Modern Trends in Structural and Solid Mechanics 1

– “The field has been brilliantly presented in book form …” (Prof. Luis A. Godoy et al., Institute of Advanced Studies in Engineering and Technology, Science Research Council of Argentina and National University of Cordoba, Argentina, Thin-Walled Structures).

– “Elishakoff is one of the pioneers in the use of the probabilistic approach for studying imperfection-sensitive structures” (Prof. Chiara Bisagni and Dr. Michela Alfano, Delft University of Technology; AIAA Journal).

– “Recently, Elishakoff et al. presented an excellent literature review on the historical development of Timoshenko’s beam theory” (Prof. Zhenlei Chen et al., Journal of Building Engineering).

Professor Isaac Elishakoff is the author or co-author of an impressive list of seminal books in the field of deterministic and non-deterministic mechanics, presented below.

Ben-Haim, Y. and Elishakoff, I. (1990). Convex Models of Uncertainty in Applied Mechanics . Elsevier, Amsterdam.

Cederbaum, G., Elishakoff, I., Aboudi, J., Librescu, L. (n.d.). Random Vibration and Reliability of Composite Structures . Technomic, Lancaster.

Elishakoff, I. (1983). Probabilistic Methods in the Theory of Structures . Wiley, New York.

Elishakoff, I. (1999). Probabilistic Theory of Structures . Dover Publications, New York.

Elishakoff, I. (2004). Safety Factors and Reliability: Friends or Foes? Kluwer Academic Publishers, Dordrecht.

Elishakoff, I. (2005). Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions of Semi-Inverse Problems . CRC Press, Boca Raton.

Elishakoff, I. (2014). Resolution of the Twentieth Century Conundrum in Elastic Stability . World Scientific/Imperial College Press, Singapore.

Elishakoff, I. (2017). Probabilistic Methods in the Theory of Structures: Random Strength of Materials, Random Vibration, and Buckling . World Scientific, Singapore.

Elishakoff, I. (2018). Probabilistic Methods in the Theory of Structures: Solution Manual to Accompany Probabilistic Methods in the Theory of Structures: Problems with Complete, Worked Through Solutions . World Scientific, Singapore.

Elishakoff, I. (2020). Dramatic Effect of Cross-Correlations in Random Vibrations of Discrete Systems, Beams, Plates, and Shells . Springer Nature, Switzerland.

Elishakoff, I. (2020). Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories . World Scientific, Singapore.

Elishakoff, I. and Ohsaki, M. (2010). Optimization and Anti-Optimization of Structures under Uncertainty . Imperial College Press, London.

Elishakoff, I. and Ren, Y. (2003). Finite Element Methods for Structures with Large Stochastic Variations . Oxford University Press, Oxford.

Elishakoff, I., Lin, Y.K., Zhu, L.P. (1994). Probabilistic and Convex Modeling of Acoustically Excited Structures . Elsevier, Amsterdam.

Elishakoff, I., Li, Y., Starnes Jr., J.H. (2001). Non-Classical Problems in the Theory of Elastic Stability . Cambridge University Press, Cambridge.

Elishakoff, I., Pentaras, D., Dujat, K., Versaci, C., Muscolino, G., Storch, J., Bucas, S., Challamel, N., Natsuki, T., Zhang, Y., Ming Wang, C., Ghyselinck, G. (2012). Carbon Nanotubes and Nano Sensors: Vibrations, Buckling, and Ballistic Impact . ISTE Ltd, London, and John Wiley & Sons, New York.

Elishakoff, I., Pentaras, D., Gentilini, C., Cristina, G. (2015). Mechanics of Functionally Graded Material Structures . World Scientific/Imperial College Press, Singapore.

Ariaratnam, S.T., Schuëller, G.I., Elishakoff, I. (1988). Stochastic Structural Dynamics – Progress in Theory and Applications . Elsevier, London.

Casciati, F., Elishakoff, I., Roberts, J.B. (1990). Nonlinear Structural Systems under Random Conditions . Elsevier, Amsterdam.

Chuh, M., Wolfe, H.F., Elishakoff, I. (1989). Vibration and Behavior of Composite Structures . ASME Press, New York.

