 # 1. Provide a general description (1-2 sentences) of the Raleigh-Rit...

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1. Provide a general description (1-2 sentences) of the Raleigh-Ritz method. Specific questions: a. What is minimized? b. Which feature of the solution is approximated? c. Which feature of the solution is exactly satisfied? d. How are the coefficients determined? e. Write an appropriate polynomial trial function (generic term of the series, let it be different as the number of the series term changes) for a beam of length L with zero displacements at the ends x=0, x=L. f. For fixed domain and boundary conditions, what kind of solution are we trying to find? Over what part of the domain? Is the solution exact? Why use this method? g. What is a variation? If we have a variation in a displacement function with displacement conditions at the ends (essential or rigid boundary conditions), what is the variation at the ends? h. Is it possible in general, for a problem in the calculus of variations (i.e. not just the Raleigh-Ritz method), to find the boundary conditions as part of the solution? i. If the variation in a displacement solution is 8(x), the trial solution is u(x), , and the exact solution is y(x), write an equation relating 8(x), u(x) and y(x). . 2. What are the 3 key characteristics of an exact solution to an elasticity boundary value problem? 3. Plane problems. a. Write the defining assumptions for plane stress (stress components) and plane strain (displacement components). Show that the same simplified equilibrium equations result. Show that the z-direction equilibrium equation is identically satisfied for both plane strain and plane stress. b. Substitute the expressions defining the Airy stress function Q(x,) y) (use Cartesian coordinate expressions) into the remaining 2 equilibrium equations and show that equilibrium is satisfied. Is there an important assumption about the specific form chosen for the function that is needed to assure that equilibrium is satisfied? 4. Consider the following state of stress in a cylindrical body with z-axis normal to its cross- sections: 0 0 [o] = 0 0 0 0 O2 (x, y) Use the 6 compatibility equations to show that, in the absence of body forces, the most general form of O E (x,y) is given by o z (x,y) = OCX + By + Y, where O, Band yare constants. 5. It can be shown that the displacements, corresponding to a stress field which is axisymmetric about the z-axis, are given as (with the rigid-body constants set to zero): u, r ] - Inr - + + no-4azro where E is the Young's modulus and A, B, Care undetermined constants. The corresponding stresses are + + 2a2 Tro =0 = Discuss why, for a thick-walled cylinder (Fig 8-8), or a pressurized hole (Fig 8-10) a3 = 0, while for a problem involving a jump discontinuity in us (e.g. a mode III crack) we would want a3 0. What are the resulting stresses and displacements for a thick-walled cylinder? (Do not evaluate the constants.) 6. Consider the stress function - where a is the radius of a solid circular torsion bar and C is a constant. Show that this stress function solves the torsion problem for a circular cylinder of radius a (work it out, don't try to change the stress function, solve for C). Use Eq. (9.3.18) (hint: this is easily accomplished with polar coordinates r2 =x2 + y2 2 ) to determine the expression for the applied torque in terms of a and the angle of twist per unit length a. Then assuming the applied torque is known, give the expression for the angle of twist per unit length a in terms of the torque and the radius a. 7. Consider the Airy stress function 0 = ox'y with zero body forces. (a) Verify that it satisfies the biharmonic equation. (b) Determine the in-plane stresses o , oy, I (c) Determine the tractions on the four rectangular boundaries X = 0, x = b, Y = 0, V = C. . (d) As a plane strain solution, determine Tx2,T,22,00 and all of the strain components. (e) As a plane stress solution, determine Tx2,Ty2,02 and all of the strain components. e Y (0,c) (b,c) ^ A - e e X X (0,0) (b,0) - ,

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