Any practitioner who takes his profession in earnest, such that daily work is not a heavy duty but part of their life, will recognize in this book the rigorousness of the analysis and the comprehensive presentation of the problems. This professional attitude is solely able to make the research and design engineer deal with strength structures and their behaviour. Indeed, the computational means that are nowadays available permit the numerical computation of whatever problem; the pro­ gram libraries are extremely rich and programs themselves have developed intensively. Howeyer, though computers are available at any moment without restrictions on the frequency with which they are employed, they finally impoYerish the creative compe­ tency of the civil engineer. Thus, he will calculate increasingly more while devising increasingly less. He will draw less and less on the experience gained in devising and implementing bearing structures because the computational process can be repeated as often as desired over a minimum time-period by means of the available programs. \Ve note that nowadays structures are no longer investigated or economically designed to comply with the requirements of the topic of interest. :Much to the contrary, the solutions are chosen so as to comply with the capabilities of the programs. A bearing structure lives as is prescribed by its initial con­ structive data.

This book is intended primarily as a teaching text, as well as a reference for individual study in the behavior of thin walled structural components. Such structures are widely used in the engineering profession for spacecraft, missiles, aircraft, land-based vehicles, ground structures, ocean craft, underwater vessels and structures, pressure vessels, piping, chemical processing equipment, modern housing, etc. It presupposes that the reader has already completed one basic course in the mechanics or strength of materials. It can be used for both undergraduate and graduate courses. Since beams (columns, rods), plates and shells comprise components of so many of these modern structures, it is necessary for engineers to have a working knowledge of their behavior when these structures are subjected to static, dynamic (vibration and shock) and environmental loads. Since this text is intended for both teaching and self-study, it stresses fundamental behavior and techniques of solution. It is not an encyclopedia of all research or design data, but provides the reader the wherewithal to read and study the voluminous literature. Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving them to the extent necessary to treat the following material. Chapter 2 presents, in a concise way, the basic assumptions and derives the governing equations for classical Bernoulli-Euler beams and plates in a manner that is clearly understood.

Thin walled structures so extensively used nowadays in industry and civil engineering are usually loaded by very complex systems of forces acting on their edges or over their surfaces. In calculating the strength of a structure we replace real loads by certain idealized loads distinguishing between typical surface loads distributed over a great area of the structure and loads acting over a small area. The latter are called concentrated loads. When the area under the load is very small in comparison with the dimensions of the surface of the structure, for example, when the diameter of the loaded area is smaller than the wall thickness, the load can be considered as a single force or a mo­ ment acting on the structure at one point only. The real loads which are met in practice can always be replaced by a combination of components such as forces normal and tangential to the wall as well as bending and twisting moments. Knowing the distribution of the stresses in the structure produced by each component, we can find it under any arbitrary load using the principle of superposition. There are two main reasons for the appearance of the concentration of stresses in the structure. It can be produced by notches, rapid changes of the cross-section, holes, cutouts, etc. on one hand and by concentrated loads resulting from the interaction of the elements of the structure on the other.

Although the theory of thermoelasticity has a long history, its foun­ dations having been laid in the first half of the nineteenth century by Duhamel and Neumann, wide-spread interest in this field did not develop until the years subsequent to World War Two. There are good reasons for this sudden and continuing revival of interest. First, in the field of aeronautics, the high velocities of modern aircraft have been found to give rise to aerodynamic heating; in turn, this produces intense thermal stresses and, by lowering the elastic limit, reduces the strength of the aircraft structure. Secondly, in the nuclear field, the extremely high temperatures and temperature gradients originating in nuclear reactors influence their design and operation. Likewise, in the technology of modern propulsive systems, such as jet and rocket engines, the high temperatures associated with combustion processes are the origin of unwelcome thermal stresses. Similar phenomena are encountered in the technologies of space vehicles and missiles, in the mechanics of large steam turbines, and even in shipbuilding, where, strangely enough. ship fractures are often attributed to thermal stres­ ses of moderate intensities. The investigations of these, and similar, problems have brol!ght forth a remarkable number of research papers, both theoretical and experimental, in which various aspects of thermal stresses in engineering structures are described.

