Abstract
In this paper, the induced stress in longitudinal reinforcement bars with cross sections subjected to concrete creep and shrinkage was evaluated. The study was performed for a reinforced concrete column, with loading varying from zero—in which the structure is exclusively subjected to selfweight—to forces close to the loss of stability by bifurcation of equilibrium. A real precast concrete pole, extremely slender, with geometry varying lengthwise, was the structure chosen for the analysis. The aspects related to the material nonlinearity, as well as creep, shrinkage and the development of characteristic concrete strength, were taken into account following Brazilian normative recommendations in NBR 6118:2014 (Design of structural concrete—Procedure) by the Brazilian Association of Technical Standards (ABNT). The critical buckling load was defined according to the structural frequency by using particularly nonlinear formulations. The results were obtained for different instants without exceeding the expected time for convergence of results. The maximum relative value found for the reinforcement stress was 1.24% of the steel yield strength, eliminating the possibility of yielding material.
Introduction
This work follows on from two previous ones, representing the consolidation of the procedures performed so far. In the first investigation (Wahrhaftig et al. [1]), the analytical solution currently presented in the mathematical modeling was built and evaluated in comparison with the finite element method (FEM). In the second one (Wahrhaftig et al. [2]), aspects related to the limit states of design were analyzed considering the concrete creep only and the secant modulus of elasticity of the concrete. In the present article, the temporal homogenization of inertias of cross sections subject to creep, as well as the shrinkage of the concrete, was added to the calculations together with the initial modulus of elasticity of the concrete. These operations allowed the assessment of the maximum stresses on the longitudinal reinforcement with the additional portions transferred by these two phenomena. Besides that, two tridimensional graphics were also elaborated, allowing a better visualization of the analyzed problem, and the workflow of the routine programming developed to calculate the critical buckling load in the dynamic way was also added.
Creep and shrinkage can be described as rheological phenomena associated with the viscoelastic behavior of some materials, such as concrete. It is important to discuss aspects related to creep and shrinkage simultaneously, since they share a direct correlation with the hydrated cement paste. Creep represents the increase of deformations over time, under constant stress, and shrinkage is related to a temporal decrease in volume. Both phenomena are assumed to be connected, mostly, to the removal of absorbed water from the hydrated cement paste (Metha et al. [3]).
The structural elements of reinforced concrete are formed by a combination of two materials—steel and concrete—that have distinctive properties, becoming a composite material. In this case, the calculation of internal forces should be performed by using homogenization techniques and proceedings that consider equations for the equilibrium of forces and the compatibility of displacements. Homogenization techniques, as mentioned, have been applied by Xavier et al. [4] in order to evaluate beams subjected to asymmetrical bending, even considering a variable adherence between the steel and concrete. However, the main hypothesis for analyzing reinforced concrete structures assumes a perfect adherence between the two materials. The bond between the steel and concrete along the zone of contact can be understood in the papers developed by Manfredi and Pecce [5], Oliveira et al. [6], Hameed et al. [7] and Zhao and Zhu [8], in which the correlation between the deformations presented by both materials becomes evident. As a consequence of this slow deformation and concrete shrinkage, a function of the material’s rheological behavior, some stresses that were initially resisted by the concrete are gradually transferred to the reinforcement steel, causing an increase in the values initially calculated. Should the steel yield, permanent deformations due to material yielding are introduced to the structural system, which can compromise its serviceability and even aesthetics. The stress transfer from concrete to steel occurs under the hypothesis of perfect adherence between the two materials (Bhang and Kang [9]). From the same hypothesis emerges the affirmation that the translation of inplane points belonging to cross sections occurs in an even way.
Although studies related to this subject show the possibility of yielding of steel bars due to stresses transferred from the concrete, there are still uncertainties about the level of induced stress. Kataoka and Bittencourt [10] and Madureira et al. [11] have investigated the problem though without being conclusive on the definition of these levels. Khan et al. [12] performed experimental tests with reinforced concrete specimens to evaluate the internal stresses caused by creep and shrinkage. The authors concluded that, as a consequence of the increase of stresses induced by both effects, microcracks appear over the hardened concrete, related to a low tensile strength.
The cracking that forms due to the tensile induced stress in the structural elements of the concrete is considered almost inevitable, even when under serviceability, since concrete has a low strength for resisting this type of load. This strength is equivalent to 10% of the compressive strength. Cracks in the concrete have been observed in investigations related to reinforced concrete on a numerical as well as an experimental basis. In order to take into account both effects, Tamayo et al. [13] considered the concrete tensile strength by mathematically establishing limits on the stiffness under the effect of local forces acting in the material. Deng et al. [14] reported the appearance of cracks as a limiting factor when performing a test with reinforced columns with incorporated glass fiber when compressed up to 21% of the strength capacity and subject to small lateral displacement. Cracks were also found in compressed concrete when induced stresses caused deformations around two per thousand in specimens during tests for obtaining the material’s stress–strain relationship.
Although examined in specimens, the basic properties of concrete can be considered for real structures. However, these are more complex systems when compared to the simple specimens, since they are subjected to several interactions. Regarding soil–structure interaction, Rosa et al. [15] experimentally investigated the deformation of some reinforced concrete columns throughout the construction process of a building and observed that there was a new distribution of the initial forces of the project, after considering creep and shrinkage.
Creep also alters the stiffness of structural systems and, for this reason, should be considered in the calculation of the vertical load capacity of slender columns, i.e., when determining critical buckling loads. Actually, the assessment of the stability of slender elements should consider all the changes the material undergoes during the lifespan of the system, including issues of deterioration, as revealed by Li et al. [16] and Lee et al. [17]. For concrete structures, this problem is aggravated by the occurrence of the cracks previously mentioned, which reduce the cross section inertia. Typically, a slender column reaches the ultimate limit state, defined by the loss of stability, without experiencing the full resisting capacity of the cross section.
In order to evaluate the stress in the reinforcement bars, including the load transfer due to creep and concrete shrinkage, a slender column was selected for study. The analysis considered a load varying from zero, i.e., with the structure subjected to selfweight only, to the critical buckling load. Therefore, a mathematical model was elaborated based on concepts of the vibration of mechanical systems to define the proximity to loss of equilibrium. In this context, the effects of concrete creep, shrinkage and cracking were included in the calculation, in accordance with the recommendations of the Brazilian Association of Technical Standards, ABNT NBR 6118:2014 [18], as well as the secondorder effects, which were linearized in the geometric stiffness portion. Finally, it was possible to evaluate the results referring to the stability loss by buckling and verify the induced stress levels in the steel of the longitudinal reinforcement bars.
Mathematical modeling
A slender reinforced concrete column may be described as a continuous system, thus with infinite degrees of freedom, subjected to axial compressive loads, including selfweight. This system, from an equilibrium analysis perspective, may be associated with another similar system, also continuous, but having only one degree of freedom. In this way, the first vibration mode, or buckling mode, becomes restricted to a configuration previously defined by a mathematical function, properly chosen to univocally represent it. From this point, general properties of interest are obtained, including the portion that composes the mass and stiffness of the system. Rayleigh [19] applied this concept to the study of prismatic elements, establishing a valid and differentiable function in all domains.
The model presented in Fig. 1 represents a column cantilevered at the bottom and free at the top end, where t indicates time; φ(x) is a mathematical function that sets the shape of the first buckling mode; L is the total length (or height) of the structure; L_{s} and L_{s−1} are the height at the upper and lower limit of a given segment s, whose length is obtained by the difference between these two positions; and q(t) is the global coordinate of the system, located at the column’s free end, whose amplitude is restricted to a state resembling the initial configuration (undeformed). For this model, the Bernoulli–Euler hypothesis is admitted, which means the cross sections remain plane even after loading, guaranteeing even translation of points. The Bernoulli–Euler hypothesis was the kinetics base for determining the critical buckling load of columns, as similarly done by Lee et al. [20].
Equation (1) refers to a trigonometric function taken as a shape function, considered valid at any point in the structure domain and that obeys the boundary conditions of the problem, where x starts at the base:
This model represents a column under axial compressive load that can have constant or variable properties along its length. These properties include geometry, elasticity or viscoelasticity and density. Translational springs are applied to represent the lateral soil–structure interaction. With these conditions, the column is under the influence of gravitational forces, originated from the distributed mass due to the selfweight of the structure and other added masses; and of a concentrated mass at the free top end, to be defined in the event of loss of stability, whose value represents the maximum vertical load that can be applied.
