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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An authoritative guide to the theory and practice of static and dynamic structures analysis Static and Dynamic Analysis of Engineering Structures examines static and dynamic analysis of engineering structures for methodological and practical purposes. In one volume, the authors noted engineering experts provide an overview of the topic and review the applications of modern as well as classic methods of calculation of various structure mechanics problems. They clearly show the analytical and mechanical relationships between classical and modern methods of solving boundary value problems. The first chapter offers solutions to problems using traditional techniques followed by the introduction of the boundary element methods. The book discusses various discrete and continuous systems of analysis. In addition, it offers solutions for more complex systems, such as elastic waves in inhomogeneous media, frequency-dependent damping and membranes of arbitrary shape, among others. Static and Dynamic Analysis of Engineering Structures is filled with illustrative examples to aid in comprehension of the presented material. The book: * Illustrates the modern methods of static and dynamic analysis of structures; * Provides methods for solving boundary value problems of structural mechanics and soil mechanics; * Offers a wide spectrum of applications of modern techniques and methods of calculation of static, dynamic and seismic problems of engineering design; * Presents a new foundation model. Written for researchers, design engineers and specialists in the field of structural mechanics, Static and Dynamic Analysis of Engineering Structures provides a guide to analyzing static and dynamic structures, using traditional and advanced approaches with real-world, practical examples.
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Table of Contents
Cover
About the Authors
Preface
Introduction
Chapter 1: Methods of Dynamic Design of Structural Elements
1.1 The Method of Separation Variables
1.2 The Variational Methods
1.3 Integral Equations and Integral Transforms Methods
1.4 The Finite Element Method
1.5 The Finite Difference Method
1.6 The Generalized Method of Integral Transformation
1.7 The Method of Delta-Transform
1.8 The Generalized Functions in Structural Mechanics
1.9 General Approaches to Constructing Boundary Equations, and Standardized Form of Boundary Value Problems
1.10 The Relationship of Green's Function with Homogeneous Solutions of the Method of Initial Parameters
1.11 The Spectral Method of Boundary Elements
1.12 The Compensate Loads Method
Chapter 2: Boundary Elements Methods (BEM) in the Multidimensional Problems
2.1 The Integral Equations of Boundary Elements Methods
2.2 The Construction of Boundary Equations by the Delta-Transformation Technique
2.3 The Equivalence of Direct and Indirect BEM
2.4 The Spectral Method of Boundary Elements (SMBE) in Multidimensional Problems
2.5 The Problems Described by the Integro-Differential System of Equations
Note
Chapter 3: Oscillation of Bars and Arches
3.1 The Nonlinear Oscillations of Systems with One Degree of Freedom
3.2 The Nonlinear Oscillations of Systems with Multiple-Degrees-of-Freedom
3.3 The Nonlinear Oscillations of Systems with Distributed Mass
3.4 The Oscillations of the Beam of the Variable Cross-sections
3.5 The Optimum Design of the Bar
3.6 The Oscillations of Flexural-Shifted (Bending-Shifted) Bars Under the Seismic Impacts
3.7 Oscillations of Circular Rings and Arches
3.8 The Free Oscillations of System “Flexible Arch-Rigid Beam”
3.9 The Results of Dynamic Testing Model of Combined System Rigid-beam and Flexible Arch
3.10 The Oscillations of the Combined System Taking into Account its Extent at a Given Harmonic Motion Base
3.11 The Determination of the Reactions of Multiple Spans Frame Bridges, Extended Buildings, and Structures Taking into Account the Initial Phase of Passing (Propagation) of the Seismic Wave
Chapter 4: Oscillation of Plates and Shells
4.1 The Design of the Cantilever Plate of Minimal Mass Working on the Shift with the Assigned Fundamental Frequency
4.2 The Experimental and Theoretical Research of Oscillation of a Cantilever Plate with Rectangular Openings
4.3 The Oscillations (Vibrations) of Spherical Shells
4.4 The Application of the Spectral Method of Boundary Element (SMBE) to the Oscillation of the Plates on Elastic Foundation
Chapter 5: The Propagation of Elastic Waves and Their Interaction with the Engineering Structures
5.1 The Propagation of Seismic Waves in the Laminar Inhomogeneous Medium
5.2 Diffraction of Horizontal Waves on the Semi-cylindrical Base of Structure
5.3 Method of Calculation of the Lining of Tunnels to Seismic Resistance
5.4 A Study of the Action of Seismic Wave on the Rigid Ring Located in the Half-plane
5.5 Calculations of Underground Structures with Arbitrary Cross-section under Seismic Action Impact
Note
Chapter 6: The Special Features of the Solution of Dynamic Problems by the Boundary Element Methods (BEM)
6.1 One Method of Calculation: The Hilbert Transform and its Applications to the Analysis of Dynamic System
6.2 Construction of Green's Function for Bases Having Frequency-Dependent Internal Friction
6.3 The Green's Functions of Systems with the Frequency-Independent Internal Friction
6.4 The Numerical Realization of Boundary Element Method (BEM)
6.5 The Construction of the Green's Function of the Dynamic Stationary Problem for the Elasto-Viscous Half-Plane
Chapter 7: The Questions of the Static and Dynamic Analysis of Structures on an Elastic Foundation
7.1 The Kernel of the Generalized Model of Elastic Foundation (Base)
7.2 The Determination of the Characteristics of the Generalized (Unified, Integrated) Model of the Elastic Foundation (Base)
7.3 Contact Problem for the Rigid Die, Lying on the Generalized Elastic Base
7.4 On One Method of Calculation of Structures on an Elastic Foundation
7.5 The Calculation of the (Non-isolated) Beams and Plates, Lying on an Elastic Foundation, Described by the Generalized Model
7.6 The Forced Oscillations of a Rectangular Plate on an Elastic Foundation
7.7 The Calculation of the Membrane of Arbitrary Shape on an Elastic Foundation
Appendix A: Certificate of Essential Building Data
Appendix B: Contact Stresses on the Sole of the Circular Die and the Sole of the Plane Die
B.1 Contact Stresses on the Sole of the Circular Die.
