Seismic vulnerability and retrofitting by damped braces of fire-damaged r.c. framed buildings

Engineering Structures 101 (2015) 179–192

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Seismic vulnerability and retrofitting by damped braces of fire-damaged r.c. framed buildings Fabio Mazza Dipartimento di Ingegneria Civile, Università della Calabria, Via P. Bucci, 87036 Rende (CS), Italy

a r t i c l e

i n f o

Article history: Received 8 February 2015 Revised 15 June 2015 Accepted 16 July 2015

Keywords: R.c. framed buildings Hysteretic damped braces Fire scenarios Time–temperature curves Thermal mappings Nonlinear dynamic analysis

a b s t r a c t There is a lack of knowledge regarding the seismic response of fire damaged reinforced concrete (r.c.) buildings; moreover, an amplification of the damage is to be expected for such structures in the event of an earthquake. To evaluate the nonlinear seismic response following a fire, a numerical investigation is carried out with reference to a five-storey r.c. framed building, which, designed according to the previous Italian seismic code for a medium risk zone now, needs to be considered as in a high-risk seismic zone in line with the current Italian seismic code. More specifically, the nonlinear seismic response of the test structure in a no fire situation is compared with that in the event of fire, at 45 (i.e. R45) and 60 (i.e. R60) minutes of fire resistance, assuming damaged (i.e. DS) and repaired (i.e. RS) stiffness in combination with damaged strength conditions. Then, the fire-damaged test structures are retrofitted by the insertion of hysteretic damped braces (HYDBs), placed only at the storey where the fire compartment is hypothesized. Five fire scenarios have been considered on the assumption that the fire compartment is confined to the first level (i.e. F1), the first two (i.e. F1/2) and the upper (i.e. Fi, i = 3–5) levels, with the parametric temperature–time fire curve in accordance with Eurocode 1. The nonlinear dynamic analysis is performed through a step-by-step procedure based on a two-parameter implicit integration scheme and an initial-stress-like iterative procedure. At each step of the analysis, plastic conditions are checked at the critical (end) sections of the girders and columns, where a thermal mapping with reduced mechanical properties is evaluated in accordance with the 500 °C isotherm method proposed by Eurocode 2, while the behaviour of a HYDB is idealized through the use of a bilinear law provided that buckling is prevented. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Assessment of the seismic vulnerability and retrofitting of existing structures where fire safety is neglected represent a far-reaching problem especially in the case of reinforced concrete (r.c.) framed buildings designed for vertical loads only or with erroneous seismic zone classifications and code provisions. Traditional retrofitting techniques are based on increasing strength and stiffness, by adding new structural elements to the system (e.g. r.c. shear wall or steel brace) and/or enlarging the existing members (e.g. steel encasing or concrete jacketing). However, the increase of seismic resistance capacity is generally combined with an increase of the seismic demand and it is possible that the retrofitted structure will be less safe than in the original condition [1]. On the other hand, it is well known that, even after a strong earthquake, greater seismic protection can be obtained by using supplementary damping devices supported by steel braces [2,3]. At E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2015.07.027 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

present, a wide variety of damped braces is available [4–12]: displacement-dependent (e.g. friction damper, FRD; hysteretic damper, HYD), velocity-dependent (e.g. viscoelastic damper, VED; viscous damper, VSD) or self-centring (e.g. shape memory alloys, SMA). In the present work the attention is focused on metallic yielding HYDs, which are characterized by a stable hysteretic behaviour independent on temperature and velocity of motion. These devices are generally manufactured from traditional materials and require little maintenance, representing a low cost and reliable solution for energy dissipation. In the conventional aseismic design it is accepted that structures can withstand strong ground motions by undergoing inelastic deformations. Consequently, fire can be a serious problem for a structure that has been partially damaged in a prior seismic event, because fire resistance will decrease [13]. More specifically, the fire response of r.c. frame members depends on the thermal (i.e. thermal conductivity, specific heat, thermal diffusivity and mass loss), mechanical (i.e. compressive and tensile strength, modulus of elasticity and stress–strain law) and deformation (i.e. thermal

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expansion and creep) properties of concrete and reinforcing steel bars, changing substantially with heating rate, strain rate and temperature gradient [14]. The combined effect of earthquake and fire has gained attention of recent studies into the nonlinear response of r.c. framed [15] and frame-wall [16] structures. On the other hand, earthquakes following fires may find structures whose seismic resistance is considerably reduced. Numerical [17–19] and experimental [20,21] studies have been carried out on the assessment of the residual seismic load capacity of r.c. structures, in terms of stiffness, strength and ductility after fire. In most cases of fire, structures experience degradation of material properties, due to high temperature, and damage to the structural members, from thermal expansion. In addition, fire induces spalling of concrete that can play a significant role in the seismic performance of r.c. frame members. However, despite the lack of knowledge on the seismic vulnerability of existing framed structures in the event of fire we can expect an amplification of the structural damage in the case of existing structures that have been exposed to fire. In the present work, the nonlinear seismic response of r.c. framed structures in a no fire situation is compared with that in which fire has occurred, at 45 (i.e. R45) and 60 (i.e. R60) minutes of fire resistance, assuming damaged (i.e. DS) and repaired (i.e. RS) stiffness of the frame members in combination with damaged strength conditions. To this end, five-storey r.c. office buildings are designed in line with the previous Italian seismic code [22] for a medium risk zone. A numerical fire investigation is preliminarily carried out considering a thermal–mechanical mapping analysis, with reduced mechanical properties evaluated in accordance with the 500 °C isotherm method proposed by Eurocode 2 [23], followed by a sequentially uncoupled nonlinear dynamic analysis [24,25]. Then, the fire-damaged test structures are retrofitted by the insertion of hysteretic damped braces (HYDBs), placed only at the storey where the fire compartment is hypothesized. In order to study the seismic response of the r.c. framed building damaged from fire, real ground motions corresponding to a high-risk zone and two topographic conditions are considered in line with the current Italian seismic code [26]. Five fire scenarios are hypothesized assuming the fire compartment confined to the area of the first level (i.e. F1), the first two (i.e. F1/2) and the upper (i.e. Fi, i = 3–5) levels, with the parametric temperature–time fire curve evaluated in accordance with Eurocode 1 [27]. 2. Test structure: Design and fire modelling A typical five-storey office building, with a r.c. framed structure shown in Fig. 1a, is considered as test structure. Non-structural elements regularly distributed in elevation, such as perimeter

(a) Plan and fire compartment.

masonry infills, are considered. For the sake of simplicity, the plane frames orientated along the horizontal ground motion direction (Y), perpendicular to the floor slab direction (X) shown in Fig. 1a, are considered as a reference scheme. The dimensions of the cross sections assumed for the columns and the girders, equal at a given level and regularly tapering in elevation, are reported in Fig. 1b. A simulated design of the test structure is carried out in line with the previous Italian seismic code [22], for a medium-risk seismic region (degree of seismicity S = 9, which corresponds to a coefficient of seismic intensity C = 0.07) and a typical subsoil class (subsoil parameter e = 1). The gravity loads are represented by a dead load of 4.48 kN/m2 on the top floor and 5.18 kN/m2 on the other floors, and a live load of 3.0 kN/m2 on all the floors; infill walls, regularly distributed in elevation along the perimeter are assumed to weigh, on average, about 2.7 kN/m2. A cylindrical compressive strength of 20 N/mm2 for the concrete and a yield strength of 375 N/mm2 for the steel are assumed for the r.c. frame members. The design complies with the ultimate limit states satisfying minimum conditions for the longitudinal bars of the girders and columns: at least two 12 mm bars are provided both at the top and bottom throughout the entire length of the frame members; for the girders, a tension reinforcement ratio not less than 0.37% (for the assumed yield strength) is provided and, at their end sections, a compression reinforcement not less than half of the tension reinforcement is placed; minimum steel geometric ratio is 1% for the symmetrically-reinforced section of each column. The dynamic properties of the six main vibration modes are reported in Table 1: i.e. vibration period (Ti); effective masses in the X (mE,X) and Y (mE,Y) directions, expressed as percentage of the total mass (mtot). Five fire scenarios are reported in Fig. 1a and b, assuming the fire compartment is confined to the area of the first level (i.e. F1), the first two (i.e. F1/2) and the upper (i.e. Fi, i = 3–5) levels. It is worth noting that F1/2 fire scenario is obtained from F1 and F2, which occur simultaneously. The geometric properties of the fire compartment are reported in Table 2, for the first and the upper

Table 1 Dynamic properties of the test structure (mtot = 9.35 kN s2/cm). Mode

Ti (s)

mE,X (%mtot)

mE,Y (%mtot)

1 2 3 4 5 6

0.808 0.749 0.326 0.302 0.195 0.183

81.6 0 13.0 0 3.4 0

0 80.4 0 13.5 0 3.8

(b) Elevation and fire scenarios.

Fig. 1. R.c. test structure (dimensions in cm).

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F. Mazza / Engineering Structures 101 (2015) 179–192 Table 2 Geometric properties of fire compartment. Fire compartment

L (m)

D (m)

H (m)

Af (m2)

At (m2)

Av (m2)

heq (m)

O (m0.5)

First level Upper levels

18 18

10 10

4.5 3.5

180 180

612 556

24 24

1.5 1.5

0.048 0.053

levels: i.e. L, D and H representing length, depth and height, respectively; Af, floor area; At, total area of enclosure (i.e. walls, ceiling and floor, including openings); Av, total area of vertical openings; heq, weighted average height of windows. From these values it is possible to define an opening factor, representing the amount of ventilation: 0:5

O ¼ Av heq =At

ð1Þ

Depending on combustible contents and relevant combustible parts of the building, the design value of the fire load density is defined as [27]:

the test structure, with reference to both R45 and R60 at the first and upper levels. Assuming closed openings once the fire scenarios have been hypothesized (Fig. 1b), the temperature is considered uniform in the selected compartment in such a way as to be the most severe condition for the fire before an earthquake (Fig. 1a). Several models can be used to simulate the time–temperature evolution during an actual fire, from the flashover to the full development: conventional fire curves (e.g. the standard ISO-834 curve [28]), in which the temperature monotonically increasing with time represents the heating phase only, when the fuel supply is assumed to be inexhaustible; natural fire curves (e.g. the EC1 curve [27]), in which temperature depends on the ventilation and fire load in the compartment and the cooling phase is represented, based on the assumption that at a certain point, either the air or the combustible material will diminish. The EC1 parametric fire curve is used in the present study, on the assumption that the fire load of the compartment is completely consumed. In the heating phase, the gas temperature hg (°C) 

qt;d ¼ qf ;d Af =At

ð2Þ





hg ¼ 20 þ 1325ð1  0:324e0:2t  0:204e1:7t  0:472e19t Þ

ð6Þ

*

related to the value qf,d corresponding to the surface area of the floor

is a function of a fictitious time t obtained by considering time t (in hours) multiplied by a dimensionless parameter equal to

qf ;d ¼ Q f ;k  dq1  dq2  dn =Af

C ¼ ðO=bÞ2 =ð0:04=1160Þ2

ð3Þ

where Qf,k is the characteristic fire load; dq1 and dq2 are dimensionless factors, taking into account the fire risk due to the size of the compartment and the type of occupancy, respectively; dn is a dimensionless factor taking into account the different fire fighting measures. With reference to 45 min (i.e. R45) and 60 min (i.e. R60) of exposure, the design parameters of the fire load reported in Table 3 are obtained, for the first and the upper levels of the test structure. Once this has been carried out, the thermal absorptivity coefficient of the surrounding surfaces in the fire compartment can be calculated as

pffiffiffiffiffiffiffiffi b ¼ qck

ð4Þ

q being the density, c the specific heat and k the thermal conductivity of enclosure boundary. More specifically, in the case the j-th enclosure surface is made of different layers of material, b1j and b2j thermal absorptivity coefficients should be adopted to define the design value bj, where labels 1 and 2 refer to the layer directly exposed to the fire and the next one, respectively. Finally, to account for different thermal absorptivity coefficients in the walls (i.e. masonry infills), ceiling (i.e. top floor slab) and floor (i.e. bottom floor slab), the thermal absorptivity coefficient is modified thus: b¼

X ðbj  Aj Þ=ðAt  Av Þ

ð5Þ

ð7Þ

The gas temperature in the cooling phase is given by

t max 6 0:5h ! hg ¼ hmax  625ðt   t max Þ 0:5h < tmax < 2h ! hg ¼ hmax  250ð3  t max Þðt   tmax Þ

ð8Þ

t max P 2h ! hg ¼ hmax  250ðt   tmax Þ where the maximum temperature hmax in the heating phase happens for

tmax ¼ ð0:2  103 qt;d =OÞC

ð9Þ

Finally, the standard ISO-834 curve is defined as [28]

hg ¼ 20 þ 345log10 ð8t þ 1Þ

ð10Þ

where t is expressed in hours. With reference to the fire scenarios shown in Fig. 1, the ISO and EC1 time–temperature curves are compared in Fig. 2 in the case where a fire compartment is confined to the first level and the upper ones. The combination of fire parameters, for the open-plan office building, results in a temperature of 918 °C (t⁄max = 1.07 h) and 942 °C (t⁄max = 1.27 h) in 45 min (i.e. fire resistance R45 in Fig. 2a) and 816 °C (t⁄max = 1.19 h) and 836 °C (t⁄max = 1.41 h) in 60 min (i.e. fire resistance R60 in Fig. 2b), at the first and upper floors, respectively. Finally, a zoomed view of the R45 and R60 fire curves is also plotted in Fig. 2c with reference to the EC1 fully developed fire range.

j

Aj being the area of the j-th enclosure surface, openings not included. In Table 4, the main parameters for defining the thermal absorptivity coefficient of the fire compartment are reported for

Table 3 Design parameters of fire load. Fire compartment

Fire resistance (min)

Qf,k (MJ)

dq1

dq2

dn

qf,d (MJ/m2)

qt,d (MJ/m2)

First level Upper levels

R45

102,904 102,904

1 1

1 1

1 1

571.7 571.7

168.1 185.1

First level Upper levels

R60

114,102 114,102

1 1

1 1

1 1

633.9 633.9

186.4 205.2

3. Numerical results: Fire loading before an earthquake Once the time–temperature curve of the fire compartment is determined with the EC1 model, it is possible to evaluate the temperature distribution in the frame members of the test structure at R45 and R60. As an example, thermal mappings of interior and perimeter r.c. frame members in the first and fifth levels of the test structure are plotted in Figs. 3 and 4, respectively, by means of the PROSAP structural program [29]. Mean values of thermal parameters are considered: i.e. a convection factor, characterizing the heat convection from air to the structure surface, equal to 25 W/m2 K [27], and an emissivity of the concrete surface, describing the material’s ability to release the thermal energy which it has absorbed, equal to 0.56 [30].

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Table 4 Thermal absorptivity coefficient of fire compartment. Fire compartment

Enclosure surface

b1j (J/m2 s1/2 K)

b2j (J/m2 s1/2 K)

bj (J/m2 s1/2 K) R45

R60

R45

R60

First level

Masonry infill Top floor slab Bottom floor slab

1131.37 1131.37 1341.64

963.00 963.00 2300.00

1034.14 1034.14 1341.64

1030.56 1030.56 1341.64

228 180 180

1128.3

1125.8

Upper levels

Masonry infill Top floor slab Bottom floor slab

1131.37 1131.37 1341.64

963.00 963.00 2300.00

1034.14 1034.14 1341.64

1030.56 1030.56 1341.64

172 180 180

1138.2

1135.8

(a) R45.

Aj (m2)

b (J/m2 s1/2 K)

(b) R60.

(c) R45 and R60. Fig. 2. Conventional (ISO 834) and natural (EC1) fire curves.

In detail, the finite thermal element cross-section model of columns, exposed to fire on four (Figs. 3a, b and 4a, b) and one (Figs. 3c, d and 4c, d) sides, and girders, exposed to fire on three (Figs. 3e, f and 4e, f) and one (Figs. 3g, h and 4g, h) sides, are considered assuming an ambient temperature of 20 °C for the unexposed cross-section sides. As expected, at the end of 45 and 60 min of exposure, the increase in temperature is greater in the case of cross-sections exposed on four or three sides compared to those exposed on one side only. Moreover, the temperature profiles highlight a decrease in the maximum value with the time increment from R45 to R60, in accordance with EC1 natural fire curves shown in Fig. 2, but the concrete zone damaged by heat is more extended at the fifth level in the case of R60 (Fig. 4). Sophisticated mathematical models of concrete [31], including hygro-thermal performance and thermo-chemical degradation, and steel [32], at high temperatures, are available in literature. More specifically, an accurate modelling of the damage evolution and distribution in case of fire is dependent on heating rate and requires many parameters to be determined [33]. However, extensive analyses of r.c. framed structures require the use of simple and reasonably accurate models to evaluate the residual seismic load

capacity after fire, so that the analysis could be carried out with a reasonable computational effort. From this point of view, the 500 °C isotherm method of Eurocode 2 [23] can be adopted to calculate conservative values [34] of residual stiffness, strength and ductility, considering a reduction of the cross-section size, with respect to a heat damaged zone defined by thermal mapping, and steel strength. More specifically, this method is applicable only if the cross-section has a minimum width [26] as function of fire resistance (for standard fire exposure) and fire load density (for parametric fire exposure). It is based on the following hypotheses: (a) concrete inside the 500 °C isotherm is not affected by fire and retains its initial values of compression strength and modulus of elasticity; (b) concrete outside the 500 °C isotherm is completely neglected in terms of strength and stiffness; (c) each longitudinal bar is considered with strength corresponding to its own temperature. The thickness of the damaged concrete, with temperatures exceeding 500 °C, is made equal to the average depth of the 500 °C isotherm in the compression and corresponds to roughly the thickness of the unconfined zone. Moreover, the reduced yield strength of each longitudinal reinforcement bar, in the tension and compression zones

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F. Mazza / Engineering Structures 101 (2015) 179–192

(a) R45-Interior column.

(b) R60-Interior column.

(c) R45-Perimeter column.

(d) R60-Perimeter column.

(e) R45-Interior girder.

(f) R60-Interior girder.

(g) R45-Perimeter girder.

(h) R60-Perimeter girder.

Fig. 3. Thermal mappings at 1st level of test structure.

(a) R45-Interior column.

(b) R60-Interior column.

(c) R45-Perimeter column.

(d) R60-Perimeter column.

(e) R45-Interior girder.

(f) R60-Interior girder.

(g) R45-Perimeter girder.

(h) R60-Perimeter girder.

Fig. 4. Thermal mappings at 5th level of test structure.

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F. Mazza / Engineering Structures 101 (2015) 179–192

of the cross-section, is evaluated with respect to the steel temperature profile in the centre of the bar [35]. It is worth noting that some of the reinforcing bars may fall outside the reduced cross-section, but they are included in the calculation of the ultimate values of load bearing capacity [26] and curvature ductility, which is evaluated in line with the provisions of Eurocode 8 for the assessment of existing buildings [36]. For the sake of simplicity, transient and creep strains of concrete at high temperatures are not considered in the present work. It should be noted that transient strain, occurring during the first time heating and independent of time, and creep strain, time-dependent plastic deformation increasing the rate of deformation at high temperatures, may be important in case of earthquakes following fires to evaluate residual deformations of fire-damaged structures [37]. In Fig. 5, flexural stiffness in the fire-exposed cross-sections is reported along the building height, assuming a direct correspondence between the examined level and the fire compartment. In particular, interior columns (Fig. 5a) and girders (Fig. 5b) are examined at the end of 45 (i.e. F.R45 structure) and 60 (i.e. F.R60 structure) minutes of exposure to fire. As expected, a local decrease in stiffness from a minimum, of about 34% and 26% at the 1st level, and a maximum, of about 52% and 30% at the 5th level, is obtained for the interior columns and girders of the F.R45 structure, respectively, in comparison with the no-fire condition. A further minimum decrease in stiffness, of about 12% and 9% at the 1st level, and maximum, of about 13% and 10% at the 5th level, is obtained after an additional 15 min of fire (i.e. F.R60 structure). Further results reported in Fig. 5c and d have confirmed limited fire effects in the perimeter frame members at all levels where the fire compartment is supposed. Thereafter, ultimate ductility of the columns, corresponding to the axial force due to the gravity loads, confirmed limited fire

effects in the perimeter columns and a significant reduction only for the interior columns (Fig. 6a), of about 22% and 33% for the F.R45 and F.R60 structures, respectively. Moreover, it is interesting to note that an increase of ductility is obtained on the bottom side of the interior girders (Fig. 6b), of about 7% (i.e. F1.R45) and 13% (i.e. F1.R60), due to the fire-reduced yield strength of the longitudinal bars in tension. On the other hand, a decrease in ultimate ductility at the top, of about 15% (i.e. F1.R45) and 29% (i.e. F1.R60), is observed because of heat damage to the compression zone of concrete and corresponding longitudinal reinforcement. Finally, analogous results are reported for the ultimate interaction domain between axial load (NRd) and bending moment (MRd) of columns (Fig. 7) and the ultimate bending moment of girders (Fig. 8). For clarity, only fire compartments at the first (i.e. F1 in Figs. 7a and 8a) and fifth (i.e. F5 in Figs. 7b and 8b) level are examined. As can be observed, the interior columns exhibit a marked narrowing of their NRd–MRd domains, especially for values of the compressive axial load greater than those corresponding to the balanced compressive load. For this axial load, a local reduction in flexural strength, in comparison with the no-fire condition, of about 26% and 39%, at the first storey, and 37% and 53%, at the fifth storey, is found in the F.R45 and F.R60 structures, respectively. On the other hand, a significant decrease in strength is observed on the bottom side of the interior girder 3–8 (see Fig. 1a) with a maximum local reduction of about 35% and 55%, at the first floor (Fig. 8a), and 28% and 45%, at the fifth floor (Fig. 8b), considering the F.R45 and F.R60 structures, respectively. Finally, to upgrade the test structures damaged by fire, diagonal steel braces with hysteretic dampers (HYDBs) are inserted at the level where the fire compartment is hypothesized only (Fig. 9a). Three fire scenarios are reported in Fig. 9b, assuming the fire

(a) Interior column n. 8.

(b) Interior girder n. 3-8.

(c) Perimeter column n. 6.

(d) Perimeter girder n. 1-6.

Fig. 5. Flexural stiffness of r.c. frame members exposed to fire.

F. Mazza / Engineering Structures 101 (2015) 179–192

(a) Interior column n. 8.

185

(b) Interior girder n. 3-8 (1st floor/F1).

Fig. 6. Ultimate ductility of r.c. frame members exposed to fire.

(a) Interior column n. 8 (1st storey/F1).

(b) Interior column n. 8 (5th storey/F5).

Fig. 7. Ultimate NRd–MRd domains of r.c. columns exposed to fire.

(a) Interior girder n. 3-8 (1st floor/F1).

(b) Interior girder n. 3-8 (5th floor/F5).

Fig. 8. Ultimate bending moments of r.c. girders exposed to fire.

compartment confined to the area of the first level (i.e. F1), the first two levels (i.e. F1/2) and the fourth level (i.e. F4). For simplicity, only HYDBs in the plane frames along the horizontal ground motion direction are shown in Fig. 9. In detail, HYDBs with stiffness and strength properties able to restore the corresponding initial values at that level, in the no fire condition, are placed in the plane frames with marked fire effects. To this end, the decrease in

stiffness and strength in the fire-exposed interior and perimeter columns of the interior and central frames are evaluated in accordance with the results above reported. It is worth noting that the limited fire effects in the lateral frames are also taken into account and assigned to the HYDBs of the interior frames. The elastic stiffness (i.e. KDB) and yield strength (i.e. NY) of the HYDBs, given that the bracing system is stiffer than the HYD which is supported

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(a) Plan and fire compartment.

(b) Elevation and fire scenarios.

Fig. 9. R.c. retrofitted test structures.

Table 5 Stiffness and strength properties of the HYDBs for the FR.60 structure. Fire scenario

Level

Interior frames 2

F1 F1/2 F4

1 1 2 4

Central frame

KDB (kN m )

NY (kN)

KDB (kN m2)

NY (kN)

4.60E+04 4.60E+04 6.59E+04 2.54E+04

1.40E+02 1.40E+02 1.15E+02 6.24E+01

1.44E+04 1.44E+04 2.12E+04 9.19E+03

5.47E+01 5.47E+01 4.29E+01 2.87E+01

(i.e. KDB ffi KD), are reported in Table 5 considering, for the sake of brevity, the FR.60 structure only. 4. Numerical results: Seismic loading following fire In order to study the seismic response of the r.c. framed building damaged by the fire loading proposed above, nonlinear dynamic analyses are carried out. To this end, the r.c. frame members are idealized by means of a two-component model (TCM), comprising an elastic–plastic component and an elastic one, assuming a bilinear moment–curvature law [38]. The elastic component is characterized by the flexural stiffness pEI, p being the hardening ratio of the moment–curvature law. The elastic-perfectly plastic component exhibits inelastic deformations, lumped at the end cross-sections (i and j), which are determined with reference to an axial force–biaxial bending elastic domain. Flat surfaces are used to describe the elastic domain, by considering a piecewise linearization of its bounding surface (Fig. 10). Specifically, a satisfactory representation of the NMyMz interaction domain has been obtained by considering 26 flat surfaces, including: 6 surfaces normal to the principal axes x, y and z (Fig. 10a); 12 surfaces normal to the bisections of the yz, xy and xz principal planes (Fig. 10b); 8 surfaces normal to the bisections of the octants (Fig. 10c). The elastic–plastic solution, represented by point P, is obtained as a projection of the elastic solution, represented by point E, on

Fig. 10. Flat surfaces approximating the elastic domain of r.c. section.

the active flat surface of the elastic domain (Fig. 11a). Otherwise, when the elastic solution lies within the fan limited by the planes normal to the boundaries of the flat surfaces, point P can be located along the active line (Fig. 11b) or at the active corner (Fig. 11c) resulting from the intersection of these surfaces. The elastic–plastic solution is evaluated in terms of the initial state and the incremental load on the basis of a holonomic law, as a solution of the Haar–Kàrmàn principle [39,40]. The numerical implementation of the Haar–Kàrmàn principle turns out to be computationally effective because it is carried out separately for each beam element of the structure. Moreover, the proposed TCM takes into account the interaction between the inelastic flexural deformations lumped at the end sections of the beam element. The TCM represents a good trade-off between the computational effort, because it is much less demanding in computer memory than a fibre model (FM), and reliability of the results. It has been implemented in a finite element code able to simulate the nonlinear response of a framed structure subjected to static (monotonic and cyclic) and dynamic loadings; comparisons with more refined FMs have proved the reliability of the proposed model [39,40]. Moreover, hysteretic damped braces (HYDBs) have a stable hysteretic behaviour and present a mechanism of energy dissipation depending on the storey drifts [11,41]; their activation happens when preset stress levels are reached or overcome. An in-series system constituted of a steel brace and a hysteretic damper (Fig. 12a) is considered to simulate the nonlinear dynamic response of a HYDB, assuming the idealized bilinear force–displacement law (N–D) shown in Fig. 12b and adopting: an elastic-linear law for the brace (B), with elastic stiffness KB, providing that the buckling be prevented (Fig. 12c); a bilinear law for the hysteretic damper (HYD), with yield load Ny, initial stiffness KD and stiffness ratio rD (Fig. 12d). In the Rayleigh hypothesis, the damping matrix of the framed structure is assumed to be a linear combination of the mass and stiffness matrices, assuming a damping ratio of 5% with reference to the frequencies of the second and sixth vibration modes in the Y direction (see Table 1). In this way, a somewhat lower value of the damping ratio is achieved for the fourth mode in the same direction, while the higher frequency modes, which do not contribute significantly to the dynamic response, are practically eliminated due to their high damping ratio. Real ground motions corresponding to a high-risk seismic region provided by the current Italian seismic code (NTC08, [26]) and a design subsoil class B (i.e. subsoil stratigraphic parameter SS = 1.08) are considered. In detail, two sets of seven recorded accelerograms, selected by the computer code REXEL [42], are considered for the topographic classes T1 and T2 (i.e. topographic

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(a) Active flat surface.

(b) Active line.

(c) Active corner.

Fig. 11. Elastic–plastic solution of r.c. frame member.

(a) Typical arrangement of a damped brace.

(b) Hysteretic Damped Brace (HYDB).

(c) Brace (B)

(d) Hysteretic Damper (HYD).

Fig. 12. Elastic–plastic solution of a hysteretic damped brace (HYDB).

Fig. 13. Acceleration (elastic) response spectra for the topographic classes T1 (a) and T2 (b).

parameters ST1 = 1.0 and ST2 = 1.2). The corresponding elastic response spectra of normalized acceleration (Sa/g) are plotted in Fig. 13, assuming an equivalent viscous damping ratio in the horizontal direction (nH) equal to 5%, considering different scale factors (SFs). On average, these spectra match the corresponding target NTC08 response spectra for PGAT1 = 0.34 g (Fig. 13a) and PGAT2 = 0.41 g (Fig. 13b), in the range of vibration periods 0.05 s4 s, which also contains the lower and upper limits of the vibration period (i.e. Tmin = 0.2T1 and Tmax = 2T1, where T1 is the fundamental vibration period of the structure). Thus, the selected accelerograms can be used for the nonlinear dynamic analysis of structures with T1 6 1 s. All the following results are obtained as an average of those separately obtained for the sets T1 and T2 of real motions. Moreover, the curvature ductility demand is calculated with reference to the two loading directions, assuming as yielding curvature for the columns that corresponding to the axial force due to the gravity loads.

Firstly, maximum ductility demand of columns and girders along the building height is plotted in Fig. 14, for the topographic class T1 (PGAT1 = 0.34 g), comparing framed structures both in the no-fire condition and at the end of 45 (i.e. R45) and 60 (i.e. R60) minutes of fire exposure. More specifically, interior columns (Fig. 14a, c, e) and girders (Fig. 14b, d, f) are examined, on the assumption that the fire compartment is confined to the area of the first level (i.e. F1, Fig. 14a and b), the first two levels (i.e. F1/2, Fig. 14c and d) and the fourth level (i.e. F4, Fig. 14e and f). Moreover, r.c. frame members are assumed to have been previously damaged by fire, with a reduced cross-section (i.e. in the Damaged Stiffness condition, DS), and later repaired, e.g. using jacketing with thin layers of concrete (i.e. in the Repaired Stiffness condition, RS). It is noteworthy that the fire damaged strength of r.c. frame members mantains also in the RS condition. As can be observed, an amplification in the structural response of fire-exposed structures is localized to the level of the fire

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(a) Interior columns.

(b) Interior girders.

(c) Interior columns.

(d) Interior girders.

(e) Interior columns.

(f) Interior girders.

Fig. 14. Global ductility demand for the topographic class T1: (a) and (b), fire scenario F1; (c) and (d), fire scenario F1/2; (e) and (f), fire scenario F4.

compartment involved. As expected, this result is more marked in the Fi.R60 structures than in the Fi.R45 ones (with i = 1, 2, 4), because the first are characterized by lower strength and ductility. The results also show that the seismic response of the test structures differs in the DS and RS conditions, with higher ductility demand when the initial stiffness is restored (i.e. RS condition). This kind of behaviour can be explained observing that jacketing with thin layers of concrete (i.e. RS condition) can be unsuitable, because the increase of stiffness is accompanied by an increase of the seismic demand. To better clarify the local damage distribution, top (i.e. lTi and lTj) and bottom (i.e. lBi and lBj) maximum ductility demand in the frame members and corresponding ultimate values are reported in Fig. 15 for the fire scenarios F1 (Fig. 15a and b) and F4 (Fig. 15c and d). In particular, girders and columns exposed to fire on three (e.g. the interior girder n. 3–8) and four (e.g. the interior column n. 8) sides are examined. Note that the maximum ductility demand and the number of sections in which the corresponding ultimate value is exceeded increase especially at the first level, with an amplification when the initial stiffness is restored (i.e. RS condition) and R60 is assumed (Fig. 15a and b). In this case, three interior columns and all the interior girders exceed the corresponding ultimate ductility demand at the first level. Moreover, a significant increase of ductility demand on the fourth level is observed only in the interior columns of the F4.R60 structure (Fig. 15c), producing a ‘‘strong beam-weak column’’ local mechanism at this level. Further results, omitted for

the sake of brevity, have confirmed this behaviour also in the case of the F3 and F5 fire scenarios. Afterwards, taking into account the symmetric plan shown in Fig. 1a, the distribution of maximum ductility demand in the perimeter and interior frame members is investigated in Fig. 16. Local maximum ductility demand in the perimeter and interior columns (Fig. 16a and c) and girders (Fig. 16b and d) for the DS (Fig. 16a and b) and RS (Fig. 16c and d) conditions are plotted at the first level where the fire compartment is hypothesized. Note that the ductility demand increases especially in the frame members exposed on three (i.e. the interior girders n. 2–7, n. 7–12, n. 3–8 and n. 8–13) and four (i.e. the interior columns n. 7 and n. 8) sides in the RS condition. Moreover, among the columns exposed on one side greater ductility demand is obtained in the perimeter ones of the exterior frames (i.e. the perimeter columns n. 1, n. 6 and n. 11). Further results, omitted for the sake of brevity, confirm an increase of maximum ductility demand and number of frame members in which the corresponding ultimate value is exceeded especially at the lower storeys. Curves analogous to those above reported are shown in Fig. 17 where fire compartment is confined to the area of the first two levels (i.e. F1/2 scenario) and R60 is assumed, with reference to the topographic classes T1 (i.e. PGAT1 = 0.34 g) and T2 (i.e. PGAT2 = 0.41 g). As can be observed, a significant increase in ductility demand is obtained in the frame members of the fire compartment moving from T1 to T2 topographic class. Moreover, the ductility demand of the interior columns (Fig. 17a) and girders (Fig. 17b) increases

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(a) Interior column n. 8 (1st storey/F1).

(b) Interior girder n. 3-8 (1st floor/F1).

(c) Interior column n. 8 (4th storey/F4).

(d) Interior girder n. 3-8 (4th floor/F4).

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Fig. 15. Local ductility for the topographic class T1: (a) and (b), fire scenario F1; (c) and (d), fire scenario F4.

(a) Columns at the 1st storey (DS condition).

(b) Girders at the 1st floor (DS condition).

(c) Columns at the 1st storey (RS condition).

(d) Girders at the 1st floor (RS condition).

Fig. 16. Local ductility demand for the topographic class T1: fire at the 1st level (fire scenario F1).

at the levels where the event of fire is hypothesized while it is slightly reduced, in comparison with the no fire condition, on the other levels. This behaviour is amplified when the RS condition and topographic class T2 are considered. In this case, the ultimate ductility demand is reached in all the interior columns of the first two levels and all the interior girders of the first one. After this, to check the effectiveness of the HYDBs for controlling the local damage undergone by critical sections of the fire-damaged r.c. frame members, in Fig. 18 the maximum values of the ductility demand attained by the interior columns (Figs. 18a, c, e) and girders (Fig. 18b, d, f) are shown along the

building height. Only the results for the topographic class T1 at the end of 60 (i.e. R60) minutes of fire exposure are reported, relating to fire scenarios F1 (Fig. 18a and b), F1/2 (Fig. 18c and d) and F4 (Fig. 18e and f). In detail, the following cases are compared: unbraced frame in the no fire situation; fire-damaged unbraced frames, in the DS and RS conditions; damped braced frames in the DS condition. As can be observed, the HYDBs effectively reduce the ductility demand, especially for the columns at the levels of the fire compartment. It is worth noting that the increased damping capacity due to the HYDBs makes the DBFi.R60 (DS) structures preferable to the Fi.R60 (RS), because the latter locally increase

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(a) Interior columns.

(b) Interior girders.

Fig. 17. Global ductility demand for the topographic classes T1 and T2: fire at the 1st and 2nd levels (fire scenario F1/2).

(a) Interior columns.

(b) Interior girders.

(c) Interior columns.

(d) Interior girders.

