Problems secs 31 through 37 31 a stress distribution

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Unformatted text preview: a are free of shearing stress and no normal stress acts on edge x = a. 3.3. In bending of a rectangular plate (Fig. P3.3), the state of stress is expressed by x = c1y + c2xy xy = c31b2 - y22 (a) What conditions among the constants (the c’s) make the preceding expressions possible? Body forces may be neglected. (b) Draw a sketch showing the boundary stresses on the plate. 3.4. Given the following stress field within a structural member, x y z = a[y2 + b1x2 - y22] xy 2 2 = ab1x + y 2 2 = - 2abxy yz = a[x + b1y - x 2] 2 = xz =0 2 where a and b are constants. Determine whether this stress distribution represents a solution for a plane strain problem. The body forces are omitted. y b x b a FIGURE P3.1. 136 FIGURE P3.3. Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 137 3.5. Determine whether the following stress functions satisfy the conditions of compatibility for a two-dimensional problem: £ 1 = ax2 + bxy + cy 2 (a) £ 2 = ax3 + bx2y + cxy 2 + dy3 (b) Here a, b, c, and d are constants. Also obtain the stress fields that arise from £ 1 and £ 2. 3.6. Figure P3.6 shows a long, thin steel plate of thickness t, width 2h, and length 2a. The plate is subjected to loads that produce the uniform stresses o at the ends. The edges at y = ; h are placed between the two rigid walls. Show that, by using an inverse method, the displacements are expressed by u=- 1E 2 o x, v = 0, w= 11 + 2 E oz 3.7. Determine whether the following stress distribution is a valid solution for a two-dimensional problem: x = - ax2y y 1 = - ay3 3 xy = axy 2 where a is a constant. Body forces may be neglected. 3.8. The strain distribution in a thin plate has the form B ax3 axy 2 axy2 R ax2y in which a is a small constant. Show whether this strain field is a valid solution of an elasticity problem. Body forces may be disregarded. 3.9. The components of the displacement of a thin plate (Fig. P3.9) are given by u = - c1y2 + x2 v = 2cxy Here c is a constant and v represents Poisson’s ratio. Determine the stresses x, y, and xy. Draw a sketch showing the boundary stresses on the plate. 3.10. Consider a rectangular plate with sides a and b of thickness t (Fig. P3.10). (a) Determine the stresses x, y, and xy for the stress function £ = px3y, where p is a constant. (b) Draw a sketch showing the boundary stresses on the plate. (c) Find the resultant normal and shearing boundary forces ( Px, Py, Vx, and V ) along all edges of the plate. y FIGURE P3.6. Problems 137 ch03.qxd 12/20/02 7:21 AM Page 138 y b x b a a FIGURE P3.9. FIGURE P3.10. 3.11. Redo Prob. 3.10 for the case of a square plate of side dimensions a and p £= 2a2 1x2y2 + 1xy32 3 where p is a constant. 3.12. Resolve Prob. 3.10 a and b for the stress function of the form p £=- b3 xy 213b - 2y2 where p represents a constant. 3.13. A vertical force P per unit thickness is applied on the horizontal boundary of a semi-infinite solid plate of unit thickness (Fig. 3.11a). Show that the stress function £ = - 1P/ 2y tan-11y/x2 results in the following stress field within the plate: 2P x =- x3 , 1x2 + y222 2P y =- xy2 , 1x2 + y222 2P xy =- yx2 1x2 + y222 Also plot the resulting stress distribution for x and xy at a constant depth L below the boundary. 3.14. The thin cantilever shown in Fig. P3.14 is subjected to uniform shearing stress o along its upper surface 1y = + h2 while surfaces y = - h and x = L are free of stress. Determine whether the Airy stress function FIGURE P3.14. 138 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 139 £= 1 4o ¢ xy - xy3 Ly 3 xy2 Ly 2 - 2+ + 2≤ h h h h satisfies the required conditions for this problem. 3.15. Figure P3.15 shows a thin cantilever beam of unit thickness carrying a uniform load of intensity p per unit length. Assume that the stress function is expressed by £ = ax2 + bx2y + cy3 + dy 5 + ex2y3 in which a, Á , e are constants. Determine (a) the requirements on a, Á , e so that £ is biharmonic; (b) the stresses x, y, and xy. 3.16. Consider a thin square plate with sides a. For a stress function £ = 1p/a2211x2y2 - 1y42, determine the stress field and sketch it along the 2 6 boundaries of the plate. Here p represents a uniformly distributed loading per unit length. Note that the origin of the x, y coordinate system is located at the lower-left corner of the plate. 3.17. Consider a thin cantilever loaded as shown in Fig. P3.17. Assume that the bending stress is given by Mzy x =- =- I p2 xy 2I (P3.17) and z = xz = yz = 0. Determine the stress components y and xy as functions of x and y. 3.18. Show that for the case of plane stress, in the absence of body forces, the equations of equilibrium may be expressed in terms of displacements u and v as follows: 1+ 02u 02u + 2+ 2 10x 0y 1+ 02v 02v + 2+ 2 10y 0x 0 0u 0v ¢ + ≤=0 0x 0x 0y 0 0v 0v ¢ + ≤=0 0y 0y 0x (P3.18) [Hint: Substitute Eqs. (3.10) together with (2.3) into (3.6).] 3.19. Determine whether the following compatible stress field is possible wi...
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