B.2 Contact Stresses on the Sole of the Plane Die.
References
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Loads and corresponding generalized functions.
Table 1.2 Unit displacements and unit reactions expressed through generalized gr...
Table 1.3 Homogeneous solutions of the initial parameters method.
Chapter 2
Table 2.1 The basic types of boundary integral equations of the spatial problem ...
Chapter 3
Table 3.1 The Ratios
T
Nonlinear
/
T
linear
for Values of
.
Table 3.2 The value of roots
λ
j
.
Table 3.3 The value of the coefficient of the modes of oscillations
γ
ij
(
n
)
.
Table 3.4 Values of
and relations on nonlinear and linear ...
Table 3.5 Values of relations of nonlinear and linear frequencies depending on c...
Table 3.6 Values of
.
Table 3.7 The frequencies of free oscillations of circular rings.
Table 3.8 Frequencies of free oscillations of circular arches.
Table 3.9 Frequency of vertical oscillations.
Table 3.10 Geometric characteristics of the bridge model.
Table 3.11 Periods of free oscillations.
Table 3.12 Angular frequency and period of free oscillations.
Table 3.13 Adjustable overpass single-span.
Table 3.14 Values of the coefficients of free oscillation modes of the bridge.
Table 3.15 Values of the maximum horizontal and vertical accelerations, acting o...
Table 3.16 Values of accelerations for the second option of the external seismic...
Table 3.17 Optimum values of the span length.
Table 3.18 Magnitude of seismic forces and number of intermediate supports.
Chapter 4
Table 4.1 The value of relative mass depending of the characteristic of the valu...
Table 4.2 Value of the fundamental frequencies of the free oscillations of canti...
Chapter 5
Table 5.1 The values of contour stresses
(
t
= 0.68 s)
.
Table 5.2 The values of contour displacements...
Table 5.3 The values of contour stresses
(
t
= 2.46 s)
.
Table 5.4 The values of contour displacements...
Table 5.5 Values of stresses on the contour of a tunnel with elliptical cross-se...
Table 5.6 Values of stresses on the contour of a tunnel with elliptical cross-se...
Table 5.7 Values of stresses on the contour of a tunnel with elliptical cross-se...
Table 5.8 Values of stresses on the contour of a tunnel with elliptical cross-se...
Chapter 6
Table 6.1 The numerical results of a problem of propagation of elastic waves in ...
List of Illustrations
Chapter 1
Figure 1.1 The scheme of a cantilever beam.
Chapter 2
Figure 2.1 The linearly deformed medium
R
, the area of contact
D
, and the boun...
Chapter 3
Figure 3.1 Dependences of the relations of periods of non-linear and linear os...
Figure 3.2 Dependences of relations of periods of nonlinear and linear oscilla...
Figure 3.3 Modes of free linear and nonlinear oscillations of the 10-story bui...
Figure 3.4 Dependencies of periods of nonlinear and linear oscillations of ben...
Figure 3.5 Scheme of the beam.
Figure 3.6 Dependencies of frequencies of free oscillation of a beam from a pa...
Figure 3.7 (a) Form of optimally shifted bar with
βl = 0.86;
...
Figure 3.8 Distribution of the maximum bending moments and shear forces with t...
Figure 3.9 Distribution of the maximum bending moments and shear forces on the...
Figure 3.10 Diagram of forces acting on the element of the ring.
Figure 3.11 The scheme of a combined system of the rigid-beam type and flexibl...
Figure 3.12 Modes of natural vertical oscillations.
Figure 3.13 Overall view of Arpa Bridge in Jermuk, Armenia.
Figure 3.14 Overall view of model of Arpa Bridge in Jermuk, Armenia.
Figure 3.15 Graphs of static (a) vertical; and (b) horizontal transverse displ...
Figure 3.16 Mode of natural lateral (transverse) oscillations of model.
Figure 3.17 The dependence of coefficient
β
on
l
0
/
l
.
Figure 3.18 Bridge model under dynamic testing.
Figure 3.19 Part of a bridge model.
Figure 3.20 (a) The motion of the test load along the bridge with various spee...
Figure 3.21 The dynamic design scheme and the scheme of seismic excitation.
Figure 3.22 Structural and design schemes of the bridge: (a) structural schema...
Figure 3.23 (a) and (b) Schemes of multi-span trestle frames; and (c) dynamic ...
Figure 3.24 Scheme of motion on extended one story, single-span frame.
Figure 3.25 Scheme of motion on extended one story, single-span frame with abs...
Figure 3.26 Spectra of the accelerations of the extent structures with harmoni...
Figure 3.27 Dependence of dynamic coefficient on the extent structures and gro...
Figure 3.28 (a) Averaged values of the accelerations of the real accelerogram ...
Chapter 4
Figure 4.1 Scheme of cantilever plate: (a) deflection diagram; (b) shear force...
Figure 4.2 Partition of the area of plate on the subdomain, in which the optim...
