Lemes et al 2022b - Revision history

Igor.lemes: Igor.lemes moved page Lemes et al 2022a to Lemes et al 2022b

2022-10-04T14:09:28Z

Igor.lemes moved page Lemes et al 2022a to Lemes et al 2022b

Rimni at 13:33, 4 October 2022

2022-10-04T13:33:21Z

Rimni at 13:27, 4 October 2022

2022-10-04T13:27:53Z

Rimni at 13:27, 4 October 2022

2022-10-04T13:27:04Z

Rimni: /* 2.6. Flexural stiffness */

2022-10-04T13:25:24Z

‎2.6. Flexural stiffness

Rimni at 13:24, 4 October 2022

2022-10-04T13:24:47Z

Rimni: /* 3.2. Steel-concrete composite section */

2022-10-04T13:20:15Z

‎3.2. Steel-concrete composite section

Rimni at 13:16, 4 October 2022

2022-10-04T13:16:15Z

Rimni at 13:08, 4 October 2022

2022-10-04T13:08:14Z

Rimni: /* 3. Numerical applications */

2022-10-04T13:04:04Z

‎3. Numerical applications

@@ Line 769: / Line 769: @@
 :* In general, regardless of the adopted residual stresses model, the FYCs were practically identical for the bare steel sections, and very similar for the composite section. Thus, it can be concluded that the adopted residual stresses model does not change the bearing capacity of the cross-sections;
 :* For the bare steel section, the IYCs in the minor axis bending analysis presented asymmetry in relation to the horizontal axis (bending moment). This asymmetry demonstrates the need to review the propositions of the RPHM for the evaluation of minor axis bending;
-:* The IYCs can be defined through specific analytically calculated points. In the present work, they were defined as points A, B, C and D;
+:* The IYCs can be defined through specific analytically calculated points. In the present work, they were defined as points <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math>;
-:* In all simulated cases, the AISC [28] approach defines a larger elastic region within the NM diagrams. A more detailed study of the UB 914x305x201 section provided a range of quantitative parameters for the validation of the above information. By comparing all the bending moment increments in the moment-curvature analysis, it was found that, on average, the analysis using the European residual stress model was 4.81% less rigid than the one using the American model;
+:* In all simulated cases, the AISC [28] approach defines a larger elastic region within the NM diagrams. A more detailed study of the UB <math>914\times 305\times 201</math> section provided a range of quantitative parameters for the validation of the above information. By comparing all the bending moment increments in the moment-curvature analysis, it was found that, on average, the analysis using the European residual stress model was 4.81% less rigid than the one using the American model;
-:* Still observing the IYC of UB914x305x201, we can see that the differences between the design code models appear for the tensiled section. In this case, for example, the average difference between the bending moments is 31.25% for the major axis bending. This cross section was chosen because the graphical responses of the moment-curvature relationships using the two residual stress models was very closed;
+:* Still observing the IYC of UB <math>914\times 305\times 201</math>, we can see that the differences between the design code models appear for the tensiled section. In this case, for example, the average difference between the bending moments is 31.25% for the major axis bending. This cross section was chosen because the graphical responses of the moment-curvature relationships using the two residual stress models was very closed;
 :* For the analysis of the steel-concrete composite section, it was noticeable the effects of residual stresses become more relevant as the axial compressive forces increase and the softening effect of the concrete is treated with greater intensity.

@@ Line 498: / Line 498: @@
 ===3.1. Steel sections===
-Zubydan [26] analyzed the flexural stiffness variation in I (UB) and H (UC) sections versus the increase of the bending moment acting in the cross-section. The author exclusively studied steel elements under minor axis bending. Thus, the author used the considerations of EC3 [29] to define the residual stresses distribution. The material yield strength is 25 kN/cm² and Young’s modulus is given by 20000 kN/cm<sup>2</sup>.
+Zubydan [26] analyzed the flexural stiffness variation in I (UB) and H (UC) sections versus the increase of the bending moment acting in the cross-section. The author exclusively studied steel elements under minor axis bending. Thus, the author used the considerations of EC3 [29] to define the residual stresses distribution. The material yield strength is 25 kN/cm<sup>2</sup> and Young’s modulus is given by 20000 kN/cm<sup>2</sup>.
 [[#img-7|Figure 7]] shows the moment-curvature and bending moment-flexural stiffness curves for both UC and UB sections using the EC3 [29] and AISC [28] considerations. Through [[#img-7|Figures 7]](b) and [[#img-7|7]](d), the good agreement between the results obtained considering the European residual stress model with the Zubydan [26] results, for both UB and UC sections is shown. Using the AISC [28] model, a good proximity to the other results for the UB sections can be observed. These cross-sections have a height that is greater than the width; thus, the residual compressive stress at the flange ends is equivalent to the EC3 [29] residual stresses, which considerably approximate the responses ([[#img-7|Figure 7]](c)).

@@ Line 631: / Line 631: @@
 ===3.2. Steel-concrete composite section===
-Chiorean [18] presented a study of the influence of residual stress models on the steel-concrete composite section behavior as shown in [[#img-12|Figure 12]]. This is a W12<math>\times</math>120 section that is fully encased in concrete and reinforced by 4 20 mm diameter bars. The section dimensions <math>h</math>, <math>b</math> and <math>c</math>, defined in [[#img-3|Figure 3]], are taken as 600 mm, 60 0mm and 30 mm, respectively. The strengths steel section, reinforcement steel and compressive concrete strengths are taken equal to 30, 40 and 2 kN/cm², respectively. The modulus of elasticity of the steel section and the reinforcement are equal to 20000 kN/cm². In addition, it is noted that the concrete limit strains, in concrete, the final strain of the second-degree parabola, <math>\varepsilon_{ci}</math>, was taken as 0.002 and the ultimate strain was defined as 0.0035. The steel ultimate strain was taken as 0.01. The concrete softening was simulated by the parameter <math>\gamma = 0.15</math>, and was considered <math>\alpha_1 = 1</math> and <math>\alpha_2 = 0.75</math> for the concrete tensile behavior [18].
+Chiorean [18] presented a study of the influence of residual stress models on the steel-concrete composite section behavior as shown in [[#img-12|Figure 12]]. This is a W12<math>\times</math>120 section that is fully encased in concrete and reinforced by 4 20 mm diameter bars. The section dimensions <math>h</math>, <math>b</math> and <math>c</math>, defined in [[#img-3|Figure 3]], are taken as 600 mm, 60 0mm and 30 mm, respectively. The strengths steel section, reinforcement steel and compressive concrete strengths are taken equal to 30, 40 and 2 kN/cm<sup>2</sup>, respectively. The modulus of elasticity of the steel section and the reinforcement are equal to 20000 kN/cm<sup>2</sup>. In addition, it is noted that the concrete limit strains, in concrete, the final strain of the second-degree parabola, <math>\varepsilon_{ci}</math>, was taken as 0.002 and the ultimate strain was defined as 0.0035. The steel ultimate strain was taken as 0.01. The concrete softening was simulated by the parameter <math>\gamma = 0.15</math>, and was considered <math>\alpha_1 = 1</math> and <math>\alpha_2 = 0.75</math> for the concrete tensile behavior [18].
 <div id='img-12'></div>

@@ Line 408: / Line 408: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
 |}
 Tangent flexural stiffness, that is, using values of the modulus of elasticity tangent to the constitutive relationships, must be calculated in relation to the updated position of the PC, <math>y_{PC}</math>. Thus, the <math>EI_T</math> stiffness is given by:
@@ Line 420: / Line 421: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)
 |}
 Since the system is discretized in fibers, the integrals of the previous equation become summations. That is:
@@ Line 432: / Line 434: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)
 |}
 Substituting the value of <math>y_{PC}</math> (Eq. (19)) in Eq. (21), we arrive at:
@@ Line 456: / Line 459: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (23)
 |}
 Correlating Eqs. (15-18) to Eq. (23), it is noted that <math>EI_T</math> can be described in terms of the constitutive matrix of the cross section. Thus, the flexural stiffness of the section can be written as:
@@ Line 468: / Line 472: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (24)
 |}
 Graphically, the previous equation refers to the tangent flexural stiffness to the moment-curvature relationship.

