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5 = Bending reinforcement =
6
7 A keystone in ultimate bending design is the ability to design required reinforcement in a section (whether tensile, compression, or both) with enough flexural capacity for specific section forces. This can be done in several ways, as a trial and error method, a coefficient method, or a direct method. The suitable method depends on section type, moment vector direction, reinforcement rules, and so on. Here we assume bending axis to be parallel to y-axis or z-axis, and then use a direct method to get the most efficient result. In case of biaxial bending, the design problem generally has more than one solution, and we then use an approximate [[method>>||anchor="HArrangingreinforcementbars"]].
8 [[image:1538248760118-971.png||height="499" width="1010"]]
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10 Conditions for the design must be set before the design calculation can begin. These include concrete and reinforcement quality, bar sizes, cover and spacing rules, as well as maximum and minimum rules for bar sizes and number of bars.
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12 Cover and spacing rules dictate possible positions of reinforcement layers. This applies equally to bottom and top of section as well as possible numbers of bars in each layer at a given bar size.
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14 From material properties we get maximum compressive strain in concrete, and maximum tensile strain and yield strain in reinforcement. Since we have assumed bending axis to be parallel to the y-axis or z-axis, it follows that the neutral axis is also parallel to the y-axis or z-axis. It is now possible to establish various strain distributions for the section within the limits given by the maximum strain values. A set of internal normal- and moment forces will correspond to each combination of strain distribution and bar positions. An iterative procedure is then mostly necessary to establish a suitable set of bars, which together with an appropriate strain distribution will give the requested capacity.
15
16 == Balanced reinforcement and balanced moment ==
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18 To begin, we choose a strain distribution corresponding to ε,,c,, = ε,,cmax,, and ε,,s,, = f,,y,, / E,,sk,,. When calculating internal forces in section, we use the method proposed in chapter 3.3.3.1. No reinforcement area is set initially, so the strain distribution will only give the compression force F,,c,, (in the concrete) and its moment M,,c,, with reference to the point **R** (the point of action for the external section forces). Equilibrium in a direction perpendicular to the section plane will then give the required tensile force in reinforcement, and thus the corresponding reinforcement area. The following expressions thus apply, with forces being positive at tension:
19
20 F,,s,, = N,,x,, - F,,c,,
21
22 and the balanced reinforcement area:
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24 A,,s0bal,, = F,,s,, / σ,,s0,,
25
26 where:
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28 * σ,,s0,, = f,,s,, (ε,,s,,) is calculated using the stress-strain relation for steel (see chapter 3.3.3.3).
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30 If F,,s,, < 0 we set Fs = 0 because a negative value does not conform to the chosen strain distribution.
31
32 The moment capacity with balanced reinforcement will be the balanced moment
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34 M,,bal,, = M,,cap,, = M,,c,, + F,,s,, (z,,R,, - z,,0,,)
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36 Thus far we have a moment capacity, which corresponds to a reinforcement area in one lower layer. If current section force M,,y,, is not equal to M,,bal,,, various steps of action have to be taken depending on whether M,,y,, < M,,bal,, or not.
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38 In later iteration steps, additional reinforcement layers (normally filled up) may be present on the tensile side (and compression side) and thus contribute to the internal forces. These layers will be regarded as part of the section, and balanced reinforcement in the lowest unfilled layer can be calculated gradually. The equilibrium equation is modified to
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40 F,,s,, = N,,x,, - F,,c,, - F,,t,,
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42 and the balanced moment in this case becomes
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44 M,,bal,, = M,,cap,, = M,,c,, + M,,t,, + F,,s,, (z,,R,, - z,,0,,)
45
46 == Reinforcement with compression bars ==
47
48 Most cases of M,,y,, > M,,bal,, will give us the greatest economy of reinforcement (= smallest total area), with reinforcement in compression zone and increased tensile reinforcement.
49
50 The additional force in upper layer //n// and lower layer //0// will then be
51
52 [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F = (M,,y,, - M,,bal,,) / (z,,n,, - z,,0,,)
53
54 Calculate strain in upper layer n and corresponding stress σ,,sn,, using the stress-strain relation for steel as above:
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56 σ,,sn,, = f,,s,, (ε,,n,,)
57
58 Calculate the concrete stress σ,,cn,, at the position of layer //n// using the stress-strain relation for concrete (see chapter 3.3.3.2).
