1 2 3 4

Displacement based design for precast concrete frames 5 with not-emulative connections 6

Andrea Belleri1 7 Department of Engineering and Applied Sciences, University of Bergamo, Italy 8

Abstract 9

The Displacement Based Design (DBD) methodology to precast concrete frame structures with not-10 emulative connections is investigated herein. The seismic design procedure is applied to both 11 single-storey and multi-storey structures. Industrial and office buildings, warehouses and 12 commercial malls with a structural layout typical of the European market are considered: cantilever 13 columns resting on isolated footings connected at the floor level to pre-stressed precast beams, 14 supporting pre-stressed precast concrete floor or roof elements. The need to control the lateral 15 seismic displacement is dictated by the high flexibility of these structures, which in turn is 16 associated to the structural scheme and to the inter-storey height. 17 Starting from the general displacement based design procedure, the paper focuses on how properly 18 taking into account the influence of column-foundation and beam-column precast connections; 19 expressions and procedures are developed to determine the yield curvature, the equivalent viscous 20 damping, the effective height and the effective mass of the single degree of freedom substitute 21 structure adopted in the DBD procedure. 22 The proposed procedure is applied to selected case studies and validated through non-linear time 23 history analyses, showing the ability of the design procedure in controlling lateral displacements. 24

Keywords: 25 precast structures; seismic design; displacement based design; not-emulative connections; grouted 26 sleeves connections; yield deflected shape; equivalent viscous damping; 27

1. Introduction 28

Precast concrete structures are widely adopted, especially in the industrial and commercial sector, 29 due to the reduced on-site construction time and cost effectiveness, to the ability of covering wide 30 spans with pre-stressed elements and to a better quality control of materials and structural elements 31 compared to traditional reinforced concrete structures. Although different typologies of lateral force 32 resisting system solutions are available in the literature and in the worldwide practice, such as 33 reinforced concrete emulative structures [1, 2], jointed ductile connections [3-5] and rocking and 34 hybrid walls [6, 7] among others, the majority of European industrial buildings, warehouses and 35 commercial malls are single-storey or few-storey buildings with a simple structural layout: 36 cantilever columns, connected at the floor and at the roof by simply supported precast and pre-37 1 Assistant professor, Ph.D.; andrea.belleri@unibg.it

Andrea Belleri. 2017. “Displacement based design for precast concrete frames with not-emulative connections”. Engineering Structures, Vol. 141, pp. 228-240. DOI:10.1016/j.engstruct.2017.03.020

Publisher’s version:http://dx.doi.org/10.1016/j.engstruct.2017.03.020

stressed beams, supporting pre-stressed concrete elements. The columns are placed and grouted on-38 site in isolated precast cup-footings or connected to shallow foundations through mechanical splices 39 or grouted sleeve solutions [8-11]. The column-to-beam connection is typically pinned [12-14] and 40 the energy dissipation is provided by the development of plastic hinges at the base of the columns. 41 The hinged-frame static scheme and the high inter-storey height lead to flexible structures in which 42 the contribution of elastic displacements is higher compared to traditional reinforced concrete 43 frames. If not appropriately considered in the design phase, this high flexibility could lead to 44 displacement incompatibility between structural elements [15, 16]; the contact between the end of 45 the beam and the column during their relative rotations may lead to a change in the boundary 46 conditions, and between structural and non-structural elements, such as precast cladding panels, 47 causing their premature failure [17-23]. The seismic performance of these structures is therefore 48 related to inter-storey drift control rather than material strain limitations. 49 The seismic design approach commonly adopted by professional engineers, as in EN 1998-1 [24], is 50 the well known force based design (FBD): the equivalent lateral inertia forces are obtained 51 considering a system with reduced flexural stiffness (an effective modulus of inertia Ieff is defined 52 as a percentage of the gross module Ig to account for concrete cracking) and an acceleration 53 spectrum scaled by a force reduction factor depending on the structural typology is used. The lateral 54 displacements are obtained at the end of the design process. For flexible structures, as those 55 considered herein, the displacements are obtained from the equal displacement approximation 56 which states that the displacement ductility is equal to the force reduction factor. 57 Following the FBD procedure [25], the results could be affected by the aforementioned sources of 58 approximations: the force reduction factor, the effective modulus of inertia and the equal 59 displacement approximation. Although these limitations could be overcome by the definition of 60 refined formulations, the displacements are evaluated at the end of the design process. Being lateral 61 displacements so important in the seismic response of the structures considered herein, a more 62 rational approach would consider the displacements as the input of the design process. Performance 63 based design methodologies, such as displacement based design (DBD), follow this approach. 64 Starting from the DBD procedure proposed by Priestley et al. [26], the paper considers how to 65 implement typical details of precast concrete structures, such as column-to-foundation and beam-to-66 column connections, in the design process. Regarding the column-to-foundation connections, the 67 influence on the system energy dissipation capacity and on the yield curvature is investigated; the 68 former affects the equivalent viscous damping formulation, while the latter affects the displacement 69 ductility formulation. Regarding the beam-to-column connections, the paper analyzes the influence 70 on the effective height and effective mass of the substitute structure used in the design process. 71 Finally, the proposed procedure is applied to selected case studies, both single and multi-storey 72 buildings, and validated by means of non linear time history analyses. 73

2. Displacement Based Design 74

The DBD procedure [26] adopts a substitute structure approach [27], which considers a single 75 degree of freedom (SDOF) elastic structure with stiffness equal to the secant stiffness at maximum 76 displacement and with damping equal to an equivalent viscous damping accounting for hysteretic 77 energy dissipation. 78 The definition of the structural deflected shape (Di) for a considered multi degrees of freedom 79 (MDOF) system is the first step of the procedure. The deflected shape represents the first inelastic 80

mode of vibration and it is associated to a particular structural typology. Priestley et al. [26] report 81 the deflected shapes for typical structural typologies, based on analytical derivations or as results of 82 non-linear time history analyses. It is worth noting that the diaphragm stiffness could alter the 83 lateral deflection, with greater lateral displacements in the central part of the diaphragm, especially 84 when the lateral force resisting system is located at the diaphragm edges. The properties of the 85 SDOF substitute structure, as the target displacement (Dd), the effective height (heff) and the 86 effective mass (meff), are obtained directly from the MDOF-system target deflected shape, which is 87 selected to limit, for instance, inter-storey drifts or material strains. Such properties are: 88

