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Influence of Confined Concrete Models on the Seismic Response of RC Frames
Hüseyin Bilgin
*
and Bredli Plaku
Department of Civil Engineering, EPOKA University, Tirana, 1039, Albania
*Corresponding Author: Hüseyin Bilgin. Email: hbilgin@epoka.edu.al
Received: 13 December 2023 Accepted: 13 March 2024 Published: 15 May 2024
ABSTRACT
In this study, the influence of confined concrete models on the response of reinforced concrete structures is inves-
tigated at member and global system levels. The commonly encountered concrete models such as Modified Kent-
Park, Saatçioğlu-Razvi, and Mander are considered. Two moment-resisting frames designed according to the
pre-modern code are taken into consideration to reflect the example of an RC moment-resisting frame in the
current building stock. The building is in an earthquake-prone zone located on Z3 Soil Type. The inelastic
response of the building frame is modelled by considering the plastic hinges formed on each beam and column
element for different concrete classes and stirrups spacings. The models are subjected to non-linear static analyses.
The differences between confined concrete models are comparatively investigated at both reinforced concrete
member and system levels. Based on the results of the comparative analysis, it is revealed that the column beha-
viour is mostly influenced by the choice of model, due to axial loads and confinement effects, while the beams are
less affected, and also it is observed that the differences exhibited in the moment-curvature response of column
cross-sections do not significantly affect the overall behaviour of the global system. This highlights the critical role
of model selection relative to the concrete strength and stirrup spacing of the member.
KEYWORDS
Non-linear static analysis; moment-curvature relationships; plastic hinges; concrete confinement models; seismic
action
1 Introduction
Concrete is the most used material in construction due to its strength, durability, and versatility. It
possesses excellent compressive strength, but it also has limited resistance to tensile forces.
To better understand and predict the behaviour of concrete, researchers have developed various concrete
models that attempt to capture the complex interactions between the concrete, reinforcement, and external
confining pressures. They are mathematical representations of the behaviour of concrete under different
loading conditions used to simulate the stress-strain relationship, the failure criteria, and the post-failure
behaviour of concrete elements in structural analysis. Some proposed concrete models are by Richart
et al. [1], Roy et al. [2], Kent et al. [3], Scott et al. [4], Sheikh et al. [5], Fafitis et al. [6], Yong et al. [7],
Mander et al. [8], Saatçioğlu et al. [9], Cusson et al. [10], El-Dash et al. [11], Hoshikuma et al. [12],
Mansur et al. [13], and Assa et al. [14].
This work is licensed under a Creative Commons Attribution 4.0 International License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
DOI: 10.32604/sdhm.2024.048645
ARTICLE
ech
T
PressScience
There has been a notable trend in recent decades, particularly in the assessment of seismic behaviour of
existing structures, towards displacement-based analysis methods instead of force-based methods. These
methods consider the non-linear behaviour of structures, enabling more realistic results to be obtained.
The structural elements, and consequently the non-linear behaviour of the structure, are modelled using
plastic hinges determined according to the element properties [15]. Therefore, determining the behaviour
of plastic hinges is a crucial part of the analysis for estimating the structure’s behaviour.
2 Reinforced Concrete
Reinforced concrete is a composite material that combines the compressive strength of concrete with the
tensile strength of steel reinforcement. When a beam is subjected to vertical loads or a column is subjected to
eccentric axial and lateral loads, it experiences flexural stresses that cause cracks to the region in tension [16]
and shear stresses that cause cracks near the supports.
By incorporating longitudinal reinforcement at the regions with high tensile stress concentration, the
capacity to withstand tensile forces is significantly enhanced, thus preventing the formation of cracks.
Unconfined concrete is mostly used in foundations, pavements, and walls, where the shear stresses are
minimal; and confined concrete is used in structural members that are subject to bending and shear stresses.
Similarly, by confining the longitudinal reinforcement using transverse reinforcement, the resistance to
shear forces is increased. In contrast to unconfined concrete, confined concrete exhibits a ductile failure due
to the presence of lateral reinforcement [17].
3 Stress-Strain Relationship
The stress-strain relationship is the fundamental concept in the mechanics of a material. It describes the
deformation of a material when a load is applied to it. Fig. 1 demonstrates the stress-strain relationship of
brittle, ductile, and plastic materials. Strain is the ratio of change in the length of the body.
The stress-strain relationship for concrete is obtained by the Concrete Compression Test. The test is
conducted 28 days after the concrete is set, due to it reaching 99% of its strength. A cylindrical (150 mm
× 300 mm) or cubic (150 mm × 150 mm × 150 mm) sample is placed inside the Compression Test
Machine and is placed under a constantly increasing load until it fails [18]. Due to the ease of drilling on-
site and flexibility, the cylindrical shape is the most widely used for the specimen. The cylindrical
specimen’s strength is usually 10%–20% less than the equivalent cube specimen.
Figure 1: The relationship for different types of materials
198 SDHM, 2024, vol.18, no.3
The stress-strain curve for unconfined concrete has three regions, the linear ascending curve that
represents the elastic phase, the peak point, the non-linear descending curve that represents the plastic
phase, and the descending curve that represents failure [19].
Additionally, the stress-strain curve for confined concrete exhibits a higher strain value before failure,
indicating greater ductility. This means that confined concrete can undergo more significant deformation
before reaching its ultimate failure point. The curve for confined concrete shows a more gradual descent
after the peak stress, unlike the sharp drop observed in the unconfined concrete section.
4 Global and Local System Levels
To analyse the safety of a structure, we need to consider the Local to Global levels relationship of the
structure. Starting with the first level of the hierarchy, material, we need to understand and predict the
behaviour of concrete from its stress-strain relationship. From the curve, the elastic limit where concrete
transitions from elastic to plastic phases can be determined. The plastic phase is non-linear and involves
cracking and crushing of concrete.
Moving to the next section level, the moment-curvature relationship describes how the cross-section
deforms in response to applied moments. It determines the location and rotation of plastic hinges, which
are the regions where concrete has cracked (determined from the stress-strain relationship) and steel has
yielded. The plastic hinges indicate the loss of stiffness and strength of the section.
The next level is the member/connection level. As previously stated, concrete will undergo elastic
deformation, reach its elastic limit, and then go to plastic deformation. This plastic deformation is
concentrated in plastic hinges, which are critical points of failure. The plastic hinges affect the ductility
and stability of the member and connection, as well as their energy dissipation capacity.
Finally, at the system level, the non-linear static pushover analysis of roof displacement vs. base shear is
performed. This displacement-based method considers the non-linear behaviour of the material for more
realistic results. It helps evaluate the global performance of the system under different load levels and
patterns and locate the plastic hinges [15].
5 Confined Concrete Models
The stress-strain curve of concrete is influenced by numerous components, making it impossible to
define a single curve for each case, especially for the non-linear phase. Thus, researchers have proposed
different concrete models that describe the behaviour of concrete using empirical equations. The stress-
strain relationship for some concrete models is shown in Table 1.
Table 1: Peak stress and strain equations for some concrete models [20]
Researcher Peak stress (fccÞPeak strain ðεcc)
Sheikh et al. Ks0:85 f uc
Ks¼1þB02
140 P0cc
1nC
02
5:5B
02
1s
2B
0
2
ffiffiffiffiffiffiffiffiffi
psfy
p
80 Ksfuc 106
Fafitis et al. fuc þ1:15 þ3048
fuc
fl1:027 107fuc þ0:0296 fl
fcc
þ0:00195
Mander et al.
fuc 2:254 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ7:94 fl
fuc
s2fl
fuc
1:254
"#
eco 1þ5fcc
fuc 1
(Continued)
SDHM, 2024, vol.18, no.3 199
5.1 Modified Kent-Park Model
The Modified Kent-Park Model is based on the original Kent-Park Model by taking into consideration
the change in concrete strength due to the confinement, as shown in Fig. 2. The maximum stress and strain
are at point B [21,22].
K¼1þrsfyh
f0
c
(1)
where fyh is the yield strength of steel hoops.
Table 1 (continued)
Researcher Peak stress (fccÞPeak strain ðεcc)
Yong et al. 1þ0:0091 1 0:245 s
B
psþnd
0
st
Bsd
s
pt
fyh
ffiffiffiffiffiffi
fuc
p
fuc 0:00265 þ0:0035 1 0:734 s
B
psfyh
2
3
ffiffiffiffiffiffi
fuc
p
Saatçioğlu et al. fuc þ6:7f
l
ðÞ
0:17 fleco 1þ5K½
K¼6:7f
l
ðÞ
0:17 fl
fuc
El-Dash et al. fuc þ5:1fuc
fyh
0:5d0
st
ps
0:25
"#
fl
fl¼0:5p
sfyh 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
1:25 ds
r
eco þ66
s
d0
st
f1:7
uc
2
6
6
43
7
7
5
fl
fuc
Cusson et al. fuc þ2:1fl
fuc
0:7
eco þ0:21 fl
fuc
1:7
Mansur et al. fuc 1þ0:6psfy
fuc
1:23
"# eco 1þ2:6psfy
fuc
0:8
"#
Hoshikuma et al. fuc 1þ0:73 psfy
fuc
0:00245 þ0:0122 psfy
fuc
Assa et al. fuc 1þ3:36 fl
fuc
eco 1þ21:5fl
fuc
Figure 2: Stress-strain relationship for concrete proposed by Kent and Park
200 SDHM, 2024, vol.18, no.3
•Region AB, where ec0:002. The unconfined and confined curves are the same in the original model.