To get the analytical solution of this problem, it was necessary to consider a trigonometric function to reduce the problem from a system with infinite degrees of freedom into an equivalent system with only one degree of freedom. Equation (1) shows the chosen trigonometric function for this study, which can be used for structures with varying geometries, as by Wahrhaftig [21] in comparison with computational analyses using the finite element method (FEM). The FEM, as an analysis technique in a similar context, was used by Alkloub et al. [22] in order to analyze the buckling of reinforced concrete columns with a truncated cone shape and square cross section; by Rodrigues et al. [23] to evaluate the buckling of reinforced concrete columns, eccentrically loaded, considering nonlinearities inherent to the system, including creep; and by Fabro et al. [24], who employed polynomial and trigonometric interpolation functions to optimize the application in real structures with low frequency.
Applying the principle of virtual work and its derivatives, as similarly done by Wahrhaftig et al. [1], the stiffnesses and masses of the system can be obtained. The portion of the conventional elastic/viscoelastic stiffness is specified as:
with
where for the segment s of the structure, E_{s}(t) is the viscoelastic module of the material as a function of time, which represents the concrete creep, calculated considering the particular characteristics of the cross section S to which the segment belongs; I_{s}(x, t) is the moment of inertia, which varies with time and along the segment about the considered buckling mode, obtained by interpolation of subsequent sections, all homogenized (if geometrically constant, it is simply going to be I_{s}(t)); k_{0s}(t) is the term for the conventional stiffness varying with time; K_{0}(t) is the timedependent portion of the total conventional stiffness, and n is the total number of segments in the analyzed geometry. It is important to make clear that if the material exclusively displays an elastic behavior, without creep, the respective equations will no longer be functions of time.
The viscoelastic behavior of the material, as established in Equation (3), followed the normative criteria for considering creep provided by NBR 61118:2014 [18], in which the longitudinal deformation module is given by:
where E_{c0} is the longitudinal deformation module of concrete at the instant of loading the structure, E_{c28} is the initial tangent modulus at 28 days, and φ(t,t_{0}) is the creep coefficient of concrete at the instant t, for loads applied at the instant t_{0}, given by:
where φ_{a} is the fast deformation coefficient (Equation (6)), φ_{f∞} is the final value of the slow irreversible deformation coefficient (Equation (7)); β_{f}(t) and β_{f}(t_{0}) are coefficients related to slow irreversible deformation, depending on the age (t or t_{0}) of the concrete (Equation (8)); φ_{d∞} is the final value of the reversible slow deformation coefficient, considered constant equal to 0.4 (value established by NBR 6118:2014 [18]); and β_{d} is the coefficient relative to the slow reversible deformation as a function of the time (t − t_{0}) elapsed after initiating the loading. In Equation (5), the parameters unfold into:
where f_{c}(t_{0}) and f_{c}(t_{∞}) are factors corresponding to the compressive strength of concrete at time t_{0} and t_{∞} respectively,
being that:
where U is the environment relative humidity (in percent), and h_{fic} the fictitious thickness of the element expressed in centimeters, calculated by the set of Equations (9), in which S indicates the cross section under consideration,
and finally, the β coefficients are calculated through
The terms A, B, C and D are given by Eqs. (12) to (15), respectively:
where h_{fic} is here expressed in meters.