In the last decade or so the theory of shells has undergone a tremendous increase in development. Formerly a subject of interest only to a few special­ ists and for which the literature was relatively smalI, the needs of structures for aerospace missions instigated a torrent of papers on all facets of the theory which also found application in the less glamorous earthbound shell struc­ tures important in everyday life. Some idea of the rapidity of the development can be gained from the fact that a bibliography* completed in 1953 listed some 1455 books and papers as the sum total ofthe literature on shell theory to that date. Three years later, however, a supplementt added another 884 papers to the list, an increase of 60 per cent in that short period of time. The number of papers published since these listings has increased to an extent that does not bear contemplation. Obviously no single volume could contain all that constitutes the theory of shells and so this book is restricted to that portion of the theory associated with small deformations of elastic shells. Plastic deformations of shells, which is hardly developed, and nonlinear deformations and stability, which would require at least aseparate volume, are thus excluded. Even with this restriction, however, the present volume represents a long overdue compro­ mise between completeness and finiteness. In making this compromise I have undoubtedly omitted discussions of many topics and references to many excellent papers which should have been included.

The present monograph deals with refined theories of elastic plates in which both bending and transverse shear effects are taken into account and with some of their applications. Generally these more exact theories result in inte­ gration problems of the sixth order; consequently, three mutually independent boundary conditions at each edge of the plate are required. This is in perfect agreement with the conclusions of the theory of elasticity. The expressions for shearing forces following from refined theories are then valid for the whole investigated region including its boundary where the corresponding boundary conditions for these shearing forces can be prescribed. Quite different seems to be the situation in the classical Kirchhoff-Love's theory in which the influence of transverse shearing strains is neglected. Owing to this simplification the governing differential equation developed by the classical theory is of the fourth order only; consequently, the number of boundary conditions appurtenant to the applied mode of support appears now to be in disagreement with the order of the valid governing equation. Then, limiting the validity of the expressions for shearing forces to the open region of the middle plane and introducing the notion of the so called fictitious Kirchhoff's shearing forces for the boundary of the plate, three actual boundary conditions at each edge of the plate have to be replaced by two approximate conditions transformed in the Kirchhoff's sense.

This book is intended primarily as a teaching text, as well as a reference for individual study in the behavior of thin walled structural components. Such structures are widely used in the engineering profession for spacecraft, missiles, aircraft, land-based vehicles, ground structures, ocean craft, underwater vessels and structures, pressure vessels, piping, chemical processing equipment, modern housing, etc. It presupposes that the reader has already completed one basic course in the mechanics or strength of materials. It can be used for both undergraduate and graduate courses. Since beams (columns, rods), plates and shells comprise components of so many of these modern structures, it is necessary for engineers to have a working knowledge of their behavior when these structures are subjected to static, dynamic (vibration and shock) and environmental loads. Since this text is intended for both teaching and self-study, it stresses fundamental behavior and techniques of solution. It is not an encyclopedia of all research or design data, but provides the reader the wherewithal to read and study the voluminous literature. Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving them to the extent necessary to treat the following material. Chapter 2 presents, in a concise way, the basic assumptions and derives the governing equations for classical Bernoulli-Euler beams and plates in a manner that is clearly understood.

Prof. W. Z. Chien was born on 9 October, 1912 and 1982 saw the 70th anniversary of his birth. Some of his friends, colleagues, and former students prepared this special volume in honour of his outstanding contribution to the field of mechanics. The volume does not contain contributions from all of his students and friends and for this we apologize. Prof. Chien's family have lived. in Qufangquiao Village, Hongshengli, Wuxi County, Jiangsu Province for generations. Many members of his family have been teachers in this village. When he was 14 years old his father died and for a time it appeared necessary to terminate his education but, fortunately, an uncle, Chien Mu, who later became a very famous historian in China, came to his aid and he was able to continue his studies. In 1931 he took entrance exams and was simultaneously admitted to five prestigious Chinese universities. Of these, he chose to enter Tsing-hau University in Beijing, with major work in physics. He received his baccaulaurate in 1935 and taught at middle school for a time until he was awarded a Sino:'British scholarship to study abroad. In the competition for this award, three of the recipients were in the field of mechanics: Prof. C. C. Lin, Prof. Kuo Yung-huai, and Prof. Chien Wei-zang. All three arrived in Toronto in August, 1940 and entered the Depart­ ment of Applied Mathematics of the University of Toronto to study under Prof. J. L. Synge.