Figure 4.3 Topography of the optimum construction with disregard for limitatio...
Figure 4.4 Topography of the optimum plate with the limitation:
h ≥ 0.7.
...
Figure 4.5 Cantilever plate with rectangular openings.
Figure 4.6 Diagram of the vibration stand.
Figure 4.7 Circuit of test measuring equipment.
Figure 4.8 Cantilever plates: (a) continuous (solid) section; (b) with 1 openi...
Figure 4.9 Record of the shift of oscillation of the plate with the continuous...
Figure 4.10 Scheme of a plate.
Figure 4.11 Element of the boundary of a plate.
Chapter 5
Figure 5.1 Diagram of laminar half-space.
Figure 5.2 Dependence of the relative displacement of free surface of the laye...
Figure 5.3 Scheme of a structure with semi-cylindrical foundation and base.
Figure 5.4 Dependencies of the relative displacement
W
2
(−
H
)/
W
*
of th...
Figure 5.5 Scheme of the action of a seismic wave on the lining of a tunnel.
Figure 5.6 Distribution is at the maximum over time with (
t
= 0.68 s) normal a...
Figure 5.7 Figure 5.7 Distribution is at the maximum over time with (
t
= 0.68 ...
Figure 5.8 Distribution of (a) normal and (b) shearing stress displacements in...
Figure 5.9 Schemas used with solutions of problems of the impact of seismic wa...
Figure 5.10 Scheme of an underground tunnel structure.
Figure 5.11 Scheme of a tunnel cross-section and designations of stresses in d...
Chapter 6
Figure 6.1 Contour of integration.
Figure 6.2 Contour of integration.
Figure 6.3 Corner point (6.4.1).
Figure 6.4 The symmetrical intervals relative to a singulatity point.
Figure 6.5 The partitioning of the boundary with the numeration of functional ...
Figure 6.6 Half-plane in Cartesian coordinates.
Figure 6.7 (a) Graphs of the amplitudes of the oscillations of the surface of ...
Chapter 7
Figure 7.1 (a) Influence function
K
0
(
r
);
(b) influence function
K
0
(
r
);
(c) inf...
Figure 7.2 Graph of function
K
(
s
)
.
Figure 7.3 Schema of the hole during of die testing.
Figure 7.4 Kernel
K
(
r
)
for silty (dusty) loams.
Figure 7.5 Approximation of contact pressure.
Figure 7.6 Diagram of contact stresses.
Figure 7.7 Diagram of contact stresses.
Figure 7.8 Diagram of contact stresses.
Figure 7.9 Scheme of loading of the non-insulated beams.
Figure 7.10 Diagram of the displacements of the non-insulated beams.
Figure 7.11 Design scheme of a plate on an elastic foundation.
Figure 7.12 Decomposition of the load on (into) symmetric and skew-symmetric c...
Figure 7.13 Diagram of deflections and bending moments.
Figure 7.14 Diagram of deflections and bending moments.
Figure 7.15 Dependence of the deflection of the center of the plate on frequen...
Figure 7.16 Schema of membrane and the element of the boundary.
Figure 7.17 Extended domain in the form of the rectangle.
Figure 7.18 Schematic of triangular membrane.
Figure 7.19 Distribution of the displacements over the diagonal of the membran...
Guide
Cover
Table of Contents
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Static and Dynamic Analysis of Engineering Structures
Incorporating the Boundary Element Method
Levon G. Petrosian
Washington District Department of Transportation, USA
Vladimir A. Ambartsumian
Structural Engineering and Structural MechanicsArmenian National University of Architecture & Construction, Armenia
This edition first published 2020
© 2020 John Wiley & Sons Ltd
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The right of Levon G. Petrosian and Vladimir A. Ambartsumian to be identified as the authors of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data
Names: Petrosian, Levon G. (Levon Gregory), 1- author. | Ambartsumian, V. A. (Vladimir Alexander), author.
Title: Static and dynamic analysis of engineering structures: incorporating the boundary element method / Levon G. Petrosian, Professor of Structural Engineering and Structural Mechanics, Washington District Department of Transportation, USA Vladimir A. Ambartsumian, Professor of Structural Engineering and Structural Mechanics, Armenian National University of Architecture & Construction, Armenia
Description: First edition. | Hoboken, NJ: John Wiley & Sons, Inc., 2020. | Includes bibliographical references and index.
Identifiers: LCCN 2019034987 (print) | LCCN 2019034988 (ebook) | ISBN 9781119592839 (hardback) | ISBN 9781119592884 (adobe pdf) | ISBN 9781119592938 (epub)
Subjects: LCSH: Structural analysis (Engineering)–Mathematics. | Boundary element methods.
Classification: LCC TA640 .P48 2020 (print) | LCC TA640 (ebook) | DDC 624.1/7–dc23
LC record available at https://lccn.loc.gov/2019034987
LC ebook record available at https://lccn.loc.gov/2019034988
Cover Design: Wiley
Cover Image: © aaaaimages /Getty Images
In Memory ofmy Teacher, Mentor, and Friend
ALEXANDER IZRAELEVICH TSEITLIN(1933 – 2011)Doctor of Technical Science, Professor,Member of the Russian & International EngineeringAcademy of Science
About the Authors
Levon G. Petrosian is a Professor of Structural Engineering and Structural Mechanics. He received his B.S. and M.S. degrees from the Armenian National University of Architecture & Construction. He received a Ph.D. from the Moscow State University of Civil Engineering (MGSU) and the degree of Doctor of Technical Science from the Moscow Research Center of Construction, Research Institute of Building Structures (TSNIISK). Dr. Petrosian is the author of more than 60 scientific works and publications, including monographs and text books. He has been associated with transportation and structural engineering industries for over 40 years as a research engineer and scientist. Prior to his move to the United States, Dr. Petrosian was the Chairman of Department of Structural Mechanics at the Armenian National University of Architecture & Construction as well as the Executive Director and the Head of the Armenian Earthquake Engineering Research Institute. Currently, Dr. Petrosian serves as the Chief of the Plan Review Division at the District of Columbia Department of Transportation.