@@ Line 292: / Line 292: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
 |}
 Starting from the supposed equilibrium at point <math>{\bf X}+ \delta{\bf X}</math>, that is, <math>{\bf F}({\bf X}+\delta{\bf X}) = {\bf 0}</math>, and knowing that <math>\delta {\bf X}={\bf X}^{k+1} -{\bf X}^{k}</math>, Eq. (10) can be rewritten as:
@@ Line 304: / Line 305: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)
 |}
 where <math>\boldsymbol{F}</math> is the Jacobian matrix of the non-linear problem stated in Eq. (8). That is:
@@ Line 320: / Line 320: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)
 |}
 The term <math>f_{11}</math> can be determined as follows:
@@ Line 344: / Line 345: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (14)
 |}
 The derivative of stress, <math>\sigma </math>, with respect to strain, <math>\varepsilon</math>, results in the tangent modulus of elasticity, <math>E_T</math>. In this way, <math>f_{11}</math> can be write as:
@@ Line 356: / Line 358: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)
 |}
 Analogous to the development of <math>f_{11}</math>, the other terms of <math>\boldsymbol{F}'</math> are given by:

← Older revision	Revision as of 14:09, 4 October 2022
(No difference)

@@ Line 68: / Line 68: @@
 To make the analysis of the deformed cross-section condition and obtain the bearing capacity and its stiffness, a section discretization is made. In plane structures study, the layer discretization is satisfactory [23]. Herein, the residual stresses will be introduced explicitly in the steel cross-section. Thus, the two-dimensional discretization will be done, as shown in [[#img-1|Figure 1]].
-The concept of sub-regions is then applied ([[#img-1|Figure 1]]b). By means of a structured mesh generator, the cross-section is divided into smaller regions. Although only a full encased I-section is shown in [[#img-1|Figure 1]], the algorithm is applied to several types of steel, reinforced concrete and composite cross-sections.
+The concept of sub-regions is then applied ([[#img-1|Figure 1]](b)). By means of a structured mesh generator, the cross-section is divided into smaller regions. Although only a full encased I-section is shown in [[#img-1|Figure 1]], the algorithm is applied to several types of steel, reinforced concrete and composite cross-sections.
 <div id='img-1'></div>
@@ Line 105: / Line 105: @@
 Although the SCM-based methodology is generalized, that is, applicable to any cross-section type, this paper focuses exclusively on the steel and steel-concrete composite I-sections’ behavior submitted to 2D eccentric loads. Thus, the residual stress models used in this research are defined by design codes for this type of cross-section.
-The EC3 [29] uses the residual stress model based on Huber and Beedle’s [3] proposition, shown in [[#img-2|Figure 2]]b. In this model, the stresses are arranged in the cross-section plates by a bilinear distribution. Furthermore, the <math> \sigma_r </math> values depend directly on the relation between the cross-section height <math> h </math> and its width <math>b  </math>, as follows:
+The EC3 [29] uses the residual stress model based on Huber and Beedle’s [3] proposition, shown in [[#img-2|Figure 2]](b). In this model, the stresses are arranged in the cross-section plates by a bilinear distribution. Furthermore, the <math> \sigma_r </math> values depend directly on the relation between the cross-section height <math> h </math> and its width <math>b  </math>, as follows:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 137: / Line 137: @@
-AISC [28] follows the proposal of Galambos and Ketter [4] that describes the residual stresses at the flanges in a similar way, but with different maximum values of tensile, <math> {\sigma }_{rt} </math>, and compression, <math> {\sigma }_{rc} </math>. In the web, the distribution is constant, as shown in [[#img-2|Figure 2]]c, and described as:
+AISC [28] follows the proposal of Galambos and Ketter [4] that describes the residual stresses at the flanges in a similar way, but with different maximum values of tensile, <math> {\sigma }_{rt} </math>, and compression, <math> {\sigma }_{rc} </math>. In the web, the distribution is constant, as shown in [[#img-2|Figure 2]](c), and described as:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 183: / Line 183: @@
