String for Same.]
Another method is to build up or _stave_ the string. There are several
ways of doing this. In one, comparatively narrow pieces are cut to the
required curve or to portions of it, and are fastened together, edge
to edge, with glue and screws, until the necessary width is obtained
(see Fig. 59). The heading joints may be either butted or beveled, the
latter being stronger, and should be cross-tongued.
Fig. 60 shows a method that may be followed when a wide string is
required, or a piece curved in the direction of its width is needed for
any purpose. The pieces are stepped over each other to suit the desired
curve; and though shown square-edged in the figure, they are usually
cut beveled, as then, by reversing them, two may be cut out of a batten.
[Illustration: Fig. 59. Fig. 60. Methods of Building Up Strings.]
Panels and quick sweeps for similar purposes are obtained in the
manner shown in Fig. 61, by joining up narrow boards edge to edge at
a suitable bevel to give the desired curve. The internal curve is
frequently worked approximately, before gluing up. The numerous joints
incidental to these methods limit their uses to painted or unimportant
work.
[Illustration: Fig. 61. Building Up a Curved Panel or Quick Sweep.]
In Fig. 62 is shown a wreath-piece or curved portion of the outside
string rising around the cylinder at the half-space. This is formed by
reducing a short piece of string to a veneer between the springings;
bending it upon a cylinder made to fit the plan; then, when it is
secured in position, filling up the back of the veneer with staves
glued across it; and, finally, gluing a piece of canvas over the whole.
The appearance of the wreath-piece after it has been built up and
removed from the cylinder is indicated in Fig. 63. The canvas back has
been omitted to show the staving; and the counter-wedge key used for
connecting the wreath-piece with the string is shown. The wreath-piece
is, at this stage, ready for marking the outlines of the steps.
Fig. 62 also shows the drum or shape around which strings may be bent,
whether the strings are formed of veneers, staved, or kerfed. Another
drum or shape is shown in Fig. 64. In this, a portion of a cylinder is
formed in the manner clearly indicated; and the string, being set out
on a veneer board sufficiently thin to bend easily, is laid down round
the curve, such a number of pieces of like thickness being then added
as will make the required thickness of the string. In working this
method, glue is introduced between the veneers, which are then quickly
strained down to the curved piece with hand screws. A string of almost
any length can be formed in this way, by gluing a few feet at a time,
and when that dries, removing the cylindrical curve and gluing down
more, until the whole is completed. Several other methods will suggest
themselves to the workman, of building up good, solid, circular strings.
[Illustration: Fig. 62. Wreath-Piece Bent around Cylinder.]
[Illustration: Fig. 63. Completed Wreath-Piece Removed from Cylinder.]
[Illustration: Fig. 64. Another Drum or Shape for Building Curved
Strings.]
[Illustration: Fig. 65. Laying Out Treads and Risers around a Drum.]
One method of laying out the treads and risers around a cylinder or
drum, is shown in Fig. 65. The line _D_ shows the curve of the rail.
The lines showing treads and risers may be marked off on the cylinder,
or they may be marked off after the veneer is bent around the drum or
cylinder.
There are various methods of making inside cylinders or wells, and of
fastening same to strings. One method is shown in Fig. 66. This gives a
strong joint when properly made. It will be noticed that the cylinder
is notched out on the back; the two blocks shown at the back of the
offsets are wedges driven in to secure the cylinder in place, and to
drive it up tight to the strings. Fig. 67 shows an 8-inch well-hole
with cylinder complete; also the method of trimming and finishing same.
The cylinder, too, is shown in such a manner that its construction will
be readily understood.
Stairs having a cylindrical or circular opening always require a weight
support underneath them. This support, which is generally made of rough
lumber, is called the _carriage_, because it is supposed to carry any
reasonable load that may be placed upon the stairway. Fig. 68 shows
the under side of a half-space stair having a carriage beneath it. The
timbers marked _S_ are of rough stuff, and may be 2-inch by 6-inch or
of greater dimensions. If they are cut to fit the risers and treads,
they will require to be at least 2-inch by 8-inch.
In preparing the rough carriage for the winders, it will be best to
let the back edge of the tread project beyond the back of the riser
so that it forms a ledge as shown under _C_ in Fig. 69. Then fix the
cross-carriage pieces under the winders, with the back edge about flush
with the backs of risers, securing one end to the well with screws, and
the other to the wall string or the wall. Now cut short pieces, marked
_O O_ (Fig. 68), and fix them tightly in between the cross-carriage
and the back of the riser as at _B B_ in the section, Fig. 69. These
carriages should be of 3-inch by 2-inch material. Now get a piece of
wood, 1-inch by 3-inch, and cut pieces _C C_ to fit tightly between
the top back edge of the winders (or the ledge) and the pieces marked
_B B_ in section. This method makes a very sound and strong job of the
winders; and if the stuff is roughly planed, and blocks are glued on
each side of the short cross-pieces _O O O_, it is next to impossible
for the winders ever to spring or squeak. When the weight is carried in
this manner, the plasterer will have very little trouble in lathing so
that a graceful soffit will be made under the stairs.
[Illustration: Fig. 66. One Method of Making an Inside Well.]
[Illustration: Fig. 67. Construction and Trimming of 8-Inch Well-Hole.]
The manner of placing the main stringers of the carriage _S S_, is
shown at _A_, Fig. 69. Fig. 68 shows a complete half-space stair;
one-half of this, finished as shown, will answer well for a
quarter-space stair.
[Illustration: Fig. 68. Under Side of Half-Space Stair, with Carriages
and Cross-Carriages.]
Another method of forming a carriage for a stair is shown in Fig. 70.
This is a peculiar but very handsome stair, inasmuch as the first
and the last four steps are parallel, but the remainder _balance_ or
_dance_. The treads are numbered in this illustration; and the plan of
the handrail is shown extending from the scroll at the bottom of the
stairs to the landing on the second story. The trimmer _T_ at the top
of the stairs is also shown; and the rough strings or carriages, _R S_,
_R S_, _R S_, are represented by dotted lines.
This plan represents a stair with a curtail step, and a scroll handrail
resting over the curve of the curtail step. This type of stair is not
now much in vogue in this country, though it is adopted occasionally
in some of the larger cities. The use of heavy newel posts instead of
curtail steps, is the prevailing style at present.
In laying out geometrical stairs, the steps are arranged on principles
already described. The well-hole in the center is first laid down
and the steps arranged around it. In circular stairs with an open
well-hole, the handrail being on the inner side, the width of tread for
the steps should be set off at about 18 inches from the handrail, this
giving an approximately uniform rate of progress for anyone ascending
or descending the stairway. In stairs with the rail on the outside, as
sometimes occurs, it will be sufficient if the treads have the proper
width at the middle point of their length.
Where a flight of stairs will likely be subject to great stress and
wear, the carriages should be made much heavier than indicated in the
foregoing figures; and there may be cases when it will be necessary
to use iron bolts in the sides of the rough strings in order to give
them greater strength. This necessity, however, will arise only in the
case of stairs built in public buildings, churches, halls, factories,
warehouses, or other buildings of a similar kind. Sometimes, even in
house stairs it may be wise to strengthen the treads and risers by
spiking pieces of board to the rough string, ends up, fitting them
snugly against the under side of the tread and the back of the riser.
