This rule shows how great the strains must be 1.1 hen the angle c A D becomes very open, approaching to l 0 degrees. But when the angle C A D becomes very small, its sine (which is Our divisor) is also very small ; and we should expect a very great quotient in this ease also. But we must that in this case the sine of E A D is also very small : and this is our multiplier. In such a ease the quotient cannot exceed unity.
Lint it is unnecessary to consider the c'alc'ulation by the tables of sines More particularly. The angles are seldom luiffwn otherwise than by drawing the figure of the frame of carpentry. In this case we can always obtain the measures of the strains from the sante scale with equal accuracy, by drawing the A F E a.
I hitherto we have considered the strains excited at A only, as they affect tlw picee,4 on which they are exerted. But the pieces, in order to sustain, or be subject to, any strain, must be supported at their ends c and n ; and we may consider them as mere intermediums, by is 'deli these strains are made to act on the points of support ; therefore A P. and A c are also measures of the forces which press or pull at C and D. The supports to be found tor these points may be infinitely various ; but We shall attend only to such as depend on the framing itself.
l'iyure 21 is a structure which very frequently occurs, where one beam, n A. is strongly pressed to the end of another, A D, which latter is prevented froth yielding, as well because it, lies on a third, as because its end, n, is hindered from sliding backwards. It is indillifrent from what this pressure arises : we have represented it as owing to a weight hung at a, is bile 12 is withheld from yielding by a rod or rope hooked to the wall. The beam A D may be supposed at full liberty to exert all its pressure on D., as if it were supported on rollers lodged in the beam tt n ; hut the • loaded Leant u A presses both on the bean] A D and on 1-1 D. We wish only to know what strain is borne by A D.
All bodies act on each other in the direction perpendicular to their totalling surfaces ; thereff ti•e the support given Ify u is in a direction perpendicular to it. We Inay therefore sup ply its place at A by a beam, A C. perpendicular to u D, and firmly supported at c. In this ease, theretbre, we may take A F., as before, to represent the pressure exerted by the
loaded beam, and draw E G petpendieular to A D, and E F parallel to it, meeting the perpendicular A C in F. Then A G is the strain compressing A D, and the pressure on the beam n D.
It may Le thought, that ftfince we assuine as a principle that the mutual pressures of solid bodies are exerted pe•pen dicular to their touching surfaees, this balance of pressures, in framings of timbers, depends on the directions of their butting joints ; but it does not, as will readily appear by con sidering the present ease. Let the joint or abutment of the two pieces It A. A D. I)C Mitred in the usual manner. in the direction/ A Therefore, if A e be dravf it perpendicular to it will be the direction the actual p•essure exerted by.
the loaded beam, n A, oil the meant A D. Batt the reaction of A a,in the opposite direction A /, Will not balance the pressure of tt A ; beeanse it is not in the direction precisely opposite. 11 A will therefore slide the joint. and press on the beam II D. A E represents the load on the mitre joint A. Draw E e I wrpendieular to A e, and Ef parallel to it. The pressure A E w ill be balanced by the reactions e A and/ s : or, the pressUre A E produces the pressures A e and Al; of which Af !mist be resisted by the beam n n, and A e by the beam A D. The pressure A f not being perpendicular to n n, cannot. be fully resisted by it ; because, by our assmned principle, it reacts only in a direction perpendicular to its surface. Therefore, f to 11 D, and perpendicular to it. The pressure Af will be resisted by n D with the force p A ; there is required :mother force, i s. to prevent the beam It A from slipping outwards. This most be furnished by the reaction of the beam a A. In like manner the other force, A e, cannot be fully resisted by the beam A I), or rather by the prop D, acting by the intervention of the beam ; for the action of that prop is exerted through the beam in the direction D A. The been A n, therefore, is pressed to the Leant n D, by the force A e as well as by Ai. To find what this pressure IM tt D draw' ey perpvntliculttr to II D, and CO parallel to it. elating E c in r. The forces g A and o A will resist, and balance A e.