Vladimir A. Ambartsumian, Professor of Structural Engineering and Structural Mechanics received B.S., M.S., Ph.D., and the degree of Doctor of Technical Science from the Armenian National University of Architecture & Construction. He was a Professor of Structural Mechanics at the Armenian National University of Architecture & Construction. He authored and co-authored numerous scientific publications.
Preface
This book presents both methodological and practical purposes of static and dynamic analysis of engineering structures. Therefore, accounting of all methods is accompanied by the solution of the specific problems, which not only are illustrative material, but also may represent independent interest in the study of various technical issues. It provides an overview and applications of all modern as well as well-known classic methods of calculation of various structure mechanics problems in one place.
The book generalizes all existing classical and modern methods of calculations of engineering problems and structural mechanics problems over the span of the last 50 years. Through comprehensive analysis, the book shows analytical and mechanical relationships between classical and modern methods of solving boundary value problems.
The book features extensive use of the generalized functions for describing the impacts on structures, and substantiations of the methods of the apparatus of the generalized functions.
The book illustrates the modern methods of static and dynamic analysis of structures and the methods for solving boundary value problems of structural mechanics and soil mechanics.
The book includes examples of solving different problems of static and dynamic calculation of beams, plates, shells, multi-body systems, regular structures, bridge structures, and underground constructions.
The book provides a wide spectrum of applications of modern technics and methods of calculation of static, dynamic, and seismic problems of engineering design.
The book shows which methods and techniques should be used for specific and complex problems. These methods are most effective for solving deferent problems of static and dynamic analysis of structures.
Finally, the book presents a new model of a cohesive elastic base covering a wide range of properties of real soils. The limiting cases of the model are the Winkler base and the isotropic elastic half-space.
The book provides numerous solutions of various static and dynamic problems of the theory of elasticity.
Levon G. Petrosian
Professor of Structural Engineering and Structural Mechanics
Department of Transportation
Plan Review Division Chief
Washington DC, USA
Introduction
The classical methods of structural mechanics widely used for static and dynamic design of structures, in particular the methods of force and displacement/deformation, are based on very transparent mechanical representations associated with the replacement of a given structure to some other, more convenient to calculate “primary system,” that is loaded in such a way that the entire primary system or one of its parts is turned into a given. In the theory of elasticity and in mathematical physics a similar method of solving the boundary-value problems is used and considered classical; the method of potential. When applying this method, the assigned domain is actually substituted by some unlimited domain, loaded with some additional load, such that the solution in the given domain and chosen unlimited domain (analog of primary system) are identical. The reason for choosing the primary system in a form of unlimited domain, for which solutions of the corresponding boundary-value problems are constructed, has proven to be very successful and has found development in such engineering methods as the method of compensating loads, extended domains, and boundary elements, as well as in purely mathematical methods of generalized solutions (GSMs), boundary integral equations, and delta-transforms. From the listed methods, which have progressed in the recent years, the greatest development was obtained in the method of boundary integral equations with the discreteness of the boundary, also called the method of boundary element by the analogy with the known finite element method (FEM). Today FEM is the main tool of calculations and analysis of construction in design and research practice. Methods of the Boundary Element Method (BEM) type have proved to be very effective to successfully compete in plane (two-dimensional) and spatial (three–dimensional) problems with FEM and Finite Difference Method (FDM), since they do not require discreteness of the entire domain, but only its boundary. This leads to per-unit reduction of the dimensionality of a problem with more effective computational algorithms. Another advantage of the BEM is the ability to solve problems in unlimited domains, which is paramount for the dynamics of constructions, soil mechanics, and other areas of the theory of elasticity and structural mechanics.
In this book, along with the general approach to the construction of boundary equations of different types, two new methods are applied; Delta Transformation Method and Spectral Method of Boundary Elements (SMBEs) which make obtaining effective solutions of various boundary-value problems of structural mechanics possible. Application of the methods is illustrated by solving a number of problems in the calculation of structures on static and dynamic impacts.
Contemporary structural mechanics and theories of construction investigate stresses on many objects, calculation of which is associated with the solution of boundary-value problems for differential operators. In this case, in contrast to the classical problems of mathematical physics, the boundary-value problems of structural mechanics have higher order and more complex structures. Therefore, both general methods of solving differential and integral equations, and specific engineering methods purely based on mechanical performances are widely used. In this book we will examine the methods of solving boundary-value problems; typical for structural mechanics and related fields. All of these methods, while different in form, can be united by one basic procedure – the Integral Transform, which is either an integral part of the method itself, or is used for its substantiation or conclusion. In the modern technical and physical literature, the Method of Integral Transforms is presented in essence as apparatus connected with the use of classical Fourier transform, Laplace, Melina, Hankel, as well as the transforms of Kantorovich-Lebedev, Meler-Fock, and others. Each of the transforms is obtained within a comparatively clearly outlined field of application.
