:1–22 11. Li M, Lu X, Ye L (2014) Influence of soil–structure interaction on collapse resistance of super-tall buildings. J Rock Mech Geotech Eng 6:477–485 12. Cruz C, Miranda E (2017) Evaluation of soil–structure interaction effects on the damping ratios of buildings subjected to earthquakes. Soil Dyn Earthq Eng 100:183–195 13. Sotiriadis D, Kostinakis K, Morfidis K (2017) Effects of nonlinear soil–structure interaction on seismic damage of 3D buildings on cohesive and friction soils. Bull Earthq Eng. https://doi. org/10.1007/s10518-017-0108-8 14. Farhadi N, Saffari H, Torkzadeh P (2018) Estimation of maximum and residual interstorey drift in steel MRF considering soil–structure interaction from fixed base analysis. Soil Dyn Earthq Eng 114:85–96 15. Sessa S, Marmo F, Rosati L, Leonetti L, Garcea G, Casciaro R (2018) Evaluation of the capacity surfaces of reinforced concrete sections: Eurocode versus a plasticity-based approach (2018) Meccanica 53(6):1493–1512. https://doi.org/10.1007/s11012-017-0791-1 16. Marmo F, Sessa S, Rosati L (2016) Analytical solution of the Cerruti problem under linearly distributed horizontal loads over polygonal domains. J Elast 124(1):27–56. https://doi.org/10. 1007/s10659-015-9560-3 17. Marmo F, Sessa S, Vaiana N, De Gregorio D, Rosati L (2020) Complete solutions of threedimensional problems in transversely isotropic media. Contin Mech Thermodyn 32(3):775– 802. https://doi.org/10.1007/s00161-018-0733-8 18. Hokmabadi AS, Fatahi B (2016) Influence of foundation type on seismic performance of buildings considering soil-structure interaction. Int J Struc Stab Dyn 16:1550043–1550129 19. Yingcai H (2002) Seismic response of tall building considering soil–pile–structure interaction. Earthq Eng Eng Vib 1(1):57–64 20. Bozorgnia Y, Bertero VV (2004) Earthquake engineering from engineering seismology to performance-based engineering. CRC Press, Washington, DC

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