CARPENTRY
BY
IRA SAMUEL GRIFFITH
_Chairman of the Manual Arts Department University of Missouri_
[Illustration]
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
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
Chapter I. Foundations 13
1. Laying out; 2. Grade line; 3. Excavation; 4. Foundations;
footings; 5. Foundation materials; 6. Forms for concrete walls;
7. Waterproofing; 8. Basement frames.
Chapter Ii. Main Frame 27
9. Methods of framing superstructure; 10. Sills and girders;
11. Bridging; 12. Trimmers and headers; 13. Walls and
partitions; joists and rough floors; 14. Openings in framework.
Chapter Iii. Roof Frame: Square Cornered Building 45
15. Roof framing; 16. Framing the common rafter; laying out
the plumb cut; 17. Finding the length of a common rafter;
18. Laying off a common rafter seat cut and end cut. 19. Ridge
piece; 20. Hip and valley rafter; 21. Framing hip and valley
rafters; 22. Side or cheek cut of hip or valley rafter;
23. Determining length of hip or valley rafter; 24. Laying off
and end cut of hip rafter; 25. Reduction of hip or valley length
seat cut because of ridge piece; 26. Backing a hip rafter;
27. Valley rafters; 28. Framing jack rafters; plumb cut; side
cut; 29. Lengths of jacks.
Chapter Iv. Roof Frame: Any Polygon 69
30. Tangents; miter cuts of plate; 31. Octagonal roofs; 32.
Common rafter for plate of any number of sides; 33. Hip and
valley rafters for octagon and other polygons; 34. Plumb cut
of octagonal and other polygonal hips and valleys; 35. Side
or cheek cuts of hip or valley rafters, any polygon; 36. Rafter
lengths of octagonal and other polygonal hips and valleys;
37. Reductions in lengths for king-post; 38. Seat cut and end
cut of octagonal and other polygonal hips and valleys; 39.
Backing octagonal and other hips; 40. Framing octagonal and
other polygonal jacks; 41. Side cut of octagonal and other
polygonal jacks; 42. Lengths of octagonal and other polygonal
jacks; 43. Framing by means of a protractor; 44. Translating
framing problems from protractor to framing square and vice
versa; 45. Framing an octagon bay; 46. Framing a roof of one
pitch to another of different pitch; 47. Framing roof of
uneven pitch; 48. Decks; chimney openings.
Chapter V. Exterior Covering and Finish 95
49. Sheathing; 50. Scaffolding; 51. Cornice; 52. Raked
mouldings; 53. Shingling; 54. Shingling hips and valleys;
55. Finishing exterior walls; 56. Setting window and door
frames; 57. Siding.
Chapter Vi. Interior Finish 115
58. Lathing; grounds; 59. Interior walls; 60. Stair building;
porch steps; 61. Risers and treads; 62. Porches; 63. Interior
finish; 64. Setting door jambs; 65. Fitting window sash;
66. Placing door, window, and other trim; 67. Hanging doors;
68. Fitting a door; 69. Hinging a door; 70. Fitting locks;
71. Laying and scraping floors; 72. Door and window frames;
73. Woodwork in masonry structures.
Chapter Vii. Estimating 142
74. Methods of estimating; 75. Table for estimating by
cubic-foot unit; 76. Grading rules; 77. Estimating lumber
quantities; 78. Estimating millwork quantities; 79. Example
of form for bill of materials; 80. Estimating labor costs;
81. Estimating quantities of nails; 82. Example of form for
carpentry costs; 83. Total building costs by percentages.
Appendix 158
I. Natural trigonometric functions; formulæ deduced. Solution
of right triangles, (brief)
II. Table of natural functions (for degrees only). Interpolation.
III. Useful tables.
Fractional equivalents for decimal values.
Wood and machine screw sizes.
Length and number of nails.
Wire brads.
Board measure table.
Strength of materials.
Stresses for structural timbers.
Contents of brick walls.
IV. Short cuts to roof framing.
Directions for Griffith's Framing Tables.
V. Estimating.
Excavations.
Masonry.
Slate.
Plaster.
Painting.
Bibliography of References
Index
[Illustration: The house used as a model for many of the illustrations
in this book.]
CARPENTRY
CHAPTER I
Foundations
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 8. American Bond]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 15. Plumbing and Leveling Cellar Frame]
CHAPTER II
Main Frame
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 18. Half-Frame House]
[Illustration: Fig. 19. Balloon Frame House]
[Illustration: Fig. 20. Plank Frame Barn.]