The method of doing this is shown in Fig. 71, in which the letter _O_
shows the pieces nailed to the string.
[Illustration: Fig. 69. Method of Reinforcing Stair.]
[Illustration: Fig. 70. Plan Showing One Method of Constructing
Carriage and Trimming Winding Stair.]
=Types of Stairs in Common Use.= In order to make the student
familiar with types of stairs in general use at the present day, plans
of a few of those most likely to be met with will now be given.
Fig. 72 is a plan of a straight stair, with an ordinary cylinder at
the top provided for a return rail on the landing. It also shows a
stretch-out stringer at the starting.
Fig. 73 is a plan of a stair with a landing and return steps.
Fig. 74 is a plan of a stair with an acute angular landing and cylinder.
Fig. 75 illustrates the same kind of stair as Fig. 74, the angle,
however, being obtuse.
Fig. 76 exhibits a stair having a half-turn with two risers on landings.
Fig. 77 is a plan of a quarter-space stair with four winders.
Fig. 78 shows a stair similar to Fig. 77, but with six winders.
[Illustration: Fig. 71. Reinforcing Treads and Risers by Blocks Nailed
to String.]
[Illustration: Fig. 72. Plan of Straight Stair with Cylinder at Top for
Return Rail.]
[Illustration: Fig. 73. Plan of Stair with Landing and Return Steps.]
Fig. 79 shows a stair having five dancing winders.
Fig. 80 is a plan of a half-space stair having five dancing winders and
a quarter-space landing.
Fig. 81 shows a half-space stair with dancing winders all around the
cylinder.
Fig. 82 shows a geometrical stair having winders all around the
cylinder.
Fig. 83 shows the plan and elevation of stairs which turn around a
central post. This kind of stair is frequently used in large stores
and in clubhouses and other similar places, and has a very graceful
appearance. It is not very difficult to build if properly planned.
The only form of stair not shown which the student may be called upon
to build, would very likely be one having an elliptical plan; but,
as this form is so seldom used—being found, in fact, only in public
buildings or great mansions—it rarely falls to the lot of the ordinary
workman to be called upon to design or construct a stairway of this
type.
[Illustration: Fig. 74. Plan of Stair with Acute-Angle Landing and
Cylinder.]
[Illustration: Fig. 75. Plan of Stair with Obtuse-Angle Landing and
Cylinder.]
[Illustration: Fig. 76. Half-Turn Stair with Two Risers on Landings.]
[Illustration: Fig. 77. Quarter-Space Stair with Four Winders.]
[Illustration: Fig. 78. Quarter-Space Stair with Six Winders.]
[Illustration: Fig. 79. Stair with Five Dancing Winders.]
[Illustration: Fig. 80. Half-Space Stair with Five Dancing Winders and
Quarter-Space Landing]
[Illustration: Fig. 81. Half-Space Stair with Dancing Winders all around
Cylinder.]
GEOMETRICAL STAIRWAYS AND HAND-RAILING
The term _geometrical_ is applied to stairways having any kind of curve
for a plan.
The rails over the steps are made continuous from one story to another.
The resulting winding or twisting pieces are called _wreaths_.
[Illustration: Fig. 82. Geometrical Stair with Winders all Around
Cylinder.]
=Wreaths.= The construction of wreaths is based on a few
geometrical problems—namely, the projection of straight and curved
lines into an oblique plane; and the finding of the angle of
inclination of the plane into which the lines and curves are projected.
This angle is called the _bevel_, and by its use the wreath is made to
twist.
[Illustration: Fig. 83. Plan and Elevation of Stairs Turning around a
Central Post.]
In Fig. 84 is shown an obtuse-angle plan; in Fig. 85, an acute-angle
plan; and in Fig. 86, a semicircle enclosed within straight lines.
=Projection.= A knowledge of how to project the lines and curves
in each of these plans into an oblique plane, and to find the angle of
inclination of the plane, will enable the student to construct any and
all kinds of wreaths.
The straight lines _a_, _b_, _c_, _d_ in the plan, Fig. 86, are known
as _tangents_; and the curve, the _central line_ of the plan wreath.
The straight line across from _n_ to _n_ is the _diameter_; and the
perpendicular line from it to the lines _c_ and _b_ is the _radius_.
A _tangent_ line may be defined as a line touching a curve without
cutting it, and is made use of in handrailing to square the joints of
the wreaths.
=Tangent System.= The _tangent system_ of handrailing takes its
name from the use made of the tangents for this purpose.
In Fig. 86, it is shown that the joints connecting the central line of
rail with the plan rails _w_ of the straight flights, are placed right
at the springing; that is, they are in line with the diameter of the
semicircle, and square to the side tangents _a_ and _d_.
The center joint of the crown tangents is shown to be square to
tangents _b_ and _c_. When these lines are projected into an oblique
plane, the joints of the wreaths can be made to butt square by applying
the bevel to them.
All handrail wreaths are assumed to rest on an oblique plane while
ascending around a well-hole, either in connecting two flights or in
connecting one flight to a landing, as the case may be.
In the simplest cases of construction, the wreath rests on an inclined
plane that inclines in one direction only, to either side of the
well-hole; while in other cases it rests on a plane that inclines to
two sides.
[Illustration: Fig. 84. Obtuse-Angle Plan.]
Fig. 87 illustrates what is meant by a plane inclining in one
direction. It will be noticed that the lower part of the figure is a
reproduction of the quadrant enclosed by the tangents _a_ and _b_ in
Fig. 86. The quadrant, Fig. 87, represents a central line of a wreath
that is to ascend from the joint on the plan tangent _a_ the height of
_h_ above the tangent _b_.
[Illustration: Fig. 85. Acute-Angle Plan.]
[Illustration: Fig. 86. Semicircular Plan.]
In Fig. 88, a view of Fig. 87 is given in which the tangents _a_ and
_b_ are shown in plan, and also the quadrant representing the plan
central line of a wreath. The curved line extending from _a_ to _h_
in this figure represents the development of the central line of the
plan wreath, and, as shown, it rests on an oblique plane inclining
to one side only—namely, to the side of the plan tangent _a_. The
joints are made square to the developed tangents _a_ and _m_ of the
inclined plane; it is for this purpose only that tangents are made use
of in wreath construction. They are shown in the figure to consist of
two lines, _a_ and _m_, which are two adjoining sides of a developed
section (in this case, of a square prism), the section being the
assumed inclined plane whereon the wreath rests in its ascent from _a_
to _h_. The joint at _h_, if made square to the tangent _m_, will be
a true, square butt-joint; so also will be the joint at _a_, if made
square to the tangent _a_.
In practical work it will be required to find the correct geometrical
angle between the two developed tangents _a_ and _m_; and here, again,
it may be observed that the finding of the correct angle between the
two developed tangents is the essential purpose of every tangent system
of handrailing.
[Illustration: Fig. 87. Illustrating Plane Inclined in One Direction
Only.]
[Illustration: Fig. 88. Plan Line of Rail Projected into Oblique Plane
Inclined to One Side Only.]