2

1

1

n

i ii

d n

i ii

m

m

=

=

DD =

D

å

å; 1

1

n

i i ii

eff n

i ii

m hh

m

=

=

D=

D

å

å; 1

n

i ii

effd

mm =

D=

D

å (1) (2) (3) 89

The following step is the evaluation of the equivalent viscous damping, which accounts for the 90 elastic (xel) and the hysteretic (xhy) damping: xel considers material viscous damping, radiation 91 damping due to the foundation system and damping due to non-structural components; xhy considers 92 the energy dissipation capacity of the system and depends on the hysteretic behaviour of the 93 structural elements. Various equivalent viscous damping formulations are available in the literature 94 [26, 28, 29]. The formulation adopted herein [29] depends on the effective period and displacement 95 ductility of the SDOF substitute structure, being the displacement ductility represented by the ratio 96 between design and yield displacement (µ∆=∆d/∆y): 97

( )

1 10.05 1 1eq el hy dbeff

aT c

x x xµD

æ öæ öç ÷= + = + - +ç ÷ç ÷+è øè ø

(4) 98

The parameters (a, b, c, d) depend on the non-linear properties (i.e. hysteretic model) of the 99 structural elements and they are obtained by regression analysis. It is worth noting that, being Teff 100 not available at the beginning of the design process, a first tentative value is necessary, for instance 101 Teff = 1s, and subsequently updated. The hysteretic model considered as a reference herein is the 102 Takeda model [30] whose force-displacement relationship (Figure 1) is defined by a = 0.3, b = 0.6, 103 r = 0.05; the corresponding parameters of Eqn. 4 are: a = 0.249, b = 0.527, c = 0.761 and d = 3.250. 104

(Dy) (Du)

(Fy)

(F)

(ki)

(rki)

(Dp)

(bDp)

(ku=kiµD )-a

f y f u

My

M

k'i

r'k'i

fp

b 'fp

k'u=k'iµf-a'

(Dres)f res Dy Du

Fy

ki

rki

DpbDp

ku=kiµD-a

105

Figure 1 – Takeda hysteretic model. 106

The yield displacement (Dy) corresponds, for single-storey hinged frames, to a linear variation of the 107 curvature from 0 to yield (fy), from the column tip to the column base; for multi storey structures, a 108

specific formulation of Dy will be defined in the following. According to Priestley et al. [26], the 109 yield curvature of rectangular reinforced concrete elements can be related to the properties of the 110 cross-section: 111

2 2

2.13 3

yy y

H HBe

fD = ´ = ´ (5) 112

B and H are the cross-section depth and the column height respectively; ey is the yield deformation 113 of the longitudinal reinforcement. 114 The equivalent viscous damping is used to scale the elastic displacement spectrum (SD_el) for 115 damping values different from 5%. According to EN 1998-1 [24], this reduction is: 116

__

( ) 0.10( 0.05) 0.05D el eq

D el eq eq

SS

xh

x x= =

= + (6) 117

The substitute structure effective period (Teff) is the period of the damped displacement spectrum 118 (SD_el(xeq)) corresponding to the target displacement (Dd). From Teff it is possible to evaluate the 119 effective stiffness (keff), associated to the substitute structure maximum response, and thereafter the 120 design base shear (Vb): 121

224eff

b eff d deff

mV k

Tp= D = D (7) 122

The obtained base shear is distributed as design forces along the height of the MDOF system 123 considering the inelastic deflected shape: 124

( )1

/n

i b i i i ii

F V m m=

æ ö= D Dç ÷

è øå (8) 125

Finally, capacity design principles are applied to inhibit fragile mechanisms. 126 It is worth noting that both Eqn. 4 parameters and the coefficients in Eqn. 6 depend on the ground 127 motion set considered, i.e. the calibration of such equations leads to different parameters and 128 coefficients if different ground motion sets are used; however, Pennucci et al. [31] showed that the 129 resulting value of h (Eqn. 6) is not dependent on the ground motions set considered, providing that 130 the same ground motion set is used in the calibration of both equations. 131

3. Column-to-foundation connections 132

The column-to-foundation connections used in precast structures influence the DBD procedure. In 133 particular, they affect the yield curvature and the energy dissipation capacity of the structural 134 system. Typical connections are represented by cup footings, mechanical splices and grouted sleeve 135 solutions. 136

3.1 Yield curvature 137

As reported in Eqn. 5, the yield curvature (fy) affects directly the yield displacement (Dy); as a 138 result, the equivalent viscous damping is also affected, being dependent on the displacement 139 ductility. The yield curvature formulation of Eqn. 5 was obtained [26] analysing square columns 140 with a cross-section size equal to 160 cm, a concrete cover equal to 5 cm and longitudinal re-bars 141 equally distributed along the section sides. This equation does not properly describe the yield 142 curvature when the effective depth is not as close to the cross-section size, as in the case of grouted 143

sleeve solutions [9]. To overcome this limitation, the cross-section size B is substituted with the 144 cross-section effective depth ds, and the constant 2.1 with the parameter α1: 145

1y

ysd

ef a= × (9) 146

α1 is obtained by a least square procedure on the results of moment-curvature analyses conducted 147 with the computer code Cumbia [32], accounting for the influence of different variables such as the 148 cross-section size (Ac), the concrete cover (cc), the concrete compressive strength (fc), the steel yield 149 strength (fy), the steel overstrength ratio (ratio between ultimate fu and yield stress), the axial load 150 ratio (ν = N/Acfck, where N is the axial load) and the ratio (ρ) between the longitudinal reinforcement 151 area and Ac. Four different sets of longitudinal reinforcement were evaluated: 4, 8, 12 and 16 re-bars 152 equally spaced along the cross-section’s sides. A sensitivity analysis has been conducted to check 153 the influence of the selected variables; the results are reported in Table 1 in terms of the maximum 154 recorded scatter to the reference case (Ac = 45x45 cm, cc = 5 cm; fc = 40 MPa; fy = 500 MPa; fu/fy = 155 1.3; ρ = 0.02; ν = 0.15). 156

Table 1. Results of the sensitivity analysis 157

Variable Range Max. difference (%)

Ac 900 – 3600 cm2 3.1%

cc 3 – 8 cm 3.7%

fc 30 – 60 MPa 4.2%

fy 450 – 550 MPa 3.1%

fu/fy 1.1 – 1.5 0%

ρ 0.005 – 0.04 21.0%

ν 0.05 – 0.30 15.6%

Among these parameters, n and r have been selected to describe α1: 158

1 1 2 3y s

y

dh h h

fa n r

e×

= = × + × + (10) 159

Table 2 shows the values of h1, h2 and h3 as a function of the total number of longitudinal re-bars. 160

Table 2. Yield curvature coefficients 161

Parameter Number of longitudinal rebars

4 8 12 16

h1 1.94 1.11 1.22 1.97

h2 9.18 6.50 6.30 4.30

h3 1.39 1.69 1.69 1.18

162

3.2 Energy dissipation capacity 163

The use of different column-to-foundation connections is generally associated to different hysteretic 164 models and to different plastic hinge lengths; as, for instance, the strain penetration associated to 165 various mechanical connectors used in precast buildings leads to differences in the plastic hinge 166 length. The difference in the energy dissipation capacity is directly related to the equivalent viscous 167 damping adopted in the DBD procedure. 168 A procedure is herein proposed in order to calibrate the hysteretic damping expression associated to 169 various types of column-to-base connections. The procedure represents an alternative to [29] and it 170 is based on the analysis of the force displacement inelastic response of SDOF systems. The 171 procedure is summarized in the following steps and graphically represented in Figure 2. 172

1. Select the hysteretic model whose hysteretic damping needs to be calibrated. In this case the 173 chosen hysteretic model is the most representative of the force-displacement relationship of 174 the considered column-to-foundation subassembly. 175

2. Select a displacement ductility value (µD). 176 3. Get the elastic spectral displacement (SD_el(T)) and the constant ductility inelastic spectral 177

displacement (SD_in(T,µD)); the latter could be obtained by finite element software or by 178 dedicated tools, such as Ruaumoko-Inspect [33]. 179 It is important to note that the inelastic spectral displacement refers to a SDOF systems with 180 a given elastic period (T0), while the hysteretic damping equation (Eqn. 4) includes the 181 effective period (Teff). The relationship between T0 and Teff, for hysteretic systems with a 182 backbone loading curve resembling a bilinear curve with post-yield stiffness ratio r, is: 183

01 ( 1)

effrT T µµ

D

D

+ -= (11) 184

Therefore, in order to allow the comparison between SD_el(T) and SD_in(T,µD), it is required to 185 consider the elastic spectral displacement of the substitute system (SD_el(Teff)), with T=Teff, 186 and the inelastic spectral displacement of the initial SDOF system (SD_in(T0,µD)), with T=T0. 187

4. Select an effective period Teff and determine the corresponding T0 from Eqn. 11. 188 5. Evaluate SD_el(Teff) and SD_in(T0,µD) from the displacement spectra previously obtained. 189 6. Determine the hysteretic damping directly from Eqn. 6, where SD_el(xeq) and SD_el(5%) are 190

substituted with SD_in(T0,µD) and SD_el(Teff) respectively. 191 ( ) ( )

2 2

_ _

_ 0 _ 0

, , 0.05

( ) ( )0.10 0.05 0.05 0.10 1

( ) ( )

hyst eff eq eff

D el eff D el eff

D in D in

T T

S T S TS T S T

x µ x µD D= - =

æ ö é ùæ ö æ öç ÷ ê ú= × - - = × -ç ÷ ç ÷ç ÷ ç ÷ç ÷ ê úè ø è øè ø ë û

(12) 192

7. The equivalent viscous damping equation parameters are obtained by means of a least 193 square regression, based on the average value (or a selected percentile) of the ground motion 194 inelastic spectra. 195

8. Repeat for different Teff. 196 9. Repeat for different µD. 197

SelectµD

START

Ground motionsselection

Elastic Spectral Displacement SD_el(T)Ineastic Spectral Displacement SD_in(T,µD)

SelectTeff

T0 = Teff 1+ r (µD-1)µD

x hy (Teff, µD) = 0.1 SD_in(T0)

2

SD_el(Teff ) -1

END

TeffT0

SD

(Eqn. 11)

SD_el(Teff )

T

SD_elSD_in

SD_in(T0)

198

Figure 2 – Flow chart for the hysteretic damping calibration 199

The proposed procedure is applied to the unbonded grouted sleeves connection (Figure 3) reported 200 in Belleri and Riva [11]. The Takeda parameters governing the hysteretic response of the reported 201 experimental test are r = 0.005, a = 0.35 and b = 0. Given the hysteretic response parameters, the 202 proposed calibration procedure has been applied. The obtained Eqn. 4 parameters are a = 2.356, 203 b = 0.027, c = 0.634 and d = 0.703, suitable for effective periods in the range (0.5s-4s) and 204 displacement ductility in the range (1-4). A conservative equivalent viscous damping estimation, 205 independent from the effective period, has been also derived: 206

0.25

10.05 0.39 1eq el hyx x x µD

æ ö= + = + -ç ÷

è ø (13) 207

a)

Section A-AA A

grout

unbonded

rebar

sleeve

region

b)

-100

-50

0

50

100

-5 -2.5 0 2.5 5

Hor

izon

tal l

oad

(kN

)

Drift (%)

ExperimentFE results

P-D

208

Figure 3 – Unbonded grouted sleeve solution: 209 a) typical details; b) experiment versus FE comparison. 210

4. Beam-to-column connections 211

Considering EN 1998–1 [24], precast connections are classified based on their position compared to 212 the energy dissipation regions of the structure: (i) connections outside critical regions; (ii) 213 connections inside critical regions but overdesigned to remain elastic and (iii) connections inside 214 critical regions with adequate ductility and dissipation capacity. For precast frames with beam-to-215 column hinged connections, the energy dissipation is provided by the development of a plastic 216 hinge at the base of the column; in this case the beam-to-column connections are identified as type 217 (i) and designed as pinned connections. 218 The effect of beam-to-column connections other than pinned is considered herein, distinguishing 219 between single-storey and multi-storey frames. 220

4.1 Single-storey frames 221

In the case of single-storey frames, a SDOF substitute structure with appropriate effective mass and 222 height is defined. A representative scheme is depicted in Figure 4 for lateral and central columns. 223 Considering beam-to-column and column-to-foundation connections with a bilinear hysteretic 224 model (elasto-plastic, Takeda or others), the first inelastic mode shape is assumed as a rigid base 225 rotation of the structure after yielding of such connections. In fact, after this condition, a mechanism 226 develops. The considered structure is reduced by static condensation to a SDOF system. 227

228

L/2 L

k kk

M

HEIc

EIb

EIc

EIb EIb

Heff

IP

A. B.

229 Figure 4 – Beam-to-column representative scheme. 230

Note: EIb and EIc are the beam and column flexural stiffness, respectively; k connection stiffness; 231 L and H are the beam and column length, respectively; 232 Heff is the effective height and IP is the inflection point. 233

