QE Topics List: Structural Analysis August 2003

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QE Topics List: Structural Analysis August 2003 General Trusses Structural idealizations Equations of static equilibrium in 2-D and 3-D Method of sections; method of joints Static determinacy (vs. indeterminacy) and stability (including Stiffness method of analysis for 2-D and 3-D configurations eigenvalue methods) Temperature loads; misfits; secondary stresses Reactions for statically determinate structures Moving loads; influence lines (Mueller-Breslau principle) Matrix algebra: Operations – multiplication, transposition, and use of submatrices Determinants; inverses; linear equations; Gauss elimination; Choleski decomposition The Finite Element Method for Two-Dimensional Problems 2-D elasticity (stress equilibrium; strain-displacement; Hooke’s law; compatibility; Cauchy relations; plane stress and plane strain idealizations) Principle of virtual displacements for 2-D continuum Constant strain triangle (CST) element formulation (B matrix; stiffness matrix) Solution of simple plane stress and plane strain linear elasticity problems using CST elements Interpretation of computed strains and stresses from CST solutions Continuous Beams and Frames Virtual Work and Energy Methods Shear and moment diagrams for beams Principle of virtual displacements for truss and beam-frame Differential equation for beam bending structures Moment-area and conjugate beam methods for deflections Interpolation concepts; beam interpolating (shape) Moment distribution method (beams and frames without functions sidesway) Finite element for beams; internal and external virtual work Stiffness method for 2-D and 3-D configurations: for beams Applications to continuous beams and simple frames with sway Beam stiffness coefficients and fixed-end forces via flexibility method Plane frame element coordinate transformation Methods to impose displacement boundary conditions Modified frame stiffness for force-moment releases Imposition of non-global displacement constraints Methods to orient space frame elements Structural symmetry; load decomposition to symmetric and anti- symmetric parts Internal hinges; elastic supports; semi-rigid connections; curved members .

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Symmetry Groups and Conservation Laws in Structural Mechanics
Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part 1, 223–230. Symmetry Groups and Conservation Laws in Structural Mechanics Vassil M. VASSILEV and Peter A. DJONDJOROV Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev St., Bl.4, 1113 Sofia, Bulgaria E-mails: [email protected] and [email protected] Recent results concerning the application of Lie transformation group methods to structural mechanics are presented. Focus is placed on the point Lie symmetries and conservation laws inherent to the Bernoulli–Euler and Timoshenko beam theories as well as to the Marguerre- von Kármán equations describing the large deflection of thin elastic shallow shells within the framework of the nonlinear Donnell–Mushtari–Vlasov theory. 1 Introduction The present paper is concerned with the invariance properties (point Lie symmetries) of three classes of self-adjoint partial differential equations arising in structural mechanics – the dynamic beam equations of Bernouli–Euler and Timoshenko type governing vibration of beams on a variable elastic foundation and dynamic stability of fluid conveying pipes, and Marguerre-von Kármán equations describing the large deflection of thin isotropic elastic shallow shells subjected to an external transverse load and a nonuniform heating. Once the invariance properties of a given differential equation are established, several impor- tant applications are available. First, it is possible to obtain classes of group-invariant solutions. For a self-adjoint equation another application of its symmetries arises since it is the Euler– Lagrange equation of a certain functional. If a symmetry group of such an equation turned out to be its variational symmetry as well, that is a symmetry of the associated functional, then Noether’s theorem guarantees the existence of a conservation law for the solutions of this equation.
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