numerical modelling of infilled rc frame s: the …

NUMERICAL MODELLING OF INFILLED RC FRAMES: THE

DETECTION OF COLUMN FAILURE DUE TO LOCAL SHEAR

INTERACTION

De Risi Maria Teresa1(*), Del Gaudio Carlo1, Ricci Paolo1, Verderame Gerardo Mario1

1 Department of Structures for Engineering and Architecture, University of Naples Federico II,

via Claudio 21, 80125, Naples, Italy

{mariateresa.derisi, carlo.delgaudio, paolo.ricci, verderam}@unina.it

(*) beneficiary of an AXA Research Fund Post-Doctoral grant

Abstract

Numerical and experimental studies highlighted that the presence of masonry infills in Rein-

forced Concrete (RC) frames certainly leads to the increase in their lateral strength and stiff-

ness. Nevertheless, post-earthquake observed damage showed that infills can also cause po-

tential brittle failures due to the local interaction with structural elements, thus producing a

limitation of deformation capacity of the surrounding frame. This detrimental effect is partic-

ularly important for existing masonry-infilled RC buildings designed for gravity loads only

without any capacity design requirements.

A proper numerical model both for infills and RC members is necessary to reliably detect the

shear failure on columns due to the local interaction with the masonry infill. In the last years,

several researches dealt with this topic, ranging from very simple models such as the equiva-

lent strut model to more complex models like the double- and triple-strut models or FEM-

approaches. Nevertheless, the choice of the proper modelling strategies and degrading shear

strength model is still a frontier issue for the most recent research works, especially when the

shear failure occurs in the degrading phase of the infilled frame response.

This paper presents a preliminary numerical investigation on column shear failure due to lo-

cal interaction between structural and non-structural elements, starting from the results of

two experimental tests on infilled frames performed by the Authors. Different shear strength

models and different strategies of macro-modelling for infills are applied and discussed, in

order to (i) to match the experimental response in terms of initial stiffness, peak strength and

corresponding displacement, and softening behaviour and (ii) to capture (or not) the column

shear failure exhibited (or not) during the test. The best modelling strategy is finally identified

to provide a support towards future necessary numerical investigations.

Keywords: Masonry infills, Reinforced concrete frames, Local interaction, Column shear

failure, Numerical simulations.

2523

COMPDYN 2019 7th ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis (eds.)

Crete, Greece, 24–26 June 2019

Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2019) 2523-2542

ISSN:2623-3347 © 2019 The Authors. Published by Eccomas Proceedia.Peer-review under responsibility of the organizing committee of COMPDYN 2019. doi: 10.7712/120119.7092.19079

M.T. De Risi, C. Del Gaudio, P. Ricci, G.M. Verderame

1 INTRODUCTION

Numerical and experimental studies highlighted that the presence of masonry infills in Re-

inforced Concrete (RC) frames leads to the increase in their lateral strength and stiffness.

Nevertheless, post-earthquake observed damage (e.g. [1]-[3]) showed that infills can also

cause potential brittle failures due to the local interaction with structural elements, thus pro-

ducing a limitation of deformation capacity of the surrounding frame. This detrimental effect

is particularly important for existing masonry-infilled RC buildings designed for gravity loads

only without any capacity design requirements.

During last decades, a growing attention has been addressed to the behaviour of masonry

infill walls in RC buildings under seismic action, also considering that damage to these ele-

ments largely affects repair costs of buildings and their loss of functionality after earthquakes

[4]. Several experimental studies investigated the seismic behaviour of RC frames with infills.

A significantly smaller number of experimental studies investigated the effects of the interac-

tion between panel and surrounding elements resulting in brittle failure mechanisms such as

shear failure in RC columns (e.g., [5]-[6]), especially for hollow clay bricks, very widespread

in the Mediterranean region.

1.1 Columns-infill shear interaction modelling from codes and literature

Experimental data have been the support for analytical modelling efforts since late 1970s'

(e.g., [7]-[9]). Infills have been generally modelled by means of quite complex FEM micro-

modelling approaches or simpler single- or multi-struts (reacting only in compression) ap-

proaches. Nevertheless, even recently, there is lack of unanimity about the best modelling ap-

proach among the various literature proposals (e.g., [10]-[12]), above all about the column-

infill shear interaction modelling strategy.

More specifically, about the shear failure modelling in non-ductile RC frames, some stand-

ards propose simplified procedures aimed at taking into account the effects of interaction be-

tween panels and surrounding elements. These procedures usually consider a concentrated

load on the column (or beam) equal to the horizontal (or vertical) component of the resultant

of the stresses along the loaded diagonal of the panel. Among codes, some practice-oriented

prescriptions are present in the American code ASCE/SEI 41-06 [13], which suggested to

model the infill as a single eccentric strut in compression as shown in Figure 1a. On the con-

trary, the most recent Italian technical standards ([14]-[15]) do not provide any indication for

the modelling and the assessment of the local interaction phenomena in the case of solid pan-

els adjacent to column/beam elements.

About research studies from literature, the issue of shear failure modelling in non-ductile

RC frames due to local interaction with infill elements has been investigated with different

approaches (Figure 1) during the last years (e.g., [16]-[19], among others). Multiple-strut ap-

proaches are generally suggested in these studies, generally proposing to model the infill by a

minimum of one-strut - eccentrically placed respect to the infill diagonal, like suggested in the

ASCE-SEI/41[13] - to a maximum of two (Figure 1b,d) or three struts (Figure 1c,e,f), basical-

ly different for the position of the loading points on the adjacent columns, the ratio of the total

infill lateral load adsorbed by each strut, and for the struts inclination. The number of struts,

their positions, and their width have to be carefully selected to reliably reproduce the stress

demand they induce and the strength and deformability contribution of the panel to the in-

filled RC frame. Currently, additional efforts seem to be still required to this aim.

2524


(a) (b)

(c) (d)

(e) (f)

Figure 1: Examples of column-infill interaction modelling from codes and literature: adapted from ASCE-

SEI/41 [13] (a); Crisafulli et al. [16] (b,c); Burton and Dierlein [17] (d); Sattar and Liel [18] (e); Jeon et al. [19]

(f)

Moreover, a key issue of the shear local interaction modelling is the adequacy and the ap-

plicability of the usually adopted shear capacity models in capturing the shear-controlled be-

haviour of a RC member adjacent to an infill panel. All the proposals from codes and

literature mentioned above suggest to use the model by Sezen and Moehle [20], also adopted

in the American code ASCE-SEI/41 (2017) [21], as explained in detail in Section 5 and

shown in Figure 2. This model assumes a shear strength degradation due to inelastic ductility

demand, as typical in “free” columns/beams. In the case of very “squat” portions of columns,

as those generated by the limited infill-to-column contact length (as shown in Figure 1), a

shear strength degradation due to an increasing flexural demand appears not totally meaning-

ful. Therefore, the model by Sezen and Moehle [20] should be applied without any strength

degradation (k=1 in Figure 2). Alternatively, a different “family” of shear strength models

should be adopted, degrading with the shear crack opening demand or, from a predictive

standpoint, with the strain demand, instead than with the flexural inelastic ductility demand.

2525


sw ywcn g

c g

A f0.50 f NV = k 1 (0.80A ) d

a / d s0.50 f A

+ +

k =1

k =0.7

V

µ2 6

VRd,max

VRd,min

Figure 2: Schematic representation of the model by Sezen and Moehle [20]

Therefore, the topic certainly deserves a deeper investigation, both about the best infill

modelling strategy and the proper shear strength model to be adopted.

1.2 Aims and objectives

This paper presents a preliminary numerical investigation on column shear failure due to

local interaction between structural and non-structural elements, starting from the results of

some experimental tests on infilled RC frames performed by the Authors. Two conforming

and non-conforming infilled frames were tested, designed according to the current Italian

seismic technical code (SLD), and according to an older Italian technical code in order to be

representative of existing RC buildings constructed between 1970s and 1990s (GLD). Infill

panels are made of hollow clay bricks, common in Mediterranean countries. Due to the differ-

ences in these two specimens, the GLD frame exhibited a column shear failure due to interac-

tion with the infill; conversely, in SLD frame no shear failures were experimentally detected.

Different strategies of macro-modelling for infills and different shear strength models are

applied and discussed, in order (i) to match the experimental response in terms of initial stiff-

ness, peak strength and corresponding displacement, and softening behaviour and (ii) to cap-

ture (or not) the column shear failure exhibited (or not) during the test. The best modelling

strategy is finally identified to provide a support towards further numerical investigations.

2 ANALYSED EXPERIMENTAL DATA

One-storey one-bay half-scaled RC frames shown in Figure 3 were tested and presented in

Verderame et al. [22]. These specimens represent the subject of the modelling strategies pre-

sented in this paper, and their results are briefly recalled in this Section. The GLD specimen

(hereinafter referred to as GI-80) was designed in order to be representative of the bottom sto-

rey of a five-storey gravity load designed RC frame, according to Italian technical codes in

force between 1970s and 1990s ([23]-[24]). The SLD specimen (hereinafter referred to as SI-

80) was designed according to the Italian seismic code [14], in compliance with all the capaci-

ty design requirements.

Mean value of 28-day cylindrical concrete strength was equal to 21.9 MPa. Deformed bars

were used for longitudinal and transverse reinforcement in both the specimens. Commercial

typology of reinforcing steel B450C [14] was used, characterized by a mean yielding strength

equal to 507 MPa, 586 MPa, 490 MPa and 481 MPa, for bar diameters of 6 mm, 8 mm, 10

mm, and 12 mm, respectively. Hollow clay units with cement mortar were used as infill mate-

rial. Dimensions of brick units were 250×250×80 mm3, with 66.3% void ratio, and the catego-

ry of the mortar was M15, with a mean compressive strength of 14.03 MPa. Compression

tests carried out on three-course masonry prisms, perpendicular and parallel to the holes, and

2526


diagonal shear test carried out on a five-course masonry prisms, lead to the mechanical prop-

erties shown in Table 1.

700 200 2100 200 700

800 2300 800500

1350

250

300

500

1475

425

2400

Ø6@150mm

Ø6@50mm

Ø6@150mm

700 200 2100 200 700

800 2300 800

50

01

350

25

03

00

50

01

475

42

5

24

00

Ø6@50mm

Ø6@50mm

Ø6@100mm

Ø6@50mm

Ø6@50mm Ø6@100mm

beam column

(3+2+3) 8

21

25

0

20

8

200

21200

21

21

20

0

15

9

(3+3) 10

beam column

(3+2+3) 12

21

25

0

20

8200

21200

23

23

20

0

15

5

(3+3) 10

(a) (b)

Figure 3: Geometry and reinforcement details of GLD (a) and SLD (b) specimens, all measures are in mm.

3-course masonry wallette

Dimension of the wallette 770×770×80 (mm3)

Compressive strength (// to holes), fw-h 4.88 (MPa)

Compressive strength (⊥ to holes), fw-v 3.19 (MPa)

5-course masonry wallette Dimension of the wallette 1285×1285×80 (mm3)

Shear strength, fv 0.36 (MPa)

Table 1: Mechanical properties of infill materials.

About the test setup, the foundation block of the specimen was anchored to the strong floor

by means of vertical post-tensioned steel rods connected to stiff steel profiles. The lateral load

is applied by means of a hydraulic actuator in displacement control. The actuator is fixed to a

steel reaction wall anchored to the strong floor. The actuator was connected to the mid-span

of the beam through steel profiles connected to steel rod passing through the transverse hole

in the mid-span of the beam. The related loading protocol applied to infilled specimens con-

sisted of 3-push-pull-cycles per each imposed drift levels, the latter assumed to be equal to

0.01%, 0.02%, 0.15%, 0.50%, 0.90%, 1.30%, 1.70%, 2.00%, 2.40%, 3.00%, and 3.6%.

The vertical load on columns is applied by hydraulic jacks in load control. The vertical

load is kept constant during the tests and it corresponds to an axial load ratio equal to 10%.

Further details about setup can be found in Verderame et al. [22].

Linear Variable Displacement Transducers (LVDTs) were used to measure cracks width

and deformations at columns’ and beam’s ends. Wire potentiometers were placed along infill

panel diagonals in infilled specimens GI-80 and SI-80.

2527


2.1 Experimental results

The experimental results related to the infilled specimens, which represent the core of the

following numerical simulations, are reported in brief in this Section. Note that the corre-

sponding “bare” frames (without infills) were also tested by the Authors, and the main related

results can be found in [22].

First, the lateral load-drift cyclic response of the SI-80 specimen is shown in Figure 4, to-

gether with the final damaged state of the specimen.

-200

-150

-100

-50

0

50

100

150

200

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

late

ral lo

ad

[kN

]

drift [%]

expected yielding (bare frame)

expected strength (bare frame)



End of the test

Figure 4: Lateral load-drift response of infilled specimen SI-80 – adapted from [22].

Basically, a ductile, flexure-controlled post-elastic behaviour of the frame, without shear

failures, was observed, thanks to the applied strength hierarchy prescriptions. The formation

of a plastic mechanism involving beam’s ends and columns’ base was observed. A corner

crushing failure of the infill, completed at the peak load of the response, was observed. The

lateral response following the collapse of the infill panel (i.e., for drift higher than ±2.00%)

perfectly matches the response of the corresponding bare specimen tested by the Authors (see

[22]). This is consistent with the development of the same plastic mechanism in the RC frame,

as described above.

Similarly to SI-80, the lateral load-drift cyclic response of the GI-80 infilled frame is

shown in Figure 5, together with the most significant photos of the damaged specimen. For

GI-80 specimen, diagonal cracking developed in the panel since very low drift values (i.e.,

between 0.15% and 0.50%). A drop in lateral force associated to the development of severe

diagonal cracking at the top of the columns (with crack inclination quite close to 45°) was ob-

served for an applied drift range between 0.50% and 1.30%. At 1.70% of drift, an abrupt in-

crease in vertical displacement of the top of the columns also occurred, highlighting the

potential for an imminent axial failure. The observation of the local behaviour showed the ev-

idence of a shear failure due to the local interaction between RC columns and infill panel [22],

as also marked on the cyclic response of the specimen in Figure 5.

3 ADOPTED NUMERICAL MODELLING APPROACH

Some preliminary calculations can been carried out for beams and columns of the analysed

frames depending on their design typology. Table 2 reports the yielding moment and the flex-

ural strength of beams and columns evaluated based on the simplified assumption of constant

axial load, equal to the initial test value (N0), i.e. Ncol,0=P=86.0 kN in columns and Nb,0=0 in

beam. Axial load - bending moment interaction is not be taken into account in such a prelimi-

2528


nary simple calculation. Moments for columns and beam at yielding and peak condition are

referred to as Mc,y, Mb,y and Mc,max, Mb,max, respectively.

-150

-100

-50

0

50

100

150

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

late

ral lo

ad [

kN]

drift [%]

right column shear crack ing

left column shear crack ing

left columnaxial load failure

right column axial load failure



End of the test

Peak load

(left column)

End test

(left column)

End test

(right column)

Figure 5: Lateral load-drift response of infilled specimen GI-80 – adapted from Error! Reference source not

found..

For the following calculations, bending moments are reported at the intersection of the

beam and columns centrelines and referred to as M*c,y, M*b,y and M*c,max, M*b,max, respec-

tively. Such values can be used for a preliminary classification of the failure mode of beams

and columns, in a simplified approach, allowing the calculation of the plastic shear (Vpl) to be

compared with the maximum-degraded shear strength (Vn,min) (evaluated according to ASCE-

SEI/41 [21], with K=0.7 in Figure 2). The plastic shear of the column is equal to Vc,pl

=2Mc,max/Hw, and plastic shear of the beam is calculated as Vb,pl =2Mb,max/Lw, where Hw and

Lw are the infill panel height and length, respectively. Since, both for beams and columns in

SLD and GLD frames, Vpl results lower than Vn.min, the elements can be defined as “ductile

elements”.

The expected yielding (Vy) and the maximum (Vmax) lateral loads for the GLD and SLD

frames are also reported in Table 2 and calculated by means of Eqs. (1) and (2), namely by

assuming the simultaneous attainment of yielding and flexural strength, respectively, in mem-

bers involved in the collapse mechanism.

Starting from these preliminary remarks, first the modelling strategy for the ductile behav-

iour of the frames is described here.

* *

c,y b,y c,y

y

M min(M ;M )V = 2

H

+ (1)

2529


* *c,max b,max c,max

max

M min(M ;M )V = 2

H

+(2)

Frame Element No My Mmax My

* Mmax* Vpl Vn,min Vy Vmax

(kN) (kNm) (kNm) (kNm) (kNm) (kN) (kN) (kN) (kN)

GLD Column 86.0 22.1 25.7 26.2 30.4 38.1 43.3

65.6 78.4 Beam 0.0 24.2 24.9 26.5 27.3 23.7 52.9

SLD Column 86.0 33.4 40.2 38.0 45.7 59.6 90.6

81.0 99.9 Beam 0.0 24.2 24.9 26.5 27.3 23.7 114.1

Table 2: Yielding and maximum lateral load of GLD and SLD specimens.

The infilled frames have been numerically reproduced by means of the numerical model

shown in Figure 6. The considered loading direction is shown with a red arrow in these fig-

ures. However, if the opposite loading direction is considered, all the remarks reported in the

following can be repeated in a specular way. Increasing horizontal displacements are imposed

in the mid-point of the beam, as in the experimental setup, thus faithfully reproducing the var-

iation of axial load acting in beam during the test. The fibre-type ForceBeamColumn element

has been adopted in OpenSees [25] for each beam/column element. Mander et al. [26]’s con-

crete law, considering confining effect, if any, has been used (Concrete04 uniaxial material).

ReinforcingSteel uniaxial material reproduces the experimental constitutive laws of the

adopted reinforcing steel, characterized by the strength values reported in Section 2. Beams

and columns have been considered as ductile members, as classified above. Beam-column

joints are considered as supplied by infinite strength and stiffness.

The modelling approaches adopted for the infill panels, the core of this work, are investi-

gated in detail in the next Section. Two different modelling approaches have been used to re-

produce the response of the infill panel for specimens SI-80 and GI-80, based on the most

common modelling strategies, namely, in a single- or three- (compressive only) strut approach.

Compressive-only struts have been introduced in Figure 6 consisting with the considered

loading direction.

fiber type nonlinear beam-column rigid offset

hc/2 hc/2Lw

L

hb/2

Hw

H

L R

ϴ


hc/2 hc/2Lw

L

hb/2

Hw

H

L R

zc

ϴ

(a) (b)

Figure 6: Numerical modelling approach adopted for infilled frames: single-(a) and three-(b) strut approaches.

2530


4 ADOPTED INFILL MODELLING STRATEGIES

First, the global infill response adopted for numerical simulations in terms of horizontal

load (Vw) -versus-horizontal displacement (D) (or, equivalently, Drift) reproduces the exper-

imental response. The latter has been derived as the difference between the infilled frame and

the corresponding bare one, thus assuming that the RC frame and the infill panel work as a

parallel system. This procedure (see [12]) implies that the RC surrounding frame exhibits the

same base shear-top displacement response in bare and infilled configurations. Such a hy-

pothesis is not totally rigorous, but often suggested and adopted in literature (e.g. [12], [27]-

[28]) to obtain the infill lateral response. This hypothesis will be “validated” by the numeri-

cal-versus-experimental comparisons themselves, as shown in Section 6. Force-drift infill re-

sponse has been finally averaged between positive and negative loading directions for both

the specimens (Figure 7a,b), by assuming that experimental results are approximately sym-

metric. The obtained averaged curve represents the experimental response of the infill panel

adopted for the simulations.

Hw(kN)

Vw(kN)

0 0.5 1 1.5 2 2.5

Drift (%)

0

20

40

60

80

100

120

Fw

(kN)

Pos

Neg

Av erage

multilin

Hw(kN)

Vw(kN)

(a) (b)

Figure 7: Average curve and multi (three)-linearized experimental response for infills in SI-80 (a) and GI-80 (b).

The experimental responses obtained in such a way clearly show a high initial stiffness un-

til first cracking occurrence and a subsequent stiffness degradation up to the peak load. After

the achievement of the maximum lateral load, a degrading branch can be easily recognised.

Therefore, some characteristic points have been recognised, namely: cracking point (Dcr,

Vw,cr); peak load point (Dpeak, Vw,peak); residual load point (Dres, Vw,res). Figure 7 shows the

three-linearization of the experimental response for GI-80 and SI-80, based on the characteris-

tic points mentioned above. The corresponding data are reported in Table 3 together with a

comparison with the predictions by the well-known model proposed by Panagiotakos and

Fardis (“P&F) [29]. Such a model well predicts the cracking point, but slightly underestimates

the peak strength (-15% on average) and significantly the corresponding displacement (-42%

on average) for both tests (see Table 3). Figure 7 finally shows that the infill response is simi-

lar for the corresponding two tests; slight differences, at the peak point and in the softening

branch, can be likely ascribable to the different failure mode exhibited by the infill panels.

Vcr,h Dcr,h Vmax,h Dmax,h αsoft= Ksoft/ Kel

(kN) (%) (kN) (%) (%)

GI-80 (exper.) 59.37 0.03 90.86 0.46 -1.56

SI-80 (exper.) 64.21 0.04 93.47 0.40 -3.43

P&F [29] 60.48 0.03 78.62 0.25 -(0.5÷10)

Table 3: Characteristic points of infill responses for tests GI-80 and SI-80.

2531


As anticipated before, the modelling of infills adopted herein is based on two strategies.

The first one models the infill as a single compressive diagonal strut (as shown in Figure

6a). Such an approach is generally adopted to catch the global behaviour of infilled frames,

being able to reproduce the contribution of the infill panel to the global strength and stiffness

of the infilled frame. In this case, the horizontal response of the infill is simply projected

along the diagonal direction to characterise the axial load-deformation behaviour of the com-

pressive strut. Nevertheless, this model is not able to detect any eventual localised shear de-

mand in the surrounding RC members, thus failing in the detection of eventual shear failures

of the adjacent columns induced by the infill-column shear interaction.

Therefore, the second modelling approach reproduces the infill contribution by means of

three compressive struts (as shown in Figure 6b) based on the proposals by Chrysostomou et

al. [30] and Jeon et al. [19]. In this case, the global (Vw – drift) response obtained before is

distributed among three compressive diagonal struts, as explained in the following. First, the

contact length z between infill panel and column is computed by following the approach pro-

posed by Stafford Smith [31]. In the critical section of column shear failure, assumed to be

located at a distance equal to the column effective depth from the beam-column interface [19],

the contact length z is divided into two portions: one in contact with the lower part of the cen-

tral strut and the other in contact with the lower off-diagonal strut. The portion of the global

lateral load absorbed by the central strut (γc) is determined based on the corresponding area of

the bearing stress distribution with respect to the total area of the same bearing stress distribu-

tion, according to the proposal by Jeon et al. [19]. On the same bases, the end-point of the

lower off-diagonal truss (zc in Figure 6b) is obtained. As a result, the central diagonal truss

absorbs 35% of the global lateral load (γc = 0.35).

In addition, elastic stiffness, secant-to-peak stiffness and lateral loads of the off-diagonal

struts have been reproduced according to the proposal by Chrysostomou et al. [30] depending

on their eccentricity with respect to central diagonal strut, and, more specifically, depending

on the parameter α = (zc /Hw)<1.

First, being K and ΔU the global stiffness of the infill panel and an assigned incremental

displacement in horizontal direction, respectively, the axial stiffness of the central diagonal

strut (kc) and related incremental displacement (Δuc) can be computed as in Eqs. (3) and (4):

c c2

Kk =

cos

(3)

cu = U cos (4)

Axial stiffness of the off-diagonal struts (koff) and related incremental displacement (Δuoff)

can be obtained in the hypothesis of linear deformed shape of the columns, as suggested by

Chrysostomou et al. [30] and shown in Eqs. (5) and (6):

2c c

off 2 2 2 2

K k cos K(1 )k =

2(1 ) cos 2(1 ) cos

− − =

− − (5)

offu = U cos (1 ) − (6)

The same “eccentricity” and lateral load-portion for the two off-diagonal struts has been

also assumed, for the sake of simplicity. As a result, the incremental axial load acting in the

central diagonal (Δpc) and off-diagonal (Δpoff) struts can be calculated, as in Eqs. (7) and (8),

respectively:

2532


( ) wc c c c c c2

VK K Up = k u U cos

cos coscos

= = =

(7)

( )c coff off off 2 2

w c

K (1 ) K (1 )p = k u U cos (1 ) U

2(1 ) cos2(1 ) cos

V (1 ) 1

cos 2 (1 )

− − = − = =

− −

− =

−

(8)

In such a way, the global (horizontal) infill action ΔVw obtained by means of a three-strut

model is equal to the infill lateral response obtained by means of a single concentric strut

model, as expected, namely:

w c c off off

cw c w w c c

V = k u 2(k u ) (1 ) cos

(1 ) V 2 H (1 ) V ( (1 ))

2(1 )

+ − =

− = + − = + −

−

(9)

By assuming that γoff = (1- γc)/2, it results: γc + 2 γoff = 1.

For both the modelling approaches, the struts have been implemented in OpenSees by

means of Truss elements defined with a Hysteretic uniaxial material and reacting only in

compression.

5 COLUMNS SHEAR FAILURE DETECTION

The numerical response described until now does not account for a potential shear failure

of the columns, particularly important for RC frames interested by a local shear interaction

with the infill panel.

Shear failures in columns can be first detected starting from a post-processing of the nu-

merical output, but a proper shear strength model should be selected, suitable for squat col-

umns, as those generated by the column-to-infill interaction (see Figure 6b). Therefore, two

models have been adopted herein as explained in the following.

