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STRESS 36.8
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
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(36.15)
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where T = torque = radius to stress element r = radius of bar J = second moment of area (polar) The total angle of twist of such a bar, in radians, is = Tl GJ (36.16)
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where l = length of the bar. For the shear stress and angle of twist of other cross sections, see Table 36.1.
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36.3.1 Principal Unit Strains For a bar in uniaxial tension or compression, the principal strains are
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=
=
(36.17)
Notice that the stress state is uniaxial, but the strains are triaxial. For triaxial stress, the principal strains are
1 2 3 E E E 2 1 3 E E E 3 1 2 E E E (36.18)
These equations can be solved for the principal stresses; the results are 1 = E 1(1 ) + E( 2 + 3) 1 2 2 E 2(1 ) + E( 1 + 3) 1 2 2 E 3(1 ) + E( 1 + 2) 1 2 2 (36.19)
2 =
3 =
The biaxial stress-strain relations can easily be obtained from Eqs. (36.18) and (36.19) by equating one of the principal stresses to zero.
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
TABLE 36.1 Torsional Stress and Angular Deflection of Various Sections
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
TABLE 36.1 Torsional Stress and Angular Deflection of Various Sections (Continued)
36.10 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
TABLE 36.1 Torsional Stress and Angular Deflection of Various Sections (Continued)
36.11 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 36.12
CLASSICAL STRESS AND DEFORMATION ANALYSIS
36.3.2 Plastic Strain It is important to observe that all the preceding relations are valid only when the material obeys Hooke s law. Some materials (see Sec. 32.9), when stressed in the plastic region, exhibit a behavior quite similar to that given by Eq. (36.11). For these materials, the appropriate equation is = K n where = true stress K = strength coefficient = true plastic strain n = strain-strengthening exponent (36.20)
The relations for the true stress and true strain are = Fi Ai = ln li l0 (36.21)
where Ai and li are, respectively, the instantaneous values of the area and length of a bar subjected to a load Fi. Note that the areas in Eqs. (36.9) are the original or unstressed areas; the subscript zero was omitted, as is customary. The relations between true and engineering (nominal) stresses and strains are = exp = ln( + 1) (36.22)
36.4 FLEXURE
Figure 36.4a shows a member loaded in flexure by a number of forces F and supported by reactions R1 and R2 at the ends. At point C a distance x from R1, we can write MC = Mext + M = 0 (36.23)
where Mext = xR1 + c1F1 + c2F2 and is called the external moment at section C. The term M, called the internal or resisting moment, is shown in its positive direction in both parts b and c of Fig. 36.4. Figure 36.5 shows that a positive moment causes the top surface of a beam to be concave.A negative moment causes the top surface to be convex with one or both ends curved downward. A similar relation can be defined for shear at section C: Fy = Fext + V = 0 (36.24)
where Fext = R1 F1 F2 and is called the external shear force at C. The term V, called the internal shear force, is shown in its positive direction in both parts b and c of Fig. 36.4. Figure 36.6 illustrates an application of these relations to obtain a set of shear and moment diagrams.
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