There is a high agreement among the
standards: the interaction with the MIs is not
considered when designing overall the frame
but the interactive forces with the MIs are
considered when verifying the columns in
shear.
2. The instructions in TCVN 9386:2012
and EN 19981:2004 are rather ambiguous,
leading to various interpretations when they
are applied, e.g. the diagonal strut width wm,
the length of contact regions lc, etc.
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hrough
the modified Takeda model and its forcedisplacement curve is taken
according to ASCE 4113 (Figure 2.2).
a) b) c)
Figure 2.2. a) Plastic b) The modified Takeda c) Generalized M–θ
deformation concentrated hysteresis rule relationship at plastic
on the frame components hinges of RC frame components
2.3. ESTABLISH THE NONLINEAR BEHAVIOR MODEL OF THE MIs IN
RC FRAMES
2.3.1. Setting up the forcedisplacement relationship of the model
The behavior of the MIs in the frame is modeled as a curve shown in
Figure 2.3. In the frame model, the MIs are shown in Figure 2.4.
7
2.3.2. Define the basic parameters of the model
2.3.2.1. The stiffness of MIs
According to Nguyen Le Ninh (1980), the width (1 ) 0
m n
m mw e w
−= (2.1)
with
( )0
m
m
h l
dw
h l kλ λ
=
+ +
(2.2); 4 42 2;4 4
m m m m m m
h l
c c m c b m
E t l E t h
E I h E I l
λ λ= = (2.3)
where n = H/Hu, H is the lateral force and Hu is the lateral force at the
time when MI reaches the ultimate strength; m and k are coefficients
depending on the type of masonry; other parameters indicate the
geometric and mechanical properties of frames and MIs (Figure 1.8).
From the width wm, determine the stiffness of the MI at the beginning
of the crack (2.4) and when reaching to the ultimate strength (2.5):
0.4
20 cos
m
m m m
my
m
e w t EK
d
θ= (2.4); 20 0.4cos
mym m m
mu m
m
Kw t EK
d e
θ∗ = = (2.5)
2.3.2.2. The strength of the MIs
1. The ultimate strength of masonry infill Vmu is determined from the
condition ( )min ,mu ms mcV V V= (2.6), where:
a) Vms is the sliding shear strength of MIs selected from approaches of
following authors: Rosenblueth (1980); Smith and Coull (1991); Paulay
and Priestley (1992); Decanini et al. (1993); Panagiotakos and Fardis
(1994), Fardis (2009); Zarnic and Gostic (1997); FEMA 356 (2000), Al
Chaar (2002), ASCE 4106, ASCE 4113; Galanti et al. (1998), EN 1998
1:2004; FEMA 306 (1998); EN 199611:2005; according to TCVN
5573:2011 (2.10).
11 0.72
bs m m
ms
f t lV
n tgµ θ
=
−
(2.10)
Figure 2.3. The forcedisplacement relationship Figure 2.4. Position of plastic
of the MI’s behavior model hinges in the model of infilled frames
8
b) Vmc is the diagonal compression strength of MIs selected from
approaches of following authors: Smith and Coull (1991); Decanini et al.
(1993); Galanti et al.
(1998); FEMA 306; Al
Chaar (2002); Tucker
(2007); ASCE 4113.
In order to select the
appropriate strengths for
MI’s model, comparative
analyses are performed on
the infilled RC frame
consistent with the object
and objectives of the
research. The results are
the curves representing
relationships of Vms and
Vmc associated with the common
hm/lm ratios of MIs in Figures 2.9
and 2.10. Since then, choose the
strength Vms according to TCVN
5573:2011 (2.10) and the
strength Vmc according to ASCE
4113:
cos
3
m
mc mc m
hV f t θ= (2.11)
2. The yielding strength of the masonry infill Vmy is selected from
approaches of following authors: Nguyen Le Ninh (1980), Dolsek and
Fajfar (2008); Decanini et al. (1993); Panagiotakos and Fardis (1994);
Saneinejad and Hobbs
(1995), FEMA 306; Tucker
(2007); Stavridis (2009).
Similarly, from the results in
Figure 2.12, choose Vmy =
0.6Vmu (2.12) as suggested by
Nguyen Le Ninh and Dolsek
and Fajfar (2008).