59
60 Reinforcement area in compression zone will be
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62 A,,sn,, = [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F / (σ,,sn,, - σ,,cn,,) and tensile reinforcement area will be
63
64 A,,s0,, = A,,s0bal,, + [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F / σ,,s0,,
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66 As the reinforcement area corresponds to a number of bars (represented by an integer), we will not get the most economical reinforcement if we put in bars both in compression zone and tensile zone corresponding to respective zone area. Instead we choose to round the compression zone area to the closest integer of bars and calculate the tensile area corresponding to this rounded-off area value. We get
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68 A,,sn,adjusted,, = round (A,,sn,, / A,,bar,,) A,,bar,,
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70 where round means we round using the common method of rounding (Half Round Up).
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72 A,,s0,adjusted,, = A,,s0,, - (A,,sn,, - A,,sn,adjusted,,) (σ,,sn,, - σ,,cn,,) / σ,,s0,,
73
74 == {{id name="Underbalanced reinforcement"/}}Underbalanced reinforcement ==
75
76 When M,,y,, < M,,bal,,, the balanced reinforcement area in the tensile zone is too great and must be decreased to provide greater economy of reinforcement.
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78 As the balanced reinforcement area is calculated with ε,,s,, ≥ ε,,y,,, a decrease in area also means a decrease of tensile force in reinforcement. The decrease of force in compression zone and tensile reinforcement will then be
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80 [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F = (M,,y,, - M,,bal,,) / (z,,c,, - z,,0,,)
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82 Required compression force will be given by F,,c,, = F,,c,bal,, + [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F (note that [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F is negative in this case). With no reinforcement in compression zone, the area of the compression zone then must be decreased to maintain equilibrium, which will influence the position of F,,c,, and thus also change z,,c,,. Consequently, an iterative procedure is required to calculate [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F.
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84 The iteration is performed by estimating a new value of the compression zone depth. A suitable estimation at each step //i// is
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86 d,,c,, (i + 1) = d,,c,, (//i//) M,,y,, / M,,cap,, (//i//)
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88 Keeping the concrete strain constant will yield a new value of the tensile strain ε,,s,, by calculating
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90 ε,,s,, = ε,,c,, (d - d,,c,,) / d,,c,,
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92 and we may calculate new forces F,,c,, and F,,s0,, (which include [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F).
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94 When a constant ε,,c,, gives ε,,s,, > ε,,s,max,, instead ε,,c,, must be decreased keeping ε,,s,, = ε,,s,max,, constant.
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96 The iteration continues until M,,cap,, (//i//) = M,,y,,.
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98 When ε,,s0,, and F,,s0,, have been calculated corresponding stress σ,,s0,, is calculated using stress-strain relation and required reinforcement area will be
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100 A,,s0,, = F,,s0,, / σ,,s0,,
101
102 == Reinforcement at tensed section ==
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104 At external tensile normal force (N,,x,, > 0) the eccentricity of the normal force must be checked to be able to decide if tensile reinforcement is needed both in bottom and top of section.
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106 The eccentricity of the normal force is:
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108 e,,N,, = M,,y,, / N,,x,,
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110 When e,,N,, > z,,R,, - z,,0,, the calculation may be performed as in the usual way as described before (see chapters 3.2.1.2 - 3.2.1.3).
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112 When e,,N,, < z,,R,, - z,,0,, equilibrium is not possible without tensile reinforcement in top as well as in bottom.
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114 In the last case, forces in top and bottom can be calculated directly. Forces in upper layer n and lower layer //0// are calculated as:
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116 F,,s0,, = [M,,y,, + N,,x,, (z,,R,, - z,,0,,)] / (z,,n,, - z,,0,,)
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118 F,,sn,, = [-M,,y,, + N,,x,, (z,,n,, - z,,R,,)] / (z,,n,, - z,,0,,)
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120 We then set the strain distribution equal to ε,,y,, both in upper and lower layer and use the stress-strain relation to calculate stresses in the layers.
