Free standing stair case (Figure 1) is among the special types of staircase whose design and construction are usually very challenging. The mechanism by which this stair case carry loads have been a subject of many research. It is one type of stair that is aesthetically appealing when done properly. In BS 8110 for instance, an approximate method based on strain energy principles were developed to determine some critical moments used in the design. Generally, the analysis process to obtain the moments is very rigorous and more suited to finite difference method and finite element method available in some top civil engineering softwares such as StaaPro V8i, SAP2000 etc.
The manual solution to the structure depends on determining accurately the moments acting at each point on the stair case due to applied loads. According to Reynolds and Steedman (1988), Cusens (1966) employed strain energy principles to determine expressions relating the horizontal restraint force (H) and moment (Mo) at the mid-span of a free-standing stair case (see Figures 3, 4 and 5). By solving the two resulting equations simultaneously, the values of Mo and H can then be substituted into some general expressions to obtain the moments and forces at any point along the structure. This method, among others such as Fuchesteiner (1954), Siev (1962) identified as analytical methods by Vahid et al (2015) idealizes the stair as ‘space plate’ or ‘space frame’. These methods are faulted first, because of their inability to predict the variation of stress resultants across any section of the flight or landing and secondly, because of certain assumptions aimed at making the structure determinate which in turn causes loss of overall structural integrity of the stair due to indeterminacy. In addition, analytical methods are not conservative and fails to simulate actual interaction of plates in three dimension. These shortcomings have made it difficult to carry out practical design of free-standing stair cases.
Having this in mind, Vahid et al (2015) developed a set of conservative empirical equations based on 3D finite element analysis (see Figure 2) carried out for many configurations of the stair cases. According to them, the methods are more suited to the practical analysis and design of free-standing stair case and are shown in this article. Below are significant conclusion from their work (I refer the reader to visit the journal article for more).
- A comparison of the proposed equations with the finite element results was performed to verify their accuracy of the proposed equations, values given by these equations are compared with the corresponding values obtained from finite element analysis.
- In all cases, the proposed equations provided reasonably accurate results on the safe side, thus establishing their acceptability. From these equations, we can determine Mmax1, Mmax2, Vmax, Tmax, for the flight and landing.
- The effect of various geometric parameters on the design forces and moments were investigated. Based on the findings of this study, guidelines for direct analysis of the staircase were developed.
- The design forces and moments can be rapidly and easily determined from the suggested equations without requiring any formal analysis. These equations yield results that are on the conservative side but are within acceptable limits of accuracy.
- The transverse reinforcing steel in the landing should be significantly concentrated in the vicinity of the line of intersection of the flights and the landing. Large torsion moments are present in the flights of free-standing stairs, and an appropriate thickness of concrete must be selected to resist these moments due to the difficulty of reinforcing shallow-wide sections against torsion.
- Analytical approaches are not practically suitable for the analysis of free-standing stairs as far as economy and efficiency in design are concerned. These methods fail to simulate the actual interaction of plates in three dimensions. In addition, these approaches cannot demonstrate the variation of stress resultants across any cross section.
The selection of stair thread, riser width and that of the landing and flight are guided by Table 1.
Table 1: Optimum dimensions for stairs according to BS 5395
Structural analysis, design, detailing and construction principles of the stair
The structural design and construction of free standing stair just like the design of other structural components can be categorized into four stages namely:
1. Analysis for stresses (forces and moments)
2. Checking for the thickness and calculation of the reinforcements
3. Detailing of the reinforcement layout
4. Construction of the stair
Analysis for stresses
Basis of analysis
The analysis of free standing staircase is based on the premise that the stairs are symmetrically loaded with ends fixed and with the mid landing portion treated as a propped cantilever giving a line support in ‘symbiotic-static’ upper and lower flights.