[Illustration: 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.
[Illustration: 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.
=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. Fig. 23. Cutting Bridging 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.
[Illustration: Fig. 24. Laying off a Stay]
[Illustration: 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.
[Illustration: Fig. 26. Placing Headers and Trimmers]
[Illustration: 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
[Illustration: Fig. 28-a. Fig. 28-b.
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.
[Illustration: Fig. 29. Headers and Trimmers in Wall Frame]
[Illustration: 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.
[Illustration: Fig. 31. Marking Joists from Pattern]
[Illustration: Fig. 32-a.]
[Illustration: Fig. 32-b. Corner Post Types]
[Illustration: Fig. 32-c.]
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.
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.
[Illustration: Fig. 33. Corner Post Being Plumbed and Stayed]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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."
[Illustration: 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.
[Illustration: Fig. 40. Roof Details]
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 43. Run, Rise and Length]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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_.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
_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.
[Illustration: Fig. 51. Finding Rafter Length by Scaling]
[Illustration: 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.
[Illustration: Fig. 53. Independent Rafter Tail]
[Illustration: 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.
[Illustration: Fig. 55. Determining Diagonal Thickness of Hip of Square
Corner.]
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 58-a. Hip Rafter.]
[Illustration: 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.
[Illustration: 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 _c″²_ =
_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.
[Illustration: 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.
[Illustration: Fig. 61. Side Cut]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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
[Illustration: 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.
[Illustration: Fig. 69-a. Table of Tangents]
[Illustration: Fig. 69-b. Rafter Table.]
_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°
[Illustration: Fig. 70-a. Fig. 70-b. Griffith's Roof Framing Tables]
Fourth, the tangent value is needed in finding the length of a side of
a polygon, the span or run of the polygon being known, and vice versa.
Length of side = span x tangent of plate, using 12" as base.
_Example:_
An octagonal silo has a span of 18'; determine the length of plate
for any side.
_Solution:_
The tangent value of the octagon = 4.97" (to each 12" of run) 18 × 4.97"
= 89.46" = 7' 5.46" = 7' 5½".
[Illustration: Fig. 71-a. Fig. 71-b Laying out Miters]
_Example:_
A side of a hexagon measures 4'; determine the run of the hexagon.
_Solution:_
Transposing the rule above: Span = length of side divided by tangent
of plate.
Tangent of hexagon = 6.92" when base = 12".
4' divided by 6.92" = 6' 6.48" = span. Run = 3' 3.24".
=31. Octagonal Roofs.=--While the square cornered building is
the most common, the octagon is frequently used in the form of a bay
attached to the side of a house. The octagon is also common upon silos
and towers. The manner of finding the run, tangent, length of hip and
valley rafter, miter cut of plate or sill, the manner of determining
the numbers to use on the square to lay out the plumb and seat cuts,
etc., will be found developed herein for both square and octagonal
roof. Having mastered the principles involved in these two forms, the
student should be able to work out framing problems for roofs of any
number of sides.
=32. Common Rafter for Plate of any Number of Sides.=--By
referring to Figs. 68 and 72 it will be seen that common rafters
have for their runs the apothem of the polygon made by the plate,
represented in Fig. 68 by the lines _b, b’, b″_. The run of the
hip is represented by the line _c, c′, c″_. The rise will be found
the same for full length common rafters and hips. Plumb and seat cuts
and lengths per foot of run of common rafters and jacks are determined
for a building of any number of sides just as for the square cornered
building. The degree of inclination of common and jack rafter is
applicable, too, Fig. 49.
=33. Hip and Valley Rafters for Octagonal and other
Polygons.=--Before the table of constants for hips and valleys for
octagonal and other polygonal roofs can be formed, it is necessary to
determine the tangent values of these polygons, as described in Sec. 30.
Proceeding with the octagon, whose tangent was found to be 4.97" when
the run of the common rafter was taken as 12", by the formula _c²_
= _a²_ + _b²_, Fig. 72, the run of the octagon hip or valley
will be found to be 12.99" for each foot of run of the common rafter.
[Illustration: Fig. 72. Run of Common Rafter of any Roof is Apothem of
Polygon]
=34. Plumb Cut of Octagonal and other Polygonal Hips and
Valleys.=--The run of an octagonal hip or valley is 12.99" for each
foot of run of common rafter. For practical purposes this is considered
as 13". To lay off a plumb cut for an octagonal hip or valley, take
13" on the tongue and the rise per foot of run of common rafter on the
blade, Fig. 73; scribe on the blade.