At the same time, the variety of problems which are encountered in the various sections of science cannot be solved only with the aid of traditional integral transforms. Therefore, further development of the method of integral transforms should proceed in the direction of expanding the quantity of practical suitable kernels, and generalization and necessary application of the methods of constructing the kernels of transforms. This book presents a general approach to the method of integral transforms based on the spectral theory of the linear differential operators, and provides new transforms which will aid us in solving various problems relevant to bars, beams, plates, and shells, in particular.
Another equally important task is to show the close relationship of integral transforms with different modern methods and with types of potentials; in particular, with the method of delta-transform (A.I. Tseitlin [370]), which may be the basis for a mathematical foundation of many methods of structural mechanics.
The book pursues both methodological and practical purposes, and the accounting of all methods is accompanied by solutions of the specific problems, which are not merely illustrative in nature, but may represent independent interest in study of various technical issues. Two special features of the book are an extensive use of the generalized functions for describing the impacts on structures and substantiations of the methods of the apparatus of the generalized functions. In structural mechanics the theory of the generalized functions does not yet apply as thoroughly as in other sections of mechanics and physics. In this book it is shown that the theory of generalized functions allows us to obtain broad generalizations of the classical methods of structural mechanics, to simplify solutions of many problems, and identify the relationships among the methods.
This book illustrates the modern methods of static and dynamic analysis of structures and the methods for solving boundary-value problems of structural mechanics and soil mechanics based on the application of boundary equations. The systems with non-linear and variable elastic characteristics are examined. The fundamentals of the general theory of oscillations are considered. Examples of solving different problems of static and dynamic calculation of beams, plates, shells, multi-body systems, regular structures, bridge structures, underground constructions, and structures on an elastic foundation are given according to the methods presented in the book. The book also analyzes the impact of seismic influences on regular structures.
Further, the book offers a method of physical realization of the dynamic systems based on the examples of an elastic-viscous foundation with internal friction. The internal friction in the foundation is described by the models of Kelvin-Voigt and Sorokin.
Chapter 1 shows the main methods used in the dynamic analysis of bars, plates, and shells. Historically, one of the first methods of solving equations of mathematical physics is the method of separation of variables. The application of this method is described in the examples of solving problems of free oscillation of rectangular and circular membranes.
The range of problems which can be solved accurately in a closed form is limited, and therefore, there are often variational methods used in practice. The application of two variational methods – the Rayleigh-Ritz method and the Galerkin method are shown in the examples of the transverse oscillations of bars. The use of integral equations, in particular the reduction of problems with oscillation of one-dimensional structures to the integral equations of Volterra of the second kind and Fredholm of the second kind, are demonstrated. The methods used for numerical solutions of problems of dynamics are described. These are the FEM and FDM. In either the FEM or FDM the whole domain of the partial differential equations (PDEs) requires discretization. The FEM is presented in an example of a plane problem of the theory of elasticity. A diagram of solving a problem of non-stationary oscillations of a cantilever plate is shown. The oscillations of a compressed bar are solved using the FDM.
The BEM originated from the works carried out by several research groups during the 1960s in the application of boundary integral equations for the solution of engineering problems. These researchers were looking for a different solution from the FEM which was starting to become more widely established for computational analysis of structural mechanics problems.
The Boundary Integral Equations Method (BIEM) is generally found in the methods of Theory of Potential and Boundary Integral Equations, but the basic features and idea of the method of boundary equations can be also found in the classical methods of structural mechanics that have developed considerably earlier than the corresponding methods of mathematical physics. BIEM in structural mechanics were known in the western countries through the work of former Soviet Union researchers and scientists such as N.I. Muskelishvili, S.G. Miklin, V.D. Kupradze, V.Z. Parton, P.I. Perlin, G.J. Popov, A.I. Tseitlin, and Y.V. Veryuzhsky. At that time these methods were considered to be difficult to implement numerically.
Successful approaches of structural mechanics to the calculation of complex systems by replacing them with simpler systems and making them more accessible for calculations carried all signs of the Methods of Boundary Equations. Since structural mechanics initially developed mainly as a science of bar systems, the application of such classical methods as methods of force, displacement, or mixed method led to the boundary equations in a discrete set of joints.
Thus, as a rule, auxiliary tasks for which one-dimensional analogues of Green's function – a unit reaction and unit displacement, were determined and corresponded to elements of the system (e.g. the horizontal and vertical elements of the frame), so that the construction of auxiliary conditions was associated with the narrowing of the domain to several sub-domains. The first one-dimensional analogues of the BEM appeared in structural mechanics with an introduction of infinite systems, which initiated the Compensating Loads Method (CLM), Extended-Domain Method (EDM), and BEM, as well as in purely mathematical methods of GSM, BIEM, and the Delta-Transform Method (DTM). Among these methods developed in recent decades, the most developed is the method of Boundary Integral Equations with the discretization of the boundary, also known as the BEM. BEM is similar to the method of finite elements, which is now the major instrument for actual calculations and analysis of structures in the design and research practice. The BEM is a technique for solving a range of engineering and physical problems. The heart of the BEM technique lies in the integral equation formulation for a given boundary value problem. The mathematical basis of this approach, of course, is classical Green's function, and as mentioned earlier it corresponds with the influence function of structural mechanics. The BEM has the distinction in and advantage of the fact that only the surfaces of the domain need to be meshed. Methods such as boundary elements have proved to be very effective, and they successfully compete in two and three-dimensional problems with the (FEM) and the (FDM), because unlike these methods, BEM does not require sampling (discreteness) of the entire area, but only its border, which decreases dimension of the problem by one unit and leads to more efficient computing algorithms. The advantages in the BEM arise from the fact that only the boundary or boundaries of the domain of the PDEs require sub-division to produce a surface or boundary mesh.