The geometric stiffness is presented as a function of the axial force, including the selfweight contribution, being calculated as follows:
with
and
where k_{gs}(m_{0}) is the geometric stiffness of segment s; N_{0}(m_{0}) is the concentrated force at the top part of the system, calculated by Eq. (17); and K_{g}(m_{0}) is the total geometric stiffness of the structure, given by Eq. (18). As can be seen, all the parameters depend on the mass m_{0} located at the free end of the structure. In Eq. (16), N_{s} represents the normal force in the segments above the considered one, which can be obtained from
The generalized mass of the system can be calculated as follows:
with
where \(\overline{m}_{s}\) is the mass per unit length, defined as
where A_{s}(x) represents the variable cross section area and ρ_{s} is the material density for the respective segments. If the cross section has a constant area along the interval, A_{s}(x) is going to be A_{s}, and consequently, the mass distribution per unit length is also going to be constant. In the same way, if the mass m_{0} does not vary, all of its dependent parameters are going to be constant. In order to consider the influence of soil in the equilibrium of the system, it is necessary to represent it as a series of vertically distributed springs along the foundation. Thereby, the soil’s contribution to the structure stiffness can be defined as
with
and
in which the parameter \(k_{s}^{so} (x)\) depends on the geometry D_{s}(x) and on the elastic property of the soil S_{s} along the depth of the foundation for each soil layer. Considering the normal compressive load to be positive, it is possible to obtain the total stiffness of the system as a function with two variables, time and concentrated mass at the free end:
Thus, the frequency of the first free vibration mode, undamped, in Hertz, can be found using Eq. (27):
Taking into account the previous postulations and assuming the concentrated mass as the independent unknown of the problem, once the instant of interest is defined, the critical buckling load N_{cr} can be determined using a mathematical concept presented in Eq. (28), which defines the proximity of the loss of equilibrium or vertical load capacity of the column at the moment the structure loses its stiffness, making the frequency go to zero:
Complementary details of the presented analytical process can be found in Wahrhaftig et al. [25]. It is worth mentioning that this method can be applied to any structure that can be converted into a unidimensional model that is sufficiently long to suffer bending. Moreover, the formulations used to obtain the generalized properties of the system have nonlinear characteristics regarding the geometry and material, which is essential to the examination of reinforced concrete systems, as affirmed by Koç and Emiroğlu [26].
Case study
In order to apply the discussed concept and determine the stress in the reinforcement, a practical problem was selected for the study. The analyzed structure is a real pole, extremely slender, made from reinforced concrete, that is 46 m high, including a 40 m superstructure, with a hollow circular cross section, and a belled shaft foundation with 6 m depth. The dimensions, heights and structural arrangement of the column and column segments are shown in Fig. 2a and b, where g is the gravitational acceleration; S1, S2, S3, S4 and S5 are cross sections from 1 to 5; D and th indicate, respectively, the external diameter and the wall thickness; and “Tap” represents the tapered section at the segment.
The dimensions and respective reinforcement arrangements of the cross sections are presented in Table 1, where d_{b} and n_{b} represent the diameter and the number of reinforcement bars, respectively, and c´ is the concrete cover.
The initial tangent deformation moduli (or moduli of elasticity) of concrete 28 days after casting, for the superstructure and foundation, are, respectively, 37,566 MPa and 25,044 MPa, calculated according to ABNT NBR 6118:2014 [18] considering the characteristic strength of concrete f_{ck} equal to 45 MPa and 20 MPa, as shown in Eq. (29):
Additional devices are also installed along the length of the superstructure, configuring a distributed mass of 40 kg/m. The foundation has a 140 cm diameter base and 20 cm depth, with an 80 cm shaft diameter and length of 580 cm. The lateral soil–structure interaction is represented by an elastic parameter with property equal to 2669 kN/m^{3}, considered constant at each soil layer in this case, although there is no need for this to be the case. It is worth mentioning that the springs coefficients (or settlement coefficients) were provided by a geotechnical specialist. Different methods are applied by different specialists to determine these coefficients.
In this case study, the physical (or material) nonlinearity of concrete has been calculated following the recommendations in ABNT NBR 6118:2014 [18], assuming a 50% reduction in the moment of inertia. To justify this assumption, it is worth mentioning that, in ABNT NBR 6118:2014 [18], a possibility is presented to use a multiplying factor equal to 0.5 to the product of bending stiffness EI for structural elements that may be predominantly under bending, in which the reinforcement distribution at the section is symmetrical, i.e., A_{s}ʹ = A_{s}, where A_{s}ʹ is the compression reinforcement and A_{s} the tension reinforcement. Generally, for columns, the recommended value for the inertia reduction in a nonlinear analysis is 80%. However, the chosen value is based on the fact that the behavior of an element of this nature and purpose, when loaded, is going to be similar to a cantilevered beam, mostly when there are lateral loads due to wind that may cause bending. Therefore, the structure is going to be under similar circumstances to the normative references.
The reinforced concrete density was defined as 2600 kg/m^{3} for the superstructure, since the reinforced concrete was centrifugated and made in an industrial environment, and 2500 kg/m^{3} for the foundation, since it is made with reinforced concrete produced using conventional methods. Homogenization factors were calculated, which multiply the moment of inertia and the area of each cross section, seeking to take into account the presence of the steel bars. Creep and shrinkage, due to the rheological behavior of concrete, were considered in the superstructure in accordance with the recommendations in ABNT NBR 6118:2014 [18], and separately according to the geometric properties of each cross section. Due to the temporal nature of the moduli of elasticity of concrete, originated by the viscoelastic behavior, the homogenization factors become also timedependent. The mathematical base used for calculating the homogenizing factors can be found in Wahrhaftig et al. [2].
