# Solids

 The flashcards below were created by user chloe_h on FreezingBlue Flashcards. Describe St Venant's principle and its importance to the derived loading scenario St Venant's principle states that the stress distribution is independent of the exact mode of loading except in the immediate vicinity of the load application Explain what conditions would apply if plane strain can be assumed for analysis of the stress distribution Conditions of plane strain apply when the loaded body is very long in the z direction and where there is no displacement in the z direction. The cross-sections are all under the same conditions. For such problems, γ_yz = 0, γ_xy = 0 and E_z = 0. Since σ_z = ν(σ_x + σ_y), the problem reduces to three unkowns, σ_x, σ_y, and Τ_xy. Body force Acts per unit volume (eg inertia) Surface force Acts per unit area (eg pressure) Briefly explain the typical scenario that can be described as plane stress, detailing what additional info would be required to confirm that the assumption made for a square thin plate to be modelled as plane stress is valid. Plane stress refers to a thin body where the loads are applied in the plane of the body and act over the thickness. Information on the loads acting on the plate needs to be known to confirm that conditions of plane stress apply. Determine what fundamental principles are satisfied if the biharmonic equation is satisfied The conditions of compatibility and the equations of equilibrium are satisfied if the biharmonic equation is satisfied. Outline the main features of St Venant's theory of the torsion of prismatic bars of arbitrary cross-section and show that the distribution of shear stresses in such a bar may be obtained by solving the following eqn subject to BCs: Provides exact solutions to the shear stress distributions and warping deformation for a variety of non-circular sections Semi inverse methodDirect methods = find stress function that satisfies geometry and BCsInverse methods = find geometry and BCs that satisfy assumed stress functionSemi-inverse: make assumptions regarding deformations and stresses, then find coefficients for a 'special' stress function Has direct analogy with other physical phenomena (eg soap film technique) hence the method can be easily visualised St Venant's torsional stress functionSatisfies Poisson's eqnIs constant (usually zero) along boundariesVolume under the surface proportional to the applied coupleShear stresses are given by the spatial derivatives of this function St Venant's torsional stress function Satisfies Poisson's eqnIs constant (usually zero) along boundariesVolume under the surface proportional to the applied coupleShear stresses are given by the spatial derivatives of this function Describe the mathematical princples behind the Finite Element Method for analysis of stresses and deformations in linear elastic solides Principles of FEM:Numerical method for the solution of PDEs (partial differential eqns) and related boundary value problems (BVP)Based on the "variational" formulation of the problem, i.e. its "weak form" with ENERGY being the "functional" of the displacement functionThus in solid mechanics it is defined as an ENERGY methodThe "trial" functions of this variational method are the interpolated FE displacements given by the "shape functions"Both interpolation (of displacements) and integration (of stresses) are performed within the finite elementsSuitable for the analysis of complex geometries due to its "element-by-element" nature Describe the physical meaning of the Principle of Virtual Work and explain how it can be used to derive finite element equations PVW - work done by "poking a node" or adding displacementUsed to derive RE eqnsWeak (variational) statement of equilibrium and BCsAt the element level: W[external] - W[internal] = 0 Authorchloe_h ID273241 Card SetSolids Descriptionsolid Updated2014-05-08T17:12:42Z Show Answers