Diagram 8 Shear - Diagram for a Four - Span, Continuous Beam with Simply- Fig. 2. Supported 1 10 P Diagram 9. Shear - Diagram for a Four - Span, Continuous Beam with Simply- Diagram 10. Shear - Diagram for a Four - Span, Continuous Beam with Simply- the reactions are determined after the negative bending moments at the supports are found by the THEOREM OF THREE MOMENTS, and the positive and negative shears at the supports computed. 2. Shear as a Measure of Diagonal Tension. As the horizontal fiberstresses in a beam in a state of flexure under a system of loading vary in magnitude along its length from the points of support to the mid-span, there is developed on the surfaces of every horizontal layer of concrete a force or INTERNAL STRESS which tends to cause a slipping of adjacent layers past each other. Coincident with this HORIZONTAL SHEAR, at any cross-section, there also exists a VERTICAL SHEAR which is equal to it in amount at that point. The formulas used in general practice consider only the vertical shear, but it should be remembered that although our codes are drawn and designs made on this basis, it is actually the stresses due to the DIAGONAL TENSION to resist which a member must be designed. 3. Working Formulas. The following formulas relate to SHEAR as affecting the design of rectangular and T beams of reinforced-concrete. Let V = total vertical shear at the section considered, in pounds; v = total unit shearing-stress at the section in pounds per square inch; j = 0.875 (an approximation used for shear-computations); b = width of beam for rectangular beams, and width of web for T beams, in inches; d = effective depth of beam in inches. The intensity of the unit shearing-stress on any vertical section taken through a beam varies from a value of zero at the extreme fiber on the compressionface, to a maximum value at the neutral surface. As the tensile value of the concrete is neglected in computation, the intensity of the shearing-stress is considered as if it were constant from the neutral surface of the beam to the centroid of the reinforcement. Fig. 3 illustrates this condition. Fig. 3. Diagram of Shearing Stresses in a Reinforced-Concrete Beam The portion of the beam shown in the illustration is considered to be cut by two vertical planes normal to the side of the beam. These planes are separated by only a short distance, x, which is so small that the value of the external shear, V, is practically the same for each section. As the reinforcement is assumed to resist all tensile stress, the resultants of all tension on the two adjacent planes may be represented by T and T', acting in the plane of the steel. Simi larly, C and C' may represent the resultants of all compressive stresses acting above the neutral surface. On any horizontal section between the neutral surface and the reinforcement, the increase in stress between the two vertical planes, x distance apart, is T - T'. This horizontal shear is resisted by the concrete over an area equal to xb, when b is the breadth of the beam. T'/bx. Since the section must be in equilibrium, the sum of the moments about the point o must equal zero and (TT)jd Vx, or TT' Vx/jd. Substituting in the equation, above, The unit shearing stress is then, v = T = = · v = Vx/bxjd Or, cancelling the x, ย = V/jbd (1) This is the general formula for determining SHEAR AS A MEASURE OF THE DIAGOFor example, using the data on page 36, to compute the vertical NAL TENSION. In the usual specifications, the ALLOWABLE UNIT SHEARING-STRESS in concrete, without web-reinforcement, as computed by Formula (1), is given a limiting value of 2 or 3% of the ULTIMATE COMPRESSIVE* STRENGTH, or, for a 2 000 lb concrete, 40 lb or 60 lb per sq in. When this value is exceeded webreinforcement must be provided, either by bending up a part of the main tensile steel, or by using stirrups, or both. Besides the LIMITING UNIT SHEARINGSTRESS permitted for plain concrete without web-reinforcement, all codes set a MAXIMUM VALUE for the allowable stress when adequate reinforcement is provided. This maximum value, computed by Formula (1), is usually taken. as 120 lb or 150† lb per sq in for concrete developing by test, at an age of 28 days, an ultimate compressive strength of 2 000 lb per sq in. Besides the limiting values quoted in the footnotes the Joint Committee, 1924, recommends two special formulas for further limiting the maximum permissible values of the UNIT SHEARING STRESS as computed by Formula (1). Let f'e the ultimate compressive strength of concrete at 28 days; fo A, = = = S= tensile unit stress in web-reinforcement; total area of web-reinforcement in tension within a distance s or the total area of all bars bent up in any one plane; spacing of web bars or stirrups measured at the plane of the lower reinforcement and in the direction of the longitudinal axis of the beam expressed in inches; a = angle between web bars and longitudinal bars. *When special anchorage is provided as described on page 63, the Joint Committee, 1924, permits 0.03f', or 60 lb per sq in for a 2 000 lb concrete, otherwise 0.02 f'c. † When special anchorage is provided as described on page 63, the Joint Committee, 1924, permits 