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5 = End zone reinforcement =
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7 [[image:end zone 1.jpg]]
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9 Figure 1: Linear strain distribution at l,,disp,,
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12 [[image:end zone 2.jpg]]
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14 Figure 2: Fracture modes
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17 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.
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20 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.
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23 For hollow core slabs (which has no stirrups) only a [[Spalling check>>doc:PRE-Stress.Theory PRE-Stress.Resistance to spalling (hollowcore elements).WebHome]] is carried out here.
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26 In addition to the mentioned phenomena PRE-Stress also (regardless of calculation settings) calculate required anchorage reinforcement in the end zones.
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29 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.
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32 [[image:end zone 3.jpg]]
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35 = Design of Spalling reinforcement =
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37 [[image:end zone 4.jpg]]
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39 Figure 3: Fracture mode - spalling
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41 [[image:end zone 5.jpg]]
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43 Figure 4: Equilibrium model - 'strut-and-tie'
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46 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.
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48 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.
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51 Methodology: (this is done for both ends)
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53 1. Calculate the base transfer length, l,,pt,,, according to EN 1992-1-1 8.10.2.2 (2)
54 1. 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.
55 1. From the strain distribution and the shape of the section a load distribution is derived.
56 1. From the load distribution the resulting forces are derived through 'fit calculation' by interpolation i.e.
57 a) F,,t,, = F,,c,, will give the position of F,,c,, and the upper limit for A,,2,,.
58 b) The location of P,,1,, is set to match the COG (centre of gravity) of the strands. The upper limit of A,,1,, is derived so that momentum, of the loading, below P,,1,, is equal to the momentum above P,,1,,.
59 c) The location of P,,2,, is defined by same principal of equilibrium i.e. momentum, of the loading between P,,2,, and the lower limit of A,,2,,, should be equal to the momentum, of the loading between P,,2,, and the upper limit of A,,2,,.
60 1. l,,ptB,, = P,,2,, / [[image:Sigma.png]]P * l,,pt,, i.e. part of transfer length that will be dedicated to P,,2,,.
61 1. l = [[image:Delta.png||height="13" width="12"]]x * cot(θ). The angle of q should be limited so that 1 ≤ cot(θ) ≤ 2.5 i.e 21.8° ≤ θ ≤ 45°. [[image:Delta.png||height="13" width="12"]]x is given by the position of P,,1,, and P,,2,,.
62 1. a,,b,, represents half the width of the group of stirrups needed to resist the tensional force, F,,stir,,, and is calculated as;
63 a,,b,, = (c + 0.5 * Ø) + 0.5 * ((n,,b,, - 1) * (Ø + s)) where
64 c = concrete cover to first stirrup from end of beam
65 Ø = diameter of stirrup
66 n,,b,, = number of stirrups
67 s = spacing between stirrups
68 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.
69 1. Begin the iterative design process by setting n,,b,, = 1. For each iteration the following checks are performed;
70 Check 1: a,,b,, ≤ 0.5 * l,,ptB,,
71 Check 2: a,,b,, + l + a,,c,, ≤ l,,disp,, where
72 a,,c,, = 0.5 * minimum sectional width.
73 Check 3: A,,b,, * f,,yd,lim,, ≤ F,,stir,, = P,,2,, / cot(q) where
74 A,,b,, = (stirrup area) = n,,b,, * n / mod(2) * π * Ø^^2^^ / 4.
75 n / mod(2) = number of legs in even pairs
76 f,,yd,lim,, = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
77 Check 4: w,,req,, ≤ w,,avail,, where
78 w,,req,, = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
79 w,,avail,, = available width = minimum sectional width
80 \\If Check1 ≠ OK: increase number of legs per stirrup i.e. n = n + 2.
81 If Check 2 ≠ OK: increase q within the given range.
82 If Check 3 ≠ OK: increase number of stirrups i.e. nb = nb + 1.
83 If Check 4 ≠ OK: it's not possible to design the stirrups needed.
84 1. After designing the stirrups the top reinforcement has to be controlled so that it can withstand the tensional force, Ft.
85 1. The force in the reinforcement is derived as;
86 F,,s,, * (h,,t,, - a,,s,,) = F,,t,, * 2/3 * h,,t,, where
87 h,,t,, is the height of the tensional prism
88 a,,s,, is the distance from the top to the centre of the top reinforcement
89 F,,s,, is the corresponding force in the reinforcement
90 1. Check if A,,s,, * f,,yd,lim,, ≤ F,,s,, where
91 A,,s,, = reinforcement area.
92 1. 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.
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95 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)).
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98 [[image:MicrosoftTeams-image (7).png||height="334" width="387"]]
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101 = Design of Bursting reinforcement =
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103 If activated in the calculation settings, this calculation determines required stirrups in the end zone due to the Bursting phenomenon.
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106 [[image:end zone 6.jpg]]
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108 Figure 5: Fracture mode - bursting
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110 [[image:end zone 7.jpg]]
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112 Figure 6: Symmetric prism analogy
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115 The bursting force, N,,bs,, , is calculated using the symmetric prism analogy, see Figure 6.
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117 The methodology for calculating N,,bs,, , according to Modelcode 90: 7.13.6.5.2, is slightly adjusted and described in short below.
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120 Methodology: (this is done for both ends)
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122 1. Calculate the length of the prism, l,,bs,, = l,,pt,, - l,,spal,,.
123 1. The upper limit of the prism, h,,bs,, = 2 * y , where y is set equal to COG (centre of gravity) of the strands.
124 1. Calculate P,,1,, and P,,0,, (prestress force above and below COG respectively).
125 1. Distribute the total prestress force P,,TOT,, = P,,0,, + P,,1,, with respect to sectional area above and below COG i.e. P,,2,, = P,,TOT,, * A,,2,, / (A,,1,, + A,,2,,).
126 1. Calculate e,,1,, and e,,2,, where
127 e,,1,, = the distance from y to COG of strands above y
128 e,,2,, = the distance to centre of pressure block above y = y / 2 = h,,bs,, / 4
129 1. Calculate bursting force, N,,bs,, , by the moment equilibrium around point 'A' in Figure 7 i.e.
130 \\[[image:end zone 8.jpg]]
131 Figure 7: Equilibrium model
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133 1. If σ,,bs,, = N,,bs,, / (w,,bs,, * l,,bs,,) ≥ f,,ctd,, there's a need for extra stirrups.
134 σ,,bs,, = the stress due to bursting
135 w,,bs,, = minimum sectional width
136 f,,ctd,, = design tensional concrete stress = 'crack' stress
137 1. Number of stirrups required, n,,b,, = A,,s,, / A,,b,, where
138 A,,s,, = (total reinforcement area) = N,,bs,, / f,,yd,lim,,
139 f,,yd,lim,, = limited yield value due to crack width control. This can be modified under 'Calculation settings' tab.
140 A,,b,, = (stirrup area) = n / mod(2) * π * Ø^^2^^ / 4.
141 n / mod(2) = number of legs in even pairs
142 1. Check 1: l,,req,, ≤ l,,bs,, where:
143 l,,req,, = n,,b,, * (Ø + s) > l,,bs,,
144 Check 2: w,,req,, ≤ w,,bs,, where:
145 w,,req,, = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
146 w,,bs,, = available width = minimum sectional width
147 If Check1 ≠ OK: increase number of legs per stirrup i.e. n = n + 2 and re-calculate n,,b,,.
148 If Check 2 ≠ OK: it's not possible to design the stirrups needed.
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151 = Design of Anchorage reinforcement =
152
153 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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156 [[image:end zone 9.jpg]]
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158 Figure 8: Anchorage model
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161 Methodology: (this is done for both ends)
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163 1. Find the first position (from the end), xcrit, where σct > fctd i.e. where the beam is cracked.
164 1. Calculate the base transfer length, lpt, according to EN 1992-1-1 8.10.2.2 (2)
165 1. Calculate the anchorage force, F,,td,, = M,,Ed,, / z + [[image:Delta.png]]F,,td,, where:
166 M,,Ed,, = design moment in position x,,crit,,
167 z = internal lever = 0.9 * d
168 d = effective height
169 [[image:Delta.png]]F,,td,, = additional force due to shear = V,,Ed,, * a,,1,, / z where:
170 a,,1,, = d (without shear reinforcement), or
171 a,,1,, = 0.5 * z * cot(θ) (with shear reinforcement)
172 see EN 1992-1-1 9.2.1.3 (2)
173 1. Calculate σ,,pd,, = F,,td,, / A,,p,, where
174 A,,p,, = area of the strands
175 1. Calculate σ,,pm∞,, at position l,,pt2,, = 1.2 * l,,pt,, (see EN 1992-1-1 8.10.2.3 Fig. 8.17)
176 σ,,pm∞,, = initial prestress after any long-term effect
177 1. Calculate anchorage length , l,,bpd,, according to EN 1992-1-1 8.10.2.3 (4).
178 1. If x,,crit,, - c < l,,bpd,, anchorage reinforcement is needed.
179 1. It's only the excess force that is not anchored by bondage that has to be anchored by anchorage reinforcement i.e.
180 [[image:Delta.png]]L = l,,bpd,, - x,,crit,,
181 If [[image:Delta.png]]L ≤ l,,pt2,, : [[image:Delta.png]]F = σ,,pm∞,, * ([[image:Delta.png]]L / l,,pt2,,) * A,,p,,
182 If [[image:Delta.png]]L > l,,pt2,, : [[image:Delta.png]]F = (σ,,pm∞,, + (σ,,pd,, - σ,,pm∞,,)) * ([[image:Delta.png]]L - l,,pt2,,) / ( l,,bpd,, - l,,pt2,,)) * A,,p,,
183 1. Number of anchorage stirrups required, n,,b,, = As / A,,b,, where:
184 A,,s,, = (total anchorage reinforcement area) = [[image:Delta.png]]F / f,,yd,,
185 A,,b,, = (stirrup area) = n / mod(2) * π * Ø^^2^^ / 4.
186 n / mod(2) = number of legs in even pairs
187 1. Check 1: w,,req,, ≤ w,,avail,, where:
188 w,,req,, = required width of the section needed to fit the stirrups in depth = 2 * c + Ø + (n - 1) * (Ø + s)
189 w,,avail,, = available width = minimum sectional width
190 If Check 1 ≠ OK: it's not possible to design the anchorage stirrups needed.
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197 **Contents**
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199 {{toc/}}
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Copyright 2020 StruSoft AB
PRE-Stress Documentation