Considerations about the determination of γz coefficient

Considerations about the determination of γ_z coefficient

D. M. Oliveira^I; N. A. Silva^II; C. F. Bremer^III; H. Inoue^IV

^IUniversidade Federal de Minas Gerais, Escola de Engenharia, Deptº de Engenharia de Materiais e Construção, [e-mail], Av. Antônio Carlos 6627, bl. 1, sala 3315, Pampulha, 31270-901, Belo Horizonte, MG, Brasil

^IIUniversidade Federal de Minas Gerais, Escola de Engenharia, Deptº de Engenharia de Estruturas, [e-mail], Av. Antônio Carlos 6627, bl. 1, Pampulha, 31270-901, Belo Horizonte, MG, Brasil

^IIIUniversidade Federal de Minas Gerais, Escola de Arquitetura, Deptº da Tecnologia da Arquitetura e do Urbanismo, [e-mail], Rua Paraíba 697, Funcionários, 30130-140, Belo Horizonte, MG, Brasil

^IVUniversidade Federal de São João del-Rei, Campus Alto Paraopeba, Deptº Multidisciplinar de Tecnologia, Ciências Humanas e Sociais, [e-mail], Rod. MG 443, km 7, caixa postal 131, 36420-000, Ouro Branco, MG, Brasil

ABSTRACT

In this work, the γ_z coefficient, used to evaluate final second order effects in reinforced concrete structures, is studied. At the start, the influence of the structural model in determination of γ_z coefficient is evaluated. Next, a comparative analysis of γ_z and B₂ coefficient, usually employed to evaluate second order effects in steel structures, is performed. In order to develop the study, several reinforced concrete buildings of medium height are analysed using ANSYS-9.0 [1] software. The results show that simplified analysis provide more conservative values of γ_z. It means that, for structures analysed by simplified models, large values of γ_z don't imply, necessarily, in significant second order effects. Furthermore, it was checked that γ_z can be determinated from B₂ coefficients of each storey of the structures and that, for all the analysed buildings, the average values of the B₂ coefficients are similar to γ_z.

Keywords: reinforced concrete, structural model, γ_z Coefficient, B₂ Coefficient.

RESUMO

Neste trabalho apresenta-se um estudo do coeficiente γ_z, empregado para indicar a necessidade ou não de se considerar os efeitos de segunda ordem globais na análise das estruturas de concreto armado. Inicialmente, procura-se avaliar a influência do modelo estrutural adotado no cálculo de γ_z. Em seguida, realiza-se uma análise comparativa do coeficiente γ_z e do coeficiente B₂, comumente empregado para avaliar os efeitos de segunda ordem em estruturas de aço. Para conduzir o estudo, diversos edifícios de médio porte de concreto armado são processados utilizando o programa computacional ANSYS-9.0 [1]. Os resultados obtidos permitem verificar que análises menos refinadas tendem a fornecer valores de γ^z mais conservadores. Isto significa que, para estruturas analisadas por meio de modelos simplificados, a obtenção de altos coeficientes γ_z não implica necessariamente em efeitos de segunda ordem significativos. Além disso, mostra-se que o γ_z pode ser calculado a partir dos coeficientes B₂ determinados para cada pavimento das estruturas, e que, para todos os edifícios analisados, os valores médios dos coeficientes B₂ apresentam boa proximidade em relação ao γ_z.

Palavras-chave: concreto armado, modelo estrutural, coeficiente γ_z, coeficiente B².

1. Introduction

Of late, erecting more economical, slender structures and taller, bolder buildings has become increasingly common.

The taller and more slender the building, the greater the strains present, particularly those resulting from lateral actions. In these cases the stability analysis and evaluation of second order effects start taking on fundamental importance in the structural project.

Second order effects arise when the structure equilibrium considering the deformed configuration study is done. In this way, existing forces interact with displacements, thereby producing additional efforts. Second order efforts introduced by the structure joints moving horizontally, when subject to vertical and horizontal loads, are referred to as global second order effects.

It is well known that all structures are displaceable. However, horizontal joint displacements are small in some more stiff structures and, as a result, second order global effects have little influence on total efforts, and so can be ignored. These structures are referred to as nonsway structures. In these cases, bars can be sized separately, with their extremities tied, where efforts obtained by the first order analysis are applied.

On the other hand, some more flexible structures have significant horizontal displacements and therefore global second order effects depict an important part of final efforts and cannot be ignored. This is the case of sway structures for which a second order analysis must be done.

According to NBR 6118:2007 [2], if global second order effects are less than 10% of the respective first order efforts, the structure can be classified as being nonsway structure. Otherwise (that is, when global second order effects are over 10% higher than first order effects), the structure is classified as being sway structure.

NBR 6118:2007 [2] also establishes that structures can be classified using two approximate processes, the α instability parameter and the γ_z coefficient. However, the γ_z coefficient goes beyond the α parameter, since it can also be utilized to evaluate final efforts, which include second order efforts, as long as their value does not exceed 1.3. However, it is obvious that, for second effects to be evaluated satisfactorily, the γ_z coefficient needs to be calculated accurately.

It is worth noting that the γ_z coefficient must be employed in reinforced concrete structures. To assess second order effects on steel structures, the B₂ coefficient must be utilized. As with γ_z, this coefficient is also able to provide an estimate of a structure's final efforts, as long as their value does not go beyond a certain threshold.

Within this context, this paper's primary intention is to ascertain the adopted structural model's influence in calculating the γ_z coefficient. Thus, the γ_z values for two medium height reinforced concrete buildings are determined, considering five distinct three-dimensional models developed utilizing ANSYS-9.0 [1] software. The results obtained make it possible to identify the more adequate models for putting the project into practice, as well as those whose utilization could prove disadvantageous and uneconomical.

Moreover, the attempt has been made to carry out a comparative study of coefficients γ_z and B₂. To conduct the study, first of all an expression associating these parameters is developed. Next, the γ_z and B₂ values for several medium height reinforced concrete buildings are calculated, utilizing ANSYS-9.0 [1] software.

2. Coefficient γ_z

NBR 6118:2007 [2] ordains that the γ_z coefficient, valid for reticulated structures at least four stories high, can be determined from a first order linear analysis, by reducing the structural elements' stiffness, in order to consider the physical non-linearity approximately.

For each load combination, the γ_z value is calculated using the following expression:

- M_1,tot,d (first order moment) being: a sum of the all the horizontal force moments (with their design values) of the considered combination relative to the structure base, which can be written as:

F_hid being the horizontal force applied to storey i (with its design value), and h_i being the height of storey i.

-ΔM_tot,d (increase in moments after the first order analysis) being: a sum of the products of all the vertical forces working on the structure (with their design values), in the considered combination, by the horizontal displacements of their respective application points:

P_id being the vertical force working on storey i (with its design value), and u_i being the horizontal displacement of storey i.

Bearing in mind that second order effects can be ignored as long as they do not show a greater than 10% increase in the respective first order efforts, a structure may be classified as being nonsway structure if its γ_z< 1.1.

NBR 6118:2007 [2] establishes that final efforts (first order + second order) can be evaluated from the additional 0.95γ_z horizontal efforts magnification of the considered loading combination, as long as γ_z does not exceed 1.3. However, according to the NBR 6118:2000 [3] Revision Project, final efforts values could be obtained by multiplying the first order moments by 0.95γ_z, also on the condition that γ_z< 1.3. It is therefore understood that γ_z ceased to be the first order moment magnifier coefficient and became the horizontal loads magnifier coefficient.

According to Franco &Vasconcelos [4], utilizing γ_z as a first order moments magnifier provides a good estimate for the second order analysis results; the method was applied successfully on tall buildings with γ_z in the region of 1.2 or more. Vasconcelos [5] adds that this process is valid even for γ_z values lower than 1.10, in which cases technical norms allow second order effects to be disregarded.

It is also noted that, according to Vasconcelos [6], the process of evaluating second order effects by multiplying first order moments by γ_z is based on the assumption that the successive elastic lines produced by vertical force action on the structure with displaced joints follow in geometric progression. Indeed, it was seen in countless cases that up to the value γ_z = 1.3 this assumption is valid with less than 5% error. However, there are some particular situations where the assumption formulated in developing the method does not apply or applies with greater errors. As examples of these exceptional cases, Vasconcelos [6] quotes: when there is a sudden change in inertia between stories (in particular between the ground and first floor), where ceiling heights from one floor to the next are very different, cases of column transition in beams, when there is torsion in the spatial frame or uneven settling in the foundations, and others.

