SEISMIC ANALYSIS OF BUILDINGS WITH PUNCTUAL DISCONTINUITY

Transcription

1 SEISMIC ANALYSIS OF BUILDINGS WITH PUNCTUAL DISCONTINUITY OF VERTICAL ELEMENTS José António Varandas Ferreira Brito ABSTRACT In this study we investigate the behaviour of a reinforced concrete structure under seismic action, clarifying the implications of the vertical elements discontinuity in the global and local behaviour of the structure. Discontinuity mainly occurs in large dimension spans, and prestress systems. With the evolution of the prestress steels (increased extension at maximum load), their use in earthquake zones is allowed, no longer being an impediment to the ductile behavior of prestressed sections. However, the vertical component of the seismic action may not be ignored in prestress beams supporting discontinued vertical elements since it may result in an inadequate design. The analysis of the effects of discontinuities of a vertical element was carried out a generic irregular structure, used to define the three study cases. The comparison of the effects global (frequencies, modes of vibrations, shear base forces and horizontal displacements) leads us to conclude that the change in continuity and geometry of the vertical element does not compromise the dynamic behaviour and the seismic efficiency of the structure. Locally we intended to analyse, in all three cases, the influence of the design forces in the structural elements located in zone of discontinuity, especially in prestressed beams. Ductility was ensured by the control of the level of effort and reinforcement area. In prestress elements the difficulty to limit the position of the neutral axis was compensated by the careful detailing of the reinforcement. An adequate energy hysteric dissipation capacity and the concrete confining reinforcement in critical regions enables structures to deform without substantial reduction of its overall resistance to vertical and horizontal seismic action. Thus the accurate structural conception and design based on the EN contributes to the seismic efficiency of the structure, limiting the influence of the removal of vertical elements in the global and local behaviour. Key words: Ductility, Reinforce Concrete, EN (Eurocode 8), Capacity Design, Discontinued, Prestress 1. INTRODUCTION The scope of this study arises from the need to analyze the discontinuity of vertical elements, evaluating the behavior of the global and local structure, when under a seismic action. Admitting that the architectural and functional limitations lead to the growing need for the buildings to have, on the lower floor, a larger height and a larger span, resulting from the suppression of the continuity of certain vertical elements, we intend to analyze the effects of this conception in the distribution of the forces resulting from seismic action and the ductility capacity of the prestress beams, needed for the functionality of the structure. From the global behavior point of view, the use of prestress systems allows multiple and differentiated structural solutions. In this study we compare three prestress structural solutions, i.e., a traditional

2 solution, removing the column and prestressing the beams on all the floors (15 metres beams and the adjacent, if any. A second solution, similar to the previous but adding a vertical elements between the first and the last floor, resulting in a element with reduced axial force, and a third solution, similar to the second solution, but only prestressing the beams in the first floor. In the overall analysis, we intended to compare the dynamic behavior of the different models, which depends on the mass and inertia characteristics of the structural elements, and discuss the results of the comparison response of the models to dynamic characterization parameters. In the analysis of the local behavior under seismic action we need to guaranty the necessary energy dissipation capacity in the critical regions, i.e. ductility, to avoid the risk of fragile hinges. When seismic phenomenon occurs, exploring ductility is an essential condition for the correct function of a structure. The macroscopic behavior of the structural materials in association with the mechanical and chemical phenomenon under repeated and alternate cycles of charge guarantees the existence of inelastic deformations that allow energy dissipation. To analyze and simulate da response of the models to seismic action we have used: the SAP2000 program and the response spectrums contained in the EU rule EN STRUCTURAL MATERIALS BEHAVIOR AND SEISMIC DESIGN OF STRUCTURES The analysis of the behavior of reinforced concrete and prestressed structures in conditions of service and rupture, and also the rules of measurement of the limits states demands a strict evaluation of materials properties more directly related to the structure's response to the actions applied. The structures in seismic regions must be designed and constructed to meet the design requirements recommended in EN , with an adequate degree of reliability. 