Another advantage of BEM is a possibility of solving problems in an unbounded domain, which is very important for the structural dynamics, soil mechanics, and other fields of the theory of elasticity and structural mechanics.
This book describes both a general approach to the construction of various types of boundary equations and a spectral BEM, which allows us to obtain effective solutions of various boundary value problems of structural mechanics. The application of SMBE is illustrated by solving a number of problems on the structural analysis of the design of construction on the static and dynamic impacts.
The issue of the equivalence of boundary and initial value problems with the inhomogeneity in the boundary (initial) conditions and the right side of the equation in respect to the design of structures on the static and dynamic loads are considered. This book shows a “standard” form of the problem, obtained by using the delta transform and allowing replacement of the boundary or initial conditions by a certain corresponding load on the structures in the form of a delta-function and its derivatives. The construction of the “standard” form is illustrated on the simplest examples of the boundary value problem for beams and the Cauchy problem for the system with one degree of freedom.
There is a general scheme for solving the problems of the theory of elasticity by the CLM in the first chapter.
Chapter 2 examines the application of BEM in the multidimensional problems. The integral equations of BEM are considered; construction of the boundary equations using the method of delta-transform is given; and three basic schemes for constructing boundary integral equations using the DTM are analyzed. Simple proof of the complete equivalence of two basic versions of BEM – direct and indirect – is given. The SMBE, based on expansion of differential operators considering in the extended domain is proposed. In particular, for this purpose the Fourier and Hankel transforms, as well as expansion of beam functions widely used in structural mechanics can be used. Problems described by the integro-differential system of equations are examined. As an example, a contact problem for a plane structure with stress-strained conditions in a certain domain is described by a linear differential operator, some given boundary conditions, and contact conditions with a linear-deformed medium.
Chapter 3 outlines the issues of vibrations of bars, arches, and combined systems. The free nonlinear vibration systems with one and many degrees of freedom, and the systems with distributed mass at various dependencies of the restoring force from moving are examined. The values of the periods of free vibration are obtained by various methods, such as direct integration of the equations of motion and the use of asymptotic methods.
A study of free vibrations of a beam with an arbitrary law of variation of the cross section is also discussed in the third chapter. Classical results for the oscillation problems of a cantilever beam of minimum mass are shown. The outlined solutions have specific applications in the dynamic structural analysis. In particular, the chapter shows the application of the solution of the problem of forced oscillations of the shifted flexural-cantilever beam under the seismic analysis of the structures.
Along with the study of the oscillations of elements with a straight axis, the oscillations of curvilinear elements, in particular circular arches and rings, are also examined. These include the equation of oscillations of S.P. Timoshenko for circular arches and rings, as well as some simple results on values of natural frequencies of these elements.
The third chapter further describes free vibrations of combined systems such as “rigid beam-flexible arch” considering the dynamic thrust when the axis of the arch is outlined by a square parabola and a circular arch.
The chapter shows an experimentally-theoretical method for calculating the combined systems taking into account the joint dynamic work of the basic span to adjacent bridge trestles using the factor “increasing” frequency.
We show a method for seismic analysis of combined systems with regard to their length using real seismograms of powerful earthquakes. There are general equations of motion for a system with a finite number of degrees of freedom in a spatial deformation structures.
Chapter 4 is devoted to the vibrations of plates and shells. We show the optimum design of the shear cantilever plate. The minimum of the total mass of the plate is considered as the optimality criterion. The problem is solved considering the limitation of the thickness of the plate.
This chapter gives the experimental results of free vibration of cantilever plates with rectangular holes. We also show a satisfactory match between the results of theoretical calculations and the experiment, and present the asymmetric vibrations of thin elastic spherical shells and equations for determining the frequency of a hemisphere with free ends.
The application of the SMBE to the oscillation of the plates on an elastic foundation is examined. In particular a spectral BEM to the oscillations or free vibration of plate of arbitrary shape lying on Winklerian foundation and being under a load is provided.
Chapter 5 examines issues of elastic waves spread and their interaction with engineering structures. The chapter examines the problem of propagation of elastic waves in inhomogeneous layered medium and analyzes cases of change of shear modulus and the density of the exponential and power law. This chapter solves the problem of interactions of elastic waves with a semi-cylindrical base structures. The issues of interaction of seismic waves on the tunnel lining are examined. An approach to the study of stress around a circular lining, located near the surface of the half-space, is outlined. The chapter proposes a method for studying the dynamic stresses affecting the cross-section lining. Numerical results are stated.
Chapter 6 describes features of solving dynamic methods of boundary equations. It is known that most building materials have damping properties, characterized by the frequency's independent loss. However, there is a certain dependency of damping parameters (coefficients loss, decrement) on the frequency of the deformation for soils in the experiments. Frequency-dependent losses are typical for a number of modern materials, such as certain plastics. It is therefore, a very important task to describe the dynamic behavior of soil and structural materials possessing both frequency-independent and frequency-dependent internal friction. We discuss the types of problems in relation to the construction of the Green function for the corresponding environment. From the theory of dynamic systems, it follows that between the real and imaginary parts of the transmission function of the casual system, certain integral dependencies which define the formulas of Hilbert transform, must exist. This chapter presents a new way of calculation of Hilbert transform through Fourier transformation of corresponding functions, which simplifies the problem, due to extensive Fourier transforms tables. The presented method of calculating the Hilbert transform allows to comparatively easily analyze the aprioristic models of the dynamic systems based on the creation of complex rigidity or complex pliability of system. This method makes it possible to analytically determine one of the components of the complex rigidity in accordance with the experimentally defined other component. A few examples of the calculation of the Hilbert transform of delta-functions, Heaviside function, and a power function are examined.
