School of Civil EngineeringSpring 2007 CE 595: Finite Elements in Elasticity Instructors: Amit Varma, Ph.D. Timothy M. Whalen, Ph.D.

Review of Elasticity -2- Section 1: Review of Elasticity 1.Stress & Strain 2.Constitutive Theory 3.Energy Methods

Review of Elasticity -3- Section 1.1: Stress and Strain Stress at a point Q : Stress at a point Q :

Review of Elasticity : Stress and Strain (cont.) Stresses must satisfy equilibrium equations in pointwise manner: Stresses must satisfy equilibrium equations in pointwise manner: “Strong Form”

Review of Elasticity : Stress and Strain (cont.) Stresses act on inclined surfaces as follows: Stresses act on inclined surfaces as follows:

Review of Elasticity : Stress and Strain (cont.) Strain at a pt. Q related to displacements : Strain at a pt. Q related to displacements :

Review of Elasticity : Stress and Strain (cont.) Normal strain relates to changes in size : Normal strain relates to changes in size :

Review of Elasticity : Stress and Strain (cont.) Shearing strain relates to changes in angle : Shearing strain relates to changes in angle :

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

Review of Elasticity -10- Section 1.2 : Constitutive Theory For linear elastic materials, stresses and strains are related by the Generalized Hooke’s Law : For linear elastic materials, stresses and strains are related by the Generalized Hooke’s Law :

Review of Elasticity : Constitutive Theory (cont.) For isotropic linear elastic materials, elasticity matrix takes special form: For isotropic linear elastic materials, elasticity matrix takes special form:

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

Review of Elasticity : Constitutive Theory (cont.) Other constitutive relations: 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, …

Review of Elasticity : Constitutive Theory (cont.) Strain Energy Strain Energy – Energy stored in an elastic material during deformation; can be recovered completely.

Review of Elasticity : Constitutive Theory (cont.) Strain Energy Density : strain energy per unit volume. Strain Energy Density : strain energy per unit volume. In general, In general,

Review of Elasticity -16- Section 1.3 : Energy Methods Energy methods are techniques for satisfying equilibrium or compatibility on a global level rather than pointwise. Energy methods are techniques for satisfying equilibrium or compatibility on a global level rather than pointwise. Two general types can be identified: 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 1 st theorem, …) Most important for FEA!

Review of Elasticity : Energy Methods (cont.) Principle of Virtual Displacements (Elastic case): (aka Principle of Virtual Work, Principle of Minimum Potential Energy) 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.

Review of Elasticity : Energy Methods (cont.) External and Internal Work: External and Internal Work: So, PVD for an elastic body takes the form So, PVD for an elastic body takes the form

Review of Elasticity : Energy Methods (cont.) Recall: Integration by Parts Recall: Integration by Parts In 3D, the corresponding rule is: In 3D, the corresponding rule is:

Review of Elasticity : Energy Methods (cont.) Take a closer look at internal work: Take a closer look at internal work:

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

Review of Elasticity : Energy Methods (cont.) Rayleigh-Ritz Method : a specific way of implementing the Principle of Virtual Displacements. 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.

Review of Elasticity : Energy Methods (cont.) Rayleigh-Ritz Method: Example 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 F 1 and F 2 act at x = 0 and x= L, respectively, and the corresponding displacements are u 1 and u 2. 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.

Review of Elasticity : Energy Methods (cont.) Solution : 1)Treat u 1 and u 2 as unknown parameters. Thus, the virtual displacement is given by 2)Calculate internal and external work:

Review of Elasticity : Energy Methods (cont.) (Cont) : 2) 2) 3)Equate internal and external work: