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    Home»Structures»Structural analysis and design of sawtooth or slabless staircase
    Structures

    Structural analysis and design of sawtooth or slabless staircase

    Mezie EthelbertBy Mezie EthelbertJuly 9, 2021Updated:March 30, 20222 Comments
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    Slabless, sawtooth staircase (Figure 1) is one type of the stair that produces a lot of aesthetics appeal. Aesthetics and function are prime considerations in the design of buildings so if you learn how to design and build this type of stair, it would be good to actualize it. Different codes have their way of designing this type of stair but I would dwell on the method outlined in the book by Reynolds and Steedman (1988) which is in consonance with BS 8110 and Eurocode 2 (EC 2).

    Figure 1; Slabless stair case

    According to Reynolds and Steedman (2010), Cusens (1966) showed that if axial shortening is neglected and the strain energy due to bending only is considered, the midspan moment for the sawtooth stair case can be given by the general expression.

    Where k = stiffness of thread/stiffness of riser and j is the number of threads

    I refer my reader to study principles of strain energy to understand more the concept defined here.

    Steps in the design of slabless stair case

    Step 1; Carry out the functional design of the staircase to determine length of the span based on the stairwell and using Blondel formula to determine the suitable length of thread (going) and riser height.

    The Blondel formula states thus, 2R+T = 600 mm where R = Riser and T = Thread or Going (See table 1).

    Table 1; Optimum dimensions of stair case (mm) – BS 5395

    Step 2; Carry out the load analysis based on EC 2 guide to determine the design load, nd. The design load is considered as concentrated load at the mid span as shown below (Figure 2)

    Figure 2; Load distribution on slabless stair case

    Step 3; Determine k from the expression below, (See Figure 3 for explanation of expressions)

    Figure 3; Description of important stair parameters

    Where Lt = length of going/thread; Lr = length of riser; ht = height of thread; hr = height of riser

    Step 4; Determine support moment coefficient from the chart in Figure 4 based on k and j

    Figure 4; Support moment coefficient chart

    Using the support moment coefficient, calculate the support moment based on the expression below,

    Support moment, Ms = coefficient x ndL2

    Otherwise, follow step 4a

    Step 4a; Determine the parameters, k11, k12, k13 and k14 from j, which is the number of goings/threads and determine the support moment from the expression.

    Step 5; Determine the free bending moment from the expression

    Step 6; Determine the maximum moment at the midspan (Mo) with the expression

    Mo = M – Ms

    Where M = free bending moment and Ms = support moment

    Step 7; Draw the bending moment and shear force diagrams if required. They should be as shown

    Figure 5; Bending moment diagram for slabless staircase
    Figure 6; Shear force diagram for slabless staircase

    Step 8; Determine the reinforcement required at mid span and support based on the mid span moment and support moments and detail the slab as shown below. Due to the stair profile, concentrations of stress occur in the re-entrant corners, and the actual stresses to be resisted will be larger than those calculated from the moment. Note that re-entrant corner is any inside corner that forms an angle of 1800 or less. In a solid object subjected to internal or external loads, re-entrant corners create high stress concentrations. To resist such stresses, Cusens recommends providing twice the reinforcement theoretically required (at the re-entrant corners) unless suitable fillets or haunches are incorporated at these junctions. If this can be done, the actual steel provided should be about 10% more than the theoretically necessary. The possible reinforcement patterns are shown below (Figure 7)

    Figure 7; Arrangement of rebars in slabless staircase

    Figure 7 is very suitable but practically, if haunches are provided otherwise the bars should be arranged for wall-to-wall corners as shown below (Figure 8).

    Figure 8; Arrangement of rebars at re-entrant corners

    If Figure 7 would not be possible, Figure 9 can be adopted for the reinforcement

    Figure 9; Arrangement of rebar in slabless staircase

    Example

    A proposed slabless stair case has the following geometrical details and design information

    Length of going/thread, Lt= 300 mm

    Length of Riser, Lr = 150 mm

    Height of thread, ht = 125 mm

    Height of riser, hr = 125 mm

    Number of threads = 6

    Width of stair = 1200 mm

    Length of stair, L = 300 x 6 = 1800 mm

    Variable action (public access) = 3.0 kN/m2

    Weight due to finishes as 1.5 kN/m2

    Unit weight of concrete = 25 kN/m3

    Characteristic strength of concrete, fck = 25 N/mm2;

