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 1998-1 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 1998-1 (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

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 EN1998-1.

2. 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 1998-1, with an adequate

degree of reliability.

2.1 STRUCTURAL MATERIALS BEHAVIOR

EN 1992-1-1 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.

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 1998-1, 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

10138-1, 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 φu is the curvature at ultimate of a section and

φy 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, EN1998-1 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 1998-1, 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 εsy,d takes the average value of 1,810-

3, and replacing the previous equation for the ρ for

As/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 As,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 1998-1 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

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 (Ap) 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 ρmax:

(

)

( 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 EN1998-1

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 1998-1. 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 1998-1 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.

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 1998-1 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.

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

1998-1 and in line with the stipulations of the

structural conception of buildings in seismic zones

(Figure 3).

Figure 2 — Example of the changes made to the case study

a ) Case 1 b ) Case 2

c ) Case 3

PEB1

CE

PEB1 PEB2

DC

RCB

Table 1 – Periods, frequencies and modal participation mass of the first six vibration modes

Mode Period Frequency Modal Participation Mass

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

1998-1 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 Structural Type

X 33,8 % 66,2 % Dual System

Equivalent to Wall

Y 38,8 % 61,2 % 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 1998-1, 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.

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 4.3.3. 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

Study Case X 0,122

Y 0,116

1 X 0,121

Y 0,115

2 X 0,122

Y 0,115

3 X 0,123

Y 0,117

Figure 5 — Location of the discontinued vertical

elements (alignment D-7)

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.

Table 4 — Frequencies and modal configurations of the first vibration modes

Modo

Case Study Case 1 Case 2 Case 3

f (Hz) 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

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.

Figure 6 — Interstorey drift along alignment C-8

a) Direction X

b) Direction Y

Figure 7 — Interstorey drift along alignment C-1:

a) Direction X

b) Direction Y

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 pre-

stress 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 = 0.27.

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

1998-1. 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

Section Case 1 Case 2 Case 3

Support D-6 Span D-7 Support D-6 Span D-7 Support D-6 Span D-7

As,

lon

gi Upper

3φ25 14,73 cm2

3φ25 14,73 cm2

3φ25 + 2φ16 18,75 cm2

3φ25 14,73 cm2

4φ25 19,63 cm2

4φ25 29,45 cm2

Lower 2φ25 + 3φ20

19,24 cm2 2φ25 + 3φ20

19,24 cm2 4φ25 + 1φ20

22,77 cm2 4φ25 + 1φ20

22,77 cm2 6φ25

29,45 cm2 6φ25

29,45 cm2

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

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 1998-1 (and not EN 1992-1-

1) 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 five-

storey 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 6 — Reinforcement proposals and ductility verifications for the compatibility element

Direction Longitudinal Reinforcement Shear Reinforcement

Detailing

Local Ductility

Detailing = % ωwd,min ωwd,adopted

X 3 φ 20 / side 8 φ 20

25,13 cm2 2,05%

Critical Region φ 8 // 0,125 0,08 0,19

Y 3 φ 20 / side Outside Critical

Region φ 8 // 0,20

Table 7 — Reinforcement proposals and ductility verifications for column D-6 (Case 2)

Direction Longitudinal Reinforcement

Shear Reinforcement Detailing Local Ductility

Detailing = % ωwd,min ωwd,adopted

X 5 φ 25 / side 16 φ 25

78,56 cm2 1,23%

Critical Region φ 12 // 0,125

(4 hoops)

0,15 0,29

Y 5 φ 25 / side Outside Critical

Region φ 12 // 0,20

(4 hoops)

Table 8 — Reinforcement proposals and ductility verifications for column D-6 (Case 3)

Direction Longitudinal Reinforcement

Shear Reinforcement Detailing Local Ductility

Detailing = % ωwd,min ωwd,adopted

X 14 φ 25 / side 18 φ 25

88,38 cm2 1,38%

Critical Region φ 12 // 0,125

(4 hoops)

0,14 0,29

Y 10 φ 25 / side Outside

Critical Region φ 12 // 0,20

(4 hoops)

(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 high-

energy 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 1998-1, also

ensuring the safety limit states.

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