A programming routine, as shown in Fig. 3, was elaborated to obtain the critical buckling loads. The processing, which used an Intel CPU (R) 2.70–2.90 GHz, i7 (7th generation), Core (7 M) 7500U, running Windows 10 (64 bits), 8 GB RAM, needs approximately 14 h to be completed, when using Δm_{0} increments of 0.1 kg for the mass, and Δt of 100 days for time, adopting initial values equals to 14,000 kg for m_{0} and 0 for t, with a final time equal to 5000 days.
Results and discussion
The total stiffness and the frequency of the structure, obtained from Eqs. (26) and (27) for different instants of time within the assumed convergence period, can be found in Fig. 4a and b, respectively. Figure 4 reveals the proximity of the loss of equilibrium and the corresponding load in the system, for null stiffness, thereby null structural frequency, in accordance with Eqs. (28a and 28b). Additionally, it is possible to observe in Fig. 5 the frequency (Hz) and structural stiffness (kN/m) variations as functions of the mass at the upper end (kg) and time (days).
Besides the instants of time arbitrarily selected, the analysis could have been performed for any other instant of interest during the structural system’s lifespan. However, 5000 days is an appropriate design horizon, since technology changes and maintenance tend to happen every 15 years. Even for a 10,000 day timeframe (about 27 years), the critical buckling load would vary by only 0.96%. Should the design horizon be 15,000 days (about 42 years), a 0.46% reduction would be found when compared to the previous time frame. The mass at the upper end can be obtained by simply dividing the critical load by the gravitational acceleration.
Although the deformation field could be established for the whole structure, only the segments subjected to creep and shrinkage were analyzed. The deformation imposed by the load in a homogenized concrete section S, as defined by Eq. (30), has been calculated using variations of properties over time, such as: strength development; homogenization factors of sections, considering a modulus of elasticity of steel equal to 205 GPa; creep and shrinkage.
In Eq. (30), for a structural segment s, \(\varepsilon_{s}^{cc}\) is the specific strain due to creep, \(\varepsilon_{s}^{cs}\) is the specific strain due to concrete shrinkage. For this condition, the strain due to creep is obtained with:
with
where φ_{S}(t) is the creep coefficient that was calculated considering the geometry of the analyzed section S; E_{ci0} and E_{ci28} are moduli of concrete at the instant of loading at 28 days, considered equal in the present simulation; and N_{S}(m_{0}) is the normal force acting in the section, which includes the selfweight of the segment s of the section, obtained as follows in Eq. (33),
where
with s varying from 3 to 5, n = 5 and N_{s} defined by Eq. (19) considering that for t = 0, η = 4.50130; t = 90 days, η = 4.08859; t = 500 days, η = 3.84525; t = 1000 days, η = 3.76617; t = 2000 days, η = 3.70880; t = 3000 days, η = 3.68531; t = 4000 days, η = 3.67245; t = 5000 days, η = 3.66436. If the value of η for t = 0 is adopted for the whole time interval, a maximum difference of less than 1.5% is found. From the normal forces calculated by Eq. (33), the induced stresses in sections 3 to 5 reach a 10% maximum. In order to evaluate the induced stresses in concrete,\(\nu_{S}^{c}\), Eq. (35) was applied:
where f_{cd} is the design strength of the concrete, f_{cd} = fc_{k}/γ_{c}, with γ_{c} = 1.4; \(A_{S}^{h} \left( t \right)\) is the homogenized area of the concrete with time. The deformation portion due to shrinkage does not depend on the applied load. For considering the shrinkage, the NBR 6118:2014 [18] considers the aspects related to the relative humidity of the environment, the consistency of the concrete and the fictitious thickness of the structure. Between moments t_{0} and t, the total shrinkage strain is determined by:
and the term \(\varepsilon_{s}^{cs\infty }\) is calculated by Equation (37):
where \(\varepsilon_{s}^{cs\infty }\) is the coefficient that is related to the consistency of the concrete and the relative humidity of the environment, determined by Equation (38), and \(\varepsilon_{s}^{cs\infty }\) is dependent on the fictitious thickness of the element, calculated by Equation (39):
and β_{s}(t) or β_{s}(t_{0}) is the coefficient relative to the shrinkage at the instant t or t_{0}, according to Equation (40):
where the parameter A is considered constant equal to 40, and the terms B, C, D and E are calculated by the following expressions, with h_{fic} expressed in meters:
In a similar condition, i.e., when there are no external forces in the structure, the strain due to creep represents roughly one tenth of the strain due to shrinkage, as shown in Fig. 6.