The maximum stress at B is taken from the cylinder’s compressive strength test.
fc¼Kf0
c2ec
0:002K ec
0:002K
2
(2)
a¼ec
0:002K 1ec
0:006K
(3)
g¼1
2
3ec
0:008K
1ec
0:006K
(4)
•Region BC, where 0:002K ece20m;c:
fc¼Kf0
c1Zmec0:002K
ðÞðÞ
0:2Kf0
c(5)
a¼1
ec0:004K
3þec0:002KðÞ
Z
20:002KðÞ
2
(6)
g¼11
ec
e2
c
20:002KðÞ
2
12
!
Ze3
c
30:001K e2
cþ0:002KðÞ
3
6
!
ec0:002K
3
Ze2
c
20:002K ecþ0:002KðÞ
2
2
!
0
B
B
B
B
@
1
C
C
C
C
A
(7)
Z¼0:5
e50u þe50h 0:002K (8)
e50u ¼3þ0:29f0
c
145f0
c1;000 (9)
e50h ¼3
4rsffiffiffiffiffi
b00
sh
s(10)
e20c ¼0:8
Zþ0:002K ¼ecu (11)
where f0
cis the concrete compressive strength from a 150 mm × 300 mm cylinder test; rsis the reinforcement
volumetric ratio; bnis the width of the confined core; shis the centre-to-centre tie spacing.
•Region CD, where ec>e20m;c:
fc¼0:2Kf0
c(12)
a¼1
ec0:004K
3þ0:32
Zþ0:2K ec0:0004K
(13)
SDHM, 2024, vol.18, no.3 201
g¼11
ec
1:2667 106
Kþ0:00064K
Zþ0:83
6Z2þ0:1e2
c
0:004K
30:32
Zþ0:2K ec0:0004K
0
B
B
@1
C
C
A
(14)
The stress–strain relationship will change due to the hysteresis behaviour of concrete when it
experiences repeated load cycles.
5.2 Saatçioğlu-Razvi Model
Saatçioğlu and Razvi proposed a model based on uniform confinement pressure caused by transverse
reinforcement. This model can be applied to both circular and rectangular sections [23]. This model
consists of a parabolic rising curve and a linear decreasing curve up to 20% of the concrete’s strength, as
shown in Fig. 3.
The parabolic curve can be expressed by Eqs. (15) and (16):
fc¼f0
cc 2ec
ecc
ec
ecc
2
!
1
1þ2K
(15)
f0
cc ¼fco þk1f0
l(16)
ecc ¼eco 1þ5KðÞ (17)
eco ¼0:002 (18)
e20c ¼0:2f0
cc (19)
K¼k1f0l
fco
(20)
k1¼6:7f0
l
0:17 (21)
fl¼k2fl(22)
k2¼0:26 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bc
s
bc
sl
1
fl
s(23)
Figure 3: Stress-strain relationship of confined concrete proposed by Saatçioğlu and Razvi [9]
202 SDHM, 2024, vol.18, no.3
For circular sections:
fl¼2Ash fyh
dcs(24)
For rectangular sections:
fl¼P2Ash fyh sin a
bcs(25)
For square sections:
fl¼f0
lx bcx þf0
ly bcy
bcx þbcy
(26)
where f0
lis the effective lateral pressure; f0
lx is the effective lateral pressure perpendicular to bcx;f
0
ly is the
effective lateral pressure perpendicular to bcy;ais the angle of stirrup with bc;b
c;bcx;bcy are the core
dimensions from stirrup centre to stirrup centre; s is the distance between transverse reinforcement; slis
the distance between longitudinal reinforcement; k2¼1 for circular and rectangular sections with small-
spaced transverse reinforcement.
The descending linear curve can be expressed from the strain corresponding to 85% of the concrete’s
strength:
e85c ¼260rsh ecc þe85u (27)
rsh ¼PAsh
sb
cx þbcy
(28)
e85c ¼0:85f0
cc (29)
where rsh is the volumetric ratio of transverse reinforcement; Ash is the cross-sectional area of transverse
reinforcement.
5.3 Mander Model
One of the most widely used confined concrete models is the Mander model. It is used as the default
model in a lot of structural engineering software, such as SAP2000 [24]. Mander proposed a unified
stress-strain model that includes both circular and rectangular confined concrete sections. The stress-strain
curve is derived from Popovics’Equation [25], while the confinement coefficient is like the approach of
Sheikh et al. [5]. This model incorporates the effect of arching pressure as the source of confining stress,
as shown in Fig. 4.
