from the axis of inertia; then, if M be the moment of the straining forces on the section in question, we have, by Art. 22,

This is the tension per unit of area on the dangerous section arising from the transverse strain.

And if P be the compression on each rafter and A the area of the section,

is the tension on each unit of area produced by the compression along each rafter. Wherefore if S be the whole tension per unit of area, on the dangerous section, the dimensions of each rafter will be found from the equation

Let, as before, W

two rafters; then

W

W 2

=

Α

2pl be the total load sustained by the

is the load supported by each. Hence,

as cos a is the component of the straining weight on each 2

rafter in a direction perpendicular to it, the moment M of the strain on the middle of the rafter is, by Art. 35,

In this formula W is to be taken in pounds avoirdupois, and l and A in inches, because the coefficient S is expressed in pounds avoirdupois per square inch.

STRENGTH AND DIMENSIONS OF TIE-BEAMS.

82. The tie-beam is strained by its own weight, and, in certain cases, by the weight of a floor and its load. It is also strained by the action of the foot of each rafter.

In addition to the notation of the preceding article, let H be the horizontal component of the compression on each rafter BA or BC (fig. 38); then if S be the tension per unit of area on the dangerous section of the tie-beam AC, we have, as in the last article,

S =

Μα H
I

+

Α

The value of H is given in (2) Art. 80, and M = {WI; W being the weight in pounds avoirdupois of the tie-beam and floor, and the length of the tie-beam in inches.

Section 2.

BRACED RAFTERS.

83. In order to prevent the bending of long rafters, they are sometimes braced together by a cross beam DF (fig. 39), called

a stay.

B

D

F

E

Let AB and BC represent two rafters of a roof inclined to the horizon at an angle a, and braced by the cross beam DF. Let the length of each rafter be 7, the equally distributed load upon each unit of length of the rafter P, and the length BD or BF, l. The part BD has the load pl, and the part DA the load p (7), to bear.

Fig. 39.

The vertical pressure produced at B by the load plı, distributed along BD, is pli; and as an equal pressure is produced

at B by the load distributed along BF, the whole vertical pressure at B by the loads distributed along BD and BF is pl. Let P be the component of this pressure along BA or BC; then, by the resolution of forces,

pl, 2 P sin a,

or Ppl, cosec a.

(1).

Again: the vertical pressure at D produced by the load p (1 − 1), distributed along DA, isp (1-7), and that produced by pl along BD at D is pl1; hence the whole vertical pressure at D, produced by AB, is

The pressure at D may be resolved into two components, the one along DA, and the other along the horizontal line DF. Let Q be the component along DA. Then, because the three forces at D, viz., Q, the vertical pressure DF, are proportional to the sides DA, DE, EA, of the triangle DEA (DE being a vertical line, and AE a horizontal one), we have,

pl, and the component along

and consequently the whole horizontal thrust at D or A is

H = (P + Q) cos a = p(+) cot a (4).

We see by (3) and (4) that the compression along BA or BC, and the horizontal thrust at D or A, become less and less as we diminish the value of . Hence, when = 0, or when the stay is removed, the minimum compression and the minimum thrust become respectively,

It would appear from these results that it is advantageous to leave out the cross-beam or stay DF. In this case, however, the rafters would sustain a bending whereby the angle a would be diminished, and consequently the thrust would become greater. The object, then, of the cross-beam is to diminish the transverse strain in long rafters.

STRENGTH AND DIMENSIONS OF BRACED RAFTERS.

84. In deducing a formula to determine the strength of braced rafters, we will suppose the stay to connect the middle points of the rafters, as it is in the middle of the rafter, where the transverse strain is the greatest. Then each rafter will be strained half-way between its middle-point and each extremity.

Let S be the tension per unit of area on this cross section of the rafter, M the moment of the straining forces on the same at right angles to the beam, I the moment of inertia of the section, A its area, and x the distance of the extreme fibre from the neutral axis, as in the preceding articles; then

P being the compression along each rafter, as in (1) Art. 83. But W being the load on both rafters, we have, by Art. 35,

Obs. In deducing the formulæ of the preceding sections, we have not taken into account the fixing of the part of the structure under consideration. Hence the error is on the side of safety;

for the resistance of a beam when fixed is greater than when it is merely supported. (Art. 33.) This part of the subject is not very satisfactorily established by experimentalists. Different writers give different coefficients for fixing.

Examples.

1. The part of the weight of a roof which is supported by two rafters AB, BC, as in fig. 38, is 100 lbs. on each foot of length of the rafters, the length of the tie-beam is 30 feet, the height of the roof is 4th of the length of the tie-beam; find the compression on each rafter.

By Art. 80, if P be the compression on each rafter, then

W being the total load, and a the inclination of each rafter to the horizon.

From the data we find

7 16.77 feet 201.24 inches, and cosec a = 2.236;

=

hence W = 2 pl = 2 × 100 × 16·77 = 3,354 lbs.

Substituting these values in (1),

P1874-886 lbs.

2. In the last example find the horizontal thrust on the tie-beam.

Let H be the horizontal thrust; then (Art. 80)

3. In Example 1, find the dimensions of each square rafter in order that the load may be sustained with safety, the coefficient of safety being taken at 1000.