Assuming that both the beam-to-column and column-to-foundation connections have yielded, the 234 roof yield displacement is obtained from double integration of the column curvature according to 235 the moment distribution in Figure 4, considering a bending moment at the column tip equal to Mycon 236 and 2Mycon for configurations A and B, respectively: 237

2

2 3

conyroof col

y A yc

M HEI

fæ ö

D = -ç ÷ç ÷è ø

; 2

3

conyroof col

y B yc

M HEI

fæ ö

D = -ç ÷ç ÷è ø

(14) (15) 238

where fycol is the column curvature at yield, Mycon is the yield moment of the beam-to-column 239 connection, EIc is the column flexural stiffness. 240 The resulting substitute structure is characterized by an effective mass (meff) equal to the whole roof 241 mass, being the original system reduced to a SDOF by static condensation, and an effective height 242 (Heff) corresponding to the column inflection point (IP in Figure 4). The target displacement is 243 associated to a target inter-storey drift (b), as typically governing the considered precast concrete 244 structures. Referring to configuration B (Figure 4), the plastic roof displacement, defined as the 245 roof displacement associated to the rotation of the plastic hinge at the column base, is obtained by 246 subtracting the roof yield displacement (Eqn. 15) to the target displacement: 247

( )2

target 213roof roof roof colplast y y

HHb f aD = D -D = - - (16) 248

where a2 is the ratio between the yield moment of the beam-to-column (Mycon) and column-to-249 foundation (Mycol) connection. For sake of clarity, it is worth noting that a2 refers to the value of a 250 single beam-to-column connection of configuration B (Mycon) and not to the bending moment at the 251 top of the column (2Mycon). Therefore, the substitute structure target displacement, evaluated at a 252 height equal to the column inflection point, is: 253

( )( )

22 2

222

2 13 1 21 2

colyIP IP IP

d y plast

H Hf a a baa

-D = D +D = +

++ (17) 254

The associated displacement ductility is: 255

( ) ( )22 23 1 2

2 1IPdIP coly y H

b aµ a a

fD+D

= = - +D

(18) 256

The derivation of Eqn. 17 and 18 is reported in Appendix A. Based on these data, it is possible to 257 apply the DBD procedure shown before. 258

As already mentioned, the previous formulas have been derived for central columns, 259 configuration B in Figure 4. In the case of portal frames or perimeter columns, configuration A in 260 Figure 4, a2 needs to be substituted by 0.5a2 in the previous equations. For multiple bays it is 261 herein considered the weighted value: 262

2, 20.5 per col int col

weightedper col int col

n nn n

a a× +

=+

(19) 263

where nper col and nint col is the number of perimeter and interior columns, respectively. 264 The resulting DBD procedure needs iterations, being a2 unknown at the beginning. In order to get a 265 first estimation of a2, it is suggested to apply the DBD procedure neglecting the beam-to-column 266 connection contribution, i.e. Mycon = 0 and a2 = 0, then evaluate a2 at the end of the DBD 267 procedure and iterate. It is worth noting that the presence of beam-to-column connections is 268 associated to a shear load (Vi) at each beam end which modifies the axial load in the columns; this 269 contributes to resist the total overturning moment as highlighted in Figure 5c for multi-storey 270 frames. Therefore, the total overturning moment due to the lateral seismic loads is counteracted by 271 the bending moment developed at the columns base (Mbi) and the overturning moment associated to 272 such axial load (T·Ltot, in the case of equal connections and equal spans). In order to estimate the 273 design moment at the column bases, the contribution of the overturning moment T·Ltot is to be 274 detracted from the total overturning moment obtained following the DBD approach. 275

4.2 Multi-storey frames 276

In the case of multi-storey frames, three situations are identified based on the beam-to-column 277 connections (Figure 5). Figure 5a and 5b show similar hinged connections differing from each 278 other by the gap at the beam-to-column interface, which results in different connection rotation 279 capacity before the contact between the column and the beam. Figure 5a represents a connection 280 able to ensure the rotation compatibility between the connected elements, being the rotation demand 281 concentrated at the joint region due to its lower stiffness compared to the connected precast 282 concrete elements. The static scheme is therefore a hinged-frame. Figure 5b shows a connection 283 with different behaviours in the clockwise and counter-clockwise rotations: in the former, the 284 connection is actually a hinge; in the latter, the free rotation is available until closure of the gap 285 between the two structural elements, then the sub-assemblage gains rotational stiffness. The 286 resulting static scheme depends on the direction of the lateral loads and it is represented by a hinge 287 connection at one beam end and by a degree of fixity at the other end. Figure 5c considers a 288 connection specifically designed to provide a rotational degree of fixity and to dissipate seismic 289 energy; this type of connection involves mechanical devices, such as buckling inhibited bars [34], 290 and it is compatible with precast pre-stressed elements being dry installed after the floor erection. 291 The resulting static scheme is represented by a degree of fixity at both beam ends. As mentioned 292 before for single-storey frames, a shear load (Vi) develops at each beam end (Figure 5b,c) as a 293 consequence of beam-to-column connections which modifies the axial load in the columns. In the 294 case of equal connections and equal spans, as represented in Figure 5 and considered herein, only 295 the axial load of the lateral columns is affected by Vi; in fact the sum of Vi at each side of the inner 296 columns is zero. This contributes to resist the total overturning moment by an amount equal to 297 T·Ltot. Therefore, in order to estimate the design moment at the column base (Mbi), the contribution 298

of the overturning moment T·Ltot is to be detracted from the total overturning moment obtained 299 following the DBD approach. 300

Buckling inhibiteddevice

Hingedconnection

Precastbeam

Precastcolumn

gap gapa) b) c)

k

k k

k k

k k

kk

k k

k

Vi

ViT = S

L

Mbi

Fi

tot

Vi

ViT = S

L

Mbi

Fi

tot

301 Figure 5 – Beam-to-column connections and resulting static schemes 302

An important aspect of the DBD procedure is the definition of the inelastic deflected shape. In the 303 case of hinged-frames, the available formula [26] for shear walls could be applied, considering the 304 column yield curvature expression proposed herein (Eqn. 10): 305

2 2

, 11 12 3 2 3yi i i i

y i yn s n

H H H HH d H

ef a

æ ö æ öD = - = × -ç ÷ ç ÷

è ø è ø (20) 306

Where Hi and Hn are the height of the ith and roof level respectively. The formula is obtained from a 307 triangular distribution of bending moment along the column height. Considering instead a triangular 308 distribution of lateral forces at each floor and applying the fundamental properties of series, the 309 yield displacement at the ith floor is (derivation reported in Appendix B): 310

( ) ( )5 3 2

2 2 2, 2 3 3 3 1 2 3 12 3 1 20 6 2

yhinged i i iy i

n n

H H Hn n n n nn n H Hf é ù

D = - + + + + +ê ú+ + × ×ë û (21) 311

It is worth noting that Eqn. 21 becomes the formula presented in [35] for n, total number of floors, 312 tending to infinite and it is valid also for shear walls. 313 In the case of partially fixed beam-to-column connections, the yield deflected shape becomes 314 (derivation reported in Appendix C): 315

( )3

23, , 3

112 3 2 6

connection hinged i i ny i y i y i

n

n H H Hn n HH n

aa fé ù× ×æ öD = D + × - - + + -ê úç ÷× ×è øë û

(22) 316

where a3 is equal to a2 for the static scheme of Figure 5b and the lateral columns of Figure 5c, and 317 equal to twice a2 for the interior columns of Figure 5c, being a2 the ratio between the yield 318 moment of beam-to-column and column-to-foundation connection as mentioned before. A weighted 319 value of a3 could be considered for the static scheme depicted in Figure 5c similarly to Eqn. 19. 320 Eqn. 22 is also suitable for coupled shear walls. The rotations at the connection level are obtained 321 deriving Eqn 20-22 with respect to Hi (Appendix C). Post yield displacements are obtained from 322 Eqn 20-22 by adding the displacements associated to the plastic hinge rotation at the column base. 323

4.3 Equivalent viscous damping 324

For both cases, single-storey and multi-storey frames, it is possible to evaluate the substitute 325 structure equivalent viscous damping [26] by a weighted average of the hysteretic damping 326 associated to the columns and connections, in which the weights are the respective dissipated 327 energies (Ediss col, Ediss con): 328

hy col diss col hy con diss con

eq eldiss col diss con

E E

E E

x xx x

× + ×= +

+ (23) 329

To evaluate the dissipated energy at the column base, according to Takeda hysteresis, it is possible 330 to consider the following approximated formulas, valid respectively for Takeda parameters a = 0.3 331 b = 0.6 r = 0.05, as in [29], and a = 0.35 b = 0 r = 0.005, as for the grouted sleeve solution 332 mentioned before: 333 0.76 0.1 6diss col el colE Eµ µ

-D Dé ù= - - ×ë û ;

0.654 1diss col el colE Eµ-

Dé ù= - ×ë û (24, 25) 334

Eel col is the half product of the maximum column bending moment times the maximum base 335 rotation. 336 The dissipated energy at the beam-to-column connection depends on the actual hysteresis, which 337 varies based on the inelastic mechanism. In the case of unknown hysteresis, it is suggested to 338 neglect the connection contribution in the substitute structure equivalent viscous damping. 339

5. DBD procedure application to selected case studies 340

Two case studies are considered representing single-storey and multi-storey precast buildings. A 341 scheme of the finite element models is shown in Figure 6. In both cases, the concrete 28-day 342 cylindrical strength and the steel reinforcement yield stress are assumed equal to 40 MPa and 343 450 MPa, respectively. Non-linear time history (NLTH) analyses are conducted [36] considering a 344 set of seven ground motions2 selected and scaled from the European strong motion database [37] in 345 order to be spectrum compatible with EN 1998–1 [24] type 1 spectrum, soil type C, and peak 346 ground acceleration on rock equal to 0.30 g. 347

2 Record code [37] and scale factor in brackets: 000333xa (1.75), 000333ya (1.68), 001726xa (1.83), 001726ya (1.49), 000133xa (3.70), 000335ya (3.36), 000348ya (12.93)

ColumnInverted T

beam

Beam-to-column connection

Lumped mass

Column plastic hinge

ColumnInverted T

beam

b) Multi-storey frame structure

a) Single-storey frame structure

348 Figure 6 – Finite element scheme: a) single-storey and b) multi-storey frame structure 349

Considering the single-storey case study, a precast concrete building with plan dimensions 350 87.5x76.3 m is selected. The columns, 7.65 m high, are connected to the foundation through 351 grouted sleeves and placed at the corners of a 17.5 x 10.9 m net. The columns support inverted T 352 and L pre-stressed beams in the short direction, supporting double-T pre-stressed roof elements 353 spanning in the other direction. Figure 7a and Figure 7b represent the double-T roof-to-beam and 354 the beam-to-column connections, respectively: the former is constituted by arch-shape ductile 355 connections, reported in Belleri et al. [15], placed at each double-T stem, while the latter is 356 constituted by two grouted sleeves with 28 mm diameter bolts, 640 MPa yield stress and 800 MPa 357 ultimate stress, anchored in the column top. The arch-shape device increases the rotational stiffness 358 of the beam-to-beam connection and provides energy dissipation. Being the roof elements designed 359 as pin-supported structures for gravity loads, their geometry is known from the gravity loads design, 360 leading to known values of the bending moment capacity of the connections, herein taken as 361 210 kNm for the beam-to-column connection and 210 kNm for the sum of the bending moment 362 capacities of the double-T to beam connections present in the column tributary area. It is worth 363 noting that the bending moment capacity associated to each arch-shape device (Figure 7a) has been 364 evaluated as the product between the axial capacity of the arch shape device and the distance 365 between such device and the centre of the topping. The seismic mass corresponding to a single 366 column tributary area is 86’700 kg. 367

a) b)Topping

Arch shape device

Double-T element

Beam

368 Figure 7 – a) double-T to beam connection; b) beam-to-column connection 369

The DBD procedure is applied to the selected case study; a target inter-storey drift of 2.5% is 370 chosen for demonstration purpose representing damage control [38, 39]. The inter-storey drift is 371 defined as the ratio between relative storey displacement and inter-storey height. For comparison 372 sake the DBD procedure is applied to the same case study with pinned beam-to-column 373 connections. Table 3 reports the results of the DBD procedure and the NLTH analyses; the latter 374 expressed in terms of mean, maximum and standard deviation of the set of values constituted by the 375 maximum drift obtained for each ground motion. The finite element model scheme is shown in 376 Figure 6a; the properties of elements and connections are reported in Appendix D. The results in 377 Table 3 show a general good agreement between the target and the obtained drift values in terms of 378 mean values. In the case of not-emulative beam-to-column connections, the effective height is 379 lower and the displacement ductility is higher; in addition, more conservative results are obtained. 380 This is related to the computation of the equivalent viscous damping based on global ductility; in 381 fact at a local ductility level the connections experience a higher ductility demand and therefore a 382 higher contribution in viscous damping. It is worth noting that in the hinged-frame case, which 383 directly resembles a SDOF system, the DBD target is well reflected by the mean results of NLTH 384 analyses. This is a direct consequence of the calibration of the equivalent viscous damping (step 7 385 of the procedure presented in chapter 3), based on mean values in the present paper; a calibration of 386 the equivalent viscous damping based on a lower percentile would lead to more conservative 387 results. The choice of such percentile is a topic of ongoing research. 388

389

Table 3 – DBD and NLTH analyses results for the single-storey case study: 390 hinged and not-emulative frame 391

DBD – hinged frame Iteration Heff (m) µD Vbase (kN) Mcol (kNm) a2 Teff (s)

1 7.65 2.12 173 1321 0 1.99 2 7.65 2.12 173 1318 0 1.95

Column 70x70 cm – 16 26mm diam. rebars – Mu = 1334 kNm – T = 0.97s

NLTH analyses results (in terms of drift) DBD target NLTH mean NLTH max NLTH std

2.50 % 2.44 % 4.38% 1.15 %

DBD – not-emulative frame (Mycon = 210 kNm) Iteration Heff (m) µD Vbase (kN) Mcol (kNm) a2 Teff (s)

1 5.60 2.54 224 939 0.197 1.43 2 5.49 2.59 222 894 0.207 1.43 3 5.41 2.62 222 885 0.209 1.41 4 5.40 2.63 222 882 0.210 1.41

Column 65x65 cm – 16 22mm diam. rebars – Mu = 952 kNm – T = 1.03s

NLTH results (in terms of drift) DBD target NLTH mean NLTH max NLTH std

2.50 % 2.10 % 3.76% 0.92 % 392 Considering the multi-storey case, a 3-storey precast concrete building with plan dimensions 393 24x24 m is selected. The columns are continuous along the building height and connected to the 394 foundation through grouted sleeves. The inter-storey height is 4 m and the bay length is 6 m in both 395 directions. The floors are constituted by inverted T and L pre-stressed beams and double-T pre-396 stressed elements. The three static schemes of Figure 5 are considered. For demonstration purposes, 397 only the plane constituted by the inverted T and L beams is analysed. The seismic mass is 398 400 kg/m2 and 800 kg/m2 for intermediate and roof level respectively. The DBD procedure is 399 applied to the selected case study considering a design drift of 2.5%. The procedure results are 400 reported in Table 4 as a function of the considered static scheme. 401

402

Table 4 – DBD results: multi-storey case study. 403 Note: OTM stands for overturning moment 404

Iteration Heff (m) µD Vbase (kN) Mcol (kNm) a3 Teff (s)

Case A (Figure 5a) 1 9.13 1.34 172 1578 0 1.31 2 9.13 1.32 175 1598 0 1.30 3 9.13 1.32 175 1598 0 1.30

Column 80x80 cm – 16 26mm diam. rebars – Mu = 1600 kNm – T = 0.55s

Case B (Figure 5b – Mycon = 125 kNm) 1 9.13 1.24 184 1442 0.065 1.26 2 9.09 1.35 168 1284 0.073 1.34 3 9.09 1.36 166 1266 0.074 1.35 4 9.09 1.37 166 1264 0.074 1.35

Column 75x75 cm – 16 24mm diam. rebars – Mu = 1296 kNm – T = 0.62s OTM taken by change of axial load in the column (Figure 5) is 16% of total OTM

Case C (Figure 5c – Mycon = 125 kNm) 1 9.13 0.93 243 1747 0.107 1.10 2 9.07 1.06 202 1363 0.137 1.24 3 9.04 1.12 189 1240 0.151 1.29 4 9.03 1.15 183 1188 0.159 1.31 5 9.03 1.16 181 1162 0.161 1.33 6 9.02 1.17 179 1149 0.163 1.33 7 9.02 1.18 179 1142 0.164 1.34 8 9.02 1.18 178 1137 0.165 1.34 9 9.02 1.18 178 1135 0.165 1.34

Column 60x60 cm – 16 26mm diam. rebars – Mu = 1146 kNm – T = 0.67s OTM taken by change of axial load in the column (Figure 5) is 29% of total OTM

405 Figure 6b shows the finite element model scheme, whose properties are reported in Appendix D. 406 Table 5 reports the results of the NLTH analyses for all the considered static schemes, expressed in 407 terms of mean, maximum and standard deviation of the set of values constituted by the maximum 408 inter-storey drift obtained for each ground motion. Figure 8 provides a graphic representation of the 409 results in terms of mean drift and mean deflected shape compared to the DBD predictions. A 410 general good agreement between the target and the obtained values is observed, particularly for 411 Case A and Case B. Case C, which is characterized by the lowest scatter of the results, presents a 412 slight underestimation both in terms of drift and deflected shape. This is related to the computation 413 of the equivalent viscous damping based on global ductility, as in the case of single-storey frames. 414 Beside this, the presented conservative formulation is herein suggested for the considered structural 415 typology. Further research is required to highlight the influence of various connections 416 configuration in the equivalent viscous damping formulation. 417

418

Table 5 – NLTH analyses results (inter-storey drift): multi-storey case study 419

Static scheme

Floor DBD result

(%) NLTH mean

(%) NLTH max

(%) NLTH std

(%)

Case A (Figure 5a)

1 0.87 1.24 2.65 0.71 2 2.04 2.08 3.54 0.76 3 2.50 2.54 3.92 0.77

Case B (Figure 5b)

1 0.94 1.14 2.37 0.64 2 2.12 2.10 3.32 0.77 3 2.50 2.53 3.70 0.75

Case C (Figure 5c)

1 1.05 0.89 1.23 0.18 2 2.26 1.82 2.60 0.41 3 2.50 2.17 3.13 0.45

420

0.00

0.33

0.67

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0

H/H

tot

drift (%)

DBDNLTH

Case A

0.00

0.33

0.67

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0

H/H

tot

drift (%)

DBDNLTH

Case B

0.00

0.33

0.67

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0

H/H

tot

drift (%)

DBDNLTH

Case C

421

0.00

0.33

0.67

1.00

0.00 0.05 0.10 0.15 0.20 0.25

H/H

tot

deflected shape (m)

DBDNLTH

Case A

0.00

0.33

0.67

1.00

0.00 0.05 0.10 0.15 0.20 0.25

H/H

tot

deflected shape (m)

DBDNLTH

Case B

0.00

0.33

0.67

1.00

0.00 0.05 0.10 0.15 0.20 0.25

H/H

tot

deflected shape (m)

DBDNLTH

Case C

422 Figure 8 – Comparison between DBD predictions and NLTH analyses results in terms of mean values of 423

drift (top row) and deflected shape (bottom row) 424

425 To estimate the safety of the designed buildings, pushover analyses have been conducted in 426 accordance to EN 1998-1 [24]. The results are expressed in terms of the ratio between the collapse 427 and design peak ground acceleration (PGA); such ratios are 1.43, 1.39, 1.47, 1.63, 1.71 for the 428 single-storey hinged frame, not-emulative frame and multi-storey Case A, B and C, respectively. 429 The pushover analysis considered the failure of the plastic hinge at the column base. It is observed 430 that beam-to-column connections contribute to increase the PGA associated to structural failure in 431 the multi-storey case, provided that such connections have been detailed to accommodate the 432 required rotation demand. The opposite happens in the single-storey case. In general, the difference 433 in the PGA ratios could partially be related to the displacement ductility demand: the lower the 434

displacement ductility demand the higher the collapse PGA. Finally, it is worth noting that the same 435 DBD approach could be used also to design the structure at the collapse prevention limit state. In 436 the case of emulative connections, the general procedure developed by Priestley et al. [26] for 437 reinforced concrete frames could be adopted. 438

Conclusions 439

The Displacement Based Design (DBD) procedure was herein adapted for the application to precast 440 concrete frames typical of the European practice. New expressions for the DBD were developed 441 considering peculiar aspects of precast structures as column-to-foundation and beam-to-column 442 connections for both single-storey and multi-storey buildings. In particular, regarding column-to-443 foundation connections, a new formula was proposed for the yield curvature estimation; such 444 formulation is able to capture for instance the difference between the cross-section depth and the 445 effective depth typical of some precast connections, as in the case of grouted sleeves. A novel 446 algorithm was developed to calibrate the hysteretic damping expression associated to different types 447 of connections, allowing for a faster solution by means of inelastic spectra. 448 Regarding beam-to-column connections, the effects of a degree of fixity was eventually considered 449 for both single-storey and multi-storey structures. New expressions were derived for the target 450 displacement and displacement ductility in single-storey frames. Refined yield displacement 451 formulas were derived for multi-storey frames; such formulas are also suitable for shear wall and 452 coupled shear wall structures. 453 The proposed procedure was validated by means of non-linear time history analyses considering 454 single-storey and multi-storey buildings with hinged or not-emulative connections. A general good 455 agreement between the DBD target values and the obtained results was observed. The highest 456 scatter of the results was associated to hinged frames. Not-emulative connections generally 457 provided more conservative results both in terms of drift and deflected shape. This is related to the 458 computation of the equivalent viscous damping based on global ductility; in fact, at a local ductility 459 level the connections experience a higher ductility demand and therefore a higher contribution in 460 viscous damping. Beside this, the presented conservative formulation is herein suggested for the 461 considered structural typology. Further research is required to highlight the influence of various 462 connections configuration in the equivalent viscous damping formulation. 463 It is worth noting that eventually the results are affected by the choice made in the definition of the 464 percentile used in the equivalent viscous damping calibration. In the present paper the calibration 465 was based on average results; a calibration based on a lower percentile would lead to more 466 conservative results. The choice of such percentile is a topic of ongoing research. 467

Acknowledgements 468

The author gratefully acknowledges the financial support of the Italian RELUIS project and the 469 careful review and constructive suggestions by the anonymous reviewers. 470

References 471

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559

APPENDIX A: Derivation of Eqn. 17 and 18 560

Considering the moment distribution of Figure 4 with a bending moment 2Mycon and Mycol at the 561 column tip and base respectively, Heff is: 562

22 1 2

coleff ycol con

y y

H HH MM M a

= =+ +

(A.1) 563

where a2 = Mycon / Mycol. 564 The yield displacement at the inflection point is: 565

( )

2 2

22

13 3 1 2

col coly eff yIP

y

H Hf f

aD = =

+ (A.2) 566

The displacement at the inflection point associated to the rotation of the plastic hinge at the column 567 base is: 568

( )2

22 2

1 11 2 1 2 3

roofeff plastIP roof col

plast plast y

H HHH

b f aa a

D é ùD = D = = - -ê ú+ + ë û

(A.3) 569

Therefore Eqn. 17 is obtained: 570

( )( )

( )( )

2 2

2222

22 2

222

1 1 13 1 2 31 2

2 1

3 1 21 2

colyIP IP IP col

d y plast y

coly

H HH

H H

fb f a

aa

f a a baa

é ùD = D +D = + - - =ê ú++ ë û

-= +

++

(A.4) 571

Eqn. 18 is obtained directly as: 572

( )( )

( )

( ) ( )

22 2

222 2

2 22

22

2 13 1 21 2 3 1 2

2 11

3 1 2

coly

IPd

colIP colyy y

H H

H H

f a a baa b a

µ a af f

a

D

-+

++ +D= = = - +D

+

(A.5) 573

APPENDIX B: Derivation of Eqn. 21 574

Considering the triangular distribution of lateral forces according to Figure B.1a, the bending 575 moment associated to the ith floor is: 576

( )( )11

1n p

i n h ih

M F n h H H-

- +=

= - + -å (B.1) 577

Substituting iH i H= ×D and nH n H= ×D where DH is the inter-storey height: 578

( )( )

( )

1

2 2

1

1 1

2 1 2 2

n p

ih

n p

h

M F H n h n h i

F H n n n i i h h n i

-

=

-

=

= ×D - + - + - =

é ù= ×D + - × + - + + - - +ë û

å

å (B.2) 579

From the fundamental properties of series 580

( ) ( )3 2

2

1 1

( 1) 2 3; 2 6

m m

z z

m m m m mz z= =

+ + += =å å (B.3; B.4) 581

the following expression of Mi is obtained: 582

( )( ) ( ) ( ) ( )

( ) ( )( )

