Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/337

] of 1 ton per foot run on a span of 50 feet. The upper curve in tig. 156 shows the curve of bending moments when curve those Case 2 the loads of case 1 and 2 both occur at once. This is obtained by adding the ordinates in case 1 to in case 2. Fig. 15c shows the four separate lines of bending moment for four separate weights, and the broken upper line is the line of bending moments for the case when the four loads all rest at once on the beam; it is got by simply adding at each point of the span the four ordinates due to the four loads considered separately. The curve in fig. 15 and the curve ABCD in fig. 15f are para bolas. Fig. 15a. Fig. 15b. The bending moment at any section is reckoned as posi tive when the external forces on the beam tend to turn the right hand side of the beam in a left handed direction (or in the direction opposite to that followed by the hands of a watch), in other words, when the bending moment tends to bend the beam downwards between the supports. We shall hereafter see that in beams with more than two sup ports the bending moment is at places negative, tending to bend the beam up; the curve of bending moments is drawn below the datum line where the moments are nega tive. Fig. 25a gives an example of a curve of bending moments of this class. Fig. 15c.

§17. Moment of Elastic Forces.—On the hypothesis above stated, viz., that for any one material the modulus of elasticity may be taken as constant for all stresses, and assuming that our investigation is to be confined to those cases in which the cross section of the beam has a vertical axis of symmetry, and in which the centre of gravity lies at a point equidistant from the top and bottom of the beam, the general equation—

tt = als=2 ^ ^ d (3, 14), allows simple expressions of the value of /u, to be obtained for all practical cases by substituting for the ratio 21 V its value in terms of the dimensions of the cross section. d Thus, for a rectangle of the depth d and breadth b,

21 _ 9 Id* Id? _ Sd where S is the area of the cross section. The following are the values for the commonest forms of cross sections. 21 IX.—Values of -r for various Cross Sections. &quot; d &quot; 21 &quot;d Form of Cross Section. In Terms of Linear In Terms of the Aresi of Dimensions of the the Cross Section (S) iiiul the Cross Section. Depth. I 3 Si Square, side b (j IT Uectangle, breadth b Id* Sd depth d 6 6 Circle, radius r T Sd 8&quot; Hollow circle, extcr- } nal radius r, in- &amp;gt; Sd -T- when TI differs lit ternal 1 } ir tle from r. s Sd . i~n when moment of Hollow rectangle, in- tcnul depth d lt breadth b L bd -b^d* web is neglected, and S is taken as area of cross section of flan Gd ( ges only, and d 1 I differs little from d. 