 ===2.4. Uniaxial behavior of concrete===
-Concrete has distinct mechanical properties with respect to tensile and compression. With respect to tensile, this material exhibits maximum strength <math> f_{cr} </math> ([[#img-4|Figure 4]]b). It is also emphasized that when the strength <math> f_{cr} </math> is reached, the cracking process begins. The concrete loses strength and stiffness in case of strains greater than <math>\varepsilon_{cr} </math>. Thus, several researchers and even design codes disregard their contribution when tensioned. In the present study, under tensile, the constitutive relationship proposed by Vecchio and Collins [33], shown in [[#img-4|Figure 4]]b, is used. The compressed concrete is modeled as defined in Chiorean [18], as it is the second-degree parabola presented in a reduced form. Thus, to describe the concrete behavior we have:
+Concrete has distinct mechanical properties with respect to tensile and compression. With respect to tensile, this material exhibits maximum strength <math> f_{cr} </math> ([[#img-4|Figure 4]](b)). It is also emphasized that when the strength <math> f_{cr} </math> is reached, the cracking process begins. The concrete loses strength and stiffness in case of strains greater than <math>\varepsilon_{cr} </math>. Thus, several researchers and even design codes disregard their contribution when tensioned. In the present study, under tensile, the constitutive relationship proposed by Vecchio and Collins [33], shown in [[#img-4|Figure 4]](b), is used. The compressed concrete is modeled as defined in Chiorean [18], as it is the second-degree parabola presented in a reduced form. Thus, to describe the concrete behavior we have:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 617: / Line 617: @@
 It can be stated that for this norm, the elastic regime is higher than in EC3 [29], as seen in [[#img-9|Figure 9]]. Point <math>C </math> is analyzed in the same manner as point <math>B </math>, except for using American code considerations.
-Point <math>D </math>, as shown in [[#img-11|Figure 11]]a, follows the same approach shown in the UB sections curves. Note that the UC sections squarer in shape, modifying the value of <math>\sigma_{r}</math> given by the European design code. In this situation, <math>\sigma_{r} = 0.5f_y</math>. Thus, since the axial tensile force responsible for overriding the residual compressive stresses at the webs’ ends is <math>N_d = A_g (0.5f_y</math>), the maximum bending moment is obtained as described for point <math>A</math>, discussed above. However, by canceling these compressive stresses, the tensile stresses in the PC increase, which then reach the <math>f_y</math> value. This description is very clear in [[#img-11|Figure 11]], where the EC3 [29] residual stress model is used.
+Point <math>D </math>, as shown in [[#img-11|Figure 11]](a), follows the same approach shown in the UB sections curves. Note that the UC sections squarer in shape, modifying the value of <math>\sigma_{r}</math> given by the European design code. In this situation, <math>\sigma_{r} = 0.5f_y</math>. Thus, since the axial tensile force responsible for overriding the residual compressive stresses at the webs’ ends is <math>N_d = A_g (0.5f_y</math>), the maximum bending moment is obtained as described for point <math>A</math>, discussed above. However, by canceling these compressive stresses, the tensile stresses in the PC increase, which then reach the <math>f_y</math> value. This description is very clear in [[#img-11|Figure 11]], where the EC3 [29] residual stress model is used.
 With the present study, the need to review the initial yield curves for steel I-sections subjected to minor axis bending is evidenced. With a brief description of the ordered pairs referring to points <math>A </math>, <math>B </math>, <math>C</math> and <math>D</math>, it is possible to draw simplified curves, where the non-linear procedure located in the cross-section can be avoided by reducing the processing time in the numerical structural analyses via RPHM. It is worth mentioning in the study by Gonçalves ''et al.'' [37], for the application of the RPHM for the analysis of elements under minor-axis bending, the authors had to use the tangent elastic modulus for the flexural stiffness degradation, since the initial yield curves did not portray the reality.