3 22

3 2 2

2 32 1

61

2 22

(3 3 1) (2 3 1)6

i

n i n i n in i n n n i i

M F Hn i n i

n i

F H i i n n n n n

é ù- + - + -- + - × + - + +ê ú

ê ú= ×D =ê ú- - ++ - + -ê úë û

×D é ù= - + + + + +ë û

(B.5) 583

584

Hi

DH

n floorth

i floorth

F

2F

i F

n Fn floorth

i floorth

Mc

Mc

Mc

Mc n floorth

i floorth

McF

2F

i F

n F

Mc

Mc

Mca) b) c)D

585 Figure B.1 – Static scheme considered for lateral deflection evaluation. 586

Substituting /ii H H= D , /nH H nD = and nn H H×D = : 587

3

2 2 22 (3 3 1) (2 3 1)6i

i i nn

HFM n H n n H n nHé ù

= - + + + + +ê úë û

(B.6) 588

Considering the base moment (Hi=0): 589

2(2 3 1)6b nFM H n n= + + (B.7) 590

From which F is obtained: 591

2

6(2 3 1)

b

n

MFH n n

=+ +

(B.8) 592

Substituting back in Eqn. A2.6: 593

3

2 2 22 (3 3 1) (2 3 1)2 3 1

b i ii

n n

M H HM n n n n nn n H H

é ùæ öê ú= - + + + + +ç ÷+ + ê úè øë û

(B.9) 594

Considering the curvature along the column height, /i iM EIf = , the column rotation ( iq ) and lateral 595 displacement ( iD ) at the i

th floor are respectively: 596

( )

4 22 2 2

32

1 1 (3 3 1) (2 3 1)4 22 3 1

b i ii i

n n

M H Hn n n H n n AH Hn n EI

qé ù

= - + + + + + +ê ú+ + ë û (B.10) 597

( )

5 3 22 2 2

32

1 1 (3 3 1) (2 3 1)20 6 22 3 1

b i i ii i

n n

M H H Hn n n n n AH BH Hn n EI

é ùD = - + + + + + + +ê ú+ + ë û

(B.11) 598

Where A and B (integration’s constants) are both equal to 0 being 0iq = and 0iD = at the base (i.e. 599 Hi=0). Eqn. 21 is obtained considering yielding at the column base (i.e. /b yM EI f= ): 600

( ) ( )5 3 2

2 2 2, 2 3 3 3 1 2 3 12 3 1 20 6 2

yhinged i i iy i

n n

H H Hn n n n nn n H Hf é ù

D = - + + + + +ê ú+ + × ×ë û (B.12) 601

APPENDIX C: Derivation of Eqn. 22 602

To derive Eqn. 22 it is first necessary to consider the deflected shape associated to yielding of all 603 column-to-beam connections (Figure B.1b). The bending moment distribution on the column is 604 stepped, with a value at the ith floor equal to: 605 ( 1)i cM n i M= - + (C.1) 606 The deflected shape is obtained by double integration of the column curvature: 607

( ) ( )

( ) ( )

( )

( )

11

1

11

1

1

2

1

2 32

2

( 1)2

( 1)2

( 1) 2 2 12

1 12 3 2 6

ih h h

i h h i hh

ic h h

h h i hh

ic

h

ic

h

c

M H HH H H HEI

M H Hn h H H H HEIM Hn h H H i hEI

M H n h i hEI

M H i i n iEI

--

=

--

=

=

=

-é ùD = - - + - =ê úë û-é ù= - - + - + - =ê úë û

Dé ù= - - + D +D - =ê úë ûD

= - - + - + =

é ùD æ ö= - - + + -ç ÷ê úè øë û

å

å

å

å

(C.2) 608

Substituting /ii H H= D and /nH H nD = : 609

21 1 1

2 3 2 6c i n

i i in

M H HH n H nEI H n

é ùæ öD = - - + + -ê úç ÷è øë û

(C.3) 610

The rotation at the ith floor is: 611

2 1 12

2 2 6c i n

i in

M H Hn H nEI H n

qé ùæ ö= - - + + -ê úç ÷

è øë û (C.4) 612

Considering the static scheme depicted in Figure B.1c, the lateral force required to obtain the same 613 base moment as in Figure B.1a is: 614

2

1

c ii i n

jj

M HF F nH

=

= +

å (C.5) 615

Following the same procedure adopted to derive Eqn. 21, the lateral displacement associated to such 616 lateral force distribution is: 617

( ) ( )

5 3 22 2 2

, 2 3

,

1 3 3 1 2 3 12 3 1 20 6 2

1

yc i i iy i

b n n

hingedcy i

b

M H H Hn n n n n nM n n H H

MnM

fæ ö é ùD = + - + + + + + =ç ÷ ê ú+ + × ×è ø ë û

æ ö= + Dç ÷è ø

(C.6) 618

Finally, the total lateral displacement is obtained adding Eqn. A3.3 to Eqn. A3.6: 619

3

2, ,

1 112 3 2 6

connection hinged c c b i i ny i y i i

b b n

M M M n H H Hn n HM M EI H n

æ ö é ù× ×æ öD = D + × - - + + -ç ÷ ê úç ÷× ×è øè ø ë û (C.7) 620

The corresponding rotation is 621

2

, ,1 1 11 22 2 6

connection hinged c c b i ny i y i i

b b n

M M M H Hn n H nM M EI H n

q qæ ö é ùæ ö= + × - × × - + + -ç ÷ ê úç ÷

è øè ø ë û (C.8) 622

APPENDIX D: Properties of finite element models 623

Considering the finite element models, the beam-to-column connection has been modelled with the 624 same Takeda hysteresis used for the column-to-foundation connection (r = 0.005, a = 0.35 and 625 b = 0 – Figure 2) for both single-storey and multi-storey structures, owing the demonstrative 626 purpose of the present study. Table D.1 contains the parameters used in the finite element models. 627

Table D.1 – finite element model properties 628

Common data

Concrete Strength fc = 40 MPa; elastic modulus Ec = 35’000 MPa

Steel Strength fy = 450 MPa; elastic modulus Es = 210’000 MPa

Beam Inverted-T, Area A = 0.595 m2, Second moment of area I = 0.04674m4

Beam-to-column connection

stiffness kcon = 200’000 kNm/rad

Takeda hysteresis r = 0.005, a = 0.35 and b = 0

Specific data

Single-storey Hinged frame

Single-storey Connections

Multi-storey Case A

Multi-storey Case B

Multi-storey Case C

Static scheme Figure 6a Figure 6a Figure 5a Figure 5b Figure 5c

Column cross-section

70x70cm 65x65cm 80x80cm 75x75cm 60x60cm

Nodal mass (kg) 86’700 86’700 28’800 (1st - 2nd storey); 14’400 (3rd storey)

Mycol (kNm) 1334 952 1600 1296 1146

Mycon (kNm) 0 210 0 125 125 629