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There are mathematical and graphical methods of computation of winder stairs. Mathematical methods are rather volume and complicated and therefore graphical methods are most often applied in practice. The most widespread of graphical methods are the method of proportions and the method of principal lines. All of them give approximately equal results of laying out the treads in the flight.

Hereafter we suggest you to review them:

**Method of Proportions**

Let us describe the calculation process of 90 Degree Winder Stairs with closed strings as an example.

1) At first you shall plot two pairs of parallel lines at an angle of 90 degrees to each other. The distance between the parallel lines shall be equal to the width between the closed strings of the would-be flight of stairs.

Fig. Calculation stairs according to the method of proportions - step 1

2) Then we mark out the thickness of the closed strings.

Fig. Calculation stairs according to the method of proportions - step 2

3) Then we plot the pitch line and draw the axis C - C1.

Fig. Calculation stairs according to the method of proportions - step 3

4) Now we shall mark the tread depths at the pitch line.

Fig. Calculation stairs according to the method of proportions - step 4

5) We determine the required number of winder treads in the flight. In our case we will have 9 winder treads.

We draw the lines at the edges of the winder treads until they intersect the axis C - C1. If we know the number of winder treads we can fashion the overall view of the flight of stairs. Plot the desirable number of square treads before and after winder treads.

Fig. Calculation stairs according to the method of proportions - step 5

6) Now let us draw the lines parallel to the axis C- C1 at the distance of 76 mm in each direction. In the aggregate we will receive the value 152 mm which constitutes the minimal size of the going.

Fig. Calculation stairs according to the method of proportions - step 6

7) The points X and X1, received in the result of intersection of these lines with the edge of the closed string, are connected with the tread depth marks of the central tread, and the received lines are prolonged until they intersect the edge of the closed string. In result we receive the central winder tread.

Fig. Calculation stairs according to the method of proportions - step 7

8) Now let us prolong one of the sides of the central winder tread down until it intersects the axis C - C1.

Fig. Calculation stairs according to the method of proportions - step 8

9) From the received point Р we draw a line at an acute angle and divide it into segments in ratio 1:2:3:4. The number of segments shall be equal to the number of winder treads after the central winder tread.

Fig. Calculation stairs according to the method of proportions - step 9

10) Then we join the points C1 and L4 with a line and draw the lines parallel to it through the points L3, L2, L1.

Fig. Calculation stairs according to the method of proportions - step 10

11) The points S1, S2, S3, which were formed as consequence, are joined with the tread depth marks with the help of lines, and these lines are prolonged until they intersect the edge of the closed string. As the result we received the arrangement of winder treads from the one side of the central winder tread.

Fig. Calculation stairs according to the method of proportions - step 11

12) So far as the central winder tread is located symmetrically to the axis C - C1 we can symmetrically represent the arrangement of winder treads for the other side.

Fig. Calculation stairs according to the method of proportions - step 12

13) The layout of 90 Degree Winder Stairs with closed strings calculated according to the method of proportions.

Fig. 90 Degree Winder Stairs

The calculation of 180 Degree Winder Stairs is executed the same way as that for 90 Degree Winder Stairs. For illustration purposes we adduce a drawing. Please, be aware that the minimal distance between the flights makes 100 mm. It should be mentioned that when designing 180 Degree Winder Stairs you shall try to make the distance between the flights as wide as possible. Otherwise, it will be very difficult to do the lay-out in compliance with norms of safety concerning the depth of treads at the narrowest point and at the distance of 305 mm from the edge.

**Method of Principal Lines**

90 Degree Winder Stairs with closed strings

To do the lay-out of treads according to the method of principal lines repeat steps 1 - 7 from the procedure of lay-out of treads according to the method of proportions.

8) Now we shall prolong both sides of the central winder tread until they intersect the axis L - L1. As a result we shall receive a segment Q-Q1 at the axis L - L1.

Fig. Calculation according to the method of principal lines step 8

9) Plot to the right the segment Q-Q1 as many times as the number of treads before the central winder tread.

Fig. Calculation according to the method of principal lines step 9

10) Join the ends of the segments received with the tread depth marks. As the result we received the form of winder treads located from the one side of the central winder tread.

Fig. Calculation according to the method of principal lines step 10

11) To receive the form of winder treads located at the other side of the central winder tread, represent them symmetrically to the axis C - C1.

Fig. Calculation according to the method of principal lines step 11

12) The layout of 90 Degree Winder Stairs with closed strings calculated according to the method of principal lines.

Fig. Calculation according to the method of principal lines step 12

The calculation of 180 Degree Winder Stairs is executed the same as that for 90 Degree Winder Stairs.

Now that we have the form and sizes of treads in the flight, we have to make lateral developed views of closed strings and to plot down the sizes of treads onto them. At that both inner and outer closed strings shall have a deflected position. When forming a curve try to make the line balanced without sharp bends, in addition do not forget to adhere to parallel and perpendicular alignment of the steps, and take minimal sizes of closed and open strings into account (read about it in the section Stairs Models).

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