The effective cylinder compressive strength of confined concrete is 0.8 times the cube compressive
strength of confined concrete. The thickness and spacing of the transverse reinforcement affect the
confining stress and the effective lateral pressure determines the peak cylinder strength [26]. When
fl¼0)f0
cc ¼f0
c.
f0
cc ¼f0
c2:254 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ7:94 f0
l
f0
c
s2f0
l
f0
c1:254
! (30)
fc¼f0
cc xr
r1þx0(31)
SDHM, 2024, vol.18, no.3 203
f0
c¼0:8fck (32)
x¼ec
ecc
(33)
ecc ¼0:002 1þ5f0
cc
f0
c1
(34)
r¼Ec
EcEsec
(35)
Ec¼5000 ffiffiffiffi
f0
c
q(36)
Esec ¼f0
cc
ecc
(37)
where f0
cis the cylinder compressive strength of unconfined concrete; f 0
cc is the cylinder strength of confined
concrete; f0
lis the confining stress.
When experiencing repeated load cycles, the unloading curves follow the same pattern as the monotonic
curve until the maximum stress, but the unloading modulus is modified by the stress, fun, and strain, eun,at
the unloading point [27].
6 Moment-Curvature Relationship
In the analysis of reinforced concrete structures, the deformation of its members is an important factor to
consider. The displacement of a member reflects its ability to resist stresses caused by external loads.
Therefore, the displacement must be determined accurately and efficiently. Moment-curvature relationship
describes how the cross-section deforms in response to applied forces.
Depending on the beam and loading conditions, the equation of the elastic curve can be derived and take
various forms. Some examples are shown in Table 2.
By integrating the equation of the elastic curve successively, the Shear VðÞto Moment MðÞto Curvature
f
ðÞto Rotation uðÞrelationship can be obtained as shown in Table 3. This linear relationship is valid only for
the elastic range of concrete.
Figure 4: Stress-strain relationship of concrete proposed by Mander
204 SDHM, 2024, vol.18, no.3
The Moment-Curvature relationship is analysed at the section system level, which means that only the
section properties, such as the geometry, material, and reinforcement, are required to be known to perform the
analysis.
Table 2: Specific cases of beam deflection and slopes [28]
Beam and loading Elastic curve Equation of elastic curve
•For x <a:
yxðÞ¼ P
6EIL x3L2b2
x
•For x ¼a:
yxðÞ¼
Pa2b2
3EIL
yxðÞ¼ w
24EI x42Lx3þL3x
yxðÞ¼ P
6EI x33Lx2
yxðÞ¼ w
24EI x44Lx3þ6L2x2
Table 3: P to V to M to
f
to uto drelationship
Load pxðÞ
Shear VxðÞ¼
RpxðÞdxðÞ
Moment MxðÞ¼
RVxðÞdxðÞ
Curvature
f
xðÞ¼
MxðÞ
EI
Rotation ux
ðÞ
¼R
f
x
ðÞ
dx
ðÞ
Deflection dxðÞ¼
RuxðÞdxðÞ
SDHM, 2024, vol.18, no.3 205
6.1 The Presence of Compressive Reinforcement
Let us consider the following cross-section and calculate the moment-curvature according to the Kent-
Park model when compressive reinforcement is absent and present. The data is given in Fig. 5.
The moment and curvature values for each critical point are calculated and demonstrated in Table 4.
The shape of the moment-curvature curve can be observed from Fig. 6. The presence of compressive
longitudinal reinforcement increases the moment capacity slightly compared to the case without
compressive reinforcement, as stated earlier.
Figure 5: Example of an unconfined concrete section
Table 4: Calculated moment-curvature values
Compressive RC Not present Present
Point Moment (kN m) Curvature (1/m) Moment (kN m) Curvature (1/m)
0 0.00 0.000000 0.00 0.000000
Before cracking 39.73 0.000713 32.56 0.000713
Cracking 39.73 0.000713 32.56 0.000668
Yield 219.61 0.008434 220.32 0.006000
Ultimate 205.01 0.031003 219.28 0.036500
Figure 6: Graphic representation of the moment-curvature data
206 SDHM, 2024, vol.18, no.3
6.2 The Presence of Transverse Reinforcement
To improve the ductility and shear performance of the member the longitudinal reinforcement is
confined with transverse reinforcement, such as steel stirrups. A confinement zone will be formed in the
cross-section, as shown in Fig. 7. The role of confinement is especially evident in enhancing the shear
strength of columns [29].
The procedure for determining the moment-curvature relationship of confined concrete sections depends
on the stress-strain relationship, which is determined by the selection of the proposed concrete models. The
geometry, confinement of the section, time when the structure was built, the software package used and more
as shown in Table 5, can be some of the factors affecting this choice.
Figure 7: Confined beam section
Table 5: A comparison between the considered concrete models
Concrete
model
Pros Cons
Modified
Kent-Park
Considers change in concrete strength due to
confinement.
Assumes equal strength for confined and
unconfined concrete, which may not always
be true.
Utilizes a second-degree parabola for the
ascending branch, representing a more
realistic material behaviour.
Confinement effect on rectangular tie
strength is considered very small.
Descending branch accounts for lateral steel,
tie spacing, and core concrete area.
