Machinery's Handbook, 31st Edition
274
Curved Beams s Z Wl 4 =
The units of s are newtons per square millimeter; W is the load in newtons; l is the length in mm; and Z = section modulus of the cross section of the beam = I ÷ dis tance in mm from neutral axis to extreme fiber. Curved Beams.— The formula S = Mc / I used to compute stresses due to bending of beams is based on the assumption that the beams are straight before any loads are applied. In beams having initial curvature, however, the stresses may be considerably higher than predicted by the ordinary straight-beam formula because the effect of initial curvature is to shift the neutral axis of a curved member in from the gravity axis toward the center of curvature (the concave side of the beam). This shift in the position of the neutral axis causes an increase in the stress on the concave side of the beam and decreases the stress at the outside fibers. Hooks, press frames, and other machine members which as a rule have a rather pro nounced initial curvature may have a maximum stress at the inside fibers of up to about 3 1 ∕ 2 times that predicted by the ordinary straight-beam formula. Stress Correction Factors for Curved Beams: A simple method for determining the maximum fiber stress due to bending of curved members consists of 1) calculating the maximum stress using the straight-beam formula S = Mc / I ; and; and 2) multiplying the calculated stress by a stress correction factor. Table 4 on page 275 gives stress cor- rection factors for some of the common cross sections and proportions used in the design of curved members. An example in the application of the method using English units of measurement is given at the bottom of the table. A similar example using metric SI units is as follows: The fiber stresses of a curved rectangular beam are calculated as 40 newtons per millimeter 2 , using the straight beam formula, S = Mc / I . If the beam is 150 mm deep and its radius of curvature is 300 mm, what are the true stresses? R / c = 300 ∕ 75 = 4. From Table 4 on page 275, the K factors corresponding to R / c = 4 are 1.20 and 0.85. Thus, the inside fiber stress is 40 3 1.20 = 48 N/mm 2 = 48 megapascals (MPa); and the outside fiber stress is 40 3 0.85 = 34 N/mm 2 = 34 MPa. Approximate Formula for Stress Correction Factor: The stress correction factors given in Table 4 on page 275 were determined by Wilson and Quereau and published in the University of Illinois Engineering Experiment Station Circular No. 16, “A Simple Method of Determining Stress in Curved Flexural Members.” In this same publication the authors indicate that the following empirical formula may be used to calculate the value of the stress correction factor for the inside fibers of sections not covered by the tabular data to within 5 percent accuracy, except in triangular sections where up to 10 percent de- viation may be expected. However, for most engineering calculations, this formula should prove satisfactory for general use in determining the factor for the inside fibers. . . K bc I R c R 100 05 1 1 2 = + − + : D (Use 1.05 instead of 0.5 in this formula for circular and elliptical sections.) I = moment of inertia of section about centroidal axis b = maximum width of section c = distance from centroidal axis to inside fiber, i.e., to the extreme fiber nearest the center of curvature R = radius of curvature of centroidal axis of beam
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