[Illustration: Fig. 73. Table for Octagon Hip or Valley]
In a similar manner, having determined the tangent of any polygon under
consideration, the run of hip or valley per foot of run of common
rafter may be figured. The result will give the number to take on the
tongue when the rise per foot of run of common rafter is taken on the
blade, for laying out plumb cuts.
=35. Side or Cheek Cuts of Hip and Valley Rafters for Roofs of Any
Number of Sides.=--Fig. 74 illustrates the principles involved and
method used in determining side cuts whatsoever the pitch and number of
sides involved. (1) Lay the square across the jack or hip or valley,
whatever is to be framed, at any convenient place, using on the blade
the tangent (with 12" as base) of the polygon being framed, and on the
tongue the constant of the run of the common rafter, 12". Scribe along
the tongue _A-C_, Fig. 74. (2) With the square, carry the line
_B-C_ across the edge as indicated. (3) Lay off the plumb cut for
the required pitch, taking upon the blade the rise, and upon the tongue
the constant of the run of the hip or valley or jack, according to the
requirements for that particular member as determined by the style of
roof. (4) Measure square back from this plumb cut line the distance
_A-B_ of Fig. 74. (5) Thru point A scribe a line _D-A_
parallel to that of the plumb cut line and (6) square this across the
edge as at _D-E_. (7) Adjust the bevel to pass thru _E_ and
_F_, Fig. 74. (8) Scribe a plumb cut line upon the reverse surface
of the stock.
[Illustration: Fig. 74-a. Fig. 74-b.
Laying off Side Cut of Jack, Hip or Valley, any Polygon]
It will be observed that in one case the square is laid across the
edge with 12" on the tongue and 12" on the blade, Fig. 74-a. This, as
might be supposed, is for finding the cheek or side cut for jack, hip,
or valley where the junction angle is 45. In the case of the octagonal
hip, valley, or jack 5" must be taken upon the blade, since that is
the tangent value of 22½°, 12" as base being taken on the tongue, Fig.
74-b. This tangent value will vary, then, according to the change in
the junction angle.
[Illustration: Fig. 75-a. Fig. 75-b.
Securing Value of A-B of Fig. 74, Various Angles]
The reason for using the tangent and run for this work is indicated
by the position of the square on the plan of the roof, Fig. 75. These
figures are for use only when the timbers lie in the plane of the
plate, or any parallel plane. When rafters take on pitch or rise,
however, the upward projection of the plan of the miter cut, Fig. 74,
will determine the side cut as just described.
=36. Rafter Lengths of Octagonal and other Polygonal Hips and
Valleys.=--_First Method:_ Knowing the run of a hip or valley
for the polygon under consideration (17" for the square, 13" for the
octagon, etc.), by assuming the respective rises for the various
pitches and solving _c′²_ = _a′²_ + _b²'_, Fig. 76, data
pertaining to hip or valley unit rafter lengths, such as that for the
octagon in Fig. 73, is obtained.
To determine a rafter length, having available such a table, multiply
the hip or valley length per foot of run of common rafter as given in
the table by the run of the common rafter of that roof. Reduce to feet.
Such lengths will be laid off by measurement from the side or cheek
cut, which will have been laid off, down the top edge of the rafter.
Lengths of hip or valley tails will be determined in a similar manner
from the same table.
[Illustration: Fig. 76. King-Post]
_Second Method:_ This consists in successive placings of the
square, using the same numbers on tongue and blade as will be used
in laying out the plumb cut for this particular roof. The successive
advances will be determined, as in the hip or valley of a square
cornered building, by the number of feet in the run of the common
rafter of this roof. A fractional part of a foot in run will be treated
in a manner similar to that described for the square cornered building.
Suitable allowance must be made for the fact that the length of rafter
is along the middle of the top edge of the rafter, when this latter
method is used.
_Example:_
An octagonal roof of ¼ pitch has a span of 25′.
The run of common rafter = 12' 6".
Taking 13" on the tongue and 6" on the blade lay off successively
12 measurements. Take 6/12 or ½ of 13" or 6½" on the tongue and
6/12 or ½ of 6" or 3" on the blade for the fractional foot of run.
=37. Reductions in Lengths for King-Post.=--Suitable reduction
must be made for king-post, should there be one, Fig. 76, or for rafter
thicknesses should no king-post be used, Fig. 75-b. Where a king-post
is used the reduction will be ½ the width of the square out of which
the king-post is formed, the measurement being made square back from
the line of the plumb cut, as in reducing common rafter lengths for
ridge piece, Fig. 56.
Where an apex is formed as in Fig. 75-b, one pair of hips is framed
each with a run equal to ½ the octagon's diagonal, with cuts at the top
the same as those of common rafters. The second pair will be similarly
framed but the lengths will each be reduced an amount equal to ½ the
thickness of the first pair, measuring straight back from the plumb
cut. The third and fourth pairs will be reduced an amount equal to ½
the diagonal thickness of the rafters already framed, measured straight
back from the plumb cut, Figs. 74-a and 75-b. These rafters will have
to have double side cuts as indicated in Fig. 75-b. It will be noted
that these latter rafters meet the others at an angle of 45 degrees,
making the framing of the side cut similar to that of a square cornered
building except that these are given double cheeks.
=38. Seat and End Cut of Octagonal and other Polygonal Hips and
Valleys.=--The method of procedure in laying out the seat and end
cuts of octagonal and other polygonal hips and valleys is similar to
that described for the square hip, except of course, the numbers to use
on the square will differ. These numbers will be determined by the run
of the hip for that particular roof, and by the rise in inches per foot
of run of common rafter. In the case of the octagon the numbers will be
13" on the tongue, and the rise on the blade. The length of tail may
be figured by the tables, if such are available, or the framing square
may be advanced successively as many times as there are feet in the
run of the tail or the lookout of the common rafter of that roof, as
heretofore described.
The miter cut on the end of a hip rafter and in the crotch of a valley
rafter will vary with the tangent value of the plate. For the octagon,
whose tangent is 5" the measurement from the end cut and at right
angles to it will be 5/12 that used on the square cornered building.
This same ratio will hold for the measurement at the crotch of the
valley, Fig. 65-b.
=39. Backing Octagonal and other Hips.=--The principle involved in
determining the amount of backing on hip rafters for the octagon, as
well as that of other polygons, is similar to that for backing hips for
the square cornered building.
[Illustration: Fig. 77. Laying out Backing for Octagon Hip]
(1) Place the framing square upon the hip as for making the seat cut of
the octagon hip rafter, Fig. 77.
(2) Referring to Fig. 78 it will be seen that the more sides a roof
possesses the less will be the backing required. Fig. 78 represents the
hips on square, octagon, and hexagon as they would appear upon the plan
if made to stand straight up at the corners of the plates. The backing
in each case is determined by the tangent of the angle whose adjacent
side is ½ the rafter thickness, and whose angle is equivalent to ½ the
central angle as _A, A′, A″_, Fig. 78. The fact that the hips are
inclined and not vertical in their final position in a roof makes no
difference in the principle of determining backing because of the fact
that our measurements are made in a horizontal plane, or in the plane
of the plate, Figs. 64 and 77.
The ratio of the backing of the octagon or any other polygonal hip to
that of the square of equal rise will be proportional to their tangent
values. For illustration, on the square roof we measured along the
tongue of the square, from the top edge of the hip, Fig. 64, ½ the
thickness of the rafter, for reasons made plain in Fig. 78. In laying
off the octagon backing we should lay off along the tongue in the line
of the octagon seat cut, 5/12 of ½ the thickness of the rafter. In the
case of a 2" rafter, 5/12" will be set off, using that side of the
square containing graduations in 12ths.
(3) Where backing is not desired the rafter will be dropped a plumb
distance equal to the plumb distance of the backing of the rafter, Fig.
77. Consult Fig. 73.
[Illustration: Fig. 78. Backing for Hip of any Polygon]
=40. Framing the Octagonal and other Polygonal Jacks.=--Plumb
cuts of jacks are determined by the rise and run of the common rafter
of that particular roof. The seat cuts will be determined by the same
numbers. Side cuts, end cuts, and lengths of jacks are determined as
described in Sections 41 and 42.
=41. Side Cut of Octagonal and other Polygonal Jacks.=--_First
Method:_ The method described in Section 35 is applicable to jacks
as well.
_Second Method:_ If a table of common rafter lengths per foot of
run is available, the side cut of any polygonal jack may be laid out
by placing the framing square across the top of the rafter, taking the
tangent value of the polygon on the tongue, and the length of common
rafter per foot of run on the blade, and scribing on the blade. The
tangent of the polygon forms the opposite side of a triangle in which
the adjacent side is the length of common or jack rafter per foot of
run. The angle formed being the angle of side cut of jack.