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122 In the lower layer the tensile reinforcement area will be
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124 A,,s0,, = F,,s0,, / σ,,s0,,
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126 And in the upper layer,
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128 A,,sn,, = F,,sn,, / σ,,sn,,
129
130 == Without tensile reinforcement ==
131
132 When M,,y,, < M,,bal,, the balanced reinforcement area is too large and (in some cases) unnecessary for establishing equilibrium between internal and external forces.
133
134 The eccentricity of the normal force is
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136 e,,N,, = M,,y,, / N,,x,,
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138 If the balanced reinforcement area is ignored, we can verify that if e,,N,, < M,,c,bal,, / F,,c,, and M,,y,, < M,,c,bal,, , no tensile reinforcement is needed. As N,,x,, < F,,c,, , it is always possible to find a strain distribution that satisfies equilibrium.
139
140 When e,,N,, > M,,c,bal,, / F,,c,, or M,,y,, > M,,c,bal,, we start with assuming that underbalanced reinforcement is a correct solution. If we then at iteration find that for a suitable εs equilibrium requires that ε,,s,, < 0 no tensile reinforcement is needed and the iteration is changed to find a compression zone depth that satisfies equilibrium.
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142 The iterative procedure is as follows: the iteration is performed by estimating a new value of the compression zone depth. A suitable estimation at each step i is
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144 d,,c,, (i + 1) = d,,c,, (//i//) N,,x,, / F,,c,, (//i//)
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146 Keeping ε,,c,, constant will give a new value of the tensile strain ε,,s,, (although no reinforcement is assumed at the lower layer) by calculating
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148 ε,,s,, = ε,,c,, (d - d,,c,,) / d,,c,, and we can calculate a new value of the force F,,c,,.
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150 The iteration continues until F,,c,, (//i//) = N,,x,, .
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152 When M,,y,, < M,,c,, the capacity is sufficient without reinforcement for the obtained strain distribution.
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154 When M,,y,, > M,,c,, , compression reinforcement (and perhaps tensile reinforcement) is needed in order to get sufficient moment capacity.
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156 The added force in upper layer //n// required to get M,,y,, = M,,cap,, will then be
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158 [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F = (M,,y,, - M,,c,,) / (z,,n,, - z,,R,,)
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160 Calculate strain in upper layer //n// and corresponding stress σ,,sn,, using the stress-strain relation as above:
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162 σ,,sn,, = f,,s,, (ε,,n,,)
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164 Finally, you may calculate the concrete stress σ,,cn,, at the position of layer //n// using the concrete stress-strain relation (see chapter 4.4.3.2).
165
166 Compression reinforcement area will be
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168 A,,sn,, = [[image:https://wiki.pre-stress.strusoft.com/xwiki/wiki/prestresswiki/download/PRE-Stress/Theory%20PRE-Stress/Inelastic%20strain%20at%20cracking/WebHome/1537735242649-279.png?width=12&height=13||alt="1537735242649-279.png"]]F / (σ,,sn,, - σ,,cn,,)
169
170 == Arranging reinforcement bars ==
171
172 When calculating the required reinforcement area for layers in bottom and top, it must be possible to arrange the corresponding number of bars in the layer according to rules for cover and spacing. If this is possible, the design of reinforcement is finished. If the requisite number of bars is greater than the maximum possible number of bars in a layer, we must accept this number and increase the number of layers to provide space for the missing bars. Since the new layer is not as favourably situated as the one used for calculation, the calculation must be repeated to get a more correct value for the added layer.
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174 When repeating the calculation, the layers that are filled up will be treated as fixed, and the added layers are treated as bottom and top layer.
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176 The calculation is repeated as long as reinforcement must be moved to new layers. When there is no space for a new layer on the proper side of the neutral axis, we have exhausted the possibility of applying more reinforcement, and this sets a limit to the capacity of the section.
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178 [[image:1538249202943-752.png||height="446" width="416"]]
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185 **Contents**
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187 {{toc/}}
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Copyright 2020 StruSoft AB
PRE-Stress Documentation