3. The residual strength of
the masonry infill Vmr:
0 0.1mr myV V≤ ≤ (2.13)
Figure 2.9. Variation of Vms determined by different
approaches associated with hm/lm
Figure 2.10. Variation of Vmc determined by
different approaches associated with hm/lm
Figure 2.12. Variation of Vmy determined by
different approaches associated with hm/lm
9
2.3.2.3. Steps to establish the forcedisplacement curve of the model
Step 1. Determine Kmy using (2.4). Step 2. Determine Vmu using (2.6).
Step 3. Determine mu mu muV K ∗∆ = (2.14). Step 4. Determine Vmy using
(2.12). Step 5. Determine my my myV K∆ = (2.15). Step 6. Determine Vmr
using (2.13). Step 7. Determine
( ) /mr mu mr mu mrV V K∆ = ∆ + − (2.16)
2.3.2.4. Axial nonlinear response of
equivalent diagonal strut
Using the stressdeformation
relationship of masonry proposed by
Kaushik, Rai and Jain (2007) (Figure
2.13).
2.3.3. Calibrate the behavior model of
the MI
Calibration of the proposed model is performed based on the experimental
data of infilled RC frame models designed according to EC8 and EC2 of
Kakaletsis and Karayannis (2008) and Morandi, Hak and Magenes (2014
2018).
2.3.3.1. Kakaletsis and Karayannis (2008)
Based on the parameters of the
experimental models, we establish the
behavior models of the MIs based on the
steps in section 2.3.2.3 (Figure 2.16).
Using these models together with the
behavior models of the RC materials and
structural elements selected in section
2.2, performing a nonlinear pushover
analysis of the experimental frame
models. The capacity curves obtained
from analyses are compared with the
experimental forcedisplacement
envelopment (Figure 2.18). The results
show a good fit between them.
Figure 2.18. Comparison between
experimental results and analytical
results using the proposed method
a) Weak MI (S) b) Strong MI (IS)
Figure 2.16.
Force 
displacement
relationship of
the proposed
MIs’ models
Figure 2.13. Idealized stressstrain
relationship for masonry under
uniaxial compression
10
2.3.3.2. Morandi et al. (20142018)
Figure 2.20. Force  displacement
relationship of the proposed MI’s
model
Similarly, set up a behavior
model of the MI using the
proposed method (Figure 2.20)
and perform a nonlinear pushover
analysis of the experimental frame models. The results show that the capacity
curves obtained from the analyses are quite consistent with the experimental
envelopment (Figure 2.21).
2.4. REMARKS ON CHAPTER 2
A simple model is established to simulate the behavior of the MIs in
RC frames taking into account the decrease in strength and stiffness of
frame and MIs. The verification results on the infilled RC frame models
designed according to the current seismic conception exhibit good results.
So, the calibration of the model is not necessary.
CHAPTER 3
EFFECTS OF MASONRY INFILLS TO THE CONTROL OF THE FAILURE
MECHANISM OF RC FRAME STRUCTURES UNDER SEISMIC ACTIONS
3.1. MODERN CONCEPTION AND DESIGN RULES FOR FRAMES IN THE
CURRENT SEISMIC DESIGN STANDARDS
3.1.1. Modern conception in design of structures for earthquake
resistance
According to the current seismic conception, the design purpose of a
building is to protect directly both human life and social properties. When
a strong earthquake occurs, the buildings are allowed to work beyond the
elastic limit, but they are not collapsed suddenly.
3.1.2. Basic design principles according to modern seismic conception
From the aforementioned goals, the structure must be designed to
experience plastic failure, and shear failure must happen after flexural
failure when a strong earthquake occurs.
Figure 2.21. Comparison between
experimental results and analytical results
using the proposed method
11
3.1.3. Design RC frames according to current seismic standards
To carry out the above design principles, the capacity design method
is used. By using this method, the forces used to design a frame must be
as follows, for example, according to TCVN 9386:2012 (the “socalled”
basic design principle of strong columns  weak beams):
a) Beam: The bending moment M and the axial force N are taken from
the results of structural analysis, while the shear force Q is determined
from the bending resistance of the beam.
b) Column: The bending moment M is redefined from the following
condition:
1.3Rc RbM M≥∑ ∑ (3.1)
in which: ΣMRc is the sum of the minimum design values of the moment
resistances of the columns framing to the joint, taking into account the
column axial force N in the seismic design situation; ΣMRb is the sum of
the design values of the moment resistances of the beams framing to the
joint.
