THE STEEL SQUARE
=Introductory.= The Standard Steel Square has a _blade_ 24 inches
long and 2 inches wide, and a _tongue_ from 14 to 18 inches long and 1½
inches wide. The blade is at right angles to the tongue.
The _face_ of the square is shown in Fig. 1. It is always stamped with
the manufacturer’s name and number.
The reverse is the _back_ (see Fig. 2).
The longer arm is the _blade_; the shorter arm, the _tongue_.
In the center of the tongue, on the face side, will be found two
parallel lines divided into spaces (see Fig. 1); this is the _octagon
scale_.
The spaces will be found numbered 10, 20, 30, 40, 50, 60, and 70, when
the tongue is 18 inches long.
To draw an octagon of 8 inches square, draw an 8 inch square and then
draw a perpendicular and a horizontal line through its center. To find
the length of the octagon side, place one point of a compass on any of
the main divisions of the scale, and the other point of the compass on
the eighth subdivision; then step this length off on each side of the
center lines on the side of the square, which will give the points from
which to draw the octagon lines.
The diameter of the octagon must equal in inches the number of spaces
taken from the square.
On the opposite side of the tongue, in the center, will be found the
_brace rule_ (see Fig. 3). The fractions denote the _rise_ and _run_ of
the brace, and the decimals the _length_. For example, a brace of 36
inches run and 36 inches rise, has a length of 50.91 inches; a brace of
42 inches run and 42 inches rise, has a length of 59.40 inches; etc.
[Illustration: Fig. 1. Face Side of Tongue of Steel Square, Showing
Octagon Scale.]
[Illustration: Fig. 2. Back of Blade of Steel Square, Showing Rafter
Table.]
[Illustration: Fig. 3. Back of Tongue of Steel Square, Showing Brace
Measure.]
[Illustration: Fig. 4. Back of Blade of Steel Square, Showing Essex
Board Measure.]
On the back of the blade (Fig. 4) will be found the _board measure_,
where eight parallel lines running along the length of the blade are
shown and divided at every inch by cross-lines. Under 12, on the outer
edge of the blade, will be found the various lengths of the boards,
as 8, 9, 10, 11, 12, etc. For example, take a board 14 feet long and
9 inches wide. To find the contents, look under 12, and find 14; then
follow this space along to the cross-line under 9, the width of the
board; and here is found 10 feet 6 inches, denoting the contents of a
board 14 feet long and 9 inches wide.
=To Find the Miter and Length of Side for any Polygon, with the Steel
Square.= In Fig. 5 is shown a pentagon figure. The miters of the
pentagon stand at 72 degrees with each other, and are found by dividing
360 by 5, the number of sides in the pentagon. But the angle when
applied to the square to obtain the miter, is only one-half of 72, or
36 degrees, and intersects the blade at 8-23/32, as shown in Fig. 5.
[Illustration: Fig. 5. Use of Steel Square to Find Miter and Side of
Pentagon.]
By squaring up from 6 on the tongue, intersecting the degree line at
_a_, the center _a_ is determined either for the inscribed or the
circumscribed diameter, the radii being _a b_ and _a c_, respectively.
The length of the sides will be 8-23/32 inches to the foot.
[Illustration: Fig. 6. Use of Steel Square to Find Miter and Side of
Hexagon.]
If the length of the inscribed diameter be 8 feet, then the sides would
be 8 × 8-23/32 inches.
The figures to use for other polygons are as follows:
Triangle 20-25/32
Square 12
Hexagon 7
Nonagon 4⅜
Decagon 3⅞
In Fig. 6 the same process is used in finding the
miter and side of the hexagon polygon.
To find the degree line, 360 is divided by 6, the number
of sides, as follows:
360 ÷ 6 = 60; and
60 ÷ 2 = 30 degrees.
Now, from 12 on tongue, draw a line making an angle of 30 degrees with
the tongue. It will cut the blade in 7 as shown; and from 7 to _m_, the
heel of the square, will be the length of the side. From 6 on tongue,
erect a line to cut the degree line in _c_; and with _c_ as center,
describe a circle having the radius of _c_ 7; and around the circle,
complete the hexagon by taking the length 7 _m_ with the compass for
each side, as shown.
[Illustration: Fig. 7. Use of Steel Square to Find Miter and Side of
Octagon.]
In Fig. 7 the same process is shown applied to the octagon. The degree
line in all the polygons is found by dividing 360 by the number of
sides in the figure:
360 ÷ 8 = 45; and 45 ÷ 2 = 22½ degrees.
This gives the degree line for the octagon. Complete the process as was
described for the other polygons.
By using the following figures for the various polygons, the miter
lines may be found; but in these figures no account is taken of the
relative size of sides to the foot as in the figures preceding:
Triangle 7 in. and 4 in.
Pentagon 11 " " 8 "
Hexagon 4 " " 7 "
Heptagon 12½ " " 6 "
Octagon 17 " " 7 "
Nonagon 22½ " " 9 "
Decagon 9½ " " 3 "
The miter is to be drawn along the line of the first column, as shown
for the triangle in Fig. 8, and for the hexagon in Fig. 9.
In Fig. 10 is shown a diagram for finding degrees on the square. For
example, if a pitch of 35 degrees is required, use 8-13/32 on tongue
and 12 on blade; if 45 degrees, use 12 on tongue and 12 on blade; etc.
[Illustration: Fig. 8. Use of Square to Find Miter of Equilateral
Triangle.]
In Fig. 11 is shown the relative length of run for a rafter and a hip,
the rafter being 12 inches and the hip 17 inches. The reason, as shown
in this diagram, why 17 is taken for the run of the hip, instead of
12 as for the common rafter, is that the seats of the common rafter
and hip do not run parallel with each other, but diverge in roofs of
equal pitch at an angle of 45 degrees; therefore, 17 inches taken on
the run of the hip is equal to only 12 inches when taken on that of the
common rafter, as shown by the dotted line from heel to heel of the two
squares in Fig. 11.
[Illustration: Fig. 9. Use of Square to Find Miter of Hexagon.]
In Fig. 12 is shown how other figures on the square may be found for
corners that deviate from the 45 degrees. It is shown that for a
pentagon, which makes a 36-degree angle with the plate, the figure to
be used on the square for run is 14⅞ inches; for a hexagon, which makes
a 30-degree angle with the plate, the figure will be 13⅞ inches; and
for an octagon, which makes an angle of 22½ degrees with the plate, the
figure to use on the square for run of hip to correspond to the run of
the common rafters, will be 13 inches. It will be observed that the
height in each case is 9 inches.
[Illustration: Fig. 10. Diagram for Finding Pitches of Various Degrees
by Means of the Steel Square.]
Fig. 13 illustrates a method of finding the relative height of a hip or
valley per foot run to that of the common rafter. The square is shown
placed with 12 on blade and 9 on tongue for the common rafter; and
shows that for the hip the rise is only 6-7/16 inches.
[Illustration: Fig. 11. Square Applied to Determine Relative Length of
Run for Rafter and Hip.]
=The Steel Square as Applied in Roof Framing.= Roof framing at
present is as simple as it possibly can be, so that any attempt at a
new method would be superfluous. There may, however, be a certain way
of presenting the subject that will carry with it almost the weight
assigned to a new theory, making what is already simple still more
simple.
The steel square is a mighty factor in roof framing, and without doubt
the greatest tool in practical potency that ever was invented for the
carpenter. With its use the lengths and bevels of every piece of timber
that goes into the construction of the most intricate design of roof,
can easily be obtained, and that with but very little knowledge of
lines.
[Illustration: Fig. 12. Use of Square to Determine Length of Run for
Rafters on Corners Other than 45°.]
In roofs of equal pitch, as illustrated in Fig. 14, the steel square is
all that is required if one properly understands how to handle it.
What is meant by a _pitch_ of a roof, is the number of inches it rises
to the foot of run.
[Illustration: Fig. 13. Method of Finding Relative Height of Hip or
Valley per Foot of Run to that of Common Rafter.]
In Fig. 15 is shown the steel square with figures representing the
various pitches to the foot of run. For the ½-pitch roof, the figures
as shown, from 12 on tongue to 12 on blade, are those to be used on the
steel square for the common rafter; and for ⅜ pitch, the figures to be
used on the square will be 12 and 9, as shown.
[Illustration: Fig. 14. Diagram to Illustrate Use of Steel Square In
Laying Out Timbers of Roofs of Equal Pitch.]
To understand this figure, it is necessary only to keep in mind that
the pitch of a roof is reckoned from the span. Since the run in each
pitch as shown is 12 inches, the span is two times 12 inches, which
equals 24 inches; hence, 12 on blade to represent the foot run, and 12
on tongue to represent the rise over ½ the span, will be the figures on
the square for a ½-pitch roof.
For the ⅜ pitch, the figures are shown to be 12 on tongue and 9 on
blade, 9 being ⅜ of the span, 24 inches.
The same rule applies to all the pitches. The ⅙ pitch is shown to
rise 4 inches to the foot of run, because 4 inches is ⅙ of the span,
24 inches, the ⅓ pitch is shown to rise 8 inches to the foot of run,
because 8 inches is ⅓ of the span, 24 inches; etc.
[Illustration: Fig. 15. Steel Square Giving Various Pitches to Foot of
Run.]
The roof referred to in Figs. 16 and 17 is to rise 9 inches to the foot
of run; it is therefore a ⅜-pitch roof. For all the common rafters, the
figures to be used on the square will be 12 on blade to represent the
run, and 9 on tongue to represent the rise to the foot of run; and for
all the hips and valleys, 17 on blade to represent the run, and 9 on
tongue to represent the rise of the roof to the foot of run.
Why 17 represents the run for all the hips and valleys, will be
understood by examining Fig. 19, in which 17 is shown to be the
diagonal of a foot square.
In equal-pitch roofs the corners are square, and the plan of the hip or
valley will always be a diagonal of a square corner as shown at 1, 2,
3, and 5 in Fig. 14.
In Fig. 18 are shown ⅙ pitch, ⅜ pitch and ½ pitch over a square corner.
The figures to be used on the square for the hip, will be 17 for run
in each case. For the ⅙ pitch, the figures to be used would be 17
inches run and 4 inches rise, to correspond with the 12 inches run and
4 inches rise of the common rafter. For the ⅜ pitch, the figures to be
used for hip would be 17 inches run and 9 inches rise, to correspond
with the 12 inches run and 9 inches rise of the common rafter; and for
the ½ pitch, the figures to be used on the square will be 17 inches run
and 12 inches rise, to correspond with the 12 inches run and 12 inches
rise of the common rafter.
[Illustration: Fig. 16. Method of Laying Out Common Rafters of a
⅜-Pitch Roof.]
It will be observed from above, that in all cases where the plan of the
hip or valley is a diagonal of a square, the figures to be used on the
square for run will be 17 inches; and for the rise, whatever the roof
rises to the foot of run. It should also be remembered that this is
the condition in all roofs of equal pitch, where the angle of the hip
or valley is a 45-degree angle, or, in other words, where we have the
diagonal of a square.
It has been shown in Fig. 12 how other figures for other plan angles
may be found; and that in each case the figures for run vary according
to the plan angle of the hip or valley, while the figure for the height
in each case is similar.
[Illustration: Fig. 17. Method of Laying Out Hips and Valleys of a
⅜-Pitch Roof.]
In Fig. 14 are shown a variety of runs for common rafters, but all have
the same pitch; they rise 9 inches to the foot of run. The main roof
is shown to have a span of 27 feet, which makes the run of the common
rafter 13 feet 6 inches. The run of the front wing is shown to be 10
feet 4 inches; and the run of the small gable at the left corner of the
front, is shown to be 8 feet.
The diversity exhibited in the runs, and especially the fractional part
of a foot shown in two of them, will afford an opportunity to treat
of the main difficulties in laying out roof timbers in roofs of equal
pitch. Let it be determined to have a rise of 9 inches to the foot
of run; and in this connection it may be well to remember that the
proportional rise to the foot run for roofs of equal pitch makes not
the least difference in the method of treatment.
To lay out the common rafters for the main roof, which has a run of 13
feet 6 inches, proceed as shown in Fig. 16.
Take 12 on the blade and 9 on the tongue, and step 13 times along the
rafter timber. This will give the length of rafter for 13 feet of run.
In this example, however, there is another 6 inches of run to cover.
For this additional length, take 6 inches on the blade (it being ½ a
foot run) for run, and take ½ of 9 on the tongue (which is 4½ inches),
and step one time. This, in addition to what has already been found
by stepping 13 times with 12 and 9, will give the full length of the
rafter.
[Illustration: Fig. 18. Method of Laying Out Hips and Rafters for Roofs
or Various Pitches over Square Corner.]
The square with 12 on blade and 9 on tongue will give the heel and
plumb cuts.
Another method of finding the length of rafter for the 6 inches is
shown in Fig. 16, where the square is shown applied to the rafter timber
for the plumb cut. Square No. 1 is shown applied with 12 on blade and 9
on tongue for the length of the 13 feet. Square from this cut, measure
6 inches, the additional inches in the run; and to this point move the
square, holding it on the side of the rafter timber with 12 on blade
and 9 on tongue, as for a full foot run.
It will be observed that this method is easily adapted to find any
fractional part of a foot in the length of rafters.
In the front gable, Fig. 14, the fractional part of a foot is 4 inches
to be added to 10 feet of run; therefore, in that case, the line shown
measured to 6 inches in Fig. 16 would measure only 4 inches for the
front gable.
=Heel Cut of Common Rafter.= In Fig. 16 is also shown a method to
lay out the heel cut of a common rafter. The square is shown applied
with 12 on blade and 9 on tongue; and from where the 12 on the square
intersects the edge of the rafter timber, a line is drawn square to the
blade as shown by the dotted line from 12 to _a_. Then the thickness
of the part of the rafter that is to project beyond the plate to hold
the cornice, is gauged to intersect the dotted line at _a_; and from
_a_, the heel cut is drawn with the square having 12 on blade and 9 on
tongue, marking along the blade for the cut.
The common rafter for the front wing, which is shown to have a run of
10 feet 4 inches, is laid out precisely the same, except that for this
rafter the square with 12 on blade and 9 on tongue will have to be
stepped along the rafter timber only 10 times for the 10 feet of run;
and for the fractional part of a foot (4 inches) which is in the run,
either of the two methods already shown for the main rafter may be used.
The proportional figures to be used on the square for the 4 inches will
be 4 on blade and 2¼ on tongue; and if the second method is used, make
the addition to the length of rafter for 10 feet, by drawing a line 4
inches square from the tongue of square No. 1 (see Fig. 16), instead of
6 inches as there shown for the main rafter.
=Hips.= Three of the hips are shown in Fig. 14 to extend from
the plate to the ridge-pole; they are marked in the figure as 1, 2,
and 3 respectively, and are shown in plan to be diagonals of a square
measuring 13 feet 6 inches by 13 feet 6 inches; they make an angle,
therefore, of 45 degrees with the plate.
In Fig. 18 it has been shown that a hip standing at an angle of 45
degrees with the plate will have a run of 17 inches for every foot run
of the common rafter. Therefore, to lay out the hips, the figures on
the square will be 17 for run and 9 for rise; and by stepping 13 times
along the hip rafter timber, the length of hip for 13 feet of run is
obtained. The length for the additional 6 inches in the run may be
found by squaring a distance of 8½ inches, as shown in Fig. 17, from
the tongue of the square, and moving square No. 1 along the edge of the
timber, holding the blade on 17 and tongue on 9, and marking the plumb
cut where the dotted line is shown.
In Fig. 18 is shown how to find the relative run length of a portion
of a hip to correspond to that of a fractional part of a foot in the
length of the common rafter. From 12 inches, measure along the run of
the common rafter 6 inches, and drop a line to cut the diagonal line in
_m_. From _m_ to _a_, along the diagonal line, will be the relative run
length of the part of hip to correspond with 6 inches run of the common
rafter, and it measures 8½ inches.
[Illustration: Fig. 19. Diagram Showing Relative Lengths of Run for
Hips and Common Rafters In Equal-Pitch Roofs.]
[Illustration: Fig. 20. Method of Determining Run of Valley for
Additional Run in Common Rafter.]
The same results may be obtained by the following method of figuring:
As 12 : 17 :: 6
6
____
12 )102
————
8 - 6 = 8½
In Fig. 19 is shown a 12-inch square, the diagonal _m_ being 17 inches.
By drawing lines from the base _a b_ to cut the diagonal line, the part
of the hip to correspond to that of the common rafter will be indicated
on the line 17. In this figure it is shown that a 6-inch run on _a b_,
which represents the run of a foot of a common rafter, will have a
corresponding length of 8½ inches run on the line 17, which represents
the plan line of the hip or valley in all equal-pitch roofs.
In the front gable, Fig. 14, it is shown that the run of the common
rafter is 10 feet 4 inches. To find the length of the common rafter,
take 12 on blade and 9 on tongue, and step 10 times along the rafter
timber; and for the fractional part of a foot (4 inches), proceed as
was shown in Fig. 16 for the rafter of the main roof; but in this case
measure out square to the tongue of square No. 1, 4 inches instead of 6
inches.
[Illustration: Fig. 21. Corner of Square Building, Showing Plan Lines
of Plates and Hip.
Fig. 22. Corner of Square Building, Showing Plan Lines of Plates and
Valley.]
The additional length for the fractional 4 inches run can also be found
by taking 4 inches on blade and 3 inches on tongue of square, and
stepping one time; this, in addition to the length obtained by stepping
10 times along the rafter timber with 12 on blade and 9 on tongue, will
give the full length of the rafter for a run of 10 feet 4 inches.
[Illustration: Fig. 23. Use of Square to Determine Heel Cut of Valley.]
In the intersection of this roof with the main roof, there are shown
to be two valleys of different lengths. The long one extends from the
plate at _n_ (Fig. 14) to the ridge of the main roof at _m_; it has
therefore a run of 13 feet 6 inches. For the length, proceed as for the
hips, by taking 17 on blade of the square and 9 on tongue, and stepping
13 times for the length of the 13 feet; and for the fractional 6
inches, proceed precisely as shown in Fig. 17 for the hip, by squaring
out from the tongue of square No. 1, 8½ inches; this, in addition to
the length obtained for the 13 feet, will give the full length of the
long valley _n m_.
The length of the short valley _a c_, as shown, extends over the run of
10 feet 4 inches, and butts against the side of the long valley at _c_.
By taking 17 on blade and 9 on tongue, and stepping along the rafter
timber 10 times, the length for the 10 feet is found; and for the 4
inches, measure 5⅝ inches square from the tongue of square No. 1, in
the manner shown in Fig. 17, where the 8½ inches is shown added for the
6 inches additional run of the main roof for the hips.
[Illustration: Fig. 24. Steel Square Applied to Finding Bevel for
Fitting Top of Hip or Valley to Ridge.]
The length 5⅝ is found as shown in Fig. 20, by measuring 4 inches from
_a_ to _m_ along the run of common rafter for one foot. Upon _m_ erect
a line to cut the seat of the valley at _c_; from _c_ to _a_ will be
the run of the valley to correspond with 4 inches run of the common
rafter, and it will measure 5⅝ inches.
=How to Treat the Heel Cut of Hips and Valleys.= Having found the
lengths of the hips and valleys to correspond to the common rafters, it
will be necessary to find also the thickness of each above the plate to
correspond to the thickness the common rafter will be above the plate.
In Fig. 21 is shown a corner of a square building, showing the plates
and the plan lines of a hip. The length of the hip, as already found,
will cover the span from the ridge to the corner 2; but the sides of the
hip intersect the plates at 3 and 3 respectively; therefore the
distance from 2 to 1, as shown in this diagram, is measured backwards
from _a_ to 1 in the manner shown in Fig. 17; then a plumb line is
drawn through 1 to _m_, parallel to the plumb cut _a_-17. From _m_ to
_o_ on this line, measure the same thickness as that of the common
rafter; and through _o_ draw the heel cut to _a_ as shown.
In like manner the thickness of the valley above the plate is found;
but as the valley as shown in the plan figure, Fig. 22, projects beyond
point 2 before it intersects the outside of the plates, the distance
from 2 to 1 in the case of the valley will have to be measured outwards
from 2, as shown from 2 to 1 in Fig. 23; and at the point thus found
the thickness of the valley is to be measured to correspond with that
of the common rafter as shown at _m n_.
[Illustration: Fig. 25. Steel Square Applied to Jack Rafter to Find
Bevel for Fitting against Side of Hip or Valley.]
In Fig. 24 is shown the steel square applied to a hip or valley timber
to cut the bevel that will fit the top end against the ridge. The
figures on the square are 17 and 19¼. The 17 represents the length of
the plan line of the hip or valley for a foot of run, which, as was
shown in previous figures, will always be 17 inches in roofs of equal
pitch, where the plan lines stand at 45 degrees to the plates and
square to each other.
The 19¼ taken on the blade represents the actual length of a hip or
valley that will span over a run of 17 inches. The bevel is marked
along the blade.
The cut across the back of the short valley to fit it against the side
of the long valley, will be a square cut owing to the two plan lines
being at right angles to each other.
In Fig. 25 is shown the steel square applied to a jack rafter to
cut the back bevel, to fit it against the side of a hip or valley.
The figures on the square are 12 on tongue and 15 on blade, the 12
representing a foot run of a common rafter, and the 15 the length of
a rafter that will span over a foot run; marking along the blade will
give the bevel.
[Illustration: Fig. 26. Finding Length to Shorten Rafters for Jacks per
Foot of Run.]
The rule in every case to find the back bevel for jacks in roofs of
equal pitch, is to take 12 on the tongue to represent the foot run, and
the length of the rafter for a foot of run on the blade, marking along
the blade in each case for the bevel.
In a ½-pitch roof, which is the most common in all parts of the
country, the length of rafter for a foot of run will be 17 inches;
hence it will be well to remember that 12 on tongue and 17 on blade,
marking along the blade, will give the bevel to fit a jack against a
hip or a valley in a ½-pitch roof.
In a roof having a rise of 9 inches to the foot of run, such as the one
under consideration, the length of rafter for one foot of run will be
15 inches. The square as shown in Fig. 25, with 12 on tongue and 15 on
blade, will give the bevel by marking along the blade.
To find the length of a rafter for a foot of run for any other pitch,
place the two-foot rule diagonally from 12 on the blade of the square
to the figure on tongue representing the rise of the roof to the foot
of run; the rule will give the length of the rafter that will span over
one foot of run.
[Illustration: Fig. 27. Finding Length of Jack Rafter in ½-Pitch Roof.]
[Illustration: Fig. 28. Finding Length of Jack Rafter in ⅜-Pitch Roof.]
The length of rafter for a foot of run will also determine the
difference in lengths of jacks. For example, if a roof rises 12 inches
to one foot of run, the rafter over this span has been found to be 17
inches; this, therefore, is the number of inches each jack is shortened
in one foot of run. If the rise of the roof is 8 inches to the foot of
run, the length of the rafter is found for one foot of run, by placing
the rule diagonally from 12 on tongue to 8 on blade, which gives 14½
inches, as shown in Fig. 26. This, therefore, will be the number of
inches the jacks are to be shortened in a roof rising 8 inches to the
foot of run. If the jacks are placed 24 inches from center to center,
then multiply 14½ by 2 = 29 inches.
In Fig. 27 is shown how to find the length with the steel square. The
square is placed on the jack timber rafter with the figures that have
been used to cut the common rafter. In Fig. 27, 12 on blade and 12 on
tongue were the figures used to cut the common rafter, the roof being ½
pitch, rising 12 inches to the foot of run. In the diagram it is shown
how to find the length of a jack rafter if placed 16 inches from center
to center. The method is to move the square as shown along the line of
the blade until the blade measures 16 inches; the tongue then would be
as shown from _w_ to _m_, and the length of the jack would be from 12
on blade to _m_ on tongue, on the edge of the jack rafter timber as
shown.
This latter method becomes convenient when the space between jacks is
less than 18 inches; but if used when the space is more than 18 inches
it will become necessary to use two squares; otherwise the tongue as
shown at _m_ would not reach the edge of the timber.
[Illustration: Fig. 29. Method of Determining Length of Jacks Between
Hips and Valleys; also Bevels for Jacks, Hips, and Valleys.]
In Fig. 28 the same method is shown for finding the length of a jack
rafter for a roof rising 9 inches to the foot of run, with the jacks
placed 18 inches center to center. The square in this diagram is shown
placed on the jack rafter timber with 12 on blade and 9 on
tongue; then it is moved forward along the line of the blade to _w_.
The blade, when in this latter position, will measure 18 inches. The
tongue will meet the edge of the timber at _m_, and the distance from
_m_ on tongue to 12 on blade will indicate the length of a jack, or, in
other words, will show the length each jack is shortened when placed 18
inches between centers in a roof having a pitch of 9 inches to the foot
of run.
[Illustration: Fig. 30. Method of Finding Bevels for All Timbers in
Roofs of Equal Pitch.]
[Illustration: Fig. 31. Method of Finding Bevel 5, Fig. 30, for Fitting
Hip or Valley Against Ridge when not Backed.]
When jacks are placed between hips and valleys as shown at 1, 2, 3, 4,
etc., in Fig. 14, a better method of treatment is shown in Fig. 29,
where the slope of the roof is projected into the horizontal plane.
The distance from the plate in this figure to the ridge _m_, equals
the length of the common rafter for the main roof. On the plate _a n
n_ is made equal to _a n n_ in Fig. 14. By drawing a figure like this
to a scale of one inch to one foot, the length of all the jacks can be
measured and also the lengths of the hip and the two valleys. It also
gives the bevels for the jacks, as well as the bevel to fit the hip and
valley against the ridge; but this last bevel must be applied to the
hip and valley when backed.
It has been shown before, that the figures to be used on the square for
this bevel when the timber is left square on back as is the custom in
construction, are the length of a foot run of a hip or valley, which is
17, on tongue, and the length of a hip or valley that will span over 17
inches run, on blade—the blade giving the bevel.
Fig. 30 contains all the bevels or cuts that have been treated upon so
far, and, if correctly understood, will enable any one to frame any
roof of equal pitch. In this figure it is shown that 12 inches run and
9 inches rise will give bevels 1 and 2, which are the plumb and heel
cuts of rafters of a roof rising 9 inches to the foot of run. By taking
these figures, therefore, on the square, 9 inches on the tongue and 12
inches on the blade, marking along the tongue will give the plumb cut,
and marking along the blade will give the heel cut.
[Illustration: Fig. 32. Method of Finding Back Bevel 6, Fig. 30, for
Jack Rafters, and Bevel 7, for Roof-Board.]
[Illustration: Fig. 33. Determining Miter Cut for Roof-Board.]
Bevels 3 and 4 are the plumb and heel cuts for the hip, and are shown
to have the length of the seat of hip for one foot run, which is 17
inches. By taking 17 inches, therefore, on the blade, and 9 inches on
the tongue, marking along the tongue for the plumb cut, and along the
blade for the heel cut, the plumb and heel cuts are found. Bevel 5,
which is to fit the hip or valley against the ridge when not backed, is
shown from _o w_, the length of the hip for one foot of run, which is
19¼ inches, and from _o s_, which always in roofs of equal pitch will
be 17 inches and equal in length to the seat of a hip or valley for one
foot of run.
These figures, therefore, taken on the square, 19¼ on the blade, and 17
on the tongue, will give the bevel by marking along the blade as shown
in Fig. 31, where the square is shown applied to the hip timber with
19¼ on blade and 17 on tongue, the blade showing the cut.
Bevels 6 and 7 in Fig. 30 are shown formed of the length of the rafter
for one foot of run, which is 15 inches, and the run of the rafter,
which is 12 inches. These figures are applied on the square, as shown
in Fig. 32, to a jack rafter timber; taking 15 on the blade and 12 on
the tongue, marking along the blade will give the back bevel for the
jack rafters, and marking along the tongue will give the face cut of
roof-boards to fit along the hip or valley.
[Illustration: Fig. 34. Laying Out Timbers of One-half Gable of
⅜-Pitch Roof.]
It is shown in Fig. 30, also, that by taking the length of rafter 15
inches on blade, and rise of roof 9 inches on tongue, bevel 8 will give
the miter cut for the roof-boards.
In Fig. 33 the square is shown applied to a roof-board with 15 on
blade, which is the length of the rafter to one foot of run, and with 9
on tongue, which is the rise of the roof to the foot run; marking along
the tongue will give the miter for the boards.
[Illustration: Fig. 35. Finding Backing of Hip in Gable Roof.]
Other uses may be made of these figures, as shown in Fig. 34, which
is one-half of a gable of a roof rising 9 inches to the foot run. The
squares at the bottom and the top will give the plumb and heel cuts
of the common rafter. The same figures on the square applied to the
studding, marking along the tongue for the cut, will give the bevel to
fit the studding against the rafter; and by marking along the blade we
obtain the cut for the boards that run across the gable. By taking 19¼
on blade, which is the length of the hip for one foot of run, and
taking on the tongue the rise of the roof to the foot of run, which is
9 inches, and applying these as shown in Fig. 35, we obtain the backing
of the hip by marking along the tongue of the two squares, as shown.
It will be observed from what has been said, that in roofs of equal
pitch the figure 12 on the blade, and whatever number of inches the
roof rises to the foot run on the tongue, will give the plumb and heel
cuts for the common rafter; and that by taking 17 on the blade instead
of 12, and taking on the tongue the figure representing the rise of the
roof to the foot run, the plumb and heel cuts are found for the hips
and valleys.
By taking the length of the common rafter for one foot of run on blade,
and the run 12 on tongue, marking along the blade will give the back
bevel for the jack to fit the hip or valley, and marking along the
tongue will give the bevel to cut the roof-boards to fit the line of
hip or valley upon the roof.
With this knowledge of what figures to use, and why they are used,
it will be an easy matter for anyone to lay out all rafters for
equal-pitch roofs.
[Illustration: Fig. 36. Laying Out Timbers of Roof with Two Unequal
Pitches.]
In Fig. 36 is shown a plan of a roof with two unequal pitches. The main
roof is shown to have a rise of 12 inches to the foot run. The front
wing is shown to have a run of 6 feet and to rise 12 feet; it has thus
a pitch of 24 inches to the foot run. Therefore 12 on blade of the
square and 12 on tongue will give the plumb and heel cuts for the main
roof, and by stepping 12 times along the rafter timber the length of
the rafter is found. The figures on the square to find the heel and
plumb cuts for the rafter in the front wing, will be 12 run and 24 rise,
and by stepping 6 times (the number of feet in the run of the rafter),
the length will be found over the run of 6 feet, and it will measure 13
feet 6 inches.
If, in place of stepping along the timber, the diagonal of 12 and 24 is
multiplied by 6, the number of feet in the run, the length may be found
even to a greater exactitude.
Many carpenters use this method of framing; and to those who have
confidence in their ability to figure correctly, it is a saving of
time, and, as before said, will result in a more accurate measurement;
but the better and more scientific method of framing is to work to a
scale of one inch, as has already been explained.
[Illustration: Fig. 37. Finding Length of Rafter for Front Wing in Roof
Shown in Fig. 36.]
According to that method, the diagonal of a foot of run, and the number
of inches to the foot run the roof is rising, measured to a scale, will
give the exact length. For example, the main roof in Fig. 36 is rising
12 inches to a foot of run. The diagonal of 12 and 12 is 17 inches,
which, considered as a scale of one inch to a foot, will give 17 feet,
and this will be the exact length of the rafter for a roof rising 12
inches to the foot run and having a run of 12 feet.
[Illustration: Fig. 38. Laying Out Timbers of Roof Shown in Fig. 36, by
Projecting Slope of Roof into Horizontal Plane.]
The length of the rafter for the front wing, which has a run of 6 feet
and a rise of 12 feet, may be obtained by placing the rule as shown in
Fig. 37 from 6 on blade to 12 on tongue, which will give a length of
13½ inches. If the scale be considered as one inch to a foot, this will
equal 13 feet 6 inches, which will be the exact length of a common
rafter rising 24 inches to the foot run and having a run of 6 feet.
It will be observed that the plan lines of the valleys in this figure
in respect to one another deviate from forming a right angle. In
equal-pitch roofs the plan lines are always at right angles to each
other, and therefore the diagonal of 12 and 12, which is 17 inches,
will be the relative foot run of valleys and hips in equal-pitch roofs.
In Fig. 36 is shown how to find the figures to use on the square for
valleys and hips when deviating from the right angle. A line is drawn
at a distance of 12 inches from the plate and parallel to it, cutting
the valley in _m_ as shown. The part of the valley from _m_ to the
plate will measure 13½ inches, which is the figure that is to be used
on the square to obtain the length and cuts of the valleys.
[Illustration: Fig. 39. Method of Finding Length and Cuts of Octagon
Hips Intersecting a Roof.]
It will be observed that this equals the length of the common rafter as
found by the square and rule in Fig. 37. In that figure is shown 12 on
tongue and 6 on blade. The 12 here represents the rise, and the 6 the
run of the front roof. If the 12 be taken to represent the run of the
main roof, and the 6 to represent the run of the front roof, then, the
diagonal 13½ will indicate the length of the seat of the valley for 12
feet of run, and therefore for one foot it will be 13½ inches. Now, by
taking 13½ on the blade for run, and 12 inches on the tongue for rise,
and stepping along the valley rafter timber 12 times, the length of the
valley will be found. The blade will give the heel cut, and the tongue
the plumb cut.
In Fig. 38 is shown the slope of the roof projected into the horizontal
plane. By drawing a figure based on a scale of one inch to one foot,
all the timbers on the slope of the roof can be measured. Bevel 2,
shown in this figure, is to fit the valleys against the ridge. By
drawing a line from _w_ square to the seat of the valley to _m_, making
_w_ 2 equal in length to the length of the valley, as shown, and by
connecting 2 and _m_, the bevel at 2 is found, which will fit the
valleys against the ridge, as shown at 3 and 3 in Fig. 36.
[Illustration: Fig. 40. Showing How Cornice Affects Valleys and Plates
in Roof with Unequal Pitches.]
In Fig. 39, is shown how to find the length and cuts of octagon hips
intersecting a roof. In Fig. 36, half the plan of the octagon is shown
to be inside of the plate, and the hips _o_, _z_, _o_ intersect the
slope of the roof. In Fig. 39, the lines below _x y_ are the plan
lines; and those above, the elevation. From _z_, _o_, _o_, in the plan,
draw lines to _x y_, as shown from _o_ to _m_ and from _z_ to _m_; from
_m_ and _m_, draw the elevation lines to the apex _o″_, intersecting
the line of the roof in _d″_ and _c″_. From _d″_ and _c″_, draw the
lines _d″ v″_ and _c″ a″_ parallel to _x y_; from _c″_, drop a line to
intersect the plan line _a o_ in _c_. Make _a w_ equal in length to _a″
o″_ of the elevation, and connect _w c_; measure from _w_ to _n_ the
full height of the octagon as shown from _x y_ to the apex _o″_; and
connect _c n_. The length from _w_ to _c_ is that of the two hips shown
at _o o_ in Fig. 36, both being equal hips intersecting the roof at an
equal distance from the plate. The bevel at _w_ is the top bevel, and
the bevel at _c_ will fit the roof.
Again, drop a line from _d″_ to intersect the plan line _a z_ in _d_.
Make _a_ 2 equal to _v″ o″_ in the elevation, and connect 2 _d_.
Measure from 2 to _b_ the full height of the tower as shown from _x y_
to the apex _o″_ in the elevation, and connect _d b_. The length 2 _d_
represents the length of the hip _z_ shown in Fig. 36; the bevel at 2
is that of the top; and the bevel at _d_, the one that will fit the
foot of the hip to the intersecting roof.
