CARPENTRY

BY

IRA SAMUEL GRIFFITH

Chairman of the Manual Arts Department
University of Missouri

THE MANUAL ARTS PRESS

PEORIA, ILLINOIS

Copyright 1916 by
Ira Samuel Griffith
Fourth Edition, 1919

ACKNOWLEDGMENTS

To my father, whose patient instruction and forbearing oversight during the period of carpentry apprenticeship has made possible the practical aspect of this present volume, grateful acknowledgment is made.

Acknowledgment is also made of assistance derived from the various trade magazines and from the few books on carpentry.

Credit is due Mr. franklin G. Elwood, Peoria, for most of the excellent drawings which accompany and clarify the text. A number of the drawings were penciled by Gordon Kellar, Boston. The photographs are the work of James F. Barham, Columbia, Mo-.

I. S. G.

PREFACE

I

IT is the author's hope that the following text may be of service to apprentices to the trade, to vocational and trade school students, and to manual training students. The author's experience as a carpenter leads him to feel that not a few journeyman carpenters may find their horizon widened and their usefulness as framers of the unusual roof increased by a study of Chapter IV where an effort has been made to indicate how the principles involved in framing the square and octagonal roof may be "generalized" so as to make possible their application to roofs of any number of sides. Beyond this, the book makes claims to being nothing more than an elementary treatise of the essentials of carpentry.

No apology is offered for making use of trigonometric solutions of plane right triangles as a basis for developing generalized roof framing principles in Chapter IV. There is absolutely nothing in the use of natural trigonometric functions to prevent their introduction early in the mathematical experience of a boy, except academic tradition. The author has made use of this mathematical tool with upper grammar grade boys with less effort upon their part in mastering the principles than was expended in mastering square root. The ease with which roof framing problems lend themselves to solution by the use of natural trigonometric functions and the readiness with which problems may be generalized thereby has emboldened the author to make use of it in a text as elementary as this. No previous knowledge of trigonometry is presupposed, the Appendix provides all the information required for the solution of any problem given herein.

Should a reader, because of lack of time or for any other cause, not care to consider more than roof framing of the square cornered building, he will find a complete treatise in Chapter III without reference to solutions other than by common arithmetic. Appendix IV offers a still more abbreviated approach to both square and octagonal roof framing.

The greatest good in studying the chapter on "Estimating" will come only when each student is provided with a set of plans and specifications completely drawn, as by a practicing architect. Plans and specifications, such as will serve the purpose, can be purchased at small cost from architectural companies, should local architects be unwilling to provide sets for the schools.

Also, there must be provided for each student, catalogs of lumber and millwork specifications and prices. These can be obtained from mail order lumber and millwork companies. As a rule, local lumber and millwork companies are glad to provide such data, but it must be in a form complete, and readily accessible to be of the greatest value.

Ira S. Griffith.

Columbia, Missouri,
September, 1916.

CONTENTS

The house used as a model for many of the illustrations in this book.

CARPENTRY

CHAPTER I
Foundations

Fig. 1. Transit.

1. Laying out Foundations.—In most communities it is customary for the carpenter to be present and to assist the mason in the laying out of the foundations. Where buildings are large and important, this work is done by an engineer with a steel tape and a surveyor's instrument, [Fig. 1.] This instrument is known as a builder's transit, and consists of a tripod upon which rests a small telescope with crossed hair wires within, by means of which the observer may fix the line of sight very accurately. A circular dial contains a magnetic needle which enables the fixed dial to be set with reference to the true north and south line of the observer. After the fixed dial has been adjusted, the telescope may be swung to the right or the left until the circular graduations indicate that it points in the direction wanted, after which stakes may be set. A level upon the telescope enables the observer to sight grades or levels; a helper carrying the leveling rod, [Fig. 2.]

Fig. 2 Leveling Rod

[Fig. 3] shows a more common instrument. This is an architect's Y-level and differs from the other in that it is less complete. It has no attachment for measuring vertical angles. This is not serious, however, since the builder seldom needs such an attachment, the level being the most essential part. Y-levels are made both with and without compass attachments.

Fig. 3 Y-Level

Upon ordinary residence work a surveyor is employed to locate lot lines. Once these lines are located the builder is able to locate the building lines by measurement. Suppose it is desired to locate a building by means of the side lot line: (1) Measure from the side lot line, along the front and along the back lot lines, a distance equal to that which it is desired the house shall hold relative to the lot side line. Drive stakes here. (2) While sighting from one of these stakes to the other, have an assistant locate two other stakes in the line of sight, a distance apart sufficient to guarantee the placing of the cross-lines for the back and front of the house without restaking these, A-B, [Fig. 4.] The process of laying out lines for a house is almost identical with that used in laying out a rectangle on a drawing board. (3) Having located a line of indefinite length for one side of the house, a second line of indefinite length, preferably for the front of the building, may next be located. To do this, first locate a front corner stake upon the first line just located. This is done by measurement from the street line. Having located and driven in this stake, A , [Fig. 4], drive a nail in the top of the stake to more accurately locate this corner.

Fig. 4. Batter Boards

If an instrument is available it will be located over this stake and the front line A-C, [Fig. 4], located by laying it off at 90 degrees from the side line already located. If no instrument is available, the front line may be laid off at right angles to A-B by holding a framing square at their intersection. This angle should be verified by the 6-8-10 method. This consists in measuring from the intersection at A along one line a distance of 6 feet and sticking a pin in the line at that point; a pencil mark may be used when the cord is white. In a similar manner, measure off 8 feet along the other line and then measure the hypotenuse of the triangle so formed. It should measure 10 feet. If it does not, the front building line must be shifted until it does. (4) With these two lines located, the remaining two lines may be located by measurement from them, the nail of stake A giving the starting point. Before this is attempted, however, the batter boards should be placed. Batter boards are variously constructed. Those shown are common types. They should be placed free of the foundation proposed by at least 3 or 4 feet. (5) Test the squareness of the whole lay-out by measuring the diagonals A-D and B-C. If the building lay-out is square the diagonals should be equal. If they are not equal, shift the cords at C and D, retaining their parallelism, until the diagonals become equal. (6) Once the lay-out is correct, saw kerfs should be made in the batter boards where the cords are placed. These kerfs will permit the cords being removed and replaced without further measuring.

2. Grade Line.—A properly drawn set of plans will show both the present lay of the ground upon which the building is to be erected and the new grade line which is to be established after the building is completed. The most convenient method of determining old grade lines and of establishing new ones is by means of the transit, [Fig. 1], or the Y-level, Figs. [3] and [5], with the rod, [Fig. 2.] Both instruments operate upon the same principle in grade work. The telescope is set level and sights taken thru it to the target upon the rod. The reading of the target's position upon the rod compared with the height of the telescope above the base, usually the street walk, determines the difference in grade of that particular placing of the target.

Fig. 5. Taking Sights with Y-Level

To locate levels for the masonry, (1) set the instrument at some convenient place and level the dial. (2) Having determined the height of the instrument above some predetermined base, such as the street walk, swing the telescope about and, making allowance for the difference in level as shown by the drawings, place successively stakes at each corner of the building with the required level marked thereon. As a rule, the mason has his own Y-level and uses it freely as the wall is constructed, especially where levels are to be maintained as the layers of material are placed.

Fig. 6. Leveling with Straight-edge

In a similar manner the earth grade about the building may be located, stakes being driven into the ground at frequent intervals and the amount of "fill" or reduction indicated thereon. Grade levels are established usually only after the builders are thru, except that the mason will have the grade indicated for him where the wall above the grade is to be differently finished from that below.

Where no surveyor's level is at hand, the mason or carpenter will secure the levels by means of a straight-edge of some 14 feet in length. A common level is placed upon this plank as shown in [Fig. 6.] By successive levels with stakes driven to indicate the successive levelings, a grade may be carried quite a distance without very great variations.

Fig. 7. Foundation Detail

3. Excavations.—Excavations should be made enough larger than the proposed foundation that the mason may have room to wield his trowel in pointing the outer joints, and for waterproofing. An extra foot of excavation upon each side will usually be required.

All foundations must be carried well below the frost line. Excavations should be made accordingly.

4. Foundations; Footings.—Because of the tendency of a building to settle unevenly, due to variations in the strength of the supporting ground or the unequal weight placed upon this ground, foundations must be constructed of some non-yielding material such as brick or stone, and of such thickness and so bonded that the weight of the building may be evenly distributed.

The thickness of wall will depend upon the weight to be supported and upon the character of the soil.

Unless rock or gravel is encountered, every foundation should have a footing, [Fig. 7.] The amount of footing used is usually twice the thickness of the foundation wall. In brick walls this footing draws into the wall by "stepped" courses of brick, each layer being narrower than the one just preceding. For ordinary residence work with ordinary soil conditions a 10- or 12-inch wall resting upon a footing 2 feet wide and 8 or 10 inches deep will suffice.

A safe footing for supporting posts of 66" × 6" yellow pine, for most soils, will be 10 inches deep by 18 inches square. Partition walls carrying no unusual load need not be over 8 inches in thickness.

Fig. 8. American Bond

Fig. 9. English Bond

In many communities the use of concrete is supplanting that of stone or brick, especially below the grade line. Such a wall should be composed of 5 parts of crushed stone or gravel, 3 parts sand, and 1 part cement. The footing may be formed by tamping the mixture in a form made by spading out of the earth a depth and width sufficient for the wall to be supported.

5. Foundation Materials; Construction.—Of the materials commonly used in the construction of foundations monolithic concrete is becoming the most common for that part of the wall which lies below the ground or grade level. Brick and stone are sometimes used.

Where brick or stone is made use of, some device is required to "tie" the material together, due to the fact that the mortar used in filling the voids or spaces between the members has little strength as compared with that of the stone or brick itself. This bonding is secured by placing the brick or stone so that they shall overlap one another, both along the faces of the wall and across the wall.

Bricks laid with their lengths in the same direction as that of the wall are known as stretchers; those laid with their lengths across the wall are known as headers, [Fig. 8.] The manner of placing these headers among the stretchers determines the type of bond. The American, English and Flemish are the more common types. Of these the American, [Fig. 8], is the most used upon ordinary work. It consists of a course of headers placed every sixth course. The English bond, [Fig. 9], is much stronger, having every other course a header course. It is used mainly upon very important work where unusual strength is required. Flemish bond is illustrated in [Fig. 10.]

Fig. 10. Flemish Bond

Of the various types of stone work, rubble work and ashlar predominate, [Fig. 11.] Rubble work is most frequently used for that part of the wall below the grade line, and ashlar for the remainder of the wall. In either case, thru stones are placed every 4 or 5 feet in the length of the wall and every 18 inches in the height, to provide bonds.

Fig. 11. Types of Stone Work

In rubble work the stones are rough and unhewn. They must be laid upon a good bed of stiff mortar with their stratifications in a horizontal position. Otherwise, the face of the wall might "peel" from the effects of frost and moisture, making an unsightly as well as a weaker wall. The term "ashlar" refers to a wall builded of stones having finished faces. When either rubble work or ashlar is laid up in courses it is known as coursed rubble or coursed ashlar. When the horizontal joints are not continuous the wall is known as random rubble or broken ashlar.

Not infrequently a wall will be constructed with an ashlar facing attached to a brick backing by means of metal bonds. In such a wall, the faced ashlar, unless more than 8 inches in thickness and well bonded into the wall, should not be considered in estimating the strength of the wall.

Fig. 12. "Form" for Concrete

In the construction of both brick and stone walls the work should be carried up as nearly as possible at the same levels. In both brick and stone walls the corners are run up with stepped courses, the corners being plumbed as the wall is carried upward. A line is then stretched between the corners and, layer by layer, the rest of the wall filled in. No corner should, ordinarily, be carried more than 3 feet above the rest of the wall. In the case of uncoursed stone work the wall is leveled every 15 to 18 inches in its height.

6. Forms for Concrete Walls.—The economical building of forms for concrete walls is a matter of importance in building construction. [Fig. 12] shows a type of form suitable for foundation work. Such forms should be made of semi-seasoned stock. Thoroughly seasoned stock will warp badly when the wet concrete is placed. Spruce, Norway pine, etc., are better woods to use than hard or Georgia pine.

For ordinary foundation work 1-inch boards may be used, the studs being placed not over 2 feet apart. These studs may be assisted materially in holding the forms in position, by wires placed as in [Fig. 12], and by props placed against the dirt wall of the excavation.

In placing the concrete a 4-inch layer is laid and then "spaded" or "worked" well into place, a "wet mix" being used. The smoothness of the resulting faces is increased by an additional spading of the mixture away from the form. A good spading tool is made by straightening out an ordinary garden hoe. This allows the cement and mortar to flow next to the form and hold this place while the filling proceeds.

Where forms are placed to give finished walls, that is, walls to which no plaster is to be applied, they should be aligned with no greater variation than ⅜" from the lines specified.

Forms should be allowed to remain until the concrete will resist indentation with the thumb, upon ordinary walls.

There is no limit to the ingenuity one may make use of in form building. The illustration given is merely suggestive.

7. Waterproofing.—The extent to which a wall should be waterproofed will depend upon the location of the building. Foundations near running water must naturally be better protected than those in well drained locations. [Fig. 7] illustrates a treatment which will prove quite safe for almost all localities. The exterior face of the wall is covered with several layers of asphaltum or tar. By coating the top of the footing and the top of the concrete floor just before the finish floor of cement is placed, little water will enter. A drain tile carried about the house as shown in [Fig. 7], especially if gravel is placed against the wall above it, will meet every emergency.

There are other ways of waterproofing basement walls, but this is typical of the external wall treatments. In monolithic construction waterproofing may be secured by appropriate additions to the mixture of waterproofing materials such as slacked lime, just before the mixture is placed, no external applications being required.

Fig. 13. Cellar Frame with Sash

8. Basement Frames.—[Fig. 13] illustrates one successful form of basement window frame construction, with sash. In this type the sash is hinged to the top of the frame, and a catch or button at the bottom of the frame secures the sash when closed. The construction is such as to best shut out wind and water when the sash is closed.

Fig. 14. Basement Door Frame

[Fig. 14] illustrates a basement door frame. Frames such as this, and the window frame of [Fig. 13], are made of heavy stock and are known as plank frames.

Basement frames are held in place by means of wooden blocks nailed to the sides of the frame, as well as by the projecting "lugs" of the frame itself. The frame is set and plumbed by the carpenter as soon as the mason has prepared the sill. [Fig. 14] shows a frame plumbed and stayed, ready for the mason to lay the adjacent wall. [Fig. 15] indicates the position of plumb and level in the setting of a frame. The edges of a door frame are "sighted" for wind.

Where it is necessary to attach frames or other woodwork to brick walls, it is customary to have the mason insert wooden "bricks" as the wall is constructed. Wooden bricks are of the same size as other bricks, and should be constructed with the edge which is to be laid back in the wall thicker than the front edge, so that a dovetailed effect is secured.

Fig. 15. Plumbing and Leveling Cellar Frame

CHAPTER II
Main Frame

Fig. 16. Full Frame House

9. Methods of Framing the Superstructure.—In the early days when lumber was plentiful, houses and barns were framed in what is known as "full frame." Such frames consisted of heavy and solid timbers mortised and tenoned and pinned together, Figs. [16] and [17]. With the growing scarcity of lumber the "half frame" of [Fig. 18] became common. This latter type, it will be seen, makes less use of heavy timbers and wooden pins, and more use of planks and nails. To-day the vast majority of buildings, where wood is the material used, are constructed by what is known as "balloon framing" in houses and "plank framing" in barns, Figs. [19] and [20]. In view of this, attention will be directed to balloon framing only. One who is able to frame a house should have no trouble with plank barn framing, where drawings show the details.

Fig. 17. Heavy Timber Barn

It must be understood, too, that there are quite a variety of ways of framing a balloon and a plank frame. It will be possible in this chapter to treat of but one type. A mastery of this one type should enable the student to work out other types, with suitable detailed drawings provided him.

10. Sills and Girders.—In [Fig. 21] will be found illustrated three types of box sill construction. Whatever the sill used, care must be taken to so plan that mice may not have free access to the various parts of the building. If the sill does not inhibit, then blocks should be spiked between the studs. Such blocks serve as fire breaks.

Fig. 20. Plank Frame Barn.

Fig. 21. Three Types of Box Sills

The bed plate of the box sill should be selected from stock with straight edges. In the framing of joists, plan so that the crowning edges shall be up when in position, and in placing the joists see that the most crowning are in the middle of a room. Joists are fastened to their sills as in [Fig. 21.]

[Fig. 22-a] illustrates a built up girder, and the manner of framing the joists to it. Three 2" × 10"'s with a 2" × 4" attached to each side, the whole thoroughly spiked together, form the girder. The advantage of this type of girder lies mainly in the fact that it leaves the headroom of a basement clear, which is not the case in the type shown in [Fig. 22-b]. This second type is somewhat easier to frame, and is therefore greatly used where the owner does not object. It is better where furnace stacks must be placed in a partition above it.

Fig. 22-a Fig. 22-b. Girder Types

First floor joists, like second floor joists and studs, should be spaced 16 inches from center to center, beginning at one side or end of a room. Not to make such provision would cause a waste in lathing, since the lath are all 4 feet in length, a multiple of 16 inches. Any remainder after such a spacing should be allowed to come at the side or & end of the room.

Fig. 23. Cutting Bridging

11. Bridging.—To add to the carrying power of floor joists, bridging is cut in between them as shown in [Fig. 23.] For ordinary dwellings 1" × 3" stock will serve. On large work, stock two inches thick should be made use of. Bridging should be spaced not more than 8 feet apart. A miter-box, set at the appropriate angle, may be used in cutting bridging, all the pieces being cut at one time with the exception of those for the odd spacings at the side or end of a room. A more common practice is to take a piece of stock, and, after cutting a bevel on one end, place it as in [Fig. 23] with the beveled end above the lower edge of the joist against which it rests, a distance slightly in excess of the thickness of the stock; then saw as indicated, sawing vertically and along the joist.

Fig. 24. Laying off a Stay

Fig. 25-a-b. Headers and Trimmers in Floor Frame

Before placing bridging, the joist must be spaced and properly fastened in place. This is done by placing a piece of stock, 1" × 6" or 2" × 4", as in [Fig. 24.] With a try square, mark the locations of the joists. This board may then be transferred to the center of the room and the joists there spaced according to the marks, and held in place by being "tacked." A second method consists in placing the spacing board in the center of the room and having a second person sight the joists for straightness while the first party places them as directed and tacks them. This tacking consists in driving the nails only partially in, leaving the heads project enough that they may later be withdrawn with a claw hammer. Still another method is to lay off the "stay" by measurement with the framing square so that it corresponds with the spacings of the joists at the side walls.

Bridging should be nailed with two nails at each end of the piece.

Fig. 26. Placing Headers and Trimmers

Fig. 27. Floor Frame and Rough Floor

12. Trimmers and Headers.—In the making of stair and chimney openings it becomes necessary to support the ends of joists other than in the usual manner. This is done by cutting in headers as in Figs. [25], [26] and [27]. Where the span is not great, such as at an ordinary chimney in residence work, in which but one or two tail beams are to be carried, headers are not doubled and are merely spiked in place. Where many joists are to be carried, headers or trimmers, or carrying joists must be doubled. Iron stirrups or hangers should be used instead of spikes in joining headers to carrying joists where spikes would weaken the carrying joist and would not give

Fig. 28-a.	Fig. 28-b.
Fig. 28-a. Headers and Trimmers in Wall Frame

sufficient strength to the joint. Except upon long spans, tail beams are usually fastened to the header by spiking only. On long spans they should be framed to the header as joists are framed to a girder, a 2" × 4" being spiked firmly to the header as a support.

In determining the amount of space to allow for head room in framing about a well hole for a stair, determine the run and rise of the stair from the plan and elevation, and then plan to allow at least 6' 6", measured from the proposed nosing line of the treads up to the proposed location of the trimmer, or carrying joist, or header, as the case may be, at the ceiling level, [Fig. 121.]

Fig. 29. Headers and Trimmers in Wall Frame

Fig. 30. Stud and Joist Patterns

The term "header" is also used to designate the studding, or joist in the case of double doors, placed horizontally over window and door openings, [Fig. 28.] Studding cut in below window openings forms the stool, also known as header. The illustration shows the manner of framing for openings of different widths. A small single window may require but one thickness of 2" × 4". A medium sized opening will have a header of two pieces of 2" × 4". Where the opening is rather large, as in the case of double door openings, two joists will be set on edge over the opening as header.

13. Walls and Partitions; Joists and Rough Floors.—A study of Figs. [16], [17], [18], [19], [20] and [29] should give an understanding of the essential members of the framed wall of a building, and their relations one to another.

Whether side walls shall be framed and raised before the rough or false floor of the first story is laid will depend upon the type of sill construction made use of. In laying off studs, joists, etc., a pattern is first framed. These patterns are afterward used in the building and are therefore counted in with the total number of pieces to be framed. To these patterns, stops and fences are attached near the two ends and at the middle, [Fig. 30.] The other studs or joists of similar dimensions are laid off one at a time by superimposing these patterns and marking about them with pencil, [Fig. 31.]

Fig. 31. Marking Joists from Pattern

Ribband or ribbon boards and plates are laid off by placing them alongside the "lay-out" for the studs made upon the sills, and transcribing the marks to the ribband board and plate by means of try-square and pencil. Sometimes ribband boards and plates are laid off by measurement, as are sills.

Fig. 33. Corner Post Being Plumbed and Stayed

Corner posts are constructed first and placed. [Fig. 32-a] shows a section of a corner post which has much to commend it. [Fig. 32-b] illustrates a more common type of construction. The most serious objection to this type is the fact that the post must be furred after the lather has placed the lath upon one side of the room. Corner posts are plumbed and stayed in two directions, after being raised, [Fig. 33.] Either 2" × 4" or 1" × 6" stock will be used for stays. With the corner posts set, the ribband boards are placed. Where the span is too long for any available length of ribband board, in laying out the ribband boards provision must be made for their "breaking" joints upon studs. These studs will be raised immediately after the corner posts, the ribband board attached to corner post and stud, after which the stud will be plumbed and stayed, [Fig. 34.] Studs are framed before being raised so that ribband boards may be "let into" them as shown in [Fig. 34.] Second and third floor joists will be notched to slip over these boards and will be spiked to the studs in addition. Remaining studs are placed one at a time, one man setting up and nailing the foot while another fastens the ribband board to the stud at the second floor line, [Fig. 35.]

Fig. 34. Side Wall Stayed

With the completion of the raising of the two outside walls which are to bear the joist ends, the middle partition, should there be one, paralleling these walls should be framed and raised. A slightly different procedure from that just described is followed, that is, instead of raising one stud at a time the whole partition is framed and nailed together upon the floor, even to the cutting in of headers, etc. When a section such as the number of men available can raise is ready, the same is raised, and stayed after being plumbed. The studs of partitions are framed but one story high and "plated" at such a height that second floor joists may be placed thereon in splicing. Just as far as possible first and second floor joists should be spaced to rest one directly above another and in line with the supporting studs of partitions so that furnace stacks may be placed with ease. If joists rest upon partition plates and not directly above studs, a double plate must be made use of.

Having placed the second floor joists, the studs at the ends of the house may be set up. Their locations will be marked upon sill and upon second floor joist which is to be placed at the end of the house. This marking is best done by placing the joist upon the sill and transcribing the marks laid out upon the sill to the joist, after which it is to be raised into place.

Fig. 35. Setting up Studs and Attaching to Ribbon Board

Double plates will next be framed. They should break upon studs and be marked by transcribing the marks for the studs from the sills. At the corners the plates will be framed with butt joints, the second set lapping over the joints made by the first plate.

Next, the sustaining middle partition of the second story is raised as was that of the first story. The attic floor joists are placed as were those for the second floor.

All walls and partitions are now "lined up," that is, any irregularities are taken out by additional stays.

Fig. 36. Estimating Window Openings

False or rough floors are laid in the various stories where not already placed, bridging being placed and openings for stairs and chimneys framed. Such floors are laid either diagonally or straight across the joists. The diagonal floor is considered better, [Fig. 27.]

14. Openings in Framework.—Studs in outside walls are set without reference to openings for doors and windows. Such openings are cut and headers and stools placed after the walls are up and ready for sheathing. The seeming waste occasioned by this method is slight since the cut-out material is available for headings, etc. Most carpenters make a story pole to be used in laying off window and door heights in cutting out studs. This is nothing more than a piece of 1" × 2" or 1" × 3" stock with the heights of the openings from the rough floor or from the joists, where the rough floor is not laid, marked plainly thereon. This pole is placed alongside the stud to be cut and the mark transcribed from pole to stud.

Beginners are frequently troubled in determining the proper opening, even when the size of the window is specified. In general, carpenters plan to have the studs on either side of an opening, either door or window, so set that the outer edges of the exterior casings will break upon their centers. Windows are specified by the width and height of their glass and the number of divisions or lights, width always being specified first. The distribution of excess measurement due to the meeting rail, top and bottom rails, side rails or stiles is shown in [Fig. 36.] Rail and stile widths and sash thicknesses will vary from those given when any very great increase in size of window is made. Manufacturers of sash and doors provide catalogs in which stock sizes are listed.

Fig. 37. Framing Wall Openings

Estimate an opening vertically, [Fig. 36], thus: Sill, 2"; subsill, where frame is made with one, 1"; bottom rail, from edge to bottom of rabbet, 3"; glass in lower sash, 34"; meeting rail, from rabbet to rabbet, 1"; glass in upper sash, 34"; top rail, 2"; space for head jamb and lugs of side jambs, 2" or 3"; total, 79". A carpenter would say, "Add 11" to the glass measurement to get vertical height between stool and header." Window sashes with muntins require an addition of ¼" for each muntin. The thicknesses of header and stool must be considered in addition to the measurement just mentioned when studs are sawed, [Fig. 37.]

The width between studs would be estimated thus: Width of glass, 28"; width of stiles, from rabbeted edge to outer edge, 4"; width of casings, 8"; total 40", distance from center of stud to center of stud. Comparing this with the width of glass it will be seen that the difference is 12". A carpenter, therefore, makes use of a general rule: Add 10" to the glass measurement to get distance between studs, where a 4" or 4½" casing is used with this type of window frame.

Fig. 38. Threshold Detail

For the 3' x 7' door, Figs. [37] and [38], estimate the opening as follows: Height of door, 7'; allowance for rough floor, ¾"; finish floor, ¾"; threshold, ⅝" to ¾", head jamb and space for lugs of side jambs, 2" to 3"; total from joist, may be 7' 5".

For the width of opening estimate: Width of door, 3'; width of casings, at 4½" each, 9"; total spacing of studs center to center, 3' 9". Distance between studs will be 3' 7". This will leave space enough to put the doubling studs on each side between header and floor. Since locations of openings in the main frame, both window and door, are dimensioned to the centers of the openings, it is easiest in laying off to estimate from the center each way rather than to estimate total width.

After these openings are made, the frame of the house may be covered with sheathing, or the roof may be framed; both orders of procedure are common.

CHAPTER III
Roof Frame: Square Cornered Buildings

15. Roof Framing.—The problem of framing the various members of a roof is not a difficult one provided the underlying principles are understood, and dependence placed upon this understanding rather than upon mere knowledge of what figures to use upon the square to get the cuts, without knowing why those figures are used. An effort will be made in this treatment to indicate the "why."

GABLE HIP SHED GAMBREL
Fig. 39. Roof Types

In [Fig. 39] are illustrated four types of roof. Figs. [40], [41], and [42] illustrate the rafter forms and the names of the various cuts to be made in framing the members to place. The common rafter, it will be seen, has three cuts—plumb or ridge cut, seat or heel or plate cut, and end cut. The hip, valley, and jack have four cuts each; a side cut or cheek cut is possessed by each in addition to the three cuts belonging to the common rafter.

