**5R18. Analysis and Design of Elastic Beams: Computational Methods. - ** WD Pilkey *(Dept of Mech and Aerospace Eng, Univ of Virginia). Wiley, New York. 2002. 461 pp. ISBN 0-471-38152-7. $95.00.*

*Reviewed by SN Krivoshapko (Dept of Strength Mat, Peoples Friendship Univ of Russia, 6, Micklukho-Maklaya Str, Moscow, 117198, Russia).*

The book is printed on good acid-free paper. It has a hard cover of nice-looking color and considered design. Text, figures, and tables are well read.

The book begins with an introduction to the classical theory of linear elasticity. The six linearized strain-displacement relations, the six stress-strain equations for linearly elastic isotropic materials, and the three static relations with 15 unknown parameters (ie, the three components of the displacement vector, the three normal and three shear stress components, and the three normal and three shear strains) are presented without deduction. The author is punctual in notations and descriptions of properties, functions, constants, and coefficients, but it might also be good to present alternative descriptions for some equations and relations. For example, the strain compatibility conditions are known as the B de Saint-Venant’s conditions. The stress-strain equations are also known as the generalized Hooke’s law, but this is mentioned only in passing in the middle of the book (p 237). Formulas, in which the normal stresses are expressed through the dilatation and the normal strains, are also called as Hooke’s law in Lame´’s form. All basic relations and equations are written in matrix form. A beam in pure bending is studied in detail, and several numerical examples of analysis of thin-walled horizontal cantilevered beam with an asymmetrical cross section loaded with a vertical concentrated force at the free edge or subjected to joint action of the vertical concentrated force and the horizontal compressive force are presented. As a variant, a beam with nonhomogeneous material properties on the cross section is considered. The presented brief list of references dealing with the contents of Chapter 1 defines the status of the linear theory of elasticity in our time, but it would be desirable to see the book by SP Timoshenko and JN Goodier, *Theory of Elasticity* (McGraw-Hill, NY, 1970) in this list.

Engineering beam theory, presented in Chapter 2, is an approximate theory because it neglects the normal stresses $\sigma y$ and $\sigma z$ and assumes that Poisson’s ratio is zero, but the error of the calculation will be insignificant because the normal stresses rejected are much smaller than the axial stresses $\sigma x.$ This theory gives good results if a beam has two general dimensions of cross section, much smaller than a length of the beam element. The author presents fundamental engineering theory equations in analytical and in matrix form illustrating their application for planar engineering beam theory. Examples of determination of the deflections of the beams with symmetric and asymmetrical cross sections through use of integration of the differential equation of the elastic axis of the beam are given. Cantilevered beams and statically indeterminate beams are examined. But in the given examples, the author does not mention in the text and does not show in the figures that concentrated forces subjected to the beams must pass through a shear center of the section to avoid the appearance of torque. The solution of the first-order form of the differential equations is demonstrated, and examples of the determination of a transfer matrix for a general beam element based on planar engineering beam theory are adduced. After that, WD Pilkey goes on to the determination of stiffness matrices for a Bernoulli-Euler beam element and for a beam element on an elastic foundation. He assumes that the deflection of a beam element can be approximated by a polynomial of the third order. In one example, torsion is considered too. The chapter is ended by the consideration of exact and lumped mass matrices and dynamic stiffness matrices for dynamic problems of an undamped structure.

Complex beams and planar frameworks are studied in Chapter 3 with the application of the element matrices of Chapter 2, and that is why local and global coordinate systems are introduced into use. The discussion is directed primarily to the static analysis based on the displacement method. Each node of the beam element has three degrees of freedom. A two-element framework with the three nodes subjected to nodal loading and a beam on elastic foundation under action of concentrated forces were chosen for demonstration of the theoretical proposition of the chapter. The free vibration analysis and forced response is discussed briefly. The form of cross-sections of the beam elements is not specified.

In the short Chapter 4, the author describes the idea of an isoparametric element in which the functions used for presentation of behavior under deforming are also applied for the description of the cross-sectional properties (ie, the cross-sectional area, the first moments of area, the area moments of inertia, and the product of inertia). The constructing of an isoparametric element represents transformation of nondimensional square element, called a reference domain, with nine nodes into the real element curved with the same number of nodes. An example of determination of properties of an asymmetric cross section with the help of the finite-element mesh definitions of Chapter 4 has shown that the values calculated here are the same as those of the analytical method presented in Chapter 1. For those who want to have additional information on finite elements for cross-sectional analysis, Pilkey listed three textbooks where detailed accounts of the FEM are available.

