358 



GRAPHIC STATICS 



Suppose a table is drawn up, in one column of 

 which are the months of the year, and in the other 

 the corresponding average temperatures of the air, 

 at some particular place, during these months ( the 

 average temperature for eacli month being the mean 

 of the daily temperatures). Let two lines, OX and 

 OY, be drawn from O, one horizontally, the other 

 vertically; let the successive months of the year 

 be represented on any convenient scale along OX, 

 and let temperature be measured along OY, also 

 on a convenient scale. Corresponding to each 

 month in the year there will be a length along 

 OX, and to each temperature there will corre- 

 spond a point on OY. At the middle point cor- 

 responding to each month draw perpendicular to 

 OX a line representing the temperature on the 

 scale of OY. A series of lines will thus be obtained, 



65 Y 



45 

 40 

 86 



30 



Jan. Feb. Mar. Ap. May Ju. Jul. Aug. Sep. Oct. Nov. Dec. 



through the upper ends of which there may be 

 drawn, freehand, a smooth curve. The points on 

 the curve in the figure represent the upper ends 

 of these lines. A general glance at such a curve 

 will reveal certain features regarding the tempera- 

 ture of the whole year ; at what dates maxima 

 and minima occurred ; when the temperature rose 

 or fell quickest, and so on. Such a curve, repre- 

 senting the gradual change of daily temperature, 

 may be produced automatically by photographic 

 representation : a sheet of sensitised paper passes 

 uniformly, by means of clockwork, behind the 

 thermometer stem, in front of which is placed 

 a source of light ; the paper above the mercury 

 column is blackened, that below being left 

 unaffected ; the curve separating the black and 

 white portions represents the temperature at 

 different times. The same principle is used in the 

 thermograph, barograph, and tide-gauge recording 

 machine. 



Instead of time and temperature any other two 

 variable quantities may be taken. When the 

 curve obtained by such graphical methods has 

 some regular geometrical features the mathe- 

 matical Taw of the phenomenon may be found ; 

 and many qualitative and quantitative 

 results in physics are obtained in this 

 way. It must be remembered that such 

 ' graphical representations do no more than 

 embody the results of observation o 

 experiment, and cannot be made more 

 accurate than the data themselves. 



The graphic method is so largely em- 

 ployed in physical science, and also in 

 statistics, that only a few instances of its 

 application may be given. Watt's Indi- 

 cator Diagram shows the amount of 

 work done in a complete (double) stroke of the 

 piston ; it acts on the principle that the force 

 applied multiplied by the distance through which 

 it acts is a measure of the work done. Pressure 

 and volume are therefore the variables here involved. 

 'The temperature of a body at different times may 

 be given by a curve, from which may be found the 

 rate of cooling ; a curve may also represent the 



temperature at different points of a body, and from 

 it may be deduced, if its thermal conductivity be 

 known, the flux of heat across any section of it. 

 The thermo-electric diagram (see Tait's Heat) 

 is also a valuable application of the method. 

 Andrew's diagram of the volume of carbonic acid 

 gas under varying pressure may be mentioned as 

 another (see Andrew's Collected Scientific Papers, 

 Lond., Macmillan, 1889). The method has also 

 many applications in electricity e.g. the 'arrival' 

 curve in a submarine cable; and in sound, where 

 acoustic vibrations, beats, and harmonics may be 

 graphically represented. 



Graphic Statics. When forces simultane- 

 ously act on a particle which remains at rest they 

 are in equilibrium, and, if there be three of them, 

 lines drawn so as to represent the respective forces 

 in magnitude and direction may be so arranged as 

 together to form the well-known Triangle of Forces. 

 Problems in which trigonometrical methods of find- 

 ing the magnitude and direction of the third side of 

 such a triangle (the resultant) are applied, when 

 those of the other two (the components ) are known, 

 or of resolving any given force in any given direction 

 into two ' components ' in any two assigned direc- 

 tions, are of common occurrence in text-books. For 

 practical purposes, however, it is very useful actu- 

 ally to draw to scale the triangle of forces appro- 

 priate to the data of any particular case ; two sides 

 being thus drawn to scale, the third side can be 

 laid down by simply joining two points, and then 

 the line so drawn can be measured with respect to 

 its length and its direction. Similarly the resultant 

 of a number of simultaneous forces can be usefully 

 ascertained by drawing the corresponding Polygon 

 of Forces, and ascertaining the lie and the length 

 of the missing side. The utility of this graphic 

 method is, however, most fully seen in the recent 

 extensions of this method to engineering work. 

 The subject of Graphic Statics is a large one, and 

 we can do little more here than refer the reader to 

 Cotterill's Applied Mechanics, which gives, incident- 

 ally, full references to the literature of the subject ; 

 but in order to give an idea of the nature of the 

 method one il- 

 lustration may 

 here be sup- 

 plied. Suppose 

 a bridge-girder 

 (weightless) 

 made up of two 



Fig. 1. 



N girders in ten divisions (fig. 1), the diagonals 

 being all so arranged as to be in tension ; it is 100 

 feet long, and a load of 100 tons is distributed over 

 it so as to rest uniformly upon the lower booms. 

 Find the stress in each bar. First draw the girder 

 to scale, and mark the bars as in fig. 2 : The 

 lower boom of each division may, so far as the 



Fig. 2. 



girder at large is concerned, be considered as hav- 

 ing its proportion of the uniform load (10 tons) 

 arranged in 5-ton loads at its two ends ; hence at 

 the angle between 1 and 22, and also at 12-13, there 

 are imaginary loads of 5 tons; at be, de, fff, &c., 

 imagine 10-ton loads. The supporting piers each 

 exert an upward pressure of 50 tons. There is 

 equilibrium, and this equilibrium may be traced out 



