End zone reinforcement

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Figure 1: Linear strain distribution at ldisp

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Figure 2: Fracture modes

In Figure 2 the three different fracture modes for a prestressed beam is presented. According to Modelcode 90: 7.13.6.5.4, the beam has to fulfill some pre-specified cover and spacing values in order to avoid splitting i.e. there are no stirrup design methodologies for this fracture mode. The design process for the remaining fracture modes are carried out in the order in which they affect the beam, from beam end and inwards i.e. beginning with spalling, affecting the end of the beam, and then bursting,having its effect further in.

If the calculation setting 'Check vert.stresses in the anchor zone (Splitting, Spalling and Bursting)' is set, required stirrup reinforcement is calculated and presented in the 'End reinforcement' table (Design mode). Only the controls for spalling and bursting can result in stirrups reinforcement here.

For hollow core slabs (which has no stirrups) only a Spalling check is carried out here.

In addition to the mentioned phenomena PRE-Stress also (regardless of calculation settings) calculate required anchorage reinforcement in the end zones.

In regard to design of stirrups here, a window for end stirrups parameters is available in the 'Stirrups reinforcement' tab. The spalling and bursting calculations uses a reduced steel strength value to avoid crack width check (fyd,lim, see calculation settings). For calculation of anchorage reinforcement the actual chosen material is used.

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Design of Spalling reinforcement

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Figure 3: Fracture mode - spalling

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Figure 4: Equilibrium model - 'strut-and-tie'

If activated in the calculation settings, this calculation determines required stirrups in the end zone due to the Spalling phenomenon. The required reinforcement, due to fracture mode spalling (Figure 3), is designed in accordance with the 'strut-and-tie' methodology, see Figure 4.

There are many ways to set up a 'strut-and-tie' model. The model in Figure 4 and the methodology is proposed in 'Design and analysis of prestressed concrete structures' by B. Engström. The methodology is slightly adjusted and described in short below.

Methodology: (this is done for both ends)

  1. Calculate the base transfer length, lpt, according to EN 1992-1-1 8.10.2.2 (2) 
  2. Determine the length of the discontinuity region i.e. dispersion length, ldisp, according to EN 1992-1-1 8.10.2.2 (4). At the end of the discontinuity region the strain distribution can be assumed to be linear, see Figure 1. The dispersion length is calculated with the lower value of lpt i.e. lpt1 = 0.8 * lpt since shorter transfer length is unfavorable in this case. 
  3. From the strain distribution and the shape of the section a load distribution is derived. 
  4. From the load distribution the resulting forces are derived through 'fit calculation' by interpolation i.e.
    a) Ft = Fc will give the position of Fc and the upper limit for A2.
    b) The location of P1 is set to match the COG (centre of gravity) of the strands. The upper limit of A1 is derived so that momentum, of the loading, below P1 is equal to the momentum above P1.
    c) The location of P2 is defined by same principal of equilibrium i.e. momentum, of the loading between P2 and the lower limit of A2, should be equal to the momentum, of the loading between P2 and the upper limit of A2
  5. lptB = P2 / Sigma.pngP * lpt i.e. part of transfer length that will be dedicated to P2
  6. l = Delta.pngx * cot(θ). The angle of q should be limited so that 1 ≤ cot(θ) ≤ 2.5 i.e 21.8° ≤ θ ≤ 45°. Delta.pngx is given by the position of P1 and P2
  7. ab represents half the width of the group of stirrups needed to resist the tensional force, Fstir, and is calculated as;
    ab = (c + 0.5 * Ø) + 0.5 * ((nb - 1) * (Ø + s)) where
    c = concrete cover to first stirrup from end of beam
    Ø = diameter of stirrup
    nb = number of stirrups
    s = spacing between stirrups
    c, Ø and s are set in either 'Reinforcement details' tab or via 'End stirrups' button in 'Stirrup reinforcement' tab while nb is the result of an interactive design process.
  8. Begin the iterative design process by setting nb = 1. For each iteration the following checks are performed;
    Check 1: ab ≤ 0.5 * lptB
    Check 2: ab + l + ac ≤ ldisp where
     ac = 0.5 * minimum sectional width.
    Check 3: Ab * fyd,lim ≤ Fstir = P2 / cot(q) where
     Ab = (stirrup area) = nb * n / mod(2) * π * Ø2 / 4.
             n / mod(2) = number of legs in even pairs
     fyd,lim = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
    Check 4: wreq ≤ wavail where
     wreq = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
     wavail = available width = minimum sectional width

    If Check1 ≠ OK: increase number of legs per stirrup i.e. n = n + 2.
    If Check 2 ≠ OK: increase q  within the given range.
    If Check 3 ≠ OK: increase number of stirrups i.e. nb = nb + 1.
    If Check 4 ≠ OK: it's not possible to design the stirrups needed.
  9. After designing the stirrups the top reinforcement has to be controlled so that it can withstand the tensional force, Ft. 
  10. The force in the reinforcement is derived as;
    Fs * (ht - as) = Ft * 2/3 * ht where
    ht is the height of the tensional prism
    as is the distance from the top to the centre of the top reinforcement
    Fs is the corresponding force in the reinforcement 
  11. Check if As * fyd,lim ≤ Fs where
    As = reinforcement area. 
  12. If the given reinforcement is not enough an iterative design process begins by increasing As. If more layers are needed as has to be updated.
     