    Characteristic strength of steel, fyk = 460 N/mm2

    Determine the support moments and midspan moment and design the stair

    Step 1; The functional design parameters are outlined above

    Step 2; Loading

    Differences between BS 8110 and Eurocode 2 (EC 2) based on design notations

    Self-weight of the staircase = {[(Lt x ht) + (Lr x hr)]/Lt} x fck = {[(0.3 x 0.125) + (0.15 x 0.125)]/0.3 x 25 =

    0.1875 x 25 = 4.69 kN/m2

    Weight of finishes = 1 kN/m2

    Total permanent action = 4.69 + 1 = 5.69 kN/m2

    Variable action = 3 kN/m2

    Design action, nd = 1.35gk + 1.5qk = 1.35 (5.69) + 1.5 (3) = 12.18 kN/mper m width, say 13 kN/mper m width

    Step 3; Determination of k

    K = (ht3lr)/ (hr3lt) = (1253 x 150) (1253 x 300) = 150/300 = 0.5

    Step 4; Determine support moment coefficient from the chart in Figure 4 based on k and j

    Since k = 0.5 and j = 6 the support moment coefficient from chart = -0.875

    Support moment, Ms = -0.0875 x 13 x 1.82 = -3.69 kNm

    Using formula to estimate this value, since j is even,

     = ((1/48) x 6 x (6-1) x (6-2) = 2.5

     = (1/48) x (6-1) x (6-2) x (6-3) = 1.25

     = (1/2) x (6-1) = 2.5

     = (1/2) x (6-2) = 2

    Ms = [13 x 1.82 (2.5 + (0.5 x 1.25))]/62[2.5 + (0.5 x 2)] = 131.625/126 = 1.044 kNm (the value is lower than one obtained from chart). We would proceed with the value from chart.

    Step 5; Determine the free bending moment. Since j is even, free bending moment, M = (1/8) x 13 x 1.82 = 5.265 kNm

    Step 6; Determine the maximum moment at the midspan (Mo) with the expression

    Mo = 5.265 – 3.69 = 1.575 kNm

    Step 7; Draw the bending moment and shear force diagrams if required. They should be as shown

    Step 8: Estimation of area of reinforcement: In slabless stair case, the links are designed while the main bars are provided. It is usually 6 number of bars per link and of the same size as the link.

    MEd = 1.575 kNm

    b = 1000; fck = 25 N/mm2

    Assuming bar size of 12 mm and concrete cover of 25 mm, d = 150 – 25 – (12/2) = 119 mm

    k = 1.575 / (25 x 1000 x 1192) = 0.00445 < 0.167 Ok

    z = d (0.5 + √0.25 – 0.882k) = 0.99d ˃ 0.95d, use 0.95d

    z = 0.95 x 119 = 133 mm

    As1 = 1.575/ (0.87 x 460 x 113) = 34.83 mm2/m provide H12@ 300 mm c/c (377 mm2/m)

    Check for shear

    Maximum design shear force, MEd = 11.7 kN/m

    k = 1 + √ (200/d) = 1 + √ (200/119) = 2.30 ˃2.0, use 2.0

    ρl = As1/bd = 34.83/(1000 x 119) = 0.000293 ≤ 0.02

    VRd,c = [0.12k(100ρlfck)1/3] bd = [0.12 x 2 (100 x 0.000293 x 25)1/3 x 1000 x 119 = 25.75 kN/m

    Since VEd (11.7 kN/m) ˂ VRd,c (25.75 kN/m), shear is satisfied

    Reinforcement details

    Figure 10; Reinforcement details

    References

    Cusens, A. R. (1966): Analysis of slabless stairs. Concrete and Constructional Engineering 61(10), pp. 359—64.

    Reynolds, C.E. and Steedman, J.C. (1988): Reinforced Concrete Designer’s Handbook, 10th  edition. E & FN Spon, Taylor & Francis Group 11 New Fetter Lane, London.

    Treatment for slab and beam subject to excessive deflection

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    Mezie Ethelbert

    An inquisitive civil engineer with wide interests in different aspects of civil engineering.

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