sive. A depth equal to or greater than -j^-th of the span is certain to give sufficient stiffness, and the usual method is to assume the depth and then to observe whether experiment gives a deflection agreeing with that found by calculation. The calculation is made by finding the radius of curvature of the beam at a series of sections, and then determining the curve assumed by the whole beam either by integration or by an approximate graphic method. When the curve is known the deflection or versed sine is found either from the equations of the curve, or by actual measurement on the diagram to be presently described. Let R be the radius of curvature of the neutral surface of a beam at a given section, under a load producing a maximum stress PI at the outer elements of the section at a distance y l from the neutral surface. Consider a short length x of the beam fie:. 20. Before deflection the length of the outer element is equal to x, but after deflection (if v;e con sider the upper member) the length of the outer element will be shortened to x lt while the length of the element in the neutral surface remains equal to x. By similar triangles we have x : x, = R : R - y ,. or R = - , but x-x, X-Xi is the extent to which the most compressed element is shortened, and by the definition of the modulus of elasticity we have E = p l
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Pi from which the radius of curvature at any section can be obtained in terms of E, y v and p^ all known quantities with a given cross section, material, and load. Another form of the same expression, which is sometimes more con venient, is obtained by remembering that M, = ^- , hence
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v, 2. R- E J M t where M is the. bending moment which will produce p r f Pi is made equal to / x the maximum safe stress on the material, equation 1 or 2 gives the minimum safe radius of curvature.
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Fig. 20.
We will first consider the special case in which the beam under consideration is of equal depth and uniform strength at all cross sections, and which we will call a beam of class 1 ; these conditions can only be fulfilled by any one beam for a given constant distribution of load ; the beam of uniform strength for a load at the centre, for instance, will clearly not be the beam of uniform strength for a uniformly distributed load. In beams of class 1 for a given load, both y l and p 1 are constant, and there fore R, by equation 1, will also be constant throughout the whole length of the beam, or, in other words, the beam will bend into a circular arc. The approximate expression for the versed sine of an arc having a chord L and radius R will therefore give the deflection, and we have
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and employing for R the value given by equation 1, we have
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and if y l = d, we have
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5 V= ~FT
If for p 1 we substitute f v the maximum safe stress for the given materials, equation 4 or 5 will give the maxi mum safe deflection, which may be called v l ; we observe, then, that the safe deflection for a beam of this class will be proportional to the square of the span, and inversely proportional to the depth of the beam.
In beams of class 1 , the deflections which different loads produce will be simply proportional to the values of p l pro duced by those loads ; thus, for a given distribution of load, the deflection will be simply proportional to the load ; if we change the distribution of the load, keeping the total load constant, the same rule will give the solution ; for in stance, since a load uniformly distributed produces only half the stress p 1 which would be produced by the same load at the centre of a beam of the same span and cross section, we see that the uniformly distributed load will produce only half the deflection that would be produced by the same load at the centre of a beam of the same span and same section at the centre (both beams belonging to class 1). It would not be correct to say that a load uni formly distributed would produce half the deflection pro duced by the same load at the centre of the same beam, because the same beam cannot be uniformly strong through out its length for two different distributions of load.
We may compare the deflections produced by the same load on various beams of similar cross section as follows : By equation 4, 14, we see that for a given moment M, p l is proportional to - ; moreover, for equal loads, the moment M at any cross section similarly placed will be proportional to L ; hence we may write
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Ld L Substituting this expression for p^ in equation 5, we have
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6 an equation which expresses the fact that, for beams of class 1, the deflection produced by a given total load similarly applied will be proportional to the cube of the length, and inversely proportional to the breadth and to the cube of the depth. (In I girders the expression breadth must be understood as proportional to S, the section of the flange.)