The result from this design procedure is the required number of stirrups as well as the distribution length, lspal = (c + 0.5 * Ø) + (nb * (Ø + s)).

MicrosoftTeams-image (7).png
 

Design of Bursting reinforcement

If activated in the calculation settings, this calculation determines required stirrups in the end zone due to the Bursting phenomenon.

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Figure 5: Fracture mode - bursting

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Figure 6: Symmetric prism analogy

The bursting force, Nbs , is calculated using the symmetric prism analogy, see Figure 6.

The methodology for calculating Nbs , according to Modelcode 90: 7.13.6.5.2, is slightly adjusted and described in short below.

Methodology: (this is done for both ends)

  1. Calculate the length of the prism, lbs = lpt - lspal
  2. The upper limit of the prism, hbs = 2 * y , where y is set equal to COG (centre of gravity) of the strands. 
  3. Calculate P1 and P0 (prestress force above and below COG respectively). 
  4. Distribute the total prestress force PTOT = P0 + P1 with respect to sectional area above and below COG i.e. P2 = PTOT * A2 / (A1 + A2). 
  5. Calculate e1 and e2 where
    e1 = the distance from y to COG of strands above y
    e2 = the distance to centre of pressure block above y = y / 2 = hbs / 4 
  6. Calculate bursting force, Nbs , by the moment equilibrium around point 'A' in Figure 7 i.e.

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    Figure 7: Equilibrium model
     
  7. If σbs = Nbs / (wbs * lbs) ≥ fctd there's a need for extra stirrups.
    σbs = the stress due to bursting
    wbs = minimum sectional width
    fctd = design tensional concrete stress = 'crack' stress 
  8. Number of stirrups required, nb = As / Ab where
    As = (total reinforcement area) = Nbs / fyd,lim
             fyd,lim = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
     Ab = (stirrup area) = n / mod(2) * π * Ø2 / 4.
             n / mod(2) = number of legs in even pairs 
  9. Check 1: lreq ≤ lbs where:
    lreq = nb * (Ø + s) > lbs
    Check 2: wreq ≤ wbs where:
    wreq = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
    wbs = available width = minimum sectional width
    If Check1 ≠ OK: increase number of legs per stirrup i.e. n = n + 2 and re-calculate nb.
    If Check 2 ≠ OK: it's not possible to design the stirrups needed.
     

Design of Anchorage reinforcement

Anchorage of strands are designed in accordance with EN 1992-1-1: 8.10.2.3 and the methodology is presented in 'Svenska Betongföreningen Betongrapport nr. 15, volym II, utgåva 2 - Beräkningsexempel B1'. This methodology is slightly adjusted and described in short below.

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Figure 8: Anchorage model

Methodology: (this is done for both ends)

  1. Find the first position (from the end), xcrit, where σct > fctd i.e. where the beam is cracked. 
  2. Calculate the base transfer length, lpt, according to EN 1992-1-1 8.10.2.2 (2) 
  3. Calculate the anchorage force, Ftd = MEd / z + Delta.pngFtd where:
    MEd = design moment in position xcrit
    z = internal lever = 0.9 * d
    d = effective height
    Delta.pngFtd = additional force due to shear = VEd * a1 / z where:
    a1 = d (without shear reinforcement), or
    a1 = 0.5 * z * cot(θ) (with shear reinforcement)
    see EN 1992-1-1 9.2.1.3 (2) 
  4. Calculate σpd = Ftd / Ap where
    Ap = area of the strands 
  5. Calculate σpm∞ at position lpt2 = 1.2 * lpt (see EN 1992-1-1 8.10.2.3 Fig. 8.17)
    σpm∞ = initial prestress after any long-term effect 
  6. Calculate anchorage length , lbpd according to EN 1992-1-1 8.10.2.3 (4). 
  7. If xcrit - c < lbpd anchorage reinforcement is needed. 
  8. It's only the excess force that is not anchored by bondage that has to be anchored by anchorage reinforcement i.e.
    Delta.pngL = lbpd - xcrit
    If Delta.pngL ≤ lpt2 : Delta.pngF = σpm∞ * (Delta.pngL / lpt2) * Ap
    If Delta.pngL > lpt2 : Delta.pngF = (σpm∞ + (σpd - σpm∞)) * (Delta.pngL - lpt2) / ( lbpd - lpt2)) * Ap
  9. Number of anchorage stirrups required, nb = As / Ab where:
    As = (total anchorage reinforcement area) = Delta.pngF / fyd
    Ab = (stirrup area) = n / mod(2) * π * Ø2 / 4.
    n / mod(2) = number of legs in even pairs 
  10. Check 1: wreq ≤ wavail where:
    wreq = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
    wavail = available width = minimum sectional width
    If Check 1 ≠ OK: it's not possible to design the anchorage stirrups needed. 
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PRE-Stress Documentation