## A Variational Approach to Structural AnalysisAn insightful examination of the numerical methods used to develop finite element methods A Variational Approach to Structural Analysis provides readers with the underpinnings of the finite element method (FEM) while highlighting the power and pitfalls of virtual methods. In an easy-to-follow, logical format, this book gives complete coverage of the principle of virtual work, complementary virtual work and energy methods, and static and dynamic stability concepts. The first two chapters prepare the reader with preliminary material, introducing in detail the variational approach used in the book as well as reviewing the equilibrium and compatibility equations of mechanics. The next chapter, on virtual work, teaches how to use kinematical formulations for the determination of the required strain relationships for straight, curved, and thin walled beams. The chapters on complementary virtual work and energy methods are problem-solving chapters that incorporate Castigliano's first theorem, the Engesser-Crotti theorem, and the Galerkin method. In the final chapter, the reader is introduced to various geometric measures of strain and revisits straight, curved, and thin walled beams by examining them in a deformed geometry. Based on nearly two decades of work on the development of the world's most used FEM code, A Variational Approach to Structural Analysis has been designed as a self-contained, single-source reference for mechanical, aerospace, and civil engineering professionals. The book's straightforward style also provides accessible instruction for graduate students in aeronautical, civil, mechanical, and engineering mechanics courses. |

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antisymmetry applied load axis beam shown becomes bending boundary conditions centroid coefﬁcient of thermal complementary virtual components compute consider constraint coordinates cross section curvature curved beams deﬁned deﬁnition deﬂection deformable body derivative determine differential equation distribution elastic Engesser-Crotti theorem equation of equilibrium equilibrium paths exact differential example expression Figure ﬁnal ﬁrst ﬁxed ﬂexural rigidity ﬂexural rigidity E1 force frame function Galerkin method horizontal integral left end Legendre transformation nondimensional obtain potential energy principal pole principle of virtual Problem radius reaction relationship represents right end rotation sectorial centroid shear center shear stress shown in Fig simply supported solution spring of modulus stability stationary stiffness matrix strain energy stress resultants structure symmetry temperature thermal expansion thin walled beams tion tip deﬂection torsion variables variation vector vertical virtual displacement virtual load virtual strain yields zero

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Page 9 - the derivative of the variation": /(*) ] = - t*(*) = *'(*). (29.1) dx dx dx In the second case we have "the variation of the derivative": S £ /(*)-/^-/(*)-(/+ «*')-/-«*'(*). (29.2) dx This gives Aiy-a^?. (29.3) dx dx This shows that the derivative of the variation is equal to the variation of the derivative. In a similar way we may be interested in the variation of a definite integral. This means that we take the definite integral evaluated for the modified integrand minus the definite integral...