2D PLANE PROBLEMS (0, Airy Stress Function (No body forces, i.e. B x = B y = 0) An example (in polar coordinates) is given in Section below. AE APPLIED ELASTICITY IN AEROSTRUCTURES Talha MUTLU Monday, November 16, Ebru BAYRAKTAR Although it is easier to solve for than it is to solve for stress directly, this is still not. Consider the Airy stress function φ = α(x 1 4 − x 2 4). (a) Verify that it satisfies the biharmonic equation. (b) Determine the in-plane stresses T 11, T 12 and T (c) Determine and sketch the tractions on the four rectangular boundaries x 1 = 0, x 1 = b, x 2 = 0, x 2 = c. (d) As a plane strain solution, determine T 13, T 23, T 33 and all the strain components. Note that, unlike stress and displacement, the Airy stress function has no obvious physical physical meaning. fo rces are The reason for writing the stresses in the form is that, provided the body forces zero, the equilibrium equations are automatically satisfied, which can be seen by substituting Eqns. into Eqns. { Problem 1}.

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Series solution of ode: Airy's equation, time: 14:16

() Note that, unlike stress and displacement, the Airy stress function has no obvious physical meaning. The reason for writing the stresses in the form is that, provided the body forces are zero, the equilibrium equations are automatically satisfied, which can . Airy Stress Function Solutions to plane strain and plane stress problems can be obtained by using various stress function techniques which employ the Airy stress function to reduce the generalized formulation to the governing equations with solvable unknowns. The stress function formulation is based on the idea representing the stress fields that satisfy the equilibrium equations. 2D PLANE PROBLEMS (0, Airy Stress Function (No body forces, i.e. B x = B y = 0) An example (in polar coordinates) is given in Section below. AE APPLIED ELASTICITY IN AEROSTRUCTURES Talha MUTLU Monday, November 16, Ebru BAYRAKTAR Although it is easier to solve for than it is to solve for stress directly, this is still not. Stress Functions. • Relate six stresses to (fewer) functions defined in such a manner that they identically satisfy the equilibrium conditon • Can be done for 3-D case • Can be done for anisotropic (most often orthotropic) case --> See: Lekhnitskii, Anisotropic Plates, Gordan & Breach, Note that, unlike stress and displacement, the Airy stress function has no obvious physical physical meaning. fo rces are The reason for writing the stresses in the form is that, provided the body forces zero, the equilibrium equations are automatically satisfied, which can be seen by substituting Eqns. into Eqns. { Problem 1}. Consider the Airy stress function φ = α(x 1 4 − x 2 4). (a) Verify that it satisfies the biharmonic equation. (b) Determine the in-plane stresses T 11, T 12 and T (c) Determine and sketch the tractions on the four rectangular boundaries x 1 = 0, x 1 = b, x 2 = 0, x 2 = c. (d) As a plane strain solution, determine T 13, T 23, T 33 and all the strain components.Mar 2, Note that, unlike stress and displacement, the Airy stress function has no number of examples including non-zero body forces are examined. function. The Airy stress function is determined so that the prescribed boundary condition at a far Example Problem: Circular Inclusion Problem. Consider finding located at thekeep.online pdf. Relate six stresses to (fewer) functions defined in such a manner Define the “ Airy” Stress Function = φ(x, y) . example later) and in plane strain as well. Nov 16, If we replace the strains in the compatibility eqns. by the stress from Hooke's Law. It leads below 2) Define Airy stress function by: Example. problems can be treated using stress functions which are polynomials in x, y. In for example, that we consider just those terms in a general polynomial whose . Airy stress function is so defined that whatever stress function is used, the cor-. -

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