HIP

498

HIP

This ascertains the angle which the jack-rafter makes withthe hip. Since all the sides are now given, we shall have,by another well-known property, of the sines of the sidesbeing as the sines of the opposite angles, the following pro-

portion :—

As the sine of the hypothenuse 56° 44'. = 9.92227

Is to the sine of a right angle, or 90°. = 10.00000

So is the sine of the side pch, 52° . = 9.89653

19.89653

9.92227

To the sine of the opposite angle 70° 28'. = 9.97426

Therefore the backing is twice 70° 28'. == 140° 56'

In finding the angle opposite the side f c h, it was notnecessary that the bypothenusal side should have first beenfound. It might have been found independently thus :—Thesine of either of the sides about the right angle is to radius,as the tangent of the remaining side is to the tangent of theangle opposite to that side ; therefore,

As the sine of the side ret, 27°. = 9.65705

Is to the tangent of the side fch 52°. == 10.10719

So is radius, sine of 90°. = 10.00000

20.10719

9.65705

To the tangent of the angle opposite the side

f c h, 70° 28'. = 10.45014

, In the same manner may other bevels be found by trigo-nometrical calculations ; but as such extreme exactness is notnecessary, the geometrical constructions ought to be wellunderstood.

Example II.— The figure ABCDf Figure 2) of the wall-plate of a hip span-roof and the height of the roof being given;to find the backing of the hips, the angles made upon the sidesof the purlins by their longitudinal arrises, and the anglesmade upon the sides of the jach-rafters ; the roof being equallyinclined to the different sides of the building, except at theoblique end, A b.

Figure 2. —Let the two sides, a b, a d, and d c of the wall-plate be at right angles to each other, and the end c b atoblique angles to A b and c d ; draw the seat, e f, of the ridgein the middle of the breadth, parallel to a b and d c ; makeA G and dh equal to half the breadth of the building ; joinG h, which will be the seat of the common rafters adjoiningthe hips ; make e i equal to the height of the roof; and drawl G and I H, which are the length of the common rafters.Draw E d and e a, the seats of the hips; make e k equal toe i ; and draw K A, which gives the length of each hip.Through any point, l, in the seat of the hip a e, draw m nperpendicular to ae, cutting the adjacent sides of the wall-plate at m and N ; take the nearest distance from l to therafter A K, and make L o equal to it ; and draw o M and O N ;and mon is the backing of the hips, represented by their seatsa e and d e.

This operation is the same as having the two legs of aright-angled solid angle to find the angle opposite to one ofthe legs ; the angle mon being exactly the double of theangle so found ; for the hip angle of the roof consists of twoequal solid angles.

Suppose the bevel end at C b to be inclined at a differentangle to the other sides, and let f c and f b be the seats ofthe hips; draw F Q perpendicular to F C, and f p perpendicu-

lar to f b, each equal to the height of the roof; then drawQ c and pb, which are the'lengths of the hip-rafters.

The backings sui and v w x, are found in the samemanner as above, and may be described in the same words.

From a, with the distance a k, describe an arc cutting G Bat J, and join A j ; then g J a will be the side bevel, whichthe jack-Yafters make with the hips ; and if a right angle beadded to G J a, it will form an obtuse angle, which is thatmade by the upper arris of the side of a purlin placed in theinclined side of the roof with the hip-rafter.

Let a be the position of a purlin in the rafter hi; in a Btake any point, b, and draw be parallel to the inward direc-tion of the purlin a ; from b, with any distance, b c, describean arc cd, cutting g h at d; draw b e, cf and dg, parallel* 0e f ; the former two cutting e n at e and f-, draw f g parallelto G h, and join eg ; produce be to h; and he g, or b eg willbe the angle required, according to which side it is applied •this will be found synonymous to one of the legs, and theadjacent angle of a right-angled solid angle being given, tofind the hypothenuse. In the same manner, if neither sideof the purlin should be parallel to the inclined side of theroof, as at h in the rafter g i, the bevel or angle upon eachside may be found, as shown.

Plate II.— Figure 3, shows half the angle of the backingof the hips, the length of the common and hip-rafters, thebevel of the jack-rafters on their upper sides in an equalinclined roof, without laying down or drawing any more thanthe necessary seats; and this is all that is necessary wheneach side of the roof is alike; ab being the wall-platebetween the hip and the rafter which joins the top of the hip)A c the seat of the rafter which joins the top of the hip, B cthat of the hip, a f the length of the rafter which joins thehip, b e the length of the hips, c h g half the backing, ad®the angle which the jack-rafters form with the upper sides °*the hips, and, consequently, with the addition of a right*angle, the side bevel of the purlin.

Figure 4, shows the same bevels, exeept that the side join*of the purlin is found by a different process, thus: from ®>with the distance b a, describe an arc at d ; from g, with thedistance A c, describe another arc, cutting the former at D jjoin bd, and the angle gb d will be the angle in the plane 0 *the roof, made by the lower arris of the purlin and the jointagainst the hip-rafter.

Besides the angles already mentioned, A f c Figure 3, showsthe angle formed by the upper side of the rafter and the ridg e *piece, and the angle bec, the angle which the top side °*the hips makes with a vertical or plumb-line: also the angl®fac shows the form of the heel of the common rafters, ante b c that of the hips.

Figtire 5, is a diagram showing the length of the parts

and

angles concerned in the roof, in the same manner as above)but the plan of the building, or form of the wall-plate, is aquadrilateral, which has neither part of its opposite sides p 8 *'allel; the method of executing the roof in this case is to f° r ®]a level on the top, from the top of the hips at the narrow ento the other extremity, as otherwise the roof must eitherwind, or be brought to a ridge forming a line inclined to t®horizon; and either of the two last cases is very unsign* v'But, that nothing should be wanting, the construction is give®in the next figure. .

Plate III.— Figure 6.— To lay out an irregular roof 1ledgment, with all its beams bevel upon the plan, so that tnridge may be level when finished; the plan and height ofroof being given. g

The lengths of the common and hip-rafters are found 8usual. From each side in the broadest end of the r° ’through c and d, draw two lines parallel to the ridge-l llie ’