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CE 595: Finite Elements in Elasticity

CE 595: Finite Elements in Elasticity. Section 1: Review of Elasticity. Stress & Strain Constitutive Theory Energy Methods. Section 1.1: Stress and Strain. Stress at a point Q : . 1.1: Stress and Strain (cont.). Stresses must satisfy equilibrium equations in pointwise manner:.

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CE 595: Finite Elements in Elasticity

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  1. CE 595:Finite Elements in Elasticity

  2. Section 1: Review of Elasticity • Stress & Strain • Constitutive Theory • Energy Methods

  3. Section 1.1: Stress and Strain • Stress at a point Q :

  4. 1.1: Stress and Strain (cont.) • Stresses must satisfy equilibrium equations in pointwise manner: “Strong Form”

  5. 1.1: Stress and Strain (cont.) • Stresses act on inclined surfaces as follows:

  6. 1.1: Stress and Strain (cont.) • Strain at a pt. Q related to displacements :

  7. 1.1: Stress and Strain (cont.) • Normal strain relates to changes in size :

  8. 1.1: Stress and Strain (cont.) • Shearing strain relates to changes in angle :

  9. 1.1: Stress and Strain (cont.) • Sometimes FEA programs use elasticity shearing strains : • Strains must satisfy 6 compatibility equations: (usually automatic for most formulations)

  10. Section 1.2 : Constitutive Theory • For linear elastic materials, stresses and strains are related by the Generalized Hooke’s Law :

  11. 1.2 : Constitutive Theory (cont.) • For isotropic linear elastic materials, elasticity matrix takes special form:

  12. 1.2 : Constitutive Theory (cont.) • Special cases of GHL: • Plane Stress : all “out-of-plane” stresses assumed zero. • Plane Strain : all “out-of-plane” strains assumed zero.

  13. 1.2 : Constitutive Theory (cont.) • Other constitutive relations: • Orthotropic : material has “less” symmetry than isotropic case. FRP, wood, reinforced concrete, … • Viscoelastic : stresses in material depend on both strain and strain rate. Asphalt, soils, concrete (creep), … • Nonlinear : stresses not proportional to strains. Elastomers, ductile yielding, cracking, …

  14. 1.2 : Constitutive Theory (cont.) • Strain Energy • Energy stored in an elastic material during deformation; can be recovered completely.

  15. 1.2 : Constitutive Theory (cont.) • Strain Energy Density : strain energy per unit volume. • In general,

  16. Section 1.3 : Energy Methods • Energy methods are techniques for satisfying equilibrium or compatibility on a global level rather than pointwise. • Two general types can be identified: • Methods that assume equilibrium and enforce displacement compatibility. (Virtual force principle, complementary strain energy theorem, …) • Methods that assume displacement compatibility and enforce equilibrium.(Virtual displacement principle, Castigliano’s 1st theorem, …) Most important for FEA!

  17. 1.3 : Energy Methods (cont.) • Principle of Virtual Displacements (Elastic case): (aka Principle of Virtual Work, Principle of Minimum Potential Energy) • Elastic body under the action of body force b and surface stresses T. • Apply an admissible virtual displacement • Infinitesimal in size and speed • Consistent with constraints • Has appropriate continuity • Otherwise arbitrary • PVD states that for any admissible is equivalent to static equilibrium.

  18. 1.3 : Energy Methods (cont.) • External and Internal Work: • So, PVD for an elastic body takes the form

  19. 1.3 : Energy Methods (cont.) • Recall: Integration by Parts • In 3D, the corresponding rule is:

  20. 1.3 : Energy Methods (cont.) • Take a closer look at internal work:

  21. 1.3 : Energy Methods (cont.) • By reversing the steps, can show that the equilibrium equations imply • is called the weak form of static equilibrium.

  22. 1.3 : Energy Methods (cont.) • Rayleigh-Ritz Method : a specific way of implementing the Principle of Virtual Displacements. • Define total potential energy ; PVD is then stated as • Assume you can approximate the displacement functions as a sum of known functions with unknown coefficients. • Write everything in PVD in terms of virtual displacements and real displacements. (Note: stresses are real, not virtual!) • Using algebra, rewrite PVD in the form • Each unknown virtual coefficient generates one equation to solve for unknown real coefficients.

  23. 1.3 : Energy Methods (cont.) • Rayleigh-Ritz Method: Example Given: An axial bar has a length L, constant modulus of elasticity E, and a variable cross-sectional area given by the function , where β is a known parameter. Axial forces F1 and F2 act at x = 0 and x= L, respectively, and the corresponding displacements are u1 and u2 . Required: Using the Rayleigh-Ritz method and the assumed displacement function , determine the equation that relates the axial forces to the axial displacements for this element.

  24. 1.3 : Energy Methods (cont.) Solution : • Treat u1 and u2as unknown parameters. Thus, the virtual displacement is given by • Calculate internal and external work:

  25. 1.3 : Energy Methods (cont.) (Cont) : • Equate internal and external work:

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