The calculations developed for determining the axial stresses in the longitudinal reinforcement \(\sigma_{S}^{st}\) were performed according to the deformations evaluated for the concrete, assuming the compatibility of displacements and a valid hypothesis of perfect adherence between the materials, as shown in Eq. (45):
where E^{st} is the modulus of elasticity of steel. On the other hand, the normalized stresses \(\nu_{S}^{st}\) induced in the reinforcement were obtained using Eq. (46):
in which n_{b} is the number of bars in the considered section, f_{yd} is the design yield strength of steel, taken as f_{yd} = f_{yk}/γ_{st}, where f_{yk} is the yield strength of the material (500 MPa), and γ_{st} is the safety factor applied to take into account the variability of the material, usually 1.15. It is possible to observe that, as a result of concrete creep and shrinkage, there is a gradual stress transfer to the reinforcement steel bars, as shown in Fig. 7, which reveals a relationship between the nondimensional axial load in the concrete and the normalized stress in the reinforcement, following Eqs. (35) and (46) for the value of critical buckling loads in Table 2. This transfer, though, does not cause yielding of the material. A particular comparison of the results for creep plus shrinkage, and creep only with time for section S3 is depicted in Fig. 7c.
Indeed, the stress transferred to the steel, according to Eq. (45), shows a linear growth, reaching the maximum value at the critical buckling load. In this situation, m_{0} = N_{cr}/g. The results were obtained considering two analysis hypotheses: (1) with creep and shrinkage effects simultaneously; and (2) with creep effects only. In this order, the stresses found were 108.20 MPa and 35.84 MPa for section S3; 105.62 MPa and 33.26 MPa for section S4; and 98.24 MPa and 25.88 MPa for section S5. The evolution of the relative stresses in each reinforcement bar for case (1) can be seen in Fig. 8 and for case (2) in Fig. 9. The inflexion points in both graphs indicate the proximity to loss of equilibrium. An analysis for section S3 considering both cases (1) and (2) at the instant zero and 5000 days is presented in Fig. 10.
Table 2 summarizes the results obtained in this study, with critical buckling load calculations defined by Eq. (28b). The option to define the critical buckling load at null frequency can be explained by the fact that this option reduces approximation issues, even when minimal, coming from the increments chosen for the programming routine. In Table 2 “Δ” indicates variation; N_{cr} is the critical buckling load; \(\nu_{S}^{st}\) represents the reduced normal force at each longitudinal reinforcement bar; ω is the steel ratio; S3, S4, S5 denote cross sections 3, 4, 5; and columns (1) and (2) represent the results for creep and shrinkage, but also for creep only, respectively. The following considerations were made during the analyses performed: concrete production under standard conditions, humidity of 70%, and gravitational acceleration of 9.807 m/s^{2}.
To represent the stress evolution on the reinforcement given by the conditions assessed, the results for section S3 are shown in Fig. 11, representing columns (1) and (2) of Table 2.
Comparing the results obtained in the previous paper (Wahrhafig et al. [2]), it was possible to note that the critical buckling load grew by 14.04% without the creep effect, (t = 0) and 26.52% when the creep effect is taken into account for the final time interval (t = 5000 days). The current adoption of the initial modulus of the concrete instead of the secant modulus contributed to that. The difference between the two moduli is around 9%. Another contributing aspect, in that context, was the inclusion of the temporal variation of the moment of inertia of cross sections subjected to creep in the calculation of the flexural stiffness, as can be seen in Equation (3). The elevation found for that parameter is around 36% at the end of the considered time.