Before any rafter can be framed, the rise and run of the common rafter, in other words, the pitch of the roof, must be known.

In roof framing, the "run" of a rafter when in place is the horizontal distance measured from the extreme end of the seat to a point directly below the ridge end of the rafter, [Fig. 43.] The "rise" is the vertical distance from the ridge end of the rafter to the level of the seat. The "pitch" of a roof or rafter is the ratio of the rise of the rafter to the span or whole width of the building.

Fig. 40. Roof Details

Fig. 41. Plan of Roof Rafters

The terms rise, run, and rafter length have still another set of meanings—they may be used to designate "unit" lengths. In all such cases 12" of run of the common rafter is assumed as the base, and the other unit lengths or constants are computed from this constant. The numerical values of these constants will be computed as the development of the subject of roof framing makes their use necessary.

Fig. 42. Raising the Rafters

It will be noted in [Fig. 44] that the constant of run, or 12", is taken along the tongue and the rise per foot of run along the blade of the square. It is not essential that this order be followed; the beginner will generally find it easier to visualize his work, however, if he keeps the tongue for either rise or run, and the blade for the opposite. There are occasions when the reverse order is necessary no matter which form is followed, so that it is unwise to insist upon only one way.

Fig. 43. Run, Rise and Length

Fig. 44. Unit Length of Common Rafter

The variation in terminology in roof framing is so general that the beginner will do well to familiarize himself with the most common. Hereafter an effort will be made to confine the text to the following: plumb cut, seat cut, end cut, side cut.

The value to a beginner of a carefully made plan of a roof to be framed with necessary data such as rafter lengths and positions indicated thereon, cannot be too strongly emphasized. Architects not infrequently prepare elaborate and complete framing plans for the use of the carpenter. Upon intricate plans, experienced men prepare plans before attempting to frame the same. [Fig. 43] illustrates a framing plan ready for the placing thereon of the necessary data, such as measurements along the plate for spacing the rafters, lengths of rafters, ridge pieces, etc.

Fig. 45-a-b. Laying off Common Rafter

16. Framing the Common Rafter; Laying out the Plumb Cut.—While in this discussion the plumb cut is first described, it should be understood that it is equally as convenient and more common among carpenters to begin the framing of the members of a square cornered roof frame with the end and seat cuts. In framing other than a square cornered roof it is somewhat more convenient to begin with the plumb cut.

The method of framing of the common rafter is the same for all buildings, whether the buildings have four sides or more or less. (1) Place the framing square as in [Fig. 45-b], taking 12" on the tongue as the run, and upon the blade the rise in inches per foot of run. Keep these numbers against the crowning, or what is to become the top edge of the rafter, and scribe along the blade. This gives the plumb cut. Occasionally a carpenter will be found who frames to a center line rather than the top edge of a rafter.

Fig. 46. Position in Laying off Plumb Cut when Laid off before Seat Cut

Figs. [45], [46] and [47] illustrate the proper position of the worker relative to his work. Such a position will seem awkward to the beginner but he should learn to visualize his work while in this position that the efficiency of framing may not be reduced thru the awkward position first likely to be assumed.

Fig. 47. Laying off Plumb Cut when Seat Cut is First Laid off

17. To Find the Length of a Common Rafter.—First Method: The theoretic length of a rafter is indicated by the center lines in Figs. [45-a] and [48]. In estimating the total length of stock for a rafter having a tail, the run of tail or length of lookout must be considered.

The pitches most commonly used are the half, third, and quarter. From an examination of [Fig. 43] it will be seen that the length of a common rafter is the hypotenuse of a right triangle whose legs are the rise and the run of the roof. The problem, then, of finding the length of a common rafter when the rise and run are known is merely that of solving the equation c² = a² + b².

Fig. 48. Rafter Length

Practical carpenters would not consider it economy to take time to solve for rafter lengths in this manner, for every variation in rise or run would necessitate a rather long solution. Instead, they have discovered that for every foot of run of a rafter the length of the rafter increases proportionately, the ratio of rise to run remaining the same, [Fig. 44.] With a table, therefore, in which the length of rafter for each foot of run, for each of the common pitches is given, the length of rafter for any given pitch can be found by merely multiplying the constant given by the amount of run for that particular rafter.

[Fig. 49] shows such a table worked out for a rather extended number of pitches. From this table it will be seen that the number to take as a constant for the run is 12", and that the rise in inches per foot of run is taken upon the other member of the framing square. A jack rafter as will be illustrated later is but a shortened common rafter, therefore, what is said of the common rafter is also true of the jack rafter. The jack, however, has an additional cut which will be discussed in another section.

Example:
Determine the length of a common rafter of a house with a 25' span
and a quarter pitch, without tail.

Fig. 49. Framing Table for Common Rafter

Solution:
Run = 12#'
Length per foot of run for quarter pitch = 13.42"
12.5 × 13.42" = 167.75" = 13.98'
(Looking for the nearest fractional value of .98 in the Table of Decimal
Equivalents in Appendix III, 63/64 or practically 1')
The rafter would be framed 14' in length.

When a tail is a part of the rafter, proceed in the manner described adding the run of the tail, or length of lookout, to the run of the rafter.

Fig. 50. Framing Square Detail

[Fig. 50] shows a framing square, containing among other data, the rafter lengths per foot of run. To use the data pertaining to common or jack rafter lengths, (1) consider the run as 12" taken on the tongue; (2) select upon the blade along its outer edge the inch mark which represents the rise of the roof per foot of run required to give the pitch specified; (3) the number directly below this mark, reading across the blade in the space marked "Length of Common Rafter Per Foot of Run" gives the length per foot for that particular rise or pitch.

As a check for rafter length computations, the following procedure is suggested: Selecting the run as 12" on the tongue and the rise in inches per foot of run on the blade, place one square upon another as shown in [Fig. 51], using that side of the square divided into inches and twelfths. Do not use the end of the blade, the rounded corner makes it impossible to secure the accuracy demanded. Extreme accuracy is required if the constant is to be used for rafters of considerable length of run. Read the diagonal length between the numbers representing the run and rise. Read the whole number of inches as feet, and the fractions as inches, and take off any fractional remainder upon a very sharp pointed pair of dividers. Read this divider spacing by means of the hundredths scale on the framing square. The result should, if the work is very accurately done, be the same as that obtained by computation from the tables, even to the hundredths place decimal. Upon ordinary work where great accuracy is not required carpenters sometimes determine this constant for a given pitch by placing the framing square as in [Fig. 46] or 47, taking upon the tongue the run and on the blade the rise, marking along both tongue and blade. The distance between these marks is then read on a square placed along the edge.

Fig. 51. Finding Rafter Length by Scaling

Second Method: In determining rafter length, an equally common practice is to lay the framing square as is shown in [Fig. 45-a]. While in this position the seat cut is scribed, cf. Section 18, and also a short sharp line scribed along the other member of the square at the top edge of the rafter. The square is moved along, using the same numbers, and another advance mark scribed. This operation is repeated just as many times as there are feet in the run of the common rafter. With a span of 24' the operation would be repeated 12 times.