(e) Interior columns.

(f) Interior girders.

Fig. 18. Global ductility demand in the primary and retrofitted fire-damaged structures: (a) and (b), fire scenario F1; (c) and (d), fire scenario F1/2; (e) and (f), fire scenario F4.

the lateral stiffness but, due to their low damping capacity, produce an increase in the ductility demand. To further compare the local damage distribution in the Fi.R60 (RS) and DBFi.R60 (DS) structures, top and bottom maximum ductility demand in the interior column n. 8 and interior girder n. 3–8 are shown in Fig. 19a, b and c, d, respectively. For the sake of brevity, only the fire scenarios F1 (Fig. 19a and c) and F4 (Fig. 19b and d) are examined. The results show that the ultimate ductility demand of the Fi.R60 (RS) structures is always exceeded in the columns (Fig. 19a and b) and only on the bottom side in

the girder on the first floor (Fig. 19c). On the other hand, the ductility threshold imposed by NTC08 is not reached for any of the frame members of the DBFi.R60 (DS) structures. Finally, in terms of distribution of maximum ductility demand in the perimeter and interior frame members (Fig. 20), note that the ductility demand in the interior girders of the DBFi.R60 (DS) structures increases in comparison with the no fire condition in both the F1 (Fig. 20b) and F4 (Fig. 20d) fire scenarios. However, in all the frame members a reduction in ductility demand due to the insertion of the HYDBs is obtained in comparison with the

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(a) 1st storey/F1.

(b) 4th storey/F4.

(c) 1st floor/F1

(d) 4th floor/F4

Fig. 19. Local ductility in the primary and retrofitted fire-damaged structures: (a) and (b), interior column n. 8; (c) and (d) interior girder n. 3–8.

(a) Columns at the 1st storey.

(b) Girders at the 1st floor.

(c) Columns at the 4th storey.

(d) Girders at the 4th floor.

Fig. 20. Local ductility demand in the primary and retrofitted fire-damaged structures: (a) and (b), fire scenario F1; (c) and (d), fire scenario F4.

cases F1.R60 and F4.R60 in the RS condition, especially for the columns (Fig. 20a and c). 5. Conclusions The nonlinear seismic response of five-storey r.c. framed buildings, primarily designed in line with the previous Italian seismic code for a medium risk zone, is studied under two sets of seven real motions whose response spectra match, on average, NTC08 spectra with regard to a high-risk seismic zone and two topographic classes. More specifically, the nonlinear dynamic analysis of framed structures in a no fire situation is compared with that in the event of fire, at 45 (i.e. R45) and 60 (i.e. R60) minutes of exposure to fire, assuming damaged (i.e. DS) and repaired (i.e. RS) stiffness in combination with damaged strength conditions. Then, for the seismic retrofitting of the test structures, HYDBs are inserted at the level where the fire compartment is hypothesized only, assuming stiffness and strength properties to restore the undamaged values to that level. Five fire scenarios are considered: the fire compartment confined to the area of the first level (i.e. F1), the first two (i.e. F1/2) and the upper (i.e. Fi, i = 3–5) levels. The following conclusions can be drawn from fire results before earthquake.

– A greater increase in temperature is obtained in the interior columns and girders whose cross-sections are exposed to fire on four and three sides, respectively. The temperature profiles highlight a decrease in the maximum value with the time increment from R45 to R60, but a more extended concrete zone damaged from heat in the case of R60. – A significant decrease of stiffness is obtained at all levels in the structural members exposed to fire, in comparison with the no-fire condition. – A marked narrowing of the axial load and bending moment interaction domain is exhibited by the interior columns, especially for values of the compressive axial load greater than those corresponding to the balanced compressive load, while a significant decrease in flexural strength is observed on the bottom side of the interior girders. – An increase in ductility is obtained on the bottom side of the interior girders, due to the fire-reduced yield strength of the longitudinal bars in tension, with a decrease on the top side value because of damage to the compression zone of concrete and the corresponding longitudinal reinforcement. The following conclusions can be drawn from seismic results following fire.

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– An amplification in the structural response is localized to the level of the fire compartment, with a slight reduction on the other levels in comparison with the no fire condition. – The ductility demand increases especially in the frame members exposed on three and four sides in the RS condition, while negligible effects are found in the perimeter frame members. Maximum ductility demand and number of sections in which the corresponding ultimate value is exceeded increase especially on the lower levels while fire scenarios at the upper ones produce ‘‘strong beam-weak column’’ local mechanisms. – The seismic vulnerability of the test structure differs in the DS and RS conditions, with higher ductility demand in the case where the initial stiffness has been restored. – To retrofit fire-exposed r.c. frame members, traditional jacketing with thin layers of concrete (i.e. RS condition) can be unsuitable, because the increase of stiffness is accompanied by an increase of the seismic demand, and modern retrofitting techniques based on additional damping should be preferred. Seismic retrofitting with HYDBs represents a satisfactory means of reducing the ductility demand below the NTC08 threshold, especially for the interior columns at the levels of the fire compartment and in comparison with RS condition. In detail, HYDBs with stiffness and strength properties able to restore the corresponding initial values at that level, in the no fire condition, should be placed in the plane frames with marked fire effects. To this end, the decrease in stiffness and strength in the fire-exposed interior and perimeter columns of the interior and central frames needs to be evaluated.

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