CHAPTER XVI. STIFFNESS AND DEFLECTION OF BEAMS. IN Chaps. XIV. and XV. we have considered the strength of beams to resist breaking only; but in all first-class buildings it is desired that those beams which show, or which support a ceiling, should not only have sufficient strength to carry the load with safety, but should do so without bending enough to present a bad appearance to the eye, or to crack the ceiling: hence, in calculating the dimensions of such beams, we should not only calculate them with regard to their resistance to breaking, but also to bending. Unfortunately, we have at present no method of combining the two calculations in one operation. A beam apportioned by the rules for strength will not bend so as to strain the fibres beyond their elastic limit, but will, in many cases, bend more than a due regard for appearance will justify. The amount which a beam bends under a given load is called its deflection, and its resistance to bending is called its stiffness: hence the stiffness is inversely as the deflection. The rules for the stiffness of beams are derived from those for the deflection of beams; and the latter are derived partly from mathematical reasoning, and partly from experiments. We can find the deflection at the centre, of any beam not strained beyond the elastic limit, by the following formula: Def. in inches = load in lbs. x cube of span in inches x c The values of c are as follows: Beam supported at both ends, loaded at centre . 0.021 66 fixed at one end, loaded at the other. (1) By making the proper substitutions in Formula 1, we derive the following formula for a rectangular beam supported at both ends, and loaded at the centre: load x cube of span × 1728 Def. in inches = 4 × breadth × cube of depth x E' (2) the span being taken in feet. From this formula the value of the modulus of elasticity, E, for different materials, has been calculated. Thus beams of known dimensions are supported at each end, and a known weight applied at the centre of the beam. The deflection of the beam is then carefully measured; and, substituting these known quantities in Formula 2, the value of E is easily obtained. Formula 2 may be simplified somewhat by representing 1728 4× E by F which gives us the formula Def. in inches = W × L8 (3) For a distributed load the deflection will be five-eighths of this. NOTE. The constant F corresponds to Hatfield's F, in his Transverse Strains. If we wish to find the load which shall cause a given deflection, we can transpose Formula 2 so that the load shall form the lefthand member. Thus : 4× breadth x cube of depth × def. in ins. × E (4) Now, that this formula may be of use in determining the load to put upon a beam, the value of the deflection must in some way be fixed. This is generally done by making it a certain proportion of the span, Thus Tredgold and many other authorities say, that, if a floorbeam deflects more than one-fortieth of an inch for every foot of span, it is liable to crack the ceiling on the under side; and hence this is the limit which is generally given to the deflection of beams in first-class buildings. Then, if we substitute for "deflection" the value, length in feet 40, in the above formula, we have, Many engineers and architects think that one-thirtieth of an inch per foot of span is not too much to allow for the deflection of floor beams, as a floor is seldom subjected to its full estimated load, and then only for a short time. If we adopt this ratio, we shall have as our constant for deflec In either of the above cases, it is evident that the values used for E, F, e, or e1, should be derived from tests on timbers of the same size and quality as those to be used. It has only been within the last three or four years that we have had any accurate tests on the strength and elasticity of large timbers, although there had been several made on small pieces of various woods. The values of the various constants for the first three woods in the following table have been derived from tests made by Professor Lanza and his students at the Massachusetts Institute of Technology, and the values for the other woods are about six-sevenths of the values derived from Mr. Hatfield's experiments. The author believes that the values given in this table may be relied upon for timber such as is used in first-class construction. TABLE I. Values of Constants for Stiffness or Deflection of Beams. E = Modulus of elasticity, pounds per square inch. F= Constant for deflection of beam, supported at both ends, and e loaded at the centre. = Constant, allowing a deflection of one-fortieth of an inch per foot of span. e1 = Constant, allowing a deflection of one-thirtieth of an inch per Rules for Stiffness of Beams. Knowing the deflection caused by a weight at the centre of a beam, and the ratio of other deflections, caused by different modes of loading and supporting, we can easily deduce the formulas for the different cases considered under the strength of rectangular beams. These cases are BEAMS SUPPORTED AT BOTH ENDS. Loaded at a point other than the centre, m and ʼn being the segments into which the beam is divided, Inclined beam, loaded at the centre,' breadth cube of depth xe Safe load = length hor. dist. between supports' (12) Tables II., III., and IV. have been prepared so as to show at a glance the greatest load that a beam one inch thick will support without exceeding the limit of deflection or the safe strength. They give the same results as would be obtained by using the above formulas. Ratio of the Stiffness of Beams. If the stiffness of a beam, supported at both ends, and loaded at the centre be called. 1 Then that of the same beam, with the same load uniformly distributed, will be . Firmly fixed at both ends, and loaded at the centre, according Firmly fixed at both ends, and uniformly loaded Fixed at one end, and loaded at the other. Fixed at one end, and uniformly loaded 5 8 The stiffest rectangular beam containing a given amount of material is that in which the ratio of depth to breadth is as 10 to 6: hence, in designing beams, the depth and breadth should be made to approach as near this ratio as is practicable. EXAMPLE 1.- What is the greatest distributed load that an 8 by 10 inch white-pine girder of 12 foot clear span will support, without deflecting at the centre more than of an inch per foot of span ? Ans. This girder comes under the case of a beam supported at both ends, and loaded with a uniformly distributed load, and hence should be calculated by Formula 10. Substituting the given dimen sions in Formula 10, we have, 8 × 8 × 1000 × 62 Safe load = 5511 pounds. 5 x 144 |