Chapter 5 begins with the description of fundamentals of Saint-Venant’s pure torsion. The fundamental assumption in this analysis is that cross sections are free to warp without restraint. The displacement formulations are illustrated by examples of analysis of bars with circular, elliptical, and rectangular cross sections. The force method relations are used to solve analytically torsion problems for two bars with elliptical and equilateral triangle cross-sectional shapes. Subsequently, Pilkey addresses classical formulas for thin-walled open and closed cross sections of bars subjected to pure torsion. The examples presented and comparisons illustrating the opportunities of closed and open cross sections help the reader to learn the theoretical material better. The formulas in this part are extended to apply to an $n$-celled tube, a hollow section with fins, a wing section, and a composite cross section. The analysis of the composite cross section, transformed to an equivalent cross section with the help of the modulus weight ratio for the shear modulus, is presented without any concrete example. Using the analytical expressions described before, the author goes over to a finite element formulation for the linear Saint-Venant torsion problem and shows that the principle of virtual work and Galerkin’s method lead to the same element stiffness relations. Chapter 5 concludes with brief information on alternative computational methods such as the boundary element method and the direct integration method. The presented references can provide additional information on the examined problem. Those who want to know more on torsion can also read the book by NH Arutyunyan and BL Abramyan, *Torsion of Elastic Bodies*, (1963, 688 pp.).

Shear stresses generated by shear forces on straight beams are considered in Chapter 6. First, the standard methods of determination of shear and normal stresses in homogeneous and nonhomogeneous beams subjected to transverse shear loads are described with the help of engineering beam theory and theory of elasticity. After that, the same problems are solved in finite element solution formulation. One section is devoted to the determination of shear center for thin-walled cross sections and traditional formulas for the location of shear centers for common cross sections are provided. Having briefly described a history of the study of shear deformation, Pilkey goes over the determination of shear deformation coefficients for various cross-sections, assuming that the strain energy for a beam is equal to the strain energy for a 1D beam based on technical beam theory and using the finite element solution formulation and traditional approximate analytical formulas.

In Chapter 7, Restrained Warping of Beams, the traditional analytical method for studying thin-walled beams within the limits of linear theory is described. The method was devised in general by VZ Vlasov. New concepts such as the warping function, moment of warping, sectorial characteristics, principal pole, warping torque, warping constant, a bimoment, and so on are introduced into practice. It is shown that the shear center and the principal pole are the same point. Having assembled governing equations, the author expresses them in first-order form. Analytical formulas for the determination of normal and shear stresses due to restrained warping are presented. The theory is illustrated by several examples.

In Chapter 8, additional information from the theory of elasticity is presented. Principal stresses, extreme shear stresses, and three failure theories (maximum stress theory, maximum shear theory, and distortion energy theory) are discussed. Such failure theories as maximum normal strain theory (Mariott, 1682), simplified theory of strength of O Mohr for brittle materials, and some others are not considered.

Chapter 9 gives elementary information necessary for understanding the idea of an optimal cross-sectional shape design of thin-walled beams with the help of rational B-spline curves.

Chapter 10, Shape Optimization of Thin-Walled Sections, is one of the important parts of the book because the problem designated in the chapter’s title often becomes the main object of investigation. A shape design method described in this chapter is assumed to be defined by nonuniform rational B-spline curve.

In one appendix, Pilkey describes how to prepare input data files for some of the computer programs in Fortran 90 presented on his web site.

*Analysis and Design of Elastic Beams: Computational Methods* can be a training appliance for students and post-graduate students learning the theory of elasticity and matrix calculus. This book will be a good reference for mechanical and civil engineers and designers working in corresponding fields of industry where the thin-walled bars are used. Lecturers and instructors dealing with beam analysis also should take an interest in this book.

In this book, the author undertook describing the very wide circle of the theory of elasticity, and he carried out the stated aims not repeating other published manuals and reference books. That is why this book can be recommended for libraries as well as individuals.