End zone reinforcement

Figure 1: Linear strain distribution at l_disp

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 (f_yd,lim, see calculation settings). For calculation of anchorage reinforcement the actual chosen material is used.

Design of Spalling reinforcement

Figure 3: Fracture mode - spalling

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, l_pt, 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. l_pt1 = 0.8 * l_pt 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) F_t = F_c will give the position of F_c 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. l_ptB = P₂ / P * l_pt i.e. part of transfer length that will be dedicated to P₂. 
6. l = x * cot(θ). The angle of q should be limited so that 1 ≤ cot(θ) ≤ 2.5 i.e 21.8° ≤ θ ≤ 45°. x is given by the position of P₁ and P₂. 
7. a_b represents half the width of the group of stirrups needed to resist the tensional force, F_stir, and is calculated as;
   a_b = (c + 0.5 * Ø) + 0.5 * ((n_b - 1) * (Ø + s)) where
   c = concrete cover to first stirrup from end of beam
   Ø = diameter of stirrup
   n_b = 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 n_b = 1. For each iteration the following checks are performed;
   Check 1: a_b ≤ 0.5 * l_ptB
   Check 2: a_b + l + a_c ≤ l_disp where
    a_c = 0.5 * minimum sectional width.
   Check 3: A_b * f_yd,lim ≤ F_stir = P₂ / cot(q) where
    A_b = (stirrup area) = n_b * n / mod(2) * π * ز / 4.
            n / mod(2) = number of legs in even pairs
    f_yd,lim = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
   Check 4: w_req ≤ w_avail where
    w_req = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
    w_avail = 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;
    F_s * (h_t - a_s) = F_t * 2/3 * h_t where
    h_t is the height of the tensional prism
    a_s is the distance from the top to the centre of the top reinforcement
    F_s is the corresponding force in the reinforcement 
11. Check if A_s * f_yd,lim ≤ F_s where
    A_s = reinforcement area. 
12. If the given reinforcement is not enough an iterative design process begins by increasing A_s. 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, l_spal = (c + 0.5 * Ø) + (n_b * (Ø + s)).

Design of Bursting reinforcement

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

Figure 5: Fracture mode - bursting

Figure 6: Symmetric prism analogy

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

The methodology for calculating N_bs , 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, l_bs = l_pt - l_spal. 
2. The upper limit of the prism, h_bs = 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 P_TOT = P₀ + P₁ with respect to sectional area above and below COG i.e. P₂ = P_TOT * 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 = h_bs / 4 
6. Calculate bursting force, N_bs , by the moment equilibrium around point 'A' in Figure 7 i.e.
   
   
   Figure 7: Equilibrium model
    
7. If σ_bs = N_bs / (w_bs * l_bs) ≥ f_ctd there's a need for extra stirrups.
   σ_bs = the stress due to bursting
   w_bs = minimum sectional width
   f_ctd = design tensional concrete stress = 'crack' stress 
8. Number of stirrups required, n_b = A_s / A_b where
   A_s = (total reinforcement area) = N_bs / f_yd,lim
            f_yd,lim = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
    A_b = (stirrup area) = n / mod(2) * π * ز / 4.
            n / mod(2) = number of legs in even pairs 
9. Check 1: l_req ≤ l_bs where:
   l_req = n_b * (Ø + s) > l_bs
   Check 2: w_req ≤ w_bs where:
   w_req = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
   w_bs = available width = minimum sectional width
   If Check1 ≠ OK: increase number of legs per stirrup i.e. n = n + 2 and re-calculate n_b.
   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.

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, F_td = M_Ed / z + F_td where:
   M_Ed = design moment in position x_crit
   z = internal lever = 0.9 * d
   d = effective height
   F_td = additional force due to shear = V_Ed * 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 σ_pd = F_td / A_p where
   A_p = area of the strands 
5. Calculate σ_pm∞ at position l_pt2 = 1.2 * l_pt (see EN 1992-1-1 8.10.2.3 Fig. 8.17)
   σ_pm∞ = initial prestress after any long-term effect 
6. Calculate anchorage length , l_bpd according to EN 1992-1-1 8.10.2.3 (4). 
7. If x_crit - c < l_bpd 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.
   L = l_bpd - x_crit
   If L ≤ l_pt2 : F = σ_pm∞ * (L / l_pt2) * A_p
   If L > l_pt2 : F = (σ_pm∞ + (σ_pd - σ_pm∞)) * (L - l_pt2) / ( l_bpd - l_pt2)) * A_p
9. Number of anchorage stirrups required, n_b = As / A_b where:
   A_s = (total anchorage reinforcement area) = F / f_yd
   A_b = (stirrup area) = n / mod(2) * π * ز / 4.
   n / mod(2) = number of legs in even pairs 
10. Check 1: w_req ≤ w_avail where:
    w_req = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
    w_avail = available width = minimum sectional width
    If Check 1 ≠ OK: it's not possible to design the anchorage stirrups needed.