Oliveira [7] did an evaluation of the γ_z coefficient's efficiency as a first order efforts magnifier (for bending moments, axial and shearing forces) and as a horizontal loads magnifier, to obtain final, including second order, efforts. The study was carried out for structures with maximum γ_z values in the region of 1.3, that is, for which, according to NBR 6118:2007 [2], the simplified final efforts evaluation process employing the γ_z coefficient is still valid. It was found that the γ_z coefficient must be utilized as magnifier of first order moments (and not for horizontal loads) to obtain final moments. In the case of axial force on columns and shearing force on beams, magnification by the γ_z coefficient was not necessary, since the first and second order efforts values obtained in these cases were practically the same.

3. Coefficient B₂

To evaluate second order effects on steel structures, AISC/LRFD [8] adopts the approximate method of amplifying the first order moments by magnification factors B₁ and B_2. So the second order bending moment, M_Sd, must be determined by means of the following expression:

M_nt being the design bending moment, assuming there is no side sway in the structure, M_lt being the design bending moment due to the frame's side sway; both M_nt and M_lt are obtained by first order analyses. The B₁ amplification coefficient depicts the P-δ effect, relating to the instability of the bar, or to local second order effects; B₂ considers the P-Δ effect, relating to the instability of the frame, or to global second order effects.

The B₂ coefficient can be calculated for each storey of the structure, as:

with ΣN_Sd as the summation of the design axial compression forces on all the columns and other elements resistant to the storey's vertical forces; ∆_0h as the relative horizontal displacement; L as the storey's length and ΣH_Sd as the summation of all the design horizontal forces on the storey producing ∆_0h.

According to Silva [9], if the B₂ coefficient does not exceed the value of 1.1 on all storeys, the structure can be considered almost insensitive to horizontal movement and, in this case, global second order effects can be ignored. When the greater B₂ is situated between 1.1 and 1.4, the approximate B₁-B₂ method can be utilized for the bending moment, with the other efforts (axial and shearing forces) being directly obtained from the first order analysis. Lastly, when B₂ > 1.40, the recommendation is that a rigorous second order elastoplastic analysis be performed. Silva [9] also adds that, in the event 1.1 < B₂< 1.2, the bending moments can alternatively be based on a first order analysis performed with the magnified horizontal efforts by the greater B₂.

So it can be seen that, like the γ_z coefficient, the B₂ coefficient is an "indicator" of the importance of global second order effects on a structure. In this way, in the next item, an expression capable of relating these parameters will be obtained.

4. Relation between coefficients γ_z and B₂

Figure [1] shows a structure consisting of three storeys of equal length (L). In this figure the vertical (P_id) and horizontal (F_hid) design forces working on each storey i, along with their respective horizontal displacement (u_i) are also shown.

To calculate γ_z, equation (1), the values of M_1,tot,d and ∆M_tot,d need to be determined. Through equations (2) and (3), we get, respectively:

The B₂ coefficient, given by equation (5), shows distinct values for each storey of the structure. Thus, referring to the B₂ coefficient of storey i as B_2,i and the parts (L.ΣH_Sd) and (∆_0h.ΣN_Sd) as M_i and ∆M_i, respectively, we get:

• 1st storey:

• 2nd storey:

• 3rd storey:

Adding up M₁, M₂ and M_3, equations (8), (11) and (14), and ∆M₁, ∆M₂ and ∆M₃, equations (9), (12) and (15) gives:

Comparing equations (17) and (18) with equations (6) and (7) we can write:

By substituting equations (19) and (20) in equation (1), the γ_z coefficient becomes defined as:

Inverting equation (21) gives:

Substituting equations (10), (13), (16) and (19) in equation (22), gives:

Finally equation (23) can be written as:

with constants c₁, c₂ and c₃ being given respectively through:

As such, for a structure consisting of n storeys, the γ_z coefficient can be calculated by reference to the B₂ coefficient as:

5. Influence of the structural model adopted to calculate γ _z

As commented previously, NBR 6118:2007 [2] establishes that the γ_z coefficient can be determined from a first order structure analysis. However, this analysis can be carried out utilizing various types of structural models. For example, a building can be modelled considering the slabs as rigid diaphragms or depicting them by means of shell elements. Additionally, the eccentricity existing between the beam axis and the average slab plane may or may not be taken into account. In this way, in order to evaluate the possible influence of the structural model on the value of γ_z, the γ_z coefficients will be determined for two reinforced concrete buildings, considering five distinct three-dimensional models developed utilizing ANSYS-9.0 [1] software. The results of these models will then be analyzed and compared.