Example
Analyse a free standing staircase with the geometrical properties and loading outlined below
Geometrical dimensions of stair:
Tread, G = 300 mm
Riser, R = 150 mm
Distance between two flights, A = 200 mm
Width of landing, B = 1200 mm
Width of flight, C = 900 mm
Length of landing = 2000 mm
Horizontal length of going of stairs, L = 3000 mm
Head room of stair, H = 3000 mm
Depth of flight, T1= 150 mm
Depth of landing, T2= 150 mm
Loading on stair:
Assuming:
Permanent action, gk = 1.2 kN/m2 (excluding self- weight)
Variable action, qk = 3.0 kN/m2
Actions of different parts of the stair
Flight
Slab self-weight = 25 x 0.243 = 6.08 kN/m2 = 0.00608 N/mm2
Permanent action (excluding self-weight) = 1.2 kN/m2 = 0.0012 N/mm2
Characteristic permanent action, gk = 6.08 + 1.2 = 7.28 kN/m2 = 0.00728 N/mm2
Characteristic variable action, qk = 3.00 kN/m2 = 0.003 N/mm2
Landing
Slab self-weight = 25 x 0.15 = 3.75 kN/m2 = 0.00375 N/mm2
Permanent action (excluding self-weight) = 1.2 kN/m2 = 0.0012 N/mm2
Characteristic permanent action, gk = 3.75 + 1.2 = 4.95 kN/m2 = 0.00495 N/mm2
Characteristic variable action, qk = 3.00 kN/m2 = 0.003 N/mm2
Determination of moments
The method developed by Vahid et al (2015) for the determination of the different external forces acting on a free standing staircase at worst case situations are applied in this article. The equations developed are as shown below:
In flights,
i. Mmax1 = −0.00343 A + 0.002692 B − 0.00296 C −0.00077 H − 0.01073 T −0 .01023 L − 1198.04 gk − 1819.47 qk + 30.91564 kNmm/mm
ii. Mmax2 = − 0.00224 A − 0.00208 B − 0.00092 C + 0.00074 H − 0.00464 T + 0.005625 L + 100.6893 gk + 808.1385 qk − 11.681 kNmm/mm
iii. Vmax = − 5.8E-07 A − 2.4E-06 B + 1.59E-06 C − 6.5E-08 H + 3.02E-06 T + 8.06E-06 L + 1.697078 gk + 2.678156 qk −0.01935 kN/mm
iv. Tmax = 0.005544 A + 0.001997 B + 0.003612 C + 5.11E-05 H + 0.005184 T − 0.00032 L + 917.2982 gk + 586.8903 qk − 9.98993 kNmm/mm
In landing,
i. Mmax1 = − 0.00306 A − 0.01656 B −0 .02909 C − 0.00112 H +0 .051901 T − 0.00504 L − 7320.07 gk − 4574.51 qk + 76.70828 kNmm/mm
ii. Mmax2 = − 0.0088 A + 0.005846 B − 0.01279 C − 0.00018 H + 0.001066 T − 0.00267 L − 2148.38 gk − 1362.52 qk + 21.66841 kNmm/mm
iii. Vmax = −0 .00014 A − 2.8E-05 B − 3.8E-05 C − 9.5E-06 H +0 .000183 T − 6.4E-06 L − 13.7538 gk − 8.80052 qk + 0.16955 kN/mm
iv. Tmax = 0.00331 A + 0.005331 B + 0.006495 C + 0.000533 H− 0.01922 T + 0.001563 L + 1855.02 gk + 1177.53 qk − 21.4839 kNmm/mm
Where,
Mmax1 = Maximum bending moment in support flight
Mmax2 = Maximum bending moment in span flight
Vmax = Maximum shear force
Tmax = Maximum torsion moment
A = space between two flights (mm)
B = width of landing (mm)
C = width of flight (mm)
H = head room of stair (mm)
T = thickness of flight and lading (mm)
L = horizontal length of flight (mm)
gk = Permanent action (kPa)
qk = Variable action (kPa)
By the application of the equations developed by Vahid et al (2015), the following moments where obtained:
In flights
Mmax1 = -15.2 kNmm/mm
Mmax2 = 6.1 kNmm/mm
Vmax = 0.024 kNmm
Tmax = 5.18 kNmm/mm
In landing
Mmax1 = -30.61 kNmm/mm
Mmax2 = -7.7 kNmm/mm
Vmax = -0.0413 kNmm
Tmax = 7.541 kNmm/mm
Design and Detailing
Based on the moments determined above, the actual design of the free standing staircase can be carried out following the usual process of design of stair as applicable to different codes.
Important preliminary detailing rules
1. The deflection response of the staircase in different loading conditions is strictly considered in the detailing process.
