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DEFLECTION 37.3
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DEFLECTION
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37.2 DEFLECTION DUE TO BENDING
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The relations involved in the bending of beams are well known and are given here for reference purposes as follows: q d 4y = EI dx4 V d 3y = EI dx3 M d2y = EI dx2 = dy dx (37.4) (37.5) (37.6) (37.7) (37.8)
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These relations are illustrated by the beam of Fig. 37.1. Note that the x axis is positive to the right and the y axis is positive upward. All quantities loading, shear force, support reactions, moment, slope, and deflection have the same sense as y; they are positive if upward, negative if downward.
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37.3 PROPERTIES OF BEAMS
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Table 37.1 lists a number of useful properties of beams having a variety of loadings. These must all have the same cross section throughout the length, and a linear relation must exist between the force and the deflection. Beams having other loadings can be solved using two or more sets of these relations and the principle of superposition. In using Table 37.1, remember that the deflection at the center of a beam with offcenter loads is usually within 2.5 percent of the maximum value.
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37.4 ANALYSIS OF FRAMES
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Castigliano s theorem is presented in Chap. 38, and the energy equations needed for its use are listed in Table 38.2. The method can be used to find the deflection at any point of a frame such as the one shown in Fig. 37.2. For example, the deflection C at C in the direction of F2 can be found using Eq. (38.2) as C = U F2 (37.9)
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where U = the strain energy stored in the entire frame due to all the forces. If the deflection is desired in another direction or at a point where no force is acting, then a fictitious force Q is added to the system at that point and in the direction in which
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DEFLECTION 37.4
CLASSICAL STRESS AND DEFORMATION ANALYSIS
FIGURE 37.1 (a) Loading diagram showing beam supported at A and B with uniform load w having units of force per unit length, R1 = R2 = w /2; (b) shear-force diagram showing end conditions; (c) moment diagram; (d) slope diagram; (e) deflection diagram.
the deflection is desired. After the partial derivatives have been found, Q is equated to zero, and the remaining terms give the wanted deflection. The first step in using the method is to make a force analysis of each member of the frame. If Eq. (a) is to be solved, then the numerical values of F1 and F2 can be used in the force analysis, but the value of F2 must not be substituted until after each
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DEFLECTION
TABLE 37.1 Properties of Beams
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DEFLECTION
TABLE 37.1 Properties of Beams (Continued)
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DEFLECTION
TABLE 37.1 Properties of Beams (Continued)
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DEFLECTION
TABLE 37.1 Properties of Beams (Continued)
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