82 REPORT—1862. 
On the Strains in the Interior of Beams. 
By Grorce Bropett Arry, F.R.S., Astronomer Royal. 
[A communication ordered to be printed among the Reports.] 
Tux author states that he had long desired to possess a theory which should 
enable him to compute numerically the strains on every point in the interior 
of a beam or girder, but that no memoir or treatises had given him the least 
assistance*. He had therefore constructed a theory which solved completely 
the problems for which he wanted it, and which appears to admit of applica- 
tion at least to aJl ordinary cases. 
The theory contemplates forees acting in one plane. A beam therefore is 
considered as a lamina in a vertical plane, the same considerations applying 
to every vertical lamina of which a beam may be conceived to be composed. 
The author remarks that it is unnecessary to recognize every possible strain 
inabeam. Metallic masses are usually in a state of strain, from circum- 
stances occurring in their formation ; but such strains are not the subject of 
the present investigation, which is intended to ascertain only those strains 
which are created by the weight of the beam and its loads. The algebraical 
interpretation of this remark is, that it is not necessary to retain general 
solutions of the equations which will result from the investigation, but only 
such solutions as will satisfy the equations. 
After defining the unit of force as the weight of a square unit of the lamina, 
and the measure of compression-thrust or extension-pull as the length of 
the ribbon of lamina, whose breadth is the length of the line which is subject 
to the transverse action of the compression or tension, and whose weight is 
equal to that compression or tension, the author eonsiders the effect of tension, 
&e. estimated in a direction inclined to the real direction of the tension, and 
shows that it is proportional to the square of the cosine of inclination. He 
then considers the effect of compounding any number of strains of compression 
or tension which may act simultaneously on the same part-of a lamina, and 
shows that their compound effect may in every case be replaced by the com- 
pound effect of two forces at right angles to each other, the two forces being 
both compressions or both tensions, or one compression and one tension. 
Succeeding investigations are therefore limited to two such forees. 
Proceeding then to the general theory of beams, it is remarked that if a 
curve be imagined, dividing a beam into any two parts, the further part of 
the beam (as estimated from the origin of coordinates) may be considered to 
be sustained by the forces which act in various directions across that curve, 
taken in combination with the weight of the further part of the beam, the 
load upon that part, the reaction of supports, &ce. Expressing the forces in 
conformity with the principles already explained, and supposing that there is 
one compression-force B making an angle 6 with y (in the direction of y 
diminishing for increase of x), and another compression-force C making an 
angle 90°+ 3 with y, it is easily seen that the element ds of the curve, sup- 
posed to make the angle @ with y, sustains the forces 
Inw, B.désx sin(8+60)x smn B+C.ésx sin(8+90°+6)x sin (8+90°). 
In y, —B.ésx sin(§+6)x cosB—C.¢sx sin (B+90°+6) x cos (8+90°). 
The weight of lamina bounded by y and y+éy, and estimated as acting 
* Subsequently to the communication of this Report, the author learned that one in- 
stance (the second) of those given here had been treated by Professor Rankine, by methods 
peculiar to that instance. 