2.1 STRUCTURAL MATERIALS BEHAVIOR EN presents the main basic characteristics of concrete and steel. See below some parameters used to explain the structural materials behavior. CONCRETE Structural concrete does not have the same behavior characteristics as traction (particles spacing) and compression (particles approach), which is related with his composition. Concrete is a fragile material that has an inelastic deformation under compression due to a progressive loss of stiffness related to cracking phenomena s. This phenomenon originates a massive loss resistance capacity to major deformations due to transverse cracking and expansion. However the behavior of concrete can become more ductile if we confine to this lateral expansion. When that happens, compression radial tensions are generated which causes an augmentation of the traction strength under, especially the ultimate extension, which translates into better flexible capacity. The importance of confinement is associated with the hysterical behavior of concrete. Hysteresis cycles are associated with energy dissipation phenomena s in the concrete, by inelastic deformation. A progressive decreasing of inclination of a tension-extension curve, for various cycles, is then related to the evolution of damages in the concrete. STEEL The main characteristics of structural steel can be obtained from tests of traction/compression until rupture. One of the principal properties of ordinary steel is shown on an identical behavior of traction and compression. The importance of the elongation at maximum force is connected to ductility properties (energy dissipation) which is what we meant steel will react to after the yielding. This means that the section of reinforce concrete could deform without a considerable resistance capacity loss. The hysterical behavior of steel has an important effect on reinforced concrete and is characterized by the existence of several loading and unloading cycles. The first loading cycle corresponds to the monotonic model response of steel in uniaxial compression. During the performance of hysteresis cycles, the spread of material degradation leads to increased inelastic deformation with the number of cycles. In each cycle, steel elongation increases until the rupture. The degree of ductility can be analyzed indirectly by the number of cycles of loading and unloading the material can support up to rupture and the area contained in the hysteresis cycles.

3 PRESTRESS STEEL Pre-stress steels exhibit a similar behavior to that of cold worked steel, characterized by a high resistance and a plastic deformation without a clear level of lending. This material can be proportionally cheaper than ordinary steels normally employed in construction, since their resistance can be approximately three to four times higher, despite the costs associated with anchoring systems. The application of prestress steel in seismic zones has been limited by the EN , for failure to submit the required strain of prestressing steel at maximum load (minimum of 5% for steel class B) and limiting section ductility. Recent studies from the LNEC pointed out that a wide range of prestressing steels, when subjected to tensile tests, present values of elongation at the maximum load about 5%, allowing their use in structures in seismic regions. In pren , Table 2, are available prestressing steels classified as Class B. Although the material has an elongation that allows the dissipation of energy, the behavior of the section is also influenced by level of tensile stress and the position of the neutral axis. DUCTILITY AND ENERGY DISSIPATION IN THE BEHAVIOR OF RC ELEMENTS At the measurement of some structures with resistant capacity to the seismic action is usually accepted the deformation of those structures further the elastic limit taking advantage of the capacity of energy dissipation by hysteresis. It s important that all the structural elements present ductility that means the capacity of deformation after cession without a considerable loss of capacity resistance. Ductility is one of the most important parameters for reinforced concrete (RC) structures in resisting seismic loads. The ductility is often defined as the ratio of curvature at a given response level to curvature at yield response. The global ductility ratio of any structure can be expressed as (equation 1), ( 1 ) where φ u is the curvature at ultimate of a section and φ y is the curvature when the tension reinforcement first reaches the yield strength. The structural ductility capacity must be known to determine the appropriate force-reduction factor in the force based design approach. The structural ductility capacity, in combination with other parameters, is used to determine the appropriate force-reduction factor in the force based design approach (non-linear analysis). In a ductile structure, the inelastic energy dissipation can be achieved in a somewhat regulated manner without jeopardizing the integrity or stability of the structural system. The design seismic force can be reduced, and The allocation of the results is performed by dividing the value of each quantity is obtained from the analysis by the coefficient of elastic behavior, q. In the new seismic regulation, control levels of ductility and curvature are no longer liable to direct measurement. For this reason, EN presents a set of rules and expressions, some of them referred in this work, in order to ensure a ductile behavior of the elements and sections during seismic events. This check is carried out indirectly by limiting the amount of reinforcement and neutral axis position of the sections. Among the set of rules given in EN , stand out in this study the verifications of the local ductility of the beams to ensure the formation of plastic hinges. In reinforced concrete beam elements the maximum percentage of reinforcement in the area pulled forward, ρmax, you should take the value from the following equation: ( 2 ) Despising the compression reinforcement in section, assuming that ε sy,d takes the average value of 1,8 10-3, and replacing the previous equation for the ρ for A s /bd, we obtain the following simplification: ( 3 ) in which ω represents the percentage of mechanical reinforcement and k (=x/d) the position of the neutral line. It should be noted that the analysis of the equations 2.2 and 2.3 concludes that if you want to increase the reinforcement area of traction beyond A s,max, you must be considering equal amount of compression reinforcement, keeping the position of the neutral axis of the section at failure. With the adoption of prestressing, some of the conditions given in EN relating to the beams need to be modified to include this new variable. The regulation informs us about the need for a lower amount of reinforcement compression at least 50% of the top reinforcement in critical regions, to ensure

4 the verification of the conditions of ductility. The equation 4 translates this need and considers the amount of prestress infinite time (P ) and the amount of prestressed reinforcement (A p ) in these sections: ( ) ( 4 ) The percentage of reinforcement in the area of the critical sections should take the value from the equation 5, ensuring that it is lower than the value de ρ max : ( ) ( 5 ) For the columns and resistance walls, local ductility depends directly to the level of axial force and global ductility admitted to the structure. 2.2 SEISMIC BEHAVIOUR OF STRUCTURES A set of dynamic displacements or a quantity of energy transmitted to the structure can translate the effects or demands of the seismic action. The structure must have the ability to sustain these seismic demands and if not, the structure is considered vulnerable. However, the concept of vulnerability is not absolute, as a building may be vulnerable to a type of seismic action and seismic resistant to another. It all depends on the properties of the structural system and on the characteristics of the seismic action. It is possible to classify the reasons for the collapse of a building in two groups; one regarding the internal conditions of the building and another concerning the external conditions of the building. The external conditions concern the type of foundation, the type of soil and the interaction with adjacent buildings. The internal conditions concern the regularity in terms of mass and stiffness, the ability to dissipate energy, the concentration of stresses in singular zones, the deficient detailing and design of structural elements and non-structural elements. 3. APPLICATION OF EN The scope of the legislation in earthquake-resistant plans and design of buildings and civil engineering works in seismic regions is to ensure, in the event of earthquakes, the protection of human lives, the limited structural damage and ensure the operability of important structures civil protection. 3.1 PERFORMANCE REQUIREMENTS Serviceability and safety are the two basic performance objectives adopted in performance of structures in EN Serviceability is the performance objective so as to keep functional use without repair normally under moderate earthquakes. Therefore, the serviceability limit state shall be corresponded to minor or no damage levels. Safety is the performance objective so as to protect human life, and corresponds to the ultimate limit state or the safety limit state. Therefore, the design objective may be selected so that the structure can bear gravity loads and would not collapse. In terms of structural damage, the state may be just before collapse at the loss of gravity load carrying capacity or P-δ deformation limits. Ideally, the criteria should be established by quantifying the damage level of structural and nonstructural members such that economically allowable repair is possible, i.e., by taking into account estimated cost for restoration after earthquakes, where the diminished basic performance of safety and serviceability caused by the earthquake shall be restored to the required levels. To verify the requirement of limited damage (Damage Limitation states), EN admits a new concept of seismic action, the seismic service. This new action has the same configuration of the earthquake plan spectrum, less than a reduction coefficient, which reflects the different level of risk between different seismic events. This factor should not only reflect the national choice for the protection of buildings but also the risk seismic region. In this context it is noteworthy that the earthquake is related to service evaluation to determine the strain limit states design. Spectrum of service does not applying any coefficient of behavior, which can lead to spectral values larger than the viewer project. This situation can only make sense if in such results are only used deformations.

5 3.2 METHODS OF ANALYSIS The seismic effects, combined with the effects of the other actions, included in the seismic design situation may be determined by the following methods: a. Linear static (lateral force method )analysis; b. Dynamic static (modal response spectrum) analysis. c. Non-linear static (pushover) analysis; d. Non-linear time history (dynamic) analysis, The reference method for the seismic analysis in the EN is the modal response spectrum analysis with the design spectrum and be applied to buildings which do not satisfy the criteria for regularity in plan and in elevation. With the available tools and the advancement in finite elements analysis, the linear dynamic analysis of three-dimensional models have become the most common method of analysis based on the behavior of structures under vertical and horizontal seismic actions, at new building and/or rehabilitation and strengthening of existing structures. 3.3 SEISMIC ACTION The basic representation of the seismic action is performed by spectrum elastic response of the structure to certain characteristics of the seismic action transmitted by the soil, commonly known as spectrum elastic response. The shape of the spectrum elastic response is identical whatever the performance requirements and their correlation with the limit states. The spectrum response of the project, instead of the elastic spectrum are affected in a reduction coefficient - coefficient of behavior, which depends on the type of structural system, defined by the level of ductility class DCM (class structure considers to case study) and the behavior of structural materials. In buildings with large spans, the analysis for determining the effects of the vertical component of the seismic action can be realized locally, including the elements on the vertical component which is considered to act. 3.4 DESIGN PHILOSOPHY The structure must be designed to resist seismic events without local or global collapse, maintaining its structural integrity and a residual load capacity after the earthquake. The Capacity Design allows us to define how the structure will behave during a seismic event, regardless of his characteristics. With this type of design, the designer defines the areas where they will form the plastic hinges, as well as, eventually, the order of their formation. In practice, the designer imposes the values of strength and ductility to the different structural elements through different arrangements depending on the area to be considered, ensuring first, the existence of resistance in areas where you do not want to form plastic hinges. And on the other hand, there are areas in the structure - critical areas - where the actuating effort equals the value of effort resistant, allowing the formation of plastic hinges. The plastic hinges should be designed and detailed so as to possess ductility and energy dissipation measures. Generally it can be said that a structure presents good behavior during a seismic event when develop mechanisms to effectively dissipate, and with a hysteretic mode, the seismic forces. 4. BUILDINGS IN ANALYSIS 4.1 CASE STUDY The base structural configuration of a common concrete building to be built in Portimão is represented in Figure 1. The structure has 5 floors with 5, 5 meters spacing between floors on the lower level and of 4 meters on the reaming floors. All frames are resistant in both main second moment of area directions, separated by 7,5m. On floors 4 and 5 an area reduction of the plan implantation region and the interruption of the respective vertical support elements have been considered. The beams and the columns present geometrical sections of varied dimensions depending on the plan location and the vertical charges they are subject to.

6 Figure 1 Three-dimensional overview of the case study and first floors plant 4.2 VARIANTS OF THE CASE STUDY To analysis the effects of discontinuity in elevation we have considered current situations of conception when there is the need to interrupt a vertical alignment. The first situation, Case 1 (Figure 2.a.) we totally supress the column to increase the span dimension between columns to 15 meters in the directions according to axis X and Y, applying prestress to all beams (PEB1) with 15 meters span and with continuity to the 7,5 meters adjacent spans, if existent, in the directions X and Y. Case 2 (Figure 2.b.) is identical to Case 1. The only difference is that a vertical element linking the beams of floors 1 to 5 has been considered. This element, defined as compatibility element (CE) intends to create compatibility of the span deformations, unifying the vertical displacements of each floor. The last Case, Case 3 (Figure 2.c.) presents a discontinued column (DC) between the foundations and the first floor. In this case only the beams with 15 m span and with continuity for adjacent spans of 7.5 meters, if existent, are prestressed (PEB2). All other resistant elements, vertical and horizontal, keep the same geometry as the model case study. With the changes in the case study we aim to compare the influence of discontinuity and/or the suppression of an interior vertical element in the local and overall behaviour under the seismic action of project and service. 