May not effectively capture the hysteresis
behaviour in repeated load cycles.
Saatçioğlu-
Razvi
Models stress-strain relationship based on
equivalent uniform confinement pressure.
May oversimplify post-peak behaviour with a
linear descending branch.
Considers tri-axial state of stress, leading to
increased concrete strength.
Assumes constant residual strength, which
may not represent all concrete behaviours.
Utilizes a second-order parabola for the
ascending branch.
Mander Unified stress-strain model for both circular
and rectangular sections.
Assumption of arching action as the sole
source of confining stress may not be
universally applicable.
(Continued)
SDHM, 2024, vol.18, no.3 207
7 Moment-Rotation Relationship
Plastic hinges are lumped masses where the concentration of the plastic deformation is accumulated due
to bending. These regions are formed when steel reaches its yield point, and their location is determined by
the moment-curvature relationship of the cross-section. The moment-rotation relationship describes the
behaviour of the member due to bending.
The rotation of the member is determined by the integration of the curvature along the length of the
member, including the elastic and plastic regions.
u¼
f
‘p(38)
where uis the rotation angle;
f
is the curvature length; ‘pis the plastic hinge length.
There is still no compromise on the best way to estimate the length of plastic hinges, thus researchers
have proposed empirical formulas for different types of reinforced concrete beams and columns, as
shown in Table 6. The variables that affect the plastic hinge length include the degree of confinement,
amount of shear stresses across the supports, support to point of contraflexure distance, longitudinal
reinforcement, axial forces at the section, and the maximum acceptable strain of concrete [21].
Considering the axial load performance in vertical members, it is imperative to acknowledge the effect of
the cross-sectional size in the load-carrying capacity, which affects the moment-curvature relationship and
consequently the moment-rotation of the member [30].
Table 5 (continued)
Concrete
model
Pros Cons
Accounts for effective confining pressure and
confinement effectiveness coefficient.
Complexity might require calibration for
specific cases.
Recognised as default model in many
structural engineering software packages.
Table 6: Empirical equations for calculating the length of plastic hinges [31]
Researcher Plastic hinge length, ‘p
Baker (1956) kz
d
1
4d
Sawyer (1964) 0:25d þ0:075z
Corley (1966) 0:5d þ0:2ffiffiffiffi
d
pz
d
Mattock (1967) 0:5d þ0:05z
Priestley et al. (1987) 0:08z þ6db
Paulay et al. (1992) 0:08z þ0:022dbfy
Sheikh et al. (1993) 1:0h
Coleman et al. (2001) Gc
f
0:6f0
ce20 ecþ0:8f0
c
Ec
(Continued)
208 SDHM, 2024, vol.18, no.3
Where Agis the gross area of the concrete section; Asis the area of tension reinforcement; d is the
effective depth; dbis the diameter of longitudinal reinforcement; Ecis the Modulus of Elasticity of
concrete; fcis the compressive strength of concrete; f yis the yielding stress of the rebars; Gc
fis the
concrete fracture energy in compression; h is the overall depth of section; p is the applied axial force.
p0¼0:85 f0
cAgAs
þfyAs(39)
where p0is the nominal axial load capacity per ACI 318-05 [32]; ecis the peak compressive strain; z is the
shear span, thus the distance from the critical point to the point of contraflexure. The point of contraflexure is
the point where the moment changes sign, and therefore the moment is zero. In simple terms, z is the length
of the member segment from the critical point to the point where the moment is zero.
8 Case Study
8.1 Four-Storey Residential Building
The first case study involves the non-linear analysis of a 3D frame of a four-storey residential building
located in a seismic zone. It has four bays in both directions with a width of 4 m in the x-direction and 3 m in
the y-direction, as shown in Figs. 8 and 9. The frame is modelled using the three concrete models to compare
their performance under earthquake loading. The layout of the frame is based on pre-modern building design.
The nominal concrete strength is 16 MPa and the steel reinforcement grade is 220 MPa. However, due to the
age and quality of construction, the actual concrete strength and stirrup spacing vary throughout the building.
Therefore, four scenarios are considered: 16 MPa concrete and 100 mm stirrup spacing; 10 MPa concrete and
250 mm stirrup spacing; 16 MPa concrete and 250 mm stirrup spacing; 10 MPa concrete and 100 mm stirrup
spacing. These values are obtained from the site surveys conducted on such buildings [33]. The additional
information is shown in Table 7.