5.1 Buildings and models analyzed

The first building analyzed, shown in figure [2], consists of sixteen storeys (with a 2.9 m ceiling height) and is symmetrical in both X and Y directions. 20 MPa for the characteristic strength of the concrete to compression and a Poisson coefficient equal to 0.2 were adopted.

The second building, depicted in figure [3], consists of eighteen storeys (with a ceiling height of 2.55 m) and has no symmetry. The concrete presents characteristic strength to compression and a Poisson coefficient equal to 30 MPa and 0.2, respectively.

Each building was analyzed utilizing five distinct three-dimensional models. In the first model the columns and beams are depicted by means of bar elements (defined in ANSYS-9.0 [1] as "beam 4"and "beam 44"respectively) and the slabs by means of shell elements (called "shell 63"). The "beam 4" and "beam 44" elements show six degrees of freedom at each node: three translations and three rotations, in directions X, Y and Z. The "shell 63" element has four nodes, each node presenting six degrees of freedom, the same as the bar elements. The "beam 44"element, utilized to represent the beams, enables the eccentricity existing between the beam axis and the average slab plane to be taken into account. Thus, this model simulates the real situation between the slabs and the beams, as depicted in figure [4]. It is worth commenting that, when their axes did not coincide, the connection between the beams and the columns was carried out using rigid bars, as figure [5] shows.

The second model only differs from the previous one by replacing the "beam 44" element with the "beam 4" element to depict the beams. In this way, in this model the average slab plane coincides with the beam axis, figure [6], since the "beam 4" element does not allow eccentricities to be considered.

In the third model, the columns and beams are depicted by means of the "beam 4" element and the slabs are treated as rigid diaphragms, that is, it is accepted that they have infinite stiffness on their own plane and nil stiffness crosswise. In the ANSYS-9.0 software [1], the hypothesis of a rigid diaphragm is embodied in the model by means of a specific command which relates the degrees of freedom of the nodes making up the slab plane. Thus, a "master" node, corresponding to the point representing all the storey's nodes is defined. The remaining nodes, called "slaves", have their own degrees of freedom and those represented by the "master" node.

The fourth model, like the previous one, is also made up of bars (depicting the columns and beams by means of the "beam4" element), but without considering the hypothesis of a rigid diaphragm.

Finally, the last model only differs from the previous one because the "beam4" element is replaced by the "beam44" element to depict the beams, whereby the eccentricity existing between the beam axis and the average slab plane can be considered.

It can be seen, then, that in models 3,4, and 5 the structural system just consists of bars, since the slabs are not modelled (unlike models 1 and 2 in which the slabs are depicted by means of shell elements). In all the models, the beams' torsional stiffness was reduced, by reproducing the cracking effect.

Table [1] sums up the main characteristics of the models employed.

5.2 Design considerations

The actions working on the buildings are divided into two groups: vertical actions and horizontal actions.

Vertical actions consist of permanent loads and the accidental load. The permanent loads considered were the own weights of structures, the masonry loads and the slab coatings and finishings. The accidental loads were determined in accordance with the precepts of NBR 6120:1980 [12].

The chief horizontal actions that must be taken into account in the structural project are the forces due to the wind and those relating to geometric imperfections (out-of-plumb). However, according to NBR 6118:2007 [2], these loadings do not need to be overlapped and only the most unfavorable (the one causing the greatest total moment at the structure base) may be considered. According to Rodrigues Junior [13], for tall buildings, just as with the main variable load choice, it is possible to prove that, in most practical cases, the wind corresponds to the most unfavorable situation. In this way, in this paper, the horizontal loading applied to the structures was that corresponding to the action of the wind, considered more unfavorable than out-of-plumb, both for direction X and for direction Y. It is worth pointing out that the drag forces were calculated in accordance with the precepts of NBR 6123:1988 [14].

The coefficients applied to the actions, defined from the ultimate normal combination that considers the wind to be the main variable action, were determined as recommended by NBR 6118:2007[2].

5.3 Results obtained

The γ_z coefficient was calculated from the first order linear analysis of the structures, for the vertical loads acting simultaneously with the horizontal actions. In this analysis the physical non-linearity was considered in a simplified way, as established by NBR 6118:2007[2], reducing the stiffness of the structural elements.

The γ_z values (in directions X and Y) obtained for both buildings and considering all the models utilized, are shown in table [2].