@@ Line 634: / Line 634: @@
-Chiorean [18] performed several analyses using both EC3 [29] and AISC [28] residual stress models. Additionally, results were also generated disregarding the effect of such residual stresses (WRS data). All these analyses were also done in the present work in order to calibrate the models, and to verify the initial (IYCs) and full (FYCs) yield curves of this composite section. Another aim of this analysis is the evaluation of the influence of the elastic limit strain of the concrete elastic regime (<math>\varepsilon_{ci}/2</math>) [21] on definitions of IYCs. The moment-curvature relationship ([[#img-13|Figure 13]]a) and flexural stiffness-bending moment curves ([[#img-13|Figure 13]]b) for the major axis are presented below. The results in all cases were compared with literature. For each curve, the axial force values were set as: 4000 kN, 8000 kN and 12000 kN. The numerical procedure described in item 2.5 was, thus, used.
+Chiorean [18] performed several analyses using both EC3 [29] and AISC [28] residual stress models. Additionally, results were also generated disregarding the effect of such residual stresses (WRS data). All these analyses were also done in the present work in order to calibrate the models, and to verify the initial (IYCs) and full (FYCs) yield curves of this composite section. Another aim of this analysis is the evaluation of the influence of the elastic limit strain of the concrete elastic regime (<math>\varepsilon_{ci}/2</math>) [21] on definitions of IYCs. The moment-curvature relationship ([[#img-13|Figure 13]](a)) and flexural stiffness-bending moment curves ([[#img-13|Figure 13]](b)) for the major axis are presented below. The results in all cases were compared with literature. For each curve, the axial force values were set as: 4000 kN, 8000 kN and 12000 kN. The numerical procedure described in item 2.5 was, thus, used.
 <div id='img-1'></div>
@@ Line 657: / Line 657: @@
 Results of the analyses were observed to be in good agreement with the results obtained by Chiorean [18]. It is worth mentioning that in some cases, the moment-curvature relationships found in the present study are less ductile than those in the literature. Although procedures used are similar, the stopping criteria are different as they directly influence the moment-curvature relationship’s critical bending moment.
-All the curves in [[#img-13|Figure 13]]b demonstrate a marked decrease in flexural stiffness. This decrease is related to the cracking start in the most tensioned fibers and the consequent decrease of the concrete tangent modulus of elasticity. Note that in the literature for approaches to steel-concrete composite structures [21,23,24], this fact is considered in a simplified form, for example, the 40% reduction in concrete section flexural stiffness. Thus, to eliminate the approximation, there should be three curves within the NM diagram, i.e., in addition to the FYC and IYC, there is a need for an initial cracking curve for a more realistic simulation [22].
+All the curves in [[#img-13|Figure 13]](b) demonstrate a marked decrease in flexural stiffness. This decrease is related to the cracking start in the most tensioned fibers and the consequent decrease of the concrete tangent modulus of elasticity. Note that in the literature for approaches to steel-concrete composite structures [21,23,24], this fact is considered in a simplified form, for example, the 40% reduction in concrete section flexural stiffness. Thus, to eliminate the approximation, there should be three curves within the NM diagram, i.e., in addition to the FYC and IYC, there is a need for an initial cracking curve for a more realistic simulation [22].
-The comparison of the moment-curvature relationships ([[#img-13|Figure 13]]a) and the bending moment-flexural stiffness curves ([[#img-13|Figure 13]]b) shows a slight divergence in the EC3 [29] and AISC [28] responses. In these figures, it is clear how the AISC [28] approach allows for greater stiffness than the EC3 [29] approach. This is because the W12x120 section is almost shaped like a square, that is, <math>h/b <1.2</math>. Thus, the European code defines that the maximum residual stresses acting in the section reach 0.5<math>f_y</math>, which differs from Galambos and Ketter’s [4] proposal.