Table 6 (continued)
Researcher Plastic hinge length, ‘p
Panagiotakos et al. (2001) 0:18z þ0:02ldbfy
Bae et al. (2008) lp
h¼0:3p
p0
þ3As
Ag
1
z
h
þ0:25 0:25
Figure 8: Top view of the frame in xy plan
SDHM, 2024, vol.18, no.3 209
The longitudinal reinforcement shown in Fig. 10 is:
•8ϕ12 for the 200 × 500 beams
•6ϕ14 for the 300 × 300 side columns
•10ϕ14 for the 250 × 500/500 × 250 columns
To evaluate the non-linear behaviour of the structural members, the moment-curvature relationship
of each member is calculated using the SEMAp section analysis tool, by the Scientific and Technical
Research Council of Türkiye (TÜBİTAK) under Project No. 105M024 [34]. This section analysis tool
can calculate the strength-deformation relationships, including stress-strain, force-deformation, and
moment-curvature, and generate diagrams for a given reinforced concrete section. It does this by
utilising the Hognestad model for unconfined concrete [35], and the Modified Kent-Park, Saatçioğlu-
Razvi, and Mander models for confined concrete. The section properties are inserted into SEMAp as
shown in Fig. 11, and then the plastic hinge properties are calculated according to concrete compressive
strength, section moment capacity and reinforcement capacity in accordance with FEMA-356 [36],
depending on whether the section is a column or a beam. This plastic hinge properties can then be
transferred to SAP2000.
Figure 9: Side views of the frame in xz and yz plans respectively
Table 7: Data for the first case
Case 1.1 1.2 1.3 1.4
Type Residential building
Number of floors 4
Floor height (m) 2.80
Bay width in x-dir (m) 4
Bay width in y-dir (m) 3
Frame weight (kN) 6,830.04
Concrete cover (mm) 25.00
Concrete grade (MPa) 16 10 16 10
Steel grade (MPa) 220
Stirrup spacing (mm) 100 250 250 100
210 SDHM, 2024, vol.18, no.3
The initial data for the structural analysis of the building is obtained by running SAP2000 linear analysis
with only the dead and live loads (DL + 0.3LL) applied to the model. The results show that the average shear
span of the beams in the xz-direction is 0.85 m, and the beams in the yz-direction have an average shear span
of 0.70 m. The columns have different shear spans for each floor, 1.91 m for the ground floor, 1.44 m for the
first and second floors, and 1.50 m for the third floor.
For this study, two representative members are selected for detailed analysis: one of the middle beams in
the xz plan (both are symmetric) on the second floor, and the middle column in the xy plan on the ground
floor, both belonging to the middle frame of the building, as demonstrated. These members have been chosen
because they are the most critical in terms of their demand and their influence on the global response of the
building. The curves of stress-strain and moment-curvature for each model are generated as shown in
Figs. 12 and 13 and the location of the critical members is shown in Fig. 14.
The main difference between the beam and column members in this analysis is the axial force applied to
the cross-section. The beam is assumed to have zero axial force, while the column has an axial force of
358 kN, which corresponds to the gravity load on the ground floor.
Figure 10: Member cross-sections for the four-storey frame
Figure 11: Configuring the section in SEMAp
SDHM, 2024, vol.18, no.3 211
Figure 12: Generated stress-strain curves for the C250 × 500 column in Case 1.1
Figure 13: Generated moment-curvature curves for the C250 × 500 column in Case 1.1
212 SDHM, 2024, vol.18, no.3
The moment-rotation relationship of all members is obtained from the moment-curvature relationship
using Eq. (38). The software provides you the option on which equation to use, as shown in Fig. 15, for
the plastic hinge length. In these case studies, it has been calculated for each member using Paulay and
Priestley’s equation from Table 6.
The force-deformation and moment-rotation data that is generated for each member is manually
assigned to the hinges of the corresponding member in SAP2000, as demonstrated in Figs. 16 and 17.
The beams have two degrees of freedom: shear (V2) and moment (M3). The columns have five degrees
of freedom: axial (P), shear (V2 and V3), and moment (M2 and M3). The suffixes 1, 2, and 3 denote the
local axis of the cross-section of the member, which determines the direction and orientation of the forces
and moments acting on the member.
Figure 14: The most critical beam and column
Figure 15: Plastic hinge equations provided in SEMAp
SDHM, 2024, vol.18, no.3 213
Figure 16: Generated moment-rotation curve for the Modified Kent-Park model of the C250 × 500 column
in Case 1.1
Figure 17: Defining the moment-rotation parameters for moment, M2, of column
214 SDHM, 2024, vol.18, no.3
The force-displacement and moment-rotation values can be either entered directly into the sheet or
divided by safety factors for a clearer presentation.
The next step is performing the non-linear pushover analysis at the global system level, after assigning
all the hinges to the frame members. The frame has a natural period of 0.55409 s in the x-direction and
0.54321 s in the y-direction, which reflects its dynamic characteristics and stiffness. The modal load case,
which applies a proportional load distribution based on the mode shapes of the frame, is used for
conducting the pushover analysis in both directions. The load application control is displacement-
controlled, meaning that the load is increased until the target displacement is reached. The target
displacement is set to 4% of the frame’s height, which is equivalent to 448.0 mm.