In table [2] it can be seen that, with the exception of model 1, all the models provided practically the same γ_z values, for both buildings I and II. Therefore, the presence or lack of symmetry did not have any influence on the results obtained. Furthermore, the γ_z values calculated based on model 1, the most sophisticated (for it is the only one, among all the models adopted, that considers simultaneously the representation of the slabs as shell elements and the eccentricity existing between the beam's axis and the slab's average plane), are considerably inferior to the other models. This means that more simplified analyses tend to provide more conservative results. In this way, it can be claimed that, for structures analyzed by means of simplified models, obtaining high γ_z values does not necessarily mean significant second order effects: considering the results for model 1, building 1 would be classified as being nonsway structure in both directions, and building II in the direction of Y. However, according to the other models, both the structures would be classified as being sway structures in the directions of X and Y. So, from this point of view, utilization of less refined models proves disadvantageous and uneconomical, since it can result in quite relevant second order effects, when in fact they should not be so.

It is important to mention that, obviously, the smaller the γ_z coefficient value is, the more stiff the structure, which is easily found by analyzing equation (1). If the structure's horizontal displacements are fairly big, so that the increase in moments ΔM_tot,d becomes approximately equal to the M_1,tot,d moment, that is, ΔM_tot,d/ M_1,tot,d≅1, the γ_z coefficient will tend to infinity. This would be the case of an infinitely flexible structure. On the other hand, for an infinitely stiff structure, that is, that does not shift under the action of loads, the ΔM_tot,d would be nil and consequently, the γ_z coefficient would be equal to 1. Based on these considerations, it can be stated that, on observation of the γ_z values shown in table [2], the buildings, if analyzed utilizing model 1, appear much more stiff than if analyzed considering the other models. Furthermore, it can be seen that this considerable increase in stiffness is due to the representation of the slabs as shell elements associated with the consideration of the eccentricity existing between the beam axis and the average slab plane, and it is not sufficient to take only one of these factors into account, as can be found by observing the results of models 2 and 5. Thus, from tables [1] and [2], it can also be stated that the representation of the slabs by means of shell elements (model 2) or the consideration of the hypothesis of a rigid diaphragm (model 3) did not themselves contribute to the increase in stiffness of the structures, observed in model 1. In the same way, considering the eccentricity existing between the beam axis and the average slab plane in the bar model (model 5) did not alter the results previously obtained (model 4), indicating that substituting the "beam 4" element for the "beam 44" element to represent the beams did not prove advantageous in the absence of slabs.

Finally, based on the principle that model 1, the most sophisticated and which involves the most computer work, is not generally adopted by the technical medium, including calculating the γ_z coefficient, and considering that all the other models provide practically identical results, in the next item of this paper the buildings will be analyzed utilizing model 4, the simplest one. However, it is worth commenting that, in putting the project into practice, model 1 must be utilized for preference, since it represents the actual behaviour of the structure more accurately and provides much lower γ_z values to those obtained by the other models, which leads to greater savings and, in many cases, dispenses with carrying out analyses which consider, in a simplified way or otherwise, the second order effects.

6. Comparative study of the γ _z and B₂ coefficients

With the purpose of carrying out a comparative study of the γ_z and B₂ coefficients, the values of these parameters were calculated for several reinforced concrete buildings of medium height, including those that were the object of study in item 5.

The buildings were then first order processed, utilizing three-dimensional models on ANSYS-9.0 [1] software, with the columns and beams depicted by means of the "beam 4" element (according to model 4, described in the previous item).

As already mentioned, the actions working on the buildings are divided into two groups: vertical actions (consisting of permanent loads and accidental load) and horizontal actions (corresponding to the action of the wind in directions X and Y). The coefficients applied to the actions were defined from the ultimate normal combination considering the wind to be the main variable action, and determined according to NBR 6118:2007 [2] recommendations.

6.1 Results obtained

Table [3] shows the values of γ_z (the only one for the whole structure) and B₂ (determined for each storey) obtained for the first building analyzed ("building I"), in directions X and Y.

It can be seen in table [3] that, on several storeys of building I, the B₂ coefficient exceeds the value of 1.1 both in direction X and direction Y. In this way, the structure can be considered very sensitive to horizontal movement and, in this case, the global second order effects cannot be ignored. The γ_z coefficient provides a like classification, that is, it considers the structure as being sway structure in both directions X and Y.