First, shear strength is evaluated according to ASCE/SEI 41 [21] (see Eq. (10)), as pro-

posed by Sezen and Moehle [20] (“S&M”) and very commonly adopted in previous studies

from literature for similar applications (e.g. [17]-[19], [32]):

sw ywcn g

c g

A f0.50 f NV = k 1 (0.80A ) d

a / d s0.50 f A

+ +

(10)

In Eq. (10), a/d is the shear span-to column effective depth ratio, Ag the gross area of the

column, Asw, fyw, and s are the transverse reinforcement area, yielding strength and stirrups

spacing, respectively. N is the column axial load at the interface section with the joint and

changes for increasing applied horizontal displacement (reaching a minimum value in the left

column and a maximum value in the right column, given the assumed loading direction). Ac-

tually, ASCE/SEI 41 [21] suggests to conservatively assume N=0 in Eq. (10) to assess the

shear capacity in columns subjected to the local shear interaction with the surrounding infills.

2533


Such an assumption can result too much conservative for the numerical-versus-experimental

comparison. Therefore, the actual axial load acting on the column (and varying step-by-step)

is assumed herein in Eq. (10). Factor k takes into account the strength degradation due to ine-

lastic ductility demand. Since the lack of meaning of a strength degradation assessment due to

an increasing flexural demand in such a squat columns, the strength model by ASCE/SEI-41

[21] is applied herein always with k=1. The shear span a in Eq. (10) has been assumed as

equal to the distance between the null point of the moment diagram in the squat portion of

column and the beam-column interface (variable during the analysis), limiting the a-to-d ratio

between 2 and 4, as prescribed by Sezen and Moehle [20] and in ASCE/SEI-41 [21].

Additionally, as anticipated in Section 1.1, a different “family” of shear models could be

adopted for the squat columns of the infilled frames. A model providing a strength value that

degrades with the crack opening demand or, from a predictive standpoint, with the strain de-

mand (numerically evaluable) - instead than with the flexural inelastic ductility demand -

should be more suitable for the very “squat” portions of columns due to the limited infill-to-

column contact length. Therefore, the modified compression field theory (MCFT) model [33]

has been analysed and, in particular, its (more practical) simplification [34] is adopted herein.

According to the Simplified MCFT (SMCFT) proposed by Bentz et al. [34], shear strength

can be evaluated according to Eq. (11):

sw yw

Rd c w

A fV = f (b d) 0.9d cot

s +

(11)

where coefficient β represents the capacity of cracked concrete to transfer shear stress, and ϴ

is the inclination of the principal compressive stress. Both β and ϴ depend on the shear crack

opening demand, namely, on the longitudinal strain demand (εx) (according to Eq.s (12) and

(13)).

x xe

0.4 1300

1 1500 1000 s =

+ +

(12)

( ) xex

s29 7000 0.88 75

2500

= + +

(13)

The parameter sxe in above equations – representing the expected horizontal distance

among shear cracks – is assumed equal to 300 mm, according to Bentz et al. [34]. The longi-

tudinal strain at the mid-depth of the web (εx) is evaluated as equal to one-half of the strain in

the longitudinal tensile reinforcing steel [34] in a “critical section” located at a distance d

from the null point of the bending moment (see Figure 8), also taking into account the transla-

tion of the bending moment diagram, as suggested by Model Code [35], namely:

x *s s

1 M= V 0.5N

2A E d

+ −

(14)

where As and Es are tensile longitudinal steel area and Young modulus of steel, respectively; V,

M and N represent the shear, moment and axial compression demand, respectively, acting on

the (squat) column at the “critical section” (see Figure 8), and d* is assumed equal to 0.9 times

the effective depth of the column. Therefore, looking at Figure 9, a shear failure is detected if

2534


the column shear demand-drift curve (black curve) intersects the column shear capacity-drift

curve (green curve), for a shear value hereinafter referred to as Vcol,SF.

a

cz

w,offV

d

M V

critical section

oN

N

x

Figure 8: Definition of the critical section for shear safety check.

sw yw

Rd c w


s +

V

d (f(εx))

SHEAR FAILURE

Vcol,SF

Figure 9: Shear failure detection.

The shear failure could be also directly detected by properly adding to the numerical model

an explicit shear spring in the squat column, as shown in Figure 10 (similarly to Jeon et al.

[19], among others). This spring could be defined by means of a sort of “limit state material”

(as defined in [32]), which monitors the strain demand in the critical section of the column

(according to Bentz et al. [34]) instead of the inter-story drift demand (as originally suggested

by Elwood [32]) during the analysis. Such a monitoring possibility is currently not imple-

mented in OpenSees; therefore, herein, the strength of this spring is defined as the shear de-

mand for which the shear failure is achieved (Vcol,SF).

The shear spring introduced in the model is characterized by an infinite elastic stiffness up

to Vcol,SF, and by a softening constant stiffness up to a residual strength (Vc,res), if any (see

Figure 11). The softening stiffness adopted is defined according to Elwood and Moehle [36]

proposal (referred to as “E&M” in the following). On a mechanical base, Elwood and Moehle

[36] proposed to estimate the displacement δa,E&M at which column begins to lose its axial

load carrying capacity after the occurrence of a shear failure as in Eq. (15). Such a displace-

ment coincides with the total loss of lateral load carrying capacity (Vc,res=0).