8.2 Seven-Storey Residential Building
The second case study involves the non-linear analysis of a 7-storey frame. The top view of the plan in
the xy direction is the same as in Fig. 8 and the side views are as shown in Fig. 18. The additional information
is shown in Table 8.
Figure 18: Side views of the frame in xz and yz plans respectively
Table 8: Data for the second case
Case 2.1 2.2 2.3 2.4
Type Residential building
Number of floors 7
Floor height (m) 2.80
Bay width in x-dir (m) 4
Bay width in y-dir (m) 3
Frame weight (kN) 15,100.96
Concrete cover (mm) 25.00
Concrete grade (MPa) 16 10 16 10
Steel grade (MPa) 220
Stirrup spacing (mm) 100 250 250 100
SDHM, 2024, vol.18, no.3 215
The longitudinal reinforcement shown in Fig. 19 is:
•8ϕ14 for the 250 × 600 beams
•12ϕ14 for the 400 × 400 side columns
•8ϕ16 for the 300 × 600/600 × 300 columns
For this case, two representative members are selected as shown in Fig. 20: one of the side beams in the
xz plan (both are symmetric) on the sixth floor, and the middle column in the xy plan on the ground floor,
both belonging to the middle frame of the building.
The frame has a natural period of 0.76456 s in the x-direction and 0.75352 s in the y-direction. The target
displacement is set to 4% of the frame’s height, which is equivalent to 784.0 mm.
9 Results and Discussions
9.1 Influence of Concrete Models
A comparison of the moment-curvature relationships of the beam and column sections for different
concrete models reveals that the concrete behaviour in the beam case, in Fig. 21, is less sensitive to the
choice of the model than in the column case, in Fig. 22. This happens due to the presence of axial loads
and confinement effects of the column, which affect the concrete behaviour and strength. The axial load
reduces the curvature and increases the ultimate moment capacity of the column, while the confinement
effect enhances the concrete strength and ductility by preventing lateral expansion and cracking.
The observation made on the capacity curve of the frames, which is obtained from the non-linear
pushover analysis, shows the base shear vs. the roof displacement of the frame under lateral loads. The
results show that the capacity curve is not sensitive to the choice of the concrete model, as all the models
produce similar curves for both frames.
9.2 Influence of Concrete Quality and Stirrup Spacing
Another factor that influences the moment-curvature of the members is the stirrup spacing and concrete
quality. As expected, the members with smaller stirrup spacing and better concrete quality have higher
moment capacity than the members with larger stirrup spacing and poor concrete quality. This is because
the stirrups provide confinement and shear resistance to the concrete, while the concrete quality affects
the compressive strength and stiffness of the material. Moreover, the strain capacity of the members,
which represents their ductility and deformation ability, is lower when the concrete quality is poor and
the stirrup spacing is large, especially in the seven-story case. This indicates that these members are more
prone to brittle failure and less able to dissipate energy under seismic loads.
Figure 19: Member cross-sections for the seven-storey frame
216 SDHM, 2024, vol.18, no.3
Figure 20: The most critical beam and column in the 7-storey frame
SDHM, 2024, vol.18, no.3 217
At the global level, as shown in Figs. 23 and 24, the concrete grade and the stirrup spacing of the
members influence the capacity curve as well. The frames with better concrete grade and smaller stirrup
spacing have a higher base shear and roof displacement than the frames with poor concrete grade and
larger stirrup spacing. This means that these frames have higher strength and deformation capacity. The
effect of the concrete grade and stirrup spacing is more evident in the seven-story frame than in the
0
20
40
60
80
0.00 0.20
Moment (kN m)
Curvature (1/m)
4-storey / B200x500 /
C16@100
0
20
40
60
80
0.00 0.10 0.20 0.30
Curvature (1/m)
4-storey / B200x500 /
C10@250
0
20
40
60
80
0.00 0.20
Curvature (1/m)
4-storey / B200x500 /
C10@100
0
20
40
60
80
0.00 0.20
Curvature (1/m)
4-storey / B200x500 /
C16@250
0
40
80
120
0.00 0.20
Moment (kN m)
Curvature (1/m)
7-storey / B250x600 /
C16@100
0
40
80
120
0.00 0.20
Curvature (1/m)
7-storey / B250x600 /
C10@250
0
40
80
120
0.00 0.20
Curvature (1/m)
7-storey / B250x600
/C10@100
0
40
80
120
0.00 0.20
Curvature (1/m)
7-storey/ B250x600
/ C16@250
Figure 21: Moment-curvature data for the selected beam sections
0
40
80
120
0.00 0.10
Moment (kN m)
Curvature (1/m)
4-storey / C250x500 /
C16@100
0
40
80
120
0.00 0.10
Curvature (1/m)
4-storey / C250x500 /
C10@250
0
40
80
120
0.00 0.10
Curvature (1/m)
4-storey / C250x500 /
C10@100
0
40
80
120
0.00 0.10
Curvature (1/m)
4-storey / C250x500 /
C16@250
0
50
100
150
200
250
300
0.00 0.05 0.10
Moment (kN m)
Curvature (1/m)
7-storey / C300x600 /
C16@100
0
50
100
150
200
250
0.00 0.05 0.10
Curvature (1/m)
7-storey / C300x600 /
C10@250
0
50
100
150
200
250
0.00 0.05 0.10
Curvature (1/m)
7-storey / C300x600 /
C10@100
0
50
100
150
200
250
300
0.00 0.05 0.10
Curvature (1/m)
7-storey / C300x600 /
C16@250
Figure 22: Moment-curvature data for the selected column sections
218 SDHM, 2024, vol.18, no.3
four-story frame, as the seven-story frame has more members and more gravity load than the four-story
frame. Therefore, these factors are important for designing and evaluating the seismic performance of
reinforced concrete frames.