=42. Lengths of Octagonal and other Polygonal Jacks.=--The methods
of procedure in determining the lengths of jacks for other than square
cornered buildings differs from that described for the square cornered
building only in the fact that the runs, hence lengths, of the jacks
must be determined differently. An examination of Figs. 68 and 72 will
show that the runs, hence the lengths, of jacks for the square, the
octagon, etc., when the run of common rafter is the same, will vary
inversely as the tangent ratio. For example, the tangent of the square
is 12" when its run of common rafter is 12", while that of the octagon
is 4.97" or practically 5". The runs, and, therefore, the lengths of
octagon jacks will differ one with reference to another, 12/5 of those
of the jacks for the square building. If an octagon jack rests 24" from
the corner, its run, and therefore its length, will be 12/5 times that
of a jack on a square cornered building similarly placed.
_Second Method:_ Difference in lengths of jacks may be determined
by counting the spacings along the plate and dividing the length of
common rafter by that number.
=43. Framing by Means of a Protractor.=--By means of a protractor
used in connection with the columns containing degree measurements,
Figs. 49, 60, and 73, roof framing may be greatly simplified.
To lay off a seat cut, it is merely necessary to look in the table of
hip, valley, jack, or common rafter, whatever is being framed, and read
the degree of inclination of rafter for the pitch required. The blade
of a T-bevel is set by means of the protractor to this angle; or a
combination tool may be used, and the tool applied as in Fig. 79.
The plumb cut and seat cut are complementary. Since a protractor
is made to read in either of two directions, the plumb cut setting
may be got by adding to or subtracting from 90 degrees the angle of
inclination of that rafter. With a combination tool, one setting of the
tool serves to lay out both seat and plumb cut.
[Illustration: Fig. 79. Laying off Seat Cut with Fig. 80. Laying off
Plumb Cut with Framing Tool Framing Tool]
The degree of setting for a side or cheek cut, the gage setting for
backing, and the amount of drop where no backing is used will be found
under appropriate columns in the tables referred to above. The manner
of applying the combination tool for laying out plumb cut, seat cut,
side cut and end cut is indicated in Figs. 79, 80, 81 and 82. The
manner of determining the data found in these tables is a matter of
trigonometric solution, no more difficult than that already given, and
omitted for lack of space. Such problems may well form a part of the
pupil's work in mathematics.
For Example: Data for side cuts of jack, hip and valley for square and
octagonal roofs will be found in Figs. 49, 60, and 73. In these tables
will be found a column marked "Degree of side cut of hip or valley,"
also of jack. To secure the angle of side cut it is only necessary to
solve by simple trigonometric formulæ, the triangle _a b c_ and
then _a′ b′ c′_ of Fig. 62. The angle _A_ is the angle of
inclination of hip or jack or valley, and will be found for each inch
of rise in the tables; the manner of determining the same having been
described. The side _b_ is _A-B_ of Fig. 74, the value of
which is also easily determined once the principles of Fig. 74 are
mastered. With this data the student may find the angle _A'_ of
Fig. 62 as given in the tables.
Rafter lengths are determined as previously described in connection
with framing with the steel square.
[Illustration: Fig. 81. Side Cut]
[Illustration: Fig. 82. End Cut]
=44. Translating Framing Problems from Protractor to Framing Square
and Vice Versa.=--Frequently it is desirable to translate framing
problems from degrees to numbers to be used upon the steel square, and
vice versa. To change from degree framing to steel square framing it is
only necessary to remember that the numbers to use on the square must
be numbers such that their ratio one to the other shall give a tangent
value equal to that given in the Table of Natural Functions, Appendix
II, for the angle under consideration.
_Example:_
Given: Angle of inclination of common rafter or of roof = 30 degrees.
Find the numbers to take on the square to frame seat and plumb cuts.
_Solution:_
Tan 30° = .577 (by Table, Appendix II).
By agreement, run of common rafter takes 12" on one member of the
square for constant of run.
Therefore, rise must be 6.93", or 611/12", must be taken on the other
member of the square. (The base of the table is 1 so that for
12" run we must have 12" x .5774.)
In a similar manner the number to be taken on the blade, when the
inclination of any other common or jack rafter is given, may be
determined. In the case of the hip or valley inclination, however,
it must be remembered that 17" is to be taken on the tongue for the
run in the square cornered house, with 13" in the octagon, which will
necessitate multiplying the tangent value of the angle of inclination
by 17 and 13 respectively to find the number to take on the blade when
one or the other of these is taken on the tongue.