Should the run not happen to be in even feet, the square would be placed as many times as there were full feet in the run. In addition it would be advanced that fractional part which the fraction of the run was of 12". For example, in a run of 12' 7", with a roof of ¼ pitch, the square would be advanced 12 times using the number 12 on the tongue and 6 on the blade. In addition to this the square would be advanced using 7/12 of 12" or 7" on the tongue and 7/12 of 6" or 3½" on the blade. As these numbers do not allow enough of the square to rest on the rafter to give a full line, as soon as the advance limit of rafter length is indicated the square may be moved up, using the set of numbers first used, that is 12" and 6". On common rafters, this last operation is simplified by noting that the fractional run, divided by 12, times 12, always equals itself. The final position of the square, therefore, may be obtained by simply sliding the member, used in laying out the last full foot line which parallels the seat cut, an additional distance equal to the fractional foot of total run, [Fig. 44.] The tail length is obtained similarly, [Fig. 44.]

Fig. 52. Laying out Rafter

18. Laying off Common Rafter Seat Cut and End Cut.—First Method: Having determined the rafter length as directed in [Sec. 17,] first method, (1) lay off this length along the upper edge beginning at the plumb cut. The whole number of feet is more safely "taken off" by means of a pole marked in feet, and of good length. The rule or square may be used to transmit fractional parts of a foot. (2) Place square as at "b," [Fig. 52], standing as in [Fig. 45-b], and scribe a plumb line as indicated at 1-2, [Fig. 52.] (3) From the point 1, [Fig. 52], measure along the line marked 1-2 a distance equal to one-half that of 1-2. The distance 1-3 may be increased or decreased somewhat when an extreme pitch makes it advisable. As a rule this should be 2½" to 3". (4) Place the square as at c, [Fig. 52], with the edge of the tongue resting on 3 and scribe a line for the seat cut, as 3-4. These last marks give the bird's mouth joint which is to fit over the plate.

Fig. 53. Independent Rafter Tail

Fig. 54. Length of Ridge Piece

While many carpenters allow end cutting of the rafter tails to wait until the rafters are set in place so that they may be lined and cut while in position, certain kinds of work permit the ends to be cut at the same time the remainder of the rafter is framed. In this latter method the square is placed as in [Fig. 44] and (5) the end cut scribed. The point of cutoff on the tail is determined in the same manner as that used in determining rafter length, the run of the tail being considered and the tail length being measured from the point 1, [Fig. 52.]

Where a cornice is of unusual width, tails are usually framed independent of the rafters and are then spiked to the ends of the rafters either above or below the plate, [Fig. 53.]

Second Method: Where the second method of finding length, Section 17, is employed, the end cut and seat cut will be laid out before the plumb cut. The operator will stand as in [Fig. 45-a].

When one rafter has been laid out it is cut and used as a pattern by which to cut similar rafters.

Fig. 55. Determining Diagonal Thickness of Hip of Square Corner.

Fig. 56. Reduction of Common Rafter for Ridge Piece.

19. Ridge Piece.—Roofs may be framed with or without a ridge piece. The use of a ridge piece makes the assembly or raising of a roof somewhat easier, especially a hip roof. Upon an ordinary dwelling a ridge piece is usually a 1" × 6" board. Upon a gabled roof the length of ridge piece will be the same as that of the plate which it is to parallel, and will be laid off by placing the ridge board alongside the plate after the rafter positions have been marked upon the plate. These marks are transcribed upon the ridge board by means of the square and pencil.

On a hip roof, [Fig. 54], the length of a ridge piece will be equal to the length of the parallel plate diminished by the length of the plate at right angles to this. This, however, is the theoretic length of ridge as measured from center to center. Enough extra stock must be left on the ridge when framing it to allow full contact of hip cheeks. This additional measurement at each end of the ridge will be equal to ½ the diagonal thickness of the hip plus ½ the thickness of the ridge, [Fig. 54], making a total addition equal to the diagonal thickness of the hip plus the thickness of the ridge. [Fig. 55] illustrates the placing of the square to determine the diagonal thickness of a hip rafter which strikes the ridge at an angle of 45 degrees.

Fig. 57. Hip or Valley Rafter is Diagonal of Square Prism

In reckoning the length of a common rafter which is to rest against a ridge, the total length must be reduced by an amount equal to one-half the thickness of the ridge measured at right angles to the plumb cut, [Fig. 56.]

Fig. 58-a. Hip Rafter.

Fig. 58-b. Valley Rafter

20. Hip and Valley Rafters of Square Cornered Buildings.—First Method: The line of measurement for length of a hip and valley rafter is along the middle of the back or top edge, as on common and jack rafters. The manner of determining the number to use on the tongue of the square as a constant for the run, in terms of the 12" constant run of the common rafter, when the rise of the hip or valley rafter per foot of common run is taken on the blade; and the manner of constructing a table of unit lengths of hip and valley rafter, per foot of run of common rafter, are illustrated in Figs. [57], [58], [59] and [60]. From a study of these illustrations it will be seen that a hip or valley rafter of a square cornered building is in either case the diagonal of a square prism which has for its base dimensions the tangent and run of the roof, and for its height the rise of the roof, [Fig. 57.] On a square cornered building the run and tangent are always equal.

Fig. 59. Determining Unit Length of Hip or Valley Rafter.

The length of the diagonal of the base of such a prism, which is the run of the hip or valley rafter, is found by the formula c'² = a'² + b'², [Fig. 58.] When tangent and run are equal and each taken as 12", the run of the hip or valley equals 16.97", which for practical purposes of carpentry is considered as 17". In laying on the square, then, in framing a hip or valley rafter of a square cornered building, 17" will be taken upon the tongue, the rise of the roof per foot of run of common rafter or per 17" of run of hip or valley rafter, being taken on the blade.

The table of hip and valley lengths per foot of run of common rafter, [Fig. 60], will be formed by solving the right triangle c² = a′² + b′², [Fig. 59], for each of the pitches represented.

The positions to be assumed by the worker in framing a hip or valley rafter are similar to those to be assumed in framing the common rafter.

In measuring the length of a hip or valley rafter by the first method, the plumb cut may be laid off first. The upper end of the hip rafter will have to be framed with a side cut as shown in [Fig. 61.] The measurement for length will be made from a point along the middle of the top arris. Where the second method is employed, the end and seat cuts are laid off first.

21. Laying off Plumb Cut of Hip or Valley Rafter for Square Cornered Buildings.—Assuming a position with reference to the rafter similar to that in framing the common rafter, lay off the plumb cut using 17" on the tongue, and on the blade the rise per foot of run of the roof, or common rafter, which is also the rise of a hip or valley on that roof per 17" of hip or valley run. Scribe along the blade.

Fig. 60. Framing Table for Hip or Valley Rafters

22. Side or Cheek Cut of Hip or Valley Rafter.—First Method: There are a number of ways to lay out a side cut on a square cornered building. The simplest to remember, where no framing tables are at hand, consists in measuring square back from the plumb cut line a distance A-B, [Fig. 62], equal to the thickness of the rafter being framed. Thru this point lay off another line parallel to the plumb cut line and "carry" this across the top edge of the rafter with the square, as at D-E. Now adjust the bevel to pass thru E and F, [Fig. 62], and the setting is obtained for all side cuts of hip or valley rafters of that pitch of roof. Scribe this line on the top edge of the rafter. Carry it down the remaining side using the same numbers on the square as were used in laying off the plumb cut on the first side.

