Structural Concrete - Hassoun

18.3 Examples 713
2. The width of stairs that normally ranges from 4 to 8 ft. All other parameters are constant.
The increase of stair width by 100%, from 4 to 8 ft, increases the torsional moment by about
700%.
3. Variation in the interior and exterior radii (R i and R e ) keeping the stair width of 6 ft constant.
The increase in R e (from 9 to 12 ft) and R i (from 3 to 6 ft) with a ratio of R e /R i that varies
between 3 and 2, increases the lateral moment by about 230%.
4. The thickness of stair slab is not as critical as the other parameters. For a variation in slab
thickness between 6 and 12 in., the lateral moment increases by about 70%, while the torsional
moment increases by about 170%.
5. The total height of the helical stair, h, has the least effect on all forces (for h between 9 and
15 ft). The increase in lateral moment is about 70% and in torsional moment is about 40%.
Other forces decrease by about 80%.
6. Based on this study, the possible practical dimensions may be chosen as follows: Total
subtended arc between 120 ∘ and 320 ∘ , stair width between 4 and 6 ft, stairs slab thickness
between 6 and 10 in., and stair height between 10 and 15 ft.
The above information can be used as a guide to achieve a proper and economical design of
helical staircase.
An alternative method of providing a helical stair is to use a central helical girder located
at the midwidth of the stairs and have the steps project equally on both sides of the girder. Each
step is analyzed as a cantilever, and the reinforcement bars extend all along the top of the run.
Precast concrete steps may be used and can be fixed to specially prepared horizontal faces at the
top surfaces of the girder.
18.3 EXAMPLES
Example 18.1
Design the cantilever stairs shown in Fig. 18.18 to carry a uniform live load of 100 psf. Assume the
rise of the steps equals 6.0 in. and the run equals 12 in. Use normal-weight concrete with f c ′ = 3ksi and
f y = 60 ksi.
Solution
1. Loads: Assume the thickness of the slab (waist) is 4.0 in. Weight of the assumed slab (areas A 1
and A 2 )is
( )
Trapezoidal area mnn ′ m ′ 4.9 + 10.9
=
(1)(150) =98.8lb∕ft
2 × 12
Refer to Fig. 18.18b. Assume the weight of the step cover is 5 lb/ft. Total DL = 103.8 lb/ft.
W u = 1.2D + 1.6L = 1.2 × 103.8 + 1.6 × 100 = 285lb∕ft
2. Maximum bending moment per step is W u l 2 /2.
M u = 0.285
2 (6)2 = 5.13K ⋅ ft
Average thickness of a step is (10.9 + 4.9)/2 = 7.9 in.