4.3 CASE STUDY DESIGN CONCEPT In this section the dynamic characterization of the case study is presented. In order to clearly understand the behavior of the structure, the periods, frequencies and the percentages of mass participation factor for each direction for the first 6 vibration modes, are presented in Table 1. From the analysis of the vibration modes presented in Figure 3 we conclude that there is a predominance of translation in modes 1 and 2, respectively, in Y and X direction without any torsion and the existence of a third mode, rotation, without any associated mass. When translation occurs in inferior modes and torsion in superior modes without mobilized mass, we are in presence of the concept defined by EN and in line with the stipulations of the structural conception of buildings in seismic zones (Figure 3). RCB CE DC PEB1 PEB1 PEB2 a ) Case 1 b ) Case 2 c ) Case 3 Figure 2 Example of the changes made to the case study

7 Table 1 Periods, frequencies and modal participation mass of the first six vibration modes Modal Participation Mass Mode Period Frequency UX (%) UY (%) Σ UX (%) Σ UY (%) 1 1,176 0,850 0,00 81,97 0,00 81,97 2 1,099 0,910 80,71 0,00 80,71 81,97 3 0,993 1,007 0,59 0,17 81,30 82,14 4 0,379 2,637 0,10 12,18 81,40 94,33 5 0,347 2,879 12,33 0,27 93,73 94,60 6 0,292 3,427 1,00 0,37 94,73 94,97 Mode 1: f = 0,850 Hz Mode 2: f = 0,910 Hz Mode 3: f = 1,007 Hz Figure 3 Frequencies and modals configurations 4.4 CASE STUDY DESIGN CONCEPT In this section the dynamic characterization of the case study is presented. In order to clearly understand the behavior of the structure, the periods, frequencies and the percentages of mass participation factor for each direction for the first 6 vibration modes, are presented in Table 1. From the analysis of the vibration modes presented in Figure 3 we conclude that there is a predominance of translation in modes 1 and 2, respectively, in Y and X direction without any torsion and the existence of a third mode, rotation, without any associated mass. When translation occurs in inferior modes and torsion in superior modes without mobilized mass, we are in presence of the concept defined by EN and in line with the stipulations of the structural conception of buildings in seismic zones (Figure 3). CHARACTERIZATION OF THE BUILDING STRUCTURAL TYPE The classification of the of the structural type is effected by the numeric evaluation of the percentage of seismic base shear transmitted to the different structural elements walls and columns in relation to the total force of basal cut supported by the structure. We present in Table 2 the percentages of the force supported by each type of resistant element and the type of structural system used for this study. Table 2 Classification of the study case structural type by direction Direction Columns / FBS FWalls/ FBS X 33,8 % 66,2 % Y 38,8 % 61,2 % Structural Type Dual System Equivalent to Wall Dual System Equivalent to Wall BEHAVIOUR FACTORS AND DESIGN RESPONSE SPECTRUM Considering the structural geometry (Figure 1), the lack of regularity in elevation and other requirements stated in EN , the behaviour factor for horizontal seismic actions assumes a value of 2,9. In respect of the vertical behaviour factor to consider in the variants of the case study, the limited number of energy dissipation zones in prestress beams reduces the possibility for plastic deformations. Conservatively was used a value of behaviour factor for vertical actions equals to 1. Figure 4 shows the evolution of the project spectrum acceleration in respect of the frequency in the most conditioning seismic action, seismic action type 1.

