Chapter IV: Part I: Stair-Building (3)

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.

_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.

_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.

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″_.

_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.

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.

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.

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_.

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_.

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.

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.

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.

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.

=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.

Tracy & Swartwout, Architects; Ballantyne & Evans, Associated.

_Reproduced by courtesy of “The Architectural Review.”_]