In Fig. 89 is shown the geometrical solution—the one necessary to find
the angle between the tangents as required on the face-mould to square
the joints of the wreath. The figure is shown to be similar to Fig. 87,
except that it has an additional portion marked “Section.” This section
is the true shape of the oblique plane whereon the wreath ascends, a
view of which is given in Fig. 88. It will be observed that one side of
it is the developed tangent _m_; another side, the developed tangent
_a″_ (= _a_). The angle between the two as here presented is the one
required on the face-mould to square the joints.
In this example, Fig. 89, owing to the plane being oblique in one
direction only, the shape of the section is found by merely drawing the
tangent _a″_ at right angles to the tangent _m_, making it equal in
length to the level tangent _a_ in the plan. By drawing lines parallel
to _a″_ and _m_ respectively, the form of the section will be found,
its outlines being the projections of the plan lines; and the angle
between the two tangents, as already said, is the angle required on the
face-mould to square the joints of the wreath.
The solution here presented will enable the student to find the correct
direction of the tangents as required on the face-mould to square
joints, in all cases of practical work where one tangent of a wreath is
level and the other tangent is inclined, a condition usually met with
in level-landing stairways.
Fig. 90 exhibits a condition of tangents where the two are equally
inclined. The plan here also is taken from Fig. 86. The inclination of
the tangents is made equal to the inclination of tangent _b_ in Fig.
86, as shown at _m_ in Figs. 87, 88, and 89.
[Illustration: Fig. 89. Finding Angle between Tangents.]
In Fig. 91, a view of Fig. 90 is given, showing clearly the inclination
of the tangents _c″_ and _d″_ over and above the plan tangents _c_
and _d_. The central line of the wreath is shown extending along the
sectional plane, over and above its plan lines, from one joint to the
other, and, at the joints, made square to the inclined tangents _c″_
and _d″_. It is evident from the view here given, that the condition
necessary to square the joint at each end would be to find the true
angle between the tangents _c″_ and _d″_, which would give the correct
direction to each tangent.
In Fig. 92 is shown how to find this angle correctly as required on the
face-mould to square the joints. In this figure is shown the same plan
as in Figs. 90 and 91, and the same inclination to the tangents as in
Fig. 90, so that, except for the portion marked “Section,” it would be
similar to Fig. 90.
To find the correct angle for the tangents of the face-mould, draw the
line _m_ from _d_, square to the inclined line of the tangents _c′_
_d″_; revolve the bottom inclined tangent _c′_ to cut line _m_ in _n_,
where the joint is shown fixed; and from this point draw the line _c″_
to _w_. The intersection of this line with the upper tangent _d″_ forms
the correct angle as required on the face-mould. By drawing the joints
square to these two lines, they will butt square with the rail that is
to connect with them, or to the joint of another wreath that may belong
to the cylinder or well-hole.
[Illustration: Fig. 90. Two Tangents Equally Inclined.]
[Illustration: Fig. 91. Plan Lines Projected into Oblique Plane Inclined
to Two Sides.]
[Illustration: Fig. 92. Finding Angle between Tangents.]
Fig. 93 is another view of these tangents in position placed over and
above the plan tangents of the well-hole. It will be observed that
this figure is made up of Figs. 88 and 91 combined. Fig. 88, as here
presented, is shown to connect with a level-landing rail at _a_. The
joint having been made square to the level tangent, _a_ will butt
square to a square end of the level rail. The joint at _h_ is shown to
connect the two wreaths and is made square to the inclined tangent
_m_ of the lower wreath, and also square to the inclined tangent _c″_
of the upper wreath; the two tangents, aligning, guarantee a square
butt-joint. The upper joint is made square to the tangent _d″_, which
is here shown to align with the rail of the connecting flight; the
joint will consequently butt square to the end of the rail of the
flight above.
The view given in this diagram is that of a wreath starting from a
level landing, and winding around a well-hole, connecting the landing
with a flight of stairs leading to a second story. It is presented
to elucidate the use made of tangents to square the joints in wreath
construction. The wreath is shown to be in two sections, one extending
from the level-landing rail at _a_ to a joint in the center of the
well-hole at _h_, this section having one level tangent _a_ and one
inclined tangent _m_; the other section is shown to extend from _h_ to
_n_, where it is butt-jointed to the rail of the flight above.
[Illustration: Fig. 93. Laying Out Line of Wreath to Start from
Level-Landing Rail. Wind around Well-Hole, and Connect at Landing with
Flight to Upper Story.]
This figure clearly shows that the joint at _a_ of the bottom
wreath—owing to the tangent _a_ being level and therefore aligning
with the level rail of the landing—will be a true butt-joint; and
that the joint at _h_, which connects the two wreaths, will also be a
true butt-joint, owing to it being made square to the tangent _m_ of
the bottom wreath and to the tangent _c″_ of the upper wreath, both
tangents having the same inclination; also the joint at _n_ will butt
square to the rail of the flight above, owing to it being made square
to the tangent _d″_, which is shown to have the same inclination as the
rail of the flight adjoining.
As previously stated, the use made of tangents is to square the joints
of the wreaths; and in this diagram it is clearly shown that the way
they can be made of use is by giving each tangent its true direction.
How to find the true direction, or the angle between the tangents _a_
and _m_ shown in this diagram, was demonstrated in Fig. 89; and how to
find the direction of the tangents _c″_ and _d″_ was shown in Fig. 92.
[Illustration: Fig. 94. Tangents Unfolded to Find Their Inclination.]
Fig. 94 is presented to help further toward an understanding of
the tangents. In this diagram they are unfolded; that is, they are
stretched out for the purpose of finding the inclination of each one
over and above the plan tangents. The side plan tangent _a_ is shown
stretched out to the floor line, and its elevation _a′_ is a level
line. The side plan tangent _d_ is also stretched out to the floor
line, as shown by the arc _n′ m′_. By this process the plan tangents
are now in one straight line on the floor line, as shown from _w_ to
_m′_. Upon each one, erect a perpendicular line as shown, and from
_m′_ measure to _n_, the height the wreath is to ascend around the
well-hole. In practice, the number of risers in the well-hole will
determine this height.
[Illustration: Fig. 95. Well-Hole Connecting Two Flights, with Two
Wreath-Pieces, Each Containing Portions of Unequal Pitch.]
Now, from point _n_, draw a few treads and risers as shown; and along
the nosing of the steps, draw the pitch-line; continue this line over
the tangents _d″_, _c″_, and _m_, down to where it connects with the
bottom level tangent, as shown. This gives the pitch or inclination
to the tangents over and above the well-hole. The same line is shown
in Fig. 93, folded around the well-hole, from _n_, where it connects
with the flight at the upper end of the well-hole, to _a_, where it
connects with the level-landing rail at the bottom of the well-hole. It
will be observed that the upper portion, from joint _n_ to joint _h_,
over the tangents _c″_ and _d″_, coincides with the pitch-line of the
same tangents as presented in Fig. 92, where they are used to find the
true angle between the tangents as it is required on the face-mould to
square the joints of the wreath at _h_.