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
thgiewcimsie
S
/r
a
e
h
sesaB
Drift (%)
4-storey / C16@100 / x-
direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C10@250 / x-
direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C10@100 /
x-direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C16@250 / x-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
thgiewcimsieS/raehsesaB
Drift (%)
7-storey / C16@100 / x-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C10@250 / x-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C10@100 / x-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C16@250 / x-
direction
Figure 23: Capacity curves for the x-direction span
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
th
gi
ew
cim
si
e
S/
ra
e
h
sesaB
Drift (%)
4-storey / C16@100 / y-
direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C10@250 / y-
direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C10@100 / y-
direction
0.000
0.040
0.080
0.120
0.160
0.00% 2.00%
Drift (%)
4-storey / C16@250 / y-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
thg
i
e
w
c
i
ms
i
e
S
/
r
aehses
a
B
Drift (%)
7-storey / C16@100 / y-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C10@250 / y-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C10@100 / y-
direction
0.000
0.040
0.080
0.120
0.00% 2.00%
Drift (%)
7-storey / C16@250 / y-
direction
Figure 24: Capacity curves for the y-direction span
SDHM, 2024, vol.18, no.3 219
10 Conclusions
The main objective of this study was to compare the different concrete models in SAP2000 for the non-
linear analysis of reinforced concrete frames under seismic loads. The concrete models were applied to the
cross-sections of the beam and column members of two frames: a four-story frame and a seven-story frame.
The moment-curvature and moment-rotation relationships of the members were calculated using the SEMAp
section analysis tool, which was developed by the Scientific and Technical Research Council of Türkiye
(TÜBİTAK) under Project No. 105M024. The force-deformation and moment-rotation data were then
manually assigned to the hinges of the corresponding members in SAP2000. The non-linear pushover
analysis was performed on the frames using the modal load case and the displacement-controlled load
application.
The moment-curvature relationships of column sections show distinct behaviours across the concrete
models. The column’s response is significantly sensitive to the choice of the model, due to the effects
of axial load and confinement.
The axial load notably reduces the curvature and increases the ultimate moment capacity, especially in
the Modified Kent-Park and Saatçioğlu-Razvi concrete models. The confinement effect enhances the
strength and ductility of the columns by limiting lateral expansion and preventing cracking.
The cases with smaller stirrup spacing and higher concrete quality show increased moment capacity of
the members, indicating the effect and importance of confinement in shear resistance, and the role of
concrete quality in the compressive strength of the members.
The capacity curves of both buildings are not sensitive to the choice of concrete model, due to them
producing similar base shear, seismic weight, and drift values. This underlines that while the choice of
a concrete model is crucial for localized member behaviour, the global system behaviour is mostly
influenced by the concrete strength and stirrup spacing, crucial for the strength and deformation
capacity of the frames.
The impact of concrete quality and stirrup spacing is more pronounced in the seven-storey building,
due to the greater number of members and the higher gravity loads, suggesting a more significant role
of them in the seismic assessment of taller structures.
For future research, more concrete models can be investigated and validated using experimental data and
other software packages. Additionally, more frames with different geometries, heights, and boundary
conditions, and more concrete grades and stirrup spacings can be considered for further research. These
aspects can provide more insight into understanding the non-linear behaviour and performance of
reinforced concrete structures under seismic loads to help improve the design and assessment of these
structures.
Acknowledgement: None.
Funding Statement: The authors received no specific funding for this study.
Author Contributions: The authors confirm contribution to the paper as follows: study conception and
design: Prof. Hüseyin Bilgin; data collection: Bredli Plaku; analysis and interpretation of results: Prof.
Hüseyin Bilgin and Bredli Plaku; draft manuscript preparation: Bredli Plaku. All authors reviewed the
results and approved the final version of the manuscript.
Availability of Data and Materials: The data generated and analysed during this study are available from
the corresponding author upon reasonable request.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the
present study.
220 SDHM, 2024, vol.18, no.3
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