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STRESS 36.5
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STRESS
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FIGURE 36.2 Mohr s circle diagram for plane stress. (From Applied Mechanics of Materials, by Joseph E. Shigley. Copyright 1976 by McGraw-Hill, Inc. Used with permission of the McGrawHill Book Company.)
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36.1.1 Programming To program a Mohr s circle solution, plan on using a rectangular-to-polar conversion subroutine. Now notice, in Fig. 36.2, that ( x y)/2 is the base of a right triangle, xy is the ordinate, and the hypotenuse is an extreme of the shear stress. Thus the conversion routine can be used to output both the angle 2 and the extreme value of the shear stress. As shown in Fig. 36.2, the principal stresses are found by adding and subtracting the extreme value of the shear stress to and from the term ( x + y)/2. It is wise to ensure, in your programming, that the angle indicates the angle from the x axis to the direction of the stress component of interest; generally, the angle is considered positive when measured in the ccw direction.
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36.2 TRIAXIAL STRESS
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The general three-dimensional stress element in Fig. 36.3a has three normal stresses x, y, and z, all shown as positive, and six shear-stress components, also shown as positive. The element is in static equilibrium, and hence
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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STRESS 36.6
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
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FIGURE 36.3 (a) General triaxial stress element; (b) Mohr s circles for triaxial stress.
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xy = yx
yz = zy
zx = xz
Note that the first subscript is the coordinate normal to the element face, and the second subscript designates the axis parallel to the shear-stress component. The negative faces of the element will have shear stresses acting in the opposite direction; these are also considered as positive. As shown in Fig. 36.3b, there are three principal stresses for triaxial stress states. These three are obtained from a solution of the equation
2 2 2 3 ( x + y + z) 2 + ( x y + x z + y z xy yz zx ) 2 2 2 ( x y z + 2 xy yz zx x yz y zx z xy ) = 0
(36.7)
In plotting Mohr s circles for triaxial stress, arrange the principal stresses in the order 1 > 2 > 3, as in Fig. 36.3b. It can be shown that the stress coordinates for any arbitrarily located plane will always lie on or inside the largest circle or on or outside the two smaller circles. The figure shows that the maximum shear stress is always max = 1 3 2 (36.8)
when the normal stresses are arranged so that 1 > 2 > 3.
36.3 STRESS-STRAIN RELATIONS
The stresses due to loading described as pure tension, pure compression, and pure shear are = F A = F A (36.9)
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
STRESS 36.7
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where F is positive for tension and negative for compression and the word pure means that there are no other complicating effects. In each case the stress is assumed to be uniform, which requires that
The member is straight and of a homogeneous material. The line of action of the force is through the centroid of the section. There is no discontinuity or change in cross section near the stress element. In the case of compression, there is no possibility of buckling.
Unit engineering strain , often called simply unit strain, is the elongation or deformation of a member subjected to pure axial loading per unit of original length. Thus = where = total strain l0 = unstressed or original length l0 (36.10)
Shear strain is the change in a right angle of a stress element due to pure shear. Hooke s law states that, within certain limits, the stress in a material is proportional to the strain which produced it. Materials which regain their original shape and dimensions when a load is removed are called elastic materials. Hooke s law is expressed in equation form as =E = G (36.11)
where E = the modulus of elasticity and G = the modulus of ridigity, also called the shear modulus of elasticity. Poisson demonstrated that, within the range of Hooke s law, a member subjected to uniaxial loading exhibits both an axial strain and a lateral strain. These are related to each other by the equation = lateral strain axial strain (36.12)
where is called Poisson s ratio. The three constants given by Eqs. (36.11) and (36.12) are often called elastic constants. They have the relationship E = 2G(1 + ) By combining Eqs. (36.9), (36.10), and (36.11), it is easy to show that = Fl AE (36.14) (36.13)
which gives the total deformation of a member subjected to axial tension or compression. A solid round bar subjected to a pure twisting moment or torsion has a shear stress that is zero at the center and maximum at the surface. The appropriate equations are
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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