8 Figure 5 Location of the discontinued vertical elements (alignment D-7) Figure 4 Design response spectrum (type 1) 5. INFLUENCE OF THE DISCONTINUED ELEMENT IN STRUCTURE BEHAVIOR This section describes and discusses the overall behavior of the variants of the case study on seismic regulations considered in Section In order to check the impact in the global dynamic behaviour, the adoption of the possibilities considered in the variants of the case study (Figure 2). For the complete characterization of the missing cases set the vertical alignment of the discontinuous element. To maximize the rotation effects of the structure it was decided to change the continuity of the column in alignment D-7. Table 3 Seismic coefficient β para os diferentes casos Case Directon Seismic Coefficient β X 0,122 Study Case Y 0,116 X 0,121 1 Y 0,115 X 0,122 2 Y 0,115 X 0,123 3 Y 0,117 The behaviour of the three cases will be compared in terms of displacement and base shear force, which measures of the importance of global seismic action in the struture, and frequency and modal configuration, which measures the dynamic behavior of the structure. To this end we considered the horizontal displacement of the column and resistant wall in alignment E-1 and C-8, respectively. The frequencies of the first modes of vibration of the cases 1 to 3 are very similar to the case study (Table 1). The vibration modes have the same predominated movements, and the percentages of mass participation remain approximately equal, suggesting that changing the continuity of the vertical element does not influence the total stiffness of the structure (Table 4). For higher values of frequency, vibration modes appear characterized by deformation localized in areas of greater design flexibility. This is the case, for example, the phenomena of local vibration in beams prestressed in Cases 1 through 3 to frequencies in the range of 3-4 Hz. In these cases due to its vertical component, those modes are important for the analysis of effects of the vertical component of the earthquake. Modo f (Hz) Table 4 Frequencies and modal configurations of the first vibration modes Case Study Case 1 Case 2 Case 3 Modal Configuration f (Hz) Modal Configuration f (Hz) Modal Configuration f (Hz) Modal Configuration 1 0,8501 Translation Y-Y 0,8385 Translation Y-Y 0,8405 Translation Y-Y 0,8520 Translation Y-Y 2 0,9102 Translation X-X 0,9022 Translation X-X 0,9043 Translation X-X 0,9193 Translation X-X 3 1,0075 Rotation Z-Z 0,9932 Rotation Z-Z 0,9954 Rotation Z-Z 1,0100 Rotation Z-Z 4 2,6372 Translation Y-Y 2,5811 Translation Y-Y 2,5892 Translation Y-Y 2,6261 Translation Y-Y 5 2,8785 Translation X-X 2,8620 Translation X-X 2,8658 Translation X-X 2,8824 Translation X-X 6 3,4273 Rotation Z-Z 3,3980 Rotation Z-Z 3,4028 Rotation Z-Z 3,4237 Rotation Z-Z 7 5,2596 Translation Y-Y 3,4897 Local 3,8042 Local 3,9486 Local

9 A global measure of the seismic behaviour of structures is the seismic coefficient, which represents the relationship between base shear and weight of the structure, i.e., the ratio of the seismic forces and gravity loads. Table 3 presents the different seismic coefficients for X and Y directions, concluding that the variations are nonexistent, which is structurally favourable. The value of this parameter allows the designer, as mentioned above, an overall assessment of the magnitude of the effects of seismic action. Figure 6 shows the horizontal displacements on the column in alignment E-1 and Figure 7 the displacements on the resistant wall in alignment D- 8. A brief analysis of Figures 6.3 to 6.6 can identify a less uniformity in the different levels of displacement in the direction Y. This is due to the lower bending stiffness in the Y direction (direction of the fundamental frequency) and therefore more sensitive to the removal of a column. For the alignment of C-8 (Figure 6) the assumptions of Cases 1 and 2 leads to decrease in local stiffness, resulting in increased relative displacements between floors in the direction Y. For the E-1 alignment (Figure 7), located at the opposite end of the structure, there is a reduction in amplitude of the displacements of Cases 1 and 2 in comparison with the study case and the Case 3, results in decreasing the prevalence of the rotation component. Note that the differences in the X direction are minor and are due to the increase of displacements resulting from the rotation component. The analysis carried out confirm that the structural design adopted for the case study, with the placement of walls along alignments and resistant exterior leads to an effective dynamic behavior, which results on a limited impact on the dynamic behavior of the global adoption of the chances of Case 1 to 3. a) Direction X b) Direction Y Figure 6 Interstorey drift along alignment C-8 a) Direction X b) Direction Y Figure 7 Interstorey drift along alignment C-1:

10 6. ANALYSIS AND DESIGN OF STRUCTURAL ELEMENTS In this chapter we analyzed the ultimate limit states of different resistance elements and detail of critical sections in order to ensure the necessary ductility conditions. The design values were obtained for the combination defined from the seismic response spectra of the project (Figure 4). BEAM (ALIGNMENT D-8) In prestressed beams, the level of tensions affects the design and its detailing. The existence of prestress modifies the evolution of forces along the beam, especially in sections of the span. In these sections, the distribution of stress has negative values for the vertical seismic environment, but mainly the horizontal. The strain imposed by the prestress in conjunction with the deformation caused by vertical acceleration of the seismic action leads to negative design values in the span, wich requires a speciaç attention to the values of upper longitudinal reinforcement. Table 5 shows the area of longitudinal reinforcement, upper and lower sections of the support and range from prestressed beams of three cases for verifying the conditions of ductility and strength of the local sections. Based on the upper limit of k imposed by equation 3, it s possible to ensure the condition of local ductility of prestressed section support D-6 of Case 3, k = It should be noted that conditions in equation 3 depends of nonlinear characteristics of the structural materials that sometimes are difficult to define. The limit of k is an approximate value that can vary depending on the behavior considered for the structure. The resistance condition is satisfied in all sections of beam in Case 3. For Cases 1 and 2, the values of k obtained are well below the limit imposed by the equation 3, ensuring the ductility of the section and the other conditions outlined in EN A proper detail, with increased longitudinal reinforcement and consideration of compression reinforcement, would improve the behaviour of a section, reducing the neutral axis position. These assumptions were considered in Case 3, in particular. COMPATIBILITY ELEMENT (CASE 2) With the compatibility element (CE) we aim to standardize the deflections of the prestress beams (Figure 8). The process of calculating the prestressing is demanding and errors may occur in the process of laying of cables that can lead to increased vertical displacements. These situations can happen especially in slabs or beams more slender. With the EC element any such trend is offset by the remaining beams, getting the same strain in all floors, with advantages for masonry and other non-resistant elements. Figure 8 Vertical deflection in prestress span of quasi-permanent combination Table 5 Area of longitudinal reinforcement and conditions of ductility in critical sections of prestress beams As,longi Section Upper Lower Case 1 Case 2 Case 3 Support D-6 Span D-7 Support D-6 Span D-7 Support D-6 Span D-7 3φ25 14,73 cm 2 2φ25 + 3φ20 19,24 cm 2 3φ25 14,73 cm 2 2φ25 + 3φ20 19,24 cm 2 3φ25 + 2φ16 18,75 cm 2 3φ25 14,73 cm 2 4φ25 4φ25 19,63 cm 2 29,45 cm 2 4φ25 + 1φ20 4φ25 + 1φ20 6φ25 6φ25 22,77 cm 2 22,77 cm 2 29,45 cm 2 29,45 cm 2 k 0,15 0,11 0,15 0,13 0,27 0,24 ρ' 0,0063 0,0045 0,0075 0,0048 0,0049 0,0064 ρ 0,0104 0,0134 0,0117 0,0146 0,0109 0,0136 ρmax 0,0143-0,0154-0,0128 0,0144

11 Table 6 Reinforcement proposals and ductility verifications for the compatibility element Longitudinal Reinforcement Shear Reinforcement Local Ductility Direction Detailing = % Detailing ωwd,min ωwd,adopted X 3 φ 20 / side 8 φ 20 Y 3 φ 20 / side 25,13 cm 2 2,05% Critical Region φ 8 // 0,125 Outside Critical Region φ 8 // 0,20 0,08 0,19 Although quantitatively subjected to fewer efforts than the existing in the columns, the design and detailing of the CE was performed as primary seismic elements to ensure the necessary ductility (Table 5). In this element the mechanical percentage of longitudinal reinforcement could be significantly reduced, because it was intended that it may ultimately presents more ductile capacity (critical confinement section) rather than resistance. The design and detailing were performed based on the requirements of EN (and not EN ) in order to provide higher ductility to this element with the purpose of ensuring the horizontal displacements between floors and the transfer of gravity loads. COLLUMN (ALIGNMENT D-6) For the analysis of a vertical element was selected the column in the alignment D-6, as the nearest column of the zone of discontinuity. The bending efforts do not present substantial s differences along the vertical alignment of the column. Although there s minimum difference on the bending moment on Cases 1 and 2, that led Case 2 to be selected for bending analysis and detailing effects. In Case 3 was expected the existence of increased bending moments to the first floor level resulting from the addition of the moments on the prestressed beams when compared to other cases. In Table 7 and 8 are presented in summary the areas of longitudinal and transversal reinforcement to adopt in Case 2 and 3 in the lower floors. Local ductility was also checked at columns edges. The result confirms the conclusions presented in Section 5, where the removal or discontinuance of the column didn t change the local and global behavior of the structures. 