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This method is cumbersome, however, and is not used. As has already been pointed out the moment of resistance equals the total tension times the lever arm of the resisting couple; or BM=MR= T.a=f.A5.a using the notation already familiar. The arm, a, varies so little for varying tension steel areas that it may be assumed constant without serious error. With this assumption it becomes possible to lay down the principle that the area of steel required varies directly with the bending moment and the same curve by proper choice of scale mar serve both as moment curve and area-required curve. Usually practical considerations as to commercial size of bars result in the maximum area furnished being larger than that required. It is common to neglect this difference and compute bar lengths as though the maximum area furnished equaled that required. This reinforced concrete beam (Fig. 24) carries a uniform load. What are the minimum possible values of the dimensions a and b. The bending moment curve for this loading is a parabola with maximum ordinate at the center and the area-required curve accordingly is the same. The parabola of Fig. 24b is drawn with the center ordinate representing 6 sq. in. One bar may be bent up when 5 sq. in. only are required; so a = 9'4 = 3.7 ft. Similarly�� a + b = 9v' = 6.4 ft.

Shearing Stresses in Homogeneous Concrete Beams

A brief review of the shearing stresses in homogeneous beams is desirable in order that a clear picture may be obtained of the web stresses in beams of all kinds. For rigorous demonstration of these matters the reader should consult the standard treatises on the strength of materials are equal. Knowledge of the variation of horizontal shear intensity, therefore, gives also that of vertical shear. In Fig. 25b is shown a portion of a rectangular beam lying between any two sections, AA' and BB'. The variation of normal tension intensity at each section is indicated by the partial stress diagrams, the abscissas on AA' being shown larger than those on BB' on the assumption that the moment at AA', is the larger. Considering the stability of the small piece of beam cdBA, the pull on the cA face is larger than that on the dB face and the only force available to balance the difference is the horizontal shear on the plane ed. A brief consideration of the problem shows that the nearer the cd plane is to the neutral plane ab, the larger is the difference between the two tensions, and the larger the horizontal shear. Therefore the horizontal shear, and accordingly also the vertical shear, increase in intensity at a decreasing rate from zero at the extreme fiber to a maximum at the neutral plane. For a rectangular section the law of this variation is a parabola (Fig. 25a) with a maximum intensity of 317. The resultant intensity of stress at any point away from the extreme fibers, as, for example, on the vertical faces of the elementary prisms 1 and 2, Fig. 25a, must be inclined in direction, acting somewhat as shown. Referring again to the elementary prism shown, the shearing forces there shown may be resolved into components along the diagonals, and these components may be combined to give inclined tensile and compressive forces acting at 45 degrees (Fig. 25d) with an intensity (v) equal to that of the shear. This illustrates the case when the prism lies at the neutral plane where there is no direct stress. When it lies in the face of the beam there are no horizontal nor vertical shearing stresses and the resultant tension is horizontal, being that given by the usual formula for fiber stress. A more detailed study of the state of stress at any point in this cross-section would show that passing through it are two inclined planes, 90 degrees apart, on which there is no shear, the resultant stress being compression on one and tension on the other, of intensity greater than on any other plane through the point. These stresses are called the principal stresses at the point. Midway between these planes are those of maximum shear intensity. In the web of a plate girder the action of the inclined tension is easily resisted by the steel but the diagonal compression tends to cause buckling and it is necessary to limit the minimum thickness of the web or to provide suitably spaced stiffeners, or both. In a concrete beam, on the other hand, the material easily resists the diagonal compression but is weak in tension. The wooden beam resists both.