The above-described method of calculating the Hilbert transform is applied to the analysis of dynamic models bases of internal friction, described by the hypothesis of Voigt, Sorokin, and Schlippe-Boc. Further, construction of Green's function for the unlimited plate, which lies on Winklerian type elasto-viscous base with the arbitrary dependence of the parameters of the complex coefficient of bed on the frequency is examined. The importance, from a practical point of view, of the task on oscillations of the system with one degree of freedom, whose elastic-viscous properties (parameters) depend on frequency of oscillations, is considered.
The question of construction of the Green's function for one-dimensional dynamical systems with damping described by different model frequency-independent and frequency-dependent internal friction is examined. We examine stationary one-dimensional wave tasks, based on an example of the longitudinal free vibrations of the isotropic uniform unlimited bar with unit harmonic load in the central section or the lateral oscillations of string. The following basic models of medium are investigated:
common linear model;
elastic-viscous model of Voigt; and
the model of the frequency-independent elastic-viscous resistance.
The construction of the Green's function for the non-stationary transverse free vibration of a bar and plate taking into account the frequency-independent elastic-viscous resistance is considered. A numerical realization of a direct BEM for the solution of plane stationary problems of the linear theory of elasticity, taking into account the internal friction, is presented.
An urgent problem of structural mechanics, connected in particular, with the development and improvement of foundation engineering, a major trend in the construction business, is the theory of analysis of structures on elastic foundation. This theory provides for reliability and efficiency of some of the most popular and important structures, such as: foundations of buildings, road and airport paving, coating slopes of hydraulic structures, dams, base layers of floors of industrial buildings, power floors of test cases, and more.
Improving the methods of analysis of structures based on actual soil properties is one of the major hurdles in creating a more efficient and reliable design solution, and preventing undesirable consequences in construction on weak and subsiding soils, and in seismically active areas near mining and blasting. This problem has become very significant after the devastating earthquakes in Armenia, China, Chili, Italy, Japan, Haiti, Pakistan, and Philippines in connection with large-scale reconstruction work and the revaluation intensities of a seismically active zone.
Chapter 7 is devoted to the static and dynamic analysis of structures on elastic foundation. Description of a new model of cohesive elastic foundation, covering a wide range of properties of real soils, is examined. The limiting cases for this model are the Winkler foundation and isotropic elastic half-space. The proposed model, along with the versatility, has several advantages over the earth foundation models currently applied in practical calculations. In particular, under a concentrated load the model gives the final displacements and stresses, allows for jumps of displacements on the surface of the base, and does not result in endless movement solutions of plane problems. The proposed model has solved a number of problems in the analysis of structures on the elastic foundation, such as; membranes, hard foundation, bare beams, and slabs; and free oscillations of rectangular foundation slabs are also examined. This chapter shows techniques for determining the model parameters under the results of experimental studies of soils.
The application of SMBE to the calculation of structures on the elastic foundation is further examined. The equilibrium of the membrane, bend of the non-isolated beams and plates with the free ends, lying on the basis with the proposed base kernel, are examined.
The numerical solution of contact task for the plane and axisymmetrical (axisymmetric) rigid die, which rests on the basis with the proposed kernel and a new iteration technique for solving contact problems, are given. The proposed generalized model of elastic foundation due to the regularity of its kernel paves the way for the application of effective methods of solving contact problems and problems of design structures on elastic foundation, associated with different ways of directly reducing the formulated problem to the solution of the integral equation of Fredholm of the second kind. The effectiveness of this method is determined by the ability to apply a simple iterative procedure for obtaining the most important characteristic (variable) – contact pressure. Then the calculation of structures can be produced with the combined action of external load and obtained contact pressure. To illustrate the method of calculation a solution to the problem of the bending of the beam of finite length, free lying on the elastic foundation, described by the generalized model is given.
The authors offer methods and applications (not always a simple task considering boundary-value problems), in a sufficiently simple and clear way in order to make it easily accessible for readers with mathematical and technical background at university level. However, for comprehension of the book, initial knowledge in such areas of mathematics as the theory of the generalized functions, transforms, and theory of linear integral equations, is necessary. The specified mathematical apparatus in recent decades has widely penetrated the technical sciences, especially structural mechanics, and knowledgeable and sophisticated readers should be familiar with it. Within the framework of the limited size of this publication, dedication was made to emphasize BEM calculations of constructions, however it is not possible to illustrate all aspects of the rapidly developing areas of BEM. One of the purposes of this publication is to familiarize the readers with the different methods of building the boundary equations, including those based on the orthogonal expansions that lead to the boundary algebraic equations.
The techniques of solving boundary integral equations, the choice of boundary elements, approximation of functions, etc. remain beyond the scope of this work. The readers who are interested in these issues should pursue monographs and numerous articles by M.H. Aliabadi, P.K. Banerjee, R. Butterfield, C.A. Brebbia, S. Walker, T.A. Cruse, F.J. Rizzo, P. Fedelinski, S. Hirose, S. Mellings, V.Z. Parton, P.I. Perlin, U.V. Veryuzhsky, J.O. Watson, and others.