In Fig. 89 the same pitch is shown given to tangent _m_ as in Fig. 94;
and in both figures the pitch is shown to be the same as that over and
above the upper connecting tangents _c″_ and _d″_, which is a necessary
condition where a joint, as shown at _h_ in Figs. 93 and 94, is to
connect two pieces of wreath as in this example.
In Fig. 94 are shown the two face-moulds for the wreaths, placed upon
the pitch-line of the tangents over the well-hole. The angles between
the tangents of the face-moulds have been found in this figure by the
same method as in Figs. 89 and 92, which, if compared with the present
figure, will be found to correspond, excepting only the curves of the
face-moulds in Fig. 94.
The foregoing explanation of the tangents will give the student a
fairly good idea of the use made of tangents in wreath construction.
The treatment, however, would not be complete if left off at
this point, as it shows how to handle tangents under only two
conditions—namely, first, when one tangent inclines and the other is
level, as at _a_ and _m_; second, when both tangents incline, as shown
at _c″_ and _d″_.
[Illustration: Fig. 96. Finding Angle between Tangents for Bottom
Wreath of Fig. 95.]
[Illustration: Fig. 97. Finding Angle between Tangents for Upper Wreath
of Fig. 95.]
In Fig. 95 is shown a well-hole connecting two flights, where two
portions of unequal pitch occur in both pieces of wreath. The first
piece over the tangents _a_ and _b_ is shown to extend from the
square end of the straight rail of the bottom flight, to the joint in
the center of the well-hole, the bottom tangent _a″_ in this wreath
inclining more than the upper tangent _b″_. The other piece of wreath
is shown to connect with the bottom one at the joint _h″_ in the
center of the well-hole, and to extend over tangents _c″_ and _d″_ to
connect with the rail of the upper flight. The relative inclination
of the two tangents in this wreath, is the reverse of that of the two
tangents of the lower wreath. In the lower piece, the bottom tangent
_a″_, as previously stated, inclines considerably more than does the
upper tangent _b″_; while in the upper piece, the bottom tangent _c″_
inclines considerably less than the upper tangent _d″_.
The question may arise: What causes this? Is it for variation in the
inclination of the tangents over the well-hole? It is simply owing to
the tangents being used in handrailing to square the joints.
The inclination of the bottom tangent _a″_ of the bottom wreath is
clearly shown in the diagram to be determined by the inclination of the
bottom flight. The joint at _a″_ is made square to both the straight
rail of the flight and to the bottom tangent of the wreath; the rail
and tangent, therefore, must be equally inclined, otherwise the joint
will not be a true butt-joint. The same remarks apply to the joint at
5, where the upper wreath is shown jointed to the straight rail of
the upper flight. In this case, tangent _d″_ must be fixed to incline
conformably to the inclination of the upper rail; otherwise the joint
at 5 will not be a true butt-joint.
[Illustration: Fig. 98. Diagram of Tangents and Face-Mould for Stair
with Well-Hole at Upper Landing.]
The same principle is applied in determining the pitch or inclination
over the crown tangents _b″_ and _c″_. Owing to the necessity of
jointing the two wreaths, as shown at _h_, these two tangents must have
the same inclination, and therefore must be fixed, as shown from 2 to
4, over the crown of the well-hole.
The tangents as here presented are those of the elevation, not of the
face-mould. Tangent _a″_ is the elevation of the side plan tangent
_a_; tangents _b″_ and _c″_ are shown to be the elevations of the plan
tangents _b_ and _c_; so, also, is the tangent _d″_ the elevation of
the side plan tangent _d_.
[Illustration: Fig. 99. Drawing Mould when One Tangent Is Level and One
Inclined over Right-Angled Plan.]
If this diagram were folded, as Fig. 94 was shown to be in Fig. 93, the
tangents of the elevation—namely, _a″_, _b″_, _c″_, _d″_—would stand
over and above the plan tangents _a_, _b_, _c_, _d_ of the well-hole.
In practical work, this diagram must be drawn full size. It gives the
correct length to each tangent as required on the face-mould, and
furnishes also the data for the layout of the mould.
Fig. 96 shows how to find the angle between the tangents of the
face-mould for the bottom wreath, which, as shown in Fig. 95, is to
span over the first plan quadrant _a b_. The elevation tangents _a″_
and _b″_, as shown, will be the tangents of the mould. To find the
angle between the tangents, draw the line _a h_ in Fig. 96; and from
_a_, measure to 2 the length of the bottom tangent _a″_ in Fig. 95;
the length from 2 to _h_, Fig. 96, will equal the length of the upper
tangent _b″_, Fig. 95.
From 2 to 1, measure a distance equal to 2-1 in Fig. 95, the latter
being found by dropping a perpendicular from _w_ to meet the tangent
_b″_ extended. Upon 1, erect a perpendicular line; and placing the
dividers on 2, extend to _a_; turn over to the perpendicular at _a″_;
connect this point with 2, and the line will be the bottom tangent as
required on the face-mould. The upper tangent will be the line 2-_h_,
and the angle between the two lines is shown at 2. Make the joint at
_h_ square to 2-_h_, and at _a″_ square to _a″_-2.
[Illustration: Fig. 100. Plan of Curved Steps and Stringer at Bottom of
Stair.]
The mould as it appears in Fig. 96 is complete, except the curve, which
is comparatively a small matter to put on, as will be shown further
on. The main thing is to find the angle between the tangents, which is
shown at 2, to give them the direction to square the joints.
In Fig. 97 is shown how to find the angle between the tangents _c″_
and _d″_ shown in Fig. 95, as required on the face-mould. On the line
_h_-5, make _h_-4 equal to the length of the bottom tangent of the
wreath, as shown at _h″_-4 in Fig. 95; and 4-5 equal to the length of
the upper tangent _d″_. Measure from 4 the distance shown at 4-6 in
Fig. 95, and place it from 4 to 6 as shown in Fig. 97; upon 6 erect a
perpendicular line. Now place the dividers on 4; extend to _h_; turn
over to cut the perpendicular in _h″_; connect this point with 4, and
the angle shown at 4 will be the angle required to square the joints of
the wreath as shown at _h″_ and 5, where the joint at 5 is shown drawn
square to the line 4-5, and the joint at _h″_ square to the line 4 _h″_.
Fig. 98 is a diagram of tangents and face-mould for a stairway having
a well-hole at the top landing. The tangents in this example will be
two equally inclined tangents for the bottom wreath; and for the top
wreath, one inclined and one level, the latter aligning with the level
rail of the landing.
[Illustration: Fig. 101. Finding Angle between Tangents for Squaring
Joints of Ramped Wreath.]
The face-mould, as here presented, will further help toward an
understanding of the layout of face-moulds as shown in Figs. 96 and
97. It will be observed that the pitch of the bottom rail is continued
from _a″_ to _b″_, a condition caused by the necessity of jointing the
wreath to the end of the straight rail at _a″_, the joint being made
square to both the straight rail and the bottom tangent _a″_. From
_b″_ a line is drawn to _d″_, which is a fixed point determined by the
number of risers in the well-hole. From point _d″_, the level tangent
_d″_ 5 is drawn in line with the level rail of the landing; thus the
pitch-line of the tangents over the well-hole is found, and, as was
shown in the explanation of Fig. 95, the tangents as here presented
will be those required on the face-mould to square the joints of the
wreath.
