Machinery's Handbook, 31st Edition
270 BENDING STRESS AND OBLIQUE FORCE Bending Stress Due to an Oblique Transverse Force.— The following illustration shows a beam and a channel being subjected to a transverse force acting at an angle φ to the center of gravity. To find the bending stress, the moments of inertia I around axes 3-3 and 4-4 are computed from the following equations: I 3 = I x sin 2 φ + I y cos 2 φ , and I 4 = I x cos 2 φ + I y sin 2 φ . The computed bending stress f b is then found from sin cos f M I y I x b x y z z = + c m where M is the bending moment due to force F .
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Beams of Uniform Strength Throughout Their Length.— The bending moment in a beam is generally not uniform throughout its length, but varies. Therefore, a beam of uniform cross section which is made strong enough at its most strained section will have an excess of material at every other section. Sometimes it may be desirable to have the cross section uniform, but at other times the metal can be more advantageously distrib- uted if the beam is so designed that its cross section varies from point to point, so that it is at every point just great enough to take care of the bending stresses at that point. Table 3a and Table 3b are given showing beams in which the load is applied in different ways and which are supported by different methods, and the shape of the beam required for uniform strength is indicated. It should be noted that the shape given is the theoretical shape required to resist bending only. It is apparent that sufficient cross section of beam must also be added either at the points of support (in beams supported at both ends), or at the point of application of the load (in beams loaded at one end), to take care of the vertical shear. It should be noted that the theoretical shapes of the beams given in the two tables that follow are based on the stated assumptions of uniformity of width or depth of cross sec- tion, and unless these are observed in the design the theoretical outlines do not apply without modifications. For example, in a cantilever with the load at one end, the outline is a parabola only when the width of the beam is uniform. It is not correct to use a strictly parabolic shape when the thickness is not uniform, as, for instance, when the beam is made of an I- or T-section. In such cases, some modification may be necessary; but it is evident that whatever the shape adopted, the correct depth of the section can be obtained by an investigation of the bending moment and the shearing load at a number of points, and then a line can be drawn through the points thus ascertained, which will provide for a beam of practically uniform strength whether the cross section be of uniform width or not.
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