In the elementary theory the lateral stress y given

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Unformatted text preview: cts of P and F results in the following expression for combined stress in a pivot or in a wedge–cantilever: r =- r1 P cos 1 2 + sin 2 2 - F cos r1 - 1 2 1 sin 2 2 , = 0, r =0 (3.44) The foregoing provides the local stresses at the support of a beam of narrow rectangular cross section. Concentrated Load on a Straight Boundary (Fig. 3.11a) By setting = /2 in Eq. (3.40), the result r =- 2P cos , r = 0, r =0 (3.45) is an expression for radial stress in a very large plate (semi-infinite solid) under normal load at its horizontal surface. For a circle of any diameter d with center on the x axis and tangent to the y axis, as shown in Fig. 3.11b, we have, for point A of the circle, d # cos = r. Equation (3.45) then becomes r =- 2P d (3.46) We thus observe that, except for the point of load application, the stress is the same at all points on the circle. The stress components in Cartesian coordinates may be obtained readily by following a procedure similar to that described previously for a wedge: 120 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 121 FIGURE 3.11. (a) Concentrated load on a straight boundary of a large plate; (b) a circle of constant radial stress. x =- 2P cos4 x y =- 2P sin2 x xy =- 2P sin cos3 x x3 1x2 + y222 xy 2 2P =2 1x + y222 x2y 2P =1x2 + y222 2P =cos2 (3.47) The state of stress is shown on a properly oriented element in Fig. 3.11a. 3.10 STRESS DISTRIBUTION NEAR CONCENTRATED LOAD ACTING ON A BEAM The elastic flexure formula for beams gives satisfactory results only at some distance away from the point of load application. Near this point, however, there is a significant perturbation in stress distribution, which is very important. In the case of a beam of narrow rectangular cross section, these irregularities can be studied by using the equations developed in Sec. 3.9. Consider the case of a simply supported beam of depth h, length L, and width b, loaded at the midspan (Fig. 3.12a). The origin of coordinates is taken to be the FIGURE 3.12. Beam subjected to a concentrated load P at the midspan. 3.10 Stress Distribution Near Concentrated Load Acting on a Beam 121 ch03.qxd 12/20/02 7:20 AM Page 122 center of the beam, with x the axial axis as shown in the figure. Both force P and the supporting reactions are applied along lines across the width of the beam. The bending stress distribution, using the flexure formula, is expressed by œ x =- My 6P L = ¢ - x≤y I bh3 2 where I = bh3/12 is the moment of inertia of the cross section. The stress at the loaded section is obtained by substituting x = 0 into the preceding equation: œ x 3PL y bh3 = (a) To obtain the total stress along section AB, we apply the superposition of the bending stress distribution and stresses created by the line load, given by Eq. (3.45) for a semi-infinite plate. Observe that the radial pressure distribution created by a line load over quadrant ab of cylindrical surface abc at point A (Fig. 3.12b) produces a horizontal force /2 L 0 1 r sin 2r d = /2 2P L 0 sin cos d = P (b) and a vertical force /2 L - /2 1 /2 r cos 2r d = L - 2P cos2 d = P (c) /2 applied at A (Fig. 3.12c). In the case of a beam (Fig. 3.12a), the latter force is balanced by the supporting reactions that give rise to the bending stresses [Eq. (a)]. On the other hand, the horizontal forces create tensile stresses at the midsection of the beam of P bh (d) Ph y 6P =y 2I bh2 (e) – x = as well as bending stresses of – x =- Here Ph/2 is the bending moment of forces P/ about the point 0. Combining the stresses of Eqs. (d) and (e) with the bending stress given by Eq. (a), we obtain the axial normal stress distribution over beam cross section AB: x = 3P 2h L≤y + 3¢ bh P bh 3PL 4h 3PL P 1- 0.637 ≤= 2¢ 2 3L bh 2bh 2bh (3.48) At point B(0, h/2), the tensile stress is 1 x2B = 122 Chapter 3 (3.49) Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 123 The second term represents a correction to the simple beam formula owing to the presence of the line load. It is observed that for short beams this stress is of considerable magnitude. The axial normal stresses at other points in the midsection are determined in a like manner. The foregoing procedure leads to the poorest accuracy for point B, the point of maximum tensile stress. A better approximation [see Ref. 3.7] of this stress is given by 1 x2B = 3PL P - 0.508 bh 2bh2 (3.50) Another more detailed study demonstrates that the local stresses decrease very rapidly with increase of the distance (x) from the point of load application. At a distance equal to the depth of the beam, they are usually negligible. Furthermore, along the loaded section, the normal stress x does not obey a linear law. In the preceding discussion, the disturbance caused by the reactions at the ends of the beam, which are also applied as line loads, are not taken into account. To determine the radial stress distribution at the supports of the beam of narrow rectangular cross section, Eq. (3.44) can be utilized. Clearly, for the beam under consideration, we use F = 0 and replace P by P/2 in this expressio...
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This note was uploaded on 01/09/2011 for the course ME 201 taught by Professor Yok during the Spring '08 term at Boğaziçi University.

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