+The comparison of the moment-curvature relationships ([[#img-13|Figure 13]](a)) and the bending moment-flexural stiffness curves ([[#img-13|Figure 13]](b)) shows a slight divergence in the EC3 [29] and AISC [28] responses. In these figures, it is clear how the AISC [28] approach allows for greater stiffness than the EC3 [29] approach. This is because the W12<math>\times</math>120 section is almost shaped like a square, that is, <math>h/b <1.2</math>. Thus, the European code defines that the maximum residual stresses acting in the section reach 0.5<math>f_y</math>, which differs from Galambos and Ketter’s [4] proposal.
 It is also worth mentioning that, in the analysis without residual stresses (WRS results), it is observed, as expected, a more rigid behavior for all the moment-curvature relationships as well as presented in bending moment-flexural stiffness curves.
-The same observations made for analysis around the major axis can now be made for the minor axis ([[#img-13|Figures 13]]c and [[#img-13|13]]d), which, although when presenting curves with different values, the observed behavior is quite similar.
+The same observations made for analysis around the major axis can now be made for the minor axis ([[#img-13|Figures 13]](c) and [[#img-13|13]]d), which, although when presenting curves with different values, the observed behavior is quite similar.
-[[#img-14|Figures 14]]a and [[#img-14|14]]b show both IYCs and FYCs obtained using the EC3 [29] and AISC [28] requirements for the major and minor axes bending. In addition, the results found disregarding residual stresses are also presented. It is worth mentioning that there was good accuracy between the results obtained in the present study and those presented by Chiorean [18]. Although practically equal procedures have been used to obtain the moment-curvature relationship, the stopping criterion is different. This criterion is precisely the one responsible for defining the FYCs points, which validates the use of the Jacobian matrix singularity in the definition of the cross-section-bearing capacity. As can be observed in [[#img-14|Figures 14]]a and [[#img-14|14]]b, there is a slight difference in the FYCs using the residual stresses of EC3 [29] and AISC [28]. The main factor of comparison is in the IYCs. There is a significant divergence for the major and minor axes, and in both situations, the AISC [28] model provided a higher elastic region within the <math>NM</math> diagrams. It is also noted that when neglecting the effect of residual stresses, the elastic region of the <math>NM</math> diagram is amplified, as expected. Furthermore, it can be verified for intermediate values of <math>N</math>, the IYCs are coincident for the three results, defining that for these values, the strain <math>\varepsilon_{ci}</math> is responsible for defining the referred curve.
+[[#img-14|Figures 14]](a) and [[#img-14|14]](b) show both IYCs and FYCs obtained using the EC3 [29] and AISC [28] requirements for the major and minor axes bending. In addition, the results found disregarding residual stresses are also presented. It is worth mentioning that there was good accuracy between the results obtained in the present study and those presented by Chiorean [18]. Although practically equal procedures have been used to obtain the moment-curvature relationship, the stopping criterion is different. This criterion is precisely the one responsible for defining the FYCs points, which validates the use of the Jacobian matrix singularity in the definition of the cross-section-bearing capacity. As can be observed in [[#img-14|Figures 14]](a) and [[#img-14|14]](b), there is a slight difference in the FYCs using the residual stresses of EC3 [29] and AISC [28]. The main factor of comparison is in the IYCs. There is a significant divergence for the major and minor axes, and in both situations, the AISC [28] model provided a higher elastic region within the <math>NM</math> diagrams. It is also noted that when neglecting the effect of residual stresses, the elastic region of the <math>NM</math> diagram is amplified, as expected. Furthermore, it can be verified for intermediate values of <math>N</math>, the IYCs are coincident for the three results, defining that for these values, the strain <math>\varepsilon_{ci}</math> is responsible for defining the referred curve.
 <div id='img-14'></div>

