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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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″.
Fig. 96. Finding Angle between Tangents
for Bottom Wreath of Fig. 95.
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.
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.
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.
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.
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″.
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.