For example, phenomena such as buckling can only be modelled when the interaction between the axial loads and the lateral deformations are coupled. However, for large deformations, and normal force in an axially loaded beam can affect the lateral deformation as well. This uncoupling is only valid for small deformations. i.e., the axial loading is only related to the axial deformation, and the lateral loading is only related to the lateral deformation. It is possible to augment the Euler Bernoulli and the Timoshenko beams so that they also account for axial deformations of the neutral axis, but because of the assumption of small deformations, the resulting equations of axial-loading deformations and bending deformations are uncoupled. The Euler Bernoulli and the Timoshenko beam formulations described account only for the deformations due to bending, without considering any axial deformation in the neutral axis of the beam. Compute the normal force, the stress distribution, and the strain distribution in a beam by solving the differential equation of equilibrium.Describe the three basic assumptions for the equilibrium equation of beam under axial load.Reproduce the derivation of the equilibrium equation of a beam under axial loading.One and Two Dimensional Isoparametric Elements and Gauss Integration.Euler Bernoulli Beams under Lateral Loading.Approximate Methods: The Rayleigh Ritz Method.The Principle of Minimum Potential Energy for Conservative Systems in Equilibrium.Illustrative Examples for the Principle of Virtual Work.Applications of the Principle of Virtual Work.Expressions For the Strain Energy in Linear Elastic Materials.Principle of Material Frame-Indifference.A Method for Estimation of the Material Parameters of Hyperelastic Material Models in Relation to the Linear Elastic Material Model.Principal Stresses of Isotropic Hyperelastic Materials.Examples of Isotropic Hyperelastic Potential Energy Functions.Frame-Indifferent Isotropic Hyperelastic Potential Energy Functions.Plane Isotropic Linear Elastic Materials Constitutive Laws.Matrix of Material Properties of Linear Elastic Materials.Classification of Material Mechanical Response. ![]() First and Second Piola Kirchhoff Stress Tensors.The Deformation and the Displacement Gradients.Description of Motion and Simple Examples.Vector Calculus in Cylindrical Coordinate Systems.Additional Definitions and Properties of Linear Maps.Additional Structure for Linear Vector Spaces.
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