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Structures, Models, & Assumptions

Terms like Euler-Bernoulli beam or Timoshenko beam are often used in the literature or technical communications. I personally think these terms are confusing because they use a model along with its associated assumptions to name a structure. Actually a beam structure is a solid with its axis much larger than the two other dimensions. They can be modeled using 3D continuum mechanics, Euler-Bernoulli model, Timoshenko model, Vlasov model, and other higher order models depending on the behavior you want to capture. Usually these models were derived using a set of assumptions, for example, the well-known Euler Bernoulli assumptions and Timoshenko assumptions. I prefer to consider models completely separated from the structure and associated assumptions. Following the standard formulism of continuum mechanics, each model contains kinematics, equilibrium equations and constitutive equations. For example, the Euler-Bernoulli beam model has four displacement variables including three displacements and the twist angle, it has four strain variables including tension and twist rate, and curvatures which are related to beam displacements using four strain-displacement relations, the strain conjugates are axial forces and moments in three directions which are governed by the four equilibrium equations, and related with beam strains using four constitutive equations. In other words, the Euler-Bernoulli beam model contains a total of 12 equations for 12 unknowns. Similarly, the Timoshenko beam model has six displacement variables including three displacements and three rotations, it has six strain variables including two additional transverse shear strains. Beam strains are defined in terms of beam displacements using six strain-displacement relations, the strain conjugates are three forces and three moments governed by six equilibrium equations, and related with beam strains using six constitutive equations. In other words, the Timoshenko beam model contains a total of 18 equations for 18 unknowns. Assumptions due to Euler-Bernoulli or Timoshenko can help evaluate the stiffness terms EA, GJ, etc in for isotropic beams. However, these assumptions are not absolutely needed. Particularly when it comes to composite beams, the stiffness matrix in the constitutive relations could be fully populated and we lose most of the intuition we had about isotropic homogeneous beams, such assumptions should be avoided. MSG is created with this motivation: obtaining constitutive relations for simple engineering models such as the Euler-Bernoulli model without using its originally associated assumptions.  

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