The book is intended mainly for professionals and specialists in the field of structural mechanics and related areas. In preparing the book, the authors tried to make it accessible not only for scientists, researchers, and graduate students in the field of structural mechanics and related areas, but also for engineers working in design centers and organizations.
In the appendix to the book printouts are given of the programs of solution of contact problem for the base with the regular kernel, and tables of the contact stresses obtained with solution of plane and axisymmetrical (axisymmetric) contact problems.
Additionally, the book can serve as a guide for students of technical colleges in the study of the relevant sections for the course of structural mechanics.
This book was written in collaboration with Dr. Ambartsumian, who passed away several years prior to its completion and publication. I would like to acknowledge his contribution to this monograph and his dedication to science and research.
I wish to take this opportunity to express my appreciation to my former teachers and colleagues who over the years have so graciously advised and encouraged discussions that led to preparation of this text. In particular, I gratefully acknowledge indebtedness to professors A.I. Tseitlin, A.R. Rzhanitsin, B.G. Korenev, N.N. Leontiev, V.I. Travush, D.N. Sobolev, A.A. Babloyan, V.A. Ilichev, E.E. Khachiyan, M.A. Dashevsky, V.A. Smirnov, and A.W.Taylor.
I would especially like to thank my wife, Natalie, and my daughters, Galina and Irina, for their support, patience and love.
Finally, I want to thank John Wiley and Sons Inc. and the staff for their effective cooperation and their great care in preparing this edition of the book.
The work was spread between authors the following way:
Paragraphs 1.1, 1.2, 1.3, 1.4, and 1.5 of chapter 1; and paragraph 5.1 of chapter 5 are written together. Paragraphs 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, and 1.12 of chapter 1; paragraphs 2.1, 2.2, 2.3, 2.4, and 2.5 of chapter 2, paragraphs3.8, 3.9, 3.10, and 3.11 of chapter 3; paragraph 4.4 of chapter 4, chapter 6, and chapter 7 of the book are written by L.G. Petrosian:
Paragraphs 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7 of chapter 3; paragraphs 4.1, 4.2, and 4.3 of chapter 4; paragraphs 5.2, 5.3, 5.4, and 5.5 of chapter 5 of the book are written by V.A. Ambartsumian.
I am deeply grateful to those users of this book who have been kind enough to write me their impressions and criticisms. Any further comment and suggestion for improvement of the book will be gratefully received.
Chapter 1Methods of Dynamic Design of Structural Elements
1.1 The Method of Separation Variables
Let us briefly present the substance of the Fourier Method or a method of separation of variables. The separation of variables is the process in which solutions are found for separable differential equations. The method of separation of variables relies upon the assumption that a function of the form u(x, t) = X(x)T(t) will be a solution to linear homogeneous partial differential equations (PDEs) in x and t. This is called product solution, and provided the boundary conditions are also linear and homogeneous, this will also satisfy the boundary conditions. This technique works because if the product of functions of independent variables is a constant, each function must separately be a constant as well. Success requires the choice of an appropriate coordinate system and may not be attainable at all depending on the equation. Consider the problem of oscillations if the string is fixed at the ends. The problem is reduced to the solution of the equations of hyperbolic type:
Let u(x, t) denote the vertical displacement of the string from the x axis at position x and time t. In the derivation of (1.1.1) it was assumed that the extension of individual sections of the string during the oscillation does not occur and, therefore, according to the Hooke's law, the tension T0 = |T| does not depend on time or on x. The axis x coincides with the direction of a string in equilibrium position. The string is a thin string that does not resist bending, and is not associated with a change in its length.
The string has length l: (0 < x < l). The function u(x,t) satisfies the initial conditions.
The conditions that specify the initial state of the system are:
The conditions at the boundary of the problem are:
The idea of separation of variables is simple. The idea is to assume the solution to the PDE (1.1.1) has a special form of solution, namely (x, t) = X(x)T(t). We attempt to convert the given PDE into several ordinary differential equations. When a problem is posed, such as our problem for u(x, t), one can look for a product solution in the form of u(x, t) = X(x)T(t). The solution can be done by inserting X(x)T(t) into the PDE for the variable u, and then separating the variables so that each side of the equation depends on only one variable. Once the equation has been broken up into separate equations of one variable, the problem can be solved like an ordinary differential equation. We will first seek particular solutions of Eq. (1.1.1) that are not equal to zero and satisfying the boundary conditions (1.1.3), in the form of:
Substituting (1.1.4) in (1.1.1), we come to equations:
where λ is constant. The value of λ is not specified in the equation, finding the values of λ for which there exists a non-trivial solution of (1.1.1) satisfying the boundary conditions. For obtaining the non-trivial solutions of (1.1.4) it is necessary to find non-trivial solutions satisfying the conditions:
This brings us to the problem of Sturm-Liouville or eigenvalue problem. In mathematics, a certain class of PDEs are subject to extra constraints, known as boundary values, on the solution.
The eigenvalues of this problem are the numbers:
These eigenvalues correspond to the normalized unique eigenfunctions:
when λ = λκ, Eq. (1.1.5) has the general solution
Therefore the function
satisfies Eq. (1.1.1) and the boundary conditions (1.1.3) for any ak and bk. The solution of Eq. (1.1.1), satisfying the conditions (1.1.2)–(1.1.3), is found in the form of a series
If this series converges uniformly and can be twice differentiated term by term, the sum of the series will satisfy Eq. (1.1.1) and the boundary (1.1.3). Determining constants ak and bk, such that the sum of the series (1.1.8) satisfies the initial condition (1.1.2), the following equations are obtained:
