The extra exertion required in ascending a staircase over that for walking on level ground is due to the weight which a person at each step is required to lift; that is, the weight of his own body. Hence the difficulty of ascent will be in proportion to the height of each step, or to the rise, as it is termed. To facilitate the operation of going up stairs, therefore, the risers should be low. The grade of a stairs, or its angle of ascent, depends not only upon the height of the riser, but also upon the width of the step; and this has a certain relation to the riser; for the width of a step should be in proportion to the smallness of the angle of ascent.

The distance from the top of one riser to the top of the next is the distance travelled at each step taken, and this distance should vary as the grade of the stairs; for a person who in climbing a ladder, or a nearly vertical stairs, can travel only 12 inches, or less, at a step, will be able with equal or greater facility to travel at least twice this distance on level ground. The distance travelled, therefore, should be in proportion inversely to the angle of ascent; or, the dimensions of riser and step should be reciprocal: a low rise should have a wide step, and a high rise a narrow step.

252. - Pitch-Board: Relation of Rise to Tread. - Among the various devices for determining the relation of the rise to the tread, or net width of step, one is to make the sum of the two equal to 18 inches.

For example, for a rise of 6 inches the tread should be 12, for 7 inches the tread should be 11; or -

 6 + 12 = 18 6 1/2 + 11 1/2 = 18 7 + 11 = 18 7 1/2 + 10 1/2 = 18
 8 + 10 = 18 8 1/2 + 9 1/2 = 18 9 + 9 = 18 9 1/2 + 8 1/2 = 18

This rule is simple, but the results in extreme cases are not satisfactory. If the ascent of a stairs be gradual and easy, the length from the top of one rise to that of another, or the hypothenuse of the pitch-board, may be proportionally long; but if the stairs be steep, the length must be shorter.

There is a French method, introduced by Blondel in his Cours d' Architecture. It is referred to in Gwilt's Encyclopedia, Art. 2813.

This method is based upon the assumed distance of 24 inches as being a convenient step upon level ground, and upon 12 inches as the most convenient height to rise when the ascent is vertical. These are French inches, old system. The 24 inches French equals about 25 7/12 inches English.

With these distances as base and perpendicular, a right-angled triangle is formed, which is used as a scale upon which the proportions of a pitch-board are found. For example, let a line be drawn from any point in the hypothenuse of this triangle to the right angle of the triangle; then this line will equal the length of the pitch-board, along the rake, for a stairs having a grade equal to the angle formed by this line and the base-line of the scale.

##### Proportions Of The Pitch-Board

In the absence of the triangular scale, the lengths of the pitch-boards, as found by this rule, may be computed by this expression -

W=25 7/12 - 2 h; (107.)

in which W equals the tread, or base of the pitch-board, and h the riser, or its perpendicular height. For example, let h = 6; then -

W = 25 7/12 - 2x6 = 13 7/12.

This result is greater than would be proper in some cases.

The length of the hypothenuse of the pitch-board should be proportional not only to the angle of ascent (Art. 251), but also to the strength and height of the class of people who are to use the stairs. Tall and strong persons will take longer steps than short and feeble people. The hypothenuse of the pitch-board should be made in proportion to the distance taken at a step on level ground by the persons who are to use the stairs.

If people are divided into two classes, one composed of robust workmen and the other of delicate women and infirm men, then there may be two scales formed for the pitch-boards of stairs - one to be used for shops and factories, and the other for dwellings. The distance on level ground travelled per step, by men, varies from about 26 to 32 inches, or on an average 28 inches. The height to which men are accustomed to rise on ladders is from 12 to 16 inches at each step, or on the average 14 inches.

With these dimensions, therefore, of 14 and 28 inches, a scale may be formed for pitch-boards for stairs, in buildings to be used exclusively by robust workmen. And with 12 and 24 inches another scale may be formed for pitch-boards for stairs, in buildings to be used by women and feeble people. These two scales are both shown in Fig. 126. They are made thus: Let C A B be a right angle. Make A B equal to 28 inches, and A C equal to 14; then join B and

C. At right angles to C B, from A, draw A F; then with A F for radius describe the arc F G. Then a line, as A K or A L, drawn from A at any angle with A B and limited by the line G F B will give the length of the hypothenuse of the pitch-board, for shop stairs of a grade equal to the angle which said line makes with A B. From K, perpendicular to A B, draw K N; then K N will be the proper riser for a pitch-board of which A N is the tread. So, likewise, L M will be the appropriate riser for the tread A M. The arc F G is introduced to limit the rake-line of pitch-boards occurring between F and C, in order to avoid making them longer than the one at F. The scale for the stairs for dwellings is made in the same manner; A D = 24 inches being the base, A E = 12 inches the rise, and J H D the line limiting the rake-lines of pitch-boards.

Fig. 126.

To compute the length of risers and treads, we have for the scale for shops, for those occurring between F and B -

r = 1/2(28-t): (108.)

t = 28 - 2r; (109.)

and for those between F and G, we have -