7. CONCLUSION In this study was characterized and modelled a fivestorey structure and three variants. The main objectives of analyse were the effects in the local and global behaviour of discontinuities in a column and the application of prestress in beams, under the seismic action. The asymmetries of the case study were opposed by the careful placement of resistant walls, ensuring, along with the frames, two predominant modes of vibration in translation and a third in torsion, without any associated mass percentage. The decrease of rigidity of the system, resulting from the possibilities of Case 1 to 3 does not affect the dynamic behaviour. Locally it was necessary to meet the requirements of ductility, ensuring that ductile ruptures could move forward with sufficient reliability, to modes of fragile rupture. For the beam in alignment D, the ductility of the critical sections was compared using k parameter Table 7 Reinforcement proposals and ductility verifications for column D-6 (Case 2) Longitudinal Reinforcement Local Ductility Direction Shear Reinforcement Detailing Detailing = % ωwd,min ωwd,adopted φ 12 // 0,125 X 5 φ 25 / side 16 φ 25 Critical Region (4 hoops) 1,23% 0,15 0,29 Y 5 φ 25 / side 78,56 cm 2 Outside Critical φ 12 // 0,20 Region (4 hoops) Table 8 Reinforcement proposals and ductility verifications for column D-6 (Case 3) Longitudinal Reinforcement Local Ductility Direction Shear Reinforcement Detailing Detailing = % ωwd,min ωwd,adopted φ 12 // 0,125 X 14 φ 25 / side 18 φ 25 Critical Region (4 hoops) 1,38% 0,14 0,29 Y 10 φ 25 / side 88,38 cm 2 Outside φ 12 // 0,20 Critical Region (4 hoops)

12 (position of the neutral axis) and the areas of longitudinal reinforcement in the three cases. Cases 1 and 2 show better energy dissipation capacity, confirmed by a lower value of k, and a more efficient confinement of critical areas. In case 3, the high values of k limit the curvature of the sections and ductile capacity. Nevertheless, all measures of local ductility were observed. For the measurement of the column D-6 and CE (compatibility element), were admitted performance requirements of a primary seismic column to ensure the required deformation capacity for horizontal and vertical actions. Regarding the CE element, also stressed the importance of concrete confinement as a necessary condition to give ductility to the critical section. To this element, the ductility in the critical zones needs to override the section bending resistance. Analyses made leds to conclude that the structural design of Case 2 has the best overall performance locally and globally, ensuring more efficiently highenergy dissipation and curvature. The idea that the interruption of a vertical element is necessarily a negative factor in the behaviour of a structure under seismic action is questioned in this work. In Case 3, which represents the currently most adopted structural solution, presents a less ductile behavior when compared with Case 2, due to a major depth of the neutral line in sections. In the end, Case 3 checks all seismic design criteria given in EN , also ensuring the safety limit states. REFERENCES [1] CEN, Eurocódigo 2 (EN ) Projecto de estruturas de betão, Parte 1-1: Regras gerais e regras para edifícios, Instituto Português da Qualidade, [2] J. JIRSA e I. D. KARSAN, Behavior of Concrete under Compressive, Journal of the Structural Division, [3] CEN, pren Prestressing steels, Part 1: General requirements, CEN, [4] CEN, Eurocódigo 8 (EN ) Projecto de estruturas para resistência aos sismos, Parte 1: Regras gerais, acções sísmicas e regras para edifícios, Instituto Português da Qualidade, [5] CEN, pren Prestressing steels, Part 3: Strand, CEN, [6] Mário LOPES, Sismos e Edifícios, Edições Orion, [7] C. MARCHÃO e J. APPLETON, Folhas de Apoio às Aulas de Betão Armado e Pré-esforçado I: Volume 2 - Verificação da segurança aos estados limites últimos de elementos com esforço axial desprezável, Instituto Superior Técnico. [8] The Institution of Civil Engineers, Manual for the design of reinforced concrete building structures to EC2, Institution of Structural Engineers, [9] M. N. FARDIS e G. TSIONIS, Specific rules for design and detailing of concrete buildings: Design for DCM and DCH (Illustration of elements design), Lisbon, [10] CEN, Eurocode 1 (EN 1991): Actions on structures - Part 1-1: General actions - Densities, self-weight, imposed loads for buildings, CEN, 2001.