In Fig. 98 the tangents of the face-mould for the bottom wreath are
shown to be _a″_ and _b″_. To place tangent _a″_ in position on the
face-mould, it is revolved, as shown by the arc, to _m_, cutting a line
previously drawn from _w_ square to the tangent _b″_ extended. Then, by
connecting _m_ to _b″_, the bottom tangent is placed in position on the
face-mould. The joint at _m_ is to be made square to it; and the joint
at _c_, the other end of the mould, is to be made square to the tangent
_b″_.
[Illustration: Fig. 102. Bottom Steps with Obtuse-Angle Plan.]
[Illustration: Fig. 103. Developing Face-Mould, Obtuse-Angle Plan.]
The upper piece of wreath in this example is shown to have tangent _c″_
inclining, the inclination being the same as that of the upper tangent
_b″_ of the bottom wreath, so that the joint at _c″_, when made square
to both tangents, will butt square when put together. The tangent _d″_
is shown to be level, so that the joint at 5, when squared with it,
will butt square with the square end of the level-landing rail. The
level tangent is shown revolved to its position on the face-mould,
as from 5 to 2. In this last position, it will be observed that its
angle with the inclined tangent _c″_ is a right angle; and it should
be remembered that in every similar case where one tangent inclines
and one is level over a square-angle plan tangent, the angle between
the two tangents will be a right angle on the face-mould. A knowledge
of this principle will enable the student to draw the mould for this
wreath, as shown in Fig. 99, by merely drawing two lines perpendicular
to each other, as _d″_ 5 and _d″ c″_, equal respectively to the level
tangent _d″_ 5 and the inclined tangent _c″_ in Fig. 98. The joint at 5
is to be made square to _d″_ 5; and that at _c″_, to _d″ c″_. Comparing
this figure with the face-mould as shown for the upper wreath in Fig.
98, it will be observed that both are alike.
In practical work the stair-builder is often called upon to deal with
cases in which the conditions of tangents differ from all the examples
thus far given. An instance of this sort is shown in Fig. 100, in which
the angles between the tangents on the plan are acute. In all the
preceding examples, the tangents on the plan were at right angles; that
is, they were square to one another.
[Illustration: Fig. 104. Cutting Wreath from Plank.]
[Illustration: Fig. 105. Wreath Twisted, Ready to be Moulded.]
Fig. 100 is a plan of a few curved steps placed at the bottom of a
stairway with a curved stringer, which is struck from a center _o_. The
plan tangents _a_ and _b_ are shown to form an acute angle with each
other. The rail above a plan of this design is usually ramped at the
bottom end, where it intersects the newel post, and, when so treated,
the bottom tangent _a_ will have to be level.
[Illustration: Fig. 106. Twisted Wreath Raised to Position, with Sides
Plumb.]
In Fig. 101 is shown how to find the angle between the tangents on
the face-mould that gives them the correct direction for squaring the
joints of the wreath when it is determined to have it ramped. This
figure must be drawn full size. Usually an ordinary drawing-board will
answer the purpose. Upon the board, reproduce the plan of the tangents
and curve of the center line of rail as shown in Fig. 100. Measure the
height of 5 risers, as shown in Fig. 101, from the floor line to 5; and
draw the pitch of the flight adjoining the wreath, from 5 to the floor
line. From the newel, draw the dotted line to _w_, square to the floor
line; from _w_, draw the line _w m_, square to the pitch-line _b″_.
Now take the length of the bottom level tangent on a trammel, or on
dividers if large enough, and extend it from _n_ to _m_, cutting the
line drawn previously from _w_, at _m_. Connect _m_ to _n_ as shown by
the line _a″_. The intersection of this line with _b″_ determines the
angle between the two tangents _a″_ and _b″_ of the face-mould, which
gives them the correct direction as required on the face-mould for
squaring the joints. The joint at _m_ is made square to tangent _a″_;
and the joint at 5, to tangent _b″_.
[Illustration: Fig. 107. Finding Bevel, Bottom Tangent Inclined, Top
One Level.]
[Illustration: Fig. 108. Application of Bevels in Fitting Wreath to
Rail.]
In Fig. 102 is presented an example of a few steps at the bottom of a
stairway in which the tangents of the plan form an obtuse angle with
each other. The curve of the central line of the rail in this case
will be less than a quadrant, and, as shown, is struck from the center
_o_, the curve covering the three first steps from the newel to the
springing.
In Fig. 103 is shown how to develop the tangents of the face-mould.
Reproduce the tangents and curve of the plan in full size. Fix point 3
at a height equal to 3 risers from the floor line; at this point place
the pitch-board of the flight to determine the pitch over the curve
as shown from 3 through _b″_ to the floor line. From the newel, draw
a line to _w_, square to the floor line; and from _w_, square to the
pitch-line _b″_, draw the line _w m_; connect _m_ to _n_. This last
line is the development of the bottom plan tangent _a_; and the line
_b″_ is the development of the plan tangent _b_; and the angle between
the two lines _a″_ and _b″_ will give each line its true direction as
required on the face-mould for squaring the joints of the wreath, as
shown at _m_ to connect square with the newel, and at 3 to connect
square to the rail of the connecting flight.
[Illustration: Fig. 109. Face-Mould and Bevel for Wreath, Bottom
Tangent Level, Top One Inclined.]
The wreath in this example follows the nosing line of the steps without
being ramped as it was in the examples shown in Figs. 100 and 101. In
those figures the bottom tangent _a_ was level, while in Fig. 103 it
inclines equal to the pitch of the upper tangent _b″_ and of the flight
adjoining. In other words, the method shown in Fig. 101 is applied to
a construction in which the wreath is ramped; while in Fig. 103 the
method is applicable to a wreath following the nosing line all along
the curve to the newel.
The stair-builder is supposed to know how to construct a wreath under
both conditions, as the conditions are usually determined by the
Architect.
[Illustration: Fig. 110. Finding Bevels for Wreath with Two Equally
Inclined Tangents.]
The foregoing examples cover all conditions of tangents that are likely
to turn up in practice, and, if clearly understood, will enable the
student to lay out the face-moulds for all kinds of curves.
=Bevels to Square the Wreaths.= The next process in the
construction of a wreath that the handrailer will be called upon to
perform, is to find the bevels that will, by being applied to each end
of it, give the correct angle to _square_ or _twist_ it when winding
around the well-hole from one flight to another flight, or from a
flight to a landing, as the case may be.
[Illustration: Fig. 111. Application of Bevels to Wreath Ascending on
Plane Inclined Equally in Two Directions.]
[Illustration: Fig. 112. Finding Bevel Where Upper Tangent Inclines More
Than Lower One.]
The wreath is first cut from the plank square to its surface as shown
in Fig. 104. After the application of the bevels, it is twisted, as
shown in Fig. 105, ready to be moulded; and when in position, ascending
from one end of the curve to the other end, over the inclined plane of
the section around the well-hole, its sides will be plumb, as shown in
Fig. 106 at _b_. In this figure, as also in Fig. 105, the wreath _a_
lies in a horizontal position in which its sides appear to be out of
plumb as much as the bevels are out of plumb. In the upper part of the
figure, the wreath _b_ is shown placed in its position upon the plane
of the section, where its sides are seen to be plumb. It is evident, as
shown in the relative position of the wreath in this figure, that, if
the bevel is the correct angle of the plane of the section whereon the
wreath _b_ rests in its ascent over the well-hole, the wreath will in
that case have its sides plumb all along when in position. It is for
this purpose that the bevels are needed.