Shear force Q is redefined from the flexural strength of columns.
Remarks: (i) The frame design process must follow a very strict
process; (ii) Frame design rules do not take into account frameMI
interaction.
3.2. EFFECTS OF MIs TO THE BEAM RESPONSE
Experimental studies on the infilled frames show that the interactive
forces with the MIs make the beams behave more stiffly than that of bare
frames. To clarify this phenomenon, consider a RC frame without MI
(bare frame) as shown in Figure 3.2a. The external force H causes the
bending moment at the ending section C of the beam:
,
3
2 6 1bC H
HhM ω
ω
=
+
(3.2) where: b
c
I h
I l
ω = (3.3)
a) Bare frame; b) Infilled frame; c) Equivalent infilled frame
Figure 3.2. Models for calculation of the frame
The curvature of the beam at the end C has the following value:
,,
3
2 6 1
bC H
bC H
c b c b
M Hh
E I E I
ωρ
ω
= =
+
(3.4)
12
When MI is available, the model to calculate an infilled frame is as
shown in Figure 3.2b, where Rm is the compression force in the diagonal
strut with the area of crosssection of wmtm. Replace the model in Figure
3.2b with the equivalent model in Figure 3.2c (Vm is the horizontal
projection of the compression force Rm in the diagonal strut). With this
model, we have the moment and curvature of the beam when taking into
account the interactive force with MIs:
( )
, 
 3
2 6 1m
m
bC H V
H V h
M ω
ω
=
+
(3.5)
, 
,  ,
(  ) 3
2 6 1
m
m
bC H V m
bC H V bC H
c b c b
M H V h
E I E I
ω
ρ ρ
ω
= = <
+
(3.6)
Thus, the interaction with MIs makes the beam stiffer. Let Ibm ( >Ib) be
the equivalent moment of inertia of the beam when considering
interaction with MIs. Similarly (3.4), we will get the curvature of the
beam in this case:
*
,*
,
3
2 6 1
bC H m
bC H
c bm c bm m
M Hh
E I E I
ω
ρ
ω
= =
+
(3.7), where: bmm
c
I h
I l
ω = (3.8)
Considering (3.3) and (3.8), we obtain the coefficient bm mIb
b
Ik
I
ω
ω
= =
(3.9) which indicates the increase in moment of inertia (flexural stiffness)
of the beam when interacting with MIs.
Balancing the curvatures (3.6) and (3.7), we establish the relationship:
6 1
6 1
m
m
H
H V
ω
ω
+
=
+ −
(3.11)
From the relationship between horizontal force H and Vm established
on the basis of the calculation diagrams in Figures 3.2b, 3.2c, and from
(3.11), we set the ratio ωm/ω. With this result, determine the coefficient
kIb (3.9) at the ultimate time (wm = wm0 when n = 1.0 see Chapter 2) when
considering the interaction with MIs:
3 2
0w cos 3 21
72
bmu m m m
Ibu
b c c m
I h t Ek
I E I d
θ ω
ω
+
= = + (3.18)
Equation (3.18) shows that, when considering the interaction with
MIs, the moment of inertia of the beam Ibmu is increased by kIbu times:
Ibmu = kIbuIb. This means that the crosssection height of the beam is
increased to 3 bmu b Ibuh h k= (3.19) called the equivalent crosssection
13
height. The increase in the crosssection height of the beam leads to an
increase in its bending resistance both positive RbM
+ and negative RbM
− for
the considered sense of the seismic action. In the general case, at any
columnbeam joint:
Rbmu Rbmu Rbmu Rb Rb RbM M M M M M
− + − += + > = +∑ ∑ (3.21)
where ΣMRbmu and ΣMRb are the sums of the design values of the moments
of resistance of the beams framing the joint when considering and not
considering the interaction with MIs for the considered sense of the
seismic action, respectively.