_Example:_
Given: Side cut of jack rafter, or hip or valley = 38 degrees.
Find the numbers to use on the framing square to lay off this
same angle.
_Solution:_
Tan 38° = .7813
By agreement, we shall take 12" on the tongue.
The number to be taken on the blade must be, therefore,
12" x .7813 = 9⅜
(In the case of the side cut, any number other than 12" might have
been assumed on the tongue.)
_Example:_
Given: A hip rafter on a square roof which is framed with 17" on the
tongue and 8" on the blade.
To find the angle of inclination of the hip.
_Solution:_
8/17 = .4706
The angle whose tan = .4706 is 25° 19′
=45. Framing an Octagon Bay.=--Whatever may be one's opinion as to
the propriety of the octagon bay architecturally, its very common use
makes it obligatory upon the builder to know how to properly frame it.
Referring to Figs. 72 and 83-a it will be seen that the octagon bay is
but a portion of a full octagon set up against the side of a house.
From this it follows that the framing of the plate, sills, the laying
out of plumb and seat cuts, and side or cheek cuts of jacks that rest
against hips, etc., will be according to principles discussed in
previous sections of the text. The chief thing which needs explanation
is the matter of laying out side cuts for hips and jacks which are to
rest against the side of the building.
[Illustration: Fig. 83-a. Fig. 83-b.
Framing Rafters of Octagon Bay]
Consider the bay as cut along the line _I-J_, Fig. 83-a. To frame
the hip nearest the building: (1) Determine the angle which would
give the side or cheek cut of the rafter involved when the rafter has
no pitch, that is, when it lies in the plane of the plate. Fig. 83-a
indicates the manner of determining this angle, which will be found to
be 22½°. If we wish to use a framing square instead of a framing tool
we may readily translate the 22½°. We shall find that 12" taken on the
tongue of the square will require 4.97" or 5" on the blade. (2) Place
the square as in Fig. 83-b and scribe along the tongue. Remember that
this gives the miter when the rafter lies in the plane of the plate.
This would give the cut to be used where ceiling joist or floor joist
of octagon bay are run parallel with the rafters. On small bays, joists
are seldom run thus but are run parallel with other joists of the
house. No special directions are needed for setting the protractor, or
the combination tool, the manner of setting and placing is so obvious.
(3) Once having this miter when the rafter lies in the plane of the
plate, proceed as directed in Sec. 35, (2) _et seq._ With one such
rafter framed the mates may be laid out by transposition.
To frame the remaining hips of Fig. 83-a proceed in a manner similar
to that just described. In this case the angle made by the hip when it
lies in the plane of the plate and the side of the house will be 67½°
degrees. Why? Fig. 84-a. The framing tool or protractor and T-bevel
provide the simplest means of framing. If one wishes to use the framing
square he will find that by using the tangent value of 67½° he gets
28.97" to take on the blade when 12" is taken on the tongue. This he
cannot find, of course. He may either take other smaller numbers having
the same ratio, or he might better take the cotangent value of 67½°.
Cot of an angle of a right triangle is the ratio of the adjacent side
to the opposite side.
_Solution:_
Angle A′ = 67½°, Fig. 84-a.
Cot 67½° = .414 when _b′_ = 1.
When _b′_ = 12" cot _A′_ = 4.97" or 5".
[Illustration: Fig. 84-a. Fig. 84-b.
Framing Octagon Hip Intercepted at 67½°]
(1) Place the square as in Fig. 84-b and scribe along the blade. By
this time the student should have observed that the tangent value of
any angle increases as the cotangent value of the angle decreases and
vice versa. Note that when a cotangent value is used, 12" is still
taken on the tongue and the same number as for the tangent value of
that number of degrees is taken on the blade, but that the scribing
is done along the blade and not along the tongue as is the case when
tangent values are used. (2) Proceed from this point as in Sec. 35, (2)
_et seq._
To frame the jack intercepted by the side of the house. Fig. 85-a; (1)
Proceed to find the miter for the intercepted jack when it lies in the
plane of the plate. It is 45 degrees. Tan 45 degrees = 1 when _b″_
= 1. Tan 45 degrees = 12" when _b″_ = 12". (2) Place the square as
in Fig. 85-b and scribe along the tongue. (3) From this on, continue as
in Sec. 35, (2) _et seq._
[Illustration: Fig. 85-a. Fig. 85-b.