@@ Line 581: / Line 581: @@
-Three points are highlighted in [[#img-9|Figure 9]](a): <math>A</math>, <math>B</math> and <math>C</math>. Point A represents the situation where the cross section shows the greatest bending moment within the elastic regime. For the UB sections, it is apparent that this point is coincident for both residual stress models, EC3 [29] and AISC [28]. This is because both these models present compression stresses at the flanges ends equal to 0.3<math>f_y</math>. For minor axis bending, these stresses are responsible for determining the maximum possible elastic bending moment in the section. To reach this moment, an axial tensile force, <math>N_a</math>, capable of canceling the value of the residual compression stresses (0.3<math>f_y</math>) is required, that is:
+Three points are highlighted in [[#img-9|Figure 9]](a): <math>A</math>, <math>B</math> and <math>C</math>. Point <math>A</math> represents the situation where the cross section shows the greatest bending moment within the elastic regime. For the UB sections, it is apparent that this point is coincident for both residual stress models, EC3 [29] and AISC [28]. This is because both these models present compression stresses at the flanges ends equal to 0.3<math>f_y</math>. For minor axis bending, these stresses are responsible for determining the maximum possible elastic bending moment in the section. To reach this moment, an axial tensile force, <math>N_a</math>, capable of canceling the value of the residual compression stresses (0.3<math>f_y</math>) is required, that is:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 592: / Line 592: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (25)
 |}
 where <math>A_g</math> is the steel section area and <math>f_y</math> is the material yield strength.
@@ Line 618: / Line 617: @@
 It can be stated that for this norm, the elastic regime is higher than in EC3 [29], as seen in [[#img-9|Figure 9]]. Point <math>C </math> is analyzed in the same manner as point <math>B </math>, except for using American code considerations.
-Point <math>D </math>, as shown in [[#img-11|Figure 11]] 11(a), follows the same approach shown in the UB sections curves. Note that the UC sections squarer in shape, modifying the value of <math>\sigma_{r}</math> given by the European design code. In this situation, <math>\sigma_{r} = 0.5f_y</math>. Thus, since the axial tensile force responsible for overriding the residual compressive stresses at the webs’ ends is <math>N_d = A_g (0.5f_y</math>), the maximum bending moment is obtained as described for point A, discussed above. However, by canceling these compressive stresses, the tensile stresses in the PC increase, which then reach the <math>f_y</math> value. This description is very clear in [[#img-11|Figure 11]], where the EC3 [29] residual stress model is used.
+Point <math>D </math>, as shown in [[#img-11|Figure 11]] 11(a), follows the same approach shown in the UB sections curves. Note that the UC sections squarer in shape, modifying the value of <math>\sigma_{r}</math> given by the European design code. In this situation, <math>\sigma_{r} = 0.5f_y</math>. Thus, since the axial tensile force responsible for overriding the residual compressive stresses at the webs’ ends is <math>N_d = A_g (0.5f_y</math>), the maximum bending moment is obtained as described for point <math>A</math>, discussed above. However, by canceling these compressive stresses, the tensile stresses in the PC increase, which then reach the <math>f_y</math> value. This description is very clear in [[#img-11|Figure 11]], where the EC3 [29] residual stress model is used.
 With the present study, the need to review the initial yield curves for steel I-sections subjected to minor axis bending is evidenced. With a brief description of the ordered pairs referring to points <math>A </math>, <math>B </math>, <math>C</math> and <math>D</math>, it is possible to draw simplified curves, where the non-linear procedure located in the cross-section can be avoided by reducing the processing time in the numerical structural analyses via RPHM. It is worth mentioning in the study by Gonçalves ''et al.'' [37], for the application of the RPHM for the analysis of elements under minor-axis bending, the authors had to use the tangent elastic modulus for the flexural stiffness degradation, since the initial yield curves did not portray the reality.