When a cornice of any considerable width runs around a roof of this
kind, it affects the plates and the angle of the valleys as shown in
Fig. 40. In this figure are shown the same valleys as in Fig. 36; but,
owing to the width of the cornice, the foot of each has been moved the
distance _a b_ along the plate of the main roof. Why this is done is
shown in the drawing to be caused by the necessity for the valleys to
intersect the corners _c c_ of the cornice.
The plates are also affected as shown in Fig. 41, where the plate of
the narrow roof is shown to be much higher than the plate of the main
roof.
The bevels shown at 3, Fig. 40, are to fit the valleys against the
ridge.
[Illustration: Fig. 41. Showing Relative Position of Plates in Roof
with Two Unequal Pitches.]
[Illustration: Fig. 42. Method of Finding Bevels for Purlins in
Equal-Pitch Roofs.]
In Fig. 42 is shown a very simple method of finding the bevels for
purlins in equal-pitch roofs. Draw the plan of the corner as shown, and
a line from _m_ to _o_; measure from _o_ the length _x y_, representing
the common rafter, to _w_; from _w_ draw a line to _m_; the bevel shown
at 2 will fit the top face of the purlin. Again, from _o_, describe an
arc to cut the seat of the valley, and continue same around to _S_;
connect _S m_; the bevel at 3 will be the side bevel.
INDEX
PART PAGE
B
Bevel I, 44
Bevels, application of in fitting wreaths to rail I, 58
Bevels to square wreaths I, 60
Bottom steps with obtuse-angle plan I, 56
Bullnose stair I, 33
Bullnose steps I, 32
Bullnose tread I, 29
C
Carriage I, 16
Carriage pieces I, 28
Circular stairs I, 25
Close stairway I, 2
Cove I, 28
Curved steps and stringer, plan of I, 54
Cylinder I, 35
D
Dog-legged stairs I, 25
F
Face-mould I, 68
Face-mould developing (obtuse-angle plan) I, 56
Flyer I, 25
Foci I, 70
G
Geometrical stair I, 44
Geometrical stairways and handrailing I, 43
K
Kerfing I, 36
N
Newel post I, 14
O
Open-newel stairs I, 32
P
Pitch-board I, 10
Plan lines I, 48
Platform I, 20
Platform stairs I, 26
Projection I, 44
Q
Quarter-space stair with six winders I, 43
Quick sweep I, 37
R
Rail, plan line of I, 46
Rise and rim I, 3
Riser I, 3
arrangement of I, 74
Rough brackets I, 28
S
Springing the plank I, 74
Stair with five dancing winders I, 43
Stair-building I, 1-75
definitions I, 2
introductory I, 1
laying out I, 22
open-newel stairs I, 32
platform I, 20
stairs with curved turns I, 34
strings I, 7
housed I, 7
notched I, 8
open I, 7
rough I, 7
staved I, 8
well hole I, 18
Stairs, plans and elevation turning around central post I, 44
Stairs, setting of I, 8
Stairs, types of in common use I, 41
Steel square II, 1-27
applied in roof framing II, 7
applied to jack rafter II, 17
back II, 1
back of blade, showing essex board measure II, 3
back of blade, showing rafter table II, 2
back of tongue, showing brace measure II, 3
bevels, method of finding for all timbers in roofs
of equal pitch II, 20
blade II, 1
board measure II, 1
brace rule II, 1
common rafters, method of laying out II, 11
face II, 1
face side of tongue, showing octagon scale II, 2
finding backing of hip in gable roof II, 22
finding length of rafter for front wing in roof shown
in Fig. 36 II, 24
finding length to shorten rafters for jacks per foot
of run II, 18
giving various pitches to foot of run II, 10
heel cut of common rafter II, 13
hips II, 13
hips and rafters for roofs of various pitches over
square corner, method of laying out II, 12
hips and valleys, how to treat the heel cut of II, 16
hips and valleys of a ⅜-pitch roof, method of laying out II, 11
jack rafter, finding length of II, 18
jacks between hips and valleys, method of
determining length II, 19
laying out timbers of one-half gable of ⅜-pitch roof II, 22
laying out timbers of roof with two unequal pitches II, 23
method of finding bevels for purlins in equal-pitch roofs II, 27
method of finding length and cuts of octagon hips
intersecting a roof II, 25
octagon scale II, 1
rise and run II, 1
tongue II, 1
use of to find miter of equilateral triangle II, 6
use of to find miter of hexagon II, 7
use of to find miter and side of pentagon, hexagon
and octagon II, 4, 5
Story rod I, 8
Straight flights I, 25
String-board I, 3
Strings I, 7
housed I, 7
methods of building up I, 37
notched I, 8
open I, 7
rough I, 7
staved I, 8
T
Tangent inclined-lower tangent level I, 63
Tangent system I, 44
Tangents and face-mould, diagram of I, 53
Tangents unfolded I, 50
Tread I, 3
Treads and risers, housing of I, 13
Trimming joists I, 19
Twisted wreath raised to position I, 57
W
Well-hole I, 18
Well-hole connecting two flights I, 51
Winder I, 25
Winding stairs I, 25
Wreath, cutting from plank I, 57
Wreath with two equally inclined tangents,
finding bevels for I, 59
Wreath twisted, ready to be moulded I, 57
Wreaths I, 44