0.12f'e or 240 lb per sq in for a 2000 lb concrete, otherwise 0.06ƒ'c. If ORDINARY ANCHORAGE only is provided, as described on page 63, the value of v shall not exceed that given by the following formulas: IF SPECIAL ANCHORAGE is provided, as described on page 63, the permissible value of v is obtained by substituting 0.03 f'e in the above formulas in place of 0.02f'c. It is further specified that not less than one-fourth of the total shearing resistance required at either end of the span shall be provided at the section where the computed shearing stress is zero; from that section to the ends of the span the required shearing resistance shall be assumed to vary uniformly. In the design of RECTANGULAR BEAMS the sectional area of the concrete is usually governed by the limiting strength fe of the concrete in compression. Consequently, the shear should not be considered until after b and d have been determined, at which time the unit shearing stress v is checked by Formula (1). Then, if v is less than 40 lb per sq in, no web-reinforcement is required. If it is between 40 and 120 lb, or between 40 and whatever high limit is set by the particular code or specification in use, web-reinforcement must be used to resist the stress due to diagonal tension. If v is greater than the allowable limit, the cross-sectional area of the beam must be enlarged by increasing b ord, so that the unit shearing-stress will not exceed the limiting value. See page 65 for the complete procedure in rectangular beam design. In the case of T BEAMS, where the wide flange is available to resist the compression due to positive moment, and where both the 15% additional compression-value in the concrete over the supports and the value of the straight bars combine to resist the compression due to the negative moment, the consideration of shear usually governs the depth of the member. The procedure in such cases is to make b of sufficient width properly to accommodate the reinforcement (see page 13), and not less than about one-third the probable depth, based on an estimate of 1 in for every foot of clear span. Then the exact depth is computed by Formula (1) by transposing d and v and using the MAXIMUM ALLOWABLE value of the latter. This gives The depth is then checked by Formula (10), Chapter II, in which the COMPRESSIVE STRENGTH of the concrete is the governing factor. See page 78 for the complete procedure in T-BEAM DESIGN. 4. Provision for the Diagonal Tension. Some codes require that all the shearing-stress (the total value of v as computed above) be resisted by the steel in those sections of the beam in which the shear exceeds the value allowed for concrete without diagonal reinforcement; some permit the concrete to take its full unit shearing-stress throughout and require the steel to carry the excess amount only; while still another method is to proportion the DIAGONAL-TENSION reinforcement to carry two-thirds of the SHEARING-STRESS wherever the shear exceeds the limit for plain concrete. As it is usually expedient to raise approximately 50% of the main tensile reinforcement of continuous beams at about the fifth-point of the span, in order to resist the negative bending moment over the support, the inclined portions of the double-bend rods are available to resist a part of the diagonal tension, and they consequently reduce the number of stirrups. The length of beam through which a stirrup, or rods bent-up in a single plane, may be considered effective, varies in different recommendations and for different conditions from a value of about one-half d to somewhat over the full depth d. The Joint Committee, 1924, specifies that when stirrups or bent-up bars are properly anchored and the shearing stress is not greater than 0.06f'c, the distance s measured in the direction of the axis of the beam between two successive stirrups, or between two successive points of bending up of bars, or from the point of bending up of a bar to the edge of the support, shall not be greater than where the angle a is in degrees. When the shearing-stress is greater than 0.06f'c, the distance s shall not be greater than two-thirds of the values given by this formula. The use of INCLINED STIRRUPS in members subjected to vertical forces is not practical, owing to the difficulty of placing them at the theoretically correct angle; and most designers now use a VERTICAL STIRRUP which should be anchored by a hook having a radius of not less than four times the diameter of the web bar, or, where more convenient, the ends may be bent around the longitudinal reinforcement, see Fig. 4, the Joint Committee, 1924, specifies that web bars shall be anchored at both ends by: (a) providing continuity with the longitudinal reinforcement; or (c) a semicircular hook which has a radius not less than four times the The specification further requires that stirrup anchorage shall be so provided in the compression and tension regions of a beam as to permit the development of the safe working tensile stress in the stirrup at a point 0.3d from either face. The end anchorage of a web member not in bearing on the longitudinal reinforcement shall be such as to engage an amount of concrete sufficient to |