Second Method: This method of laying off side or cheek cut consists in laying the framing square across the top edge of the rafter, taking 17" on the tongue and the length of hip or valley rafter per foot of run of common rafter for the pitch required on the blade, and scribing along the blade.

Fig. 61. Side Cut

Fig. 62. Laying off Side Cut

23. Determining Length of Hip or Valley Rafter.—First Method: If a table of unit lengths of hip or valley per foot of run of common rafter is available, [Fig. 60], the total rafter length may be determined by multiplying the unit of hip or valley rafter length per foot of run of common rafter by the total run of common rafter. Do not make the mistake of trying to multiply by the run of the hip or valley rafter. Remember that these tables are all worked out with the 12" run of the common rafter as the base. This is true no matter whether the house is four sided, eight sided, or any other number of sides. The respective tables are based in every case upon 12" run of the common rafter.

Measurements for lengths of hip or valley are to be made along the top edge of the stock beginning at the line for side cut and midway between the point and heel, [Fig. 61.]

Second Method: This method of determining length of a hip or valley rafter is not unlike the second method described for the common rafter. Here, the numbers are 17 on the tongue, and the rise per foot of run of roof or of common rafter, on the blade. The end and seat cuts are scribed, after which the square is advanced step by step, using these same numbers, as many times as there are feet of run of common rafter. Should there be a fraction of a foot in the run of common rafter an additional and proportional advancement must be made. For example, to frame a hip for a square roof of ¼ pitch, having a run of common rafter of 12' 7". Advance the framing square 12 times, using 17" on the tongue and 6" on the blade. For the fractional advance take 1/12 of 17" or 9-11/12" (the framing square is laid off in twelfths on one side) on the tongue and 7/12 of 6" or 3½" on the blade, and scribe the limit. Fractional foot length of tail will be determined in a similar manner, the run or horizontal extension, or the lookout, of the common rafter determining the number of times the square must be advanced using 17" and 6" for the above given pitch.

Fig. 63. Miter Cut of Hip Rafter End

24. Laying off Seat and End Cut of Hip Rafter for Square Cornered Building.—The seat cut and end cut of a hip rafter will be laid off in a manner quite similar to that used in laying off the seat and end cuts of the common rafter as described in [Sec. 18]. There will be this difference, of course; the numbers to be used on the square will be 17" on the tongue instead of 12" as in the case of the common rafter. The rise per foot of run will be the same as for the common rafter. The run of the tail of the common rafter determines the length of lookout or the number of times the square will be advanced. The distance 1-3, [Fig. 52], must be the same on hip and valley as on common rafter of the same pitch of roof. The end cut of a hip rafter must be mitered to receive the fascia. The amount to be taken off for a square cornered building will be indicated by laying off lines a distance equal to one-half the thickness of the rafter, measured straight back from the lay-out of the end cut, [Fig. 63.] Since these cuts are identical with the side cut at the upper end of hip or valley, the square may be used as in laying off a side cut, cf. Section 22, second method.

Fig. 64. Backing the Hip Rafter

25. Reduction of Hip or Valley Rafter Length Because of Ridge Piece.—If a hip rafter of a square cornered building is to be framed against a ridge piece, [Fig. 40], its length must be reduced correspondingly. To make such allowance, measure square back from the line of plumb cut a distance equal to ½ the diagonal thickness of the ridge, [Fig. 61-A-B].

26. Backing a Hip Rafter for Square Cornered Building.—First Method: Since the line of measurement of a hip rafter is along the center of the top edge, if the rafter is framed with the same plumb distance as was given the common rafters, 1-3, [Fig. 52], it stands to reason that the roof boards will not fit the top edge of the hip properly until the arrises of the hip have been removed as in the cross-section of [Fig. 64.] The laying out and removal of these arrises is known as backing the hip.

The amount of backing for a hip rafter will depend upon the rafter thickness, the pitch of the roof, and the number of sides to the plate, and is indicated by gage lines on either side and one on the top edge of the rafter. To determine the location of these gage lines on the sides of the rafter, (1) place the square on the hip as in laying out the seat cut for the hip on which the backing is to be placed, the constant, 17", on the tongue and the rise on the blade, if the house is rectangular, [Fig. 64.] (2) Measure from the edge of the hip back along the tongue a distance equal to % the thickness of the rafter, and mark. This point gives the setting for the gage. (3) Gage both sides of the rafter and then remove the arrises as shown in the cross-section. Carpenters more frequently frame a hip without backing, allowing the roof boards to rest upon the arrises of the hip, forming a small triangular space between the roof boards and the top edge of the hip. In order to keep these arrises in the same planes as the tops of the common rafters, they must reduce the plumb height 1-3, [Fig. 52], of the hip. The amount of reduction, that is, the amount of drop the hip must make is equal to the plumb height of the backing, [Fig. 64.]

Fig. 65-a.	Fig. 65-b.
Framing Valley Rafter at Plate

Second Method: Take the rise in inches per foot of run of common rafter on the tongue, and the length of hip or valley per foot of run of common rafter on the blade; scribe along the tongue to get the angle of backing.

27. Valley Rafters.—As has been indicated in previous sections of the text, valley rafters have their lengths, plumb cuts, and seat cuts determined like hip rafters.

There is one difference; the valley rafter at its seat must be framed as in [Fig. 65] in order that the plumb line may come directly over the corner of the building. The ends of roof boards will rest upon the valley rafter at its center line, which line is in the same plane as that of the common rafters.

Like the hip rafter, the upper end may be laid out first, after which the rafter length is measured from this, the measurement being made along the middle of the back of the rafter, the top edge.

To lay out the cuts shown in [Fig. 65-a], proceed as in laying out the end of a hip rafter, as described in [Sec. 24], [Fig. 63.] In the case of an octagon the amount would be 5/12 of that used for the square, Fig. 65-b.

In Figs. [40] and [41] is shown a valley rafter framed thru to the ridge. This is done to give the valley support, for a valley, unlike a hip, is not self supporting when the jacks are attached. Against this valley rafter is framed a second valley rafter. The upper end of this second valley rafter is framed with a plumb cut such as would be given a hip or valley of the same rise and run; the end, however, is cut square across as in the case of a common rafter resting against a ridge.

28. Framing the Jack Rafter for Square Cornered Buildings; Plumb Cut; Side Cut.—Jack rafters which have their top ends framed against a hip are known as hip jacks; those having the lower ends framed against a valley are known as valley jacks; those which are framed in between hip and valley are known as cripple jacks.