Any practitioner who takes his profession in earnest, such that daily work is not a heavy duty but part of their life, will recognize in this book the rigorousness of the analysis and the comprehensive presentation of the problems. This professional attitude is solely able to make the research and design engineer deal with strength structures and their behaviour. Indeed, the computational means that are nowadays available permit the numerical computation of whatever problem; the pro­ gram libraries are extremely rich and programs themselves have developed intensively. Howeyer, though computers are available at any moment without restrictions on the frequency with which they are employed, they finally impoYerish the creative compe­ tency of the civil engineer. Thus, he will calculate increasingly more while devising increasingly less. He will draw less and less on the experience gained in devising and implementing bearing structures because the computational process can be repeated as often as desired over a minimum time-period by means of the available programs. \Ve note that nowadays structures are no longer investigated or economically designed to comply with the requirements of the topic of interest. :Much to the contrary, the solutions are chosen so as to comply with the capabilities of the programs. A bearing structure lives as is prescribed by its initial con­ structive data.

This book is intended primarily as a teaching text, as well as a reference for individual study in the behavior of thin walled structural components. Such structures are widely used in the engineering profession for spacecraft, missiles, aircraft, land-based vehicles, ground structures, ocean craft, underwater vessels and structures, pressure vessels, piping, chemical processing equipment, modern housing, etc. It presupposes that the reader has already completed one basic course in the mechanics or strength of materials. It can be used for both undergraduate and graduate courses. Since beams (columns, rods), plates and shells comprise components of so many of these modern structures, it is necessary for engineers to have a working knowledge of their behavior when these structures are subjected to static, dynamic (vibration and shock) and environmental loads. Since this text is intended for both teaching and self-study, it stresses fundamental behavior and techniques of solution. It is not an encyclopedia of all research or design data, but provides the reader the wherewithal to read and study the voluminous literature. Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving them to the extent necessary to treat the following material. Chapter 2 presents, in a concise way, the basic assumptions and derives the governing equations for classical Bernoulli-Euler beams and plates in a manner that is clearly understood.

Thin walled structures so extensively used nowadays in industry and civil engineering are usually loaded by very complex systems of forces acting on their edges or over their surfaces. In calculating the strength of a structure we replace real loads by certain idealized loads distinguishing between typical surface loads distributed over a great area of the structure and loads acting over a small area. The latter are called concentrated loads. When the area under the load is very small in comparison with the dimensions of the surface of the structure, for example, when the diameter of the loaded area is smaller than the wall thickness, the load can be considered as a single force or a mo­ ment acting on the structure at one point only. The real loads which are met in practice can always be replaced by a combination of components such as forces normal and tangential to the wall as well as bending and twisting moments. Knowing the distribution of the stresses in the structure produced by each component, we can find it under any arbitrary load using the principle of superposition. There are two main reasons for the appearance of the concentration of stresses in the structure. It can be produced by notches, rapid changes of the cross-section, holes, cutouts, etc. on one hand and by concentrated loads resulting from the interaction of the elements of the structure on the other.

Although the theory of thermoelasticity has a long history, its foun­ dations having been laid in the first half of the nineteenth century by Duhamel and Neumann, wide-spread interest in this field did not develop until the years subsequent to World War Two. There are good reasons for this sudden and continuing revival of interest. First, in the field of aeronautics, the high velocities of modern aircraft have been found to give rise to aerodynamic heating; in turn, this produces intense thermal stresses and, by lowering the elastic limit, reduces the strength of the aircraft structure. Secondly, in the nuclear field, the extremely high temperatures and temperature gradients originating in nuclear reactors influence their design and operation. Likewise, in the technology of modern propulsive systems, such as jet and rocket engines, the high temperatures associated with combustion processes are the origin of unwelcome thermal stresses. Similar phenomena are encountered in the technologies of space vehicles and missiles, in the mechanics of large steam turbines, and even in shipbuilding, where, strangely enough. ship fractures are often attributed to thermal stres­ ses of moderate intensities. The investigations of these, and similar, problems have brol!ght forth a remarkable number of research papers, both theoretical and experimental, in which various aspects of thermal stresses in engineering structures are described.