( )2

a,E&M

sw yw

1 tan4=

N sL 100tan

A f d tan

+

+

(15)

2535



hc/2 hc/2Lw

L

hb/2

Hw

H

L R

ϴ

z c

Shear Spring

Figure 10: Shear spring modelling.

In Eq. (15), L is imposed equal to zc, defining the displacement δa,E&M as the relative dis-

placement between the two end points of the squat column. Additionally, the column axial

load N is assumed equal to its initial value (N0), in the hypothesis that, when the column axial

load failure occurs, the infill does not provide anymore a strength contribution, so that no axi-

al load variation is induced in the adjacent column with respect to N0.

Additionally, the shear critical angle θ in Eq. (15) is assumed by Elwood and Moehle [36]

equal to 65°. Nevertheless, the values of θ should be coherent with Eq. (13). Therefore, Eq.

(15) is implemented by assuming both the values of θ: the one suggested by Elwood and

Moehle (2005) and θSMCFT.

In both cases, the shear spring has been introduced in the numerical model in OpenSees as

shown in in Figure 10, namely by means of a ZeroLength Element (between two nodes, in red,

geometrically coincident) defined with a Hysteretic Uniaxial Material.

( )2

a,E&M

v

sw yw

1 tan4=

N sL 100tan

A f d tan

+

+

V

d

Elwood & Moehle (2005)

sw yw

Rd c w


s +

Vcol,SF

Figure 11: Shear spring backbone.

6 NUMERICAL VERSUS EXPERIMENTAL COMPARISONS

The results of the described modelling approach shown in Figure 6, with a single- (dotted

red line) or a three-strut (solid red line) infill model, is shown in Figure 12a and Figure 12b

for test SI-80 and GI-80, respectively. Note that Vb in Figure 12 is the base shear, which coin-

cides with the applied lateral load.

2536


Vy

Vmax

-Vy

-Vmax

Vy

Vmax

-Vy

-Vmax

(a) (b)

Figure 12: Numerical (red) versus experimental (grey) response: test SI-80 (a); test GI-80 (b) single-strut (dotted

red line) and tri-strut (solid red line) modelling approaches.

In both numerical simulations for SLD infilled frame (test SI-80), the resulting numerical-

versus-experimental comparison shows a very good matching in terms of maximum load and

also initial stiffness and frame deformability in the first loading steps. Note that the missing of

the fixed-end-rotation and of the joint panel deformability contribution in the numerical mod-

elling does not affect this comparison, because the weight of the infill strut lateral stiffness is

clearly predominant over the numerically missing deformability contributions. The maximum

lateral load finally achieved during the simulation is equal to 159.0 kN, very close to the cor-

responding experimental value (-158.6 kN). Moreover, at about 2.00% drift, lateral load drops

up to the expected flexural strength (Vmax) of the bare frame only (calculated as in Eq. 2), in

tune with the experimental evidence. Such an outcome can be useful to validate the assump-

tions that allowed the experimental definition of the infill panel response, as explained in Sec-

tion 4. Additionally, the agreement between the results of the modelling approach based on a

single concentric strut and a tri-strut modelling approach (shown in Figure 12a) confirms the

validity of the proposals by Chrysostomou et al. [30] for the off-diagonal struts. It can be con-

cluded that the linear deformed shape of columns, which is the base of this proposal, results

realistic for this struts configuration.

Regarding test GI-80 (see Figure 12b), the agreement between modelling results and ex-

perimental findings appears very good in terms of maximum load and also initial stiffness and

frame deformability, especially in the ascending branch of the lateral behaviour up to shear

cracks in the column start to significantly increase their width (namely until the penultimate

cycle of the response). The maximum lateral load achieved during the simulation is equal to

140.9 kN, that is very close to the corresponding experimental value (-140.1 kN). Similarly to

SI-80 test, the agreement between the results of the modelling approach based on a single

concentric strut and a tri-strut modelling approach confirms the validity of the proposal by

Chrysostomou et al. [30] and the assumption of the linear deformed shape of the columns that

is the base of this proposal.

As stated before, the modelling approach shown in Figure 6, and resulting in Figure 12, is

not able to detect any eventual shear failure. Therefore, first this eventuality is checked by

means of a post-processing of the analysis results (as explained in Section 5). In particular, the

attention will be focused on the left (L) side column, where axial load decreases during the

simulation thus producing a reduction in shear strength, and which clearly reached the crack-

ing condition during the test GI-80 (where local shear interaction exhibited the most signifi-

2537


cant experimental evidence). Both shear strength models, by ASCE/SEI 41 [21] and by Bentz

et al. [34] (SMCFT) are applied herein for both the infilled specimens (see Figure 13). As a

results, no shear failure has been detected by ASCE/SEI 41 [21]’s strength model (see

“S&M” curve in Figure 13a). Note that VRd by “S&M” model is reducing for increasing drift

levels due to the action of the lower off-diagonal strut that produces a progressive reduction of

the column axial load. On the contrary, for test GI-80, Bentz et al. [34]’s proposal leads to a

column shear failure at about 1.0 % of applied lateral drift (red star in Figure 13a), namely on

the descending branch of the global lateral response (red curve), as experimentally observed.

