Fig. 12a. Maximum negative or minimum positive moment, Fig. 126. Maximum negative or minimum positive moment, Fig. 12c. Maximum negative moment or minimum positive moment, Fig. 12d. Maximum negative moment or minimum positive moments, Fig. 12. Arrangement of Loads on Continuous Beams Causing Maximum Positive Bending Moments-(continued) These equations may be used without appreciable error for any series of more than three spans. In applying the graphs and equations discussed in the preceding chapters and paragraphs, it should be borne in mind that for those in Chapter III no distinction is made between the live and dead loads, although conditions are considered resulting from no loading over certain spans of a continuous beam. The graphs and equations of this Chapter VI, however, give the absolute maximum values for the critical conditions of the live loads alone, entirely separated from those caused by the dead loads. From a study of the maximum moments that can be caused by critical loading, the DESIGN-COEFFICIENTS given on page 22 have been determined for uniformly-distributed loads. In the design of reinforcements it is also necessary to consider the effect of variable live loads in determining those sections of the beam in which the bars should be raised in order to provide for the changing moments. 5. Deflection of Concrete Beams. Turneaure and Maurer Method.* The formulas developed by Turneaure and Maurer are based upon the deflectionformulas for homogeneous beams, modified according to the following assumptions: (1) The representative or mean section has a depth equal to the distance from the top of the beam to the center of the steel; (2) It sustains tension as well as compression, both following the linear law; (3) The proper mean modulus of elasticity of the concrete equals the average or secant modulus up to the working compressive stress; (4) The allowance for steel in computing the moment of inertia of the mean section should be based upon the amount of steel in the mid-sections. the maximum deflection in inches: the total load in pounds; the numerical coefficient in the formula for the deflection of homo = the span in inches; geneous beams, c1 I = E W13 EI' ing and the supports; depending upon the character of the load the moment of inertia of the cross-section of the beam, in inches'; = the modulus of elasticity of the concrete, in pounds per square inch; For a simple beam loaded at the middle, c1 = * Principles of Reinforced Concrete Construction, Third Edition, page 197. For a beam fixed at one end and supported at the other, and with a load at the middle, c1 = 0.00932; For a beam fixed at one end and supported at the other, and uniformly loaded, C1 0.0054; = 1 For a beam with fixed ends, and loaded at the middle, c1 = 192' For a beam with fixed ends, and uniformly loaded, c1 = 1 384' Es = the modulus of elasticity of the reinforcement, in pounds per square inch; a value of 30 000 000 lb per sq in may be used; n = the ratio of the moduli of elasticity of steel and concrete; a = a numerical coefficient depending upon p and n; β b = a numerical coefficient depending upon p and n; = the width of beam for rectangular beams and the width of flange for T beams, in inches; b' = the width of web for T beams, in inches; d = the depth of beam to the center of the steel, in inches; k the proportionate depth of the neutral axis; = = the steel-ratio, or the area of steel-section divided by the area of the section bd; t = the total flange-thickness of a T beam, in inches. A value of from 8 to 10 is recommended for n in the following formulas. For rectangular beams Knowing the steel-ratio p and the value of n, the value of k is computed and the coefficient a or ẞ is determined and substituted in the appropriate equation for the deflection. 6. Moments in Building-Frames. Although the theorem of three moments is adequate for the solution of most problems ordinarily encountered in the design of continuous beams, its use involves relatively laborious computations, and it can never be applied when it is necessary to determine the DEGREE OF RESTRAINT developed by the supporting members. This latter fact is obvious from the character of Formula (1), page 98, which merely gives the relation between the moments at consecutive supports, as depending upon the sections, loads, and spans of the beams themselves. It must be borne in mind, also, that in previous discussions the moments at the terminations of spans have been assumed to be due to either a condition of free support or of complete fixity, except when some arbitrary value has been given to a moment, such as WL 16 at a wall-column. The computation of the exact degree of restraint at any particular joint of a structure in the members of which there are indeterminate stresses, as in a monolithic building-frame, is a relatively long operation; but by means of certain assumptions it is possible to obtain a close approximation to the exact value of this restraint, which is sufficiently accurate for purposes of building-design. The method preferred by the author is that of SLOPEDEFLECTIONS, developed by the application of the principle of AREA-MOMENTS.* This method of solution as applied to any problem in monolithic frames consists in writing two or more equations for the moments at each support, determined by considering the adjacent members. These equations are then combined with one or more equations for equilibrium and solved for the slopevalues of the tangents to the elastic curve at each support (Fig. 14). Having determined the slopes, the moments are computed by substituting their values in the moment-equations. 7. Demonstration of the Two Principal Propositions of the Slope-Deflection and Area-Moment Method. These basic propositions may be found in any modern text-book on mechanics and are expressed as follows: M (1) When a member is subjected to flexure, the difference in the slope of the elastic curve between any two points is equal in magnitude to the area of the diagram (Diagram 1, Fig. 14), for the portion of the member between the two points. ΕΙ (2) When a member is subjected to flexure, the distance of any point Q (Diagram 1, Fig. 14), on the elastic curve, measured normal to the initial position of the member, from a tangent drawn to the elastic curve at any other point P, is equal in magnitude to the first or statical moment of the area of the M ΕΙ diagram between the two points, about the point Q. *G. A. Maney, University of Minnesota Bulletin No. 1; W. M. Wilson and G. A. Maney, University of Illinois Bulletin No. 80; W. M. Wilson, F. E. Richart, and Camillo Weiss, University of Illinois Bulletin No. 108. |