MEG
158
MEC
formly diffused over any part, e f, is the same as if the twoparts, ed,df, of the load were collected at the middle points,e and f Therefore the strain on any part, d, occasioned bya load uniformly distributed over the whole beam, is one-halfof the strain that is produced when the same load is laid onat D ; and the strain on the middle point, c, occasioned by aload uniformly distributed over the whole beam, is the samewhich half that load would produce if laid on at c.
Figure 24.—7. A beam supported at both ends on twoprops, b and c, will carry tvvice as much when the endsbeyond the props are kept from rising, as it will carry whenit rests loosely on the props.
8. Lastly, the transverse strain on any section, occasionedby a force applied obliquely, is diminished in the proportionof the sine of the angle which the direction of the forcemakes with the beam. Thus, if it be inclined to it in anangle of thirty degrees, the strain is one-half of the strainoccasioned by the same force acting perpendicularly.
On the other hand, the relative strength of a beam, or itspower in any particular section to resist any transverse strain,is proportional to the absolute cohesion of the section directly,to the distance of its centre of effort from the axis offracture directly, and to the distance from the strained pointinversely.
Thus, in a rectangular section of the beam, of which b isthe breadth, d the depth (that is, the dimension in the direc-tion of the straining force), measured in inches, and f thenumber of pounds which one square inch will just supportwithout being torn asunder, we must have/ - X b X cP, pro-portional to w X C b. Or, f x b X cP, multiplied by somenumber, m, depending on the nature of the timber, must beequal to tv X c b. Or, in the case of the section, c, of Figure46, that is strained by the force, w, applied at D, we must
AC X D B
have m X fbd=w X-Thus, if the beam is of
J AB
sound oak, m is very nearly — J. Therefore we have
fbd* ACXCB
—;r-:= w X -. Hence we can tell the precise force,
y a b
w, which any section, c, can just resist when that force isapplied in any way whatever. For the above-mentioned
formula gives w = for the case represented by Figure
25. But the case represented in Figure 26 having thestraining force applied at d, gives the strain at C (= w)
„ Wxab
= f x ——-.
J 9 A C X c B
Example .—Let an oak beam, four inches square, restfreely on the props a and b, seven feet apart, or 84 inches.What weight will it just support at its middle point, C, onthe supposition that a square inch rod will just carry 16,000pounds, pulling it asunder ?
^ , . , 16000 X 4 X 16 X 84
The formula becomes w = -, or
9 X 42 x 42
86016000 etlo , m . .
w — --- = 5418 pounds. Ibis is very near what was
15876
employed in Buffon’s experiment, which was 5312.
Had the straining force acted on a point, d, half waybetween c and B, the force sufficient to break the beam at c16000 X 4 X 16 X 84
would be equal to ---—-—-= 10836 lbs.
H 9 X 42 X 21
Had the beam been sound red fir, we must have takeny10,000 nearly, and m nearly 8; for although fir ^ ^jcohesive than oak in the proportion of 5 to 8 nearly, 11 ls t0compressible, and its axis of fracture is therefore nea fethe concave side. ,^ e .
Having considered at sufficient length the strains of dl erent kinds which arise from the form of the parts of a ir . il j c |,of carpentry, and the direction of the external forces w 11 „act on it, whether considered as impelling or as supp° rt n jits different parts, we must now proceed to consider the & e ^by which this form is to be secured, and the connectio 119which those strains are excited and communicated.
The joinings practised in carpentry are almost infij’^ j,ivarious, and each has advantages which make it prefer 11 '
ble 1rely *
some circumstances. Many varieties are employed inere ,', s e, !please the eye. We do not concern ourselves with 1 ' \
nor shall we consider those which are only employed 1° ;
necting small works, and can never appear on a great sc ^yet, even in some of these, the skill of the carpenter '
discovered by his choice ; for, in all cases, it is wise to ®' g ,every part of his work, even in the smallest details, as s , fSas the materials will admit. He will be particularly att . el ].iiig jto the changes which will necessarily happen by the shr"’ 0 f jof timber as it dries, and will consider what dimens 1 ® j |his framings will be affected by this, and what will not > jwill then dispose the pieces which are less essential to ^strength of the whole, in such a manner that their ten Jto shrink shall be in the same direction with the shrink 1 ’ 1 ® ,the whole framing. If he do otherwise, the seams will''j-and parts will be split asunder. He will dispose his b ( |,e 1ings in such a manner as to contribute to the stiffness o\^ 9 ;whole, avoiding at the same time the giving them P® 81 ^which will produce lateral strains on truss-beams whieh^fiJ jgreat pressures : recollecting, that although a singl e ® 9 y:has little force, yet many united have a great deal, anfrequently perform the office of very powerful struts. \
Our limits confine us to the joinings which ^^o> e >essential for connecting the parts of a single piece of 3when it cannot be formed of one beam, either for wan {necessary thickness or length ; and the joints for conthe different sides of a trussed frame. g0 tbf
Much ingenuity and contrivance has been bestowed ft jidmanner of building up a great beam of many thickness j,ymany singular methods are practised, as great nostradifferent artists ; but when we consider the manner j n c j e aii/ |the cohesion of the fibres performs its office, we shall ^tsee that the simplest are equally effectual with 1 '^tii)! 19refined, and that they are less apt to lead us into fal seof the strength of the assemblage.
Thus, were it required to build up a beam for a g rea gl ior a girder, so that it may act nearly as a beam of j 1 $ & 1size of one log ; it may either be done by plain jogghtVj. ji9 9Figure 27, a, or by scarfing, as at b or c. If it I s t ° e tl ,slever, having the gudgeon on the lower side, we be , Ji0
tb > 9
most artists will prefer the form b and c ; at least - pi 1 -been the case with nine-tenths of those to whom we ,. b"posed the question. The best informed only hesjta . nathe ordinary artists were all confident in its super 101 1 'flwwe found their views of the matter very coinciden^ j )0 ok^considered the upper piece as grasping the lowerand several imagined that, by driving the one ver ^ t jfe 1°? 6the other, the beam would be stronger than an f n re in t tbut if we attend carefully to the internal proced®loaded lever, we shall find the upper one clearly the tl>If they are formed of equal logs, the upper one is 4 ^
the other by the depth of the joggling or scarfing> col‘ eSlsuppose to be the same in both; consequently, if 4