David, H. and Elishakoff, I. (1990). Impact and Buckling of Structures . ASME Press, New York.

Elishakoff, I. (1999). Whys and Hows in Uncertainty Modeling . Springer, Vienna.

Elishakoff, I. (2007). Mechanical Vibration: Where Do We Stand? Springer, Vienna.

Elishakoff, I. and Horst, I. (1987). Refined Dynamical Theories of Beams, Plates and Shells and Their Applications . Springer Verlag, Berlin.

Elishakoff, I. and Lin, Y.K. (1991). Stochastic Structural Dynamics 2 – New Applications . Springer, Berlin.

Elishakoff, I. and Lyon, R.H. (1986), Random Vibration-Status and Recent Developments . Elsevier, Amsterdam.

Elishakoff, I. and Seyranian, A.P. (2002). Modern Problems of Structural Stability . Springer, Vienna.

Elishakoff, I. and Soize, C. (2012). Non-Deterministic Mechanics . Springer, Vienna.

Elishakoff, I., Arbocz, J., Babcock Jr., C.D., Libai, A. (1988). Buckling of Structures: Theory and Experiment . Elsevier, Amsterdam.

Lin, Y.K. and Elishakoff, I. (1991). Stochastic Structural Dynamics 1 – New Theoretical Developments . Springer, Berlin.

Noor, A.K., Elishakoff, I., Hulbert, G. (1990). Symbolic Computations and Their Impact on Mechanics . ASME Press, New York.

Figure P.11. Elishakoff with his wife, Esther Elisha, M.D., during an ASME awards ceremony

On behalf of all the authors of this book, including those friends who were unable to contribute, we wish Prof. Isaac Elishakoff many more decades of fruitful works and collaborations for the benefit of world mechanics, in particular.

Modern Trends in Structural and Solid Mechanics 1 – the first of three separate volumes that comprise this book – presents recent developments and research discoveries in structural and solid mechanics, with a focus on the statics and stability of solid and structural members.

The book is centered around theoretical analysis and numerical phenomena and has broad scope, covering topics such as: buckling of discrete systems (elastic chains, lattices with short and long range interactions, and discrete arches), buckling of continuous structural elements including beams, arches and plates, static investigation of composite plates, exact solutions of plate problems, elastic and inelastic buckling, dynamic buckling under impulsive loading, buckling and post-buckling investigations, buckling of conservative and non-conservative systems, buckling of micro and macro-systems. The engineering applications concern both small-scale phenomena with micro and nano-buckling up to large-scale structures, including the buckling of drillstring systems.

Each of the three volumes is intended for graduate students and researchers in the field of theoretical and applied mechanics.

Prof. Noël CHALLAMEL

Lorient, France

Prof. Julius KAPLUNOV

Keele, UK

Prof. Izuru TAKEWAKI

Kyoto, Japan

February 2021

For a color version of all the figures in this chapter, see www.iste.co.uk/challamel/mechanics2.zip.

Accurate solutions of the linear elasticity equations governing three-dimensional static deformations of fiber-reinforced composite laminates are needed to efficiently design them for structural applications, by considering failure modes in them. In this chapter, the governing equations are written as first-order partial differential equations for three displacements, three transverse stresses and three in-plane strains. The mixed formulation facilitates the satisfaction of continuity conditions at an interface between two adjacent plies. These nine equations are numerically solved by minimizing residuals in them and in the boundary conditions by using the least-squares method. The functional of the residuals is evaluated by expressing the unknown quantities as the product of complete polynomials of different orders in the three independent coordinates and using appropriate quadrature rules. Minimization of the functional, with respect to coefficients appearing in the polynomials for the solution variables, provides a system of simultaneous linear algebraic equations that are numerically solved. It is shown that polynomial functions of degree, at most, 8 in the in-plane coordinates and 3 in the thickness coordinate for each layer of a laminate, provide accurate solutions for stresses and displacements when compared with the analytical solutions of problems. Through-the-thickness stress distributions are found to agree very well with those found analytically. It provides an efficient numerical scheme for accurately finding stresses and displacements in a laminate.