It is worth remembering that the γ_z coefficient can be calculated from the values of B₂, utilizing equation (28). Thus, it is enough to determine the c_i constants for each storey, given by equation (29).

In this equation, the portion can be written as:

Substituting the F_hid values (design horizontal forces working on each storey of the structure), given in table [4] and ^[5], in equation (30), gives:

Also considering equation (29), the must be calculated for each storey of the structure; the results obtained are shown in table [4] and ^[5], together with all the data needed to determine the c_i constants and γ_z coefficient, in directions X and Y.

It can be seen in table [4] and ^[5] that, as expected, the γ_z values calculated from the B₂ coefficients coincide with those previously obtained, shown in table [3].

Table [6] shows the γ_z and B₂ parameter values for other buildings analyzed (whose characteristics can be found in Oliveira [7]), together with the classification of the structures, in directions X and Y. However, in the case of the B₂ coefficient, only the average (B_2,avg) and maximum (B_2,max) values of the storeys are shown. Note that, according to Silva [9], a structure can be considered almost insensitive to horizontal movement if, on all its storeys, the B₂ coefficient does not exceed the value of 1.1. If B₂ is greater than this value on at least one storey, the structure will be considered very sensitive to horizontal movement. In this way, classification of the buildings is carried out by analyzing the B_2,max value obtained.

Table [6] shows that, in all cases, the γ_z and B₂ coefficients provide the same classification for the structures. Furthermore, the γ_z and B_2,avg proved to be extremely close, the major difference, corresponding to direction X of building I, being around 3.4%. It is also worth commenting that, in the large majority of cases B_2,avg was lower than γ_z.

7. Final considerations

This paper sought to carry out a study of the γ_z coefficient, employed to indicate the need or otherwise to consider the global second order effects in the analysis of reinforced concrete structures. To conduct the study, several reinforced concrete buildings of medium height were processed utilizing ANSYS-9.0 [1] software.

Initially, the influence of the structural model adopted in calculating γ_z was evaluated. On the basis of the studies done, it was ascertained that less refined analyses tend to provide more conservative γ_z values. This means that, for structures analyzed by means of simplified models, obtaining high γ_z values does not necessarily mean significant second order effects. As such, on adopting simplified models, it is up to the technical medium to be aware that using them can, in many cases, prove disadvantageous and uneconomical, resulting in quite relevant second order effects, when in fact they should not be so.

On putting the project into practice, more sophisticated models (in which the slabs are depicted as shell elements and the eccentricity existing between the beam axis and the average slab plane is considered), although they involve more computer work, should be preferably be utilized, since they depict the actual behaviour of the structures more accurately and provide much lower γ_z values than those obtained by more simplified models, which leads to greater savings and, in many cases, dispenses with carrying out analyses that consider second order effects approximately or otherwise.

Next, a comparative analysis was done of the γ_z coefficient and the B₂ coefficient, commonly employed to evaluate second order effects on steel structures. To conduct the study, initially an equation relating these parameters was developed. Later, the values of γ_z and B₂ for several reinforced concrete buildings of medium height were calculated. From the results obtained, it was observed that the average values of the B₂ (B_2,avg) coefficients showed close proximity in relation to γ_z and that, in all cases, the γ_z and B₂ parameters provided the same classification as the structures.

However, an important aspect deserves to be highlighted concerning the γ_z coefficient: contrary to the B₂ coefficient, it presents a single value for the entire structure, although, as found in several works (Carmo [15], Lima & Guarda [16] and Oliveira [17]), second order effects suffer variations along the height of the buildings. This means that, should the γ_z coefficient be utilized as magnifier of first order moments, as Oliveira [7] suggests, the final moments at some storeys could be underestimated, and overestimated at others.

Thus, a better estimate of the final moments could be made utilizing both coefficients γ_z and B₂, which is calculated for each storey of the structure and whose average value is approximately γ_z. The magnifier of the first order moments would then be differentiated for each storey i of the structure, and given as (B_2,i/B_2,avg).γ_z. Although more specific studies on the subject have not been done, we believe this to be very logical and rational alternative for taking into account how second order effects vary according to how high storeys in reinforced concrete buildings are.

8. Bibliographical references

Received: 20 Jan 2012

Accepted: 25 Sep 2012

Available Online: 08 Feb 2013

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Publication Dates

• Publication in this collection
  28 Feb 2013
• Date of issue
  Feb 2013

History

• Received
  20 Jan 2012
• Accepted
  25 Sept 2012