Vcol,SF is equal to 52.4 kN in this case. Note that Figure 13 also shows axial load (N) and

shear demand (V) variation during the simulation (in grey and black solid line, respectively),

together with the global (Vb-drift) response (red line). It is also noteworthy that, as expected,

the same safety checks for SI-80 specimen do not provide any column shear failure, since

shear strength capacity evaluated by both SMCFT theory and “S&M” proposal is higher than

the maximum column shear demand (see Figure 13b).

0 0.5 1 1.5 2

drift (%)

0

50

100

150

For c

e( k

N)

Vcol

Ncol

Vbase

VRd,S&M

VRd,SMCFT

numer. SF

0 0.5 1 1.5 2

drift (%)

0

50

100

150

200

Forc

e(k

N)

Vcol

Ncol

Vbase

VRd,S&M

VRd,SMCFT

(a) (b)

Figure 13: Shear failure (SF) detection for left-side squat column for test GI-80 (a) and SI-80 (b).

In a second step of modelling, for the specimen GI-80, where the shear failure is detected,

an explicit shear spring has been added to the model, as shown in Figure 10 and explained in

Section 5. Such a spring is linear elastic up to Vcol,SF, and then characterised by a softening

constant stiffness up to a null residual strength (Vc,res = 0). The softening stiffness adopted is

defined according to Elwood and Moehle [36] proposal, namely by means of Eq. (15), where

the shear critical angle θ should be first defined. If θ is evaluated by Eq. (13) at the onset of

the shear failure (θSMCFT), the value of 41.2° is obtained. Otherwise, according to Elwood and

Moehle [36] proposal, θ is equal to 65°. Both these hypotheses on θ (65° and 41.2°) have been

analysed. The corresponding values of δa,E&M are equal to 30 mm and 8.2 mm, respectively.

Note that θSMCFT value is closer to the experimental main shear crack angle shown in Section

2. Therefore, a better prediction of the numerical response is expected when δa,E&M is evaluat-

ed with θ = θSMCFT, as shown immediately below.

The results of these simulations for the specimen GI-80 are shown in Figure 14a and the

related responses of the shear spring are reported in Figure 14b. It can be observed that shear

failure – meant as the initiation of a shear-controlled softening response – begins at a drift

value roughly equal to 1% (see Figure 14a). The global softening branch changes depending

on the assumption adopted for θ. The numerical response better reproduces the experimental

response if the θSMCFT value is adopted (red solid lines in Figure 14a).

2538


SF onset

SF onset

(a) (b)

Figure 14: Numerical-versus-experimental comparison for test GI-80 including column shear spring with θSMCFT

(solid red line) or θ=65° (dotted red line) (black line is related to the simulation without shear spring) (a); related

shear spring responses (b).

In conclusion, Figure 15 shows the axial load (Nw)-axial deformation (εw) response of the

three compressive struts without (Figure 15a) and with (Figure 15b) the explicit shear spring

in the numerical model. At the end of the test all the struts are still contributing to the lateral

response of the frame (Nw>0) if no shear spring is modelled (Figure 15a). On the contrary, the

lower off-diagonal strut starts unloading at the onset of the shear failure in the adjacent col-

umn, as expected, when the shear spring is explicitly modelled; after the column shear failure,

the remaining struts still contribute to the lateral response since the axial displacement de-

mand can still increase in these two (Diagonal and Upper Off-Diagonal) struts (Figure 15b).

SF onset

(a) (b)

Figure 15: Response of the three infill struts during the analysis in terms of axial load (Nw)-axial deformation

(εw) without shear spring (a) and with shear spring (and δa,E&M with θSMCFT) (b).

7 CONCLUSIONS

A preliminary numerical investigation on column shear failure due to local interaction be-

tween structural and non-structural elements has been presented, starting from the results of

some experimental tests on infilled frames performed by the Authors. Different shear strength

models and different strategies of macro-modelling for infills have been applied and discussed,

in order to (i) capture (or not) the column shear failure exhibited (or not) during the test and

2539


(ii) to match the experimental response in terms of initial stiffness, peak strength and corre-

sponding displacement, and softening behaviour.

The analysed experimental data have been briefly described and numerically reproduced

by means of proper models carefully taking into account the local shear interaction between

infill panel and RC columns observed during one of the analysed tests. It resulted that the

shear demand on the surrounding columns due to the interaction with the infill panels could

be well caught by means of a three-compressive struts modelling approach, properly identify-

ing the location and the contribution of each strut according to the proposals by Jeon et al. [19]

and Chrysostomou et al. [30]. On the other side, the column shear strength could be well es-

timated based on the modified compression field theory [34], more suitable model for the

squat columns produced by this interaction with respect to shear strength models degrading

due to the increasing cyclic ductility demand.

The numerical results shown in the paper appear to be very close to the experimental ones.

Nevertheless, future further efforts will be devoted to the extension of this modelling strategy

to other similar specimens experimentally tested in the literature, to finally provide a wider

validation of the adopted numerical models.

ACKNOWLEGMENTS

This work was developed under the support of ReLUIS-DPC 2014-2018 Linea CA -WP6

Tamponature funded by the Italian Department of Civil Protection (DPC), and of the AXA

Research Fund Post-Doctoral Grant “Advanced nonlinear modelling and performance as-

sessment of masonry infills in RC buildings under seismic loads: the way forward to design or

retrofitting strategies and reduction of losses” funded by AXA Research Fund. These sup-

ports are gratefully acknowledged.

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