Another aspect worth mentioning, although it has little influence in the results, was the application of the creep criteria for the geometry of each cross section, which leads to having three different moduli of elasticity, Equation (4) associated with (9), instead of using a unique temporal modulus for the entire column, calculated using a weighted criterion for the geometry, as done in the previous paper. When adopting a unique modulus, differences of less than 1% at the end of the period are found. If all the moduli are plotted together, as seen in Fig. 12, it is not possible to distinguish the difference existing among them. There, E(t) (unique) is the temporal modulus of elasticity for the entire structure; E_{S3}(t), E_{S4}(t) and E_{S3}(t) are the temporal modulus of elasticity calculated for sections S3, S4 and S5, respectively.
Conclusions
In this study, the stresses in the longitudinal reinforcement for cross sections subjected to concrete creep and shrinkage were evaluated. In order to do that, a structural element with great slenderness, made from reinforced concrete, with cross section varying lengthwise, was used. The analysis was defined for loads applied at the freeend of the column, from zero, in which the structure was subjected exclusively to selfweight, to the ultimate load capacity, calculated with an analytical method based on vibration of structural system concepts, with nonlinear properties. All necessary parameters for the calculations were considered, such as: geometric imperfections, evaluated in the geometric stiffness; physical, or material, nonlinearity, concrete creep and shrinkage, following recommendations in ABNT NBR 6118:2014. It is worth mentioning that no safety factor was set for the critical buckling load in the present analysis. However, for calculating deformations, the most commonly applied safety factors in actual designs were applied to the materials.
The following conclusions were obtained from the study:

A maximum stress of 1.24% of the design yield strength (435 MPa) was found for steel. This evidence reveals the impossibility of yielding of the longitudinal reinforcement, even when considering the load transfer portion due to concrete creep and shrinkage.

An increase of 36% of the moment of inertia of cross sections under creep at the end of the considered time is noted. Therefore, introducing the temporal variation of the inertias into the calculation of the flexural stiffness increased the loading capacity of the system.

The critical buckling load depends entirely on the modulus of elasticity of the concrete. When adopting the initial tangent modulus, an increase of 9% in relation to the secant one is found, with the structure becoming stiffer.

When the temporal variation of the homogenizing factors of inertias is not introduced in the calculation of the flexural stiffness; the creep effect is calculated for the weighting average of the geometry; and the modulus of elasticity considered in the calculation is the secant one; the maximum stress on the reinforcement corresponds to 7.49% of the design yield strength, as observed in previous investigations (Wahrhaftig et al. [2]).

By employing the initial modulus of elasticity of the concrete; introducing the temporal variation of inertias in the calculation of the flexural stiffness; and with the creep calculated for the geometry of each section; the critical buckling load grows by 14% for t = 0, and 27% when the creep effect is taken into account at the end of the time interval (t = 5000 days), in comparison with these criteria assumed as mentioned at the precedent item.

Adopting a unique temporal modulus of elasticity for the whole structure by considering a weighted criterion to the geometry does not produce significant changes to the results when compared to the adoption of an individual viscoelastic modulus for each section under creep.

Calculating creep only, the maximum steel stress was 0.41% of the design yield strength, i.e., 0.33% of the value obtained in the simultaneous analysis (creep and shrinkage).

Induced stresses in the steel showed maximum variations of 505% and 95% in the calculated values without creep and shrinkage. Moreover, material yielding does not occur for any of the analysis hypotheses.

A 52% reduction of the vertical load capacity of the column was observed. This is related to the fact that a critical buckling load of 293 kN was determined at the initial instant and another one of 142 kN was found after 5000 days from the loading phase, a period of time assumed for results convergency.
For future studies, a comparative analysis with other calculation methods and design criteria from different codes should be performed, as well as the evaluation of the influence of mechanical and thermal deformations. Experimental studies to evaluate the influence of cracking on concrete creep and shrinkage phenomena should also be developed. Moreover, comparisons with the verification method for columns by iteration diagrams should be completed. Besides, additional ways to turn the present analysis into a nondimensional one, in terms of slenderness, should also be proposed.
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de Macêdo Wahrhaftig, A., Magalhães, K.M.M. & Nascimento, L.S.M.S.C. Stress assessment in reinforcement for columns with concrete creep and shrinkage through Brazilian technical normative. J Braz. Soc. Mech. Sci. Eng. 43, 6 (2021). https://doi.org/10.1007/s40430020027316
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Keywords
 Stresses
 Reinforcement
 Bifurcation of equilibrium
 Analytical solution
 ABNT NBR 6118:2014