In the last decade or so the theory of shells has undergone a tremendous increase in development. Formerly a subject of interest only to a few special­ ists and for which the literature was relatively smalI, the needs of structures for aerospace missions instigated a torrent of papers on all facets of the theory which also found application in the less glamorous earthbound shell struc­ tures important in everyday life. Some idea of the rapidity of the development can be gained from the fact that a bibliography* completed in 1953 listed some 1455 books and papers as the sum total ofthe literature on shell theory to that date. Three years later, however, a supplementt added another 884 papers to the list, an increase of 60 per cent in that short period of time. The number of papers published since these listings has increased to an extent that does not bear contemplation. Obviously no single volume could contain all that constitutes the theory of shells and so this book is restricted to that portion of the theory associated with small deformations of elastic shells. Plastic deformations of shells, which is hardly developed, and nonlinear deformations and stability, which would require at least aseparate volume, are thus excluded. Even with this restriction, however, the present volume represents a long overdue compro­ mise between completeness and finiteness. In making this compromise I have undoubtedly omitted discussions of many topics and references to many excellent papers which should have been included.

The present monograph deals with refined theories of elastic plates in which both bending and transverse shear effects are taken into account and with some of their applications. Generally these more exact theories result in inte­ gration problems of the sixth order; consequently, three mutually independent boundary conditions at each edge of the plate are required. This is in perfect agreement with the conclusions of the theory of elasticity. The expressions for shearing forces following from refined theories are then valid for the whole investigated region including its boundary where the corresponding boundary conditions for these shearing forces can be prescribed. Quite different seems to be the situation in the classical Kirchhoff-Love's theory in which the influence of transverse shearing strains is neglected. Owing to this simplification the governing differential equation developed by the classical theory is of the fourth order only; consequently, the number of boundary conditions appurtenant to the applied mode of support appears now to be in disagreement with the order of the valid governing equation. Then, limiting the validity of the expressions for shearing forces to the open region of the middle plane and introducing the notion of the so called fictitious Kirchhoff's shearing forces for the boundary of the plate, three actual boundary conditions at each edge of the plate have to be replaced by two approximate conditions transformed in the Kirchhoff's sense.

This book is intended primarily as a teaching text, as well as a reference for individual study in the behavior of thin walled structural components. Such structures are widely used in the engineering profession for spacecraft, missiles, aircraft, land-based vehicles, ground structures, ocean craft, underwater vessels and structures, pressure vessels, piping, chemical processing equipment, modern housing, etc. It presupposes that the reader has already completed one basic course in the mechanics or strength of materials. It can be used for both undergraduate and graduate courses. Since beams (columns, rods), plates and shells comprise components of so many of these modern structures, it is necessary for engineers to have a working knowledge of their behavior when these structures are subjected to static, dynamic (vibration and shock) and environmental loads. Since this text is intended for both teaching and self-study, it stresses fundamental behavior and techniques of solution. It is not an encyclopedia of all research or design data, but provides the reader the wherewithal to read and study the voluminous literature. Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving them to the extent necessary to treat the following material. Chapter 2 presents, in a concise way, the basic assumptions and derives the governing equations for classical Bernoulli-Euler beams and plates in a manner that is clearly understood.

Prof. W. Z. Chien was born on 9 October, 1912 and 1982 saw the 70th anniversary of his birth. Some of his friends, colleagues, and former students prepared this special volume in honour of his outstanding contribution to the field of mechanics. The volume does not contain contributions from all of his students and friends and for this we apologize. Prof. Chien's family have lived. in Qufangquiao Village, Hongshengli, Wuxi County, Jiangsu Province for generations. Many members of his family have been teachers in this village. When he was 14 years old his father died and for a time it appeared necessary to terminate his education but, fortunately, an uncle, Chien Mu, who later became a very famous historian in China, came to his aid and he was able to continue his studies. In 1931 he took entrance exams and was simultaneously admitted to five prestigious Chinese universities. Of these, he chose to enter Tsing-hau University in Beijing, with major work in physics. He received his baccaulaurate in 1935 and taught at middle school for a time until he was awarded a Sino:'British scholarship to study abroad. In the competition for this award, three of the recipients were in the field of mechanics: Prof. C. C. Lin, Prof. Kuo Yung-huai, and Prof. Chien Wei-zang. All three arrived in Toronto in August, 1940 and entered the Depart­ ment of Applied Mathematics of the University of Toronto to study under Prof. J. L. Synge.