[Illustration: Fig. 113. Finding Bevel Where Upper Tangent Inclines
Less Than Lower One.]
A method of finding the bevels for _all wreaths_ (which is considered
rather difficult) will now be explained:
_First Case._ In Fig. 107 is shown a case where the bottom tangent of a
wreath is inclining, and the top one level, similar to the top wreath
shown in Fig. 98. It has already been noted that the plane of the
section for this kind of wreath inclines to one side only; therefore
one bevel only will be required to square it, which is shown at _d_,
Fig. 107. A view of this plane is given in Fig. 108; and the bevel _d_,
as there shown, indicates the angle of the inclination, which also
is the bevel required to square the end _d_ of the wreath. The bevel
is shown applied to the end of the landing rail in exactly the same
manner in which it is to be applied to the end of the wreath. The true
bevel for this wreath is found at the upper angle of the pitch-board.
At the end _a_, as already stated, no bevel is required, owing to the
plane inclining in one direction only. Fig. 109 shows a face-mould and
bevel for a wreath with the bottom tangent level and the top tangent
inclining, such as the piece at the bottom connecting with the landing
rail in Fig. 94.
[Illustration: Fig. 114. Finding Bevel Where Tangents Incline Equally
over Obtuse-Angle Plan.]
[Illustration: Fig. 115. Same Plan as in Fig. 114, but with Bottom
Tangent Level.]
_Second Case._ It may be required to find the bevels for a wreath
having two equally inclined tangents. An example of this kind also is
shown in Fig. 94, where both the tangents _c″_ and _d″_ of the upper
wreath incline equally. Two bevels are required in this case, because
the plane of the section is inclined in two directions; but, owing
to the inclinations being alike, it follows that the two will be the
same. They are to be applied to both ends of the wreath, and, as shown
in Fig. 105, in the same direction—namely, toward the inside of the
wreath for the bottom end, and toward the outside for the upper end.
[Illustration: Fig. 116. Finding Bevels for Wreath of Fig. 115.]
In Fig. 110 the method of finding the bevels is shown. A line is drawn
from _w_ to _c″_, square to the pitch of the tangents, and turned over
to the ground line at _h_, which point is connected to _a_ as shown.
The bevel is at _h_. To show that equal tangents have equal bevels, the
line _m_ is drawn, having the same inclination as the bottom tangent
_c″_, but in another direction. Place the dividers on _o′_, and turn to
touch the lines _d″_ and _m_, as shown by the semicircle. The line from
_o′_ to _n_ is equal to the side plan tangent _w a_, and both the
bevels here shown are equal to the one already found. They represent
the angle of inclination of the plane whereon the wreath ascends, a
view of which is given in Fig. 111, where the plane is shown to incline
equally in two directions. At both ends is shown a section of a rail;
and the bevels are applied to show how, by means of them, the wreath is
_squared_ or _twisted_ when winding around the well-hole and ascending
upon the plane of the section. The view given in this figure will
enable the student to understand the nature of the bevels found in Fig.
110 for a wreath having two equally inclined tangents; also for all
other wreaths of equally inclined tangents, in that every wreath in
such case is assumed to rest upon an inclined plane in its ascent over
the well-hole, the bevel in every case being the angle of the inclined
plane.
[Illustration: Fig. 117. Upper Tangent Inclined. Lower Tangent Level,
Over Acute-Angle Plan.]
_Third Case._ In this example, two unequal tangents are given, the
upper tangent inclining more than the bottom one. The method shown in
Fig. 110 to find the bevels for a wreath with two equal tangents, is
applicable to all conditions of variation in the inclination of the
tangents. In Fig. 112 is shown a case where the upper tangent _d″_
inclines more than the bottom one _c″_. The method in all cases is to
continue the line of the upper tangent _d″_, Fig. 112, to the ground
line as shown at _n_; from _n_, draw a line to _a_, which will be the
horizontal trace of the plane. Now, from _o_, draw a line parallel to
_a n_, as shown from _o_ to _d_, upon _d_, erect a perpendicular line
to cut the tangent _d″_, as shown, at _m_; and draw the line _m u o″_.
Make _u o″_ equal to the length of the plan tangent as shown by the
arc from _o_. Put one leg of the dividers on _u_; extend to touch the
upper tangent _d″_, and turn over to 1; connect 1 to _o″_; the bevel at
1 is to be applied to tangent _d″_. Again place the dividers on _u_;
extend to the line _h_, and turn over to 2 as shown; connect 2 to _o″_,
and the bevel shown at 2 will be the one to apply to the bottom tangent
_c″_. It will be observed that the line _h_ represents the bottom
tangent. It is the same length and has the same inclination. An example
of this kind of wreath was shown in Fig. 95, where the upper tangent
_d″_ is shown to incline more than the bottom tangent _c″_ in the top
piece extending from _h″_ to 5. Bevel 1, found in Fig. 112, is the real
bevel for the end 5; and bevel 2, for the end _h″_ of the wreath shown
from _h″_ to 5 in Fig. 95.
[Illustration: Fig. 118. Finding Bevels for Wreath of Plan, Fig. 117.]
_Fourth Case._ In Fig. 113 is shown how to find the bevels for a
wreath when the upper tangent inclines less than the bottom tangent.
This example is the reverse of the preceding one; it is the condition
of tangents found in the bottom piece of wreath shown in Fig. 95. To
find the bevel, continue the upper tangent _b″_ to the ground line, as
shown at _n_; connect _n_ to _a_, which will be the horizontal trace
of the plane. From _o_, draw a line parallel to _n a_, as shown from
_o_ to _d_; upon _d_, erect a perpendicular line to cut the continued
portion of the upper tangent _b″_ in _m_; from _m_, draw the line _m
u o″_ across as shown. Now place the dividers on _u_; extend to touch
the upper tangent, and turn over to 1, connect 1 to _o″_; the bevel at
1 will be the one to apply to the tangent _b″_ at _h_, where the two
wreaths are shown connected in Fig. 95. Again place the dividers on
_u_; extend to touch the line _c_; turn over to 2; connect 2 to _o″_;
the bevel at 2 is to be applied to the bottom tangent _a″_ at the joint
where it is shown to connect with the rail of the flight.
_Fifth Case._ In this case we have two equally inclined tangents over
an obtuse-angle plan. In Fig. 102 is shown a plan of this kind; and in
Fig. 103, the development of the face-mould.
In Fig. 114 is shown how to find the bevel. From _a_, draw a line to
_a′_, square to the ground line. Place the dividers on _a′_; extend to
touch the pitch of tangents, and turn over as shown to _m_; connect _m_
to _a_. The bevel at _m_ will be the only one required for this wreath,
but it will have to be applied to both ends, owing to the two tangents
being inclined.
_Sixth Case._ In this case we have one tangent inclining and one
tangent level, over an acute-angle plan.