Thus, when considering the interactive forces with MIs, the moments
of resistance of the beams are increased by the following coefficient:


1
Rbmu Rbmu Rbmu
Mb
Rb Rb Rb
M M Mk
M M M
+
+
+
= = >
+
∑
∑
(3.22)
3.3. METHODS TO DESIGN THE RC FRAMES FOR EARTHQUAKE
RESISTANCE WHEN CONSIDERING THE INTERACTION WITH THE MIs
3.3.1. Condition to control the failure mechanism of the RC frames
From the above mentioned research results, in cases taking into
account of the interaction with MIs, the condition to control the plastic
failure mechanism (3.1) of a frame in TCVN 9386:2012 may not be
accurate, because RbM∑ in the righthand side is increased via kMb. This
also means that the columns may be failed before the beam and the soft
story failure mechanism may appear unintentionally.
Therefore, to let the infilled frames be failed plastically as the design
purpose, the design conditions (3.1) shall be rewritten as follows:
1.3Rcmu Mb RbM k M≥∑ ∑ (3.23)
in which ΣMRcmu is the sum of the minimum design values of the moment
resistances of the columns framing to the joint, taking into account the
column axial force N in the seismic design situation at the ultimate limit
state of MIs. With this condition, whether the MIs are available or not,
the design principle of "strong columns  weak beams" will be guaranteed
and the frames will be failed in plastic mechanisms under strong
earthquakes.
3.3.2. The method to design RC frame structures under seismic actions
when considering the interaction with the MIs
Step 1. Design and detail of RC beams in accordance with current
seismic design standards.
14
Step 2. Determine kIbu in (3.18) and the equivalent crosssection height
of beam hbmu in (3.19). Then determine the moment resistances of the
equivalent beams when considering the interaction with MIs RbmuM
− and
RbmuM
+ . Determine kMb in (3.22).
Step 3. Determine the bending moment to design the columns
RcmuM∑ in the proposed condition (3.23). Then design and detail the
longitudinal reinforcement of columns according to the rules of the
current seismic design standard.
3.4. CALCULATION EXAMPLES
3.4.1. The calculation data
A 3storey castinplace RC frame building with dimensions as shown
in Figure 3.4. The exterior beams of 25x45 cm, the interior beams of
25x50 cm, the slab thickness of 15 cm.
Materials: concrete B30, longitudinal reinforcement type CB400V,
stirrup reinforcement type CB240T.
The KB and KE frames are filled with solid brick masonry 20 cm
thick, burnt clay bricks M100, cement mortar M75.
Vertical load (permanent load g and imposed load q) at each floor
(including roof): g + ψ2q = 9 kN/m2.
The building is built in a region with the reference peak ground
acceleration on type A ground (rock) agR = 0.1097g, ground type D,
importance factor γI = 1.2; ductility class medium (DCM) according to
TCVN 9386:2012.
a) Plan view of the typical floor b) Elevation of the frame
Figure 3.4. Models for the frame structure
3.4.2. Design the RC frame structures according to the regulations of
TCVN 9386:2012
The reinforcement details of typical frame KE are shown in Figure
3.5.
15
Figure 3.5. Reinforcement details of frame KE
3.4.3. Determine responses of frame KE designed according to TCVN
9386: 2012
Using nonlinear pushover
analysis is to determine
responses of frame KE.
Behavior models of
materials and frame
components are taken from
EC2 and ASCE 4113. The
analysis results show that the
frame is failed in agreement
with the plastic mechanism
as the design goal set out
(Figure 3.6). The capacity
curve is shown in Figure 3.7
(solid line).
a) Step 6 b) Step 22 c) Step 48 d) Step 102
Figure 3.6.
Behavior of frame
KE designed
according to
TCVN 9386:2012
Figure 3.7. Capacity curves of frame KE
in different cases
16
3.4.4. Determine responses of frame KE designed according to TCVN
9386:2012 when considering the interaction with MIs
Calculation results for forcedisplacement relations of the MIs are
shown in Figure 3.8.
a) 1stfloor b) 2nd to 3rd floors
Figure 3.8. Forcedisplacement relationship in the behavior model of MIs
The pushover analysis in Figure 3.10 shows that the plastic
deformation process starts from the MIs to beams and the bases of
columns on the first floor. The capacity curve (dashed line) in Figure 3.7
shows that the frame stiffness drops suddenly and varies irregularly when
the base shear force reaches its maximum value of V = 626.27 kN and ∆
= 0.023 m in step 10 because the MIs on the first and second floors are
failed considerably. Until the target displacement ∆ = 4% H = 0.36 m
(step 108) is achieved, the plastic deformations are almost focused on the
bases of columns on the foundation surface and the top of columns on the
first floor, the MIs on the first floor are no longer capable of bearing
(Figure 3.10). The infilled RC frame is failed in agreement with the “soft
storey” mechanism.
Comparing the capacity curves of frame KE in Figure 3.7 (without
considering (solid lines) and considering (dashed lines) the interaction
with MIs) shows that the interaction with MIs has greatly increased the
stiffness, horizontal bearing capacity, and energy dissipation capacity of
the frame in the initial elastic phase.
3.4.5. Design and detail the RC frame structures considering the
interaction with MIs using the proposed method
The design of the frame structure shown in Figure 3.4 is implemented
using the method proposed in section 3.3.2.
Figure 3.10.
Behavior of
frame KE when
considering
the interaction
with MIs
a) Step 3 b) Step 10 c) Step 15 d) Step 108
17
Step 1: Calculate beam reinforcement of frame KE and calculate their
flexural resistance MRb as design results in section 3.4.2 (Figure 3.5).
Step 2: Determine kIbu = 2.508 and the equivalent crosssection height
hbmu = 680 mm, thereby determine bending resistance of the beams and
the coefficient kMb = 1.14.
Step 3: Determine the required bending moment to design the columns
yc
RcmuM∑ from the proposed condition (3.23), from which design and
arrange longitudinal reinforcement of the columns. Compared with the
standard design results (Figure 3.5), the crosssection height of columns
on the first floor (C1 and C4) must be increased by 50mm while the
reinforcement of all columns remains the same.
Performing pushover analysis is to determine responses of infilled frame
KE with behavior models of materials, structural members, and MIs used in
the calculation example in section 3.4.4. The analysis results show that the
frame designed using the proposed method isn’t failed corresponding to the
soft storey mechanism (Figure 3.11). The capacity curves in Figure 3.7 show
that frame KE designed using the proposed method with the condition (3.23)
(dashed doubledot line) has superior behavior compared with the case
designed according to the condition (3.1) of TCVN 9386:2012.
3.5. REMARKS ON CHAPTER 3
1. The increasing coefficients of flexural stiffness kIbu and flexural
resistance kMb of beams are quantified when considering the interaction
with MIs.
2. On this basis, the condition that controls failure mechanism (3.23)
is proposed to replace the condition (3.1) of TCVN 9386:2012 which is
no longer accurate when considering the interaction with MIs. Then a
method to design RC frame structures for earthquake resistance is
proposed.
3. Specific calculation examples have demonstrated the reliability of
the performed theoretical research: the model of MIs, the method to
design RC frame structures for earthquake resistance when considering
the interaction with MIs, etc.
Figure 3.11.
Behavior of frame
KE designed using
the proposed
condition (3.23)
a) Step 11 b) Step 17 c) Step 113
18
CHAPTER 4
CONTROL OF LOCAL FAILURES OF RC FRAMES UNDER SEISMIC
ACTIONS CONSIDERING THE INTERACTION WITH THE MIs
4.1. CONTROL OF LOCAL FAILURES OF RC FRAMES IN CURRENT
SEISMIC DESIGN STANDARDS
4.1.1. Control of shear failure in RC frames
Shear failure is a brittle failure mode, so
it must be prevented, and is not allowed to
occur before flexural failure. For frame
columns, according to TCVN 9386:2012,
the design shear force is determined from
the bending resistance of the column (called
the capacity shear force) (Figure 4.1).
,1 ,2
1 2
,
,
Rb Rb
Rd Rc Rc
Rc Rc
CD c
cl c
M M
M M
M M
V
l
γ
+ =
∑ ∑
∑ ∑
(4.3)
where lcl,c is the clear length of the column; MRc,i is the design value of
the column moment resistance at the end i (i = 1, 2) in the sense of the
seismic bending moment under the considered sense of the seismic action;
( / )Rb Rc iM M∑ ∑ ≤ 1 where ∑MRc and ∑MRb are the sums of the design
values of the moment resistances of the columns and the sum of the design
values of the moment resistances of the beams framing into the joint,
respectively; γRd is the factor accounting for overstrength. The values of
MRc,i and ∑MRc should correspond to the column axial force in the seismic
design situation for the considered sense of the seismic action.