Framing an Intercepted Jack]
From these solutions the student should be able to generalize
sufficiently to care for rafters of any angle of intersection with
the side of a building, and of any pitch. The use of the framing tool
or protractor and T-square is strongly recommended upon such work as
this. Nothing but tradition prevents its more general use in carpentry.
Having determined the lay-out for the side cut of these rafters, the
length of each must next be determined. To determine the length of
hip or jack intercepted by the side of the house and the plate: (1)
Determine the run of the intercepted part of the common or jack rafter,
as _b_, Fig. 83-a. (2) By means of the table of rafter lengths per
foot of run compute the length of rafter under consideration.
In determining the run of an intercepted part it is necessary to have
data concerning the size of the octagon and the amount cut off by
the building. The lines _E-D_ and _F-D_, Fig. 83-a, when
dimensioned, give this data. The actual lengths of the various hips,
jacks, etc., in practical carpentry, upon small bays, are usually
determined by actual measurement from the plate or sill to the proper
point of intersection on the building as indicated by a stick or
extension rule held at the proper angle. Sometimes a large scale
drawing is made and the runs taken from this. Upon large work where
accuracy is necessary and measurements impossible, trigonometric
solutions should be used.
To determine the length of the intercepted hip rafter over _c_ of
triangle _a b c_ Fig. 83-a, the value of _b_ must first be
secured.
This value represents common rafter run for hip rafter rising over
side _c_, and run of intercepted common rafter rising over side
_b_.
Length of intercepted hip rising over side _c_ = _b x_ unit
length of hip. (Table of lengths of octagon hips in terms of run of
common rafter, Fig. 73.)
Length of intercepted common rafter or jack nearest house = _b x_
unit length of common rafter. (Table of lengths of common rafter in
terms of run of common rafter, Fig. 49.)
It must be remembered that these are theoretic lengths measured down
the middle of the top edge of the rafters. In framing, suitable
reductions must be made. For example, in framing the intercepted common
rafter nearest the building, the rafter must be set over one-half its
thickness that it may be nailed against the side of the building. The
allowance necessary in order to do this may be secured by measuring
the length from the long point instead of the middle of the side or
cheek cut. Suitable reduction, too, must be made for the hip thickness
in this case one-half the diagonal across the top edge of the hip,
laid off at an angle of 22½°, or 5" and 12" on the square. This amount
will be laid off straight back from the plumb cut, Sec. 35, (2) _et
seq._
Length of hip must be measured along the middle of the top of the
rafter.
=46. Framing a Roof of One Pitch to Another of Different
Pitch.=--Occasionally one must frame a roof of one pitch against a
roof of another pitch. To determine the cut, if the combination tool
is to be used, merely find the difference in degrees of the angles of
inclination of the two rafters, and apply the tool as indicated in the
tables for making any other plumb cut, Figs. 49, 60 and 73. The seat
cut will be determined by the rise and run of the shed rafter in the
usual manner.
[Illustration: Fig. 86. Laying off Cut of Shed Rafter]
Where the framing square is to be used, lay the square as in framing a
plumb cut on the shed rafter, as at _A_, Fig. 86, taking on the
tongue, the run, and on the blade the rise of the shed rafter. Next lay
a second square as at _B_, Fig. 86, taking on the blade the rise
and on the tongue the run of the rafter of the main roof, using the
blade of _A_ as a reference edge. Blade of _B_ gives the cut.
=47. Framing a Roof of Uneven Pitch.=--Not infrequently a roof
must be framed in which several pitches are involved. All of the
principles necessary for framing such a roof have been developed. It
remains for the student to make the applications to uneven pitches. It
is advisable to prepare a framing plan as shown in Fig. 87. From such
a plan it may be seen that the seat and plumb cuts of common and jack
rafters are determined in the usual manner, being different upon the
different pitches, of course, but determined as for any given pitch.
Lengths of common rafters will be determined for any pitch by the
tables already made use of, the run being known or determined.
In selecting the numbers to use on the tongue and the blade of the
square, in laying out seat and plumb cuts of hip or valley rafters
of intersecting roofs of different pitches, any numbers may be used
providing they have a ratio equal to that of the run and rise of the
rafter being framed. Since the angle of intersection changes with every
change of pitch, it is hardly worth while developing a constant to
be used on the tongue in framing hip and valley rafters on irregular
pitches.
Side or cheek cuts for valley, hip or jacks are determined according to
principles developed in Sec. 35.
[Illustration: Fig. 87. Plan of Uneven Pitches]
The backing of hips on roofs of uneven pitches, while the amount to
be removed on each side of the hip will vary, is determined by the
principles developed in Sec. 39. The amount to be removed from each
side must be separately determined according to the angle the hip makes
with the plate.
Lengths of hip and valley jacks are determined as in Sec. 42. Lengths
of cripple jacks will not be of uniform length, as in even pitched
roofs. The runs for such jacks may be obtained with sufficient accuracy
by measurements taken from an accurately made scale drawing.
The following example will make clear the method of attack where it is
desired to develop a constant for hip or valley in terms of the common
rafter of one of the pitches.
_Example:_
Given: Main roof, Rise = 8', Run = 12' = ½pitch.
Minor roof, Rise = 5', Run = 6' = 5/12 pitch.
(1) Find the run of valley rafter over _c_, Fig. 88-a, in terms of
the run of the common rafter over _b_.
_Solution:_
_c²_ = _a²_ + _b²_ = (7½)² + 6² (_a_:12::5:8,
whence _a_ = 7½)
_c_ = 9.60'
Expressing this run of valley in terms of 12" of run of common rafter
over _b_.
9.60':6'::_x_:12".
_x_ = 19.20". (Check this value by scale drawing.)
Length of valley rafter, then, is found by taking 19.20" on the tongue
with 10" (5/6 of 12") on the blade, advancing the setting as many times
as there are feet in the run of the common rafter over _b_.
[Illustration: Fig. 88-a. Uneven Pitches]
The numbers just given will give the plumb cut and the seat cut of this
valley rafter.
(2) Find the side cut of the valley rafter when it rests against the
ridge of the minor roof.
_Solution:_ The side cut of the valley when allowed to rest in the
plane of the
plate = angle _B_, Fig. 88-a, whose tangent =
_b_/_a_ = 6/7½ (= 12/15 = .800 = 38° 40')
Therefore, take 7½" on the blade (always the run of intercepted common
rafter of major roof), and 6" on the tongue (always the run of the
common rafter of the minor roof); scribe on the blade.
For side cut of valley rafter when it is to be fitted to a ridge of the
major roof, the lay-out when in the plane of the plate is obtained by
means of these same numbers, but the scribing is done on the tongue.
This lay-out in either case is for the rafter when lying in the plane
of the plate. Having secured this, proceed as in Sec. 35, (2).
Side cuts of jacks for the minor roof are determined by the length
of the ridge over _a_, Fig. 88-a, and the length of the rafter
over _b_. But the length of the ridge over _a_= the run of
the intercepted common rafter of the major roof, 7%' here. A general
rule may be derived, therefore: For the side cut of a jack on a minor
roof take on the blade of the square the run of the intercepted common
rafter of the major roof (inches for feet), and on the tongue take the
length of rafter of the minor roof; scribe on the tongue, Fig. 88-b.
For the side cut of jacks on a major roof, take on the tongue of the
square the run of the common rafter of the minor roof, and on the blade
the length of the intercepted common rafter of the major roof; scribe
on the blade, Fig. 88-c.
[Illustration: Fig. 88-b. Fig. 88-c.
Side Cuts of Jacks for Uneven Pitches]
A new problem arises in connection with uneven or irregular pitches,
the problem of making the projecting cornice member one with another.
Manifestly, if one part of a roof is steeper than another and the plate
the same height all around the building, the cornice cannot be made
to meet in the same plane. This difficulty is overcome by raising the
plate of the steeper roof an amount equal to the difference in the
rises of the two pitches for a run equal to that of the projecting
cornice. For example, in a half-pitch the rise would be 24" for a run
of 24" in the cornice. In a quarter-pitch rise for a 24" run of cornice
would be 12", a difference in rises of 12". The plate of the steeper
roof must be raised that much higher than that for the lower pitch.
[Illustration: Fig. 89. Deck Frame]
=48. Decks; Chimney Openings.=--Decks are sometimes used in roof
framing to prevent some part of the roof from rising too high above the
remainder. Such decks are framed of joist stock spiked together with
butt joints. Upon the top of such deck joists are placed feathering
strips so sawed as to give the deck a slight "fall" in each direction,
two to three inches in eight to ten feet. Upon these feathering strips
flooring is laid.
The deck is framed, raised and braced, being held up by studding placed
under each corner. Next, the hip rafters are placed, then the flooring
on the deck, and finally the common and jack rafters. Should common
rafters be placed before the deck flooring, the joists must be braced
to withstand the pressure against their sides. On large decks the outer
deck frame is doubled.
Chimney openings are framed as shown in Fig. 89.