[Illustration: Fig. 119. Laying Out Curves on Face-Mould with Pins and
String.]
In Fig. 115 is shown the same plan as in Fig. 114; but in this case the
bottom tangent _a″_ is to be a level tangent. Probably this condition
is the most commonly met with in wreath construction at the present
time. A small curve is considered to add to the appearance of the stair
and rail; and consequently it has become almost a “fad” to have a
little curve or stretch-out at the bottom of the stairway, and in most
cases the rail is ramped to intersect the newel at right angles instead
of at the pitch of the flight. In such a case, the bottom tangent _a″_
will have to be a level tangent, as shown at _a″_ in Fig. 115, the
pitch of the flight being over the plan tangent _b_ only.
To find the bevels when tangent _b″_ inclines and tangent _a″_ is
level, make _a c_ in Fig. 116 equal to _a c_ in Fig. 115. This line
will be the base of the two bevels. Upon _a_, erect the line _a w m_ at
right angles to _a c_; make _a w_ equal to _o w_ in Fig. 115; connect
_w_ and _c_; the bevel at _w_ will be the one to apply to tangent _b″_
at _n_ where the wreath is joined to the rail of the flight. Again,
make _a m_ in Fig. 116 equal the distance shown in Fig. 115 between _w_
and _m_, which is the full height over which tangent _b″_ is inclined;
connect _m_ to _c_ in Fig. 116, and at _m_ is the bevel to be applied
to the level tangent _a″_.
[Illustration: Fig. 120. Simple Method of Drawing Curves on Face-Mould.]
[Illustration: Fig. 121. Tangents, Bevels, Mould-Curves, etc., from
Bottom Wreath of Fig. 95, In which Upper Tangent Inclines Less than
Lower One.]
_Seventh Case._ In this case, illustrated in Fig. 117, the upper
tangent _b″_ is shown to incline, and the bottom tangent _a″_ to be
level, over an acute-angle plan. The plan here is the same as that in
Fig. 100, where a curve is shown to stretch out from the line of the
straight stringer at the bottom of a flight to a newel, and is large
enough to contain five treads, which are gracefully rounded to cut the
curve of the central line of rail in 1, 2, 3, 4. This curve also may be
used to connect a landing rail to a flight, either at top or bottom,
when the plan is acute-angled, as will be shown further on.
[Illustration: Fig. 122. Developed Section of Plane Inclining Unequally
in Two Directions.]
[Illustration: Fig. 123. Arranging Risers around Well-Hole on Level
Landing Stair, with Radius of Central Line of Rail One-Half Width of
Tread.]
To find the bevels—for there will be two bevels necessary for this
wreath, owing to one tangent _b″_ being inclined and the other tangent
_a″_ being level—make _a c_, Fig. 118, equal to _a c_ in Fig. 117,
which is a line drawn square to the ground line from the newel and
shown in all preceding figures to have been used for the base of a
triangle containing the bevel. Make _a w_ in Fig. 118 equal to _w o_
in Fig. 117, which is a line drawn square to the inclined tangent _b″_
from _w_; connect _w_ and _c_ in Fig. 118. The bevel shown at _w_ will
be the one to be applied to the joint 5 on tangent _b″_, Fig. 117.
Again, make _a m_ in Fig. 118 equal to the distance shown in Fig. 117
between the line representing the level tangent and the line _m′_ 5,
which is the height that tangent _b″_ is shown to rise; connect _m_ to
_c_ in Fig. 118; the bevel shown at _m_ is to be applied to the end
that intersects with the newel as shown at _m_ in Fig. 117.
The wreath is shown developed in Fig. 101 for this case; so that, with
Fig. 100 for plan, Fig. 101 for the development of the wreath, and
Figs. 117 and 118 for finding the bevels, the method of handling any
similar case in practical work can be found.
=How to Put the Curves on the Face-Mould.= It has been shown how
to find the angle between the tangents of the face-mould, and that
the angle is for the purpose of squaring the joints at the ends of
the wreath. In Fig. 119 is shown how to lay out the curves by means
of pins and a string—a very common practice among stair-builders. In
this example the face-mould has equal tangents as shown at _c″_ and
_d″_. The angle between the two tangents is shown at _m_ as it will be
required on the face-mould. In this figure a line is drawn from _m_
parallel to the line drawn from _h_, which is marked in the diagram as
“Directing Ordinate of Section.” The line drawn from _m_ will contain
the minor axes; and a line drawn through the corner of the section at
3 will contain the major axes of the ellipses that will constitute the
curves of the mould.
[Illustration: Fig. 124. Arrangement of Risers Around Well-Hole with
Radius Larger Than One-Half Width of Tread.]
[Illustration: Fig. 125. Arrangement of Risers around Well-Hole, with
Risers Spaced Full Width of Tread.]
[Illustration: Fig. 126. Plan of Stair Shown in Fig. 123.]
[Illustration: Fig. 127. Plan of Stair Shown in Fig. 124.]
[Illustration: Fig. 128. Plan of Stair Shown in Fig. 125.]
[Illustration: Fig. 129. Drawing Face-Mould for Wreath from
Pitch-Board.]
The _major_ is to be drawn square to the _minor_, as shown. Place, from
point 3, the circle shown on the minor, at the same distance as the
circle in the plan is fixed from the point _o_. The diameter of this
circle indicates the width of the curve at this point The width at each
end is determined by the bevels. The distance _a b_, as shown upon the
long edge of the bevel, is equal to ½ the width of the mould, and is
the hypotenuse of a right-angled triangle whose base is ½ the width of
the rail. By placing this dimension on each side of _n_, as shown at
_b_ and _b_, and on each side of _h″_ on the other end of the mould,
as shown also at _b_ and _b_, we obtain the points _b_ 2 _b_ on the
inside of the curve, and the points _b_ 1 _b_ on the outside. It will
now be necessary to find the elliptical curves that will contain these
points; and before this can be done, the exact length of the _minor_
and _major_ axes respectively must be determined. The length of the
minor axis for the inside curve will be the distance shown from 3 to 2;
and its length for the outside will be the distance shown from 3 to 1.
To find the length of the major axis for the inside, take the length of
half the minor for the inside on the dividers: place one leg on _b_,
extend to cut the major in _z_, continue to the minor as shown at _k_.
The distance from _b_ to _k_ will be the length of the semi-major axis
for the inside curve.
[Illustration: Fig. 130. Development of Face-Mould for Wreath
Connecting Rail of Flight with Level-Landing Rail.]
To draw the curve, the points or _foci_ where the pins are to be fixed
must be found on the major axis. To find these points, take the length
of _b k_ (which is, as previously found, the exact length of the
semi-major for the inside curve) on the dividers; fix one leg at 2,
and describe the arc _Y_, cutting the major where the pins are shown
fixed, at _o_ and _o_. Now take a piece of string long enough to form
a loop around the two and extending, when tight, to 2, where the pencil
is placed; and, keeping the string tight, sweep the curve from _b_ to
_b_.
[Illustration: Fig. 131. Arranging Risers in Quarter-Turn between Two
Flights.]