4.1.2. Verification of column shear failure in seismic standards
TCVN 9386:2012 and EN 19981:2004 require checking and detailing
of columns in shear considering the interaction with MIs through the
condition: , , ,Rd c Ed c lcV V≥ (4.4)
Figure 4.1. Diagram of
determining column shear force
Figure 4.2.
Acting shear on
the columns
due to MIs
19
in which VRd,c is the shear resistance at the ends of the columns designed
according to the standard; VEd,c,lc is the increased design shear due to the
horizontal strut force acting at the column ends (Figure 4.2):
( ), , , , , ,min ;Ed c lc Ed c ms Ed c MV V V= (4.5)
where: (i) , ,Ed c ms m m mvV V A f= = (4.6)
with Am = tmlm and fmv is the shear strength of the MIs;
(ii) , , ,2Ed c M Rd Rd c cV M lγ= (4.7)
Other countries' standards are the same.
4.1.3. Remarks on the rules for verification of shear failure
1. There is a high agreement among the
standards: the interaction with the MIs is not
considered when designing overall the frame
but the interactive forces with the MIs are
considered when verifying the columns in
shear.
2. The instructions in TCVN 9386:2012
and EN 19981:2004 are rather ambiguous,
leading to various interpretations when they
are applied, e.g. the diagonal strut width wm,
the length of contact regions lc, etc.
4.2. FRAME  MI INTERACTIVE FORCES AND LOCAL RESPONSE OF RC
COLUMNS UNDER INTERACTIVE FORCES
According to Nguyen Le Ninh, the contact
lengths zh and zl between the MI and the frame
change when the infilled frame is subjected to
lateral load. At the ultimate time of MI (n =
1.0):
0 0 2h hz β π λ= and 0 0 2l lz β π λ= (4.14)
with:
( )0
m
mk h l
d
w h l k
β
λ λ
=
+ +
(4.13)
Along the contact regions zh and zl, interactive stresses which are
assumed to be linearly distributed appear, causing the force Rm in the
equivalent diagonal strut (Figure 4.3). According to Tassios et al. (1988),
it is possible to divide Rm into 3 parts as shown in Figure 4.4. At the
ultimate state of the MI (n = 1.0), the interactive forces at the contact
regions of column and beam with the MI are determined by the following
expressions (Figure 4.5):
Figure 4.4. The
distribution of strut’s
force on frame’s elements
Figure 4.3. The interactive
forces between frame and MI
20
0 00.8h mu hq V z= (4.17)
0 00.4l mu lq V tg zθ= (4.18)
where Vmu is the horizontal projection of
the force in the diagonal strut Rmu. From
the force qh0, determine the column shear
force due to the local interaction with the
MI (Figure 4.5):
3 4
0 0 0 0 0 0
, 2 3
, ,2 4 10
h h h h h h
c mA
cl c cl c
q z q z q zV
l l
= − + (4.19)
3 4
0 0 0 0
, 2 3
, ,4 10
h h h h
c mB
cl c cl c
q z q zV
l l
= − −
(4.20)
4.3. METHOD TO DESIGN THE RC COLUMNS IN SHEAR WHEN
CONSIDERING THE FRAMEMI INTERACTIVE FORCES
4.3.1. Condition to control the column shear failure
When considering the interactive forces with the MIs, the capacity
design shear of columns, VCD,c,m is determined in (4.3) in which ∑MRcmu
is determined by increasing kMb times using the proposed method in
section 3.3, Chapter 3. So it will be greater than VCD,c determined
according to TCVN 9386:2012. However, this increase in shear force is
only caused by the stiffening effect of the beams, not counting the
interactive forces between the MIs and the columns. Therefore, the design
shear of columns will be determined from the following proposed
condition:
, , , , , ,max( ; )Ed c m CD c m c pt mV V V= (4.23)
where , , , ,c pt m c pt c mV V V= + (4.24) is the column shear force determined
from structural analysis considering the local interaction with the MIs;
Vc,pt is the column shear force determined from the st
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