The jack rafter, being but a portion of a common rafter, is framed in a manner quite like that used in framing the common rafter. The chief difference is in the fact that the jack rafter has a side or cheek cut, and that the lengths of jacks vary with their position along the plate. The order of procedure may be: (1) To lay off the plumb cut, just as for a common rafter having the same rise, that is, using 12" on the tongue, and the rise per foot of run on the blade; scribe along the blade. (2) Lay off the side cut or cheek cut. This is done just as in laying off the side cut of the hip rafter on a square cornered building, first method only, [Fig. 62.] Where a table of common rafter lengths per foot of run is available, [Fig. 49], a second method of laying out the side cut of a jack rafter consists in taking 12" on the tongue of the framing square, and the common rafter length per foot of run for the pitch given, on the blade; laying the square across the edge of the rafter and scribing along the blade. (3) Lay off the length of the jack as determined in the next section. (4) Lay off the seat cut just as in laying off the seat cut of the common rafter for the same pitch of roof, Section 18. Equally common is the practice of beginning with the end and seat cuts.

The framing square of [Fig. 50] contains data which makes possible the laying out of side cuts for the square cornered building by means of numbers taken upon tongue and blade.

While the ratios of the numbers used upon the tongue and the blade are always the same for any given pitch, different makers of squares use different numbers for side cuts. The student will have to have special directions for each different make of square. These may be gotten from the manufacturers.

29. Lengths of Jack Rafters for Square Cornered Roofs.—First Method: The framing table for common rafters and jack rafters, [Fig. 49], may be made use of in determining lengths of jacks. To make use of this table we shall need to know the run of each separate jack. An examination of [Fig. 66] shows that in a rectangular house the run of a jack is the same as the length of plate or of ridge which forms the angle. This is true of hip jack, valley jack, or cripple jack. However, such measurements are along the centers of the top edges of the rafters and allowance must be made in the length of the jacks for the thickness of hip or valley rafter. In the case of the cripple jack this amount of reduction will be equal to ½ the diagonal thickness of the hip plus ½ the diagonal thickness of the valley, measured at right angles to the plumb cut, [Fig. 61], or measured in the plane of the plate, or a parallel plane.

Fig. 66. Lengths of Jack Rafters.

Top and bottom ends of cripples are alike, but in nailing them in place the lower ends must be held up so that their center lines will strike the center of the valley rafter. Their tops will be kept even with the outer arrises of the hip whether the hip is backed or not.

In determining the true length of hip jack and valley jack we should know that a reduction of ½ the diagonal thickness of hip or valley, measured straight back from the plumb cut, is to be made. In the case of a valley jack resting against a ridge piece, an additional reduction must be made as described in Section 19, [Fig. 56.] In actual practice carpenters usually measure the length of hip or valley jack from the long point, along the arris, instead of along the center of the top edge, no reduction being made for ½ the diagonal thickness of hip or valley. Cripple jacks are measured from long point to long point, no reduction being made for thickness of hip and valley.

Fig. 67. Determining Length of Jack Rafters

Second Method: Where jacks are framed so that equal spacings may be laid off, beginning with a full length common rafter, as in [Fig. 67], the simplest method of determining lengths of jacks is to first count the number of spaces between jacks, which must be laid off on ridge or on plate, and divide the length of common rafter by this number. The result will be the common difference between lengths of jacks. The longest jack will be framed first by reducing the length of common rafter by the common difference. The next, by reducing the jack just framed by the common difference, etc. This method is applicable to roofs of any number of sides.

Third Method: If we begin to frame with the shortest jack instead of the longest, we first determine the length of the shortest jack, remembering that its run in the square cornered building will be the same as its spacing from the corner along the plate, or along the ridge in case of a valley jack. In a similar manner the second jack can be framed. The difference in the lengths of these two is the common difference. To the length of this second jack, and to each succeeding jack add the common difference, to get the length of the next.

Fourth Method: As rafters are usually spaced either 16" or 24" apart, a table consisting of the common differences in lengths for the various pitches will be found convenient, [Fig. 49.] The steel square of [Fig. 50] also shows such a table for the square roof.

CHAPTER IV
Roof Frame: any Polygon

Fig. 68. Tangents

30. Tangents; Miter Cuts of the Plate.—Before the principles involved in the laying out of rafters on any type of roof can be understood, a clearer idea of the term tangent as used in roof framing must be had. A tangent of an angle of a right triangle is the ratio or fractional value obtained by dividing the value of the side opposite that angle by the value of the adjacent side. The tangent at the plate, to which reference was made is the tangent of the angle having for its adjacent sides the run of the common rafter and the run of the hip or valley. By making use of a circle with a radius of 12" we may represent the value of this tangent graphically in terms of the constant of common rafter run, [Fig. 68.] By constructing these figures very carefully and measuring the line marked tangent, we may obtain the value of the tangent for the polygon measured in inches to the foot of run of the common rafter. Such measurements, if made to the 1/100 of an inch will serve all practical purposes. A safer way, however, is to make use of values secured thru the trigonometric solutions described in Appendix I, using the graphic solutions as checks. The values of tangents at intervals of one degree are given in the Table of Natural Functions, Appendix II. By interpolation, fractional degree values may be found.

Example:
Find the value of the tangent for an octagonal plate.
Solution:
Angle A′ of [Fig. 68] = 22½°
(1/16 of the sum of all the angles about a point)
Tan 22½° = .4143
Tables are builded with 1 as a base. In roof framing 1" or 12" is
taken as the constant or base, or unit of run of common rafter.
.4143 may be considered as feet, which equals 4.97".
In a similar manner tangents may be found for plates of buildings
of any number of sides.

In [Fig. 69] is illustrated a handy device one side of which, by the twirling of one disk within the other, can be made to give tangent values, in terms of a 12" base, for any number of degrees. The reverse side of this "key" gives data to be used in the framing of square cornered and octagonal roofs. Such a key will be found a convenient way in which to carry needed data and should be easily understood and intelligently used, once the principles discussed in this chapter are mastered. An explanation of the author's key, [Fig. 70], will be found in Appendix IV.

Now as to some of the uses for tangent values: First, by taking 12" on the tongue and the tangent value in inches per foot of common rafter run upon the blade of the square, we are able to get the lay-out for the miter joint of the plate.

[Fig. 71-b] illustrates the square placed for the lay-out of the octagonal plate or sill miter. Five inches is taken as tangent since the real value 4.97" is equivalent to 5" for all practical purposes.

For the square cornered building 12" and 12" would be used in making the plate miter lay-out, since the tangent of 45 is 1 according to the Table, Appendix II. Any other like numbers would give a tangent value of 1, of course, but it is best to consider 12" on the tongue, in which case 12" must be taken on the blade.

Second, this tangent value is needed in determining the cheek or side cut of hip, valley and jack rafters, as will be shown in [Sec. 35].

Third, this tangent value is needed in determining the amount of backing to be given hip rafters. This is discussed in [Sec. 39].

Not infrequently the plate miter in degrees is required. This is determined for any regular polygon by the proposition: The plate or miter angle of any regular polygon = 90 - (central angle/2)

Example:
Find the value of the plate miter of the octagon.
Solution:
The octagon has 8 sides; therefore central angle = 45°
45° ÷ 2 = 27½°
90° - 22½° = 67½°

Fig. 70-a.	Fig. 70-b.
Griffith's Roof Framing Tables