The same method, for finding the _major_ and _foci_ for the outside
curve, is shown in the diagram. The line drawn from _b_ on the outside
of the joint at _n_, to _w_, is the semi-major for the outside curve;
and the points where the outside pins are shown on the major will be
the _foci_.
[Illustration: Fig. 132. Arrangement of Risers around Quarter-Turn
Giving Tangents Equal Pitch with Connecting Flight.]
[Illustration: Fig. 133. Finding Bevel for Wreath of Plan, Fig. 132.]
To draw the curves of the mould according to this method, which is a
scientific one, may seem a complicated problem; but once it is
understood, it becomes very simple. A simpler way to draw them,
however, is shown in Fig. 120.
The width on the minor and at each end will have to be determined by
the method just explained in connection with Fig. 119. In Fig. 120, the
points _b_ at the ends, and the points in which the circumference of
the circle cuts the minor axis, will be points contained in the curves,
as already explained. Now take a flexible lath; bend it to touch _b_,
_z_, and _b_ for the inside curve, and _b_, _w_, and _b_ for the
outside curve. This method is handy where the curve is comparatively
flat, as in the example here shown; but where the mould has a sharp
curvature, as in case of the one shown in Fig. 101, the method shown in
Fig. 119 must be adhered to.
[Illustration: Fig. 134. Well-Hole with Riser in Center. Tangents of
Face-Mould, and Central Line of Rail, Developed.]
With a clear knowledge of the above two methods, the student will be
able to put curves on any mould.
The mould shown in these two diagrams, Figs. 119 and 120, is for the
upper wreath, extending from _h_ to _n_ in Fig. 94. A practical
handrailer would draw only what is shown in Fig. 120. He would take the
lengths of tangents from Fig. 94, and place them as shown at _h m_ and
_m n_. By comparing Fig. 120 with the tangents of the upper wreath in
Fig. 94, it will be easy for the student to understand the remaining
lines shown in Fig. 120. The bevels are shown applied to the mould in
Fig. 105, to give it the twist. In Fig. 106, is shown how, after the
rail is twisted and placed in position over and above the quadrant _c
d_ in Fig. 94, its sides will be plumb.
[Illustration: Fig. 135. Arrangement of Risers in Stair with
Obtuse-Angle Plan.]
[Illustration: Fig. 136. Arrangement of Risers in Obtuse-Angle Plan,
Giving Equal Pitch over Tangents and Flights. Face-Mould Developed.]
In Fig. 121 are shown the tangents taken from the bottom wreath in Fig.
95. It was shown how to develop the section and find the angle for the
tangents in the face-mould, in Fig. 113. The method shown in Fig. 119
for putting on the curves, would be the most suitable.
[Illustration: Fig. 137. Arrangement of Risers in Flight with Curve at
Landing.]
Fig. 121 is presented more for the purposes of study than as a method
of construction. It contains all the lines made use of to find the
developed section of a plane inclining unequally in two different
directions, as shown in Fig. 122.
[Illustration: Fig. 138. Development of Face-Moulds for Plan, Fig. 137.]
=Arrangement of Risers in and around Well-Hole.= An important
matter in wreath construction is to have a knowledge of how to arrange
the risers in and around a well-hole. A great deal of labor and
material is saved through it; also a far better appearance to the
finished rail may be secured.
In level-landing stairways, the easiest example is the one shown in
Fig. 123, in which the radius of the central line of rail is made equal
to one-half the width of a tread. In the diagram the radius is shown
to be 5 inches, and the treads 10 inches. The risers are placed in the
springing, as at _a_ and _a_. The elevation of the tangents by this
arrangement will be, as shown, one level and one inclined, for each
piece of wreath. When in this position, there is no trouble in finding
the angle of the tangent as required on the face-mould, owing to that
angle, as in every such case, being a right angle, as shown at _w_;
also no special bevel will have to be found, because the upper bevel of
the pitch-board contains the angle required.
The same results are obtained in the example shown in Fig. 124, in
which the radius of the well-hole is larger than half the width of a
tread, by placing the riser _a_ at a distance from _c_ equal to half
the width of a tread, instead of at the springing as in the preceding
example.
In Fig. 125 is shown a case where the risers are placed at a distance
from _c_ equal to a full tread, the effect in respect to the tangents
of the face-mould and bevel being the same as in the two preceding
examples. In Fig. 126 is shown the plan of Fig. 123; in Fig. 127,
the plan of Fig. 124; and in Fig. 128, the plan of Fig. 125. For the
wreaths shown in all these figures, there will be no necessity of
_springing_ the plank, which is a term used in handrailing to denote
the twisting of the wreath; and no other bevel than the one at the
upper end of the pitch-board will be required. This type of wreath,
also, is the one that is required at the top of a landing when the rail
of the flight intersects with a level-landing rail.
In Fig. 129 is shown a very simple method of drawing the face-mould for
this wreath from the pitch-board. Make _a c_ equal to the radius of the
plan central line of rail as shown at the curve in Fig. 130. From where
line _c c″_ cuts the long side of the pitch-board, the line _c″ a″_ is
drawn at right angles to the long edge, and is made equal to the length
of the plan tangent _a c_, Fig. 130. The curve is drawn by means of
pins and string or a trammel.
In Fig. 131 is shown a quarter-turn between two flights. The correct
method of placing the risers in and around the curve, is to put the
last one in the first flight and the first one in the second flight
one-half a step from the intersection of the crown tangents. By this
arrangement, as shown in Fig. 132, the pitch-line of the tangents will
equal the pitch of the connecting flight, thus securing the second
easiest condition of tangents for the face-mould—namely, as shown, two
equal tangents. For this wreath, only one bevel will be needed, and it
is made up of the radius of the plan central line of the rail _o c_,
Fig. 131, for base, and the line 1-2, Fig. 132, for altitude, as shown
in Fig. 133.
The bevel shown in this figure has been previously explained in Figs.
105 and 106. It is to be applied to both ends of the wreath.
The example shown in Fig. 134 is of a well-hole having a riser in the
center. If the radius of the plan central line of rail is made equal
to one-half a tread, the pitch of tangents will be the same as of
the flights adjoining, thus securing two equal tangents for the two
sections of wreath. In this figure the tangents of the face-mould are
developed, and also the central line of the rail, as shown over and
above each quadrant and upon the pitch-line of tangents.
The same method may be employed in stairways having obtuse-angle and
acute-angle plans, as shown in Fig. 135, in which two flights are
placed at an obtuse angle to each other. If the risers shown at _a_
and _a_ are placed one-half a tread from _c_, this will produce in the
elevation a pitch-line over the tangents equal to that over the flights
adjoining, as shown in Fig. 136, in which also is shown the face-mould
for the wreath that will span over the curve from one flight to another.
In Fig. 137 is shown a flight having the same curve at a landing. The
same arrangement is adhered to respecting the placing of the risers,
as shown at _a_ and _a_. In Fig. 138 is shown how to develop the
face-moulds.
[Illustration: FINISHED ROOF TRUSS IN FIRST PRESBYTERIAN CHURCH,
SYRACUSE, N. Y.
Tracy & Swartwout, Architects; Ballantyne & Evans, Associated.
_Reproduced by courtesy of “The Architectural Review.”_]