2. The most critical portion of the staircase is the top landing because it is subjected to severe torsion. Experimental studies suggest that the mid-landing should be strengthened like beam element across the junction of the flights and the landing because stresses along the junction are non-linear and occur in high concentrations at the corner.
3. Normal torsional reinforcement should be provided in the form of closed loops for half the width of the mid landing and the cantilever reinforcements in the landing can be advantageously carried into the flight.
4. The torsion members of the staircase should have top and bottom reinforcement and closed stirrups which enhance the torsional capacity of the members.
As noted earlier, the analysis of a free-hanging staircase is primarily based on the premise that the stairs are symmetrically loaded with ends fixed, and with the midlanding portion treated as a propped cantilever, giving a line support in ‘symbiotio-state’ to upper and lower flights. Considering the support offered by mid-landing as a propped cantilever, the bending moment (hogging) in the midlanding is -30.61 kNmm/mm.
Since the midlanding slab, under uniform loading of the entire staircase, suffers bi-axial bending in the X-Y plane, it is necessary to provide top reinforcement in the slab, parallel to X-axis. For this purpose the mid-landing shall be assumed as cantilevering out from the line of flights, moment is -7.7 kNmm/mm
Detailing of reinforcement in flights: The negative moments at supports due to vertical loads, causing flexural moments, will be the algebraic sum of moments WL/8 treated as a propped cantilever plus the carry over moment from the centilevered midlanding. Therefore, maximum hogging moment in flight is -15.2 kNmm
The above moment will be compounded with torsional moments arising from unsymmetrical live load conditions on the flights.
The torsional stresses will be maximum in any one of the flights, when the other flight is fully loaded with live load. For this condition, the torsional moment at flight is 5.18 kNmm. Reinforcement for torsion when required, shall consist of longitudinal and transverse reinforcement (Rao, 1983).
Maximum positive moment in span flight is 6.1 kNmm.
Detailing of reinforcement in top landing: The top landing is subjected to the reactions from the flights emanating/culminating at it, causing flexural moments and also torsional moments due to the eccentric loading of the flights (Rao, 1983).
Foundation design: The design of foundation is to be principally based on statical considerations. The reactions from the upper flight, mid-landing and from the bottom flight, which constitute the unbalanced loads, will have to be resisted by the inertial mass of the foundation block (Rao, 1983).
Construction rules according to Rao (1983)
1. There should be proper quality control to ensure that the reinforcements are kept at their place.
2. The entire staircase above construction joint at the beginning of the flight at ground level up to and including the landing should be concreted in one operation.
3. All concretes should be C 16/20 grade except the concrete in foundation that should be C 10/15.
4. Props and formwork should not be removed before 28 days.
5. During concreting, the flight above first floor level, the flights in ground floor should be properly supported if they are to be used for supporting top flights.
6. Stripping of formwork should start from the free edge of mid-landing and proceed towards both supports.
7. The staircase should be treated as cantilever at every stage of construction.
8. Necessary holes for fixing balusters should be made during concreting and no holes should be left for after the concreting operation.
References
Cusens, A. R. and Kuang, J.-G. (1966). “Experimental study of a free standing staircase.” Journal of The American Concrete Institute, Vol. 63, No. 5, pp. 587-604.
Fuchesteiner, W. (1954). “Die freitragende wendeltreppe free standing circular stair.” Beton-und Stahlbetonbau, Berlin, Germany, Vol. 59, No. 11, pp. 256-258
Rao, P.K. (1983): ‘‘Analysis, detailing and construction of a free-standing staircase.’’ Indian Concrete Journal. pp 111 – 116.
Reynold, C.E. and Steedman, J.C. (1988): Reinforced Concrete Designers Handbook. 10th edition. E & FN Spon, Taylor & Francis Group 11 New Fetter Lane, London EC4P 4EE
Siev, A. (1962). “Analysis of free straight multi-flight staircases.” Journal of the Structural Division, ASCE, Vol. 88, No. 3, pp. 207-230.
Vahid, S.Z., Sadeghian, M.A., Osman, S.A. and Abdul Rashid, A.K.B. (2015): ‘‘Formulation for free-standing staircase.’’ KSCE Journal of Civil Engineering. pp 1-7.
